Physics of Solid-State Laser Materials

Richard C. Powell

Physics of Solid-State Laser Materials

AlP

I?&�

Springer

Physics of Solid-State Laser Materials

Richard C. Powell University of Arizona Optical Sciences Center Tucson, AZ 85721 USA

Library of Congress Cataloging-in-Publication Data Powell, Richard C. (Richard Conger), 1939 Physics of solid state laser materials p.

em.

I Richard C. Powell.

(Atomic, molecular, and optical physics series)

Includes bibliographical references and index. ISBN 1 56396 658 1 (alk. paper) 1. Solid state lasers TA1705.P69

Materials.

I. Title.

II. Series.

1998

621.36'61 -{!c21

97 27736

Printed on acid-free paper. AlP Press is an imprint of Springer-Verlag New York, Inc..

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ISBN 1-56396-658-1 Springer-Verlag New York Berlin Heidelberg

SPIN 10639940

Atomic, Molecular, and Optical Physics EDITOR-IN-CIDEF:

Gordon W .F. Drake, Department of Physics, University of Windsor, Windsor, Ontario, Canada

EDITORIAL BOARD: W.E. Baylis, Department of Physics, University of Windsor, Windsor, Ontario, Canada Robert N. Compton, Oak Ridge National Laboratory, Oak Ridge, Tennessee M.R. Flannery, School of Physics, Georgia Institute of Technology, Atlanta, Georgia Brian R. Judd, Department of Physics, The Johns Hopkins University, Baltimore, Maryland Kate P. Kirby, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts Pierre Meystre, Optical Sciences Center, The University of Arizona, Tucson, Arizona

PUBLISHED VOLUMES:

Richard C. Powell, Physics of Solid-State Laser Materials

Series Preface

Throughout this century, unraveling the physics of atoms and molecules has provided a rich source of new ideas and techniques. A vast accumulation of knowledge is now available with wide applications in chemistry, surface sci­ ence, aeronomy, condensed matter physics, and engineering. Laser physics is also closely intertwined and provides much of the continuing impetus for new development. The need to unify the field and provide a guide to the literature motivated the recent publication of the Atomic, Molecular, and Optical Physics Hand­ book, published in

1995 by AlP Press, now an imprint of Springer-Verlag. The

Handbook's aim and objectives were to provide a concise summary of the

principal ideas, methods, techniques, and results of the field, within the bounds of a single volume, and presented at a level accessible to a researcher new to the field. Although these aims were well achieved, many of the

88 chapters

can, and should, be expanded to a full-length book. Much more remains to be said than what could be covered within the restricted space available in the Handbook. The aim of the AlP Atomic, Molecular, and Optical Physics Series of books is therefore to maintain the same philosophy as for the Handbook itself, but to cover the material in greater depth, together with related material not directly addressed there. Each book should be reasonably self-contained, and written in a concise and authoritative manner. Rather than research monographs, the aim is to publish books that span the gap between standard undergraduate textbooks and the research literature. The emphasis should be on the basic ideas, meth­ ods, techniques, and results of the field, but presented in greater detail than in the Handbook. Gordon W.F. Drake

University ofWindsor

Preface

The invention of the laser has had a profound impact on the quality of our lives. Lasers are commonly being used in a multitude of diverse applications ranging from price scanners at the checkout counters of retail stores to a variety of medical surgery procedures. The laser has been the catalyst for developing new technologies that have revolutionized important industries. One example of this is fiber optics, which has become the basis of our com­ munications industry. Another example is compact disks, which has become an important technique for data storage used for both the computer industry and the entertainment industry. As more and more applications of lasers are identified, the demand for lasers with very specific output characteristics is increasing. Driven by these applications requirements, the field of laser re­ search and development continues to be an active area of science and en­ gineering. Although many different types of lasers are used in today's appli­ cations, solid-state lasers are always preferable if they are available with the desired operating characteristics. For use outside the laboratory, it is desir­ able to have rugged, low-maintenance, compact systems, and solid-state lasers are generally better in these categories than their gas and liquid coun­ terparts. Thus the development of new and better solid-state lasers with specific operating characteristics is currently one of the most important areas of scientific research. Research and development in the field of solid-state lasers has spanned over thirty five years. It has become a mature scientific discipline with all of the obvious ideas and easy experiments already attempted. There have been tremendous accomplishments in obtaining tunable emission, ultrafast pulses, ultranarrow linewidths, high efficiencies, high power, and other spe­ cial laser operating characteristics. However, it is still not always possible to tailor-make a solid-state laser with the exact operating characteristics required for a specific application. The main goal of this field of research is to change the current modus operandi of building an application around an existing laser system to one of building an optimum laser for a specific application. Accomplishing this goal will require significant breakthroughs in several areas. Solid-state laser development is a multidisCiplinary field IX

x

Preface

involving material scientists, solid-state physicists, optical engineers, electrical engineers, and end users who may range from medical doctors to systems engineers. Innovative ideas in the growth of new laser materials, artificially structured nonlinear optical materials, resonator architecture with uncon­ ventional optical components, and improved schemes for optical pumping may all play a role in future advances in solid state lasers. The purpose of this book is to present the fundamental physical concepts relevant to solid-state lasers. There are currently several excellent books on lasers in general, on the linear and nonlinear optical spectroscopy of solids, and on solid-state laser engineering. However, none of these focus on the direct relationship between fundamental physical processes and specific solid­ state laser systems as is done in this book. There are three distinct parts to the book. The introductory chapter summarizes the basic concepts in solid state laser operation including optical pumping and material requirements. The equations that describe the output characteristics of a solid-state laser are derived and expressed in terms of the spectroscopic properties of the laser material and the conditions of optical pumping. The next part of the book consists of four chapters describing the fundamental physics underlying the operation of solid-state lasers. Chapter 2 discusses the origin of the electronic energy levels of ions in solids that are associated with optical pumping and lasing transitions. This includes a description of the energy levels of free ions and how they are altered when the ions are doped into a solid host material. The next chapter discusses the interaction of light with a material doped with optically active ions. The various concepts of optical spectroscopy including transition strengths, fluorescence lifetimes, and line shapes are discussed. Chapter 4 describes the effects of thermal vibrations on the optical spectra including radiationless and vibronic transitions, line broadening, line shift­ ing, and lifetime shortening. In Chap. 5 the interaction between optically active ions is discussed. This includes a description of how the presence of exchange-coupled ion pairs and energy transfer between ions can affect the spectral properties of ions in solids. The final section consists of five chapters in which the fundamental physical concepts treated in Chaps. 2-5 are applied to specific laser systems. Chapter 6 presents a complete, detailed treatment of ruby as a prototype solid-state laser system involving a transi­ tion metal as the active ion. Chapter 7 then discusses other transition-metal­ ion lasers in comparison with the properties of ruby. This includes other chromium-doped materials as well as lasers based on each of the other first­ row transition metals between titanium and nickel. Chapters 8 and 9 pro­ vide the same treatment for standard solid-state lasers based on lanthanide ions. The former presents a complete, detailed treatment of Nd-YAG as a prototype rare-earth laser system. The latter discusses other rare-earth ion lasers in comparison with Nd-YAG. This includes other neodymium-based systems as well as lasers based on all of the trivalent lanthanide ions between praseodymium and ytterbium. Chapter 10 summarizes the properties of other types of solid-state lasers. This includes lasers based on divalent lanthanide

Preface

xi

ions, trivalent cerium, and one actinide ion, uranium, as well as a brief summary of color-center lasers and other special systems such as solid-state dye lasers. In addition, an overview is given of solid-state lasers based on nonlinear optical processes. This is not meant to be a handbook of laser properties. The book contains several tables of laser properties, but these are presented to show sample comparisons of properties from one host to another or one active ion to another. They are not meant to be a complete compilation of characteristics. These can be found in any of a number of laser handbooks that have been published. This book should be useful both as a text book and as a reference book for those interested in understanding solid-state lasers. An attempt has been made to make this book as self contained as possible. Some knowledge of quantum mechanics is assumed but the specific concepts relevant to the topics of interest here such as angular momentum coupling and quantiza­ tion of the photon and phonon fields are reviewed in some detail. Similarly, some knowledge of solid-state physics is assumed but the concepts of sym­ metry as associated with group theory and crystal-field theory are treated in detail. This should make the book useful to a multidisciplanary audience having different levels of education in laser physics. This includes laser sys­ tems engineers, material scientists involved in the synthesis of new materials for solid-state lasers, and program managers and end users of laser systems wishing to have a better understanding of the physics of solid-state lasers. This book is based on the class notes I used in lecturing on this topic for over twenty years at Oklahoma State University and the University of Arizona. During this time I was fortunate to have many outstanding students in my class and working in my research group. As with every pro­ fessor, I learned a great deal from my students and much of their thoughts and contributions are reflected in this book. It is impossible to mention all of those who contributed in this way, but one of them deserves special rec­ ognition, James Murray, who spent many hours reading this manuscript and suggesting valuable corrections and changes. In addition, I have been fortunate to collaborate with colleagues throughout the world on research projects dealing with solid-state laser physics. All of these collaborations have added significantly to my knowledge of this field. The greatest of these learning experience was the year I spent working with Stephen Payne and the laser group at Lawrence Livermore National Laboratory. The contents of this book represent the cumulative experience gained from all of these collaborations. This book could not have been written without the strong support, love, and understanding of my wife Gwen. Richard C. Powell Tucson, Arizona December 1997

Contents

Series Preface Preface

vn

IX

1 . Introduction 1 . 1 Solid-State Laser Operation and Design Parameters 1 .2 Material Requirements for Laser Hosts and Active Ions 1 .3 Material Preparation and Optical Quality References

2 23 27 30

2. Electronic Energy Levels 2. 1 Free-Ion Energy Levels 2.2 Elements of Group Theory 2.3 Crystal-Field Splitting of Energy Levels References

31 32 49 57 82

3. Radiative Transitions 3.1 The Photon Field 3.2 Selection Rules 3.3 Properties of Spectral Lines 3.4 Nonlinear Optical Properties References

84 84 94 98 107 115

4. Electron-Phonon Interactions 4. 1 The Phonon Field 4.2 Weak Coupling: Radiationless Transitions 4.3 Weak Coupling: Vibronic Transitions 4.4 Weak Coupling: Spectral Linewidth and Line Position 4.5 Example: Spectral Properties of SrTi0 3 : Cr3 + 4.6 Strong Coupling 4.7 Jahn-Teller Effect References

1 16 1 17 127 1 37 140 147 1 56 1 69 1 74 Xlll

XIV

Contents

5. Ion-Ion Interaction 5. 1 Exchange-Coupled Ion Pairs 5.2 Nonradiative Energy Transfer: Single-Step Process 5.3 Phonon-Assisted Energy Transfer 5.4 Nonradiative Energy Transfer: Multistep Process 5.5 Connection with Experiment: Rate Equation Analysis References

175 177 1 82 1 92 195 204 213

6. Ah0 3 : Cr3 + Laser Crystals 6. 1 Energy Levels of Cr3+ 6.2 Crystal-Field Splitting 6.3 Spin-Orbit Splitting and Selection Rules 6.4 Strong-Field Laser Materials References

215 215 224 233 237 252

7. Transition-Metal-Ion Laser Materials 7. 1 Broad-Band Cr3 + Laser Materials: Alexandrite 7.2 Spectral Properties of Cr3 + in Different Hosts and Their Laser Characteristics 7.3 Transition-Metal Ions and Host Crystals 7.4 Laser Materials Based on Ti3 + Ions 7.5 Laser Materials Based on Ions with 3d2 Configurations 7.6 Laser Materials Based on Ions with 3d3 Through 3d8 Configurations References

254 254 260 270 271 282

8. Y3 Als 0 1 2 : Nd3 + Laser Crystals 8 . 1 Energy Levels of Nd3 + 8.2 Crystal-Field Splitting 8.3 Radiative Transitions: Judd-Ofelt Theory 8.4 Example: Y3 Al s 0 1 2 :Nd3 + References

294 294 301 309 319 337

9. Rare-Earth-Ion Laser Materials 9. 1 Nd3 + Lasers 9.2 Other Trivalent Lanthanide Lasers References

339 339 359 377

287 292

1 0. Miscellaneous Laser Materials 10. 1 Other Rare-Earth-Ion Lasers 10.2 Nonlinear Optical Lasers 10.3 Color-Center Lasers 10.4 Other Solid-State Lasers References

380 381 384 407 41 1 415

Index

417

1

Introduction

Optical technology is rapidly emerging out of the laboratory and into use in a wide variety of practical applications ranging from fiber optic communi­ cations to laser surgery. The catalyst driving this technology revolution is the laser. After the invention of the ruby laser in 1 960, there followed a decade of research and development focused on obtaining laser action from as many materials as possible. This led to the demonstration of lasers based on solid, liquid, and gas media with a wide variety of different opera­ tional characteristics. The research activity during the next decade focused on laboratory demonstrations of the use of lasers. During the 1 980s the practical use of laser-based systems began to appear and escalated through­ out the decade. Now that we have learned how to make lasers and seen what they can do for us, the thrust of research and development in the 1 990s is to make lasers with operational characteristics that are optimized for spe­ cific applications. In order to accomplish this, it is necessary to have a detailed understanding of the fundamental physical processes taking place in laser media. Gas and liquid lasers have played an important role in the development of laser-based technology, as have special systems such as chemical lasers, free­ electron lasers, and x-ray lasers. These classes of lasers will always have specialized areas of application. However, solid-state lasers are preferable for most applications because of their ruggedness, relative simplicity, and ease of operation. The problem is that solid-state lasers are not currently available with specific operational parameters required for some applica­ tions. Since no single solid-state laser will have the optimum parameters required for all applications, it is important to develop as many materials and device configurations as possible. The requirements for solid-state lasers having a wide variety of operational parameters for different types of appli­ cations has provided the stimulus for significant technology advancements in this field over the past several years. There are two major classes of solid-state lasers: those based on insulating host materials containing optically active point defects, and semiconductors. The fundamental physics and the device configurations of these two classes

2

1 . Introduction

of lasers are significantly different. The issues concerning research and development for these two types of systems are so different that they are best treated in two different books. This book will focus on the laser physics of ions in insulating hosts. Readers interested in the physics of semiconductor lasers are referred to recent books on this subject. 1 Following the general convention currently in use, the term solid-state lasers will be used to refer to systems based on optically active centers in insulator host materials while the term semiconductor lasers will be used explicitly when referring to this class of lasers. The critical components of a solid-state laser include the laser material, the mechanism of pumping, and the cavity configuration. The operational characteristics of a specific laser system are determined by the properties of these individual components and how they are integrated into the overall system design. During the past several years, there have been significant technological advances for all three components: new materials with wave­ length tunability have been developed; diode array pump sources have become available; slab and guided wave configurations have been demon­ strated; and new methods of mode locking have been discovered. This book focuses on the physical processes taking place within the lasing material. Readers who are interested in the engineering design of solid-state laser sys­ tems are referred to appropriate books on this subject. 2 An historical per­ spective of this field can be gained from the milestone reprint volume on solid-state lasers. 3 In the following sections, a brief summary of solid-state laser operation is presented and the relevance of optical spectroscopic properties and other material characteristics are discussed. The fundamental physical processes relevant to determining the spectroscopic properties of laser materials are then described in detail. Understanding these properties requires a basic knowledge of group theory and quantum mechanics, especially perturbation theory, plus some background in solid-state physics. The relationship between these processes and laser parameters is discussed, and examples of all of these processes in specific laser materials are presented in subsequent chapters. 1.1

Solid-State Laser Operation and Design Parameters

A solid-state laser material is a physical system consisting of an ensemble of optically active ions dispersed in a host crystal or glass. While operating as a laser, this material absorbs and emits optical radiation in a controlled ther­ mal environment, and thus its optical spectroscopic properties are critical to its performance. To understand the properties of this system, it is necessary to have extensive knowledge of the physical properties of the free ions, the host material, and the interaction mechanisms affecting the system. This problem is approached by determining the electronic states of the optically

1 . 1 . Solid-State Laser Operation and Design Parameters

5

lifetime of level 2. Using these concepts in Eq. ( 1 . 1 . 1 ) gives

c5(I(v)dv) I(v) c5x

= hvc B2! ( gg2, n, - n2) s(v) =

cta (v) .

For a specific frequency, the intensity through a distance x changes as

m

the beam after transmission

I (x) /(0) exp( ctax) , ( 1 . 1 .2 ) which is known as the Beer-Lambert law for absorption. The absorption coefficient in units of cm 1 is defined as =

( 1 . 1 .3) The total absorption is found by integrating this expression across the line­ shape function. It is generally assumed that S(v) is normalized and varies rapidly with v compared to the v2 factor that appears explicitly in Eq. ( 1 . 1 .3). At low temperatures when most of the atoms are in the ground state, this results in a total absorption of 2 ( 1 . 1 .4) cta (v)dv = ;_ g2 n, . nr, g , Here;. is the wavelength of the transition in the material. The shape of the spectral line for an electronic transition is discussed in detail in Chap. 3. Depending on the specific conditions of the physical system, the line shape can be approximated by either a Gaussian or a Lorentzian mathematical expression. The most general condition is a combination of these expressions, called a Voigt profile. All three of these types of expressions are tabulated mathematical functions. The integral over the absorption line for a Lorentzian line shape can be approximated by the linewidth for the transition Av multi­ plied by the peak absorption coefficient ct, with a factor of 2/n on the right­ hand side from product of the width and peak of a Lorentzian line shape,

J

_!_ g2 !!.!._ (l _ g , nz 8nr, g , Av

).

g2 n, For a Gaussian line shape there is an additional factor of (ln 2/n) 1/2 . For birefringent host crystals, the line-shape function must be written to account for n and () polarizations (electric field vector parallel to or perpendicular to

6

1 . Introduction

the major symmetry axis, respectively), s

"·"

v (-)

I.,-,,(v) f[2I.,-(v) + I,(v) J dv '

( 1 . 1 .6)

where this expression is written in terms of wave number v instead of fre­ quency v. The absorption cross section a 1 2 is defined as ( 1 . 1 .7) or for conditions of low excitation, n2 « n 1 ,

CT J 2 (v)

o:a ( v ) . n1

The integrated absorption cross section for these conditions is given by

J

a

()

A2 g l 2 ( v )dv 8 nr, g2 1

and the peak cross section is a12 �

=

( 1 . 1 .8)

·

( 1 . 1 .9)

Note that Eq. ( 1 . 1 .9) is for a Lorentzian line shape and the additional factor discussed above must be included for a Gaussian line shape. In the last expression, the refractive index has been included so the wavelength is now the wavelength in air instead of in the material. Also, the linewidth has been expressed in wave numbers. Note that under these low excitation conditions, the absorption cross section is a property of a transition for an individual ion while the absorption coefficient is a property of the entire ensemble of ions. The stimulated emission cross section is related to the absorption cross section by

gl ( 1 . 1 . 10) 0'2 ! -0' g2 ! 2 · The oscillator strength (or f number) for an absorption transition is

derived in Chap. 3. It is defined with respect to the absorption properties of an ensemble of quantum-mechanical dipole oscillators as me 9n fa 2 ne (n2 + 2) 2 an(v)dv me A� 9n g2 � ne2 8nn2 r, 2 2 (n + 2) g 1 '

J

()

where the conditions for low levels of excitation have been used in the final

7

1 . 1 . Solid-State Laser Operation and Design Parameters

expression. This leads to the useful relationship farr

�

( )·

g2 9n n (n2 + 2) 2 g 1

A.� ( 1 .5 1 sjcm2 ) 2

(1 . 1 . 1 1 )

The expression for the oscillator strength of an emission transition is the same as the expression for an absorption transition except that the ratio of the degeneracies of the levels (g 1 / g2 ) is now an additional factor. In addi­ tion, it is common to replace the radiative lifetime in the cross-section and oscillator-strength expressions with the fluorescence lifetime and include the quantum efficiency 1'fqe of the metastable state as an additional factor since 1'fqe

= r,'J

.

( 1 . 1 . 1 2)

The key to light amplification in a laser material can be seen from Eqs. ( 1 . 1 .2) and ( 1 . 1 .3). For low excitation conditions, (g 1 n2 )/(g2 n J ) « 1, so Q: is a positive parameter and the beam of photons decreases exponentially as it moves through the material. If conditions exist such that (g 1 n2 )/(g2 n 1 ) 1 , absorption is balanced by emission and the system i s saturated. The material is essentially transparent to the beam of photons. Finally, for the conditions (g 1 n2 )/(g2 n 1 ) > 1 , the absorption coefficient is a negative parameter and Eq. ( 1 . 1 .2) predicts an exponential increase in the beam of photons as it is transmitted through the material. This results in light amplification by stimulated emission of radiation and hence the acronym laser. The expressions derived above show that the most important physical parameters determining the light amplification by a laser material are the strength of the atomic transition designated by the radiative lifetime or the oscillator strength [related by Eq. ( 1 . 1 . 1 1 )], the difference in population density of ions in the excited state versus the terminal state of the transition, and the line-shape function. The physical conditions reflected by the differ­ ent line shapes result in different laser properties related to saturation, power extraction, and frequency agility. The strengths of electronic transitions are determined by the selection rules resulting from the matrix elements of the electron-photon interaction Hamiltonian. These depend on the wave func­ tions describing the initial and final electronic states of the ion in the local symmetry environment of the host material. This is discussed in detail in Chap. 3. The laser cavity properties are related to the spectroscopic properties of the lasing material through its amplification factor. When more ions are in the excited state than the ground state, the system is said to have a pop­ ulation inversion and the absorption coefficient becomes an amplification coefficient. At the peak frequency position of the line-shape function, the above expressions show the amplification factor to be given by

=

Q:a

= -ae!l.n,

( 1 . 1 . 1 3)

8

1 . Introduction

where ae is the peak stimulated emission cross section and the population inversion factor is given by

( 1 . 1 . 14) Substituting Eqs. ( 1 . 1 . 14) and ( 1 . 1 . 1 3) into Eq. ( 1 . 1 .2) results in an expres­ sion for the exponential gain experienced by a beam of photons traveling through a material with a population inversion. It is common to use the emission cross section as derived above 'to define the small-signal gain

coefficient,

go(..l.) = ae(..l.)dn.

( 1 . 1 . 1 5)

In order to maintain laser operation, the amount of round-trip gain expe­ rienced by a light wave in a laser cavity must be greater than the round-trip losses. When the gain and loss are equal, the laser is at threshold. The con­ ditions for achieving threshold for laser operation can be found by writing the round-trip gain in the cavity, including losses, as

( 1 . 1 . 1 6) where g is the gain coefficient, rx includes scattering, ground- and excited­ state absorption, and other passive losses, and l is the length of the cavity. Rt and Rz are the power reflectivities of the cavity mirrors and account for the scattering, absorption, and transmission losses. Since R 1 is the output coupler, it has a designed transmission loss for the emission of the laser beam. At threshold the gain per round-trip is equal to one, so

(1 . 1 . 1 7) This expression can be rewritten as

2gl = T + L,

( 1 . 1 . 1 8)

where T represents the active cavity output and L represents the passive losses of the cavity, T = ln R 1 0 ( 1 . 1 .19) L = 2 rxl -- ln( 1 - Lm ) , ( 1 . 1 .20) where Lm accounts for absorption, scattering, and diffraction losses of both mirrors as well as leakage from the high-reflector cavity mirror while R 1 0 is the reflectivity of the output coupler. To derive specific expressions for the gain coefficient, the population inversion at threshold must be determined for different pumping conditions. It is necessary to consider specific cases three- and four-level atomic systems with either pulsed or cw excitation. The schematic energy-level diagrams, transition rates, and level populations for these systems are shown in Fig. 1.4. The rate equations describing the time evolution of the populations of

10

1 . Introduction

and

( 1 . 1 .22) Here ifJ is the density of photons in the cavity mode, a is the cross section for stimulated emission between levels 2 and 1 , and Wp is the pumping rate. Taking the time derivative of Eq. ( 1 . 1 . 14) and substituting Eqs. ( 1 . 1 .21) and ( 1 . 1 .22) gives -

dt

=

-

( 1 -92) Anrftac +

91

-

92 An + n T 91

TJ

+ W":p (nT - An) ,

( 1 . 1 .23)

where n T = n 1 + n2 and r1 = r 2 1 . Similar rate equations can be written for a four-level system under the assumption of fast, efficient relaxation from the pump level to the meta­ stable state,

dn3 dt

-

=

(

( 1 . 1 .24)

)

n3 93 - n 3 - -n 2 rftac - T3 2 + T3 + Wpn 1 .

For "ideal conditions" r21

dAn dt

-

=

92

1

( 1 . 1 .25)

0 so n T n 1 + n 3 and An n3 . Thus An W": (nT An) . ( 1 . 1 .26) + p - Anrftac - T =

�

=

J

The equation describing the photon density in the cavity for both three­ and four-level systems is

dr/J rftacAn ifJ S - + , dt rc

-

=

( 1 . 1 .27)

where rc is the lifetime of a photon in the cavity mode and S is the rate of spontaneous emission into the cavity mode, which is generally quite small. The cavity decay time is related to the cavity losses expressed in Eqs. ( 1 . 1 . 19) and ( 1 . 1 .20) by

( 1 . 1 .28) At threshold and above, the photon density in the lasing mode increases so drft/dt � 0. From Eq. ( 1 . 1 .27) the population inversion at threshold is

1 . Anth care =

--

( 1 . 1 .29)

Substituting into this equation the expression in Eq. ( 1 . 1 .9), modified for

1 . 1 . Solid-State Laser Operation and Design Parameters

emission, gives

11

(nu - 9u nt) = 'lf83 nv2 91 th rc c S(v)'lqe

( 1.1 .30)

= �v_3_

( 1 . 1 .3 1 )

where subscripts u and I indicate the upper and lower energy levels asso­ ciated with the transition, and the line-shape function (discussed in Sec. 3.3) is given by

S( vo) for a Gaussian line and

S(vo)

=

2

( 1 . 1 .32)

for a Lorentzian line. Thus for a laser with low threshold and high gain, it is necessary to have a narrow-band optical transition (small �v) and a long cavity mode lifetime (low losses). For the continuous wave threshold conditions of rp 0 and d�n/dt = 0, the fractional population inversions for three- and four-level systems are found from Eqs. ( 1 . 1 .23) and ( 1 . 1 .26) to be

=

92 Wp r1 - 9! three-level system ( 1 . 1 .33) WpTf + 1 ' Wp rf ( 1 . 1 .34) four-level system. Wp rf + 1 ' For small pumping rates, Wp rf 1 , these results show that there is a popu­ �nth nr

«

lation inversion for any pump rate in a four-level system, while in a three­ level system the pump rate at threshold is given by

Wp (th) = J!l:_. 9JTf

( 1 . 1 .35)

Thus the threshold for laser operation is lower for a four-level system than for a three-level system. To have a low threshold for a three-level system it is important that the metastable-state lifetime be long, whereas the fluo­ rescence lifetime of the metastable state does not affect the threshold of a four-level system. Next, consider the balance between input and output power in order to determine the efficiency of laser operation. At threshold, the fluorescence power density is hvon;(th) ( 1 . 1 .36) PJ (th) = ,

'f

where i is the initial state of the laser transition. To maintain laser operation it is necessary for the pump power to compensate for the fluorescence

12

1 . Introduction

power, Pab

P Pj = Vp f = . Vo'lo

( 1 . 1 .37)

'1 1

The ratio of the pump photon energy hvp to the laser photon energy hv0 is known as the quantum defect, and the pump efficiency 'lo is defined as Wp = 11o W1p, where p denotes the level being pumped and

•321

three-level system •321 + •3/ '

(1 . 1 .38 )

four-level system. r431 + r421 + r4 11 '

The quantity 11 1 is the pump efficiency with a factor for the quantum defect included. The difference between the pump power absorbed and the fluo­ rescence power emitted is dissipated as heat in the laser material. Note that equating the pump power and fluorescence power at threshold gives 1 1 PJ ( th ) Pab(th) = hvp n 1 ( th ) Wp ( th ) = -- = hvo n ; ( th ) --

'1 1 TJ

'1 1

.

For a four-level system where n2 « n 1 at threshold, this shows that the pump­ ing rate at threshold is much smaller than the fluorescence decay rate, Wp ( th) « 1 / r!. This is significantly different from the result given in Eq. ( 1 . 1 .35) for a three-level system, which shows that the pump rate at threshold, where the level populations are approximately equal, is almost equal to the fluorescence decay rate within the degeneracy factor. Also, it is important to note that substituting Eqs. ( 1 . 1 .28) and ( 1 . 1 .29) into Eqs. ( 1 . 1 .36) and ( 1 . 1 .37) for the population inversion at threshold shows that the pump power at threshold for a four-level system depends on the product of u and r1. This can pose interesting problems in the choice of optimum laser mate­ rials as discussed in later chapters. Well above threshold, the cavity mode has a large photon density due to stimulated emission. Steady-state conditions are reached when the pop­ ulation inversion is balanced between the pumping rate and the sum of the stimulated and spontaneous emission rates. For dfl.n/dt = 0 and large values of rjJ , Eqs. ( 1 . 1 .23) and ( 1 . 1 .26) give

fin( sat) = nr

(wP _J3_) [ ( 1 gg21 ) curjJ g 1 rf +

three-level system

(

= nr Wp curjJ + Wp +

1 TJ

)1

,

+

Wp +

_!_]TJ

1

,

four-level system.

( 1 . 1 .39)

1 . 1 . Solid-State Laser Operation and Design Parameters

13

The gain for this saturated population inversion gives the saturated gain

coefficient

g ( sat)

(

)

= D"An sat { ( + g 2 ) c mp = go \ pg+1 -1 + 1 l ( lTCr/J ) = go WP + r:/ + 1 , \

1

�

, three-level system

1f

( 1 . 1 .40)

four-level system,

where the small signal gain coefficient is given by Eq. ( 1 . 1 . 1 5),

go

�p TJ-

g2 gl , three-level system

= lTnT Wpr:f + 1 = lTnT pW<Jp<J+ 1 , �

(1 . 1 .41 ) ( 1 . 1 .42)

four-level system,

with the use of Eqs. ( 1 . 1 .33) and ( 1 . 1 .34). Note that small-signal gain depends only on material parameters and the pump rate while saturation gain has an additional dependence on the concentration of photons in the cavity mode. The power density in the beam is given by chvr/J, so the gain can be written as

go g=-1 + ­I, Is

I=

( 1 . 1 .43)

where the saturated power density is defined as the power density at which the gain is one-half of the small-signal gain. From Eq. ( 1 . 1 .40)

Is= (wP + _!_TJ) lT( 1 hv+ g2) , three-level system gl 1_ ) hV four-level system. ( TJ lT -

+

l

( 1 . 1 .44)

For a four-level system with Wp « r:j 1 , the saturated power density reduces to

Is=.!. lTTJ

( 1 . 1 .45)

The rate of stimulated emission is given by

( 1 . 1 .46)

14

1 . Introduction

or in terms of the saturated power density given in Eqs. ( 1 . 1 .44) or ( 1 . 1 .45), 1 gz Is + g 1 (1 . 1 .47) 7:st = 1 , three-level system

I fV:p + Is I

1

I

'J

1

fV:p + -

�

'J s , four-level system. I

( 1 . 1 . 48)

I(z ) ·

( 1 . 1 .49)

'J Thus when the power density of a four-level system reaches the level of saturation, the stimulated emission lifetime is equal to the fluorescence life­ time. In a three-level system this is an approximation with the exact value of the stimulated emission lifetime at saturation dependent on the pump rate and the degeneracy ratio. The characteristics of saturation also depend on the broadening mechanisms affecting the shape of the spectral line. If the transition is homogeneously broadened, saturation occurs uniformly across the spectral line. On the other hand, if the transition is inhomogeneously broadened, subunits of the spectral line can saturate independently of the rest of the line resulting in spectral hole burning . The expressions derived above for threshold population inversion, gain, and cavity losses can all be combined into expressions describing laser oper­ ation parameters. For a beam of photons passing through a gain media in an optical resonator, the intensity increases as oi (z ) oz

=

I(z) + Is

goi(z )

1

-

_

Ct:

For intensities much less than the saturation intensity the single-pass gain in the material is

Gm = II(!) (O) = exp [ ( go

Ct:

)� .

( 1 . 1 .50)

Combining Eqs. ( 1 . 1 . 1 8) and ( 1 . 1 .43) gives the intensity after a round-trip in a cavity as 2/go I=I 1 . ( 1 . 1 .51 )

(

s L +T

)

The fluorescence power can be expressed as2

( 1 . 1 .52) where Pin is the electrical input to the pump source, 11J is the pumping effi­ ciency including the quantum defect, 172 is the efficiency of the pump source, 173 is the efficiency of coupling the pump light into the gain medium, and 174

1 . 1 . Solid-State Laser Operation and Design Parameters

15

i s the efficiency of power absorption. Converting Eq. ( 1 . 1 .36) from power density to fluorescence power at threshold for population inversion allows the small-signal gain coefficient for an ideal four-level system given by Eqs ( 1 . 1 .42) and ( 1 . 1 .34) to be expressed in terms of the electrical input power at threshold,

(1 . 1 .53) where V = AI is the volume of the cavity with length I and cross section A. The cavity gain can be expressed as In G =go! = KPin(th) . ( 1 . 1 .54) Using the expression given in Eq. ( 1 . 1 .45) for the saturation intensity of a four-level system, the efficiency K is defined as

(1 . 1 .55) Using Eqs. ( 1 . 1 . 17)-( 1 . 1 . 19) the electrical power input needed to achieve threshold can be expressed in terms of the cavity losses and efficiency param­ eter K as . _ L - In R 10 Pm ( th) ( 1 . 1 .56) 2K or ln Rw = 2KPin ( th) L. ( 1 . 1 .57) The expression in Eq. ( 1 . 1 .57) is useful for comparing with experimental results to obtain the properties of a laser system. Measurements can be made of the input power required to achieve laser threshold for output cou­ plers having different values of Rw. An example plot of the data from these types of measurements is shown in Fig. 1 .5. According to Eq. ( 1 . 1 .57), the results should fall on a straight line with a slope given by d ( -ln Rw) / dPin ( th) = 2 K. This gives a value for the efficiency parameter, and com­ bined with the saturation intensity determined from spectral measurements this can be used in Eq. ( 1 . 1 .55) to obtain the product of the efficiencies. Extrapolating the line to the intercept at Pin(th) = 0 gives a value for the cavity loss parameter L. This type of procedure was first developed by Findley and Clay4 and is sometimes referred to as the Findley-Clay analysis.

In order to determine the values for the threshold and efficiency of laser operation experimentally, the laser power output can be plotted versus power input. An example of this type of plot is shown in Fig. 1 .6. There is no laser power output until threshold input power is reached, and above threshold Eq. ( 1 . 1 .56) is modified by Eq. ( 1 . 1 .43) to give the power input as L- ln Rw 1 + II = Pin(th) 1 + II . Pin = ( 1 . 1 .58) 2K s s

( )

( )

16

1 . Introduction

FIGURE 1 .5. Variation in laser-power threshold with output coupler reflectivity.

The power output above threshold is described by the expression Pout= 11s[Pin

( 1 . 1 .59)

Pin(th)],

where 11s is the slope efficiency, defined as 11s = KlsA17s

( 1 . 1 .60)

= 11t11211311411s·

Here the parameter 175 is an effective efficiency for output coupling/ 115

=

-Rw) �(L-lnRw) 2( 1

( 1 . 1 . 61 )

.

Pout(kW)

0

4

FIGURE 1 .6. Laser-power output versus power input.

1 . 1 . Solid-State Laser Operation and Design Parameters

17

Figure 1 .6 and Eq. ( 1 . 1 .59) show that the power output of a laser above threshold increases linearly with input power and the slope of the line is given by Yls· From Eqs. ( 1 . 1 .56) and ( 1 . 1 .61) it can be seen that an increase in cavity losses given by L will cause an increase in threshold power and a decrease in slope efficiency of laser operation. The threshold and efficiency also depend on the reflectance of the output coupler. By combining Eqs. ( 1 . 1 .58) and ( 1 . 1 .61) with Eq. ( 1 . 1 .59), the optimum value of the output coupler can be determined by maximizing the output power through requir­ ing dPoutfdRw 0. This condition gives 1 /2 L R 10 ( opt ) 1 - (2KPinL) (1 . 1 .62) . 1 +L For pulsed operation the flashlamps generally provide high levels of input power so the optimum value of Rw will be small. On the other hand, cw operation involves low values of Pin, which requires a high value of the output coupler reflectivity for optimum performance. For a three-level system the equations derived above are modified slightly. The small-signal gain with the expression for the population inversion for a three-level system from Eqs. ( 1 . 1 . 7) and ( 1 . 1 .41) has the form -

ao

go

g g1

WP r2 1 -2

- Wp'r2 J + 1 '

( 1 . 1 .63)

gl

(nr n1 n2

where a0 is the absorption coefficient when all atoms are in the ground state since 0),

(1 . 1 .64)

ao

For no pumping, go -ao . Assuming that the rate of pumping is a linear function of lamp input power,

( 1 . 1 .65) Combining this with Eq. ( 1 . 1 .63) and substituting into the expression bal­ ancing round-trip cavity gain and loss given by Eqs. ( 1 . 1 . 1 8) and ( 1 . 1 . 19) gives g

ln Rw

2 21 IXO KPin(th) - g1 L. KPin(th) + 1

(1 . 1 .66)

gl

This equation can be used to analyze data such as those represented in Fig. 1 .5. The absorption coefficient is generally known from independent spec­ troscopic measurements. The slope of the curve determines K while the intercept at the extrapolated value of Pin(th) 0 gives L.

18

1 . Introduction

With Eq. ( 1 . 1 .63) for go for a three-level system, the expression for the saturation intensity in Eq. ( 1 . 1 .44) becomes Is=

hv (JTJ 1

g g oco g l

__ __

.

( 1 . 1 .67)

The expressions for power out versus power in and the slope efficiency used to analyze data such as those shown in Fig. 1 .6 for a 3-level system are still given by Eqs. ( 1 . 1 .59-60), but the output coupling efficiency is now given by2 ( 1 LM) (l Rw) ( 1 . 1 .68) '15 VRiQ(L ln Rw) ' where _ L ln Rw LM ( 1 . 1 .69) 2/oco

The input power threshold for a three-level system is2 1 + LM Pin(th) = ( 1 . 1 .70) K ( l _ LM) These expressions can be used with Fig. 1 .6 to find values for K and L from the measured values of the threshold and slope efficiency. To do this it is convenient to combine these expressions to give L = A + ln Rw, ( 1 . 1 .7 1 ) where PJ(l

Rw)

2/oco

( 1 . 1 .72)

and K=

2/oco +A Pin (th)(2/oco A)

( 1 . 1 .73)

The maximum output fluence that can be obtained for a laser is deter­ mined by the saturation fluence of the laser transition of a three-level and an ideal four-level system, r = l!:! Wp rf + 1 ' three-level system AL (JL 1 + g2

he

gl

four-level system. ( 1 . 1 .74) AL (JL ' This must be less than the threshold fluence for optical damage of the host

1 . 1 . Solid-State Laser Operation and Design Parameters

19

material. The bulk damage properties of the host are improved by improv­ ing the purity and optical properties of the material. Surface damage prop­ erties are improved by careful polishing techniques. Lensing effects of the light beam traveling through the laser material can be caused by thermal effects and nonlinear processes. This can lead to self-focusing, which may cause the ftuence to exceed the optical damage threshold. For high-energy lasers, thermal management is especially important. Along with providing external air or water cooling, there are several other methods of minimizing thermal effects. One approach is to optimize the cavity design. This involves new geometries such as slab lasers instead of cylindrical rod lasers and new configurations to couple the pump light more uniformly into the material. A second approach is to minimize the heat generated during optical pumping. This can be realized by using monochromatic pumping into a level with a small quantum defect. Pumping solid-state lasers with bars of diode laser arrays is an important way of doing this. Diode-pumped solid-state lasers are important for a variety of technological applications. However, there is a limited number of wavelengths available from diode-pump sources. Therefore some special schemes have been developed for pumping active ion energy levels that are not directly resonant with available pump wave­ lengths. These techniques include avalanche pumping and up-conversion pumping of laser transitions. Finally, it is important to mention confined cavity configurations that make optimum use of the length of the gain medium. These include fiber lasers for optical communications systems and channel waveguide lasers for integrated optics applications in photonic devices. Both of these types of systems can be configured to obtain specific laser output characteristics. It is possible to determine the value of the cross section for the laser tran­ sition from data obtained by measuring laser gain. To do this, Eqs. ( 1 . 1 .45), ( 1 . 1 .55), and ( 1 . 1 .56) are combined to give an expression for the threshold power, P.m ( th)

-

(L + C)hvA 2

rrrtp

'

( 1 . 1 .75)

where rtp is the pumping efficiency, C represents the output coupling loss, and it has been assumed that the radii of the resonator mode and the pump mode are equal. Figure 1 .5 shows a linear relationship between the thresh­ old power and the cavity losses, G p Pin ( th ) = L + C , ( 1 . 1 .76) where Gp is the laser gain per absorbed power. Combining the expressions in Eqs. ( 1 . 1 .75) and (1 . 1 .76) gives an expression for the cross section in terms of the laser gain,

O"etr

=

C)hvA

Gp,

( 1 . 1 . 77)

20

1 . Introduction

where the cross section obtained in this way O"eff is called the effective cross section to differentiate it from the cross section measured directly from spectroscopic data. Differences between the two types of cross sections may be associated with processes such as excited-state absorption that contribute to losses in the laser-gain measurement but do not directly effect the spec­ troscopic measurement. One common technique of generating high-peak-power laser pulses is called Q switching. The quality factor of a laser cavity, Q, is the ratio of the energy stored in the cavity to the energy loss per cycle in the cavity. This can be controlled by inserting a variable loss element in the cavity. If this ele­ ment can change the losses from very high (low Q) to very low (high Q) in a short time, it is called a Q switch. For the ideal condition of a change in Q that is much faster than the lifetime of the metastable state in the presence of strong optical pumping, the Q switch can be approximated as a step function in time. The device can be a passive element such as a saturable dye or an active element based on electro-optic or acousto-optic effects. For Q-switched operation, the laser material is optically pumped under high-loss (low-Q) conditions. This allows the population inversion to build up to a value well above threshold conditions while the cavity losses pro­ hibit laser oscillation from occurring. The energy storage time is of the order of the metastable-state lifetime. Thus the pumping time for maximum energy storage should be of the order of this lifetime. When the Q of the cavity is switched to a high value, the large population inversion results in a high gain and the stored energy is emitted in a short time. Since the Q­ switched pulse is so short, the theory of Q-switched laser operation can be described by the rate equations ( 1 . 1 26) and ( 1 . 1 . 27) with the pumping and spontaneous emission terms neglected,

.

( g) o!1.n 1 z !1.mpO"c , at 91 =

+-

=

three-level system

!1.mpO"c , four-level system , ot/J = I !1.nfjJO"c -e at I' tR ifJ '

- -

( 1 . 1 .78) ( 1 . 1 .79)

where the photon lifetime in the cavity is now expressed in terms of the cavity round-trip time tR = 21' / c and the fractional loss per round-trip e, where

tR

e=-= 'l"c

ln Rw + L + ( ( t) ,

( 1 . 1 .80)

where ((t) is the loss introduced by the Q switch [for a step function ((t < 0) = Cmax' ((t � 0) = 0]. Here I is the length of the active material while I' is the cavity length. The peak output occurs when ot/Jfot = 0 at which time 11np ej (20"1) = n 1 • Taking the quotient of the two rate equa=

1 . 1 . Solid-State Laser Operation and Design Parameters

21

tions gives ( 1 . 1 .8 1 ) for a three-level system and the same expression without the degeneracy factor for a four-level system. Integrating this expression gives

] gg2l ) I' is the initial population inversion. Setting � (t) �

(1

I

n,

+

+ n;

( 1 . 1 .82)

An(t) ,

where n; An(t) = np gives the maximum photon flux cftm · The peak output power is derived from this pho­ ton flux in the cavity lifetime, tc = tR / e, with the fraction ln ( 1 / R 1 ) / e going into the laser emission, 1 ln p � eft Alhv Rw

( )

p - m lc

e

Vhv In

I

�

( 1 . 1 .83)

Again, this is specifically for a three-level system with the equivalent expression for a four-level system found be dropping the degeneracy factor. After the pulse emission occurs, cp(t) = 0 in Eq. ( 1 . 1 .82), and the remaining population inversion n1 is found from the expression n1

n;

=

( 1 . 1 .84)

where An in Eq. ( 1 . 1 .82) has been set equal to n1 . Therefore the higher the pumping (n;) the greater the extracted power (n; - n1) . The total energy in the Q-switched pulse is approximated by the simplified expression 2 1 ln Rw Vhv (n; - n1) E -, ( 1 . 1 .85) 1 1 +In - +L g1 R10 where the cavity length has been set equal to the gain length. Note that E, Pp , and time of extraction are all dependent on initial population inversion and therefore on pumping strength. Repetitively Q-switched lasers provide a means of producing high-peak­ power, high-average-power radiation, which is useful in many applications.

=

�

( ) ( )

22

1. Introduction

The radiative lifetime of the laser material, the round-trip cavity time, and the pulse repetition rate all play an important role in determining the overall performance of such a system. The upper lasing level is pumped con­ tinuously in a repetitively Q-switched laser, while the intracavity loss is modulated repetitively to produce Q-switched pluses at a specific repetition rate fr. Because the recharging cycle required to repopulate the upper level has a dependence on the radiative lifetime of the laser medium, the energy obtained per Q-switched pulse will depend on the repetition rate of the modulated loss. A detailed analysis of these interdependencies2 shows that for low repetition rates fr :s; l j r, , the Q-switched output pulse energy will be a constant fixed by the cw pumping rate. In this regime the average power will increase linearly with fr. At high repetition frequencies fr > 1 / r, , the output pulse energy will decrease approximately linearly with fr and the average power will asymptotically approach Pew, the power obtained when the equivalent system is operated under cw conditions. The energy buildup time and the cavity length photon travel time limit Q-switched pulse widths to a minimum of around 10 ns. To generate sub­ nanosecond pulses, mode-locking or cavity-dumping techniques are used. In a typical free-running laser, many longitudinal and transverse modes oscil­ late without any correlation of their amplitudes and phases. Thus the laser emission is randomly distributed over a range of frequencies and time, determining the spectral and temporal widths of the output pulse. There is a large number of longitudinal modes separated by c/(2/) and these can be locked in fixed phase relationships resulting in a Gaussian pulse with a temporal width fl.t related to the frequency bandwidth fl. v through a time­ bandwidth product !!.vAt � K, where K is a constant of the order of unity. This pulse repeats itself with a repetition rate of twice the cavity transit time. The frequency linewidth is the gain bandwidth of the laser material and the number of longitudinal modes within this bandwidth is N fl.vtR. Thus At � KtR/N. Since N can be of the order of 104 even for fairly narrow-line lasers and cavity transient times are a few nanoseconds, it is easily possible to generate trains of mode-locked pulses with widths of the order of a few picoseconds or less. These pulses have high peak powers since the free-running laser emission energy is now emitted in a very short time. Some examples of time-bandwidth products for important solid-state laser materials are Y3 Als 0 1 2 : Nd3 + (J.v 120 0Hz; At 8 ps), ruby (fl.v 60 GHz; At 17 ps); Nd3 + : glass (fl.v 3 THz; At 333 fs), and Ti-sapphire (fl.v l OO THz; fl. t = lO fs) . Mode locking can be achieved by putting an element in the cavity that controls the loss or the phase of the electromagnetic waves in the cavity. For cw pumping, active mode locking with acousto-optic elements can produce a train of pulses with a high repetition rate. Saturable absorption dyes give passive mode locking in pulse-pumped lasers. These modulate loss at a fre­ quency fm c/21 to give gain at a carrier frequency vo and sidebands at frequencies vo ± fm , thus locking modes with vo , vo + fm , and vo - fm in

1 .2. Material Requirements for Laser Hosts and Active Ions

23

amplitude and phase. These sidebands then lock with modes at frequencies v0 ± 2fm , etc., until all axial modes are coupled. In some materials mode locking occurs automatically under certain conditions. A typical cause of this "self-mode-locking" is the Kerr effect. This is a third-order nonlinear optical response of a material to high-peak-power light beams. The light induces a change in the refractive index of the material that can cause the material to act like a lens that produces self-focusing. With the appropriate cavity design, this acts like an intensity-modulated loss with a specific geo­ metric pattern that favors a specific cavity mode. From the discussion in the preceding paragraphs of this section, it is clear that the important laser operating characteristics include appropriate wave­ length, low threshold, high gain, high efficiency, and high power extraction. A low-threshold and high-gain laser requires high values of o-81, Tf, and IJp ­ On the other hand, Q-switched operation requires maximizing the stored energy and thus a small value of o-81 but a value of Tf longer than the pumping pulse time. Since these parameters are related by o-81 oc (n2r1�v) - 1 , a large value of 'f and small O"st are compatible with high-energy storage but not high gain and efficient energy extraction. Thus there is generally a compromise between these two parameters depending on the mode of laser operation desired. Note that the host can affect this relationship through its refractive index value that appears in the denominator. Lower values of n2 can increase o-81 • Another complication is the requirement for �v. The typi­ cal desire for monochromatic emission requires small � v and thus results in a large o-81 • However, for tunable output or mode-locked operation a broad gain bandwidth is required, and this reduces o-81 • It is clear that different types of materials will be appropriate for different modes of laser operation. The important spectroscopic properties of a solid-state laser material that directly affect laser operation characteristics (2, �v, o-81 1 Tf , and IJp ) are all determined by the interaction of the lasing ions with the radiation field, the static host material environment, the phonon field, and the interaction with other ions. These physical processes are discussed in Chaps. 2-5. 1 .2

Material Requirements for Laser Hosts and

Active Ions

In order for a material to be useful for solid-state laser applications, it must possess appropriate chemical, mechanical, thermal, and optical properties. These are determined by a combination of the inherent properties of the host material, the properties of the optically active ions, and the mutual in­ teraction between the host and the dopant ions. These are summarized in Tables 1 . 1 and 1 .2. The most fundamental requirement for a laser material is that it can be easily and economically produced with high quality in large sizes. This aspect of laser materials is discussed in the following section. It should also have a high enough hardness to allow for good optical polishing.

