Laser Diodes and Their Applications to Communications and Information Processing

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P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010

9:45

LASER DIODES AND THEIR APPLICATIONS TO COMMUNICATIONS AND INFORMATION PROCESSING

Printer Name: Yet to Come


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WILEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING KAI CHANG, Editor Texas A&M University

A complete list of the titles in this series appears at the end of this volume.


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LASER DIODES AND THEIR APPLICATIONS TO COMMUNICATIONS AND INFORMATION PROCESSING

TAKAHIRO NUMAI Ritsumeikan University

A JOHN WILEY & SONS, INC., PUBLICATION


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C 2010 by John Wiley & Sons, Inc. All rights reserved. Copyright

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Numai, Takahiro Laser diodes and their applications to communications and information processing / Takahiro Numai. ISBN 978-0-470-53668-1 (cloth) Printed in the Singapore 10 9 8 7 6 5 4 3 2 1


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To my grandparents in the United States Kenichiro and Asano Kanzaki


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CONTENTS

PREFACE

PART I

ix

PHYSICS REQUIRED TO DESIGN LASER DIODES

1 Energy Bands in Bulk and Quantum Structures 1.1 1.2 1.3 1.4

3

Introduction, 3 Bulk Structure, 4 Quantum Structures, 14 Superlattices, 19 References, 21

2 Optical Transitions 2.1 2.2 2.3 2.4 2.5

1

23

Introduction, 23 Direct and Indirect Transitions, 24 Light-Emitting Processes, 25 Spontaneous Emission, Stimulated Emission, and Absorption, 26 Optical Gains, 27 References, 37

3 Optical Waveguides

39

3.1 Introduction, 39 3.2 Two-Dimensional Optical Waveguides, 41 3.3 Three-Dimensional Optical Waveguides, 52 References, 54 vii


viii

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CONTENTS

4 Optical Resonators 4.1 4.2 4.3 4.4

Introduction, 57 Fabry–Perot Cavity, 59 Waveguide Grating, 65 Vertical Cavity, 82 References, 91

5 pn- and pnpn-Junctions 5.1 5.2 5.3 5.4

57

93

Intrinsic Semiconductor, 93 Extrinsic Semiconductor, 97 pn-Junction, 103 pnpn-Junction, 117 References, 121

PART II CONVENTIONAL LASER DIODES

123

6 Fabry–Perot Laser Diodes

125

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Introduction, 125 Rate Equations, 128 Current versus Voltage Characteristics, 130 Current versus Light Output Characteristics, 131 Polarization of Light, 146 Transverse Modes, 148 Longitudinal Modes, 157 Modulation Characteristics, 160 Noises, 172 References, 189

7 Quantum Well Laser Diodes

191

7.1 Introduction, 191 7.2 Features of Quantum Well LDs, 191 7.3 Strained Quantum Well LDs, 201 References, 211 8 Single-Mode Laser Diodes 8.1 8.2 8.3 8.4

Introduction, 213 DFB LDs, 213 DBR LDs, 222 Vertical Cavity Surface-Emitting LDs, 224 References, 229

213


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CONTENTS

9 Semiconductor Optical Amplifiers 9.1 9.2 9.3 9.4

ix

233

Introduction, 233 Signal Gain, 233 Polarization, 238 Noises, 239 References, 241

PART III ADVANCED LASER DIODES AND RELATED DEVICES

247

10 Phase-Controlled DFB Laser Diodes

249

10.1 10.2 10.3 10.4 10.5

Introduction, 249 Theoretical Analysis, 249 Device Structure, 251 Device Characteristics and Discussion, 252 Summary, 254 References, 255

11 Phase-Shift-Controlled DFB Laser Diodes 11.1 11.2 11.3 11.4 11.5


12 Phase-Controlled DFB Laser Filter 12.1 12.2 12.3 12.4

271

Introduction, 271 Device Structure, 272 Device Characteristics and Discussion, 272 Summary, 276 References, 277

13 Phase-Shift-Controlled DFB Filter 13.1 13.2 13.3 13.4 13.5

257


279


x

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CONTENTS

14 Passive Phase-Shifted DFB Filter 14.1 14.2 14.3 14.4 14.5


15 Two-Section Fabry–Perot Filter 15.1 15.2 15.3 15.4 15.5

289

305


16 Optical Functional Devices with pnpn-Junctions

317

16.1 Introduction, 317 16.2 Edge-Emitting Optical Functional Device, 318 16.3 Surface-Emitting Optical Functional Device, 321 References, 336

PART IV SYSTEM DEMONSTRATIONS USING ADVANCED LASER DIODES AND RELATED DEVICES

339

17 Photonic Switching Systems

341

17.1 17.2 17.3 17.4

Introduction, 341 Wavelength Division Switching, 344 Wavelength- and Time-Division Hybrid Switching, 345 Summary, 350 References, 350

18 Optical Information Processing 18.1 18.2 18.3 18.4 18.5 18.6 18.7

Introduction, 353 Serial-to-Parallel Data Conversion, 354 Optical Self-Routing Switch, 355 Optical ATM Switch, 356 Optical Interconnection, 359 Optical Memory, 362 Optical Bus, 365 References, 366

353


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CONTENTS

xi

Appendix A: Density of States

367

Appendix B: Density of States Effective Mass

381

Appendix C: Conductivity Effective Mass

383

INDEX

385


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PREFACE

With the rapid increase in Internet users, photonics will be more important in the future, because photonics contributes to building an infrastructure for the Internet. Laser diodes, which are used commercially as light sources for optical fiber communications and the read/write ability of compact disks (CDs) and digital video disks (DVDs), have the potential to expand photonics technology if their operating principles are applied to photonic devices such as optical filters and optical functional devices. In this book, the potentials of laser diodes and their applications to optical functional devices and photonic systems are explained. To develop excellent photonic devices we have to fully understand the physics behind the operation of photonic devices. Therefore, the physics behind energy bands of semiconductors, optical transitions, optical waveguides, and semiconductor junctions is explained in detail. In addition, the physical characteristics of laser diodes, optical filters, and optical functional devices are reviewed. Using these photonic devices, photonic systems are demonstrated and some experimental results are described. The book consists of four parts: Part I, Physics Required to Design Laser Diodes; Part II, Conventional Laser Diodes; Part III, Advanced Laser Diodes and Related Devices; and Part IV, System Demonstrations Using Advanced Laser Diodes and Related Devices. First, the physics behind the operating principles of laser diodes is explained in detail. Second, concepts in the design of laser diodes, optical filters, and optical photonic devices are presented, and their characteristics and experimental results in system applications are reviewed.

xiii


xiv

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PREFACE

Features highlighted in the book are as follows: 1. We remove a large gap between journal articles and textbooks for universities and graduate schools; the reader will find that journal articles are readable after he or she has finished the book. However, only knowledge of electromagnetism and quantum mechanics obtained in an undergraduate course is required to read the book. 2. We give the reader analytical tools for Fabry–Perot laser diodes (LDs), distributed feedback (DFB) LDs, and vertical cavity surface-emitting LDs (VCSELs). If the reader follows the equations in this book, he or she will be enabled to analyze the characteristics of Fabry–Perot LDs, DFB LDs, and VCSELs. 3. We describe the types of potentials in laser diodes used for photonic devices, such as optical filters and optical functional devices. In addition, differences in the specifications required for light sources and those required for optical filters and optical functional devices are discussed. 4. We describe experimental results for system applications of laser diodes, optical filters, and optical functional devices. The reader is introduced to subjects that must be dealt with in the future, with the goal of stimulating research and development in photonics technology. Finally, I would like to thank Professor Emeritus of the University of Tokyo, Koichi Shimoda (former professor of Keio University); Professor Emeritus of Keio University, Kiyoji Uehara; Professor Tomoo Fujioka of Tokai University (former professor of Keio University); and Professor Minoru Obara of Keio University for their warm encouragement and helpful advice when I was a student. I am also indebted to NEC Corporation, where I began research on laser diodes after graduation from the Graduate School of Keio University. Thanks are extended to Mr. George Telecki of Wiley for his kind help and to Professor Kai Chang of Texas A&M University, Editor of Wiley’s Microwave and Optical Engineering Series, for giving me the opportunity to write the book. I am especially grateful to Springer Science+Business Media for allowing me to adapt material used in Chapters 1 and 8, published originally in my book Fundamentals of Semiconductor Lasers (Springer-Verlag Series in Optical Sciences, Vol. 93, 2004). Takahiro Numai

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PART I PHYSICS REQUIRED TO DESIGN LASER DIODES

1


August 9, 2010

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1 ENERGY BANDS IN BULK AND QUANTUM STRUCTURES

1.1

INTRODUCTION

Transitions of Electrons The emission and absorption of light are generated by the transitions of electrons. Light is emitted because electrons transit from high-energy states to lower-energy states, and light is absorbed in the reverse process. When electrons transit from highenergy states to lower-energy states, nonradiative transitions, which do not emit light, may exist as well as radiative transitions, which accompany light emissions.

Energy Bands When the atomic spacing is so large that mutual interactions of atoms may be neglected, the electron energies are discrete and energy levels are formed. With a decrease in the atomic spacing, the positions of the electrons of neighboring atoms start to overlap. Therefore, the energy levels begin to split to satisfy the Pauli exclusion principle. With a further decrease in atomic spacing, the number of electrons whose positions overlap with each other increases. As a result, the number of split energy levels goes up, and the energy differences in the adjacent energy levels are reduced. In semiconductor crystals, the number of atoms per cubic centimeter is on the order of 1022 , where the atomic spacing is about 0.2 nm. As a result, the spacing of energy levels is much narrower than the bandgap energy, on the order of Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright

3


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ENERGY BANDS IN BULK AND QUANTUM STRUCTURES

electron volts. Therefore, the constituent energy levels are considered to be almost continuous, and energy bands are formed.

1.2 BULK STRUCTURE Bulk Semiconductors in which constituent atoms are placed periodically at a sufficiently long range compared with lattice spacing are called bulk semiconductors. In this section, the energy bands in bulk semiconductors are calculated.

k· p Perturbation Semiconductors have free electrons and holes only in the vicinity of band edges. As a result, the band shapes and effective masses of carriers near band edges often give us sufficient information about optical transitions. To analyze the energy bands in the neighbor of band edges, k · p perturbation theory [1–4] is often employed. The wave functions and energies of the bands are calculated with k = k − k0 as a perturbation parameter, where k is a wave vector near a band edge and k0 is a wave vector at a band edge. For simplicity, k0 = 0 is selected in the following.

Schrödinger Equation The Schrödinger equation in the steady state is given by [5, 6] 2 2 − ∇ + V (r) ψn (k, r) = E n (k)ψn (k, r), 2m

(1.1)

where = h/2π = 1.0546 × 10−34 J · s is Dirac’s constant, h = 6.6261 × 10−34 J · s is Planck’s constant, m = 9.1094 × 10−31 kg is the electron mass in vacuum, V (r) is a potential, ψn (k, r) is a wave function, E n (k) is an energy eigenvalue, n is a quantum number, and k is a wave vector. In single crystals where the atoms are placed periodically, the potential V (r) is also spatially periodic. Therefore, as a solution of (1.1), we can consider a Bloch function, such as ψn (k, r) = exp(i k · r)u n (k, r), u n (k, r) = u n (k, r + R),

(1.2) (1.3)

where R is a translational vector which represents the periodicity of the crystal. Equations (1.2) and (1.3) constitute the Bloch theorem. Substituting (1.2) into (1.1)


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BULK STRUCTURE

5

leads to 2 2 − ∇ + V (r) + H u n (k, r) = E n (k)u n (k, r), 2m

(1.4)

where 2 k 2 + k· p, 2m m p = − i ∇.

H =

(1.5) (1.6)

Note that the k· p perturbation theory, whose name is derived from the second term on the right-hand side of (1.5), is valid only for small k, and we solve (1.4) by regarding (1.5) as the perturbation. First-Order Perturbation Theory For an energy band with n = 0, the wave equation for an unperturbed state with k = 0 is expressed as 2 2 − ∇ + V (r) u 0 (0, r) = E 0 (0)u 0 (0, r). 2m

(1.7)

In the following, for simplicity, the energy E 0 (0) is represented as E 0 . In first-order perturbation theory, the wave function u 0 (k, r) for a nondegenerate case is given by u 0 (k, r) = u 0 (0, r) +

− i (2 /m)k · α|∇|0 α=0

α|∇|0 =

E0 − Eα

u α (0, r),

u α ∗ (0, r)∇u 0 (0, r) d3 r.

(1.8) (1.9)

Here u 0 (k, r) and u α (k, r) are assumed to be orthonormal functions and α| and |0 are the bra and ket vectors, respectively, which were introduced by Dirac. Second-Order Perturbation Theory In second-order perturbation theory, an energy eigenvalue is obtained as E(k) = E 0 +

0| pi |αα| p j |0 2 k 2 2 + 2 ki k j . 2m m i, j E0 − Eα α=0

(1.10)


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The reciprocal effective mass tensor is defined as

1 m

ij

⎛

⎞ 0| pi |αα| p j |0 ⎠ 1 ∂ E 1 2 ≡ 2 = ⎝δi j + . ∂ki ∂k j m m α=0 E0 − Eα 2

(1.11)

Using (1.11), (1.10) is reduced to E(k) = E 0 +

2 1 ki k j . 2 i, j m i j

(1.12)

Equations (1.11) and (1.12) indicate that the effect of the periodic potential of the crystal is included in the effective mass of the electron, which makes analysis easier. In a cyclotron resonance experiment, the rest mass in vacuum in not measured, but the effective mass is measured. sp3 Hybrid Orbitals Next, we consider the energy bands of semiconductor crystals with zinc blende structures, which are used widely as material for light sources. In zinc blende structures, the atomic bonds are formed via sp3 hybrid orbitals. Therefore, the wave functions for electrons in zinc blende or diamond structures are expressed as superpositions of s- and p-orbital functions. We assume that the bottom of a conduction band and the tops of valence bands are placed at k = 0, as in direct transition semiconductors. When spin-orbit interaction is neglected, the tops of the valence bands are threefold degenerate, corresponding to the three p-orbitals ( px , p y , pz ). Here the s-orbital wave function for the bottom of the conduction band is u s (r), and the p-orbital wave functions for the tops of the valence bands are u x = x f (r), u y = y f (r), and u z = z f (r),where f (r) is a spherical function. Since the energy bands are degenerate, a perturbed wave equation is given by a linear superposition of u s (r) and u j (r) ( j = x, y, z), such as u n (k, r) = Au s (r) + Bu x (r) + Cu y (r) + Du z (r),

(1.13)

where A, B, C, and D are coefficients. To obtain the energy eigenvalues, (1.4) is rewritten −

2 2 2 k 2 ∇ + V (r) + Hd u n (k, r) = E n (k) − u n (k, r), 2m 2m i 2 Hd = k· p = − k·∇. m m

(1.14) (1.15)


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BULK STRUCTURE

7

By setting k = 0 in (1.14), an unperturbed equation is obtained. For a conduction band, we set E n (0) = E c , which is the energy of the bottom of a conduction band, and u 0 (0, r) = u s (r); for valence bands, we set E n (0) = E v , which is the energy of the top of each valence band, and u 0 (0, r) = u j (r) ( j = x, y, z). Substituting (1.13) into (1.14); multiplying u s ∗ (r), u x ∗ (r), u y ∗ (r), and u z ∗ (r) from the left-hand side; and then integrating with respect to a volume over the space leads to (Hss + E c − λ)A + Hsx B + Hsy C + Hsz D = 0, Hxs A + (Hx x + E v − λ)B + Hx y C + Hx z D = 0, Hys A + Hyx B + (Hyy + E v − λ)C + Hyz D = 0, Hzs A + Hzx B + Hzy C + (Hzz + E v − λ)D = 0,

(1.16)

where Hi j = u i |Hd |u j =

u i ∗ (r)Hd u j (r) d3 r λ = E n (k) −

(i, j = s, x, y, z),

2 k 2 . 2m

(1.17) (1.18)

Note that the orthonormality of u s (r) and u j (r) ( j = x, y, z) was used to derive (1.16). The condition used to obtain solutions A, B, C, and D other than A = B = C = D = 0 is E c − λ Pk x Pk y Pk z P ∗kx Ev − λ 0 0 = 0, (1.19) P ∗k y 0 Ev − λ 0 P ∗ kz 0 0 Ev − λ where P = −i

2 m

us ∗

∂u j 3 d r, ∂r j

P∗ = − i

2 m

u j∗

∂u s 3 d r ∂r j

( j = x, y, z, r x = x, r y = y, r z = z).

(1.20)

From (1.19) we obtain Ec + Ev 2 k 2 E 1,2 (k) = + ± 2 2m

Ec − Ev 2

E 3,4 (k) = E v + where (1.18) was used.

1/2

2

2 k 2 , 2m

+ k |P| 2

2

,

(1.21) (1.22)


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Electron

Nucleus

Electron

Nucleus

(a)

(b) FIGURE 1.1 Motions of an electron.

Spin-Orbit Interaction In addition to k· p perturbation, we consider spin-orbit interaction and second-order perturbation. First, let us consider spin-orbit interaction semiclassically. As shown in Fig. 1(a), an electron with electric charge −e = −1.6022 × 10−19 C rotates about the nucleus with electric charge +Z e. The velocity of the electron is v and the position vector of the electron is r, with the position of the nucleus as the initial point. If we see the nucleus from the electron as shown in Fig. 1.1(b), the nucleus seems to rotate about the electron with a velocity −v. As a result, a magnetic flux density B is produced at the position of the electron, which is written B=

µ0 r ×v µ0 Z e 1 Ze 3 = l. 4π r 4π m r 3

(1.23)

This equation is known as Biot–Savart’s law. In (1.23), µ0 is magnetic permeability of vacuum, and l is the orbital angular momentum, which is given by l = r × p = r × mv.

(1.24)

The spin magnetic moment µs is expressed as µs = −

e 2µB s=− s, m

(1.25)

where s is the spin angular momentum and µB is the Bohr magneton, which is defined as µB ≡

e = 9.2732 × 10−24 A · m2 . 2m

(1.26)

As a result, the magnetic field, which is generated at the position of the electron due to the orbital motions of the nucleus, interacts with the electron’s spin magnetic moment. The interaction energy HSO between the magnetic flux density B and the


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BULK STRUCTURE

9

spin magnetic moment µs is obtained as HSO = −µs · B =

µ0 Z e 2 1 l · s. 4π m 2 r 3

(1.27)

Note that (1.27) is obtained using classical electromagnetism. From Dirac’s relativistic quantum mechanics, the interaction energy HSO is given by HSO =

µ0 Z e2 1 l · s, 4π 2m 2 r 3

(1.28)

which is half of (1.27). Pauli’s Spin Matrices Pauli’s spin matrices σ are defined as s = σ, 2 01 0 −i 1 0 σx = , σy = , σz = . 10 i 0 0 −1

(1.29) (1.30)

Using Pauli’s spin matrices, the spin-orbit interaction Hamiltonian HSO can be rewritten HSO =

µ0 Z e2 1 l · σ. 4π 2m 2 r 3 2

(1.31)

If the up-spin ↑ (sz = /2) and down-spin ↓ (sz = −/2) are expressed as α and β, respectively, they are written in matrix form: 1 0 α= , β= . 0 1

(1.32)

As a result, operations of σz on α and β are written σz α = α, σz β = −β.

(1.33)

When a spherical polar coordinate system is used, the spin-orbit interaction Hamiltonian HSO is expressed as HSO

l+ σ− + l− σ+ = ξ (r) l · σ = ξ (r) l z σz + , 2 2 2

(1.34)


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where µ0 Z e 2 1 , 4π 2m 2 r 3 l+ = l x + i l y , l− = l x − i l y , σ+ = σx + i σ y , σ− = σx − i σ y . ξ (r) =

(1.35)

When the spin-orbit interaction Hamiltonian HSO is added to (1.14) as a perturbation term, the Schrödinger equation is written

2 2 2 k 2 − ∇ + V (r) + Hd + HSO u n (k, r) = E n (k) − u n (k, r). 2m 2m

(1.36)

It should be noted that l operates on exp(i k · r) in the Bloch function, but this operation is neglected because the result is much smaller than the other terms. To solve (1.36), the wave functions are represented in the spherical polar coordinate system as us = us , u+ = −

ux + u y ux − u y x+y x−y ∼ − √ , u− = √ ∼ √ , u z ∼ z. (1.37) √ 2 2 2 2

In (1.37), √ the spherical function f (r) is omitted after ∼ to simplify expressions. Note that 2 is introduced in the denominators to normalize the wave functions. Using the spherical harmonic function Ylm , the wave functions u + , u − , and u z are expressed as 1 =− 2

u+ =

Y11

u− =

Y1−1

1 = 2

uz =

Y10

3 x +iy 1 =− 2 2 2 2π x + y + z 2 3 x −iy 1 = 2 2 2 2π x + y + z 2

1 = 2

3 exp(i φ) sin θ, 2π

3 exp(−i φ) sin θ, 2π

3 z 1 = 2 2 2 π x +y +z 2

(1.38)

3 cos θ, π

where x = r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ . Including the up- and down-spins α and β, the following eight wave functions are present: u s α, u s β, u + α, u + β, u z α, u z β, u − α, u − β. Therefore, we have to calculate the elements of the 8 × 8 matrix to obtain energy eigenvalues from (1.36).


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11

BULK STRUCTURE

For brevity, we assume that k = k x , k y , k z is a vector in the positive direction of the z-axis and express the elements of k as k z = k, k x = k y = 0.

(1.39)

In this case we only have to solve the determinant for the 4 × 4 matrix on four elements of u s α, u + β, u z α, u − β or those of u s β, u − α, u z β, u + α because of the symmetry in the 8 × 8 matrix. This determinant for the 4 × 4 matrix is written Ec − λ 0 0 0 Ev − λ − 3 √ 2 P ∗k 0 3 0 0

0 = 0, Ev − λ 0 0 0 Ev − λ + 3 √Pk 2 0 3

0

(1.40)

where the terms including 0 are the matrix elements of HSO , and the other terms are those of Hd . Here, using ξ (r) in (1.35), 0 is expressed as 0 2 2 = u + ∗ u + ξ (r) d3 r = u − ∗ u − ξ (r) d3 r 3 2 2 2 2 = (u x 2 + u y 2 )ξ (r) d3 r = u z 2 ξ (r) d3 r. 4 2

(1.41)

From (1.40), the energy of valence band 1 is obtained as

E v1 (k) = E v +

0 2 k 2 + . 3 2m

(1.42)

When |P|2 k 2 is small enough, the energy of the conduction band E c is reduced to 2 k 2 |P|2 k 2 E c (k) = E c + + 2m 3

2 1 + Eg E g + 0

,

(1.43)

where Eg = Ec − Ev −

0 . 3

(1.44)


12

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Similarly, the energies of valence bands 2 and 3 are given by E v2 (k) = E v +

0 2 k 2 2|P|2 k 2 + − , 3 2m 3E g

2 2 k 2 |P|2 k 2 E v3 (k) = E v − 0 + − . 3 2m 3(E g + 0 )

(1.45) (1.46)

Note that these results were obtained under first-order k· p perturbation. Valence Bands Under second-order perturbation, the energies of the valence bands are given by 0 E v1,2 (k) = E v + + A2 k 2 3 1/2 ± B2 2 k 4 + C2 2 k x 2 k y 2 + k y 2 k z 2 + k z 2 k x 2 (1 → +, 2 → −), (1.47) 2 E v3 (k) = E v − 0 + A2 k 2 . (1.48) 3 Equations (1.43), (1.47), and (1.48) are shown in Fig. 1.2. From the definition of effective mass in (1.11), the band with energy E v1 (k) is referred to as a heavy hole band and that with E v2 (k) is called a light hole band. It should be noted that the heavy and light hole bands are degenerate at k = 0. The band with energy E v3 (k) is called the split-off band, and 0 is called the split-off energy. The coefficients A2 , B2 , and C2 in (1.47) and (1.48) are determined experimentally by cyclotron

Conduction Band

Heavy Hole Band Light Hole Band Split-off Band FIGURE 1.2 Energy bands of a bulk structure when the spin-orbit interaction is considered under a second-order perturbation.


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BULK STRUCTURE

13

TABLE 1.1 Relations Between Operators and Eigenvalues Operator 2

l lz s2 sz j2 jz

Eigenvalue l(l + 1) (l = 0:s-orbital, l = 1: p-orbitals) m l , m l = 1, 0, −1 s(s + 1)2 , s = 1/2 m s , m s = 1/2, −1/2 j( j + 1)2 , j = 3/2, 1/2 m j , m j=3/2 = 3/2, 1/2, −1/2, −3/2, m j=1/2 = 1/2, −1/2 2

resonance. In general, the effective masses depend on the direction of k, and the energy bands are more complicated. Note that in the preceding analysis, the energy bands of direct transition semiconductors, in which the bottom of the conduction band and the tops of the valence bands are placed at k = 0, are calculated. In indirect transition, the k’s of the bottom of the conduction band and the k’s of the tops of the valence bands are different. Due to the spin-orbit interaction, the quantum states are indicated by j = l + s, where l is the angular momentum operator and s is the spin operator. Therefore, as indexes of the wave functions, we can use the quantum numbers j and m j , which represent the eigenvalues of operators j and jz , respectively. The relations between the operators and the eigenvalues are summarized in Table 1.1. When we express the wave functions as | j, m j , the wave functions of the valence bands under the second-order perturbation are expressed as follows: For a heavy hole band, 3 3 1 , 2 2 = √2 |(x + i y)α, 3 , − 3 = √1 |(x − i y)β, 2 2 2

(1.49)

for a light hole band, 3 1 1 , 2 2 = √6 |2zα + (x + i y)β, 3 , − 1 = √1 |2zβ − (x − i y)α, 2 2 6

(1.50)


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and for a split-off band, 1 1 1 , 2 2 = √3 |zα − (x + i y)β, 1 , − 1 = √1 |zβ + (x − i y)α. 2 2 3

(1.51)

1.3 QUANTUM STRUCTURES Quantum Effects Semiconductor structures whose sizes are small enough that their quantum effects, such as splitting of energy bands and the tunneling effect, may be significant are called quantum structures. Square Well Electrons in quantum structures feel both the periodic potential of crystals and the quantum well potential. First, the energy eigenvalues and wave functions of a particle in a square well potential are reviewed briefly. As shown in Fig. 1.3, we assume that a carrier is present in a square potential well V (r) as V (r) =

0 ∞

inside the well, at the boundaries

(1.52)

When the potential well is a cube with side L, the boundary conditions for a wave function ϕ(x, y, z) are given by ϕ(0, y, z) = ϕ(L , y, z) = 0, ϕ(x, 0, z) = ϕ(x, L , z) = 0, ϕ(x, y, 0) = ϕ(x, y, L) = 0.

ϕ

ϕ

FIGURE 1.3 Square well potential.

(1.53)


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QUANTUM STRUCTURES

15

FIGURE 1.4 Wave function ϕ and energy eigenvalues E in a one-dimensional square well potential.

Under these boundary conditions, the wave function ϕ(x, y, z) and energy eigenvalue E are obtained as

8 sin k x x · sin k y y · sin k z z, L3 2 E= (k x 2 + k y 2 + k z 2 ), 2m nyπ nx π nz π kx = , ky = , kz = (n x , n y , n z = 1, 2, 3, . . .). L L L ϕ(x, y, z) =

(1.54)

Figure 1.4 shows the wave function ϕ and energy eigenvalues E for a one-dimensional square well potential. The energy eigenvalues E are discrete and their values are proportional to a square of the quantum number n x . In addition, with a decrease in L, the energy separation between the energy levels increases.

Potential Well and Energy Barrier Figure 1.5 shows the energies of the conduction band and valence bands at k = 0 for GaAs, which is sandwiched by AlGaAs layers. The low-energy regions for electrons in the conduction band and holes in the valence bands are called potential wells. Note that in Fig. 1.5, the vertical line shows the energy of the electrons, and the energy of the holes decreases with an increase in the height of the vertical line. When the width of the potential well L z is on the order of less than several tens of nanometers, the potential well is called the quantum well. The bandgaps of AlGaAs layers are higher than those of GaAs. As a result, these AlGaAs layers become the energy barriers for GaAs and are called energy barrier layers. At the interfaces of the quantum well and the barriers, energy differences exist in the conduction band E c , and in the valence bands, E v , and are called band offsets.


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Band Offset

Band Offset

Energy Barrier Potential Energy Barrier Layer Well Layer FIGURE 1.5 Quantum well structure.

Effective Mass Approximation The periods of the potential for semiconductor crystals are represented by lattice constants which are on the order of 0.5 nm. In contrast, the thickness of potential wells or barriers in quantum structures is between an order of nanometers and that of several tens of nanometers. Hence, in quantum structures, electrons and holes feel both periodic and quantum potentials. If we use effective mass, the effect of the periodic potential is included in the effective mass, as shown in (1.12), and we only have to consider the quantum potential, referred to as the effective mass approximation. Under effective mass approximation, a wave function in the quantum structure is obtained as a product of base function ψ and envelope function ϕ. As base function ψ we use a wave function for the periodic potential: ψn (k, r) = exp(i k · r)u nk (r),

u n (k, r) = u nk (r + R).

(1.55)

As the envelope function ϕ, we use a wave function for the quantum potential. For example, for a cube with a side length of L and infinite potential at the boundaries, the envelope function ϕ is given by ϕ(x, y, z) =

8 sin k x x · sin k y y · sin k z z. L3

(1.56)

Figure 1.6 shows one-, two-, and three-dimensional quantum wells. A sheet in which only L z is of quantum size, as shown in Fig. 1.6(a), is called a one-dimensional quantum well or simply, a quantum well. A stripe in which only L y and L z are quantum sizes, as shown in Fig. 1.6(b), is called a two-dimensional quantum well


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QUANTUM STRUCTURES

(a)

(b)

17

(c)

FIGURE 1.6 (a) One-, (b) two-, and (c) three-dimensional quantum wells.

or a quantum wire. A box whose L x , L y , and L z are all quantum sizes, as shown in Fig. 1.6(c), is called a three-dimensional quantum well or a quantum box. The energies of the carriers, which are confined completely in the sheet shown in Fig. 1.6(a), are written

Ex y

E = Ex y + Ez , 2 π 2 2 2 π 2 2 = (n x + n y 2 ), E z = nz , ∗ 2 2m L 2m ∗ L z 2

(1.57)

where is Dirac’s constant; m ∗ is the effective mass of the carrier; and n x , n y , and n z are quantum numbers. If n x , n y , and n z are of the same order, we have E x y E z . Density of States As an example, let us calculate the density of states in a one-dimensional quantum well for n z = 1. The density of states is determined by the number of combinations of n x and n y . When n x and n y are large enough, the combinations (n x , n y ) for constant energy E x y are represented by the points on the circumference of a circle with radius r , which is given by r 2 = nx 2 + n y 2 =

2m ∗ L 2 Ex y . 2 π 2

(1.58)

Because both n x and n y are positive numbers, the number S of combinations (n x , n y ) is given by the area of a quarter circle with radius r . As a result, S is expressed as S=

1 2 π π 2m ∗ L 2 m∗ L 2 πr = (n x 2 + n y 2 ) = E = Ex y . x y 4 4 4 2 π 2 22 π

(1.59)

Considering the up- and down-spins, the number of states N is twice as large as S, which is written N = 2S =

m∗ L 2 Ex y . 2 π

(1.60)


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Substituting E x y = E − E z=1 into (1.60), the electron concentration n for the energy between zero and E is obtained as n=

N L2 L

= z

m∗ 2 π L

(E − E z=1 ).

(1.61)

z

When we define the density of states per volume for the energy between E and E + dE as ρ1 (E), we have ρ1 (E) dE ≡ n.

(1.62)

From (1.61) and (1.62), we obtain ρ1 (E) ≡

dn m∗ = 2 . dE π Lz

(1.63)

The densities of states for n z = 2, 3, . . . are calculated similarly, and the results are shown in Fig. 1.7(a). Here L z is 3 nm; m ∗ is 0.08m, where m is the electron mass in vacuum; and the ρ1 (E) for n z = 1, 2, and 3 are indicated as ρ11 , ρ12 , and ρ13 , respectively. It should be noted that the density of states for a one-dimensional quantum well is a step function. In contrast, the bulk structures have a density of states such that ρ0 (E) =

(2m ∗ )3/2 1/2 E , 2π 2 3

(1.64)

which is proportional to E 1/2 as shown by a dashed line, because the number of states is represented by the number of the points existing in one-eighth of a sphere with radius r .

Bulk

5

5

4

4

/

4

/

/

5

3

3

3

2

2

2

1

1

1

0

0

0

2 4 6 Energy (eV)

(a)

8

0

2 4 6 Energy (eV)

(b)

8

0

0

2 4 6 Energy (eV)

8

(c)

FIGURE 1.7 Density of states for (a) one-, (b) two-, and (c) three-dimensional quantum wells.


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19

SUPERLATTICES

If we set L y = L z = L, the energies of the carriers, which are confined completely in the wire shown in Fig. 1.6(b), are written E = E x + E yz , 2 π 2 2 2 π 2 2 Ex = n x , E yz = (n y + n z 2 ). 2 ∗ 2m L x 2m ∗ L 2

(1.65)

For a pair of quantum numbers (n y , n z ), the density of states ρ2 (E) is obtained as √ √ 2m ∗ −1/2 2m ∗ ρ2 (E) = E = (E − E yz )−1/2 . x π L 2 π L 2

(1.66)

The result calculated for (1.66) is shown in Fig. 1.7(b). When the energy E is equal to E yz , the density of states ρ2 (E) is infinity. When E exceeds E yz , ρ2 (E) decreases in proportion to (E − E yz )−1/2 , which leads to a density of states ρ2 (E) with a sawtoothed shape. If we set L x = L y = L z = L, the energies of the carriers, which are confined completely in the box shown in Fig. 1.6 (c), are written E = Ex + E y + Ez , 2 π 2 2 2 π 2 2 2 π 2 2 Ex = nx , E y = n y , Ez = nz . ∗ 2 ∗ 2 2m L 2m L 2m ∗ L 2

(1.67)

It should be noted that the energy eigenvalues are completely discrete. The density of states ρ3 (E) is a delta function, which is written ρ3 (E) = 2

δ(E − E x − E y − E z ).

(1.68)

n x ,n y ,n z

Figure 1.7(c) shows the number of states per volume and the density of states in a three-dimensional quantum well. With an increase in the dimension of the quantum wells, the energy bandwidths of the densities of states decrease. Therefore, the energy distribution of the electron concentrations narrows with an increase in the dimension of the quantum wells, as shown in Fig. 1.8. Therefore, the optical gain concentrates on a certain energy (wavelength). As a result, in quantum well lasers, a low threshold current, a high speed modulation, low chirping, and a narrow spectral linewidth are expected.

1.4

SUPERLATTICES

Array quantum structures and solitary structures are called superlattices. From the viewpoint of the potential, superlattices are classified as follows. Figure 1.9 shows three kinds of potentials of superlattices. The horizontal direction indicates the position of the layers, and the vertical direction represents the energy of the electrons.


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10

10

8

8

8 /

/

/

10

6

6

6

4

4

4

2

2

2

0

0 20 40 60 80 100

0

0 20 40 60 80 100

0

0 20 40 60 80 100

(b)

(a)

(c)

FIGURE 1.8 Energy distribution of electron concentrations in quantum wells: (a) bulk structure; (b) one-dimensional quantum; (c) two-dimensional quantum structure.

With an increase in height, the energy of electrons increases and that of holes decreases. As shown in Fig. 1.9(a), in a type I superlattice , the position of the potential well for electrons in the conduction band is the same as that for holes in the valence band. Therefore, both electrons and holes are confined in semiconductor layer B, which has a narrower bandgap than that of semiconductor layer A. In the type II superlattice in Fig. 1.9(b), the electrons in the conduction band are confined in semiconductor layer B, and the holes in the valence band are confined in semiconductor layer A. In the type III superlattice in Fig. 1.9(c), the energy of the conduction band of semiconductor layer B overlaps that of the valence band of semiconductor layer A, which results in a semimetal. Note that in the literature, types II and III are sometimes called types I and II, respectively. From the perspective of the period, superlattices are classified as follows. Figure 1.10 shows the relationships between the characteristics of superlattices and the thickness of barriers and wells. When each layer thickness is larger than several tens of nanometers, only the bulk characteristics are observed. If the barrier thickness is less than several tens of nanometers, the quantum mechanical tunneling effect appears. When the barriers are thick and only the wells are thin, quantum energy levels are formed in the wells. If such wells are used as the active layers in light-

BABAB

BABAB

BABAB

(a)

(b)

(c)

FIGURE 1.9 Classification of super lattices by potential: (a) type I; (b) type II; and (c) type III.


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REFERENCES

21

Electron Well Thickness (nm)

Bulk Region Tunneling Effect

Minizone

Quantum Level

Bending of Brillouin Zone

Quantum Level

Barrier Thickness (nm) FIGURE 1.10 Classification of superlattices by period.

emitting devices, narrow light emission spectra are obtained. When both barriers and wells are thinner than about 10 nm, the wave functions of a well start to penetrate adjacent wells. As a result, the wave functions of each well overlap each other, which produces minizones and induces Bloch oscillations or negative resistances. When the thickness of both barriers and wells decreases further, down to the order of atomic layers, bending of Brillouin zones appears, which will transform indirect transition materials into direct transition materials.

REFERENCES 1. E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids 1, 249 (1957). 2. C. Kittel, Quantum Theory of Solids, 2nd ed., Wiley, New York, 1987. 3. G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Halsted Press, New York, 1988. 4. S. Datta, Quantum Phenomena, Addison-Wesley, Reading, MA, 1989. 5. L. I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill, New York, 1968. 6. A. Messiah, Quantum Mechanics, Vols.1 and 2, North-Holland, Amsterdam, 1961. 7. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004.


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2 OPTICAL TRANSITIONS

2.1

INTRODUCTION

Excitation and Relaxation The electrons in semiconductors tend to stay in the lowest-energy state, because the lowest-energy state is most stable. Electrons that are excited by thermal energy, light, or electron beams absorb these energies and transit to high-energy states. Transitions of the electrons from low-energy states to high-energy states are called excitations. Electrons in high-energy states, however, transit to low-energy states in certain lifetimes, because high-energy states are unstable. Transitions of excited electrons from high-energy states to low-energy states are called relaxations. Excitation and relaxation processes between a valence band and a conduction band are shown in Fig. 2.1. Recombination In semiconductors, transitions of electrons from high-energy states to low-energy states are interpreted as recombinations of electrons and holes. Such recombinations are both radiative and nonradiative. Radiative recombinations emit photons, and the energies of photons are given by differences in the energies between the initial and final energy states related to the transitions. Nonradiative recombinations do not emit photons. When phonons are emitted to crystal lattices or electrons are trapped in defects, the transition energy is Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright

23


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OPTICAL TRANSITIONS

Conduction Band

Electron

Electron

Electron

Hole

Hole

Electron Valence Band

Stable State

Excitation

Relaxation

Stable State

FIGURE 2.1 Excitation and relaxation.

transformed into heat. When three-body collisions happen among two electrons and one hole, or one electron and two holes, the transition energy is used to excite an electron, and these three-body collisions do not emit light. These processes, known as Auger processes, are also categorized as nonradiative recombinations. To obtain high light-emitting efficiency, nonradiative recombinations have to be minimized. However, to enhance modulation characteristics, nonradiative recombination centers may be induced intentionally in the active layers, because the carrier lifetimes are reduced.

2.2 DIRECT AND INDIRECT TRANSITIONS Wave Vector In a conduction band, electrons tend to reside at the bottom, because the energy of the electron is lowest at the bottom. In a valence band, holes tend to reside at the top, because the energy of the hole is lowest at the top. Therefore, recombinations of electrons and holes occur most frequently between the bottom of the conduction band and the top of the valence band. So, let us consider transitions of the electrons from the bottom of a conduction band to the top of a valence band. A semiconductor in which the bottom of the conduction band and the top of the valence band are placed at a common wave vector k is called a direct transition semiconductor. A semiconductor in which the bottom of the conduction band and the top of the valence band have different k-values is called an indirect transition semiconductor. Direct and indirect transitions are shown schematically in Fig. 2.2. Phonon In electron transitions, the energy and momentum are conserved. Therefore, phonons with momentum k do not take part in direct transitions, because a difference in the wave vectors k for the bottom of the conduction band and the top of the valence band is 0. In contrast, transitions of phonons with momentum k = 0 occur in indirect transitions to satisfy the momentum conservation law. Hence, in direct transitions,


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LIGHT-EMITTING PROCESSES

(a)

25

(b)

FIGURE 2.2 (a) Direct and (b) indirect transition semiconductors.

the transition probabilities are determined by electron transition probabilities alone; in indirect transitions, the transition probabilities are given by the product of electron and phonon transition probabilities. As a result, the transition probabilities of direct transitions are much higher than those of indirect transitions. Consequently, direct transition semiconductors such as InGaAs, InGaAsP, and AlGaAs are superior to indirect transition semiconductors such as Si and Ge for light-emitting devices.

2.3

LIGHT-EMITTING PROCESSES

Luminescence Light emission due to radiative recombinations is called luminescence. Lightemitting processes are classified according to the lifetime, excitation methods, and energy states related to the transitions. Lifetime With regard to lifetime, there are two light emissions: fluorescence, with a short lifetime of 10−9 to 10−3 s, and phosphorescence, with a long lifetime of 10−3 s to 1 day. Excitation Luminescence that is accompanied by optical excitation or optical pumping, called photoluminescence, is used widely to characterize materials. Optical excitation is also used to pump dye lasers and solid-state lasers. When the photon energy of pumping light is ω1 and that of the luminescence is ω2 , luminescence with ω2 < ω1 is called Stokes luminescence and that with ω2 > ω1 is called anti-Stokes


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OPTICAL TRANSITIONS

luminescence. Luminescence that is generated by electrical excitation, called electroluminescence, has been used for panel displays. In particular, luminescence by injection of electric current, called injection-type electroluminescence, has been used for light-emitting diodes (LEDs) and semiconductor lasers or laser diodes (LDs). In LEDs and LDs, the carriers are injected into the active layers by forward bias across the pn-junctions. Luminescence due to electron beam irradiation, called cathodoluminescence, has been adopted to characterize materials. Luminescence induced by mechanical excitation using stress is called triboluminescence, and that induced by thermal excitation is called thermoluminescence. Luminescence during a chemical reaction, referred to as chemiluminescence, has not been reported in semiconductors.

2.4 SPONTANEOUS EMISSION, STIMULATED EMISSION, AND ABSORPTION Emissions and Absorption Figure 2.3 shows spontaneous emission, stimulated emission (or induced emission), and absorption. In the spontaneous emission shown in Fig. 2.3(a), an excited electron decays in a certain lifetime and a photon is emitted irrespective of incident light. In contrast, in the stimulated emission shown in Fig. 2.3(b), incident light induces a radiative transition of an excited electron. Light emitted due to stimulated emission has the same wavelength, phase, and direction as those of incident light. Therefore, the light generated by stimulated emission is highly monochromatic, coherent, and directional. In stimulated emission, one incident photon generates two photons: the incident photon itself and a photon emitted due to stimulated emission. As a result, the incident light is amplified by stimulated emission. In the absorption shown in Fig. 2.3(c), an electron transits from a lower-energy state to a higher-energy state by absorbing energy from the incident light. Because this transition is induced

Electron

Electron

Electron Conduction Band

Valence Band Hole (a)

Hole (b)

Hole (c)

FIGURE 2.3 (a) Spontaneous emission; (b) stimulated emission; (c) absorption.


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OPTICAL GAINS

27

by the incident light, it is sometimes called induced absorption. It should be noted that spontaneous absorption does not exist. Thermal Equilibrium When light is incident on a material, stimulated emission and absorption take place simultaneously. In thermal equilibrium, there are more electrons in a lower-energy state than in a higher-energy state. Therefore, in thermal equilibrium, the absorption exceeds the stimulated emission, and only the absorption is observed when light is incident on a material. Population Inversion To obtain a net optical gain, we have to make the number of electrons in a higherenergy state greater than the number in lower-energy state. This condition is called inverted population, or population inversion, because the electron population is inverted compared with that in thermal equilibrium. In semiconductors, the population inversion due to optical pumping or electric current injection generates many electrons at the bottom of the conduction band and many holes at the top of the valence bands. Laser Laser oscillators use fractions of spontaneous emission as optical input and amplify the fractions by stimulated emission under population inversion. Once the optical gains in laser oscillators exceed the optical losses, laser oscillations take place. The term laser is an acronym for “light amplification by stimulated emission of radiation” and is used alone as a noun with a meaning laser oscillator. Semiconductor light-emitting devices employ spontaneous emission and are used in remote-control transmitters, switch lights, brake lights, displays, and traffic signals. In contrast, laser diodes are oscillators of light using stimulated emission and are used as light sources in lightwave communications, compact disks (CDs), magnetooptical disks (MOs), digital videodisks (DVDs), laser beam printers, laser pointers, and bar-code readers.

2.5

OPTICAL GAINS

Laser Oscillator Figure 2.4 shows a laser oscillator where a fraction of the spontaneous emission is used as input and the optical gain is produced by the stimulated emission. To feed back light, optical resonators or optical cavities, which are composed of reflectors or mirrors, are adopted. Due to this configuration, characteristics of lasers are affected by the optical gains and optical resonators. It should be noted that all LDs use optical resonators.


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OPTICAL TRANSITIONS

Optical Feedback by Reflectors Fraction of Spontaneous Emission

Optical Gain

Laser Light

FIGURE 2.4 Laser oscillator.

Spontaneous Emission As explained earlier, a fraction of the spontaneous emission is used as the input of a laser. However, note that all the light emitted spontaneously cannot be used as input, because it has different wavelengths, phases, and propagation directions. Among these, only light that has wavelengths within the optical gain spectrum and that satisfies the resonance conditions of optical resonators can be a source of laser oscillations. Other spontaneous emission, which does not satisfy the resonance conditions of optical resonators, is readily emitted outward without obtaining sufficient optical gain for laser oscillation. The light amplified by stimulated emission has the same wavelength, phase, and propagation direction as those of the input light. Therefore, the laser light is highly monochromatic, bright, coherent, and directional. Quasi-Fermi Levels When a lot of electrons are excited in the conduction band and a lot of holes are generated in the valence band by optical pumping or electrical current injection, the carrier distribution is in nonthermal equilibrium. As a result, one Fermi level E F cannot describe the distribution functions of the electrons and holes. In this case it is useful to determine the distribution functions by assuming that the electrons in the conduction band and the holes in the valence band are governed separately by Fermi–Dirac distribution. For this purpose we introduce quasi-Fermi levels E Fc and E Fv , which are defined as E c − E Fc n = Nc exp − , kB T E Fv − E v p = Nv exp − , kB T 2πm e kB T 3/2 Nc = 2 Mc , h2 2π m h kB T 3/2 Nv = 2 . h2

(2.1)


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OPTICAL GAINS

29

Here n and p are the electron and hole concentration, respectively; E c and E v are the energy at the bottom of the conduction band and that at the top of the valence band, respectively; kB = 1.3807 × 10−23 J/K is the Boltzmann constant; T is the absolute temperature; Nc and Nv are the effective density of states for the electrons and holes, respectively; m e and m h are the effective mass of the electrons and holes, respectively; Mc is the number of the band edges of the conduction band; and h is Planck’s constant. From (2.1), these quasi-Fermi levels E Fc and E Fv are written as E Fc = E c + kB T ln E Fv

n , Nc

p = E v − kB T ln . Nv

(2.2)

Using E Fc and E Fv , we can express a distribution function for the electrons in the valence band f 1 with the energy E 1 and that for the electrons in the conduction band f 2 with the energy E 2 as f1 =

1 , exp [(E 1 − E Fv ) /kB T ] + 1 (2.3)

1 f2 = . exp [(E 2 − E Fc ) /kB T ] + 1 It should be noted that the distribution function for the holes in the valence band is given by [1 − f 1 ]. Interaction Between Light and a Direct Transition Semiconductor We assume that light, which has a photon energy of E 21 = E 2 − E 1 and a photon concentration of n ph (E 21 ), interacts with a direct transition semiconductor as shown in Fig. 2.5, where E c is the energy at the bottom of the conduction band and E v is the energy at the top of the valence band. Stimulated Emission Rate The concentration n 2 of an electron, which occupies a state with energy E 2 in a conduction band, is expressed as n 2 = ρc (E 2 − E c ) f 2 ,

(2.4)

where ρc (E 2 − E c ) is the density of states, which is a function of E 2 − E c , and f 2 is the distribution function in (2.3).


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OPTICAL TRANSITIONS


Valence Band Hole FIGURE 2.5 Schematic model for radiation and absorption.

The concentration p1 of a hole, which occupies a state with energy E 1 in the valence band, is written p1 = ρv (E v − E 1 )[1 − f 1 ],

(2.5)

where ρv (E v − E 1 ) is the density of states, which is a function of E v − E 1 , and f 1 is the distribution function in (2.3). When we express the transition probability per unit time for the transition from E 2 to E 1 as B21 , the stimulated emission rate per unit volume r21 (stim) is given by r21 (stim) = B21 n 2 p1 n ph (E 21 ) = B21 n ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 ) f 2 [1 − f 1 ].

(2.6)

Absorption Rate The concentration p2 of an empty state, which is not occupied by electrons with energy E 2 in the conduction band, is expressed as p2 = ρc (E 2 − E c )[1 − f 2 ].

(2.7)

The concentration n 1 of the electron, which occupies a state with energy E 1 in the valence band, is written n 1 = ρv (E v − E 1 ) f 1 .

(2.8)

When we express the transition probability per unit time for the transition from E 1 to E 2 as B12 , the absorption rate per unit volume r12 (abs) is given by r12 (abs) = B12 p2 n 1 n ph (E 21 ) = B12 n ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 ) f 1 [1 − f 2 ].

(2.9)


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OPTICAL GAINS

31

Spontaneous Emission Rate When we express the transition probability per unit time for the transition from E 2 to E 1 as A21 and use (2.4) and (2.5), the spontaneous emission rate per unit volume r21 (spon) is given by r21 (spon) = A21 n 2 p1 = A21 ρc (E 2 − E c )ρv (E v − E 1 ) f 2 [1 − f 1 ].

(2.10)

Blackbody Radiation First, we consider the emissions and absorption in thermal equilibrium. In thermal equilibrium, the carrier distributions are described by one Fermi level E F , and the emissions balance the absorption. Therefore, we obtain r21 (stim) + r21 (spon) = r12 (abs), E F1 = E F2 = E F .

(2.11)

Substituting (2.6), (2.9), and (2.10) into (2.11) results in n ph (E 21 ) =

A21 , B12 exp E 21 /(kB T ) − B21

(2.12)

and the blackbody radiation theory gives n ph (E 21 ) =

8π n r 3 E 21 2 . h 3 c3 exp(E 21 /kB T ) − h 3 c3

(2.13)

Here n r is the effective refractive index of a material, h is Planck’s constant, and c = 2.99792458 × 108 m/s is the speed of light in vacuum. Comparing (2.12) with (2.13), we have A21 = m(E 21 )B, B21 = B12 = B, 8πn r 3 E 21 2 m(E 21 ) = , h 3 c3

(2.14)

where A21 is called Einstein’s A coefficient, B is called Einstein’s B coefficient, and m(E 21 ) is the mode density. Net Stimulated Emission The third equation in (2.14) suggests that spontaneous emission takes place for all the modes with an energy of E 21 , whereas stimulated emission and absorption occur only for the mode corresponding to the incident light.


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OPTICAL TRANSITIONS

In nonthermal equilibrium conditions, in which many free carriers exist, radiations do not balance with absorption. When light exists in a material, stimulated emission and absorption take place simultaneously. Therefore, the net stimulated emission rate r 0 (stim) is given by r 0 (stim) = r21 (stim) − r12 (abs) = Bn ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 )[ f 2 − f 1 ] A21 = n ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 )[ f 2 − f 1 ]. m(E 21 )

(2.15)

From (2.15), to obtain the net stimulated emission r 0 (stim) > 0, it is found that we need f2 > f1,

(2.16)

which indicates that population inversion is required in the semiconductors. With the help of (2.3), (2.16) is reduced to E Fc − E Fv > E 2 − E 1 = E 21 ,

(2.17)

which is known as the Bernard–Duraffourg relation. In LDs, a typical carrier concentration for r 0 (stim) > 0 is on the order of 1018 cm−3 . Optical Power Gain Coefficient per Unit Length The power optical gain coefficient per unit length g is defined as dI = g I, dz

(2.18)

where I is the light power per unit area and z is a coordinate for a propagation direction of the light. Note that the field optical gain coefficient gE , which is the gain coefficient for the electric field, is given by gE = g/2. The light power per unit area I is expressed as I = vg E 21 n ph (E 21 ) = vg ωn ph (E 21 ), dω c vg = = , dk nr dn ph (E 21 ) dI = vg E 21 = vg E 21 r 0 (stim), dt dt

(2.19)

where vg is a group velocity of the light in the material, E 21 = ω is the photon energy, and n ph (E 21 ) is the photon concentration.


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OPTICAL GAINS

33

Because the position z is a function of the time t, we have dI dt dI = = dz dz dt

dz dt

−1

dI 1 dI = , dt vg dt

(2.20)

dI = g I = g vg E 21 n ph (E 21 ). dz From (2.19) and (2.20), we have r 0 (stim) = g vg n ph (E 21 ).

(2.21)

As a result, a large r 0 (stim) leads to a large g. Using (2.15), (2.19), and (2.21), we can also express g as g=

r 0 (stim) n r nr = Bρc (E 2 − E c )ρv (E v − E 1 )[ f 2 − f 1 ]. n ph (E 21 ) c c

(2.22)

Einstein’s A and B Coefficients From time-dependent quantum mechanical perturbation theory, Einstein’s B coefficient in (2.14) is given by B=

e2 h 1| p|22 . 2m 2 ε0 n r 2 E 21

(2.23)

Here e is the elementary electric charge; h is Planck’s constant; m is the electron mass; ε0 is permittivity in vacuum; n r is the refractive index of the material; E 21 is the photon energy; p is the momentum operator; and 1| and |2 are the bra and ket vectors, corresponding to the wave functions of the valence band and the conduction band in a steady state, respectively. Substituting (2.23) into (2.14), we can write Einstein’s A coefficient as A21 =

4πe2 n r E 21 1| p|22 . m 2 ε0 h 2 c3

(2.24)

Einstein’s A and B coefficients can be written B=

π e2 ω πω 2 1|r|22 = µ , 2 ε0 n r ε0 n r 2 2nr ω3 2 A21 = µ , ε0 hc3 µ2 = 1|er|22 .

(2.25)


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OPTICAL TRANSITIONS

Here µ is the dipole moment, and the following relations on the momentum operator p are used: p=m

dr , dt

(2.26)

1 d 1| p|2 = 1|r|2 = i ω1|r|2, m dt

(2.27)

where r ∝ e i ωt is assumed. In the explanations above, Einstein’s B coefficient is defined to be in proportion to the photon concentration. Therefore, the quantum mechanical transition rate for the stimulated emission is equal to Einstein’s B coefficient. Transitions Between Various Energy States In semiconductors, transitions take place between various energy states in the conduction and valence bands, as shown in Fig. 2.6. If we set E 21 = E and E 2 − E c = E , the electron energy in the valence band E for the transition allowed is given by E = E − E. Therefore, integrating (2.15) with respect to E gives the net stimulated emission rate r 0 (stim) as

e2 h n ph (E) r (stim) = 2m 2 ε0 n r 2 E 0

∞

1| p|22 ρc (E )ρv (E )[ f 2 (E ) − f 1 (E )] dE , (2.28)

0

where (2.23) is used. From(2.22) and (2.28), the power optical gain coefficient g(E) is given by g(E) =

e2 h 2 2m ε0 n r cE

∞

1| p|22 ρc (E )ρv (E )[ f 2 (E ) − f 1 (E )] dE .

0


Valence Band Hole FIGURE 2.6 Transition with a constant photon energy.

(2.29)


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OPTICAL GAINS

35

k-Selection Rule In (2.29), ρc (E )ρv (E ) is considered to be the density- of-state pairs related to optical transitions. The density-of-state pairs are also expressed using the law of momentum conservation (k-selection rule) and the law of energy conservation (E 21 = E 2 − E 1 ). Under the k-selection rule, optical transitions take place for k = k1 = k2 only in direct transition semiconductors. Therefore, we can define the density-of-state pairs or the reduced density-of-states ρred (E 21 ) as ρred (E 21 ) dE 21 ≡ ρv (E 1 ) dE 1 = ρc (E 2 ) dE 2 ,

(2.30)

dE 21 = dE 1 + dE 2 .

(2.31)

where

From (2.30) and (2.31), we have

1 1 ρred (E 21 ) = + ρv (E 1 ) ρc (E 2 )

−1

.

(2.32)

When the conduction and valence bands are assumed to be parabolic, the energies E 1 and E 2 are written 2 k 2 , 2m h 2 k 2 E2 = Ec + , 2m e

E1 = Ev −

(2.33) (2.34)

where m h and m e are the effective masses of the holes and electrons, respectively. As a result, E 1 and E 2 are rewritten me me + mh mh E2 = Ec + me + mh

E1 = Ev −

E 21 − E g ,

(2.35)

E 21 − E g .

(2.36)

The number of state pairs per volume associated with the optical transitions for photon energies between E 21 and E 21 + dE 21 is given by ρred (E 21 ) [ f 2 (E 21 ) − f 1 (E 21 )] dE 21 .

(2.37)


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OPTICAL TRANSITIONS

FIGURE 2.7 Power optical gain coefficient under the k-selection rule.

We assume that the energy states related to optical transitions have a Lorentzian spectrum with a width of /τin : L(E 21 ) =

1 /τin , π (E 21 − E)2 + (/τin )2

(2.38)

where τin is the relaxation time due to electron scatterings and transitions. In this case, the power optical gain coefficient g(E) is expressed as [1] e2 h g(E) = 2m 2 ε0 n r cE

∞

1| p|22 ρred (E 21 ) [ f 2 (E 21 ) − f 1 (E 21 )] L(E 21 ) dE 21 ,

0

(2.39) which is plotted in Fig. 2.7 for bulk structures with τin = 10−13 s. It is found that relaxation is equivalent to band tailing, which contributes to optical transitions in E < E g . With an increase in the carrier concentration n, the gain peak shifts toward a higher energy. This shift in the gain peak is caused by the band filling effect [2], in which the electrons in the conduction band and the holes in the valence band occupy each band from the band edges. Because higher-energy states are more dense than lower-energy states, the optical gain in the higher-energy states is larger than that in the lower-energy states. From (2.10) and (2.24), the spontaneous emission rate r21 (spon) is expressed as 4π e2 n r E r21 (spon) = 2 2 3 m ε0 h c

∞

1| p|22 ρc (E )ρv (E ) f 2 (E )[1 − f 1 (E )] dE . (2.40)

0

Because f 2 and [1 − f 1 ] are always positive, spontaneously emitted light distributes at a higher energy level than that of the gain peak. Figure 2.8 shows optical power for stimulated emission (laser light) and spontaneous emission as a function of the photon energy.


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Optical Power

REFERENCES

37

Stimulated Emission Spontaneous Emission

Photon Energy FIGURE 2.8 Stimulated and spontaneous emission.

REFERENCES 1. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004. 2. J. I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1975. 3. H. C. Casey, Jr., and M. B. Panish, Heterostructure Lasers A, Academic Press, San Diego, CA, 1978. 4. L. I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill, New York, 1968.


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3 OPTICAL WAVEGUIDES

3.1

INTRODUCTION

Free Space and Optical Waveguides An optical beam that propagates in free space expands its beam width due to diffraction. For example, if a light beam with a wavelength of 1 µm, which is emitted from a slit with a diameter of 1 mm, propagates by 100 m, its beam width widens to about 10 cm. In contrast, optical waveguides confine light to themselves during the propagation of light [1–6] and contribute to efficient amplification of light for LDs. In typical double heterostructure LDs with bulk active layers, the thickness of the active layers is on the order of 0.1 µm, the beam width is 0.3 µm, and the cavity length is 300 µm. If there are no optical waveguides, the beam width expands up to 500 µm by a single-way propagation, where a wavelength of light is assumed to be 0.5 µm. In LDs, only the active layer has an optical gain, and it is extremely inefficient to amplify a 500 µm-wide laser beam by a 0.1 µm-thick active layer. Therefore, to amplify light efficiently, optical waveguides are indispensable for LDs. Index and Gain Guiding Figure 3.1 shows schematic cross-sectional views of optical waveguides. The operating principles of the optical waveguides are divided into index guiding and gain guiding. In index guiding, light is confined in a shaded region with a high refractive index n f , surrounded by regions with low refractive indexes n c and n s , as shown in Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright

39


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OPTICAL WAVEGUIDES

g

g

g

g g g

(a)

(b)

FIGURE 3.1 Cross sections of optical waveguides: (a) index guiding; (b) gain guiding.

Fig. 3.1(a). In gain guiding, light is confined in a shaded region with a high optical gain coefficient gf , surrounded by regions with low optical gain coefficients gc and gs , as shown in Fig. 3.1(b). Complex Refractive Index Using the complex refractive index n˜ = n r − i κ,

(3.1)

where n r is the refractive index and κ is the extinction coefficient, the field optical gain coefficient gE is expressed as ω gE = − κ. c

(3.2)

Note that the power optical gain coefficient g is equal to 2gE , because the optical power is proportional to the square of the electric field. From (3.1) and (3.2) it can be said that index guiding and gain guiding use the distribution of the refractive index n r and the extinction coefficient κ, respectively. Two- and Three-Dimensional Optical Waveguides From the viewpoint of shape, optical waveguides are classified into a twodimensional optical waveguide (planar optical waveguide) and a threedimensional optical waveguide (strip optical waveguide). A two-dimensional waveguide has a plane much larger than that of a wavelength of light and confines light one-dimensionally, as shown in Fig. 3.2(a). The size of its confinement direction is on the order of only a light wavelength or less. A three-dimensional waveguide confines light two-dimensionally, and the sizes along these two confinement directions are on the order of a light wavelength or less, as shown in Fig. 3.2(b). The remaining direction is the propagation direction of light.


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TWO-DIMENSIONAL OPTICAL WAVEGUIDES

Propagation Direction

41


(a)

(b)

FIGURE 3.2 (a) Two- and (b) three-dimensional optical waveguides.

3.2


Three-Layer Waveguides Figure 3.3 shows a two-dimensional waveguide where film is sandwiched between the cladding layer and the substrate. The refractive indexes of the film, the cladding layer, and the substrate are n f , n c , and n s , respectively. To confine light to the film, we need n f > n s ≥ n c , and n f − n s is usually on the order of 10−3 to 10−1 to confine only a fundamental mode. Snell’s Law Figure 3.4 shows propagation directions of light when light enters from the substrate to the cladding layer through the film. From Snell’s law, the angles θs , θf , and θc , which are formed by the interface normals and the directions of the light, and the refractive indexes n s , n f , and n c have the relation n s sin θs = n f sin θf = n c sin θc .

(3.3)

Critical Angle When θs is equal to π/2, light cannot propagate in the substrate, and the total reflection, with a power reflectivity of 100%, takes place at the interface of the

Cladding Layer Film Substrate FIGURE 3.3 Cross section of a two-dimensional optical waveguide.


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OPTICAL WAVEGUIDES

FIGURE 3.4 Snell’s law.

substrate and the film. When θc is equal to π/2, the total reflection takes place at the interface of the cladding layer and the film. The minimum value of θf to obtain the total reflection is called a critical angle. According to θs = π/2 and θc = π/2 we have two critical angles, θfs and θfc , which are expressed as θfs = sin−1

ns nc , θfc = sin−1 . nf nf

(3.4)

Here θfs ≥ θfc is satisfied under the assumption of n f > n s ≥ n c . Propagation Modes Corresponding to a value of θf , there are three propagation modes, as shown in Fig. 3.5. (a) Radiation mode (θf < θfc ≤ θfs ). The light is not confined in the optical waveguide at all. (b) Substrate radiation mode (θfc < θf < θfs ). Total reflection occurs at the interface of the cladding layer and the film, while refraction takes place at the interface of the film and the substrate. As a result, a fraction of light is radiated to the substrate. (c) Guided mode (θfs < θf < π/2). At both interfaces, total reflections occur. Light is completely confined in the film during the propagation of light.

(a)

(b)

(c)

FIGURE 3.5 Propagation modes: (a) radiation mode; (b) substrate radiation mode; (c) guided mode.


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43

The guided mode shown in Fig. 3.5(c) is indispensable for LDs and photonic integrated circuits. Note that the guided mode does not exist in Fig. 3.5(a) and (b). We now examine the guided mode in detail.

TE and TM Modes In the following we consider a plane wave and define the plane of incidence as a plane on which all the directions of incident light, reflected light, and refracted light coexist. The transverse electric (TE) mode represents linearly polarized light whose electric field E is normal to the plane of incidence. The transverse magnetic (TM) mode represents linearly polarized light whose magnetic field H is normal to the plane of incidence. Figure 3.6 shows the TE and TM modes where the plane of incidence is parallel to the surface of the page. Note that only incident and reflected light are illustrated. From the Fresnel formulas, the amplitude reflectivities are given by

rTE,c =

rTE,s =

rTM,c

n f cos θf − n f cos θf + n f cos θf − n f cos θf +

n c 2 − n f 2 sin2 θf n c 2 − n f 2 sin2 θf

, (3.5)

n s 2 − n f 2 sin2 θf n s 2 − n f 2 sin2 θf

,

n c 2 cos θf − n f n c 2 − n f 2 sin2 θf = , n c 2 cos θf + n f n c 2 − n f 2 sin2 θf

rTM,s =

(a)

(3.6)

n s cos θf − n f n s − n f θf . 2 2 2 2 n s cos θf + n f n s − n f sin θf 2

2

2

sin2

(b)

FIGURE 3.6 (a) TE mode; (b) TM mode.


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OPTICAL WAVEGUIDES

Here the subscripts TE and TM represent the TE and TM modes, respectively; and the subscripts c and s represent the reflection at the cladding layer–film interface and that at the substrate–film interface, respectively. Total Reflection When total reflections take place, |r |2 = 1 is satisfied, where r represents the amplitude reflectivities in (3.5) and (3.6), which are expressed by complex numbers. As a result, the amplitude reflectivity r can be written r = exp( i 2φ),

(3.7)

where φ is the phase shift, which is added to the reflected wave at the interface. Note that in the total reflection, the phase of the reflected wave is shifted at the interface of the reflection. When we rewrite (3.5) and (3.6) in the form of (3.7), the phase shifts φs are expressed as tan φTE,c = tan φTE,s =

tan φTM,c

n f 2 sin2 θf − n c 2 , n f cos θf (3.8) n f 2 sin2 θf − n s 2 , n f cos θf

n f n f 2 sin2 θf − n c 2 = , n c 2 cos θf

tan φTM,s =

(3.9)

nf nf θf − n s , 2 n s cos θf 2

sin2

2

where the subscripts are the same as in (3.5) and (3.6). Monochromatic Coherent Plane Wave Let us consider a monochromatic coherent plane wave that propagates in an optical waveguide with a film thickness of h, as shown in Fig. 3.7. When a time-dependent factor is neglected, the propagating electric field E is expressed as E = E 0 exp[−i k0 n f (±x cos θf + z sin θf )],

(3.10)

where k0 = ω/c is a wave number in vacuum, and the ± sign preceding x represents the positive and negative propagation directions of light along the x-axis, respectively.


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45

FIGURE 3.7 Coordinate system for a guided mode.

FIGURE 3.8 Propagation constant.

As shown in Fig. 3.8, the propagation constant β and phase velocity vp along the z-axis are written β = k0 n f sin θf =

ω . vp

(3.11)

Phase Shifts If we see light from a coordinate system moving toward the positive z-axis direction with phase velocity vp , we observe that light is multireflected along the x-axis. The phase shifts during a round trip of the light along the x-axis are given by ⎧ 0 → h (the first half): ⎪ ⎪ ⎨ h (at the interface of the reflection): x= h → 0 (the second half): ⎪ ⎪ ⎩ 0 (at the interface of the reflection):

k0 n f h cos θf , −2φc , k0 n f h cos θf , −2φs .

Transverse Resonance Condition To obtain a lightwave propagating throughout the optical waveguide without decay, a total phase shift in the round trip must satisfy 2k0 n f h cos θf − 2φc − 2φs = 2 mπ

(m = 0, 1, 2, . . .),

(3.12)

where m is the mode number. Equation (3.12) shows a resonance condition along the x-axis which is normal to the z-axis and is called the transverse resonance condition.


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OPTICAL WAVEGUIDES

In summary, the guiding condition is given by both the transverse resonance condition (3.12) and the total reflection condition θfs < θf < π/2.

Goos–Hänchen Shift When a total reflection takes place, a phase of a lightwave shifts by −2φ at the interface of the reflection. In geometrical optics, this phase shift is considered equivalent to a shift in the optical path at the reflection interfaces, as shown in Fig. 3.9. This shift in the optical path Z , called the Goos–Hänchen shift, is given by

Z=

dφ , dβ

(3.13)

where β is the propagation constant along the z-axis.

Evanescent Wave At each reflection interface, a lightwave propagates along the interface and has a component with a decay constant 1/ X along the x-axis, where X is called the penetration depth. This exponentially decaying wave represents the evanescent wave, and energy does not flow along the x-axis. Using (3.8) and (3.9), the Goos–Hänchen shifts and penetration depths are expressed as k0 Z TE,c = (n f 2 sin2 θf − n c 2 )−1/2 · tan θf , k0 Z TE,s = (n f 2 sin2 θf − n s 2 )−1/2 · tan θf , Z TE,c Z TE,s X TE,c = , X TE,s = , tan θf tan θf

FIGURE 3.9 Goos–Hänchen shift.

(3.14)


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−1 N2 N2 + − 1 , nc2 nf 2 2 −1 N N2 2 2 2 −1/2 = (n f sin θf − n s ) · tan θf + 2 −1 , ns2 nf Z TM,c Z TM,s X TM,c = , X TM,s = . tan θf tan θf

k0 Z TM,c = (n f 2 sin2 θf − n c 2 )−1/2 · tan θf k0 Z TM,s

47

(3.15)

Effective Refractive Index Using a wave number k0 = ω/c in vacuum and the propagation constant β, we define the effective refractive index N as N≡

β = n f sin θf . k0

(3.16)

In LDs, the effective refractive index N is used to analyze the resonance conditions.

Optical Confinement Factor The optical confinement factor is defined as a ratio of the optical power of the light existing in the relevant layer to the total optical power. Because the light distributes as shown in Fig. 3.10, the optical confinement factor f for the film, which is shown as a hatched area, is given by

h

f = 0∞ −∞

|E(x)|2 dx . |E(x)|2 dx

Light Intensity

FIGURE 3.10 Distribution of optical power.

(3.17)


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OPTICAL WAVEGUIDES

Similarly, the optical confinement factors c for the cladding layer and s for the substrate are written

∞

c = h ∞ −∞

|E(x)|2 dx |E(x)| dx 2

0

, s = −∞ ∞ −∞

|E(x)|2 dx .

(3.18)

|E(x)|2 dx

The optical confinement factor is important in the design of the optical losses or optical gains in optical waveguides.

Normalized Expressions for an Eigenvalue Equation The equation for the transverse resonance condition (3.12) is called an eigenvalue equation. Normalizing (3.12) gives us dispersion curves, which are common to all steplike two-dimensional optical waveguides. To obtain a normalized expression of an eigenvalue equation, the asymmetry measure a, normalized wave-guide refractive index b, and normalized frequency, or normalized waveguide thickness V , are introduced as follows: ns2 − nc2 , nf 2 − ns2 N 2 − ns2 b= 2 , nf − ns2 V = k0 h n f 2 − n s 2 . a=

(3.19) (3.20) (3.21)

Substituting (3.8), (3.9), and (3.19)–(3.21) into (3.12), the normalized eigenvalue equation is obtained as √ V 1 − b = mπ + tan−1 χs

b a+b + tan−1 χc , 1−b 1−b

(3.22)

where

χi =

1: (n f /n i )2 :

TE mode TM mode (i = s, c).

