Terahertz Spectroscopy: Principles and Applications (Optical Science and Engineering Series)

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Terahertz Spectroscopy Principles and Applications

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OPTICAL SCIENCE AND ENGINEERING Founding Editor Brian J. Thompson University of Rochester Rochester, New York

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108. Optics in Magnetic Multilayers and Nanostructures, Sˇtefan Viˇsˇnovsky’ 109. Optical Inspection of Microsystems, edited by Wolfgang Osten 110. Applied Microphotonics, edited by Wes R. Jamroz, Roman Kruzelecky, and Emile I. Haddad 111. Organic Light-Emitting Materials and Devices, edited by Zhigang Li and Hong Meng 112. Silicon Nanoelectronics, edited by Shunri Oda and David Ferry 113. Image Sensors and Signal Processor for Digital Still Cameras, Junichi Nakamura 114. Encyclopedic Handbook of Integrated Circuits, edited by Kenichi Iga and Yasuo Kokubun 115. Quantum Communications and Cryptography, edited by Alexander V. Sergienko 116. Optical Code Division Multiple Access: Fundamentals and Applications, edited by Paul R. Prucnal 117. Polymer Fiber Optics: Materials, Physics, and Applications, Mark G. Kuzyk 118. Smart Biosensor Technology, edited by George K. Knopf and Amarjeet S. Bassi 119. Solid-State Lasers and Applications, edited by Alphan Sennaroglu 120. Optical Waveguides: From Theory to Applied Technologies, edited by Maria L. Calvo and Vasudevan Lakshiminarayanan 121. Gas Lasers, edited by Masamori Endo and Robert F. Walker 122. Lens Design, Fourth Edition, Milton Laikin 123. Photonics: Principles and Practices, Abdul Al-Azzawi 124. Microwave Photonics, edited by Chi H. Lee 125. Physical Properties and Data of Optical Materials, Moriaki Wakaki, Keiei Kudo, and Takehisa Shibuya 126. Microlithography: Science and Technology, Second Edition, edited by Kazuaki Suzuki and Bruce W. Smith 127. Coarse Wavelength Division Multiplexing: Technologies and Applications, edited by Hans Joerg Thiele and Marcus Nebeling 128. Organic Field-Effect Transistors, Zhenan Bao and Jason Locklin 129. Smart CMOS Image Sensors and Applications, Jun Ohta 130. Photonic Signal Processing: Techniques and Applications, Le Nguyen Binh 131. Terahertz Spectroscopy: Principles and Applications, edited by Susan L. Dexheimer

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Terahertz Spectroscopy Principles and Applications

Edited by

Susan L. Dexheimer

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑0‑8493‑7525‑5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Dexheimer, Susan L. Terahertz spectroscopy : principles and applications / Susan L. Dexheimer. p. cm. ‑‑ (Optical science and engineering) Includes bibliographical references and index. ISBN 978‑0‑8493‑7525‑5 (hardcover : alk. paper) 1. Terahertz spectroscopy. I. Title. II. Series. QC454.T47D49 2007 543’.5‑‑dc22

2007024954

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Contents Preface.......................................................................................................................xi Editor..................................................................................................................... xiii Contributors............................................................................................................ xv

Section I Instrumentation and Methods Chapter 1 Terahertz Time-Domain Spectroscopy with Photoconductive Antennas..................1 R. Alan Cheville Chapter 2 Nonlinear Optical Techniques for Terahertz Pulse Generation. and Detection — Optical Rectification and Electrooptic Sampling......................... 41 Ingrid Wilke and Suranjana Sengupta Chapter 3 Time-Resolved Terahertz Spectroscopy and Terahertz Emission. Spectroscopy............................................................................................................. 73 Jason B. Baxter and Charles A. Schmuttenmaer

Section II A pplications in Physics and Materials Science Chapter 4 Time-Resolved Terahertz Studies of Carrier Dynamics in Semiconductors, Superconductors, and Strongly Correlated Electron Materials.............................. 119 Robert A. Kaindl and Richard D. Averitt Chapter 5 Time-Resolved Terahertz Studies of Conductivity Processes. in Novel Electronic Materials................................................................................. 171 Jie Shan and Susan L. Dexheimer

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Chapter 6 Optical Response of Semiconductor Nanostructures in Terahertz Fields. Generated by Electrostatic Free-Electron Lasers...................................................205 Sam G. Carter, John Cerne and Mark S. Sherwin

Section III A pplications in Chemistry and Biomedicine Chapter 7 Terahertz Spectroscopy of Biomolecules............................................................... 269 Edwin J. Heilweil and David F. Plusquellic Chapter 8 Pharmaceutical and Security Applications of Terahertz Spectroscopy................. 299 J. Axel Zeitler, Thomas Rades and Philip F. Taday Index....................................................................................................................... 325