24

1 . Introduction

TABLE 1 . 1 . Criteria for laser materials. Total system Economic production and fabrication in large size Ion host compatibility Valence and size of substitutional ion similar to host ion Uniform distribution of optical centers in the host Host material Rugged and stable with respect to operational environment Chemical: stability against thermal, photo, and mechanical changes Mechanical High stress fracture limit Small thermal expansion and stress optic coefficients to stop lensing High threshold for optical damage Hardness for good polishing Optical Minimum scattering centers Minimum parasitic absorption at lasing and pump wavelengths Low index of refraction to maximize the stimulated emission cross section Optically active centers Efficient absorption of pump radiation Efficient internal conversion to metastable state population with small quantum defect Appropriate energy storage time in the metastable state to utilize all pump energy Efficient radiative emission at the laser wavelength with high quantum efficiency No absorption at the lasing wavelength (either ground or excited state) Emission linewidth compatible with desired tunability and stimulated emission probability Ion ion interaction compatible with maximum pumping and minimum quenching

TABLE 1 .2. Properties of some important host crystals ( Data from Ref. 10.) IXe

K

Material

( l o- 6 /"C)

(W/m 0C)

Y3 Als0 1 2 LiYF4 LiSrAlF6 Cas ( P04h F LaMgAl 1 1 0 1 9 Gd3 Sc2 Ga30 1 2 La2 Be2 0s Al2 03 BeAh04 MgF2

6.7 13, 8 1 9, 1 0 10.0, 9 . 4 7.5 8 4.8, 5.3 4.4, 6.8 6.9 1 3 . 1 , 8.8

10 5.8, 7.2 3.1 2.0 4, 6 6.0 5 28 23 21

n

1 .823 1 .634, 1 .4 1 1 .63 1 .777, 1 .942 1 .964, 1 .762, 1 .746, 1 .38

1 .63 1

dnjdt ( l o- 6 /"C) 8.9 2.0, 4.3 2.5, 4.0 1 0, 8

1 .769 1 .997, 2.035 1 .755 1 .748, 1 .756

1 0. 1 6.2, 1 . 5, 2.9 1 1 .7, 12.8 9.4, 8.3 0.9, 0.3

Discussed in Chap. 7, 8 9 6 9 9 6, 9 9 6, 7 6 7

1 .2. Material Requirements for Laser Hosts and Active Ions

25

In order for the material to be useful in systems applications outside the laboratory, it should also be rugged and stable with respect to local environ­ mental changes such as temperature, humidity, and stress. Since the oper­ ation of the laser requires exposing the material to both light and heat, chemical instabilities in the material can be either photo-induced or ther­ mally induced. Thus, once a material is in use, it must be chemically stable in terms of the valance state of the optically active ions, ion diffusion, the formation of second phases, and the formation of color centers and other defects. In addition, internal stress created thermally or optically in the material can distort its shape or in extreme cases cause it to fracture. Both of these properties depend on parameters such as the thermal expansion coefficient rxe and the thermal conductivity K. These are listed in Table 1 .2 for several common laser crystal host materials. The details of these thermal effects depends on the pumping geometry. It is interesting to note that if the pumping results in uniform heating with a temperature rise of the change in optical path length is given by

11L = ( (n - 1 ) + :;)LI1T. rxe

11T,

( 1 .2. 1 )

Thus it is possible for lensing due to thermal expansion to be offset by the thermal change in the !refractive index for materials with negative values of dn/dt. Several of these are listed in Table 1 .2. A typical thermal lens length in a Y3 A1 5 0 1 2 : Nd3 + laser is about 20 em while in a LiYF4 : Nd3 + laser it is about -4 m. Ideal materials have a high stress-fracture limit, small thermal expansion, and small stress-optic coeffic1e�is. Optical damage due to the presence of defects or laser-induced electric breakdown can limit the useful power levels for laser operation. Therefore it is important for a material to have a high threshold for optical damage. The ions useful for providing the optical dynamics of laser materials must be able to absorb pump radiation efficiently and to emit radiation efficiently at the desired laser wavelength. Some types of ions have excellent absorp­ tion properties but poor emission properties, or vice versa. In this case it is possible to put two types of ions in the same host material, one to absorb the pump energy (called sensitizer ions) and the other to provide the laser emission (called activator ions). The key to making this scheme work is having efficient nonradiative energy transfer from the sensitizers to the acti­ vators. This is achieved through having strong overlap of the emission spec­ trum of the sensitizers and the absorption spectrum of the activators. How­ ever, if the coupling interaction between the two types of ions is too strong, they no longer have the properties of independent ions, but instead form a coupled ion pair with its own spectral properties. The optical spectral properties of a laser material are determined by the electronic transitions of the active ions in the local ligand field environment of the host. The types of ions that are useful for laser emission in the near ultraviolet, visible, and near-infrared spectral regions are transition-metal

26

I . Introduction

ions and rare-earth ions. Both of these types of ions have electron configura­ tions that include unfilled shells and thus have electron transitions between energy levels within a specific shell (d-d transitions for transition metal ions and fd transitions for rare-earth ions). These give rise to absorption and emission transitions in the appropriate spectral range. These are parity­ forbidden transitions that produce weak, narrow spectral lines in the free­ ion spectra. When the ions are placed in a host material, the electrostatic interaction with its surrounding ligand ions can split some of the degenerate free-ion energy levels and cause some of the transitions to be much stronger. The key to having efficient absorption of pump radiation is having a strong absorption transition at the wavelength of the pump radiation. If the pump source is a broad-band spectral emitter, then the absorption band of the ion should be broad in order to absorb the maximum number of pump photons. If narrow-line pump sources are used (such as a laser), then the absorption band of the ion can be narrow but must be exactly matched in frequency with the pump emission. Generally the terminal state of the absorption is not the level from which laser emission occurs. Thus another important aspect of pump efficiency is that the transition absorbing the pump energy must result in populating the metastable state of the laser transition. This requires efficient radiationless relaxation to the desired level without loss of excitation energy to other emission transitions. For max­ imum efficiency of laser operation the difference between absorption energy and emission energy (quantum defect) should be small. Finally, it is important that no excited-state absorption ( ESA) of pump photons occurs. That is, no pump absorption transitions should take place from the lasing metastable state or from any excited level where relaxation to the metastable state normally occurs. The key to having efficient emission of radiation for laser applications is having a strong emission transition at the wavelength of the desired laser output. The quantum efficiency for radiative emission should be high, i.e., small probability for radiationless decay processes. The branching ratio should be favorable so a large fraction of the emitted radiation is in the laser transition compared to the other possible transitions from the same initial level. The lifetime of the metastable state should be long enough to store all of the pump radiation and the emission transition linewidth should be nar­ row enough to have a high stimulated emission probability. Note that the lifetime and linewidth are related by Eq. ( 1 . 1 .9), so there may be a tradeoff with these two conditions. Also, for some applications it is important to have a broad emission band so that tunable laser output can be obtained. The interaction between active ions should be small enough to minimize concentration quenching processes. However, in some cases up-conversion interactions are required to pump the desired metastable state. Finally, there should be minimum absorption of the fluorescence emission, both ground­ state and excited-state absorption. The host material must be transparent to both the pump light and the

1 .3 . Material Preparation and Optical Quality

27

lasing light. Absorption by the host not only limits the availability of pump and lasing photons, it can also result in the production of color centers that further degrade laser operation. It should have a low refractive index to maximize the stimulated emission cross section according to Eq. ( 1 . 1 .9). The interaction between the host lattice and the optically active ions is critical in determining the spectral properties of the material as discussed in the preced­ ing paragraphs. The strength and symmetry of the static crystal field along with the polarizability of the anions determine the energy-level splittings and the strengths of the radiative transitions. The level splittings control the Boltzmann population distributions, which can be critical in determining threshold conditions and transition strengths. The dynamic crystal field gives rise to electron-phonon interactions that result in radiationless and vibronic transitions and in temperature-dependent positions and widths of spectral lines. There is no single laser material that meets all of the criteria listed above. In fact, some of these criteria are mutually contradictory, and there are different types of laser operating parameters that require different types of material characteristics. Thus in designing solid-state laser systems, it is important to have a wide variety of materials available for use and to understand the optical properties of these materials thoroughly. In the fol­ lowing four chapters the electronic energy levels, ion-photon, ion-phonon, and ion-ion interactions are discussed in detail. In the remaining chapters, these basic concepts are applied to specific types of laser materials. 1 .3

Material Preparation and Optical Quality

Although laser operation has been demonstrated in a wide variety of mate­ rials, only a very few types of solid-state lasers have been developed for commercial applications. In many cases the development of a specific type of laser has been limited due to the lack of availability of high-optical-quality material. This can be due to a number of reasons including the expense of exotic materials and the difficulty in producing large-size synthetic materials with the appropriate properties. The standard techniques for growing laser crystals 5-10 are pulling from the melt (Czochralski) and melt growth ( Bridgman-Stockbarger). The former is generally used for oxide materials that must be grown at high temperatures, while the latter has been most useful for fluoride crystals. In the Czochralski growth . technique, into the melted mate­ in boules- of­ sl()wly in . a crucible rial . . Both ��� temperature gradients are critically important in determining crystal size and quality. The development of �uto�J

1

(2. 1 .2)

where the sum is over all of the optically active electrons, i.e., those in partially filled shells. The presence of the other electrons in filled shells is accounted for in the second term describing the Coulomb interaction of the optically active electrons with the nucleus by using the effective nuclear charge eZ; . The third term describes the electron-electron interaction. The Coulomb interaction Hamiltonian can be rewritten in terms of one-electron operators

(2. 1 .3) and two-electron operators

(2. 1 .4) giving Ho =

L fi + L gif . i>j

(2. 1 .5)

The second term in the total Hamiltonian accounts for the spin-orbit interaction, Hso =

L �(r;)l;

·

s; ,

(2. 1 .6)

where �(r;) is the spin-orbit coupling parameter and I; and s; are the orbi­ tal and spin angular momenta vectors of the ith optically active electron, respectively. The energy levels of the electronic states can be found by calculating the matrix elements of the Hamiltonian between the eigenstates of the system that are described in terms of the wave functions of the different electron orbitals. Because of the different magnitudes of the three types of physical interactions described by the Hamiltonian, this can be done in successive steps using the techniques of perturbation theory. The wave functions for the electronic states of the ion can be approximated by linear combinations of products of single-electron wave functions. These wave functions must be constructed to be antisymmetric with respect to the interchange of electrons in two orbitals since the electrons obey Fermi-Dirac statistics and thus must obey the Pauli exclusion principle, which states that no two electrons can have exactly the same set of quantum numbers (including electron spin). This physical condition is enforced mathematically by constructing a wave

34

2. Electronic Energy Levels

function that is antisymmetric with respect to an interchange of the elec­ trons. Antisymmetry guarantees that the wave function vanishes identically if two of the spin orbitals are identical. Before constructing the multielectron wave functions, the properties of the relevant operators acting on single-electron wave functions must be reviewed. The operators of interest are the single-electron Hamiltonian and the orbital and spin angular momentum operators. The time-independent single-electron wave functions 1/J can be expressed as the product of a spatial component u and a spin component X·

(2. 1 .7)

1/J; (j) = u;(j)x; (j),

where the subscript i refers to a specific wave function while the index in parentheses labels a specific electron. The spin component is a two-valued function describing electron states with "spin up" or "spin down." The spa­ tial and spin components of these wave functions obey the orthogonality relations

J u;(j) * uk(j)drj = J;,k ,

(2. 1 .8)

Lx; (j) *xk U) = J;,k ,

(2. 1 .9)

where the integration is over all space for the jth electron and the sum is over the two possible spin states. The energy of the electron orbitals is quantized and the single-electron Coulomb interaction Hamiltonian operating on the single-electron wave functions gives the energy eigenvalues En , where n is an integer called the principal quantum number, which designates the energy state of the system. In addition, only specific shapes and orientations of electron orbitals are allowed, and this is reflected in quantized values for the total orbital angular momentum and the z component of the orbital angular momentum vector, respectively. The angular momentum operator for the kth electron is and is quantized such that has eigenvalues equal to [/ (! + 1) ] h2 , where the angular momentum quantum number l can take on integer values between 0 and n - 1 . The angular momentum is determined by the shape of the elec­ tron's orbit. The orbital shapes for the lowest four angular momentum quantum numbers are shown in Fig. 2. 1 . The z component of the angular momentum operator\ has values equal to m1h, where the orientational quantum number mi takes on integer values ranging from -l to +l. Similarly, the spin angular momentum is quantized such that has eigenvalues i h2 and the z component has eigenvalues msh, where the spin orienta­ tiona! quantum number is ms = ± !, since the electron is a spin-! particle. The eigenstates can .be expressed in terms of their complete sets of quan­ tum numbers and the effects of the operators for the kth electron in a

lk

lk

lkz

Skz

Sk

d electrons 1=2

s electrons 1=0 m=O

o

p electrons 1=1 m=O

f

m=1

2. 1 . Free-Ion Energy Levels

m=1

:�-� * m=o

35

m=2

m=1

m=2

m=3

FIGURE 2. 1 . Shapes of the s- , p-, d-, and /-electron orbitals.

specific orbital can be written explicitly as

Hk l rxn l m, ms) = En l rxn! m, ms) , li l rxnl m, ms) = l(l + l )n2 l rxnl m, ms) , lkz l rxnl m, ms) = m,n l rxnl m, ms) , Skz l rxnl m, ms) = ms h l rxnl m, ms) ,

(2. 1 . 10) (2. 1 . 1 1 ) (2. 1 . 12) (2. 1 . 1 3)

where the total spin quantum number has been omitted since it always has the same value, and a includes the quantum numbers for other observables required to make a complete set of commuting operators. Two other types of operators of interest are the orbital and spin angular momentum raising and lowering operators

(2. 1 . 14) (2. 1 . 1 5) These are not part of the complete set of commuting operators for the sys­ tem so their operation changes an eigenstate into a new eigenstate with a different orientational orbital or spin angular momentum quantum number. Thus,

It l rxnl m, ms) = n [( l + m,) ( l ± m, + 1 )] 1 /2 l rxnl m, ± 1 ms) , st l rxn l m,ms) = li [ (s + ms)(s ± ms + 1 ) ] 1 /2 l rxn l m, ms ± 1 ) .

(2. 1 . 16) (2. 1 . 17)

The interaction between the magnetic moments arising from the spin and orbital parts of the electron's motion acts as a small perturbation on

36

2. Electronic Energy Levels

the energy levels and eigenfunctions determined by considering only the Coulomb interaction. These changes are accounted for by

(2. 1 . 1 8) (2. 1 . 1 9) The spin-orbit interaction Hamiltonian is given by Eq. (2. 1 .6). This inter­ action couples the spin and orbital motion so that the orientations of these angular momenta are no longer independent and thus and ms are no longer good quantum numbers. Instead the total angular momentum oper­ ator j = I + and its z component are quantized according to the same form described above for I and Since I · = (f - 12 - 2 )/2 and the wave func­ tions are designated by the quantum numbers and instead of and ms, the relevant matrix elements for the spin-orbit operators are

m1

s

s.

s

s

j m1

mt

(nl}mJIC, (r) l · s l nl'j'm)' = [J (j + 1 ) 1(1 + 1 ) - s(s + 1 ) ]bssbll'bJJ'bm1mj >

(2. 1 .20)

t,

where depends on the radial parts of the wave functions,

For a Coulomb potential,

� ou ( r) , t,(r) = � c m2 2 2 r or Ze 2 u ( r) = , r

with hydrogen wave functions,

e 2 h2

z4

C.nt = 2m2 c 2 5 n3 1(1 + !) (! + 1 ) · For a 3d electron C.nt = 1 .44 w 2 Z4 em-] This perturbation splits the unperturbed energy levels into levels with total angular momentum quan­ a

X

0

tum numbers ranging in the interval I I s l :: I + s in steps of + 1 . Next consider ions having more than one electron. This requires two extensions of the above discussion: Coulomb interactions between pairs of electrons must be considered, and interactions between the spin and orbital magnetic moments of the optically active electrons must be taken into account. The wave functions describing the states of a multielectron ion can be expressed as linear combinations of products of the single-electron wave functions of the optically active electrons. Again, the Pauli exclusion princi-

j ::

j

2. 1 . Free-Ion Energy Levels ANJULAR MOMENTUM VECTORS

L

s

J

s

(A)

I I

37

EXAMPLE MULTIPLETS

3Po, soo. 7 Fo 2 2 2 .. p3/ 2 · o5/ 2 • F7/ 2 .. 3p2 · 3o , 3F4 3 < 2p112· 2o3/2· 2f'-Jt2 3o i , 3F2 --

-sl :S j :S l+s (B)

FIGURE 2.2. Vector addition of angular momentum.

ple must be satisfied and only the antisymmetric combinations are allowed. The electronic configuration of the optically active electrons is designated by nzm, where m is the number of optically active electrons, n is the principal quantum number of their orbitals, and the orbital angular momentum quan­ tum number is designated by spectroscopic notation where s , p, d , j, g , . . . refers to l = 0 , 1 , 2, 3, 4 , . . . , respectively. The angular momenta of the orbital and spin components of the electron motion for a multielectron ion can be described as the vector sum of the individual angular momenta components of each electron. As long as the spin-orbit interaction is small compared to the electron-electron inter­ action, the procedure for the addition of angular momenta can be done as shown in Fig. 2.2. Here the orbital angular momenta vectors of all of the electrons are summed to give the total orbital angular momentum vector for the ion,

and the spin angular momenta vectors of all of the electrons are summed to give the total spin angular momentum for the ion,

These angular momenta vectors can then be coupled to give the total an­ gular momentum vector for the ion, J L + S. A second alternative for addition of angular momenta is first to sum the individual orbital and spin angular momenta for each electron to form the total i; I; + s;, and then to sum the i; over all electrons to give

38

2. Electronic Energy Levels

Note that the size of the spin-orbit interaction determines which coupling scheme gives a better zero-order approximation to the actual spectrum, but it plays no role in the mathematical transformations involved. The first approach is more appropriate for ions of interest to solid-state laser appli­ cations. The properties of the angular momenta operators and the raising and lowering operators for the multielectron ion are exactly the same as those of the single-electron ion discussed above with the quantum numbers designating the eigenstate of the multielectron ion given by L, S, ML, Ms, J, and MJ. Since the addition of angular momentum plays an important role in describing the eigenstates of a multielectron ion, the quantum-mechanical formalism for this procedure is summarized here. Consider two general angular momenta operators h and h to be added to give j. The eigen­ functions of the coupled state are linear combinations of the products of the individual eigenfunctions, ljt m t ) lh m2 ) = ljth m t m2 ) . A linear combination of these products is needed for each coupled eigenstate with j = Ut - h i , . . . , j1 + h - These combinations are expressed as

ljthjm ) = L ljth m t m2 ) Uth m t m2 ljthjm ) . m t ,m2 The expansion coefficients, called Clebsch-Gordan or Wigner coefficients, are given by

(jtj2 m 1 m2 ljtj2jm ) = J(m, m 1 + m2 ) x [(j + j1 - h ) ! (j - h + h )!(jt + h - j) ! (j + m) ! (2j + 1 ) ] 1 /2 x [(j + h + h + 1 ) ! (h - m t )!(jt + m t )!(h - m2 ) ! (h + m2 ) !r 1 1 2 x L { [(- 1 ) '+h +m2 (j + h + m t - r) ! (jt - m t + r) ! ] r x [(j - j1 + h - r) ! (j + m - r) !r!(r + j1 - h m) !r 1 }, (2. 1 .21 ) r

where the sum over r includes all integers that leave the factorials ranging over non-negative integers. These coefficients have been tabulated and are available from computer routines. Table 2. 1 lists some of the values most important to the spectroscopy of active laser ions. The Clebsch-Gordan coefficients C/,:/�3 m3 can be expressed interms of 3j symbols through

( ].I

.

.)

]2 ]3 m t m2 m 3 .

(2. 1 .22) (2j3 + 1 ) The advantage of using 3j symbols is their symmetry properties. They are invariant to an even permutation of their columns while an odd permutation of columns introduces a phase factor of ( - 1 /1 +h +13 • Changing the sign of =

2. 1 . Free-Ion Energy Levels

39

TABLE 2. 1 . Selected values of Clebsch Gordan coefficients (Reprinted from Ref. 1 with the permission of Cambridge University Press.) (j1 � m1 mz lji �jm)

j= j] + 1 jl j] - 1

mz = 1 (j1 + m ) + (j1 + m + 1 ) ( 2jl + 1 ) (2j] + 2) - m + 1 ) + (jl + m) 2ji (h + 1 ) - m + 1 ) + (j1 - m) 2j] (2jl + 1 )

mz = 0

mz = - 1

(h - m + 1 ) + (h + m + 1 ) (2jl + 1 ) (jl + 1 ) m + 1) - m) + (h + m) jl (2jl + 1 )

m + 1 ) + (j1 - m) (2jl + 1 ) (2j] + 2) U1 + m + 1 ) + (j1 - m) 2jl (jl + 1 ) U1 + m) + (j1 + m + 1 ) 2j] (2jl + 1 )

j= (h + m - �) (ji + m + �) (ji + m + �) (2j l + 3 ) (2jl + 2) (2jl + 1 ) 3 (j] + m - �) (ji + m + �) (h - m + �) (2jl + 3 ) (2ji ) (2j] + 1 )

3 (jl + m + �) (ji - m + �) (j] + m + �) (2j] + 3) (2jl + 2) (2j] + 1 ) - (h - 3m + �)

3 (jl + m - �) (j] - m + �) (ji - m + �) ( 2jl - 1 ) (2jl + 2) (2jl + 1 ) (j1 - m - �) (j] - m + �) (j] - m + �) ( 2j1 - 1 ) (2M (2h + 1 ) (h + m - �) (j] + m + �) (j1 + m + �) ( 2jl + 3 ) (2j] + 2) (2jl + 1 ) 3 (j] + m - �) (ji + m + �) (h - m + �) ( 2jl + 3) (2M (2j] + 1 ) 3 (jl + m - �) (h - m + �) (ji - m + �) (2jl - 1 ) (2j] + 2) (2j] + 1 ) (j1 - m - �) (ji - m + �) (j1 - m + �) (2j] - 1 ) (2j] ) (2j] + 1 )

(h + m + � (2j] + 3 ) (2ji ) (2j] + 1 ) (j1 - m + �) (2jl + 2 ) (2jl - 1 ) (2j] + 1 )

3 (jl + m - �) (ji - m - !) UI - m + !) (2j] ) (2j] - 1 ) (2j] + 1 ) 3 (j] + m + !) UI - m + �) (ji + m + �) (2j] + 3) (2j] + 2) (2j] + 1 ) - (j1 - 3m + �)

(j1 + m + �) (2j] + 3 ) (2ji ) ( 2jl + 1 ) (j1 - m + �) (2jl + 2) (2j] - 1 ) (2j] + 1 )

3 (j] + m - �) (ji - m - �) (ji - m + �) (2ji ) ( 2jl - 1 ) (2jl + 1 )

40

2. Electronic Energy Levels

Table 2. 1 (Cont.) i=

m2 = 2 + m l ) (j, + m) (j, + m + l ) (ii + m + 2) (2j, + 1 ) (2}, + 2)(2}I + 3 ) (2}, + 4)

}I + 2

(j, + m

}I + I

I

}I

2

+m

I ) (}I 2}I (j1

m) (}I m + ! ) (}I I ) (}I + 1 ) (2}1 + I )

jl + 1 j] j] - 1 j] - 2

i=

m + 2)

Ut

2m)

.

( ; , + 2m +

}I

( Zm + I ) (j,

m

3(j, m + l ) (j, + m) 1 ) (2j, + 2)(2j, + 2)(2j, + 3)

(}I (}I ) (ii m + 1 ) (}1 (ji ) ( }I

m + I ) (}I m) 1 ) (2j, + 1 ) (2}1 + 2)

m)(j1 1 ) (2}1

m 1 ) (}1 + m 1 ) (2}1 + I )

I)

3(j] - m + 2) (ji - m + 1 ) ( ji + m + 2) ( j] + m + 1 ) (2jl + 1 ) (2jl + 2) (2jl + 3) ( ji + 2) 3( j] - m + 1 ) ( ji + m + 1 ) m ( ji ) (2j] + 1 ) ( ji + 1 ) ( ji + 2) 3m2 -ji ( h + 1 ) 1 )j! ( j] + 1 ) (2j] + 3) 3 ( jt - m) ( ji + m) ( ji ) (2j] + 1 ) ( ji - 1 ) ( ji + 1 ) - m - 1 ) (ji + m) ( ji + m - 1 ) (2jl - 2) (2j] - 1 ) (2j] + 1 ) ( ji ) I

m2 =

(}I m + I ) ( }I m) (2j, + 1 J (2J, + 2 J (j, + 2)

m m

(j, + m + l ) (j, + m) i t( j, 1 ) (2}1 + 1 ) (2j, + 2) 1 ) (}1 + m + 1 ) (}1 + m) (j1 + m j , u , 1 ) (2}1 1 ) (2j, + I )

(}I I)

2

m + l ) (j, m) ( Jt m + 2) (2}1 + 1 ) (2}1 + 2)(2j, + 3) (2j, + 4)

3(j1 + m + l ) (j1 m) j1 (2}1 1 ) (2}1 + 2)(2j, + 3)

2m

Ut + m + I ) (}I + m) (2ji ) (j1 + 1 ) (}1 + 1 ) (}1 + 2)

(ji ) (2}I

( }I'

m + 2) (j, m + l ) (jt m) ( Jt + m + 2) (2j, + l ) (j, + 1 ) (2}1 + 3) (}1 + 2)

}I + I

2

Zm +

mz = 0

m2 =

}I + 2

}I

(1

m + 2) (JI + m + 2) (ii + m + l ) (j, + m) (2}I + 1 ) (2}, + 1 ) (2j, + 3)(j, + 2)

l ) (j, m) (j, m + l ) (j, m + 2) (2j, 2)(2j, 1 ) (2}I ) (2j, + 1 )

j] + 2

I

( ;t

l ) (ii + m) (j1 m + I ) (}I m + 2) (2ji ) (2j, ! ) (}I + 1 ) (2j, + 3)

j=

}I

.

l ) (j, + m) ( }I + m + l ) (j, m + 2) (2ji ) (j1 + 1 ) (}1 + 2) (2}1 + I )

3(}1 + m

}I }I

m2 = I

1 ) (}1 m + I ) (}I m) (j1 + m + 2) }I (2j, + 1 ) (2j, + 1 ) (2}1 + 4) 1 ) (}1 + m + 1 ) (}1 m) ( }I + m + 2) }I (2j, 1 ) (2j, + 2) (2j, + 3)

m

1 ) (}1 + m + ! ) (}I + m) (}I + m + 2) }I (}1 1 ) (2}1 + 1 ) (2}1 + 2)

+m

l ) (j, + m + l ) (j, + m)(j, + m + 2) 2}1 (}1 2)(2}1 1 ) (2}1 + I )

2. 1 . Free-Ion Energy Levels

41

all elements of the bottom row introduces the same phase factor. The 3j symbols are zero unless the j's obey the triangle rule lj1 h i s j3 s j1 + jz. Also, l m i l < ji and m 1 + mz + m3 = 0. 3j symbols are also tabulated and available from computer routines. Using the principles described above, the procedure for finding the eigen­ values and eigenfunctions of the system are first summarized and then the details presented through specific cases. The first step in determining the eigenvalues and eigenfunctions for a multielectron ion with a specific elec­ tronic configuration is to consider only the Coulomb interaction of each electron with the nucleus. The eigenstates found in this way all have the same energy. This degeneracy is partially lifted by treating the Coulomb interaction between the electrons as a perturbation that leads to a splitting of levels with different values of L and S. Including the spin-orbit inter­ action further lifts the degeneracy by splitting the levels into states with spe­ cific values of J. The final degeneracy due to the 2J + 1 different possible spatial orientations for the angular momentum can be partially lifted only through the presence of an external perturbation such as an applied electric field (Stark effect) or magnetic field (Zeeman effect). There are two sets of commuting operators that can be used to describe eigenstates of the system, I rxLSMLMs ) and I rxLSJM, ) , where rx represents the quantum numbers of all other observables necessary to form a complete set. These two sets of eigenstates are related by the Wigner formula and Clebsch-Gordan coefficients discussed above. The first set of eigenstates can be grouped into subsets having the same values of L and S with different values of ML and Ms. The (2L + 1 ) (2S + 1 ) states in each subset have the same energy since the Hamiltonian for the system does not contain oper­ ators involving Lz or Sz . In other words, the system is spatially invariant and the energy levels do not depend on the direction of orientation of the electron orbits (unless an external field is applied as discussed below). Each L, S subset is called a spectroscopic term, designated by ZS+ l L. The super­ script is called the multiplicity of the term. The spectroscopic notation S, P, D, F, . . . is used for L = 0 , 1 , 2, 3, . . . and the nomenclature singlet, doublet, triplet, etc., is used for multiplicities of 1 , 2, 3, etc., respectively. When the spin-orbit interaction is taken into account, each term is split into states having specific values of J designated by ZS+ I L,, called multiplets. In the Russell-Saunders (or LS coupling) approximation, L and S are still treated as good quantum numbers. This is appropriate for the ions of inter­ est here since the spin-orbit interaction is small compared to the Coulomb interactions. In this case the spin-orbit interaction becomes Hso = ).L · S so the interaction matrix element similar to expression ( 2 . 1 . 20 ) is

( LSJM, I Hso i L'S'J' MJ' )

= 2 [J(J + 1 ) - L (L + 1 ) ;.

S(S + 1 )]Jwc5 ss'JJJ'�M1M1,

42

2. Electronic Energy Levels

which leads to the Lande interval rule for the energy splitting of the multiplets

(2 .1. 23 ) Here A is a radial integral dependent on L and S and is generally found as an adjustable parameter in fitting theoretical predictions with experimental data. Within an LS term, the matrix element of 2.:; ¢(r;) l; · is proportional to the matrix element of L · S. However, L S can mix terms when the mul­ tiplets have the same value of J. As a first approximation, this mixing can be neglected and A is given by

s;

·

A M 1M L S

.

;

.

¢nl m[ m�.

The ground state of a multielectron ion can be determined by a set of general empirical rules known as (a) The lowest energy term will be one with the maximum allowed multi­ plicity. (b) Of the terms satisfying rule (a), the one with the greater value of L will be the lowest in energy. (c) For configurations with less than half filled shells, the multiplet with the smallest value of J is lowest in energy; for configurations with greater than half filled shells, the multiplet with the largest value of J is lowest in energy. To further lift the degeneracy and split the multiplets into energy levels with different M, quantum numbers requires an external field such as a magnetic field (Zeeman effect) or electric field (Stark effect). A local crystal field is a specific type of Stark effect that will be considered in detail below, and the Zeeman effect is described here. The energy of a magnetic dipole moment M in a magnetic field B is En -M · B . The magnetic dipole moment of an orbiting, spinning electron is M ML Ms x where the gyromagnetic ratio for an electron is gs approximately Since M precesses about J, the average value of M is equal to the component of M parallel to Mav (M · there is a unit vector in the direction of Then using and ! the interaction energy of the magnetic moment of an orbiting electron in a magnetic field is

Hund's rules:

+

(ej2me)(j + s) 2.

(I + s)

j,

j.

j =I+s

(ej2me)

jfj)jfj, jfj s . j (j2 + s2 12 ) , (2 .1. 24)

where M B ·

MzB,

eh 2me is the Bohr magneton and j(j + l ) + s(s + l) 1(1 + 1 ) _ 1 + !_j gj2 _- 1 + 2j(j + 1 ) f.n

2. 1 . Free-Ion Energy Levels

43

is the Lande factor. The magnetic dipole moments of the electrons in multi­ electron ions are added vectorially to obtain the total magnetic moment of the ion. Thus each M, component of a specific multiplet may have a differ­ ent energy if an external perturbation such as a magnetic field is present. The detailed treatment for determining the eigenvalues and eigenfunctions can now be discussed. In order to determine the energy of the spectral terms, the unperturbed eigenfunctions are expressed as Slater determinants,

1/1

1/1 1 (1 ) I/J 2 ( l ) (n!) 1 /2 1/1 1 (2) 1/12 (2)

1/Jn ( l ) l/ln (2)

1/Jn (n) 1/1 1 (n) 1/12 (n) (n!) 1 /2 L ) - 1 ) P 1/1 1 ( 1 ) 1/1 2 (2) · · · 1/Jn (n) p 1 2 / (n!) A ll .

(2. 1 .25)

The sum is over all possible permutations P of the electrons in the different orbitals, A is the antisymmetrizing operator, and ll is the product of single­ electron orbitals. The factor (n!) 1 /2 will normalize ljJ if each of the products of the single-electron eigenfunctions is normalized. The secular determinant for the terms is formed by the matrix elements of the Coulomb interaction Hamiltonian (neglecting the spin-orbit interaction) with these wave func­ tions. It has the form

HI ! E HI2 H21 H22 - E

0.

Expanding the determinant gives

(-E) N + ( - Et- I

Lk Hkk + . . .

0.

This is an algebraic equation of the Nth degree, which in general can be written as C(E - E1 ) (E - E2 ) · · · (E - EN ) 0 , and expanded to give

CE N - CE N I

Lk Ek + . . .

0,

where C is a constant and the E; are the roots of the equation. For the sec­ ular determinant expansion to be identical to this expression, C ( - 1 ) N .

44

2. Electronic Energy Levels

Equating terms in E N J gives

diagonal sum rule,

This is the which states that the sum of the roots of a secular determinant is equal to the sum of the diagonal elements of the determinant. The matrix elements that must be evaluated can be written in simplified form as ( II1 I Ho I II;) where Ho can be written as the sum of one-electron and two-electron operators. 3 Because of the orthogonality of the spin and orbital parts of the single-electron wave functions, the matrix elements of Ho be­ tween product wave functions differing by more than two single-electron wave functions are identically zero. The single-electron term in Ho connects product wave functions differing by only one single-electron wave function and therefore can have nonzero diagonal matrix elements. The two-electron operator in Ho connects product wave functions differing by two, one, or no single-electron wave functions. Since the single-electron orbitals can differ in either their orbital or spin parts, the possible nonzero matrix elements are ( II; I Ho i iiJ ) =

( ab l g l a'b' )# - ( ab l g l b'a' )#,

( II; I Ho i iiJ) =

( a l f l a' )# + L ( ( ab l g l a'b )# ( ab l g l ba' )#) , a

a Gu E3Jz 9 , G9 EljZu> £3/Z u, Gu Elfz 9 , £3/Zg , G9

used in optical spectroscopy such as oscillator strengths, transition cross sections, and line shapes are described here. In the preceding section, the strength of a radiative transition was dis­ cussed. It is common practice to characterize the strength of a spectral line by a dimensionless parameter called the oscilator strength or f number. The name is derived from the analogy between the quantum-mechanical expres­ sion for an harmonic oscillator transition strength and the classical expres­ sion for the strength of a radiating electric dipole. The classical expression for the energy emitted by a three-dimensional radiating dipole treated as a simple harmonic oscillator is E = 3mw6 x6 , where wo is the resonant oscil­ lation frequency and x0 is the equilibrium position. 4 For a system with f oscillators, the power radiated is directly proportional to the energy per oscillator and the number of oscillators, P = yfE, where y is a proportion­ ality constant given by y = ( e2 w6 ) / ( 6nmc3 ) . From the preceding section, the TABLE 3.2. Selection rule for the Eg to Tzu transition

(r1 r; => rEo). x

oh

E

8C3

3Cz

6C4

6q

Eg T2u T1u £9 x T2u

2 3

0 0 0

2 I I

0 I I

0 I I

0

2

0

0

3 6

x, y, z (ED) T1u + T2u =

100

3 . Radiative Transitions

energy of a quantum mechanical oscillator is E 3hw and the power radi­ ated is P hwA 2 I · Using the classical expression to equate power and en­ ergy, hwA 2 I yf3hw, orf A 2 1 /(3y) . Using the expression for the Einstein coefficient given in Eq. (3.2.6) divided by 4n to account for the angular dis­ tribution of the emission, the expression for the oscillator strength of a pho­ ton emission transition between states 2 and I is given by !zi

2mw M I 2 . 3he2 I 2 I

(3.3. 1 )

Here m and e are the mass and charge of an electron, w is the frequency of the transition, and M2 1 is the transition matrix element. If there is more than one emission transition from level 2 to different terminal levels, it is necessary to sum over all such transitions. If the initial level is degenerate, the degeneracy factor g2 must be included in the denominator of the oscil­ lator strength. The sum of the oscillator strengths from a given initial state to all possible final states equals the number of electrons in the system that can take part in these transitions. The direct relationship between h i and the Einstein's spontaneous emis­ sion coefficient from Eq. (3.2.6) is

A21

8nw3 M I 2 2e2 w2 3h c3 I 2 I m e3 h i ·

(3.3.2) Since the emission rate is the inverse of the radiative lifetime, r2} A 2 I ,

there is a useful expression for the product of the radiative lifetime and the oscillator strength of a transition,

hi r2 I

mc3 2 e2 w 2

-

1 .5 U22 I .

(3.3.3)

Note that the wavelength is for the light in the host material and is ex­ pressed in centimeters in this expression. [Equation ( 1 . 1 . 1 1 ) gives this ex­ pression for f with the factor for the refractive index included.] Since the Einstein coefficient for stimulated emission Bz 1 is related to the A 2 I coefficient by Eq. (3. 1 .36), the oscillator strength of the transition can be expressed in terms of the B2 I coefficient as

2n2 e2 mw

Bz i

(3.3.4)

The absorption and stimulated emission coefficients are related by the ratio of the degeneracies of the two levels of the transition, B 1 2 / B2 I gzl g i . Thus the oscillator strengths for absorption and emission transitions between the same levels are related by !zi

gi /!2 gz

(3.3.5)

In addition to the strength of a transition, the shape of the spectral line is important. The shape is determined by the sum of the energy (or frequency)

3 . 3 . Properties of Spectral Lines cr(v)

101

FIGURE 3 . 2 . Gaussian (inhomogeneous) and Lorentzian (homogeneous) line shapes.

11v single ion c �s vo (A) Gaussian

v

v

(B) Lorentzian

widths of the initial and final energy levels for the transition. There are three major types of line shapes: Lorentzian, Gaussian, and Voigt. Different types of physical processes give rise to Gaussian or Lorentzian line shapes as dis­ cussed below. Any physical process that has the same probability of occur­ rence for all atoms of the system produces a Lorentzian line shape, while a physical process that has a random distribution of occurrence for each atom produces a Gaussian line shape. The former is known as homogeneous broadening and the latter inhomogeneous broadening. Figure 3.2 shows a schematic representation of the difference of these two types of line shapes. If both types of broadening processes are present, the line shape is the con­ volution of Lorentzian and Gaussian contributions, and this is called a Voigt profile. Lorentzian broadening is sometimes referred to as lifetime broadening be­ cause the physical processes that produce Lorentzian line shapes are gen­ erally ones that shorten the lifetimes of the energy levels involved in the transition. The most basic process of this type is associated with the Heisen­ berg uncertainty relationship relating time and energy. This contribution is referred to as the natural line width for a transition. The derivation of the expression for the spectral shape of a radiative transition can be found in most quantum-mechanics textbooks5 and the procedure is summarized here. Consider an atomic system with a time-dependent perturbation H', causing a transition from a specific state n to a continuum of energy states m. The

102

3. Radiative Transitions

expressions describing this system are

ihcm (t)

=

ihcn (t)

=

cn H'mn eiwmnl ' cm H'nm e- iwm•1 dm

J

'

where the c; are the probability amplitudes for finding the system in state i. At time t = 0, cn (O) = 1 and cm (O) = 0. The probability of finding the sys­ tem in state n decreases exponentially with time as

Cn (t)

=

e y l/ 2 ,

where y is a constant. This is introduced as a phenomenalogical damping term. Integrating the first expression over time gives h

i H e Uwm. r/2)1 �n y lWmn - 2

Cm =

_

.

1

Substituting these two expressions for en and em into the second expression above and solving for y gives

2i h

J

I Y = 2 I Hmn I 2P (Wnm )

1

e Uwnm +Y/2)1 · dwnm , zy Wnm - 2

where the density of final states is derived from

dm

=

dm dw ) dw Wnm nm p(Wnm nm · =

Since y is the indeterminacy of the initial state, it is much smaller than the transition frequency Wnm · Thus,

2i

J

2 Y�h I H�n I P ( Wnm )

1

- eiOJnml dw nm · Wnm

The exponent can be expressed in terms of trigonometric functions. For large times the cos (wt) function gives no contribution to the integral except when w = 0 at which point it is necessary to use the principal value of the integral. The definition of the J, function can be used to evaluate the integral of the second term at long times. The results give y = R e ( y ) + i lm (y) , where Re ( y )

=

lm ( y )

=

I H l 2p(wn = Wm ) , h �n (w ) P I H�n 1 2 p n Wm dwnm · Wnm

Wnm h

=

J

=

The expression for Wnm in Eq. (3.3.6) is the transition rate.

(3.3.6) (3.3.7)

3 . 3 . Properties of Spectral Lines

103

Substituting these expressions into the em expression and taking the limit of long times gives

l cm ( oo ) I 2

�h 1 H�n l 2

W2 [Wm + 21 Im ( Y ) - Wn ] 2 + nm

•

(3.3.8)

The intensity of a spectral line is directly proportional to the steady state probability of finding the atom in state m, which is given by I em ( oo) 1 2 . As seen in Eq. (3.3.8), this has the form of a Lorentzian function

I(w ) �

11w (w - wo ) 2 +

(

2,

(3.3.9)

where wo is the frequency of the line peak and 11w is the full width at half maximum of the line. Comparing Eqs. (3.3.8) and (3.3.9) shows that the linewidth is equal to the transition rate. w 1 is the lifetime r of the state and related to the width of the energy level through the uncertainty principal r/1£ h . Im(y) is a self-energy shift of the energy levels of the atom in the photon field, called the Lamb shift. It is due to the continual absorption and emission of virtual photons. In general, the observed spectral linewidth is significantly greater than the natural linewidth due to the presence of additional broadening mechanisms that shorten the lifetime of the initial state of the transition. An example of this type of mechanism in gas systems is collision broadening. In solids, radiationless relaxation processes produce Lorentzian broadening. These are discussed in the following chapter. Gaussian line shapes are produced when different subsets of atoms have different peak frequencies. Each subset produces a spectral line with a Lor­ entzian shape, but the envelope produced by the addition of a random dis­ tribution of the lines from these subsets has a Gaussian shape. The Gaussian function describing the superposition of random events is

I (w)

--4n ( n ) (-4 11w

In 2 ' /2 - exp

In 2(w - w0) 2

(11w ) 2

)

(3.3 . 1 0)

where the linewidth and peak position are the same as defined previously. In gas systems, Gaussian broadening is associated with the Doppler shifts in the frequencies due to a Maxwellian distribution of velocities. In solids Gaussian shapes arise due to random distributions of local crystal fields at the sites of ions due to microscopic strains. The differences between Gaussian and Lorentzian line shapes can be im­ portant in determining laser characteristics. For transitions with Lorentzian shapes, all of the ions can participate in laser emission at a specific fre­ quency and single longitudinal mode operation can be obtained. For tran­ sitions with Gaussian line shapes, several subsets of ions may lase simulta­ neously if the Gaussian linewidth covers several free spectral ranges of the

1 04

3 . Radiative Transitions

cavity, and multimode laser operation will ensue. Inhomogeneously broad­ ened lines exhibit spectral hole burning while homogeneously broadened lines exhibit only spatial hole burning. The magnitude of a line-broadening contribution generally depends on temperature and concentration of atoms. In Sec. 1 . 1 the Beer-Lambert Law was derived, demonstrating the ex­ ponential decay in the intensity of a beam of light traveling through an absorbing medium. The dynamics of the energy density of photons in the beam Pv , is linked to the transition probabilities and the absorption spec­ trum of a sample by the equation

- dp

n 1 W12 hvS(v) - nz Wz 1 hvS(v) + AnzhvS(v), where S(v) i s the normalized line-shape function and n; represents the den­ sity of ions in level i. The induced transition rates are related to the Einstein coefficients by W1 2 B 1 2pv , and the spontaneous emission coefficient is related to the radiative lifetime of the level by A c 1 . For absorption of

Tt

a directional beam of photons, the spontaneous emission term can be neglected and dt replaced by dxj c. Then the absorption expression can be integrated to give (3.3. 1 1 ) Pv (x) Pv (O) e a ( v)x where the absorption coefficient is given [as in Eq. ( 1 . 1 .3)] by

a(v)

(

)

hv 92 B n 9 1 nz c 9 1 21 1 - 92 S(v) 92 n S(v) . � 8nr2 9 1 n 1 - � 92 2

(

)

(3.3 . 1 2)

Here Eqs. (3. 1 . 8 ) and (3. 1 .20) have been used to relate the Einstein B2 1 co­ efficient to the radiative lifetime of the excited state. The integrated absorp­ tion coefficient is found from (3.3 . 1 3 )

where the normalized nature of the line-shape function has been used and I is the average wavelength (centroid) of the spectral line in the material. Under normal conditions of weak pumping n2 /n 1 « 1 , so

2 n A. 92 1 . (3.3. 14) a(v) dv � 8n 9 1 r2 This expression is known as the fundamental formula for absorption. Another useful parameter is the absorption cross section defined by [see

I

Eq. ( 1 . 1 .7)]

(3.3 . 1 5)

3 . 3 . Properties of Spectral Lines

105

For a Lorentzian line shape, the peak absorption cross section is given by

;. 92 1 a(vp) 4n2; v g; � r2 '

(3.3. 1 6)

while for a Gaussian line shape the peak absorption cross section is given by

;.; 92 _!_ _ a(vp) �v 8n 9 1 r2 � The integrated absorption cross section is given by 2 1 ). 92 , a(v) dv 8n 9 1 r2

(3.3. 1 7)

I

for both types of line shapes. Thus, the radiative lifetime and the sponta­ neous transition rate can be expressed in terms of the integrated absorption cross section as

-1 A 2 1 '2

8n 9 1 a(v) dv. =/ 92

I

(3.3. 1 8)

The Einstein B coefficients can also be expressed in terms of the integrated absorption cross section,

e f!.!. h v 92 a(v) dv B1 2 a(v) dv. B2 1

I

(3.3. 19)

Finally, the oscillator strength and the integrated absorption cross section are related by

2 I (v) dv _ neme (]"

+

9n

1 2·

(3.3.20 )

The intensity of a spectral line associated with a fluorescence transition of an ensemble of atoms with concentration n0 in length l of sample and ab­ sorption transition cross section a(va ) excited by a beam of photons with intensity Io is described by

Ij(VJ) Pj (va , VJ )hvt dva dvf nolo(va ) la(va ) w2 1 ( vJ )r2 dva dvj . (3.3.21 ) p1 (va, VJ ) is the number of photons emitted per second at frequencies be­ tween VJ and VJ + dv1 after absorption of photons with frequencies between Va and Va + dva . Multipling the photon emission rate by hv1 gives the energy emitted per second. The total integrated intensity in the spectrum is found by integrating over dva and dv1 . Rate equations are very useful in understanding the intensity and time evolution of a fluorescence transition. The expressions describing the con-

106

3 . Radiative Transitions

centration of ions in each level of a three-level system such as the one shown in Fig. 1 .4 (A) are

n 1 - w1 3 n 1 + r2 11 n2 , n2 r321 n 3 - r2 11 n 2 , n3 w1 3 n 1 - r321 n3 , n n 1 + n2 + n3 . 0

0

0

(3.3.22)

The transitions characterized by the rates used in these equations are shown in Fig. 1 . 4 (A) . For steady-state conditions (continuous pumping and con­ stant loss, that is, no Q switching), the time derivatives can be set equal to zero and the set of equations solved for the population of the metastable state n2 ,

'!'321 -

l m> I I

�

I

�

FIGURE 3.4. Schematic diagram for a beam of photons interacting with a four-level atomic system (after Ref. 8).

versus the ground state, 8

(xm( l )

Xe(3rr)

_

Nm Xg( 1 ) ) N '

(3.4. 1 6)

where Nm represents the number of ions in the metastable state. Using a rate-equation approach to describe the transitions and energy-level pop­ ulations as described in Sec. 1 . 1 , the fraction of ions in the metastable state under steady-state pumping conditions is

Nm N

W W+r 1 '

(3.4. 17)

where W and r 1 are the pump and decay rates for populating the meta­ stable state, respectively. Below saturation conditions W < , 1 so the frac­ tion of excited ions is proportional to the pump rate. Since W depends on the intensity of the light beam and thus the square of the optical electric field, this has the same optical electric field dependence as the normal x ( 3 ) term. For optically pumped laser materials, this term can produce intensity­ dependent absorptive, dispersive, and thermal changes in the complex re­ fractive index. The effective x ( 3 ) term can be understood by considering the four-level atomic system shown schematically in Fig. 3.4. l g ) and l m ) represent the ground and metastable states, respectively, of an ensemble of ions such as the active ions of a laser material. These are split by an energy difference hA. I a) and l b ) represent all of the other excited states of the system. Optical pumping distributes some of the ions in the metastable state and leaves some in the ground state with Pam and Pmu representing the relaxation rates from the pump band to the metastable state and from the metastable state to the ground state, respectively. A beam of photons of frequency w interacts with the ions in the ground state through a dipole moment operator flua and with

3.4. Nonlinear Optical Properties

1 13

the ions in the excited state through a dipole moment operator flmb · The components of the density matrix p for the system can be found from the solution of the Liouville-Schrodinger equation [H , p] + ihPdecay·

ihp

(3.4. 1 8 )

The Hamiltonian for the system is H

Ho + Hint ,

where

� �} c 0 0 a 0 0 m 0 0

f.