(3.23)

The guided modes for the gain guiding waveguides can also be calculated using (3.12) or (3.22) if we replace the refractive index n i (i = s, f, c) with the complex refractive index n˜ i .


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Normalized Waveguide Index


49

From the top

Normalized Frequency FIGURE 3.11 Normalized dispersion curves for guided TE modes.

Cutoff Figure 3.11 shows normalized dispersion curves for guided TE modes. If the refractive index and thickness of each layer are given, we can design optical waveguides using Fig. 3.11. To design optical waveguides, the cutoff condition, in which guided modes do not exist, is important. When the angle of incidence θf is equal to the critical angle θfs , the lightwave is no longer confined in the film, and a fraction of the optical power is radiated to the substrate. In this case, we have N = n s , which results in b = 0 from (3.20). Hence, from (3.22), a normalized frequency for the cutoff condition Vm is given by √ Vm = mπ + tan−1 χc a.

(3.24)

For Vm < V < Vm+1 , guided modes from zeroth to mth order exist as shown in Fig. 3.11, and a lower-order mode has a larger b for a common V value. Therefore, from (3.20), a lower-order mode has a larger effective refractive index, and the fundamental mode (m = 0) has the largest N among the guided modes. Figure 3.12 shows electric fields for fundamental (m = 0), first-order (m = 1), and second-order (m = 2) TE modes, in which the mth-order TE mode is indicated by TEm .


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OPTICAL WAVEGUIDES

FIGURE 3.12 Distributions of electric fields.

Multiple-Layer Waveguide To analyze multiple-layer optical waveguides with more than three layers, as shown in Fig. 3.13, it is convenient to use matrices. When light propagates along the z-axis and the layers are stacked along the x-axis, the relationship between the light at x and xi is expressed by a matrix G, defined as

ψ y (x) ψ y (xi ) = G(x − xi ) . ψz (x) ψz (xi )

(3.25)

...

Here an electric field E and a magnetic field H of the light are assumed to be expressed by a separation-of-variables procedure. As a result, we have ψ y (x) = E y (x), ψz (x) = iZ 0 Hz (x) for the TE mode and ψ y (x) = Z 0 Hy (x), ψz (x) = −iE z (x) for the TM√mode, where a subscript indicates a component along each coordinate and Z 0 = µ0 /ε0 is the impedance of vacuum.

FIGURE 3.13 Optical waveguide with multilayers.


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51

We introduce parameters such as

κi = k0 2 n i 2 − β 2 ,

γi = β 2 − k0 2 n i 2 , ζi =

1: ni 2:

TE mode TM mode

(3.26) (3.27)

(3.28)

where k0 is the wave number of the light in vacuum, n i is the (complex) refractive index of the ith layer, and β is the propagation constant, which is calculated by solving the eigenvalue equation described below. For the ith layer with β < k0 n i , matrix G is written ⎡

⎤ k0 ζi sin κi x ⎥ ⎢ cos κi x κi G(x) = ⎣ κi ⎦. − sin κi x cos κi x k0 ζi

(3.29)

For the ith layer with β > k0 n i , matrix G is written ⎡

⎤ k0 ζi cos γ x sin γ x i i ⎥ ⎢ γi G(x) = ⎣ γi ⎦. sin γi x cos γi x k0 ζi

(3.30)

Because both ψ y (x) and ψz (x) are continuous at the boundaries, ψ y (x0 ), ψz (x0 ), ψ y (x N ), and ψz (x N ) have the relation

N −1 ψ y (x0 ) ψ y (x N ) A B ψ y (x N ) = G(−di ) = ψz (x0 ) ψz (x N ) ψz (x N ) C D

(3.31)

i

where di is the thickness of the ith layer. Because the propagation modes should have a field distribution of exp(γ0 x) in the region 0 and exp (−γ N x) in the region N , we have the eigenvalue equation k0 ζ N k 0 2 ζ0 ζ N k0 ζ0 A−B− C+ D = 0. γN γ0 γ N γ0

(3.32)

By solving (3.32), we can obtain the propagation constant of the multilayer optical waveguide β and the field distribution.


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Refractive Index

Horizontal Position

(a)

(b)

FIGURE 3.14 Antiguiding optical waveguides.

Antiguiding Figure 3.14 shows two examples of multilayer optical waveguides. The optical waveguide shown in Fig. 3.14(a) was proposed by Kawakami, and it was found that negative dispersion is obtained in an optical fiber [7]. In addition, the optical waveguide shown in Fig. 3.14(a) confines a fundamental mode and radiates higherorder transverse modes (antiguiding effect for higher-order transverse modes). As a result, it is expected that the kink level increases in LDs [8–10, 12]. The optical waveguide shown in Fig. 3.14(b) also confines a fundamental mode and has an antiguiding effect for higher-order transverse modes, which leads to a high kink level in LDs [11].

3.3 THREE-DIMENSIONAL OPTICAL WAVEGUIDES Confinement of Light Two-dimensional optical waveguides can confine light to an area sandwiched between two parallel planes. In contrast, three-dimensional optical waveguides confine light to an area surrounded by two axes other than the propagation direction. When the three-dimensional optical waveguides are adopted in LDs, the guided modes are efficiently amplified, which leads to low-threshold, high-efficiency laser operations. In an analysis of three-dimensional optical waveguides, however, we cannot obtain exact analytical solutions. As a result, approximate analytical methods or numerical analyses are used to design three-dimensional optical waveguides.

Approximate Analytical Methods As approximate analytical methods, the effective refractive index method and Marcatili’s method are well known. They can be applied to guided modes only for optical waveguides with an aspect ratio of a/ h > 1, where h is the film thickness and a is the width of the optical waveguide, under guiding conditions, which is far from cutoff conditions.


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THREE-DIMENSIONAL OPTICAL WAVEGUIDES

53

Top View

Cross-Sectional View

FIGURE 3.15 Effective refractive index method.

Effective Refractive Index Method Figure 3.15 shows a ridge optical waveguide where the upper figure is a top view and the lower figure is a cross-sectional view. Here a is the width of the optical waveguide, h is the film thickness of the core region, and f is the film thickness of the surrounding regions. First, we separate the core and surrounding regions, and we regard each region as a two-dimensional optical waveguide. Therefore, the normalized frequency V and the normalized waveguide refractive index b for each region are written Vh = k0 h n f 2 − n s 2 , Vf = k0 f n f 2 − n s 2 , bh =

Nh − n s 2 Nf − n s 2 , bf = 2 , 2 2 nf − ns nf − ns2 2

(3.33)

2

(3.34)

where the subscripts h and f correspond to the core and surrounding regions, respectively; and Nh and Nf are the effective refractive indexes for each region. Second, we assume that film with a refractive index Nh and thickness a is sandwiched between layers with refractive index Nf , as shown in the top view of Fig. 3.15. As a result, the normalized frequency Vy of a three-dimensional optical waveguide is obtained as

Vy = k 0 a Nh 2 − Nf 2 . (3.35) If we express the effective refractive index of a three-dimensional optical waveguide as Ns , the normalized waveguide refractive index bs of a three-dimensional optical waveguide is given by bs =

Ns 2 − Nf 2 . Nh 2 − Nf 2

(3.36)

Substituting Vy and bs into (3.22) leads to dispersion curves for a three-dimensional optical waveguide.


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OPTICAL WAVEGUIDES

III, V,

I,

IV,

II,

FIGURE 3.16 Marcatili’s method.

Marcatili’s Method Figure 3.16 shows a cross-sectional view of a three-dimensional optical waveguide, where light propagates along the z-axis. If most guided modes are confined in region I, light field amplitudes decay drastically with increased distance from the interfaces. Therefore, the light distributed in the shaded areas in Fig. 3.16 can be neglected, which is referred to as Marcatili’s method [13].

REFERENCES 1. T. Tamir, ed., Integrated Optics, 2nd ed., Springer-Verlag, Berlin, 1979. 2. T. Tamir, ed., Guided-Wave Optoelectronics, 2nd ed., Springer-Verlag, Berlin, 1990. 3. R. G. Hunsperger, Integrated Optics: Theory and Technology, 3rd ed., Springer-Verlag, Berlin, 1991. 4. D. Marcuse, Theory of Dielectric Waveguides, 2nd ed., Academic Press, San Diego, CA, 1991. 5. K. J. Ebeling, Integrated Optoelectronics, Springer-Verlag, Berlin, 1992. 6. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004. 7. S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a lowindex inner cladding,” IEEE J. Quantum Electron. 10, 879 (1974). 8. N. Shomura, M. Fujimoto, and T. Numai, “Fiber pump semiconductor lasers with optical antiguiding layers for horizontal transverse modes,” IEEE J. Quantum Electron. 44, 819 (2008). 9. N. Shomura, M. Fujimoto, and T. Numai, “Fiber-pump semiconductor lasers with optical antiguiding layers for horizontal transverse modes: dependence on mesa width,” Jpn. J. Appl. Phys. 48, 042103 (2009). 10. N. Shomura and T. Numai, “Ridge-type semiconductor lasers with optical antiguiding layers for horizontal transverse modes: dependence on step positions,” Jpn. J. Appl. Phys. 48, 042104 (2009).


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55

11. H. Takada and T. Numai, “Ridge-type semiconductor lasers with antiguiding cladding layers for horizontal transverse modes,” IEEE J. Quantum Electron. 45, 917 (2009). 12. H. Yoshida and T. Numai, “Ridge-type semiconductor lasers with antiguiding layers for horizontal transverse modes: dependence on space in the antiguiding layers,” Jpn. J. Appl. Phys. 48, 082105 (2009). 13. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071 (1969).


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4 OPTICAL RESONATORS

4.1

INTRODUCTION

As shown in Fig. 4.1, lasers use a fraction of spontaneously emitted light as input and amplify the fraction by stimulated emission. To feed back the light, optical resonators, or optical cavities, which consist of reflectors, are adopted. Threshold gain and oscillation wavelength are determined by the resonance conditions of the optical resonators. Optical resonators are divided into three groups, whose constituent components are mirrors, diffraction gratings, and periodic multilayers. Figure 4.2 shows a Fabry–Perot cavity, a ring cavity with mirrors, and a ring cavity with a circular waveguide. In the Fabry–Perot cavity, which consists of two parallel mirrors, light propagates backward and forward between the two mirrors. In the ring cavity, which has more than two mirrors or a circular waveguide, light propagates clockwise or counterclockwise. In the circular waveguide shown in Fig. 4.2(c), the waveguide–air interface functions as a mirror, as explained in Chapter 3. Figure 4.3 shows distributed feedback (DFB) and a distributed Bragg reflector (DBR). The DFB has an active layer, which generates light, and an optical gain in its corrugated region, which functions by itself as an optical resonator. In contrast, the DBR does not have an active layer in its corrugated region and functions as a reflector, not as a resonator. Therefore, the DBR is combined with other DBRs or cleaved facets to form optical resonators. Note that both DFBs and DBRs use periodic modulation of the refractive index to reflect light.

Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright

57


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OPTICAL RESONATORS

Optical Feedback by Reflectors Fraction of Spontaneous Emission

Laser Light

Optical Gain FIGURE 4.1 Laser.

Mirror Laser Light

Mirror

Mirror

Mirror

Laser Light

Laser Light

Circular Waveguide

Laser Light Mirror

(a)

(b)

(c)

FIGURE 4.2 Optical resonators with mirrors: (a) Fabry–Perot cavity with parallel mirrors; (b) ring cavity with mirrors; (c) ring cavity with a circular waveguide.

Active Layer

Laser Beam

Active Layer

Laser Beam

Laser Beam

Laser Beam

(a)

(b) FIGURE 4.3 (a) DFB; (b) DBR.

DBR Active Layer DBR Laser Beam FIGURE 4.4 Vertical cavity.


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FABRY–PEROT CAVITY

Cleaved Facet

Laser Beam

59

Active Layer

Laser Beam

FIGURE 4.5 Fabry–Perot LD.

FIGURE 4.6 Analytical model for a Fabry–Perot cavity.

Figure 4.4 shows a vertical cavity, which consists of periodic multilayers and is used as a surface-emitting laser. Each set of multilayers is called DBR, because the operating principle of the periodic multilayers is common to that of diffraction grating without optical gain.

4.2


Cleaved Facets In Fabry–Perot LDs, cleaved facets are used as mirrors, as illustrated in Fig. 4.5, because light is reflected at the semiconductor–air interface due to a difference in the refractive indexes. Cleaved facets are very flat, on the order of atomic layers, and they are much smoother than light wavelengths. Cleaved facets are often coated with dielectric films to control the reflectivities or to prevent the facets from being oxidized. Transmission Characteristics We assume that the amplitude reflectivities of the two mirrors are r1 and r2 and the amplitude transmissivities of the two mirrors are t1 and t2 , as shown in Fig. 4.6. The reflectivities and transmissivities depend on the angle of incidence and the polarization of light. Here the angle of incidence θ0 is supposed to be small enough, and r1 , r2 , t1 , and t2 are regarded as constant.


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OPTICAL RESONATORS

If a time-dependent factor is neglected, the incident electric field E i through the surface at z = 0 is written in ω rt E i = t1 E 0 exp − (x sin θ + z cos θ ) × exp [gE (x sin θ + z cos θ )] . c

(4.1)

Here E 0 is an amplitude of the electric field of incident light, n rt is a real part of the complex refractive index of material placed between the two mirrors, ω is an angular frequency of the light, c is the speed of light in vacuum, and gE is a field optical gain coefficient. The electric field E t transmitted through the Fabry–Perot cavity is expressed as gE L i n rt ω E t = t1 t2 E 0 exp − (x sin θ + L cos θ ) + c cos θ 2gE L 2gE L × 1 + r1r2 exp −i δ + + (r1r2 )2 exp 2 −i δ + cos θ cos θ + ··· i n rt ω gE L t1 t2 E 0 exp − (x sin θ + L cos θ ) + c cos θ = , (4.2) 2gE L 1 − r1r2 exp −i δ + cos θ where δ=

2n rt ωL cos θ . c

(4.3)

The incident light intensity I0 and the transmitted light intensity It are related to the electric fields E 0 and E t as I0 ∝ E 0∗ E 0 , It ∝ E t∗ E t .

(4.4)

Therefore, from (4.2), we obtain It =

(t1 t2 )2 G s I0 , 1 + (r1r2 )2 G s 2 − 2r1r2 G s cos δ

(4.5)

where

2gE L G s = exp cos θ

.

(4.6)


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61

Because the power reflectivities R1 and R2 and the power transmissivities T1 and T2 are given by R1 = r1∗r1 , R2 = r2∗r2 , T1 = t1∗ t1 , T2 = t2∗ t2 ,

(4.7)

we can rewrite (4.5) as It =

T1 T2 G s I0 . √ 1 + G s R1 R2 − 2G s R1 R2 cos δ 2

(4.8)

Using the angle θf in the optical waveguide as shown in Fig. 3.7, the angle θ is written as θ = π/2 − θf . As a result, (4.3) becomes δ=

2n r ωL ω = 2n r k0 L , k0 = , c c

(4.9)

where cos θ = sin θf and the effective refractive index N = n rt sin θf ≡ n r are used. When the angle of incidence θ0 is small, as in a fundamental mode, (4.8) leads to It =

T1 T2 G s0 I0 , √ √ (1 − G s0 R1 R2 )2 + 4G s0 R1 R2 sin2 (n r k0 L)

(4.10)

where G s0 = exp (2gE L) = exp(gL).

(4.11)

Here (4.9) is used and g is the power optical gain coefficient. Figure 4.7 shows It /I0 as a function of n r k0 L, where G s0 = 1, and R1 = R2 = R, T1 = T2 = T = 0.98 − R with an optical power loss of 2% at each mirror. The transmissivities have maximum values at n r k0 L = nπ and minimum values at (n + 1/2)π , where n is a positive integer. With an increase in R, the transmission spectra narrow and the transmissivities decrease. If the optical power loss at the mirror is null, we have T = 1 − R, which results in a maximum power transmissivity of 1 (100%), irrespective of R. Resonance Condition The resonance condition is the condition at which the power transmissivity has a peak. From (4.10), the resonance condition is given by n r k0 L =

n r ωL n rt ωL = cos θ = nπ, c c

(4.12)

where n is a positive integer. From (4.12) it is found that the effective refractive index n r is useful to express the resonance condition. At normal incidence with


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OPTICAL RESONATORS

Free Spectral Range

FIGURE 4.7 Transmission characteristics for a Fabry–Perot cavity.

θ0 = θ = 0, the resonance condition is written n rt ωL = n rt k0 L = nπ. c

(4.13)

Using a wavelength in vacuum λ0 , (4.13) results in L=n

λ0 . 2n rt

(4.14)

Because λ0 /n rt is a wavelength in a material, a product of a positive integer and a half-wavelength in a material is equal to the cavity length L at the resonance condition. Free Spectral Range A Fabry–Perot cavity is used as an optical filter because its transmissivity depends on a light wavelength. However, two light sources whose n r k0 L value is different by nπ , in which n is a positive integer, cannot be resolved by a Fabry–Perot cavity because there is a common transmissivity for the two lights. When (n − 1/2)π < n r k0 L < (n + 1/2)π is satisfied, the light is resolved, and the frequency or wavelength of the light becomes single-valued. This region of n r k0 L is called the free spectral range, because the light is resolved free from other light. The free spectral range in an angular frequency ωFSR is given by ωFSR =

c π. nr L

(4.15)


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63

Using the free spectral range λFSR in a wavelength, ωFSR is written as

ωFSR

1 1 = 2πc − λ0 λ0 + λFSR

.

(4.16)

As a result, we have λFSR

λ0 2 λ0 2 = ωFSR . 2n r L 2πc

(4.17)

Spectral Linewidth Half width at half maximum (HWHM) and full width at half maximum (FWHM) are often used as spectral linewidths. HWHM is the difference between the wavelength or frequency for maximum transmissivity and that for half of the maximum transmissivity. FWHM is the difference between two wavelengths or frequencies for half of the maximum transmissivity and is twice as large as HWHM. When the resonance condition (4.13) is satisfied, the transmissivity takes a maximum value. As a result, in the resonance condition, the denominator in (4.10) results in (1 − G s0 R1 R2 )2 . When the transmissivity is half-maximum, the denominator in (4.10) is twice as large as that in the resonance condition, and we obtain (1 − G s0 R1 R2 )2 + 4G s0 R1 R2 sin2 (n r k0 L) = 2(1 − G s0 R1 R2 )2 ,

(4.18)

which leads to 1 sin(n r k0 L) = ± (1 − G s0 R1 R2 ). 4 2 2 G s0 R1 R2

(4.19)

Expressing k0 = km on resonance and k0 = km ± k0 at half-maximum, then substituting k0 = km ± k0 into (4.19), we have 1 (1 − G s0 R1 R2 ). sin(n r k0 L) = 4 2 2 G s0 R1 R2

(4.20)

When n r k0 L 1 is satisfied, (4.20) results in k0 =

√ 1 − G s0 R1 R2 . 4 2n r L G s0 2 R1 R2

(4.21)


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OPTICAL RESONATORS

Using k0 = ω/c, HWHM in an angular frequency ωH is expressed as √ c(1 − G s0 R1 R2 ) . ωH = ck0 = 4 2n r L G s0 2 R1 R2

(4.22)

Accordingly, FWHM ωF , which is twice HWHM, is given by ωF = 2ωH =

√ c(1 − G s0 R1 R2 ) . 4 n r L G s0 2 R1 R2

(4.23)

Using a wavelength in vacuum, FWHM in a wavelength λF is written √ λ0 2 λ0 2 (1 − G s0 R1 R2 ) λF ωF = . 4 2πc 2πn r L G s0 2 R1 R2

(4.24)

Note that the spectral linewidth narrows with an increase in the optical gain below the threshold of laser oscillation. Electric Field Inside a Fabry–Perot Cavity When the amplitude reflectivities are r1 = r2 = r and the amplitude transmissivities are t1 = t2 = t, the electric field E inside a Fabry–Perot cavity is written E = t E 0 exp(keff x sin θ ) {exp(keff z cos θ ) + r exp[keff (2L − z) cos θ ]}

× 1 + r 2 exp(2keff L cos θ ) + r 4 exp(4keff L cos θ ) + · · · =

t E 0 exp(keff x sin θ ) {exp(keff z cos θ ) + r exp[keff (2L − z) cos θ ]} , 1 − r 2 exp(2keff L cos θ )

(4.25)

where keff = −

i n rt ω + gE . c

(4.26)

In (4.25), exp(keff z cos θ ) represents a forward running wave along the z-axis, and exp[keff (2L − z) cos θ ] expresses a backward running wave along the z-axis after reflection at a plane with z = L. If we introduce a = r exp(2keff L cos θ ), (4.25) is reduced to E = t E 0 exp(keff x sin θ )

(1 − a) exp(keff z cos θ ) + 2a cosh(keff z cos θ ) . 1 − r 2 exp(2keff L cos θ )

(4.27)

In (4.27), the first term in the numerator represents a forward running wave, and the second term in the numerator shows a standing wave. From (4.25)–(4.27) it is found


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WAVEGUIDE GRATING

65

that the light intensity inside a Fabry–Perot cavity takes a maximum value only when n rt ωL n r ωL cos θ = = n r k0 L = nπ. c c

(4.28)

Here n is a positive integer which shows the number of nodes existing between z = 0 and z = L for a standing wave. From (4.12) and (4.28) it is revealed that both the transmissivity and the internal light intensity of a Fabry–Perot cavity have the largest values at the resonance condition.

4.3

WAVEGUIDE GRATING

The DFB and DBR are optical waveguides which have diffraction gratings. They feed light back by spatially modulating the complex refractive indexes of optical waveguides. The difference between a DFB and a DBR is that the former has optical gain in the corrugated region and the latter does not. As described earlier, a DFB functions as an optical resonator, and a DBR forms an optical resonator with other DBRs or cleaved facets. Coupled Wave Theory When a lightwave is assumed to propagate along the z-axis, a propagation constant k in the optical waveguide is given by k = ω µ ε˜ = ω µ (εr − i εi ) k0 n r (z) 2

2

2

2

2

2α(z) 1+i , k0 n r (z)

(4.29)

where α(z) 2 ε˜ = εr − i εi = ε0 n r (z) + i . k0

(4.30)

Here ω is an angular frequency of the light, µ is the permeability of a material, ε˜ is a complex dielectric constant, k0 = ω/c = 2π/λ0 is a wave number in vacuum, λ0 is a wavelength of the light in vacuum, ε0 = 8.854 × 10−12 F/m is the permittivity of vacuum, n r (z) is a real part of a complex refractive index, and α(z) = gE is a field optical gain coefficient. For the usual optical materials, µ is almost equal to the permeability of vacuum, µ0 = 4π × 10−7 H/m. Because of |α(z)| k0 , a second-order term of α(z) is neglected in (4.29). Figure 4.8 shows a schematic cross-sectional view of diffraction grating. The effective refractive index of an optical waveguide is periodically modulated with a grating pitch of by the corrugations formed at the interface between the two layers with refractive indexes n A and n B (n A = n B ).


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FIGURE 4.8 Diffraction grating.

We assume that n r (z) and α(z) are sinusoidal functions of z: n r (z) = n r0 + n r1 cos(2β0 z + ), α(z) = α0 + α1 cos(2β0 z + ).

(4.31)

Here is a phase at z = 0 and β0 is given by β0 =

π ,

(4.32)

where is the grating pitch. Under the assumption that n r1 n r0 and α1 α0 , substituting (4.31) into (4.29) leads to k(z) = k0 n r0 + i 2k0 n r0 α0 + 4k0 n r0 2

2

2

π n r1 α1 +i λ0 2

cos(2β0 z + ).

(4.33)

When the refractive index is uniform (n r1 = 0) and the material is transparent (α0 = α1 = 0), the propagation constant k(z) in (4.33) is reduced to k(z) = β = k0 n r0 .

(4.34)

In optical waveguides with corrugations, a forward running wave and a backward running wave are coupled, due to reflections. To express this coupling, the coupling coefficient κ of the diffraction gratings is defined as κ=

πn r1 α1 +i , λ0 2

(4.35)

which is important in describing the resonance characteristics of DFBs and DBRs. Using (4.34) and (4.35), (4.33) is reduced to k(z)2 = β 2 + i 2βα0 + 4βκ cos(2β0 z + ).

(4.36)


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WAVEGUIDE GRATING

67

A wave equation for the electric field E is given by d2 E + k(z)2 E = 0. dz 2

(4.37)

Substituting (4.36) into (4.37) gives us d2 E 2 + β + i 2βα0 + 4βκ cos(2β0 z + ) E = 0. dz 2

(4.38)

Coupled Wave Equation The electric field E(z), which is a solution of (4.38), is represented by superposition of a forward running field Er (z) and a backward running field E s (z), such as E(z) = Er (z) + E s (z), Er (z) = R(z) exp(−i β0 z),

(4.39)

E s (z) = S(z) exp( i β0 z ), where R(z) and S(z) are the field amplitudes of the forward and backward running waves, respectively [1]. Note that both R(z) and S(z) are functions of z. Inserting (4.39) into (4.38) gives the wave equations for R and S: −

dR + (α0 − i δ)R = i κ S exp(−i), dz

(4.40)

dS + (α0 − i δ)S = i κ R exp( i ), dz where δ is defined as δ≡

β 2 − β0 2 β − β0 . 2β0

(4.41)

Here R and S were assumed to be slowly varying functions of z, and the second derivatives with respect to z are neglected. Because the forward running wave R and the backward running wave S are coupled by the coupling coefficient κ, (4.40) is called a coupled wave equation, and a theory based on (4.40) is referred to as coupled wave theory [2]. Bragg Wavelength The Bragg wavelength in vacuum λB , which satisfies δ = 0, is defined as λB =

2n r0 , m

(4.42)


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OPTICAL RESONATORS

where (4.32), (4.34), and (4.41) are used and m is a positive integer called the order of diffraction. Transfer Matrix Using constants a and b, which are determined by the boundary conditions, general solutions of (4.38) are given by Er (z) = [a exp(γ z) + ρ exp(−i) · b exp(−γ z)] exp(−i β0 z), E s (z) = [ρ exp( i) · a exp( γ z) + b exp(−γ z)] exp( i β0 z),

(4.43)

where γ 2 = (α0 − i δ)2 + κ 2 , −γ + (α0 − i δ) ρ= . iκ

(4.44)

It is useful to introduce a transfer matrix F i [3], which is defined as

Er (0) Er (L i ) = Fi . E s (0) E s (L i )

(4.45)

Here L i is the length of a corrugated region and F i is written Fi =

F11 F21

F12 , F22

(4.46)

where

F11 F12 F21 F22

α0 − i δ = cosh (γ L i ) − sinh (γ L i ) exp ( i β0 L i ), γ iκ = sinh (γ L i ) exp [−i (β0 L i + )], γ iκ =− sinh (γ L i ) exp [ i (β0 L i + )], γ α0 − i δ = cosh (γ L i ) + sinh (γ L i ) exp (−i β0 L i ). γ

(4.47)

When multiple regions are connected in series, as shown in Fig. 4.9, the total transfer matrix F is given by a product of transfer matrices in all regions, such as F=

Fi .

(4.48)

i

If the power reflectivities of both facets are R1 and R2 without optical loss, the electric fields at the facets in the air, Er (0), E s (0), Er (L), and E s (L), and the electric


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69

WAVEGUIDE GRATING

Air

Semiconductor

Air

FIGURE 4.9 Analytical model for a diffraction grating.

fields at the facets inside the semiconductor, Eri (0), E si (0), Er i (L), and E si (L), are related as Eri (0) =

1 − R1 Er (0) +

R1 E si (0), E s (0) = R1 exp(iπ )Er (0) + 1 − R1 E si (0) = − R1 Er (0) + 1 − R1 E si (0), Er (L) = 1 − R2 Eri (L) + R2 exp(iπ )E s (L) = 1 − R2 Eri (L) − R2 E s (L), E si (L) = 1 − R2 E s (L) + R2 Eri (L).

(4.49)

(4.50)

(4.51) (4.52)

Here the phase shift of π at the reflection of light is considered when the light is incident on a semiconductor from the air, because the refractive index of the semiconductor is larger than that of the air. From(4.49) and (4.50) we have

√ 1 Er (0) 1 − R1 Eri (0) √ =√ . E s (0) E si (0) 1 − R1 − R1 1

(4.53)

From(4.51) and (4.52) we have

√ 1 Eri (L) R2 Er (L) √1 =√ . E si (L) E s (L) R2 1 1 − R2

(4.54)

Combining (4.53) and (4.54) gives us the total transfer matrix F R as follows: √ √ 1 1 − R1 1 R2 √ √ FR = √ ×F× . R2 1 (1 − R1 )(1 − R2 ) − R1 1

(4.55)

When we assume that the corrugated region length is L and the input is Er (0) with E s (L) = 0, the output for the transmission is Er (L) and that for the reflection is E s (0). From the definition of a transfer matrix, the power transmissivity T and power


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OPTICAL RESONATORS

FIGURE 4.10 Calculated transmission and reflection spectra of a diffraction grating.

reflectivity R are given by T =

1

, ∗ F11 F11

R=

∗ F21 F21 . ∗ F11 F11

(4.56)

Figure 4.10 shows calculated transmission and reflection spectra of a diffraction grating. The horizontal line is δ × L = δL and the vertical line is the power transmissivity T and power reflectivity R. Here it is assumed that the optical waveguide is transparent (α0 = α1 = 0), κ L = 2, and R1 = R2 = 0. A low-transmissivity (high-reflectivity) region that is symmetrical about δL = 0 is called a stopband. Application of a Transfer Matrix to a Fabry–Perot Cavity A Fabry–Perot cavity can also be analyzed by the total transfer matrix F R . Because a Fabry–Perot cavity does not have corrugations, κ = 0 is satisfied and (4.55) is simplified as 1 FR = √ (1 − R1 )(1 − R2 )

FR11 FR21

FR12 , FR22

(4.57)

where FR11 = e(−α+iβ)L − R1 R2 e(α−iβ)L , FR12 = R2 e(−α+iβ)L − R1 e(α−iβ)L , FR21 = − R1 e(−α+iβ)L + R2 e(α−iβ)L , FR22 = − R1 R2 e(−α+iβ)L + e(α−iβ)L .

(4.58)


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71

The power transmissivity T is given by T =

1 ∗ FR11 FR11

=

(1 − R1 )(1 − R2 ) exp(2αL) . √ √ [1 − exp(2αL) R1 R2 ]2 + 4 exp(2αL) R1 R2 sin2 (β L) (4.59)

When exp(2αL) = G s0 , 1 − R1 = T1 , 1 − R2 = T2 , and β = n r k0 is satisfied, (4.59) agrees completely with It /I0 given by (4.10), which explains the power transmissivity of a Fabry–Perot cavity. Category of Diffraction Gratings From the viewpoint of the pitch and depth, the diffraction gratings are divided into four groups. A diffraction grating with uniform pitch and depth, which is shown in Fig. 4.11(a), is a uniform grating. Figure 4.11(b) shows a phase-shifted grating [4, 5], whose corrugations shift in its optical waveguide; this grating is especially important for longitudinal single-mode operations in DFB LDs. Figure 4.11(c) shows a tapered grating whose corrugation depth is spatially modulated along the propagation direction of light. Figure 4.11(d) shows a chirped grating whose pitch varies along the propagation direction. Phase-Shifted Grating Optical fibers have dispersion. As a result, if LDs operate with multiple longitudinal modes, optical pulses broaden in the time domain, and finally, adjacent optical pulses overlap with each other, which limits the transmission of signals. This overlap of adjacent pulses becomes serious with increases in the transmission distance and signal speed. Therefore, longitudinal single-mode LDs are required for long-haul, large-capacity optical fiber communication systems. To achieve stable single-mode operations, DFB LDs with phase-shifted gratings or gain-coupled gratings have been

(a)

(b)

(c)

(d)

FIGURE 4.11 Diffraction gratings used for DFBs and DBRs: (a) uniform; (b) phase-shifted; (c) tapered; (d) chirped.


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developed. In the following, phase-shifted gratings, which have been used commercially, are explained.

Transmission and Reflection Characteristics Figure 4.12 shows the structure of phase-shifted grating, in which the corrugation phase is shifted by − along the z-axis and the corrugations in uniform gratings are represented by a dashed line. It should be noted that the negative sign of the phase shift is based on a definition of the spatial distribution of the complex refractive index in (4.31). Figure 4.13 shows an analytical model in which the diffraction gratings consist of two regions, and the phase shift is introduced as a phase jump at the interface of the two regions. It is also assumed that both the pitch and depth are uniform in the two regions. When the phase at the left edge of region 1 is θ1 , the phase θ2 at the right edge of region 1 is given by θ2 = θ1 + 2β0 L 1 .

Phase Shift

Pitch

FIGURE 4.12 Phase-shifted grating.

Region 1

Region 2

FIGURE 4.13 Analytical model for phase-shifted grating.

(4.60)


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Power Reflectivity

WAVEGUIDE GRATING

FIGURE 4.14 Reflection spectrum.

Due to the phase shift , the phase θ3 at the left edge of region 2 is obtained as θ3 = θ2 + = θ1 + 2β0 L 1 + .

(4.61)

By using (4.56), the transmission and reflection characteristics of the phase-shifted gratings are given by the transfer matrix F = F 1 × F 2 , where F 1 and F 2 are the transfer matrices of regions 1 and 2, respectively. Figure 4.14 shows calculated power reflectivity as a function of δ × L = δL for κ L = 2. Here it is assumed that a material is transparent and that the facet reflectivity is null. The solid and dashed lines represent phase-shifted grating with − = π and uniform grating with − = 0, respectively. Note that the phase-shifted grating has a passband within the stopband, and the phase-shifted DFB LDs oscillate at this transmission wavelength. The transmission wavelength depends on the value of the phase shift −. When the phase shift − is π, the transmission wavelength agrees with the Bragg wavelength, which is represented by δ = 0.

Comparison of Phase-Shifted Grating and Fabry–Perot Cavities Comparing phase-shifted grating with Fabry–Perot cavities, we can understand the physical meaning of the relationship between the transmission wavelength and the phase shift [6]. In addition, it is found that the operating principle of a vertical cavity is the same as that of phase-shifted grating. Figure 4.15 shows a Fabry–Perot cavity where the power reflectivity of the mirror is R0 and the cavity length is L. From (4.14), the resonance condition of the Fabry–Perot cavity is given by L=n

λ0 . 2n r

(4.62)


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Mirror

Mirror

FIGURE 4.15 Fabry-Perot cavity.

(b)

(a)

FIGURE 4.16 Resonance characteristics of a Fabry–Perot cavity: (a) power reflectivity R0 of a mirror; (b) power transmissivity T of a cavity.

Here n is a positive integer, n r is the effective refractive index of a material, and λ0 is a wavelength of light in vacuum. Figure 4.16 shows the power reflectivity R0 of a mirror and the power transmissivity T of a Fabry–Perot cavity as a function of wavelength λ. It should be noted that R0 is independent of λ if the dispersion is neglected, and sharp peaks are present for all wavelengths that satisfy the resonance condition (4.62). Figure 4.17 shows a phase-shifted grating with sawtoothed corrugations and one with rectangular corrugations. Note that the grating shapes affect the value of κ, but the concepts regarding the grating pitch are common to both structures. In the following, to focus solely on the grating pitch, we consider the rectangular shape. The phase shift − is defined as shown in Fig. 4.17. Such a phase-shifted grating can be regarded as a Fabry–Perot cavity with length L and two mirrors whose reflectivities depend on a wavelength. In Fig. 4.17(b), L and are related as || || L= + = 1+ , 2 2β0 2 π

(4.63)

where (4.32) is used. From (4.63) it is found that the cavity length L changes with the phase shift −, which varies a resonance (transmission) wavelength according to (4.14). When the phase-shift is π , (4.42) and (4.63) give

L= =m

λB , 2n r0

(4.64)


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75

Phase Shift

(a) Phase Shift

Mirror

Mirror (b)

FIGURE 4.17 Phase-shifted gratings with (a) sawtoothed corrugations and (b) rectangular corrugations.

Stopband

Stopband

(a)

(b)

FIGURE 4.18 Resonance characteristics of phase-shifted grating: (a) power reflectivity R1 of a mirror; (b) power transmissivity T of a cavity.

where m is a positive integer called the order of diffraction, and n r0 , the average refractive index, is shown in (4.31). For first-order grating (m = 1), the resonant wavelength is the Bragg wavelength. Figure 4.18 shows the power reflectivity R1 of the mirror shown in Fig. 4.17(b) and the power transmissivity T of the phase-shifted grating with − = π as a function of a wavelength. The power reflectivity R1 of the mirror depends on a wavelength λ, and only the wavelength region within the stopband has high reflectivity. Therefore, only a resonant wavelength, which is located in the stopband, is selectively multireflected in the cavity. Also, because the cavity length L is on the order of a wavelength of light, the free spectral range is so large that only one transmission peak is present within the stopband. As described earlier, phase-shifted grating can be explained qualitatively as a Fabry–Perot cavity. The only difference between phase-shifted grating and a Fabry– Perot cavity is the dependence of the mirror reflectivities on a wavelength of light. In addition, from the viewpoint of distribution of the refractive index in the phase-shifted grating shown in Fig. 4.19, the operating principle of the phase-shifted grating is the same as that of the vertical cavity.


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Refractive Index

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Position FIGURE 4.19 Distribution of a refractive index in phase-shifted grating.

Definition of the Phase Shift There are two definitions of the phase shift in a phase-shifted grating, as shown in Fig. 4.20 [7]. In Fig. 4.20(a), the phase shifts in region 1 in z < 0 and region 2 in z > 0 are symmetrical with respect to z = 0. In Fig. 4.20(b), the phase shift in region 1 is not present; only the phase shift in region 2 in z > 0 is present. Using the definition in Fig. 4.20(a), the refractive index n r (z) in region 1 is expressed as n r (z) = n r0 + n r1 cos(2β0 z + θ ),

(4.65)

whereas the refractive index in region 2 is written as n r (z) = n r0 + n r1 cos(2β0 z − θ ).

(4.66)

Using the definition in Fig. 4.20(b), the refractive index in region 2 is written n r (z) = n r0 + n r1 cos(2β0 z + θ + ),

Region 1

Region 2

(a)

(4.67)

Region 2

Region 1

(b) FIGURE 4.20 Definition of phase shift.


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77

where (4.65) is used. Because (4.66) and (4.67) represent the same refractive index, the two phase shifts and θ are related as θ + = 2mπ − θ,

(4.68)

2θ + = 2mπ,

(4.69)

which is reduced to

where m is an integer. These two definitions of the phase shift are used in various journal articles and books, and we should be careful in discussing a value of the phase shift. Fabrication of Diffraction Gratings In DFB LDs and DBR LDs, oscillation wavelengths are at or in the vicinity of Bragg wavelengths. The oscillation wavelength for the light sources of optical fiber transmission systems is 1.3 µm, in which dispersion of non-dispersion-shifted optical fiber is the lowest, or 1.55 µm, in which the absorption loss of an optical fiber is the lowest. Because the effective refractive index n r0 of LDs is nearly 3.2, pitches of the first-order grating are approximately 0.2 µm for an oscillation wavelength of 1.3 µm and 0.24 µm for an oscillation wavelength of 1.55 µm from (4.42). The corrugation depth is about 0.1 µm = 100 nm just after grating fabrication and is reduced to several tens of nanometers after the epitaxial growth of semiconductor layers on diffraction grating, due to thermal decomposition of the grating surface during heating prior to epitaxial growth. To fabricate such fine diffraction gratings with high accuracy, holographic exposure [8–16], electron beam exposure [17], and x-ray exposure [18] systems have been developed. Holographic Exposure In holographic exposure, interference of two coherent laser beams is used to generate interference fringe patterns. The interference fringe patterns obtained are transferred to the photoresist coated on the substrate via developing the photoresists. Finally, the substrate is etched with a patterned photoresist as the etching mask. Figure 4.21 shows the principle of holographic exposure, where solid lines show wave fronts of two lightwaves and filled circles represent points at which the light intensity is enhanced due to interference. When wave fronts propagate toward the substrate, filled circles connected by dashed lines end in arrows. Consequently, the photoresist regions which are pointed to by the arrows are exposed to light, and the grating pitch is determined by the spacing of the dashed lines. Figure 4.22 shows the angles of incidence θ1 and θ2 , which are formed by the normals of the substrate plane and the propagation directions of the two laser beams.


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Pitch

Wave Front

Photoresist Substrate FIGURE 4.21 Principle of holographic exposure.

Propagation Direction of the Laser Beam Wavelength

Photoresist Substrate FIGURE 4.22 Angle of incidence in holographic exposure.

The grating pitch is given by =

λe , sin θ1 + sin θ2

(4.70)

where λe is a wavelength of the incident laser beams. Figure 4.23 shows a holographic exposure system. He-Cd lasers with λe = 441.6 nm, or 325 nm, or Ar ion lasers with λe = 488 nm, are widely used as light sources. A single laser beam emitted from the light source is divided into two laser beams by a beamsplitter, and these two laser beams are expanded by beam expanders. These expanded beams are collimated by collimating lenses and are finally incident on the photoresist coated on the substrate, which is represented as a sample in Fig. 4.23. Electron Beam Exposure In electron beam exposure, electron beams are scanned on the photoresist in vacuum, as shown in Fig. 4.24. Scanning the electron beams and transfer of samples are controlled by computers, and flexible patterns are easily fabricated. The problem in electron beam exposure is that the exposure time is very long.


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Mirror

Shutter

He-Cd laser Ar ion laser

79

Mirror

Collimating Lens

Beamsplitter

Sample

Spatial Filter

Mirror Beam Expander

FIGURE 4.23 Holographic exposure system.

Electron Beam

Photoresist Substrate FIGURE 4.24 Electron beam exposure.

X-Ray Exposure In x-ray exposure, photoresist is irradiated by x-rays through a photomask, as shown in Fig. 4.25. Because wavelengths of the x-rays are short, diffraction angles are small. As a result, x-rays are suitable to transfer fine patterns of the photomask to the photoresist. To obtain sufficient x-ray intensity for exposure, synchrotron radiation, which needs a huge plant, is often used. In addition, it is difficult to obtain highly reliable photomasks for x-ray exposure. Fabrication of Phase-Shifted Grating Electron beam and x-ray exposure systems are suitable for fabricating various patterns. These systems have problems, however, such as high costs and low productivity. As a result, holographic exposure, which has high productivity with low costs, has attracted a lot of interest. The most stable single-longitudinal-mode laser operations are obtained in DFB LDs with a phase shift of || = π , which corresponds to a shift in length /2 in first-order grating. Therefore, in first-order phase-shifted grating, the top and bottom of corrugations are reversed in an optical waveguide. From the viewpoint of the reverse of the corrugations, positive and negative photoresists are exposed simultaneously. By development, exposed areas are removed in a positive photoresist; unexposed areas are removed in a negative photoresist. Therefore, by selectively forming the positive and negative photoresists on the substrate,


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X-ray Photomask Photoresist Substrate FIGURE 4.25 X-ray exposure. Laser Beam

Laser Beam

Cyclized-Rubber Negative Photoresist

Novolak Negative Photoresist SiN

Novolak Positive Photoresist

(a)

Novolak Positive Photoresist

(b)

FIGURE 4.26 Holographic exposure using positive and negative photoresists.

a pattern and its reverse can be obtained on the same plane. Figure 4.26 shows examples of this method. In Fig. 4.26(a), novolak-type positive photoresist is formed selectively, and cyclized-rubber negative photoresist, which does not react chemically with novolak positive photoresist, is coated on the entire surface [8]. Novolak positive photoresist has high resolution, but cyclized-rubber negative photoresist has low resolution, owing to imbibition during developing. As a result, it is difficult to fabricate phase-shifted grating with fine patterns. In Fig. 4.26(b), both positive and negative novolak photoresists with high resolution are used, and SiN is inserted between novolak positive and negative photoresists to prevent chemical reactions between positive and negative photoresists [9]. In this method it is difficult to control etching time for SiN, and the production yield is low. From the viewpoint of the shift in the pitch, wave fronts of laser beams are shifted, as shown in Figs 4.27 and 4.28. In Fig. 4.27, a material with a larger refractive index than the refractive index of the air is placed selectively on a surface of a photoresist in order to shift both wave fronts of the two incident laser beams. The light incident on this material is refracted by Snell’s law, and the exposed positions are shifted. In Fig. 4.27(a) [11], the phase-shift plate is placed on the photoresist. If there is a tiny air gap, which is on the order of 1 µm between the phase-shift plate and the photoresist, exposed patterns are heavily degraded due to multireflections of the laser beams. As a result, precise position control of the phase-shift plate and the photoresist is required. In contrast, in Fig. 4.27(b) [12], the phase-shift layer is coated on a buffer layer, which results automatically in no air gap between the phase-shift and buffer layers. In Fig. 4.28, an optical element such as a phase-shift plate or hologram is inserted in the optical path for one laser beam in order to shift only one of the wave fronts of the two incident laser beams. In Fig. 4.28(a) [13], the wave fronts are disturbed due


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Laser Beam

81

Laser Beam Phase-Shift Layer

Phase-Shift Plate

Buffer Layer

Photoresist

Photoresist

(a)

(b)

FIGURE 4.27 Holographic exposure shifting both wave fronts of the two incident laser beams.

Laser Beam

Laser Beam

Laser Beam

Laser Beam Phase-Shift Plate

Hologram

(a)

(b)

FIGURE 4.28 Holographic exposure shifting one of the wave fronts of the two incident laser beams.

to diffraction at steps in the phase-shift plate, and the grating formed area is limited to only a small region. Figure 4.28(b) [14] uses a volume hologram to generate a required phase shift on the photoresist, and distortions are not present in the wave fronts. However, a highly reliable volume hologram has not yet been obtained. Figure 4.29 shows a replica method [15] where one laser beam is incident on a replica of a master phase-shifted grating, which is placed above a photoresist. The fringe patterns, which are formed by interference of the light transmitted and the light diffracted from the replica, are transferred to the photoresist. Because of reproducibility, productivity, a large tolerance in lithography conditions, and costs, the phase-shift method shown in Fig. 4.27(b) [12] was first applied to manufacture phase-shifted gratings. Later, the replica method shown in Fig. 4.29 [15] was also used commercially. Recently, electron beam exposure systems have also been used in some factories.

Laser Beam Replica Photoresist FIGURE 4.29 Replica method.


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FIGURE 4.30 Analytical model for the discrete approach.

4.4 VERTICAL CAVITY Discrete Approach As shown in Fig. 4.4, the vertical cavity consists of periodic multilayers and is used for a surface-emitting laser. Each set of the periodic multilayers is also called a DBR, because the operating principle of periodic multilayers is common to that of diffraction grating without optical gain. Periodic multilayers can be analyzed using the discrete approach [19]. Figure 4.30 shows a model for analysis where a region with a complex refractive index n 2 and length h 2 and a region with n 3 and h 3 are placed alternately. The angles formed by the interface normal and the light propagation directions are supposed to be θ2 and θ3 in the former and latter regions, respectively. Characteristic Matrix The relationship between input and output light is expressed by a characteristic matrix M 2 , which is defined as

U (0) U (z) = M2 . V (0) V (z)

(4.71)

Here it is assumed that an electric field E and a magnetic field H are expressed by a separation-of-variables procedure. Dependence of E and Z 0 H on z are written √ as U (z) and V (z), where Z 0 = µ0 /ε0 is the impedance of vacuum, µ0 is the permeability of vacuum, and ε0 is the permittivity of vacuum. As shown in Fig. 4.30, the light propagates along the z-axis and the plane of incidence is the x z-plane. As a result we have U (z) = E y (z) and V (z) = Z 0 Hx (z) for the TE mode (E x = E z = 0) and U (z) = Z 0 Hy (z) and V (z) = −E x (z) for the TM mode (Hx = Hz = 0), where a subscript represents a component along each axis.


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83

Derivation of the Characteristic Matrix A characteristic matrix M 2 is derived in the following. When the electric current does not flow, Maxwell’s equations are written rotE = ∇ × E = −µ0

∂H ∂E , rotH = ∇ × H = ε0 n r 2 , ∂t ∂t

(4.72)

where n r is the refractive index of a material. Each component of (4.72) is expressed as ∂ Ey ∂ Hy ∂ Ez ∂ Hx ∂ Hz ∂ Ex − = −µ0 , − = ε0 n r 2 , ∂y ∂z ∂t ∂y ∂z ∂t ∂ Hy ∂ Ey ∂ Ex ∂ Ez ∂ Hx ∂ Hz − = −µ0 , − = ε0 n r 2 , ∂z ∂x ∂t ∂z ∂x ∂t ∂ Ey ∂ Hy ∂ Ex ∂ Hz ∂ Hx ∂ Ez − = −µ0 , − = ε0 n r 2 . ∂x ∂y ∂t ∂x ∂y ∂t

(4.73)

It is assumed that light propagates along the z-axis, and the dependence of the electric field E and magnetic field H on time is expressed as exp(iωt). In this case, we obtain ∂/∂t = i ω. Therefore, (4.73) is reduced to ∂ Ey ∂ Hy ∂ Ez ∂ Hz k0 − = −ik0 Z 0 Hx , − = i nr2 E x , ∂y ∂z ∂y ∂z Z0 ∂ Ex ∂ Ez ∂ Hx ∂ Hz k0 − = −ik0 Z 0 Hy , − = i nr2 E y , ∂z ∂x ∂z ∂x Z0 ∂ Ey ∂ Ex ∂ Hy ∂ Hx k0 − = −ik0 Z 0 Hz , − = i nr2 Ez , ∂x ∂y ∂x ∂y Z0

(4.74)

√ where k0 = ω/c is a wave number in vacuum and c = 1/ ε0 µ0 is the speed of light in vacuum.

TE Mode We assume that the lightwave is uniform along the y-axis, which results in ∂/∂ y = 0. For the TE mode with E x = E z = 0 and Hy = 0, (4.74) is reduced to ∂ Ey = ik0 Z 0 Hx , ∂z

∂ Ey = −ik0 Z 0 Hz , ∂x

∂ Hx ∂ Hz k0 − = i nr2 E y . ∂z ∂x Z0

(4.75)


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From (4.75), we obtain a wave equation as ∂2 Ey ∂2 Ey + = ik0 Z 0 ∂x2 ∂z 2

∂ Hx ∂ Hz − ∂z ∂x

= −k0 2 n r 2 E y .

(4.76)

Here we assume that E and H are expressed by a separation-of-variables procedure. Substituting E y = X (x)U (z) into (4.76) and then dividing both sides by E y leads to 1 ∂2 X 1 ∂ 2U + = −k0 2 n r 2 . X ∂x2 U ∂z 2

(4.77)

From (4.77), if we set 1 ∂2 X = −k0 2 n r 2 sin2 θ, X ∂x2

1 ∂ 2U = −k0 2 n r 2 cos2 θ, U ∂z 2

(4.78)

where θ is the angle formed by the direction of of light and the z-axis, we can express E y as E y = U (z) exp [ i(ωt − k0 n r x sin θ ) ] .

(4.79)

Z 0 Hx = V (z) exp [ i(ωt − k0 n r x sin θ ) ] , Z 0 Hz = W (z) exp [ i(ωt − k0 n r x sin θ ) ] .

(4.80)

Similarly, we have

Substituting (4.79) and (4.80) into (4.75) results in dU = i k0 V, dz n r sin θ · U = W, dV + i k0 n r sin θ · W = i k0 n r 2 U. dz

(4.81) (4.82) (4.83)

From (4.82) and (4.83) we obtain dV = i k0 n r 2 cos2 θ · U. dz

(4.84)

Differentiating both sides of (4.84) with respect to z with the help of (4.81), we have d2 V + k0 2 n r 2 cos2 θ · V = 0. dz 2

(4.85)


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85

In summary, U and V are related as d2 U d2 V 2 2 2 + k n cos θ · U = 0, + k0 2 n r 2 cos2 θ · V = 0, 0 r dz 2 dz 2 dU dV = i k0 V, = i k0 n r 2 cos2 θ · U. dz dz

(4.86) (4.87)

From (4.86) and (4.87), solutions for U and V are expressed as U = A cos(k0 n r z cos θ ) + B sin(k0 n r z cos θ ),

(4.88)

V = i n r cos θ [ A sin(k0 n r z cos θ ) − B cos(k0 n r z cos θ )],

(4.89)

where A and B are constants, which are determined by boundary conditions. Because (4.86) is a linear differential equation of second order, we can express U and V as

U (z) F(z) = V (z) G(z)

f (z) g(z)

U (0) , V (0)

(4.90)

where U1 = f (z), U2 = F(z), V1 = g(z), V2 = G(z).

(4.91)

Also, from (4.87) we have the relations dU1 dU2 ≡ U1 = i k0 V1 , ≡ U2 = i k0 V2 , dz dz dV1 dV2 ≡ V1 = i k0 n r 2 cos2 θ · U1 , ≡ V2 = i k0 n r 2 cos2 θ · U2 , dz dz

(4.92) (4.93)

from which we obtain U1 V2 − V1 U2 = 0, V1 U2 − U1 V2 = 0.

(4.94)

d (U1 V2 − V1 U2 ) = 0, dz

(4.95)

As a result, we have

which leads to

U1 U2

V1 V2 = constant.

(4.96)


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As the matrix elements satisfying (4.96), we use f (0) = G(0) = 0, F(0) = g(0) = 1, (4.97) F(z)g(z) − f (z)G(z) = 1. From (4.90) and (4.97), the transfer matrix M is defined as

U (0) U (z) g(z) =M , M= V (0) V (z) −G(z)

− f (z) . F(z)

(4.98)

In addition, from (4.88), (4.89), and (4.97), we set f (z) =

i sin(k0 n r z cos θ ), n r cos θ

F(z) = cos(k0 n r z cos θ ),

(4.99)

g(z) = cos(k0 n r z cos θ ), G(z) = i n r cos θ · sin(k0 n r z cos θ ). If we place βi = k0 n r z cos θ,

pi = n r cos θ,

(4.100)

and substitute (4.99) and (4.100) into (4.98), the transfer matrix M can be expressed as M=

cos βi −i pi sin βi

−

i sin βi m 11 pi = m 21 cos βi

m 12 . m 22

(4.101)

The tangent of the electric field E y and that of the magnetic field Hx are continuous at each interface. Therefore, if we write the tangent components of electric fields for the incident, the reflected, and transmitted waves as A, R, and T , respectively, the boundary condition is expressed as A + R = U (0), T = U (z 1 ),

(4.102)

p0 (A − R) = V (0), p1 T = V (z 1 ),

(4.103)

p0 = n 0 cos θ, p1 = n 1 cos θ.

(4.104)

where


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87

Here n 0 and n 1 are the refractive indexes of the first and last media, respectively. From (4.98) and (4.101)–(4.104), we obtain A + R = (m 11 + m 12 p1 )T, p0 (A − R) = (m 21 + m 22 p1 )T.

(4.105) (4.106)

From (4.105) and (4.106), the amplitude reflectivity r and the amplitude transmissivity t are given by R (m 11 + m 12 p1 ) p0 − (m 21 + m 22 p1 ) = , A (m 11 + m 12 p1 ) p0 + (m 21 + m 22 p1 ) T 2 p0 t= = . A (m 11 + m 12 p1 ) p0 + (m 21 + m 22 p1 )

r=

(4.107) (4.108)

Using the amplitude reflectivity r and amplitude transmissivity t, the power reflectivity R and power transmissivity T are expressed R = r ∗r, T =

p1 ∗ t t. p0

(4.109)

TM Mode For the TM mode with Hx = Hz = 0 and E y = 0, Hy , E x , and E z are written Z 0 Hy = U (z) exp [ i (ωt − k0 n r x sin θ ) ] , E x = −V (z) exp [ i (ωt − k0 n r x sin θ ) ] , E z = −W (z) exp [ i (ωt − k0 n r x sin θ ) ] .

(4.110)

∂ Hy k0 n r 2 = −i Ex , ∂z Z0 ∂ Hy k0 n r 2 =i Ez , ∂x Z0 ∂ Ex ∂ Ez − = −i k0 Z 0 Hy . ∂z ∂x

(4.111)

From (4.74) we have

Therefore, we obtain the wave equation ∂ 2 Hy ∂ 2 Hy k0 n r 2 + = −i ∂x2 ∂z 2 Z0

∂ Ex ∂ Ez − ∂z ∂x

= −k0 2 n r 2 Hy .

(4.112)


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Substituting (4.110) into (4.112) gives d2 U + (k0 n r cos θ )2 U = 0. dz 2

(4.113)

Inserting (4.110) into (4.111) results in dU = i k0 n r 2 V, dz sin θ · U = n r W, dV + i k0 n r sin θ · W = i k0 U. dz

(4.114) (4.115) (4.116)

From (4.115) and (4.116) we obtain dV = i k0 cos2 θ · U. dz

(4.117)

Differentiating (4.117) with respect to z with the help of (4.114) results in d2 V + (k0 n r cos θ )2 V = 0. dz 2

(4.118)

In summary, a relationship between U and V for the TM mode is written d2 U d2 V 2 2 2 + k n cos θ · U = 0, + k0 2 n r 2 cos2 θ · V = 0, 0 r dz 2 dz 2 dU dV = i k0 n r 2 V, = i k0 cos2 θ · U. dz dz

(4.119) (4.120)

For the TM mode, the transfer matrix M, amplitude reflectivity r , amplitude transmissivity t, power reflectivity R, and power transmissivity T are obtained by replacing pi for the TE mode with qi =

cos θ . nr

(4.121)

Elements of the Characteristic Matrix When we introduce parameters such as β2 =

2π 2π n 2 h 2 cos θ2 , β3 = n 3 h 3 cos θ3 , λ0 λ0 p2 = n 2 cos θ2 ,

p3 = n 3 cos θ3 ,

(4.122)


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VERTICAL CAVITY

89

the characteristic matrix M 2 for the TE mode is written M2 =

cos β2

−

−i p2 sin β2

i sin β2 cos β3 p2 cos β2 −i p3 sin β3

−

i sin β3 p3 . cos β3

(4.123)

Using the number of periods N , the total characteristic matrix of the optical waveguide M is given by M = M2 N =

m 11 m 21

m 12 . m 22

(4.124)

If θ2 = θ3 = 0, we have common characteristic matrices for the TE and TM modes. From (4.124), when the outside of the vertical cavity is the air, the power transmissivity T and power reflectivity R are obtained as

2

2

, T =

(m 11 + m 12 ) + (m 21 + m 22 )

(m 11 + m 12 ) − (m 21 + m 22 ) 2

. R= (m 11 + m 12 ) + (m 21 + m 22 )

(4.125)

Comparison of Coupled Wave Theory and the Discrete Approach The results of the coupled wave theory and the discrete approach are compared in the following. For simplicity, it is assumed that a material is transparent (α0 = α1 = 0), and reflectivity at the Bragg wavelength (δ = 0) in first-order diffraction gratings (m = 1) is considered. Coupled Wave Theory From (4.44), the assumption of α0 − i δ = 0 leads to γ = ±κ. Substituting this result into (4.46), (4.47), and (4.56), the power reflectivity R is obtained as R = tanh2 (κ L),

(4.126)

where L is the corrugated region length. Discrete Approach useful to set

To analyze DFBs and DBRs using the discrete approach, it is n 2 = n r0 + n, n 3 = n r0 − n, h2 = h3 =

, θ2 = θ3 = 0, 2

(4.127)


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OPTICAL RESONATORS

where n r0 is an average refractive index of a material, n is a shift of refractive index from n r0 ; is the grating pitch; and L = N is the corrugated region length. When n n r0 is satisfied, the power reflectivity R is obtained as R = 1−

n r0 − n n r0 + n

2N 2

1+

n r0 − n n r0 + n

2N −2 ,

(4.128)

where (4.123)–(4.125) are used. The second terms in both sets of brackets are approximated as

n r0 − n n r0 + n

2N

4N n/n r0 n 4N n n r0 /n 1− = 1− n r0 n r0 n = exp −4N . (4.129) n r0

By using N = L and (4.42), the exponent of the right-hand side in (4.129) is written 4N

n 8n 8n = L= L. n r0 2n r0 λB

(4.130)

If we set 8n = 2πn r1

(4.131)

and substitute (4.131) into (4.130), we have 4N

n πn r1 =2 L = 2κ L , n r0 λB

(4.132)

where (4.35) and the assumption of α1 = 0 are used. Substituting (4.132) into (4.129) results in

n r0 − n n r0 + n

2N

e−2κ L .

(4.133)

= tanh2 (κ L).

(4.134)

Inserting (4.133) into (4.128) leads to

1 − e−2κ L R 1 + e−2κ L

2

From (4.126) and (4.134), it is found that the result of coupled wave theory agrees with that of the discrete approach when n n r0 is satisfied.


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REFERENCES

91

It should be noted that coupled wave theory assumes that complex refractive indexes vary sinusoidally, while the discrete approach presumes that complex refractive indexes change abruptly. Also, in the example of the discrete approach, two layers were alternated. Therefore, with an increase in n, the results of coupled wave theory and those of the discrete approach differ. The diffraction grating has n, which is on the order of 10−3 ; the vertical cavity has n, which is on the order of 10−2 . Therefore, for analysis of a vertical cavity, the discrete approach has to be used.

REFERENCES 1. T. Numai, “1.5 µm phase-shift-controlled distributed feedback wavelength tunable optical filter,” IEEE J. Quantum Electron. 28, 1513 (1992). 2. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327 (1972). 3. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474 (1987). 4. H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. 12, 532 (1976). 5. H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, Englewood Cliffs, NJ, 1984. 6. T. Numai, A study on semiconductor wavelength tunable optical filters and lasers. Ph.D. dissertation, Keio University, Yokohama, Japan, 1992. 7. T. Numai, “1.5-µm wavelength tunable phase-shift-controlled distributed feedback laser,” IEEE/OSA J. Lightwave Technol. 10 199 (1992). 8. K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP/InP DFB lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008 (1984). 9. K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP/InP DFB lasers,” IEEE J. Quantum Electron. 22, 1042 (1986). 10. M. Okai, S. Tsuji, M. Hirao, and H. Matsumura, “New high resolution positive and negative photoresist method for λ/4-shifted DFB lasers,” Electron. Lett. 23, 370 (1987). 11. M. Shirasaki, H. Soda, S. Yamakoshi, and H. Nakajima, “λ/4-shifted DFB-LD corrugation formed by a novel spatial phase modulating mask,” European Conference on Optical Communications/Integrated Optics and Optical Communications, 25 (1985). 12. T. Numai, M. Yamaguchi, I. Mito, and K. Kobayashi, “A new grating fabrication method for phase-shifted DFB LDs,” Jpn. J. Appl. Phys. Pt. 2 26, L1910 (1987). 13. S. Tsuji, A. Ohishi, M. Okai, M. Hirao, and H. Matsumura, “Quarter lambda shift DFB lasers by phase image projection method,” 10th Int. Semiconductor Laser Conf. 58 (1986). 14. Y. Ono, S. Takano, I. Mito, and N. Nishida, “Phase-shifted diffraction-grating fabrication using holographic wavefront reconstruction,” Electron. Lett. 23, 57 (1987). 15. M. Okai, S. Tsuji, N. Chinone, and T. Harada, “Novel method to fabricate corrugation for a λ/4-shifted distributed feedback laser using a grating photomask,” Appl. Phys. Lett. 55, 415 (1989).


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OPTICAL RESONATORS

16. H. Sugimoto, Y. Abe, T. Matsui, and H. Ogata, “Novel fabrication method of quarterwave-shifted gratings using ECR-CVD SiNx films,” Electron. Lett. 23, 1260 (1987). 17. K. Sekartedjo, N. Eda, K. Furuya, Y. Suematsu, F. Koyama, and T. Tanbun-ek, “1.5-µm phase-shifted DFB lasers for single-mode operation,” Electron. Lett. 20, 80 (1984). 18. T. Nishida, M. Nakao, T. Tamamura, A. Ozawa, Y. Saito, K. Nishimura, and H. Yoshihara, “Synchrotron radiation lithography for DFB laser gratings,” Jpn. J. Appl. Phys. Pt. 1 28, 2333 (1989). 19. M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press, Cambridge, UK, 1999. 20. T. Numai, Fundamentals of Semiconductor Lasers, Springer-verlag, New York, 2004.


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5 PN- AND PNPN-JUNCTIONS

5.1

INTRINSIC SEMICONDUCTOR

Resistivity Semiconductors are solids whose resistivity at room temperature is in the range 10−2 to 109 · cm. Element semiconductors such as Si and Ge and compound semiconductors such as GaAs, InP, and GaN are widely used. Si and Ge have diamond structures and GaAs and InP have zinc blende structures, which are shown in Fig. 5.1. Thermal Excitation Intrinsic semiconductors are pure semiconductors without containing impurities. As shown in Fig. 5.2, the electrons existing in a valence band are excited to a conduction band by receiving thermal energy. As a result, holes are generated in the valence band. Both the holes in the valence band and the electrons in the conduction band contribute to electric conduction. Therefore, the electrons in the conduction band are called conduction electrons. With an increase in temperature, thermal excitation becomes active, and the conduction-electron and hole concentrations increase, which leads to a decrease in the electrical resistance of the intrinsic semiconductors.


93


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PN- AND PNPN-JUNCTIONS

(a)

(b)

FIGURE 5.1 Crystal structure of semiconductors: (a) diamond structure; (b) zinc blende structure.

Conduction Band Conduction Electron Hole Valence Band FIGURE 5.2 Thermal excitation of electrons from a valence band to a conduction band.

Carrier Concentration In the intrinsic semiconductors, the conduction electrons and holes are generated by thermal excitation. As a result, the conduction-electron concentration n and the hole concentration p are equal, which is written n = p.

(5.1)

Both the conduction electrons and the holes are called carriers, because they carry electric charges. Conduction-Electron Concentration The electrons are Fermi particles with spin 1/2 and occupy the energy states according to Fermi statistics. As a result, the conduction-electron concentration n is given by n=

E0

g(E − E c ) f (E) dE Ec Ec − EF = Nc exp − . kB T

∞

g(E − E c ) f (E) dE

Ec

(5.2)


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INTRINSIC SEMICONDUCTOR

95

Here E c is the energy of the bottom of the conduction band, g(E − E c ) = De (E − E c )/L 3 is the density of states per unit volume of the conduction band, f (E) is the Fermi–Dirac distribution function, E 0 is the vacuum level of a semiconductor, kB is Boltzmann’s constant, and T is the absolute temperature. Nc is the effective density of states for the conduction band, which is defined as Nc ≡ 2

2πm e kB T h2

32

Mc ,

(5.3)

where m e is the effective mass of the conduction electron, h is Planck’s constant, and Mc is the number of band edges of the conduction band. In general, the effective mass of the conduction electron has different values according to the direction of the wave vectors. When the effective masses of a conduction electron along the principal axes of wave vectors are expressed as m 1 ∗ , m 2 ∗ , and m 3 ∗ , the density-of-states effective mass of the conduction electron m e is given by 1 m e = m de ≡ m 1 ∗ m 2 ∗ m 3 ∗ 3 .

(5.4)

In Si and Ge, by using the transverse effective mass m t and the longitudinal effective mass m l , the density-of-states effective mass m e is written 1 m e = m de = m t 2 m l 3 , (5.5) where

m1∗ = m2∗ = mt, m3∗ = ml.

(5.6)

Hole Concentration The hole concentration p is given by EF − Ev g(E v − E)[1 − f (E)] dE Nv exp − . kB T −∞

p=

Ev

(5.7)

Here E v is the energy of the top of a valence band and g(E v − E) is the density of states per unit volume for the valence band. Nv is the effective density of states for the valence band, which is defined as Nv ≡ 2

2π m h kB T h2

32

.

(5.8)

Here, by using the effective mass of the heavy hole m hh and the effective mass of the light hole m lh , the effective mass of the hole m h is expressed as 23 3 3 m h = m dh ≡ m hh 2 + m lh 2 , where m dh is called the density-of-states effective mass for the valence band.

(5.9)


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Intrinsic Carrier Concentration The carrier concentration in the intrinsic semiconductor, that is, the intrinsic carrier concentration n i , has a relation such as n i = n = p.

(5.10)

Using the energy gap E g = E c − E v , we have Eg Ec − Ev n i 2 = np = Nc Nv exp − = Nc Nv exp − . kB T kB T

(5.11)

As a result, the intrinsic carrier concentration n i is written ni =

Ec − Ev Nc Nv exp − 2kB T

=

Eg Nc Nv exp − . 2kB T

(5.12)

In thermal equilibrium, both the intrinsic and extrinsic semiconductors satisfy (5.11) and (5.12). Intrinsic Fermi Level From (5.2), (5.7), and (5.10), the intrinsic carrier concentration n i is given by Ec − Ei Ei − Ev n i = Nc exp − = Nv exp − , kB T kB T

(5.13)

where E i is the Fermi level in the intrinsic semiconductor, that is, the intrinsic Fermi level. Therefore, by using the intrinsic carrier concentration n i and the intrinsic Fermi level E i , the effective density of states Nc and Nv can be rewritten as

Ec − Ei Nc = n i exp , kB T Ei − Ev Nv = n i exp . kB T

(5.14) (5.15)

If we put E F = E i in (5.2) and (5.7), the intrinsic Fermi level E i is expressed as 1 1 Nv (E c + E v ) + kB T ln 2 2 Nc 1 1 Nv = E v + E g + kB T ln . 2 2 Nc

Ei =

(5.16)

From (5.16) it is found that the intrinsic Fermi level E i shifts slightly from the center of the energy gap E g /2.


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EXTRINSIC SEMICONDUCTOR

5.2

97


Impurities Extrinsic semiconductors are semiconductors that contain impurities. To control the electric conductivity, impurities are added intentionally to the semiconductors, and the added impurities are called dopants. Dopants that give conduction electrons to the conduction band are called donors; dopants that accept electrons from the valence band are called acceptors. A semiconductor in which the conduction-electron concentration n is larger than the hole concentration p (n > p) is called an n-type semiconductor; a semiconductor in which n is smaller than p (n < p) is called a p-type semiconductor. Equations of Motion for the Carriers Using the elementary electric charge e, the electric charge of a carrier q is −e for the conduction electron, e for the hole. When the magnetic flux density B(= 0) is a vector in the positive direction of the z-axis, the x-, y-, and z-components of the equations of motion for the carriers in the semiconductor are given by d 1 + vx = q E x + Bv y , dt τ d 1 m∗ + v y = q E y − Bvx , dt τ d 1 m∗ + vz = q E z . dt τ m∗

(5.17)

In a steady state (d/dt = 0), (5.17) is reduced to q τ E x + ωc τ v y , m∗ q v y = ∗ τ E y − ωc τ v x , m q vz = ∗ τ E z . m vx =

(5.18) (5.19) (5.20)

Here ωc is the cyclotron angular frequency, which is defined as ωc ≡

qB . m∗

(5.21)

Hall Effect Let us consider a case in which the electric field E = (E x , 0, 0) and the magnetic flux density B = (0, 0, B) are applied to a semiconductor sample. When the electric current flows only along the x-axis under this condition, we have v y = 0 because of


98

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the y-component of the current density j y = 0. As a result, from (5.18), (5.19), and (5.21), we obtain E y = ωc τ E x =

q Bτ Ex . m∗

(5.22)

The x-component of the current density jx is given by jx = n c qvx =

ncq 2τ Ex , m∗

(5.23)

where n c is the carrier concentration. Using the carrier concentration n c and the x-component of the current density jx , the Hall coefficient RH is defined as RH ≡

Ey 1 = , jx B ncq

(5.24)

where n c is the carrier concentration. We have RH = −1/n c e < 0 for the conduction electron and RH = 1/n c e > 0 for the hole. Donor When a group V element such as P is added to a group IV element such as Si, the group V element combines with the group IV element. In this case, an outermost electron of the group V element is left, because the group V element has five outermost electrons while the group IV element has four outermost electrons. If the group V element receives energy such as heat, the group V element donates the excess outermost electron to the conduction band and the group V element becomes a positive ion. The electron donated to the conduction band from the group V element functions as a conduction electron. Therefore, the group V element can play a role as a conductionelectron donor and contributes to generating carriers. The energy level of the group V element is called the donor level. As shown in Fig. 5.3, the energy at the donor level E d is lower than the energy at the bottom of the conduction band E c . To obtain Conduction Electron

Conduction Band

Ionized Donor

Neutral Donor

Valence Band FIGURE 5.3 Donor level.