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Preface The development of new sources and methods in the terahertz, or far-infrared, spectral range has generated intense interest in terahertz spectroscopy and its application in a wide range of fields. Terahertz frequencies, broadly defined as 0.1–30 THz, span the range of low-energy excitations in electronic materials, low-frequency vibrational modes of condensed phase media, and vibrational and rotational transitions in molecules, making this a key spectral range for probing fundamental physical interactions as well as for practical applications. Moreover, the short pulse durations achievable with terahertz techniques based on femtosecond laser sources allow timeresolved measurements on femtosecond timescales, which were previously inaccessible in this spectral range. This volume provides an up-to-date reference on state-of-the-art terahertz spectroscopic techniques, focusing in particular on time-domain methods based on femtosecond laser sources, and reviewing important recent applications of terahertz spectroscopy in physics, chemistry, and biology. Chapters are contributed by leading experts in each topic. The focus of this book specifically on spectroscopy and the range of applications presented set it apart from the few other recent volumes in the emerging terahertz field. The reader is referred to Sensing with Terahertz Radiation edited by Daniel Mittleman (Springer, 2003) and to Terahertz Sensing Technology edited by Dwight L. Woolard et al. (World Scientific, 2003) for applications including ranging and imaging, and Terahertz Optoelectronics edited by Kiyomi Sakai (Springer, 2005) for a review focusing specifically on work on generation of terahertz radiation carried out in Japan. The first section of this book is devoted to instrumentation and methods for time-domain and time-resolved terahertz spectroscopy. The first chapter provides a comprehensive discussion of time-domain terahertz spectroscopic measurements using photoconductive antenna sources and detectors, including methods for determining optical constants from time-domain measurements. The second chapter provides a review of time-domain terahertz techniques based on the nonlinear optical processes of optical rectification for terahertz pulse generation and free-space electrooptic sampling for terahertz pulse detection, and the third chapter reviews current state-of-the-art femtosecond time-resolved techniques and terahertz emission spectroscopy using femtosecond laser sources. Measurements using free electron laser sources and the established far-infrared frequency-domain spectroscopic techniques of FTIR and photomixing are also discussed in the context of specific applications in Sections II and III. Applications of terahertz spectroscopy to key areas of current interest, both fundamental and applied, are presented in Sections II and III of the book. The second section is devoted to applications in physics and materials science, including studies of carrier processes in state-of-the-art semiconductor materials, superconductors, novel

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electronic materials, and semiconductor nanostructures. The third and final section of the book is devoted to applications of terahertz spectroscopy to the biological and chemical fields. This first chapter of this section reviews studies of biologically important molecules, and the final chapter reviews applications of terahertz spectroscopy to pharmaceutical analysis and to the detection of security hazards. Susan L. Dexheimer Editor

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Editor Susan L. Dexheimer received an S.B. degree in Physics from the Massachusetts Institute of Technology and a Ph.D. in Physics from the University of California, Berkeley. She has a broad research background at the interfaces of experimental condensed matter physics, chemical physics, and molecular biophysics. She is currently on the faculty at Washington State University, where her research includes the application of femtosecond optical and terahertz spectroscopic techniques to study ultrafast dynamics in electronic materials and molecular systems.

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Contributors Richard D. Averitt Department of Physics Boston University Boston, Massachusetts Jason B. Baxter Department of Chemistry Yale University New Haven, Connecticut Sam G. Carter JILA University of Colorado Boulder, Colorado John Cerne Department of Physics State University of New York at Buffalo Buffalo, New York R. Alan Cheville Department of Electrical and. Computer Engineering Oklahoma State University Stillwater, Oklahoma Susan L. Dexheimer Department of Physics Washington State University Pullman, Washington Edwin J. Heilweil National Institute of Standards. and Technology Gaithersburg, Maryland

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Robert A. Kaindl Lawrence Berkeley National. Laboratory Berkeley, California David F. Plusquellic National Institute of Standards. and Technology Gaithersburg, Maryland Thomas Rades School of Pharmacy University of Otago Dunedin, New Zealand Charles A. Schmuttenmaer Department of Chemistry Yale University New Haven, Connecticut Suranjana Sengupta Department of Physics, Applied. Physics and Astronomy Rensselaer Polytechnic Institute Troy, New York Jie Shan Department of Physics Case Western Reserve University Cleveland, Ohio Mark S. Sherwin Department of Physics University of California, Santa Barbara Santa Barbara, California Philip F. Taday TeraView Ltd. Cambridge, United Kingdom