Hint

e

e

b

�

V:,g

Vga 0 0 0 (3.4. 19 ) 0 0 0 Vbm -pij Eo , where Eo is the mag­

�}

and the interaction operator is given by Vij nitude of the electric field of the optical beam e; represents the energy of the ith state. The susceptibility for this system is calculated by evaluating the trace of the product of the density matrix operator and the dipole moment operator, 8

x

4nN Tr

E

(pp ) .

(3.4.20 )

In most cases of interest the relaxation rates between excited states are large compared to the rate of decay of the metastable state, and the metastable­ state decay rate is large compared to the pump rate below saturation con­ ditions. Under these conditions the expressions given above can be solved to give the real and imaginary components of the susceptibility,

( (

)

2n 2Wmb lflmb 1 2 _ 2Wga lflga 1 2 2 - h wm2 b w2 wga2 - w2 - L lflga I 2 T�1 lflmd Tu,1 � I) 2n WN , �Xlm gm h (wmb - w ) 2 + T'f.b (wga - w ) 2 + T�2 n2Nm �(J/4n , ( I)

_

)

(3.4.21 )

where the inverse of the metastable-state decay rate has been replaced by the lifetime of the state Tm , the T2; parameters are the dephasing times of the ith level, and � r:xp and � (J are the differences in the polarizability and absorption cross section of the ion in the excited state versus the ground state. For most of the materials of interest for solid-state lasers, the real part of this effective susceptibility is greater than the imaginary part. Using the ex­ pression in Eq. (3.4.8), the polarizability difference given in Eq. (3.4.21) can be converted into a refractive-index difference. If the population distribution is not uniform throughout the material this can lead to a spatial variation in

1 14

3. Radiative Transitions

!J.n that alters the optical path of a laser beam propagating in the material. A radial distribution of !J.n results in a population lens. The importance of this effect is discussed further for specific laser materials in Chaps. 7 and 9.

It is important to note that the radiationless relaxation processes occurring in the optical pumping dynamics of the active laser ions is one major source for generating heat that leads to a thermal change in the refractive index as described by Eq. (3.4.8). Since the same pumping dynamics are associated with producing this heat and producing the metastable-state population, the shapes of the spatial distribution of the heat generated and the metastable­ state population are the same. Thus the spatial patterns of !J.n for the ther­ mal lens and the population lens are the same. If these have opposite signs, one effect can be offset by the other. The processes shown schematically in Figs. 3.3( B) and 3.3(C) both in­ volve absorption of light by ions that are already in an excited electronic state. Thus both of these types of processes are actually excited-state ab­ sorption processes. It is usual to distinguish between these two types of pro­ cesses because of the different ways they are involved in optical pumping dynamics. The distinguishing feature between these processes is that in one case the terminal state of the first photon absorption transition is the initial state for the second absorption transition, while in the other case this is not true. In general, the radiationless relaxation processes between excited states of optically active ions in solids is very fast. Thus under normal conditions for optically pumped laser materials, the sequential two-photon excitation process (STEP) is negligible. However, for pumping with ultrafast laser pulses ( picoseconds or faster), STEP transitions can be important. The cross section for the second absorption transition for a STEP mechanism can be determined experimentally from the fluorescence spectrum by using the ex­ pression9

p hv P2 !J.t CT23 p�; hh hv2 0.375/p ' 3

(3.4.22)

where h and h are the fluorescence intensities of the emission from the in­ termediate and terminal states of the STEP transition immediately after a Gaussian pump pulse of intensity lp and temporal width !J.t. hvi represents the energy of the photons emitted from the ith level, p; represents the radi­ ative emission rate for the ith level, and pi is the total decay rate of the ith level. Thus the cross section for a STEP transition depends on the ratio of the excitation pulse width to the intermediate-state lifetime. The STEP mechanism has been found to be important when high-power, short-pulse lasers are used to pump ions at a wavelength in resonance with a two­ photon transition having a real intermediate state. This can be very useful as a spectroscopic tool to elucidate the radiative and radiationless relaxation properties of excited states of ions in laser materials. Examples of this are discussed in Chaps. 8 and 9.

References

1 15

The third type of two-photon absorption processes is excited state ab­ sorption ( ESA), which generally occurs from the metastable state in lasing ions. Since a significant number of ions can be in the metastable state during the optical pumping of laser materials, this can be a significant process in the pumping dynamics. Two types of ESA processes are important, one in­ volving the absorption of laser photons and the other involving the absorp­ tion of pump photons. The first of these can act as a significant loss mecha­ nism that degrades laser performance. It is a major factor in limiting the tuning range of vibronic laser materials and can prevent some materials from lasing at all. These effects are discussed for specific laser materials in Chaps. 6-10. The second type of ESA process can be especially effective for monochromatic pump beams if they are at a wavelength in resonance with an ESA transition. This can be very important for diode laser-pumped solid­ state lasers. The ESA of pump photons results in decreased pumping effi­ ciency for the laser. However, in some cases there are higher-energy meta­ stable states of the ion that can laser. In this case ESA processes are termed up-conversion pumping for the upper laser level. The effects of these pro­ cesses on specific solid-state lasers are discussed in Chaps. 8 and 9. As discussed above, the nonlinear optical effects associated with the re­ fractive index can be important in determining the beam quality of solid­ state laser operation, while the effects associated with multiphoton absorp­ tion provide both loss mechanisms and alternate pumping schemes for laser operation. Both types of effects will be discussed further in Chaps. 6-10 where specific types of lasers are considered. References

1 . W. Heider, Quantum Theory of Radiation (Oxford University Press, London, 1 944) . 2. J.D. Jackson, Classical Electrodynamics, ( Wiley, New York, 1 975) . 3. C. Coen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and A toms ( Wiley, New York, 1 989). 4. H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron A toms (Springer-Verlag, Berlin, 1 957) . 5. E. Merzbacher, Quantum Mechanics ( Wiley, New York, 1 96 1 ) . 6. R. Adair, L.L. Chase, and S.A. Payne, J. Opt. Soc. Am. B 4 , 875 ( 1 987); Phys. Rev. B 39, 337 ( 1 989) . 7. M.J. Weber, D. Milam, and W.L. Smith, Opt. Eng. 17, 463 ( 1 978) . 8. R.C. Powell, S.A. Payne, L.L. Chase, and G.D. Wilke, Phys. Rev. B 4 1 , 8593 ( 1 990) . 9. G.J. Quarles, G.E. Venikouas, and R.C. Powell, Phys. Rev. B 31, 6935 ( 1 985).

4

Electron-Phonon Interactions

The ions in a crystalline or glass solid are never completely at rest. The thermal vibrations of these ions modulate the local crystal field at the site of an optically active point defect. This modulation can have several types of effects on the optical properties of the defect. For example, it can modulate the position of the electronic energy levels, thus leading to a broadening and shifting in peak position of the spectral transition. Also it can cause tran­ sitions to occur between electronic energy levels accompanied by the ab­ sorption or emission of thermal energy but with or without the emission or absorption of photons. Since the motion of the host ions is due to thermal vibrations, all of these effects depend on the temperature and the thermal conductivity of the material. In addition, they depend on the strength of the coupling between the electrons of the optically active ion and the local crys­ tal field. Ions with unshielded optically active electrons exhibit stronger thermal effects than those with their optically active electrons in inner-shell orbitals shielded by outer-shell electrons. Finally, these effects depend on the frequency distribution and density of states of the host vibrational modes. For example, hosts with high-frequency vibrational modes can be more effective in causing radiationless transitions between electronic energy levels with a large energy separation. When the optically active defect differs in mass or charge from the host ion for which it substitutes, local modes of vibration can be introduced that have frequencies that are different from the lattice modes. These local modes can be effective in contributing to the thermal properties of the optical spectrum before dissipating into the host lattice modes. Thermal effects can improve laser performance through processes such as enhanced optical pumping, through efficient depopulation of the terminal level of the laser transition, and through provision of broad emission bands for tunable laser emission. However, thermal effects can also be detrimental to laser performance through processes such as decreasing the quantum efficiency of the laser transition, decreasing the gain cross section through spectral line broadening, and thermally populating the terminal level of the laser transition. In the following sections thermal effects on the optical 1 16

4. 1 . The Phonon Field

1 17

spectra of ions in solids are described in terms of the weak coupling and strong coupling cases. Thermal effects on specific laser systems are discussed in Chaps. 6-10. 4. 1

The Phonon Field

The thermal vibrations of the atoms or ions in a solid are best described as quantized harmonic oscillators. This is similar to the treatment of the radia­ tion field outlined in Sec. 3. 1 . The quantized thermal vibrations provide a field of phonons instead of the field of photons associated with electro­ magnetic radiation. The atoms in the solid vibrate about their equilibrium positions, and since the atoms are bound to each other, their vibrational characteristics are coupled to each other. If there are N atoms in the solid, there are 3N modes of vibration grouped into two branches, acoustic pho­ nons and optic phonons. The problem is thus to find the solutions of the coupled harmonic oscillator equations in terms of the normal modes of vibration and normal coordinates of the solid. For molecular solids and glasses having only short-range order, local modes of vibration are impor­ tant, whereas for crystals with long-range order the extended lattice vibra­ tional modes provide the critical contribution to the thermal properties of interest. In general, the term phonon will be taken to include both types of vibrational modes, even though it is usually reserved for vibrational modes where the wave vector is a good quantum number. Placing dopant ions in the solid can alter the vibrational modes of the host as discussed below. The mathematical approach to any of these cases begins with the fundamental procedure for describing the vibrations of ions in a simple crystal lattice. This can be found in any solid-state physics textbook1 and is outlined below. The first step in the problem is to determine the Hamiltonian for the sys­ tem. This can be expressed as H

Hion + H,

+ H? ,

where the three terms on the right-hand side of the equation are the Hamil­ tonians for the ion, the lattice, and the electron-phonon interaction. To de­ rive the lattice Hamiltonian, consider a crystal lattice with basis vectors a , , a2 , and a . The position of an ion in the lattice is given by the primitive 3 lattice vector (4. 1 . 1 ) where the n; are integers. The crystal is assumed to have one atom per unit cell and contains N N1N2 N3 cells, where N; is the number of cells in the a; direction. Next, consider the vibration of atoms on this crystal lattice. This is done by expanding the expressions for the kinetic and potential energies of the vibrating atoms about their equilibrium positions. In the harmonic approx-

118

4. Electron Phonon Interactions

imation, only the quadratic terms of the expansions are retained. Assuming that the interatomic forces are proportional to the relative displacements of the atoms, the kinetic energy T and potential energy V of the atomic motion are expressed as

T = 2I m V=4

N 3

i=l !i.=l

U' ;2a '

( 4. 1 .2)

N N 3 3 L L L L A ;a,jp U;a �p ,

( 4. 1 .3)

i= l j=l a= ! P=l

where U;a is the displacement of the ith atom in the et. direction at time t, and the A ;a,jft are constants describing the interatomic forces. The Hamil­ tonian for the lattice is then given by H=

N N 3 3 1 N 3 T+ V=l ! f + L L L L L A;a,jp U;a �p , L P a m

(4. 1 .4)

where P ia = m U;a . Because the Hamiltonian is intrinsically harmonic, the complex normal coordinates can be introduced with the following Fourier transform pairs, i= l a=!

Ur.a QqA =

i=l j=l a=! P=l

q,A l, !i.

A eiq · r; QqA eqa ,

(4. 1 .5)

iq r; Ul·!i. i• ' qa e

( 4. 1 .6)

where q ranges from 1 to N and A ranges over the different acoustic and optic branches. Since Q� q = Q;* , the coordinates associated with q and - q are not independent. In order to deal with only the 3N-independent real co­ ordinates, Eq. (4. 1 .5) can be rewritten as U!!1.·

=

1

__

N/2 q> O A

q qa e

iq · r,

+ QqM iqa• e

iq · r, )

·

( 4. 1 . 7)

Using Eqs. (4. 1 .5)-(4. 1 .7) in Eqs. (4. 1 .2) and (4. 1 .3), the kinetic and po­ tential energies can be expressed in terms of the normal coordinates. The kinetic energy becomes

=_

1 2N

z,rx

q,A q',A'

)

. . A eA' . QqA QqA'' ei(q+q' ) · r, eqa q'a

1 19

4. 1 . The Phonon Field

The sums over i and IJ. can be evaluated using the orthogonality conditions, 1

which reduces this expression to

T=2

(4. 1 . 8)

q,.l.

Similarly, the potential energy becomes V = l2

i, rx j,p

1 A· · rrx , JP m

N

q,.l.

Qq.l. e.l.qrx e iq·r,

)

q ' , .l.'

Qq.l.'' eq.l.''P eiq' ·r1

)

This can be simplified by defining a function Grxp (q' ) through /

1

L. A trx,JP erq -r1 m

_

L. A trx,. J.P e - r"q' (r,-r1 ) erq ·r; m 1

•

·

J

J

I

. Grxp (q' ) erq r, · •

I

Therefore the potential energy can be written as

=2

q, .l.

w2q.l. Qq.l. Q .l. q ·

(4. 1 . 9)

In deriving the above expression, the following relationship was used 1 :

3

p

' GrxP ( - q ) e.l._qp

w2q.l.' e.l.' qrx

_

1 m

p

j

A

i ·(r, r1 ) e.l.' irx,jp e q qp ·

(4 . 1 . 1 0)

The Lagrangian for the system is (4 . 1 . 1 1 )

1 20

4. Electron Phonon Interactions

Thus the momentum conjugate to Q; is ).

aL

Pq = a . ). = Qq

=

PA*q ·

(4. 1 . 1 2)

The Hamiltonian is then given by

H = L P;Q; - L = ! L p� q p; + ! L W�;. Q� q Q: q, ). q, ). q, ). 2 ). ). M ! =2 q pq + Wq). Qq Qq ) . q, ).

(4. 1 . 1 3 )

For a quantum-mechanical system the form o f the Hamiltonian i s the same as Eq. (4. 1 . 1 3) except that the normal coordinates and conjugate momenta are operators,

H = ! L (P;t p; + W�;_�t Q: ) . q, ).

(4. 1 . 14)

This has the form of the Hamiltonian of an ensemble of harmonic oscil­ lators. The usual creation and annihilation operators can be formed (see Sec. 3 . 1 ), ( 4. 1 . 1 5)

These expression can be inverted to express the coordinates and momenta in terms of the creation and annihilation operators: (4. 1 . 1 6)

These operators create and annihilate phonons in the normal way for har­ monic oscillator models

b; I n; ) = b;t 1 n; ) =

;; I n; - 1 ) ,

+ l i n; + 1 ) ,

(4. 1 . 1 7) (4. 1 . 1 8 )

where the n� are the occupation numbers for phonons with wave vector q and branch A.. The Hamiltonian for the phonon field can now be written in the form

3N H1 = L hwq;. (b;tb; + !) . q, ).

(4. 1 . 19)

The electron-phonon interaction Hamiltonian can be determined from the expressions derived above. Substituting the expression for Q; from Eq. (4. 1 . 1 6) into Eq. (4. 1 . 5) gives an expression for the displacement of the

4. 1 . The Phonon Field

121

atom from its equilibrium position in terms of the creation and annihilation operators UIIX

=

=

2,; m

�" eiq · r• (bAq + bAtq ) -

q, A

;:; q ,.
l

w�� where

-

(4 . 2. 10 ) (4.2. 1 1 )

( 4.2. 1 2)

4.2. Weak-Coupling: Radiationless Transitions

131

theoretical calculation from first principles of the electron-phonon cou­ pling coefficient fJ is generally difficult to do because the exact expressions for V1 are not known. Thus the values of these parameters are determined from fitting theory to experimental data. However, it is possible to use group theory to determine when the matrix element in Eq. (4.2. 12) is zero or nonzero for a specific vibrational mode. Since the electronic eigenfunctions for the initial and final states of the transition transform as irreducible rep­ resentations of the group r ; and r1 and the coupling parameter V1 trans­ forms as rq for the qth vibrational mode, the reduction of the direct product r; X rq X r/ must contain the totally symmetric irreducible representation A1g for the matrix element to be nonzero. Thus only certain vibrational modes can cause nonradiative transitions between specific electronic states. Note that the rates of direct phonon emission and absorption between two specific states differ only in their temperature-dependent factors. If the transitions are fast enough between the two levels, the populations of the levels are said to be in thermal equilibrium. This condition can be described by writing the rate equations for the populations of the levels shown in Fig. 4.2 ( A ) , A

(4.2. 1 3)

where n , + n2 = N. In thermal equilibrium the time derivatives can be set equal to zero and these equations solved to give

n,

=

hw/kB T N +e ehw/kB T ' 1

(4.2. 14) (4.2. 1 5)

and thus

n2 hw/kB T . n, e

(4.2. 1 6)

The time it takes for a system to reach this equilibrium population distribu­ tion can be determined by defining a system relaxation time r ,

, 1

=

wnrem + wnrab

( )

3w3 ( el v, I el 2 coth hw 2npv5h I t/11 l t/1; ) 1 2knT The rate equations in Eq. (4.2. 1 3) can be solved to give n, ( t) n� e - tfr + NW��r( 1 e tfr ) , n2 (t) n�e -tfr + Nw:; r(1 e -tfr ) , =

·

(4.2. 17)

(4.2. 1 8)

1 32

4. Electron Phonon Interactions

where the quantities n? and ng are the initial populations of the levels, and the equilibrium populations given in Eqs. (4.2. 14) and (4.2. 1 5) can be expressed as

n1 ( ) NW�:,n r, n2( ) oo

oo

NW�� r .

(4 . 2 . 19)

Next consider radiationless transitions involving two phonons. There are two contributions to these processes, one that is described by the single­ phonon term in the electron-phonon interaction Hamiltonian and second­ order perturbation theory, and the other that is described by the two­ phonon term in the interaction Hamiltonian and first-order perturbation theory. These contributions add coherently. In general it is assumed that higher-order terms in the crystal-field expansion are very small and only the first contribution is considered. Since both contributions involve the same change in phonon occupation numbers, they have the same temperature dependence. The temperature dependence of the transition rate is the main measurable parameter, and thus it is difficult to distinguish between these two types of contributions. The physical processes of this type can be sepa­ rated into those involving two phonons being simultaneously absorbed or emitted, and those involving the absorption of one phonon and the emission of another phonon. These different cases are shown schematically in Fig.

4.3.

First consider processes involving the absorption or emission of two pho­ nons. In many cases, the ion couples strongly to one specific type of phonon mode, and in this case both phonons involved are of the same mode. The contribution to the transition rate for a two-phonon emission process de-

-'Vj'M f,l 'Vf.f

ljlel

(A) Direct Processes

6.

(B) Ram an Processes

(C) Orbach Processes

FIGURE 4.3. Two-phonon radiationless transitions.

4.2. Weak-Coupling: Radiationless Transitions

1 33

scribed by the first-order interaction Hamiltonian in Eq. (4. 1 .24) and second-order perturbation theory is

I

1 (2) ( em ) _ 2n (lfr%1 i (nw + 2 1 HfP i nw + 1 ) i lfr�) (lfr: i (nw + 1 IHfP I!fr�) l nw ) h Ea - (Ea + hw) 1 n + 1 lfrt) \lfr%1 l (nw + 1 IHfP I !fr� ) l nw ) 2 n +2 + \lfr% 1 ( w IHfP i wE -) l(E P! a b + hw) M;lM;� MglM;l 2 2n 1 n n + ) ( 2) + w h 2Mw w -hw + hw (4.2.20)

w nr

l

I

1

Note that the contribution described by the second-order interaction Ham­ iltonian and first-order perturbation must be added coherently to this ex­ pression. A similar expression can be written for the rate of absorption of two phonons of the same mode. For transitions involving three or more phonons, the procedure described above leading to Eq. (4.2.20) for two-phonon processes can be extrapolated by using higher-order perturbation theory. For a p-phonon process the expression for the transition rate will involve 2P l matrix elements. If the coupling is to a large number of phonons m, terms involving the emission of one of each type of phonon dominate terms involving the emission of many phonons into one highly excited mode. The general rate for a p-phonon emission transition of this type is given by

�r (em )

w

w; = x

� ,wm

+

1 )p / 2

( lfr%1 1 HfP llfr:1) ( lfr%1 1 H�P llfr:1) ( lfr�1 1 H�P llfr�1) (Ea - Eab - hw;) (Ea - Eab - 2hw;) [Ea - Eab - (p - 1 )hw;] ·

·

·

·

·

·

(4.2.21 ) Since the transition is sharply peaked about a particular frequency wo this expression can be simplified by defining an average matrix element - 1 such that = I M;l i P /(hwo)P

p ( em )

wnr

2n h 2Mwo

2 ) nw0 + 1 )P 22 (p - 1) m2p I M;l2 (p1 P l ) t5 (Ea _ ( Eb + hwo ) . ( hw0 ) (4.2.22)

Although it is difficult to calculate theoretical values for the radiationless decay rates using these expressions, it is possible to estimate the number of

1 34

4. Electron Phonon Interactions

phonons involved in the process. This can be done by comparing the transi­ tion rates of the p and p - 1 processes. From the preceding equations,

2 4m2 I M;i l ' n 1 ) + w o 2MWo ( hwo) 2

WKr( em) WKr I (em)

where the assumption has been made that the average matrix elements for the two processes are the same. For small values of the electron-phonon interactions,

wgr( em) wgr- I (em)

1

e «

.

Under these weak coupling conditions, the transition rate can be expressed in the form of an exponential,

WKr (em) = wgr l ( em ) = wgr- 2 ( em ) 2 = Wo el'ln (e) = Wo e ( !!E/ hw) ln ( ) . e

e

.

.

.

=

w�r (em)lf

e

(4.2.23)

In the final expression, the assumption has been made that the nonradiative decay process involves p phonons of equal energy hw crossing an electronic energy level gap of AE. This result predicts the well-known energy gap law for radiationless relaxation processes in rare-earth ions. This will be applied to specific cases in Chaps. 8 and 9. The prefactor to the exponent in Eq. (4.2.23) is difficult to calculate be­ cause of the lack of knowledge about the nature of the electron-phonon interactions involved in the matrix element. However, the temperature de­ pendence of the transition rate is contained in this factor through the pho­ non occupation numbers. From Eqs. (4.2.22) and (4.2.23), the temperature dependence of the process can be expressed explicitly as

WKr (T)

(

)p _

WKr (O) (nw + 1 )f ehw/kn T WKr (O) ehw jk8 T 1 '

(4.2.24)

where Eq. (4.2.7) has been used for the phonon occupation numbers. This expression shows that radiationless decay processes involving different numbers of phonons exhibit distinctly different temperature dependences. These can be measured experimentally and compared to the predictions of Eq. (4.2.24) to determine the value of p for the process of interest. This can then be used to determine the frequency of the phonon involved in the pro­ cess from the relationship phw AE and the known value of the energy gap. The treatment outlined above involving one effective phonon mode has been very useful in understanding the multiphonon radiationless decay pro­ cesses affecting the optical pumping dynamics of many rare-earth-doped =

4.2. Weak-Coupling: Radiationless Transitions

135

1 .0 0. 8

D ensity of S tates

10

20

Phonon

30

40

Frequency (x lo12 sec-1)

50

FIGURE 4.4. Density occupied phonon states in the Debye approximation.

laser materials. However, it must be remembered that the frequency spec­ trum of phonons for any specific type of solid has a significant amount of structure in its density of states, and the symmetry properties of different types of phonons can cause the matrix elements for electron-phonon inter­ actions to be quite different. Thus the above treatment is a very rough approximation to the real physical situation. For physical systems involving stronger coupling, the specific nature of the phonon spectrum must be taken into account as discussed in the following sections. The other types of two-phonon processes shown schematically in Fig. 4.3 involve the absorption of one phonon of frequency Wa and the emission of another phonon of frequency We . These processes can be important in non­ radiative transitions when the energy splitting between the electronic levels is so small that the density of states of phonons with the frequency necessary for a direct one-phonon transition is very small. Figure 4.4 shows a typical situation for the density of occupied phonon states, which can be seen from Eqs. (4.2.6) and (4.2.7) to be proportional to w2 [exp(hw/kBT) 1r 1 . The energy difference between the two electronic states is then made up by the difference in the energies of the two phonons. If the intermediate state of the process is a virtual state, these are called Raman processes; if the intermediate state is a real electronic energy level, these are called Orbach

processes.

Consider first the contribution of Raman processes to nonradiative relaxation between two electronic energy levels. In Sec. 4.4 the mathematical description of these processes is outlined in detail since they make the dom­ inant contribution to the broadening of many sharp spectral lines. For line­ broadening effects, the initial and final electronic states of the system are the

1 36

4. Electron Phonon Interactions

same, whereas for radiationless relaxation transttlons they are different. Other than this difference in the electronic matrix elements, the expressions for the Raman transition rates for line-broadening and nonradiative decay processes are the same. From Sec. 4.4, the Raman process transition rate is found to be

(I._)7JTD/T

WnRr A TD

o

X6 e x dx, ( ex - 1 ) 2

where the coupling coefficient is given by

(4.2.25)

1

Vt l l/1�1 ) (l/1}1 1 Vt l l/1�1 ) 2 (4.2.26) E;"l E1el In these expressions, a Debye distribution of phonons has been a,ssumed with a cutoff frequency of WD. This defines a Debye temperature TD hwD/kB. The integral in Eq. (4.2.25) is called the Debye integral and it can be found tabulated in tables of functions. For the condition T « TD the integral becomes 6 ! so the rate for the Raman relaxation transition is pro­ portional to T7 . Since the same set of phonons is involved in the absorption and emission parts of the transition, the Raman rate for excitation is ap­ proximately the same as the Raman rate for relaxation. Thus the rate at which the pupulations of the two electronic levels reach an equilibrium dis­ tribution through Raman processes is given by the characteristic time wRex + wdRec 2 WnrR · (4.2.27)

I

1 v A = 4n9w 3p2 v l 0 ( l/lJel l 2

+

_

� �

Thus the Raman relaxation rate is found from combining Eqs. (4.2.25)­ (4.2.27) and generally obeys a T7 temperature dependence around room temperatures. The final type of nonradiative decay processes to be considered are Orbach processes. These two-phonon processes are similar to Raman pro­ cesses except that the intermediate state is a real electronic level as shown in Fig. 4.3. The interest is in relaxation and excitation processes between levels 1 and 2 in the case when direct phonon processes are not effective because of selection rules, density of states, or other problems. For the situation of greatest interest, f5 « � < hwD, where f5 and � are the differences between the energy levels as shown in Fig. 4.3. The rate equations for the dynamics of the populations of ions in energy levels 1 and 2 are

(4.2.28) where the single-phonon transition rates given in Eqs. ( 4.2. 10) and (4.2. 1 1 ) can be written using Eq. (4.2. 12) as wl3 p l3 n(� + c5) ' w3 1 p l3 [n(� + c5) + 1 ] , W23 P23 n(�) , W32 P23 [n(�) + 1 ] .

4.3. Weak Coupling: Vibronic Transitions

1 37

For the case p 13 � P23 P and for temperatures kBT « Ll so that n 3 can be neglected with respect to n , and n2 giving N n 1 + n2 , the rate equations can be rewritten as

= pe- !!.fkB T (n, e-ofkB T n2) , where Eq. (4.2.7) has been used for n(E) . Setting this equation equal to zero n2

_

_

h,

gives the equilibrium populations

ne1

1

(4.2.29) 1' The solutions to the rate equations can be written in terms of these equilib­ rium populations as +

(4.2.30) where the characteristic time for relaxation to equilibrium is (4.2. 3 1 ) Thus the temperature dependence of radiationless relaxation by Orbach processes varies exponentially with the energy splitting to the intermediate electronic energy level. The measured temperature dependence of the radiationless relaxation can be used to determine the dominant type of transition involved in the pro­ cess. All three types of two-phonon processes, direct, Raman, and Orbach, have been found to be important in the optical pumping dynamics of differ­ ent types of solid-state laser materials. It should be noted that when the matrix elements of the electron-phonon interaction Hamiltonians repre­ sented by Eqs. (4. 1 .24) and (4. 1 .25) are zero, higher-order interactions or the presence of an external perturbation such as a magnetic field are required for the nonradiative transitions to occur. This can lead to a different tem­ perature dependence for the radiationless relaxation rate of Raman processes. 4.3

Weak Coupling: Vibronic Transitions

The sharp lines in the absorption and emission spectra of ions in solids due to electronic transitions with weak electron-phonon coupling are generally accompanied by low-intensity sidebands that display a distinct frequency structure. These are associated with phonon-assisted transitions and are re­ ferred to vibronic sidebands while the line associated with the purely elec­ tronic transition is referred to as a zero-phonon line. The vibronics on the high-energy side of the zero-phonon line in the fluorescence spectrum in­ volve phonon annihilation while those on the low-energy side correspond to phonon-creation transitions. The opposite is true in the absorption spectrum

1 38

4. Electron Phonon Interactions

;

- -

;

- -

HIGH ENERGY IBRO NICS

(A) ABSORPTION SP ECT RA

- -

nI

- -

HIGH ENERGY VIBRONICS

(B) FLUORESCENC E SPECTRA

FIGURE 4.5. Diagrams for vibronic transitions.

where the high-energy vibronics are associated with phonon creation of a phonon and the low-energy vibronics are involved with the annihilation of a phonon. The schematic energy-level diagrams for vibronic transitions are shown in Fig. 4.5. The mathematical expressions describing the transition rates for vibronic processes can be derived using second-order perturbation theory along with the electron-photon and electron-phonon interaction Hamiltonians given in Eqs. (3.2. 1 0), (4. 1 .24) and (4. 1 .25), where the strain Hamiltonian is used to describe the latter interaction since lattice phonons of various wave vectors and different branches can be involved. The quantum-mechanical diagrams describing the different types of vibronic processes are shown in Fig. 4.5. As an example, consider the low-energy vibronic transition in a fluorescence

4.3. Weak Coupling: Vibronic Transitions

1 39

spectrum that involves the emission of a photon of frequency Wt and polar­ ization nf along with the emission of a phonon of frequency wk . The matrix element describing this process is

M

nk + l , nt+ l iHe r lt/1}1 , nk + 1 , nt) (t/lj1 , nk + 1 , nti H? lt/1�1 , nb nt) E,.e1 _ ( �e1 +hwk ) (t/1/1 , nk + 1 , nt+ I I H? lt/1}1 , nb nt+ 1 ) (t/1}1 , nk , nt+ I I He r lt/1�1 , nk , + E,.e1 (Et + hwt) (4.3 . 1 ) ·

The sum runs over all possible intermediate states of the system. Substitut­ ing the expressions for the interaction Hamiltonians into Eq. (4.3. 1) and evaluating the nonzero contributions to the matrix element leads to

.

em x

Wk {f v

�

e

( t/1/1 1 2: e

)

f I t/1}1 ( t/JY I Vt I t/1�1 ) ' Ej1 (Et + hwk ) ik r P

·

1t

(4.3.2)

where the second term in Eq. (4.3. 1 ) has been neglected since the photon energy is much greater than the phonon energy. A similar expression can be obtained for the matrix element describing high-energy emission vibronic transitions involving the creation of a photon and annihilation of a phonon,

ff ffv

e em z. Wk2 nh MHE 2Mv Wt m t/J;t 1 2: e- rkr p . 1tf I t/1}1 ( t/1}1 1 Vt I t/1�1 ) x Ej1 - (Et - hwk )

�(

)

·

(4.3.3)

The expressions obtained for vibronic absorption transitions involving the annihilation of a photon and the creation or annihilation of phonons are

(4 . 3.4)

(4.3.5)

140

4. Electron Phonon Interactions

The matrix elements in Eqs. (4.3.2)-(4.3.4) can be used in the golden-rule equation (4. 1 . 1 ) to determine the transition rates for vibronic processes. Since the frequency structure of vibronic sidebands is due to the frequency distribution of the phonon density of states, the vibronic peaks will be sym­ metric in position about the zero-phonon line. The intensity of a specific peak depends on the occupation number of the phonon involved, and the temperature dependence of the transition rate is contained in the factor nk through the expression given in Eq. (4. 1 .8). Whether or not a specific vibra­ tional mode appears in a vibronic sideband of an electronic transition can be determined by group theory. The reduction of the direct product of the irreducible representations of the initial electronic state, the vibrational mode, and the radiation multipole operator must contain the irreducible representation of the final electronic state for the vibronic matrix element to be nonzero, ri X rq X r, ::J r/ . At very low temperatures, only single-phonon emission vibronics are present. As temperature is raised, single-phonon absorption vibronics also appear. At high temperatures two-phonon and higher multiphonon vibronics appear. The multiphonon spectrum can be treated as the convolution of single-phonon vibronics. An example of the analysis of vibronic spectra is given in Sec. 4.5.

4.4

Weak Coupling: Spectral Linewidth and

Line Position

There are several types of physical processes that contribute to the width and position of a spectral line associated with an electronic transition and how they change with temperature. Line broadening and line shifting mech­ anisms are described below. The width of a spectral line associated with a specific ion undergoing an electronic transition is the combined widths of the initial and final energy levels involved in the transition. At least one of these levels will be an excited state of the system and thus have a finite lifetime. Due to the ucer­ tainty principle relating the energy of a quantum-mechanical system with the time the system remains in the same energy state, a long-lived energy state will give rise to a very narrow energy level. This radiative lifetime con­ tribution to the width of a spectral line is called the natural linewidth for the transition. The measured width of a spectral line is almost always signif­ icantly greater than the natural linewidth due to the presence of other broadening mechanisms. When the ion is placed in a crystalline or glass environment, direct radiationless transitions and ion-ion interactions can shorten the lifetime of the energy levels and thus broaden the spectral lines. This type of contribution to the spectral linewidth is called lifetime broad­ ening. By measuring the fluorescence lifetime of a transition and using the

4.4. Weak Coupling: Spectral Linewidth and Line Position

141

usual transformation to energy, the lifetime broadening to the linewidth of the transition can be determined. Generally the width of a spectral line is broader than that predicted from lifetime measurements, implying that line-broadening mechanisms are pres­ ent that do not affect the lifetime of the levels involved in the transition. One mechanism of this type is the two-phonon Raman scattering process dis­ cussed in Sec. 4.2. In such processes, phonons of different frequencies are absorbed and emitted but the ion remains in the same electronic state and thus the lifetime of the energy level is not altered. The quantum-mechanical diagrams for phonon Raman scattering processes are shown in Fig. 4.3. These can be used with the electron-phonon interaction Hamiltonian in Eq. ( 4 . 1 . 2 5) to write the matrix element describing the Raman scattering of phonons:

MR (t/1�1 , nk - l , nk' + l i HR it/1�1 , nk , nk') nk - 1 , nk' + l i H? lt/1}1 , nk - 1 , nk') (t/1}1 , nk - 1 , nk' I H? lt/1�1 , nk , nk' ) J

Elel - (Pi -hWk ) j

(t/1�1 , nk - l , nk' + l i H? It/lj1 , nk , nk' + l ) (t/lj1 , nk , nk' + E�1 - (E�J 1 + hQ)k' ) + (t/1�1 , nk - 1 , nk' + l i H? l t/1�1 , nk > nk' )

+

[2.: (I

l

(t/1�1 1 Vi lt/1}1 ) ! 1 (nk - 1 , nk' + l l bl, bk I nk , nk') - 2Mv2 E;ei - ( El - hwk ) j (t/1�1 1 V! l t/1}1 ) ! 1 ( nk - 1 , nk' + l l bk bl, I nk > nk' + Efi - ( J0el + hwk' ) + (t/1�1 1 V2 lt/lj1 ) (nk - 1 , nk' + l l 2bk bl, l nk , nk')] _

h

I

(4 . 4 . 1 )

In the last step, the phonon energy has been neglected with respect to the photon energy. This expression can be rewritten in more compact form by factoring out the phonon frequency and occupation nubmer parts and de­ fining a coupling coefficient rx that contains the remaining constants and matrix elements, ( 4 . 4 . 2) where

(4.4.3)

142

4. Electron Phonon Interactions

The next step is to substitute this expression for the transition matrix ele­ ment into the golden rule transition rate expression given in Eq. (4.2.2). This calculation also reuqires an expression for the density of final states. For the case of the Raman scattering of phonons broadening a sharp electronic en­ ergy level, this can be expressed as the product densities of states of the two phonons involved in the process and a J, function to conserve energy,

(4.4.4) Thus the transition rate for Raman scattering of phonons is found from in­ tegrating over the phonon spectra to give

WR �� I I [ MR[ 2p (wk )p(wk')J (wk Wk') dwk dwk' 2n 2 h2 [ 0! [ I [p (wk )] 2wk2 nk(nk + 1 ) dwk .

If a Debye density of states is assumed for the phonons as discussed above, and Eq. (4.2.7) is used for the phonon occupation numbers, this ex­ pression becomes

2n 2 ( 3 v2 ) OJD w% ehwk /kB T d R W - h2 [0![ 4Jtlv6 Jo ehwk/kB T 1 wk [ 0! [ 2 2n9V3 h22v6 (kBTD)7 (TDT )7 ITn/T (exx6�1 ) 2 dx. _

0

(4.4.5)

Here the expression in Eq. (4.2.5) has been used for the phonon density of states and the Debye temperature is defined in terms of the Debye cutoff frequency as above, TD hwD/kB. Converting this rate to units of energy gives the contribution to the linewidth due to the Raman scattering of phonons,

a. (-TTD)7 JTn/T (exx6ex1 ) 2 dx,

ilv ( cm 1 )

(4.4.6)

0

where the new coupling coefficient has been defined to contain all the con­ stants in expression (4.4.5),

0!

(3.34 x x

1 0 ) 2n3p92 vl

(I: I

-II

0

(kBTD)7

1 2 + (t/1�11 V2 [ t/1�1)

)2

0

(4.4. 7)

Since the coupling constant is intrinsically positive, an increase in temper­ ature will cause an increase in the linewidth. Note that the Debye integral appearing in Eq. (4.4.6) is tabulated, and its values can be found in tables of functions covering a wide range of temperatures.

4.4. Weak Coupling: Spectral Linewidth and Line Position

143

Both the Raman scattering mechanism and the lifetime broadening mechanism have the same probability of occurrence for all ions in the en­ semble. Therefore they produce homogeneous broadening and result in a Lorentzian line shape. This is similar to collisional broadening of spectral lines in gases. In addition to these mechanisms, it is possible to have in­ homogeneous broadening mechanisms that produce a Gaussian contribution to the line shape similar to Doppler broadening of spectral lines in gases. These mechanisms have a different probability of occurrence at the site of each ion. For ions in solids, this is due to different local crystal-field envi­ ronments associated with a random distribution of microscopic strain fields in the material. Thus the same spectral transition has a slightly different fre­ quency position for each ion, and the observed spectral line is the super­ position of all of the individual ion lines. As discussed in Chap. 2, the superposition of Gaussian and Lorentzian contributions to the width of a spectral line results in a Voigt profile for the line shape. These concepts on the width of a spectral line of an ion in a solid can be summarized by the following expression:

+ L jJ ehwo /ka T

1

f>z

+ L jJ ehwo /ka T

1 f

•• • 0

..

J: 10

1 01 � 2

:. 1 o •

.. z �

!5 2 !5

I0

20

!50

100 200

TEMPERATUR E ( • K )

FIGURE 4.9. Variation of the width of the R 1 line of Cr3 + in SrTi03 . The circles are experimental points; the solid line is the best fit using a Debye distribution of pho­ nons, the dotted line is the fit obtained using one-phonon density of states obtained from the computer analysis of the vibronic sideband, and the dashed and dotted line is obtained using only the lowest-energy peak in the computer analysis of the vi­ bronic sideband for the effective phonon density of states [taken from Ref. 6(a)].

1 56

4. Electron Phonon Interactions

ature TD and the coupling parameters treated as adjustable parameters. As seen by the solid line in the figure, this procedure can give a reasonable fit to the data. However, the value of TD required to obtain this good fit is only 1 1 5 K, which is significantly less than the value of 400 K obtained from specific-heat measurements. In this model such a low value of the Debye temperature implies that the lower-frequency phonons make a greater con­ tribution to the thermal line broadening than the higher-frequency phonons in the density of states. The two broken lines in the figure were obtained from models using the effective density of states for phonons obtained from the vibronic spectra as described above. The results obtained using the entire one-phonon sideband (dotted line) give a poor fit to the data at high tem­ peratures. However, the results obtained using only the lowest-energy peak (dashed-dotted line) in the vibronic structure as the effective density of pho­ non states give an excellent fit to the experimental data at all temperatures. This again implies that there is much stronger coupling to the low-energy phonons for the line-broadening processes than there is to the high-energy phonons. Similar results are obtained for the thermal broadening of the Rz zero-phonon line and for treating the thermal shifts in the positions of these lines. 6 4.6

Strong Coupling

When the coupling between the electrons on the optically active ion and the lattice vibrations is strong, many phonons can be involved in the optical transitions. This leads to broad bands with strong temperature dependences of the intensities and decay times of the optical spectra of ions in solids. One method for theoretically treating this strong-coupling case is to use the N­ order perturbation theory approach leading to Eq. (4.2.21 ) for the emission rate of N phonons. This can be combined with the treatment of vibronic transitions given in Sec. 4.3. Although this works well for transitions in­ volving only a few phonons as was shown in Sec. 4.2, it becomes very cum­ bersome for transitions involving many phonons. Also, the characteristics of multiphonon radiationless transition rates such as their temperature de­ pendences may be accurately predicted by this approach, but the theoretical predictions of absolute transition rates are very inaccurate. In addition, this treatment utilized the harmonic approximation to describe the phonons and assumed the phonon modes to be the same for both electronic states in­ volved in the transition. The importance of allowing for anharmonic inter­ actions has been pointed out 7 but this is difficult to treat from first principles and will not be included here. However, some of the effects of anharmonic interactions on spectral characteristics will be mentioned below. One impor­ tant extension of the electron-phonon coupling theory is to allow for differ­ ent types of vibrational modes in the initial and final electronic states. This allows states to be connected whose quantum numbers differ by other than

4.6. Strong Coupling

1 57

± 1 . Several different approaches have been developed to treat the case of strong electron-phonon coupling, and one of the approaches commonly used is outlined below. However, before presenting this theoretical treat­ ment, the concept of configuration-coordinate models is discussed. This is helpful in obtaining a qualitative understanding of the effects of lattice vibra­ tions on transitions between electronic states of ions in solids. Configuration-coordinate diagrams are often used to describe transitions between electronic energy levels coupled to lattice vibrations. Although this model involves a one-dimensional displacement of a single vibrational mode, which is a great oversimplification of the true situation, it is still quite useful in explaining some aspects of the optical spectra of ions in solids. Schematic configuration-coordinate diagrams are shown in Fig. 4. 10. These depict the variation in the electronic state energy with respect to the dis.�

T

·� "

" .Q

Q (A)

S10kes Shift

j_

l'3 0: � Il

i

Qo Qo'

Q

(B)

" .Q fj

>-

"5

� Il

§

...

Qo Qo' Q

§' �

(C)

>"

�

(D)

Q

FIGURE 4. 10. Configuration-coordinate diagrams. (A) The case of no displacement between the excited-state and ground-state potential wells leads to sharp zero­ phonon lines in the absorption and emission spectra. ( B) Intermediate displacement between the excited- and ground-state potentials gives zero-phonon lines and strong vibronic sidebands with a Stokes shift between the absorption spectrum and emission spectrum peaks. The Huang-Rhys energies and vibrational wave functions are shown. (C) Large offset of the potential wells leads to a crossover between the ground and excited electronic states at an activation energy !l.E. This leads to a broad absorption band with no emission due to radiationless quenching. ( D) Anharmonic potentials enhance the Frank-Condon overlap factors and leads to enhanced radiationless quenching.

1 58

4. Electron Phonon Interactions

placement of the normal vibrational coordinate away from its equilibrium position. Results are shown for the ground state and excited state of the system for cases with different magnitudes of electron-phonon coupling. Since the electron charge cloud distribution of the excited state can be significantly different from that of the ground state, the equilibrium posi­ tions of the configuration coordinates can be different for the ground and excited states. In the harmonic approximation, the potential curves are de­ scribed by parabolas,

( 4.6. 1 ) The vibrational wave functions of the ground state and excited state are given by the usual harmonic oscillator expressions discussed further below, Xn9 ( Q )

= Nn e - ( Q/ag ) 212 Hn

Xme C Q )

= Nm e - [( Q

Qo ) fa,] 212 Hm

(4.6.2) where the Hn are Hermite polynomials with normalizing factors Nn, m · The force constants k; and zero-point vibrational amplitude factors a; are given by

h ag2 = -VJ(;M ' h 2 ae = v'J(;M "

(4.6.3)

Because the Frank-Condon approximation assumes that the vibrational motion of the ions is much slower than the motion of the electrons, elec­ tronic transitions involving absorption and emission of photons appear as vertical lines on configuration-coordinate diagrams. Since vibrational relax­ ation within an electronic state is very fast, electronic transitions at low temperatures will initiate from the ground vibrational level of the initial electronic state. Transitions initiating from higher levels can be present at higher temperatures when the presence of thermal energy causes these levels to be populated according to a Boltzmann distribution. It should be noted that the harmonic oscillator wave functions for the lowest vibrational level are peaked in the center of the electronic potential well while the wave functions for the higher-lying vibrational levels are peaked at the turning points. The Frank-Condon factor for the vibrational wave-function overlap appearing in the expression for nonradiative transition probability (dis­ cussed below) implies that nonzero transitions can occur from one specific vibrational level of the initial electronic state to several vibrational levels of the final electronic state. This leads to a spread in the allowed transition en­ ergies and thus to the appearance of broad spectral bands as shown in Fig. 4. 10. For the typical case of transitions initiating from the lowest vibrational level of an electronic state, the photons involved in an absorption transition

4.6. Strong Coupling

1 59

have higher energies than those involved in an emission transition. The en­ ergy difference of the absorption and emission spectral bands is called the

Stokes shift.