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99

as many conduction electrons as possible from donors, we usually select the group V element whose donor ionization energy, E d = E c − E d , is at most several tens of millielectron volts, which is on the same order as the thermal energy at room temperature. Acceptor When a group III element such as B is added to a group IV element such as Si, the group III element combines with the group IV element. In this case, an outermost electron of the group III element is lacking, because the group III element has three outermost electrons whereas the group IV element has four outermost electrons. If the electron in the valence band receives energy such as heat, the group III element accepts the electron in the valence band and the group III element becomes a negative ion. The hole generated in the valence band can carry the electric charge. Therefore, the group III element functions as an electron acceptor and contributes to generating carriers. The energy level of the group III element is called the acceptor level. As shown in Fig. 5.4, the energy at the acceptor level E a is higher than the energy at the top of the valence band E v . To obtain as many holes as possible from acceptors, we usually select the group III element whose acceptor ionization energy, E a = E a − E v , is at most several tens of millielectron volts, which is on the same order as the thermal energy at room temperature. Figure 5.5 shows the Fermi level E F of n-type Si and that of p-type Si as a function of carrier concentration. Figure 5.6 shows the Fermi level E F of n-type Si and that of p-type Si as a function of absolute temperature with a difference in the carrier concentrations n = n − p as a parameter. Drift Current Density The carriers in semiconductors move through the semiconductors by colliding with the constituent atoms or impurities. Using the average collision time τ , the equation of motion for the carrier is expressed as m∗

dv m∗v = qE − . dt τ Conduction Band

Ionized Acceptor

Electron

Neutral Acceptor

Valence Band

Hole FIGURE 5.4 Acceptor level.

(5.25)


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n

p

FIGURE 5.5 Fermi levels of Si as a function of carrier concentration.

FIGURE 5.6 Fermi levels of Si as a function of absolute temperature with a difference in the carrier concentrations n = n − p as a parameter.

Here m ∗ is the effective mass of the carrier, v is the velocity of the carrier, t is a time, q is the electric charge of the carrier, and E is the electric field. Using the effective masses along the principal axes of the wave vector, m 1 ∗ , m 2 ∗ , and m 3 ∗ , the conductivity effective mass of the carrier m c is defined as 1 1 ≡ mc 3

1 1 1 + + m1∗ m2∗ m3∗

.

(5.26)

For example, in Ge and Si, from (5.6), the conductivity effective mass of the conduction-electron m c is given by mc =

3m t m l . m t + 2m l

(5.27)

In a steady state (d/dt = 0), the velocity of the carrier v is written v=

qτ E ≡ µE. m∗

(5.28)


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101

Here µ is the mobility of the carrier, which is defined as µ≡

qτ . m∗

(5.29)

Using the carrier concentration n c , the drift current density j d , which is the electric current per unit area generated by the electric field E, is expressed as j d = n c qv = n c qµE =

ncq 2τ E ≡ σ E, m∗

(5.30)

where σ is the electric conductivity of the semiconductor, which is written σ =

ncq 2τ . m∗

(5.31)

When both conduction electrons and holes are present in a semiconductor, the drift current density jd is given by jd = e(nµn + pµ p )E.

(5.32)

Here n and p are the conduction-electron and hole concentrations, respectively, and µn and µ p are the mobility of the conduction electron and that of the hole, respectively. Diffusion Current Density When there is a gradient in the carrier concentration, electric current flows due to the diffusion. For example, if a gradient in the carrier concentration exists along the x-axis, the diffusion current density due to the conduction electron jn and the diffusion current density due to the j p are written dn dn jn = −e −Dn = eDn , dx dx dp dp j p = e −D p = −eD p . dx dx

(5.33) (5.34)

Here n is the conduction-electron concentration, p is the hole concentration, and Dn and D p are the diffusion coefficients of the conduction electron and the hole, respectively. The diffusion coefficients and mobilities are related as Dp Dn kB T = = , µn µp e which is known as Einstein’s relation.

(5.35)


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Quasi-Fermi Level If the conduction-electron concentration n and hole concentration p are much larger than the values in equilibrium due to optical excitation or injection of the electric current, we cannot describe the Fermi–Dirac distribution function with one Fermi level. As a result, we assume that the conduction and valence bands are expressed separately by Fermi–Dirac distribution functions with different Fermi levels, such as 1 , E − E Fc 1 + exp kB T 1 , f v (E) = E − E Fv 1 + exp kB T f c (E) =

(5.36)

(5.37)

where E Fc and E Fv are quasi-Fermi levels. When the quasi-Fermi level of the conduction band E Fc is uniform in a semiconductor, the conduction electrons are in a diffusive equilibrium state. Therefore, electric current due to the conduction electron does not flow. If the quasi-Fermi level has a gradient, electric current flows with electric current density j n such as j n = µn n grad E Fc .

(5.38)

In an extrinsic semiconductor with n i n Nc ,

(5.39)

replacing Fermi level E F with quasi-Fermi level E Fc in (5.2) results in E Fc = E c + kB T ln

n . Nc

(5.40)

Inserting (5.40) into (5.38) leads to j n = µn n grad E c + µn kB T grad n.

(5.41)

Using the electric potential ϕ, the gradient of the energy of the bottom of the conduction band E c is expressed as grad E c = − e grad ϕ = e E,

(5.42)

where E is the electric field. From (5.35), the diffusion coefficient Dn is given by Dn =

µn kB T . e

(5.43)


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103

PN-JUNCTION

From (5.41)–(5.43), the electric current density j n is written j n = eµn n E + eDn grad n,

(5.44)

where the first term on the right-hand side, eµn n E, represents the drift current density and the second term on the right-hand side, eDn grad n, represents the diffusion current density. Using a procedure similar to the calculations above, the hole current density j p is obtained as j p = µ p p grad E Fv = eµ p p E − eD p grad p.

5.3

(5.45)

PN-JUNCTION

Depletion and Space-Charge Layers The pn-junction is a junction of p- and n-type semiconductors. At the interface of a pn-junction, the conduction electrons diffuse from an n-type semiconductor to a p-type semiconductor; the holes diffuse from a p-type semiconductor to a n-type semiconductor, so that Fermi levels in p- and n-type semiconductors may have a common value. Figure 5.7(a) shows the diffusion of conduction electrons and holes. By diffusion of the carriers, the carriers are depleted at the interface of the pn-junction, and this region is called the depletion layer. Space charges such as ionized acceptors and ionized donors are left in the depletion layer, as shown in Fig. 5.7(b), and the depletion layer is also called the space-charge layer. Diffusion and Built-in Potential Using the intrinsic Fermi level of an n-type semiconductor, E in , and that of a p-type semiconductor, E ip , the electric potential of an n-type semiconductor, φn , and that of

Conduction Electron

Hole

n

p Depletion Layer (a)

Ionized Donor

Ionized Acceptor

n

p

Depletion Layer (Space-Charge Layer) (b)

FIGURE 5.7 pn-junction diode: (a) carrier diffusion at the interface; (b)depletion layer (space-charge layer).


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a p-type semiconductor, φp , are defined as E F − E in kB T Nd = ln , e e ni E F − E ip kB T Na φp ≡ =− ln . e e ni φn ≡

(5.46) (5.47)

Here E F is the common Fermi level of the pn-junction, e is the elementary electric charge, kB is Boltzmann’s constant, T is the absolute temperature, Nd is the donor concentration in an n-type semiconductor, Na is the acceptor concentration in a ptype semiconductor, and n i is the intrinsic carrier concentration. For simplicity it is assumed that only donors are doped in an n-type semiconductor and only acceptors are doped in a p-type semiconductor. In addition, all the donors and acceptors are assumed to be ionized completely. At the interface of the pn-junction, the potential difference φD is generated and is written φD = φn − φp =

Eg kB T Na Nd kB T Na Nd ln = + ln , 2 e ni e e Nc Nv

(5.48)

where (5.12) is used. This potential difference φD is called the diffusion potential or built-in-potential. Owing to the diffusion potential φD , an electric field is generated at the interface of the pn-junction. In thermal equilibrium, the drift current due to the electric field balances the diffusion current, and net electric current does not flow in the pn-junction. Abrupt pn-Junction A pn-junction whose impurity concentration (Na − Nd ) changes abruptly at the interface, as shown in Fig. 5.8, is called an abrupt pn-junction. In Fig. 5.8 the interface of a pn-junction is placed at x = 0, an n-type semiconductor with donor concentration Nd is located at x ≤ 0, and a p-type semiconductor with acceptor concentration Na is located at 0 ≤ x. For simplicity it is assumed that only donors are doped in an n-type semiconductor and only acceptors are doped in a p-type semiconductor. Note that n and p in Fig. 5.8 represent the regions where n- and p-type semiconductors are located, respectively. The spatial distribution of the carrier concentrations n and p is shown in Fig. 5.9, where −ln and lp are the x-coordinates of depletion layer interfaces in n- and p-type semiconductors, respectively. Here, it is assumed that all donors and acceptors are ionized completely and the interface region of a pn-junction with −ln ≤ x ≤ lp is depleted completely. The spatial distribution of the electric charge density at the interface of a pnjunction ρ is shown in Fig. 5.10. Here it is assumed that the concentration of the carriers and that of the ionized impurities have a common value, and that an electrically neutral condition is satisfied in both x ≤ −ln and lp ≤ x.


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PN-JUNCTION

n

p

0

FIGURE 5.8 Impurity concentration distribution in an abrupt pn-junction.

n

p

Depletion Layer FIGURE 5.9 Carrier concentration in an abrupt pn-junction.

n

p

Depletion Layer FIGURE 5.10 Electric charge density in an abrupt pn-junction.

105


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Poisson Equation are written

Poisson equations in the depletion layer of an abrupt pn-junction d2 φ eNd =− (−ln ≤ x ≤ 0), 2 dx εn ε0 d2 φ eNa = (0 ≤ x ≤ lp ). 2 dx εp ε0

(5.49) (5.50)

Here φ is the electric potential, εn is the relative dielectric constant of an n-type semiconductor, εp is the relative dielectric constant of a p-type semiconductor, and ε0 = 8.854 × 10−12 F/m is the permittivity of vacuum. Electric Field In electrically neutral regions with x ≤ −ln and lp ≤ x, the electric field is E x = −dφ/dx = 0. As a result, as the boundary condition for the electric field, E x = −dφ/dx = 0 is satisfied at both x = −ln and x = lp . Under this boundary condition, integrating (5.49) and (5.50) with respect to x results in ⎧ ⎪ eNd ⎪ ⎪ (x + ln ) ⎨− dφ εn ε0 = ⎪ eNa dx ⎪ ⎪ x − lp ⎩ εp ε0

(−ln ≤ x ≤ 0), (0 ≤ x ≤ lp ).

(5.51) (5.52)

From (5.51) and (5.52), the electric field E x = −dφ/dx is shown in Fig. 5.11. The electric field E x = −dφ/dx should have a common value at the interface of the pn-junction x = 0. As a result, from (5.51) and (5.52) we obtain Nd Na ln = lp . εn εp

n

(5.53)

p

Depletion Layer FIGURE 5.11 Electric field in an abrupt pn-junction.


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PN-JUNCTION

107

If εn = εp is satisfied, we have Ndln = Nalp . Accordingly, when Nd > Na , we have ln < lp , which indicates that the depletion layer expands to a region with lower impurity concentration. Electric Potential Under the assumption of the electric potential φ = 0 at x = 0 as the boundary condition, integrating (5.51) and (5.52) with respect to x leads to ⎧ ⎪ eNd 2 ⎪ ⎪ ⎨− x + 2ln x (−ln ≤ x ≤ 0), 2εn ε0 φ= ⎪ eNa 2 ⎪ ⎪ ⎩ x − 2lp x (0 ≤ x ≤ lp ). 2εp ε0

(5.54) (5.55)

The electric potential φ and diffusion potential φD are shown in Fig. 5.12. The energy of the electron is represented by −eφ with the elementary electric charge e. Therefore, according to the electric potential shown in Fig. 5.12, the conduction and valence bands bend in the interface of the pn-junction, as shown in Fig. 5.13. Thickness of the Depletion Layer If the relative dielectric constants of n- and ptype semiconductors are common, such as εn = εp = εs , the electric field at x = 0, which is written as E m , is given by Em =

eNd eNa σ ln = lp ≡ . εs ε0 εs ε0 εs ε0

(5.56)

Here σ is the electric charge, which is stored in the depletion layer per unit area, and is expressed as σ = eNd ln = eNa lp .

n

(5.57)

p

Depletion Layer FIGURE 5.12 Electric potential in an abrupt pn-junction.


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n

p

Depletion Layer FIGURE 5.13 Energy in an abrupt pn-junction.

When a forward bias voltage V > 0 is applied to a pn-junction, the electric potential φ(x), which is a function of a position x, is related to the diffusion potential φD as φD − V = φ (−ln ) − φ lp = φn − φp e σ2 1 1 = Nd l n 2 + Na l p 2 = + . 2εs ε0 2eεs ε0 Nd Na

(5.58)

From (5.57) and (5.58), ln and lp are expressed as σ ln = = eNd σ lp = = eNa

2εs ε0 Na (φD − V ) , eNd Na + Nd

(5.59)

2εs ε0 Nd (φD − V ) . eNa Na + Nd

(5.60)

The thickness of the depletion layer lD is given by lD = ln + lp .

(5.61)

Junction and Depletion Layer Capacitance In an abrupt pn-junction, the positive and negative electric charges are localized, as shown in Fig. 5.10, and it can be interpreted that a capacitor has been formed. The capacitance per unit area is called the junction capacitance or depletion layer capacitance, and its value CJ is given by dσ eεs ε0 Na Nd εs ε0 = CJ = · = , dV 2 (φD − V ) Na + Nd lD where (5.58) is used.

(5.62)


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109

FIGURE 5.14 CJ −2 as a function of bias voltage V .

From (5.62) we have CJ −2 =

2 (φD − V ) Na + Nd · . eεs ε0 Na Nd

(5.63)

Figure 5.14 shows CJ −2 as a function of the reverse bias voltage V . From Fig. 5.14 it is found that CJ −2 = 0 is obtained at V = φD , which makes it possible to determine the diffusion potential φD experimentally. Graded pn-Junction A pn-junction whose impurity concentration (Na − Nd ) changes gradually at the interface, as shown in Fig. 5.15, is called a graded pn-junction. Here x = 0 is located at the interface of the pn-junction and the (Na − Nd ) is written Na − Nd = ax (−l0 ≤ x ≤ l0 ),

(5.64)

where a is positive and −l0 and l0 are the x-coordinates of the depletion layer interfaces in n- and p-type semiconductors, respectively. The pn-junction shown in Fig. 5.15 is called a linearly graded pn-junction. An n-type semiconductor with donor concentration Nd is located in x ≤ 0; a p-type semiconductor with acceptor concentration Na is located in 0 ≤ x. For simplicity it is assumed that only donors are doped in an n-type semiconductor and only acceptors are doped in a p-type semiconductor. Note that n and p in Fig. 5.15 represent the regions where the n- and p-type semiconductors are located, respectively. The spatial distribution of carrier concentrations n and p is shown in Fig. 5.16. Here it is assumed that all donors and acceptors are ionized completely and that the interface region of a pn-junction with −l0 ≤ x ≤ l0 is depleted completely. Poisson Equation The spatial distribution of the electric charge density at the interface of a pn-junction ρ is shown in Fig. 5.17. Here it is assumed that the


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n

p

Depletion Layer FIGURE 5.15 Impurity concentration in a graded pn-junction.

n

p

Depletion Layer FIGURE 5.16 Carrier concentration in a graded pn-junction.

n

p

Depletion Layer FIGURE 5.17 Electric charge density in a graded pn-junction.


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PN-JUNCTION

111

concentration of the carriers and that of the ionized impurities have a common value, and an electrically neutral condition is satisfied in both x ≤ −l0 and l0 ≤ x. The Poisson equation in the depletion layer of a linearly graded pn-junction is written d2 φ eax = (−l0 ≤ x ≤ l0 ). dx 2 εs ε0

(5.65)

Here φ is the electric potential, εs is the relative dielectric constant of the semiconductor, and ε0 = 8.854 × 10−12 F/m is the permittivity of vacuum. For simplicity, the relative dielectric constants of n- and p-type semiconductors are assumed to have a common value. Electric Field In an electrically neutral region in x ≤ −l0 and l0 ≤ x, the electric field is E x = −dφ/dx = 0. As a result, E x = −dφ/dx = 0 is satisfied at both x = −l0 and x = l0 as the boundary condition for the electric field. Under this boundary condition, integrating (5.65) with respect to x results in ea x 2 − l0 2 dφ = (−l0 ≤ x ≤ l0 ). dx 2εs ε0

(5.66)

Figure 5.18 shows an electric field E x = −dφ/dx as a function of x. The electric field at x = 0 is given by Em =

eal0 2 . 2εs ε0

n

(5.67)

p

Depletion Layer FIGURE 5.18 Electric field distribution in a graded pn-junction.


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Electric Potential Under the assumption of an electric potential φ = 0 at x = 0 as the boundary condition, integrating (5.66) with respect to x leads to ea x 3 − 3l0 2 x φ= (−l0 ≤ x ≤ l0 ). 6εs ε0

(5.68)

The electric potential φ and the diffusion potential φD are shown in Fig. 5.19. The energy of the electron is represented by −eφ with elementary electric charge e. Therefore, according to the electric potential shown in Fig. 5.19, the conduction and valence bands bend in the interface of the pn-junction, as shown in Fig. 5.20.

n

p

Depletion Layer FIGURE 5.19 Electric potential in a graded pn-junction.

n

p

Depletion Layer FIGURE 5.20 Energy in a graded pn-junction.


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113

Thickness of the Depletion Layer The electric charge σ , which is stored in the depletion layer per unit area, is written σ =

0

−l0

−eax dx =

0

l0

1 −eax dx = eal0 2 . 2

(5.69)

When the forward bias voltage V > 0 is applied to the pn-junction, the electric potential φ(x), which is a function of a position x, is related to the diffusion potential φD as √ 2eal0 3 4 2 σ3 φD − V = φ (−l0 ) − φ (l0 ) = φn − φp = = . 3εs ε0 3εs ε0 ea

(5.70)

As a result, l0 is expressed as l0 =

13 3εs ε0 (φD − V ) . 2ea

(5.71)

The thickness of the depletion layer lD is given by lD = 2l0 .

(5.72)

Junction and Depletion Layer Capacitance In a graded pn-junction, the positive and negative electric charges are localized, as shown in Fig. 5.17, and it is considered that a capacitor has been formed. From (5.70), the junction capacitance CJ is given by 1 dσ ea (εs ε0 )2 3 εs ε0 εs ε0 CJ = = = = . dV 12 (φD − V ) 2l0 lD

(5.73)

Diffusion Equations for the Carriers In thermal equilibrium, a pn-junction has a space-charge layer, which is sandwiched between the electrically neutral p- and n-layers. If these layers are placed along the x-axis as shown in Fig. 5.9, the diffusion equations for the carriers are expressed by ∂ pn pn − pn0 ∂ 2 pn =− + D pn ∂t τ pn ∂x2

(in an electrically neutral n-region),

(5.74)

∂n p n p − n p0 ∂ 2np =− + Dn p ∂t τn p ∂x2

(in an electrically neutral p-region).

(5.75)

Here t is the time, pn is the hole concentration in the n-region, pn0 is the steady-state value of pn , τ pn is the lifetime of the hole in the n-region, and D pn is the diffusion


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coefficient of the hole in the n-region. In addition, n p is the conduction-electron concentration in the p-region, n p0 is the steady-state value of n p , τn p is the lifetime of a conduction electron in the p-region, and Dn p is the diffusion coefficient of a conduction electron in the p-region. Diffusion Length In a steady state (∂/∂t = 0), the excess carrier concentrations pn

and n p are written pn = pn − pn0 , n p = n p − n p0 .

(5.76)

We assume that all donors and acceptors are ionized completely. In this case, the hole concentration in the p-region in a steady state pp0 and the conduction-electron concentration in the n-region in a steady state n n0 are written pp0 = Na , n n0 = Nd .

(5.77)

Substituting (5.77) into (5.48) results in φD =

pp0 n n0 kB T ln , e ni2

(5.78)

where n i 2 = pp0 n p0 = pn0 n n0 .

(5.79)

From (5.78) and (5.79) we have

pn0 n p0

eφD = pp0 exp − , kB T eφD = n n0 exp − . kB T

(5.80) (5.81)

Under the forward bias (V > 0), by replacing φD with (φD − V ) in (5.80) and (5.81), the hole concentration in the n-region pn and the conduction-electron concentration in the p-region n p are obtained as

e (φD − V ) eV pn = pp0 exp − = pn0 exp , kB T kB T e (φD − V ) eV n p = n n0 exp − = n p0 exp , kB T kB T

(5.82) (5.83)


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PN-JUNCTION

115

where (5.80) and (5.81) are used. Substituting (5.82) and (5.83) into (5.76) leads to eV pn = pn0 exp −1 , kB T eV n p = n p0 exp −1 . kB T

(5.84) (5.85)

The diffusion equations for these excess carrier concentrations in a steady state are expressed as ∂ 2 pn

pn

pn

= ≡ (in an electrically neutral n-region), ∂x2 D pn τ pn L pn 2 ∂ 2 n p ∂x2

=

n p Dn p τn p

≡

n p L np 2

(in an electrically neutral p-region).

(5.86) (5.87)

Here L pn is the diffusion length for a hole in the n-region and L n p is the diffusion length for a conduction electron in the p-region, which are defined as L pn ≡ L np ≡

D pn τ pn ,

(5.88)

Dn p τn p .

(5.89)

Excess Carrier Concentration From (5.86) and (5.87), the excess conductionelectron concentration in the electrically neutral p-region n p and the excess concentration of the hole in the electrically neutral n-region pn are obtained as eV x + ln pn (x) = pn0 exp − 1 exp , kB T L pn x − lp eV n p (x) = n p0 exp − 1 exp − . kB T L np

(5.90) (5.91)

Here, −ln and lp are the x-coordinates of the depletion layer interfaces in n- and p-type semiconductors, respectively. From (5.91) and (5.90), the conduction electron current density in an electrically neutral p-region due to diffusion Jn p (x) and the hole current density in an electrically neutral n-region due to diffusion J pn (x) are given by Dp d pn

eV x + ln = e n pn0 exp − 1 exp , dx L pn kB T L pn dn p Dn p x − lp eV Jn p (x) = −eDn p =e n p0 exp − 1 exp − . dx L np kB T L np

J pn (x) = eD pn

(5.92) (5.93)


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Saturation Current Density The total electric current density J is given by the sum of Jn p (lp ) and J pn (−ln ) and is expressed as J = Jn p (lp ) + J pn (−ln ) = Js

eV exp kB T

−1 ,

(5.94)

where the saturation current density Js is given by Js = e

Dn p D pn n p0 + pn0 . L np L pn

(5.95)

Diffusion Capacitance When the forward bias is applied to the pn-junction, conduction electrons are injected from the n-region to the p-region and holes are injected from the p-region to the n-region. Both conduction electrons, which are injected to the p-region, and holes, which are injected to the n-region, are minority carriers. Due to the spatial distribution of these minority carriers, the forward electric current flows in the pn-junction and the carriers are accumulated at the interfaces of the depletion layer between the electrically neutral p- and n-regions. For example, from (5.90), the electric charge per unit area in the n-region σd p due to excess concentration of the hole in the n-region pn is given by σd p = e

ln −∞

eV pn (x) dx = eL pn pn0 exp −1 . kB T

(5.96)

From (5.96), the diffusion capacitance per unit area due to the holes Cd p is defined as dσd p e2 L pn pn0 eV = Cd p ≡ exp − 1 . dV kB T kB T

(5.97)

Similarly, the diffusion capacitance per unit area due to the conduction electrons Cdn is defined as Cdn ≡

e2 L n p n p0 eV exp −1 . kB T kB T

(5.98)

Breakdown When the absolute value of the reverse bias voltage exceeds a certain value, electric current flows abruptly from the n-region to the p-region in a pn-junction, as shown in Fig. 5.21. The direction of this electric current at reverse bias is opposite that at forward bias, a phenomenon called breakdown. In avalanche breakdown, collisions between the carriers and atoms occur in succession, and in Zener breakdown,


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117

PNPN-JUNCTION

Reverse Bias

Forward Bias

FIGURE 5.21 Electric current I as a function of bias voltage V in a pn-junction.

electrons in the valence band penetrate the energy barrier due to the quantum mechanical tunneling effect, and electrons in the valence band transit to the conduction band. Ionization Coefficient The ionization coefficient α is defined as the number of electron–hole pairs that are generated when a carrier travels a unit length. Assuming that collisional ionizations occur in a region with −ln < −lc ≤ x ≤ la < lp , the current multiplication factor M is given by

M = 1−

la −lc

−1 α dx

.

(5.99)

Avalanche breakdown occurs at M = ∞, and this condition is expressed as

la

−lc

5.4

α dx = 1.

(5.100)

PNPN-JUNCTION

Shockley Diode A Shockley diode is a pnpn diode, shown in Fig. 5.22. A Shockley diode is a thyristor in a broad sense and has three pn-junctions: J1 , J2 , and J3 . When the anode voltage V is positive (V > 0), junctions J1 and J3 are forward biased and junction J2 is reverse biased. As a result, when the anode voltage V is lower than the switching voltage Vs , most of the voltage V is applied to junction J2 , and the depletion layer of junction J2 broadens. The energy band under this condition is shown in Fig. 5.23. In junction J2 , there is an energy barrier on the order of eV ,


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Anode

Cathode

FIGURE 5.22 Shockley diode.

Conduction Electron Reverse Biased

Forward Biased Hole Forward Biased FIGURE 5.23 Energy band of a Shockley diode when anode voltage V is in the range 0 < V < Vs .

where e is the elementary electric charge and V is the anode voltage. In junctions J1 and J3 there are energy barriers on the order of diffusion potentials. Therefore, in the condition band shown in Fig. 5.23, electric current scarcely flows, and this condition is called the OFF-state. When the anode voltage V becomes higher, the electric field that is applied to the depletion layer of junction J2 is higher. When the anode voltage V reaches the switching voltage Vs , avalanche breakover occurs in the depletion layer of junction J2 , leading to switching on, and electric current starts to flow. In this condition, the carriers are injected to junction J2 and the space-charge density in junction J2 goes low, due to the screening effect. As a result, the voltage between the anode and cathode decreases. In junctions J1 and J3 , electric current flows due to diffusion; in junction J2 , electric current flows due to drift. This condition is called the ON-state, and the energy band in this condition is shown in Fig. 5.24. When the anode voltage V is negative (V < 0), junction J2 is forward biased; junctions J1 and J3 are reverse biased. As a result, depletion layers are formed in junctions J1 and J3 , and electric current scarcely flows. When the absolute value of the bias voltage |V | increases further and avalanche breakdown occurs in the depletion layers of junctions J1 and J3 , the carriers just cross junctions J1 and J3 . As a result, space-charge densities of junctions J1 and J3 hardly change. Therefore, the


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PNPN-JUNCTION

Diffusion

Drift

119

Diffusion Conduction Electron

Hole FIGURE 5.24 Energy band of a Shockley diode after switching.

Avalanche Diffusion Avalanche

Hole

Conduction Electron Reverse Biased Forward Biased Reverse Biased FIGURE 5.25 Energy band of a Shockley diode at breakdown.

voltage between the anode and the cathode is almost equal to the breakdown voltage. The energy band under this condition is shown in Fig. 5.25. Figure 5.26 shows the electric current versus voltage characteristics of a Shockley diode, where Vs is the switching voltage. Punch-Through When a positive voltage is applied to the anode with the cathode grounded in the Shockley diode in Fig. 5.27, let us calculate avalanche breakover voltage VBO and punch-through voltage VPT . The avalanche breakover voltage VBO is the voltage required to start avalanche breakover. The punch-through voltage VPT is the voltage at which the depletion layer of junction J2 contacts the depletion layer of junctions J1 . If VPT is smaller than VBO , the Shockley diode is never switched on. Therefore, we have to design the layer parameters of a Shockley diode so that VBO may be smaller than VPT . Here it is assumed that the avalanche breakover electric field is


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FIGURE 5.26 Electric current versus voltage characteristics of a Shockley diode.

Anode

Cathode

FIGURE 5.27 Depletion layer in the n1 region of a Shockley diode.

E BO and the thickness of the n1 region is W1 . In addition, it is supposed that the impurity concentration in p2 is much higher than that in n1 , and the depletion layer at the interface between n1 and p2 is formed in the n1 region. Here −l0 and 0 are the x-coordinates of the edges of the depletion layer in the interface between n1 and p2 . In addition, we assume that the impurity in the n1 region consists of donors only, that all donors are ionized completely, and that the depletion layer is depleted completely. The Poisson equation for the depletion layer is written d2 φ(x) eNd =− , 2 dx εn1 ε0

(5.101)

where φ(x) is the electric potential at x, e is the elementary electric charge, Nd is the donor concentration in the n1 region, εn1 is the relative dielectric constant in the n1 region, and ε0 is the permittivity of vacuum. Assuming E x (−l0 ) = 0 as the boundary condition, the x-component of the electric field E x (x) is expressed as E x (x) = −

dφ(x) eNd (x + l0 ) . = dx εn1 ε0

(5.102)


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121

REFERENCES

Assuming φ (−l0 ) = 0 as the boundary condition, the electric potential φ(x) is obtained as φ(x) = −

eNd 2 eNd (x + l0 )2 . x + 2l0 x + l0 2 = − 2εn1 ε0 2εn1 ε0

(5.103)

If the avalanche breakover electric field E BO is set equal to E x (0), we have l0 =

εn1 ε0 εn1 ε0 E x (0) = E BO , eNd eNd

(5.104)

where (5.102) is used. When the avalanche breakover voltage VBO is applied only to the depletion layer at the interface between n1 and p2 , the avalanche breakover voltage VBO is given by VBO = φ(−l0 ) − φ(0) =

eNd 2 εn1 ε0 l0 = E BO 2 , 2εn1 ε0 2eNd

(5.105)

where (5.103) and (5.104) are used. Note that the switching voltage Vs is almost equal to the avalanche breakover voltage VBO , because most of the voltage applied between the anode and cathode is applied to junction J2 in the OFF-state. The punch-through voltage VPT , which is the voltage at l0 = W1 , is expressed as VPT = φ(−W1 ) − φ(0) =

eNd W1 2 , 2εn1 ε0

(5.106)

where (5.103) is used and the thickness of the depletion layer in junction J1 is neglected. It should be noted again that VBO has to be smaller than VPT to switch on a Shockley diode.

REFERENCES 1. W. Shockley, Electrons and Holes in Semiconductors, D. Van Nostrand, New York, 1950. 2. S. M. Sze, Physics of Semiconductor Devices, 2nd ed., Wiley, New York, 1981. 3. S. M. Sze, Semiconductor Devices: Physics and Technology, 2nd ed., Wiley, New York, 2002. 4. S. M. Sze, Semiconductor Devices: Pioneering Papers, World Scientific, Singapore, 1991. 5. A. S. Grove, Physics and Technology of Semiconductor Devices, Wiley, New York, 1967. 6. J. I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1975. 7. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004.


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PART II CONVENTIONAL LASER DIODES

123


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6 FABRY–PEROT LASER DIODES

6.1

INTRODUCTION

Fabry–Perot Cavity ¯ surfaces are used as two In Fabry–Perot LDs, cleaved facets such as {011} or {011} parallel mirrors, as shown in Fig. 6.1. When light is normally incident on a facet, the power reflectivity R0 is given by R0 =

n rt − 1 n rt + 1

2 ,

(6.1)

where n rt and 1 are the refractive indexes of the semiconductor and the air, respectively. When n rt is 3.5, R0 is about 31%. It should be noted that cleaved facets are flat on the order of atomic layers, much smaller than the wavelength of light. Therefore, cleaved facets can be used as mirrors with very high quality. To control the reflectivities or to protect the facets, dielectric films are often coated on cleaved facets. pn-Junction When carriers are injected into active layers, spontaneous emission is generated, and a fraction of the spontaneous emission, which satisfies the resonance condition Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright

125


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FABRY–PEROT LASER DIODES

Cleaved Facet

Active Layer

Laser Beam

Laser Beam

FIGURE 6.1 Fabry–Perot LD.

p-Cladding Layer Active Layer n-Cladding Layer FIGURE 6.2 Cross section of a pn-junction in an LD.

with the wavelength within the optical gain spectrum, is amplified by the stimulated emission. To inject carriers into the active layer, the active layer is placed inside the pnjunction. Therefore, the active layer is sandwiched by the p-cladding layer and the n-cladding layer, as shown in Fig. 6.2. Applying a forward bias voltage which is positive on the p-side and negative on the n-side across this pn-junction, conduction electrons are injected from the n-cladding layer to the active layer and holes are injected from the p-cladding layer to the active layer. When population inversion is generated by carrier injection, net stimulated emission is obtained. Note that the impurities are often undoped in the active layer to achieve high radiation efficiency. However, background carriers whose concentration depends on epitaxial growth methods are present in the active layer. Therefore, the active layer is not an ideal intrinsic semiconductor layer. If impurities are doped in the active layer, the injected carriers combine with the impurities. Therefore, the carrier lifetime is reduced, and the modulation speed is enhanced. However, recombinations of injected carriers and impurities do not contribute to laser transitions, which decreases the radiation efficiency. As a result, the active layer is sometimes doped intentionally to achieve high-speed modulations as long as the radiation efficiency is not highly degraded.

Double Heterostructure A heterojunction is a junction consisting of different semiconductors; a homojunction is a junction composed of common semiconductors. Note that the different semiconductors include semiconductors with different compositions.


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INTRODUCTION

p-Cladding Layer

Active Layer

127

n-Cladding Layer

(a)

(b) FIGURE 6.3 Double heterostructure: (a) energy of electrons; (b) distribution of refractive index.

The bandgap energies in semiconductors depend on the constituent elements and compositions. As a result, the heterostructures have energy barriers at the junction interfaces, and the carriers are confined in the well layers, due to the energy barriers. To achieve efficient recombinations of conduction electrons and holes, the conduction electrons and holes have to be confined in the active layer. Therefore, heterostructures are formed at both interfaces of an active layer, so the active layer may be the well layer. This structure is called a double heterostructure because double heterojunctions are present. Figure 6.3 shows the distributions of energy and the refractive index of a double heterostructure. At the junction interfaces, band offsets E c for the conduction band and E v for the valence band are present, as shown in Fig. 6.3(a). Under a forward bias, the holes are injected from the p-cladding layer to the active layer, and the energy barrier for the holes is E v at the interface of the n-cladding layer and the active layer; the conduction electrons are injected from the n-cladding layer to the active layer, and the energy barrier for the conduction electrons is E c at the interface of the p-cladding layer and the active layer. The refractive indexes of many semiconductors increase with a decrease in the bandgap energies. Hence, the refractive index of the active layer n a is usually greater than that of the p-cladding layer n p and that of the n-cladding layer n n . As a result, light is efficiently confined in the active layer, which results in a high light amplification rate. A double heterostructure confines both the carriers and light to the active layer. Therefore, a double heterostructure is indispensable to achieving continuous-wave (CW) laser oscillation at room temperature in LDs [1, 2].


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6.2 RATE EQUATIONS In the following it is assumed that the conduction-electron concentration in an active layer n is equal to the hole concentration in an active layer p. The rate equations for the carrier concentration n and the average photon concentration S = Nph /Vm of the laser light can be written dn J n = − G(n)S − , dt ed τn dS S n = a G(n)S − + βsp . dt τph τr

(6.2) (6.3)

Here Nph is the total number of photons; Vm is the mode volume; J is the injected current density, which is an electric current flowing through a unit area; e is the elementary electric charge; d is the active layer thickness; G(n) is the amplification rate due to the stimulated emission; τn is the carrier lifetime; a is the optical confinement factor of the active layer; τph is the photon lifetime; βsp is the spontaneous emission coupling factor; and τr is the radiative recombination lifetime for a spontaneous emission. Note that the optical confinement factor of the active layer a is given approximately by Va /Vm , where Va is the volume of the active layer. In (6.2), J/ed is an increasing rate of carrier concentration in the active layer due to injection of the carriers, −G(n)S shows a decay rate of the carrier concentration due to the stimulated emission, and −n/τn expresses a decay rate of the carrier concentration in the carrier lifetime τn . In (6.3), a G(n)S shows an increasing rate of photon concentration S due to the stimulated emission, −S/τph is a reduction rate of the photon concentration inside the optical cavity due to the absorption and light emission toward the outside of the optical cavity, and βsp n/τr represents a coupling rate of the spontaneous emission to the lasing mode, which is a resonant mode of the cavity. In the following, G(n), τn , and βsp are explained in detail. When the carrier concentration n is low, the active layer absorbs light. With an increase in n, the active layer has an optical gain, and at n = n 0 the active layer becomes transparent to light for the lasing mode. Using the transparent carrier concentration n 0 , in which a semiconductor is transparent, the amplification rate due to the stimulated emission G(n) can be written approximately as G(n) = g0 (n − n 0 ),

(6.4)

where g0 is the differential gain coefficient. Using the radiative recombination lifetime τr and the nonradiative recombination lifetime τnr , the carrier lifetime τn can be expressed as 1 1 1 = + . τn τr τnr

(6.5)


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RATE EQUATIONS

129

Note that the radiative recombination lifetime τr is determined only by spontaneous emission and is not affected by stimulated emission. The nonradiative recombination lifetime τnr depends on collisions between the injected carriers and the defects or impurities, and these collisions do not contribute to emit light. The spontaneous emission coupling factor βsp is defined as βsp =

spontaneous emission coupling rate to the lasing mode . total spontaneous emission rate

(6.6)

When the spontaneous emission spectrum is assumed to be Lorentzian with center angular frequency ω0 and the FWHM ω, the spontaneous emission coupling rate to the lasing mode per unit time and unit volume rsp is given by rsp = rsp0

(ω/2)2 , (ω − ω0 )2 + (ω/2)2

(6.7)

where rsp0 is a coefficient. To calculate the total spontaneous emission rate, we consider the number of modes dN with two polarizations that exist in a volume V : a solid angle for propagation direction d and an angular frequency range dω. When the distribution of the modes is continuous, as in a free space, dN is given by d n r 3 ω2 d = V 2 3 dω , 4π π c 4π n r 3 ω2 m(ω) = 2 3 , π c

dN = V m(ω) dω

(6.8) (6.9)

where m(ω) is the mode density. From (6.7) and (6.8), the total spontaneous emission rate Rsp is obtained as Rsp =

rsp dN = rsp0

V n r 3 2 ω0 ω. 2π c

(6.10)

Using (6.6), (6.7), and (6.10), the spontaneous emission coupling factor βsp for the angular frequency ω0 (a wavelength in vacuum λ0 ) can be expressed as βsp = a

rsp 2π = a Rsp V

c nr

3

1 a λ0 4 = , ω0 2 ω 4π 2 n r 3 V λ

(6.11)

where a is the optical confinement factor of the active layer and λ is the FWHM in units of wavelength. From (6.11) it is found that the spontaneous emission coupling factor βsp increases with a decrease in the volume V and the spectral linewidth λ.


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6.3 CURRENT VERSUS VOLTAGE CHARACTERISTICS In a steady state, (6.2) and (6.3) are reduced to n 1 S = −βsp , τr a G(n) − 1/τph n I = eVa G(n)S + . τn

(6.12) (6.13)

Here I is the injected current and Va = Sa d is the volume of the active layer, where Sa is the area of the active layer and d is the thickness of the active layer. A flowing current in an LD consists of the diffusion current, the drift current, and the recombination current. Here it is assumed that the radiative recombination is dominant and the diffusion and drift current are neglected, as shown in (6.13). Under the forward bias (V > 0), the hole concentration in the n-region, pn , exceeds the steady-state value, pp0 . Using pn and the conduction-electron concentration in the n-region n n n n0 , where n n0 is the steady-state value, we have np = n n pn = n i 2 exp

eV kB T

,

(6.14)

where (5.79) and (5.82) are used. As a result, the carrier concentration n is given approximately by n = n i exp

eV 2kB T

.

(6.15)

From (6.12)–(6.15), the injected current I is related to the voltage V as I = eVa

n i eeV /2kB T G(n i eeV /2kB T ) n i eeV /2kB T −βs + , τr a G(n i eeV /2kB T ) − 1/τph τn

(6.16)

which is shown in Fig. 6.4, where it is assumed that the photon lifetimes τph is 1 ps; the sum of a contact resistance and a bulk resistance is 4 , which is typical in conventional LDs. Other physical parameters are βs = 10−5 , n i = 2.7 × 1011 cm−3 , τr = τn = 1 ns, T = 293.15 K (20◦ C), a = 0.1, (∂ G/∂n)n=n th = 2.5 × 10−6 cm3 /s, and Va = 40 µm3 = 4 × 10−11 cm3 . It is also supposed that the transparent carrier concentration is n 0 = 0.6 n th , where n th is the threshold carrier concentration, the effective refractive index is 3.5, and the group velocity vg is 8.57 × 109 cm/s.


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131

Injected Current

CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS

Applied Voltage FIGURE 6.4 Current versus voltage (I –V ) characteristics.

6.4


With an increase in the current injected into an LD, the carrier concentration n in the active layer is enhanced. When the carrier concentration n exceeds the threshold carrier concentration n th , laser oscillation starts and the light output increases drastically over that below the threshold. This change in light output is considered to be an abrupt increase in the photon concentration. The current versus light output (I –L) characteristics can be analyzed by the rate equations on the carrier concentration n and the photon concentration S in the active layer.

Threshold Current Density Using the rate equations, the threshold current density Jth is calculated in the following. At first, the rate equations below the threshold are considered. For simplicity, it is assumed that net stimulated emission is negligible and S = 0 below the threshold. Under this assumption, (6.2) is reduced to J n dn = − . dt ed τn

(6.17)

From (6.17), the carrier concentration n in a steady state (d/dt = 0) is given by n=

J τn . ed

(6.18)


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When the carrier concentration n increases from 0 to the threshold carrier concentration n th , it is expected that (6.18) is still valid at the threshold. As a result, the threshold current density Jth is expressed as Jth =

ed n th . τn

(6.19)

Because the optical confinement factor a of the active layer depends on the active layer thickness d, the threshold current density n th is a function of d and an optimum d value is present to achieve the lowest Jth . Second, the threshold carrier concentration n th is calculated using rate equations above the threshold. In most LDs the spontaneous emission coupling factor βsp is on the order of 10−5 . Therefore, as a first approximation, the term βsp n/τr in (6.3) is neglected. Because (6.3) is valid for any S value in a steady state, we have a G(n) = a g0 (n − n 0 ) =

1 , τph

(6.20)

where (6.4) is used.

Threshold Gain The amplification rate due to the stimulated emission G(n) at the threshold is expressed by the threshold gain, which is the optical gain required for laser oscillation. The threshold gain is obtained by using the resonance condition of the Fabry–Perot cavity. From (4.10), the ratio of the transmitted light intensity It to the incident light intensity I0 is written It T1 T2 G s0 = , √ √ I0 (1 − G s0 R1 R2 )2 + 4G s0 R1 R2 sin2 (n r k0 L)

(6.21)

G s0 = exp(2gE L) = exp(gL).

(6.22)

where

Here T1 and T2 are the power transmissivities of the facets, R1 and R2 are the power reflectivities of the facets, n r is the effective refractive index, L is the cavity length, gE is the field optical gain coefficient for the electric field, and g is the power optical gain coefficient. Oscillation represents a state in which an output is present without an input from outside. As a result, the oscillation condition is given by I0 = 0 and It > 0 in (6.21). Therefore, at the oscillation condition, the denominator in (6.21) is 0. Hence, the


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133

oscillation condition for Fabry–Perot LDs is expressed as Resonance condition: Gain condition:

sin(n r k0 L) = 0, √ 1 − G s0 R1 R2 = 0.

(6.23)

Resonance Condition From (6.23), the resonance condition is written n r ωL = n r k0 L = nπ, c

(6.24)

where n is a positive integer. Using a wavelength in vacuum λ0 , (6.24) is reduced to L=n

λ0 , 2n r

(6.25)

which is the same as the resonance condition of a Fabry–Perot cavity. Note that laser oscillation starts at the resonant wavelength nearest the gain peak. Gain Condition From (6.23), the gain condition is obtained as 1 − G s0 R1 R2 = 1 − R1 R2 exp(gL) = 0.

(6.26)

As a result, the power optical gain coefficient g is written g=

1 1 ln √ , L R1 R2

(6.27)

where the right-hand side is called the mirror loss. Guided modes propagate in optical waveguides while they are confined in the film (active layer), and the fields of guided modes penetrate the p- and n-cladding layers, as shown in Fig. 6.5. As a result, the light sees optical losses in the p- and n-cladding layers. Therefore, the power optical gain coefficient g for the entire region consists of optical gain in the active layer and optical losses in the active and cladding layers. Using optical confinement factors, the power optical gain coefficient, and optical power optical loss coefficients, the power optical gain coefficient g of the optical waveguide is written approximately as g = a ga − a αa − p αp − n αn .

(6.28)

Here a , p , and n are the optical confinement factors of the active, p-cladding, and n-cladding layers, respectively; ga is the power optical gain coefficient of the


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Light Intensity p-Cladding Layer Optical Waveguide

Active Layer n-Cladding Layer

FIGURE 6.5 Distribution of light intensity in an optical waveguide.

active layer; and αa , αp , and αn are the power optical loss coefficients of the active, p-cladding, and n-cladding layers, respectively. Note that the exact g value is obtained by solving eigenvalue equations, which include complex refractive indexes. The internal loss αi is defined as αi = a αa + p αp + n αn .

(6.29)

Substituting (6.29) into (6.28) leads to g = a ga − αi ,

(6.30)

where a ga is the modal gain. Inserting (6.30) into (6.27) results in a ga = αi +

1 1 1 1 ln √ = αi + ln , L 2L R1 R2 R1 R2

(6.31)

which is the threshold gain of Fabry–Perot LDs. If the light propagates along the z-axis, a derivative of the photon concentration S with respect to a time t is given by dS dz dS c dS = = . dt dt dz n r dz

(6.32)

From (6.31) and (6.32), G(n) is expressed as G(n) =

c ga , nr

(6.33)

where n r is the effective refractive index and c is the speed of light in vacuum. From (6.20) and (6.33), the photon lifetime τph is expressed as 1 c = τph nr

1 1 αi + ln 2L R1 R2

.

(6.34)


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135

From (6.20), the carrier concentration n in a steady state is given by n=

1 + n0. a g0 τph

(6.35)

Because (6.35) holds even at the threshold, the threshold carrier concentration n th is written n th =

1 + n0. a g0 τph

(6.36)

In LDs, changes in the cavity length, facet reflectivities, and refractive indexes during laser operation are small. As a result, the right-hand sides of (6.35) and (6.36) are considered to be constant. Therefore, above the threshold, the carrier concentration n is clamped on the threshold carrier concentration n th when coupling of the spontaneous emission to the lasing mode is neglected. Accordingly, G(n) is constant above the threshold as long as the gain saturation and coupling of the spontaneous emission to the lasing mode are neglected. Substituting (6.36) into (6.19) results in Jth =

ed n th = J A + J B , τn

(6.37)

where JA =

ed n0, τn

JB =

ed 1 . τn a g0 τph

(6.38)

From (6.37) and (6.38) it is found that the threshold current density Jth depends explicitly on the optical confinement factor a . Figure 6.6 shows the results calculated for the optical confinement factor a for Alx Ga1−x As/GaAs double heterostructures. With an increase in the active layer thickness d, a is enhanced. Note that a is proportional to d 2 when d is small. Figure 6.7 shows the threshold current density Jth as a function of the active layer thickness d. The threshold current density Jth takes a minimum value when d is approximately 0.1 µm. In Fig. 6.7, J A is a current density which is required to obtain the population inversion and J B is a current density in which the optical gain balances the loss in the optical cavity. The current density J A is proportional to the active layer thickness d as in (6.38); for a thin active layer, J B is inversely proportional to d, because a is proportional to d 2 . Because Jth is given by J A + J B , an optimum d value is present for a minimum Jth .


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Optical Confinement Factor

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Active Layer Thickness

Threshold Current Density

FIGURE 6.6 Optical confinement factor a as a function of active layer thickness d.

Active Layer Thickness FIGURE 6.7 Threshold current density Jth as a function of active layer thickness d.

I–L Characteristics in CW Operation Without Coupling of Spontaneous Emission to the Lasing Mode When the spontaneous emission coupling factor βsp is small, coupling of the spontaneous emission to the lasing mode can be neglected. Under this condition, I –L characteristics in CW operation are analyzed using the rate equations (6.2) and (6.3). Below the threshold, the carrier concentration n increases with an increase in J , according to (6.18); the photon concentration S is 0. Above the threshold, n no longer increases and remains at the threshold carrier concentration n th ; S increases with J , because the excess carriers are converted to photons.


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137

The steady-state photon concentration S above the threshold is obtained by setting dn/dt = 0 in (6.2) and is written 1 S= G(n)

J n th − ed τn

.

(6.39)

Substituting (6.19) and (6.20) into (6.39) results in S=

a τph (J − Jth ). ed

(6.40)

From the results above, the dependence of the carrier concentration n and photon concentration S on the injected current density J is summarized as follows: For J < Jth , n=

J τn , ed

Jth τn , ed

S=

S = 0;

(6.41)

and for J ≥ Jth , n=

a τph (J − Jth ). ed

(6.42)

The results calculated for (6.41) and (6.42) are shown in Fig. 6.8, where it is clearly indicated that the carrier concentration n is clamped on n th above the threshold current density Jth . With Coupling of Spontaneous Emission to the Lasing Mode For simplicity it is assumed that the nonradiative recombination is negligible, which leads to τr ≈ τn . In a steady state (d/dt = 0), when the coupling of spontaneous emission to the lasing

FIGURE 6.8 Carrier concentration n and photon concentration S as a function of injected current density J when coupling of the spontaneous emission to the lasing mode is neglected.


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FIGURE 6.9 Carrier concentration n and photon concentration S as a function of the injected current density J when coupling of the spontaneous emission to the lasing mode is included.

mode is included, (6.2) and (6.3) are reduced to J n = g0 (n − n 0 )S + , ed τn S n = a g0 (n − n 0 )S + βsp , τph τn

(6.43) (6.44)

where (6.4) is used. The results calculated for (6.43) and (6.44) are shown in Fig. 6.9, where the solid and dashed lines correspond to βsp > 0 and βsp = 0, respectively. Coupling the spontaneous emission to the lasing mode lowers n and enhances S, which results in a vague threshold. Note that the light emitted below the threshold is incoherent or amplified spontaneous emission. Radiation Efficiency According to Figs. 6.8 and 6.9, the current versus light output (I –L) characteristics of LDs look like Fig. 6.10, where Ith is the threshold current. To evaluate the radiation

FIGURE 6.10

I –L characteristics.


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139

efficiency of LDs, slope efficiency and external differential quantum efficiency are often used. Slope Efficiency The slope efficiency per facet Sd j is defined as the ratio of the increase in light output power P j ( j = 1, 2) to the increase in injected current I , which is given by Sd j =

P j . I

(6.45)

From (6.45), Sd j is expressed in units of mW/mA or W/A. The slope efficiency for the total light output Sd,tot is obtained as Sd,tot =

P1 + P2 P = , I I

(6.46)

where P is an increase in the total light output power. External Differential Quantum Efficiency The external differential quantum efficiency ηd is defined as the number of photons emitted outward per carrier injected. The number of photons emitted outward per second is P/ω, where ω is an angular frequency of the light and is Dirac’s constant. The carrier injected per second is I /e, where e is the elementary electric charge. Therefore, the external differential quantum efficiency ηd for the total light output is given by ηd =

P I P e e ÷ = = Sd,tot . ω e I ω ω

(6.47)

From(6.47), ηd is expressed in no units. As shown in (6.31), the total loss is a sum of the internal loss and the mirror loss. When the light output is measured outside the optical cavity, the mirror loss indicates the light emission rate from the optical cavity. Using the internal quantum efficiency ηi , which is defined as the number of photons emitted inside the optical cavity per carrier injected, the external differential quantum efficiency ηd is expressed as

mirror loss ηd = ηi = ηi total loss

1 1 1 ln ln 2L R1 R2 R1 R2 = ηi , 1 1 1 αi + ln 2αi L + ln 2L R1 R2 R1 R2

(6.48)

where the optical losses at the facets due to absorption or scattering are assumed to be negligible.


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FIGURE 6.11 Optical powers inside and outside a Fabry–Perot cavity.

Light Output Ratio from Facets Optical powers, which are emitted from the two facets, are P1 and P2 . As shown in Fig. 6.11, it is assumed that the optical powers in the vicinity of the facets inside a Fabry–Perot cavity are Pa , Pb , Pc , and Pd ; the power reflectivities of the facets are R1 and R2 ; and the power transmissivities of the facets are T1 and T2 . Here, the arrows indicate the propagation directions of the light. In a steady state, the optical powers are related as Pa = R1 Pd , Pb = egL Pa , Pc = R2 Pb , Pd = egL Pc , P1 = T1 Pd , P2 = T2 Pb ,

(6.49)

where g is the power optical gain coefficient and L is the cavity length. Deleting Pk (k = a, b, c, d) from (6.49) results in √ P1 T1 R2 = √ . P2 T2 R1

(6.50)

From (6.50), to extract a large light output P j from a facet j, the power transmissivity of the facet T j should be high; the power reflectivity of the facet R j should be low. Substituting P = P1 + P2 into (6.49) leads to √ T1 R2 P1 = √ √ P, T1 R2 + T2 R1

√ T2 R1 P2 = √ √ P. T1 R2 + T2 R1

(6.51)

As a result, the external differential quantum efficiencies for each light output ηd1 and ηd2 are written ηd1

√ T1 R2 = √ √ ηd , T1 R2 + T2 R1

ηd2

√ T2 R1 = √ √ ηd , T1 R2 + T2 R1

(6.52)

where ηd is the external differential quantum efficiency for the total light output. When the optical losses at the facets are negligibly small, we can use T1 = 1 − R1 and T2 = 1 − R2 .


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141

Light Output

AlGaAs/GaAs LD InGaAsP/InP LD

Pulsed

Injected Current FIGURE 6.12

I –L characteristics with temperature as a parameter.

Dependence of I–L on Temperature With a rise in temperature, the threshold current density Jth usually increases and the external differential quantum efficiency ηd decreases, as shown in Fig. 6.12. The reason for a decrease in the external quantum efficiency ηd with an increase in Jth is that the threshold carrier concentration n th increases with Jth , which enhances the free carrier absorption, which is derived below. An equation of motion for the electron in a crystal without a magnetic field is given by m∗

d2 x 1 dx + m∗ = −eE, dt 2 τ dt

(6.53)

where m ∗ is the effective mass of the electron; τ is the relaxation time, such as the mean free time of collision; e is the elementary electric charge; and E is the electric field. If we assume x, E ∝ e i ωt , where ω is the angular frequency of light, a position x of the electron is obtained as x=

m∗

eE

. − i ω/τ

ω2

(6.54)

The polarization P of a semiconductor is written P = P0 + Pi ; Pi = −nex,

(6.55)


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where P0 is the polarization caused by ionization of atoms constituting the crystal and Pi is the polarization induced by the motion of the electrons. Here we express the electric flux density D as D = ε0 E + P = (ε0 E + P0 ) + Pi = εε0 E + Pi = ε ε0 E,

(6.56)

where ε0 is the permittivity of vacuum and ε is the dielectric constant of the crystal based on ionization of the atoms. The dielectric constant ε , which is modified by the motion of the electrons, is written ε = εr − i εi = ε −

nex , ε0 E

(6.57)

where εr and εi are real and imaginary parts of ε , respectively. Substituting (6.54) into (6.57) leads to ne2 , m ∗ ω2 ε0 ne2 εi ∗ 3 , m ω ε0 τ

εr − ε −

(6.58) (6.59)

where ω 1/τ was used. Using the complex refractive index n r − i κ, we can express εr and εi as εr = n r 2 − κ 2 , εi = 2n r κ.

(6.60) (6.61)

Therefore, the optical power absorption coefficient α due to free carrier absorption is obtained as α=

2ω ω εi ne2 κ= = ∗ 2 , c c nr m ω ε0 n r cτ

(6.62)

where c is the speed of light in vacuum. Characteristic Temperature Dependence of the threshold current density Jth on temperature is expressed empirically as

Tj Jth = Jth0 exp T0

,

(6.63)


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143

InGaAsP/InP LD AlGaAs/GaAs LD Pulsed

Temperature FIGURE 6.13 Dependence of threshold current density on temperature.

where Jth0 is a coefficient; Tj is the temperature in the active layer or junction temperature; and T0 , called the characteristic temperature, represents the dependence of the threshold current density on the temperature. A large characteristic temperature T0 seems to result in a small dJth /dTj value, which indicates a high-quality LD. However, T0 cannot always be used as an appropriate figure of merit of LDs, because a larger Jth leads to a greater T0 when dJth /dTj is constant. Therefore, only LDs with a common Jth at the same temperature can be compared with each other by using T0 . Figure 6.13 shows the dependence of the threshold current density on the temperature for an AlGaAs/GaAs LD and an InGaAsP/InP LD in pulsed operations. In an AlGaAs/GaAs LD with an oscillation wavelength of 0.85 µm, T0 is approximately 160 K from 25 to 80◦ C and 120 K above 80◦ C. It is considered that an increase in Jth with Tj is caused by broadening of the gain spectrum and the overflow of the carriers over the heterobarriers. To reduce the overflow of the carriers over the heterobarriers, the band offset E g between the active layer and the cladding layers has to be increased. In general, the band offset E g should be larger than 0.3 eV to suppress an increase in Jth with Tj . In an InGaAsP/InP LD with an oscillation wavelength of 1.3 µm, T0 is approximately 70 K from 25 to 65◦ C and 50 K above 65◦ C. These T0 values are lower than those of the AlGaAs/GaAs LD because of efficient overflow of the carriers due to the light effective mass of the conduction electrons and nonradiative recombinations due to Auger processes and valence band absorptions. The effective masses of the conduction electrons are 0.070m 0 for AlGaAs with a bandgap wavelength of 0.85 µm, and 0.059m 0 for InGaAsP with a bandgap wavelength of 1.3 µm, where m 0 is the electron mass in vacuum. Auger processes are shown in Fig. 6.14, where C, H, L, and S represent the conduction, heavy hole, light hole, and split-off bands, respectively. In Auger processes, two processes, such as CHSH and CHCC, are present. In the


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Conduction Band (C)

Heavy Hole Band (H) Light Hole Band (L) Split-off Band (S) (a)

(b)

FIGURE 6.14 Auger processes: (a) CHSH; (b) CHCC.

CHSH process, the energy emitted due to a recombination of conduction electron 1 in the conduction band (C) and hole 2 in the heavy hole band (H) excites valence electron 3 in the split-off band (S) to energy state 4 in the heavy hole band (H). In the CHCC process, the energy emitted due to a recombination of conduction electron 1 in the conduction band (C) and hole 2 in the heavy hole band (H) excites conduction electron 3 in the conduction band (C) to higher-energy state 4 in the conduction band (C). These processes are three-body collision processes, and a recombination rate RA for Auger processes is given by RA = Cp np 2 + Cn n 2 p,

(6.64)

where Cp and Cn are Auger coefficients for the CHSH and CHCC processes, respectively. The valence band absorptions are shown in Fig. 6.15, where an electron in the split-off band absorbs light which is generated by a recombination of a conduction electron in the conduction band and a hole in the heavy hole band. As a result, the electron in the split-off band is excited to the heavy hole band or the acceptor level. With an increase in temperature, the internal quantum efficiency ηi is decreased due to Auger processes and the external quantum efficiency ηd is reduced due to the valence band absorptions and free carrier absorption. Therefore, light emission efficiency is lowered with an increase in temperature. Derivative Light Output To detect the threshold current precisely, derivative measurements can be used. Using the photon concentration S in the active layer, the internal optical power P can be written P = ωvg SB S,

(6.65)


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145

Conduction Band (C)

Acceptor Level Heavy Hole Band (H) Light Hole Band (L) Split-off Band (S) (a)

(b)

FIGURE 6.15 Valence band absorption: an electron is excited to (a) the heavy hole band and (b) the acceptor level.

where = h/2π is Dirac’s constant, h is Planck’s constant, ω is the angular frequency of light, vg is a group velocity of the light, and SB is a beam area. From (6.65), the derivative optical power with respect to the current injected, dP/dI , is written dP dS ∂ S ∂n = ωvg SB = ωvg SB , dI dI ∂n ∂ I

(6.66)

Light Output

where I is the current injected and n is the carrier concentration. Figure 6.16 shows calculated light output and derivative light output as a function of the current injected where (6.12), (6.13), (6.65), and (6.66) are used. The threshold current is indicated clearly by a sharp rise in dP/dI .

Injected Current FIGURE 6.16 Current versus light output and derivative light output.


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Light Output

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August 9, 2010

Injected Current (a)

Injected Current (b)

FIGURE 6.17 (a) I –L and (b) I –dV /dI characteristics.

Derivative Electrical Resistance The derivative resistance dV /dI is expressed as dV ∂ V ∂n = , dI ∂n ∂ I

(6.67)

where V is the voltage that is applied to the LD. Figure 6.17(a) shows calculated I –L characteristics and Fig. 6.17(b) shows calculated I –dV /dI characteristics where (6.15), (6.16), and (6.67) are used. As shown in Fig. 6.17, each I –dV /dI curve has a kink at the threshold current. Therefore, the threshold current Ith can be determined by the derivative electrical resistance dV /dI , and this method will be especially useful for ring or disk LDs with extremely low light output. 6.5 POLARIZATION OF LIGHT Fabry–Perot LDs with bulk active layers oscillate in the TE mode. Because the bulk active layers do not have particular quantum mechanical axes, the optical gains for


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POLARIZATION OF LIGHT

147

Active Layer Effective Refractive Index

Refractive Index

Refractive Index

(a)

Refractive Index

(b) FIGURE 6.18 Reflection at a facet.

bulk active layers are independent of the polarization of light. However, the facet reflectivities depend on the polarization of light. Figure 6.18(a) shows the reflection at a facet of an LD using a zigzag model in geometrical optics. We suppose that the effective refractive index of the optical waveguide is n A and that the refractive index of the outside of the optical waveguide is n B . This reflection is also considered as the reflection at the interface of a semiconductor with refractive index n A = n A / cos θ and that with the refractive index n B , as shown in Fig. 6.18(b), where θ is the angle of incidence. Fresnel formulas give us the power reflectivities RTE for the TE mode and RTM for the TM mode:

RTE

RTM

n cos θ − n 2 − n 2 sin2 θ 2 B A A = , n A cos θ + n B 2 − n A 2 sin2 θ n 2 cos θ − n n 2 − n 2 sin2 θ 2 A B A B = , n B 2 cos θ + n A n B 2 − n A 2 sin2 θ

(6.68)

(6.69)

Figure 6.19 shows the power reflectivity R when laser beams are emitted from GaAs with the refractive index n A = 3.6 to the air with n B = 1. As shown in Fig. 6.19, it is found that RTE ≥ RTM is kept for all values of θ . Therefore, from (6.31), the threshold gains for TE modes are lower than those of TM modes, which leads to lasing in the TE modes with lower threshold current than for the TM modes. With an increase in θ (i.e., with an increase in the order of vertical transverse modes), RTE is enhanced. Therefore, the threshold gains of higher-order vertical transverse modes are smaller than those of lower-order vertical transverse modes. However, from Fig. 6.7, to minimize the threshold current density, the active layer thickness d should be approximately 0.1 µm or less, in which the higher-order vertical transverse modes are cut off and only the fundamental vertical transverse mode oscillates. Note that the optical power ratio of the TE and TM modes is approximately 1 : 1 below the threshold, because most of the light emitted below the threshold is not guided light but spontaneously emitted light due to amplified spontaneous emission.


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Power Reflectivity

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GaAs Air

TE

TM

Angle of Incidence FIGURE 6.19 Reflectivities for TE and TM modes.

6.6 TRANSVERSE MODES Transverse modes, also called lateral modes, show the light intensity distributions along the axes perpendicular to the cavity axis, and represent the shapes of the laser beams. Transverse modes are highly dependent on the structures of optical waveguides. Vertical transverse modes display light intensity distributions along the axes perpendicular to the active layer plane; horizontal transverse modes exhibit light intensity distributions along the axes parallel to the active layer plane. To examine transverse modes, near-field patterns (NFPs) and far-field patterns (FFPs), which are shown in Fig. 6.20, are generally used. An NFP is a light intensity Vertical NFP

Near-Field Pattern (NFP)

Beam Size

Horizontal NFP Beam Size

Vertical FFP

Facet of a LD

Radiation Angle

Far-Field Pattern (FFP) Horizontal FFP Radiation Angle

FIGURE 6.20 Near- and far-field patterns.


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149

distribution on a facet of an LD. The sizes of the NFP are measured in units of the length of the emission regions W and W⊥ , illustrated in Fig. 6.20. Usually, the active layer thickness d is approximately 0.1 µm or less to achieve low threshold current; the active layer width is on the order of 2 µm or more to obtain high light output. As a result, the near-field pattern is asymmetric, which is long along the parallel axis and short along the perpendicular axis to the active layer plane. An FFP is a light intensity distribution at a position far enough from a facet of an LD. As shown in Fig. 6.20, the sizes of the FFP are measured in units of radiation angles θ and θ⊥ , which are independent of the distance between a facet of an LD and a photodetector. An FFP is considered a diffracted pattern of an NFP if the NFP is regarded as light emitted from a slit. With decreases in the sizes of the slit, the sizes of the diffracted patterns increase. Therefore, an FFP is large for a small NFP and small for a large NFP. Because of asymmetry in an NFP, an FFP is also asymmetric, with a small horizontal transverse mode and a large vertical transverse mode. To achieve a large coupling efficiency of a laser beam to a lens or an optical fiber, a symmetric laser beam with a narrow radiation angle is required. These requirements can be accomplished using vertical cavity surface-emitting lasers. Vertical Transverse Modes Guided Modes Because double heterostructures are adopted in LDs, light is confined in an area sandwiched by the planes, which are parallel to the active layer plane, due to index guiding. The guiding condition of vertical transverse modes is represented by the eigenvalue equation for the transverse resonance condition. Laser Oscillation in Higher-Order Modes Active Layer Thickness d hc The optical confinement factors m for the mthorder mode and m−1 for the (m − 1)th-order mode are related as m−1 > m .

(6.70)

When the active layer thickness d is slightly larger than the cutoff guiding layer thickness h c , the difference in the reflectivities between the adjacent higher-order modes is small. Therefore, from (6.37) and (6.38), the threshold current densities Jth,m for the mth-order mode and Jth,m−1 for the (m − 1)th-order mode are related as Jth,m−1 < Jth,m ,

(6.71)

which leads to a laser oscillation in the (m − 1)th-order mode, not the mth-order mode. Active Layer Thickness d hc As shown in Fig. 6.19, higher-order TE modes have larger reflectivities than those of lower-order TE modes. Therefore, when the active layer thickness d is much larger than the cutoff guiding layer thickness h c , a laser


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oscillation does not take place in an (m − 1)th-order mode but does in an mth-order mode. For example, in an AlGaAs/GaAs LD with n f = 3.6, (n f − n s )/n f = 5%, and λ0 = 0.85 µm, the cutoff guiding layer thickness h cm for the mth-order mode is obtained as h c1 = 0.38 µm, h c2 = 0.76 µm, h c3 = 1.13 µm. In this case, according to the active layer thickness d, the following laser oscillations in higher-order modes take place: ⎧ ⎨ 0.66 µm: 0.98 µm: ⎩ 1.30 µm:

first-order mode second-order mode third-order mode

In conventional LDs, the active layer thickness d is 0.1 µm or less, which leads to laser oscillations in the fundamental vertical transverse mode (m = 0). NFP and FFP When the active layer thickness d is larger than the wavelength of light in a semiconductor, the full width at half maximum (FWHM) of the beam size of a vertical NFP W⊥ decreases with a reduction in d because the light emission region narrows. However, when d is less than the wavelength of light in a semiconductor, W⊥ increases with a reduction in d because the guided light penetrates deeply into the cladding layer and the substrate. Figure 6.21 shows the results calculated for the FWHM of the beam size of a vertical NFP W⊥ and the FWHM of the radiation angle of a vertical FFP θ⊥ for the fundamental TE mode (m = 0) as a function of the active layer thickness d with = (n f − n s )/n f as a parameter. Here n f and n s are the refractive indexes of the active layer and substrate, respectively. Because the FFP is considered to be

Active Layer Thickness FIGURE 6.21 Sizes of NFP and FFP.


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151

a diffracted pattern of the NFP, the FFP is large for a small NFP and small for a large NFP.

Horizontal Transverse Modes Horizontal transverse modes show light intensity distributions along the axes parallel to the active layer plane. To control horizontal transverse modes, gain guiding and index guiding have been used. In gain guiding, light propagates only in the optical gain region. In index guiding, light propagates in a high-refractive-index region which is surrounded by low-index regions. Gain guiding structures can be fabricated by a simpler method compared with index guiding structures, but the horizontal mode is unstable and the threshold current is higher than in index guiding structures. Gain Guiding In gain guiding structures, optical gain regions are formed by restricting the current flowing area. For example, electrodes are formed selectively, as shown in Fig. 6.22(a), which is a cross-sectional view of a gain guiding LD seen from a facet. The current injected flows from the selectively formed electrode along the arrows by diffusion. As a result, the carrier concentration is largest at the center of the stripe, and it decreases with an increase in distance from the center, as shown in Fig. 6.22(b). Therefore, the center region of the stripe has optical gain and the stripe edges have optical losses, as shown in Fig. 6.22(c). With an increase in the current injected, the refractive index changes due to the free carrier plasma effect, Joule heating in the active layer, and spatial hole burning.

Anode Active Layer Cathode (a) Carrier Concentration

Position (b)

Optical Gain Position Optical Loss (c) FIGURE 6.22 Gain guiding LD: (a) cross-sectional view; (b) distribution of carrier concentration; (c) distribution of optical gain.


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Free Carrier Plasma Effect When light is present in a semiconductor, free carriers in the semiconductor vibrate with the frequency of the light; the phases of the free carrier vibrations cancel out the electrical polarizations of the lattice atoms. This phenomenon is called the free carrier plasma effect. A change in the refractive index n rf due to the free carrier plasma effect is given in the following. When the carrier concentration increases by n, it is assumed that the real part of the dielectric constant εr and the real part of the complex refractive index n r change to εr + εr and n r + n r , respectively. In this case we have 2n r n r = εr .