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Ingrid Wilke Department of Physics, Applied. Physics and Astronomy Rensselaer Polytechnic Institute Troy, New York

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J. Axel Zeitler Department of Chemical. Engineering University of Cambridge Cambridge, United Kingdom

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Section I Instrumentation and Methods

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1

Terahertz Time-Domain Spectroscopy with Photoconductive Antennas R. Alan Cheville Oklahoma State University

Contents 1.1 Introduction to Terahertz Spectral Region........................................................1 1.2 Brief Theoretical Background for Photoconductive Terahertz Generation.........3 1.3 Terahertz-Generation Process........................................................................... 6 1.4 Detecting Terahertz Radiation Using Photoconductive Antennas.................. 10 1.5 Experimental Considerations of Terahertz Spectroscopy............................... 19 1.6 Terahertz Beam Propagation and Optical Systems.........................................24 1.7 Adaptations and Extensions of Terahertz Spectroscopy Systems................... 30 References................................................................................................................. 37

1.1 Introduction to TerahertZ Spectral Region A recent report from the National Academies outlines that electronic and optics are combining to open up a new “Tera-Era.”1 This vision includes computers performing operations at rates of teraflops, terabit-per-second communication systems, and electronic devices on the picosecond time scale. From a spectroscopic point of view, the terahertz (THz) spectral region from 300 GHz (λ = 1 mm) to 10 THz (λ = 30 µm) has not yet seen the technological development of optical or microwave frequencies with the result that commercial spectrometers covering the entire spectral range are not yet widely available. This is primarily because of the difficulty of generating and detecting THz frequencies compared with the better-established technologies of optics and electronics. The so-called “THz gap” arises from the nature of the sources and detectors used in spectroscopy both at the optical (high-frequency) side and electronic (lowfrequency) side of the gap. One terahertz corresponds to a photon energy of 4 meV (33.3 cm–1). These energies are much less than the electronic state transitions of

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Terahertz Spectroscopy: Principles and Applications Wavelength (m) 10

0

10–2

10–1

10–3

+

1010

10–5

10–6

10–7

300 K

– – Rotational Transitions

109

10–4

+

– – Vibrational Transitions 1011

1012

1013

Electronic Transitions 1014

1015

1016

Frequency (Hz)

Figure 1.1 The electromagnetic spectrum from microwave to optical region. The area easily accessible by THz spectroscopy using photoconductive sources is shown shaded in gray. Rotational, vibrational, and electronic transitions are shown along with the blackbody radiation curve at 300 K.

atoms and molecules commonly used in lasers or other light sources (Figure 1.1). Semiconductor materials also have bandgaps on the order of an electron volt, orders of magnitude larger than THz energies. Vibrational frequencies, such as the 10.6 µm CO2 laser, and rotational transition frequencies do fall in the far infrared. Light sources based on these transitions exist, but typically cover limited frequency ranges.2 Thermal emission from incoherent blackbody radiation is one of the most commonly used sources since the thermal energy, kT, is about 25 meV at room temperature (300 K). The blackbody radiation curve and the relative spectral power emitted are shown as a black line in Figure 1.1. Scaling electronics to operate in the THz gap is limited by the frequency response of devices. The high-frequency limit is determined by the time it takes electrons to traverse the device, which is determined by the physical size and carrier mobility. Although feature sizes are continually shrinking, material properties ultimately limit high-frequency response. New semiconductor materials and the ability to confine carriers at the quantum level are leading to high-speed devices pushing into the THz gap. Some exciting recent developments are laser diodes,3 high electron mobility transistors, and quantum cascade lasers.4 The same physical limitations that arise in scaling optical sources down (or electronic sources up) to the THz spectral region also hold for detectors of THz radiation. THz photons cannot excite electrons into the conduction band in a photodiode (without multiphoton absorption, which would require unreasonable intensities) or provide enough energy to cause photoelectric emission of electrons. Although bolometers can detect radiation, they are limited by incoherent thermal background signals unless cooled to very low temperatures. The far infrared spectrum and the points made previously are summarized in Figure 1.1. There is a great deal of interest in this spectral region ranging from creating faster electronic devices to the wealth of spectroscopic information available.

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

THz spectroscopy provides information on the basic structure of molecules and is a useful tool of radio astronomy. Rotational frequencies of light molecules fall in this spectral region, as do vibrational modes of large molecules with many functional groupings, including many biologic molecules that have broad resonances at THz frequencies. This chapter provides an overview of how the technologic difficulties inherent to THz generation and detection can be overcome through a synthesis of ultrafast optical and electronic techniques. By gating micron scale antennas on photoconductive substrates with femtosecond pulses, it is possible to generate and detect electromagnetic pulses with THz bandwidths.