Both the widths of the spectral bands and the magnitude of the Stokes shift depend on the amount of offset of the excited-state potential well mini­ mum Q� with respect to the ground-state potential well minimum Qo . A large offset results in a very broad band that approximates a Gaussian shape, while a medium offset gives a band with a Pekarian shape, and a zero offset produces a sharp zero-phonon line. Many times it is possible to see both a zero-phonon line and a vibronic sideband as shown in Fig. 4. 1 0( B). One important situation is when there is a possibility for the two electronic potential wells to cross as shown in Fig. 4. 10(C). This leads to nonradiative decay from the excited state to the ground state. 8 If the potential crossover point occurs at an energy flE above the lowest vibrational state, the radia­ tionless decay rate will have an activation energy of this amount, leading to an exponential temperature dependence of both the fluorescence intensity and lifetime of the excited state. It should be emphasized that any anhar­ monicity associated with the potential curves will enhance the vibrational wave-function overlaps as shown in Fig. 4. 10( D) and thus increase the radi­ ationless decay rates. 7 The discussion of configuration-coordinate diagrams can be related back to the crystal-field model for transition-metal ions. Consider the case of an ion with a dn+m configuration in an octahedral crystal field with n electrons in the t2g level and m electrons in the e0 level. As shown in Fig. 4. 1 1 , the energy of an excited state with respect to the ground state depends on the magnitude of the crystal field 1 0Dq and on the difference in ground- and excited-state electron configurations. This can be expressed as (4.6.4)

_

E

_

_

_

E

lODq

FIGURE 4 . 1 1 . Crystal-field modulation by lattice vibration mode Q.

1 60

4. Electron Phonon Interactions

so the variation of the transition energy with respect to a normal vibrational coordinate Q is

8E9e 8Dq 8E9e 8Dq 6(m (4.6.5) a Q = a Q aDq = a Q [ e - m9) - 4(ne - n9)] . Here the partial derivative of the crystal-field strength Dq with respect to normal mode Q represents the electron-phonon coupling parameter. This expression and Fig. 4. 1 1 show that states having the same electron config­

urations have the same slopes on the crystal-field energy level diagram and therefore transitions between these states result in sharp spectral lines. If the electron configurations are different for the two states, the corresponding energy versus crystal-field strength slope of each state will be different and thus the phonon modulation of the crystal field results in a broad spectral band for the transition. Examples of these cases will be given in Chap. 6. One of the standard theoretical approaches for treating transitions be­ tween electronic states ions in solids that are strongly coupled to the host lattice vibrations was developed by Huang and Rhys. 9 In this treatment, it is assumed that the Born Oppenheimer approximation is valid so the wave functions can be expressed as products of the electronic part and the vibra­ tional part. Further, it is assumed that there is only a weak dependence of the electronic wave functions on the nuclear coordinates so the transition matrix elements can be factored into an electronic part and a vibrational part (Condon approximation). Finally, to make the calculations tractable, it is assumed that there is one dominant phonon mode in the electron-phonon interaction and that this mode has the same frequency but different normal coordinates in the initial and final electronic states of the transition. This last simplification implies linear coupling in the harmonic approximation. The kinetic energy of the phonons is initially neglected while the eigenstates of the Hamiltonian are found for fixed phonon coordinates. This produces the usual adiabatic potential surfaces. The transitions between these surfaces are due to the term containing the phonon kinetic energy or nonadiabatic term in the full Hamiltonian. With these assumptions, the wave functions for the system are

(4.6.6) where r is the coordinate of an optically active electron and Q is the coordi­ nate describing the positions of the surrounding nuclei. O;, v ( Q ) is the vibra­ tional wave function and r/J; (r, Q ) is the electronic wave function for a fixed position of the nuclei. This implies that the motion of the electron is very rapid compared to the nuclear motion. For linear electron-phonon coupling, the interaction Hamiltonian is ex­ pressed from Eq. (4. 1 .27) as

(4.6.7)

4.6. Strong Coupling

161

where Vs(r) is the electron-phonon coupling parameter for the s mode of vibration. The Schrodinger equations for the electronic and vibrational parts of the system are

[H;(r) + H?(r, Q ) ] l ¢; (r, Q )) = W;(Q) I Q>Jr, Q) ) , [H1 ( Q) + W;( Q)] I O;v ( Q ) ) = E;v i O;v ( Q ) ) ,

(4.6.8) (4.6.9)

where H; is the Hamiltonian for the electronic states of the ion in a static crystal field discussed in Chap. 2, and H1 is the lattice vibration Hamiltonian given in Eq. (4. 1 . 19). The solutions of Eq. (4.6.8) are the energy levels and wave functions for the electrons for a fixed value of the lattice vibrational coordinate. The electronic energy found in this way is treated as an effective potential for the lattice vibrations in solving Eq. (4.6.9). Since the electronic and vibrational wave functions are determined inde­ pendently from Eqs. (4.6.8) and (4.6.9) resulting in electronic wave functions that depend on Q , their product resulting in l l/1; v (r, Q ) ) given by Eq. (4.6.6) is not a stationary state of the total system. Th� vibrational state energy E;v given in Eq. (4.6.9) is the difference between the energy of the stationary states of the entire system and the energy for the nonstationary states. This can be used to construct a "nonadiabatic Hamiltonian" that provides the effective interaction needed to produce radiationless transitions between two vibronic states of an ion. The total Hamiltonian operating on the product wave functions gives

Hl l/J;, v (r, Q)) = ( H�E + H�E + He + Hev ) I Q>; (r Q ) O;, v ( Q) ) = ( H� + H�E + W;) I Q\; (r Q ) O;, v ( Q )) = I Q>Jr Q) )H�E I O;, v ( Q ) ) - I Q\; (r Q) )H� I O;, v ( Q )) + (H�E + H�E + W;) I Q\; (r Q) O;, v ( Q) ) = [H�E I Q>; (r Q) O;, v ( Q ) ) ( q); (r Q ) I H�E I O;, v ( Q ) )] + ( q); (r Q) I ( H�E + H�E + W;) I O;, v ( Q) ) = [H�E I Q>; (r Q) O;, v ( Q ) ) - ( Q\; (r Q ) I H�E I O;, v ( Q ) )] + E;v l l/J;, v (r, Q ) ) . This leads to the definition of a nonadiabatic Hamiltonian

HNA i l/J;, v (r, Q ) ) = H�E ( Q ) I Q\; (r, Q ) O;, v ( Q )) - ( ¢; (r, Q ) I H�E ( Q ) I O;, v (r, Q)) .

(4.6. 10)

Thus it is the kinetic energy of the lattice vibrations that is the key to non­ adiabatic Hamiltonian. Using the normal expression for quantum-mechan-

1 62

4. Electron Phonon Interactions

ical kinetic energy gives

Since the Condon approximation assumes that the electronic state varies slowly with tespect to the vibrational coordinate, the second term in this expression can be dropped. This leaves the expression for the nonadiabatic Hamiltonian as

HNA

( r , Q) )

=

h2

M

s

o i O;, v) o Qs o Qs .

(4.6. 1 1 )

The transition rate between two electronic states strongly coupled to the lattice is given by Fermi's golden rule with HNA used for the perturbation Hamiltonian causing the transition. Assuming thermal equilibrium so that there is a Boltzmann distribution for the population of the vibrational levels of the initial state, the transition rate is there is a Boltzmann distribution for the population of the vibrational levels of the initial state, the transition rate is

wif

=

Piv I (fv' IHNA i iv ) j 2t5 (EJv' - E;v ) , �L v, v'

(4.6. 12)

where Piv is the distribution function for the Boltzmann population of initial vibrational levels and a t5 function is used for the density of final states to ensure conservation of energy. Using the expressions for the phonon popu­ lations given in Eqs. (4.2. 14)-(4.2. 1 6) the Boltzmann distribution function can be written as exp

( -E )

(4.6. 13)

In order to evaluate the transition matrix elements for the case of strong coupling, it is necessary to have an exact expression for the vibrational wave functions instead of using the second quantized occupation number formal­ ism. It was shown in Sec. 4. 1 that the normal modes of vibration of the host material can be treated as an ensemble of harmonic oscillator. Thus the normal wave functions for quantum-mechanical harmonic oscillators can be used for the vibrational wave functions. These cna be found in any quantum

4.6. Strong Coupling

1 63

mechanics textbook and are given by1 0

(4.6.14) where the Hv, are Hermite polynomials. The eigenfunctions for the vibra­ tional system are product functions for N phonon modes given by

( Qs i B;v( Qs))

B;v

N

N

II X;v, ( Qs) II Xv, ( Qs - Qs(i)) ,

(4.6. 1 5)

where the changes in the normal coordinates are due to the electron-phonon coupling in the initial and final states. These are discussed below. The electronic wave functions can be expanded with respect to the normal coordinates Qs . Assuming that the electron-phonon interaction that mixes the pure electronic states can be represented by linear coupling as given in Eq. (4.6.7), time-independent perturbation theory gives

l �;( r, Q ))

� ��t o) (r)) + L Ni ) 1 �)0l (r)) + L I �J0 (r)) . (O) s,j-1-i E; - Ej

l �)ol (r)) (4.6. 1 6)

Here Ef0l represents the energy of the unperturbed electronic state and the interaction energy is given by

() - () Vsji - (�j0 (r) I Vs(r) l �i 0 (r)) .

(4.6. 17)

Similarly, the eigenvalues in Eq. (4.6.8) can be expanded to give

W; ( Q ) = EJO) + L Vsu Qs + L s s,s'j -1-i

- Ej

(4.6. 18)

The term in brackets on the left-hand side of Eq. (4.6.9) can be divided into the lattice kinetic energy Hamiltonian plus an effective adiabatic potential for the lattice vibrations. The latter is given by

U; ( Q ) = H�E ( Q ) + W; ( Q )

= Ei(O)

+2

s

2 2 Ws Qs +

L w; Q; + W; ( Q) s

s

V.

sii Qs +

Vsij Vs'ji Qs Q; + () ( ) ···' s,s',j -1-i E; 0 - Ej 0

(4.6. 19) where Eq. (4.6. 1 8) has been used. The first two terms in Eq. (4.6. 1 9) predict the frequency and equilibrium positions of each s normal mode as indepen-

1 64

4. Electron Phonon Interactions

dent of the electronic state i. This is consistent with the results of the Born­ Oppenheimer approximation. The electron-phonon coupling represented by the higher-order terms results in a shift in equilibrium position and fre­ quency of the normal modes that depends on the specific electronic state. The effect of the third term on the equilibrium position can be seen by grouping it with the second term to give

U;( Q) El0) + = E; + where

� w; ( Qs +

�

2:= w; [Qs - Qs (i)f, s

(4.6.20) V,;; Qs ( l") = 4� ws ·

(4.6.21 )

The second of these expressions gives the shift in the equilibrium for the ith electronic state. Note that this is linearly proportional to the diagonal elec­ tronic matrix element of the electron-phonon coupling interaction. The third term in Eq. (4.6. 19) does not introduce a change in vibrational mode frequencies. Including the fourth term in the equation for the effective potential does produce a change in frequencies. This term is generally negli­ gible for rare-earth ions but can be important for some transition-metal ions. Higher-order terms in the equation are needed to describe anharmonic effects. Substituting the nonadiabatic Hamiltonian and the wave functions given above into the expression for the transition rate in Eq. (4.6. 1 2) gives

Here the second term in the matrix element has been neglected with respect to the first term and the derivative of the electronic wave function with re­ spect to the normal coordinate has been assumed to be independent of the specific Qs . The validity of these assumptions can be seen by inspection of Eq. (4.6. 1 6). Taking the derivatives of the wave functions given in Eqs. (4.6. 1 5) and (4.6. 1 6) with respect to Qs , the expression for the transition rate

4.6. Strong Coupling

1 65

becomes

The electronic part of the matrix element has been written as

Rs (

=

h2 -M

(O)

V,if

Ei - Ef(OJ

(4.6.23)

with the interaction energy given in Eq. (4.6. 1 7). The specific phonons in­ volved in the electron-phonon coupling that are responsible for the elec­ tronic transition are called the promoting modes. They have nonzero matrix elements

The vibrational matrix elements not including the modes involved in the electron-phonon coupling are called the Franck Condon factors. The phonons that become excited through the conversion of electronic energy to vibrational energy are called accepting modes. They have nonzero matrix elements contributing to the Franck-Condon factor. The Franck-Condon factor, describing the overlap of vibrational wave functions in the initial and final states, determines the number of vibrational quanta of a specific mode that are excited by the transition. This depends on the modification of the normal · coordinates in the two electronic states given by Eq. (4.6.21 ) . The essential difference be­ tween promoting and accepting modes is that the former have an "allowed" vibrational matrix element and a nonzero off-diagonal electronic matrix element V,if while the latter have a "forbidden" vibrational matrix element and a nonzero diagonal electronic matrix elements ( Vsii - V,JJ ) . The next problem is evaluating the summation over all of the vibrational modes of the initial and final states with the Boltzmann distribution function for the population of the initial states. Several methods have been devised for doing this. Huang and Rhys9 used a series expansion of the harmonic oscillator wave functions while Miyakawa and Dexter1 1 used a generating function approach1 2 to evaluating the double sum. The latter procedure will be outlined here. This approach begins by defining a function FNA such that

FNA (E)

=

L I (Jv' IHNA i iv) I 2Pivo(E - Efv' + Eiv ) v, v'

(4.6.24)

1 66

4. Electron Phonon Interactions

with p;v given in Eq . (4.6. 1 3). Comparing this to the expression for the tran­ sition rate given in Eq. (4.6. 12) shows that (4.6.25) The generating function fNA ( A. ) is defined as the Laplace transform of the spectral function FNA (E) ,

J�oo FNA (E) e J.EdE.

/NA ( A. )

(4.6.26)

Recalling that the density operator is defined as

(4.6.27) where fJ

1 /kBT and the trace of P; ( fJ) is Tr [p; ( /J)]

Lv e pE,, ,

these expressions can be combined to calculate the generating function fNA ( A. )

J�oo �, I (fv' I HNA i iv) j 2p;vb(E - Ejv' + E;v )e J.EdE 1 j (fv' I HNA i iv) j 2e J.(EJ,' E,, ) e pE,, _ r[p; /J)] L

T

(

v, v '

This expression can be manipulated using the properties of the density op­ erator in order to evaluate the sum as follows: 1 i) l v) e ( P J.) E,, ( v i HNA ( if ) l v')e J.EJ, ' v' /NA ( A. ) = Tr [p; ( /3)] � ( I HNA ( / 1 Tr[HNA ( /i)p; ( /J - A. ) HNA ( if )p1 ( A. )] . [ Tr p; ( /J)]

(4.6.28)

The trace is over all possible vibrational wave functions. From Eq. (4.6.23), the electronic matrix can be expressed in terms of the electron-phonon inter­ action strength . HNA ( if )

= - Mh2

s

(0)

Vsif

E1

(0)

E;

a a Qs

=

.

z

a

s

Ssif a Q ' s

( 4.6.29)

where Ssif is known as the Huang-Rhys parameter,

Ssif -=

Vsif M Ef(0) - E;( 0) ·

. h2 - !-

(4.6.30)

Using the vibrational wave functions and interaction Hamiltonian dis-

4.6. Strong Coupling

1 67

cussed above it is possible to evaluate the traces of the density operator­ electronic matrix element products and thus determine the function fNA ( A. ) from Eq. (4.6.28). Then an inverse Laplace transform can be performed to obtain the spectral function 1 A.E (4.6.3 1 ) FNA (E) . /NA ( A. )e dA.. 211:1 oo Once the spectral function is known, Eq. (4.6.25) can be used to calculate the nonradiative transition rate. This procedure is a long mathematical pro­ cess and the details can be found in Ref. 1 1 . With the assumption that the density of states has only one singularity at a frequency Ws , the generating function for a N-phonon process is found to be iw N 2 N! [g2 n(n + 1 )] k ( A. ) exp l �h Eo g R [g (n + 1 ) e A. ] N! (N + 1 ) !k!

J

-

(·

X

_

+

oo

)

n(n + 1 ) + (2n + 1 ) 2

-

2n + 1

(n + 1 ) (2n + 1 ) 2 + (N + k + 1 ) (N + k + 2) gn

1 (N + k + 1 ) g 2 (N + k) (2n + 1 ) 2g + (2n + 1 ) + 2n + 1

]

)

·

At temperatures well below the Debye temperature, n « 1 , and this expres­ sion reduces to f/!A ( A. )

=

;y exp (iA.

)

iNwA. .

The inverse Laplace transform of this function is

This leads to the final expression for the transition rate at low temperatures as (4.6.32)

1 68

4. Electron Phonon Interactions

where the parameters in the above expressions are given by

and Eo is the electronic energy gap of the transition. Note that at zero tem­ perature the parameter g is equal to the Huang-Rhys parameter. The result given in Eq. (4.6.32) is identical to the result obtained by Huang and Rhys using the wave-function expansion approach to the problem. The temper­ ature dependence of the transition rate is contained in the phonon occupa­ tion numbers. The factor ( n + I t appears in the expression as expected, but there will be additional contributions to the temperature dependencies through the parameter R and g. For low temperatures and very weak coupling (g So < 1 ) , it is possible to manipulate Eq. (4 . 6.32) in such a way that it predicts an exponential en­ ergy gap law similar to the one obtained previously from the N-order per­ turbation theory approach to the problem. For these conditions 2n 2 So(N 2) wifN (0) � T R (Eo - Nhw) . (N 2) ! t5

Using Sterling's formula for N » 1 , In N! � N ln N N, this expression can be rewritten as (4.6.33)

(

where

)

1 N 2 AEeff Eo - 2hw, ct. hw In -- - 1 . This form of the exponential energy gap law is somewhat different from the form derived previously. The fact that the effective energy gap in this ex­ pression is less than the electronic energy gap by the energy of two phonons has been attributed to the fact that these phonons act as promoting modes and the other phonons as accepting modes. The temperature dependence of the nonradiative transition rate is con­ tained in the factor for the population of phonons. However, this not only appears explicitly in Eq. (4.6.32) but also in the R and g factors. If the ex­ pression for R is expanded and the assumption of small So is made, the ex­ pression for the decay rate becomes S 2n N 2 e - o N Wif (T) "' h So N!

()

S e - 2n o ( n +

l )N'

( 4 .6.34)

where the temperature dependence is in the last two factors. For small

4.7. Jahn Teller Effect s. =

Ss

0.1

=

1 69

10

E (mits of f! roJ

FIGURE 4. 12. Pekarian distribution function for different values of s• .

values of So the last factor dominates. This expression has been successful in predicting the relative change in the nonradiative decay rate with temper­ ature in a number of different materials. However, it should be recognized that many assumptions have been made in deriving this expression and the full expression and its variation with temperature is much more complicated. The treatment of multiphonon emission processes can also be applied to the vibronic sidebands of zero-phonon lines. The expression for the tran­ sition given in Eq. (4.6.32) approximates a Pekarian distribution in energy. The shape of the emission bands predicated by this expression varies signif­ icantly as a function of the Huang-Rhys parameter S. General examples of plots of this function are shown in Fig. 4. 1 2 for different values of s•. For strong coupling, multiphonon vibronic sidebands appear in the spectrum as smooth, broad bands, whereas for weak coupling the transition appears as a sharp line with weak one-phonon vibronic sideband. These spectral band shapes can be correlated with the configuration-coordinate diagrams in Fig. 4. 1 0. A large offset between the ground- and excited-state potential wells leads to a large value for the Huang-Rhys factor and a broad band that approximates a Gaussian shape. A medium offset gives a smaller value of S and a band with a Pekarian shape. For zero offset S = 0 and the transition is a sharp zero-phonon line. A detailed mathematical treatment of non­ radiative and vibronic transitions based on configuration-coordinate dia­ grams has been developed by Struck and FongerY 4.7

Jahn-Teller Effect

One important result of strong electron-phonon coupling can be a lifting of electronic state degeneracy through a splitting of the vibronic energy levels.

1 70

4. Electron Phonon Interactions

This is essentially a breakdown of the Born-Oppenheimer approximation and is described by the Jahn- Teller theorem. This states that any complex occupying an energy level with electronic degeneracy is unstable against a distortion that removes that degeneracy in first order. The vibronic coupling of ions in solids can cause a local distortion of the lattice in which the atoms move in the direction of normal-mode displacements to lift the electronic degeneracy. A new equilibrium position of the atoms is reached in which the local symmetry is lower than the point-group symmetry of the crystal. This splits the electronic energy levels to higher and lower levels with unchanged center of gravity. Thus the electronic degeneracy is replaced by vibronic de­ generacy. For ions such as the first-row transition-metal ions that have strong vibronic coupling and weak spin-orbit coupling, the J ahn-Teller splitting can be larger than spin-orbit splitting. The distortion can be due to either static or dynamic vibronic coupling, and thermal energy can allow the complex to jump between different equilibrium configurations at high temperatures. These processes have significant effects on the optical spectral features of materials with strong electron-phonon coupling. The most im­ portant feature in terms of solid-state laser materials is the splitting of the electronic energy levels due to the static Jahn-Teller effect. The theoretical description of this effect is outlined below, while a more detailed discussion plus a description of the dynamic Jahn-Teller effect can be found in the re­ view article by Sturge. 14 The effective Hamiltonian describing the vibronic states can be written as the sum of the harmonic oscillator Hamiltonian describing the normal modes of vibrations, Hvib , and the effective electronic part including both the static and dynamic contributions of the crystal field, Hij, H = Hij + Hvib

= E;Jij + L Gij(k ) Qk + Hvib, k

(4.7. 1 )

where (4.7.2) The first term represents the static crystal-field Hamiltonian for the elec­ trons. If it is g -fold degenerate with eigenvalue E and eigenfunctions trans­ forming as the symmetry representation r, the effective Hamiltonian for level r is a g X g matrix. The middle term is the electron-phonon inter­ action that describes the dynamic part of the crystal field given in Eq. (4. 1 .26) with the quantity Gij (k) being a square matrix of the order of the electronic degeneracy of the level. The Qk are the normal coordinates of the complex made up of the central ion and its surrounding ligands. They can be expressed in terms of creation and annihilation operators and ex­ panded in terms of lattice phonons as described in Sec. 4. 1 . The presence of the dynamic crystal-field term is what causes the breakdown of the Born­ Oppenheimer approximation so the eigenfunctions of the system can no

4.7. Jahn Teller Effect

171

longer be described as products of electronic and vibrational wave functions. Instead vibronic wave functions that are combinations of Born-Oppenheimer functions must be used. To illustrate the static Jahn-Teller effect, consider a dopant ion strongly coupled to its ligands in an octahedral configuration. The 1 5 normal-mode coordinates of an octahedral complex and their symmetry properties were discussed in Sec. 4. 1 . Since this complex is centrosymmetric and the matrix elements in Eq. (4.7.2) are between states of the same parity, only even­ parity vibrational modes will make nonzero contributions to GiJ(k) . In addition, group-theory considerations require that the irreducible repre­ sentation of the normal vibrational mode must be contained in the reduction of the direct-product representation of the square of the irreducible repre­ sentation of the electronic state being considered for GiJ(k) to be nonzero. For transition-metal ions of interest in solid-state laser applications, the most important degenerate electronic states in octahedral complexes trans­ form as doubly degenerate E or triply degenerate T1 or T2 representations. As an example, consider a 2Eg electronic state, which leads to eEg 2Eg) ::J 2A 1 g + 2Eg. Therefore, of the 1 5 vibrational modes of this complex, those transforming as OC J g and eg can couple with the 2Eg electronic state to give a nonzero contribution to GiJ(k) . Since the totally symmetric vibrational mode OCJ g only shifts the energy of the state and does not produce splitting, the vibronic coupling with the eg mode is the only one that must be considered to explain the Jahn-Teller splitting of the 2Eg level. The components of the 2 x 2 matrix GiJ(k) for the 2Eg level can be calcu­ lated using the single-electron d wave functions. To do this the Hamiltonian in Eq. (4.7. 1 ) is expanded as H Ho + GiJ( l ) Q 1 + GiJ(2) Q2 + (4.7 . 3) where Q 1 and Q2 are the two vibrational modes transforming as Eg. The operator equivalent technique can be used to evaluate the matrix elements in this equation. This technique takes advantage of the ability to express the matrix elements of interest in terms of angular momentum operators that have the same symmetry transformation properties. It is discussed further in Sec. 8.2. Since G(i) has to transform like Q; so the Hamiltonian will be totally symmetric, the G( i) can be replaced by a constant times their angular symmetry factors expressed in terms of Cartesian coordinates. Using the functions in Eq. (2.3. 1 3) with the common numerical factors absorbed in the constant A gives ·

·

H

Ho

+ Q1 A

(�

·

·

- /) + Q2 A ! (3z2 - r ) .

(4.7.4)

Using the angular momentum operators given in Eq. (2. 1 . 14), this can be rewritten as (4.7.5)

1 72

4. Electron Phonon Interactions

where /+ and /_ are the usual creation and annihilation (ladder) operators. The secular determinant for the perturbation Hamiltonian H (I) H - Ho is (4.7.6) The matrix elements are given by

J J

Hu = d;Ly1H (I) dx1 y1 dr , H22 = d;1H ( 1 ) dz1 dr , H12 = H21

J d;1_y1H( 1 ) dz1 dr

(4.7.7)

where the symmetrized e9 orbitals for the d electron are given in Eq. (2.3. 1 3) . The secular determinant is then given by c

-2 Q 1 - AE

c

- -2 Q 1 AE

0,

(4.7.8)

where c is a proportionality constant between the real matrix elements and the operator equivalent matrix elements. Solving this gives the perturbation energy as (4.7.9) which must be added to the harmonic potential energy to give the total en­ ergy of the system as (4.7. 1 0) Changing to polar coordinates by defining Q 1 energy of the system is given by

E = 2I

± l r. C

r cos rp and Q2 r sin rp, the (4.7. 1 1 )

This expression shows that the original twofold-degenerate potential energy surface is now double-valued and cylindrically symmetric. It is split into an upper potential surface represented by the plus sign in Eq. (4.7 1 1 ) and a lower potential surface represented by the minus sign. A cross section of the surface appears as two parabolas offset in opposite directions from the cen­ tral axis. This is rotated about the central axis to give the three-dimensional "Mexican hat" potential surface shown in Fig. 4. 1 3 . The energy minima are

4.7. Jahn Teller Effect

(A)

(B)

173

Oj

Potential energy curve for Jahn-Teller coupled 2E8 - '• vibrooic level.

Qj

Potential energy curve for Jahn-Teller coupled 2T 8 - '• vibronic level. 2

FIGURE 4. 1 3 . Schematic representation of the potential energy surfaces of (A) 2E9 and ( B) 2 T29 electronic states split by Jahn Teller coupling to a e9 vibrational mode.

found at

lei . ro = 2k

(4.7. 12)

The energy required to cross from one potential to the other is the Jahn­ Teller energy EJT. The triply degenerate electronic 2 T29 level is also split by the Jahn-Teller effect as shown in Fig. 4. 13. For this case the reduction of the direct-product representation of 2 T29 with itself results in selection rules that allow coupling to IXJg, e9, and 0 while for antiferromagnetic coupling K < 0. The complete set of commuting operators for the couples pair is Hpair , Si , S� , S2 , and Sz. Thus the exchange interaction couples the spin angular momentum states of the individual ions to give pair states with total spin S. As discussed in Chap. 2, the results of the angular momentum coupling produces states with a range of total spin quantum numbers (5. 1 .5) The energy levels of the exchange-coupled pair can be found from the matrix element of the Hamiltonian

( l( I

E�:?r = ( 1/Jrir I Hpair 1 1/Jrir ) = 1/Jl 1/1] H1 + H2 -

· 1 ) 1 1/1] ) .

S 1 S 2 1/Jl

(5. 1 .6)

This can be evaluated by using (1/1/ I HI I I/1/ ) = Eiil , along with the relation

(1/1/ I Si i i/IJ ) = liS1 ( S1 + 1 ) ,

sI . s2 =

s2 - si - s�

2

Substituting these expressions into Eq. (5. 1 .6) gives

E�:fr = Ej + Ej

� [S ( S + 1 )

S1 ( S1 + 1 ) S2 ( S2 + 1 )] .

(5. 1 .7)

The ground state of the pair occurs when both individual ions are in their ground states 1 1/Jgair ) = 1 1/JJ ) 1 1/15) with energy Eo and spins So so

K ( 5 . 1 .8) Epa(O)ir 2Eo KSo ( So + 1 ) 2 S ( S + 1 ) with S = 2So, 2So - 1 , 2So 2 , . . . , 0. The first two terms shift the position _

of the energy level from its unperturbed position while the last term splits

1 80

5. Ion Ion Interactions LEYELS •

ENElil SflN •

•

•

•

•

•

•

•

•

LEVELS ENERGY

(6S0 3)K

2s0-3

(4So- l)K

2s0 2

2SoK

2s0 1

•

•

•

•

SPIN • •

•

•

6K

3

3K

2

K 2s0

0

0

(B) Antiferromagnetic Coupling (Z A given by Eq.

(A) Ferromagnetic Coupling (Zp given by Eq.

5. 1 . 1 5)

5. 1 . 14)

FIGURE 5.2. Energy-level splittings of exchange-coupled pairs.

the unperturbed level into a set of levels characterized by different values of S. The splitting between consecutive levels increases with increasing S value as (0)

LlEpair ( S)

0, K, 3K, 6K, . , So ( 2So + l )K. .

.

(5. 1 .9)

For ferromagnetic coupling K is positive so the state of lowest energy is the S 2So state. Energy splittings between successive levels are given by ilE( n + 1 ; n ) K ( 2So - n ) . For antiferromagnetic coupling K is negative so the S 0 state is the lowest in energy and the splitting of successive levels is given by il E (n + 1 ; n ) Kn. Figure 5.2 shows a schematic diagram of the ground-state splittings of coupled pairs. Since the exchange splitting of the energy levels is generally not large compared to thermal energy at ambient temperature, some of the higher levels of the exchange-split manifold are thermally populated. As discussed in Chap. 3, for a group of energy levels in thermal equilibrium the popula­ tion residing in the ith level is given by the Boltzmann distribution =

(5. 1 . 1 0) where E; is the energy of the ith level above the lowest level of the manifold, g; is the degeneracy of the level, T is the temperature, and kB is Boltzmann's constant. The factor A can be determined from the expression for the total occupation of the manifold (5. 1 . 1 1 )

5. 1 . Exchange-Coupled Ion Pairs

where Z is the partition function 1 z

L g; e i

E, fksT .

181

(5. 1 . 1 2)

Thus, (5 . 1 . 1 3)

For ferromagnetic and antiferromagnetic coupling the partition functions are given by z F zA

=

=

( 4So + 1 ) + ( 4So 1 ) e 2So K/kaT + ( 4So + e [So (2So+ l)] K/kaT , _

_

3) e (4So i ) K/kaT +

. . .

(5. 1 . 14)

1 + 3 e Kfka T + 5 e 3K/kaT + . . . + (4So + 1 ) e [So ( 2So+l )] K/ksT .

(5. 1 . 1 5)

Using Eqs. (5. 1 . 1 0) and (5 . 1 . 1 1 ) along with either Eq. (5. 1 . 1 4) or (5. 1 . 1 5) , the temperature dependence of the population of each of the levels of the exchange-split manifold can be determined. The population is important in determining the intensity of any transition originating on the level. The lowest excited state of a coupled pair occurs when one of the pair ions is in the ground state and the other is in the excited state l l/l fair ) l l/IJ ) l l/l t ) . The determination of the energy-level splitting of the excited state follows the procedure outlined above but the full expressions in Eq. (5. 1 .7) must be used since Ef , S1 and E] , Sz now have different values. Also it has been proposed that other phenomenological forms of the exchange inter­ action besides that given in Eq. (5. 1 . 3) might better describe pair interaction in some excited states. These include biquadratic exchange k(S, Sz) 2 , and individual electron spin coupling, (K' j h2 ) � i,j sf sJ . These can generally be treated by the same procedure described above. The magnitude of the exchange interaction is generally found by treating K as an adjustable pa­ rameter in fitting the observed energy-level splittings. First-principle calcu­ lations of K are generally not possible since evaluating the exchange matrix element requires knowledge of the exact expressions for the electronic wave functions of the two ions as discussed in Chap. 2, as well as knowing the exact positions and spatial orientations of the orbitals of the electrons on the two ions so the wave-function overlap can be determined. None of these parameters are generally known with the degree of precision required for accurate calculations. In addition, the exchange interaction between the two dopant ions can be significantly affected through the polarizability of the intermediate host ligands. This so-called superexchange further complicates fundamental calculations of the interaction strength. Ion pairs have been observed in the optical spectra of several solid-state laser materials. As the concentration of optically active ions increases to the level that they begin forming pairs, they decrease the lasing potential of the isolated ions. However, the pair centers themselves can provide new transi-

·

·

1 82

5. Ion Ion Interactions

tions for laser operation. The level at which ion-pair effects become important depends on whether the dopant ions enter the lattice with spatial random­ ness or whether they tend to be aggregated. If pairing takes place between two different types of ions, it has been found that aggregation is enhanced if the sum of the ionic radii of the two types of dopant ions equals the ionic radii of two host ions that they replace. 2 This can be used to enhance energy­ transfer pumping as discussed later. Ion pairs interact with the photon and phonon fields in the same way as isolated ions. Transitions between levels in an exchange-split multiplet generally take place by radiationless transitions and both direct and two-phonon Orbach processes have been found to be effective in relaxing a system to an equilibrium population distribution. Photon absorption and emission processes have been observed between the ground and excited state of pair systems and the widths of the observed spectral lines have been attributed to both Raman scattering of phonons and lifetime broadening processes. The most extensive investigation of pair spectra and lasing in solid-state laser materials has been performed on ruby and the results are summarized in Chap. 6. 5.2

Nonradiative Energy Transfer: Single-Step Processes

For the case of weak ion-ion interaction, the energy levels of the individual ions involved are the same as those of isolated ions. In this case a single ion is excited by the photon field and the interaction causes a nonradiative transfer of the electronic excitation energy to another ion that subsequently emits the energy. This energy-transfer phenomenon occurs in many different types of materials and plays an important role in a wide variety of physical properties. As stated in the introduction to this chapter, there are two funda­ mentally different types of energy transfer: photoconductivity, which involves the simultaneous transfer of electronic charge and energy, and energy trans­ fer with no accompanying charge transfer. The latter case is the one of importance to laser materials and is discussed below. There is no consistent convention for energy-transfer terminology. In this discussion the ion that absorbs the energy from the photon field is called the sensitizer and the ion that emits the energy is called the activator. In some discussions of energy transfer the terms donor and acceptor are used, but this terminology can be confused by the use of these terms for semiconductor dopants. If the sensi­ tizer is part of the host material, the term host-sensitized energy transfer is used, while impurity-sensitized applies to the case where the sensitizer is a dopant ion. If the energy moves from one sensitizer to another several times before emission occurs, the process is referred to as energy migration as opposed to single-step energy transfer directly from sensitizer to activator. Multistep energy migration is discussed in Sec. 5.4. The first step in treating energy transfer is to derive an expression for the ion-ion interaction Hamiltonian causing the processes to occur. A simple

'>

5.2. Nonradiative Energy Transfer: Single-Step Processes

Wra SENSI1ZER

(A)

ACTIVATOR

1 83

(B)

FIGURE 5.3. Single-step energy transfer between sensitizer and activator ions. (A) Energy-level model for a single-step nonradiative energy-transfer process. ( B) Feynman diagram for energy transfer.

energy-level diagram depicting the process of energy transfer between two ions is shown in Fig. 5.3(A). In the simplest analysis this process can be treated as a quantum-mechanical resonant interaction involving the exchange of a virtual photon as depicted by the diagrams in Fig. 5.3(B). The mecha­ nism of ion-ion interaction can be either an exchange interaction or an electromagnetic multipole-multipole interaction. As noted in the preceding section, exchange interactions can be very strong over a short range (a few fmgstroms), whereas electric dipole interactions can be effective over dis­ tances of tens of angstroms. The initial development of the theoretical treat­ ment of energy transfer through electric dipole-dipole interaction was done by F6rster3 and later extended by Dexter4 to include higher-order multipole interactions and exchange. Because of this, single-step nonradiative energy transfer is sometimes referred to as a Forster-Dexter process. As with any quantum-mechanical transition processes, the rate for energy transfer to occur is described by the golden rule of time-dependent pertur­ bation theory, given by ( 5.2. 1 ) The matrix element for the transition can be expanded as

(5.2.2) where Hf�t is the ion-ion interaction Hamiltonian and the l/1; represent the antisymmetrized product wave functions of the optically active electrons on

1 84

5. Ion Ion Interactions

the sensitizer and activator ions. The first term describes a resonant inter­ action while the higher-order terms are used to describe phonon-assisted energy transfer. The former is discussed below and the latter are discussed in the next section. There are two types of mechanisms for ion-ion interaction leading to energy transfer. The first is the exchange interaction discussed in Sec. 5. 1 . This occurs if the ions are close enough together that there is overlap of the orbitals of electrons on the sensitizer and activator ions. For the simplest case of isotropic Heisenberg exchange the Hamiltonian has a form similar to that in Eq. (5. 1 .3),

·

ntf�� = - L Kus; s1 , i,j

(5.2.3)

where KiJ is the exchange integral and the s; are the spins of the interacting electrons. The second type of interaction that can be responsible for energy transfer is electromagnetic interaction. The Hamiltonian in this case is expressed as the sum of all Coulomb interactions between the charge distributions on the two ions as shown in Fig. 5.4,

Here Ra and Rs are the positions of the activator and sensitizer ions, respectively, and ra and rs are the position vectors of the activator and sensitizer optically active electrons, respectively. The Coulomb interactions between the two nuclei and between the nuclei and the electrons result in zero matrix elements due to the orthogonality of the wave functions. This

FIGURE 5.4. Spatial geometry for the interaction between sensitizer and activator ions.

5.2. Nonradiative Energy Transfer: Single-Step Processes

1 85

leaves the Coulomb interactions between the electrons as the effective inter­ action. There are several useful ways of expressing this Hamiltonian. The ion-ion interaction can be expressed as a multipole expansion using a Taylor's series about the sensitizer-activator separation Rsa · The electric multipole part of the interaction Hamiltonian is given by mt

2 = :. e

Rsa + fa - fs , , e2 = eR3 [fs · fa - 3(fs · R sa) (fa · R sa)] sa , , e2 - · R, sa)] - · Rsa + R 4 [ 2s ( R' sa · Qs ) (fs · R sa) - (fa · Qs sa ' , e2 , + RS [( Rsa · Qa · Qs · Rsa ) + 4! ( Rsa · Qa · RsaRsa · Qs · Rsa ) sa + � ( Q a : Qs ) ] + . . . = Hi;�o + Ht!�Q + Hi�?D + Hi�?Q + . . . , e

_

_

_

,

,

_

,

e

(5.2.4)

where these Hamiltonians represent electric dipole-dipole, electric dipole­ quadrupole, electric quadrupole-dipole, and electric quadrupole-quadrupole interactions. The first part of each of these interactions refers to the transition on the sensitizer while the second part refers to the transition on the activa­ tor. The dielectric constant of the host crystal is given by e, the vectors fa and fs designate the positions of the electrons on the sensitizer and activator as shown in Fig. 5.4, and Qa is the quadrupole moment operator given by

(5.2.5)

Here Ms = I: fs is the dipole moment operator and i is the unit matrix. This interaction Hamiltonian can also be expanded in terms of spherical harmonics using the addition theorem for these functions. This results in the expression -

_

mt

e2 e

4n(ls + Ia ) ! ( - 1 ) 1" r�' r� ! [(21s + 1 ) (2/a + 1 ) ] 1 /2 , a l l R + + 1, 1 la O 00

00

��

( 5.2.6) where the dipole terms are found from setting I = 1 and the quadrupole terms from I = 2. The magnetic part of the ion-ion interaction can also be expressed as

1 86

5. Ion Ion Interactions

a multipole expansion by assuming each electron has a magnetic dipole moment J.l; ( ej2mc) (I; + 2s;) associated with its spin and orbital angular momenta. In this case, the lowest order magnetic dipole-dipole interaction term is the only one of importance. This is given by · J.lj 3 ( J.1; Rsa ) (J.lj · Rsa) HmMDD (5.2.7) · t 3 Rsas i,j Rsa

·

)

The terms in Eqs. (5.2.3), (5.2.4), (5.2.6), or (5.2.7) can be used in Eq. (5.2.2) to determine the matrix element for the energy-transfer transition for different types of interactions. Order-of-magnitude comparisons of the interactions show that the magnetic dipole-dipole interaction energy is w 9 smaller than the electric dipole-dipole interaction energy and therefore it will be neglected in the following discussion. The wave functions for the system of two ions must be expressed in anti­ symmetrized form, l l/1; )

[ l l/l�(rs, w�)) l l/la (ra, Wa)) lx�(s)) lxa (a)) - l l/l�(ra, w�)) l l/la (rs, Wa)) I X�( a)) lxa (s) )] ,

l l/11 ) =

( 5.2.8)

Ws)) l l/l� (ra, w�)) I Xs(s)) lia(a) )

- l l/ls(ra , Ws)) l l/l� (rs, w�)) I Xs(a)) lia(s) ) ] , where the subscript on the position vector refers to the ion where the elec­ tron is initially found. The parameter w represents the energy of the state. The matrix element has the form Msa

\ l/11 I Hi�t l l/1; ) x

( l/ls(r., Ws) I (l/l�(ra , w�) I Hi�t l l/l�(r., w:) ) l l/la (ra, Wa))

(xs(s) lx�(s)) (x� (a) lxa (a))

- ( l/ls(ra, Ws) I (l/l�(rs, w�) I Hi�t l l/l�(rs, w�)) l l/la (ra, Wa)) ( 5.2.9) The second term represents exchange interaction and the first term electro­ magnetic interaction. The first term will be considered here. Since it involves the selection rules x's Xs and x� Xa , the spin functions will no longer be written explicitly. To find the total energy-transfer rate, it is necessary to sum over all pos­ sible combinations of sensitizer and activator states for which energy is con­ served, w� - Ws w� - Wa E This can be accomplished by introducing probability distribution functions for the initial states of the sensitizer and activator ions, p� (w� ) and Pa (wa) . These distribution functions vary con­ tinuously over the energy of the excited state of the sensitizer and ground state of the activator ions. It is also important to include degeneracy factors

.

5.2. Nonradiative Energy Transfer: Single-Step Processes

1 87

for the excited state of the sensitizers g� and ground state of the activators 9a· The eigenfunctions for the sensitizer and activator ions are normalized as usual to an integral over all space. The probability distributions are nor­ malized over all possible energies as

I: p�(w�) dw� I: Pa (wa ) dwa

1.

(5.2. 10)

With the above considerations, the total transition rate is given by

I dw� I dws I Pa (wa ) dwa I p�(w�) dw�

ws�M X

I (t/lj I Hi�tt l t/li) 1 2 c5( (w� - Ws) - (w� - Wa )) ·

(5.2. 1 1 )

where the sum is over all quantum numbers other than the energy quantum number, and the c5 function is included in the density of final states to ensure conservation of energy. Integrating over w� gives

I dE I Pa (wa )dwa I p�(w�) dw� l ( t/IJ I Hi�� l t/li) l 2 ·

ws�M

(5.2. 12)

Using the first term in Eq. (5.2.4), the electric dipole-dipole interaction matrix element is

where the expectation value of the electron position vector is given by (r)

I t/J'* rt/1 dr .

This matrix element can be averaged over all possible orientations using Fig. 5.4. Assuming that there is no preferred orientation of the dipoles, the average over all orientation angles results in

Substituting these results into Eq. (5.2. 12) gives

w;oo

(g�ga ) 1

� � I dE I Pa ( Wa ) dwa I (ra) 1 2 I p� ( w� ) dw� I (rs) 1 2 .

(5.2. 1 3)

It is useful to reexpress Eq. (5.2. 13) using parameters related to the experimentally observed absorption and fluorescence spectra. For example,

1 88

5. Ion Ion Interactions

the Einstein A coefficient from Eq. (3.2.6), including a thermodynamic sta­ tistical average over all initial states, is expressed as

A(E)

2

L i

Ec ) l (ri[) l 2p' (w') dw', 3h c gi ( E J

f

(5.2. 14)

where the screening factor (Eel E) is the ratio of the electric field seen by the ion in the crystal to that "seen" by the ion in a vacuum. Because of the relationship between the Einstein A coefficient and the radiative lifetime, a normalized spectral function g is defined as (5.2. 1 5) g = rA. This is related to the integrated emission intensity of an ion as discussed in Chap. 3 . Similarly, the Einstein B coefficient from Eq. (3. 1 .37) can be used to express the absorption properties of the ions,

B(E) =

i

f

2ne2 Ec 2 i[ 2 I (r ) I p(w) dw. 3h g, E

( )J

(5.2. 1 6)

Since this is related to the absorption cross section by O"(E) = B(E)/(cjn), another normalized spectral function G can be defined as

G O"(E) ' Q

(5.2. 17)

where Q is the integrated absorption cross section, Q = J O"(E)dE. Using the definitions given in Eqs. (5.2. 14)-(5.2 . 1 7) in Eq. (5.2. 1 3) gives the energy­ transfer rate as (5.2. 18) where the oscillator strength has been used to describe the absorption tran­ sition instead of the parameter Q as given in Eq. (3.3.20). The expression in Eq. (5.2. 1 8) can be simplified by assuming that the electric field screening factor is close to unity and that the wave number in the region of spectral overlap does not vary significantly so an average value of Vsa can be used. The results of these simplifications are (5.2. 19) where the spectral overlap integral Q is given by (5.2.20)

5.2. Nonradiative Energy Transfer: Single-Step Processes

1 89

To simplify the expression further, it is useful to define a critical inter­ Ro given by

action distance

( 5.2.2 1 ) and a critical concentration Co given by (5.2.22) Substituting the expression for Ro into Eq. (5.2. 1 9) allows the energy trans­ fer rate for electric dipole-dipole interaction to be written as wsaEDD

(� RRo )6

r

= _.!._

sa

(5.2.23)

.