(6.72)

Hence, we obtain n r =

εr εr − ε e2 = =− ∗ 2 n, 2n r 2n r 2m ω ε0 n r

(6.73)

where (6.58) is used. Equation (6.73) is proportional to the injected carrier concentration n. When the carrier concentration n is on the order of 1018 cm−3 , a decrease in the refractive index is on the order of 10−3 . Because the carrier concentration is greatest at the center of the stripe, the refractive index in the center is lower than that of the surrounding regions, according to (6.73). As a result, the light is not confined completely to the active layer but is radiated to the surrounding regions, which is called the antiguiding effect. Joule Heating With an increase in current injected, the active layer is heated by Joule heating. Therefore, the refractive index increases; from experimental results, this change n rT is given by n rT = (2 ∼ 5) × 10−4 T,

(6.74)

where T is an increase in the temperature of the active layer, which is expressed in kelvin units. As opposed to the free carrier plasma effect, an increase in the refractive index due to Joule heating of the active layer contributes to confining the light to the active layer, which is called the guiding effect. Spatial Hole Burning When the injected current increases further and a large light output is obtained, a lot of carriers recombine, due to the stimulated emission. Because the stimulated emission takes places efficiently in a large optical gain region, which is the center of the stripe, the carrier concentration at the center of the stripe is lower than that at its surrounding regions. This phenomenon, called spatial hole burning, increases the refractive index at the center region and enhances the guiding effect, resulting in a large confinement of the light to the active layer. The horizontal distribution of the refractive index is determined by both the antiguiding effect, which is due to the free carrier plasma effect, and the guiding effect,


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153

Kink Kink

FIGURE 6.23 Kink in an I –L curve.

which is due to Joule heating of the active layer and spatial hole burning. Hence, according to the current injected, the horizontal transverse modes show complicated behaviors, such as changes in positions or multiple peaks. When the horizontal transverse modes change, kinks are observed in I –L curves, as shown in Fig. 6.23. In gain guiding structures, the wave fronts of horizontal transverse modes bend convexly in the propagation direction, while those of vertical transverse modes are close to plane waves. As a result, a minimum beam diameter, which is called the beam waist of a horizontal transverse mode, is placed inside the optical cavity; the beam waist of vertical transverse modes is placed on a facet of an LD. Such a difference in the positions of the beam waists for vertical and horizontal transverse modes is called astigmatism. Because of astigmatism, the vertical and horizontal transverse modes cannot be focused on a common plane by an axially symmetric convex lens, so only defocused images are obtained. Index Guiding Index guiding structures have an intentionally formed refractive index distribution, as shown in Fig. 6.24. In index guiding structures, horizontal and vertical transverse modes are both close to plane waves, which does not result in astigmatism. To obtain stable horizontal transverse modes, we need the following: (1) n r > |n rf |, where n r is the difference in the refractive indexes between the active layer and its surrounding layers and n rf is a change in the refractive index due to the free carrier plasma effect; (2) the cutoff condition for the higher-order modes is satisfied; and (3) the active layer width is shorter than the carrier diffusion length, which is 2 to 3 µm. Condition (1) is required to achieve index guiding even in a high injected current. Condition (2) is introduced to obtain a single horizontal transverse mode by confining the fundamental transverse mode to the optical waveguide. If multiple higher-order horizontal transverse modes are present in the optical waveguide, mode hopping or mode competition takes place according to operating conditions, which leads to unstable horizontal transverse modes. To satisfy conditions (1) and (2), n r /n r is approximately 10−2 . Finally, condition (3) is needed to achieve uniform


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Anode Active Layer Cathode (a) Refractive Index Position (b) FIGURE 6.24 Index guiding LD: (a) cross-sectional view; (b) distribution of refractive index.

distributions of the carrier concentration in the active layer, which reduces spatial hole burning. To date, many index guiding structures have been developed to confine the carriers and the light efficiently to the active layer, and the structures are classified into three categories: rib waveguides, ridge waveguides, and buried heterostructures (BHs). Rib Waveguide Rib waveguides are optical waveguides with convex or concave regions, which are suitable for LDs whose active layers are easily oxidized in the air during etching, and their emission efficiencies decrease drastically, due to oxidization. These active layers (e.g., AlGaAs layers) cannot be etched in the air, and the rib waveguides can be formed without etching the active layers in the air. As an example of a rib waveguide, Fig. 6.25 shows a planoconvex waveguide (PCW) structure in which the semiconductor layers are grown on a preetched substrate. Here arrow a, with a solid line, and arrow b, with a dashed line, represent the flowing paths of the electric current. Path a shows the direction of a forward current

p-diffused Region Anode

n p Active Layer n n-substrate

Cathode FIGURE 6.25 Rib waveguide.


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TRANSVERSE MODES

(a) FIGURE 6.26

155

(b)

I –V characteristics in (a) pn- and (b) pnpn-junctions.

across the pn-junction, and the electrical resistance along this path is low; path b shows the direction of a forward current across the pnpn-junction. Figure 6.26 shows I –V characteristics in the pn- and pnpn-junctions. When the voltage V applied is below the switching voltage Vs , a pn-junction inside the pnpnjunction is reverse biased and electric current barely flows. Once the applied voltage V exceeds Vs , the pnpn-junction switches on and shows an I –V characteristic similar to that of a pn-junction in forward bias. Therefore, by designing a pnpn-junction so that Vs may be larger than the voltage applied across the pn-junction, little electric current flows through path b; the current flows mainly through path a. Note that the electric current flowing regions are broad, because a current constriction structure is not formed below the p-diffused region. As a result, it can be said that the rib waveguides are not optimized for confinement of the carriers to the active layers. Therefore, the horizontal transverse mode is not as stable with respect to the injected current, and the threshold current is relatively high. Ridge Waveguide Ridge waveguides are optical waveguides with a convex region. Because they are easily fabricated by etching after epitaxial growth, low-cost LDs are expected. Etching is stopped above the active layer, as shown in Fig. 6.27, and the easily oxidized layers are not exposed to the air.

Anode

p Active Layer n-substrate

Cathode FIGURE 6.27 Ridge waveguide.


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Anode p-Cladding Guiding Active Guiding n-Cladding n-Substrate Cathode (a)

(b)

FIGURE 6.28 Ridge LDs with antiguiding waveguides: (a) antiguiding layers; (b) antiguiding cladding layers.

Note that the electric current–flowing regions are broad, because a current constriction structure is not formed below the p-diffused region. As a result, it can be said that the ridge waveguides are not optimized for confinement of the carriers to the active layers. Therefore, the horizontal transverse mode is not as stable with respect to the injected current, and the threshold current is relatively high. As shown in Fig. 6.28, ridge LDs with antiguiding layers [3–6] and antiguiding cladding layers [7] have been proposed, and it is shown theoretically that the horizontal transverse mode becomes stable. Buried Heterostructure In buried heterostructures (BHs) the active layer is surrounded by regrown regions. At first, epitaxial layers are grown on a semiconductor substrate, which is followed by etching to form a stripe. This stripe is buried by the second epitaxial growth, and the buried regions prevent the injected current from flowing. Although fabrication processes are complicated, as described above, low-threshold, high-efficiency laser operations are obtained because of efficient confinement to the active layer of both the carriers and the light. However, because the active layers are exposed to the air during etching, only active layers insensitive to oxidization can be used for BHs. For example, in InGaAsP/InP LDs, which are the light sources of longhaul optical fiber communication systems, BHs are frequently adopted. Figure 6.29 shows a BH [8] and a double-channel planar buried heterostructure (DC-PBH) [9].

p-Diffused Region Anode p

n p

Insulator

p p

Active Layer n-Substrate

(a)

Cathode

(b)

FIGURE 6.29 Buried heterostructures: (a) BH; (b) DC-PBH.

n p p


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157

To constrict the current-flowing region, the surrounding regions of the stripe are pnpn-junctions. Therefore, electric current flows efficiently through path a but flows hardly at all through path b. In the DC-PBH shown in Fig. 6.29(b), path c is a pnpin-junction because the active layer is undoped and regarded as nearly intrinsic. Therefore, electric current through path c is smaller than that through path b. As a result, the DC-PBH has a lower threshold current and higher efficiency than those of the BH.

6.7

LONGITUDINAL MODES

Longitudinal modes, also called axial modes, determine the resonant wavelengths of the cavity and show the light intensity distributions along the cavity axes. Figure 6.30 shows examples of oscillation spectra for multimode and a single-mode operation. LDs use interband transitions to obtain the optical gain, and the optical gain spectrum has a width of about 10 nm. In addition, Fabry–Perot cavities have a lot of resonant modes, which leads to low mode selectivity. For these two reasons, Fabry–Perot LDs tend to oscillate in multimodes, as shown in Fig. 6.30(a). However, single-mode LDs are not always needed for applications such as compact disks, laser printers, bar-code readers, laser pointers, or short-haul optical fiber communication systems in which Fabry–Perot LDs are used as light sources. In long-haul, large-capacity optical fiber communication systems, single-mode LDs are needed because optical fibers demonstrate dispersion in that their refractive indexes depend on the wavelengths and modes of light. Due to the dispersion, the propagation speed of light changes according to the wavelengths and modes of the light. If LDs show multimode operations, the optical pulses broaden in the time domain during propagating through the optical fibers. With an increase in the transmission distance and a decrease in the pulse spacing, adjacent optical pulses tend to overlap each other. Finally, the photodetectors cannot resolve sequentially transmitted optical pulses, as shown in Fig. 6.31.

Wavelength (a)

Wavelength (b)

FIGURE 6.30 Oscillation spectra for (a) multimode and (b) single-mode operation.


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Optical Power

Optical Power

Time

Time (b)

(a)

FIGURE 6.31 Light pulses in (a) a transmitter and (b) a photodetector.

Note that the longitudinal modes change with the transverse modes, because the effective refractive indexes of the optical waveguides depend on the transverse modes. Single-transverse-mode operation is indispensable to achieving single-longitudinalmode operation. Static Characteristics of Fabry–Perot LDs Let us calculate the wavelength spacing of the longitudinal modes λ = λm − λm+1 in the Fabry–Perot LD shown in Fig. 6.32. Using a cavity length L, a positive integer m ( 1), and a refractive index n r (λ) that is a function of a light wavelength λ, resonant wavelengths λm and λm+1 are written λm =

2L 2L n r (λm ) , λm+1 = n r (λm+1 ) . m m+1

(6.75)

As a result, we have λ = λm − λm+1

λm dn r 1 λ + n r (λm ) dλ 2L

λm n r (λm )

2 n r (λm ) ,

(6.76)

where m 1 is used. Therefore, we obtain λ =

λm 2 , 2n rλ L

(6.77)

Wavelength FIGURE 6.32 Wavelength spacing of the longitudinal modes in a Fabry–Perot LD.


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LONGITUDINAL MODES

Oscillation Spectra

159

A

B

Optical Gain A B Wavelength FIGURE 6.33 Relationship between longitudinal modes and an optical gain spectrum.

where n rλ is the equivalent refractive index, which is an effective refractive index with dispersion and is given by dn r n rλ = n r (λm ) 1 − λm . dλ

(6.78)

Generally, we have dn r /dλ < 0, and the equivalent refractive index n rλ is larger than n r . Figure 6.33 shows a relationship between the longitudinal modes and the optical gain spectrum. Fabry–Perot LDs oscillate at the resonant wavelength that is closest to the gain peak. As a result, the oscillation wavelength for optical gain A is λ0A and that for optical gain B is λ0B . If the nonlinear effect and the coupling of the spontaneous emission to the lasing mode are negligibly small, the carrier concentration in the active layer is constant above the threshold, the free carrier plasma effect does not change the refractive index, and Joule heating of the active layer enhances the refractive index with an increase in the injected current I . Hence, the resonant wavelengths become longer with I (> Ith ) according to (6.25). Above the threshold, the band-filling effect is not dominant, and the peak wavelength of the optical gain spectrum becomes longer due to Joule heating of the optical waveguide with an increase in I . With an increase in I above Ith , both the resonant wavelengths and the peak wavelength of the optical gain spectrum become longer, due to Joule heating of the optical waveguide. Therefore, the oscillation wavelength increases with I . Here it should be noted that the change rate of the resonant wavelength and that of the peak


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Peak Wavelength

-th -th

Peak Wavelength

Mode Intensity


(nm)

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(d)

(d)

(c)

Temperature

(c)

Injected Current

(a)

(b) (b)

d m dT j

Main Mode (Peak Wavelength d dT j

)

Optical Gain

Submode

(a) Wavelength

Injected Current (mA) FIGURE 6.34 Dependence of longitudinal modes on injected current.

wavelength of the optical gain spectrum with I differ from each other. As a result, with an increase in I , the lasing mode sometimes jumps, because the resonant wavelength closest to the peak wavelength of the optical gain spectrum at I=Ith departs from the peak wavelength of the optical gain spectrum at I>Ith , and other resonant wavelengths approach the peak wavelength of the optical gain spectrum at I>Ith . Figure 6.34 shows the dependence of the longitudinal modes on the injected current I . It is found that the wavelengths of longitudinal modes become longer, and the mode jumps to the other mode at several values of I . In addition, according to an increase or decrease in I , hysteresis loops are present. Similar phenomena are observed when the junction temperature Tj is changed while I is kept constant. The reason for these hysteresis loops is that the optical gain concentrates on the oscillating longitudinal mode, and the optical gains for the other modes are suppressed due to coupling of modes and intraband relaxation of the carriers. Carriers with the transition energy of the lasing mode are highly consumed by stimulated emission. These consumed carriers, however, are compensated by the intraband relaxation of the carriers, and the number of carriers with the transition energy of the nonlasing mode decreases. Therefore, the optical gain is concentrated on the lasing mode. 6.8 MODULATION CHARACTERISTICS Lightwave Transmission Systems and Modulation In lightwave transmission systems, an intensity modulation/direct detection system and a coherent system are present. With regard to the modulations of laser beams, direct modulation of LDs and external modulation using optical modulators exist.


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161

Optical Signals Optical Fiber LD

Photodetector

FIGURE 6.35 Intensity modulation/direct detection system.

Intensity Modulation/Direct Detection System Figure 6.35 shows an intensity modulation/direct detection system. In this system, a transmitter sends optical signals by modulating the light intensity; a photodetector directly detects changes in the light intensity and transforms the optical signals into electric signals. This system is simpler and more cost-effective than a coherent system. Therefore, the intensity modulation/direct detection system has been used in conventional optical fiber communication systems. A problem with transmission distance, which was inferior to that of a coherent system, was solved by the advent of optical fiber amplifiers. Coherent System In a coherent system, modulation schemes such as amplitude shift keying (ASK), frequency shift keying (FSK), and phase shift keying (PSK) are present according to modulations of the laser beam. As shown in Fig. 6.36, modulated light from a master light source LD1 and light from a slave (local) light source LD2 are simultaneously incident on a photodetector. In the photodetector, interference of these two laser beams generates an optical beat signal, which is converted to an electric signal. With an increase in the light output of LD2 , the signal-to-noise (S/N) ratio is improved, which leads to long-haul optical fiber communication systems. However, to obtain an optical beat, we have to prepare two LDs whose laser light has narrow spectral linewidths, almost common wavelengths, and the same polarizations. In addition, we need polarization controllers to achieve the same polarizations, because polarization of the lightwave changes due to contortion of the optical fibers. Also,

Optical Fiber

Coupler LD1

Photodetector LD2 FIGURE 6.36 Coherent system.


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the electronic circuits used to process electric signals from a photodetector are complicated. As a result, the cost of a coherent system is much higher than that of an intensity modulation/direct detection system. Direct Modulation In direct modulation of LDs, the current injected to LDs is modulated. During direct modulations, multimode operations, chirping, and changes in the turn-on delay time take place. Fabry–Perot LDs can be used as light sources for short-haul optical fiber communication systems with a transmission distance of 1 km or so, as in a building, because these problems are not serious. In contrast, Fabry– Perot LDs with a transmission distance of more than several tens of kilometers cannot satisfy the system specifications for long-haul optical fiber communication systems. Therefore, DFB LDs with stable single-longitudinal-mode operations are used as light sources for long-haul optical fiber communication systems. External Modulation In external modulation of the laser beams, the current injected to LDs is kept constant, and the laser beams emitted from LDs are modulated by the optical modulators. Because the current injected into LDs is constant, relaxation oscillations do not take place, in contrast to direct modulation. Therefore, multimode operations can be avoided. In addition, chirping is low, because the change in the refractive index in optical modulators is small. To reduce costs and to achieve high optical coupling efficiencies between the LDs and optical modulators, integrated light sources of DFB LDs and optical modulators have been developed and used in conventional long-haul, large-capacity optical fiber communication systems. In the following we focus on the direct modulation of LDs because it is helpful to understand the peculiar characteristics of LDs. To examine the direct modulation of Fabry–Perot LDs, a turn-on delay time and a relaxation frequency are derived.

Direct Modulation Turn-on Delay Time It is assumed that a step pulsed current is injected into an LD. As shown in Fig. 6.37(a), a bias current density Jb is below the threshold current density Jth (Jb < Jth ), and a pulsed current density Jp is injected to the LD at time t = ton = 0. Here it is assumed that the pulse width is much larger than the carrier lifetime τn . The carrier concentration n increases from a biased value n b with a time constant τn . As shown in Fig. 6.37(b), when the carrier concentration n reaches the threshold carrier concentration n th at the turn-on delay time t = td , laser oscillation starts. Once oscillation begins, the carrier concentration n and photon concentration S show relaxation oscillations, as shown in Fig. 6.37(b) and (c). Note that Fig. 6.37 was drawn by analyzing (6.2) and (6.3) numerically, because exact analytical solutions are not present for the rate equations (6.2) and (6.3). The turn-on delay time td can be calculated using rate equations. For simplicity, the coupling of spontaneous emission to the lasing mode is neglected, and the photon


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163

(a)

(b)

(c) FIGURE 6.37 Turn-on delay time and relaxation oscillations: (a) current density; (b) carrier concentration; (c) photon concentration.

concentration is S = 0 for n < n th . As a result, (6.2) is reduced to dn J n = − . dt ed τn

(6.79)

The current density J is assumed to be J = Jp · u(t) + Jb ,

(6.80)

where u(t) =

0 1

(t < 0), (t ≥ 0).

(6.81)

We substitute (6.80) into (6.79), then take the Laplace transform of both sides. If we express a Laplace transform of n(t) as N (s) and let n(0) = n b = τn Jb /ed, we


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obtain s N (s) − n(0) = s N (s) −

Jp + Jb 1 τn Jb 1 = − N (s). ed ed s τn

(6.82)

τn (Jp + Jb ) τn Jb 1 + . ed ed s + τn −1

(6.83)

As a result, N (s) is written N (s) =

1 1 − s s + τn −1

From (6.83), the inverse Laplace transform of N (s) is written τn (Jp + Jb ) τn (Jp + Jb ) −t/τn τn Jb −t/τn u(t) − e + e ed ed ed τn (Jp + Jb ) τn Jp −t/τn = u(t) − e . ed ed

n(t) =

(6.84)

Because of u(t) = 1 in t ≥ 0, (6.84) is expressed as n(t) =

τn Jp −t/τn τn J − e , ed ed

(6.85)

Jp + Jb = J.

(6.86)

where

At t = td , the carrier concentration n reaches the threshold carrier concentration n th , and we have n(td ) = n th =

τn Jth , ed

(6.87)

where (6.19) is used. Using (6.85)–(6.87), the turn-on delay time td is obtained as td = τn ln

J − Jb . J − Jth

(6.88)

To generate high-speed optical signals by modulating the current injected to LDs, the turn-on delay time td should be as short as possible. From (6.88) it is found that a large bias current density Jb , a low threshold current density Jth , and a short carrier lifetime τn are suitable for direct high-speed modulations. Relaxation Oscillation If we use small-signal analysis, approximate analytical solutions can be obtained from (6.2) and (6.3), and the solutions clearly give us their


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165


physical meanings. To proceed in small-signal analysis, the carrier concentration n, the photon concentration S, and the current density J are expressed as n = n c0 + δn, S = S0 + δS, J = J0 + δ J > Jth , n c0 δn, S0 δS, J0 δ J,

(6.89)

where n c0 , S0 , and J0 are steady-state values of the carrier concentration, photon concentration, and current density, respectively; δn, δS, and δ J are deviations from each steady-state value. If Jb Jp is assumed and several initial sharp peaks in the relaxation oscillation are excluded, the conditions for the small-signal analysis are satisfied. We set J0 = Jb and δ J = Jp in accord with Fig. 6.37. Neglecting coupling of the spontaneous emission to the lasing mode, (6.2) and (6.3) are reduced to dn J n = − G(n)S − , dt ed τn dS S = a G(n)S − . dt τph

(6.90) (6.91)

In a steady state (d/dt = 0), substituting (6.89) into (6.90) and (6.91) leads to J0 n c0 − G(n c0 )S0 − = 0, ed τn 1 . a G(n c0 ) = τph

(6.92) (6.93)

Substituting (6.89) into (6.4) results in G(n) = G(n c0 + δn) = g0 (n c0 + δn − n 0 ) = g0 (n c0 − n 0 ) + g0 δn = G(n c0 ) +

∂G δn, ∂n

(6.94)

where the differential gain, which is defined as g0 ≡

∂G , ∂n

(6.95)

is introduced. Inserting (6.89) into (6.90) and(6.91) with the help of (6.92)–(6.94) and then neglecting the second-order small term δn · S, the rate equations on the deviations δn and δS are written d δJ δS ∂G δn δn = − − S0 δn − , dt ed τph ∂n τn d ∂G δS = a S0 δn. dt ∂n

(6.96) (6.97)


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By removing δn from (6.96) and (6.97), we have d2 δS + dt 2

∂G 1 S0 + ∂n τn

d ∂G S0 ∂ G S0 δS + a δS = a δ J, dt ∂n τph ∂n ed

(6.98)

which indicates that the deviation of the photon concentration δS shows a relaxation oscillation. From (6.98), the decay coefficient γ0 , decay time τre , and oscillation angular frequency ωr are written ∂G 1 1 S0 + = , ∂n τn τre ∂G S0 ωr 2 = a . ∂n τph

γ0 =

(6.99) (6.100)

From (6.100), the relaxation oscillation frequency f r is given by 1 fr = 2π

a

∂G S0 . ∂n τph

(6.101)

To generate high-speed optical signals by modulating the current injected to LDs, the decay coefficient γ0 and the relaxation oscillation frequency f r have to be large. From (6.99) and (6.101) it is found that a large differential gain a ∂ G/∂n, a large photon concentration in a steady state S0 , a short carrier lifetime τn , and a short photon lifetime τph are required for high-speed modulations. High-speed modulations and other characteristics often have trade-offs. As shown in (6.37), (6.38), and (6.95), a large a ∂G/∂n leads to a low Jth ; short τn and τph increase Jth . As a result, to achieve simultaneous low-threshold and high-speed operations, we need to obtain a large a ∂G/∂n, which is accomplished in the quantum well LDs. With regard to system applications, if S0 is large, optical power is detected by a photodetector even when optical pulses are not transmitted. As a result, the extinction ratio decreases and the S/N ratio degrades with an increase in S0 . Therefore, S0 should be limited to satisfying the extinction ratio and the S/N ratio, which are specified in applications. The decay coefficient γ0 and the relaxation oscillation frequency f r can also be expressed using the current density J . From (6.18), the threshold carrier concentration n th and the transparent carrier concentration n 0 are written n th =

τn τn Jth , n 0 = J, ed ed 0

(6.102)

where J0 is the current density, which is required for a semiconductor to be transparent. From (6.36) we have n th − n 0 =

1 . a g0 τph

(6.103)


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167

Substituting (6.95) and (6.102) into (6.103), we obtain a

∂G ed = . ∂n τn τph (Jth − J0 )

(6.104)

Inserting (6.40) and (6.104) into (6.99) and (6.101), the decay coefficient γ0 and relaxation oscillation frequency f r are expressed as 1 J − J0 γ0 = , τn Jth − J0 a J − Jth 1 fr = . 2π τn τph Jth − J0

(6.105) (6.106)

Relationship Between Relaxation Oscillation and Longitudinal Modes As shown in Fig. 6.37(b), during relaxation oscillation, the carrier concentration n is modulated around n th . As a result, the optical gain and refractive index change simultaneously and the longitudinal modes are altered. The optical gain g exceeds the threshold optical gain gth when the carrier concentration n is larger than n th . In this case, longitudinal modes with optical gain g ≥ gth show laser operations. Therefore, during relaxation oscillation, multimode laser oscillations are often observed. When the relaxation oscillation is decayed, the number of lasing longitudinal modes decreases. The refractive index is modulated by the free carrier plasma effect. When a deviation in the carrier concentration δn is 2 × 1017 cm−3 , the wavelengths of the longitudinal modes become shorter by about 0.4 nm from their steady-state values. These dynamic changes in longitudinal modes, which are called chirping, broaden the timeaveraged light output spectra, as shown in Fig. 6.38. Multimode laser operations and chirping should be suppressed for long-haul, large-capacity optical fiber communication systems, because the optical fibers have dispersions, as described earlier. Note that the effect of Joule heating with a time scale of microseconds or more is negligible, because the decay time of the relaxation oscillation is on the order of nanoseconds. Dependence on Modulation Frequency When an absolute value of the deviation in the current density δ J is much smaller than the steady-state current density J0 (|δ J | J0 ), small-signal analysis can be used. It is assumed that the deviation δ J is expressed as δ J (ω) e i ωt . Accordingly, the deviation in the carrier concentration δn and that in the photon concentration δS are expressed as δn = δn(ω) e i ωt and δS = δS(ω) e i ωt , respectively. Substituting δ J , δn, and δS into (6.96) and (6.97), we have i ω δ J (ω) , D(ω) ed τph ωr 2 δ J (ω) δS(ω) = − , D(ω) ed δn(ω) = −

(6.107) (6.108)


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Average Longitudinal Mode During (1)−(3): Chirping

Longitudinal Mode at (1)

(1) (2)

Optical Power

Optical Gain

Longitudinal Mode at (3)

(3)

(1) (2) (3)

Wavelength FIGURE 6.38 Multimode operation during relaxation oscillation. The inset shows an optical pulse.

where D(ω) = ω2 − ωr 2 − i ωγ0 .

(6.109)

Modulation efficiencies δ(ω) and δ( f ), which are defined as the number of photons generated per injected electron, are written δS(ω) τph ωr 2 τph ωr 2 = δ(ω) = = , δ J (ω)/ed |D(ω)| (ω2 − ωr 2 )2 + ω2 γ0 2

(6.110)

2π τph f r 2 , δ( f ) = 4π 2 ( f 2 − f r 2 )2 + f 2 γ0 2

(6.111)

where (6.108) and (6.109) are used, and the modulation frequency f is ω/2π. From (6.110) we obtain δ(ω) ωr 2 = , δ(0) (ω2 − ωr 2 )2 + ω2 γ0 2

(6.112)


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Modulation Frequency

169

(Hz)

FIGURE 6.39 Resonance phenomena.

which indicates that the modulation efficiency δ(ω) shows resonance characteristics, and the resonant angular frequency equals the relaxation oscillation angular frequency ωr . Figure 6.39 shows the modulation efficiency δ( f ) as a function of the modulation frequency f = ω/2π with the injected current density J as a parameter. The resonant frequency f r represents the highest limit in the modulation frequency. As shown in (6.106), with an increase in J , f r is enhanced, which results in a large modulation bandwidth. Note that the electrical resistance and capacitance of LDs also affect the modulation bandwidth. Analog and Digital Modulation In direct modulations of LDs, analog modulation and digital modulation are present, as shown in Fig. 6.40. In analog modulation, a change in the injected current I is transformed into a change in optical power P, and high linearity in the I –L curve is required. The upper limit in the modulation frequency for analog modulation is the resonant frequency f r , and nonlinearity in the I –L curve and Joule heating during large amplitude modulations cause highfrequency distortions. Typical noises are optical feedback noises, which are induced by light reflected from edges of optical fibers, and modal noises, which are generated in the optical fibers. Compared with digital modulation, analog modulation can transmit more information with lower modulation frequencies, but in contrast to digital modulation, it is easily affected by distortions of optical signals during transmissions. As a result, analog modulation is used for short-haul, large-capacity optical fiber communication systems such as cable television (CATV). In digital modulation, the signals 1 and 0 are assigned to the ON and OFF states of laser light. To obtain optical pulses, which immediately follow the pulsed injected current, dc bias current is injected into LDs to shorten the turn-on delay time. Compared with analog modulation, digital modulation is less affected by distortions of the optical signals during transmission. As a result, digital modulation is used


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Digital Modulation

Analog Modulation

Digital Modulation

Analog Modulation

FIGURE 6.40 Analog modulation and digital modulation.

for long-haul, large-capacity optical fiber communication systems. However, higher modulation frequency than that of analog modulation is required. According to the original signals 1 and 0, return-to-zero (RZ) and nonreturnto-zero (NRZ) signals, which are shown in Fig. 6.41, are often used. In an RZ signal, a signal level is returned to zero after the original signal 1 is transmitted, which leads to a large S/N ratio. However, a short pulse width is needed to generate an RZ signal, and a high modulation frequency is required. In an NRZ signal, a signal level is not

Original Signal

RZ Signal

NRZ Signal FIGURE 6.41 RZ and NRZ signals.


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171

returned to zero after the original signal 1 is transmitted. The S/N ratio of an NRZ signal is inferior to that of an RZ signal, but the modulation speed is not as high as that of an RZ signal. When the modulation speed is higher, the pulse width is shorter, and a steady state, which is indicated as (3) in Fig. 6.38, does not exist within each optical pulse. In this case, the relaxation oscillation is sharp, the optical pulses have deep valleys, and the average optical power decreases. The carrier concentration n in the active layer of an LD changes according to preceding optical pulses, a phenomenon called the pattern effect. With a change in carrier concentration n in the active layer of an LD, the bias level of the LD is altered and the turn-on delay time td is modified, as shown in Fig. 6.42. To examine changes in the optical pulses, the pulses are intentionally overlapped with electrical pulses as the reference, as shown in Fig. 6.42(d), which is called the eye pattern. When large deviations in the turn-on delay time are present, resolutions in the photodetector degrade. In this case the eye patterns collapse and a lot of jitters appear. If deviations in the turn-on delay time are small, the eye patterns are wide open. Therefore, the quality of the signals transmitted can be evaluated using eye patterns.

Signal Reference Time (a)

(b)

a

b

c (c)

Reference Time

a c b Time Scale (d)

FIGURE 6.42 Pattern effect: (a) pulsed injected current; (b) carrier concentration n; (c) photon concentration S; (d) eye pattern.


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TABLE 6.1 Material Dispersion and Optical Loss in Quartz Optical Fibers Wavelength (µm)

˚ · km) Material Dispersion (ps/A

Optical Loss (dB/km)

0.85 1.30 1.55

9 ∼0 2

2–3 0.5 < 0.2

When the relaxation oscillation takes place, Fabry–Perot LDs oscillate with multiple longitudinal modes, which restrict the transmission distance due to dispersions in the optical fibers. The optical fibers have mode dispersion, material dispersion, and structural dispersion. In mode dispersion the effective refractive index of the optical fiber changes according to a mode which corresponds to a distribution of light fields. In material dispersion, or chromatic dispersion, the refractive index of the optical fiber depends on a wavelength of light. In structural dispersion the effective refractive index of the optical fiber for a common mode changes with a wavelength of light because the dependence of the refractive index of the core and that of the cladding of the optical fiber on a wavelength of light are different. In long-haul, large-capacity optical fiber communication systems, a fundamental transverse mode is used, and material dispersion is dominant. Table 6.1 shows the material dispersion and optical loss of quartz optical fibers for typical wavelengths of LDs. If five longitudinal modes are present during relaxation oscillation, the difference ˚ for LDs with a wavein the longest and shortest wavelengths is approximately 50 A length of 1.55 µm. When this optical signal is transmitted through the optical fibers by 10 km, the maximum delay in the optical signal is ˚ · km × 50 A ˚ × 10 km = 1 ns, 2 ps/A

(6.113)

and the optical pulse broadens.

6.9 NOISES Noises of LDs fall into two categories: noises with origins existing in and outside LDs. The former noises are quantum noises and noises on longitudinal modes. The latter noises are optical feedback noise and noises from the environment and from the driving circuits. Quantum Noises Fundamental Equations Quantum noises such as amplitude-modulating (AM) and frequency-modulating (FM) noise are caused by spontaneous emission with random amplitudes, frequencies, and phases of the light. FM noise is caused directly by


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NOISES

173

spontaneous emission and is also affected by AM noise. Due to AM noise, the amplitude of a light field fluctuates and the carrier concentration in the active layer is modulated, leading to carrier noise. As a result, through the free carrier plasma effect, the refractive indexes of semiconductors are fluctuated, and FM noise is generated. In addition, the carrier noise induced by AM noise generates electric current noise, which changes Joule heating in the active layer. Therefore, the refractive indexes of semiconductors fluctuate and FM noise is generated. Note that AM noise is also altered by the carrier noise induced by AM noise itself. To analyze quantum noises, semiclassical theory, in which electromagnetic fields are treated classically and atomic systems in the fields are considered quantum mechanically, is often used [10]. From Maxwell’s equations, an equation for the electric field E inside an LD is written ∇ 2 E − µσ

∂E ∂2 E ∂2 − µε 2 = µ 2 ( P + p). ∂t ∂t ∂t

(6.114)

Here the assumption ∇ · E = 0 is used; µ is permeability; σ is conductivity, which represents optical loss; ε0 is the permittivity of vacuum; n r is a refractive index of the semiconductor (ε ≡ ε0 n r 2 ); P is polarization of a medium, contributing to the laser transition; and p is a polarization source for spontaneous emission, which is regarded as a Langevin source. It is also supposed that the electric field E and polarizations P and p are expressed as E = Re P = Re p = Re

m

m

E m (t)em (r) , Pm (t)em (r) ,

(6.115)

pm (t)em (r) .

m

Here a spatial distribution function em (r) satisfies both the boundary conditions of the LD and the relation ∇ 2 em (r) + ωm 2 µε em (r) = 0.

(6.116)

In addition, em (r) is orthonormalized as

em ∗ · en dV = δmn Vm , all volume

(6.117)


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where Vm is a mode volume. Substituting (6.115) into (6.114) and then taking the inner product with en ∗ (r) results in 1 ˙ 1 E¨ n + E n + ωn 2 E n = − ( P¨ n + p¨ n ), τph ε

(6.118)

where (6.116) and (6.117) are used. In (6.118), a dot and a double dot above E n , Pn , and pn represent the first and second derivatives with respect to a time t, respectively; ωn is the resonant angular frequency of the nth mode; τph = ε/σ is the photon lifetime of the optical cavity; and µ = µ0 , which is usually satisfied in the optical materials. Using the electric field E n (t), the polarization of a medium Pn (t) is written Pn (t) = ε0 X (1) + X (3) |E n (t)|2 E n (t).

(6.119)

Here X (1) and X (3) are expressed as X (1) = X (3) =

χ (1) Vm (3)

χ Vm

l , L medium 3l (en ∗ · en )2 dV ≈ χ (3) , 2L medium en ∗ · en dV ≈ χ (1)

(6.120) (6.121)

where χ (1) and χ (3) are the first and third optical susceptibilities, respectively; l is the crystal length; and L is the cavity length. Under single-longitudinal-mode laser operation, the electric field E n (t) and the polarization source for spontaneous emission, which is a Langevin source, pn (t), are expressed as E n (t) = [A0 + δ(t)] e i [ωm t+φ(t)] , −

1 ∂ pn = (t) e i [ωm t+φ(t)] , ε ∂t 2

(6.122)

2

(6.123)

where A0 is the average amplitude of the electric field, δ(t) is a deviation in the amplitude of the electric field from A0 , φ(t) is an instantaneous phase, (t) is a random function representing spontaneous emission, and ωm is the average angular frequency of the laser light. It is assumed that δ(t), φ(t), and (t) are slowly varying functions compared with ωm . For simplicity, the carrier fluctuations are neglected, and it is supposed that δ(t) A0 , δ(t) = φ(t) = (t) = 0, where · · · shows a time average or ensemble average.

(6.124)


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NOISES

175

Substituting (6.119), (6.122), and (6.123) into (6.118) leads to

∂δ ∂φ 3A0 2 X (3) ∂δ 2 2 i ωm + i A0 + 2 i ωm − ωm δ ∂t ∂t nr2 ∂t 2 ωm 2 2 (1) 2 (3) ωm + ωn − ωm + i − (X + A0 X ) 2 A0 = (t), τph nr

(6.125)

where ε = ε0 n r 2 is used. A real part and an imaginary part of the steady-state solution of (6.125) are given by −1 X (1) + A0 2 X r(3) ωm 2 = ωn 2 1 + r : real, nr2 1 nr2 A0 2 = − (3) X i(1) + : imaginary, ωm τph Xi

(6.126) (6.127)

where X (1,3) = X r(1,3) − i X i(1,3) . Therefore, (6.125) is reduced to ∂φ 3A0 2 X i(3) ∂δ 3A0 2 ωm X r(3) r (t) A0 − + δ=− : real, 2 2 ∂t nr ∂t 2n r 2ωm 3A0 2 X r(3) ∂δ 3A0 2 ωm X i(3) i (t) 1+ + δ= : imaginary, 2 2 nr ∂t 2n r 2ωm

(6.128) (6.129)

where (t) = r (t) + i i (t). In (6.128), the second term on the left-hand side is usually neglected, because it is much smaller than the other terms. In this case, the third term on the left-hand side relates the amplitude fluctuation δ and the phase fluctuation φ. Spectra of Laser Light To analyze the spectra of laser light, a power fluctuation spectrum expressing AM noise, a frequency fluctuation spectrum indicating FM noise, and a field spectrum are often used. Figure 6.43 shows a measurement system for these spectra. Note that a linewidth of the field spectrum is used as a spectral linewidth of laser light emitted from an LD. To obtain the spectra of laser light, the autocorrelation functions of the fluctuations are first calculated using (6.128) and (6.129). Then, using the Wiener– Khintchine theorem, spectral density functions are obtained. Autocorrelation Function of the Amplitude Fluctuation δ(t) For the usual laser field, A0 2 X r(3) /n r 2 1 is satisfied. As a result, (6.129) is reduced to ∂δ i (t) + ω1 δ = , ∂t 2ωm

(6.130)


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Test Laser

Spectrum Analyzer

Detector

Power Fluctuation Sectrum

Reference Laser Optical Mixer

Spectrum Analyzer

Limiter Discriminator Spectrum Analyzer

Frequency Fluctuation Spectrum

Field Spectrum FIGURE 6.43 Measurement system for spectra of laser light.

where ω1 =

3A0 2 ωm X i(3) > 0. 2n r 2

(6.131)

A Laplace transform of (6.130) is written ˜ ˜ + ω1 δ(s) ˜ = i (s) , −δ(0) + s δ(s) 2ωm

(6.132)

where

∞

˜ ≡ δ(s) 0

˜ i (s) ≡

∞

δ(t) e−st dt,

(6.133)

i (t) e−st dt.

(6.134)

0

Supposing that δ(0) = 0, (6.132) results in ˜ = δ(s)

˜ i (s) . 2ωm (s + ω1 )

(6.135)

Taking an inverse Laplace transform of (6.135) leads to δ(t) =

1 2ωm

t 0

i (λ)e−ω1 (t−λ) dλ.

(6.136)


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177

From (6.136), the autocorrelation function δ(t + τ )δ(t) of the amplitude fluctuation δ(t) is given by 1 δ(t + τ )δ(t) = 4ωm 2

t+τ

t

dλ2 i (λ1 )i (λ2 ) 0 −ω1 (t+τ −λ1 ) −ω1 (t−λ2 )

dλ1 0

×e

e

.

(6.137)

Here it is assumed that the correlation functions of the Langevin sources are written i (λ1 )i (λ2 ) = r (λ1 )r (λ2 ) = W · D(λ1 − λ2 ), i (λ1 )r (λ2 ) = r (λ1 )i (λ2 ) = 0,

(6.138)

where D(x) is a δ function and W is a coefficient representing spontaneous emission. Substituting (6.138) into (6.137) leads to δ(t + τ )δ(t) =

W e−ω1 |τ | (1 − e−2ω1 t ). 8ωm 2 ω1

(6.139)

From (6.139), in a steady state where t is long enough, we have δ(t + τ )δ(t) =

W e−ω1 |τ | . 8ωm 2 ω1

(6.140)

Autocorrelation Function of the Phase Fluctuation φ(t) Neglecting the second term on the left-hand side of (6.128), which is small compared with the other terms, results in A0

∂φ 3A0 2 ωm X r(3) r (t) + δ=− . ∂t 2n r 2 2ωm

(6.141)

Under the assumption that φ(0) = 0, integrating (6.141) with respect to t gives φ(t) = −

3 X r(3) A0 4 n r 2 ω1

t

t

i (λ) dλ −

0

i (λ)e−ω1 (t−λ) dλ

0

−

1 2A0 ωm

t

r (λ) dλ,

(6.142)

0

where (6.136) is used. From (6.130), (6.138), and (6.142), the autocorrelation function φ(t1 )φ(t2 ) of the phase fluctuation φ(t) is given by φ(t1 )φ(t2 ) =

W 4ωm 2 A0 2

(1 + α ) × 2

t1 (t1 < t2 ), t2 (t1 > t2 ),

(6.143)


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where α≡

X r(3)

(6.144)

X i(3)

is the α-parameter or the spectral linewidth enhancement factor. In X = X r − i X i , the real part X r and imaginary part X i are written X r = X r(1) + X r(3) |E n |2 ,

(6.145)

X i(3) |E n |2 .

(6.146)

Xi =

X i(1)

+

As a result, X r and X i are given by ∂ Xr , ∂|E n |2 ∂ Xi = . ∂|E n |2

X r(3) =

(6.147)

X i(3)

(6.148)

From (6.144), (6.147), and (6.148), the α-parameter is expressed as α=

∂ Xr ∂|E n |2

∂ Xi ∂|E n |2

−1

=

∂ Xr ∂n

∂ Xi ∂n

−1

,

(6.149)

where |E n |2 is assumed to be linearly proportional to the carrier concentration n. Using the complex refractive index n˜ = n r − i κ, the complex dielectric constant ε˜ is written ε˜ = ε0 n˜ 2 = ε0 [(n r 2 − κ 2 ) − i 2n r κ] = ε0 (X r − i X i ),

(6.150)

X r = n r 2 − κ 2 , X i = 2n r κ.

(6.151)

where

In the vicinity of the bandgap in semiconductors, n r κ is satisfied, and we obtain ∂ Xr ∂n r ∂κ ∂n r = 2n r − 2κ ≈ 2n r , ∂n ∂n ∂n ∂n ∂ Xi ∂κ ∂n r ∂κ = 2n r + 2κ ≈ 2n r . ∂n ∂n ∂n ∂n

(6.152) (6.153)

Substituting (6.152) and (6.153) into (6.149) leads to ∂n r α= ∂n

∂κ ∂n

−1

.

(6.154)


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NOISES

179

Using the extinction coefficient κ, the power optical gain coefficient g is expressed as g=−

2ωm κ. c

(6.155)

Using G(n) in the rate equations (6.2) and (6.3), g is written a G(n) =

c g. nr

(6.156)

From (6.155) and (6.156), we have ∂κ nr ∂G =− a . ∂n 2ωm ∂n

(6.157)

Substituting (6.157) into (6.154) results in 2ωm ∂n r α=− n r ∂n

∂ G −1 a . ∂n

(6.158)

When the optical gain increases with carrier injection (∂G/∂n > 0) and Joule heating of the active layer is negligibly small, the refractive index decreases with carrier injection (∂n r /∂n < 0), due to the free carrier plasma effect. As a result, the α-parameter shown in (6.158) is positive, and the measured values are between 1 and 7. From (6.158) it is found that we should increase a ∂ G/∂n to reduce the value of α, which is achieved in quantum well LDs. Note that the α-parameter is an important parameter to characterize the spectral linewidth and the optical feedback noise in LDs. Autocorrelation Function of the Angular Frequency Fluctuation ω(t) Using the instantaneous angular frequency of the laser light ω(t), the average angular frequency of the laser light ωm , and the phase fluctuation φ(t), the fluctuation in the angular frequency ω(t) is written ω(t) ≡ ω(t) − ωm =

∂φ . ∂t

(6.159)

Therefore, the autocorrelation function ω(t1 )ω(t2 ) of the angular frequency fluctuation ω(t) is obtained as ˙ 1 )φ(t ˙ 2 ) ω(t1 )ω(t2 ) = φ(t W α2 −ω1 |t1 −t2 | = D(t1 − t2 ) + ω1 e , 2 4ωm 2 A0 2

(6.160)


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where (6.140), (6.141), and (6.149) are used, and the dot above φ represents the first derivative with respect to time. Spectral Density Function Applying the Wiener–Khintchine theorem to (6.140), the spectral density function Wδ () of the amplitude fluctuation δ(t) is obtained as Wδ () = =

1 π

∞

δ(t + τ )δ(t) e−iτ dτ

−∞

W 4π ωm

2 (2

+ ω1 2 )

.

(6.161)

Applying the Wiener–Khintchine theorem to (6.160), the spectral density function Wω () of the angular frequency fluctuation ω(t) is given by 1 π

∞

ω(t + τ )ω(t) e−iτ dτ W α 2 ω1 2 = 1 + . 2 + ω1 2 4π ωm 2 A0 2

Wω () =

−∞

(6.162)

Coefficient W The generation rate of the spontaneous emission is expressed as ωm E cv , where E cv is the number of photons, that are spontaneously emitted to the lasing mode per time. The dissipation rate of the spontaneous emission is given by Φs /τph , where Φs is the energy of spontaneous emission coupled to the lasing mode and τph is the photon lifetime. In a steady state, the generation rate is balanced with the dissipation rate, and we have Φs = ωm E cv τph .

(6.163)

The electric field of the spontaneous emission is given by solving 1 ˙ E¨ n + E n + ωn 2 E n = [r (t) + i i (t)] e i ωm t , τph

(6.164)

where ωn ≈ ωm is assumed, and the dielectric polarization due to the stimulated emission in (6.118) is neglected because only the spontaneous emission is considered. ˙ Assuming that E(0) = E(0), we have E(t) =

1 ωm

0

t

dτ [r (τ ) + i i (τ )] e i ωm τ e−(t−τ )/(2τph ) sin[ωm (t − τ )]

(6.165)


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181

NOISES

as a solution of (6.164). Hence, E ∗ (t)E(t) is given by t t 1 E (t)E(t) = dλ1 dλ2 2W D(λ1 − λ2 ) e−t/τph e(λ1 +λ2 )/(2τph ) ωm 2 0 0 × sin[ωm (t − λ1 )] sin[ωm (t − λ2 )] W τph = , (6.166) ωm 2 ∗

where (6.138) is used. Using (6.166), a steady-state spontaneous emission energy Φs is expressed as Φs = εVm E ∗ (t)E(t) =

εVm W τph , ωm 2

(6.167)

where the mode volume Vm is given by Vm ≡

|en (r)|2 dV.

(6.168)

Here ε is the dielectric constant of a semiconductor and Vm is the mode volume. From (6.163) and (6.167), W is obtained as W =

ωm 3 E cv . εVm

(6.169)

Intensity Fluctuation Spectrum (AM Noise) The light emission rate γ from the optical cavity through the mirrors is given by γ =

c 1 1 ln , nr L R

(6.170)

where c is the speed of light in vacuum, n r is the refractive index of a semiconductor, L is the cavity length, and R is the power reflectivity of a facet where both facets are assumed to have a common reflectivity. Using the electric field of the laser light E n , the light output power P is expressed as P = εE n 2 Vm γ .

(6.171)

From (6.122), (6.123), and (6.171), the average light output power P0 and the fluctuation in light output power P = P − P0 are written P0 = ε A0 2 Vm γ , P = 2ε A0 δVm γ .

(6.172)


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From (6.161), (6.169), (6.170), and (6.172), the spectral density function WP () of the light output power fluctuation P is obtained as WP () =

1 π

∞

P(t + τ )P(t) e−iτ dτ

−∞

ωm ε A0 2 Vm E cv γ 2 π(2 + ω1 2 ) ωm E cv P0 γ ωm cE cv P0 ln(1/R) = = . 2 2 π ( + ω1 ) π (2 + ω1 2 )n r L = 4ε2 A0 2 Vm 2 γ 2 Wδ () =

(6.173)

Frequency Fluctuation Spectrum (FM Noise) From (6.162) and (6.169)–(6.172), the spectral density function Wω () of the angular frequency fluctuation ω(t) is given by ωm E cv α 2 ω1 2 1 + 2 + ω1 2 4π εVm A0 2 ωm cE cv ln(1/R) α 2 ω1 2 = 1+ 2 . 4π P0 n r L + ω1 2

Wω () =

(6.174)

Field Spectrum The field spectrum is most frequently used as the laser light spectrum. Because the contribution of the amplitude fluctuation to the correlation function of the electric field is negligibly small, the correlation function E(t + τ )E(t) is given by 1 [E(t + τ ) + E ∗ (t + τ )][E(t) + E ∗ (t)] 4 A0 2 −i ωm τ iφ = [e e + c.c.], 4

E(t + τ )E(t) =

(6.175)

where the phase fluctuation φ is defined as φ ≡ φ(t + τ ) − φ(t).

(6.176)

The phase fluctuation φ is caused by a lot of independent spontaneous emission processes. As a result, the distribution of φ is given by the Gaussian distribution function g(φ) as g(φ) =

1 2π(φ)2

e−(φ)

2

/2(φ)2

.

(6.177)

Therefore, we obtain e

iφ

=

∞

−∞

g(φ) e iφ d(φ) = e− 2 (φ) . 1

2

(6.178)


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NOISES

183

By assuming that t1 = t2 = τ in (6.143), (φ)2 on the right-hand side of (6.178) is written (φ)2 =

W 4 ωm 2 A0 2

(1 + α 2 )|τ |.

(6.179)

Substituting (6.178) and (6.179) into (6.175) results in A0 2 W 2 E(t + τ )E(t) = exp − (1 + α )|τ | cos(ωm τ ). 2 8ωm 2 A0 2

(6.180)

Using the Wiener–Khintchine theorem, the spectral density function W E (ω) of the electric field is given by 1 W E (ω) = π ≈

∞ −∞

E(t + τ )E(t) e−i ωτ dτ

A0 2 ω0 , 4π (ω − ωm )2 + (ω0 /2)2

(6.181)

ωm cE cv ln(1/R) (1 + α 2 ). 4P0 n r L

(6.182)

where ω0 ≡

From (6.181), the field spectrum is Lorentzian with a FWHM of ω0 in (6.182). Equation (6.182), called the modified Schawlow–Townes linewidth formula, gives the spectral linewidth for LDs. In solid-state or gas lasers, the term α 2 can be neglected because α 2 1. In contrast, in LDs, the term α 2 is required because α 2 > 1. Noises on Longitudinal Modes Mode Partition Noise Mode partition noise is observed when a longitudinal mode is selected during laser operations with multiple longitudinal modes. This noise is large at low frequencies. For example, in Fabry–Perot LDs under pulsed operations or gain guiding LDs, noise for the total light output power is comparable to noise in single-longitudinal-mode LDs. However, noise for each longitudinal mode in multiple longitudinal modes is much larger than the noise in single-longitudinal-mode LDs, because the optical gain is randomly delivered to each longitudinal mode among multiple longitudinal modes. The mode partition noise causes a serious problem in mode selective systems such as optical fiber communication systems. To prevent mode partition noise, we need single-longitudinal-mode LDs. Mode Hopping Noise Mode hopping noise is generated when a longitudinal mode in single-mode LDs jumps to other modes. Mode hopping depends on driving conditions such as temperature and injected current. When mode hopping occurs, random


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oscillations between multiple longitudinal modes are repeated, and the noise increases due to a difference in light output powers between the relevant modes. When two competing modes are present, the noise is large in a low frequency range below 50 MHz; when three or more competing modes are present, the noise is large up to higher frequencies. Note that the mode partition noise is also large during mode hopping. For analog systems such as videodisks, the relative intensity noise (RIN) has to be lower than −140 dB/Hz, and for digital systems such as compact disks, the RIN should be lower than −120 dB/Hz. Mode hopping noise is caused by fluctuations of spontaneous emission and optical gain’s inclination to concentrate on the lasing mode. To avoid mode hopping noise, two opposite processes, such as laser operation with a single longitudinal mode and laser operation with multiple longitudinal modes are used. For systems with mode selectivity, such as optical fiber communication systems, single-longitudinal-mode LDs such as the DFB LDs are adopted. Bistable LDs, which contain saturable absorbers and have hysteresis in I –L curves, can also suppress mode competition. However, it is difficult for bistable LDs to keep stable single-longitudinal-mode operations with a large extinction ratio during modulation. As a result, bistable LDs are not used in conventional optical fiber communication systems. For systems without mode selectivity, such as video and compact disks, multiplelongitudinal-mode operations are widely used to reduce mode hopping noise. In these applications, optical systems have to be set up in a small space, and the optical feedback noise has to be reduced without using optical isolators. Therefore, multiple-longitudinal-mode operations are suitable for video and compact disks. Multiple-longitudinal-mode operations have a higher noise level than that of singlelongitudinal-mode operations, but the noise level is stable with changes in the temperature or injected current. Therefore, the maximum noise level is lower than the mode hopping noise. To obtain multiple-longitudinal-mode operations, high-frequency modulations and self-pulsations are adopted. High-frequency modulations are obtained when electric current pulses with frequencies above 600 MHz are injected into Fabry–Perot LDs and the minimum current is set below the threshold current. Selfpulsations, which are pulsed operations under dc bias, are obtained in Fabry–Perot LDs with saturable absorbers or combined structures of index and gain guidings. Due to the high-frequency modulations or self-pulsations, multiple-longitudinal-mode operations with low coherence take place. As a result, stable laser operations against optical feedback noise are obtained.

Optical Feedback Noise Optical feedback noise [11] is generated when laser light emitted from an LD is fed back to the LD itself. An external cavity is formed by a facet of the LD and reflective external objects, such as optical components, optical fibers, and optical disks. The external cavity and internal cavity of the LD produce a coupled cavity, which induces optical feedback noise.


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NOISES

185

Optical feedback noise is noticeable even when the relative feedback optical power is on the order of 10−6 . Due to the optical feedback noise, the light output characteristics of an LD change intricately, according to the distance between reflective external optical objects and the LD, the feedback optical power, and driving conditions. In static or time-averaged characteristics, the light output power, the number of lasing modes, and light output spectra are modified. In dynamic characteristics, the noise level and shape of the light pulse are altered. Fundamental Equations Figure 6.44 shows a coupled cavity and its equivalent model in which the effect of the external cavity is expressed by the equivalent amplitude reflectivity reff . It is assumed that the power reflectivities for the facets of the LD are R1 and R2 , and the power reflectivity for the external reflector is R3 . Using an angular frequency of laser light and the round-trip time in the external cavity τ , the electric field of the reflected light E r e it is expressed as E r e it = E i e it

R2 + (1 − R2 ) R3 e−iτ + (1 − R2 ) R2 R3 e−i2τ + · · · ,

(6.183)

where E i e it is the electric field of the incident light. The third- and higher-order terms in the [· · · ] of (6.183) correspond to multireflections of light in the external cavity. Usually, the power reflectivity of the external reflector R3 is on the order of 10−2 or less. Therefore, multireflections of light in the external cavity are neglected in the following. As a result, reff is obtained as reff =

Er = R2 1 + a e−iτ , Ei

(6.184)

where a = (1 − R2 )

R3 1. R2

Reflective End Internal Cavity External (LD) Cavity (a)

(6.185)

LD

(b)

FIGURE 6.44 (a) Coupled cavity and (b) its equivalent model.


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Using the power decay rate in an LD itself, γ0 , the decay rate for the electric field γ0 /2 is given by 1 1 c γ0 = 2 2 n r0

1 1 αi + ln , 2L R1 R2

(6.186)

where c is the speed of light in vacuum, n r0 is the effective refractive index of the LD in a steady state, αi is the internal loss in the LD, and L is the length of the internal cavity. Using (6.186), the decay rate for the coupled cavity γ can be written 1 1 c γ = 2 2 n r0

αi +

1 1 ln 2L R1 reff 2

=

1 γ0 − κ e−iτ , 2

(6.187)

where c c κ= a= (1 − R2 ) 2n r0 L 2n r0 L

R3 . R2

(6.188)

Here κ is the coupling rate of feedback light to the LD. For R2 = 32%, R3 = 1%, n r0 = 3.5, and L = 300 µm, we have κ = 1.7 × 1010 s−1 . This value is between the decay rate for the carrier concentration, 1/τn ∼ 109 s−1 , and that for the photon concentration, 1/τph ∼ 1012 s−1 . Using the decay rate γ , an equation for the electric field E can be expressed as d E e it = dt

1 i ω N (n) + [a G(n) − γ ] 2

E e it ,

(6.189)

where ω N (n) is an angular frequency for the N th-order resonant mode, which is a function of the carrier concentration n. Substituting (6.187) into (6.189) results in d E(t) = dt

1 i [ω N (n) − ] + [a G(n) − γ0 ] 2

E(t) + κ E(t − τ ) e−iτ , (6.190)

where the final term on the right-hand side, κ E(t − τ ) e−iτ , represents a contribution of the feedback light. Note that (6.190) includes the phase of the laser light in contrast to (6.3), because the optical feedback noise is highly dependent on the phase of the feedback light. Effect of the Feedback Light on Static Characteristics It is assumed that the electric field E takes a steady-state value. From a real part in (6.190), the amplification rate at the threshold G th is given by a G th ≡ a G(n th ) = γ0 − 2κ cos(τ ).

(6.191)


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NOISES

187

From an imaginary part in (6.190), the oscillation angular frequency at the threshold is obtained as ω N (n th ) = + κ sin(τ ).

(6.192)

Because the steady-state values ω N (n c0 ), 0 , and κ0 satisfy (6.191) and (6.192), we have a G(n c0 ) = γ0 − 2κ,

(6.193)

where it is supposed that n th = n c0 and ω N (n c0 ) = 0 . Because of the carrier lifetime τn ∼ 10−9 s and the photon lifetime τph ∼ 10−12 s, the carrier concentration n does not always take a steady-state value, even though the electric field E is in a steady state. Therefore, the threshold carrier concentration n th can be written n th = n c0 + n,

(6.194)

and the amplification rate at the threshold G th is expressed as G th = G(n c0 ) +

∂G n. ∂n

(6.195)

Substituting (6.195) into (6.191) results in

∂G n = 2κ a ∂n

−1

[1 − cos(τ )],

(6.196)

where (6.193) is used, and it is assumed that κ = κ0 because 1/τn ∼ 109 s−1 and κ ∼ 1010 s−1 . The resonant angular frequency at the threshold ω N (n th ) is expressed as ω N (n th ) = ω N (n c0 ) −

ω N (n c0 ) ∂n r n. n r0 ∂n

(6.197)

Inserting (6.158) and (6.196) into (6.197) leads to ω N (n th ) = ω N (n c0 ) + ακ[1 − cos(τ )].

(6.198)

From the right-hand side of (6.198), it is found that the spectral linewidth enhancement factor α plays an important role in the optical feedback noise as well as the spectral linewidth of the LDs. Substituting (6.198) into (6.192) gives ω N (n c0 ) = + κ sin(τ ) − ακ[1 − cos(τ )].

(6.199)


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These results are summarized as follows: When the electric field E takes a steadystate value, the amplification rate at the threshold G th is given by a G th ≡ a G(n th ) = γ0 − 2κ cos(τ ),

(6.200)

and the oscillation angular frequency is expressed as ω N (n c0 ) = + κ sin(τ ) − ακ[1 − cos(τ )].

(6.201)

The optical feedback noise in LDs is larger than that in other lasers because LDs have a larger α, a lower R2 , and a shorter L than other lasers, which leads to a larger κ. For example, a gas laser with α 1, R2 = 98%, and L ∼ 1 m has κ ∼ 105 s−1 ; an LD with α = 1 to 7, R2 = 32%, and L ∼ 300 µm has κ ∼ 1010 s−1 . The terms including trigonometric functions in (6.200) and (6.201) indicate interference between the light in an LD and the feedback light. Due to this interference, hysteresis is present in both the oscillation angular frequency and the light output power. Figure 6.45 shows the resonant angular frequency ω N (n c0 ) as a function of the oscillation angular frequency for κτ = 1 and α = 3. Here the arrows represent the points where jumps. Figure 6.46 shows I –L curves calculated in continuous-wave (CW) operations with the spectral linewidth enhancement factor α as a parameter. Here Joule heating in the active layer is also considered. Hysteresis in the light output, which is observed experimentally, is reproduced theoretically as shown in Fig. 6.46.

FIGURE 6.45 Resonant angular frequency ω N (n c0 ) as a function of oscillation angular frequency .


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189

Light Output (arb. units)

REFERENCES

Injected Current (mA) FIGURE 6.46

I –L characteristics in CW operation.

According to the phase of the feedback light, the dynamic characteristics are complicated; chaos and enhancement/suppression of the relaxation oscillation are observed. Enhancement of Noise Due to Feedback Light Due to feedback light, quantum noises increase in a certain frequency region. In addition, laser oscillations become unstable, and the noise increases in a low-frequency range which is less than several hundred megahertz by the feedback light. The reason for this is random mode hopping between the longitudinal modes in the internal cavity and those in the external cavity. Reducing Optical Feedback Noise To stabilize a longitudinal mode in the internal cavity, single-longitudinal-mode LDs such as DFB LDs or bistable LDs are needed. To suppress interference between the feedback light and the internal light in an LD, coherence of the laser light should be reduced by high-frequency modulation or selfpulsation. To decrease the optical power of feedback light, a low-coupling-rate κ is required, which is achieved by a large facet reflectivity and a long cavity, as shown in (6.187). However, a large reflectivity leads to a low light output, and a long cavity results in a large threshold current. Therefore, optical isolators are generally used to decrease the optical power of the feedback light in optical fiber communication systems, but the cost and size of the optical systems increase. REFERENCES 1. Z. I. Alferov, V. M. Andreev, E. L. Portnoy, and M. K. Trukan, “AlAs–GaAs heterojunction injection lasers with a low room-temperature threshold,” Sov. Phys. Semicond. 3, 1107 (1970).


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2. I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17, 109 (1970). 3. N. Shomura, M. Fujimoto, and T. Numai, “Fiber pump semiconductor lasers with optical antiguiding layers for horizontal transverse modes,” IEEE J. Quantum Electron. 44, 819 (2008). 4. N. Shomura, M. Fujimoto, and T. Numai, “Fiber-pump semiconductor lasers with optical antiguiding layers for horizontal transverse modes: dependence on mesa width,” Jpn. J. Appl. Phys. 48, 042103 (2009). 5. N. Shomura and T. Numai, “Ridge-type semiconductor lasers with optical antiguiding layers for horizontal transverse modes: dependence on step positions,” Jpn. J. Appl. Phys. 48, 042104 (2009). 6. H. Yoshida and T. Numai, “Ridge-type semiconductor lasers with antiguiding layers for horizontal transverse modes: dependence on space in the antiguiding layers,” Jpn. J. Appl. Phys. 48, 082105 (2009). 7. H. Takada and T. Numai, “Ridge-type semiconductor lasers with antiguiding cladding layers for horizontal transverse modes,” IEEE J. Quantum Electron. 45, 917 (2009). 8. M. Hirao, A. Doi, S. Tsuji, M. Nakamura, and K. Aiki, “Fabrication and characterization of narrow stripe InGaAsP/InP buried heterostructure lasers,” J. Appl. Phys. 51, 4539 (1980). 9. I. Mito, M. Kitamura, K. Kobayashi, S. Murata, M. Seki, Y. Odagiri, H. Nishimoto, M. Yamaguchi, and K. Kobayashi, “InGaAsP double-channel-planar-buried-heterostructure laser diode (DC-PBH LD) with effective current confinement,” IEEE J. Lightwave Technol. 1, 195 (1983). 10. A. Yariv, Quantum Electronics, 3rd ed., Wiley, New York, 1989. 11. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347 (1980). 12. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004.


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7 QUANTUM WELL LASER DIODES

7.1

INTRODUCTION

Quantum well (QW) LDs [1, 2] are LDs that have quantum well active layers. A QW LD with one QW active layer is called a single quantum well (SQW) LD, and one with multiple QW active layers is called a multiple quantum well (MQW) LD. The density of states of a one-dimensional QW is a step function of the energy. Therefore, a low threshold current, high differential quantum efficiency, high-speed modulation, low chirping, and narrow spectral linewidth are obtained simultaneously in QW LDs.

7.2

FEATURES OF QUANTUM WELL LDs

Configurations of Quantum Wells Figure 7.1 shows the configurations of various QWs at a band edge. The horizontal and vertical directions represent a position and the energy of an electron, respectively. Here E c and E v are the band-edge energies of the conduction and valence bands, respectively. Figure 7.1(a) illustrates an SQW active layer in which the optical confinement factor a is small because the active layer thickness L z is as thin as 10 nm or less. This small a leads to a large threshold current density Jth . Therefore, to obtain a larger a than that of the SQW, structures (b)–(e) were developed. Figure 7.1(b) shows a MQW which has multiple QW active layers in the optical confinement region. However, due to the energy barriers between the adjacent QW Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright

191


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QUANTUM WELL LASER DIODES

(a)

(b)

(c)

(d)

(e)

FIGURE 7.1 Configurations of quantum wells: (a) SQW; (b) MQW; (c) modified MQW; (d) SCH; (e) GRIN-SCH.

active layers, the carrier injection efficiency decreases with propagation of the carriers. Hence, it is difficult to achieve uniform carrier distribution all over the multiple QW active layers. The modified MQW shown in Fig. 7.1(c) can improve the carrier injection efficiency and the uniformity of the carrier distribution in an MQW. The energy barriers between the adjacent QW active layers are lower than those in the cladding layers, which results in a high carrier injection efficiency and uniform carrier distribution all over the multiple QW active layers. A separate confinement heterostructure (SCH), which has two energy steps, as shown in in Fig. 7.1(d), is able to obtain a large optical confinement factor a in the SQW active layer. In the materials used conventionally for LDs, the refractive index increases with a decrease in the bandgap energy. The outer potential confines light to the QW active layer by the refractive index distribution; the inner potential confines the carriers by the energy barriers. Because the potentials to confine the light and the carriers are separate, this structure is called an SCH. Figure 7.1(e) illustrates a graded index SCH (GRIN-SCH) whose potential and refractive index distributions in the outside of the active layer are parabolic. When the active layer thickness L z is small, the optical confinement factor a of a GRIN-SCH is proportional to L z and a of an SQW is proportional to L z 2 . Therefore, when the active layer is thin, the optical confinement factor a of the GRIN-SCH is larger than that of the SQW. Characteristics of QW LDs Low Threshold Current The density of states per unit area ρ(E) in an SQW is written ρ(E) =

∞ m∗ H (E − n ), π 2 n=1

(7.1)

where E is the energy of the carrier, m ∗ is the effective mass of the carrier, is Dirac’s constant, and H (x) is the Heaviside function or step function, which is defined as H (x) =

0 1

(x < 0), (x ≥ 0).

(7.2)


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193

When the barrier height or the barrier thickness is large so that each QW can be independent, the energy level of the nth subband n is expressed as n =

(nπ )2 . 2m ∗ L z 2

(7.3)

In this case, the density of states per unit area ρ(E) in an MQW is obtained as

ρ(E) = N

∞ m∗ H (E − n ), π 2 n=1

(7.4)

where N is the number of QWs. When the barrier height or barrier thickness is small, the wave functions for the carriers penetrate adjacent QWs. In this case the QWs are coupled to each other and degeneracy is removed. As a result, N quantum levels are generated per quantum level degenerated, and ρ(E) is given by

ρ(E) =

∞ N m∗ H (E − nk ), π2 n=1 k=1

(7.5)

where nk (k = 1, 2, . . . , N ) is the energy of each quantum level, which is produced by the removal of degeneracy. Because the densities of states in QW LDs are step functions, QW LDs have narrow optical gain spectra. Hence, the optical gain concentrates on a certain energy, and the peak optical gain is enhanced. As a result, a threshold current density Jth is lower than that of a bulk double heterostructure (DH) LD. Under the k-selection rule and the assumption of a negligibly low nonradiative recombination rate, the linear optical gain g of the QW LD is written c ω g(E, n) = 2 χI (E, n). nr nr

(7.6)

Here c is the speed of light in vacuum, n r is the effective refractive index of the QW LD, E is the photon energy, ω is the angular frequency of light, and χI is an imaginary part of the relative electric susceptibility. The imaginary part of the relative electric susceptibility χI is expressed as ω χI (E, n) = nr2

∞ n=0 j=l,h

j

n, j

j ρred,n [ f c (E c,n ) − f v (E v,n )]χˆ I (E, ) d.

(7.7)


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QUANTUM WELL LASER DIODES n, j

The imaginary part of the relative electric susceptibility χˆ I (E, ) for a photon with energy E and an electron–hole pair with energy is given by n, j

χˆ I (E, ) =

e2 h 2m 0 2 ε0 n r 2 E g

|Mn, j ()|2ave

/τin , (E − )2 + (/τin )2

(7.8)

where j = h and l represents the heavy hole (h) and the light hole (l), respectively; e is the elementary electric charge; h is Planck’s constant; m 0 is the electron mass in vacuum; ε0 is the permittivity of vacuum; E g is the bandgap energy; |Mn, j ()|2ave is a square of the averaged momentum matrix element; and τin is the intraband j relaxation time. The reduced density of states for the nth subband ρred,n is defined as j ρred,n

≡

1 1 + j ρc,n ρv,n

−1 ,

(7.9)

j

where ρc,n and ρv,n are the densities of states for the nth subband in the conduction j and valence bands, respectively; f c (E c,n ) and f v (E v,n ) are Fermi–Dirac distribution j functions of the conduction and valence bands, respectively. Here E c,n and E v,n , which are the energies of the conduction and valence bands, respectively, are expressed as j

E c,n =

j

j

m c v,n + m v E + m v c,n j

mc + mv j

j E v,n =

,

(7.10)

,

(7.11)

j

m c v,n − m c E + m v c,n j

mc + mv

j

where m c and m v are the effective masses of the conduction electron and hole, j respectively; c,n and v,n are the energies of the nth subband in the conduction and valence bands, respectively. Figure 7.2(a) shows the modal gain gmod = a g as a function of the injected current density J with the number of QWs N as a parameter. Here a is the optical confinement factor of the active layer. According to the optical loss in the optical cavity, which is equal to the threshold gain, the number of QWs N to minimize Jth are present. Also, in the SQW (N = 1), gain flattening is observed with an increase in the injected current, as shown in Fig. 7.2(b). The cause of the gain flattening is that the density of states is a step function H (E − E L ) of the photon energy E with a lasing photon energy E L . Characteristic Temperature Because the densities of states in QWs are step functions, changes in characteristics of QW LDs with temperature are expected to be small. However, even in a bulk DH LD, a characteristic temperature T0 as high as 200 K is obtained, and an advantage of QW LDs over bulk DH LDs has not yet been proved.


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195

g

Modal Gain g

g

g

Current Density (a)

/ (b)

FIGURE 7.2 (a) Modal gain; (b) gain flattening.

Anisotropic Optical Gain A difference in the optical gains for TE and TM modes is about 20 cm−1 in bulk DH LDs and about 140 cm−1 in QW LDs. In bulk DH LDs, an anisotropic optical gain is generated because the optical confinement factor a of the TE mode is larger than that of the TM mode, due to the configurations of the optical waveguides. In QW LDs, anisotropic optical gain is also produced by the selection rule of optical transitions, which is explained in the following. In bulk structures, the heavy hole (hh) band with m j = 3/2 and the light hole (lh) band with m j = 1/2 are degenerate at k = 0; in QWs this degeneracy is removed. Figure 7.3 shows the energies of the valence band. Here solid lines represent the energies of the heavy hole bands E hh1 and E hh2 and dashed lines represent the

FIGURE 7.3 Valence band in a one-dimensional QW.


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FIGURE 7.4 Quantum well.

energies of the light hole bands E lh1 and E lh2 , where the subscripts 1 and 2 are the values of the principal quantum number n. The effective masses of the hole in QWs depend on the directions of the axes. As shown in Fig. 7.4, if the quantization axis is the z-axis, the effective mass along the z-axis of the heavy hole m z,hh is larger than that of the light hole m z,lh and the effective mass on the x y-plane of the heavy hole m x y,hh is smaller than that of the light hole m x y,lh . This result is summarized as m z,hh > m z,lh: along the z-axis, m x y,hh < m x y,lh: on the x y-plane. According to the px - or p y -like orbital, a wave function of the heavy hole distributes on the x y-plane. Therefore, the heavy hole moves on the x y-plane more easily than along the z-axis, which leads to m x y,hh < m z,hh . According to the pz -like orbital, a wave function of the light hole distributes along the z-axis. Therefore, the light hole moves along the z-axis more easily than on the x y-plane, which results in m z,lh < m x y,lh . For example, in GaAs we have m z,hh = 0.377m 0 , m z,lh = 0.09m 0 , m x y,hh = 0.11m 0 , and m x y,lh = 0.21m 0 , where m 0 is the electron mass in vacuum. Wave Function It is assumed that a wave vector k is along the z-axis and that a contribution of the split-off band is negligibly small. If the up-spin and down-spin are expressed as α and β, respectively, the wave functions of the conduction band are written |sα, |sβ.