1.2 Brief Theoretical Background for Photoconductive TerahertZ Generation Because THz radiation generated and detected through photoconduction arises from the flow of charge, it is theoretically described by Maxwell’s equations. Maxwell’s equations do not, however, describe the processes from which charge flow occurs on a femtosecond scale in semiconductors. This is a topic worthy of a complete book,5 and the description of processes from which charge flow arises are not gone into in detail here. From the perspective of THz generation, the source terms of Maxwell’s equations are written to show time-dependent terms explicitly by adding a (t) to them.   ∂m (t )H (t ) ∇ × E (t ) = ∂t

   ∂e (t )E (t ) ∇ × H (t ) = s (t )E (t ) + ∂t

(1.1)

Here, m(t)H(t) = B(t) is the magnetic flux, H(t)= s(t)E(t) is the current density, and e(t)E(t) = D(t) is the electric displacement. Any process that creates a time-dependent change in the material properties m, s, or e can act as a source term that can result in emission of THz radiation. For THz generation using photoconductive switches, an ultrashort optical pulse incident on a semiconductor causes rapid transient changes to the macroscopic material properties represented by s(t), m(t), and e(t). For the discussion that follows, the major change caused by the optical pulse is assumed to be in the conductivity, s. The rapid, optically induced change of s on a femtosecond time scale is the origin of ultrafast THz pulses generated through photoconduction as well as the physical mechanism by which the THz pulses are detected. It is important to note that many physical processes occur when ultrafast optical pulses are absorbed by semiconductors, and, from an experimental viewpoint, they are not easily separable. Both resonant (absorption of a photon to create charge carriers, s) and nonresonant (nonlinear optical difference frequency generation, e) effects can contribute to THz pulse generation. A common example of a material in which both processes contribute to THz generation is gallium arsenide (GaAs). The driving term, s(t), in Maxwell’s equations results from an ultrafast optical pulse that impulsively excites the semiconductor. The resulting electric and magnetic

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Terahertz Spectroscopy: Principles and Applications

Electric Field

1

0

–1 –2

–1

0 Time (ps)

1

2

Spectral Amplitude

1

0 0.1

1.0

10 Frequency (THz)

100

Figure 1.2 Electric field of a measured THz pulse and the simulated field of the exciting optical pulse are shown in the upper figure, whereas the amplitude spectrum is shown in the lower figure.

fields created through the time-dependent conductivity have fast transients with correspondingly broad spectral bandwidths. The spectral bandwidth can be estimated from the uncertain relation [between the bandwidth, Dw, required to support a transient signal] Dt. Because DtDw ≥ ½, a 1-ps excitation pulse requires, and can create, spectral bandwidths on the order of 0.3 THz, whereas a 100-fs transient has a bandwidth of at least 3 THz. With sub-10-fs pulses readily available from current commercial ultrafast lasers, very broad spectral coverage of the THz region has been achieved. An example of a THz measurement from the author’s laboratory is shown in Figure 1.2. The upper figure compares the measurement of the electric field of a THz pulse in the time domain with the numerically calculated electric field of the 50-fs optical pulse at 800 nm used to generate the THz pulse. The rapidly oscillating carrier frequency of the optical pulse makes it appear to be a filled Gaussian envelope. The spectral amplitude of the THz pulse and optical pulse are shown in the lower figure. Although the actual bandwidth of the optical pulse (10.6 THz) is considerably larger than that of the THz pulse (0.7 THz), the THz pulse spans a greater fraction of the electromagnetic spectrum in terms of Dn/no: 0.7 for the THz pulse compared to 0.003 for the optical pulse. Recent developments in THz generation and detection

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have extended the frequency range spanned by THz pulses by an additional order of magnitude or more. One of the great advantages of ultrafast THz measurements is that this bandwidth allows a broad range of energy scales (frequencies) to be probed in a single measurement with high temporal resolution. Figure 1.2 highlights some of the differences between short pulses in the THz and optical spectral regions. The electric field of the THz pulse is a nearly single cycle oscillation, whereas the optical pulse is described as a Gaussian pulse envelope superimposed on a sinusoidal carrier frequency. The optical pulse thus meets the slowly varying envelope approximation, whereas the THz pulse does not. Thus many of the simplifying assumptions used in quantum and nonlinear optics do not necessarily apply to THz systems.6,7 From a theoretical perspective, an accurate description of the pulse propagation requires the solution of the coupled Maxwell-Bloch equations. Because THz pulses used for spectroscopy have very low peak powers compared with optical pulses, nonlinearities can generally be neglected and propagation is treated using linear dispersion theory.8 Perhaps the most significant difference between THz pulses and visible, infrared, or ultraviolet light more familiar to most researchers is THz spectroscopy directly measures the electric field of the pulse rather than the intensity. The direct measurement of field is possible simply because of the much slower variation of the field as can be seen in Figure 1.2. Intensity, I, and electric field, E, are related by:

I=

1 m EE * 2 e

(1.2)

Equation 1.2 illustrates the well-known fact that intensity measurements contain no phase information unless techniques such as holography or interferometry are used to measure phase relative to another optical beam. The time-resolved electric field of the THz pulse does, however, contain complete phase information that can be extracted directly from THz measurements. For the more familiar intensity measurements, energy conservation dictates that reflection and transmission coefficients are real and positive with magnitude less than one in almost all commonly encountered cases. Because THz experiments measure the electric field amplitude the Fresnel coefficients can be complex and have values less than zero or greater than one. For example, the amplitude transmission coefficient from air to silicon is t12 = 0.45, whereas that from silicon to air is t21 = 1.55. Examples will be discussed later. The final commonly encountered difference between experimental measurements at optical and THz frequencies is the effect of phase shifts on the measured pulse shape. Small changes in the phase of an optical pulse generally have little to no effect on the measured pulse shape. Larger frequency-dependent phase shifts, such as those caused by dispersive media, do result in pulse broadening. In contrast, small-frequency, independent-phase changes result in significant reshaping of THz pulses, as shown in Figure 1.3. Numeric phase changes of 0, p/16, p/8, p/4, p/2, and p result in pulse reshaping up to a complete inversion of the pulse. Such phase shifts can occur on reflection,9 when the pulse propagates through a caustic,10 or in propagating through materials with a complex index.

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π

0π

π/16

π/8

π/4 π/2

Figure 1.3 Effect of frequency-independent phase shifts from 0 to π/2 radians on the Terahertz pulse shape.

1.3 TerahertZ-Generation Process Generation and detection of THz pulses occur through nonlinear interaction of the driving optical pulse with a material with fast response. There are two major classifications, of which only one, photoconductivity, is discussed in this chapter. Photoconductive THz generation occurs when optical excitation induces conductivity changes in a semiconductor. This is a resonant interaction and the photons are absorbed through interband transitions. The second mechanism is nonresonant interaction arising from nonlinear frequency mixing or optical rectification. Transient photoconductivity is used for both the generation and detection of THz radiation, with THz generation discussed first. A simplified representation of the process of photoconductive THz generation is illustrated in Figure 1.4. The six frames of Figure 1.4 represent different steps in pulse generation and are discussed below. A great deal of research continues to be done on understanding the physics underlying THz pulse generation from transient photoconductivity. This area is highly multidisciplinary, integrating knowledge of optical generation of hot carriers, their subsequent rapid thermalization, ballistic transport on a femtosecond time scale, rapid screening of generated fields, design of high bandwidth antenna structures, and propagation of single cycle pulses in dispersive media. Because this chapter focuses primarily on using the pulses in spectroscopic application, only the basic physical fundamentals of THz generation are discussed.11 The geometry of a basic photoconductive THz source is shown in a top-down view in Figure 1.4a. A coplanar transmission line is fabricated on a semiconductor substrate with high mobility, m, usually GaAs or low temperature-grown GaAs.

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EDC EDC

– –– – –

–

(a)

t = 0++

(d) EDC

J(t)

–+ –+ –+ +– +– +–

t=0

ETHz(t)

t>0

(b)

(e) EDC

–

+

P(t)

t> 0 (f)

Figure 1.4 Illustration of the process of photoconductive Terahertz generation. Frame (a) shows the coplanar transmission lines before excitation. In (b), the optical pulse generates charge carriers which are accelerated in the static field (c), and create an opposing field in the semiconductor (d). The induced polarization creates a transient current element which radiates a Terahertz pulse (e) followed by recombination back to steady state (f).