From this expression it is clear that the critical interaction distance is the sensitizer-activator separation for which the energy-transfer rate is equal to the intrinsic decay rate of the sensitizer. Similar expressions can be derived for electric dipole-quadrupole and quadrupole-quadrupole interactions. These energy-transfer rates can be expressed in terms of the electric dipole-dipole rate as3.4 2 / WsaEDQ = WsaEQD = Q � WsaEDD (5.2.24) fn

and WsaEQQ

=

(R ) sa

( /Q)2 (�R )4 fn

sa

WsaEDD '

(5.2.25)

where A. is the average wavelength of the transition in the spectral overlap region and/Q is the oscillator strength for a quadrupole transition given by 2m

!ifEQ - 3he 2 wiJ I ( (k · r) ( er)) if l 2 . Energy transfer by these higher-order multipole interactions can be impor­ tant when electric dipole transitions are forbidden on the sensitizer or acti­ vator ions. The energy-transfer rate for exchange interaction can be derived by using the interaction Hamiltonian given in Eq. (5.2.3) in the first term of Eq. (5.2.2) to evaluate the matrix element. In this case explicit expressions for the electronic wave functions must be used. To obtain a general expression, it is customary to use hydrogenlike wave functions, 4 which leads to

[(

)]

R 1 WsaEX = r exp 1 R o J ' sa �

(5.2.26)

1 90

5. Ion Ion Interactions

where

(5.2.27) and = 2n h 's

(5.2.28)

In these expressions, L is an effective Bohr radius and K is a constant involving the spatial overlap of the electron wave functions. For the important case of rare-earth ions in solids, expressions for ion­ ion interaction can be rewritten explicitly in terms of the Hamiltonian given in Eq. (5.2.6) and wave functions expressed as spherical harmonic func­ tions. 5 The Judd-Ofelt theory is used to express the strengths of the elec­ tronic transitions on the sensitizer and activator ions in terms of special pa­ rameters and reduced matrix elements (defined in Sec. 8.2). This theory is outlined in detail in Sec. 8.3. The results for the rate of electric dipole-dipole interaction are

w;oo

=

1 (2ls + 1 ) (2Ia + 1 ) x

(�

(;;J (� n;.s (ls II u(J.) I I J;) z) z

)

il;.a (Ja II U (J.) I I J�) 2 n,

(5.2.29)

which takes the place of Eq. (5.2. 19). Here the il;.s are the Judd-Ofelt intensity parameters, ls is the total angular momentum quantum number for the energy level involved, U is the unitary operator, and the double bars represent reduced matrix elements. Similar expressions can be written for the other multipolar interactions. The energy-transfer rates derived above describe a process occurring over a fixed sensitizer-activator separation Rsa· In a typical laser material in­ volving energy transfer, there are spatially random distributions of ensem­ bles of sensitizer ions and activator ions. Any specific sensitizer ion in the excited state has a probability of interacting with each unexcited activator ion in the host with the strength of the interaction varying with distance in accordance with the type of interaction mechanism. The effective energy­ transfer rate for the entire system of ions must reflect this probability distri­ bution of transfer rates for a given sensitizer ion, along with the random nature of the activator environment for each sensitizer site. This can be ac­ complished by dividing the sensitizers into classes having the same activator environment and then finding the average of the result for all classes. As an example of this procedure, consider the electric dipole-dipole energy transfer rate for a specific class of sensitizer designated as j. Using

5.2. Nonradiative Energy Transfer: Single-Step Processes

Eq. (5.2.23), the transfer rate is

w;nn =

� t (RRo )6 , 's i=l

,

191

(5.2.30)

where R; is the distance from the sensitizer of class j to the ith activator and the sum is over all Na activator ions. One experimental method for determin­ ing the energy-transfer rate of the system of ions is through measuring the fluorescent lifetime. This is associated with the time-dependent decay of the population of the excited state of the sensitizer ions. The rate equation describing the time rate of change of the population of the excited state of sensitizers in class j (designated nsj ) is

dnsj = 1 nsj - UJEDD nsj , dt 0 's -

assuming J-function excitation at time zero. The solution of this equation is

(5.2.3 1 ) To proceed further it is necessary to know something about the distribu­ tion of ions. If a spatially random distribution of activators is assumed, the number of sensitizers belonging to class j is given by

Na 4nR� dR; , Nsj _ Ns IT i=l V

(5.2.32)

where V is the volume of the crystal. Using this as the probability distribu­ tion function for a specific class of sensitizers with a time varying excited­ state population given by Eq. (5.2.30), the average of nsj over all classes then gives

(5.2.33) where Co is the critical concentration defined in Eq. (5.2.22). Differentiating this equation with respect to time gives the rate equation of the total pop­ ulation of excited sensitizers. The result has the same form as Eq. (5.2.30) with the total energy-transfer rate given by

-

wEDD 'J.n3 f2 R5 Na - 3 ( r� t) l /2

·

(5.2.34)

Thus the energy-transfer rate for electric dipole-dipole interaction among randomly distributed ions varies with time as r 1 12 and depends directly on the concentration of activator ions and on the cube of the critical interaction distance Ro .

192

5. Ion Ion Interactions

Other expressions can be derived in a similar way for different types of energy-transfer mechanisms. For electric multipole interactions the results can be expressed by the general equation ( 5.2.35)

where q 6, 8, or 1 0 for electric dipole-dipole, electric dipole-quadrupole, and electric quadrupole-quadrupole interactions, respectively. Similarly, for exchange interaction the sensitizer population is given by (5.2.36) where the function g (z) is given by (5.2.37) One important assumption in the derivation described above is that the distributions of sensitizers and activators are uniformly random throughout the sample with no correlation effects as reflected in Eq. (5.2.32) . As dis­ cussed in the previous section, effects such as local strains can enhance cor­ relation effects and cause preferential pairing of sensitizer and activator ions.6•7 This can be treated mathematically by modifying Eq. (5.2.32) to include correlation factors. The distribution of ions can be separated into several regions of space where nearest-neighbor shells have a high degree of correlation, whereas at long separation distances the random distribution of Eq. (5.2.32) is used. This result in much more efficient energy transfer at short times due to an enhanced concentration of near-neighbor sensitizer­ activator pairs. At long times the energy-transfer rate evolves toward the value obtained by assuming an uncorrelated, random distribution of both types of ions. 5.3

Phonon-Assisted Energy Transfer

The energy-transfer rates derived in the preceding section depend critically on having resonance between the emission transition of the sensitizer and the absorption spectrum of the activator. This is explicitly reflected in the spectral overlap integral factor appearing in the expressions for the energy­ transfer rates. Phonons play an important role in ensuring the conservation of energy. For resonant electronic transitions, phonons affect the widths of the spectral lines and thus the magnitude of the spectral overalap integral. Also the temperature dependences of the resonant energy transfer rates are contained in the spectral overlap integrals. For the case when the electronic

5.3. Phonon-Assisted Energy Transfer

•

.1Esa

1 93

FIGURE 5.5. Typical phonon-assisted energy-transfer process .

'

PA W sa

transitions are out of resonance with each other resulting in a value of zero for the spectral overlap integral, the rate of resonant energy transfer is negli­ gibly small. In this situation the energy mismatch between the sensitizer and activator transitions can be made up by the contributions of phonons re­ sulting in phonon-assisted energy transfer. A typical situation for phonon­ assisted energy transfer between two ions with a transition energy mismatch AEsa is shown in Fig. 5.5. The electron-phonon interaction can occur on either ion and in either the ground or excited state. The transition rate for phonon-assisted energy transfer can be calculated using the expressions in Eqs. (5.2. 1 ) and (5.2.2) with the interaction Hamil­ tonian containing factors for both ion-ion and ion-phonon interactions. Also the wave functions must now contain a factor for the occupation number of the phonon involved. Thus the matrix element associated with a one-phonon-assisted, ( PA) energy-transfer process is given by

MsaPA

(l/lsl/l:njk ± l i Hsa l l/l;l/la nik ± l ) ( l/l;l/lanik ± l i H? (m) l l/l;l/lanJk ) Es - (Es ± hWjk ) m ,a ( l/1sl/l:njk ± I I H? ( m) l l/1sl/l: njk) ( l/1sl/l: njk I Hsa l l/1; l/1anik ) + ' Es - Ea m=s,a

L =s

(5.3 . 1 ) where the sum accounts for the phonon emission or absorption occurring at either the sensitizer or activator site. Here Hsa is one of the resonant interaction Hamiltonians discussed in the previous section and H? is the electron-phonon interaction Hamiltonian expressed in terms of the strain e and crystal field V in Eqs. (4. 1 .22)-(4. 1 .24) . Since the spatial extent of the phonon is important in this case, the exponential factor appearing in Eq. (4. 1 .20) describing the ion displacement due to the phonon must be retained explicitly. ( This was suppressed in Chap. 4 where spatially localized pro-

194

5. Ion Ion Interactions

cesses were being considered.) This expression can be simplified to8

( 5.3.2)

k Msa

where is the ion-ion matrix element independent of the phonon state and is the phonon wave vector. The strain factor in the electron-phonon interaction can be expressed in terms of the phonon creation and annihila­ tion operators, and f and g are the electronic matrix elements of the crystal­ field operator in the ground and excited states, respectively. The difference in coupling strengths in the ground and excited states (f - g) is assumed to be equal for the sensitizer and activator ions. The exponential factors result in a phase factor exp( ± ik Rm) for the ion at position Rm. In addition, conservation of energy requires that the phonon energy hw1k is equal to With these conditions, the the electronic transition energy mismatch phonon-assisted energy transfer rate is expressed as ·

!:iEsa ·

(5.3.3) where the coherence factor is given by h( k,

Rsa )

Ie

ik R,.

1 12

4 sin2

(k .

(5.3.4)

This factor describes the degree to which a particular phonon mode causes the sensitizer and activator ions to move in phase with each other. This is a key consideration in determining how effective the phonon is in bringing the energy levels of the two ions into resonance. There are two different cases that can be considered. The first is the case when the energy mismatch between the electronic transitions is small com­ pared to the available phonon energies so that the relevant phonon modes are those with small wave vectors. Thus, in this case « 1 so the pho­ non wavelength is large compared to the sensitizer-activator separation. Using a Debye distribution of phonons to evaluate the sum over k in Eq. (5.3.3) and averaging the coherence factor over all angles, the phonon­ assisted energy transfer rate for this case becomes

k · Rsa

(5.3.5) where p is the density of the host, v1 is the phonon velocity, and r:x1 is the angular average of the strain parameter. The upper term in the last factor is for phonon emission and the lower term is for phonon absorption. The temperature dependence is contained in the phonon occupation numbers. The factor of in the numerator shows that the closer the sensitizer and activator are to each other, the less effective the phonon is in bringing their

R;a

1 95

5.4. Nonradiative Energy Transfer: Multistep Process

transitions into resonance. This is because, for the conditions of this case, the phonon tends to modulate the energy levels of both ions together . The second case is when the energy mismatch between the electronic transitions is large enough that the relevant phonons are those with large wave vectors so k R sa > 1 . In this case the wavelength of the phonons is shorter than the sensitizer-activator separation. Following the same proce­ dure described above, the energy-transfer rate for this case is given by ·

wPA sa ( 2 )

M}a (f -

nph4

J

r:l.j v5J

X

{

njk + nJ'k

1

},

( 5.3.6)

The energy-transfer rate for this case has the same temperature dependence as that for the first case. However, the dependence on the transition energy mismatch is quite different. In addition, there is no longer an explicit de­ pendence of the phonon modulation term on Rsa · This is because the pho­ non in this case modulates the transition energy of the sensitizer differently than it modulates the transition energy of the activator. For cases involving very small energy mismatches where the density of states of available phonons with the appropriate energy is also very small (as shown in Fig. 4.4), two-phonon-assisted energy-transfer processes may be­ come important. In this situation it is the difference in energy between one phonon that is absorbed and another phonon that is emitted that makes up the energy difference between the two electronic transitions. The energy­ transfer rate for this situation can be derived using the same steps outlined above but using the appropriate two-phonon matrix elements for the radia­ tionless decay processes discussed in Sec. 4.2. Similarly, for very large energy mismatches (that is, much greater than available phonon energies), multi­ phonon processes become important and the N-order perturbation approach to radiationless decay processes must be used. This again leads to an energy­ transfer rate exhibiting an "exponential energy gap law" and a temperature dependence described by [n (w ) + l] N as discussed in Sec. 4 . 2 . 5.4

Nonradiative Energy Transfer: Multistep Process

For some materials the concentration of sensitizers is high enough that ex­ citation energy can be transferred from one sensitizer to another several times before the final transfer to an activator occurs. In this type of multi­ step transfer process, the excitation energy can be viewed as a quasiparticle migrating on a lattice of sensitizers, and the mathematical description of the process is the same as that for exciton migration. This case involves a lo­ calized exciton with the electron and hole both located on the same ion and moving together. It is referred to as a Frenkel exciton. This view of multistep energy migration is especially applicable to host-sensitized energy transfer where the sensitizer is a constituent of the host lattice.

196

5. Ion�Ion Interactions s

s

s

s s

s

s

s

s

s

s

s

FIGURE 5.6. Multistep energy migration and trapping.

Figure 5.6 shows a schematic representation of multistep energy migra­ tion. Each energy-transfer step between sensitizers involves one of the ion­ ion interaction mechanisms discussed in the previous section. The final step between sensitizer and activator ions also involves one of these ion-ion in­ teractions mechanisms, but it does not have to be the same mechanism as the one for sensitizer-sensitizer interaction. The dynamics of the total energy­ transfer process are characterized by two distinct contributions: the migra­ tion of energy among the sensitizer ions and the trapping of the energy at activator sites. One of the difficulties in analyzing the effects of multistep energy transfer is separating the characteristics due to the migration process and those due to the trapping process. There are two mathematical ap­ proaches used to describe localized exciton migration with trapping, one based on a random-walk model and the other based on a diffusion model. In the limit of many steps in the random walk on a uniform three-dimensional lattice, the two approaches are equivalent. Both of these approaches are described below. The simplest situation is one in which the sensitizer-activator interaction is equivalent to the sensitizer-sensitizer interaction so the migrating exciton becomes trapped only when it happens to hop onto an activator site. As an example, consider a simple-cubic lattice arrangement of active ions (sensi­ tizers and activators) with electric dipole-dipole interaction between pairs of ions. Let the time for an excitation step be represented by the hopping time th, and the probability of sensitizer fluorescence per hopping time be repre­ sented by a. Also let the probability of activator trap fluorescence per hop­ ping time by represented by p, and the fraction of active ion lattice sites that are activator traps be represented by Ca. With these assumptions and nota­ tion, the probability for sensitizer fluorescence to occur on the nth step in a random walk is given by9 • 1 0 where Vn is the number of distinct sites visited on a walk before the nth step. The last factor in this equation is therefore the probability that none of the

5.4. Nonradiative Energy Transfer: Multistep Process

1 97

sites visited before the nth step are traps. For large n on a three-dimensional lattice, random-walk theory shows that Vn -+ ( 1 - F) n, where F is the probability the exciton eventually returns to the origin. The latter quantity varies with lattice symmetry and is found to be about 0.34 for a simple-cubic lattice. 8 Thus the sensitizer fluorescence intensity after n steps and an initial excitation intensity of ls(O) is

( 5.4 . 1 ) The number of distinct sites visited before step n has been approximated by

0.66n.

A similar expression can be written for the probability for fluorescence from an exciton trapped at an activator site after the nth step in the random walk. This can be divided into four time periods. First there are n 1 - 1 steps on normal lattice sites, then one step onto a trap, then a waiting period of n - n 1 - 1 steps, and finally emission of trap fluorescence. Thus the fluo­ rescence intensity of activator emission after n steps is r a

=

)

(

n ( 1 1X ) ( 1 - Ca ) (1 - F) ' P ( l - F) Ca ( 1 - Pt 1 -P ( 1 - IX) ( 1 - P) 0.66P Ca ( 1 - Ca ) 0.66 {( 0.66 n . n [( 1 IX 1 1 (1 - P) [1X - p + ( 1 - 1X)0.66Ca] - p) - - ) ( - Ca ) ] } ( 5.4.2)

In the limit of many steps, Eqs. (5.4. 1 ) and (5.4.2) can be rewritten as func­ tions of time expressed in terms of number of steps, t = n th ,

where

ls(t)

=

Ia (t)

=

-

a ls(O) 1 - a e (a+0.66 Ca) t/ th '

-0.66Ca + 1

( 5.4.3)

(e -btf th - e- (a+0 .66 Ca) t/ th ) ,

(5.4.4)

The exponential decay rate in Eq. (5.4.3) can be used to define the fluo­ rescence decay time of the sensitizers with and without any activator ions being present. These decay times are

rs 1

=

(a + 0.66Ca ) th

'

rsO1

a. th

= -

From these expressions the rate of energy transfer for a multistep random-

198

5. Ion Ion Interactions

walk process can be defined in the usual way,

rs 1

rs(}1 + wrw sa '

(5.4.5)

where

( 5.4.6) The ratio of the sensitizer fluorescence decay time in the undoped and doped samples is T s() 1 + sO Wrw . ( 5.4. 7) T sa Ts Since the ion-ion interaction mechanism has been assumed to be electric dipole-dipole, the expressions from the preceding section can be used to obtain an equation for the hopping time,

(

)6

R0 ( 5.4.8) rsO1 Rss ' where the critical interaction distance Ro was defined in Sec. 5.3 and Rss th 1

represents the nearest-neighbor sensitizer-sensitizer separation. The sensitizer and activator fluorescence intensities can be evaluated by integrating Eqs. (5.4.3) and (5.4.4) over time. This gives

Is Ia

oo J ls(t)dt oo Ia (t)dt Jo 0

a th ls(O) l _ a a + 0_66Ca ,

b -b

th - -; · + 0 66C a a . +l th

( 5.4.9)

0.66Ca From the first of these expressions, the ratio of the sensitizer fluorescence with and without traps present is / (0)

1 + T s() Wsr:' ·

(5.4. 10)

Using the same assumptions, an expression for energy transfer can be derived treating the energy migration as a diffusion processes. 1 1 If Pk is defined as the probability of finding an exciton on site k, then the time evo­ lution of this probability is given by

dPk dt

I

1 Pk , L Wkt (Pt - Pk ) - T I

s() where the summation runs over all active ion lattice sites. A continuum approximation can be used to change this summation to an integral and the probability can be expanded in a Taylor's series. Keeping only the lowest-

5.4. Nonradiative Energy Transfer: Multistep Process

1 99

order nonzero, rotationally invariant terms gives

(5.4. 1 1 ) The difusion coefficient is defined as

l Rkt 2 6 th '

_!_ Ro6

kl

I

(5.4. 12)

where Rkt represents the average step length in the random walk. This allows Eq. (5.4. 1 1 ) to be rewritten as

aPk DV2 1 . Pk - at rs0 Pk

(5.4. 1 3)

-

The average displacement from the ongm during the lifetime of an exciton is called the difusion length and is given by

Ln

�.

(5.4. 14)

The root mean square of the hopping distance can be found from Eq. (5.4. 13) to be l .07Rkt , which justifies the assumption of nearest-neighbor hops in the random walk. 9 To calculate the total rate of multistep energy transfer in the diffusion model, the rate equation for an ensemble of migrating excitons must be solved. If No excitons are created at time t 0 by a J-function excitation pulse W( t) , Eq. (5.4. 13) can be modified to give

oN(r ' t) at

W( t)

t

+ DV2 N(r , t) - r� N (r, t)

(5.4. 1 5)

with the boundary conditions

N(r , t 0) No, N(r RT, t)

0

=

(5.4. 1 6)

where R T is the effective trapping radius around an activator ion and N(r, t) NoPk (r, t) . This equation can be solved by making the substitution

u(r , t) r

N(r, t)

e tfr., ,

r > RT

(5.4. 17)

and then using Laplace transforms to obtain

[

( )]

RT erfc - RT N(r , t) No 1 --;:2 .Jl5i -

'

! e 1 r., .

( 5.4. 1 8)

The rate of energy transfer in the diffusion model is related to the flux of

200

5. Ion Ion Interactions

excitons crossing the effective trapping surface surrounding each activator:

F1 ( t) 4rr:DR2T aN

i

8, r=Rr

.

(5.4. 19)

Assuming a spherical trapping surface and a concentration of Ca non­ interacting traps, the total flux of excitons being trapped at time t is given by

(

FN (t) 4rr:DCa No e � tjr"' Rr + The total flux of excitons into traps can be viewed as the product of the exciton concentration far from a trap, No exp ( t/rsO), and the rate of energy transfer to traps,

(

Wfa 4rr:DRrCa 1 +

(5.4.20)

Any exciton created at time t 0 within the radius of a trap has an infinite rate of being trapped. For typical cases of interest in solid-state laser mate­ rials, the term Rr / VnJ5i « 1, which results in a time-independent energy­ transfer rate given by

Wfa 4rr:DRrCa .

( 5.4.21 )

In some situations the geometry of the sensitizer interactions and the ori­ entational dependence of the ion�ion interaction restricts the energy migra­ tion to one or two dimensions. The mathematical expressions for random walk and diffusion of excitons are quite different for these cases. Using the same procedure outlined above results for one- and two-dimensional sys­ tems can be analyzed. The point-trapping approximation used in the preceding mathematical development does not always adequately describe the true physical situa­ tion. Several models have been developed to account for the characteristics of exciton trapping for different limiting situations. One example is the extension of the random-walk formalism to include the effects of trapping regions of various sizes and geometries. 1 2 This can be important for mate­ rials involving large organic molecules, but is generally not significant for inorganic solid-state laser materials. A second approach to treating the effects of trapping is to explicitly include a term for sensitizer�activator interaction in the diffusion equation

t) -Psns(r, t) + DV2 ns(r, t) L Wsa(r - R; )ns(r, t) , ( 5.4.22) t i where R; is the position vector for a given activator and Wsa(r - R;) is the interaction rate for a given sensitizer-activator pair. No general solution to this equation has been obtained. However, solutions for limiting cases have been derived with the assumption of electric dipole-dipole interaction

201

5.4. Nonradiative Energy Transfer: Multistep Process

between sensitizer ions and between sensitizers and activators. For the case of weak diffusion perturbing a strong sensitizer-activator interaction, this equation has been solved using an operator expansion with a Pade approx­ imant technique to obtain1 3

ns(t)

_

exp

(

-

)

21 3 + 1 5.74x3 t413 f3s t Na13n3 /2 Ro3 .Jf 1 + l0.910xt 8_76xt ' 2/ 3 + (5.4.23) _

where X

nps l f 3 R0 2

(5.4.24)

·

As x __ 0 this reduces to the expression for single-step electric dipole-dipole energy transfer. The solution of Eq. (5.4.22) has also been obtained for the opposite case of fast diffusion perturbed by weak sensitizer-activator interaction. In this case an approximate potential approach with a propagator expansion in the first Born approximation was used. 14 The resulting expression for the energy-transfer rate is

( + 2nCa R} J: dr r 8nCa J: dr r

)

4nCaf3s R� Rr WsaDT 4nDRrCa 1 + v'7J5i + 3Rr =

[erfc

[erfc

2

J.

(5.4.25)

L ns(t - t')iis( t')e t '/to dt',

(5.4.26)

In order to use this expression numerical integration must be used to obtain the explicit time dependence of the energy-transfer rate. A third approach has been developed for treating the characteristics of exciton trapping based on treating the transfer rate as a random variable in a stochastic hopping process. 1 5 In this model the sensitizer luminescence is proportional to the sensitizer excited-state population density described by

ns(t)

iis(t) e tf to + t0 1

where to is the average value of the hopping time and iis( t) is given by the general expressions for the time evolution of the excited sensitizer pop­ ulation in the absence of sensitizer-sensitizer interaction. The solution of this equation in the limits of fast diffusion and no diffusion are equivalent to those obtained by solving Eq. (5.4.22). However, this formalism is ideal for computer simulations. Monte Carlo techniques have been used to simulate energy migration on a random lattice. 1 6 Instead of using an average value for the hopping time, a weighted set of random numbers is used to describe

202

5. Ion Ion Interactions

the variation in hopping time in each step in the random walk due to the randomness of the lattice site distribution and the randomness in the ion­ ion interaction rates. A standard set of random numbers is first generated and then weighted by a Hertzian distribution to account for the spatial site randomness. Then the numbers are weighted again to account for the ion­ ion interaction (such as for the electric dipole-dipole interaction) . This procedure has been used to fit experimental data on energy transfer and the results show that the average hopping time obtained using the Monte Carlo procedure to simulate the randomness of the physical situation is signif­ icantly smaller than the hopping time obtained from assuming a uniform lattice. 1 6 At high concentrations of sensitizers, percolation theory may be more accurate than this Monte Carlo approach. 1 7 In this situation the dis­ tribution of sensitizers in not truly random, but instead there are regions of high densities where energy transfer is very efficient, and weak transfer occurs from one of these regions to another. It is common to write the ion-ion interaction rate in the form Ws�) = Cs�) /Rn , where Cs�) = R0jr� is a microscopic energy-transfer parameter in­ dependent of concentration and the power n depends on the nature of the multipole interaction. For an exciton diffusing by electric dipole-dipole in­ teraction on a cubic lattice, the trapping radius can be defined as the dis­ tance at which the sensitizer-activator energy transfer rate is equal to the rate of sensitizer-sensitizer transfer on the lattice,

,-6

( )1/4

R T - 0 . 676 CDsa

( 5.4.27)

The diffusion coefficient in Eq. (5.4. 12) is then given by

( 5.4.28) and the energy-transfer rate associated with diffusion and trapping from Eq. (5.4.21 ) is

(5.4.29) This provides a means to predict the sensitizer concentration dependence of the diffusion coefficient and the sensitizer and activator concentration dependencies for the energy transfer rate. If the hopping model is used, the average value for the hopping time found by assuming a random distribu­ tion of sensitizers is1 8

( 5.4.30)

5.4. Nonradiative Energy Transfer: Multistep Process

203

If the trapping radius is defined as the distance at which the rate of sensitizer-activator transfer is equal to the hopping rate, the overall energy transfer rate becomes

(5.4.31 ) This predicts the same concentration dependencies as Eq. (5.4.29) but dif­ ferent dependencies on the ion-ion interaction parameters. As pointed out above, computer simulations have shown that the rate of energy transfer predicted by assuming an average hopping time is significantly different from the value obtained when the true randomness of the exciton migration is taken into account. The differences in Eqs. (5.4.29) and (5.4.3 1 ) may reflect this same problem. In the localized hopping model for exciton migration used in the above discussion, the excitons move incoherently. That is, phase memory is lost at each step of the random walk and the wave vector for the motion is not a good quantum number. The excitons are considered to be self-trapped due to lattice relaxation after each step. This takes the transition energy out of resonance with neighboring unexcited sensitizers and thus requires phonon activation to move from one site to another. Each step in the random walk can be treated as a phonon-assisted energy-transfer processes as described above. In this case the diffusion coefficient and thus the energy-transfer rate varies with temperature as exp ( - !:iE/kBT), where !:iE is the activation energy required for hopping. For situations where the electron-phonon coupling is weak enough that the self-trapping energy is very small, the excitons can move coherently over several lattice spacings before being scattered. In this case the diffusion coefficient is expressed in terms of the group velocity of the excitons v9 and the time between scattering events r,

( 5.4.32) where A is the mean free path of the exciton motion. The energy-transfer rate for long mean-free-path exciton motion is similar to a kinetic gas scat­ tering expression,

(5.4.33) where (J is the trapping cross section. The temperature dependence of the energy-transfer rate is determined by the exciton-phonon scattering time that limits the mean free path of the exciton motion. The exact expression for this scattering time is difficult to calculate due to the unknown details of exciton-phonon coupling. However, the most generally accepted results1 9 assume that scattering by acoustic phonons dominate other scattering mechanisms and this predicts that D oc

r-112 •

204

5. Ion Ion Interactions

5.5

Connection with Experiment: Rate Equation

Analysis

There are several specific questions that must be answered in characterizing energy transfer in a specific material. The first question is whether the transfer is a single-step or multistep process. Then it is important to know if the interactions involved in the energy transfer are resonant or phonon­ assisted. Next the types of interaction mechanisms must be identified and the strength of the interactions must be determined. The latter is usually characterized by the critical interaction distance R0 . If the energy transfer is a multistep process, the properties of both the energy migration process and the trapping processes must be determined. The former can be characterized by the diffusion coefficient, diffusion length, hopping time, and number of steps in the random walk, as well as the nature of the ion-ion interaction mechanism generating the random walk. The trapping can be characterized by parameters such as trapping cross section and trapping rate as well as the nature of the sensitizer-trap interaction mechanism. If enough information is known about the material, it is possible to obtain theoretical estimates of all of the relevant parameters from the theoretical models described in the previous two sections. Experimental measurements of spectral properties such as the fluorescence intensities and fluorescence lifetimes as functions of variables such as temperature, active ion concentration, and time can be used to obtain independent estimates of these same parameters. The com­ parison between theoretical and experimental estimates be used to answer these questions about the properties of energy transfer. The most common procedure has been to study the concentration quenching of the fluorescence intensity or lifetime of sensitizer ions. The major problem with this tech­ nique is that it requires accurate knowledge of the concentration of active ions in a series of samples and this is generally is not available. Measure­ ments of the time evolution of the fluorescence is more difficult but the ex­ periment is done on one sample, thus eliminating concentration differences from sample to sample. The most general expression for the time evolution of energy away from an initially excited ion is

where Pi ( t) is the probability of finding the excitation on the ith sensitizer ion at time t , f3 is the intrinsic fluorescence decay rate of the sensitizer ions, Wy is the energy transfer rate from sensitizer i to activator j and fV.ii is the transfer in the opposite direction (backtransfer), and Win describes the energy migration among sensitizer ions before fluorescence or transfer to an activa-

5.5. Connection with Experiment: Rate Equation Analysis

205

tor ion occurs. This equation must be solved and the results related to ex­ perimental observables such as the fluorescence intensity. This requires per­ forming a configuration average over the distribution of all possible ion-ion interactions and the inclusion of the initial conditions. Although attempts have been made to develop a general solution to this equation, 20 this is a difficult task since a double configuration average is required to account for both spatial disorder (random location of ions) and spectral disorder (vari­ ation of transition energies from site to site as reflected in inhomogeneous broadening of spectral lines) . Knowing the details of these distributions is critical in understanding the physics of energy transfer in a particular case. For example, the time dependence of the energy transfer is significantly dif­ ferent if sensitizer and activator ions are located in pairs all having the same separation, randomly separated pairs, or distributed with all of the sensi­ tizers on one side of the sample and all activators on the other side. The ini­ tial excitation conditions can excite sensitizer ions in centain spatial regions and not those in other regions of the sample. However, for most practical cases with solid-state laser materials, it is sufficient to assume a random spatial distribution of activators with the uniformly excited sensitizers either having a similar random distribution or, for host-sensitizer cases, being dis­ tributed in a known lattice configuration. Also, in general the spectral dis­ tribution is most important at low temperatures and can be ignored at room temperature where phonons are available to bring transitions of neighboring ions into resonance with each other. Some of the important exceptions to these statements are discussed below and in the following chapters. Even with the simplifying assumptions discussed above, it has proven to be very difficult to use the master equation in Eq. (5.5. 1 ) based on micro­ scopic energy-transfer parameters to analyze experimental resuHs. Although theoretically this should be the most direct method for obtaining the desired information about the physics of energy transfer in a material, in practice it has been found to be easier to use models based on macroscopic parameters for the initial step in data analysis. The primary parameter obtained from experimental data is generally the macroscopic energy transfer rate. The measured results provide information on the variation of this transfer rate with the variables mentioned above, and these properties allow the identi­ fication of the microscopic ion-ion interaction mechanism. The most com­ mon method of analyzing experimental results to obtain the energy-transfer rate under specific experimental conditions is to use a phenomenological rate equation model describing all of the energy levels and transitions in­ volved in the system. The equations describing the time evolution of the populations of the various levels of the system can then be written down and solved assuming the appropriate experimental conditions. The expressions obtained for the populations of the metastable states are directly propor­ tional to the measured fluorescence intensities from transitions originating on these levels with the proportionality constant being the radiative decay rate.

206

5. Ion Ion Interactions

Ws k

J}a

FIGURE 5. 7. Phenomenological model for the rate-equation analysis of energy transfer.

A simple example of a two-level system for both sensitizers and activators is shown in Fig. 5.7. The rate equations for the populations of the excited states are dt =

dns

Ws - flsns - kns ,

( 5.5.2)

where ns and na are the concentrations of excited states of the sensitizers and activators, fls and fla are their fluorescence decay rates, Ws is the rate of ex­ citation, and k is the energy transfer rate. For experiments involving con­ tinuous excitation Ws is a constant and the time derivatives of the pop­ ulations are zero. For pulsed excitation these derivatives are no longer zero and Ws can be expressed as a J function. The solutions to the rate equations for steady-state excitation are

k ns = fl Ws k ' na = fl (fJWs+ k) ' s+ a s

(5.5.3)

whereas the solutions for pulsed excitation conditions are

(

L

ns(t) ns(O) exp -fls t - k(t') dt' n a (t)

)

= exp( fla t) t k(t') exp(fJf)ns( t') dt'.

(5.5.4)

The expressions for the energy-transfer rate with its time dependence ex­ plicitly included must be used to obtain the final solution to Eqs. (5.5.4). The magnitude and properties of the phenomenological energy-transfer rate parameter k are determined by analyzing experimental data using the expressions derived from the rate-equation model described above. Due to the difficulty of making absolute intensity measurements, it is more common to make relative measurements. These can be relative measurements of the fluorescence properties of the sensitizers in samples with and without acti-

5.5. Connection with Experiment: Rate Equation Analysis

207

vators present, or measurements of the sensitizer fluorescence properties relative to the activator fluorescence properties. The parameters that are usually measured are the total fluorescence intensities, the fluorescence life­ times, and the time evolution of the sensitizer and activator fluorescence emissions. As an example of the use of a rate-equation analysis, consider the case of a system with a time-independent energy-transfer rate under pulsed excita­ tion. The solutions to Eqs. (5.5.4) for a constant k are

ns( t) = ns(O) exp [ (Ps + k)t] , na (t) = kns(O) k { exp [-( Ps + k)t] exp( -Pa t) } . Pa Ps

( 5.5.5)

The energy-transfer rate can be determined from these equations by mea­ suring the fluorescence decay rate of the sensitizer ions in a sample with no activators present r;o1 Ps and comparing it with the fluorescence lifetime of the sensitizers in a sample with activator ions present r_;- 1 • According to the first expression of Eq. (5.5.5), the energy-transfer rate is then given by (5.5.6) k = <s ! - - 1 7:.s{}

•

A second method for determining k is to measure the integrated fluorescence intensities of the sensitizers in samples with (Is) and without (!�) activators present. The initial expression in Eq. (5.5.3) can then be used to find the energy-transfer rate (5.5.7) A third method for obtaining the energy-transfer rate is to measure the integrated fluorescence intensities of both the sensitizers and the activators in the sample. Using the two expressions in Eq. (5.5.3), the energy-transfer rate is expressed as

k

/J� la . Is

(5.5.8)

Each of these methods for determining the energy transfer rate has advan­ tages and disadvantages depending on the characteristics of the system being investigated and the experimental capabilities that are available. However, it is generally advantageous to perform measurements on a single sample instead of making different measurements on more than one sample. One important method for obtaining the maximum amount of information from experiments on a single sample is to perform time-resolved spectroscopy measurements in which the relative intensities of the sensitizer and activator ions are monitored as a function of time after pulsed excitation. For the

208

5. Ion Ion Interactions

1ft

FIGURE 5.8. Sensitizer fluorescence decay after pulsed excitation normalized to the fluorescence decay time in the undoped (no activator) sample, r. (A) Samples with no activator ions exhibit exponential decay as predicted by Eq. (5.5.4) with k 0. ( B) Single-step electric dipole-dipole interaction between randomly distributed sen­ sitizer activator pairs resulting in an energy-transfer rate varying as t 1 12 as given in Eq. (5.4.20) . (C) Electric dipole dipole energy transfer in the presence of weak dif­ fusion among the sensitizers as predicted by Eq. (5.4.23). =

example of a time-independent k, Eqs. (5.5.5) can be solved to give p�

k { 1 - exp [(Ps + k - Pa )t] } . P� Pa Ps k _

( 5.5.9)

This experimental technique is very important when the energy-transfer rate is time dependent and the functional dependence on time must be determined. Typical examples of time-dependent energy-transfer data are shown in Figs. 5.8 and 5.9. The first of these figures shows the predicted decay curves for the fluorescence emission of the sensitizer ions as a function of normal­ ized time. With no energy transfer present the decay is exponential. In the presence of single-step energy transfer between randomly distributed pairs of ions the decay is nonexponential. In the presence of sensitizer-activator energy transfer modified by diffusion among the sensitizer ions, the initial decay is nonexponential. However, this evolves into an exponential decay at long times as the diffusion process distributes the excitation energy uniformly so all activators "see" the same excited sensitizer environment. The second of these figures shows how the ratios of the fluorescence intensities of the activator and sensitizer ions evolve with time after pulsed excitation for two typical cases. The important aspect of these examples is the effect of

5.5. Connection with Experiment: Rate Equation Analysis

209

0.8

0.6

0. 4

0. 2

t (arbitrary units)

FIGURE 5.9. Time evolution of the ratios of the fluorescence intensities sensitizer and activator transitions for two typical cases. (A) Single-step electric dipole dipole in­ teraction between sensitizer activator pairs having a fixed distance and with equal rates for transfer and back transfer. ( B) Single-step electric dipole dipole interaction between randomly distributed sensitizer activator pairs with no back transfer.

backtransfer. Without backtransfer the fluorescence intensity ratio increases continuously with time, while the presence of backtransfer causes the fluo­ rescence intensity ratio to reach an equilibrium condition that is time inde­ pendent. The types of curves shown in these figures are found· in sensitized solid-state laser materials discussed in the following chapters. The magnitude of the energy transfer rate can be determined as a function of parameters such as temperature, activator concentration, hydrostatic pressure, or uniaxial stress using one of the procedures described above. Once the properties of the phenomenological energy-transfer rate parameter are known, they can then be compared to the predictions of the various theories discussed in the last section to identify k with one of the Wsa energy­ transfer rates associated with these theories. It should be emphasized that in most cases this procedure is more complicated than indicated by the simple example used here. Effects such as backtransfer of energy from the activator to the sensitizer ion, direct excitation of some of the activator ions, non­ random spatial distributions of ions, and random distributions in transition energies from site to sight can cause the energy-transfer dynamics to vary greatly from the predictions of the simple rate-equation model. 20 However, models have been developed to include these effects and the general analysis for these cases is the same as the one described above even if the equations are more complicated. After the primary parameter, i.e., the energy-transfer rate, is determined, all of the secondary parameters characterizing energy

210

5. Ion Ion Interactions

transfer such as Ro , D, etc., can be obtained from the various equations given in the preceding section. It should be noted that the discussion thus far has focused on nonradia­ tive energy transfer. As mentioned at the beginning of this chapter, energy can also be transferred radiatively. In this case the initially excited ion simply emits a photon and another ion absorbs it. This can result in both energy migration among sensitizers and sensitizer-activator transfer. In this case the emitting ion does not alter its properties because of the presence of the absorbing ion. Thus the energy transfer does not quench the fluorescence lifetime of the sensitizer. In fact, radiative reabsorption among the sensitizer ions can produce an apparent increase in fluorescence lifetime and is a major problem in obtaining an accurate value for this parameter. In the techniques discussed so far, the sensitizer ions are assumed to be spectrally different from the activator ions. Thus the activator ions act as a probe for energy transfer through providing differences in the spectral prop­ erties of the sample's fluorescence emission. This is obviously the case when the sensitizers and activators are two different types of ions. However, in some cases there is interest in understanding the properties of transfer or migration of energy among the sensitizer ions without perturbing the system by introducing activator ions into the sample. The spectral differences of sensitizer ions can be exploited to study this type of energy transfer using high-resolution laser-excitation techniques. If there are distinctly different types of crystal-field sites occupied by the sensitizer ions, one type can be selectively excited while ions in the other type of site play the role of activa­ tors. This experimental technique is generally referred to as site-selection spectroscopy. 2 1 If the differences in transition energies in different crystal­ field sites are too small to distinctly resolve the different spectral lines, it will cause inhomogeneous broadening of the absorption or emission lines of the sensitizers. If the linewidth of the laser used for excitation is significantly smaller than the inhomogeneous linewidth, it can be used to selectively excite only a subset of ions within this line. The initial spectral characteristics then appear as hole burning in the absorption transition and fluorescence line nar­ rowing in the emission transition. The widths and intensities of the spectral hole or the narrowed emission line can be followed in time as. they evolve into the normal inhomogeneously broadened line shapes through energy transfer that randomizes the distribution of excitation energy among the distribution of sites. 22 This is an especially important technique for rare­ earth ions in glass hosts that have significant inhomogeneous broadening of their spectral lines. However, the overlap of transitions between different Stark components can complicate the interpretation of data. A typical example of site-selective, time resolved, fluorescence line­ narrowing spectroscopy is shown in Fig. 5.10. A narrow-line laser-excitation source is used to excite a specific subset of ions within the inhomogeneously broadened spectral profile. At very short times after the laser pulse the fluo­ rescence comes only from the subset of ions initially excited and thus the

5.5. Connection with Experiment: Rate Equation Analysis

21 1

Laser excited subset of ions {A)

( B)

FIGURE 5. 10. Time-dependent fluorescence line narrowing. (A) Inhomogeneously broadened absorption line with one of the homogeneously broadened subsets of ions to be selectively excited by the laser. ( B) Time evolution of the fluorescence emission after site-selective, pulsed excitation.

spectrum appears as a sharp line. As time increases, energy is transferred to ions in. other subsets within the inhomogeneous profile. This causes a rela­ tive decrease in the fluorescence intensity associated with the initially excited ions and a relative increase in the fluorescence from the other ions. This trend continues as time increases until the energy is spread uniformly to all subsets of ions across the inhomogeneous profile and the emission appears as the inhomogeneous band that is observed after normal broad-band exci­ tation. In general it has been found in these types of experiments that the entire spectral profile grows uniformly as the selected subset fluorescence decreases, as opposed to a spreading of the energy from the selected subset outward across the band profile. This implies that the transfer of energy between two ions in this case is relatively independent of small spectral dif­ ferences in transition energies. This probably indicates the dominance of nonresonant interactions involving two phonons, as discussed in Sec. 5.3. The type of behavior exhibited schematically in Fig. 5. 10 is typical of many solid-state laser materials that have strong inhomogeneously broadened lines, as discussed in the following chapters . Measuring spatial energy transfer without any spectral difference is of significant interest in understanding the physics of energy-transfer processes. For materials with high concentrations of sensitizers, one powerful experi­ mental technique that has been developed to directly monitor spatial migra­ tion of energy without spectral energy transfer is laser-induced grating spec­ troscopy. 23 In this technique, coherent laser beams are crossed inside the

212

5. Ion Ion Interactions

sample to form an interference pattern in the shape of a grating. This creates a spatial pattern of excited sensitizer ions with the same shape. As discussed in Sec. 3.4, ions in the excited state have a different polarizability than those in the ground state, and this changes the refractive index according to Eq. (3.4.8). Thus the excited-state population grating looks like a refractive index grating to light beams in the material. This grating pattern decays away by two processes: the fluorescence decay of the excited sensitizers and the migration of excitation energy from the peak to valley region of the grating. A third laser beam acting as a probe beam incident on the grating undergoes Bragg diffraction from the spatial variation in the refractive index. The measured decay of the scattered probe beam (signal beam) has the same time dependence as the decay of the excited-state population grating. Thus by monitoring the grating decay rate as a function of grating spacing, the energy diffusion coefficient can be obtained. The grating decay rate can easily be changed by changing the crossing angle of the two beams writing the grating. The theoretical description of the decay dynamics of laser-induced grat­ ings was developed from the generalized master equation by Kenkre.24 The most general solution gives the normalized signal beam strength as

S(t)

(

e Z t/r Jo(bt) e ca

+a J�

du e �> ( t u) Jo (b( r - u2 ) 1 12 )

a

y,

(5.5 . 1 0)

where r is the fluorescence lifetime, is the exciton scattering rate, lo is the Bessel function of order zero, and the parameter b is

b

=

(5.5. 1 1 )

where V is the nearest-neighbor interaction rate, a is the average distance between active ions, and the grating spacing is given by A A./ [2 sin(B/2)] where A. is the excitation wavelength and B is the crossing angle between the write beams (both in air). This general expression must be used for con­ ditions when the excitons have long mean free paths ( partially coherent migration). This predicts a grating decay that is nonexponential and with temporal oscillations. As conditions change so the migration becomes less coherent, the grating decay becomes exponential and the expression for the signal decay rate simplifies to

S(t)

�

S(O) exp

( { [ (a+�)'+ t� ) r • } ) 21

1 6 V2 'in

(5.5. 1 2)

For completely incoherent hopping motion, the expression can be written in terms of the diffusion coefficient D, (5.5. 13)

References

213

As discussed in the previous section, the exciton dynamics can be charac­ terized in terms of the diffusion coefficient, the mean free path Lm, the dif­ fusion length Ld, the coherence parameter (, and the number of sites visited between scattering events Ns. With the parameters of this theory, these are given by

Ld (

(2Dr) 1 /2 ,

(5.5. 14)

0( '

� Lm Ns a . Equation (5.5. 1 0), (5.5 . 1 2), or (5.5. 1 3) can be used to fit the experimental data to obtain the microscopic parameters V and a , and these can be used in Eq. (5.5. 14) to obtain the macroscopic parameters describing the migration. In addition, the theoretical expressions for ion-ion interaction rates devel­ oped in Sec. 5.2 can be used to interpret the value obtained for V . Energy-transfer properties have been investigated in many types of solid­ state laser materials. Single-step, diffusion-limited migration, trap-limited migration, and partially coherent long-range migration cases have all been observed. The results of using the techniques described here for some impor­ tant materials are discussed in the following chapters. References 1 . K. Huang, Statistical Mechanics, 2nd ed. ( Wiley, New York, 1 987), p. 1 44. 2. J. Rubio, H. Murrieta, R.C. Powell, and W.A. Sibley, Phys. Rev. B 31, 59 ( 1 985). 3. 4. 5. 6. 7.

T. Forster, Naturfurshung 12, 233 ( 1 950) . D.L. Dexter, J. Chern. Phys. 8, 144 ( 1 960) . T. Kushida, J. Phys. Soc. Jpn. 34, 1 3 1 8 ( 1 973). A.L.N. Stevels and J.A.W. van der Does De Bye, J. Lurnin. 18/19, 809 ( 1 979) . S.R. Rotman, A. Eyal, Y. Kalisky, A. Brenier, C. Pedrini, and G. Boulon, Opt.

Mater. 4, 3 1 ( 1 994) . 8. T. Holstein, S.K. Lyo, and R. Orbach, Phys. Rev. Lett. 36, 89 1 ( 1 976); T. Hol­ stein, S.K. Lyo, and R. Orbach, in Laser Spectroscopy of Solids, edited by W.M. Yen and P.M. Selzer ( Springer-Verlzg, New York, 1 98 1 ), Chap. 2. 9 . E.W. Montroll and G.H. Weiss, J. Math. Phys. 2, 1 67 ( 1 965). 10. A. Blumen and G. Zumofen, J. Chern. Phys. 75, 892 ( 1 98 1 ) . 1 1 . S . Chandrasekhar, Rev. Mod. Phys. 15, 1 ( 1 943) .

214 12. 13. 14. 15. 1 6. 17. 18. 19. 20. 21. 22. 23. 24.

5. Ion Ion Interactions Z.G. Soos and R.C. Powell, Phys. Rev. B 6, 4035 ( 1 972). M. Yokota and 0. Tanimoto, J. Phys. Soc. Jpn. 22, 779 ( 1 967) . H.C. Chow and R.C. Powell, Phys. Rev. B 21, 3785 ( 1 980) . A.I. Burshtein, Sov. Phys. JETP 35, 882 ( 1 972) . C.M. Lawson, E.E. Freed, and R.C. Powell, J. Chern. Phys. 76, 4 1 7 1 ( 1 982). R. Kopelman, in Radiationless Processes in Molecules and Condensed Phases, edited by F.K. Fong (Springer-Verlag, Berlin, 1 976), p. 297. R.K. Watts, in Optical Properties of Ions in Solids, edited by B. DiBartolo ( Ple­ num, New York, 1 975), p. 307. V.M. Agronovich and Yu.V. Konobeev, Phys. Status Solidi 27, 435 ( 1 968); Soviet Phys. Solid State 6, 644 ( 1 964); 5, 999 ( 1 963). D.L. Huber, in Laser Spectroscopy of Solids, edited by W.M. Yen and P.M. Selzer (Springer-Verlzg, New York, 1 98 1 ), Chap. 3. R.C. Powell, in Energy Transfer Processes in Condensed Matter, edited by B. DiBartolo ( Plenum, New York, 1 983), p. 655. W.M. Yen, in Spectroscopy of Solids Containing Rare Earth Ions, edited by A.A. Kaplyanskii and R.M. Macfarlane ( Elsevier, Amsterdam, 1 987), Chap. 4. J.R. Salcedo, A.E. Siegman, D.D. Dlott, and M.D. Payer, Phys. Rev. Lett. 41, 1 3 1 ( 1 978). V.M. Kenkre and D. Schmid, Phys. Rev. B 31, 2430, ( 1 985); V.M. Kenkre, Phys. Rev. B 18, 4064 ( 1 978) .

6

Al20 3 : Cr 3 + L aser Crystals

Trivalent chromium ions have played a central role in the development of solid-state lasers. Cr3 + was the active ion in the first laser (ruby) and has been the most successful transition-metal ion used for laser applications in other host crystals. With the proper choice of host, chromium lasers can operate either pulsed or continuous wave with either sharp line or broad­ band tunable emission between about 6900 and 12 500 A. This versatility provides the ability to study the effects of the host environment on the spec­ troscopic properties of the active ion and determine how these alter the las­ ing characteristics of the material. In the following sections the properties of the electronic energy levels and transitions of Cr3 + ions are discussed, and the effects of changes in the local crystal field, electron-phonon interactions, and ion-ion interactions are de­ scribed. The spectroscopic properties of ruby are presented as an example of a strong field Cr3 + laser material. Since ruby is a three-level laser material, it is not extensively used in practical applications. However, for historical rea­ sons, the spectroscopic and laser properties of ruby have been extensively characerized, and the results are used as a basis for comparison and under­ standing other solid-state laser materials. The properties of weak-field Cr3 + materials are discussed in Chap. 7. The fundamental concepts used in this discussion were outlined in Chaps. 2-5. 6. 1

Energy Levels of Cr3 +

The electronic configuration for the 24 electrons of a chromium atom is l s 2 2s2 2p6 3s 2 3p6 3d 5 4s. The first five sets of orbitals make up the filled core, while the optically active electrons are in the half-filled 3d orbitals. The lat­ ter are shielded by the electron in the outermost 4s orbital. The trivalent chromium ion has given up three electrons from the outer two sets of orbi­ tals, leaving three unshielded electrons in the 3d orbitals. These are the electrons that play the dominant role in determining the optical properties of the ion. The Russell-Saunders coupling approach discussed in Sec. 2. 1 215

214 12. 13. 14. 1 5. 16. 17. 18. 19. 20. 21. 22. 23. 24.