(7.12)

Quantum states of the valence bands are indicated by j = l + s where l is the angular momentum operator and s is the spin operator. The wave function | j, m j can be expressed as follows: For the heavy hole, 3 3 1 , 2 2 = √2 |(x + i y)α, 3 , − 3 = √1 |(x − i y)β, 2 2 2

(7.13)


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197

and for the light hole, 3 1 1 , 2 2 = √6 |2zα + (x + i y)β, 3 , − 1 = √1 |2zβ − (x − i y)α, 2 2 6

(7.14)

where j is the eigenvalue of j and m j is the eigenvalue of jz . Momentum Matrix Elements with k Along the z-Axis As shown in (2.29), the power optical gain coefficient g is in proportion to a square of the momentum matrix element 1| p|22 . Therefore, to examine the anisotropic optical gain, the momentum matrix elements for the optical transitions are compared. The x-, y-, and z-components of the momentum matrix elements between the conduction band and the heavy hole band Mx , M y , and Mz are given by 1 √ Mx = √ 3M, 2 1 √ M y = ±i √ 3M, 2 Mz = 0,

(7.15) (7.16) (7.17)

where M is defined as √

3M ≡ s| px |x = s| p y |y = s| pz |z 1/2 1 E g (E g + 0 ) = m0 . 2m e E g + 23 0

(7.18)

Here (1.6) and (1.43) are used, m 0 is the electron mass in vacuum, m e is the effective mass of the conduction √ electron, E g is the bandgap energy, and 0 is the split-off energy. A coefficient 3 on the left-hand side in (7.18) is introduced so that a matrix element averaged over all directions of the wave vector k may be M. Momentum Matrix Elements with k Along an Arbitrary Axis As shown in Fig. 7.5, it is assumed that a quantization axis is the z-axis and a QW layer is placed on the x y-plane. If light propagates along the x-axis, an electric field E along the y-axis is the TE mode and an electric field E along the z-axis is the TM mode. For the wave vector k with an arbitrary direction, a direction of k can be expressed in a polar coordinate system, as shown in Fig. 7.5.


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(TM)

(TE) FIGURE 7.5 Direction of a wave vector k.

According to the directions of k, the x-, y-, and z-components of the momentum matrix elements Mx , M y , and Mz are given by 1 √ Mx = √ 3M(cos θ cos φ ∓ i sin φ), 2 1 √ My = √ 3M(cos θ sin φ ± i cos φ), 2 1 √ Mz = − √ 3M sin θ. 2

(7.19) (7.20) (7.21)

Note that the square of the optical transition matrix elements is proportional to 1|E · p|22 , where E is the electric field. Momentum Matrix Elements Between the Conduction Band and the Heavy Hole Band in QWs An average of the squared momentum matrix element M 2 hh,TE for the TE mode (E//y) is expressed as 2π 3M 2 1 · (cos2 θ sin2 φ + cos2 φ) dφ 2 2π 0

3M 2 kz 2 3M 2 = (1 + cos2 θ ) = 1+ 2 4 4 k 2

3M E z,n = 1+ , 4 En

M 2 hh,TE =

(7.22)

where E z,n is the quantized energy of the nth subband and E n is the total energy of the nth subband. At the subband edge with E z,n = E n , we have M 2 hh,TE =

3M 2 . 2

(7.23)


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199

An average of the squared momentum matrix element M 2 hh,TM for the TM mode (E//z) is given by 3M 2 3M 2 sin2 θ = (1 − cos2 θ ) 2 2

3M 2 kz 2 3M 2 E z,n = 1− 2 = 1− . 2 k 2 En

M 2 hh,TM =

(7.24)

At the subband edge with E z,n = E n , we obtain M 2 hh,TM = 0.

(7.25)

As shown in (7.23) and (7.25), a selection rule is present between the conduction band and the heavy hole band at the subband edge. Only the optical transition for the TE mode is allowed, and that for the TM mode is inhibited. Momentum Matrix Elements Between the Conduction Band and the Light Hole Band in QWs An average of the squared momentum matrix element M 2 lh,TE for the TE mode (E//y) is obtained as M2 (1 + cos2 θ ) + M 2 sin2 θ 4

M2 E z,n E z,n = 1+ + M2 1 − . 4 En En

M 2 lh,TE =

(7.26)

As a result, at the subband edge with E z,n = E n , we have M 2 lh,TE =

M2 . 2

(7.27)

An average of the squared momentum matrix element M 2 lh,TM for the TM mode (E//z) is written M2 sin2 θ + 2M 2 cos2 θ 2

M2 E z,n E z,n = 1− + 2M 2 . 2 En En

M 2 lh,TM =

(7.28)

Therefore, at the subband edge with E z,n = E n , we obtain M 2 lh,TM = 2M 2 .

(7.29)

Optical Gains in QW LDs The concentration of a heavy hole is larger than that of a light hole. Therefore, optical gains for the TE mode, ge-hh,TE and ge-lh,TE , and that


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for the TM mode, ge-lh,TM , are related as ge-hh,TE > ge-lh,TM > ge-lh,TE , where the subscript e-hh represents recombination of the conduction electron and the heavy hole and the subscript e-lh represents recombination of the conduction electron and the light hole. Low-Loss Optical Waveguides In a QW, the density of states is a step function, and the absorption coefficient at the band edge changes sharply with a wavelength of light. Therefore, the absorption loss in optical waveguides with a QW is lower than that with a bulk DH. As a result, the threshold gain of a QW LD is lower than that of a bulk DH LD. High-Speed Modulation given by

From (6.101), the relaxation oscillation frequency f r is

1 fr = 2π

a

∂ G S0 . ∂n τph

(7.30)

In a QW LD, based on the steplike density of states, the differential optical gain a ∂ G/∂n is larger than that in a bulk DH LD. Hence, f r in a QW LD is larger than that in a bulk DH LD. As a result the modulation speed of a QW LD is higher than that of a bulk DH LD. Figure 7.6 shows a calculated differential optical gain (n r /c)a ∂ G/∂n

QW- LD

DH-LD

Quasi-Fermi Level

(meV)

FIGURE 7.6 Differential optical gain.


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STRAINED QUANTUM WELL LDs

201

as a function of the quasi-Fermi level of the conduction band E Fc . The differential optical gain of a QW LD is higher than that of a bulk DH LD. From (7.6)-(7.8), optical gain g can be reduced to g = |M|2ave ρ [ f c − f v ] .

(7.31)

A QW modifies the density of states ρ in (7.31); the intentionally doped active layers can change [ f c − f v ]. For example, a p-doped active layer reduces f v ; a relaxation oscillation frequency of about 30 GHz was reported. Narrow Spectral Linewidth is expressed as

From (6.182), the spectral linewidth ω0 of laser light

ω0 =

ωm cE cv ln(1/R) (1 + α 2 ), 4P0 n r L

(7.32)

where α is given by α=−

2ωm ∂n r n r ∂n

a

∂G ∂n

−1

,

(7.33)

as shown in (6.158). Here n r is the refractive index, n is the carrier concentration, and G is the amplification rate. As shown in Fig. 7.6, the differential optical gain ∂ G/∂n in a QW LD is higher than that in a bulk DH LD. From (7.33), the |α| value of a QW LD is smaller than that of a bulk DH LD. As a result, ω0 of a QW LD is narrower than that of a bulk DH LD. Figure 7.7 shows calculated values of the α-parameter as a function of the quasi-Fermi level of the conduction band E Fc . Note that a small value of the α-parameter leads to low chirping during modulation, which is suitable for long-haul, large-capacity optical fiber communication systems.

7.3


Effect of Strains In a bulk DH LD, the heavy and light hole bands are degenerate at the point with k = 0 due to their high symmetry. In a QW LD, degeneracy at k = 0 is removed due to their lower symmetry than that of a bulk DH LD, as shown in Fig. 7.3. When strains are applied to semiconductor crystals, the symmetry of the crystals is reduced, and the degeneracy of the energy bands is removed. Modification of the energy bands by intentional strains is called band-structure engineering. Introduction of the strains into the active layers can lead to a low threshold current, a high differential quantum efficiency, high-speed modulation, low chirping, and a narrow spectral linewidth in LDs. According to the compressive or tensile strains, the energy bands change in different ways. For optical gain for the TE mode, gTE , and for the


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-Parameter

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August 4, 2010

DH LD

QW LD

Quasi-Fermi Level

(meV)

FIGURE 7.7 α-Parameter.

TM mode, gTM , the compressive strains in the active layer plane lead to gTE > gTM and the tensile strains result in gTE < gTM . As examples of intentional strains in semiconductor crystals, an external stress, a difference in the thermal expansion coefficients of the materials, and a difference in the lattice constants are present. Especially, a difference in the lattice constants called lattice mismatching is frequently adopted, because the strains induced by lattice mismatching are more stable than those by other methods. Lattice mismatching takes place when the lattice constant of a semiconductor layer, which is grown epitaxially on the substrate, is different from the lattice constant of the substrate. If the thickness of the grown semiconductor layer exceeds the critical thickness, dislocations are generated in the grown layer. If the thickness of the grown semiconductor layer increases further over the critical thickness and reaches up to about 1 µm, the grown semiconductor layer is relaxed and the dislocations are sometimes reduced. Such a thick grown semiconductor layer can be used as a buffer layer. However, a thick semiconductor layer is not suitable for the active layers because the threshold current becomes high. In a QW, which is thinner than the critical thickness, dislocations are not generated. Hence, strained QWs are adopted for band-structure engineering. Band-Structure Engineering Epitaxial Growth and Strains When a grown semiconductor layer is thinner than the critical thickness, the lattice constant of the grown semiconductor layer changes so that lattice matching can take place. The thickness of the substrate is several hundreds of micrometers or more, and the thickness of a QW is 10 nm or less. As a result, the lattice constant of the substrate does not change when a semiconductor


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203

layer is grown on the substrate. Therefore, elastic strains are induced in the grown semiconductor layer. It is assumed that the lattice constant of an epitaxial layer itself is a(x) and the lattice constant of the substrate is a0 . For a(x) > a0 , a compressive strain is generated in the grown semiconductor layer; for a(x) < a0 , a tensile strain is produced in the grown semiconductor layer. Low Threshold Current Low threshold current is obtained in QW LDs with strained active layers for three reasons. First, the strains in the active layers enhance the differential optical gain ∂ G/∂n. Second, the effective masses of the carriers are modified by the strains. Using the effective mass of the conduction electron m e and that of the hole m h , the condition for population inversion is written

2π kB T np > Nc Nv = 4 h2

3 (m e m h )3/2 ,

(7.34)

where (2.17) and (2.2) are used. From (7.34), the transparent carrier concentration n 0 is given by

2πkB T n0 = 2 h2

3/2 (m e m h )3/4 ,

(7.35)

where we set n = p = n 0 . If m e and m h are lightened by the strains, a low n 0 is expected, leading to a low threshold carrier concentration n th and low Auger recombination rate. Third, by modifying the energy bands, Auger transitions will be inhibited, due to the momentum conservation law and the energy conservation law. Hence, Auger processes can be drastically reduced by the strains. As a result, a high light emission efficiency and a low threshold current density can be obtained simultaneously. Anisotropic Optical Gain The tensile strains make the energy of the light hole lower than that of the heavy hole. As a result, the concentration of the light hole is larger than that of the heavy hole. Therefore, the optical gain for the TM mode is larger than that for the TE mode, and laser oscillation in the TM mode is obtained. By controlling the tensile strains so that the optical gain for the TE mode and that for the TM mode may be common, polarization-independent semiconductor optical amplifiers were demonstrated. High-Speed Modulation The strains in the active layers enhance the differential optical gain ∂G/∂n, which leads to a high-speed modulation from (6.101). The compressive strains result in large gain saturation; the tensile strains lead to low gain saturation with a large optical gain. As a result, it is expected that the tensile strains improve the modulation speed.


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Narrow Spectral Linewidth The strains increase the differential optical gain ∂G/∂n and decrease α. Therefore, a narrow spectral linewidth can be obtained.

Analysis Fundamental Equations The effect of strains is introduced into the Schrödinger equation as a perturbation, where the unperturbed Hamiltonian is assumed to include both k · p perturbation and spin-orbit interaction. This unperturbed Hamiltonian, which has | j, m j as a base wave function, is known as the Luttinger–Kohn Hamiltonian [3]; a perturbation Hamiltonian representing the effect of the strains is called the Pikus–Bir Hamiltonian [4–6].

Luttinger–Kohn Hamiltonian bands HLK is expressed as ⎡

H11 ⎢ H21 ⎢ ⎢ H31 ⎢ ⎢ H41 ⎢ ⎣ H51 H61

The Luttinger–Kohn Hamiltonian for the valence

H12 H22 H32 H42 H52 H62

H13 H23 H33 H43 H53 H63

H14 H24 H34 H44 H54 H24

⎤ H16 H26 ⎥ ⎥ H36 ⎥ ⎥, H46 ⎥ ⎥ H56 ⎦ H66

H15 H25 H35 H45 H55 H65

(7.36)

where Hi j = i|H| j and i| and | j are written 1| =

3 2

,

3 , 2

,

1 , 2

3 2

,

1 , 2

3| =

3 2

, − 12 ,

, − 12 ,

|1 = 32 ,

3 2

|3 = 32 , − 12 , |4 = 32 , − 32 ,

|5 = 12 ,

1 2

5| =

1 2

2| = 6| =

1 2

4| =

3 2

, − 32 ,

,

|2 = 32 ,

,

|6 = 12 , − 12 .

1 2

,

(7.37)

When the split-off energy 0 is larger than 0.3 eV, as in GaAs, a contribution of the split-off band can be neglected. In this case, the Luttinger–Kohn Hamiltonian for the valence bands, HLK , is reduced to a 4 × 4 matrix from the 6 × 6 matrix in (7.36), which is given by ⎡

HLK

H11 ⎢ H21 =⎢ ⎣ H31 H41

H12 H22 H32 H42

H13 H23 H33 H43

⎤ ⎡ H14 a+ ⎢ b∗ H24 ⎥ ⎥=⎢ H34 ⎦ ⎣ c∗ H44 0

b a− 0 c∗

c 0 a− −b∗

⎤ 0 c ⎥ ⎥, −b ⎦ a+

(7.38)


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205

where a± =

2 −(γ1 ∓ 2γ2 )k z 2 − (γ1 ± γ2 )(k x 2 + k y 2 ) , 2m 2 √ b= 3 γ3 (k x − i k y )k z , m 2 √ c= 3 γ2 (k x 2 − k y 2 ) − i 2γ3 k x k y . 2m

(7.39) (7.40) (7.41)

Here γ1 , γ2 , and γ3 are Luttinger parameters, and the effective masses m hh and m lh of the heavy and light holes are written 1 (γ1 − 2γ2 ) = m0 1 (γ1 + 2γ2 ) = m0

1 (heavy hole), m hh 1 (light hole), m lh

(7.42) (7.43)

where m 0 is the electron mass in vacuum. The Luttinger–Kohn Hamiltonian for the valence bands, HLK , can also be expressed as HLK

2 5 = − γ1 + γ2 k 2 + 2γ2 (k x 2 Jx 2 + k y 2 Jy 2 + k z 2 Jz 2 ) 2m 2 + 4γ3 {k x k y }{Jx Jy } + {k y k z }{Jy Jz } + {k z k x }{Jz Jx } ,

(7.44)

where Jx , Jy , and Jz are the following matrices: √ 0 3i √ 1⎢ − 3 i 0 Jx = ⎢ −2 i 2⎣ 0 0 0 √ ⎡ 3 0 √0 1⎢ 3 0 2 Jy = ⎢ 2 √0 2⎣ 0 0 0 3 ⎡ 3 0 0 1⎢ 0 1 0 Jz = ⎢ ⎣ 0 0 −1 2 0 0 0 ⎡

⎤ 0 0 ⎥ 2i √0 ⎥ , ⎦ 0 3 i √ − 3i 0 ⎤ 0 ⎥ √0 ⎥ , 3⎦ 0 ⎤ 0 0 ⎥ ⎥. 0 ⎦ −3

(7.45)

(7.46)

(7.47)


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In (7.44), {k x k y } and {Jx Jy } are defined as {k x k y } ≡

1 1 (k x k y + k y k x ), {Jx Jy } ≡ (Jx Jy + Jy Jx ). 2 2

(7.48)

Pikus–Bir Hamiltonian The Pikus–Bir Hamiltonian Hs is given by a sum of the orbit-strain interaction Hamiltonian Hos and the strain-dependent spin-orbit interaction Hamiltonian Hss . For valence bands of the zinc blende structure, the orbit-strain interaction Hamiltonian Hos at the point (k = 0) is written Hos = −a1 (εx x + ε yy + εzz )

L2 L2 L2 − 3b1 L x 2 − εx x + L y 2 − ε yy + L z 2 − εzz 3 3 3 √ √ − 3d1 L x L y + L y L x εx y − 3d1 L y L z + L z L y ε yz √ − 3d1 (L z L x + L x L z ) εzx , (7.49) and the strain-dependent spin-orbit interaction Hamiltonian Hss is expressed as Hss = −a2 (εx x + ε yy + εzz )(L · s)

L·s L·s − 3b2 L x sx − εx x − 3b2 L y s y − ε yy 3 3

L·s − 3b2 L z sz − εzz 3 √ √ − 3d2 L x s y + L y sx εx y − 3d2 L y sz + sz L y ε yz √ − 3d2 (L z sx + L x sz ) εzx .

(7.50)

Here, L x , L y , L z , and L are orbital angular momentum operators; sx , s y , sz , and s are spin angular momentum operators; εi j (i, j = x, y, z) is a matrix element for the strain tensor; and ai , bi , and di (i = 1, 2) are deformation potentials. Because the strain-dependent spin-orbit interaction Hamiltonian Hss is smaller than the orbitstrain interaction Hamiltonian Hos , Hss is often neglected. Relationship Between Strain and Stress The strain tensors are expressed by matrices using their symmetrical properties. The matrix elements of the strains consist of the hydrostatic strains εii and the shear strains εi j (i = j). The strain tensors ε and stress tensors σ are related by σi j =

k,l

ci jkl εkl = ci jkl εkl ,

(7.51)


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207

TABLE 7.1 Relations between the Indexes for Tensors and Matrices Tensor expression Matrix element

11 1

22 2

33 3

23, 32 4

31, 13 5

12, 21 6

where ci jkl is the elastic stiffness constant. Note that is omitted on the right-hand side of (7.51) by promising that we take a sum with regard to a subscript appearing twice, which is called the Einstein summation convention. Because of symmetry in the tensors, it is useful to express the tensors by matrices. As a coordinate system for the tensors, 1, 2, and 3 are used as indexes. Table 7.1 shows the relation of the indexes for the tensor, 1, 2, and 3, to the indexes for the matrix, 1, 2, 3, 4, 5, and 6. From Table 7.1, (7.51) can be rewritten as σi =

ci j ε j = ci j ε j

(i, j = 1, 2, 3, 4, 5, 6),

(7.52)

⎤ ⎡ σ31 σ1 σ23 ⎦ = ⎣ σ6 σ33 σ5

(7.53)

j

where ⎡

σ11 ⎣ σ21 σ31

⎡

ε11 ⎣ ε21 ε31

σ12 σ22 σ32

ε12 ε22 ε32

⎤ ⎡ σ13 σ11 σ23 ⎦ = ⎣ σ12 σ33 σ31

⎤ ⎡ ε13 ε11 ε23 ⎦ = ⎣ ε12 ε33 ε31

σ12 σ22 σ23

ε12 ε22 ε23

⎤ ⎡ ε31 2ε1 1 ε23 ⎦ = ⎣ ε6 2 ε ε33 5

σ6 σ2 σ4

ε6 2ε2 ε4

⎤ σ5 σ4 ⎦ , σ3

⎤ ε5 ε4 ⎦ . 2ε3

(7.54)

¯ In zinc blende structures with a symmetry of 43m or Td , (7.51) is reduced to [7, 8] ⎡

⎤ ⎡ σ1 c11 ⎢ σ2 ⎥ ⎢ c12 ⎢ ⎥ ⎢ ⎢ σ3 ⎥ ⎢ c12 ⎢ ⎥=⎢ ⎢ σ4 ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎣ σ5 ⎦ ⎣ 0 σ6 0

c12 c11 c12 0 0 0

c12 c12 c11 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

⎤⎡ ⎤ 0 ε1 ⎢ ⎥ 0 ⎥ ⎥ ⎢ ε2 ⎥ ⎢ ⎥ 0 ⎥ ⎥ ⎢ ε3 ⎥ . ⎢ ⎥ 0 ⎥ ⎥ ⎢ ε4 ⎥ 0 ⎦ ⎣ ε5 ⎦ c44 ε6

(7.55)

Bulk Structures It is assumed that an epitaxial layer is grown along the z-axis and that the layer plane is on the x y-plane. When the lattice constant of the substrate is a0 and that of the epitaxially grown layer itself is a(x), the strains due to the lattice


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mismatching are given by a0 − a(x) = ε, a(x) εzz = 0, = ε yz = εzx = 0,

εx x = ε yy = εx y

(7.56) (7.57) (7.58)

where ε < 0 corresponds to the compressive strain and ε > 0 shows the tensile strain. Due to lattice mismatching, biaxial stresses are induced in the layer plane (x yplane); neither the stress along the growth axis (z-axis) nor the shear stresses are imposed on the epitaxial layer. Therefore, the biaxial stresses are expressed as σx x = σ yy = σ, σzz = 0,

(7.59) (7.60)

σx y = σ yz = σzx = 0.

(7.61)

Relating the indexes 1, 2, and 3 of the tensors to x, y, and z, respectively, then substituting (7.56)–(7.61) into (7.55) gives ⎡ ⎤ ⎡ σ c11 ⎢ σ ⎥ ⎢ c12 ⎢ ⎥ ⎢ ⎢ 0 ⎥ ⎢ c12 ⎢ ⎥=⎢ ⎢0⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎣0⎦ ⎣ 0 0 0

c12 c11 c12 0 0 0

c12 c12 c11 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

⎤⎡ ⎤ 0 ε ⎢ ⎥ 0 ⎥ ⎥⎢ ε ⎥ ⎢ ⎥ 0 ⎥ ⎥ ⎢ εzz ⎥ . ⎥ ⎥ 0 ⎥⎢ ⎢ 0 ⎥ ⎦ ⎣ 0 0 ⎦ c44 0

(7.62)

From (7.62) we obtain σ = (c11 + c12 )ε + c12 εzz , 0 = 2c12 ε + c11 εzz .

(7.63) (7.64)

By solving (7.63) and (7.64), we have 2c12 εzz = − ε, c11

2c12 2 σ = c11 + c12 − ε. c11

(7.65) (7.66)

Substituting (7.65) and (7.66) into (7.49) results in

c12 1 2 2c12 2 Hos = −2a1 1 − ε + 3b1 L z − L 1+ ε. c11 3 c11

(7.67)


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209

Here we consider a matrix Hsv , such as ⎡

1|Hos |1 ⎢ 2|Hos |1 Hsv = ⎢ ⎣ 3|Hos |1 4|Hos |1 ⎡ δ E hy − ζ ⎢ 0 =⎢ ⎣ 0 0

1|Hos |2 2|Hos |2 3|Hos |2 4|Hos |2

1|Hos |3 2|Hos |3 3|Hos |3 4|Hos |3

0 δ E hy + ζ 0 0

0 0 δ E hy + ζ 0

⎤ 1|Hos |4 2|Hos |4 ⎥ ⎥ 3|Hos |4 ⎦ 4|Hos |4 ⎤ 0 ⎥ 0 ⎥, ⎦ 0 δ E hy − ζ

(7.68)

where

c12 δ E hy = −2a1 1 − ε, c11

2c12 ζ = −b1 1 + ε. c11

(7.69) (7.70)

Equations (7.69) and (7.70) indicate that the energy shifts at k = 0; degeneracy of the valence bands is removed. Using the deformation potential of the conduction band C1 , the orbit-strain interaction Hamiltonian for the conduction band Hsc is written Hsc = C1 (εx x + ε yy + εzz ).

(7.71)

The energies of the heavy and light hole bands are obtained as eigenvalues of the Hamiltonian HLK + Hsv by using (7.38) and (7.68). The energy of the conduction band is obtained as an eigenvalue of the Hamiltonian HLK + Hsc using (7.38) and (7.71). Figure 7.8 shows the energy bands for the strained bulk In1−x Gax As layers grown on an InGaAsP layer, which is lattice-matched to an InP substrate. Here, it is assumed that C1 = −2a1 and that dislocations are not present. From Fig. 7.8 it is found that the compressive strains lower the energy of the heavy hole more than that of the light hole; the tensile strains lower the energy of the light hole more than that of the heavy hole. Note that the energy of the holes decreases with an increase in the height of the vertical line. Strained QWs as

In a strained QW, the Hamiltonian for the valence bands is expressed

HLK + Hsv + V (z),

(7.72)


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Compressive

Lattice Matching

Tensile

LH

HH

HH HH

LH

LH HH

LH

(a)

(b)

(c)

FIGURE 7.8 Energy bands for bulk In1−x Gax As layers grown on an InGaAsP layer which is lattice-matched to an InP substrate: (a) compressive; (b) lattice matching; (c) tensile.

where V (z) is a potential of the QW, which is written V (z) =

0 E v − δ E hy

(well), (barrier).

(7.73)

When the quantization axis is the z-axis, the Schrödinger equation can be solved by replacing k z → −i ∂/∂z under the effective mass approximations. The results obtained for the energy bands and the optical gains for the heavy hole (HH) and light hole (LH) are shown in Fig. 7.9. Here the oscillation wavelength is 1.3 µm and the InGaAsP well is 10 nm thick for 1.9% tensile strain, lattice matching, and 1.4% compressive strain. As shown in Fig. 7.9, the tensile strain enhances the optical gain for the TM mode and the compressive strain increases the optical gain for the TE mode.


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211

REFERENCES

Energy (meV)

LH1

(a) TM

Optical Gain

HH1 LH2 HH2 HH3

(b)

LH1

Optical Gain

HH3 HH4 LH2

HH1

Energy (meV)

TE

(b)

TE

TM

(c)

(c)

HH2 HH3 LH1

HH4 HH5

Normalized Wave Number

Optical Gain

Energy (meV)

HH1 HH2

(a)

TE

TM

Energy (eV)

FIGURE 7.9 Energy bands and optical gains in a strained QW. Here the oscillation wavelength is 1.3 µm and the InGaAsP well is 10 nm thick for (a) 1.9% tensile strain, (b) lattice matching, and (c) 1.4% compressive strain.

REFERENCES 1. Y. Arakawa and A. Yariv, “Quantum well lasers: gain, spectra, dynamics,” IEEE J. Quantum Electron. 22, 1887 (1986). 2. P. S. Zory, Jr., ed., Quantum Well Lasers, Academic Press, San Diego, CA, 1993. 3. J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869 (1955). 4. G. E. Pikus and G. L. Bir, “Effect of deformation on the energy spectrum and the electrical properties of imperfect germanium and silicon,” Sov. Phys. Solid State 1, 136 (1959).


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5. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors: Physics and Materials Properties, 2nd ed., Springer-Verlag, Berlin, 1999. 6. S. Adachi, Physical Properties of III–V Semiconductor Compounds, Wiley, New York, 1992. 7. C. Kittel, Introduction to Solid State Physics, 7th ed., Wiley, New York, 1996. 8. J. F. Nye, Physical Properties of Crystals, Oxford University Press, New York, 1985. 9. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004.


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8 SINGLE-MODE LASER DIODES

8.1

INTRODUCTION

Single-mode LDs are LDs that operate with a single longitudinal mode. Because of dispersion in the optical fibers, single-longitudinal-mode laser operations are indispensable for long-haul, large-capacity optical fiber communication systems. Therefore, DFB LDs, DBR LDs, and vertical cavity surface-emitting LDs have been developed. Because the optical gain spectra of LDs have linewidths on the order of 10 nm, optical cavities play important roles in selecting only one longitudinal mode for laser operations.

8.2

DFB LDs

As shown in Fig. 8.1, DFB LDs have active layers in the corrugated regions. Index-Coupled DFB LDs In index-coupled DFB LDs, only the real part of the complex refractive index is modulated periodically, and the imaginary part is uniform. The real part of the complex refractive index n r (z) is expressed as n r (z) = n r0 + n r1 cos(2β0 z + ).

(8.1)


213


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SINGLE-MODE LASER DIODES

Active Layer Laser Beam

Laser Beam

FIGURE 8.1 DFB LD.

Here z is a position, is the grating phase at z = 0, and β0 is written β0 =

π ,

(8.2)

where is the grating pitch. Figure 8.2 shows index-coupled grating. If corrugations are formed on the interfaces of two layers with refractive indexes n A and n B (= n A ), the real part of the complex refractive index is modulated periodically; a forward running wave and a backward running wave are coupled, which is indicated by the grating coupling coefficient κ, which is given by κ=

π n r1 . λ0

(8.3)

Here α1 = 0 is inserted in (4.35). Uniform Grating The oscillation condition of a DFB LD with a uniform grating in which both the depth and the pitch are constant over the entire corrugated region is considered. As shown in Fig. 8.3, it is assumed that both facets are antireflection (AR)-coated, and reflections at both facets are negligibly low. The threshold condition of a DFB LD is that the transmissivity in (4.56) is infinity. From (4.46), (4.47), and (4.55), the threshold condition of a DFB LD with uniform grating is given by cosh γ L −

α0 − i δ sinh γ L = 0. γ

FIGURE 8.2 Index-coupled grating.

(8.4)


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DFB LDs

215


Laser Beam

Antireflection Films FIGURE 8.3 DFB LD with uniform grating.

Here α0 is the field optical gain coefficient, L is the cavity length, and γ 2 = (α0 − i δ)2 + κ 2 , δ=

β − β0 β − β0 , 2β0 2

(8.5)

2

(8.6)

where β is the propagation constant of light, as shown in (4.34). Figure 8.4 shows transmission spectra of a DFB LD with uniform grating, with the field optical gain coefficient α0 as a parameter, where αth is the threshold gain. As indicated in Fig. 8.4, a DFB LD with uniform grating oscillates in two longitudinal modes when the reflectivities of both facets are negligibly low. When the reflectivities of both facets are not negligible, as in cleaved facets, the threshold gain demonstrates complicated behaviors according to the grating phases

FIGURE 8.4 Transmission spectra for DFB LDs with uniform grating.


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FIGURE 8.5 Calculated αth L as a function of δ × L = δL with grating phase at a cleaved facet as a parameter for a DFB LD with uniform grating.

at the facets, as shown in Fig. 8.5. In Fig. 8.5, the vertical line shows the αth L calculated and the horizontal line shows δ × L = δL. It is found that a large difference between the lowest threshold gain and the second-lowest threshold gain is present for a certain value of . As a result, single-longitudinal-mode operations can be obtained in a DFB LD with uniform grating. However, the grating pitch is as short as about 0.2 µm, and it is almost impossible to control the grating phase at the facets in manufacturing. Therefore, it is difficult to obtain single-longitudinal-mode operations in DFB LDs with uniform grating; the yield for single-longitudinal-mode operations is less than several percent. To achieve stable single-longitudinal-mode operations with a high yield, phaseshifted and gain-coupled DFB LDs have been developed. Phase-Shifted Grating As shown in Fig. 8.6, the corrugations are shifted in phase-shifted grating [1, 2], represented by a solid line; the corrugations are repeated periodically in uniform grating, represented by a dashed line. Here the phase shift is expressed as − according to the definition of a refractive index in (8.1).

Pitch

Phase Shift

FIGURE 8.6 Phase-shifted grating.


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DFB LDs

Region 1

217

Region 2

FIGURE 8.7 Analytical model for phase-shifted grating.

Figure 8.7 shows an analytical model that consists of two regions and includes the phase shift as a phase jump at the interface of the two regions. It is assumed here that both pitch and depth of the corrugations are constant except in the phase-shifted region and that the optical gain is uniform over all the grating. It is also supposed that both facet reflectivities are negligibly low. If the grating phase at the left edge of region 1 is θ1 , the grating phase θ2 at the right edge of region 1 is given by θ2 = θ1 + 2β0 L 1 .

(8.7)

Because of the phase shift at the interface of regions 1 and 2, the grating phase θ3 at the left edge of region 2 is expressed as θ3 = θ2 + = θ1 + 2β0 L 1 + .

(8.8)

The transfer matrix for region 1 with a length of L 1 is F 1 and that for region 2 with a length of L 2 is F 2 . The threshold condition is given by F11 = 0, where F11 is a matrix element of F = F 1 × F 2 , and F11 is written cosh(γ L 1 ) −

α0 − i δ α0 − i δ sinh(γ L 1 ) cosh(γ L 2 ) − sinh(γ L 2 ) γ γ 2 κ + 2 sinh(γ L 1 ) sinh(γ L 2 ) e i = 0. (8.9) γ

When the phase-shifted position is located at the center of the optical cavity (L 1 = L 2 = L/2), (8.9) is reduced to

α0 − i δ sinh(γ L) γ 2 2 κ γL + 2 exp( i ) − 1 sinh = 0. γ 2 cosh(γ L) −

(8.10)


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FIGURE 8.8 αth L calculated as a function of δ × L = δL with the phase shift − as a parameter when the power reflectivities of the facets are zero.

The first term on the left-hand side in (8.10) is the same as the left-hand side in (8.4) for a DFB LD with uniform grating. The second term in (8.10) represents an effect of the phase shift. Figure 8.8 shows αth L calculated as a function of δ × L = δL, with the phase shift − as a parameter when the power reflectivities of the facets are zero. Laser oscillation starts at a mode with the lowest αth L. If a difference in the threshold gain of the oscillation mode and that of other modes is large enough, highly stable single-longitudinal-mode operations are obtained. For the phase shift

= 0, as in uniform grating, two modes have a common lowest threshold gain αth L, represented by filled circles. Therefore, = 0 results in two longitudinal mode operations. For the phase shift − = π/2 represented by open circles and − = 3π/2 represented by open triangles, there is only one longitudinal mode whose threshold gain is lowest, and single-longitudinal-mode operation is expected. For the phase shift − = π represented by filled triangles, there exists only one longitudinal mode whose threshold gain is lowest at the Bragg wavelength. Note that the difference between the lowest threshold gain and the second-lowest threshold gain is largest for − = π, which leads to the most stable single-longitudinalmode operations. From (4.42), the Bragg wavelength λB in vacuum, which satisfies δ = 0, is given by

λB =

2n r0 , m

(8.11)

where is the grating pitch and m is the order of diffraction, which is a positive integer.


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DFB LDs

219

FIGURE 8.9 Transmission spectra for a DFB LD with λ/4-shifted grating.

For the phase shift − = π in the first-order gratings (m = 1), the corrugations are shifted by /2. From (8.11) we have λB λm λB = = , λm = , 2 4n r0 4 n r0

(8.12)

where λm is a wavelength in a material. From (8.12), the phase shift of π corresponds to a quarter of a wavelength in a material. As a result, phase-shifted grating with − = π is often called λ/4-shifted grating or quarter-wavelength-shifted grating. Figure 8.9 shows transmission spectra of λ/4-shifted grating with α0 L as a parameter. As demonstrated in Fig. 8.9, a λ/4-shifted DFB LD oscillates at the Bragg wavelength located at the center of the stopband and shows highly stable single-longitudinal-mode operations when the reflectivities at both facets are negligibly low. According to the distribution of optical power along the cavity axis, the radiative recombinations are altered and spatial distribution of the carrier concentration is modified. In this phenomenon, called spatial hole burning, distribution of the refractive index changes with that of the carrier concentration, leading to a change in


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FIGURE 8.10 αth L calculated as a function of δ × L = δL with grating phase at a cleaved facet as a parameter for a phase-shifted DFB LD with − = π .

the phase shift. Therefore, a grating with a phase shift slightly altered from π [3] is used to achieve − = π at the operating condition, or chirped grating is adopted in the phase-shift region [4] to reduce the spatial hole burning. When reflections are present at both facets, as in cleaved facets, the threshold gains of the phase-shifted DFB LDs show complicated behaviors according to the grating phases at the facets, as shown in Fig. 8.10. The difference between the lowest threshold gain and the second-lowest threshold gain is usually lower than that of a phase-shifted DFB LD without reflections at the facets. As a result, the stability of single-longitudinal-mode operations of a phase-shifted DFB LD with cleaved facets is lower than that of a phase-shifted DFB LD without reflections at the facets. Therefore, to achieve highly stable single-longitudinal-mode operations in a phase-shifted DFB LD, AR-coated facets or window structures to reduce reflections at the facets are needed. Figure 8.11 shows αth L calculated for a phase-shifted DFB LD with AR-coated facets. Here the power reflectivity of the AR-coated facets is 2%. Experimentally, it is found that the power reflectivity of AR-coated facets should be less than 2% to obtain highly stable single-longitudinal-mode operations in phase-shifted DFB LDs [3–8]. Due to the high stability of single-longitudinal-mode operations, index-coupled DFB LDs are used in long-haul, large-capacity optical fiber communication systems. Finally, polarization of the laser light in DFB LDs is explained briefly. Because the grating coupling coefficient κTE for the TE mode is larger than that for the TM mode κTM , the threshold gain for the TE mode is lower than that for the TM mode. Therefore, DFB LDs start to lase in the TE mode with an increase in the current injected. However, a difference in the threshold gains between the TE and TM modes is smaller than that of Fabry–Perot LDs if the reflectivities at both facets are negligibly low. As a result, to stabilize the polarization of laser light in phase-shifted DFB LDs with AR-coated facets, QW active layers are frequently adopted to obtain polarization-dependent optical gains.


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DFB LDs

221

FIGURE 8.11 αth L calculated as a function of δ × L = δL with grating phase at an AR-coated facet as a parameter for a phase-shifted DFB LD with − = π . Here the power reflectivity of the facets is 2%.

Gain-Coupled DFB LDs In gain-coupled DFB LDs [9–14], optical gain or optical loss is modulated periodically along the cavity axis. Their characteristic features are stable single-longitudinalmode operations even without phase-shifted gratings and AR films coated on the facets. In addition, they are less sensitive to feedback light than are index-coupled DFB LDs. However, problems in fabrication methods and the reliability of indexcoupled DFB LDs are still present. In purely gain-coupled DFB LDs [12,13], only the optical gain or loss is modulated periodically. The grating coupling coefficient κ is given by κ= i

α1 . 2

(8.13)

Here α1 is a deviation from the steady-state field optical gain coefficient α0 . Figure 8.12 shows αth L calculated as a function of δ × L = δL, with the grating phase at a cleaved facet as a parameter for a purely gain-coupled DFB LD. Here both facets are as cleaved and κ L = 0.4 i. It should be noted that the difference between the lowest threshold gain and the second-lowest threshold gain is much smaller than in a phase-shifted DFB LD with AR-coated facets. However, stable singlelongitudinal-mode operations are obtained experimentally. Probably, the reason for stable single-longitudinal-mode operations is that antinodes of the electric field of the oscillation mode fit in the peaks of the spatially modulated optical gain. In a hybrid index/gain-coupled DFB LD [14], both the real and imaginary parts of the complex refractive index are modulated periodically, and κ is given by (4.35). Figure 8.13 shows αth L calculated as a function of δ × L = δL, with the grating phase at a cleaved facet as a parameter for a hybrid index/gain-coupled DFB LD. Here both facets are as cleaved and κ L = 2 + 0.4 i. It should be noted that δL for the


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FIGURE 8.12 αth L calculated as a function of δ × L = δL with grating phase at a cleaved facet as a parameter for a purely gain-coupled DFB LD. Here both facets are as cleaved and α1 = 20 cm−1 .

FIGURE 8.13 αth L calculated as a function of δ × L = δL with grating phase at a cleaved facet as a parameter for a hybrid index/gain-coupled DFB LD. Here both facets are as cleaved and α1 = 20 cm−1 .

lowest threshold gain tends to be less than 0. This result agrees well with experimental results.

8.3 DBR LDs Grating as a Reflector In DBR LDs [15–20], optical gain regions and corrugated regions with diffraction gratings are separated from each other, and the corrugated regions function as reflectors. Figure 8.14 shows a DBR LD with a DBR and cleaved facets and a DBR LD with two DBRs.


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DBR LDs

Laser Beam

Active Layer

Laser Beam

Laser Beam

Active Layer

223

Laser Beam

(b)

(a)

FIGURE 8.14 DBR LD with (a) a DBR and cleaved facets and (b) two DBRs.

Stopband

(a)

(b)

FIGURE 8.15 Reflectivities of (a) a cleaved facet and (b) a DBR.

Figure 8.15 shows the reflectivities of a cleaved facet and a DBR. The power reflectivity R0 for the cleaved facet is considered to be almost independent of light wavelength, although R0 is slightly modified by the material dispersion. In contrast, the reflectivity R1 for a DBR is highly dependent on a light wavelength; R1 is high only within the stopband. Note that DBR LDs do not always lase at Bragg wavelengths, because the resonance condition is not always satisfied at Bragg wavelengths. Threshold Gain Figure 8.16 shows analytical models for DBR LDs. Region 1 is an optical gain region without diffraction gratings, which leads to the grating coupling coefficient κ1 = 0. Region 2 is a corrugated region with diffraction gratings, which leads to the grating coupling coefficient κ2 = 0. As an example, Fig. 8.17 shows αth L calculated as a function of δ × L = δL, with the grating phase at a cleaved facet as a parameter for the DBR LD shown in Fig. 8.14(a), which has an index-coupled grating. The stability of single-longitudinal-mode operations can be evaluated by the difference between the lowest threshold gain and the second-lowest threshold gain. From Fig. 8.17 it is expected that the stability of single-longitudinal-mode operations of DBR LDs is inferior to that of phase-shifted DFB LDs with AR-coated facets. This expectation has already been confirmed by experimental results. Phase-shifted DFB LDs with AR-coated facets are superior to DBR LDs in both light output power and the stability of single-longitudinal-mode operations. By controlling the light phase in LDs, the difference between the lowest threshold gain and the second-lowest threshold gain can be enhanced and a light wavelength can


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Region 1

Region 2 (a)

Region 2

Region 1

Region 2

(b)

FIGURE 8.16 Analytical model for a DBR LD with (a) a DBR and cleaved facets and (b) two DBRs.

FIGURE 8.17 αth L calculated as a function of δ × L = δL with grating phase at a cleaved facet as a parameter for a DBR LD with cleaved facets.

be tuned. As a result, DFB LDs or DBR LDs with phase-control sections [18–22] have been shown to act as wavelength-tunable LDs in single-longitudinal-mode operations. If these LDs are biased just below the threshold, they operate as wavelength-tunable resonant optical amplifiers (optical filters) [23–30].

8.4 VERTICAL CAVITY SURFACE-EMITTING LDs Short Cavity Figure 8.18 shows the structure of a vertical cavity surface-emitting LD (VCSEL) [31], where the active and cladding layers are sandwiched by the DBRs, which are composed of periodic multilayers. Laser light propagates along a normal to the interface of the semiconductor layers. Therefore, the length of the optical gain region is equal to the thickness of the active layer, which is on the order of tens of nanometers to several micrometers. As a result, to achieve a low threshold current in VCSELs, high power reflectivity, such as 99.5%, is needed for DBRs. The optical cavities of VCSELs are short enough to achieve single-longitudinalmode operations. For example, the free spectral range from (4.17) is 36 nm when the


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VERTICAL CAVITY SURFACE-EMITTING LDs

225

Laser Beam

DBR Active Layer DBR Laser Beam FIGURE 8.18 Vertical cavity surface-emitting LD.

refractive index is 3.5, the cavity length is 4 µm, and the oscillation wavelength is 1 µm. This value of the free spectral range is larger than the linewidth of the optical gain spectra, which is about 10 nm. As a result, only one longitudinal mode is present in the optical gain spectrum.

Special Features Compared with other single-longitudinal-mode LDs, VCSELs have the following special features: 1. Monolithic optical cavities can be fabricated without cleaving. 2. Device characteristics can be measured with the VCSELs on wafers before pelletizing. 3. The coupling efficiency between a VCSEL and an optical component is high because there is a circular beam with a small radiation angle. 4. Devices can be integrated by stacking. 5. Extremely low threshold current is expected. 6. The structures are suitable for high-density two-dimensional arrays. Based on these features, VCSELs receive a lot of attention as key devices for parallel optical information processing and parallel lightwave transmissions. In addition, optical functional devices based on VCSELs have been demonstrated.

Low Threshold Current The reduction of threshold current has been an important research theme because zero-bias modulation and low consumption of power are expected. To reduce threshold current, small optical cavities [32, 33] and photon recycling [34, 35] have been demonstrated.


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Laser Beam DBR Active Layer

DBR

Laser Beam FIGURE 8.19 Airpost VCSEL.

Small Optical Cavities As an example of a small optical cavity, a low threshold current of 190 µA was obtained in a microcavity surface-emitting laser with a 5-µm-diameter airpost [32], which is shown in Fig. 8.19, in pulsed operation at room temperature with no heat sink. This low threshold current is attributed to high-quality epitaxial layers and a dryetched smooth sidewall. In InGaAs single-quantum-well VCSELs with an intracavity p-contact which were fabricated by selective oxidation of AlAs, low threshold currents of 8.7 µA in 3-µm2 devices and 140 µA in 10-µm2 devices with maximum output powers over 1.2 mW were achieved [33]. When the optical cavity becomes as small as on the order of a light wavelength, the mode distribution becomes discrete and the spontaneous emission coupling factor βsp increases with a decrease in the mode volume Vm . Figure 8.20 shows the carrier concentration n and photon concentration S in a microcavity LD whose size is on the order of a light wavelength as a function of the current density J injected. With an increase in βsp , the threshold current density Jth decreases and becomes indistinct.

FIGURE 8.20 Characteristics of a microcavity LD.


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VERTICAL CAVITY SURFACE-EMITTING LDs

227

High High Low Low

FIGURE 8.21 Trade-off between threshold current and light output.

Trade-off A VCSEL with a small optical cavity has a trade-off: low threshold current but high electrical resistance and low light output. Even though the threshold current is low, the power consumptions are not always low, due to large electrical resistance. In addition, the light output is low due to a small light emission region, as shown in Fig. 8.21.

Photon Recycling When electric current is injected into LDs, spontaneous emission takes place. Only a fraction of the spontaneous emission is used as the seed of laser light; spontaneous emission other than the seed of the laser light is emitted outward from the optical cavities. Therefore, the injected carriers consumed for spontaneous emission other than as the seed of the laser light are considered to be useless. If spontaneous emission other than the seed of the laser light is absorbed by confining the spontaneous emission to the optical cavity, carriers are generated [36]. In this phenomenon, called photon recycling, the carrier concentration in the active layer is larger than that without photon recycling at the same injected current. As a result, a low threshold current is obtained. This concept of photon recycling is similar to the confinement of resonant radiation in gas lasers. Figure 8.22 shows the light emission spectra and optical gain spectra of LDs. A peak wavelength in the optical gain spectra is longer than that in spontaneous emission spectra. Therefore, the active layer can absorb the spontaneous emission located in a shorter-wavelength region, which is indicated by slanted lines. Therefore, by confining spontaneous emission to the optical cavity, photon recycling can take place efficiently. As shown in Fig. 8.23, with a decrease in the threshold optical gain gth , an absorption wavelength region increases. As a result, LDs with low threshold gain, as shown in Fig. 8.23(b) are suitable for efficient photon recycling.


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Spontaneous Emission

Light Intensity

Laser Light

Wavelength

(a) Gain

Wavelength

Absorption Loss (b)

FIGURE 8.22 (a) Light emission and (b) optical gain spectra.

Figure 8.24 shows the structure of a VCSEL that uses photon recycling. To confine spontaneous emission efficiently to the cavity, the sidewalls of the cavity are covered with reflectors with high reflectivity. In photon recycling, unlike in microcavity LDs, there are no limitations in the size of optical cavities. Consequently, when photon recycling is employed in a relatively large optical cavity, on the order of several micrometers or larger, an increase in electrical resistance and a decrease in light output can be avoided. Gain

Wavelength

Absorption Loss (a) Gain

Wavelength

Absorption Loss (b)

FIGURE 8.23 Threshold gain and photon recycling.


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REFERENCES

229

Reflector DBR Active layer DBR Laser Beam FIGURE 8.24 Vertical cavity surface-emitting LD with photon recycling.

Two-Dimensional Array With the growing interest in high-capacity lightwave transmission and optical information-processing systems, two-dimensional light sources, which are required for parallel transmission and processing systems, have been studied extensively. A VCSEL is expected to be a key element for such a two-dimensional array. To achieve a large number of channels in a two-dimensional VCSEL array, it is necessary to determine which device parameters are important. The number of channels in a two-dimensional VCSEL array was analyzed from the viewpoint of electric power consumption and light output [37]. It had often been said that low threshold current is important in achieving a high-density two-dimensional VCSEL array. However, it has been shown that the contribution of threshold current reduction to an increase in the number of channels is small if the threshold current is less than 100 µA. It was found that the key parameters for a large number of channels are high slope efficiency and low electrical resistance. It has also been shown that if each channel consists of plural elements, the number of channels increases. REFERENCES 1. H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. 12, 532 (1976). 2. H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, Englewood Cliffs, NJ, 1984. 3. T. Numai, M. Yamaguchi, I. Mito, and K. Kobayashi, “A new grating fabrication method for phase-shifted DFB LDs,” Jpn. J. Appl. Phys. Pt. 2 26, L1910 (1987). 4. M. Okai, N. Chinone, H. Taira, and T. Harada,“Corrugation-pitch-modulated phase-shifted DFB laser,” IEEE Photon. Technol. Lett. 1, 200 (1989). 5. K. Sekartedjo, N. Eda, K. Furuya, Y. Suematsu, F. Koyama, and T. Tanbun-ek, “1.5-µm phase-shifted DFB lasers for single-mode operation,” Electron. Lett. 20, 80 (1984). 6. K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP/InP DFB lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008 (1984).


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24. T. Numai, S. Murata, and I. Mito, “1.5 µm tunable wavelength filter with wide tuning range and high constant gain using a phase-controlled distributed feedback laser diode,” Appl. Phys. Lett. 53, 1168 (1988). 25. T. Numai, S. Murata, and I. Mito, “1.5 µm tunable wavelength filter using a phase-shiftcontrolled distributed feedback laser diode with wide tuning range and high constant gain,” Appl. Phys. Lett. 54, 1859 (1989). 26. L. G. Kazovski, M. Stern, S. G. Menocal, Jr., and C. E. Zah, “DBR active optical filters: transfer function and noise characteristics,” IEEE J. Lightwave Technol. 8, 1441 (1990). 27. N. Tessler, R. Nager, G. Eisenstein, J. Salzman, U. Koren, G. Raybon, and C. A. Burrus, Jr., “Distributed Bragg reflector active optical filters,” IEEE J. Quantum Electron. 27, 2016 (1991). ¨ 28. O. Sahlén, M. Oberg, and S. Nilsson, “Two-channel optical filtering of 1 Gbit/s signal with DBR filter,” Electron. Lett. 27, 578 (1991). 29. T. L. Koch, F. S. Choa, F. Heismann, and U. Koren, “Tunable multiple-quantum well distributed-Bragg-reflector lasers as tunable narrowband receivers,” Electron. Lett. 25, 890 (1989). 30. T. Numai, “1.5 µm optical filter using a two-section Fabry–Perot laser diode with wide tuning range and high constant gain,” IEEE Photon. Technol. Lett. 2, 401 (1990). 31. K. Iga, F. Koyama, and S. Kinoshita, “Surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 24, 1845 (1988). 32. T. Numai, T. Kawakami, T. Yoshikawa, M. Sugimoto, Y. Sugimoto, H. Yokoyama, K. Kasahara, and K. Asakawa, “Record low threshold current in microcavity surface-emitting laser,” Jpn. J. Appl. Phys. Pt. 2 10B, L1533 (1993). 33. G. M. Yang, M. H. MacDougal, and P. D. Dapkus, “Ultralow threshold current verticalcavity surface-emitting lasers obtained with selective oxidation,” Electron. Lett. 31, 886 (1995). 34. T. Numai, H. Kosaka, I. Ogura, K. Kurihara, M. Sugimoto, and K. Kasahara, “Indistinct threshold laser operation in a pnpn vertical to surface transmission electro-photonic device with a vertical cavity,” IEEE J. Quantum Electron. 29, 403 (1993). 35. T. Numai, K. Kurihara, I. Ogura, H. Kosaka, M. Sugimoto, and K. Kasahara, “Effect of sidewall reflector on current versus light-output in a pnpn vertical to surface transmission electro-photonic device with a vertical cavity,” IEEE J. Quantum Electron. 29, 2006 (1993). 36. F. Stern and J. M. Woodall, “Photon recycling in semiconductor lasers,” J. Appl. Phys. 45, 3904 (1974). 37. T. Numai, “Analysis of a high density two-dimensional vertical-cavity surface emitting laser array,” Jpn. J. Appl. Phys. Pt. 1 36, 6393 (1997). 38. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004.


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9 SEMICONDUCTOR OPTICAL AMPLIFIERS

9.1

INTRODUCTION

If the injected current to LDs is just below the threshold current, LDs can operate as semiconductor optical amplifiers (SOAs). Fabry–Perot SOAs [1–10], DFB SOAs [11–24], and DBR SOAs [25–28] are resonant-type optical amplifiers which can selectively amplify light that satisfies the resonance condition of the optical cavities of SOAs. Fabry–Perot SOAs that have facets with negligibly low reflectivity, called traveling-wave optical amplifiers (TWAs) [29–48], can amplify light whose frequency or wavelength is located in the optical gain spectrum. In this chapter, signal gain, dependence on light polarization, and noises are explained using Fabry–Perot SOAs and TWAs.

9.2

SIGNAL GAIN

Rate Equation From (6.2) and (6.4), the rate equation for carrier concentration n in the active layer of an SOA is written dn J n = − g0 (n − n 0 )S − , dt ed τn

(9.1)


233


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SEMICONDUCTOR OPTICAL AMPLIFIERS

where J is the injected current density, e is the elementary electric charge, d is the active layer thickness, g0 is the differential gain coefficient, n 0 is the transparent carrier concentration, S is the average photon concentration, and τn is the carrier lifetime. Optical Power Gain Coefficient From (9.1), the carrier concentration n in a steady state is written n=

J/ed + g0 n 0 S . 1/τn + g0 S

(9.2)

From (9.2), (6.30), and (6.33), the power optical gain coefficient per unit length g is given by g=

1 J/ed − n 0 /τn 1 a g0 (n − n 0 ) − αi = a g0 − αi , vg vg 1/τn + g0 S

(9.3)

where vg = c/n r is the group velocity of light, a is the optical confinement factor of the active layer, and αi is the internal loss. From (9.3) it is found that g decreases with an increase in the average photon concentration S in an SOA. Signal Gain Signal gain, defined as the ratio of transmitted optical power Pt to incident optical power P0 , has been widely studied experimentally [1–10] and theoretically [29–55]. If the power optical gain coefficient per unit length g is uniform, the signal gain of a Fabry–Perot SOA is obtained by substituting (9.3) into (6.21). Figure 9.1 shows the signal gain 10 log10 Pt /P0 calculated for a Fabry–Perot SOA as a function of the light output power Pt . The parameters are the power reflectivity R of the facets and the maximum optical gain. The solid and dashed lines represent R = 0.04 × 10−2 and 31%, respectively. The values of the maximum optical gain are 15, 20, and 25 dB. With an increase in the light output power Pt , the signal gain decreases, because a large number of carriers are consumed for light amplification with an increase in the optical power in the SOA. The cutoff light output power Pc is defined as the light output power at which the signal gain decreases by 3 dB from the maximum signal gain. By comparing the results for R = 0.04 × 10−2 and 31%, Pc for R = 31% is lower than that for R = 0.04 × 10−2 % by about 20 dB, for several reasons. 1. With an increase in R, the light propagates repeatedly in the optical cavity and a larger number of carriers are consumed for light amplification. 2. The carrier concentration n to sustain the signal gain for R = 31% is lower than that for R = 0.04 × 10−2 % and the carrier lifetime τn for R = 31% is longer than that for R = 0.04 × 10−2 %.


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235

Signal Gain

SIGNAL GAIN

Light Output Power FIGURE 9.1 Calculated signal gain 10 log10 Pt /P0 of a Fabry–Perot SOA as a function of light output power Pt . The Parameters are the power reflectivity of the facets and the maximum optical gain.

3. The distribution of optical power in an SOA for R = 0.04 × 10−2 % is highly asymmetric; the optical power at the output facet is much higher than that at the input facet, as shown in Fig. 9.2(a), where facets are located at z = 0 and z = L. In contrast, the distribution of optical power in an SOA for R = 31% is relatively uniform, as shown in Fig. 9.2(b). As a result, the reduction of g by light is dominant only in the vicinity of the output facet for R = 0.04 × 10−2 %; reduction of g by the light takes place in the entire cavity for R = 31%. Therefore, g for R = 31% is reduced much more than that for R = 0.04 × 10−2 %.

Optical Power (arb. units)

Optical Power (arb. units)

Figure 9.3 shows signal gain spectra for TE modes where the maximum signal gain is 20 dB. The parameter is the power reflectivity R of the facets. The solid, dashed, dashed-dotted, and dotted lines represent R = 0.02, 0.2, 2, and 31%, respectively.

(a)

(b)

FIGURE 9.2 Distributions of optical power in an SOA for (a) R = 0.04 × 10−2 % and (b) R = 31%.


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Signal Gain

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Wavelength

Gain Ripple (dB)

FIGURE 9.3 Signal gain spectra for TE modes.

Signal Gain FIGURE 9.4 Gain ripple.

With an increase in R, the gain ripple, which is the ratio of the maximum signal gain to the minimum signal gain, increases. Figure 9.4 shows the gain ripple as a function of the signal gain 10 log10 Pt /P0 with the power reflectivity R of the facets as a parameter. It is found that the gain ripple increases with an increase in the signal gain 10 log10 Pt /P0 and the power reflectivity R.

Reduction in Facet Reflectivity To obtain a large cutoff light output power Pc and a low gain ripple, the power reflectivity R of the facets should be as low as possible. Figure 9.5 shows top views of facet structures to reduce the reflectivity of the facets: AR-coated facets [56], tilted facets [47], and window structures [57].


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237

SIGNAL GAIN

AR Film

AR Film

Tilted Facet

Tilted Facet

Window

Window

Optical Waveguide

Optical Waveguide

Optical Waveguide

(a)

(b)

(c)

FIGURE 9.5 Top views of facet structures to reduce the reflectivity of facets: (a) AR-coated facets; (b) tilted facets; (c) window structures.

When a plane wave propagating in a uniform material is normally incident upon a facet, the thickness d and refractive index n AR of the AR film are given by 2m − 1 , 4n AR √ n AR = n r ,

d=

(9.4) (9.5)

where m is a positive integer and n r is a refractive index of the material [58]. However, AR-coating conditions for guided modes in the optical waveguide of the SOAs are different from (9.4) and (9.5), because the guided modes consist of elementary plane waves with various wave vectors. To obtain AR-coating conditions for SOAs, the plane-wave angular spectrum of an incident field is derived from the Fourier transform of the incident field [56, 59, 60]. Figure 9.6 shows reflections of light at a tilted facet and a window structure from the viewpoint of geometrical optics. In tilted facets the light reflected at the facet goes to other than the optical waveguide, as shown in Fig. 9.6(a). As a result, the reflected light is not coupled to the optical waveguide, so the reflectivity of the facets is reduced. In window structures, the optical waveguide is not present near the facets; the light is diffracted at the interface of the window region and the optical waveguide, as shown in Fig. 9.6(b). As a result, a large part of the reflected light at the facet is not coupled to the optical waveguide. Tilted Facet

Facet

Window

Optical Waveguide

Optical Waveguide

Light Ray Light Ray (a)

(b)

FIGURE 9.6 Reflections at (a) a tilted facet and (b) a window structure.


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TE Mode

Signal Gain

TM Mode

Injected Current (mA) FIGURE 9.7 Signal gains for TE and TM modes.

9.3 POLARIZATION TE and TM Modes Figure 9.7 shows signal gains for TE and TM modes as a function of the current injected. The solid and dashed lines represent TE and TM modes, respectively. The signal gain for the TE mode is larger than that for the TM mode; the difference between the signal gain for the TE mode and that for the TM mode is enhanced with an increase in the injected current. The reason for this is that the optical confinement factor for the TE mode TE is larger than that for the TM mode TM [44], because the shape of an active layer is asymmetric, as shown in Fig. 9.8.

Resonant Wavelength The effective refractive index for the TE mode, n rTE , is larger than that for the TM mode, n rTM , because the shape of an active layer is asymmetric, as shown in Fig. 9.8. As a result, resonant wavelengths for the TE mode are different from those for the TM mode. Figure 9.9 shows a signal gain spectrum for TE and TM modes for R = 31%. Here the solid and dashed lines represent the signal gain for TE and TM modes, respectively.

Width Thickness

FIGURE 9.8 Cross-sectional view of an active layer.


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NOISES

TM Mode

Signal Gain

TE Mode

239

Wavelength FIGURE 9.9 Signal gain spectrum for TE and TM modes for R = 31%.

Polarization Insensitive Amplification As shown in Figs. 9.7 and 9.9, the signal gain of an SOA depends on the polarization of light. To suppress the dependence on light polarization, the difference between TE and TM should be as small as possible, which leads simultaneously to a small difference between n rTE and n rTM . For this purpose, one method is to reduce asymmetry in the shape of the active layer. The signal gain difference between the TE and TM modes when using a narrow stripe active layer was 1.3 dB [57]; when using thick active layers it was within 1 dB [61, 62]. Another method is to reduce asymmetry in the shape of the near-field pattern by modifying distribution of the refractive index of the optical waveguide in an SOA [63,64], with a resulting signal gain difference between the TE and TM modes of less than 1 dB [64]. To decrease the dependence of the optical gain coefficient on light polarization, strained QW active layers were used, and polarization sensitivity below 0.5 dB was obtained at a wavelength of 1.56 µm [65].

9.4

NOISES

Fundamental Equations Noises in SOAs are caused by spontaneous emission. From quantum mechanics, the spontaneous emission rate per mode is equal to the stimulated emission rate with one photon. As a result, the rate equation for the optical power of spontaneous emission per unit volume Psp is given by dPsp Psp = a g0 nω + a g0 (n − n 0 )Psp − , dt τph where ω is the angular frequency of spontaneously emitted light.

(9.6)


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On the right-hand side of (9.6), the first term, a g0 nω, shows a generation rate for the optical power of spontaneous emission per mode; the second term, a g0 (n − n 0 )Psp , shows an amplification rate for the optical power of spontaneous emission per mode; and the third term, −Psp /τph , shows a decay rate for the optical power of spontaneous emission per mode. From (9.6), dPsp (z)/dz is written Psp d 1 Psp (z) = a g0 nω + a g0 (n − n 0 )Psp − . dz vg τph

(9.7)

For the TWA, by solving (9.7) under the assumption of Psp (0) = 0, the optical power of spontaneous emission per unit volume at the output facet Psp (L) is obtained as Psp (L) =

a g0 nω exp(gsp L) − 1 = n sp G sp − 1 , a g0 (n − n 0 ) − 1/τph

(9.8)

where L is length of the optical cavity of the TWA and gsp =

1 1 a g0 (n − n 0 ) − , vg τph

G sp = exp(gsp L), a g0 nω n sp = . a g0 (n − n 0 ) − 1/τph

(9.9) (9.10) (9.11)

For FP SOAs, Psp (L) is given by Psp (L) = χ n sp G sp − 1 ,

(9.12)

where the excess noise coefficient χ is expressed as 1 + R1 G sp T2 χ= . √ √ (1 − G sp R1 R2 )2 + 4G sp R1 R2 sin2 (n r k0 L)

(9.13)

Here R1 and R2 are the power reflectivity of the input and output facets, respectively; T2 is the power transmissivity of the output facet; and k0 is a wave number in vacuum. Noise Characteristics The electric field E at the output facet of an SOA can be written E = E s exp [i (ωs t + θs )] +

n

(n) (n) (n) E sp exp i ωsp t + θsp .

(9.14)


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REFERENCES

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Here E s , ωs , and θs are the amplitude, angular frequency, and signal phase of a signal (n) (n) (n) light, respectively; and E sp , ωsp , and θsp are the amplitude, angular frequency, and signal phase of a spontaneously emitted light, respectively. In direct detection systems, the photocurrent i observed is given by ηe |E|2 ω

2

ηe (n) |E s |2 + =

E sp

ω n

(n) (n) (n) + 2 |E s | − ωs t + θsp − θs

E sp

cos ωsp

i=

n

(n) (m) (n) (m) (n) (m) +2 ,

E sp E sp cos ωsp − ωsp t + θsp − θsp

(9.15)

n<m

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(9.16)

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59. R. H. Clarke, “Theoretical performance of an anti-reflection coating for a diode laser amplifier,” Int. J. Electron. 53, 495 (1982). 60. G. Eisenstein, “Theoretical design of single layer antireflection coatings,” Bell Syst. Tech. J. 63, 357 (1984). 61. N. A. Olsson, R. F. Kazarinov, W. A. Nordland, C. H. Henry, M. G. Oberg, H. G. White, P. A. Garbinski, and A. Savage, “Polarisation-independent optical amplifier with buried facets,” Electron. Lett. 25, 1048 (1989). 62. B. Mersali, G. Gelly, A. Accard, J.-L. Lafragette, P. Doussiere, M. Lambert, and B. Fernier, “1.55 µm high-gain polarisation-insensitive semiconductor travelling wave amplifier with low driving current,” Electron. Lett. 26, 124 (1990). 63. T. Saitoh and T. Mukai, “Structural design for polarization-insensitive travelling-wave semiconductor laser amplifiers,” Opt. Quantum Electron. 21, S47 (1989). 64. S. Cole, D. M. Cooper, W. J. Devlin, A. D. Ellis, D. J. Elton, J. J. Isaac, G. Sherlock, P. C. Spurdens, and W.A. Stallard, “Polarisation-insensitive, near-travelling-wave semiconductor laser amplifiers at 1.5 µm,” Electron. Lett. 25, 314 (1989). 65. K. Magari, M. Okamoto, Y. Suzuki, K. Sato, Y. Noguchi, and O. Mikami, “Polarizationinsensitive optical amplifier with tensile-strained-barrier MQW structure,” IEEE J. Quantum Electron. 30, 695 (1994). 66. T. Mukai and Y. Yamamoto, “Noise characteristics of semiconductor laser amplifiers,” Electron. Lett. 17, 31 (1981). 67. T. Mukai and Y. Yamamoto, “Noise in an AlGaAs semiconductor laser amplifier,” IEEE J. Quantum Electron. 18, 564 (1982). 68. J. C. Simon, J. L. Favennec, and J. Charil, “Comparison of noise characteristics of Fabry– Perot-type and travelling-wave-type semiconductor laser amplifiers,” Electron. Lett. 19, 288 (1983). 69. J. Arnaud, J. Fesquet, F. Coste, and P. Sansoetti, “Spontaneous emission in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 21, 603 (1985). 70. R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21, 766 (1985). 71. N. A. Olsson, “Heterodyne gain and noise measurement of a 1.5 µm resonant semiconductor laser amplifier,” IEEE J. Quantum Electron. 22, 671 (1986). 72. T. Mukai and T. Saitoh, “5.2 dB noise figure in a 1.5 µm InGaAsP travelling-wave laser amplifier,” Electron. Lett. 23, 216 (1987).


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PART III ADVANCED LASER DIODES AND RELATED DEVICES

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10 PHASE-CONTROLLED DFB LASER DIODES

10.1

INTRODUCTION

The longitudinal modes of DFB LDs depend on the grating phases of facets [1, 2]. To stabilize longitudinal modes by controlling the grating phase of a facet due to the free carrier plasma effect, a phase-controlled (PC)-DFB LD [3] was developed. Kitamura et al. [3] describe a PC region that does not have corrugations but has an active layer. Therefore, with a change in the tuning current to the PC region, It , optical gain changes considerably, in addition to a change in the grating phase. As a result, a lasing mode jumps between TE−1 and TE+1 modes with a change in It . If a change in the optical gain of the PC region is small enough, a lasing wavelength can be tuned by It , owing to the free carrier plasma effect. Accordingly, a PC-DFB LD, which does not have an active layer in the PC region [4], was developed. In this chapter, theoretical analysis, device structure, and device characteristics of PC-DFB LDs are explained.

10.2

THEORETICAL ANALYSIS

Analytical Model Figure 10.1(a) shows an analytical model of a PC-DFB LD for the transfer matrix method. Here R1 and R2 are power reflectivities of the facets. The optical cavity is divided into two regions: has a PC region and a DFB region. The PC region does not Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright

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PC Region

DFB Region (a)

DFB Region (b)

FIGURE 10.1 Analytical model of a PC-DFB LD for the transfer matrix method.

have corrugations and a grating coupling coefficient κ1 = 0; the DFB region has an active layer and corrugations with a grating coupling coefficient κ2 = 0. If the PC region has neither optical gain nor optical loss, the analytical model is simplified as shown in Fig. 10.1(b), with phase at the left facet. Threshold Gain and Wavelength of a Mode Figure 10.2 shows the αth L calculated as a function of δ × L = δL with grating phase as a parameter. Here R1 = 0%, R2 = 31%, and κ L = 2 for the DFB region are assumed. When is 0, several longitudinal modes, denoted by closed circles, can lase. Among them, the longitudinal mode with the lowest threshold gain lases first with an increase in the current injected to the DFB region. When is π/2 or 3π/2, two longitudinal modes have the same lowest threshold gain; the two longitudinal modes lase simultaneously. By controlling by the current injected to the PC region due to the free carrier plasma effect, the lasing wavelength of a PC-DFB LD can be tuned. When is close to π/2 or 3π/2, the lasing mode jumps from a short-wavelength mode (TE+1 mode) to a long-wavelength mode (TE−1 mode), or vice versa.

FIGURE 10.2 αth L calculated as a function of δ × L = δL with grating phase as a parameter.


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DEVICE STRUCTURE

Anode 1

Anode 2

Laser Beam

251


Cathode PC Region

DFB Region

FIGURE 10.3 Structure of a PC-DFB LD.

10.3

DEVICE STRUCTURE

Structure Figure 10.3 shows the structure of a PC-DFB LD, which consists of DFB and PC regions. The DFB region is 250 µm long and the PC region is 130 µm long. The composition and thickness of the guiding layers in the PC region are the same as those in the DFB region. Note that the guiding layers are almost transparent for laser light. The DFB and PC regions were isolated from each other electrically by 20 µm-wide etched grooves, which were formed on both sides of the center mesa stripe area, and the isolation resistance was 280 . Both facets are as cleaved.

Fabrication Process Figure 10.4 shows a fabrication process for a PC-DFB LD. First, the first-order grating is formed partially on an n-InP substrate by the conventional holographic method, as shown in Fig. 10.4(a). On an n-InP substrate with a grating, an n-InGaAsP guiding layer (a bandgap wavelength λg = 1.29 µm, 0.16 µm thick, Sn doped at 5 × 1017 cm−3 ), an undoped InP etching-stop layer (0.08 µm thick), an undoped InGaAsP active layer (λg = 1.53 µm, 0.16 µm thick), an undoped anti-meltback layer (λg = 1.29 µm), and a p-InP cladding layer (0.2 µm thick, Zn doped at 1 × 1018 cm−3 ) are grown by liquid-phase epitaxy (LPE), as shown in Fig. 10.4(b). Then the cladding and active layers in the PC region are selectively wet etched; the wet etching is stopped at the InP etching-stop layer, as shown in Fig. 10.4(c). After etching, a p-InP cladding layer (1.0 µm thick, Zn doped at 1 × 1018 cm−3 ) is grown over the entire wafer, as shown in Fig. 10.4(d). A DC-PBH [5] was adopted for efficient current confinement and transverse-mode control. Finally, Ti/Au was partially formed as anodes and a cathode, as shown in Fig. 10.4(e).


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(a) Active Layer

(b)

(c)

(d) Anode 1

Anode 2

(e)

Cathode

FIGURE 10.4 Fabrication process for a PC-DFB LD.