Because the mobility gives the ratio of carrier drift velocity to applied electric field, carriers in high-mobility compounds accelerate rapidly, leading to fast rise times. The coplanar transmission lines are DC-biased to produce a field on the order of 106 V/m near the breakdown threshold for air. The electric field near the surface of the semiconductor is represented in the figure by parallel arrows between the lines. Energy is stored by this capacitive structure, with a capacitance on the order of several pF. The semiconductor used has a high dark resistivity to minimize current flow and the resultant local heating and device failure due to electromigration.12 The first step in generating THz radiation is optical excitation of the region of high electric field. It is important that the semiconductor bandgap energy, Egap, is less than the photon energy, hν, so that electron-hole pairs are formed. Carrier generation is shown in a cross-sectional view in Figure 1.4b, where dimensions are shown not to scale to better illustrate the physical processes occurring. A focused ultrafast optical pulse incident between the metal lines generates a thin conductive region down to approximately the absorption depth 1/α. Typical absorption coefficients of semiconductors above the band gap range from α = 103 cm–1 to α = 105 cm–1. The optically generated electron-hole pairs form an electrically neutral plasma near the semiconductor surface as illustrated in Figure 1.4b. The time evolution of the charge

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density is given by N(t) with the total carrier density the sum of that of the electrons and holes: N(t) = Ne(t) + Nh(t). At low-excitation fluences, the charge density is proportional to the intensity profile of the optical pulse I(t). The time evolution of the carrier density is given by:11

N (t ) =

∫

t

0

G (t)δt − N 0e

−

t tc

(1.3)

where the first term describes N(t) at time scales of the laser pulse and the second describes the evolution of the carrier density at longer times. Here tc is the carrier lifetime, G(t) is the optical carrier generation rate determined by the optical pulse profile, and No is the total number of carriers generated. The free carriers affect the conductivity by s(t) = N(t)em. The current density is J(t) = s(t)E, which can be written as J(t) = N(t)ev(t) with v(t) the carrier velocity. Immediately after generation of the carriers in the semiconductor the accelerating field is E(t = 0+) = EDC. The time dependence has been written explicitly to illustrate the transient nature of the photoconductivity on subpicosecond time scales. It is the transient current, J(t), that generates the THz pulse. The carriers are accelerated in the applied electric field, EDC, as shown in Figure 1.4c. The time evolution of carrier velocity is determined by the initial acceleration of carriers with effective mass m* and by rapid carrier scattering with a characteristic time ts: δv (t ) v (t ) e =− + E (t ) δt ts m* The carrier scattering time is determined by phonon scattering, carrier scattering, scattering from dopants, and the photon energy, or where in the conduction band carriers are generated. The spatial separation of the photo-induced electron and hole plasma create a macroscopic polarization, P(t), in the semiconductor, shown in Figure 1.4d. The polarization is given by P(t) = N(t)er(t), where r(t) is the spatial separation of charges. The time evolution of the induced polarization is calculated from:

δP (t ) P (t ) =− + N (t )ev (t ) δt tR

(1.4)

where the time derivative of the polarization is N(t)ev(t). As shown in Figure 1.4d, the induced polarization creates an electric field that opposes the static field of the biased coplanar stripline. The electric field responsible for carrier acceleration then evolves as:

E (t ) = E DC +

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P (t ) eg

(1.5)

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where g is a numerical factor that depends on the geometry. The four coupled differential equations describing the carrier density N(t), the velocity v(t), the polarization P(t), and the electric field seen by the carriers E(t) describe the time evolution of the current created by photo-excitation of a carrier.11 The THz pulse is generated from the transient current element, J(t), shown in Figure 1.4e on a greatly expanded spatial scale. Generation of a pulse follows directly from Maxwell’s equations because time varying current serves as a source term:

  ∂H THz (t ) ∇ × ETHz (t ) = − m ∂t    ∂e((t )ETHz (t ) ∇ × H THz (t ) = J (t ) + ∂t

(1.6)

where the subscript THz distinguishes the time resolved THz field from the field in the semiconductor. Maxwell’s equations with these terms and those described in the next paragraphs are generally not amenable to analytic solution. Because many experimental techniques of THz generation focus the optical spot to micron diameters to generate large carrier densities, the current element is typically much smaller than one wavelength of the generated THz radiation and thus can be described as a Hertzian dipole.13 The calculations are complicated by the fact that the current element is on the surface of a dielectric half space, but both analytic and numerical solutions exist for dipole antennas on dielectric half spaces.14 The major fraction of the laser-generated burst of THz radiation is emitted into the semiconductor by a factor proportional to n3 where n is the index of refraction of the semiconductor. Because most semiconductors have refractive indices higher than three at THz frequencies, this is a significant enhancement in directivity and aids in collection and collimation of the THz radiation. When the optical excitation spot is not tightly focused so the current distribution is on the same order of a wavelength or larger, the problem is often treated as a summation (array) of small radiating elements. Because the current rises and falls on a time scale of one picosecond, it generates a well-defined pulse front that propagates outward from the optical excitation point. The radiation pattern is changed by the proximity of the dipole to the air–semiconductor interface and will be discussed later. All these processes happen on time scales of picoseconds or less. On longer time scales, carrier recombination within the semiconductor or at the metal electrodes causes the induced polarization to decay and the static bias electric field to reestablish itself, This is illustrated in Figure 1.4f. The recovery of the semiconductor to the state shown in Figure 1.4a typically takes hundreds of picoseconds or longer.11 In summary, a DC-biased semiconductor (a) has an electron-hole plasma created on femtosecond time scales by an optical pulse (b). The rapid acceleration and separation of the electrons and holes create a current transient (c). The macroscopic polarization creates an opposing field that screens the DC field (d). The transient electric field determined (approximately) by the time derivative of the current results in a radiated pulse front (e). In times of hundreds of picoseconds, the semiconductor recovers (f).