5. Ion Ion Interactions Z.G. Soos and R.C. Powell, Phys. Rev. B 6, 4035 ( 1 972). M. Yokota and 0. Tanimoto, J. Phys. Soc. Jpn. 22, 779 ( 1 967) . H.C. Chow and R.C. Powell, Phys. Rev. B 21, 3785 ( 1 980). A.I. Burshtein, Sov. Phys. JETP 35, 882 ( 1 972) . C.M. Lawson, E.E. Freed, and R.C. Powell, J. Chern. Phys. 76, 4 1 7 1 ( 1 982) . R. Kopelman, in Radiationless Processes in Molecules and Condensed Phases, edited by F.K. Fong (Springer-Verlag, Berlin, 1 976), p. 297. R.K. Watts, in Optical Properties of Ions in Solids, edited by B. DiBartolo ( Ple­ num, New York, 1975), p. 307. V.M. Agronovich and Yu.V. Konobeev, Phys. Status Solidi 27, 435 ( 1 968); Soviet Phys. Solid State 6, 644 ( 1 964); 5, 999 ( 1 963). D.L. Huber, in Laser Spectroscopy of Solids, edited by W.M. Yen and P.M. Selzer (Springer-Verlzg, New York, 1 98 1 ), Chap. 3. R.C. Powell, in Energy Transfer Processes in Condensed Matter, edited by B. DiBartolo ( Plenum, New York, 1 983), p. 655. W.M. Yen, in Spectroscopy of Solids Containing Rare Earth Ions, edited by A.A. Kaplyanskii and R.M. Macfarlane ( Elsevier, Amsterdam, 1 987), Chap. 4. J.R. Salcedo, A.E. Siegman, D.D. Dlott, and M.D. Fayer, Phys. Rev. Lett. 41, 1 3 1 ( 1 978). V.M. Kenkre and D. Schmid, Phys. Rev. B 31, 2430, ( 1 985); V.M. Kenkre, Phys. Rev. B 18, 4064 ( 1 978).

6

Al20 3 : Cr 3 + Laser Crystals

Trivalent chromium ions have played a central role in the development of solid-state lasers. Cr3 + was the active ion in the first laser (ruby) and has been the most successful transition-metal ion used for laser applications in other host crystals. With the proper choice of host, chromium lasers can operate either pulsed or continuous wave with either sharp line or broad­ band tunable emission between about 6900 and 12 500 A. This versatility provides the ability to study the effects of the host environment on the spec­ troscopic properties of the active ion and determine how these alter the las­ ing characteristics of the material. In the following sections the properties of the electronic energy levels and transitions of Cr3+ ions are discussed, and the effects of changes in the local crystal field, electron-phonon interactions, and ion-ion interactions are de­ scribed. The spectroscopic properties of ruby are presented as an example of a strong field Cr3 + laser material. Since ruby is a three-level laser material, it is not extensively used in practical applications. However, for historical rea­ sons, the spectroscopic and laser properties of ruby have been extensively characerized, and the results are used as a basis for comparison and under­ standing other solid-state laser materials. The properties of weak-field Cr3 + materials are discussed in Chap. 7. The fundamental concepts used in this discussion were outlined in Chaps. 2-5. 6. 1

Energy Levels of Cr3 +

The electronic configuration for the 24 electrons of a chromium atom is l s2 2s2 2p6 3s2 3p6 3d 5 4s. The first five sets of orbitals make up the filled core, while the optically active electrons are in the half-filled 3d orbitals. The lat­ ter are shielded by the electron in the outermost 4s orbital. The trivalent chromium ion has given up three electrons from the outer two sets of orbi­ tals, leaving three unshielded electrons in the 3d orbitals. These are the electrons that play the dominant role in determining the optical properties of the ion. The Russell-Saunders coupling approach discussed in Sec. 2. 1 215

216

6. A h0 3 : Cr3 + Laser Crystals

can be used to determine the electronic terms and multiplets of the free ion with three electrons, each having quantum numbers n = 3 and l = 2. The types of spectroscopic terms available to the three optically active electrons can be determined by making use of the Pauli exclusion principle, which states that no two of the electrons can have the same values for the set of four quantum numbers n, l, mt , ms . For Cr3 +, all of the optically active electrons have the same values of n and /, and thus there are only certain combinations of the values of mt and ms that are allowed. 1 In general, the values of the total angular momentum quantum number of the multielectron terms can run from L = 0 to L = 'I.; l; = 6 in integral steps. However, L = 6 is not allowed by the Pauli principle since this would re­ quire that at least two of the electrons have a multiplet with identical sets of quantum numbers. Thus the largest range for the ML quantum number is between +5 and 5 in integral steps. Similarly, the total spin quantum number of the multielectron term can run from S = ! to S = 'L;si in integral steps. For Cr3 + this gives two allowed spin states, S = � or ! · Since the sum of the mt quantum numbers for the three electrons must add to one of the ML quantum numbers, and the sum of the ms quantum numbers must add to one of the Ms quantum numbers, a table of single­ electron states that contribute to the multiplets of the multielectron terms can be constructed. The results are shown in Table 6. 1 . The single-electron TABLE 6. 1 . Single-electron states for a 3d 3 . electron configuration. a Ms 3

2

5 4 3

2

0

•

I

2

a(2+ , 2- , I + ) b(2+ , 2- , o+)c(2+ , I + , J - ) e(2+ , 2- , I + )J(2+ , I + , o- ) g(2+ , I - , o+ )h(2- , I + , o+) j(2+ , 2+ , 2- )k(2+ , I + , I - ) /(2+ , 1 - , I + )m(2- , I + , J + ) n(2+ , o+ , O;- )o(l + , I - , o+ ) ( I + , o+ , o- ) ( I + , I - , I + ) ( 1 + , 2+ , 2- ) (2+ , I + , o- ) (2+ , I - , o+ ) (2- , I + , o+ ) ( J + , 2- , 2+ ) ( 1 - , 2+ , 2+) ( J + , I + , o- ) ( 1 + , 1 - , o+ ) ( 1 - , J + , o+ ) (2+ , 2+ , o- ) (2+ , 2- , o+ ) (2- , 2+ , o+) (2+ , J + , I - ) ( I + , I-, 2+)

Implied terms

4p 2p

Data for table: n; = 3 , I; = 2, s; = !· Single electron states: (m�'' , m�'' , m7,'' ) . The table is

symetric for ( ML ,

Ms ) values.

6. 1 . Energy Levels of Cr3 +

217

states are represented by (m11 , m1� , m 1� ), where the + or superscript des­ ignates spin up ( s = ± !) or spin down ( s = !), respectively. All possible allowed combinations of single-electron quantum numbers are organized into single-electron states and placed in the appropriate ML row and Ms column of a multielectron term. Since the largest values of ML and Ms equal L and S, respectively, for a specific term, Table 6. 1 can be used to determine the spectroscopic terms of a 3d3 ion. The highest ML row in the table has single-electron states, occu­ pying only the cells in the Ms = ± ! columns. These must belong to a 2 H term since ML = L = 5 and Ms = S = !· One of the single-electron states in each of the other cells in the Ms = ± ! columns also will belong to this term. Subtracting one single-electron state from each cell in the center two columns of the table leaves the occupied cells with the largest values for ML those with Ms = ± ! · These must belong to a 2 G term, and again one state in each cell in the two center columns will belong to the same term. Elimi­ nating these states from the table leaves the occupied cells with the largest value for the spatial orientation quantum numbers those with ML = 3 and Ms = 1· This indicates the presence of a 4 F term. Single-electron states as­ sociated with this quartet term will be present in each of the cells with smaller quantum numbers, so one state must be subtracted from the cells in each of the four columns in the table. Two occupied cells still remain in the ML = 3 row of the table, those with Ms = ± !· These are associated with a 2 F term along with a single-electron state in each of the other cells in the two center columns. Subtracting the states belonging to this doublet from the cells in the two center columns leaves two states left in each of the two cells with ML = 2, Ms = ± !· These must be associated with two different 2 D terms, which allows the subtraction of two single-electron states from each remaining cell in the Ms = ± ! columns. The ML = I cell with Ms = 1 is now the occupied cell with the highest quantum numbers, which indicates the presence of a 4 P term. Subtracting one single-electron state from each cell leaves one state in each cell with ML = 0 , ± 1 and Ms = ± !· These are associated with a 2 P term. Thus, in this way all of the single-electron states can be accounted for in relation to the multielectron terms. The next problem is to order the eight terms with respect to their energies. The ground state can be determined by Hund's rules. These require that the ground-state term have the greatest multiplicity possible and greatest orbital angular momentum value consistent with this multiplicity. Therefore, the ground state for the 3d3 configuration of Cr3 + is the 4 F term. In Sec. 2. 1 it was shown that for ions with Russell-Saunders coupling the sum of the roots in the secular determinant for the energy levels is equal to the sum of the diagonal elements (diagonal sum rule), and there are no matrix elements connecting states of different ML and Ms. Using this in­ formation along with the fact that in Russell-Saunders coupling the energy of the terms is independent of mt and ms , sets of linear equations in terms of

218

6 . Ah0 3 : Cr3 + Laser Crystals

single-electron functions can be written for the energy of levels belonging to each cell in Table 6. 1 :

EeH) E(2+, 2 , 1+), EeG) + EeH) E(2+, 2 , o+) + E(2+, 1+, 1 ) , E(4F) E(2+, 1 +, o+), E(4F) + EeH) + EeG) + EeF) E(2+ , r , 1 +) + £(2 +, 1 , o+) + £(2+ , 1 +, o ) + E(2 , 1+, o+) , EeH) + EeG) + E(4F) + EeF) + EeD) + E( 2D' ) E(2+, o+ , o ) + E(2+, 1 , - 1 +) + E(2+, -2+, 2 ) + E(2+ , 1 + , - 1 ) + E(2 , 1+, - 1+) + £( 1+ , 1 , o+), E(4F) + E(4 P) E(2+, - 1 +, o+) + E( 1 + , 2+ , -2+), EeH) + EeG) + E(4F) + EeF) + EeD) + EeD' ) + E(4 P) + EeP) E(1 + , o+ , o ) + £( 1+, 2+ , -2 ) + E(2+ , - 1 , o+) + £( 1 + , 2 , 2+) + £(1 +, 1 , 1+) + £(2+, - 1 +, o ) + E(2 , - 1 +, o+) + £( 1 , 2+, -2+) . Only these eight cells need be considered since all of the terms can be de­ termined from these cells. The above equations can be solved for the term energies in terms of the energies of the single-electron states:

EeH ) EeG) E(4F) EeF)

E(2+ , 2 , 1+), E(2+ , 2 , o+) + E(2+, 1 + , 1 ) - E(2+ , 2 , 1+), E(2+ , 1+ , o+), E(2+, r , 1+) + E(2+, 1 , o+) + E(2+ , 1+ , o ) + E(2 , 1+, o+) E(2+ , 1 + , o+) - E(2+ , 2 , o+) - E(2+ , 1+ , 1 ) , EeD) + EeD' ) E(2+, o+ , o ) + E(2+ , 1 , - 1 +) + E(2+ , -2+ , T ) + E(2+ , 1+, - 1 ) + E(2 , 1+ , - 1+) + £( 1+, 1 , o+) - E(2+, r , - 1 +o) - E(2+ , 1 , o+) - E(2+, 1 +, o ) - E(2 , 1 +, o+) , E(4P) E(2+ , 1+ , o+) + E(1+, 2+ , 2+) - E(2+ , 1 +, o+) ,

6. 1 . Energy Levels of Cr3 +

E( 2P)

219

£( 1 + , o+, o ) + £( 1 +, 2+ , -2 ) + E(2+, - 1 , o+) + £(1 +, 2 , -2+) + £( 1+, 1 , - 1+) + E(2+, - 1 + , o ) + E(2 , - 1+, o+) + £(1 , 2+ , -2+) - E(2+, o+, o-) - E(2+, 1 - , - 1+) - E(2+, -2+ , T ) E(2+, 1 +, - 1 ) - E(2 , 1+, - 1 +) - £( 1+, 1 - , o+) .

The energies of the single-electron states can be written in terms of the Coulomb and exchange integrals as defined in Eq. (2. 1 .32). The term en­ ergies are then combinations of these integrals:

E(2H) EeG) E(4F) EeF)

1 (2, 2) + 21 (2, 1 ) - K(2, 1 ) , 21 (2, 0) - K(2, 0) + 1 (1 , 1 ) , 1 (2, 1 ) + 1(2, 0) + 1 ( 1 , 0 ) - K(2, 1 ) - K(2, 0 ) - K( 1 , 0) , 21 ( 1 , 0) + 21 (2, - 1 ) - 1 ( 1 , 1 ) - K(2, 1 ) - K(2, 0) - 2K ( 1 , 0) - K(2, - 1 ) , E(4P) 1 (2, -2) + 1 ( -2, 1 ) + 1(2, 1 ) + 1( - 1 , 0) - 1 ( 1 , 0) - K(2, -2) - K( -2, 1 ) - K(2, - 1 ) - K( - 1 , 0) + K( 1 , 0) , EeD) av 1( 0, O) + 1 (2, - 1 ) + 31( 1 , - 1 ) + 21(2, -2) + 1 ( 1 , 1 ) - 1 (2, O) - 1 ( 1 , 0) - 1 (2, 2) - 21(2, 1 ) - 3K(2, - 1 ) - 3K( 1 , - 1 ) - K(2, -2) + 2K(2, 0) + 2K( 1 , 0) + K(2, 1 ) + K (2, - 1 ) , EeP) 1 (0, 0) + 21( - 1 , 0) + 21( -2, 1 ) - 21 (2, 2) + 1 (2, 0) - 21(2, 1 ) - 1(2, - 1 ) - 31 ( 1 , - 1 ) - 1 ( 1 , 1 ) - 2K (1 , 0) - 2K(2, 0) - 2K( - 1 , 0 ) - K(2, -2) - 2K( -2, 1 ) + K(2, 1 ) + K(2, 1 ) + 3 K( 1 , 1 ) . Note that only the average energy for the two 2D terms is obtained in this way. Using definitions given in Eqs. (2.33)-(2.41 ) and Tables 2.2. and 2.3,

the term energies can be rewritten in terms of either the Slater-Condon pa­ rameters or the Racah parameters. The results are 1 •2

EeH) 3Fo - 6F2 - 12F4 3A - 6B + 3C, EeG) 3Fo - 1 1F2 + l 3F4 3A - 1 1B + 3C, E(4F) 3F0 - 1 5F2 - 72F4 3A - 1 5B , EeF) 3Fo + 9F2 - 87F4 3A - 1 5B ,

220

6. Aiz0 3 : Cr3 + Laser Crystals

E(4P) 3F0 - 147F4 3A , EeD) av 3Fo + SF2 + 3F4 3A + 5B + 5C , EeP) 3Fo - 6F2 - 12F4 3A - 6B + 3C. Finally, the energies of the individual 2D terms must be determined. This

can only be done by determining specific linear combinations of the single­ electron states that make up the ML 2, Ms ! multiplets of these two terms along with solving the secular determinant for their eigenvalues. To do this we follow the treatment developed by Condon and Shortley, 2 which makes use of the fact that the multiplets associated with different terms having the same values of ML and Ms are orthogonal. Starting with the 2H (5, !) a multiplet in Table 6. 1, the L - , 1 - , s , and s lowering oper­ ators can be applied to determine the linear combinations of single-electron states that make up the other 2H (ML, Ms) multiplets. From Eqs. (2. 1 . 16) and (2. 1 . 17), L [2H (5, !)J 1 [a] ,

v'102H(4 , !)

so

- v'4(2+, 1 , 1 +) + v'6(2+ , 2 , o+) - v'4c + v'6b,

2H (4, �)

10 1 12 [ v/6b - 2c] .

The lowering operators applied to the other single-electron states give:

r [b] - 2h + 2g + v'6e , 1 [c] -v'6g + v'6J, 1 [e] -2m + 21 + 2j, r [!] v'6n + v'6k , r [g] 2o - v'6n + v'61 , r [h] -2o + v'6m, s [d] h + g +f, s [i] m + l + k. These expressions can be used when the lowering operator L - is applied successively to the 2H (4, !) multiplet given above to obtain expressions for the remaining 2H (ML, Ms) multiplets of interest:

2H (3, !) 2H (2 -I ) '2

1 (v'6e 2f 4g 2h) , - + 1

v'30

·

6. 1 . Energy Levels of Cr3 +

221

According to Table 6. 1 , the 2 G ( 4, !) multiplet is some linear combination of single-electron states b and c . This combination is arbitrary except that it must be orthogonal to the linear combination associated with the 2H (4, !) multiplet. One satisfactory choice is 2G (4, !) = [2b + v'6c] /.Jf0, which has been normalized. Applying the lowering operator L twice gives the other two relevant multiplets for this term, 2G ( 3, 2I )

=

1

+ 3f - g - 2h) ,

Next consider the 4 F term. According to Table 6. 1 , 4F ( 3, !) = d . Apply­ ing the s - lowering operator gives 2 F ( 3, !) = 3 1 12 (! + g + h) . Then apply­ ing the L - lowering operator gives

Finally, the 2 F term must be considered. Appropriate combinations of single-electron states for the other three terms having multiplets with ML = 3 and Ms = ! have been derived above. The linear combination of e , f , g, and h associated with 2 F ( 3, !) is arbitrary except that it must be orthogonal to the similar multiplets of the other three terms. One normal­ ized combination satisfying this is 2 F ( 3, !) = ( v'6e + f + g - 2h) / ,;ri. Ap­ plying the orbital angular momentum lowering operator gives 2F (2, !) = ( -2j + k -

l + v'6o)j,;ri.

Linear combinations of single-electron states for multiplets of four of the six terms having ML 2 , Ms = ! have now been derived. The orthogon­ ality condition then allows the determination of the appropriate linear com­ binations of the j, k , l , m, n, and o states associated with the multiplets of the remaining two terms, 2 D and 2D'. One normalized combination that fits the orthogonality requirements is 2 D ( 2, !) = (-j - k + l + n)/2. The second combination must be orthogonal to this multiplet as well as the oth­ erfour. Constructing 2D' ( 2, !) = Aj + Bk + Cl + D m + En + F o, where A , . , F are arbitrary coefficients, using en') (8L) 0 for all other terms and eD ' ) eD' ) = 1, gives six equation with six unknowns. Solving these gives one satisfactory combination 2D' ( 2, !) = ( - 5j + 3k + l - 4m 3n - 2v'6o)jv'84. Now that expressions have been derived for the two multiplets of the 2 D and 2D' terms in terms of linear combinations of single-electron states, their energies can be determined as before using the Coulomb and exchange integrals and then writing the results in terms of Slater-Condon parameters or Racah parameters. Using the Slater-Condon parameters =

.

.

=

222

6. Ah0 3 : Cr3 + Laser Crystals

gtves,

EeD) = 3Fo + ?F2 + 63F4 , EeD') 3Fo + 3F2 - S?F4 . The secular determinant for the Coulomb repulsion among the electrons can now be formed. Since the matrix for the Coulomb interaction is diagonal with respect to (L, S, ML, Ms ) and independent of ML and Ms, only a 2 x 2 matrix made up of the 2D(2, !) and 2D'(2, !) multiplets is required,

I

eD i g I 2D) eD' I g I 2D)

Expressing each matrix element in terms of Slater parameters gives

I

3J2f(F2 - SF4 ) (3Fo + 7F2 + 63F4 ) - E 3v'2f(F2 - SF4 ) (3Fo + 3F2 - 57F4 ) - E

Solving this determinant for the energy eigenvalues gives

EeD)

3Fo + SF2 + 3F4 = 3A + 5B + 5C -

I

O.

l 6SOF2 F4 + 8325Ff + SBC + 4C2 ,

l 6SOF2 F4 + 8325Ff + SBC + 4C2 • Expressions have now been derived for all eight terms of the 3d3 electron EeD') = 3Fo + SF2 + 3F4 + 3A + 5B + 5C +

configuration of a free ion in terms of the Racah parameters. It is difficult to determine these parameters from first-principle calculations. Thus they are generally determined empirically by fitting the theoretical expressions to measured experimental data. Since all energies are measured relative to the ground state, the 3A energy contribution can be subtracted from each term to give

12

8

�

)'D

6. 1 . Energy Levels of Cr3 +

223

cr3 + (B/C=9 1114 1 33=0. 222 )

4 2n -2 2 2G p, H - 4p

0 -4

0

0. 1

0.2

BIC

0.3

0.4

0.5

FIGURE 6. 1 . Free-ion terms of a 3d3 ion and energy levels for Cr3 + (reprinted from Ref. I with the permission of Cambridge University Press) .

( c) + 8 c + 4 , ( c) + 8 c + 4 . 1 93 1 93

EeD') 5 B + 5 + c = c

--

B

B

2

2

B

B

(6 . 1 . 1 )

These expressions are plotted1 in units of E/C vs B/C as shown in Fig. 6. 1 . By measuring the optical absorption and emission spectra of a specific type of 3d3 ion such as Cr3 +, the energy-level splittings shown on the right side of the figure have been found and identified with the designated terms. This fixes the value of B/C for Cr3 + at 0.222 as shown by the broken vertical line. Comparing the measured transition energies with the E/ C scale gives C 41 33 and thus B = 9 1 8 for this ion. Table 6.2 lists values for the Racah B and C parameters for the other 3dn transition-metal ions. 1 The numbers quoted for the exact values of the Racah parameters vary in the literature due to variations in the accuracy of both theoretical calculations and ex­ perimental measurements. The general trends in relative magnitudes of B and C in Table 6.2 are consistent with other reported values. These param­ eters change when the ion is placed in a crystalline environment as discussed in the following section. It is the values of B and C in a specific host envi­ ronment that are importance to solid-state laser applications. The free-ion terms are further split by spin-orbit interaction. However, for 3d3 ions this is generally a smaller effect than the crystal-field splitting,

6. Ah0 3 : Cr3 + Laser Crystals

224

TABLE 6.2. Racah parameters for 3dn ions. Ion Parameter

B (crn- 1 ) 2+ 3+ C ( crn 1 ) 2+ 3+ C/B 2+ 3+

Ti

v

Cr

Mn

Fe

Co

Ni

695

755 862

810 918

860 965

917 1015

971 1065

1030 1 1 15

29 10

3257 381 5

3565 4133

3850 4450

4040 4800

4497 5120

4850 5450

4. 19

4.31 4.43

4.40 4.50

4.78 4.61

4.41 4.73

4.63 4.8 1

4.71 4.89

and thus either the free-ion terms derived above or the strong-field d orbitals form the starting point of discussing the energy levels of Cr3 + ions in a crystal field. 6.2

Crystal-Field Splitting

The next step in understanding the optical spectral properties of Cr3 + ions in solids is to describe the crystal-field splitting. As discussed in Sec. 2.3, the effects of an octahedral crystal field are generally considered as a first-order perturbation with contributions from lower symmetry crystal fields treated as higher-order perturbations. It is useful to correlate the results of the weak-field and strong-field treatments so information can be obtained on the variation of the energy levels as a function of crystal-field strength. The re­ sults of doing this type of analysis for Cr3 + ions are shown here. Using Eq. (2.3.6), the characters of the symmetry operators for the octa­ hedral point group can be determined for the reducible representations of each of the eight free-ion terms derived in the last section. Then using the character table for the 0 group given in Sec. 2.2 and the reduction formula given by Eq. (2.2.8), the following table shows how the irreducible repre­ sentations of the free ion terms reduce in terms of the irreducible repre­ sentations of the 0 group. 0

2y 2G 4p 2p 2D 2D ' 4p 2p

E

3C2

8 C3

11 9 7 7 5 5 3 3

1 1 1 1 1

1 0 1 1 1 1 0 0

1 1

6C4

1 1 1 1 1

6q 1 1 1 1 1 1 1

2E + 22TI + 2T2 2A I + 2E + 2TI + 2 T2 4A 2 + 4T1 + 4T2 2A 2 + 2TI + 2T2 2£ + 2T2 2£ + 2T2 4T1 2T1

6.2. Crystal-Field Splitting

225

As an example of these calculations, consider the 4 F term. The value of the orbital angular momentum quantum number is L = 3 so the characters of the irreducible representation for this term are

x (E) = 2 X 3 + 1 = 7 , . 7n sm x ( C2) � - 1 , sm 2 14n .sm x ( C3 ) = . n = 1 , sm 3 x ( C4 )

. 7n sm-

n

. sm 4

-1.

Using these characters along with those of the five irreducible representa­ tions of the 0 group from Sec. 2.2 gives

n(A 1 ) -b (7 X 1 X 1 + 1 X 1 X 8 - 1 X 1 X 3 - 1 X 1 X 6 - 1 X 1 X 6) 0, n(A2) -b (7 X 1 X 1 + 1 X 1 X 8 - 1 X 1 X 3 - 1 X - 1 X 6 - 1 X 1 X 6) 1 , n(E) -b (7 X 2 X 1 + 1 X - 1 X 8 - 1 X 2 X 3 - 1 X 0 X 6 - 1 X 0 X 6) 0, n( T1 ) = -b (7 X 3 X 1 + 1 X 0 X 8 - 1 X - 1 X 3 - 1 X - 1 X 6 - 1 X 1 X 6) 1 , n( T2) -b (7 X 3 X 1 + 1 X 0 X 8 - 1 X - 1 X 3 - 1 X 1 X 6 - 1 X 1 X 6) 1 . This shows that the 4 F free-ion term splits into one orbital singlet crystal field term (A 2 ) and two orbital triplets (TI and T2 ) all with the same spin

multiplicity of the free-ion term. Similar calculations can be carried out for the other free-ion terms with the results shown in the table above. Thus the eight free-ion terms of a Cr3 + ion split into 20 crystal-field terms in an octa­ hedral crystal field. Next consider the strong-field approach to the crystal-field splitting. As seen from Fig. 2.6, four electron configurations must be considered, t�9 , t�9 e9 , t29 e�, and e�. In Sec. 2.3 the reduction of the direct-product repre­ sentations of two d electrons in an octahedral crystal field was considered. Now the effects of a third electron must be determined. First consider the t�9 configuration. In Sec. 2.3 it was shown that t29 x t29 A 1 9 + E9 + T1 9 + T29, and it is now required that the reduction of the direct-product representation formed by t29 and each of these four irreducible representa­ tions be found. Using the normal procedure,

t2g X A l g = T2g, t2g X Eg Tl g + T2g,

226

6. Ah0 3 : Cr3 + Laser Crystals

t2g X Tl g A 2g + Eg + Tl g + T2g, t2g X T2g = A ,g + Eg + T,g + T2g, so the lowest-energy configuration results in the crystal-field states t�9 = A,9 + A 29 + 2E9 + 3T, 9 + 4T2g · Similarly, the configuration t�9 e9 t2g x t29 x e9 A,9 + A 29 + 2E9 + 2T,9 + 2T29, the configuration t29e� t29 x e9 x e9 2T, 9 + 2T29, and e� = e9 x e9 x e9 A ,9 + A 29 + 3E9. Which of these crystal-field states are allowed as doublets or quartets can be determined using the statistical expression for total degeneracy given by Eq. (2.3.12). For the t�9 configuration the total degeneracy is given by 6 C3 6!/ [ 3!3! ] 20. This can be obtained from the available crystal field states with the terms 4A 29 + 2E9 + 2 T, 9 + 2T29. For the t�9 e9 configura­ tion, 6 C2 x 4 C1 6!/[4!2!] x 4!/ [3! 1 !] 60 for the total degeneracy, which can be obtained from the available crystal-field states with the terms 2A 1 9 + 2A 29 + 22 £9 + 22 T,9 + 22 T29 + 4 T,9 + 4 T29. For t29e� , 6 c, x 4 C2 36, which can be obtained by 22 T1 9 + 4 T1 9 + 22 T29. For e� , 4C3 4, which can be obtained by 2 E9. Note that the 20 crystal-field terms obtained by this strong-field analysis are the same as those obtained above using the weak­ field analysis. For each individual strong-field configuration there is rp.ore than one pos­ sible set of crystal-field terms that fulfill the total degeneracy requirement. However, there is only one complete set of these terms for all the config­ urations that is consistent with the terms obtained from the weak-field anal­ ysis. It is generally necessary to make several guesses at the right combina­ tion for each configuration until the correct set that matches the weak-field multiplets is found. It is possible to derive the correct combination of terms through a detailed consideration of the wave functions. This was described in Sec. 2.3 for a t�9 • The next step in this procedure is to couple the third electron into this configuration. Using group-theory procedures, the direct product of the single-electron t29 orbital representation with the representations of the four terms of the t�9 configuration can be formed and the result reduced in terms of irreducible representations of Oh . The possible terms with respect to symmetry considerations found in this way are 4g

1A 1 1£1 1 Tz 3 T1

__

__ __

__

t�g 2Tz 2T1 , 2 Tz 2A 1 , 2 E, 2 T1 , 2 Tz zA z , z E, z T1 , z Tz , 4A 2 , 4 E, 4 T1 , 4 Tz

Each of these must be checked to see if it is one of the terms allowed by

6.2. Crystal-Field Splitting

227

the Pauli exclusion principle. The wave function for the t�9 crystal-field terms can be written as a linear combination of products of the wave func­ tions of the G9 wave functions found in Sec. 2.3 and single-electron t29 functions, 'I' ( G9 [Soro] tz9 [Sr] ) =

L

Mom3 YoY3

I/I(G9 So roMo y ) tp (tz9m3 y3 ) ( SoMo ! m 3 I SM) (roy0 Tz y3 l ry) ,

where is the wave function for a state of the G9 configuration and

tp ( tz9m 3 Y3 ) is the wave function for the third tz9 electron. The last two factors in this

expression are the Clebsch-Gordan coefficients for the spin angular mo­ mentum coupling and the orbital angular momentum coupling in the cubic basis. These were both tabulated in Chap. 2. 'I' is required to be anti­ symmetric for the exchange of all three electrons. This can be accomplished by forming Slater determinants in the expansion 'I' ( 49 [So ro] tz9 [Sr] ) =

1

L { I 1Pt ( 1 ) tpz (2) I IP3 (3) - 1 1Pt ( 1 ) tpz (3) I IP3 (2)

Mom3 YoYJ

+ l iP I (2) tpz (3) I IP3 ( 1 ) } ( SoMo ! m 3 l SM ) (r o Yo Tz Y3 j ry) , where the subscripts on tp designate the single-electron orbital and the num­

ber in parentheses designates the electron. To simplify the notation, it is common to use the expressions for the two-electron wave functions given in Table 2.7 where the states in the terms are designated A z( e) , E(u, v ) , T1 ( fJ, y ), and Tz ( c;, 1J, () with a minus superscript designating spin down and no superscript for spin up. First consider as an example the 4A 29 term with 3 Tt g parentage. The rele­ vant Clebsch-Gordan coefficients for coupling the angular momentum are found from Table 2. 1 to be

rx,

( Mo ! m3 1 H) J(Mo 1 )J(md). The Clebsch-Gordan coefficients for the orbital angular momentum in the cubic basis are found from the section of Table 2.6 for the A z representation in the reduction of the T1 x T2 product representation. The results can be expressed in terms of the notation for the two-electron wave functions in

228

6. Ah0 3 : Cr3 + Laser Crystals

Table 2.7. The final wave function for this three-electron term is

The spins of all three electrons are aligned to give an M quartet term. This can be rearranged to give

� multiplet of a

which shows that 4 A 1g is an allowed crystal-field term for the t�g electron configuration in an octahedral environment. Next consider the 4 Eg term with 3 T1 g parentage. Using the same proce­ dure described above, the wave function is found to be

'P (�i Tig] tig 4EgM �) =

1

v'2 0,

[ 1 1!( 1 (1, 2) � (3 ) - I C� 1 (1, 2) '1 (3)] I 11C� I +

1

I I v'2 C�11

where a cyclic permutation of orbitals leaves the sign of the determinant unchanged. Thus, 4Eg is not one of the allowed terms. Now try the 1 Eg term with 3 T1 g parentage. Using the tables for spin and orbital angular momentum coupling and the expression given above for the expansion of the wave function in terms of products of single-electron orbi­ tals, the multielectron term wave function is

'P (t�g [3 Ti g] tig 1EgM = !) =

-1

(3 ) - l 'lC 1 (1, 2) � (3 ) [i 2y'3 1 '7 C ( I , 2) � + I C� I o , 2J '7 (3 ) + I CC l o , 2J '7 (3 )J 1 + v'3 [ 1 '7C I ( I , 2J C (3 ) + I C� I o , 2J 'l (3 ) J

=

1

1

( I- I v'2 1 �'7 C v'2 C11W ·

Thus 1Eg i s one of the allowed terms. As a final example, try the 2 T1 g term with 1 Eg parentage. The relevant Clebsch-Gordan coefficients are found from Table 2. 1 to be given by

Using these factors along with the orbital angular momentum coupling co-

6.2. Crystal-Field Splitting TABLE 6.3. Wave functions for the terms of the I 2Ad e2 ) = l ¢11(1

I 2E ! u)

t�g configuration lt�/SrMy) (after Ref. 4).

¢ii( I

I �11( I )

I 2 T1 !P) =

I 2 T1 ! tX) = 1

v'2

-

( 1 ¢111JI + WW

I 2 Td 11) =

1

v'2

229

(111((1 + 111¢�1)

I 2 E ! v) =

I 2 T1 !J') = I 2 Td 0 =

1

v'2

(1(¢�1 + 1(11iil)

efficients found from Table 2.6, the wave function is

'¥ (�iE9] ti9 2 T1 9M !) - v'3 1 J6 [ 1 �� 1 (1, 2) � (3) 1 '7'7 1 (1, 2) � (3) + 2 1 (( 1 (1, 2) � (3) 1 '7'1 1 (1, 2J � (3) I CC I ( I , 2J � (3) + 2 I �� 1 (1, 2J � (3) I CC I ( I , 2J � (3) 1 1

- I �C I ( I , 2J � (3) + 2 1 ,, 1 (1, 2) � (3)] - 2 -/2 [ 1 �� 1 (1, 2) � (3) 1 '7'7 1 (1, 2) � (3)

+ 1 '7'1 1 (1, 2J � (3) I CC I ( I , 2J � (3) + I CC I ( I , 2J � (3) = 0. Thus this 2 T1 9 is not one of the allowed terms.

� �� 1 (1, 2J � (3)]

This type of procedure can be followed for each of the symmetry-allowed terms and the results are those listed in Table 6.3 for the t�9 configuration. Similar procedures can be followed for the other configurations for 3d elec­ trons. Since this is a time-consuming procedure, for configurations with more than two electrons it generally is more efficient just to guess at appro­ priate combinations using the degeneracy calculation until the one con­ sistent with the weak-field analysis is found. Figure 6.2 shows a correlation diagram for a d3 ion in an octahedral crystal field. The free-ion terms are shown on the left side of the diagram and the strong-field configurations are shown on the right. The results of crystal-field splitting are shown for both of these sets of levels, and the one­ to-one correspondence among weak-field and strong-field terms is shown explicitly. The variation of the energy levels with the strength of the crystal field is not meant to be exact in this type of diagram and is discussed in de­ tail below. The first step in determining the quantitative value for the energy levels of the crystal-field terms is to determine the crystal-field energy, n ( 4Dq) + m(6Dq) for a f29 e; configuration. Thus for the configurations of a d3 ion, 2Dq, E(t29e�) 8Dq, and E(e�) l 8Dq. E(t�9 ) = l2Dq, E(t�9 e9) The next step is to add to these energies the energies due to the Coulomb and exchange interactions among the electrons. In Chap. 2 it was shown that the Coulomb and exchange energies for electrons in the (xz) and (yz)

230

6. Alz0 3 : Cr3 + Laser Crystals FREE ION TERMS

WEAK FIELD

SINGLE ELECTRON CONFIGURATIONS

2E

- Dq --

eg3 ( 1 8�

S TRONG FI ELD

FIGURE 6.2. Correlation of strong-field and weak-field energy levels of a 3d3 ion in an oh crystal field.

orbitals of the form given in Eq. (2.3. 1 3) are J(xz, yz) Fo - 2F2 - 4F4 and K(xz, yz) 3F2 + 20F4 , where the Fi are the Slater parameters. The interaction energies for electrons in the remaining combinations of orbitals can be calculated in the same way to give3

J(z2 , z2 ) J(� - i, � - i) J(xy, xy) J(xz, xz) Fo + 4F2 + 36F4 , J(x2 - i, xz) J( � - i, yz) J(xy, yz) = J(xy, xz) J(xz, yz) Fo - 2F2 - 4F4 ,

6.2. Crystal-Field Splitting

TABLE 6.4. Matrix elements for r!l for 3d electron wave functions.

a (xz) (yz) (xz) (yz) (z2 ) (z2 ) (z2 )

b

c

d

(ab 1 1 /rn l cd)

(z2 ) (z2 )

(xz) (yz) (z2 ) (z2 )

(x2 y2 ) (x2 - y2 ) (xZ - Y. l (xZ - Y. l

2 J3Fz + IO J3F4 2 J3Fz IOJ3F4 J3Fz 5 J3F4 -J3Fz + 5J3F4 J3Fz 5 J3F4 J3Fz 5J3F4 2J3Fz IO J3F4 3Fz I SF4 -Fz + I 5F4

(x' ) (y')

(x2 y2 ) (xZ - Y. l

23 1

(xy) (xy) (xz) (xy) (xy)

(x' )

(yz) (xy) (xz) (yz)

J(z2 , xy) J(z2 , x2 - i)

(yz) (xz) (yz) (yz) (xz)

Fo - 4F2 + 6F4 ,

J(x2 - i, xy) Fo + 4F2 + 34F4 ,

( 6.2. 1 )

and

K(x2 - i, xz) K(x2 - i, yz) K(xy, yz) K(xy, xz) K(xz, yz) 3F2 + 20F4 , K(z2 , xz) K(z2 , yz) F2 + 30F4 , K(z2 , xy) K(z2 , x2 - i) 4F2 + 1 5F4 , K(x2 - i, xy) 35F4 .

(6.2.2)

The other matrix elements for r1l are given in Table 6.4. These results can now be used to find the energy of each term. Consider as an example the ground-state term energy. For 4A 29 there must be one electron in each of the t29 orbitals, (xy) , (xz) , and (yz) in order to have all of the spins aligned and not violate the Pauli exclusion principle. The term energy must therefore include contributions from all possible two-electron combinations for Coulomb and exchange matrix elements obtained from expanding the Slater determinant. Using the above expressions shows this contribution to the energy to be 3J(xz, yz) - 3K(xz, yz) 3Fo 1 5F2 - 72F4 • Thus the total ground-state energy is E(4 A 29) - 12Dq 1 5F2 - 72F4 - 12Dq + 3A - 1 5B, where the Racah parameters have been used in the final expression. Similar calculations can be made for each of the terms. The most important results for chromium-doped laser materials

6. Ah0 3 : Cr3 + Laser Crystals

232

t� Configuration

Pze Configuration

E(4A 2 ) 3A I 5B I2Dq EeE) 3A 6B + 3C I2Dq Ee T1 ) 3A 6B + 3C I2Dq Ee T2 ) 3A + 5C I2Dq

E( 2A I ) E(4 T1 ) E(4 T2 )

=

=

= =

3A 3A 3A

l i B + 3C 2Dq 3B 2Dq I 5B 2Dq

=

The energies of the other crystal-field terms are so high that they are gen­ erally above the ultraviolet absorption band edge of the host material and thus do not play a role in the laser properties of Cr3 + laser materials. Since 3A is a common contribution to all of these energies, it can be subtracted out of these expressions and the resulting crystal-field term energies plotted as a function of crystal-field strength. Generally the energies are normalized with respect to the Racah B parameter so the energy levels expressed as Ej B are plotted versus Dqj B. The results of doing this are referred to as Tanabe­ Sugano diagrams and these are extremely useful tools in analyzing the opti­ cal spectra of transition-metal ion laser materials. 4 Figure 6.3 shows the Tanabe-Sugano diagram for a 3d3 ion. All three of the Racah parameters are associated with the Coulomb interaction between same configuration electrons. For a Cr3 + ion, B 9 1 8 cm- 1 for a free ion

�

40

WMWH/M

2T 2

30

2Tl 2E

WkMWAJ

10

ENERGY LEVELS

FIGURE 6.3. Tanabe Sugano diagram for a 3d3 ion with energy levels for ruby with Dq/B 1 720/765 2.25 (after Ref. 4) =

=

6.3. Spin Orbit Splitting and Selection Rules

233

and this reduces to 765 cm- 1 in an octahedral crystal-field environment. This difference is associated with the change in the radial wave function between the free ion and the ion in the crystal and is especially sensitive to covalency effects. In general C 4B is a good approximation for transition­ metal ions. The effect of A is to reduce the crystal-field lODq. For an alu­ minum oxide host crystal Dq = 1 720 cm- 1 • Thus Dq/ B 2 2 5 for ruby. B alone determines the energies of the 2E and 2T2 levels and the magnitude of configuration mixing. In this case the 2 E term is the lowest excited state that becomes the metastable state from which the laser transition is initiated. This is called a strong-field material. For host materials with smaller values of Dq, the 4T1 term becomes the lowest excited state and thus the initial state of the laser transition. These are referred to as weak-field materials. Since the laser transition for strong-field materials is a spin-flip transition between levels of the same t� configuration, it is a sharp line transition. In a weak-field material the laser transition is between different crystal-field con­ figurations t�e to t� and thus is a broad-band transition. These differences are discussed in detail in the Sees 6.4 and 6.5. .

6.3

Spin-Orbit Splitting and Selection Rules

For transition-metal ions, the energy-level splitting due to the spin-orbit in­ teraction is much smaller than the splitting due to the electrostatic inter­ action of the crystal field. Thus spin-orbit coupling can be considered as a perturbation on the octahedral crystal-field terms derived above for a 3d3 ion. To determine the qualitative nature of the energy-level splitting due to the spin-orbit interaction, it is again possible to use group theory. This is done by forming the direct-product representation of the spin and orbital parts of the crystal-field wave functions and then using Eq. (2.2.8) to reduce these results in terms of the irreducible representations of the crystal-field symme­ try group. For the Oh symmetry group, the spin functions for the doublet terms are represented by the E1 ;29 irreducible representation with the char­ acters given in the Oh character table shown in Sec. 2.2. The spin functions for the quartet terms are represented by the G9 irreducible representation. The spin-orbit splittings for six lowest-energy terms of a 3d3 ion in an Oh crystal field are shown in the following table: (3

t2 e

4A 2 2£ 2T1 2T2 4 T1 4T2

--+

--+

--+

--+ --+

G G

E112 + G £3;2 + G E112 + £3;2 + 2 G E112 + £3;2 + 2 G

6. Ah0 3 : Cr3 + Laser Crystals

234

As an example of how the results in this table were obtained, consider the 2T29 terms. The direct-product representation is found from the character table as shown below. E

oh

E1 ; 2 £3/2 G £1 ;2

x

2, 2, 4, T2 6,

i 2 2 4 6

2, 2, 4, 6,

2 2 4 6

8C3 I, 1 I, 1 1 I 0, 0 ,

8C3 i I, 1 I, 1 1, I 0, 0

3C2 0, 0 0, 0 0, 0 0, 0

3C2 i 6C4 6C4 i 6C� 0, 0 v'2, -v'2 v'2, - v'2 0, 0 v'Z, v'Z v'Z, v'Z 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 v'Z, v'Z v'Z, v'Z 0, 0

6c'2 i 0, 0 0, 0 0, 0 £3;2 + G

These results are then used with Eq. (2.2.8) and the character table to give

n(£1;2 ) fs ( 6 X 2 X 2 + 0 X 1 X 16 + 0 X 0 X 6 + - .Ji X .Ji X 12 + 0 X 0 X 12) 0, n(£3 /2 ) fs ( 6 X 2 X 2 + 0 X 1 X 16 + 0 X 0 X 6 + - .Ji X - .Ji X 12 + 0 X 0 X 12) 1 , n(G) fs (6 X 4 X 2 + 0 X - 1 X 1 6 + 0 X 0 X 6 + - .Ji X 0 X 12 + 0 X 0 X 12) 1 . The example of spin-orbit splitting of a 4 T1 9 term was described in Sec. 2.3.

This analysis shows that there is no splitting of the ground-state term as a result of spin-orbit coupling. In the strong-field limit, the lowest excited state 2E9 also remains unsplit by the spin-orbit interaction. However, all of the doublet and quartet T1 9 and T29 terms are split by spin-orbit coupling. The quantitative values for spin-orbit splitting can be found using the Lande formula, Eq. (2. 1 .23). As discussed in Sec. 2.3, the magnitude of the parameter A, is determined by comparing theoretical predictions with ex­ perimentally observed spectra. For ruby the spin-orbit coupling parameter ( 170 cm 1 . The radiative transitions for the free Cr3 + ion are electric dipole allowed only if they take place between energy levels have different electron config­ urations, such as 3d3 __ 3d1 4s. This type of transition generally falls in the ultraviolet spectral region. The transitions of interest for laser operation fall in the visible spectral region and take place between energy levels having the same 3d3 electron configuration. These Laport forbidden transitions take place through magnetic dipole or electric quadrupole interactions, and thus result in weak spectral lines. Table 6.5 lists the transitions that can take place between Russell-Saunders terms through these multipole interactions. As discussed in Sec. 3.2, the spin quantum number remains unchanged between these transitions while the orbital angular momentum quantum number changes by 1 or 0 for magnetic dipole and 2 , 1 , or 0 for electric

6.3. Spin-Orbit Splitting and Selection Rules

235

TABLE 6.5. Radiative transitions between Cr3 + terms. Electric quadrupole

4 p Cr

30 20 10

0

50

150

10

20

T (K)

450

- EXPERIMENTAL UNE

o

VOIGT PROFILE

--- GAUSSIAN UNE

-- LORENTZIAN LINE

0.5

-0.5

-0.1 0.1

C.-Uo)6w

0.5

FIGURE 6. 1 8 . Shape of the N1 line of a ruby sample with 2. 1 % chromium at 21 K (after Ref. 1 7) .

250

6. Ah0 3 : Cr3 + Laser Crystals

cal example1 7 of a measured line shape, the shapes of pure Gaussian and Lorentzian line shapes having the same linewidth, and the theoretical fit of a Voigt profile to the measured line shape. From this type of analysis the temperature-independent contribution to the linewidth due to strain broad­ ening is determined. For lightly doped ruby, the strain-broadened linewidth can be less than 0. 1 cm - 1 and it is usually significantly greater for heavily doped samples. The dominant broadening mechanism at high temperatures is Raman scattering of phonons. The theoretical lines in Fig. 6. 1 6 are de­ rived assuming a Debye distribution of phonon states as discussed in Sec. 4.4. For the R lines, the fit between theory and experiment in the temper­ ature region near 1 00 K can be improved slightly by including a term due to direct phonon processes between the split components of the 2 E level. For the R lines in heavily doped ruby it is necessary to include phonon processes between the exchange-split levels of the ground split manifold. The domi­ nant processes are Orbach processes obeying a selection rule !1S = ±2. The temperature-dependent shifts in positions of the lines can be explained by theoretical expressions for phonon scattering without including any con­ tributions for direct processes. Similar measurements have been made on the S and B lines/ 8 and both direct and Raman phonon processes have been found to contribute to the linewidths. At low temperatures, the quantum efficiency of radiative emission from the 2E level in ruby is approximately 0.96. As temperature increases, this quantum efficiency decreases and is approximately 0. 7 at room temperature. This is due to increased vibronic emission and population of the 2 T1 level. Above 450 K the total radiative quantum efficiency of ruby decreases due to radiationless decay to the ground state, which is associated with thermal activation of population to the 4 T2 level as shown in the configuration­ coordinate diagram1 9 in Fig. 6. 19. The detailed configuration-coordinate model for ruby developed in Ref. 1 8 uses Franck-Condon overlap integrals to account for the positions and shapes of the optical bands as well as the nonradiative decay processes and fluorescence quenching in ruby. The Huang-Rhys factors for the transitions from the 4A 2 level to the 4T2 and 4T1 levels are found to be 3 and 6, respectively. This analysis predicts emissions rate of 2 E -- 4.4 2 = 1 . 4 x 102 s- 1 and 4 T2 -- 4.4 2 = 2.3 x 1 04 s 1 with a non­ radiative crossover rate of about 104 s- 1 at 0 K. These rates increase with temperature. The radiationless relaxation times between excited levels in this model are found to be 4 T2 -- 2E � 10- 1 3 s and 4T1 -- 2E � 1 0- 1 2 s. It should be noted that other analyses based on configuration-coordinate models have resulted in much smaller estimates for the nonradiative decay rates. It has been difficult to measure these rates directly with any degree of accuracy. Picosecond pulse-probe experiments have set an upper limit of 7 ps on the decay time of the 4T2 level. 2 0 Another possible loss mechanism for emission from the 2 E metastable state is excited-state absorption. However, for ruby there is no good energy match for transitions from the metastable state to higher energy levels (see

6.4. Strong-Field Laser Materials

251

FIGURE 6. 19. Configuration-coordinate dia­ gram for ruby showing radiationless decay channels ( Ref. 1 9) .