10.4

DEVICE CHARACTERISTICS AND DISCUSSION

Wavelength Tuning Figure 10.5 shows lasing wavelength as a function of the tuning current It which was injected to the PC region. Filled and open circles represent single- and multiplelongitudinal-mode operations, respectively. The current injected into the DFB region Id was 55 mA, which corresponded to a light output power of 5 mW when the tuning current to the PC region It was not injected. With an increase in the tuning current It , the lasing wavelength changed periodically. In each period, the lasing wavelength shifted continuously toward a shorter wavelength, and then a lasing mode jumped to a longer wavelength. The continuous wavelength tuning range was 1 to 2 nm. Around the driving conditions where a longitudinal mode jump takes place, multimode oscillations were frequently observed. Note that the small negative value in the tuning current It was due to the leakage current from the DFB region to the PC region. When the tuning current in


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Wavelength


Tuning Current FIGURE 10.5 Lasing wavelength as a function of the tuning current.

the PC region It was changed to 100 mA, the threshold current changed from 19 mA to 24 mA.

FM Response

FM Efficiency (GHz/mA)

Figure 10.6 shows the FM response. The FM efficiency, which was measured using a Fabry–Perot interferometer, was plotted as a function of the modulation frequency. The current injected into the DFB region, Id , was fixed and the tuning current to the PC region, It , was modulated sinusoidally at the three bias levels It0 = 0, 4, and 10 mA. The FM response was flat up to 100 MHz, and the 3-dB-down cutoff

Modulation Frequency (MHz) FIGURE 10.6 FM response.


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Linewidth (MHz)

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Tuning Current FIGURE 10.7 Linewidth as a function of the tuning current.

frequency was about 200 MHz. These characteristics were independent of the biased current in the PC region It0 . A flat FM response is very important for FM light sources, to suppress waveform distortion in frequency-modulated optical signals [6]. Therefore, a PC-DFB LD is expected to be useful for transmission systems with relatively low transmission rates, such as less than several hundred Mbit/s. The cutoff frequency of 200 MHz was limited by carrier lifetime in the PC region. The FM efficiency, which was measured at 10 MHz, changed periodically from a few GHz/mA to 16 GHz/mA with an increase in the biased current in the PC region, It0 . Note that these values of the FM efficiency were more than 10 times as large as those for conventional DFB LDs, and a large FM efficiency leads to a small intensity modulation, which is favorable to FM light sources. In a PC-DFB LD, a parasitic intensity modulation for a frequency deviation of 1 GHz was less than 1%. Figure 10.7 shows a linewidth as a function of the tuning current in the PC region It , which was measured by a delayed self-homodyne detection method. The linewidth changed periodically and the minimum linewidth was 20 MHz. Although the linewidth was fairly large, it is possible to use this PC-DFB LD for FSK heterodyne envelope detection systems, because the requirement for the linewidth is not as difficult to attain [7]. The periodical change in the linewidth is probably due to a change in the threshold gain difference between the main mode and the submodes.

10.5

SUMMARY

The PC-DFB LD was explained briefly, and experimental study of FM response and linewidth was reviewed. With an increase in the tuning current in the PC region, It ,


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REFERENCES

255

light output, lasing wavelength, FM efficiency, and linewidth changed periodically. The continuous wavelength tuning range was over 1 to 2 nm. A high FM efficiency of more than 16 GHz/mA and a flat FM response up to 100 MHz were obtained. As a result, PC-DFB LDs are attractive as light sources for FSK heterodyne transmission systems.

REFERENCES 1. W. Streifer, R. D. Burnham, and D. R. Scifres, “Effect of external reflectors on longitudinal modes of distributed feedback lasers,” IEEE J. Quantum Electron. 11 154 (1975). 2. K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “Effect of mirror facets on lasing characteristics of distributed feedback InGaAsP/InP laser diode at 1.5 µm range,” IEEE J. Quantum Electron. 20 236 (1984). 3. M. Kitamura, M. Yamaguchi, K. Emura, I. Mito, and K. Kobayashi, “Lasing mode and spectral linewidth control in phase tunable DFB-DC-PBH LDs,” IEEE J. Quantum Electron. 21, 415 (1985). 4. S. Murata, I. Mito, and K. Kobayashi, “Frequency modulation and spectral characteristics for a 1.5 µm phase tunable DFB laser,” Electron. Lett. 23, 12 (1987). 5. I. Mito, M. Kitamura, K. Kobayashi, S. Murata, M. Seki, Y. Odagiri, H. Nishimoto, M. Yamaguchi, and K. Kobayashi, “InGaAsP double-channel-planar-buried-heterostructure laser diode (DC-PBH LD) with effective current confinement,” IEEE J. Lightwave Technol. 1, 195 (1983). 6. S. Yamazaki, K. Emura, M. Shikada, M. Yamaguchi, and I. Mito, “Realisation of flat FM response by directly modulating a phase tunable DFB laser diode,” Electron. Lett. 21, 283 (1985). 7. K. Emura, S. Yamazaki, S. Fujita, M. Shikada, I. Mito, and K. Minemura, “Over 300 km transmission experiment on an optical FSK heterodyne dual filter detection system,” Electron. Lett. 22, 1096 (1986).


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11 PHASE-SHIFT-CONTROLLED DFB LASER DIODES

11.1

INTRODUCTION

Wavelength-tunable LDs are very important for wavelength-division-multiplexed (WDM) lightwave transmission systems in the direct detection scheme and coherent lightwave transmission systems. Several wavelength-tunable LDs [1–6] that use injected current for tuning wavelength have been developed. However, with an increase in the current injected, spectral linewidths become several times as wide as those with no current injected [1–3]. The reason for this is that the optical gain difference between the main mode and the submodes decreases during a wavelength tuning operation [3]. For some LDs [4–6], spectral linewidth measurement in a tuning operation was not reported. As a wavelength-tunable LD which efficiently suppresses the intensity of submodes both below and above the threshold, a phase-shift-controlled (PSC) DFB LD [7, 8] was developed. In this chapter, characteristics of a PSC-DFB LD are explained. In a PSC-DFB LD, the optical gain difference between the main mode and the submodes is maintained large enough to obtain stable single-longitudinalmode laser operation as long as the wavelength is varied around the Bragg wavelength. As a result, an almost constant spectral linewidth from 24 to 29 MHz was achieved over a frequency (wavelength) tuning range as wide as 113 GHz (0.9 nm). The PSCDFB LD controls the light output and the oscillation wavelength independently by injected current; changes in the threshold optical gain and loss are small during wavelength tuning. As a result, an almost constant light output was also achieved.


257


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Fundamental Equations The oscillation wavelength is determined by the Bragg wavelength λB and the phaseshift of the grating [9]. Here λB is written λB = 2Na ,

(11.1)

where Na is the effective refractive index of the active region and is the grating pitch. Figure 11.1 shows a two- and a three-region model for analysis of a PSCDFB LD. Here θ1 , θ3 , and θ5 are grating phases at the left facets of each region. The PSC-DFB LD consists of three regions, and the phase-control (PC) region without a grating is placed between the two active regions. The active region has an active layer and a grating; the PC region has a guiding layer and does not have an active layer. The active region controls light output and the PC region controls the oscillation wavelength. By comparing two analytical models, an equation for the phase shift in a PSC-DFB LD will be derived in the following. From (4.46) and (4.47), the transfer

Region 1

Region 2

(a)

Region 1 Region 2 Region 3

(b) FIGURE 11.1 Analytical models of a PSC-DFB LD for the transfer matrix method: (a) two-region model; (b) three-region model. Here θ1 , θ3 , and θ5 are grating phases at the left facets of each region.


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259


matrix F i is written Fi =

F11 F21

F12 , F22

(11.2)

where α0 − i δ F11 = cosh (γ L i ) − sinh (γ L i ) exp ( i β0 L i ), γ iκ F12 = sinh (γ L i ) exp [−i (β0 L i + )], γ iκ F21 = − sinh (γ L i ) exp [ i (β0 L i + )], γ α0 − i δ F22 = cosh (γ L i ) + sinh (γ L i ) exp (−i β0 L i ). γ

(11.3)

Here L i is the length of a corrugated region. When the transfer matrixes of the two active regions are expressed as F a1 and F a2 , as shown in Fig. 11.1(a), the transfer matrix of the entire cavity F a is obtained as F a = F a1 × F a2 .

(11.4)

The grating phases on the left facets of two regions θ1 and θ3 are related by θ3 = θ1 + 2β0 L 1 + ,

(11.5)

where is the phase shift at the interface of the two regions. From the definition of a transfer matrix, the threshold optical gain and the oscillation wavelength of the cavity modes are given by solving a F11 = 0.

(11.6)

When F a1 equals F a2 , F a 11 is given by a F11

2 α0 − i δ (γ ) sinh L a exp ( i 2β0 L a ) γ 2 κ + sinh (γ L a ) exp (i 2β0 L a ) exp (i ) , γ

= cosh (γ L a ) −

(11.7)

where α is the field optical gain coefficient in the active region, L a is the length of each active region, and the total length of the active regions is 2L a .


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In the three-region model shown in Fig. 11.1(b), the transfer matrix of the entire cavity F b is obtained as F b = F b1 × F b2 × F b3 .

(11.8)

Here F b1 and F b3 are the transfer matrixes for each active region and F b2 is the transfer matrix for the PC region, which is written F b2

=

exp −γp + iβ0 L p 0

0

, exp − −γp + iβ0 L p

(11.9)

where γp is γ in the PC region and L p is the length of the PC region. When F b1 equals b F b3 , F11 is given by

b F11

2 α0 − i δ = cosh (γ L a ) − sinh (γ L a ) exp −γp L p exp i β0 2L a + L p γ 2 κ + sinh (γ L a ) exp −γp L p exp i β0 2L a + L p exp 2γp L p . γ (11.10)

By comparing (11.7) with (11.10), the equation for the phase shift in the three-region model, which is shown in Fig. 11.1(b), is derived as

= Im 2γp L p = 4π

Np Na − λB λ

4π Na − Np L p Lp . λB

(11.11)

Here Na is the effective refractive index of the active region, Np is the effective refractive index of the PC region, and |λ − λB | λB was used to obtain the final result. The effective refractive index Np of the PC region is decreased by the tuning current Ip injected into the PC region due to the free carrier plasma effect, and the phase shift is increased according to (11.11). It should be noted that the effective optical length Np L p in the PC region decreases with an increase in the tuning current injected into the PC region, Ip . Threshold Gain and Wavelength of a Mode Figure 11.2 shows the threshold optical gain αth as a function of the wavelength difference from the Bragg wavelength λ − λB with phase shift as a parameter. It is assumed that the power reflectivity of both facets is 0%, the grating coupling coefficient κ is 60 cm−1 , and the total length of the corrugated region L is 500 µm. With an increase in the phase shift , the wavelengths of the resonant modes shift toward the shorter-wavelength side. The maximum wavelength tuning range is slightly less than the stopband width at = 0 or 2π in order to maintain stable


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261

FIGURE 11.2 Threshold optical gain as a function of the wavelength difference from the Bragg wavelength with phase shift as a parameter. It is assumed that the power reflectivity of both facets is 0%, the grating coupling coefficient κ is 60 cm−1 , and the total length of the corrugated region L is 500 µm.

single-longitudinal-mode operation. As a result, the change in the phase shift that is required for wavelength tuning is 2π . The phase shift is modulated by changing the effective refractive index Np in the PC region. Because the effective refractive index Np is decreased by the tuning current Ip injected into the PC region due to the free carrier plasma effect, the phase shift is increased according to (11.11) and the oscillation wavelength shifts to the shorter-wavelength side, as shown in Fig. 11.2. When the optical gain difference between the main mode and the submodes decreases, the spectral linewidth broadens and two-mode oscillation occurs [3]. In the PSC-DFB LD, a large optical gain difference between the main mode and the submodes is maintained as long as the wavelength is varied around the Bragg wavelength. Therefore, a constant spectral linewidth is expected during wavelength tuning. Dependence of Phase Shift on Waveguide Structure and Tuning Current Figure 11.3 shows the dependence of the phase shift on the guiding layer thickness d with the tuning current injected into the PC region Ip as a parameter for L p = 50 µm and 100 µm. Here the width of the waveguide is assumed to be 2 µm. It is found that there is a d value where the largest phase-shift change is obtained. The reason for this is as follows: When the guiding layer thickness d increases, the contribution of the refractive index of the guiding layer N to the effective refractive index Np in the PC region increases, because confinement of light to the guiding layer increases. As a result, a change in N by the tuning current injected to the PC region Ip has a large effect on a change in Np . However, when the guiding layer thickness d increases, the carrier concentration decreases and a change in N decreases. Therefore, there is a d value related to obtaining the largest phase shift. As shown in Fig. 11.3, the largest phase shift is obtained around d = 0.3 to 0.4 µm. When d increases further, the grating coupling coefficient κ decreases, leading to a narrower stopband.


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Phase-shift

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Phase-shift

Guiding Layer Thickness (a)

Guiding Layer Thickness (b) FIGURE 11.3 Dependence of phase shift on guiding layer thickness d with the tuning current injected into the PC region Ip as a parameter. L p = (a) 50 µm, (b) 100 µm.

To obtain a large wavelength tuning range, which is determined by the stopband width, a large κ value is needed [10], which requires a thin guiding layer. From Fig. 11.3 it is found that the phase shift for L p = 100 µm is larger than that for L p = 50 µm. However, the phase shift does not change in proportion to the PC region length L p . The reason for this is as follows: The change in phase shift depends linearly on the carrier concentration and L p . In Fig. 11.3 the parameter is not the carrier concentration but the tuning current injected into the PC region Ip . The carrier concentration decreases with an increase in L p at a constant injected current. As a result, the phase shift in Fig. 11.3 is not proportional to L p . Figure 11.4 shows the phase shift as a function of the tuning current injected into the PC region Ip for L p = 100 µm. It is expected that the phase shift becomes larger than 2π even when d is less than 0.2 µm. 11.3

DEVICE STRUCTURE

Figure 11.5 shows the structure of a PSC-DFB LD. The PC region is located between the two active regions. The PC region does not have an active layer, in order to achieve


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Phase-shift

DEVICE STRUCTURE

Tuning Current FIGURE 11.4 Phase shift as a function of the tuning current injected into the PC region Ip .

Active Layer Anode Laser Beam

Laser Beam SiN

Active Region

SiN Cathode PC Region Active Region

FIGURE 11.5 Structure of a PSC-DFB LD.

a large change in the effective refractive index in the PC region. Each active region is 200-µm long, the PC region is 100 µm long, and the total device length is 500 µm. Each region is isolated electrically from the others by 20-µm-wide etched grooves, which are formed on both sides of the center mesa stripe area. The isolation resistance is 100 . The first-order grating pitch is 238 nm. The corrugation depth after LPE is 30 to 40 nm. The guiding layer is 0.16 µm thick. Both facets are AR-coated with SiN. Residual power reflectivity is several percent. The driving current injected to the active region Ia is injected by connecting the two active region electrodes. Fabrication Process Figure 11.6 shows a fabrication process for a PSC-DFB LD. First, the first-order grating is partially formed on an n-InP substrate by the conventional holographic method, as shown in Fig. 11.6(a). On an n-InP substrate with a grating, an n-InGaAsP guiding layer (a bandgap wavelength λg = 1.29 µm, 0.16 µm thick, Sn doped at 5 × 1017 cm−3 ), an undoped InP etching-stop layer (0.08 µm thick), an undoped InGaAsP active layer (λg = 1.53 µm, 0.16 µm thick), an undoped anti-meltback layer


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(a) Active Layer

(b)

(c)

(d) Anode 1

Anode 2 Anode 3

(e)

Cathode

FIGURE 11.6 Fabrication process for a PSC-DFB LD.

(λg = 1.29 µm), and a p-InP cladding layer (0.2 µm thick, Zn doped at 1 × 1018 cm−3 ) are grown by LPE, as shown in Fig. 11.6(b). Then the cladding and active layers in the PC region are selectively wet etched; the wet etching is stopped at the InP etching-stop layer, as shown in Fig. 11.6(c). After etching, a p-InP cladding layer (1.0 µm thick, Zn doped at 1 × 1018 cm−3 ) is grown over the entire wafer, as shown in Fig. 11.6(d). A DC-PBH [11] was adopted for efficient current confinement and transverse-mode control. Finally, Ti/Au was partially formed as anodes and a cathode, as shown in Fig. 11.6(e). Figure 11.7 shows a scanning electron micrograph of the cross-sectional view of the interface between the active region and the PC region. It is found that the active and PC regions are smoothly connected.

11.4


Figure 11.8 shows the light output P as a function of the driving current injected to the active region Ia and the tuning current injected to the PC region Ip . The threshold


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PC Region

265

Active Region

FIGURE 11.7 Scanning electron micrograph of the cross-sectional view of the interface between the active and PC regions.

Light Output

current was 28 mA at 20◦ C, when the electric current was not injected the PC region. When the current injected into the PC section Ip increased up to 50 mA with the active region current fixed, a decrease in the light output is less than 8%, as shown in Fig. 11.8(b). The reasons for changes in the light output are that the optical loss in the PC region increases due to free carrier absorption, and the threshold optical gain changes when the carriers are injected. The negative value for Ip which appears in Fig. 11.8(b) reflects a leakage current from the active regions to the PC region. Figure 11.9 shows light output spectra that were measured at Ip = −5.3 mA and Ip = 50 mA. The light output is 2 mW. These spectra show stable single-longitudinalmode operations with a submode suppression ratio (SMSR) larger than 35 dB, and a frequency (wavelength) tuning range of 113 GHz (0.9 nm) is obtained.

(a)

(b)

FIGURE 11.8 Light output P as a function of the driving current injected to the active region, Ia , and the tuning current injected to the PC region, Ip .


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Light Intensity

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Light Intensity

Wavelength (nm) (a)

Wavelength (nm) (b) FIGURE 11.9 Light output spectra measured at (a) Ip = −5.3 mA and (b) Ip = 50 mA. The light output P is 2 mW.

Figure 11.10 shows a wavelength shift as a function of the tuning current injected into the PC region, Ip , at the light output P = 1, 2, and 5 mW. The wavelength shifted continuously toward the shorter-wavelength side, maintaining a stable singlelongitudinal-mode operation with an increase in Ip . The maximum wavelength shifts are 0.93 nm at P = 1 mW, 0.9 nm at P = 2 mW, and 0.68 nm at P = 5 mW. The maximum wavelength shifts decrease with an increase in the light output P. The reason for this is considered to be heating in the PC region. When the light output P is high, the driving current injected into the active regions Ia is large, and the active regions are heated. The temperature of the active regions is transmitted to the PC region, and the temperature in the PC region increases. As a result, a decrease in Np due to the free carrier plasma effect is canceled out by an increase in Np due to the heating in the PC region, leading to suppression of a change in the phase shift . Figure 11.11 shows SMSR as a function of the tuning current injected into the PC region, Ip , with light output as a parameter. During wavelength tuning by controlling Ip , high SMSR is maintained. The reason for this is that a large optical gain difference is achieved between the main mode and the submodes. Figure 11.12 shows the spectral linewidth as a function of the tuning current injected into the PC region, Ip , at the light output P = 1, 2, and 5 mW. Spectral


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267

Wavelength Shift (nm)


Tuning Current

SMSR (dB)

FIGURE 11.10 Wavelength shift as a function of the tuning current injected into the PC region, Ip .

Tuning Current

Spectral Linewidth (MHz)

FIGURE 11.11 Submode suppression ratio as a function of the tuning current injected into the PC region, Ip .

Tuning Current FIGURE 11.12 Spectral linewidth as a function of the tuning current injected into the PC region, Ip .


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Spectral Linewidth (MHz)

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FIGURE 11.13 Spectral linewidth as a function of the reciprocal light output P −1 .

linewidth measurements were carried out by a delayed self-homodyne detection method. At P = 2 mW, an almost constant spectral linewidth was obtained from 24 to 29 MHz over a frequency (wavelength) tuning range as wide as 113 GHz (0.9 nm). The linewidth change in tuning operation is very small, as expected. Figure 11.13 shows the spectral linewidth as a function of the reciprocal light output P −1 with the tuning current injected into the PC region, Ip , as a parameter. When the light output P increases up to 5 mW, the spectral linewidth decreases. When the light output P becomes larger than 5 mW, the spectral linewidth broadens. Figure 11.14 shows time-resolved light output spectra. In this measurement, the driving current injected into the active region, Ia , was fixed at 44 mA, and the tuning current injected into the PC region, Ip , was modulated from −5 mA to 27 mA by an electric pulse with a rise time of 400 ps. Here the negative value of Ip = −5 mA is a leakage current from the active regions to the PC region. Wavelength switching with a wavelength separation of λ1 − λ2 = 0.76 nm was obtained with a switching time

FIGURE 11.14 Measured time-resolved light output spectra.


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REFERENCES

269

as short as 2.5 ns. This characteristic allowed us to use the PSC-DFB LD as a light source for a 32-channel time-division (TD) and wavelength-division (WD) hybrid photonic switching system with a TD multiplexity of 4 and a WD multiplexity of 8, and this TD & WD hybrid system was demonstrated successfully [12]. This TD & WD hybrid photonic switching system is described in Chapter 17. In a PSC-DFB LD, the stopband width was 1.2 nm. However, the maximum wavelength tuning range was limited to 0.93 nm. The reason for this is considered to be heating in the PC region. When the tuning current injected to the PC region Ip increases, the PC region is heated. As a result, the decrease in Np due to the free carrier plasma effect is canceled out by the increase in Np due to heating in the PC region. The spectral linewidth measured was rather wide. However, an increase in the device length will decrease the spectral linewidth. Because the PC region is relatively long in a PSC-DFB LD, we can expect that the spatial hole burning effect [13] is small and that a large κ value and a long device length will lead to a large wavelength tuning range with a narrow spectral linewidth.

11.5

SUMMARY

In a PSC-DFB LD, the light output and oscillation wavelength are controlled independently by injected current, and the submodes are efficiently suppressed. In addition, changes in threshold optical gain and loss are small during wavelength tuning. As a result, an almost constant spectral linewidth and an almost constant light output were achieved during wavelength tuning. At the light output of 2 mW, an almost constant spectral linewidth from 24 to 29 MHz over a frequency (wavelength) tuning range as wide as 113 GHz (0.9 nm) was obtained. We can expect that a larger grating coupling coefficient, a larger guiding layer thickness, and a longer device length will lead to a wider wavelength tuning range with a narrower spectral linewidth. Wavelength switching with a wavelength separation of 0.76 nm was also achieved with a switching time as short as 2.5 ns.

REFERENCES 1. S. Murata, I. Mito, and K. Kobayashi, “Over 720 GHz (5.8 nm) frequency tuning by a 1.5 µm DBR laser with phase and Bragg wavelength control regions,” Electron. Lett. 23, 403 (1987). 2. M. Kitamura, M. Yamaguchi, K. Emura, I. Mito, and K. Kobayashi, “Lasing mode and spectral linewidth control in phase tunable DFB-DC-PBH LDs,” IEEE J. Quantum Electron. 21, 415 (1985). 3. S. Murata, I. Mito, and K. Kobayashi, “Frequency modulation and spectral characteristics for a 1.5 µm phase tunable DFB laser,” Electron. Lett. 23, 12 (1987). 4. Y. Kotaki, M. Matsuda, M. Yano, H. Ishikawa, and H. Imai, “1.55 µm wavelength tunable FBH DBR laser,” Electron. Lett. 23, 325 (1987).


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5. Y. Yoshikuni, K. Oe, G. Motosugi, and T. Matsuoka, “Broad wavelength tuning under single mode oscillation with a multi electrode distributed feedback laser,” Electron. Lett. 22, 1153 (1986). 6. Y. Yoshikuni and G. Motosugi, “Multielectrode distributed feedback laser for pure frequency modulation and chirping suppressed amplitude modulation,” IEEE J. Lightwave Technol. 5, 516 (1987). 7. T. Numai, S. Murata, and I. Mito, “1.5 µm wavelength tunable phase-shift controlled distributed feedback laser diode with constant spectral linewidth in tuning operation,” Electron. Lett. 24, 1526 (1988). 8. T. Numai, “1.5-µm wavelength tunable phase-shift-controlled distributed feedback laser,” IEEE J. Lightwave Technol. 10, 199 (1992). 9. K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4 shifted InGaAsP/InP DFB lasers,” IEEE J. Quantum Electron. 22, 1042 (1986). 10. T. Numai, S. Murata, and I. Mito, “Tunable wavelength filters using λ/4-shifted waveguide grating resonators,” Appl. Phys. Lett. 53, 83 (1988). 11. I. Mito, M. Kitamura, K. Kobayashi, S. Murata, M. Seki, Y. Odagiri, H. Nishimoto, M. Yamaguchi, and K. Kobayashi, “InGaAsP double-channel-planar-buried-heterostructure laser diode (DC-PBH LD) with effective current confinement,” IEEE J. Lightwave Technol. 1, 195 (1983). 12. M. Nishio, S. Suzuki, N. Shimosaka, T. Numai, T. Miyakawa, M. Fujiwara, and M. Itoh, “An experiment on photonic wavelength-division and time-division hybrid switching,” in Proc. 2nd Topical Meeting on Photonic Switching, ThE5, 98 (1989). 13. H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, and S. Yamakoshi, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB-laser,” IEEE J. Quantum Electron. 23, 804 (1987).


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12 PHASE-CONTROLLED DFB LASER FILTER

12.1

INTRODUCTION

Wavelength-tunable optical filters, which select one signal from wavelength-divisionmultiplexed (WDM) input signals, are key devices for WDM lightwave transmission and wavelength-division (WD) photonic switching systems in the direct detection scheme. Semiconductor wavelength-tunable optical filters are especially suitable for monolithic integration with semiconductor lasers, detectors, and switches. In addition, semiconductor wavelength-tunable optical filters, which use diffraction gratings, can easily be integrated with semiconductor optical devices, because they achieve narrow bandwidth wavelength selection without facets. When DFB LDs are biased just below the threshold current, they can be used as active DFB filters [1, 2], which selectively amplify incident light. Active DFB filters have the advantages of high optical gain and narrow bandwidth; they have the disadvantages of simultaneous changes in bandwidth and transmissivity during wavelength tuning [1, 2]. Magari et al. solved this problem by using a multielectrode DFB filter [3, 4] and a frequency tuning range of 33.3 GHz with a constant optical gain, and a constant bandwidth was obtained by controlling electric current injected into the two electrodes. In this multielectrode DFB filter, each electric current affects the optical gain and transmission wavelength simultaneously. To achieve a wider tuning range, which is required to increase the number of channels, a 1.5-µm phasecontrolled (PC) DFB filter was demonstrated [5, 6].


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Active Layer

Anode

Anode

Input Laser Beam

Output Laser Beam Cathode Active Region PC Region

FIGURE 12.1 Structure of a PC-DFB filter.

The purpose of this chapter is to explain the 1.5-µm PC-DFB filter, which is the first optical filter that controls the transmissivity (optical gain) and transmission wavelength independently. A frequency (wavelength) tuning range as wide as 43 GHz (0.34 nm) with a high constant optical gain of 27 dB was achieved. A five-channel wavelength selection with less than −10 dB crosstalk is expected with this filter. This device also operates as a wavelength-tunable laser [7], and the wavelength tuning range of the PC-DFB LD is larger than that of the PC-DFB filter. The reason for this is examined and it is shown that suppression of the submodes is important to expand the wavelength tuning range. 12.2

DEVICE STRUCTURE

Figure 12.1 shows the structure of a PC-DFB filter. The filter consists of two regions: an active region and a PC region. The PC region does not have an active layer. This filter is used by biasing at slightly below the threshold current. In this filter, the optical gain and transmission wavelength are controlled independently. The optical gain is controlled by the driving current injected into the active region, Ia ; the transmission wavelength is controlled by the tuning current injected into the PC region, Ip . Although the tuning current injected into the PC region, Ip , causes a slight change in the optical gain, the change in optical gain is compensated almost independent of the transmission wavelength by the driving current injected into the active region Ia . The active region is 350 µm long and the PC region is 250 µm long. Each region is isolated electrically from the others by 20-µm-wide etched grooves formed on both sides of the center mesa stripe area. The isolation resistance is 500 . The first-order grating pitch is 238 nm. The corrugation depth after LPE is 30 to 40 nm and κ L is 1. Both facets are as cleaved. 12.3


Figure 12.2 shows an experimental setup for the measurement of transmission spectra. The light source was a wavelength-tunable DBR LD [8]. Light was coupled in and out


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273

Polariazation Controller SMF LD 1 3 dB Coupler

PC-DFB Filter

LD 2

Polariazation Controller

Optical Spectrum Analyzer

Power Meter

FIGURE 12.2 Experimental setup for transmission spectra measurement.

Optical Gain (dB)

with hemispherically ended single-mode fibers. Optical coupling loss between the filter and the single-mode fiber (SMF) for each facet was about 6 dB. A polarization controller was placed in front of the filter to obtain TE-polarized light input, because the transmissivity for TE-polarized light is the largest. The filter characteristics depend on the polarization of the light input; the transmission wavelength and transmissivity for TE-polarized light input are different from those for TM-polarized light input. Figure 12.3 shows the transmission spectra of a PC-DFB filter. The driving current injected to the active region, Ia , was 0.98 times the threshold current. This device lased at a threshold current of 23 mA at 20◦ C when the electric current was not injected to the PC region. Parameters are the electric current injected into the active region, Ia , and that to the PC region, Ip . The open circles, filled circles, and open triangles correspond to Ia , Ip = (22.5 mA, −2.6 mA), (21.6 mA, 1.6 mA),

Wavelength (nm) FIGURE 12.3 Transmission spectra at a light input power of −40 dBm.


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Wavelength (nm)

Optical Gain

Tuning Current FIGURE 12.4 Dependence of transmission wavelength on the tuning current injected into the PC region.

and (22.1 mA, 2.9 mA), respectively. Input optical power was maintained at a constant level of −40 dBm. A frequency (wavelength) tuning range as wide as 43 GHz (0.34 nm) was achieved while maintaining a constant optical gain of 27 dB. In the frequency (wavelength) tuning operation, the full width at half maximum (FWHM) of the transmission spectra is 5 GHz (0.04 nm). The noise level, which is the spontaneous emission level at the transmission peak wavelength, was lower than the output signal level by at least 20 dB. The negative value for Ip , which appears in Fig. 12.3, is a leakage current into the PC region from the active region. Figure 12.4 shows the dependence of the transmission wavelength on the tuning current injected into the PC region Ip with optical gain as a parameter. Measurements were carried out for three optical gain levels, which were controlled by the driving current injected into the active region Ia . The transmission wavelength shifted toward the shorter-wavelength side with an increase in Ip . The transmission wavelengths were determined primarily by Ip , and a difference in the frequency (wavelength) was at most 5 GHz (0.04 nm) for an optical gain of 14.5 to 27 dB. Spectrum selectivity is shown in Figure 12.5. Two-channel laser light was incident on the filter. Each optical power of the incident laser light Pin was −37 dBm. The two laser lights were not modulated at all. The wavelength of one laser light, λ1 , was set at the center peak transmission wavelength of 1544.45 nm, which is shown in Fig. 12.3, and the wavelength of the other light input was varied. Crosstalk between the two laser lights decreased with an increase in frequency separation. Crosstalk of less than −10 dB was obtained when the frequency separation between the two laser lights exceeded 9 GHz. Because the frequency tuning range was 43 GHz, the PC-DFB filter is expected to permit five-channel signal separation. Figure 12.6 shows the dependence of the light output power on the light input power. Figure 12.7 shows the dependence of the optical gain on the light input power; Figure 12.8 shows the dependence of the transmission bandwidth (FWHM) on the light input power. With an increase in the light input power, the light output power increased, as shown in Fig. 12.6. However, with an increase in the light input


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Crosstalk (dB)


Frequency Separation (GHz)

Output Power (dBm)

FIGURE 12.5 Crosstalk as a function of frequency separation between two input laser lights where λ1 = 1544.45 nm.

Bias Level

Input Power (dBm) FIGURE 12.6 Dependence of light output power on light input power.

Optical Gain (dB)

Bias Level

Input Power (dBm) FIGURE 12.7 Dependence of optical gain on light input power.


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Bandwidth (GHz)

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Input Power (dBm) FIGURE 12.8 Dependence of transmission bandwidth on light input power.

power, the optical gain decreases and the bandwidth broadens due to optical nonlinear effect, as shown in Figs. 12.7 and 12.8. From these experimental results, the light input power level for the PC-DFB filter should be lower than −30 dBm to achieve fivechannel selection. At an input power level above −30 dBm, the bandwidth broadens dramatically with an increase in the light input power, and the wavelength multiplexy decreases. Let us consider frequency (wavelength) tuning ranges of a PC-DFB LD and a PC-DFB filter. In a PC-DFB-LD, the main mode shifted by 125 GHz (1 nm) and then the main mode jumped. From this experimental result, the largest frequency (wavelength) shift of the PC-DFB LD was 125 GHz (1 nm). However, the frequency (wavelength) tuning range of a PC-DFB filter was limited to 43 GHz (0.34 nm). The reason for this is as follows: When Ip was 2.9 mA, there was a small transmission peak for a submode at the longer-wavelength side of the main mode, as shown in Fig. 12.3. With a further increase in Ip , this small transmission peak was enhanced and the PC-DFB filter selected two wavelength signals. What is the difference between laser operation and filter operation? In laser operation the main mode is large and the submodes are efficiently suppressed because coupling between the main mode and the submodes is large and the optical gain is concentrated in the main mode. However, in filter operation, where the laser is biased below the threshold current, the difference in the optical gain between the main mode and the submodes is smaller than that for laser operation. As a result, the frequency (wavelength) tuning range for an optical filter is smaller than that for a laser. Therefore, to design an active wavelength-tunable optical filter, we have to consider a structure where a difference in the optical gain between the main mode and submodes is large enough even when the device is biased below the threshold current.

12.4

SUMMARY

A PC-DFB filter has the advantages of high optical gain and narrow bandwidth. A frequency (wavelength) tuning of 43 GHz (0.34 nm) was achieved with a high


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REFERENCES

277

constant optical gain of 27 dB and a narrow constant bandwidth of 9 GHz at 10 dB down by controlling the transmissivity (optical gain) and the transmission wavelength independently. A five-channel wavelength selection with crosstalk below −10 dB is expected with a PC-DFB filter. The frequency (wavelength) tuning range of a PC-DFB filter is smaller than that of a PC-DFB LD. The reason for this is that suppression of the submodes is more crucial for optical filters than for LDs. As a result, it is important to suppress the intensity of the submodes in order to expand the frequency (wavelength) tuning range.

REFERENCES 1. T. Numai, M. Fujiwara, N. Shimosaka, K. Kaede, M. Nishio, S. Suzuki, and I. Mito, “1.5 µm λ/4-shifted DFB LD filter and 100Mbit/s two-channel wavelength signal switching,” Electron. Lett. 24, 236 (1988). 2. H. Kawaguchi, K. Magari, K. Oe, Y. Noguchi, Y. Nakano, and G. Motosugi, “Optical frequency-selective amplification in a distributed feedback type semiconductor laser amplifier,” Appl. Phys. Lett. 50, 66 (1987). 3. K. Magari, H. Kawaguchi, K. Oe, Y. Nakano, and M. Fukuda, “Optical signal selection with a constant gain and a optical gain bandwidth by a multielectrode distributed feedback laser amplifier,” Appl. Phys. Lett. 51, 1974 (1987). 4. K. Magari, H. Kawaguchi, K. Oe, and M. Fukuda, “Optical narrow-band filters using optical amplification with distributed feedback,” IEEE J. Quantum Electron. 24, 2178 (1988). 5. T. Numai, S. Murata, and I. Mito, “1.5 µm tunable wavelength filter with wide tuning range and high constant gain using a phase-controlled distributed feedback laser diode,” Appl. Phys. Lett. 53, 1168 (1988). 6. T. Numai, “1.5 µm phase-controlled distributed feedback wavelength tunable optical filter,” IEEE J. Quantum Electron. 28, 1508 (1992). 7. S. Murata, I. Mito, and K. Kobayashi, “Frequency modulation and spectral characteristics for a 1.5 µm phase-tunable DFB laser,” Electron. Lett. 23, 12 (1987). 8. S. Murata, I. Mito, and K. Kobayashi, “Over 720 GHz (5.8 nm) frequency tuning by a 1.5 µm DBR laser with phase and Bragg wavelength control regions,” Electron. Lett. 23, 403 (1987).


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13 PHASE-SHIFT-CONTROLLED DFB FILTER

13.1

INTRODUCTION

The PC-DFB filter [1, 2], the subject of Chapter 12, was the first optical filter to control transmissivity (optical gain) and transmission wavelength independently. In a PC-DFB filter [1, 2] and DBR optical filters [3–6], the wavelength tuning range was limited by the transmission at a submode. From these results it is considered that suppression of the submodes and expansion of the separation between the main mode and submodes are important in expanding the wavelength tuning range. The PSC-DFB LD [7,8], the subject of Chapter 11, satisfies these requirements and can be used as a wavelength-tunable optical filter when it is biased just below the threshold current. In a PSC-DFB filter [9, 10], the optical gain difference between the main mode and the submodes is maintained large enough to suppress the submodes as long as the wavelength is tuned around the Bragg wavelength. As a result, we can expect a large wavelength tuning range. A PSC-DFB filter controls transmissivity (optical gain) and transmission wavelength independently by injected current; changes in the threshold optical gain and loss are small during frequency (wavelength) tuning. Therefore, a frequency (wavelength) tuning range as wide as 120 GHz (0.95 nm) with constant optical gain and constant bandwidth was achieved. The optical gain is as high as 24.5 dB; the 10-dB-down bandwidth is 12 to 13 GHz during frequency (wavelength) tuning. Eighteen-channel wavelength selection with less than −10 dB crosstalk is


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expected with this filter. The effect of the sidelobe light input on the transmission spectrum is also examined.

13.2


Calculation Transmission Spectra

Optical Gain (dB)

In a filter operation, a PSC-DFB filter is biased just below the threshold current and the transmissivity (optical gain) of the main mode is the largest; the transmissivity (optical gain) of the submodes is smaller than that of the main mode. As a result, the wavelength of the main mode is used as a transmission wavelength in the filter operation. In a PSC-DFB filter, the optical gain difference between the main mode and the submodes is maintained large enough to suppress the submodes as long as the wavelength is varied around the Bragg wavelength. Figure 13.1 shows transmission spectra calculated with the phase shift as a parameter. Here a grating coupling coefficient κ of 60 cm−1 , a cavity length L of 500 µm, a field optical gain coefficient of 0.98 times that of the threshold, and both facet reflectivities of 0% were assumed. The solid, dashed-dotted, dashed, and dashed-double-dotted lines represent the transmission spectra for = 0 (2π ), π/2, π , and 3π/2, respectively. With an increase in the tuning current Ip injected into the PC region, the effective refractive index Np in the PC region decreases due to the free carrier plasma effect, and increases. With an increase in the phase shift , the wavelength of the main mode shifts toward the shorter wavelength, as shown in Fig. 13.1. As a result, with an increase in Ip , the transmission wavelength shifts to a shorter wavelength. The maximum wavelength tuning range is limited by the

FIGURE 13.1 Calculated transmission spectra with phase shift as a parameter. Here facet reflectivities of 0% were assumed, together with κ = 60 cm−1 , L = 500 µm, and α = 0.98αth , where α is the field optical gain coefficient and αth is the threshold field optical gain coefficient.


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281

Bandwidth (GHz)


Phase Shift FIGURE 13.2 Calculated 10-dB-down bandwidth as a function of phase shift .

Stopband Width (nm)

stopband width at = 0 or 2π in order to select one wavelength signal. It is found that the submodes are efficiently suppressed and that one wavelength signal can be selected when the wavelength is tuned within the stopband. Figure 13.2 shows the 10-dB-down bandwidth calculated as a function of the phase shift . The parameters used for the calculation are the same as those in Fig. 13.1. When the wavelength is tuned within the stopband, a nearly constant bandwidth is obtained. As shown in Figs. 13.1 and 13.2, constant optical gain and constant bandwidth wavelength tuning are expected. Figure 13.3 shows the stopband width calculated as a function of grating coupling coefficient κ with device length L as a parameter. A large grating coupling coefficient κ and a short device length L lead to a large stopband width (i.e., a wide wavelength tuning range).

Grating Coupling Coefficient FIGURE 13.3 Stopband width calculated as a function of grating coupling coefficient κ with filter length L as a parameter.


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Active Layer Anode Input Laser Beam

Output Laser Beam

SiN Active Region

SiN Cathode PC Region Active Region

FIGURE 13.4 Structure of a PSC-DFB filter.

13.3

DEVICE STRUCTURE

Figure 13.4 shows the structure of a PSC-DFB filter. The PC region is located between two active regions. The PC region does not have an active layer in order to achieve a large change in the effective refractive index in the PC region. Each active region is 200 µm long, the PC region is 100 µm long, and the total device length is 500 µm. Each region is isolated electrically from the others by 20-µm-wide etched grooves formed on both sides of the center mesa stripe area. The isolation resistance is 100 . The first-order grating pitch is 238 nm. The corrugation depth after LPE is 30 to 40 nm. The thickness of the guiding layer is 0.16 µm. A DC-PBH was adopted for efficient current confinement and transverse-mode control. Ti/Au was used as a metal contact for both the p- and n-sides. Both facets are AR-coated with SiN. Residual power reflectivities are several percentage points. The anodes for the two active regions were connected. A scanning electron micrograph of a PSC-DFB filter is shown in Fig. 13.5.

Active Region

PC Region FIGURE 13.5 Scanning electron micrograph of a PSC-DFB filter.


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Optical Gain (dB)


Wavelength (nm) FIGURE 13.6 Transmission spectra measured. The light input power is −40 dBm.

13.4


The experimental setup for transmission measurement is as follows: The light sources were a wavelength-tunable DBR LD [11] and a phase-shifted DFB LD [12]. Light was coupled in and out with hemispherically ended single-mode fibers. Optical coupling loss between the filter and the fiber for each facet was about 6 dB. A polarization controller was placed in front of the filter to obtain TE-polarized input light. Figure 13.6 shows the transmission spectra measured. The driving current Ia injected to the active region was 0.98 times the threshold current. This device lased at a threshold current of 26 mA at 20◦ C when the electric current was not injected into the PC region. The stopband width was 1.2 nm. The parameters were the driving current Ia injected into the active region and tuning current Ip injected into the PC region. The open circles, filled circles, open triangles, and filled triangles represent the experimental results for (Ia , Ip ) = (24.1 mA, 4.0 mA), (25.7 mA, 8.0 mA), (28.1 mA, 11.0 mA), and (31.4 mA, 14.8 mA), respectively. Light input power was maintained at a constant level of −40 dBm. A frequency (wavelength) tuning range as wide as 120 GHz (0.95 nm) was achieved while maintaining a constant optical gain of 24.5 dB. The transmission wavelength shifted toward the shorter-wavelength side with an increase in Ip . A nearly constant 10-dB-down bandwidth (full width at the level below the transmission peak by −10 dB) of 12 to 13 GHz was obtained during wavelength tuning. The filter noise level, which is a spontaneous emission level at the peak optical gain wavelength, was lower than the output level by at least 15 dB. Figure 13.7 shows the crosstalk measured between two light inputs as a function of frequency separation. Here crosstalk is defined as an extinction ratio between the light intensity at the resonant wavelength and that at an off-resonant wavelength. In this experiment, one wavelength λ1 was set at a peak transmission wavelength of 1553.57 nm, which is a resonant wavelength in Fig. 13.6, and the other wavelength was varied. The optical input power was −40 dBm and the light intensities were not modulated. With an increase in the frequency separation, the crosstalk decreased.


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Frequency Separation (GHz) FIGURE 13.7 Crosstalk measured as a function of the frequency separation.

Output Power (dBm)

For optical switching systems, crosstalk from a neighboring channel is required to be less than −10 dB [13]. To obtain a crosstalk value below −10 dB, the frequency separation must be larger than 7 GHz. As a result, an 18-channel wavelength selection is expected when using a PSC-DFB filter. Figure 13.8 shows the light output power measured as a function of the light input power with the bias level as a parameter. The open circles, filled circles, and open triangles represent bias levels of 0.98Ith , 0.96Ith , and 0.93Ith , respectively, where Ith is the threshold current. With an increase in the light input power, the light output power increases. While the light output power increases with an increase in the light input power, the optical gain decreases, as shown in Fig. 13.9; the bandwidth broadens as shown in Fig. 13.10. The effect of total light input power on the transmission spectrum was studied experimentally. In this experiment, two wavelength signals were incident upon the PSC-DFB filter. One wavelength signal is set at a transmission wavelength λ1 = 1553.57 nm and the other wavelength signal was set at λ2 = λ1 − 0.12 nm. The light extinction ratio between light at λ1 and light at λ2 was 15 dB when the input power

Bias Level

Input Power (dBm) FIGURE 13.8 Light output power measured as a function of the light input power.


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Optical Gain (dB)

Bias Level

Input Power (dBm) FIGURE 13.9 Optical gain measured as a function of the light input power.

Bandwidth (GHz)

of both lights was −40 dBm. The transmission spectrum was measured by varying the light input power Pin2 at λ2 . Figure 13.11 shows the optical gain measured at λ1 as a function of the light input power Pin2 at λ2 ; Figure 13.12 shows the 10-dB-down bandwidth measured as a function of Pin2 . In Figs. 13.11 and 13.12, the parameter is the light input power Pin1 at λ1 . Open circles, filled circles, and open triangles represent experimental results for Pin1 of −40 dBm, −35 dBm, and −32 dBm, respectively. When Pin2 increases to −25 dBm, the change in optical gain is only 0.2 dB; a change in the 10-dB-down bandwidth is only 0.2 GHz. Changes in both optical gain and bandwidth are very small. Figure 13.13 shows the optical gain measured as a function of the total light output power. The open circles show the experimental results when the light at λ2 was incident on the filter; the filled circles show the experimental results when the light at λ2 was not incident. As shown in Fig. 13.13, optical gain depends on total light output power rather than total light input power. These experimental results fit very well with the results described in Chapter 9, and these results suggest that the optical gain depends on the total light power in the filter. The optical gain at λ2 is smaller than that at λ1 . For example, the optical gain at λ2 was 9.5 dB; that at λ1 was 24.5 dB when the light input power was −40 dBm. As a result, the light power at

Input Power (dBm) FIGURE 13.10 Ten-decible-down bandwidth measured as a function of light input power.


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(dB)

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(dBm)

Bandwidth (GHz)

FIGURE 13.11 Optical gain measured as a function of light input power when two wavelength signals are incident upon a PSC-DFB filter.

Input Power at

(dBm)

Optical Gain at

(dB)

FIGURE 13.12 Ten-decible-down measured bandwidth as a function of light input power when two wavelength signals are incident upon a PSC-DFB filter.

with without

Total Output Power (dBm) FIGURE 13.13 Optical gain measured as a function of total light output power.


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287

λ2 inside the filter is small and the contribution of the light power at λ2 to the total light power is small. From these results, it is clear that the effect on the transmission spectrum of the light input, whose wavelength is in the sidelobe, is small. This means that changes in the bandwidth and optical gain will be very small in wavelength multiplexing. In the PSC-DFB filter, the stopband width was 1.2 nm. However, the maximum wavelength tuning range was limited to 0.95 nm, due to transmission by the Fabry– Perot mode. Because transmission at the Fabry–Perot mode is due to the residual reflectivity of both facets of the filter, reduction in the reflectivity of both facets will make it possible to use all of the stopband width as a wavelength tuning range. Reduction of the filter length L is also important, because the free spectral range of the Fabry–Perot mode increases. To obtain a wider wavelength tuning range, a larger κ value and a shorter L are required. From this perspective, a vertical cavity is suitable for optical filters [14– 18]. Incorporation of the filter function with vertical cavity surface-emitting optical functional devices will lead to flexible photonic switching systems [19, 20]. 13.5

SUMMARY

A PSC-DFB filter which controls the optical gain and wavelength independently was described. Because a PSC-DFB filter efficiently suppresses the intensity of the submodes, a frequency (wavelength) tuning range as wide as 120 GHz (0.95 nm) with a constant optical gain of 24.5 dB was achieved. In addition, it was shown that the optical gain depends on the total light power in the filter and that the sidelobe light power does not contribute much to the total light power. This means that changes in the bandwidth and optical gain will be very small in wavelength multiplexing. A larger grating coupling coefficient and a shorter device length will lead to a wider frequency (wavelength) tuning range, because a larger stopband width is obtained. This means that the wavelength separation between the main mode and the submodes is expanded. Independent control of the optical gain and wavelength, suppression of submodes, and a large wavelength separation between the main mode and submodes are key factor in obtaining wavelength-tunable optical filters that have a large wavelength tuning range with constant transmissivity and bandwidth. REFERENCES 1. T. Numai, S. Murata, and I. Mito, “1.5 µm tunable wavelength filter with wide tuning range and high constant gain using a phase-controlled distributed feedback laser diode,” Appl. Phys. Lett. 53, 1168 (1988). 2. T. Numai, “1.5 µm phase-controlled distributed feedback wavelength tunable optical filter,” IEEE J. Quantum Electron. 28, 1508 (1992). 3. L. G. Kazovsky, M. Stern, S. G. Menocal, Jr., and C. E. Zah, “DBR active optical filters: transfer function and noise characteristics,” IEEE J. Lightwave Technol. 8, 1441 (1990).


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4. N. Tessler, R. Nager, G. Eisenstein, J. Salzman, U. Koren, G. Raybon, and C. A. Burrus, Jr., “Distributed Bragg reflector active optical filters,” IEEE J. Quantum Electron. 27, 2016 (1991). ¨ 5. O. Sahlén, M. Oberg, and S. Nilsson, “Two-channel optical filtering of 1 Gbit/s signal with DBR filter,” Electron. Lett. 27, 578 (1991). 6. T. L. Koch, F. S. Choa, F. Heismann, and U. Koren, “Tunable multiple-quantum well distributed-Bragg-reflector lasers as tunable narrowband receivers,” Electron. Lett. 25, 890 (1989). 7. T. Numai, S. Murata, and I. Mito, “1.5 µm wavelength tunable phase-shift controlled distributed feedback laser diode with constant spectral linewidth in tuning operation,” Electron. Lett. 24, 1526 (1988). 8. T. Numai, “1.5-µm wavelength tunable phase-shift-controlled distributed feedback laser,” IEEE J. Lightwave Technol. 10, 199 (1992). 9. T. Numai, S. Murata, and I. Mito, “1.5 µm tunable wavelength filter using a phase-shiftcontrolled distributed feedback laser diode with a wide tuning range and a high constant gain,” Appl. Phys. Lett. 54, 1859 (1988). 10. T. Numai, “1.5 µm phase-shift-controlled distributed feedback wavelength tunable optical filter,” IEEE J. Quantum Electron. 28, 1513 (1992). 11. S. Murata, I. Mito, and K. Kobayashi, “Over 720 GHz (5.8 nm) frequency tuning by a 1.5 µm DBR laser with phase and Bragg wavelength control regions,” Electron. Lett. 23, 403 (1987). 12. T. Numai, M. Yamaguchi, I. Mito, and K. Kobayashi, “A new grating fabrication method for phase-shifted DFB LDs,” Jpn. J. Appl. Phys. Pt. 2 26, L1910 (1987). 13. S. Suzuki, M. Nishio, T. Numai, M. Fujiwara, M. Itoh, S. Murata, and N. Shimosaka, “A photonic wavelength-division switching system using tunable laser diode filters,” IEEE J. Lightwave Technol. 8, 660 (1990). 14. G. W. Yoffe, D. G. Schlom, and J. S. Harris, Jr., “Modulation of light by electrically tunable multilayer interference filter,” Appl. Phys. Lett. 51, 1876 (1987). 15. P. C. Kemeny, “III–V semiconductor thin-film optics:optical modulator and filter arrays,” J. Appl. Phys. 64, 6150 (1988). 16. S. Kubota, F. Koyama, and K. Iga, “Surface-emitting-laser-diode type wavelength selective filter,” Trans. IEICE E74, 1689 (1991). 17. F. Koyama, S. Kubota, and K. Iga, “GaAlAs/GaAs active filter based on vertical cavity surface emitting laser,” Electron. Lett. 27, 1093 (1991). 18. T. Numai, “Semiconductor wavelength tunable optical filters,” Int. J. Optoelectron. 6, 239 (1991). 19. T. Numai, M. Sugimoto, I. Ogura, H. Kosaka, and K. Kasahara, “Surface-emitting laser operation in vertical-to-surface transmission electrophotonic devices with a vertical cavity,” Appl. Phys. Lett. 58, 1250 (1991). 20. T. Numai, M. Sugimoto, I. Ogura, H. Kosaka, and K. Kasahara, “Current versus lightoutput characteristics with no definite threshold in pnpn vertical to surface transmission electro-photonic devices with a vertical cavity,” Jpn. J. Appl. Phys. Pt. 2 30, L602 (1991).


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14 PASSIVE PHASE-SHIFTED DFB FILTER

14.1

INTRODUCTION

DFB filters are classified as active [1–8] or passive [9–16]. Here active means that the DFB filters have active layers and amplify light input; passive means that the DFB filters do not have active layers and do not amplify light input. In passive DFB filters, we can expect large changes in the effective refractive index due to the free carrier plasma effect when carriers are injected into the guiding layer. The reason for this is that passive DFB filters do not oscillate even when a lot of carriers are injected, as opposed to active DFB filters. Because the transmission wavelength, which is a resonant wavelength, is proportional to the effective refractive index, it is expected that wide wavelength tuning ranges are obtained in passive DFB filters. In this chapter, a passive phase-shifted DFB filter [15, 16], which is the first wavelength-tunable optical filter in the passive semiconductor optical filters, is explained. In a passive phase-shifted DFB filter, the effective refractive index is controlled by injected current to tune the transmission wavelength. A passive phaseshifted DFB filter selects a signal whose wavelength is the Bragg wavelength. The number of channels and transmissivity are analyzed with respect to the grating coupling coefficient κ, filter length L, and field optical loss coefficient α in the optical waveguide. The influence of facet reflectivity and phase shift on the transmissivity and number of channels is also evaluated. A wavelength shift of 4.2 nm is obtained with the passive phase-shifted DFB filter that was fabricated.


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PASSIVE PHASE-SHIFTED DFB FILTER

Region 1

Region 2

FIGURE 14.1 Analytical model for a passive phase-shifted DFB filter.

14.2


Analytical Model Figure 14.1 shows an analytical model of a passive phase-shifted DFB filter. The transmission spectrum has a single transmission resonance band in the middle of the stopband. The transmission resonant wavelength, which is the Bragg wavelength λB , is written λB = 2n e ,

(14.1)

where n e is the effective refractive index for the optical waveguide and is the grating pitch. When the electric current is injected to a passive phase-shifted DFB filter, the effective refractive index n e decreases due to the free carrier plasma effect. Therefore, the transmission resonant wavelength shifts toward the shorter-wavelength side. The filter cavity is divided into two regions, and the z-axis is placed along the filter cavity as shown in Fig. 14.1. The field optical loss coefficient α is not spatially modulated; the grating coupling coefficient κ has a real part and does not have an imaginary part. Number of Channels and Transmissivity The number of channels and transmissivity are analyzed with respect to the grating coupling coefficient κ, filter length L, and the field optical loss coefficient α in the optical waveguide. A passive phase-shifted DFB filter has a transmission spectrum with a single transmission resonance band in the middle of the stopband. Because the crosstalk between the nearest two wavelength signals is required to be less than −10 dB in optical switching systems [17], 10-dB-down bandwidth λ and stopband width , in which the transmissivity at both ends of the stopband is 10 dB lower than that at the resonant wavelength, are considered. The number of channels is given by /λ if the optical loss in the optical waveguide does not change during wavelength tuning. If the field optical loss coefficient increases from α0 to α1 during wavelength


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291

FIGURE 14.2 /λ calculated as a function of κ L with αL as a parameter.

Transmissivity (dB)

tuning, the number of channels is given by the median value of /λ at α = α0 and that at α = α1 . Figure 14.2 shows the /λ calculated as a function of κ × L = κ L with α × L = αL as a parameter. The number of channels increases with an increase in κ L. However, when optical loss in the optical waveguide is present, the number of channels saturates with an increase in κ L. Figure 14.3 shows the transmissivity calculated at the transmission resonant wavelength, which is the Bragg wavelength, as a function of κ L with αL as a parameter.

FIGURE 14.3 Transmissivity calculated at the transmission resonant wavelength, which is the Bragg wavelength, as a function of κ L with αL as a parameter.


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Field Loss Coefficient FIGURE 14.4 /λ calculated as a function of −α with κ and L as parameters.

With an increase in κ L, light retention in the filter cavity becomes large. As a result, the optical power of the light in the filter cavity is strongly reduced by the optical loss in an optical waveguide when κ L is large. We next consider the dependence of the characteristics on a combination of κ and L with a constant value of κ L. In the following, κ L = 6 is used as an example. When electric current is injected to a passive phase-shifted DFB filter, the field optical loss coefficient −α increases due to the free carrier absorption. Figure 14.4 shows the /λ calculated as a function of the field optical loss coefficient −α with the grating coupling coefficient κ and filter length L as parameters. Figure 14.5 shows the transmissivity calculated at the transmission resonant wavelength, which is the Bragg wavelength, as a function of −α with κ and L as parameters. From Figs. 14.4 and 14.5 it can be said that a large grating coupling coefficient κ and a short filter length L lead to a large number of channels and high transmissivity. Influence of Facet Reflectivities and Phase Shift Figure 14.6 shows the transmission spectra calculated with the grating phases at the facets θ1 and θ4 as parameters. Here κ = 150 cm−1 , L = 400 µm, power reflectivity on both facets R1 = R2 = 2%, and α = 0 cm−1 were assumed. According to the grating phases at facets θ1 and θ4 , slight asymmetry is present in the transmission spectra. However, this slight asymmetry in the transmission spectra does not affect the transmissivity and number of channels very much. Figure 14.7 shows the transmission spectra calculated with power reflectivities at the facets R1 = R2 = R as a parameter. Here κ = 150 cm−1 , L = 400 µm, both grating phases at the facets θ1 = θ4 = π/2, and α = 0 cm−1 were assumed. With an increase in R, the transmissivity decreases. As a result, R should be suppressed to


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293

Transmissivity (dB)


Field Loss Coefficient

Transmissivity (dB)

Transmissivity (dB)

FIGURE 14.5 Transmissivity calculated at the transmission resonant wavelength, which is the Bragg wavelength, as a function of −α with κ and L as parameters.

(b)

Transmissivity (dB)

Transmissivity (dB)

(a)

(c)

(d)

FIGURE 14.6 Transmission calculated spectra with grating phases at the facets as parameters.


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Transmissivity (dB)


Transmissivity (dB)

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(b)

Transmissivity (dB)

Transmissivity (dB)

(a)

(c)

(d)

FIGURE 14.7 Transmission spectra calculated with power reflectivities at the facets R1 = R2 = R as a parameter.

a low value by using antireflection coating, for example. When R is 2%, which is easily achieved by AR coating, the decrease in transmissivity is only 0.35 dB. Figure 14.8 shows the dependence of the /λ calculated on the phase shift ; Fig. 14.9 shows the dependence of the transmissivity calculated at the resonant wavelength on the phase shift . The horizontal lines in Figs. 14.8 and 14.9 show deviation of the phase shift from λ/4. Here, at κ L = 6, power reflectivity at both facets was assumed to be R1 = R2 = 0%. If phase-shifted grating is fabricated by a holographic method using the phase shift layer [18], a deviation of the phase shift from λ/4 is ±10%. From Figs. 14.8 and 14.9, a deviation of 10% in the phaseshift leads to a decrease of 20% in the number of channels with a decrease in the transmissivity of 0.5 dB. 14.3

DEVICE STRUCTURE

Fabrication A first-order phase-shifted grating ( = 240 nm) was formed on an InP substrate with a holographic lithography using a phase-shift layer [18]. Figure 14.10 shows


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DEVICE STRUCTURE

295

Deviation of the Phase Shift (%) FIGURE 14.8 /λ calculated as a function of the phase shift .

the principle behind the fabrication method of a phase-shifted grating. In Fig. 14.10, the solid and dashed lines show the intensity peak paths through the phase-shift layer and air, respectively. The grating pitch and phase-shift are given by λex , 2 sin θ cos φ

(14.2)

Transmissivity (dB)

=

Deviation of the Phase Shift (%) FIGURE 14.9 Transmissivity calculated at the transmission resonant wavelength as a function of the phase shift .


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Intensity Peak Path Wave Fronts Phase-Shift Layer Buffer Layer Intensity Peak

Photoresist Substrate

FIGURE 14.10 Principle of the fabrication method of a phase-shifted grating.

=

2π t sin(θ + φ) − sin(θ − φ) tan φ − tan , nr

(14.3)

where θ is the incident angle of the laser beams for holographic exposure, φ is the angle of incline, t is the thickness of the phase-shift layer, n r is the refractive index of the phase-shift layer, and λex is the wavelength of the incident laser beam. From (14.3) it is found that the phase shift is linearly dependent on the thickness t of the phase-shift layer. As a result, diffraction gratings with a phase shift other than λ/4 can be fabricated by controlling the thickness t of the phase-shift layer. Figure 14.11 shows the fabrication process for a phase-shifted grating. Initially, a novolak type of photoresist is coated onto an InP substrate using a spin coater. The thickness of the novolak photoresist is 130 nm. After prebaking, the buffer layer and phase-shift layer are coated on the photoresist. The buffer layer is a photoresist type of compound with no light sensitivity, and the phase-shift layer is a photoresist. The refractive index of the phase-shift layer n r is 1.75 for light with a wavelength of 325 nm, which is the wavelength of laser beams for holographic exposure. Then the phase-shift layer is exposed using a conventional Hg lamp. If an exposure time is short enough, the novolak photoresist is not exposed, because the phase-shift layer absorbs the ultraviolet light emitted from the Hg lamp. After developing the phaseshift layer, the novolak photoresist is exposed using a holographic method through the phase-shift and buffer layers. The light source for the holographic exposure is a He-Cd laser with a wavelength of 325 nm. The phase-shift layer absorbs only 20% of the laser light intensity with a wavelength of 325 nm, and the novolak photoresist is exposed. The incident angle θ is 44.5◦ ; the angle of incline φ is 15◦ . The thickness of the phase-shift layer is 0.8 µm to fabricate the first-order λ/4-shifted grating. The phase-shift layer and buffer layers are removed, after which the novolak photoresist is developed. When the InP substrate is patterned by wet etching, a λ/4-shifted


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DEVICE STRUCTURE

297

Photoresist Substrate (a) Photomask

Hg Lamp Phase-Shift Layer Buffer Layer

(b) He-Cd Laser

(c)

(d)

(e) FIGURE 14.11 Fabrication process for a phase-shifted grating.

grating is formed. Figure 14.12 shows a scanning electron micrograph of the phaseshifted grating fabricated. The first-order grating pitch is 240 nm and the depth of corrugations is 150 nm.

Structure Figure 14.13 shows a structure of the passive phase-shifted DFB filter. On a firstorder phase-shifted grating, a 0.16-µm-thick InGaAsP guiding layer (λg = 1.3 µm), which is transparent to 1.5-µm light, was grown by LPE. Corrugation depth after LPE is 30 to 40 nm. A DC-PBH [19] was employed for efficient current confinement and transverse-mode control. Ti/Au was used as the metal contact for both p and n sides. The filter length L was 460 µm. Both facets were coated with SiN to reduce facet reflectivity. Residual power reflectivities at both facets R1 and R2 were less than 2%.


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FIGURE 14.12 Scanning electron micrograph of a fabricated phase-shifted grating.

Anode Input Laser Beam SiN

Output Laser Beam Cathode

SiN

FIGURE 14.13 Structure of a passive phase-shifted DFB filter.

14.4


Figure 14.14 shows measured and calculated transmission spectra. For this measurement, a wavelength-tunable DBR LD [20] was used as a light source. Light was coupled in and out with hemispherically ended single-mode fibers. Optical coupling loss between the passive phase-shifted DFB filter and the fiber for each facet was 6 dB. A polarization controller was placed in front of a passive phase-shifted DFB filter to obtain TE-polarized light input. The transmission characteristics of a passive phaseshifted DFB filter depend on polarization of the light input. When the light input is TE-polarized, the transmission spectrum has the sharpest form. When the light input is TM-polarized, the transmission spectrum is not as sharp as that of TE-polarized light input, because the grating coupling coefficient κ for TM-polarized light is lower than that for TE-polarized light. In addition, the transmission resonant wavelength of TM-polarized light is shorter than that of TE-polarized light, because the effective refractive index of TM-polarized light is smaller than that of TE-polarized light. In a passive phase-shifted DFB filter, the wavelength difference between the resonant


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299

Transmissivity (dB)

Calculated

Wavelength (nm) FIGURE 14.14 Measured and calculated transmission spectra.

wavelengths for TM- and TE-polarized light was 3 nm. When an elliptically polarized light is incident upon a passive phase-shifted DFB filter, the bandwidth broadens and the light extinction ratio degrades, because the elliptically polarized light is formed by superposition of TM- and TE-polarized light. Light input power was retained constant at −20 dBm to measure the transmission spectra. With an increase in the light input power, the number of photogenerated carriers increases [21]. As a result, the effective refractive index of the optical waveguide decreases due to the free carrier plasma effect; the transmission wavelength shifts slightly to the shorter-wavelength side. In addition, the transmissivity decreases slightly due to free carrier absorption. In Fig. 14.14, the solid line with filled circles shows a spectrum for no injected current; the dashed-dotted line with open circles shows a spectrum for an injected current of I = 6 mA. The full width at half-maximum was 0.5 nm and the stopband width was 2.5 nm. From these results, the grating coupling coefficient κ was estimated to be 55 cm−1 , leading to κ L = 2.5. The field optical loss coefficient α for a passive phase-shifted DFB filter was −10 cm−1 . The average field optical loss coefficient for the fabricated devices was −5 cm−1 . In the calculations described in Section 14.2, a 10-dB-down bandwidth was considered. However, due to the optical loss in the optical waveguide and a small κ L, the light extinction ratio was less than 10 dB, so a 10-dB-down bandwidth was not obtained experimentally. In Fig. 14.14, slight asymmetry is present in the transmission spectra. The reason for this is that the phase shift deviates from λ/4. The dashed line shows a calculated result with = 0.9 × λ/4, and closely matches the experimental result. In this calculation, κ = 55 cm−1 , L = 460 µm, θ1 = θ4 = π , and R1 = R2 = 2% were used.


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Wavelength Shift (nm)

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Injected Current (mA) FIGURE 14.15 Wavelength shift as a function of current injected.

With an increase in the current injected, the transmission wavelength becomes short due to the free carrier plasma effect. The transmissivity decreased because −α in the optical waveguide increased at a rate of 1 cm−1 /mA due to the free carrier absorption. As shown in Fig. 14.15, the maximum wavelength shift was 4.2 nm. When the current injected increased further, laser oscillation occurred at a wavelength of 1.3 µm, which is the λg of the optical waveguide. As a result, the carrier concentration in this optical filter was fixed at the threshold carrier concentration, and the transmission wavelength no longer shifted to the shorter wavelength side. It should be noted that the optical waveguide (λg = 1.3 µm) for 1.5-µm light functioned as an active layer for 1.3-µm light. The maximum wavelength tuning range as an optical filter is limited to half the stopband width, which is 1.25 nm. To expand the wavelength tuning range, a large stopband is needed. Figure 14.16 shows two-channel wavelength selection at an injected current of I = 0 mA and I = 6 mA. The wavelengths for light input were 1539.1 nm, which is the transmission wavelength at I = 0 mA and 1538.4 nm, which is the transmission wavelength at I = 6 mA, as shown in Fig. 14.14. The light input power for each was −28 dBm and the light intensity is not modulated. When the current injected was 6 mA, crosstalk was −5 dB. From Fig. 14.14, the crosstalk should be −6 dB. The reason for this is probably as follows: In Fig. 14.14, light input power was −20 dBm; in Fig. 14.16, light input power was −28 dBm. Because the light input power for Fig. 14.16 was lower than that for Fig. 14.14, the electron concentration for Fig. 14.16 was lower than that for Fig. 14.14 [21]. As a result, the transmission wavelength in Fig. 14.16 shifted to a longer wavelength than that in Fig. 14.14, leading to slightly more crosstalk.


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301

Power (arb. units)

REFERENCES

Power (arb. units)

Wavelength (nm) (a)

Wavelength (nm) (b) FIGURE 14.16 Two-channel wavelength selection at current injected at (a) I = 0 mA and (b) I = 6 mA.

14.5

SUMMARY

The passive phase-shifted DFB filter, which is the first wavelength-tunable optical filter of all the passive semiconductor optical filters, was explained in this chapter. It was shown that a large grating coupling coefficient κ and a short filter length L are necessary to obtain a wide tuning range, a narrow bandwidth, low crosstalk, and large transmissivity. For this purpose, vertical cavity optical devices are suitable. To achieve large transmissivity, it is also important to achieve low optical loss in the optical waveguide or to create optical gain. Large transmissivity allows us to cascade optical filters and to create light intensity that is intense enough to trigger photodetector. In order to cascade optical filters, low loss in the optical waveguide is not enough. As a result, wavelength-tunable optical filters should have optical gain over the transmission wavelength. From this viewpoint, active DFB filters [1–8] are attractive. REFERENCES 1. K. Magari, H. Kawaguchi, K. Oe, Y. Nakano, and M. Fukuda, “Optical signal selection with a constant gain and a gain bandwidth by a multielectrode distributed feedback laser amplifier,” Appl. Phys. Lett. 51, 1974 (1987).


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2. K. Magari, H. Kawaguchi, K. Oe, and M. Fukuda, “Optical narrow-band optical filters using optical amplification with distributed feedback,” IEEE J. Quantum Electron. 24, 2178 (1988). 3. K. Kikushima, K. Nawata, and M. Koga, “Tunable amplification properties of distributed feedback laser diodes,” IEEE J. Quantum Electron. 25, 163 (1989). 4. T. Numai, M. Fujiwara, N. Shimosaka, K. Kaede, M. Nishio, S. Suzuki, and I. Mito, “1.5 µm λ/4-shifted DFB LD optical filter and 100 Mbit/s two-channel wavelength signal switching,” Electron. Lett. 24, 236 (1988). 5. T. Numai, S. Murata, and I. Mito, “1.5 µm tunable wavelength filter with wide tuning range and high constant gain using a phase-controlled distributed feedback laser diode,” Appl. Phys. Lett. 53, 1168 (1988). 6. T. Numai, “1.5 µm phase-controlled distributed feedback wavelength tunable optical filter,” IEEE J. Quantum Electron. 28, 1508 (1992). 7. T. Numai, S. Murata, and I. Mito, “1.5 µm tunable wavelength filter using a phase-shiftcontrolled distributed feedback laser diode with a wide tuning range and a high constant gain,” Appl. Phys. Lett. 54, 1859 (1988). 8. T. Numai, “1.5 µm phase-shift-controlled distributed feedback wavelength tunable optical filter,” IEEE J. Quantum Electron. 28, 1513 (1992). 9. R. C. Alferness, C. H. Joyner, M. D. Divino, and L. L. Buhl, “InGaAsP/InP waveguide grating optical filters for λ = 1.5 µm,” Appl. Phys. Lett. 45, 1278 (1984). 10. R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, and L. L.Buhl, “Narrowband grating resonator optical filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49, 125 (1986). 11. R. C. Alferness, L. L. Buhl, M. J. R. Martyak, M. D. Divino, C. H. Joyner, and A. G. Dentai, “Narrowband GaInAsP/InP waveguide grating-folded directional coupler multiplexer/demultiplexer,” Electron. Lett. 24, 150 (1988). 12. G. Heise, R. März, and M. Schienle, “Investigation of Bragg gratings on planar InGaAsP/InP waveguides at normal and oblique incidence,” J. Lightwave Technol. 7, 735 (1989). 13. A. Y. Yan, H. Carru, P. Correc, J. P. Chandouineau, L. Dugrand, and M. Carre, “GaAs/GaAlAs rib waveguide grating optical filters for = 1.5 µm,” Electron. Lett. 23, 622 (1987). 14. C. Cremer, G. Heise, R. März, M. Schienle, G. Schulte-Roth, and H. Unzeitig, “Bragg gratings on InGaAsP/InP waveguides as polarization independent optical filters,” J. Lightwave Technol. 7, 1641 (1989). 15. T. Numai, S. Murata, and I. Mito, “Tunable wavelength optical filters using λ/4-shifted waveguide grating resonators,” Appl. Phys. Lett. 53, 83 (1988). 16. T. Numai, “1.5 µm semiconductor wavelength tunable optical filter using a λ/4-shifted passive waveguide grating resonator,” Jpn. J. Appl. Phys. Pt. 1 30, 2519 (1991). 17. S. Suzuki, M. Nishio, T. Numai, M. Fujiwara, M. Itoh, S. Murata, and N. Shimosaka, “A photonic wavelength-division switching system using tunable laser diode optical filters,” J. Lightwave Technol. 8, 660 (1990). 18. T. Numai, M. Yamaguchi, I. Mito, and K. Kobayashi, “A new grating fabrication method for phase-shifted DFB LDs,” Jpn. J. Appl. Phys. Pt. 2 26, L1910 (1987). 19. I. Mito, M. Kitamura, K. Kobayashi, S. Murata, M. Seki, Y. Odagiri, H. Nishimoto, M. Yamaguchi, and K. Kobayashi, “InGaAsP double-channel-planar-buried- heterostructure


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REFERENCES

303

laser diode (DC-PBH LD) with effective current confinement,” J. Lightwave Technol. 1, 195 (1983). 20. S. Murata, I. Mito, and K. Kobayashi, “Over 720 GHz (5.8 nm) frequency tuning by a 1.5 µm DBR laser with phase and Bragg wavelength control regions,” Electron. Lett. 23, 403 (1987). 21. M. J. Adams, H. J. Westlake, M. J. O’Mahony and I. D. Henning, “A comparison of active and passive optical bistability in semiconductors,” IEEE J. Quantum Electron. 21, 1498 (1985).