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10


Considerable research has gone into development of photoconductive THz emitters for specific applications. The simple coplanar stripline structure of Figure 1.4 can be replaced by an array of antenna structures. Log-spiral, bowtie, and a variety of dipole structures have all been developed. In this case, the photocurrent provides an impulse excitation to the antenna and the radiation pattern and THz pulse are determined both by the response of the semiconductor as well as the antenna. Biased antenna structures are not required to generate photoconductive THz radiation. The electric field that arises from depletion of carriers from recombination centers near the surface will accelerate photo-induced carriers away from or toward the surface. In this case, the induced electric dipole is normal to the surface, rather than in the plane of the surface as with antenna structures, which makes it more difficult to couple radiation out of the substrate. The photo-Dember effect15 can also give rise to THz radiation. The Dember effect arises from the difference in mobility between electrons and holes, which causes the carriers accelerate at different rates.

1.4 Detecting TerahertZ Radiation Using Photoconductive Antennas Detecting THz pulses using photoconductively gated antennas relies on the same physical principles as generation of THz pulses, namely ultrafast pulse excitation in a semiconductor causing rapid changes in conductivity. To detect THz pulses with subpicosecond resolution, a micron-scale dipole antenna is fabricated on a semiconductor substrate with a very short carrier lifetime of 1 ps or less. Both the antenna design and properties of the semiconductor critically affect THz detection. A typical antenna structure is shown in Figure 1.5d from a top-down view. The antenna is a center-excited dipole fabricated between coplanar strip lines with length, h, of 10 to 200 µm. The length is the most critical element of the antenna and will be discussed later. The antenna’s width is tens of microns with a center gap of less than ten microns corresponding to approximately the diameter of the focused gating laser pulse. Typical dark resistance (no optical excitation) of these antenna structures is on the order of megaohms and depends on the substrate resistivity and antenna and coplanar stripline dimensions. The photoconductive detection process is illustrated schematically in Figure 1.5a–c. Figure 1.5a is a side-on view that illustrates the ultrafast laser pulse (not to scale, see Figure 1.2) incident from the left on the antenna of Figure 1.5d. A THz pulse is propagating from the right, but has not yet arrived at the antenna structure. As in generation of THz pulses, the ultrafast laser pulse generates an electron and hole plasma which decreases conductivity of the antenna gap. The time resolved change in conductivity is s(t) = N(t)em, where N(t) is given by Equation 1.3 and m is the semiconductor mobility. The arrival of the laser pulse is analogous to closing a switch, which allows the antenna gap to conduct. The short recombination time tc of the semiconductor causes the gap resistance to change from nearly insulating to conducting (closing the switch) then back to insulating (opening the switch) on a picosecond time scale. The coplanar strip line is connected to a high-sensitivity current amplifier that detects any current flow through the antenna gap with subpicoamp resolution. If the

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11


ETHz

tB = 0–

tA< 0

(b)

w

h d

E(t) ~ I(t)

(a)

–4 (d)

tC = 0+ (c) tB

tA tC 0 Time (ps)

4

(e)

Figure 1.5 Illustration of the process of THz detection. The THz antenna of frame (d) is shown in cross section in frames (a–c). The electric field measurement is determined by the relative arrival times of the optical and THz pulse. In (a), the optical pulse arrives before the THz pulse. By delaying arrival of the optical pulse to coincide with the THz pulse as shown in (b) and (c), the measured current (e) maps out the THz field in time.