CONF I G U RATIONAL COORDINATE

Fig. 6.3). Thus excited-state absorption of photons emitted in the R 1 line is not an effective loss mechanism for ruby. As discussed in Sec. 3.4, there is a change in the polarizability of the Cr3 + ions in the 2 E level as compared to the 4A 2 level. However, with normal optical pumping conditions the change in the refractive index associated with populating the metastable state [see Eqs. (3.4.8) and (3.4. 16)] is not large enough to produce serious problems with optical beam distortion. Studies of the temperature and chromium concentration dependences of the fluorescence intensities and lifetimes of heavily doped ruby show that nonradiative energy transfer takes place between single ions of Cr3 + and ion pairs. Above a critical concentration of 0.4%, the excitation diffuses among the single ions via a superexchange interaction. This can be described 14 as localized exaiton migration with a hopping rate of about 106 s- 1 for samples with a chromium concentration of 1%. The final step of energy transfer from single ions to pairs also takes place through exchange interaction but the measured rate for this processes is much smaller. For a sample with 1% chromium the transfer rate i s about 103 s- 1 . This transfer step i s best de­ scribed as a two-phonon assisted process. The effect of this efficient ion-ion interaction in heavily doped ruby can be seen in Fig. 6. 10( B). At temper­ atures above about 75 K, the fluorescence lifetimes of the single ions and pairs are the same. This indicates that the distribution of excitation energy among these different types of opeically active centers is thermalized. For this to occur there must be fast energy transfer back and forth between the ions in the different types of sites. At lower temperatures the energy transfer becomes less efficient. In this region the R, N1 , and N2 lines all exhibit dif­ ferent fluorescence lifetimes, showing that there is no thermal equilibrium in

252

6. Aiz0 3 : Cr3 + Laser Crystals

the distribution of excitation energy among the different types of centers. Excitation by intense laser light can induce a photocurrent in ruby2 1 with electrons hopping between Cr3+ and Cr4+ ions. This can build up an inter­ nal Stark field that causes changes in the observed spectra of ruby. The spectroscopic properties of ruby described above make it an excellent laser material. It operates as a three-level system with sharp line laser emis­ sion at the R 1 line position of 6943 A. The energy density in the metastable sate at threshold is 2. 1 8 J/cm3 . Laser action has also been obtained from the R 2 line and the N1 and N2 lines. The strong, broad absorption bands result in efficient absorption of fl.ashlamp pump energy, and the fast relaxation to the 2E level, which has a high quantum efficiency and low excited-state absorption, results in efficient overall pumping of the laser emission. The narrow linewidth of the laser transition gives a high stimulated emission cross section, which becomes even greater at low temperatures. At room temperature the R 1 line stimulated emission cross section is fJR 1 2.5 x IQ- 20 cm2 and the absorption coefficient at the peak of the lasing transition is !XR 1 0.2 cm- 1 . The long lifetime of the metastable state provides the ability for a high level of optical storage leading to good pulsed and Q­ switched operation. For complete inversion, the maximum energy density of the 2E level is 4.52 J jcm 3 and the maximum extractable energy is 2.35 J jcm3 . The spectroscopic and lasing properties of Cr3 + ions vary significantly from host to host. Examples of the optical properties of other chromium-doped materials are given in the next chapter. References

1 . J.S. Griffith, The Theory of Transition Metal Ions (Cambridge University Press, London, 1 96 1 ) . 2 . E.U. Condon and G.H. Shortley, The Theory of A tomic Spectra (Cambridge University Press, London, 1 935). 3. C.J. Ballhausen, Introduction to Ligand Field Theory ( McGraw-Hill, New York, 1 962) . 4. Y . Tanabe and S. Sugano, J. Phys. Soc. Jpn. 9 , 766 ( 1 954); S . Sugano, Y . Tanabe, and H. Kamimura, Multiplets of Transition-Metal Ions in Crystals, (Academic, New York, 1 970) . 5. S. Geschwind and J.P. Remeika, J. Appl. Phys. 3 3 (Suppl.), 370 ( 1 962) . 6. C.J. Donnelly, S.M. Healy, T.J. Glynn, G.F. Imbusch, and G.P. Morgan, J. Lumin. 42, 1 1 9 ( 1 988). 7. H.G. Drickamer, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York, 1 965), vol. 1 7, p. 1 . 8. R.C. Powell, Doctoral thesis, Arizona State University Department of Physics, 1 967 (also published as Physical Sciences Research Paper No. 299, Air Force Cambridge Research Laboratories, Hanscom Field, Bedford, MA, 1 966) . 9. A. Misu, J. Phys. Soc. Jpn. 19 , 2260 ( 1 964) . 10. U. Rothamel, J. Heber, and W. Grill, Z. Phys. B 50, 297 ( 1 983). 1 1 . D.E. Nelson and M.D. Sturge, Phys. Rev. 137, Al 1 1 7 ( 1 965).

References

253

12. G.F. Imbusch, Phys. Rev. 153, 326 ( 1 967) . 1 3 . N.A. Tolstoi and Liu Shun'-Fu, Opt. i Spektroskopiy 13, 403 ( 1 962) . 14. (a) R.C. Powell and B. DiBartolo, Phys. Status Solidi A 10, 3 1 5 ( 1 972); (b) R.C. Powell, B. DiBartolo, B. Birang, and C.S. Naiman, Phys. Rev. 155, 296 ( 1 967); (c) R.C. Powell, B. DiBartolo, B. Birang, and C.S. Naiman, in Optical Proper­ ties of Ions in Crystals, edited by H.M. Crosswhite and H.W. Moos ( Inter­ science, New York, 1 967), p. 207. 1 5. A.L. Schawlow, D.L. Wood, and A.M. Clogston, Phys. Rev. Lett. 3, 27 1 ( 1 959) . 16. P. Kisliuk, A.L. Schawlow, and M.D. Sturge, in Advances in Quantum Elec­ tronics, edited by P. Grivet and N. Bloembergen (Columbia University Press, New York, 1 964), p. 725. 17. R.C. Powell, B. DiBartolo, B. Birang, and C.S. Naiman, J. Appl. Phys. 37, 4973 ( 1 966) . 1 8 . T. Kushida and M. Kikuchi, J. Phys. Soc. Jpn. 23, 1 333 ( 1 967) . 19. W.H. Fonger and C.W. Struck, Phys. Rev. B 1 1 , 325 1 ( 1 975). 20. S.K. Gayen, W.B. Wang, V. Petricevic, R. Dorsinville, and R.R. Alfano, Appl. Phys. Lett. 47, 455 ( 1 985). 21. A.A. Kaplyanskii, J. Lumin. 48-49, 1 ( 1 99 1 ) .

7

Transition-Metal-Ion Laser M aterials

Trivalent chromium has been made to lase in a wide variety of different oxide and fluoride host crystals. The strength of the crystal field varies sig­ nificantly from host to host and in some cases is much weaker than the crystal field for ruby. In these weak-field materials, broad-band fluorescence occurs from the 4 T2 level and this can be used for tunable lasers. Although trivalent chromium has been the dominant ion used for solid-state laser based on transition-metal ions, several other ions in the same row of the periodic table have been made to lase and several of them are beginning to find important applications. All of these have similar electronic config­ urations and their optical spectra are based on electronic transitions between levels of the unfilled d shell. As with Cr3 +, the emission spectra and lasing of these ions can either be a sharp line or a broad band, allowing for tunable operation, and can occur in the visible to near-infrared spectral regions, de­ pending on the host material. In some cases laser operation has been suc­ cessful only at temperatures well below room temperature. The availability of lasers based on this variety of ions offers an extension of operational characteristics including coverage of a broader spectral range, shorter pulse widths, and reduced excited state absorption losses compared to some of the Cr3 + -based systems. In the following two sections the general spectral properties of Cr3 + ions in a variety of different hosts are discussed and compared to those of ruby described in Chap. 6. Then the properties of other transition-metal ions in different host materials of interest for laser applications are discussed. The subsequent sections describe the details of the spectral and lasing character­ istics of several of the most important transition metal ion lasers. The fun­ damental concepts needed to understand these properties were outlined in Chaps. 2-5. 7. 1

Broad-B and Cr3 + Laser Materials: Alexandrite

In host materials with weak crystal-field strengths, the 4Tz9 is the lowest excited state as seen in the Tanabe-Sugano diagram of Fig. 6.3. This results 254

7 . 1 . Broad-Band Cr3 + Laser Materials: Alexandrite M

e

•

M

255

FIGURE 7. 1 . c-axis view of chrysoberyl structure where M denotes mirror planes [from Ref. l (a)].

Be AI

Qo

in broad-band emission and thus tunable laser output. The first solid-state laser material to reach commercial importance as a tunable laser was alex­ andrite (BeAh04 : Cr3 +) . Strictly speaking, the crystal-field strength in alex­ andrite is in the intermediate range with the 2E level lying just below the 4 T29 in energy. However, at room temperature there is significant thermal activation from the 2E level to the 4T29 level, and the spin-allowed transition from the quartet state is much stronger than the spin-forbidden transition from the doublet state. Thus the fluorescence and lasing characteristics are essentially those of a strong-field material with vibronic emission. The crystal structure of the chrysoberyl host material is hexagonal-close­ packed with the Pnma orthorhombic space group1 shown in Fig. 7. 1 . The Al3 + ions are octahedrally coordinated by the oxygen ions and occupy two inequivalent crystal-field sites, one with mirror symmetry belonging to the point site group Cs and one with inversion symmetry belonging to the point site group C;. The Cr3 + ions substitute for the Al3 + ions with 78% going into mirror sites and 22% going into inversion sites. The energy levels of the mirror-site chromium ions can be determined by considering the main con­ tribution to the crystal field from oxygen ligands with Oh symmetry that is slightly distorted to c•. The procedure for determining the energy levels is the same as that described in Sections 6. 1-6.3 for ruby and the results are shown in Fig. 7.2. The electric dipole transitions determined by symmetry selection rules are also shown in the figure, assuming that configuration mixing lifts the parity restrictions. A similar analysis can be done for the inversion-site ions that have a lower concentration. The absorption spectrum of alexandrite is dominated by ions in mirror sites. 2 It is very similar to that of ruby, consisting of two intense broad bands and three sets of sharp lines as shown in Fig. 7.3(A). The ftuo-

256

7. Transition-Metal-Ion Laser Materials

FIGURE 7 .2. Energy-level diagram and allowed transitions for Cr3 + ions in mirror sites in alexandrite.

E(lb E(la,c I

4A 2

..'

I

I

rescence spectrum consists of the two zero-phonon R lines and the broad vibronic band from the 4 T19 level. Emission from mirror-site ions dominate the spectrum. In addition, weak lines associated with the R lines from inver­ sion site ions can be observed as shown in Fig. 7.3(B). The optical pumping of alexandrite laser crystals assumes very fast non­ radiative relaxation processes between the 4T19 and 2£9 levels. Several dif­ ferent types of laser spectroscopy experiments have been used to study these processes and they have been found to have radiationless transition times of a few picoseconds. In order to explain this fast time for these processes, anharmonic interactions must be included in the theoretical treatment. 2 Be­ cause of the efficiency of these processes, the populations of the 4 T19 and 2 E9 levels are thermalized. Radiative transitions occur from both levels at room temperature, but because the transitions from the quartet level are spin allowed, they are stronger. Thus the lower-lying 2 9 level acts as a pop­ ulation storage level for the 4 T19 level. As the population of the quartet level decreases through fluorescence emission, it is replenished by radiationless transitions from the 2 9 level. The fluorescence lifetime of the emission of the mirror site ions decreases from about 2.3 ms at 10 K to about 290 J.lS at 300 K, as shown in Fig. 7.4. The long lifetime at low temperatures is the intrinsic lifetime of the 2 level and the thermal quenching of the fluorescence decay time is associated with thermal activation into the shorter lived 4 T29 level. Following the discussion in Sec. 4.2, the coupled lifetime of these two levels in thermal equalibrium is given by

E

E

E

r 1

rE 1 + < ± ¥ II + � > < ± ¥ II + ! > < ± ¥ II ± ! > < ± ¥ II ± � >

3 2 3 6

360 360 360 720 360v'IT

../7

14 28v'3 2 1 v'IO 42v's

066

2v'TI 7v'30 7 v'6 84

= (J! + J� )/2 v'462

7 v'3 4 v'273

v'429

7 v'39

y'35)(T3

8. Y3 Al501 2 : Nd3 + Laser Crystals

308 TABLE

J

2 3 4 5 6 7 8 2 3 4 5 6 7 8 3 4 5 6 7 8

8.5 (continued) F

M1

=0

±I

±2

±3

±4

±5

±6

±7

±8

02o = 3Jf J(J + I ) 2 3 I 2 2 3 3 4 5 0 17 8 7 28 -20 I 3 10 6 15 9 6 13 2 10 11 22 14 5 3 44 56 29 19 8 52 91 53 24 12 23 20 8 15 25 3 04o = 35J: 30J(J + I )J} + 25J} 6J(J + I ) + 3J2 (J + 1 ) 2 12 I 4 6 I 7 3 60 6 21 11 60 18 14 9 1 6 420 6 4 6 6 99 96 11 54 84 64 60 66 12 621 294 704 1001 756 25 1 429 869 13 420 31 24 3 36 39 17 39 060 = 23 1J: 3 1 5J(J + ! )J: + 735J: + 105J2 (J + I? J} 525J(J + ! )J} +294J} 5J3 (J + I ) 3 + 40J(J + I ) 2 60J(J + I ) I 20 1 80 6 15 20 22 1260 4 17 2520 48 36 40 12 15 29 43 40 22 22 20 55 7560 8 3780 176 143 200 55 - 125 286 197 50 1 20 2 13860 85 78 128 1 69 65 93

J

F

3 4 5 6 7 8

360 360 360 360 360 360

< 3 11 3 )

in the visible spectral region [Ref. 14(a)].

Y AG : Nd

1-b

T

=

(ass Ofo)

295 K

0.6

�

·

0.2

16

1a

20

22

24

38

40 44

).(103 nm)

y (102cni1)

42

4 r 13/2

411312

sa

eo

4 I1512

ee

•1,$1

6B

(A)

(B)

FIGURE 8.6. (A) Absorption and ( B) fluorescence spectra of Nd3 + in YAG in the infrared spectral region. ( Reprinted from Ref. 1 6 by permission of Springer-Verlag.)

�

..

.c

§

i'

0.6

al 0: 0.4

z

::;)

�

aa

-

w t-.

+

0.

z

;:

�

c:;< � 0

i

>
4 /1 1 ;2 transition. Lasing has been obtained only at temperatures of 90 K or less. The power of cw laser emission at 20 K has been measured to be as high as 1 2 mW in the violet for incident pump powers of 1 10 and 300 mW at the infrared and yellow wavelengths, respectively. An example of a second type of Nd3 + up-conversion laser is avalanche pumped Nd : YLF. 6 In this case absorption of a 603.6-nm pump photon originates from a thermally excited 4 /15 ;2 level. After radiationless relaxation to the 4F3 ;2 metastable state and cross relaxation with a neightboring ion, there are two ions in the 4 /15; 2 level. This avalanche procedure continues to increase the excited-state absorption at the pump wavelength until a significant number of ions have been excited

9. 1 . Nd3 + Lasers

345

to the 2 P3; 2 metastable state. Fluorescence and lasing then occurs at 4 1 3 nm due to the 2 P3 ;2 -- 4 11 1 ;2 transition. Again, lasing has been obtained only at low temperatures. One host crystal of special interest is gadolinium scandium gallium garnet, Gd3 Sc2 Ga3 0 1 2 , known as GSGG. Since they are both oxide garnet crystals, GSGG and YAG have many properties that are similar. As seen in Table 9 . 2, the laser transition cross section for Nd3 + in GSGG is about twice that in YAG. This implies that about twice as much energy density can be stored in the metastable state, which is advantageous for Q-switched performance. The metastable-state fluorescence lifetime and nonradiative decay properties are similar for the two hosts. However, the differences in spectral properties results in the energy transfer from Cr3 + ions to Nd3 + ions being much more efficient in GSGG than in YAG. Thus Cr; Nd-GSGG has the advantage of sensitized pumping with broad-band lamp excitation, resulting in approx­ imately a factor of 2 lower threshold and higher efficiency for laser opera­ tion than Nd-YAG. The differences in the properties of the Cr-Nd energy transfer between YAG and GSGG hosts is due to the difference in crystal-field strengths. The gallium garnet has larger crystal lattice parameters than the aluminum gar­ net and thus a smaller crystal-field strength. As described in Chap. 6, the fluorescence emission properties of Cr3 + ions are highly sensitive to the local environment. The strong crystal-field environment in YAG results in the 2 E level of Cr3 + lying well below the 4 T2 level and the fluorescence spectrum of YAG : Cr3+ is dominated by sharp, spin-forbidden R-line emission. As dis­ cussed in the last chapter, there is very poor spectral overlap between this emission line and Nd3 + absorption lines, leading to a low efficiency of en­ ergy transfer. For the weaker crystal-field environment in GSGG, the split­ ting between the 2 E and the 4T2 levels is much smaller and the Stokes shift of the latter causes the emission spectrum to be dominated by the broad, intense band associated with the spin-allowed 4 T2 -- 4 A 2 transition. This has excellent overlap with Nd3 + absorption transitions as shown in Fig. 9.4, and thus the energy transfer is very efficient. 7 The kinetics of the energy transfer from chromium to neodymium ions in GSGG have been described8 by a simplified version of the expressions for the time evolutions of the sensitizer and activator excited-state populations derived in Chap. 5 t (9 . 1 . 1 ) Ncr ( t) Ncr ( O) exp y - y yfi - Wt and

(

rcr

)

(9 . 1 . 2) where

9. Rare-Earth-Ion Laser Materials

346

0.7

j.. ,'v Emission

0.6

I I I I I I I I

0.5

.. CJ 0.4 1: ..

.c

�

.c oct

0.3

0.2

\ \ \ \ \ \ \ \

0.1 7 00

Wavelength (nm)

600

900

1 000

FIGURE 9.4. Fluorescence of Cr3 + and absorption spectrum of Cr;Nd GSGG (after Ref. 7).

r (t) and

{

[erf (B0 +

[1 - exp( -B2t - yVt)] }

J (1 (9.1.3)

1 + W - -. 1 B= (9.1.4) rcr TNd Here y is the Cr-Nd interaction parameter and Wis the migration enhanced energy-transfer parameter. This is essentially an expansion of Eq. (5.4.23) for diffusion-enhanced single-step energy transfer. It has been found8 that a reasonably good fit between these equations and the observed fluorescence decay kinetics after pulsed excitation can be obtained assuming electric dipole-dipole interaction for both the Cr-Nd transfer step and the Cr-Cr energy migration. However, an improved fit can be obtained if it is assumed that transfer from Cr ions to Nd ions in the first- and second-nearest­ neighbor positions takes place at a greater rate than predicted by electric dipole-dipole interaction. For a crystal with PNd Per 1020 cm-3; the values of the energy-transfer parameters that are obtained from theoretical2 fits to experimental data are W(Cr-Nd)8 2000s-1, y(Cr-Nd) l l o s-1 1 , W(Nd-Nd) 124 s-1, y(Nd-Nd) .5 s-112, 'Nd 280 ps, and rcr 120 ps. The effective energy-transfer efficiency from chromium ions to neo-

9. 1 . NdH Lasers

347

dymium ion in GSGG can reach levels of greater than 80%, which makes sensitized pumping an important process for this neodymium laser material. Another type of neodymium laser crystal of particular interest is neo­ dymium pentaphosphate, NdP5 0 1 4 · This is representative of a special class of laser crystals known as stoichiometric laser materials that can be useful in low-threshold, high-gain minilaser applications. In general, these materials are the high-concentration end of a compositional series NdxA I -xP5 0 1 4 where A La3 + or y3 + . The distinguishing feature about this class of materials is that there is much less concentration quenching of the emission compared to normal host crystals, and thus strong fluorescence and laser operation occurs in crystals with 1 00% Nd3 + concentrations. This is due to the fact that the positions of the energy levels in NdP5 0 1 4 are shifted in such a way that the 4 F3 ;4 -- 4 /15 ;2 transition occurs with less energy than the 4 19;2 -- 4 /15 ;2 transition. Thus the usual mechanism for concentration quenching through an ion-ion cross-relaxation process involving a resonant interaction through these two transitions is not very efficient. However, with the close spacing of the Nd3 + ions, there is strong resonant transfer of excitation energy in the metastable state. NdxA I - xP5 0 1 4 crystals exhibit spectral energy transfer associated with the short-range interaction between Nd ions in nonequivalent types of crystal­ field sites. The characteristics of this process are similar to those of Nd3 + ions in YAG and other host materials. In addition, long-range spatial mi­ gration of energy among the Nd3 + ions takes place in NdP5 0 1 4 and mixed crystal systems of this class. This process has been investigated using laser­ induced transient grating techniques. 9 The excitation diffusion coefficient for NdP5 0 1 4 was found to be in a range between x w- 6 and 2.5 x 1 2 w-4 cm s- , depending on the type of excitation (resonant or vibronic) and temperature. The large value of D was found for resonant excitation and the signal decay dynamics in this case exhibited nonexponential, oscillatory behavior. The ion-ion interaction causing the energy migration was found to be consistent with an electric dipole-dipole mechanism. The strong inter­ action due to the high concentration of Nd3 + ions results in long mean free paths for the migration process. The diffusion coefficients decreased with decreasing Nd concentration in mixed crystals and were about an order of magnitude for concentrations of 20%. Another type of process that can be important in some types of laser crys­ tals is host-sensitized energy transfer. Molecular crystals such as tungstates or vanadates are examples of host materials where this can take place. In YV04 : Nd3 + , the (V04 ) 3 - vanadate molecule has absorption transitions in the near-ultraviolet spectral region and emission bands in the visible spectral region of this material. When these molecular ions are excited, the energy can migration from molecule to molecule in the crystal like a Frenklel exciton. As the exciton gets near to a Nd3 + it becomes trapped and the energy is transferred to Nd3 + ion. This process has been investigated using

5.4

348

9. Rare-Earth-Ion Laser Materials

time-resolved spectroscopy techniques, 1 0 and the results are consistent with both the energy migration and trapping mechanisms being due to electric dipole-dipole interaction. At low temperatures, the energy transfer is due to a single-step process while at high temperatures the transfer is enhanced by multistep migration. The latter mechanism is a thermally activated hopping process with a diffusion coefficient of the order of 5.4 x 10- 7 cm2 s 1 . So far the mechanism of host-sensitized energy transfer has not been exploited in commercial laser systems. This is due mainly to the strong interest in direct diode laser pumping of the Nd3 + ions. YV04 : Nd3 + is an excellent material for a diode laser-pumped laser because it has strong, single-line transitions with cross sections of the order of five times greater than those of Nd : YAG. Due to the cross-relaxation interaction shown in Fig. 8 . 1 5(A), concentra­ tion quenching is an important limiting process Nd-doped laser materials. Figure 9.5 shows the quenching of the fluorescence of the fluorescence life­ time of Nd3 + in several host materials as a function of neodymium concen­ tration. An empirical expression that gives a good fit to these data is 1 1 ro (9. 1 .5) - = - + Wer :: rf = + W , 1 ro er rf ro where rf is the fluorescence decay rate, r0 is the decay rate with no cross­ relaxation present, and Wcr is the rate of cross relaxation. Assuming electric dipole-dipole interaction, the cross-relaxation rate can be described using the Forster-Dexter theory outlined in Chap. 5. For this application a quenching parameter Q can be defined as Q=

p

(9. 1 .6)

where p is the neodymium concentration. Substituting this into Eq. (9. 1 .5) gives the fits to the data shown in Fig. 9.5. The Q parameters obtained from these fits are listed in Table 9.3. For cases where the Q parameter has been calculated using the energy-transfer expression given in Eq. (5.4. 1 5) and the spectral overlap of the 4 F3 /4 --> 4 1, 5 ; 2 emission transition with the 4 19;2 --> 4 /15 ;2 absorption transition, a good agreement has generally been found between measured and predicted values. The exceptions to this are materials such as NdP5 0 1 4 where quenching is enhanced by migration to "killer sites." Another interesting class of host materials are nonlinear optical crystals. For some applications, specific laser wavelengths are required in spectral regions where no primary laser exists with the appropriate characteristics. For these situations it is possible to use a nonlinear optical crystal to change the wavelength of a primary laser through processes such as harmonic gen­ eration or parametric mixing. Usually the nonlinear optical crystal is a sep­ arate component from the laser crystal, either external or internal to the laser cavity. However, there have been several demonstrations of doping

9. 1 . Nd3 + Lasers

349

Dexter's Theory 't f 40

0

2

4

6

8

Nd CONCENTRATION (1020 cm·3 )

10

(A)

..

:I.

... E

600

·� >

B .. ..

"t)

> ·;: u

.. ;:

w

Nd

1 concentration, 020 ion/cm 3

(B)

FIGURE 9.5. Concentration quenching of the fluorescence lifetime of Nd3 + ions in several hosts. (A) Crystals ( Ref. I I ) . ( B) Glasses [Ref. 12 (a)].

350 TABLE

9. Rare-Earth-Ion Laser Materials 9.3. Concentration quenching parameters for some Nd-laser materials.

Crystal

Y3 A1 s Ou Gd 3 Sc 2 Ga 3 0 1 2 YV04 YLiF4

Q ( 1020 cm- 3 )

3.8 5.0 3.3 3.7

Glass Ed 2 Silicate LHG 7 Phosphate LG 8 12 Fluorophosphate

Q ( 10 20 cm- 3 )

3.89 6.88 3.99

rare-earth ions in nonlinear optical crystal to obtain a "self-doubling" laser. One important example of this is LiNb0 3 : Nd3 + . The normal 4 F3 ; 2 -- 4 /11 12 near-infrared laser transition is converted to second-harmonic green emis­ sion for this case. 1 3 Because of the phase-matching requirements for the primary- and second-harmonic wave in a nonlinear optical crystal, efficient self-doubling laser crystals are very sensitive to crystal orientation, align­ ment, optical path length, and temperature. The crystal and cavity design parameters required to maximize laser performance are generally not the same as the design requirements for optimizing nonlinear optical conversion efficiency. Lasers based on nonlinear optical materials are discussed further in Chap. 1 0. Thus far systems with separate laser and frequency conversion crystals have achieved better performance characteristics than self-doubling systems. However, with new host materials such as NdxY 1 -xA1 3 (B0 3 ) 4 ( NYAB) being developed, the properties of self-doubling lasers may be significantly enhanced in the future. This material has been operated as a free-running and a Q-switched system with flashlamp and with diode laser pumping and with Cr3 + activation. Both the 1 .34- and the 1 .06-flm emission lines have been self-doubled. NYAB has low concentration quenching rates allowing for high concentrations of Nd ions. The for a sample with a Nd con­ centration of x 0. 1 , the metastable-state lifetime is 60 flS and the 1 .06-flm laser line has an emission cross section of 2 x 10- 19 cm2 and a linewidth of 30 em- 1 . The luminescence quantum efficiency is about 1 5%. The nonlinear optical coefficient of the host material is almost four times that of KH2 P04 crystals. One of the major problems with self-doubling in NYAB is the strong absorption at the wavelength of the doubled light. Another problem is the temperature-dependent change in the phase-matching conditions that occurs due to the heat generated in the optical pumping. A new material with a nonlinear optical host that may prove to be inter­ esting is neodymium-doped potassium gadolinium tungstate KGd(W04 h ( Nd : KGW ) . This has diode laser-pumped laser properties similar to Nd : YAG. 14 In addition, KGW has very strong Raman gain. Thus the stimulated Raman properties of the host could be used to convert the laser emission to a different frequency. Glass materials are also excellent laser hosts for Nd3 + ions. Along with the comparative ease of synthesizing the material compared to many types

9. 1 . Nd3 + Lasers ::

'

N�

20

� c0

�

15

�3

c

351

3

10

�

.0

u ( 1 0 - 20 cm2 ) 4 ( F3/2 4 lt t/2 ) 0.421 0.439 0.406 0.402 0.447 0.401 0.067

p9/2

0.484 0.471 0.496 0.499 0.466 0.500 0.494

Pt t/2

0.091 0.085 0.093 0.095 0.083 0.095 0.360

pl3/2

0.004 0.004 0.005 0.005 0.004 0.005 0.079

Pts;2

TABLE 9.4. Properties of Nd 3 + ions in selected glass host materials. [Data for the first six samples from Ref. 1 2(b); data for the fluoride glass from Ref. 1 7(a) .]

w Vt w

"" "' (I) .. "'

t"

+

w

p.

..

z

�

354

9. Rare-Earth-Ion Laser Materials

z 0 ;=

� "'

::!

0 0:

• 3

� �0: � "'


tn

@

:0 ;;o P>

w V1 0'1

9. 1 . Nd3 + Lasers 5

357

00.0

400.0

.. .3 ..

�::J

3 00.0

2

200.0

1 00.0

o o .

o o

15

0 0

TEIAPERATURE

2o

o

(K)

FIGURE 9. 1 0. Temperature dependence of the fluorescence lifetime for Nd : ZBAN glass. [Reprinted from Ref. 1 7(a) with permission of iEEE, © 1 991 IEEE.]

through both hopping transport and long-range resonant energy migration. The strengths of each of these ion-ion interaction processes are similar to those found in crystal hosts. The data for several glasses are given in Fig. 9.5( B) and Table 9.4. A typical example of a fluoride glass laser is Nd : ZBAN, which has the composition 53.33 ZrF4 ; 19.84 BaFz; 3. 14 AlF3 ; 1 8 .70 NaF; S.O NdF 3 in mole percent. This has been successfully operated as a laser in both bulk and fiber configurations. The temperature dependence of the fluorescence life­ time is shown in Fig. 9 . 1 0 and the absorption and fluorescence spectra are shown1 7 in Fig. 9. 1 1 . The linewidths of the transitions are somewhat smaller than in oxide glasses but the individual Stark components still can not be resolved. The theoretical fit to the lifetime data is given by 1 . 1 = r£ 1 + r exp (9. 1 .7) 1

[ (��) r

As described in Chap. 4, the form of this equation describes the coupled decay time of two energy levels that are in thermal equilibrium due to direct phonon absorption and emission processes across the level splitting In this case, is the splitting between the two Stark components of the 4 F3 ;2 metastable state. Various spectroscopic properties of neodymium in ZBAN are given in Table 9.4. One interesting result of the investigation of this material is that excited state absorption of pump photons can be an im­ portant process for laser pumping into some of the absorption bands in the 750-nm spectral region.

flE

flE.

358

9. Rare-Earth-Ion Laser Materials

"';"' E

CJ

C') 0

.. -

> CJ a: w z w

-

(A)

·;: "

�

� :e

�

6.0

4.0

�

z

I! �

2.0

0 "0 750.0

WAVELENGTH

(nm)

1 1 50.0

(B) FIGURE 9 . 1 1 . (A) Absorption and (B) fluorescence spectra for Nd : ZBAN glass. [Reprinted from Ref. 1 7(a) with permission of IEEE, © 1 99 1 IEEE.]

9.2. Other Trivalent Lanthanide Lasers 9.2

359

Other Trivalent Lanthanide Lasers

Although no lasing ion has been as successful as Nd3 + , all of the other trivalent lanthanide ions have been made into lasers. The majority of these laser transitions are 4f-4f transitions with properties similar to that of neodymium. The same fundamental physical processes apply to these types of lasers. Concentration quenching is associated with ion-ion cross-relaxa­ tion interactions and this can be enhanced through energy migration pro­ cesses. Sensitization by Cr3 + is effective for several trivalent rare-earth ions and there are a wide variety of energy-transfer processes between different types of rare earth ions. Multiphonon radiationless relaxation is one type of physical process that has been studied through measurements of different types of ions in the same host material. The nonradative decay rate for a specific level is gen­ erally determined by measuring the fluorescence decay rate for the level and subtracting the radiative decay rate from this value. The radiative decay rate is usually calculated from the transition cross sections measured in absorp­ tion spectra or from Judd-Ofelt theory. If the initial level of the transition of interest does not fluoresce, the nonradiative decay rate can sometimes be determined from the delayed from the delayed rise time of the fluorescence from the terminal level of the transition through an appropriate kinetic rate equation model. Examples of results obtained on a variety of different types of rare-earth­ doped host materials20 - 24 , are given in Table 9.5 and Figs. 9.12 and 9. 1 3. The interpretation of these results is based on the weak electron-phonon TABLE 9.5. Parameters for multiphonon radiationless decay processes in some rare-earth-doped laser materials ( From Refs. 20 24). Host LaBr3 LaCl 3 LaF3 LiYF4 Y2 03 YA10 3 Y3Als 0 12 Fluoride Germanate Silicate Phosphate

(cm-1 )

hwerr

(s- 1 )

175 260 350 400 550 600 700

Crystals 1 .2 X 10 1 0 1 . 5 x 10 1 0 6.6 X 1 08 3.5 X 1 07 2.7 X 1 08 5.0 X 1 09 9.7 X 107

1 .9 X J.3 X 5.6 X 3.8 X 3.8 X 9.6 X 3.1 X

10- 2 10- 2 10- 3 10- 3 10- 3 10- 3 10- 3

0.037 0.037 0. 14 0.22 0. 1 2 0.063 0.045

500 900 l iOO 1 200

Glasses 1 .6 X 10 1 0 3.4 X 1 0 1 0 1 .4 X 1 0 1 2 5.4 X 1 0 12

5.2 X 4.9 X 4.7 X 4.7 X

10- 3 10- 3 10- 3 1 0- 3

O.o75 0.014 0.0057 0.0036

c

IX

(em)

e

�

w

·= ·e.. 5

10

1 02

0 o,

I

9/ 2

"'

77° K

s

2H

5/2

2

s0

2P 3/2

5F 5

0 2

2

"'

55

4s 3/2

50 3

4F 9t2

( 3 PI , 30 3 )

5r

6

• Thu l i u m

.o. Erb i u m

• Ho l m i u m

0 Euro p i u m

o Neodym i u m

(A)

E n ergy gop to next- lower level (cm· I J

YAI 0 3

40

2G 91

5r 1

:; ::

-�

c 0 c 0 �

"'

"'

c 0 · u;

2! e

"'

1 0° 0

10 1

1 02

103

104

105

i 06

1 07

1000

LaBr 3 (175cm - l )

(B)

Energy gap ( cm- 1 )

2000

LaCI 3 (260 cm-1) 3000

4000

FIGURE 9 . 1 2. Energy-gap dependences of multiphonon nonradiative decay rates for different host crystals. ( Reprinted from Ref. 22 with permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1 055 KV Amsterdam, The Netherlands.)

::E

:;

c 0 .c a.

3 10

.. 0 .. 104 5

7 10

9.2. Other Trivalent Lanthanide Lasers

361

FIGURE 9.13. Energy-gap dependences of multiphonon nonradiative decay rates for different host glasses (from Ref. 2 1 ) . !l �

5' 105

-1: g

c

� 104

coupling, energy gap law model discussed in Chap. 4. The parameters in the table come from the expressions

(9.2. 1 ) where WKr is the nonradiative decay rate for a process involving p phonons, C and rx depend on the host but not on the specific transition, and hwerr is the energy of the effective phonon involved in the transition. e is the ratio of the ratio of the decay rates of the p- and ( p 1 ) -phonon processes as described in Chap. 4. Recalling that the phonon occupation number is given by n(hwerr ) = [exp(hwerr/kBT) 1r 1 , the temperature dep�ndence of the nonradiative decay rate for a p-phonon process is given by

W�r( T)

=

W�r(O)

(

exp -- - 1

kBT

)P

,

(9 . 2.2)

where the temperature-independent factor is given by Eq. (9.2. 1 ) to be WKr(O) = ce ai!.E. In most cases the effective phonons involved in non­ radiative relaxation processes are the highest-energy phonons available since this minimizes the number of phonons required to conserve energy for a given energy gap and thus results in the lowest possible order of the decay process. However, this is not necessarily always true since the coupling strength or density of states of lower-energy phonons may be greater than these quantities for high-energy phonons. The order of the process and thus the value of the effective phonon energy can be determined from the tem­ perature dependence of the decay rate. An example of the variation of the nonradiative decay rate with temperature is shown in Fig. 9. 14.

362

9. Rare-Earth-Ion Laser Materials

- 10 ..

' u·

=

Q

!!

.. cr 0: 0 ·;; 0: ..

�

0: 0

6

..: .. ::

:IE

La �

Ho

••

F (1F1 l - E I' F. .1S1.l 6 E • 1 800cm.-• •

8 6

4

100

200 T( ° K )

300

FIGURE 9 . 14. Temperature dependence of the p 6 multiphonon nonradiative decay rate for the transition 5F3 -- 5F4 , 5S2 in Ho : LaF3 showing the experimental points and theoretical fit. ( Reprinted from Ref. 22 with permission of Elsevier Science NL, Sara Burgerhartstraat 25, 1 055 KV Amsterdam, The Netherlands.) =

The results of extensive investigations of nonradiative decay processes of rare-earth ions in crystal and glass hosts shows that the simple phenomeno­ logical energy gap law fits the observed results for energy gaps spanning over six orders of magnitude. Some of the noted exceptions to the pre­ dictions of the energy-gap model can be readily explained through selection rules applicable to specifications or phonon coupling properties that can be verified independently through the shape of vibronic sidebands. Also, for very small energy gaps that can be bridged by one or two phonons, the basic ssumption of the multiphonon transition model break down and the one- or two-phonon theories discussed in Chap. must be used. The variation of the multiphonon decay rates from host to host as shown in Figs. 9. 12 and 9. 13 is associated with differences in the phonon spectrum of the host materials and differences in the electron-phonon coupling strengths. The former is demonstrated by the value of the effective phonon energy and the latter is reflected by the magnitude of the parameter e. Despite the general success of the energy-gap law for describing the char­ acteristics of multiphonon relaxation rates of rare-earth ions in crystal and glass hosts, there are some deviations from theoretical predictions. For

4

9.2. Other Trivalent Lanthanide Lasers

363

example, 2 5 for a YLF crystal host, the energy gap law fits the nonradiative relaxation rates for one set of values of C and IX for A.E :;: 2000 cm- 1 and for a very different set of values of C and IX for transitions with A.E > 2000 cm- 1 . This has been attributed to the weakness of using a point-charge model for the ion-ligand interaction. By adding terms in the interaction Hamiltonian that account for the spatial distribution of the electron orbits including, exchange effects and the dipole moments of the ligands, the dif­ ferences in the energy-gap law parameters for transitions with small and large gaps can be explained. By investigating the temperature dependencies of spectral linewidths in the series of trivalent lanthanide ions in the same host crystal ( YLF ), it has beefi possible to determine the variation of the electron-phonon coupling strength from ion to ion. 26 Transitions were chosen that are well resolved in the spectrum and intense enough to be studied at high temperatures. The results were interpreted in terms of the theoretical expression given in Eq. (4.4.8) with the Raman scattering of phonons found to be the dominant line-broadening processes and the effective Debye temperature for YLF chosen to be 250 K. The results show that the electron-phonon coupling strength is strong in the beginning (Ce 3 + ) and end (Yb3 + ) of the lanthanide series and weak in the middle (Gd3 + ) . The explanation of this trend is com­ plicated by the contributions of several effects. As the atomic number of the ions in this series increases there is a contraction of the electron wave func­ tions, which will decrease the electron-phonon coupling strength. However, for ions beyond Gd3 + , the 5s and 5p orbitals contract more than the 4f orbi­ tals thus decreasing the screening of the electrons in the 4f orbitals and therefore enhancing the electron-phonon coupling of the optically active electrons. In addition, the relative position of the 5d levels With respect to the 4f levels can affect the coupling strength. There has been a significant amount of interest in lasers operating in the 2-3-,um region to be used for a variety of applications. The ions of major interest for these aplications are Er3 +, Tm3 +, and Ho 3 + . These can be used in several different combinations along with Cr3 +, depending on the specific pumping and lasing characteristics desired. There are relatively large energy gaps between the multiplets of Ho 3 + ions, resulting in small radiationless decay rates leading to several metastable states. Thus holmium ions lase on several different transitions in a variety of oxide and fluoride hosts. The transition of greatest interest if 5 h --+ 5 !8 which produces laser emission at about 2. 1 ,urn. This wavelength has strong laser-tissue of interest in medical applications and occurs in an "eyes safe" spectral region with good atmo­ spheric transition for lidar applications. Since the terminal level of this transition is one of the upper Stark components of the ground-state multip­ let that has significant thermal occupation at room temperature, it is difficult to operate as a four-level laser with holmium and the pump thresholds are relatively high. To minimize ground-state absorption of laser light, low con­ centrations of Ho 3 + are generally used. Since this also decreases pumping

364

9. Rare-Earth-Ion Laser Materials

5

Cross

5

I 1 1

1 1 1

1

Tm

3+

3+

5

5

Transfer

5

U p c o nversion

IJ

(8 -+I I I I

17

--

-'Tm

Energy

1

:

(8

1 1 1 I

7

6

Laser

Transition

Ho

3+

5

FIGURE 9 . 1 5 . Energy levels and optical pumping transitions in Y 3 Al501 2 : Tm3 + , Ho3 + [from Ref. 27 (a)].

efficiency, it is common to use sensitizer ions such as Cr3 + or Tm3 + or both. One important example of this type of laser material is Y 3 Al5 0 1 2 : Tm3 +, Ho 3 + . The spectroscopic and lasing properties of this material have been investigated in great detail for laser-pumped systems and flashlamp-pumped operation where Cr3 + is then included as an initial sensitizer ion. The laser­ pumped system without Cr3 + ions will be used for this example. 2 7 The energy levels and pertinent transitions for the optical pumping dynamics are shown in Fig. 9. 1 5. Laser pumping excites the 3 H4 level of Tm3 + . The excitation energy can then stay on the same ion and relax down to the 3 F4 metastable state where both radiative and nonradiative emission occurs. However, for the high concentrations of thulium that are used to make this an affective sensitizer ion, there is a strong probably for ion-ion interaction to occur between Tm3+ ions. The dominant interaction mechanism for these pumping conditions is cross relaxation involving the excited ion undergoing the transition 3 H4 __ 3 F4 , while an unexcited ion simultaneously undergoes the transition 3 H6 __ 3 F4 . Since this process leaves two Tm3 + ions in the 3 F4 metastable state for every ion originally excited, the quantum efficiency of any process involving emission from this metastable state has twice the quantum efficiency as achieved through normal linear pumping processes. In addition to this cross-relaxation process, the ion-ion interaction can cause energy migration among Tm3 + ions. This can occur in the 3 H4 level before relaxation or in the 3 F4 level after relaxation. The latter is generally the dominant energy-transfer process. The migration among Tm3 + ions in the 3 F4 level can occur of long distances which makes the final transfer to Ho 3 + ions very efficient. This takes place through the coupled pair of transitions 3 F4 __ 3 H6 ( Tm3 + ) , 5/s __ sh ( Ho 3+ ) . The energy-transfer processes for this system have been characterized in detail. 2 7 Rewriting the critical interaction distance for electric dipole-dipole

9.2. Other Trivalent Lanthanide Lasers

365

energy transfer in Eq. (5.2.2 1 ) in terms of both transition oscillator strengths and the integrated absorption cross section (9.2.3) the measured absorption and emission spectra can be used to characterize the strength of the ion-ion interaction. Here the functions fsm and Jt rep­ resent the normalized emission spectrum of the sensitizer and absorption spectrum of the activator, respectively. For example, at room temperature the value of Ro for a step in the Tm-Tm migration is calculated to be 1 6.0 A while the value of R0 for the Tm-Ho transfer step is calculated to be 1 7 .3 A. As temperature is lowered to 12 K the value of Ro for Tm-Tm interaction decreases to 1 1 .5 A. This is due to the decrease in the population of upper Stark components, which decreases the transitions available for spectral overlap. For this ion-ion interaction strength and the high concentrations of Tm3 + ions used for sensitization, the average energy-transfer time between pairs of thulium ions is of the order of microseconds while the metastable­ state lifetime is of the order of milliseconds. Thus multistep long-range energy migration occurs among the Tm3 + ions. The long-range energy migration in the 3 F4 level was investigated using laser-induced grating spectroscopy techniques. The signal decay was inter­ preted with the Kenkre model28 using Eq. (5.5. 12),

where V is the nearest-neighbor interaction rate sausing the energy to migrate, rx is the excitation scattering rate, a is the nearest-neighbor distance, A is the grating spacing, and r is the excitation lifetime. The value of r was measured, A was determined from the wavelength and crossing angle of the laser beams, and a was calculated with the assumption that the Tm dopant ions were distributed randomly in the sample. The theoretical expression was then fitted to the experimental results treating V and rx as adjustable parameters. The results of these measurements are shown as functions of temperature and Tm concentration in Figs. 9. 1 6-9. 19. The results shown in Figs. 9. 1 6 and 9. 1 7 demonstrate that the ion-ion interaction rate required to produce the measured long-range thulium energy migration in the metastable state is consistent with the predictions of an electric dipole-dipole mechanism. There are several types of scattering mechanisms that can limit the mean free path of migrating excitions. If the dominant mechanism is scattering by acoustic phonons, the scattering rate should exhibit a temperature dependence that varies as T3 12 . The solid line in Fig.9. 1 8 reflects this type of temperature dependence and the consis-

366

9. Rare-Earth-Ion Laser Materials 2!50xl03 ..

1

u ..

200

� .!l

" II:

1 !10

c 0 ., u " .. ..

1 00

£ E

.. I

!10

E

..

5

0

Tm 3+

10

Concentration

20 x 1 0

15

(em -3)

20

FIGURE 9. 1 6. Concentration dependence of the ion-ion interaction rate of Tm3 + ions for the 3 F4 energy migration process at room temperature. The circles are the experimental points and the line is the theoretical prediction [from Ref. 27(a)].

..

!

1

u

.!l

0 II:

§

., u

e .!l .5

200 1 50 1 00 50

0

50

1 00

1 50

(K)

200

Temperature

250

300

FIGURE 9. 1 7. Temperature dependence of the ion ion interaction rate of Tm3 + ions for the 3 F4 energy migration process for a sample with 14. 1 x 1 020 cm 3 thulium ions. The open circles are the experimental points and the solid circles are theoretical predictions [from Ref. 27(a)].

9.2. Other Trivalent Lanthanide Lasers 6x10

ju

..

.! .. .. 0 0:

01 c: "I:

�0

u

en

367

3

5 4 3

•

•

2

50

1 00

1 50

200

•

250

Temperature (K)

300

350

FIGURE 9. 1 8. Temperature dependence of the excitation scattering rate for the 3 F4 energy migration process of Tm3 + ions in a sample with 14. 1 x 1 020 cm 3 thulium ions. The circles are the experimental points and the line is the theoretical prediction [from Ref. 27(a)].

.. .. .. ..

E �

·u

.. c: .. -=

u

0 u

c: 0

·;;

:I

rs

l.a 2.5 2.0 1 .5 1 .0 0.5 0.0

0

2

Concentration (em-1 4

8

a

10

12

14

1 8x 1 0

20

FIGURE 9 . 1 9 . Concentration dependence of the excitation diffusion coefficient for the 3 F4 energy migration process of Tm3 + ions at room temperature. The circles are the experimental points and the line is the theoretical prediction [from Ref. 27(a)].