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15 TWO-SECTION FABRY–PEROT FILTER

15.1

INTRODUCTION

As explained in Chapters 12–14, the transmission wavelength of DFB filters is close to the Bragg wavelength, which is determined by the grating pitch and the effective refractive index of an optical waveguide. The effective refractive index of an optical waveguide depends on the thickness and width of the optical waveguide. As a result, the transmission wavelength changes according to the dimensions of the optical waveguide. For example, the transmission wavelength distributes with a standard deviation of several nanometers for optical filters grown on a common substrate. The optical gain spectrum for LDs has a full width at half maximum (FWHM) of 10 nm, and Fabry–Perot resonators have a lot of resonant modes, which can be used as transmission wavelengths. Therefore, if the Fabry–Perot resonators are utilized as wavelength-tunable optical filters, it is easy for the transmission wavelength to fit the preassigned wavelength. Let us consider the wavelength tuning range and transmission bandwidth of the DFB filters and Fabry–Perot filters. In DFB filters, the wavelength tuning range and transmission bandwidth are determined by the grating coupling coefficient κ and filter length L. As a result, the wavelength tuning range and transmission bandwidth cannot be designed independently. With an increase in the wavelength tuning range, the transmission bandwidth always decreases. In Fabry–Perot filters, the wavelength tuning range and transmission bandwidth can be designed independently. The wavelength tuning range is linearly dependent on the reciprocal of the filter length L and


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TWO-SECTION FABRY–PEROT FILTER

independent of the facet power reflectivity R; the transmission bandwidth depends on the filter length L and facet power reflectivity R. These characteristics will allow us to design wavelength-tunable optical filters flexibly. For example, a large bandwidth with a large wavelength tuning range makes it possible to obtain a large number of channels without precise control of wavelength. The idea to use a Fabry–Perot amplifier as a wavelength-tunable optical filter (a channel selector) was proposed by Kazovsky and Werner [1]. They tuned the transmission wavelength by simultaneous adjustment of injected current and temperature to achieve wavelength tuning with constant optical gain and constant bandwidth. To tune the transmission wavelength only by adjusting injected current, a two-section Fabry–Perot filter was developed, and wavelength tuning with constant optical gain and constant bandwidth was demonstrated [2,3]. In Fabry–Perot filters, expansion of the separation between Fabry–Perot modes is important to expand the wavelength tuning range. Note that suppression of the submodes and expansion of the separation between the main mode and submodes are important to expand the wavelength tuning range in DFB filters [4, 5]. In this chapter, a two-section Fabry–Perot filter is explained. This wavelengthtunable optical filter controls the transmissivity (optical gain) and transmission wavelength independently by injected current, and changes in the threshold gain and optical loss are small during wavelength tuning. A frequency (wavelength) tuning range as wide as 188 GHz (1.5 nm) with a constant optical gain of 23 dB and a constant bandwidth of 5 GHz was achieved. A 25-channel wavelength selection with less than −10 dB of crosstalk is expected with a two-section Fabry–Perot filter.

15.2


FWHM If the loss at facets is negligibly small, the signal gain G c for a Fabry–Perot filter is written [6] Gc =

(1 − R1 ) (1 − R2 ) G s , 2 √ √ 1 − G s R1 R2 + 4G s R1 R2 sin2 φ

(15.1)

where R1 is the power reflectivity of an input facet, R2 is the power reflectivity of an output facet, and φ is given by φ = 2π n e L

1 1 − λ λ0

.

(15.2)

Here n e is the effective refractive index of the optical waveguide, L is the filter length, λ is a wavelength of light, and λ0 is a resonant wavelength. In (15.1), G s is a single-pass optical gain, which is expressed as G s = exp g − αp L ,

(15.3)


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307

FIGURE 15.1 Analytical model for a two-section Fabry–Perot filter.

where is the optical confinement factor, g is the power optical gain coefficient, and αp is the power optical loss coefficient in the optical waveguide. From (15.1), FWHM λ is given by solving the equation

2 2 1 − G s R1 R2 + 4G s R1 R2 sin2 φ = 2 1 − G s R1 R2 .

(15.4)

From (15.4), FWHM λ is obtained as √ λ0 2 1 − G s R 1 R 2 λ = . 2π n e L 4 G s 2 R1 R2

(15.5)

Free Spectral Range In WDM or WD systems that use Fabry–Perot filters, WDM signals should be placed within the free spectral range of a Fabry–Perot filter to select one wavelength signal from the WDM signals. Therefore, the maximum wavelength tuning range of a Fabry–Perot filter is the free spectral range λFSR , which is given approximately by λFSR =

λ0 2 . 2n e L

(15.6)

Figure 15.1 shows the analytical model for a two-section Fabry–Perot filter. This filter consists of an amplification section and a control section, and controls the optical gain and transmission wavelength almost independently. While the optical gain is controlled by the electric current Ia injected to the amplification section, the transmission wavelength is tuned by the tuning current Ip injected to the control section. The principle behind the wavelength tuning is as follows: The transmission wavelength λ0 in the two-section Fabry–Perot filter is given by λ0 =

2 Na L a + Np L p . m

(15.7)

Here Na and Np are the effective refractive index in the amplification section and control sections, respectively; L a and L p are the length of the amplification and


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control sections, respectively; and m is a positive integer. The transmission wavelength is modulated by changing the optical length of the cavity Na L a + Np L p . With an increase in the tuning current Ip injected into the control section, the effective refractive index Np in the control section decreases due to the free carrier plasma effect. As a result, the transmission wavelength shifts toward the shorter-wavelength side. The free spectral range λFSR of a two-section Fabry–Perot filter is given approximately by

λFSR =

λ0 2 . 2 Na L a + Np L p

(15.8)

From (15.8), a decrease in the device length L a + L p leads to a wide tuning range. The tuning current Ip injected to the control section causes an increase in the optical loss in the control section due to the free carrier absorption. However, because an increase in the optical loss is small, the increase in the optical loss is compensated almost independent of the transmission wavelength by a slight increase in the electric current Ia injected to the amplification section.

Wavelength Tuning Range and Transmission Bandwidth Fabry–Perot filters are compared with DFB filters from the perspective of the wavelength tuning range and transmission bandwidth. In Fabry–Perot filters, the wavelength tuning range is limited by the free spectral range λFSR . In DFB filters, the wavelength tuning range is limited by the stopband width. Figure 15.2 shows the stopband width and transmission bandwidth (FWHM) of the DFB filter, which were calculated as a function of the grating coupling coefficient κ with filter length L as a parameter. Here a field optical gain coefficient of 0 cm−1 was assumed. As shown in Fig. 15.2, with an increase in the stopband width, the transmission bandwidth (FWHM) decreases. Figure 15.3 shows the free spectral range and transmission bandwidth (FWHM) for a Fabry–Perot filter, which were calculated as a function of a filter length L with the power reflectivity as a parameter. Here, a field optical gain coefficient of 0 cm−1 was assumed. The free spectral range is independent of the power reflectivity, and we can design the free spectral range λFSR and transmission bandwidth independently. Figure 15.4 shows the transmission spectra calculated for a two-section Fabry– Perot filter. A filter length L of 200 µm, a field optical gain coefficient of 0.98 times that of the threshold, and power reflectivities of R1 = R2 = 32% were assumed for both facets. In filter operation, a resonant wavelength whose optical gain is the largest is used as a transmission wavelength.


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309

Stopband Width (nm)

DEVICE STRUCTURE

Grating Coupling Coefficient

FWHM (nm)

(a)

Grating Coupling Coefficient (b) FIGURE 15.2 (a) Stopband width and (b) transmission bandwidth (FWHM) of a DFB filter, which were calculated as a function of the grating coupling coefficient κ with the filter length L as a parameter. A field optical gain coefficient of 0 cm−1 was assumed.

15.3

DEVICE STRUCTURE

Figure 15.5 shows the structure of a 1.5-µm two-section Fabry–Perot filter. The filter consists of an amplification section and a control section. Both sections have a 1.29-µm compound InGaAsP guiding layer which is 0.16 µm thick. The amplification section has an active layer to amplify the light input coupled to this device. The active layer is 1.53-µm compound InGaAsP and is 0.10 µm thick. The control section doesn’t have an active layer in order to obtain a large change in the effective refractive index without a large change in the optical gain or loss by the tuning current Ip injected to the control section. Each section is 100 µm long, and the total device is 200 µm long. The amplification and control sections are isolated electrically from each other by 20-µm-wide etched grooves, which were formed on both sides of the center mesa


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Free Spectral Range (nm)

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Filter Length

FWHM (nm)

(a)

Filter Length (b) FIGURE 15.3 (a) Free spectral range and (b) transmission bandwidth (FWHM) of a Fabry– Perot filter, calculated as a function of the filter length L with the power reflectivity as a parameter. A field optical gain coefficient of 0 cm−1 was ssumed.

stripe area. Isolation resistance is 100 . A DC-PBH [7] was adopted for efficient current confinement and transverse-mode control. Ti/Au was used as a metal contact for both the p and n sides. Both facets were cleaved and the facet reflectivities were R1 = R2 = 32%. The free spectral range λFSR is 1.8 nm. Figure 15.6 shows a fabrication process for a two-section Fabry–Perot filter. On an n-InP substrate, an n-InGaAsP guiding layer (λg = 1.29 µm, 0.16 µm thick, Sn doped at 5 × 1017 cm−3 ), an undoped InP etching-stop layer (0.08 µm thick), an undoped InGaAsP active layer (λg = 1.53 µm, 0.16 µm thick), an undoped antimeltback layer (λg = 1.29 µm), and a p-InP cladding layer (0.2 µm thick, Zn doped at 1 × 1018 cm−3 ) are grown by LPE, as shown in Fig. 15.6(a). Then the cladding and active layers in the control section are selectively wet etched; the wet etching is stopped at the InP etching-stop layer, as shown in Fig. 15.6(b). After etching, a p-InP


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311

Transmissivity (dB)


Relative Wavelength (nm) FIGURE 15.4 Calculated transmission spectra. A filter length L of 200 µm, a field optical gain coefficient 0.98 times that of the threshold, and power reflectivities of R1 = R2 = 32% for both facets were assumed.

Active Layer

Anode

Anode

Input Laser Beam

Output Laser Beam Cathode Amp. Section Control Section

FIGURE 15.5 Structure of a two-section Fabry–Perot filter.

cladding layer (1.0 µm thick, Zn doped at 1 × 1018 cm−3 ) is grown over the entire wafer, as shown in Fig. 15.6(c). Finally, Ti/Au was partially formed as anodes and a cathode, as shown in Fig. 15.6(d).

15.4


The experimental setup for transmission measurement is as follows: The light sources were a wavelength-tunable DBR LD [8] and a phase-shifted DFB LD [9]. Light was coupled in and out with hemispherically ended single-mode fibers. Optical coupling loss between the filter and the fiber for each facet was about 6 dB. A polarization controller was placed in front of the filter to obtain TE-polarized input light. Figure 15.7 shows the measured transmission spectra with the tuning current Ip injected to the control section as a parameter. The electric current Ia injected to the


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Active Layer

(a)

(b)

(c) Anode 1

Anode 2

(d)

Cathode

FIGURE 15.6 Fabrication process for a two-section Fabry–Perot filter.

Optical Gain (dB)

amplification section was controlled to be 0.98 times the threshold current for each value of Ip . This device lased at a threshold current of 30 mA at 25◦ C when the tuning current was not injected to the control section. Light input power was maintained at a constant level of −40 dBm. The transmission wavelength shifted toward the shorterwavelength side with an increase in Ip . A frequency (wavelength) tuning range as

Wavelength (nm) FIGURE 15.7 Transmission spectra measured for an input power of −40 dBm with the tuning current Ip injected into the control section as a parameter.


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313

Crosstalk (dB)


Frequency Separation (GHz) FIGURE 15.8 Crosstalk measured as a function of frequency separation.

Optical Gain (dB)

wide as 188 GHz (1.5 nm) was achieved while maintaining a constant optical gain of 23 dB and a constant bandwidth (FWHM) of 5 GHz. A further increase in Ip causes a mode jump. The filter noise level, which is the spontaneous emission level at the transmission wavelength, was lower than the output level by at least 20 dB. Figure 15.8 shows the crosstalk between the two light inputs as a function of the frequency separation. In this experiment, one wavelength, λ1 , was set at a transmission wavelength and the other, λ2 , was varied. Here the crosstalk is defined as an extinction ratio between the light output at λ1 and that at λ2 . Both light input power levels were −40 dBm, and the light intensities were not modulated. The crosstalk decreased with an increase in the frequency separation. For photonic switching systems, the crosstalk required between the selected channel and one neighboring channel is less than −10 dB [10]. To satisfy such a requirement, the frequency separation must be larger than 7.5 GHz. As a result, a 25-channel wavelength selection is expected with a two-section Fabry–Perot filter. With an increase in the amplified output power, the optical gain decreases as shown in Fig. 15.9, and the resonant wavelength shifts according to the total optical

Output Power (dBm) FIGURE 15.9 Optical gain measured as a function of total light output power.


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Wavelength (nm)

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Tuning Current FIGURE 15.10 Wavelength shift measured as a function of the tuning current injected into a control section with temperature as a parameter.

power inside the cavity. Light input at off-resonant wavelength, however, does not contribute to an increase in the internal optical power because the optical gains at offresonant wavelengths are suppressed to a rather small value. This means that changes in the spectral shape and resonant wavelength will be very small in wavelength multiplexing. Figure 15.10 shows the wavelength shift of a resonant mode with the largest optical gain as a function of the tuning current Ip injected into the control section with temperature as a parameter. The filled circles, open circles, filled triangles, open triangles, filled squares, and open squares represent experimental results for 40, 35, 30, 25, 20, and 15◦ C, respectively. During wavelength tuning, both Ia and Ip are controlled to achieve constant optical gain and constant bandwidth. The peak of the optical gain spectrum shifts with Ia ; the Fabry–Perot modes shift with Ip . Because the shift of the peak of the optical gain spectrum is larger than the shift of the Fabry– Perot modes, mode jump occurred. However, by adjusting temperature, we can use a wavelength tuning range of 4.5 nm. The free spectral range λFSR increases in proportion to the reciprocal of the filter length L. Therefore, a decrease in the filter length L is needed to obtain a wide wavelength tuning range. With a decrease in the filter length L, the free spectral range λFSR increases, and finally only one Fabry–Perot mode exists in the optical gain spectrum. In this case, a short cavity will make it possible to obtain continuous wavelength tuning without mode jump. From the viewpoint of a short cavity, a vertical cavity is suitable. If vertical cavity surface-emitting optical functional devices [11, 12] incorporate the filter function, they will contribute to flexible photonic switching systems. 15.5

SUMMARY

In a two-section Fabry–Perot filter, the wavelength tuning range and transmission bandwidth can be designed independently, in contrast to DFB filters. If a large


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REFERENCES

315

bandwidth with a large wavelength tuning range is obtained, it will be possible to achieve a large number of channels without precise control of wavelength. A frequency (wavelength) tuning range as wide as 188 GHz (1.5 nm) with a constant optical gain of 23 dB and a constant bandwidth of 5 GHz was achieved with a filter length of 200 µm. A 25-channel wavelength selection is expected by using a two-section Fabry–Perot filter. It is expected that wavelength-tunable optical filters with short cavities, such as vertical cavity surface-emitting optical devices, will lead to a wide wavelength tuning range. REFERENCES 1. L. Kazovsky and J. Werner, “Multichannel optical communications using tunable Fabry– Perot amplifiers,” Appl. Opt. 28, 553 (1989). 2. T. Numai, “1.5 µm optical filter using a two-section Fabry–Perot laser diode with wide tuning range and high constant gain,” IEEE Photon. Technol. Lett. 2, 401 (1990). 3. T. Numai, “1.5 µm two-section Fabry–Perot wavelength tunable optical filter,” IEEE J. Lightwave Technol. 10, 1590 (1992). 4. T. Numai, S. Murata, and I. Mito, “1.5 µm tunable wavelength filter using a phase-shiftcontrolled distributed feedback laser diode with a wide tuning range and a high constant gain,” Appl. Phys. Lett. 54, 1859 (1988). 5. T. Numai, “1.5 µm phase-shift-controlled distributed feedback wavelength tunable optical filter,” IEEE J. Quantum Electron. 28, 1513 (1992). ¨ 6. M. G. Oberg and N. A. Olsson, “Crosstalk between intensity-modulated wavelengthdivision multiplexed signals in a semiconductor laser amplifier,” IEEE J. Quantum Electron. 24, 52 (1988). 7. I. Mito, M. Kitamura, K. Kobayashi, S. Murata, M. Seki, Y. Odagiri, H. Nishimoto, M. Yamaguchi, and K. Kobayashi, “InGaAsP double-channel-planar-buried-heterostructure laser diode (DC-PBH LD) with effective current confinement,” IEEE J. Lightwave Technol. 1, 195 (1983). 8. S. Murata, I. Mito, and K. Kobayashi, “Over 720 GHz (5.8 nm) frequency tuning by a 1.5 µm DBR laser with phase and Bragg wavelength control regions,” Electron. Lett. 23, 403 (1987). 9. T. Numai, M. Yamaguchi, I. Mito, and K. Kobayashi, “A new grating fabrication method for phase-shifted DFB LDs,” Jpn. J. Appl. Phys. Pt. 2 26, L1910 (1987). 10. S. Suzuki, M. Nishio, T. Numai, M. Fujiwara, M. Itoh, S. Murata, and N. Shimosaka, “A photonic wavelength-division switching system using tunable laser diode optical filters,” IEEE J. Lightwave Technol. 8, 660 (1990). 11. T. Numai, M. Sugimoto, I. Ogura, H. Kosaka, and K. Kasahara, “Surface emitting laser operation in vertical to surface transmission electro-photonic devices with a vertical cavity,” Appl. Phys. Lett. 58, 1250 (1991). 12. T. Numai, M. Sugimoto, I. Ogura, H. Kosaka, and K. Kasahara, “Current versus lightoutput characteristics with no definite threshold in pnpn vertical to surface transmission electro-photonic devices with a vertical cavity,” Jpn. J. Appl. Phys. Pt. 2 30, L602 (1991).


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16 OPTICAL FUNCTIONAL DEVICES WITH PNPN-JUNCTIONS

16.1

INTRODUCTION

Photonics has been a great success in optical fiber communications and optical disk recordings. In optical fiber communications, low loss and no skin effect in the optical fiber are the key factors for the superiority of photonics to electronics. In optical disk recordings, high density and no contact are the key factors for the superiority of photonics to electronics. It is expected that applications of photonics are further expanded to information processing, because in contrast to electrons, photons are free from mutual interactions. For applications of optical information processing, switching and parallel operations will be required in photonic devices to overcome the limited flexibility of electrical interconnects. To meet these requirements, a self-electrooptic effect device (SEED) [1], which was based on MQW optical modulators, and spatial light modulators [2], which used ferroelectric liquidcrystals, were demonstrated. However, these optical functional devices were passive and they needed external light sources. To solve this problem, light-emitting optical functional devices [3–29] were developed. In light-emitting optical functional devices, switching was achieved by pnpn-junctions; parallel operations was obtained by two-dimensional arrays, which consisted of vertical cavity surface-emitting laser devices. In this chapter, edge- and surface-emitting optical functional devices that use pnpn-junctions are described.


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OPTICAL FUNCTIONAL DEVICES WITH PNPN-JUNCTIONS

EDGE-EMITTING OPTICAL FUNCTIONAL DEVICE

Selection of Output Ports Figure 16.1 shows the structure of an edge-emitting optical functional device that has a pnpn-junction [4]. Semiconductor layers which were grown on an n-GaAs substrate by molecular beam epitaxy (MBE) are as follows: an n-GaAs buffer layer (0.5 µm, 2 × 1018 cm−3 ), an n-Al0.4 Ga0.6 As cathode layer (1 µm, 2 × 1018 cm−3 ), a p-Al0.25 Ga0.75 As charge sheet layer (5 nm, 1 × 1019 cm−3 ), an undoped GaAs active layer (0.1 µm) sandwiched between two Al0.25 Ga0.75 As guiding layers (0.3 and 0.1 µm), an n-Al0.25 Ga0.75 As n-gate layer (0.5 µm, 1 × 1017 cm−3 ), a p-Al0.4 Ga0.6 As anode layer (1 µm, 2 × 1018 cm−3 ), and a p-GaAs cap layer (0.5 µm, 1 × 1019 cm−3 ). The undoped layers were p type and the background carrier concentration was 1 × 1015 cm−3 . This device has two regions, A and B, which are isolated electrically from each other by a 50-µm-wide etched groove. Ohmic contact for each anode was formed through a 10-µm-wide opening in the SiO2 film. The isolation resistance was about 2 k. Bias voltage, which is lower than switching voltage, is applied to both anodes. When the electric pulse is added to the anode and input light is simultaneously incident on the facet, the device is switched on and electric current flows to the

p-GaAs Cap Layer Anode Layer Gate Layer Guiding Layer i-GaAs Active Layer Guiding Layer Charge Sheet Layer Cathode Layer n-GaAs Buffer Layer

Output Light A B

Input Light

Au/Cr SiO2 n-GaAs Substrate Au/AuGe

FIGURE 16.1 Structure of an edge-emitting optical functional device.


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Wavelength (nm)

319

Light Output (mW)

Forward Voltage (V)

EDGE-EMITTING OPTICAL FUNCTIONAL DEVICE

Injected Current (mA) FIGURE 16.2

I –V and I –L characteristics of an edge-emitting optical functional device.

device. If the electric current in the ON-state is higher than the threshold current, the device emits a laser beam from channel A or B. Figure 16.2 shows I –V and I –L characteristics for one region of the device. The switching voltage Vs was 4.1 V when light was not illuminated, and the holding voltage Vh was around 1.5 V. Threshold current Ith was 160 mA and the maximum light output was greater than 12 mW. External differential quantum efficiency was 23% per facet. Threshold current density Jth was estimated to be 2 kA/cm2 . The lasing wavelength was 875 nm at room temperature. Note that no light emission from Al0.25 Ga0.75 As was observed. Figure 16.3 shows timing charts and experimental results. In the period of writing, the entire device is switched on by an input light while an electric bias pulse is below the switching voltage Vs . Then, if necessary, the device can be held in the ON-state with low electric power consumption by lowering the applied voltage to around the holding voltage Vh in the period of reading. The light output channel was changed at 400 Mbit/s with crosstalk of −30 dB. Filter Operation Figure 16.4 shows the structure of an edge-emitting optical functional device that shows filter operation [5]. The layer structure and basic functions of this device are the same as those described above [4]. The cavity was 275 µm long and both facets are as cleaved. The threshold current was 120 mA, the external differential quantum


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Write Hold

Read

Electric Pulse A

Electric Pulse B (a)

200 mV 200 mV 5 ns

A

B

(b) FIGURE 16.3 (a) Timing charts; (b) experimental results.

efficiency was 40% per facet, and the wavelength of laser light was 880 nm. The switching voltage Vs is 11.2 V, and the holding voltage Vh is 1.6 V. Figure 16.5 shows the optical switching energy, which is the optical energy required to switch on an edge-emitting optical functional device as a function of bias voltage. The optical switching energy decreased with an increase in the bias voltage. When the bias voltage was 0.83 Vs , the optical switching energy was only 6.4 pJ. If the bias voltage is slightly below the switching voltage Vs , an optical switching energy of about 1 pJ is expected. By adjusting load resistance, the electric current in the ON-state was set to be just below the threshold current. As a result, this device functioned as an optical filter with optical gain. Figure 16.6 shows transmission spectra for data signals with a resonant wavelength and an off-resonant wavelength when this device was switched on. Input power for the data signals was −25.4 dBm, and the pulse width was 20 ns. For a resonant wavelength, internal optical gain for the data signals was 21.4 dB when the flowing current was set slightly below the threshold current. For an off-resonant wavelength, transmission of the data signals was suppressed. The light extinction ratio between the resonant and off-resonant wavelengths was as large as 14.7 dB, which demonstrated the filter function successfully.


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SURFACE-EMITTING OPTICAL FUNCTIONAL DEVICE

321

p-GaAs Cap Layer Anode Layer Gate Layer Guiding Layer i-GaAs Active Layer Guiding Layer Charge Sheet Layer Cathode Layer n-GaAs Buffer Layer

Output Light Au/Cr SiO2 Input Light

n-GaAs Substrate Au/AuGe

Optical Switching Energy (pJ)

FIGURE 16.4 Structure of an edge-emitting optical functional device with filter operation.

Pulse Width

Bias Voltage FIGURE 16.5 Optical switching energy of an edge-emitting optical functional device.

16.3


LED Figure 16.7 shows the structure of a surface-emitting LED optical functional device [3] called an LED vertical-to-surface transmission electorophotonic device


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Intensity (arb. units)

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Resonant

Intensity (arb. units)

Wavelength (nm) (a)

Off-resonant

Wavelength (nm) (b) FIGURE 16.6 Transmission spectra of an edge-emitting optical functional device in the ON state: (a) resonant wavelength; (b) off-resonant wavelength.

Anode

Cathode

p-Gate

p-GaAs

n-Gate

n-GaAs p-GaAs Zn-Diffused Region

n-GaAs

SI-GaAs

FIGURE 16.7 Structure of an LED VSTEP.


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323

p n p

n Oscilloscope FIGURE 16.8 Circuit configuration for dynamic memory operation of an LED VSTEP.

(VSTEP). The epitaxial layers, which were grown on a semi-insulating (SI) GaAs substrate by MBE, are as follows: an n-GaAs buffer layer (0.5 µm, 2 × 1018 cm−3 ), an n-Al0.4 Ga0.6 As layer (1 µm, 5 × 1017 cm−3 ), a p-GaAs layer (5 nm, 1 × 1019 cm−3 ), an n-GaAs active layer (1 µm, 1 × 1017 cm−3 ), a p-Al0.4 Ga0.6 As layer (0.5 µm, 5 × 1018 cm−3 ), and a p-GaAs contact layer (0.15 µm, 5 × 1019 cm−3 ). During the turn-off process, p- and n-gates extract excess carriers, which are stored in the ONstate. The shape of the device is rectangular with a first mesa of 100 × 100 µm2 and a second mesa of 140 × 200 µm2 . Zn was diffused into the p-GaAs layer through the SiO2 window, which was opened on the n-GaAs active layer. Ohmic contacts for the anode and the p-gate were made by Au/Cr/AuZn and those for the cathode and the n-gate were Au/Cr/AuGeNi. To achieve high-speed turn-off, electrons and holes have to be extracted through the n- and p-gates, respectively. As shown in Fig. 16.8, a resistor RG and a Schottky diode DS were connected externally. The Schottky diode was reverse biased when a LED VSTEP was forwardly biased. Once the negative reset pulse is applied to the LED VSTEP, RG and DS function as the outlets for the excess electrons and holes, respectively. The switching voltage Vs was 5 V and the holding voltage Vh was 1.4 V for the LED VSTEP without light illumination. The switching voltage decreased from 5V to 2.4 V by connecting an RG of 10 k. An oscilloscope, which was terminated in 50 , was inserted between the cathode and the ground. The turn-off time for a dual-gate configuration was as short as 2.5 ns, which was shorter than that for a no-gate configuration by three to four orders of magnitude. Once the reset pulse was applied to an LED VSTEP, the Schottky diode DS and resistor RG operated as a bypass for the excess internal carriers. The optically written


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Set (a)

Set, Refresh, and Reset Pulses Refresh Reset

(b)

Input Light Power

Input Light

(c)

Output Light

Output Light Power

FIGURE 16.9 Dynamic memory operation of a LED VSTEP: (a) set, refresh, and reset pulses; (b) input light power; (c) output light power.

data would be held by the electric refresh pulses. The data can be regenerated in the form of light emission by applying an electric read pulse. Figure 16.9 shows dynamic memory operation, which involves optical writing, data retained by refresh pulses, regeneration, and erasing. Three refresh pulses with a peak height of 1.4 V and a width of 0.5 µs at an interval of about 7 µs were placed between two set pulses with a peak height of 2.0 V and a width of 1 µs at an interval of 30 µs. These set and refresh pulses were applied to the device repetitively, as shown in Fig. 16.9(a). A negative reset pulse was added just after the second set pulse. Bias voltage of the LED VSTEP Vb was 0.2 V. Optical data were written to the LED VSTEP by illuminating the device simultaneously with the first set pulse. Incident light pulses were generated every two blocks to confirm erasing of the data by the reset pulse, as shown in Fig. 16.9(b). The LED VSTEP was switched on by the trigger light. The ON state and the optical data can be retained with the refresh pulses. The ON state was read out by the second set pulse, and then the LED VSTEP was switched off by the negative reset pulse and the optical data were erased, as shown in Fig. 16.9(c). The consumed electric power of the refresh pulses was as small as 17 µW. The electric power consumed by the electric current flowing through RG was 3 µW. The electric power consumed for the dc bias was much smaller than 1 µW. As a result, the total holding power was about 20 µW. VCSEL To achieve high conversion efficiency from electric energy to optical energy, a VCSEL VSTEP (VC-VSTEP) was developed [6]. Figure 16.10 shows the structure of a


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Anode SiN

325

p-AlAs/GaAs DBR

Cathode

n-AlAs/GaAs DBR n-GaAs Substrate

Laser Beam

FIGURE 16.10 Structure of a VC-VSTEP.

VC-VSTEP. The VC-VSTEP has a pnpn-junction with two DBRs at the top and bottom of the device. The pnpn-junction leads to electrical or optical switching from the OFF-state to the ON-state. During the ON-state, the VC-VSTEP emits laser light. To achieve low oscillation threshold gain, an active region consisting of In0.2 Ga0.8 As strained MQWs is placed at the antinode (loop) of the standing wave in the optical cavity. The cavity space between the top and bottom DBRs is 3λ, and the spaces between the active region and the top and bottom DBRs are 2λ and λ, respectively. Here λ is a wavelength of light in the semiconductor. The undoped active layers function as absorption layers when the VC-VSTEP is switched on optically. Although the active layers are thin, enhancement of absorption is expected because of multireflection by the two DBRs. The epilayers, which were grown on an n-GaAs substrate by MBE, are as follows: an n-Al0.4 Ga0.6 As cathode layer (0.15 µm, Si doped at 2 × 1018 cm−3 ), a p-Al0.25 Ga0.75 As charge sheet layer (5 nm, Be doped at 1 × 1019 cm−3 , an undoped active region consisting of three 10nm-thick In0.2 Ga0.8 As strained QWs separated by two 10-nm-thick Al0.25 Ga0.75 As barriers, two undoped Al0.25 Ga0.75 As spacer layers (0.1 µm) surrounding the active region, an n-Al0.25 Ga0.75 As n-gate layer (0.3 µm, Si doped at 2 × 1017 cm−3 ), and a p-Al0.4 Ga0.6 As anode layer (0.15 µm, Be doped at 5 × 1018 cm−3 ). The undoped layers were p-type and the background carrier concentration was 1 × 1015 cm−3 . The two DBRs consist of an AlAs/GaAs quarter-wave stack with linearly graded transition layers that can reduce resistance. In the bottom n-DBR, the number of periods was 24.5, and Si was doped at 2 × 1018 cm−3 ; in the top p-DBR, the number of periods was 15, and Be was doped at 3 × 1018 cm−3 . A p-GaAs cap and phase-matching layer (0.16λ, Be doped at 1 × 1019 cm−3 ) was grown on the surface of the top DBR.


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FIGURE 16.11

I –V characteristics of a VC-VSTEP.

Wet etching was used to form square mesas with the side of 10, 20, and 30 µm; etching was stopped at the second GaAs layer from the top of the bottom DBR by direct monitoring. SiN was deposited on the wafer to isolate a cathode and an anode; the patterns for these two electrodes were formed by a lift-off technique. The cathode was alloyed but the anode was not alloyed, to avoid degradation of the surface of the top DBR. Figure 16.11 shows I –V characteristics of a VC-VSTEP. The switching voltage Vs and the holding voltage Vh were about 5 and 2.5 V, respectively. The resistances at Vs were 167, 320, and 500 for 30, 20, and 10-µm2 devices, respectively. Figure 16.12 shows the threshold current Ith and differential resistance measured at the threshold current as a function of the device size, which is the side of the device. Measured threshold current density Jth was as low as 1.4 kA/cm2 . The threshold current is linearly dependent on the second power of the side of the device. The differential resistance, which was measured at the threshold current, depends on the reciprocal of the second power of the side of the device. The resistivity of the device


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327

Threshold Current

Differential Resistance at


Device Size FIGURE 16.12 Threshold current and differential resistance of a VC-VSTEP.

ρs , which included metal–semiconductor contact resistance, was only 5.0 × 10−4 · cm2 , and a large part of this resistivity is dominated by the metal–semiconductor contact. Cascadability Optical switching characteristics of a VC-VSTEP were examined in terms of cascadability [9]. Figure 16.13 shows an experimental result for cascadability: applied voltage, which was slightly lower than the switching voltage Vs ; optical power of

(a)

Applied Voltage (5 V/div)

(b)

Incident Light Power

(c)

Current Response (8 mA/div)

FIGURE 16.13 Experimental result for cascadability: (a) applied voltage; (b) incident light power; (c) electric current.


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Switching Energy (pJ)

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Bottom DBR 24.5 pairs

14.5 pairs

Switching Time (ns) FIGURE 16.14 Measured optical switching energy of a VC-VSTEP as a function of switching time with the number of periods of the bottom DBR as a paramater.

incident light; and electric current. As shown in Fig. 16.13, the VC-VSTEP was switched on successfully by the incident light and the electric current flows to the VC-VSTEP in the ON-state. Note that even after the incident light pulse was removed, the VC-VSTEP remained in the ON-state while voltage was applied. This result indicates that the VC-VSTEP functions as an optical memory. In the ON-state, VC-VSTEP can emit a laser beam, and it is possible to cascade the VC-VSTEPs. The optical switching speed was as fast as 10 ns, which was faster than that of a LED VSTEP by two orders of magnitude. Figure 16.14 shows the optical switching energy of a VC-VSTEP measured as a function of the switching time with the number of periods of the bottom DBR as a parameter. The switching energy at a switching time of 10 ns for a VC-VSTEP with a 24.5-pair-bottom DBR was 35.8 pJ; that for a VC-VSTEP with a 14.5-pair-bottom DBR was only 2.2 pJ, which is comparable to the optical switching energy of an LED VSTEP. Another important characteristic of cascadability is the resonant bandwidth of the cavity, which determines the wavelength variations allowed for cascaded VCVSTEPs. Figure 16.15 shows the measured and calculated optical switching energy of a VC-VSTEP as a function of the relative wavelength of light. The filled circles and solid line represent measured and calculated optical switching energies, respectively. Resonant bandwidth, which was defined as the wavelength shift where the optical switching energy doubles the minimum value, was 0.3 nm for longer wavelengths and over 0.5 nm for shorter wavelengths. The wider bandwidth for shorter wavelengths was probably due to a coupling of the light to higher-order transverse modes.

Photon Recycling To achieve low consumption power in a VC-VSTEP, a laser operation with low threshold current and high slope efficiency is needed. By using photon recycling [30],


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Switching Energy (pJ)


329

Calculated

Measured

Wavelength Shift (nm) FIGURE 16.15 Measured and calculated optical switching energy of a VC-VSTEP as a function of the relative wavelength of light.

indistinct threshold pulsed laser operation of the VC-VSTEP was demonstrated [7, 20,22]. Figure 16.16 shows schematic drawings of the operating principles of photon recycling. In the conventional VCSEL structure shown in Fig. 16.16(a), spontaneous emission goes out from the sidewalls of the laser cavity and a relatively large number of the injected carriers are consumed for spontaneous emission other than stimulated emission below threshold. As a result, laser operations in conventional VCSELs have relatively large oscillation threshold currents. In the proposed structure shown in

FIGURE 16.16 Operating principles of photon recycling.


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Fig. 16.16(b), a laser cavity is covered with plated gold, which has high reflectivity, and laser light is emitted from the DBR without gold. As a result, spontaneous emission does not go out from the laser cavity and remains in the laser cavity. Therefore, spontaneous emission is absorbed and carriers are regenerated in the laser cavity. Because of this effect, called photon recycling, threshold current density decreases and indistinct threshold laser operation is obtained. The fraction of all photons emitted spontaneously which are reabsorbed in the cavity, θav , is written [30] θav =

θ (E)rsp (E) dE , rsp (E) dE

(16.1)

where θ (E) is the fraction of photons emitted spontaneously with a photon energy E which is reabsorbed in the active layer, and rsp (E) is the spontaneous emission rate. The effective current density Jeff which is required to reach a given excitation level is Jeff = Jext + ηsp θav Jeff ,

(16.2)

where Jext is the injected current density and ηsp is the quantum efficiency for spontaneous emission. The second term in (16.2) is associated with the reabsorption of photons. At the lasing threshold, Jeff = Jth is satisfied, where Jth is the threshold current density. As a result, the injected current density at the threshold Jext,th is given by Jext,th = Jth 1 − ηsp θav .

(16.3)

From (16.3), the injected current density required to reach threshold is reduced by a fraction ηsp θav . Figure 16.17 shows a cross-sectional view of the structure of a VC-VSTEP for photon recycling. Note that the mesa in the laser cavity is covered with 15 µm thick plated gold and the sidewall has a high power reflectivity of 98%. Spontaneous emission is reflected at the sidewall of the mesa and is retained in the mesa. As a result, spontaneous emission is absorbed in the active layer and the carriers are reproduced. This device has a pnpn structure with DBRs at the top and bottom of the device. Figure 16.18 is a photograph for the top view of the VC-VSTEP for photon recycling. Figure 16.19 shows the I –L characteristics for pulsed operation of a VC-VSTEP for photon recycling. The duty cycle was 1/2 and the pulse width was 0.5 µs. The parameter was temperature. Here, the light output plotted was the peak light output power. Note that origins of I –L curves, which are indicated by short vertical lines, are shifted horizontally. Below 40◦ C, indistinct threshold laser operations were observed; above 50◦ C, laser operations with threshold were obtained. At 20◦ C, the external differential quantum efficiency was as high as 1.7%, even below the threshold current of 2.4 mA in continuous-wave (CW) operation. The extemal differential


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SiN

Anode (Plated Gold)

331

Cathode (Plated Gold) p-AlAs/GaAs DBR

n-GaAs Substrate n-AlAs/GaAs DBR Laser Beam

FIGURE 16.17 Cross-sectional view of the structure of a VC-VSTEP for photon recycling.

quantum efficiency of 1.7% was five times as large as that for CW operation. With an increase in temperature, the slope efficiency in the low-current region decreased, and finally, a threshold appeared above 50◦ C. The reason for this is considered to be as follows: To achieve laser operation, carrier concentration in the active region should be high. However, with an increase in the temperature, injected carriers overflow to the cladding layers from the active region. Overflow of carriers to the cladding layers decreases the carrier concentration in the active region. As a result, an increase in the carrier concentration due to photon recycling is dissipated by the overflow of carriers to the cladding layers, and a threshold appears. Figure 16.20 shows an I –L curve and light output spectra at 0.5, 1.0, and 4.0 mA for pulsed operation of a VC-VSTEP for photon recycling. The temperature was

FIGURE 16.18 Top view of the VC-VSTEP for photon recycling.


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Duty

Light Output (mW)

Pulsed Operation

Injected Current (mA) FIGURE 16.19 I –L characteristics for pulsed operation of a VC-VSTEP for photon recycling with temperature as a parameter.

Light Output (mW)

Pulsed Operation Duty

Wavelength (nm)

Injected Current (mA) FIGURE 16.20 I –L curve and the light output spectra at 0.5, 1.0, and 4.0 mA for pulsed operation of a VC-VSTEP for photon recycling.


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Light Intensity (arb. units)


333

Pulsed Operation Duty

Radiation Angle (deg) FIGURE 16.21 Far-field patterns of a VC-VSTEP for photon recycling.

20◦ C and the duty cycle was 1/2, with a pulse width of 0.5 µs. Light output spectra were measured with an optical spectrum analyzer with a resolution of 0.1 nm. The time-averaged spectral linewidth measured was 0.25 nm. This spectral broadening was caused by a wavelength shift to the longer-wavelength side, because the device was heated within the duration of the electrical pulse. Note that the wavelength of a higher-order transverse mode was shorter than that of the adjacent lower-order transverse mode. The FWHM measured for the reflection spectrum of the DBR cavity was 6 nm. The resonance bandwidth of the photoluminescence spectrum of the VC-VSTEP wafer was also 6 nm. Compared with the resonance bandwidth of the DBR cavity, the FWHM of the light output spectrum was extremely narrow. Even at a low current of 0.5 mA, which was about one-fifth of the threshold current in the CW operation, the coupling efficiency between the light and the optical fiber was maintained large in contrast with the coupling efficiency for spontaneous emission. Figure 16.21 shows the far-field patterns measured at 20◦ C for pulsed operation. The parameter was the current injected. The duty cycle was 1/2 with a pulse width of 0.5 µs. The divergence angle was only 6.8◦ for a 10-µm2 device at injected currents of 2 and 4 mA. Figure 16.22 shows the dependence of polarization for the light output on current injected for pulsed operation at 20◦ C. The duty cycle was 1/2 with a pulse width of 0.5 µs. Polarization of light output was measured by detecting the light output through a Glan–Thomson prism, and the light output intensity ratio P⊥ /P was plotted. Here, P⊥ and P indicate the light output intensity and polarization of P⊥ and P are orthogonal. As shown in Fig. 16.22, the light output was linearly polarized even at a low current level. Below the CW threshold current of 2.4 mA, P⊥ /P was much larger than 1. Note that in conventional lasers, P⊥ /P is about 1 below the threshold current.


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Relative Light Output

Pulsed Operation

Injected Current (mA) FIGURE 16.22 Dependence of polarization for light output on the injected current of a VC-VSTEP for photon recycling.

Double Mesa To obtain low consumption power, it is important to achieve high electronic-optical (E/O) conversion efficiency. Reduction in the electrical resistance by the double mesa structure, efficient confinement of carriers to the active region by the protonimplanted structure, and photon recycling by sidewall reflectors resulted in a high E/O conversion efficiency of 11.4% in a VC-VSTEP [13]. Figure 16.23 shows the structure of a double mesa VC-VSTEP. To achieve high E/O conversion efficiency, both low electrical resistance and high efficiency

Anode

Proton-Implanted Region

p-AlAs/GaAs DBR SiN Cathode

n-GaAs Substrate

n-AlAs/GaAs DBR

Laser Beam

FIGURE 16.23 Structure of a double mesa VC-VSTEP.


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335

E/O Conversion Efficiency (%)

light-output characteristics are required. To decrease electrical resistance, a part of the p-doped DBR was removed and the electric current was injected to the active region from the lower mesa, because the p-doped DBR has high electrical resistance. In addition, the DBR in the lower mesa was highly p-doped at 1 × 1019 cm−3 to obtain ohmic contact between the anode and the top DBR. The upper mesa was 10.5 × 10.5 µm2 , the lower mesa was 52 × 52 µm2 , and the anode was 35 × 35 µm2 . To obtain high-efficiency light-output characteristics, injected carries must be efficiently confined in the active region. Therefore, the active region was surrounded by a proton (H+ )-implanted region, which allows efficient current injection into only the light-emitting region. After the proton-implantation, the VC-VSTEP wafer was annealed at 350◦ C for 15 min to decrease optical absorption in the proton-implanted region, because the propagating light in the optical cavity penetrates to the protonimplanted region. The mesa in the laser cavity region was covered with 15 µm thick plated gold and the sidewall has a high reflectivity of 98%. Spontaneously emitted photons were reftected at the sidewall of the mesa and was retained in the mesa. As a result, spontaneous emission is absorbed in the active layer and the carriers are reproduced. This photon recycling leads to high carrier concentration in the active region, and the threshold current decreases. This device has a pnpn structure between a top DBR with a 15-pair GaAs/AlAs stack and a bottom DBR with an 18.5-pair GaAs/AlAs stack. An active region consisted of three In0.2 Ga0.8 As strained MQWs. Figure 16.24 shows the dependence of E/O conversion efficiency on the device size when the light output was 1 mW. The filled circles, open circles, and filled

Annealed Double Mesa

Unannealed Double Mesa

Single Mesa

Mesa Size FIGURE 16.24 E/O conversion efficiency of a VC-VSTEP. The filled circles, open circles, and filled triangles represent an annealed double mesa, unannealed double mesa, and single mesa, respectively.


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triangles represent the annealed double mesa, unannealed double mesa, and single mesa, respectively. In the double mesa, the size of the upper mesa was plotted. It is found that an optimal size exists in the upper mesa to obtain the highest E/O conversion efficiency. The reason for this is that the driving current decreases while the electrical resistance increases with a decrease in the size of the upper mesa. In Fig. 16.24 it is revealed that double mesa structures have the advantage over single mesa structures of high E/O conversion efficiency. The reason for this is that the electrical resistance is efficiently reduced in double mesa structures compared with single mesa structures. In addition, the annealed double mesa structures show higher E/O conversion efficiency than that in unannealed double mesa structures, because the optical absorption decreases due to thermal annealing.

REFERENCES 1. A. L. Lentine, H. S. Hinton, D. A. B. Miller, J. E. Henry, J. E. Cunningham, and L. M. F. Chirovsky, “Symmetric self-electrooptic effect device: optical set-reset latch, differential logic gate, and differential modulator/detector,” IEEE J. Quantum Electron. 25, 1928 (1989). 2. G. Moddel, K. M. Johnson, W. Li, and R. A. Rice, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537 (1989). 3. K. Kasahara, Y. Tashiro, N. Hamao, M. Sugimoto, and T. Yanase, “Double heterostructure optoelectronic switch as a dynamic memory with low-power consumption,” Appl. Phys. Lett. 52, 679 (1988). 4. Y. Tashiro, N. Hamao, M. Sugimoto, N. Takado, S. Asada, and K. Kasahara, “Vertical to surface transmission electrophotonic device with selectable output light channels,” Appl. Phys. Lett. 54, 329 (1989). 5. T. Numai, I. Ogura, H. Kosaka, M. Sugimoto, Y. Tashiro, and K. Kasahara, “Optical self-routing switch using vertical to surface transmission electrophotonic devices with transmission light amplification function,” Electron. Lett. 27, 605 (1991). 6. T. Numai, M. Sugimoto, I. Ogura, H. Kosaka, and K. Kasahara, “Surface emitting laser operation in vertical to surface transmission electro-photonic devices with a vertical cavity,” Appl. Phys. Lett. 58, 1250 (1991). 7. T. Numai, M. Sugimoto, I. Ogura, H. Kosaka, and K. Kasahara, “Current versus lightoutput characteristics with no definite threshold in pnpn vertical to surface transmission electro-photonic devices with a vertical cavity,” Jpn. J. Appl. Phys. Pt. 2 30, L602 (1991). 8. M. Sugimoto, H. Kosaka, K. Kurihara, I. Ogura, T. Numai, and K. Kasahara, “Very low threshold current density in vertical-cavity surface-emitting laser diodes with periodically doped distributed Bragg reflectors,” Electron. Lett. 28, 385 (1992). 9. I. Ogura, H. Kosaka, T. Numai, M. Sugimoto, and K. Kasahara, “Cascadable optical switching characteristics in vertical-to-surface transmission electrophotonic devices operated as vertical cavity lasers,” Appl. Phys. Lett. 60, 799 (1992). 10. H. Kosaka, I. Ogura, T. Numai, M. Sugimoto, and K. Kasahara, “Dependence of laser characteristics on distributed Bragg reflector pairs in vertical-to-surface transmission electrophotonic devices,” Electron. Lett. 28, 1524 (1992).


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11. K. Kasahara, T. Numai, H. Kosaka, I. Ogura, K. Kurihara, and M. Sugimoto, “Vertical to surface transmission electro-photonic device (VSTEP) and its application to optical interconnection and information processing,” IEICE Trans. Electron. E75-C, 70 (1992). 12. M. Sugimoto, T. Numai, I. Ogura, H. Kosaka, K. Kurihara, and K. Kasahara, “Verticalto-surface transmission electro-photonic device with a pnpn structure and vertical cavity,” Opt. Quantum Electron. 24, S121 (1992). 13. T. Numai, K. Kurihara, I. Ogura, H. Kosaka, M. Sugimoto, and K. Kasahara, “High electronic-optical conversion efficiency in a vertical-to-surface transmission electrophotonic device with a vertical cavity,” IEEE Photon. Technol. Lett. 5, 136 (1993). 14. K. Kurihara, T. Numai, H. Kosaka, I. Ogura, M. Sugimoto, and K. Kasahara, “Determination of power reflectivity of quasi-graded distributed Bragg reflectors using stopband width,” IEEE Photon. Technol. Lett. 5, 333 (1993). 15. T. Numai, T. Kawakami, T. Yoshikawa, M. Sugimoto, Y. Sugimoto, H. Yokoyama, K. Kasahara, and K. Asakawa, “Record low threshold current microcavity surface-emitting laser,” Jpn. J. Appl. Phys. Pt. 2 32, L1533 (1993). 16. H. Kosaka, I. Ogura, H. Saito, M. Sugimoto, K. Kurihara, T. Numai, and K. Kasahara, “Pixels consisting of a single vertical-cavity laser thyristor and a double vertical cavity phototransistor,” IEEE Photon. Technol. Lett. 5, 1409 (1993). 17. K. Kurihara, T. Numai, I. Ogura, A. Yasuda, M. Sugimoto, and K. Kasahara, “Reduction in the series resistance of the distributed Bragg reflector in vertical cavities by using quasi-graded superlattices at the heterointerfaces,” J. Appl. Phys. 73, 21 (1993). 18. H. Kosaka, I. Ogura, M. Sugimoto, H. Saito, T. Numai, and K. Kasahara, “Pixels consisting of double vertical-cavity detector and single vertical-cavity laser sections for 2-D bidirectional optical interconnections,” Jpn. J. Appl. Phys. Pt. 1 32, 600 (1993). 19. K. Kurihara, T. Numai, I. Ogura, H. Kosaka, M. Sugimoto, and K. Kasahara, “Doublemesa-structure vertical-to-surface transmission electro-photonic device with a vertical cavity,” Jpn. J. Appl. Phys. Pt. 1 32, 604 (1993). 20. T. Numai, H. Kosaka, I. Ogura, K. Kurihara, M. Sugimoto, and K. Kasahara, “Indistinct threshold laser operation in a pnpn vertical to surface transmission electro-photonic device with a vertical cavity,” IEEE J. Quantum Electron. 29, 403 (1993). 21. M. Sugimoto, I. Ogura, H. Saito, A. Yasuda, K. Kurihara, H. Kosaka, T. Numai, and K. Kasahara, “Surface emitting devices with distributed Bragg reflectors grown by highly precise molecular beam epitaxy,” J. Cryst. Growth 127, 1 (1993). 22. T. Numai, K. Kurihara, I. Ogura, H. Kosaka, M. Sugimoto, and K. Kasahara, “Effect of sidewall reflector on current versus light-output in a pnpn vertical to surface transmission electro-photonic device with a vertical cavity,” IEEE J. Quantum Electron. 29, 2006 (1993). 23. K. Beyzavi, R. A. Linke, G. E. Devlin, I. Ogura, T. Numai, and K. Kasahara, “Observation of switching energy dependence on illuminating beam area in the VSTEP optoelectronic switch,” IEEE Photon. Technol. Lett. 6, 227 (1994). 24. H. Kosaka, K. Kurihara, A. Uemura, T. Yoshikawa, I. Ogura, T. Numai, M. Sugimoto, and K. Kasahara, “Uniform characteristics with low threshold and high efficiency for a singletransverse-mode vertical-cavity surface-emitting laser-type device array,” IEEE Photon. Technol. Lett. 6, 323 (1994). 25. M. Kajita, T. Numai, K. Kurihara, H. Saito, M. Sugimoto, H. Kosaka, I. Ogura, and K. Kasahara, “Thermal analysis of laser-emission surface-normal optical devices with a vertical cavity,” Jpn. J. Appl. Phys. Pt. 1 33, 859 (1994).


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26. K. Kurihara, T. Numai, T. Yoshikawa, H. Kosaka, M. Sugimoto, Y. Sugimoto, and K. Kasahara, “Uniformity improvement of optical and electrical characteristics in integrated vertical-to-surface transmission electro-photonic device with a vertical cavity,” Jpn. J. Appl. Phys. Pt. 1 33, 1352 (1994). 27. T. Numai, “Surface-emitting optical devices for 2-D integration,” SPIE Proc. 2145, 58 (1994). 28. T. Numai and K. Kasahara, “Low-threshold surface-emitting optical devices,” SPIE Proc. 2147, 122 (1994). 29. T. Numai, K. Kurihara, K. Kühn, H. Kosaka, I. Ogura, M. Kajita, H. Saito, and K. Kasahara, “Control of light-output polarization for surface-emitting-laser type device by strained active layer grown on misoriented substrate,” IEEE J. Quantum Electron. 31, 636 (1995). 30. F. Stern and J. M. Woodwall, “Photon recycling in semiconductor lasers,” J. Appl. Phys. 45, 3904 (1974).


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PART IV SYSTEM DEMONSTRATIONS USING ADVANCED LASER DIODES AND RELATED DEVICES

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17 PHOTONIC SWITCHING SYSTEMS

17.1

INTRODUCTION

For high-capacity lightwave transmission systems, high-speed direct modulation of LDs [1] and multiplexing techniques [2–6] have been demonstrated. The multiplexing techniques fall into three categories as follows: 1. Wavelength-division multiplexing (WDM) [2–4] 2. Time-division multiplexing (TDM) [5] 3. Subcarrier multiplexing (SCM) [6] In WDM systems, by operating each light source at a different peak wavelength, the integrity of the independent messages from each light source is maintained for subsequent conversion to electric signals at the receiving end. Conceptually, the WDM is the same as frequency-division multiplexing (FDM), used in microwave radio and satellite systems. Figure 17.1 shows a unidirectional WDM system in which WDM devices such as a multiplexer and a demultiplexer are used to combine different signal carrier wavelengths onto a single fiber at one end and to separate them into their corresponding detectors at the other end. Figure 17.2 shows a bidirectional WDM system, in which signals with a wavelength λ1 are sent in one direction; signals with a wavelength λ2 are transmitted simultaneously in the opposite direction. In both WDM systems, an optical filter (demultiplexer) is the most important device and the characteristics of the optical filter determine the performance of the WDM systems. Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright

341


342

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PHOTONIC SWITCHING SYSTEMS

Channel

Channel Detector

LD Optical Fiber

Channel LD WDM Device

Channel Detector WDM Device

Channel

Channel Detector

LD

FIGURE 17.1 Unidirectional WDM system that combines n independent input signals for transmission over a single optical fiber.

In TDM systems, a light source is operated at a different time slot and the receiver selects a signal at each time slot. In SCM systems, a number of baseband analog or digital signals are first frequency-division multiplexed by using electrical local oscillators (LOs) of different radio frequencies (RFs). Note that this frequency is not the optical frequency but the electrical frequency. The upconverted signals are then combined to drive a high-speed light source (typically, an LD). The LO frequencies are the subcarriers, in contrast to the optical carrier frequencies. At the receiving end, a user can receive any one of the FDM channels by tuning the electrical LO and downconvert the RF or microwave signals to baseband or intermediate frequencies, similar to the way we tune in radio or TV channels. Note that SCM lightwave systems can carry many more video, data, or voice channels than the radio systems. SCM systems have an advantage over TDM systems in that the services carried by different subcarriers are independent of each other and require no synchronization. In addition, the SCM systems are more cost-effective than the high-capacity TDM. Lightwave transmission systems have been used commercially in telecommunication systems. However, present switching systems, which are essential in telecommunication systems as well as transmission systems, are still composed of conventional electronic devices and cannot exchange optical signals directly. Therefore,

Input Channel Output Channel

Optical Fiber

LD WDM Device Detector

Detector WDM Device LD

Output Channel Input Channel

FIGURE 17.2 Bidirectional WDM system in which signals with different wavelengths are transmitted simultaneously in opposite directions over the same optical fiber.


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INTRODUCTION

343

Light Input Optical Matrix Switch

Light Output

(a)

Time

Light Input

Time

Light Output

T Time Switch MUX

DMUX (b)

Light Input

Wavelength

Wavelength

Light Output

Wavelength Switch MUX

DMUX (c)

FIGURE 17.3 Photonic switching systems: (a) SD; (b) TD; (c) WD.

electronic–optical (E/O) and optical–electronic (O/E) conversions are necessary between lightwave transmission and switching systems. Photonic switching systems, which can exchange optical signals directly without E/O and O/E conversions, are expected to have an advantage over conventional electronic switching systems in that they can exchange broadband signals. Photonic switching systems fall into three categories—space-division (SD), time-division (TD), and wavelength-division (WD) switching systems [7]—as shown in Fig. 17.3. In the SD photonic switching system, each optical cross point guarantees transparent connection between two optical transmission media and the switching system provides widely ranging bit-rate independence for individual channels. With an increase in line capacity, the SD switching system will encounter problems, as it requires


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a rapidly growing number of optical cross points and a huge number of optical fibers for inlets and outlets. TD and WD photonic switching systems have the potential to provide a good solution to these problems. A TD photonic switching system has the inherent advantage of providing good matching with conventional TDM lightwave transmission systems. Key devices for TD switching systems are optical memories. As optical memories, fiber delay lines [8, 9] and bistable LDs [10] were used. However, a lot of problems have to be solved to achieve a large capacity. Because a TD switching system must be aware of each channel bit rate, high-speed optical memories with optical control circuits, optical bit/frame synchronizers, and optical multiplexers and demultiplexers have to be developed to obtain a large capacity. Compared with a TD switching system, a WD switching system has two advantages. One advantage is bit-rate independence for individual wavelength channels. As a result, broadband signals of various speeds can be exchanged without difficulty. The other advantage is that high-speed operation is not required in switching control circuits. In a TD switching system, ultrahigh-speed operations are required for control circuits, in order to exchange TDM broadband signals. For example, to process 32-channel 150-Mbit/s TDM signals, the operation speed of the control circuits for optical memories reaches as high as 4.8 Gbit/s. On the other hand, the WD switching system is controlled by tuning wavelengths. High-speed tuning is not required for the control circuits. In addition, a WD switching system has a potential for extension to a wide-area network in partnership with a WDM lightwave transmission system [3, 4, 11]. The WDM lightwave transmission system also has bit-rate independence for individual channels. As a result, a WD switching system will be able to provide optical bit-rate-independent connection between subscribers. To construct a WD switching system, high-selectivity optical filters, wide-range wavelength-tunable light sources, and wavelength converters are required. To date, WD switching systems using acousto-optic switches [12, 13] were reported. A coherent WD switching system [14] and a coherent WD & TD hybrid switching system [15] were also demonstrated. In addition, optical self-routing switches [16–19], in which the light input itself determines the output port, were studied experimentally. In this chapter, applications of PSC-DFB LDs and filters to WD photonic switching systems and WD & TD hybrid photonic switching systems, which are the first system demonstrations using semiconductor wavelength-tunable optical filters, are explained.

17.2

WAVELENGTH-DIVISION SWITCHING

Figure 17.4 shows a block diagram of a WD photonic switching system [20, 21], consisting of a wavelength multiplexer, a wavelength switch, and a wavelength demultiplexer. In the wavelength-multiplexer, each light is intensity-modulated and combined to form a wavelength-multiplexed signal. In the wavelength switch, the wavelength-multiplexed signal is split and led into each wavelength-tunable optical filter. Each filter selects one wavelength signal, and the signal selected is converted


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WAVELENGTH- AND TIME-DIVISION HYBRID SWITCHING

Fixed Light Source

345

MOD: Modulator O/E: Optical-Electronic Converter

Wavelength-Tunable Optical Filter

Fixed Optical Filter

Input Channel

O/E MOD

Channel

Wavelength Multiplexer

Channel

O/E

Channel

MOD

O/E MOD

Output

O/E

MOD

Wavelength Switch

Wavelength Demultiplexer

FIGURE 17.4 Block diagram of a WD photonic switching system.

to an electronic signal by an O/E converter. A preassigned wavelength light carrier is intensity-modulated by an optical modulator according to an electronic signal from an O/E converter. A wavelength demultiplexer consists of an optical splitter, fixed-wavelength optical filters, and O/E converters. In this system the PSC-DFB LDs were used as light sources; the PSC-DFB filters were used as wavelength-tunable optical filters. Figure 17.5 shows the bit error rate as a function of the input optical power into the PSC-DFB filter. An experimental setup is also sketched at the top of Fig. 17.5. At the bottom the solid line represents the result calculated when the optical gain is assumed to be 12 dB. Eight kinds of plots show the experimental results when each wavelength signal is incident on the PSCDFB filter. Each wavelength is indicated as λ1 , λ2 , . . ., and λ8 , and the wavelength separation between λi and λi+1 (i = 1, 2, . . . , 7) is 0.08 nm. In Fig. 17.5, the filled stars and dashed line represent the bit error rate when the PSC-DFB filter selects one signal with wavelength λ4 from the five-channel wavelength-multiplexed signal whose wavelengths are λ2 to λ6 . The power penalty measured was only 0.5 dB. This small power penalty allows us construct eight-channel WD photonic switching systems. Figure 17.6 shows an output signal waveform (eye pattern) from a PSC-DFB filter for a 200-Mbit/s optical signal. It is found that one wavelength signal was selected successfully by a PSC-DFB filter without degradation.

17.3


A WD & TD hybrid photonic switching system can increase multiplexity, because the multiplexity of the system is given by the product of WD multiplexity and TD multiplexity. In these systems, fast wavelength tuning of both LDs and filters is required. Because the wavelength switching time for PSC-DFB LDs and filters is


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PHOTONIC SWITCHING SYSTEMS MOD: Modulator A.T.T.: Attenuator

Fixed PSC-DFB-LD MOD

O/E

MOD

Tunable PSC-DFB Filter A.T.T.

Bit Error Rate

MOD

With Crosstalk

Light Input Power (dBm)

FIGURE 17.5 Bit error rate as a function of the input optical power into a PSC-DFB filter.

FIGURE 17.6 Output signal waveform (eye pattern) from a PSC-DFB filter.


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347

E/O: Electronic-Optical Converter O/E: Optical-Electronic Converter TD Demultiplexer TD Multiplexer Input Output Tunable E/O Fixed Wavelength-Tunable Electronic Time Switch Optical Filter Optical Filter E/O

O/E

T

O/E

E/O

O/E

T

O/E

E/O

O/E

T

O/E

W & T Hybrid Switch WD & TD Multiplexer

WD & TD Demultiplexer

FIGURE 17.7 Block diagram of a WD & TD hybrid photonic switching system.

as short as 2.5 ns, a 32-channel WD & TD hybrid photonic switching system with a WD multiplexity of 8 and a TD multiplexity of 4 for 100-Mbit/s signals has been demonstrated [22]. Figure 17.7 shows a block diagram of the WD & TD hybrid switching system. This system consists of a WD & TD multiplexer, a wavelength and time (W & T) hybrid switch, and a WD & TD demultiplexer. In the WD & TD multiplexer, input signals of m × n channels are WD (wavelengths λ1 to λn )- and TD (time slots T1 to Tm )-multiplexed, and the multiplexed optical signals are led to the W & T hybrid switch. In a W & T hybrid switch, WD & TD multiplexed signals are split and their parts are led to individual wavelength-tunable optical filters. Each wavelength-tunable optical filter extracts a specific wavelength signal from the input signals at every timeslot. The output signal from the wavelength-tunable optical filter is then converted to an electronic signal. The converted electronic signal is applied to an electronic time switch, which interchanges timeslots on the input line. The output signal from the electronic time switch is sent to a tunable E/O converter, which changes the wavelength of the light output at every timeslot by using a wavelength-tunable LD. Finally, output signals from tunable E/O converters are combined to form an output signal from the W & T hybrid switch. The WD & TD demultiplexer, which consists of fixed optical filters and TD demultiplexers, separates the output signal from the W & T hybrid switch into output signals of m × n channels. To confirm the feasibility of 8-WD and 4-TD hybrid switching systems with a total multiplexity of 32 for 100-Mbit/s signals, fast wavelength tuning experiments were performed using PSC-DFB LDs and filters. Figure 17.8 shows an experimental setup for the fast wavelength tuning of the PSC-DFB filter. Four 100-Mbit/s electronic input signals were TD-multiplexed in a byte-interleaved form with a guard time of 6.7 ns between the adjacent channels. During the guard time, the PSC-DFB filter switches its transmission wavelength. Consequently, the total bit rate reaches 600 Mbit/s. Output lights from PSC-DFB LDs with


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Input TD Multiplexer MOD: Modulator A.T.T.: Attenuator Fixed PSC-DFB LD

Control Signal Generator 1 Output

MOD

O/E A.T.T.

Tunable PSC-DFB Filter

FIGURE 17.8 Experimental setup for fast wavelength tuning of a PSC-DFB filter.

wavelengths of λ1 , λ2 , λ3 , and λ8 were combined and led to the optical modulator. The WD-multiplexed light was intensity-modulated according to the 600-Mbit/s signal from the TD multiplexer using the optical modulator. The WD & TD-multiplexed optical signal, which is shown in Fig. 17.9(a), was incident upon a PSC-DFB filter. Figure 17.9(b) shows the currents injected into a PSC-DFB filter. The electric current Ia injected into the active region controls the optical gain; the tuning current Ip injected into the PC region controls the transmission wavelength. Figure 17.9(c) shows the waveforms of the light outputs, which were selected by a PSC-DFB filter at each time slot. At the time slot T1 , a signal with wavelength λ1 was selected; at the time slot T2 , a signal with wavelength λ2 was selected; and so on. It is found that at each time slot, each wavelength signal was selected without degradation. The bit-error rate for each wavelength channel was less than 10−8 for an output signal from a TD demultiplexer when the filter input power was −28 dBm. Fast wavelength tuning of a PSC-DFB LD was also demonstrated using the experimental setup shown in Fig. 17.10. Light output from the PSC-DFB LD is intensitymodulated according to an electronic signal from a TD multiplexer. The guard time was also set at 6.7 ns for wavelength switching of the PSC-DFB LD. The oscillation wavelength of the PSC-DFB LD was changed sequentially from λ1 to λ8 , as shown in Fig. 17.11(a). The WD & TD-multiplexed optical signal was incident on the fixed optical filters. Figure 17.11(b) to (d) show the light output signal waveforms from each fixed optical filter. From these experimental results it is found that fast wavelength switching of a PSC-DFB LD was demonstrated successfully. The bit-error rate was less than 10−8 for a filter light input power of −28 dBm. From Figs. 17.9 and 17.11, the fast wavelength tuning operations of a PSC-DFB filter and a PSC-DFB LD among four-channel wavelength signals of the eight-channel wavelength signals within a guard time of 6.7 ns were confirmed. These experimental


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349

(a)

(b)

(c)

(a) Input Signals into the PSC-DFB Filter (b) Electric Currents Injected to the PSC-DFB Filter (c) Output Signals from the PSC-DFB Filter FIGURE 17.9 Signal waveforms when the transmission wavelength of a PSC-DFB filter was switched.

Input TD Multiplexer MOD: Modulator A.T.T.: Attenuator Control Signal Generator 2 Output

MOD Tunable PSC-DFB LD

O/E A.T.T. Fixed PSC-DFB Filter

TD Demultiplexer

FIGURE 17.10 Experimental setup for fast wavelength tuning of a PSC-DFB LD.


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(a) (b) (c) (d) (e)

(a) Input Signals into the PSC-DFB Filter (b) Output Signal with the wavelength of (c) Output Signal with the wavelength of (d) Output Signal with the wavelength of (e) Output Signal with the wavelength of FIGURE 17.11 Signal waveforms when the oscillation wavelength of a PSC-DFB LD was switched.

results indicate the feasibility of 8-WD and 4-TD hybrid switching for 100-Mbit/s signals.

17.4

SUMMARY

The WD photonic switching system and WD & TD hybrid photonic switching system, which used semiconductor tunable optical filters for the first time, were demonstrated. These photonic switching systems are single-stage systems. By constructing a multistage system, a larger line capacity can be obtained. For example, using a three-stage system with eight-multiplexity, 1024-line capacity can be achieved. The performance of WD switching systems depends on the characteristics of wavelength-tunable optical filters. Further research and development of wavelengthtunable optical filters will lead to practical use in WD systems, and WD switching systems will play an important role in future broadband wide-area networks.

REFERENCES 1. K. Uomi, “Modulation-doped multi-quantum well (MD-MQW) lasers: I. Theory,” Jpn. J. Appl. Phys. Pt. 1 29, 81 (1990).


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REFERENCES

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2. W. J. Tomlinson, “Wavelength multiplexing in multimode optical fibers,” Appl. Opt. 16, 2180 (1977). 3. K. Ito, Y. Umeda, Y. Sugiyama, K. Nakajima, K. Oshima, and M. Nunoshita, “Bidirectional fibre optic loop-structured network,” Electron. Lett. 17, 84 (1981). 4. J. Conradi, R. Maciejko, J. Straus, I. Few, G. Duck, W. Sinclair, A. J. Springthorpe, and J. C. Dyment, “Laser based WDM multichannel video transmission system,” Electron. Lett. 17, 91 (1981). 5. I. M. McGregor, G. J. Semple, and G. Nicholson, “Implementation of a TDM passive optical network for subscriber loop applications,” IEEE J. Lightwave Technol. 7, 1752 (1989). 6. W. I. Way, “Subcarrier multiplexed lightwave system design considerations for subscriber loop applications,” IEEE J. Lightwave Technol. 7, 1806 (1989). 7. S. Suzuki, K. Nagashima, and B. Hirosaki, “Evolutional optical switching systems for broadband communications networks,” IEEE International Conference on Communications (ICC ’87) 3, 45.3, 1565 (1987). 8. H. Goto, K. Nagashima, S. Suzuki, M. Kondo, and Y. Ohta, “Optical time-division digital switching: an experiment,” Conference on Optical Fiber Communication (OFC ’83), MJ6, 22 (1983). 9. T. Matsunaga and M. Ikeda, “Experimental application of LD switch modules to 256 Mb/s optical time-division switching,” Electron. Lett. 21, 945 (1985). 10. S. Suzuki, T. Terakado, K. Komatsu, K. Nagashima, A. Suzuki, and M. Kondo, “An experiment on high-speed optical time-division switching,” J. Lightwave Technol. 4, 894 (1986). 11. W. J. Tomlinson, “Wavelength multiplexing in multimode optical fibers,” Appl. Opt. 16, 2180 (1977). 12. Y. Shimazu, S. Nishi, and N. Yoshikai, “Wavelength-division-multiplexing optical switch using acoustooptic deflector,” J. Lightwave Technol. 5, 1742 (1987). 13. N. Goto and Y. Miyazaki, “Wavelength-multiplexed optical switching system using acousto-optic switches,” Conference Proceedings GLOBECOM ’87 2, 1305 (1987). 14. M. Fujiwara, N. Shimosaka, M. Nishio, S. Suzuki, S. Yamazaki, S. Murata, and K. Kaede, “A coherent photonic wavelength-division switching system for broad-band networks,” IEEE J. Lightwave Technol. 8, 416 (1990). 15. N. Shimosaka, M. Fujiwara, S. Murata, N. Henmi, K. Emura, and S. Suzuki, “Photonic wavelength-division and time-division hybrid switching system utilizing coherent optical detection,” IEEE Photon. Technol. Lett. 2, 301 (1990). 16. P. R. Prucnal, D. B. Blumenthal, and P. A. Perrier, “Self-routing photonic switching demonstration with optical control,” Opt. Eng. 26, 473 (1987). 17. H. Hagishima and Y. Doi, “An optical self-routing switch using multiwavelength matching,” IEEE International Conference on Communications (ICC ’89), 23.5, 745 (1989). 18. M. Hashimoto, M. Fukui, and K. Kitayama, “Self-routing optical crossbar switch,” IEEE Photon. Technol. Lett. 2, 522 (1990). 19. T. Numai, I. Ogura, H. Kosaka, M. Sugimoto, Y. Tashiro, and K. Kasahara, “Optical self-routing switch using vertical to surface transmission electrophotonic devices with transmission light amplification function,” Electron. Lett. 27, 605 (1991).