368

9. Rare-Earth-Ion Laser Materials

tency between the experimental results and theoretical predictions implies that acoustic phonon scattering is major scattering mechanism. Using the primary energy-transfer parameters V and a determined by experimental measurements, the secondary parameters given in Eq. (5.5. 14) can be determined: V2 Diffusion coefficent: D = 2a2 ; Mean free path:

Lm =

v

IY.

v'la - ;

Diffusion length: LD = ..; . vtstte . . d per scattenng event: Ns = Lm . S ttes a For this material, the mean free path for the exciton migration is of the order of 1 0 6 em, while the diffusion length is of the order of 10- 5 em. The num­ ber of sites an exciton visits between scattering everts is of the order of 50. These values show that for the experimental conditions used, the properties of energy migration for this physical system are a < V; a < Lm « A. This is consistent with a long mean-free-path type of random walk. If energy­ transfer models based on the assumption of scattering at each step in the random walk are used for this system, the predicted values of the diffusion coefficient are more than two orders of magnitude different from those measured experimentally. The theoretical curve shown in Fig. 9. 1 8 that gives a good fit to the experimental data is based on the dependence D oc n41 3 ( Tm). This is predicted by all of the theories developed to describe a random-walk type of excitation migration (see Chap. 5). The overall energy-transfer efficiency from Tm to Ho ions in co-doped samples can be determined from a simple statistical interpretation of the energy transfer parameters described above assuming uniform distributions of the two types of ions. In this approach, the Ho3 + ions are treated as trapping sites for the migrating Tm3 + excitons. The transfer efficiency can also be determined by measuring the risetime of the fluorescence of the Ho ions as described in Chap. 5. In this case, the two methods are consistent within a factor of 2. They give an overall energy-transfer rate of the order of 104 s- 1 . Since the Ro for the Tm-Ho trapping step is greater than the Ro for the Tm-Tm migration step, the energy transfer is a "diffusion-limited" pro­ cesses. The Ho-Tm backtransfer process is not negligible in this system since thermal equilibrium of the excited-state populations of the two types of ions is measured to occur in about 200 Jl.S. In the energy migration model described here, this type of backtransfer is treated as a reduced trapping probability. Due to the significantly lower concentration of Ho 3 + ions com­ pared to Tm3 + ions, energy migration among the holmium ions is negligible. There are a variety of different upconversion processes that can occur on both Tm3 + and Ho 3 + ions resulting in luminescence from higher-energy IY.

.

(9.2.5)

-

9.2. Other Trivalent Lanthanide Lasers

369

metastable states. The significance of these processes depends critically on the concentrations of the two types of ions, the wavelengty of the pump laser, and the pump intensity. The extensive amount of spectroscopic information that has been obtained on Tm, Ho : YAG can be used as input to a rate equation model that describes the optical pumping dynamics of this system. This can be used for computer simulations of the laser-pumped laser operation of this system and the results compared to experimental observations. Using the energy levels and transitions shown in Fig. 9. 1 5, the rate equations describing the time evolutions of the populations of the eight energy levels labeled in the figure n ; plus the photon density in the cavity at the output wavelength np are27 dt

dn1

dt

dn2

dt

dn3

dt

dn4

dt

dns

dt

dn6

dt

dn7

dt

dns

dt

dnp

- U14 + fJ-41 - k42n4n1 - k62n6n1 + k26nsn2 + n2rz 1 + n4r4 1 P41 + n3r] 1 P3 1 + k468n4n6 + k261n2n6 - n 7r7 1 ,

(9.2.6)

2k42n4n1 + k62n6n1 - k26nsn2 - n2rz I + n4r4 I P42 + n3r] 1 P32 - k261n2n6, n4r4 l p43 + n 3r3 1 + n 7r7-1 ,

(9.2.7)

U14 - fJ-41 - k42n4n1 - n4r4 1 - k468n4n6,

(9.2.9)

W.

65 -

(9.2.8)

+ k62n6nl - k 26nsn2 + n6r6 1 + n 7r7- 1 + n s r8 1 , (9.2. 1 0)

1 Ws 6 - Wtis - Wtis - k62n6n1 + k26nsn2 - n6r6 - k468n4n6 - k261n2n6, _ k 261n2n6 - n7r7 1 ,

(9.2. 1 1 )

1 Wtis + k468n4n6 - n s r8 ,

(9.2. 1 3 )

_1 Wtis - Ws 6 + n6Wel - nprc .

(9.2. 14)

(9.2. 12)

Here the Wii parameters represent the rates of stimulated transitions between levels i and j, r; is the fluorescence lifetime of the ith level, kii is the rate of energy transfer from level i to level j, kiik is the rate of energy transfer from levels i and j to level k, pii represents the branching ratio for a transition between levels i and j, rc is the cavity lifetime for photons at the laser output wavelength, and Wei a6s cl/lc. The latter expression describes the stimu­ lated emission due to one photonjcm3 and is used to seed the cavity equa­ tion. In this expression, a65 is the stimulated emission cross section for the laser transition, I is the sample length, and lc is the cavity length. The vari­ ous energy-transfer transitions appearing in the rate equation are obvious

370

9. Rare-Earth-Ion Laser Materials

except the one originating on level 7. Any population of the 4 /5 level of Ho3 + (level 7) undergoes rapid radiationless relaxation to the 5 h level fol­ lowed by energy transfer to the 3 H5 multiplet of Tm3 + (level 3). This series of events is characterized only by the decay rate r:y 1 for simplicity. The stimulated transition rates must be determined from the sum over transition cross sections between individual Stark components taking into account the Boltzmann population distribution of the components of the initial level and the degeneracies of the components of the final level. The stimulated tran­ sition rates at the laser frequency are directly dependent on the concen­ tration laser photons in the cavity. For the pumping conditions and sample consentrations of interest here, the up-conversion processes have very little effect on the spectroscopic and lasing properties. However, when high con­ centration samples and the appropriate pumping conditions are used, up­ conversion can have a significant effect on the pumping dynamics of Tm3 + and Tm 3 + -Ho 3 + laser systems. Sequential up-conversion and relaxation processes can cause the populations of the metastable states to cycle in and out of inversion. From the extensive spectroscopic investigations of Tm, Ho : YAG, the values of all of the transition-rate parameters, branching ratios, cross sec­ tions, and lifetimes apearing in the rate equations are known. The laser cavity parameters are determined by experimental conditions. These equa­ tions can thus be used as a model for describing the optical pumping dynamics of Tm, Ho : YAG lasers with no adjustable parameters. This model does not account for the spatial distribution of excitation energy within the sample or specific cavity modes. Despite these simplifications, this model can be used to predict the temporal behavior of laser emission from this system for different excitation conditions. A computer simulation was performed with Eqs. (9.2.6-9.2. 14) using a fourth-order Runge-Kutta routine to predict the density of laser photons as function of time after the excitation pulse. 2 7 To simplify the model, the up-conversion loss mecha­ nisms were neglected. Examples of the results are shown in Fig. 9.20. Near threshold the numerical modeling predicts a single laser output spike about 200 f.1S after the pump pulse. At higher pump energies the time between excitation and lasing decreasing and the predicted laser emission appears as a series of relaxation oscillations. Figure 9.20 also shows the results of a laser-pumped Tm, Ho : YAG laser experiment. There is excellent agreement between the experimental results and the computer simulations in terms of the temporal characteristics of the laser operation. The threshold energy predicted by the numerical model is significantly lower than that found ex­ perimentally. This may be due to additional active and passive loss mecha­ nisms not take into account in the model. The results shown in Fig. 9.20 demonstrate the use of spectroscopic data and theoretical modeling to pro­ vide a better understanding of laser operational characteristics. In this case the results are especially useful in determining the optimum concentrations of Tm and Ho ions since these concentrations control the energy-transfer

9.2. Other Trivalent Lanthanide 15 10

Lasers

371

7.89 mJ

5 n I

�

:':

E u

0 -

._:

.:;-

ii c "' 0

c 0

0

.J: CL

0 2000

1 50 0

0

200

1 4. 2 mJ

' "' [] 0

500

3000 2000

0

1 8 . 2 mJ

200

400

400

'·::" [] 0

0

0

200

1 000 0

20

Ti m e

(A)

0

200

(11-s)

200

400

400

A L EXANDRITE

10

50

1 000

0

400

0

Ui .. z :: a:i a:

::5

)..

i.i5

z UJ ..

is:

11.4

mJ

:I

j

li

2

0

,

I

0

TIME

( B)

200

I;.<S)

400

: -1

FIGURE 9.20. Comparison of (A) numerical modeling and ( B) experimental results for a Tm, Ho : YAG laser pumped by an alexandrite laser. [Reprinted from Ref. 27(b) with permission of Elsevier Science NL, Sara Burgerhartstraat 25, 1 055 KV Amsterdam, The Netherlands.]

rates and thus are important in determining the temporal characteristics of laser operation. Similar laser measurements and computer simulations performed on the fluoride glass Tm, Ho : ZBAN give qualitatively similar results but with reduced time between the pump pulse and laser emission. 29 Thulium lasers are interesting because they provide the opportunity for continuous tuning of the emission over a broad wavelength range. 3° Figure 9.21 shows the fluorescence spectrum from the 3 F4 level in YAG : Tm3+ . At room temperature, the crystal-field Stark levels are significantly broad­ ened by phonon processes. There are 1 1 7 possible transitions between spe­ cific Stark levels. Each of these transitions has a width of approximately 1 0 nm. The combination of line broadening and the large number of closely spaced spectral lines allows the laser emission based on the 3 F4 -- 3 H6 tran­ sition to be tuned from 1 .87 to 2. 1 6 �tm. Similar tunability is observed for Tm3 + in other host crystals such as YSGG and YLF. 30 A smaller range of

372

9. Rare-Earth-Ion Laser Materials

"§

"iO.I

-e o.e

�

� 0.4 . en

�

0.2

FIGURE 9.2 1 . Room-temperature fluorescence spectrum from the 3 F4 level in YAG : Tm3 + (from Ref. 30).

pl2

N

1

azst

I

/

4 rl l /2

I

N3

a-3 +

( A)

Il S / 2

I

I

/ F

7

/

/2

FIGURE 9.22. Up-conversion pumping dynamics of Yb; Er : YAG lasers.

tunability in the 1 .04-,um spectral region can be obtained from Yb3 + in YAG and other host crystals. 3 1 This is associated with the 2 F5;2 -- 2 F7 ;2 transition and is due the same situation of phonon-troadened, closely spaced Stark levels as discussed for Tm3 + . The energy-level structures of rare-earth ions involving significant numbers of excited levels, some of which have long fluorescence lifetimes, are well suited for up-conversion lasers. These offer an alternative to pumping non­ linear optical crystals for frequence conversion and have the advantage of no phase-matching alignment problems and reduced problems with laser damage. Yb; Er : YAG is an example of a system in which the optical dynamics of up-conversion through sensitized energy transfer have been investigated in detail. 3 2 Figure 9.22 shows the relevant energy levels and transitions involved in the excitation dynamics of the green laser emission in this system. The rate equations describing the time evolution of the level

9.2. Other Trivalent Lanthanide Lasers

373

populations for this system are32

dt

dnt

dt

- P12nt + W2sn2n3 - Ws2nsnt + rx2sn2ns + A 2n2,

dn2 P12nt + Ws2nsnt - W2sn2n3 - rx2sn2ns - A 2n2, dn 3 dt = Ws2nsnt - W2sn2n3 + rx44n42 + rxssn 25 + As 3 ns + A 73n7 + A63 n6 + A s3ns + A 4 n4 , dn4 2 dt = 2rx44 n4 + A 74n7 + A64 n6 + As4 ns A 4n4 , dns 2 dt W2sn2n 3 - Ws2nsnt - 2rxssn 5 - rx2sn2ns + A6sn6 Asns, dt

dn6

dt

rx44n42 + A67n7 A6n6,

dn7 As - A7n7, 1 ns dns 2 dt rxssn s + rx2sn2ns - Asns,

(9.2. 1 5) (9.2. 1 6)

(9.2. 1 7) (9.2. 18) (9.2. 1 9) (9.2.20) (9.2.2 1 ) (9.2.22)

where n; is the population density of the ith level, PiJ is the external pumping rate from level i to level j, A; is the fluorescence decay rate for level i and A iJ is the total transition from level i to a specific level j, wij is the energy­ transfer coefficient from an ion in level i to an ion in level j, and aiJ is the up-conversion coefficient for two ions initially in levels i and j. All of the rate parameters in this model can be determined by spectro­ scopic measurements interpreted with the theories for radiative and non­ radiative transitions and energy transfer discussed previously. 33 The radia­ tive absorption and emission rates were determined through measured spedtral line strengths and Judd-Ofelt analysis. The nonradiative decay rates were determined by measuring the temperature dependencies of the fluorescence lifetimes and applying the energy-gap law using the parameters C 9.7 x 1 07 s- 1 , rx 3 . 1 x w 3 em, and an effective phonon energy of 700 cm- 1 determined previously for YAG. The energy transfer was found to involve multistep migration among the ytterbium ions before transfer to erbium. At average ion-ion separations determined with the assumption of random distributions. the dominant energy-transfer mechanism is electric dipole-dipole for both Yb-Yb and Yb-Er transfer. However, for the ions pairs that are spaced less than 1 0 A apart, higher-order multiple interaction may be important. The strength of the interaction between two Yb ions is about the same as the interaction strength between Yb and Er ions. How­ ever, the relative populations of the initial levels involved in the energy­ transfer transitions results in the Yb-Yb migration step rate being about an

374

9. Rare-Earth-Ion Laser Materials TABLE 9.6. Parameters for transitions in Yb; Er : YAG (after Ref. 33). Nonradiative decay rates Initial level

J .3 X 104 3 .9 X 104

Er: 4 S3;2 4 F7j2 4 /9/2 4 /1 1/2 4 /13/2 Yb: 2 F512

2.8 X 105 2.9 X 103 0.9 1 .4 X 10- 5

Energy transfer parameters Transition

Comment

Yb Yb migration step Yb Er up conversion Er Er up conversion

2.5 X 104 2.1

3.5

X

X

103 102

10 A separation 4 /1 1;2 --> 4 F7;2 at a 10 A separation 4 /1 112 -->4 F112 at a 10 A separation

For 6.5% Yb and 1 .0% Er: Total Yb Er transfer efficiency = 0.66 Csa (EDD) = 5.5 X 1 0-40 cm6 s- 1 Css (EDD) 1 .0 x 10- 39 cm6 s- 1 =

order of magnitude greater than the Yb-Er tansfer rate for the same ion­ ion separation. The backtransfer process from Er to Yb differs from forward transfer only in terms of spectral overlaps for the transitions and for this case the forward transfer is only 50% larger than the backtransfer proba­ bility. For the up-conversion processes, the Yb-Er transition was found to be stronger than the Er-Er process. This is due to the greater strength of the Yb transitions compared to Er transitions which offsets the larger spectral overlap of the Er-Er process. The results of these measurements are sum­ marized in Table 9.6. Figure 9.23 shows the time evolution of the up-conversion fluorescence from the 4S3 ;2 level at 550 nm after a 40 ns pump pulse at 940 nm. 33 The delay tetween the pump pulse and the maximum of the fluorescence is asso­ ciated with the temporal dynamics involving ion-ion energy transfer and is consistent with the predictions of the rate-equation model described above. The Yb; Er : YAG system described above has not been operated as an up­ conversion laser although the pumping dynamics leading to up-conversion fluorescence are similar to the excitation processes in systems that have been made to lase. Along with the characteristics of Nd3 + discussed previously, Er3 +, Pr3 +, Ho 3 +, and Tm3 + ions have been used for up-conversion lasers in both crystalline and glass hosts. The latter have been in fiber laser config­ urations, which can generally operate at room temperature. Fibers have the advantage for stepwise two-photon absorption processes of confining the

9.2. Other Trivalent Lanthanide Lasers 550 n m 50000

.. "'

40000

·�: "'

.30000

·;;

20000

..0

}; 0

:£

c:

397

le>

- - - - -

(rop. kp )

(rol .kl)

E8(ro8,k8)



-

1irov

lg>

(A) S pontaneous R aman Scattering

(B) Stimulated Raman Scattering

FIGURE 1 0.9. Energy levels and transitions for a Stokes Raman scattering process in which a pump photon of frequency Wp is destroyed while a phonon of frequency Wv and a Stokes photon of frequency Ws are created.

the induced polarizability of the material. This is given by the second tenn in a Taylor-series expansion of the polarizability a with respect to a vibra­ tional mode coordinate q, a

=

ao

+

()

oa q+ oq o

·

.. .

( 10.2.34)

The polarization of the medium resulting form this induced polarizability and acting to generate the scattered wave is ( 10.2.35) P = PnP = Pn aE , where Pn is the density of molecules and p is the molecular dipole moment. In a solid, the molecular vibrations contribute to form a vibrational wave that is described by a wave equation. This can be expressed in terms of a driven harmonic oscillator as given by Eq. ( 10.2.8), with a term added for the wave propagation 2 o q

oq

+

F + v2v v2 q + wv2 q = - '

( 10.2.36)

where r is the damping constant, V v is the group velocity of the vibrational wave, Wv is the fundamental oscillation frequency, n1 is the reduced mass of the molecule, and F is the driving force. The force of the electric field acting on the molecular dipole moment is F

=

� (p oq

·

E)

�

()

oa E E, oq o ·

( 1 0.2.37)

where only the first term in the expansion of Eq. ( 10.2.35) has been used.

398

1 0. Miscellaneous Laser Materials

The total field in the medium can be treated as being composed of copropagating pump wave and Stokes wave, ( 10.2.38)

Eq(z, t),

where the pump wave has a complex amplitude a propagation vec­ tor in the z direction with magnitude and a frequency Wp . The Stokes wave has the same properties with S replacing the p subscripts. is the unit vector in the x direction. The interference between the pump and Stokes waves produce a beat wave at the molecular vibration frequency = ks. This is a Raman-active = - ws and wave-vector magnitude optical phonon. This vibrational wave obeys Eq. ( 1 0.2.36) and has the gen­ eral form

kp,

ex

Wv

kv kp -

Wp

q

[qv(z, t) ei(k,z w,t) q� (z, t) e i(k,z w, t) ] ex .

( 10.2.39) + =! The polarization of the medium is now found by substituting Eqs. ( 10.2.34), ( 1 0.2.38), and ( 1 0.2.39) into Eq. ( 10.2.35) to give p

=

[�0 + G:) 0 ] (Epei(kpz wpt) + E; e i(kpz wpt) q

( 10.2.40)

Expanding this expression and keeping only the terms with the same phase factors for the pump and Stokes waves gives P�

[� Ep + ( 0qvEs] ei(kpz wpt) + c.c. � + [� E + ( 8 ) v Ep ] ei(ksz wst) + c . c , 8q 2 o

Pn

o s

o

q*

.

( 1 0.2.4 1 )

where c.c. stands for complex conjugate. This expression can be used as the source term in Eq. ( 10.2. 13) to give the set of coupled partial differential equations describing the pump and Stokes fields 1 ( 10.2.42) = +s,

8Ep 8Ep l. VpOJp K! qvE 8z Vp 8t VsOJs 8Es 1 8Es = lK! qv EP .

*

K! - f.loPnVsWs (88q�)o ' 4

(10.2.43)

where the coupling constant is given by _

( 1 0.2.44)

and vs and Vp are the phase velocities of the Stokes and pump waves, re­ spectively. Substituting Eqs. ( 10.2.38) and ( 1 0.2.39) for the wave amplitudes in Eq. ( 10.2.36) and applying the slowly varying amplitude approximation give an expression for the vibrational wave in terms of the pump and Stokes field

1 0.2. Nonlinear Optical Lasers

Kz: dqv dqv + rqv lKz. EpE* dt - Vv dz

399

amplitudes and coupling coefficient

(10. 2 .4 5)

S>

where

Kz 2m1wv (ooct.q) o . (10.2 .46) Equations (10. 2 . 42), (10. 2 . 4 3), and (10. 2 . 4 5) form a set of three coupled dif­ ferential equations describing Raman scattering in a dielectric medium. These are simplified by transforming to the retarded coordinate system and assuming V v Vp vs. The equations then become dqv + rqv lKzEp . ( t ) Es* , dt' aEp, . wp KJqvEs(t, ), (10. 2 .47 ) z ws 8Es lKJqv EP (t, ). oz' «

=

=

0

-

1

'

-

*

.

These equations must then be solved for different experimental conditions to determine the buildup of the Stokes wave in the material. For the case of steady state excitation (i.e . , when the excitation pulse width is much larger than the dephasing time of the optical phonon) the vi­ brational wave equation can be easily solved and the expression obtained for substituted into equations for the pump and Stokes fields

qv

aEp Wp KJKzi Esi 2Ep Ws r oz' 8Es KJKzi Epi 2Es r oz' _

(10.2.48)

The right-hand sides of these expressions involve the products of three elec­ tric field and thus can be written in terms of a third-order susceptibility and related to the third-order nonlinear polarization,

(10.2.49) where

XRaman (ws, -wp , Wp, ws) - - l. 4eoPmn wv (-aoct.q)2 -r1 . (3)

.

_

0

(10.2 . 50)

400

10. Miscellaneous Laser Materials

The electric fields of the pump and Stokes beams can be expressed in terms of the intensity (irradiance) of the light beams through the usual rela­ tionship I (eov/2) I E I 2 . The expressions in Eq. ( 10.2.48) then become dfp dz'

-

Wp

(10.2. 5 1 )

ps ws yl l ,

where the gain coefficient is given by

( )

a 2. Pn Ws oerwvm aq 0 a

y - 4e

(10.2.52)

Thus the intensity of the Stokes beam traveling through a material increases in direct proportional to r- 1 and (arxjaqf The expression for the rate of spontaneous Raman scattering of photons in a solid angle iln is given by1 5

= /A A

--

dWsp

Pn ).� a(J Is Wp ,

h vsLlvs an

(1 0.2.53)

where W with being the cross-sectional area of the pump beam, Llvs is the full width at half maximum of the Stokes spectral line, and a(J/an is the differential cross section for spontaneous Raman scattering. Comparing this expression with Eqs. ( 1 0.2.5 1 ) and (10.2.52) show that the can coeffi­ cient can be expressed as

Pn A� y ( v ) = h vs an S ( v) a(J

pn ).� a(J __, Ymax = h2vsLlvs an '

( 10.2.54)

where a Lorentzian line-shape function has been used in the latter expres­ siOn. Since measurements of gain or scattering cross section are generally made at a specific wavelength, it is important to develop expressions relating these parameters at different wavelengths. 1 6 Using second-order perturbation theory and Fermi's golden rule, an expression can be derived for the Raman transition probability per unit time per unit volume per unit energy. This can be multiplied by the density of radiation modes per solid angle at the Stokes frequency and by the population density of the initial state of the material, and then divided by the effective material excitation density to obtain an expression for the differential scattering cross section per solid angle,

a(J an

8n3 ns

2P

i-

Here the n; represent the photon occupation numbers at the pump and Stoke frequencies, Mfi is the matrix element for the transition, and P; is the population density of the initial state of the material. This can be used with Eq. ( 10.2.54) to obtain the following relationships between gain at two wavelengths and scattering cross sections at two wavelengths,

1 0.2. Nonlinear Optical Lasers

( a(JI an) Ap2 ( a(J1 an) Ap ]

401

( 10.2.55)

From a theoretical perspective, expressing the Raman gain coefficient in terms of the derivative of the molecular polarizability as in Eq. ( 10.2.48) can be useful. However, this quantity is difficult to measure directly, so for quantitative interpretation of experimental results it is more useful to use the expression for the gain coefficient in Eq. ( 10.2.50) where the Raman line­ width and differential cross section for spontaneous Raman scattering can be measured from conventional Raman spectroscopy. Solving Eqs. ( 1 0.2.5 1 ) gives an expression for the change in the intensity of the Stokes beam as it propagates through the medium 1 2 ' yloz -- e

ls(O) , Ip (O) ( 10.2.56) ls(z ) Io , (O) w + __ 1 ws lp (O) lp (z' ) + (wp lws)Is(z' ) is the conserved ( photon conservation) loz eY

,

where Io total field intensity. This expression shows that the intensity of the Stokes beam initially increases approximately exponentially through the medium, and then tends toward a saturation value. For a distance of travel L, the total gain G is given by ( 10.2.57) G yl0 L with y given by Eq. ( 10.2.52) or ( 10.2.54). For efficient Raman lasers, it is generally best to use a material with a high Raman gain G. Using the theory outlined above, the gain given in Eq. ( 10.2.57) can be expressed in terms of material parameters as

1

a(J

( 10.2.58)

where Pn is the density of states of the phonons involved in the scattering event, Vs is the frequency of the Stokes transition, and a(JI an is the Raman differential scattering cross section. In this expression a Lorentzian line shape has been assumed and this is proportional to the inverse of the tran­ sition linewidth Ll vs. The density of states is important in understanding the temperature dependence, but most laser systems are required to operate at room temperature. Thus for obtaining a specific laser output frequency from a given Raman-shifted pump laser, the criteria for a high-gain Raman mate­ rial are a narrow Raman linewidth and a high Raman-scattering cross sec­ tion. The material must also be transparent in the spectral region of interest. Table 10.3 lists the Raman gain and linewidth properties of several crystals that may be important for Raman laser applications. The width of a line in a Raman spectrum is determined by the lifetime of

402

10. Miscellaneous Laser Materials

the final state of the transition. This is generally dominated by the lifetime of the of the phonon that is produced in the Stokes emission process. The phonon generated through Raman scattering will decay into other phonons with lower energies as the system evolves toward thermal equilibrium. The phonon-phonon interaction mechanism causing this decay to occur involves anharmonic coupling of the vibrational modes. The most important contri­ bution to the lifetime of this optic phonon is a three-phonon anharmonic interaction in which the initial phonon scatters into two other phonons with conservation of energy and wave vector. The expression for the width of the Raman spectral line is given by1 7 2 nV Lhs dK L i ( nsK1 + l , ns' K2 + l , no O I HA i nsK p ns' K2 , no 1 ) 1 3 2 n (2n) s ,s

J

1

( 10.2.59)

with a similar term describing the self-energy that shifts the frequency of the Raman line due to interaction with the phonon field. The process described here involves the destruction of the initial phonon of frequency w0 and wave vector K simultaneously with the creation of two other phonons of frequen­ cies Ws, Ws , and wave vectors K ! , K2, with the requirements of conversation of energy and wave vector. At high temperatures other processes involving phonons in different branches may contribute to the line broadening and line shifting. The rate at which the Stokes phonon decay occurs depends on the strength of the anharmonic coupling of the vibrational modes and the con­ servation laws that are required for the transition to occur. The latter re­ quire that the sum of the energies of the two final phonons equals the energy of the initial phonon, and that the sum of the wave vectors of the two final phonon equals the energy of the initial phonon. This depends on the density of phonon states for the material. The photon-phonon interaction in the Raman scattering process creates an optic phonon with a near zero wave vector. Therefore the relaxation process for this phonon can involve any phonons with equal and opposite wave vectors. The key parameter in the Eq. ( 10.2.59) is the anharmonic interaction Hamiltonian HA . The strength of this operator depends on the anharmo­ nicity of the crystal which is characterized by the Gruneisen parameter Y a · This can be expressed in thermodynamics quantities as Ya

r:x VB

Cv

3 r:xL V

KCv .

( 10.2.60)

Here r:x is the volume expansivity while r:xL is the linear expansion coefficient, V is the volume, Cv is the specific heat, B is the bulk modulus, and K is the compressibility. These thermal properties have been tabulated for many materials, but unfortunately are not available for most of the materials of interest for Raman lasers.

10.2. Nonlinear Optical Lasers

403

Thus crystals having the smallest Raman linewidth will be those with small values of y0 and a vibrational density of states that is not compatible with conservation of energy and wave vector for the decay of the Raman phonon. 1 8 The temperature dependence of the stimulated Raman scattering Stokes spectral line in Ba(N0 3 ) 2 crystals has been investigated in detail and interpreted in terms of specific phonon decay processes. 1 9 The second parameter in Eq. ( 10.2.58) of importance for high gain in Raman laser materials is the scattering cross section. The cross section for Raman scattering depends on the strength of the photon-phonon coupling which is proportional to the modulation of the laser-induced polarizability. An accurate theoretical calculation requires knowledge of the electronic and vibrational wave functions as well as the local electron-phonon interaction. This information is not usually available. Therefore it is necessary to resort to qualitative arguments based on the lattice structure of the crystal. The vibrational modes of a crystal are classified according to the irredu­ cible representation of the symmetry group according to which they trans­ form. To satisfy conservation of wave vector and energy, only optic phonon modes near the center of the Brillouin zone will make significant contribu­ tions to Raman scattering. In some cases a crystal structure can be described in terms of molecular units. In this case the local mode molecular vibrations generally produce larger changes in polarizability than the lattice modes. Although the local crystal-field environment will alter the vibrational modes of a free molecule, the molecular vibrations are a good starting point for analyzing the vibrational modes of the crystal. The Raman scattering cross section is proportional to the square of the change in the polarizability of the vibrating group with respect to the normal mode of vibration, There are three aspects of the structural symmetry that are important in determining the cross section. 2° First, the symmetry classification of the vibrational mode should have allowed selection rules for Raman transitions and should give the maximum value of The totally symmetric breath­ ing modes of vibration classified as A 1 g have the greatest change in the polar­ izability tensor. Tightly bound molecular groups such as tungstates, molyb­ dates, nitrates, etc., generally have A 1g vibrational modes that maintain their integrity when the molecular group becomes a structural unit of a crystalline solid. Second, molecular groups such as the tungstates, molyb­ dates, nitrates, etc., mentioned above, all have allowed electronic transitions to charge transfer levels at low energies. Since the polarizability tensor has a resonant denominator that is dominated by a virtual transition to the lowest-lying energy level as the intermediate state, these molecular com­ plexes have high values of In a crystalline environment, these charge transfer levels generally shift to lower energies resulting in enhanced values of polarizabilities. With the appropriate symmetry conditions as discussed above, this results in high values of Third, the structural symmetry should result in chemical bonding characteristics that maximize that rate of change in polarizability during a vibration. Covalent bonding allows for a

(oaf oq) 2 .

oafoq.

a.

oafoq.

1 0. Miscellaneous Laser Materials

404

TABLE 1 0.4. Character table for D 3h and basis functions.

D 3h A 'I A 2' E' A "I A 2" E"

E

2 I I 2

2CJ

3 C2

(Jh

2SJ

3 av

1 I 1 I

I 0 I I 0

I 2 I I 2

I I I 1 I

I 0 1 I 0

Normal modes of AB 3

Basis functions

Rz

(x, y) z

ctxx + !Xyy, Cl.zz

VI

(ctxx - ctyy, ctxy)

VJ , V4

( Rx, Ry)

(ctxz , IXyz)

V2

greater change in polarizability than ionic bonding. Conjugated bonds enhance the polarizability change. Crystal structures such as diamond, zinc­ blende, and wurtzite are favorable for covalent bonding, and thus can have high Raman scattering cross sections. As an example of the group theory analysis required for interpreting Raman spectra, consider an AB3 molecular group such as N0 3 or C03 . The symmetry point group for this structure is D 3h with the characters of the irreducible representations given in Table 1 0.4. The basis functions listed in the table show how the components of a vector, a rotation operator, and the components of the polarizability tensor transform according to the irredu­ cible representations of the group. Using the group theory techniques de­ scribed in Sec. 2.2 along with the normal mode vibrational analysis tech­ niques described in Sec. 4. 1 , the four normal modes of vibration for the AB3 molecule are shown in Fig. 10. 10. By analyzing the symetry transformation properties of these vibrational modes, they can be associated with the irre­ ducible representations of the D 3h point group as shown in the character table. Note that v3 , V4 transform together as the bases of the doubly degen­ erate representation E'. From the character table it can be seen that V J , v3 , and v4 all transform in the same manner as some of the components of the

vl

?

v2

v3

FIGURE 1 0. 1 0. Normal modes of vibration of the AB3 molecule.

6

0

.

�

,0

0

G

0 v4

,P

10.2. Nonlinear Optical Lasers

405

TABLE 10.5. Character table for D 3h and basis functions. c3 A E

E

c3

c3z

e

e'

*

e

e

D3h Correlation

A; , Ar E ' , E"

polarizability tensor and therefore have allowed Raman transitions. On the other hand, v2 is not Raman active. In general, the strongest Raman tran­ sitions are those associated with the diagonal elements of the polarizability tensor. Thus, this analysis predicts that the A'1 ( v 1 ) normal mode vibration will produce the biggest peak in the Raman spectrum of the AB3 complex. When a molecular group is part of a crystal structure, the factor group method described in Sec. 4. 1 for local mode vibrational analysis is used to determine the change in symmetry properties and splitting of degenerate modes. For example, the NO .J molecular ions occupy a lattice site with C3 point-group symmetry in a Ba (N0 3 h crystal. The correlation of irreducible representations between the C3 and D3 h point groups is given in Table 10.5. The crystallographic point group for Ba ( N03 h is Th . Thus, the next step in the factor group analysis is to determine the correlation between the irredu­ cible representations of the C3 site symmetry and the Th crystallographic point-group symmetry. The most important representation under considera­ tion is the one according to which the VI vibrational mode of the molecular ion transforms. The correlation relationships for this are A'1 (D3h ) -A ( C3 ) -- A9, Au, T9 , Tu ( Th) . Thus the VJ vibrational mode of the free molecule splits into two nondegenerate and two triply degenerate vibra­ tional modes in the barium nitrate crystal. Components of the polarizability tensor transform according to two of these vibrational modes of the crystal, A9 and T9, and thus they have allowed Raman transition . The sum of the diagonal components 1Xxx + !Xyy + IXzz transforms according to the totally symmetric mode A9. This VJ (A9) vibrational mode produces a strong, nar­ row line in the Raman spectra. For the free nitrate ion the Raman shift for this line is 1 050 cm- 1 , while the position of this peak in the Ba (N0 3 h crys­ tal spectrum is 1047.3 cm- 1 • Thus crystals having the highest Raman cross section will be those having molecular structures where the molecular breathing mode acts as a zone center vibrational mode of the crystal that transforms as the A 19 irreducible representation of the crystallographic point group symmetry. If molecular polarizabilities are known, those with the highest polarizability will have the highest Raman cross section. The high Raman gains of the materials listed in Table 10.3 are associated with totally symmetric vibrational modes of molecular groups such as nitrates, tungstates, and molybdates. Because of this high gain, it is difficult to use these materials in an external cavity con­ figuration for frequency shifting a laser beam as is done with Raman gas

406

10. Miscellaneous Laser Materials

cells. Nonlinear cascade and higher-order photon interaction processes cause the first Stokes emission to be depleted and emission to occur at other frequencies. 2 1 Thus the best use of solid-state Raman shiftiing is in an intracavity configuration. An intracavity Raman laser operatin at 1 .5 11m consisting of a Nd-YAG laser operating at 1 .3 J1f pumping a Ba (N0 3 h Raman laser has recently been demonstrated. 22 The coupled cavity design used for this laser was optimized to produce an optical-to-optical conversion efficiency near to the quantum limit of 85% along with an output beam having near-diffraction­ limited beam quality. The poor beam quality of the pump laser was con­ verted to a Gaussian beam output of the Raman laser through the process of stimulated Raman scattering. This Raman beam cleanup effect is attrib­ uted to the four-wave-mixing nature of this x (J) processes. No detailed mathematical description has been derived to describe the mechanism of Raman beam cleanup in a Raman laser. The general process that is taking place can be understood by examining the driving polari­ zations responsible for stimulated Raman scattering (SRS). In the slowly varying envelope approximation, the third-order nonlinear polarizations associated with SRS are given by the expressions derived above to be ( 1 0.2.6 1 ) and

2 p ( 3 \ws , r) = x�; : I Ep(r) 1 Es (r) e-iksZ , where i� ) and x�; represent the third-order tensoral nonlinearty of the medium, and Ep (r) and Es (r) represent the slowly varying envelope of p

the three-dimensional electric field vector of the pump and Stokes beams, respectively. Applying the convolution theorem in terms of the spatial fre­ quency spectrum of the pump and Stokes fields, the polarization can be written as ( 1 0.2.62) and

respectively, where k is the wave vector. In generating the stimulated Stokes field it is evident that each plane wave component of the pump field mixes with every other pump component via the complex autocorrelation Yp (k) �p (k)** �; (k) . This apparent pump spectrum is then combined with all pos­ sible plane-wave compontens of the seeding Stokes field via the convolution Yp (k)** �s (k) . Hence the stimulated Stokes field is a smoothed version of the seeding Stokes spectrum. This smoothing operation is performed twice per

10.3. Color-Center Lasers

407

round-trip in the Raman laser resonator. The central limit theorem predicts a Gaussian spectrum to result from a multiorder spectral convolution where the order is greater than three. Therefore a Gaussian transverse field distri­ bution is produced after only a few round trips in the laser cavity. Under the conditions of adequate Raman gain per pass and cavity mode matching, a TEM00 diffraction limited Raman laser output can be achieved over a very large dynamic range without the use of intracavity apertures. This technique of beam shaping enables very efficient high-power laser oscillators to be built. The role of four-wave mixing in Raman beam cleanup becomes evident when the plane-wave approximation is applied to these equations. With �p ( k) �s ( k)

= =

�ptJ ( k - kpi )

+

�p2J ( k - kp2 ) ,

�st J ( k - ks t ) ,

as the example spectra, the driving polarization for the stimulated Stokes fields is given by p

( J) (Ws2 , r )

= �; : X

+

2 2 i r [( j �pt i �Sl + � �p2 1 �s t ) e- (ks i -ks2 ) · ei(kp2-kpl +ksl -ks2 ) ·r l ei (kpl -kp2+ks i -kS2 ) ·r +

( 10.2.63) where the first term represents the normal Raman gain and the second and third terms represent secondary Stokes radiation generated through four­ wave mixing. The normal Raman gain is shown to be independent of pump direction and is hence "phase matchless." The Stokes radiation generated by four-wave mixing, however, depends strongly on the direction of the pump radiation as a result of the phase mismatch. These are the terms responsible for the ultimate smoothing of the stimulated Stokes radiation. 10.3

Color-Center Lasers

Dopant ions such as those discussed above are one type of point defect in a host material that can act as a luminescent and lasing center. Another type of optically active point defect is a color center. A typical color center con­ sists of an electron trapped at ion vacancy site in the lattice. Lasers based on color centers have been demonstrated in both fluoride and oxide crystals. 2 3 The production of color centers in crystals generally depends on thermal or radiation treatments. Color-center lasers in the visible and near infrared spectral region generally operate on electronic transitions of trapped elec­ tron defects, while far-infrared color center lasers operate on the vibrational transitions of molecular defects such as CN - . Neutral atom defects can also produce laser emission in the near-infrared spectral region. Typical absorp­ tion and emission spectra for color centers exhibit strong, broad bands as-

408

10. Miscellaneous Laser Materials TABLE 1 0.6. Examples of common color-center lasers. ( Data from Refs. 24 30.) Type of center p2+

F2 + : o 2

p2 FA ( II )

Tl0 ( 1) eN

Host crystal

Tuning range (tml)

LiF KF NaCI KBr LiF KCI : Li RbCI : Li KCI KBr

0.82 1 .05 1 .22 1 . 50 1 .42 1 .78 1 . 86 2. 1 6 1 . 1 1 .25 2.3 3 . 1 2.5 3.65 1 .4 1 .6 4.86 (at 4 K)

sociated with allowed transitions having high oscillator strengths (f � 0.2) and radiative lifetimes of the order of 100 ns. The quantum efficiencies can be close to 1 00%. These are favorable for optical pumping with lasers and result in high gain cross sections and low thresholds. The color centers are strongly coupled to the host lattice and operate as quasi-four-level systems similar to other vibronic lasers. The homogeneously broadened emission band is useful for tunable laser emission or mode-locked operation. The large Stokes shift between the emission and absorption minimizes losses due both to ground-state and excited state absorption of laser emission. Table 1 0.6 lists some of the common color-center laser. 24 - 28 The nomenclature for designating specific types of defect centers can be found in various review articles on color centers. 29 One major problem with color-center lasers is the lack of stability of color centers at room temperature. The mobility of electrons and ions in the host crystals can cause the number of color centers to decay with time or change into aggragate centers. Since color center lasers are pumped by other lasers, photobleaching effects are especially important. Techniques including the introduction of dopant ions have been developed to stabilize specific types of color centers. However, cw color center lasers still operate generally only at low temperatures and only a few pulsed systems exhibit stable operation at room temperature. Alkali halide crystals such as LiF are common host materials for color­ center lasers. Fi defect centers have been especially useful for laser opera­ tion with these crystals. 3° Figure 10. 1 1 shows a portion of the LiF lattice with an Fi defect center. This type of color center involves an electron in an interstitial position between two fluorine ion vacancies in the lattice. The energy levels for this type of center are found by modeling it as a hydrogen molecular ion. 3 1 Using molecular orbital notation, the relevant energy levels and transitions for an Fi center is shown in Fig. 10. 1 2. Note that when the electron is excited to an upper-state energy level, the spatial extent of its orbital increases, leading to a stronger interaction with neighboring ions.

10.3. Color-Center Lasers u+

0

•

0 •

0

'

0

F2+ center

I

oO Do 0

•

•

0

•

0

0

•

•

0

0

•

•

409

FIGURE 1 0. 1 1 . Section of a LiF crystal lattice with a Fi defect center.

0

This results in a lattice relaxation that changes the position of the energy levels as shown in the figure. The operation of LiF : Fi color center lasers is associated with the lsag -2pau transition shown in Fig. 1 0. 12(A) . Due to the parity change, this tran­ sition has a high oscillator strength for strong pump absorption and a high quantum efficiency for radiative emission at room temperature. The excited state lattice relaxation effect mentioned above results in a large Stokes shift, as seen from the absorption and emission spectra shown in Fig 10. 1 2( B). This allows for laser output to be tuned across the entire emission band without ground-state absorption loss. In addition, there are no energy levels in a position that gives rise to excited state absorption. The positions of the energy levels depends on the dielectric constant of the host crystal. This can change significantly for different alkali halides, and thus the peak of the laser emission band for Fi color center lasers ranges from about 0.9 to about 2.2 fJ.m depending on the host crystal. The optical properties of this type of color center are highly anisotropic with the electric dipole transition being polarized along the line joining the two vacancies that make up the defect. The ensemble of Fi defect centers are distributed in orientation along the six different [1 1 0] crystallographic directions. Only those centers orientated in the direction compatible with the polarized pump light take part in the lasing operation. At high levels of pumping, it is possible for multiphoton excitation to take place, as shown in Fig. 10. 1 2(A). If excita­ tion to a high-lying energy level takes place at room temperature, the center can reorient itself along a different crystalographic direction. This decreases the number of Fi centers aligned in the appropriate direction for laser operation and thus results in photobleaching of the material. This process can be minimized by operating at low pump irradiances or by maintaining the crystal at low temperatures where the ionic motion required for defect

410

10. Miscellaneous Laser Materials

(A) EMISSION

ABSORPTION

0.5

0.6

0.7

0.8

A(!lm)

0.9

1.0

1.1

1.2

(B)

FIGURE 1 0. 1 2. Energy levels and spectra of LiF : Fi . (A) Energy levels for Fi center. ( B) Optical spectra for LiF : Fi -

reorientation is less probable. In addition, by putting an impurity ion such as 02 - next to the Fi center, orientational bleaching is inhibited. F2 centers in LiF crystals also provide room-temperature operation. 30 The tuning range is shifted to longer wavelengths compared to Fi centers. A color-center laser based on a different class of defect is KCl : T1° ( 1 ) . In this case the neutral thallium atom substitutes for a cation on a lattice site that is perturbed by a single nearest-neighbor anion vacancy. The perturba­ tion lifts the degeneracy of the energy levels and mixes even- and odd-parity states to give transition oscillator strengths on the order of 0.008. This is a stable defect center that gives tunable laser output between 1 .4 and 1 .6 f.i.

1 0.4. Other Solid-State Lasers

41 1

One interesting problem that occurs with lasers that have a high gain and a broad homogeneous linewidth, such as color-center lasers, is spatial hole burning. This makes it difficult to obtain single longitudinal mode opera­ tion. In a standing-wave cavity with a homogeneous gain profile, it would normally be expected that when one cavity mode begins to oscillate, all other modes would be surpressed. However, for a high-gain material the peak electric field intensity saturates the gain media in the spatial regions where it is a maximum for his lowest threshold mode. This intensity has a cosine-squared pattern. The spatial regions between these maxima are not saturated and can support gain for a longitudinal mode with a different frequency. This results in the simultaneous oscillation of two longitudinal modes. If single-mode operation is desired, it is necessary to use etalon in the cavity to select one mode and surpress the other. Color center lasers have been very useful laboratory sources for tunable laser emission in the near-infrared spectral region. However, for most prac­ tical applications outside the laboratory their usefulness is limited by their lack of stability at room temperature. One important application for mode­ locked color center lasers is coupling them to fibers lasers to produce soliton lasers. 32 The sync-pumped mode-locking produces trains of pulses having temporal widths between 5 and 1 0 ps. The dispersive qualities of fibers can be used to produce pulse compression that reduces the pulse widths to the femtosecond time regime. 1 0. 4

Other Solid-State Lasers

Several other types of optically active centers have been investigated for use as lasers. These include both organic and inorganic molecules as well as different types of ions such as those with closed-shell electronic configura­ tions. Some of these novel systems are beginning to be developed to the level of commercial laser systems. Organic dye molecules such as rhodamine 6G can exhibit very strong, broad-band fluorescence emission that make them very attractive for lasing centers. Dye lasers based on these molecules in liquid solvents such as alcohol have been very successful in the visible spectral region using either flashlamp or laser pumping. Using different dye-solvent combinations, broadly tunable laser emission can be obtained in bands between about 3 1 0 and 1 50 nm. The dye abosorption and emission bands are associated with transitions between the vibrational sublevels of the electronic states of the molecule. A generic energy-level diagram for molecular transitions is shown in Fig. 10. 1 3 . The ground-state manifold is a singlet state while there are sets of both singlet and triplet excited states. The singlet-singlet transitions are spin-allowed and result in strong absorption and emission bands. The latter have fast fluorescence lifetimes (of the order of a few nanoseconds). If intersystem crossing occurs to the triplet manifold, the molecule can pro-

412

10. Miscellaneous Laser Materials

FIGURE 1 0. 1 3. Typical energy-level diagram and optical transition for dye molecules.

RADIATIONLESS_ , DECAY (Ips)

•

I

,

I

I INrERSYSTEM ' , . :- c ROSSING (IOns)

(l ns) PHOSPHORESCENCE (lJ.IS)

STATE SINGlET

duce a long-lived emission (of the order of a few microseconds) . Excited state absorption can be a significant problem in the triplet energy-level mani­ fold. Initial attempts to incorporate organic dye molecules in solid host materials were unsuccessful as lasers because of rapid photodegradation under optical pumping. These were generally based on the same types of organic molecules used for liquid dye lasers and standard glass or polymer host materials. Recent developments of new types of dye molecules and host materials have significantly decreased the photodegradation problem. 3 3 One class of these new materials includes pyrromethene-BF2 complex dyes doped in acrylic plastic or xerogel hosts. 3 3 • 34 A typical dye molecule, optical spectra, and laser emission band are shown in Fig. 10. 14. Note that for this particular example, the tuning range is limited on the high-energy side by the ground-state absorption spectrum. These types of solid-state sys­ tems have the potential to replace liquid dye lasers in the near future. How­ ever, the average power handling capability of the host ultimately limits their utility. One spectral region that is not easily covered by primary solid-state lasers is the near ultraviolet. Since lasers in this region have important applications in areas such as medical procedures and material processing, there is signifi­ cant interest in developing solid-state sources that operate in this region. Currently it is possible to have solid-state systems operating between 190 and 400 nm by using frequency multiplier nonlinear optical crystals in con­ junction with visible and near-infrared lasers. Because of the complicated nature of these systems, it would be preferable to generate the uv light di­ rectly from the laser material. There are several types of materials that have the potential for accomplishing this task; however, it should be noted that pump sources are a limiting technology. Long-lived uv lamps are not avail­ able and pumping with uv gas lasers defeats the purpose of having a solid­ state source. One possible solution may be surface discharge sources, but their use for laser pumping devices has not yet been thoroughly explored.

1 0.4. Other Solid-State Lasers

FIGURE 1 0. 14. Spectral properties of PM-HMC solid-state dye laser (after Ref. 34) . (A) PM­ HMC molecular complex. ( B) Optical spectra of PM-HMC in xylene. (C) Broadband laser emission of PM-HMC.

(A) :r

s

>-

1.2

� ·a "

� � BCl

E:�

0.9

j

0.0 400 500

!:

>

t.tl

0 Cl �