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20. M. Nishio, T. Numai, S. Suzuki, M. Fujiwara, M. Itoh, and S. Murata, “Eight-channel wavelength-division switching experiment using wide-tuning-range DFB LD filters,” 14th European Conference on Optical Communication (ECOC ’88), Pt. 2, 49 (1988). 21. S. Suzuki, M. Nishio, T. Numai, M. Fujiwara, M. Itoh, S. Murata, and N. Shimosaka, “A photonic wavelength-division switching system using tunable laser diode filters,” IEEE J. Lightwave Technol. 8, 660 (1990). 22. M. Nishio, S. Suzuki, N. Shimosaka, T. Numai, T. Miyakawa, M. Fujiwara, and M. Itoh, “An experiment on photonic wavelength-division and time-division hybrid switching,” 2nd Topical Meeting on Photonic Switching, ThE5, 98 (1989).


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18 OPTICAL INFORMATION PROCESSING

18.1

INTRODUCTION

It is expected that introduction of optical interconnections into electronic data processing systems accelerates developments in massively parallel computers, because limitations in the flexibility of electronic interconnection are overcome. Required characteristics of optical devices for optical interconnections are different from those for optical fiber communication systems. For optical fiber communication systems, a lot of effort has been dedicated to increasing the data transmission rate and the transmission distance. For optical interconnections, data are transmitted in short distances between processor chips or circuit boards. The typical distance between circuit boards is a few centimeters, and the space permitted for connection hardware is limited. As a result, optical-switching devices, which have a threshold function, are quite suitable to optical interconnections, because a high-density optical connection can be obtained easily without integrating electronic decision circuits. If two-dimensional opticalswitching device arrays are used, it is expected that a high transmission capacity is achieved in the optical interconnections. As passive optical-switching devices, a SEED [1] based on MQW optical modulators and a ferroelectric liquid-crystal spatial light modulator [2] were developed. Using a SEED, Dickinson and Prise demonstrated an optical interconnection system between electronic circuit modules [3]. A ferroelectric liquid-crystal spatial light modulator can be used in image data-processing systems. Note that these passive optical-switching devices cannot emit light, and external light sources are required


353


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OPTICAL INFORMATION PROCESSING

to read out the status of the optical-switching devices. As a result, it is difficult to construct a compact optical system by using only passive optical-switching devices. As active optical-switching devices, which can emit light, optical functional devices with pnpn-junctions [4–7] were developed. These active optical-switching devices have multiple functions, such as light emission, light absorption, thresholding, and latching. This multifunctional feature makes it possible to construct a cascaded interconnection easily compared with passive optical-switching devices, because external light sources for read-out are not necessary. In this chapter, applications of active optical-switching devices to serial-to-parallel data conversion, an optical self-routing switch, an optical ATM switch, an optical interconnection, an optical memory, and an optical bus are described.

18.2

SERIAL-TO-PARALLEL DATA CONVERSION

For future parallel information processing it is necessary to prepare a device that can perform serial-to-parallel data conversion. In addition, a reconfigurable optical interconnection, which is shown in Fig. 18.1, will be needed [8]. In Fig. 18.1, electrical address signals are applied parallel to the anode lines of a VSTEP array. These signals are synchronized with electrical control signals, which are applied sequentially to the cathode lines. The voltages of addressing signals Va and control signals Vc are below the switching voltage Vs of each element of the VSTEP array (Va , Vc < Vs ); only when the addressing and control signals are applied simultaneously to an element of a VSTEP array and satisfies Va + Vc ≥ Vs is an element of the VSTEP array switched on. If the number of elements in a line and a column is both N , N 2 optical interconnections can be reconfigured in N time slots and 2N electronic control lines.

Electrical Control Signal 2-D VSTEP Array Electrical Address Signal PD Array Electrical Data Signal

Optical Signal

Output

FIGURE 18.1 Reconfigurable optical interconnection.


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OPTICAL SELF-ROUTING SWITCH

355

Electronic Circuit Pilot Signal Bias Voltage

Detector

Data Signals Reset Frag

Set Frag Device 2 Data Signals Light Input

Device 1 Light Output

FIGURE 18.2 Optical self-routing switch.

18.3

OPTICAL SELF-ROUTING SWITCH

An optical self-routing switch that uses an edge-emitting optical functional device has been demonstrated [6]. This switch, in which the light input itself determines the output port, has a high internal optical gain of more than 20 dB and a wavelength filtering function. Figure 18.2 shows an optical self-routing switch in which light input consists of a pilot signal and data signals. The pilot signal, which determines the destination for the data signals by switching on edge-emitting optical functional devices, is placed in front of the data signals. Switching from the OFF-state to the ON-state of edgeemitting optical functional devices depends on the timings of the bias voltage and pilot signal. To generate the bias voltage at time t1 for device 1 and at t2 for device 2, an optical set flag is placed in front of the pilot signal. When the optical set flag arrives at a detector, an electronic circuit generates the bias voltage. The timings for applying the bias voltages to device 1/2 are preassigned according to the profile of the optical set flag. Device 1 is switched on when the incident pilot signal is synchronized with the bias voltage at t1 . Note that only the pilot signal can switch on device 1; the data signals cannot. In this method, the bias voltage must be higher when the pilot signal enters the device than when the data signals enter. The bias voltage and the resistance connected to device 1 are set at certain values so that device 1 may be biased below the threshold current once it is switched on. As a result, device 1 plays a role as an optical amplifier during the ON-state. Because both facets of device 1 are as cleaved, device 1 selectively amplifies the data signals whose wavelength fits the resonant wavelength in the optical cavity of device 1. The resonant wavelength is tuned by controlling the electric current flowing through device 1 due to the free carrier plasma effect. As a result, WDM signals can be used as the data signals. Device 2, whose incident pilot signal is not synchronized with the bias voltage, remains in the OFF-state. During the OFF-state, the data signals are not transmitted but are absorbed in device 2.


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Device 1

Device 2

Bias Voltage Light Input

Light Output

Pulse Width: FIGURE 18.3 Experimental result of an optical self-routing switch.

To switch off device 1/2, an optical reset flag is placed after the data signals. When the optical reset flag is detected, an electrical reset pulse, which has a negative voltage, is applied to device 1/2. Figure 18.3 shows an experimental result of the optical self-routing switch. The light input power for the data signals was −25.4 dBm and the pulse width was 20 ns. When the pilot signal came to device 1/2, the bias voltage of device 1 is high; the bias voltage of device 2 is low. As a result, only device 1 is switched from the OFFstate to the ON-state and the data signals are transmitted thorough device 1; device 2 remains in the OFF-state and data signals are not transmitted thorough device 2. For a resonant wavelength, the internal optical gain for the data signals was 21.4 dB when the flowing current was set slightly below the threshold current. For an offresonant wavelength, transmission of the data signals was suppressed because optical signals were absorbed in device 2. The light extinction ratio between the resonant wavelength and the off-resonant wavelength was as large as 14.7 dB. In addition, the light extinction ratio at the resonant wavelength between the ON- and OFF-states was larger than 30 dB. From Fig. 18.3 it is found that the optical-self routing switch was demonstrated successfully using edge-emitting optical functional devices [6].

18.4

OPTICAL ATM SWITCH

In typical conventional photonic ATM switches, electronic circuits were used to control the optical matrix switch. As a result, it is necessary to convert the optical header to the electrical header, and also to generate the electrical control signal for the optical matrix switch by translating the electronic header. Therefore, the switching speed is limited by the operating speed of the electronic control circuit. In addition, N 2 electronic control signals are required for the N × N optical matrix switch. With an increase in the numbers of input and output ports, the electronic control circuit becomes quite complicated because of the rapidly growing number of high-speed electronic circuits.


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OPTICAL ATM SWITCH

357

Optical Buffer Memory

Optical Self-Routing Circuit FIGURE 18.4 Photonic input buffer ATM switch.

For a buffer memory, an electronic buffer memory or an optical fiber delay line were generally used in conventional photonic ATM switches. When an electronic buffer memory was used, the switching speed in the photonic ATM switch was limited by the operating speed of the electronic buffer memory. When an optical fiber delay line, which can store extremely high-speed optical signals, was used, precision of length in the delay line was strictly required with an increase in the cell signal speed. VSTEPs can be used as VCSELs, detectors, optical memories, optical amplifiers, and optical gates. In addition, the VSTEPs can easily be integrated in a twodimensional array. As a result, it is expected that the VSTEP-based optical buffer memory [9] using massively parallel optical interconnections is a solution to achieving ultrahigh throughput in the buffer. Figure 18.4 shows a photonic input buffer ATM switch, which consists of optical first-in first-out (FIFO) buffer memories and an N × N optical self-routing circuit. When the cells bound for the same output are sent to the self-routing circuit from the different FIFO buffer memories at the same time, a cell with the highest priority can be self-routed; the other cell signal is rejected to prevent cell contention. It is expected that a high-speed optical buffer memory can be constructed by converting serial cell signals to parallel signals and by using massively parallel optical interconnections. Figure 18.5 shows an optical buffer memory using VSTEPs, which consists of an optoelectronic shift register and a two-dimensional VSTEP array memory. The optoelectronic shift register consists of optoelectronic serial interfaces and electronic shift registers, as shown in Fig. 18.6. An input optical cell signal with p-bit cell length is sent to the electronic write shift register after O/E conversion and is converted to a p-bit parallel signal configuration. The electrical p-bit parallel signal is converted to the optical parallel signal by a one-dimensional VSTEP array through the electronic write latch and a driver circuit for the VSTEP. In the write period, the optical parallel signal from the one-dimensional VSTEP array is distributed to the d-column × p-row two-dimensional VSTEP array memory, and stored


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Opto-Electronic Shift Register From Input Optical Cell

To Output Optical Cell

2-D VSTEP Array Memory

FIGURE 18.5 Optical buffer memory using VSTEPs.

in one of the d columns. In the read period, the optical p-bit parallel signal is returned to the one-dimensional VSTEP array. The optical signal, which was sent to the one-dimensional VSTEP array, is converted to an electrical signal by detecting the current flowing through each VSTEP of the one-dimensional array using electric current detectors. The electrical parallel signal from the electric current detectors is sent to an electronic read latch. In Fig. 18.6 the parallel signal for the special header pulse, which is inserted for the self-routing operation, is generated by translating the header of the parallel configuration cell signal from the read latch at the header translator. The parallel signal from the read latch and that from the header translator are sent to the electronic read shift register together and are converted to a serial electrical signal. Finally, the serial signal is converted from an electrical signal to an optical signal, and subsequently is sent to the optical self-routing circuit. In the optical buffer memory shown in Fig. 18.5, the signal write and read operations are accomplished in the first and second half of the cell transfer time,

Electronic Write Shift Register

Electronic Read Shift Register

From Input Optical Cell

To Output Optical Cell

PD

LD Head Translator

Electronic Write Latch

Electronic Read Latch

Driver Circuit for VSTEP Electric Current Detector

1-D VSTEP Array

2-D VSTEP Array Memory

FIGURE 18.6 Optoelectronic shift register.


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OPTICAL INTERCONNECTION

359

respectively. As a result, the optical signal speed in the write or read operation can be reduced to about B × 2/ p, where B is the cell signal speed. The minimum duration time required for the write operation for a two-dimensional VSTEP array memory is determined mainly by the switching delay time required to switch on the VSTEP when an optical signal is sent to the two-dimensional array memory from the one-dimensional VSTEP array. The minimum duration time for the read operation is also determined mainly by the switching delay time required to switch on the VSTEPs in the one-dimensional array when the optical signal is sent to the one-dimensional array from the two-dimensional array memory. Because the switching energy of the VSTEP is on the order of a few picojoules, the cell signal speed can reach gigabits per second when the output power of each VSTEP is 1 mW.

18.5


A single VSTEP circuit and a differential VSTEP circuit for optical interconnections have been investigated [10]. The single VSTEP circuit consists of a single VSTEP in which the ON and OFF states stand for 1 and 0, respectively. The differential VSTEP circuit employs dual-rail logic, which has both normal and complementary signals. Figure 18.7 shows a differential VSTEP circuit that consists of two parallel connected VSTEPs and a common load resistance. When an input light power into VSTEP-1 is higher than that into VSTEP-2, only VSTEP-1 is switched on. Once VSTEP-1 is in the ON state, it is difficult to switch on VSTEP-2 because of the voltage drop through the common load resistance. The superiority of the differential VSTEP circuit to the single VSTEP circuit is that the optical-switching energy of the differential VSTEP circuit is lower than that of a single VSTEP circuit, because the light input should only have to make a small difference between the switching voltages of the two Voltage Supply

Electric Current 1 VSTEP-1

Electric Current 2 p n

p n

p

p

n

n

Light Output 1

Light Input 1

VSTEP-2

Light Output 2

Light Input 2

FIGURE 18.7 Differential VSTEP circuit.


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Lens

Lens

VSTEPs (Light Sources)

VSTEPs (Detectors)

FIGURE 18.8 Experimental setup for measuring the characteristics of optical interconnection.

VSTEPs. Figure 18.8 shows an experimental setup for measuring the characteristics of optical interconnection. Light outputs from VSTEPs (light sources) are imaged onto the other VSTEPs (detectors). The numerical aperture for both coupling lenses is 0.2, and the transmission efficiency for light power between the light sources and the detectors is more than 50%. Figure 18.9 shows timing diagrams for the optical interconnections. Voltages are applied to the VSTEPs repeatedly with the set and reset cycles. The set voltage for the single VSTEP circuit is about 90% of the unilluminated switching voltage (i.e., 5.4 V). The VSTEP is switched on only when Period Set

Applied Voltage

Reset Light Input Light Output (a) Set

Applied Voltage

Reset Light Input 1 Light Input 2 Light Output 1 Light Output 2 (b) FIGURE 18.9 Timing diagrams for optical interconnections: (a) single-VSTEP circuit; (b) differential VSTEP circuit.


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361

Applied Voltage Electric Current

Light Input

(a) Applied Voltage Electric Current

Light Input

(b) FIGURE 18.10 Measured waveform for a single-VSTEP circuit: (a) 10 MHz; (b) 17 MHz.

the set voltage and the light input exist simultaneously. The light input is turned off after the VSTEP is switched on. The set voltage for the differential VSTEP circuit was 130% of the unilluminated switching voltage (i.e., 8 V). If VSTEP-1 is switched on, VSTEP-2 remains in the OFF-state. The reset voltage was about −5 V for both circuits to accelerate the extraction of the remaining carriers in the VSTEPs. Figure 18.10 shows the waveforms measured for a single-VSTEP circuit. An ON-state where the electric current flows to the VSTEP was observed just after the light was input to the VSTEP. Synchronous changes in the electric current with the applied voltage were caused by both the junction capacitance of the OFF-state and parasitic capacitance. These capacitances limit the operating frequency because applied voltages are degraded, as shown in Fig. 18.10(b). Figure 18.11 shows the waveforms measured for a differential VSTEP circuit. The bottom trace in Fig. 18.11(a) represents the superposed light intensity for two light inputs; the bottom trace in Fig. 18.11(b) represents the superposed light intensity for light input and output. The change in the electric current for the VSTEP was almost the same as that for the single-VSTEP circuit. The maximum operating frequency measured for the differential circuit was 25 MHz. To extend the number of connections, multiphase driving is needed. Figure 18.12 shows a timing diagram of the driving voltages for cascaded connections. Each phase of the driving voltage for the neighboring array is shifted by a quarter of the period.


362

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Applied Voltage

Electric Current A Electric Current B Light Input (a) Applied Voltage

Electric Current A Electric Current B Light Input/Output (b) FIGURE 18.11 Waveform measured for a differential VSTEP circuit: (a) 10 MHz; (b) 20 MHz.

The optical data signals are latched at the rising point of each pulse. By changing the phase of the driving voltage, a bidirectional connection can be obtained.

18.6

OPTICAL MEMORY

A VSTEP has multiple functions, such as light emission, light absorption, thresholding, and latching. These features allow us to construct an optical pulse repeater easily. In this section, an optical fiber loop memory that uses a VSTEP as a simple optical pulse repeater [11] is described. The VSTEP can reshape a degraded optical pulse and make it possible to achieve a long storage time without any circuitry. Figure 18.13 shows an optical fiber loop memory consisting of a VSTEP, an optical fiber loop, a beamsplitter prism, and coupling lenses. Operations of this memory are as follows: To store optical pulses in the memory loop, the VSTEP is triggered by optical pulses from the input/output port. The optical output from the VSTEP is divided by the beamsplitter and is coupled to both facets of the optical fiber loop. The divided optical pulses propagate in the optical fiber in opposite directions. Half of the optical fiber outputs become the output optical pulses for the input/output port and the other half are fed back to the VSTEP. These feedback optical pulses switch on the VSTEP to repeat the optical pulses.


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OPTICAL MEMORY

363

Data Transition Array 1

Array 2

Array 3

Array 4

Array 5 Period FIGURE 18.12 Timing diagram of the driving voltages for cascaded connections.

Figure 18.14 shows an optical fiber loop memory in which the connection optics are replaced by an optical fiber coupler and a butt coupling component, which is suitable for a VC-VSTEP. In the optical fiber loop memory, all operations are synchronized by the master clock cycle. The VSTEP is driven by an electrical clock pulse, which repeats the set and reset voltages. The set voltage is about 90% of the switching voltage and the reset voltage is below 0 V. When the applied voltage is at a set level and the optical input energy exceeds the threshold, the VSTEP is switched on and an optical

Input/Output Port

VSTEP Lens Beamsplitter

Optical Fiber

FIGURE 18.13 Optical fiber loop memory which uses a beamsplitter prism and coupling lenses.


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Butt Coupling Fiber Coupler VSTEP

Input/Output Port Optical Fiber

FIGURE 18.14 Optical fiber loop memory which uses an optical fiber coupler and a butt coupling component.

pulse is emitted from the VSTEP. This optical pulse becomes a repeated pulse in the optical fiber loop memory. The VSTEP is switched off when the reset voltage is applied to the VSTEP. If the light input does not exist, the VSTEP remains in the OFF-state. Figure 18.15 shows a timing diagram for a 2-bit memory. This diagram represents a series of data storing, optical pulse repetitions, and erasing. The electrical clock pulse is applied continuously to the VSTEP, except for the erasing operation. Input optical pulses from the input/output port function as triggers to store optical pulses in loop memory. These input optical pulses become the first feedback optical pulses to the VSTEP and the first output optical pulses to the input/output port. The feedback optical pulses overlap the former part of the set voltage pulse with a time width of ts . In the latter part of the set voltage pulse, the VSTEP emits repeated optical data pulses. The repeated optical pulses become the next feedback optical pulses after the loop delay time T . In Fig. 18.15, the stored optical pulses are repeated two times. The stored optical pulses are erased by removing the applied clock pulse.

Store

Repeat

Erase Set Reset

Electric Clock for VSTEP Input Optical Pulse Output Optical Pulse/ Feedback Optical Pulse Repeated Optical Pulse Loop Delay FIGURE 18.15 Timing diagram for a 2-bit memory.


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OPTICAL BUS

365

In an optical fiber loop memory using the VSTEP, a storage time of over 30 min for a clock frequency of 12 MHz was confirmed. This means that the bit error rate for one repetition cycle is less than 10−10 . The maximum clock frequency was 20 MHz.

18.7

OPTICAL BUS

Figure 18.16 shows a free-space optical bus using VSTEP arrays [10]. By connecting the end arrays, a loop structure is given to the optical bus. The electronic processing circuits have optical interfaces, which consist of a VSTEP array. One interface VSTEP array is connected optically to a neighboring VSTEP array and functions as an optical-data transmitter, an optical-data detector, and an optical-data repeater without an electronic decision circuit and a read-out light source. For example, as operations of the free-space optical bus, output data signals from each interface are transferred to the interface one after another until all the interfaces detect data signals. In the interconnection scheme shown in Fig. 18.16, the data signals can be transmitted to distant optical interfaces beyond the diffraction limit of the free-space optical connection, because these interfaces can be used as repeaters. When the transmission distance is 20 mm, which is a typical distance between electronic circuit boards, the diameter of the coupling lens is about 2 mm. Over 100 connections can be obtained using this lens. The repeater operation is performed as follows: The interface switch detects optical-data signals, latches onto them, and then emits the same data signals. This function makes it possible to broadcast a set of data to other circuits by using low light output. Because the latched data signals are detected simultaneously by the electronic processing circuit connected to the interface switch, it is not necessary to use high light output for broadcasting. The fan-out of the interface is only one for unidirectional buses and two for bidirectional buses. Note that the timing of the interface clock has to be controlled precisely by taking into account the delay accompanying a cascaded connection.

Electronic Processing Circuits

Optical Interfaces (VSTEPs) Lenses Grating Beam splitters Mirrors FIGURE 18.16 Free-space optical bus using VSTEP arrays.


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In Fig. 18.16, grating beamsplitters whose diffraction angles are sensitive to light wavelength are used. However, the change in diffraction angle can be canceled out when the light passes through two identical grating beamsplitters. As a result, the focused spot positions on the VSTEP arrays are not affected by the change in light wavelength.

REFERENCES 1. A. L. Lentine, H. S. Hinton, D. A. B. Miller, J. E. Henry, J. E. Cunningham, and L. M. F. Chirovsky, “Symmetric self-electrooptic effect device: optical set–reset latch, differential logic gate, and differential modulator/detector,” IEEE J. Quantum Electron. 25, 1928 (1989). 2. G. Moddel, K. M. Johnson, W. Li, and R. A. Rice, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537 (1989). 3. A. Dickinson and M. E. Prise, “Free-space optical interconnection scheme,” Appl. Opt. 29, 2001 (1990). 4. K. Kasahara, Y. Tashiro, N. Hamao, M. Sugimoto, and T. Yanase, “Double heterostructure optoelectronic switch as a dynamic memory with low-power consumption,” Appl. Phys. Lett. 52, 679 (1988). 5. Y. Tashiro, N. Hamao, M. Sugimoto, N. Takado, S. Asada, and K. Kasahara, “Vertical to surface transmission electrophotonic device with selectable output light channels,” Appl. Phys. Lett. 54, 329 (1989). 6. T. Numai, I. Ogura, H. Kosaka, M. Sugimoto, Y. Tashiro, and K. Kasahara, “Optical self-routing switch using vertical to surface transmission electrophotonic devices with transmission light amplification function,” Electron. Lett. 27, 605 (1991). 7. T. Numai, M. Sugimoto, I. Ogura, H. Kosaka, and K. Kasahara, “Surface emitting laser operation in vertical to surface transmission electro-photonic devices with a vertical cavity,” Appl. Phys. Lett. 58, 1250 (1991). 8. K. Kasahara, T. Numai, H. Kosaka, I. Ogura, K. Kurihara, and M. Sugimoto, “Vertical to surface transmission electro-photonic device (VSTEP) and its application to optical interconnection and information processing,” IEICE Trans. Electron. E75-C(1), 70 (1992). 9. M. Nishio, S. Suzuki, K. Takagi, I. Ogura, T. Numai, K. Kasahara, and K. Kaede, “A new architecture of photonic ATM switches,” IEEE Commun. Maga. 31, 62 (1993). 10. Y. Yamanaka, K. Yoshihara, I. Ogura, T. Numai, K. Kasahara, and Y. Ono, “Free-space optical bus using cascaded vertical-to-surface transmission electrophotonic devices,” Appl. Opt. 31, 4676 (1992). 11. Y. Yamanaka, T. Numai, K. Kasahara, and K. Kubota, “Optical fiber loop memory using vertical to surface transmission electro-photonic device,” IEEE J. Lightwave Technol. 11, 2140 (1993).

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APPENDIX A DENSITY OF STATES

Wave Vector and the Number of States A wave function ψ and an energy eigenvalue E are solutions of the Schrödinger equation. In the x yz-coordinate system, a steady-state Schrödinger equation where the energy is independent of time is written

2 2 − ∇ + V (r) 2m ∇=

ψ = Eψ,

∂ ∂ ∂ ˆy + xˆ + zˆ , ∂x ∂y ∂z

∇2 = ∇ · ∇ =

∂2 ∂2 ∂2 + + . ∂x2 ∂ y2 ∂z 2

(A.1) (A.2)

(A.3)

Here m is the mass of a particle; V (r) is a potential energy; ∇ is a differential operator called nabla; and xˆ , ˆy, and zˆ are unit vectors in the positive directions of x, y, and z axes, respectively. Generally, the Schrödinger equation can have multiple wave functions as its solutions. Because a wave function ψ stands for a state, the number of states W is given by the number of wave functions.


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(a)

(b) FIGURE A.1 Boundary conditions for a particle that can move in three-dimensional space: (a) boundary conditions at rigid walls; (b) periodic boundary conditions.

From (A.1), √ a wave function ψ has the form of sin(K ·r), cos(K ·r), or exp(±iK ·r), where i = −1 is the imaginary unit. Here K is a wave vector; its magnitude is |K | = K = 2π/λ, where λ is a wavelength, and its direction represents the propagation direction of the wave. The wave function ψ is a function of the wave vector K , and the wave function with a different amplitude and the same K expresses a common state. As a result, the number of wave functions is the same as the number of wave vectors. Therefore, the number of wave vectors is the number of states W .

Boundary Conditions in Three-Dimensional Space Let ψ(x, y, z) represent a wave function for a particle, which can move in a threedimensional space. Figure A.1(a) shows boundary conditions for a particle, which can move in a cube with a side of L surrounded by rigid walls (boundary conditions at rigid walls); Fig. A.1(b) shows boundary conditions for a particle that can move freely in three-dimensional space (periodic boundary conditions). Note that a cube with a side of L is a virtual space in a periodic boundary condition.


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369

Boundary Conditions at Rigid Walls in Three-Dimensional Space First, under the boundary conditions at rigid walls, which are shown in Fig. A.1(a), the number of states W is calculated. The particle is completely confined in the cube; the particle cannot be present outside the cube. As a result, ψ(x, y, z) = 0 is satisfied outside the cube. By reaching the rigid walls of the cube from outside, the particles are not present. Therefore, the boundary conditions at the rigid walls are written ψ(0, y, z) = ψ(L , y, z) = 0, ψ(x, 0, z) = ψ(x, L , z) = 0, ψ(x, y, 0) = ψ(x, y, L) = 0.

(A.4)

The wave function ψ(x, y, z) that satisfies the boundary conditions given by (A.4) is expressed as ψ(x, y, z) = ψ0 sin (K x x) sin K y y sin (K z z) , π π π K x = nx , K y = n y , Kz = nz , L L L n x , n y , n z = 1, 2, 3, . . . ,

(A.5) (A.6) (A.7)

where K = K x , K y , K z and r = (x, y, z). Note that n x , n y , and n z satisfy n x = 0, n y = 0, n z = 0.

(A.8)

The reason for this is that if one of n x , n y , and n z becomes 0, ψ(x, y, z) = 0 is obtained in all positions in the cube, and the wave function is not present in the cube. Positive and negative values of n x , n y , and n z represent the same wave function because the positions of the nodes of the wave functions are common irrespective of the signs of n x , n y , and n z . For simplicity, positive values are selected for n x , n y , and n z in the following. Let us consider three-dimensional space with the axes K x , K y , and K z , and let a maximum length of the wave vector be K , as shown in Fig. A.2(a). Because positive values are selected for n x , n y , and n z , K x , K y , and K z have positive values. When the initial point of the wave vector K , which satisfies |K | = K , is placed on the origin of the K x , K y , and K z axes, the terminal point of K can move on the surface of one-eighth of the sphere with a radius K . Therefore, a terminal point of a wave vector whose length is less than K is present inside one-eighth of the sphere with a radius K . If K satisfies K π/L, the number of states W is written 1 4π 3 π 3 K3 3 · K ÷ = L , 8 3 L 6π 2 π 3 1 4π 3 = · K , VLR = , 8 3 L

W = VK R ÷ VLR =

(A.9)

VK R

(A.10)


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DENSITY OF STATES

(a)

(b)

FIGURE A.2 States for a particle that can move in three-dimensional space: (a) boundary conditions at rigid walls; (b) periodic boundary conditions.

where VK R is a volume of one-eighth of the sphere with a radius K , and VLR is the volume of a minimum cube formed by the terminal points of the wave vectors.

Periodic Boundary Conditions in Three-Dimensional Space Second, under periodic boundary conditions, which are shown in Fig. A.1(b), the number of states W is calculated. Assuming that the values of the wave functions are the same at the virtual walls, the periodic boundary conditions are written ψ(0, y, z) = ψ(L , y, z), ψ(x, 0, z) = ψ(x, L , z),

(A.11)

ψ(x, y, 0) = ψ(x, y, L). The wave function that satisfies the boundary conditions given by (A.11) is expressed as ψ(x, y, z) = ψ0 exp (iK x x) exp iK y y exp (iK z z) , 2π 2π 2π Kx = nx , Ky = ny , Kz = nz , L L L n x , n y , n z = 0, ±1, ±2, ±3, . . . .

(A.12) (A.13) (A.14)


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371

Let us consider a three-dimensional space with the axes K x , K y , and K z , and let a maximum length of the wave vector be K , as shown in Fig. A.2(b). When the initial point of K is placed on the origin of the K x , K y , and K z axes, the terminal point of K can move on the surface of the sphere with a radius K . Therefore, a terminal point of a wave vector whose length is less than K is present inside the sphere with a radius K . If K satisfies K 2π/L, the number of states W is written W = VK P ÷ VLP VK P =

2π 3 K3 3 = L , L 6π 2 3 2π = , L

4π 3 = K ÷ 3

4π 3 K , VLP 3

(A.15) (A.16)

where VK P is a volume of the sphere with radius K and VLP is a volume of a minimum cube formed by the terminal points of the wave vectors. From (A.9) and (A.15) it is found that the number of states W is the same irrespective of the boundary conditions. Density of States in Three-Dimensional Space Using the wave vector K , angular frequency ω, and energy E, the densities of states D(K ), D(ω), and D(E) are defined as dW = D(K ) dK = D(ω) dω = D(E) dE,

(A.17)

where dW is a small change in the number of states. The density of states D(K ) represents the number of states per unit wave vector, D(ω) represents the number of states per unit angular frequency, and D(E) represents the number of states per unit energy. From (A.9), (A.15), and (A.17), the density of states D(K ) for a particle, which can move in three-dimensional space, is expressed as D(K ) =

dW K2 3 = L . dK 2π 2

(A.18)

From (A.18), the density of states per unit volume DV (K ) is obtained as DV (K ) =

D(K ) K2 = . L3 2π 2

(A.19)

Boundary Conditions at Rigid Lines in Two-Dimensional Space Figure A.3(a) shows the boundary conditions at rigid lines for a particle that can move in two-dimensional space. The particle is completely confined in the square;


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DENSITY OF STATES

(a)

(b) FIGURE A.3 Boundary conditions for a particle that can move in two-dimensional space: (a) boundary conditions at rigid lines; (b) periodic boundary conditions.

the particle cannot be present outside the square. Therefore, the boundary conditions at the rigid lines are written ψ(0, y) = ψ(L , y) = 0, ψ(x, 0) = ψ(x, L) = 0.

(A.20)

The wave function that satisfies the boundary conditions given by (A.20) is expressed as ψ(x, y) = ψ0 sin (K x x) sin K y y , π π Kx = nx , K y = n y , L L n x , n y = 1, 2, 3, . . . .

(A.21) (A.22) (A.23)

When the initial point of K is placed on the origin of the K x and K y axes, the terminal point of K can move on the edge of one-fourth of a circle with radius K . Therefore, a terminal point of a wave vector whose length is less than K is present inside one-fourth of a circle with radius K , as shown in Fig. A.4(a). If K satisfies


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DENSITY OF STATES

(a)

373

(b)

FIGURE A.4 States for a particle that can move in two-dimensional space: (a) boundary conditions at rigid lines; (b) periodic boundary conditions.

K π/L, the number of states W is written π 2 1 K2 2 · πK2 ÷ = L , 4 L 4π π 2 1 = π K 2 , SLR = , 4 L

W = SK R ÷ SLR = SK R

(A.24) (A.25)

where SK R is an area of one-fourth of a circle with radius K and SLR is an area of a minimum square formed by the terminal points of the wave vectors.

Periodic Boundary Conditions in Two-Dimensional Space Figure A.3(b) shows the periodic boundary conditions for a particle that can move in a two-dimensional space. Supposing that the values of the wave functions are the same at the virtual lines, the periodic boundary conditions are written

ψ(0, y) = ψ(L , y), ψ(x, 0) = ψ(x, L).

(A.26)


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DENSITY OF STATES

The wave function that satisfies the boundary conditions given by (A.26) is expressed as ψ(x, y) = ψ0 exp (iK x x) exp iK y y , 2π 2π Kx = nx , Ky = ny , L L n x , n y = 0, ±1, ±2, ±3, . . . .

(A.27) (A.28) (A.29)

When the initial point of K is placed on the origin of the K x and K y axes, the terminal point of K can move on the edge of the circle with radius K . Therefore, a terminal point of a wave vector whose length is less than K is present inside the circle with radius K , as shown in Fig. A.4(b). If K satisfies K 2π/L, the number of states W is written

2π 2 K2 2 W = SK P ÷ SLP = π K ÷ = L , L 4π 2 2π SK P = π K 2 , SLP = , L 2

(A.30) (A.31)

where SK P is an area of a circle with radius K and SLP is an area of a minimum square formed by the terminal points of the wave vectors. Density of States in Two-Dimensional Space From (A.24) and (A.30), the number of states W is the same irrespective of the boundary conditions. With the help of (A.17), the density of states D(K ) for a particle that can move in a two-dimensional space is expressed as D(K ) =

dW K 2 = L . dK 2π

(A.32)

From (A.32), the density of states per unit area DS (K ) is obtained as DS (K ) =

D(K ) K = . 2 L 2π

(A.33)

Boundary Condition at Rigid Points in One-Dimensional Space Figure A.5(a) shows the boundary condition at rigid points for a particle that can move in one-dimensional space. The particle is completely confined in the line; the particle cannot be present outside the line. Therefore, the boundary condition at the


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DENSITY OF STATES

375

(a)

(b) FIGURE A.5 Boundary conditions for a particle that can move in one-dimensional space: (a) boundary condition at rigid points; (b) periodic boundary condition.

rigid points is written ψ(0) = ψ(L) = 0.

(A.34)

The wave function that satisfies the boundary condition given by (A.34) is expressed as ψ(x) = ψ0 sin (K x x) , π Kx = nx (n x = 1, 2, 3, . . .). L

(A.35) (A.36)

When the initial point of K is placed on the origin of K x axis, the terminal point of K can move on the line with a length K . Therefore, a terminal point of a wave vector whose length is less than K is present on the line with a length K , as shown in Fig. A.6(a). If K satisfies K π/L, the number of states W is written W = L K R ÷ L LR = K ÷

π K = L. L π

(A.37)

Periodic Boundary Condition in One-Dimensional Space Figure A.5(b) shows the periodic boundary condition for a particle, which can move in one-dimensional space. Supposing that the values of the wave functions are the same at the virtual points, the periodic boundary condition is written ψ(0) = ψ(L).

(A.38)


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DENSITY OF STATES

(a)

(b)

FIGURE A.6 States for a particle that can move in one-dimensional space: (a) boundary condition at rigid points; (b) periodic boundary condition.

The wave function that satisfies the boundary condition given by (A.38) is expressed as ψ(x) = ψ0 exp (iK x x) exp(− iωt), 2π Kx = nx (n x = 0, ±1, ±2, ±3, . . .). L

(A.39) (A.40)

When the initial point of K is placed on the origin of K x axis, the terminal point of K can move on the line with a length 2K . Therefore, a terminal point of a wave vector whose length is less than K is present on the line with a length 2K , as shown in Fig. A.6(b). If K satisfies K 2π/L, the number of states W is written W = L K P ÷ L LP = 2K ÷

K 2π = L. L π

(A.41)

Density of States in One-Dimensional Space From (A.37) and (A.41), the number of states W is the same, irrespective of the boundary conditions. With the help of (A.17), the density of states D(K ) is written D(K ) =

dW 1 = L. dK π

(A.42)

From (A.42), the density of states per unit length DL (K ) is obtained as DL (K ) =

D(K ) 1 = . L π

(A.43)

Modes of an Electromagnetic Wave in a Cavity Let us consider modes of an electromagnetic wave that exists in a cavity with blackbody walls. It is assumed that this optical cavity is a cube with a side of L, and its walls are conductors. By using positive integers n x , n y , and n z , x, y, and z components of


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DENSITY OF STATES

377

the electric field is written n πx n πy n πz y x z sin sin , L L L n π nyπ y nz π z x x E y = E y0 sin ωt sin cos sin , L L n πx n πy n Lπ z y x z E z = E z0 sin ωt sin sin cos . L L L E x = E x0 sin ωt cos

(A.44) (A.45) (A.46)

Transverse Wave When electric charges are not present in the cavity, the following equations are obtained from Maxwell equations: ∂ Ey ∂ Ex ∂ Ez + + = 0, ∂x ∂y ∂z ∂ ∂ ∂ ˆy + ∇= xˆ + zˆ , ∂x ∂y ∂z E = E x xˆ + E y ˆy + E z zˆ .

div E = ∇ · E =

(A.47) (A.48) (A.49)

A propagation direction by the direction of the electromagneticwave is represented of the wave vector k = n x π/L , n y π/L , n z π/L = n x , n y , n z π/L. Inserting (A.44)–(A.46) into the left-hand side of (A.47) leads to E x0 n x + E y0 n y + E z0 n z = E 0 · n = 0.

(A.50)

Here E 0 = E x0 , E y0 , E z0 is an amplitude vector of the electric field of the electromagnetic wave in the cavity. From (A.50) it is found that E 0 is normal to n = n x , n y , n z = (L/π )k, which satisfies the condition that the electromagnetic wave is a transverse wave. Angular Frequency of an Electromagnetic Wave From Maxwell equations, a wave equation is obtained as c

2

∂2 ∂2 ∂2 + + ∂x2 ∂ y2 ∂z 2

Ez =

∂ 2 Ez , ∂t 2

(A.51)

where c is the speed of light in vacuum. Inserting (A.44)–(A.46) into (A.51) leads to c2 π 2 n x 2 + n y 2 + n z 2 = ω2 L 2 .

(A.52)

1 n ≡ nx 2 + n y 2 + nz 2 2

(A.53)

Introducing the relation


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DENSITY OF STATES

and expressing an angular frequency ω as ωn leads to ωn =

nπ c . L

(A.54)

The Number of States for Electromagnetic Waves The number of states W for one mode is the same as the number of combinations for n x , n y , n z . Let us consider a three-dimensional space with the axes n x , n y , and n z , and let the maximum length of the wave vector be n. When the initial point of the vector n, which satisfies |n| = n, is placed on the origin of the n x , n y , and n z axes, the terminal point of n can move on the surface of one-eighth of the sphere with a radius n. Therefore, the terminal point of a vector whose length is less than n is present inside one-eighth of the sphere with a radius n. If n satisfies n 1, the number of states W is written W =

1 4π 3 · n . 8 3

(A.55)

By differentiating (A.55) with respect to n, the density of states for one mode Dm is obtained as Dm =

dW 1 1 = · 4πn 2 = πn 2 . dn 8 2

(A.56)

The electromagnetic wave is a transverse wave; its electric field and magnetic field vibrate orthogonally to the propagation direction of the electromagnetic wave. This direction for vibration is called the polarization direction, and the mode of the electromagnetic wave is expressed by the wave vector and polarization direction. Two electromagnetic waves whose polarization directions are orthogonal to each other do not interfere with each other. As a result, these two modes with a common wave vector are present simultaneously; the number of states for one wave vector of the electromagnetic wave is twice the number of states for one mode. Figure A.7 shows

Vibration Directions


FIGURE A.7 Coexistence of two modes whose polarizations are orthogonal to each other.


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DENSITY OF STATES

379

the coexistence of two modes whose polarization directions are orthogonal to each other. Note that only electric fields E 1 and E 2 are shown; the magnetic fields are omitted for simplicity. The Number of States for Free Electrons Let us consider the number of states We for free electrons where the energy is less than E. Electrons have two states, corresponding to up/down spins. As a result, We is twice the number of states W for one mode: We = 2W =

k3 3 L , 3π 2

where k is a length of the wave vector k for free electrons.

(A.57)

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APPENDIX B DENSITY-OF-STATES EFFECTIVE MASS

When effective mass has a different value according to a direction of the wave vector k, an energy E(k) is written 2 2 2 kx 2 + ky 2 + kz 2 , ∗ ∗ 2m 1 2m 2 2m 3 ∗ k 2 = kx 2 + k y 2 + kz 2 ,

E(k) = E 0 +

(B.1) (B.2)

where m 1 ∗ , m 2 ∗ , and m 3 ∗ are effective masses along x, y, and z axes, respectively. Introducing wave vectors such as

kx =

m de m1∗

12

kx , k y =

m de m2∗

12

k y , kz =

m de m3∗

12

kz ,

(B.3)

2 2 2 2 2 2 kx + k y + kz = E0 + k , 2m de 2m de

(B.4)

and inserting (B.3) into (B.1), the energy E(k) is written

E(k) = E 0 +

k = kx + k y + kz . 2

2

2

2

(B.5)


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DENSITY-OF-STATESEFFECTIVE MASS

Here m de is called the density-of-states effective mass. From (B.4), a small change in the energy dE is expressed as dE =

2 k dk . m de

(B.6)

The volume of the wave vector space should be the same irrespective of k and k . As a result, a following relation has to be satisfied:

dk x dk y dk z =

dk x dk y dk z =

m de 3 m1∗m2∗m3∗

12

dk x dk y dk z .

(B.7)

From (B.7), the density-of-states effective mass m de is obtained as 1 m de = m 1 ∗ m 2 ∗ m 3 ∗ 3 .

(B.8)

For example, in Ge and Si, m 1 ∗ , m 2 ∗ , and m 3 ∗ are written m1∗ = m2∗ = mt, m3∗ = ml,

(B.9)

where m t is transverse effective mass and m l is longitudinal effective mass. In this case, the density-of-states effective mass m de is reduced to 1 m de = m t 2 m l 3 .

(B.10)

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APPENDIX C CONDUCTIVITY EFFECTIVE MASS

When effective mass has a different value according to a direction of the wave vector k, an electric current density j is written

j = jx xˆ + j y ˆy + jz zˆ

(C.1)

ne τ Ex , m1∗ ne2 τ j y = neµ2 E y = Ey, m2∗ ne2 τ jz = neµ3 E z = Ez . m3∗ jx = neµ1 E x =

2

(C.2) (C.3) (C.4)

Here n is a carrier concentration; e is the elementary electric charge; µ1 , µ2 , and µ3 are mobilities along x, y, and z axes, respectively; τ is a mean collision time; xˆ , ˆy, and zˆ are units vectors along x, y, and z axes, respectively; and m 1 ∗ , m 2 ∗ , and m 3 ∗ are effective masses along x, y, and z axes, respectively. In crystals, the x, y, and z coordinates can be selected arbitrarily. As a result, actual electric current densities are obtained by averaging with respect to x, y, and z Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright

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CONDUCTIVITY EFFECTIVE MASS

axes as follows: 1 1 1 1 ne2 τ jx = ne τ × + + E ≡ Ex , x 3 m1∗ m2∗ m3∗ mc 1 1 1 1 ne2 τ j y = ne2 τ × + + E ≡ Ey, y 3 m1∗ m2∗ m3∗ mc 1 1 1 1 ne2 τ jz = ne2 τ × + + Ez ≡ Ez . ∗ ∗ ∗ 3 m1 m2 m3 mc 2

(C.5) (C.6) (C.7)

Here m c is the conductivity effective mass for a carrier, which is defined as 1 1 ≡ mc 3

1 1 1 + + ∗ ∗ m1 m2 m3∗

,

(C.8)

or mc ≡

m1∗m2∗

3m 1 ∗ m 2 ∗ m 3 ∗ . + m2∗m3∗ + m3∗m1∗

(C.9)

For Ge and Si, the conductivity effective mass for the conduction electron m c is expressed as mc =

3m t m l . m t + 2m l

(C.10)

P1: OTA/XYZ P2: ABC ind JWBS036-Numai August 13, 2010

9:6


INDEX

abrupt pn-junction, 104 absorption, 3, 26 absorption, induced, 27 absorption loss, 77 acceptor, 97, 99 acceptor level, 99 amplitude-modulating noise, 172 amplitude shift keying, 161 analog modulation, 169 anisotropic optical gain, 195 antiguiding effect, 52, 152 antireflection, 214 anti-Stokes luminescence, 26 AR, 214 aspect ratio, 52 astigmatism, 153 asymmetry measure, 48 Auger process, 24, 143 autocorrelation function, 175 avalanche breakdown, 116 average collision time, 99 axial mode, 157 band filling effect, 36, 159 band offset, 15, 127 bandgap energy, 3 band-structure engineering, 201

base function, 16 beam waist, 153 Bernard–Duraffourg relation, 32 BH, 154, 156 biaxial stresses, 208 Biot–Savart’s law, 8 bistable LDs, 184 blackbody radiation theory, 31 Bloch function, 4 Bloch oscillation, 21 Bloch theorem, 4 Bohr magneton, 8 Boltzmann constant, 29 bra vector, 5 Bragg wavelength, 67, 218 breakdown, 116 breakdown, avalanche, 116 breakdown, Zener, 116 Brillouin zone, bending of, 21 buffer layer, 202 built-in potential, 104 bulk, 4 buried heterostructure, 154, 156 carrier, 94 carrier concentration, intrinsic, 96 carrier concentration, threshold, 132


385


386

9:6


INDEX

carrier distribution, 28 carrier lifetime, 126, 128 carrier noise, 173 cathodoluminescence, 26 cavity, Fabry–Perot, 57 cavity, optical, 27, 57 characteristic matrix, 82 characteristic temperature, 143 chemiluminescence, 26 chirped grating, 71 chirping, 167 chromatic dispersion, 172 cladding layer, 41 cleaved facet, 59 coherent, 26, 160 complex refractive index, 40 compound semiconductor, 93 compressive strain, 203 conduction-electron, 93 conductivity effective mass, 100 confinement of resonant radiation, 227 coupled cavity, 184 coupled wave equation, 67 coupled wave theory, 67 coupling coefficient, 66 coupling rate of feedback light to an LD, 186 critical angle, 42 critical thickness, 202 current multiplication factor, 117 current versus light output, 131 cutoff condition, 49 cyclotron angular frequency, 97 cyclotron resonance, 12, 13 DBR LD, 222 decay coefficient, 166 decay rate, 186 decay time, 166 deformation potential, 206 delta function, 19, 177 density of states, 17 density of states, effective, 29, 95 density-of-states effective mass, 95 depletion layer, 103 depletion layer capacitance, 108 derivative electrical resistance, 146 derivative light output, 145 derivative measurement, 144 deviation, 165 DFB LD, 71, 213 dielectric film, 59 differential gain, 165 diffracted pattern, 149

diffraction grating, 57 diffusion, 101 diffusion capacitance, 116 diffusion coefficient, 101 diffusion length, 115, 153 diffusion potential, 104 digital modulation, 169 dipole moment, 34 Dirac’s constant, 4 direct modulation, 160, 162 direct transition, 13, 24 discrete, 3 discrete approach, 82 dispersion, 77, 157 dispersion, chromatic, 172 dispersion curve, 48 dispersion, material, 172 dispersion, mode, 172 dispersion, structural, 172 distributed Bragg reflector, 57 distributed Bragg reflector LD, 222 distributed feedback, 57 donor, 97, 98 donor level, 98 dopant, 97 double heterostructure, 127 drift current, 101 effective density of states, 29, 95 effective mass, 6 effective mass approximation, 16 effective mass, conductivity, 100 effective mass, density of state, 95 effective mass, longitudinal, 95 effective mass, transverse, 95 effective refractive index, 47 effective refractive index, method, 52 eigenvalue equation, 48 Einstein summation convention, 207 Einstein’s A coefficient, 31 Einstein’s B coefficient, 31 Einstein’s relation, 101 elastic strain, 203 electric conductivity, 101 electric current noise, 173 electroluminescence, 26 electroluminescence, injection-type, 26 electron beam exposure, 77 electron, conduction, 93 element semiconductor, 93 elementary electric charge, 97 emission, 3 emission, induced, 26


9:6


INDEX emission, spontaneous, 26 emission, stimulated, 26 energy band, 4 energy barrier layer, 15 energy barriers, 127 energy eigenvalue, 4 energy level, 3 ensemble average, 174 envelope function, 16 equivalent refractive index, 159 etching mask, 77 evanescent wave, 46 excitation, 23 external cavity, 184 external differential quantum efficiency, 139 external modulation, 160, 162 extinction coefficient, 40 extinction ratio, 166 extrinsic semiconductor, 97 eye pattern, 171 Fabry–Perot cavity, 57 far-field pattern, 148 Fermi–Dirac distribution, 28 Fermi level, 28 Fermi level, intrinsic, 96 Fermi level, quasi, 28, 102 Fermi particles, 94 field optical gain coefficient, 40 field spectrum, 175 film, 41 fluorescence, 25 forward bias, 126 free carrier absorption, 141 free carrier plasma effect, 152 free space, 39 free spectral range, 62 frequency fluctuation spectrum, 175 frequency-modulating noise, 172 frequency shift keying, 161 Fresnel formulas, 43, 147 full width at half maximum, 63 fundamental mode, 49 FWHM, 63 gain flattening, 194 gain guiding, 40, 151 gain, optical, 27 Gaussian distribution function, 182 graded index SCH, 192 graded pn-junction, linearly, 109 grating, chirped, 71 grating, phase-shifted, 71

grating, tapered, 71 grating, uniform, 71 GRIN-SCH, 192 guiding effect, anti-, 52, 152 half width at half maximum, 63 Hall coefficient, 98 Heaviside function, 192 heavy hole, 95 heavy hole band, 12 heterojunction, 126 high frequency modulation, 184 hole, 93 holographic exposure, 77 homojunction, 126 horizontal transverse mode, 148, 151 HWHM, 63 hybrid orbital, 6 hydrostatic strain, 206 hysteresis, 188 hysteresis loop, 160 impedance of vacuum, 50 incident light, 43 index guiding, 39 index-coupled DFB LD, 213 indirect transition, 13, 24 induced absorption, 27 induced emission, 26 injection-type electroluminescence, 26 intensity-modulation/direct-detection, 161 interaction energy, 8 interband transition, 157 interference fringe pattern, 77 internal cavity, 184 internal loss, 134 internal quantum efficiency, 139 intraband relaxation time, 194 intrinsic, 126 intrinsic carrier concentration, 96 intrinsic Fermi level, 96 intrinsic semiconductor, 93 inverse Laplace transform, 164 inverted population, 27 ionization coefficient, 117 junction capacitance, 108 ket vector, 5 kink, 153 Laplace transform, 163 laser, 27

387


388

9:6


INDEX

lateral mode, 148 lattice mismatching, 202 light hole, 95 light hole band, 12 linearly graded pn-junction, 109 linearly polarized light, 43 longitudinal effective mass, 95 longitudinal mode, 157 Lorentzian, 36, 129, 183 luminescence, 25 luminescence, anti-Stokes, 26 luminescence, cathodo, 26 luminescence, chemi, 26 luminescence, electro, 26 luminescence, injection-type electro, 26 luminescence, photo, 25 luminescence, Stokes, 25 luminescence, thermo, 26 luminescence, tribo, 26 Luttinger parameter, 205 Luttinger–Kohn Hamiltonian, 204, 205 magnetic flux density, 8 Marcatili’s method, 54 material dispersion, 172 Maxwell’s equations, 83 minizone, 21 mirror, 27, 57 mirror loss, 133 mobility, 101 modal gain, 194 mode competition, 153 mode density, 31 mode dispersion, 172 mode hopping, 153 mode number, 45 mode partition noise, 183 mode volume, 174 modified MQW, 192 modified Schawlow–Townes linewidth formula, 183 modulation efficiency, 168 MQW LD, 191 multimode, 157 multiple quantum well LD, 191 near-field pattern, 148 negative resistance, 21 node, 65 noise, 172 nonradiative recombination, 23 nonradiative recombination lifetime, 128 nonradiative transition, 3

nonreturn-to-zero, 170 nonthermal equilibrium, 28 normalized frequency, 48 normalized waveguide thickness, 48 NRZ, 170 optical cavity, 27, 57 optical confinement factor, 47 optical fiber amplifier, 161 optical filter, 62 optical gain, 27 optical isolator, 184 optical resonator, 27, 57 optical waveguide, 39 optical waveguide, planar, 40 optical waveguide, strip, 40 optical waveguide, three-dimensional, 40 optical waveguide, two-dimensional, 40 orbital angular momentum, 8 orbital angular momentum operator, 206 orbit-strain interaction Hamiltonian, 206 order of diffraction, 68, 218 orthonormalize, 173 oscillation, 132 overflow, 143 pattern effect, 171 Pauli exclusion principle, 3 Pauli’s spin matrices, 9 penetration depth, 46 periodic multilayer, 82 periodic potential, 14 perturbation parameter, 4 perturbation theory, first-order, 5 perturbation theory, second-order, 5 phase shift, 44 phase shift keying, 161 phase-shifted grating, 71 phase velocity, 45 phonon, 23 phosphorescence, 25 photoluminescence, 25 photon, 23 photon lifetime, 128 photon recycling, 227 photoresist, 77 Pikus–Bir Hamiltonian, 204, 206 planar optical waveguide, 40 Planck’s constant, 4 plane of incidence, 43 plane wave, 43 pn-junction, 103, 155 pn-junction, abrupt, 104


9:6


INDEX pn-junction, graded, 109 pn-junction, linearly graded, 109 pnpn-junction, 155 Poisson equation, 106 polarization controller, 161 population inversion, 27 potential well, 15 power fluctuation spectrum, 175 propagate, 39 propagation constant, 45 propagation mode, 42 punch-through, 119 quantum box, 17 quantum noise, 172 quantum number, 4 quantum structures, 14 quantum well, 15 quantum well LD, 191 quantum well, one-dimensional, 16 quantum well, three-dimensional, 17 quantum well, two-dimensional, 16 quantum wire, 17 quarter-wavelength-shifted grating, 219 quasi-Fermi level, 28, 102 QW, 15 QW, strained, 202 radiative recombination, 23 radiative recombination lifetime, 128 radiative transition, 3 rate equations, 128 reciprocal effective mass tensor, 6 recombination, 23 recombination, nonradiative, 23 recombination, radiative, 23 reflected light, 43 reflector, 27 refracted light, 43 refractive index, 40 refractive index, complex, 40 relative electric susceptibility, 193 relaxation, 23 relaxation oscillation, 162 resonance condition, 61, 133 resonant angular frequency, 169 resonator, 27 return-to-zero, 170 rib waveguide, 154 ridge, 53 ridge waveguide, 155 ring cavity, 57

running wave, 64 RZ, 170 saturable absorber, 184 saturation current density, 116 selection rule, 199 self-pulsation, 184 semiclassical theory, 173 semiconductor, 93 semiconductor, compound, 93 semiconductor, element, 93 semiconductor, intrinsic, 93 semimetal, 20 separate confinement heterostructure, 192 separation-of-variables procedure, 84 shear strain, 206 signal to noise ratio, 161 single crystal, 4 single-mode LD, 213 single-mode operation, 157 single quantum well LD, 191 slope efficiency, 139 small-signal analysis, 164 Snell’s law, 41 space-charge layer, 103 spatial hole burning, 151, 152, 219 spectral density functions, 175 spectral linewidth, 63, 175 spectral linewidth enhancement factor, 178 spherical polar coordinate system, 9 spin, angular momentum, 8 spin angular momentum operator, 206 spin magnetic moment, 8, 9 spin-orbit interaction, 8 spin-orbit interaction, Hamiltonian, 9 split-off band, 12 split-off energy, 12 spontaneous emission, 26 spontaneous emission coupling factor, 128 SQW LD, 191 standing wave, 64 steady state, 131 step function, 192 stimulated emission, 26 Stokes luminescence, 25 stopband, 70 strain, 201 strain, compressive, 203 strain, hydrostatic, 206 strain, shear, 206 strain, tensile, 203 strain-dependent spin-orbit interaction, Hamiltonian, 206

389


390

9:6


INDEX

strained QW, 202 stress, 202 strip optical waveguide, 40 structural dispersion, 172 substrate, 41 superlattice, 19 superlattice, type I, 20 superlattice, type II, 20 superlattice, type III, 20 synchrotron radiation, 79 tapered grating, 71 TE mode, 43 tensile strain, 203 tensor, 206 thermal equilibrium, 27 thermoluminescence, 26 three-dimensional optical waveguide, 40, 52 threshold, carrier concentration, 132 threshold current, density, 131 time average, 174 time-dependent quantum mechanical perturbation theory, 33 TM mode, 43 total reflection, 41, 44 transfer matrix, 68, 86 transition, direct, 13, 24 transition, indirect, 13, 24 transition, nonradiative, 3

transition, radiative, 3 transverse effective mass, 95 transverse electric mode, 43 transverse magnetic mode, 43 transverse mode, 148 transverse mode, horizontal, 148, 151 transverse mode, vertical, 148, 149 transverse resonance condition, 45 triboluminescence, 26 tunneling effect, 20 turn-on delay time, 162 two-dimensional optical waveguide, 40 undoped, 126 uniform grating, 71 valence band absorption, 143 VCSEL, 224 vertical cavity surface emitting LD, 224 vertical transverse mode, 148 wave function, 4 wave vector, 4 Wiener–Khintchine theorem, 175 x-ray exposure, 77 Zener breakdown, 116 zinc blende structure, 6

P1: OTA/XYZ P2: ABC series JWBS036-Numai

August 4, 2010

20:5


WILEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING KAI CHANG, Editor Texas A&M University FIBER-OPTIC COMMUNICATION SYSTEMS, Third Edition r Cavind P. Agrawal ASYMMETRIC PASSIVE COMPONENTS IN MICROWAVE INTEGRATED CIRCUITS r Hee-Ran Ahn COHERENT OPTICAL COMMUNICATIONS SYSTEMS r Silvello Betti, Ciancarlo De Marchis, and Eugenio lannone PHASED ARRAY ANTENNAS: FLOQUET ANALYSIS, SYNTHESIS, BFNs, AND ACTIVE ARRAY SYSTEMS r Awn K. Bhattacharyya HIGH-FREQUENCY ELECTROMAGNETIC TECHNIQUES: RECENT ADVANCES AND APPLICATIONS r Asoke K. Bhattacharyya RADIO PROPAGATION AND ADAPTIVE ANTENNAS FOR WIRELESS COMMUNICATION LINKS: TERRESTRIAL, ATMOSPHERIC, AND IONOSPHERIC r Nathan Blaunstein and Christos G. Christodoulou COMPUTATIONAL METHODS FOR ELECTROMAGNETICS AND MICROWAVES r Richard C. Booton, Jr. ELECTROMAGNETIC SHIELDING r Salvatore Celozzi, Rodolfo Araneo, and Giampiero Lovat MICROWAVE RING CIRCUITS AND ANTENNAS r Kai Chang MICROWAVE SOLID-STATE CIRCUITS AND APPLICATIONS r Kai Chang RF AND MICROWAVE WIRELESS SYSTEMS r Kai Chang RF AND MICROWAVE CIRCUIT AND COMPONENT DESIGN FOR WIRELESS SYSTEMS r Kai Chang, Inder Bahl, and Vijay Nair MICROWAVE RING CIRCUITS AND RELATED STRUCTURES, Second Edition r Kai Chang and Lung-Hwa Hsieh MULTIRESOLUTION TIME DOMAIN SCHEME FOR ELECTROMAGNETIC ENGINEERING r Yinchao Chen, Qunsheng Cao, and Raj Mittra DIODE LASERS AND PHOTONIC INTEGRATED CIRCUITS r Larry Coldren and Scott Corzine EM DETECTION OF CONCEALED TARGETS r David J. Daniels RADIO FREQUENCY CIRCUIT DESIGN r W. Alan Davis and Krishna Agarwal MULTICONDUCTOR TRANSMISSION-LINE STRUCTURES: MODAL ANALYSIS TECHNIQUES r J. A. Brandao ˜ Faria PHASED ARRAY-BASED SYSTEMS AND APPLICATIONS r Nick Fourikis SOLAR CELLS AND THEIR APPLICATIONS, Second Edition r Lewis M. Fraas and Larry D. Partain FUNDAMENTALS OF MICROWAVE TRANSMISSION LINES r Jon C. Freeman OPTICAL SEMICONDUCTOR DEVICES r Mitsuo Fukuda MICROSTRIP CIRCUITS r Fred Gardiol HIGH-SPEED VLSI INTERCONNECTIONS, Second Edition r Ashok K. Goel


August 4, 2010

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FUNDAMENTALS OF WAVELETS: THEORY, ALGORITHMS, AND APPLICATIONS r laideva C. Goswami and Andrew K. Chan HIGH-FREQUENCY ANALOG INTEGRATED CIRCUIT DESIGN r Ravender Goyal (ed.) ANALYSIS AND DESIGN OF INTEGRATED CIRCUIT ANTENNA MODULES r K. C. Gupta and Peter S. Hall PHASED ARRAY ANTENNAS, Second Edition r R. C. Hansen STRIPLINE CIRCULATORS r Joseph Helszajn THE STRIPLINE CIRCULATOR: THEORY AND PRACTICE r Joseph Helszajn LOCALIZED WAVES r Hugo E. Hernandez-Figueroa, Michel Zamboni-Rached, and ´ Erasmo Recami (eds.) MICROSTRIP FILTERS FOR RF/MICROWAVE APPLICATIONS r Jia-Sheng Hong and M. J. Lancaster MICROWAVE APPROACH TO HIGHLY IRREGULAR FIBER OPTICS r Huang Hung-Chia NONLINEAR OPTICAL COMMUNICATION NETWORKS r Eugenio lannone, Francesco Matera, Antonio Mecozzi, and Marina Settembre FINITE ELEMENT SOFTWARE FOR MICROWAVE ENGINEERING r Tatsuo Itoh, Giuseppe Pelosi, and Peter P. Silvester (eds.) INFRARED TECHNOLOGY: APPLICATIONS TO ELECTROOPTICS, PHOTONIC DEVICES, AND SENSORS r A. R. Jha SUPERCONDUCTOR TECHNOLOGY: APPLICATIONS TO MICROWAVE, ELECTRO-OPTICS, ELECTRICAL MACHINES, AND PROPULSION SYSTEMS r A. R. Jha OPTICAL COMPUTING: AN INTRODUCTION r M. A. Karim and A. S. S. Awwal INTRODUCTION TO ELECTROMAGNETIC AND MICROWAVE ENGINEERING r Paul R. Karmel, Gabriel D. Colef, and Raymond L. Camisa MILLIMETER WAVE OPTICAL DIELECTRIC INTEGRATED GUIDES AND CIRCUITS r Shiban K. Koul ADVANCED INTEGRATED COMMUNICATION MICROSYSTEMS r Joy Laskar, Sudipto Chakraborty, Manos Tentzeris, Franklin Bien, and Anh-Vu Pham MICROWAVE DEVICES, CIRCUITS AND THEIR INTERACTION r Charles A. Lee and G. Conrad Dalman ADVANCES IN MICROSTRIP AND PRINTED ANTENNAS r Kai-Fong Lee and Wei Chen (eds.) SPHEROIDAL WAVE FUNCTIONS IN ELECTROMAGNETIC THEORY r Le-Wei Li, Xiao-Kang Kang, and Mook-Seng Leong ARITHMETIC AND LOGIC IN COMPUTER SYSTEMS r Mi Lu OPTICAL FILTER DESIGN AND ANALYSIS: A SIGNAL PROCESSING APPROACH r Christi K. Madsen and Jian H. Zhao THEORY AND PRACTICE OF INFRARED TECHNOLOGY FOR NONDESTRUCTIVE TESTING r Xavier P. V. Maldague METAMATERIALS WITH NEGATIVE PARAMETERS: THEORY, DESIGN, AND MICROWAVE APPLICATIONS r Ricardo Marques, ´ Ferran Mart´ın, and Mario Sorolla OPTOELECTRONIC PACKAGING r A. R. Mickelson, N. R. Basavanhally, and Y. C. Lee (eds.) OPTICAL CHARACTER RECOGNITION r Shunji Mori, Hirobumi Nishida, and Hiromitsu Yamada ANTENNAS FOR RADAR AND COMMUNICATIONS: A POLARIMETRIC APPROACH r Harold Mott INTEGRATED ACTIVE ANTENNAS AND SPATIAL POWER COMBINING r Julio A. Navarro and Kai Chang


August 4, 2010

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ANALYSIS METHODS FOR RF, MICROWAVE, AND MILLIMETER-WAVE PLANAR TRANSMISSION LINE STRUCTURES r Cam Nguyen LASER DIODES AND THEIR APPLICATIONS TO COMMUNICATIONS AND INFORMATION PROCESSING r Takahiro Numai FREQUENCY CONTROL OF SEMICONDUCTOR LASERS r Motoichi Ohtsu (ed.) WAVELETS IN ELECTROMAGNETICS AND DEVICE MODELING r George W. Pan OPTICAL SWITCHING r Georgios Papadimitriou, Chrisoula Papazoglou, and Andreas S. Pomportsis MICROWAVE IMAGING r Matteo Pastorino ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES r Clayton R. Paul INTRODUCTION TO ELECTROMAGNETIC COMPATIBILITY, Second Edition r Clayton R. Paul ADAPTIVE OPTICS FOR VISION SCIENCE: PRINCIPLES, PRACTICES, DESIGN AND APPLICATIONS r Jason Porter, Hope Queener, Julianna Lin, Karen Thorn, and Abdul Awwal (eds.) ELECTROMAGNETIC OPTIMIZATION BY GENETIC ALGORITHMS r Yahya Rahmat-Samii and Eric Michielssen (eds.) INTRODUCTION TO HIGH-SPEED ELECTRONICS AND OPTOELECTRONICS r Leonard M. Riaziat NEW FRONTIERS IN MEDICAL DEVICE TECHNOLOGY r Arye Rosen and Harel Rosen (eds.) ELECTROMAGNETIC PROPAGATION IN MULTI-MODE RANDOM MEDIA r Harrison E. Rowe ELECTROMAGNETIC PROPAGATION IN ONE-DIMENSIONAL RANDOM MEDIA r Harrison E. Rowe HISTORY OF WIRELESS Tapan r K. Sarkar, Robert. Mailloux, Arthur A. Oliner, Magdalena Salazar-Palma, and Dipak L. Sengupta PHYSICS OF MULTIANTENNA SYSTEMS AND BROADBAND PROCESSING r Tapan K. Sarkar, Magdalena Salazar-Palma, and Eric L. Mokole SMART ANTENNAS r Tapan K. Sarkar, Michael C. Wicks, Magdalena Salazar-Palma, and Robert J. Bonneau NONLINEAR OPTICS r E. C. Sauter APPLIED ELECTROMAGNETICS AND ELECTROMAGNETIC COMPATIBILITY r Dipak L. Sengupta and Valdis V. Liepa COPLANAR WAVEGUIDE CIRCUITS, COMPONENTS, AND SYSTEMS r Rainee N. Simons ELECTROMAGNETIC FIELDS IN UNCONVENTIONAL MATERIALS AND STRUCTURES r Onkar N. Singh and Akhlesh Lakhtakia (eds.) ANALYSIS AND DESIGN OF AUTONOMOUS MICROWAVE CIRCUITS r Almudena Suarez ´ ELECTRON BEAMS AND MICROWAVE VACUUM ELECTRONICS r Shulim E. Tsimring FUNDAMENTALS OF GLOBAL POSITIONING SYSTEM RECEIVERS: A SOFTWARE APPROACH, Second Edition r James Bao-yen Tsui RF/MICROWAVE INTERACTION WITH BIOLOGICAL TISSUES r Andre´ Vander Vorst, Arye Rosen, and Youji Kotsuka InP-BASED MATERIALS AND DEVICES: PHYSICS AND TECHNOLOGY r Osamu Wada and Hideki Hasegawa (eds.) COMPACT AND BROADBAND MICROSTRIP ANTENNAS r Kin-Lu Wong DESIGN OF NONPLANAR MICROSTRIP ANTENNAS AND TRANSMISSION LINES r Kin-Lu Wong


August 4, 2010

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PLANAR ANTENNAS FOR WIRELESS COMMUNICATIONS r Kin-Lu Wong FREQUENCY SELECTIVE SURFACE AND GRID ARRAY r T. K. Wu (ed.) ACTIVE AND QUASI-OPTICAL ARRAYS FOR SOLID-STATE POWER COMBINING r Robert A. York and Zoya B. Popovic´ (eds.) OPTICAL SIGNAL PROCESSING, COMPUTING AND NEURAL NETWORKS r Francis T. S. Yu and Suganda Jutamulia ELECTROMAGNETIC SIMULATION TECHNIQUES BASED ON THE FDTD METHOD r Wenhua Yu, Xiaoling Yang, Yongjun Liu, and Raj Mittra SiGe, GaAs, AND InP HETEROJUNCTION BIPOLAR TRANSISTORS r Jiann Yuan PARALLEL SOLUTION OF INTEGRAL EQUATION-BASED EM PROBLEMS r Yu Zhang and Tapan K. Sarkar ELECTRODYNAMICS OF SOLIDS AND MICROWAVE SUPERCONDUCTIVITY r Shu-Ang Zhou

Laser Diodes and Their Applications to Communications and Information Processing

Laser Diodes and Their Applications to Communications and Information Processing

Laser Diodes and Their Applications to Communications and Information Processing

Stochastic Methods and Their Applications to Communications

Digital Information Processing and Communications, Part I

Organic Laser Diodes: Modelling and Simulation

Algorithms for Communications Systems and their Applications

Algorithms for communications systems and their applications

Algorithms for Communications Systems and Their Applications

Video Processing and Communications

Video Processing and Communications

Laser physics and applications

Temporal Information Processing Technology and Its Applications

Applications of Algebra to Communications, Control, and Signal Processing

Information and Communications Security

Laser Space Communications (Artech House Space Technology and Applications)

Signal Processing Fundamentals and Applications for Communications and Sensing Systems

Signal Processing Fundamentals and Applications for Communications and Sensing Systems

Laser Space Communications (Artech House Space Technology and Applications)

Information Processing with Evolutionary Algorithms: From Industrial Applications to Academic Speculations (Advanced Information and Knowledge Processing)

Multimedia Analysis, Processing and Communications

Laser Ablation: Effects and Applications

Distributed Feedback Laser Diodes and Optical Tunable Filters

Laser Pulse Phenomena and Applications

Laser Physics and Applications. Fundamentals

Dynamic Laser Speckle and Applications

Laser Ablation and Its Applications

Laser Scanning, Theory and Applications

GaN-Based Laser Diodes: Towards Longer Wavelengths and Short Pulses

Signal Processing for Communications (Communication and Information Sciences)

Signal Processing for Communications (Communication and Information Sciences)

Laser Diodes and Their Applications to Communications and Information Processing