resistance of the metal lines is negligible the resistance of the antenna is determined by the antenna gap. The gap resistance is R(t) = w/s(t)A, where the cross-sectional area, A, is approximately 2da with a the absorption depth in the semiconductor; typically d >> a. The current flow from Ohm’s law is I(t) = V(t)/R(t), where the antenna bias voltage is provided by the THz pulse: V(t) ≅ ETHz(t)h. In Figure 1.5a, the THz pulse has not yet arrived at the antenna, the bias voltage is zero, and there is no net current flow through the antenna. This is illustrated in Figure 1.5e at the point labeled tA, which occurs at t < 0. Figure 1.5b shows a point later in time; the time axis is determined by the delay between the laser pulse that generates the THz pulse and the pulse that gates the THz antenna. The positive gating peak of THz pulse is incident on the antenna at the same time the optical gating pulse arrives. In this case, ETHz > 0, and current flows from the lower half of the antenna to the upper are determined by the electric field direction. The measured time average current at point tB in Figure 1.5e is proportional to the THz electric field. After the current is measured, the delay between the laser generating and gating pulses is again changed to advance the THz pulse in time. Figure 1.5c and point tC in Figure 1.5e both correspond to a point on the pulse where the electric field is opposite in sign to Figure 1.5b. The time resolved

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12


measurement of the THz pulse electric field is thus made by changing the delay between the optical gating pulse and generating pulse. From this discussion, it is clear that, to measure the electric field with high temporal resolution, the response time of the semiconductor tC must be short compared with the rate of change of the THz field. Because to a first approximation σ ~ µ, semiconductors with both high mobility and short carrier lifetimes are optimal. The two semiconductor materials most commonly used for photoconductive THz detectors are low temperature-grown GaAs and ion implanted silicon on sapphire. Low temperature-grown GaAs has a thin layer grown on a GaAs substrate with the substrate temperature determining the recombination time.16 Recombination times of hundreds of femtoseconds or less have been reported. Silicon on sapphire is a thin (~1 µm) layer of silicon grown on a sapphire wafer with a reduction of carrier lifetime achieved through ion implantation. The carrier lifetime is dependent on implantation dosage to a point at which it saturates at about 500 fs.17 The response time can be estimated by the convolution of the THz pulse electric field, ETHz, with the detector response time determined by s(t). With the optical gating pulse arriving at t = 0 and the THz pulse at t = to, the time average current is:

I (t ) =

2 hr α w

∫

∞

s (t )ETHz (t − t 0 − t)δt

−∞

(1.7)

where the antenna dimensions have been used to estimate the resistance. Equation 1.7 indicates that photoconductors with infinitely short response times excited by infinitely short optical pulses (i.e., a delta function response) are required to exactly measure the time-resolved THz electric field. Although in general short carrier lifetimes allow the faster THz field transients to be measured, there is a tradeoff in actual performance because short lifetimes arise from a high density of trap or defect states that reduce the semiconductor mobility. Additionally, the average current is proportional to the time integral of s(t) so that fast recombination times can lead to lower average current measurements for a given THz field strength. In practice, measuring THz pulses with high-frequency components is more dependent on the laser excitation pulse than the carrier lifetime. This can be understood qualitatively through Equation 1.7 because both step functions and delta functions result in a non-zero average current. The extraction of accurate material parameters for non-ideal measurements of ETHz(t) is discussed later. This section concludes with a brief discussion of the THz antenna structure. Although Equation 1.7 indicates that an increase in the dipole size, h, allows high average currents, this is true only if the antenna length remains much less than the THz wavelength (Hertzian dipole). Longer antenna lengths have resonances and exhibit strong frequency dependence. Because the incoming THz pulse in most cases is incident through the semiconductor substrate, the wavelength of the THz radiation is reduced compared with free space by λo/n. Silicon, GaAs, and sapphire have high refractive indices in the THz region, so at frequencies near 1 THz, the wavelength at the antenna is on the order of 100 µm. A large number of antenna designs have been used both for generation and detection of THz radiation beyond simple dipoles. These include log-spiral18 and bow-tie19 designs.

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13

1.4.1 THz Spectroscopy Using Photoconductive Antennas This section provides an overview of spectroscopic measurements using photoconductive sources and detectors of THz pulses. Much of the discussion also applies to other ultrafast THz generation and detection techniques such as those based on nonlinear optics covered elsewhere in this book. Photoconductive THz pulses offer unique capabilities for spectroscopic measurements in the far infrared spectral region compared with other techniques such as Fourier transform infrared spectroscopy or techniques such as frequency mixing originally developed at microwave frequencies. This section is intended to provide a brief overview of both the advantages and disadvantages of ultrafast THz spectroscopy, describe the basic theory of extracting spectroscopic information from experimental measurements, and discuss experimental considerations of the technique that can affect the spectroscopic measurements. This section also is intended to provide an overview for the researcher looking to use THz pulses for their own spectroscopic measurements and is thus general rather than specific in the material covered. The first question to ask is whether THz spectroscopy is the appropriate technique for a given measurement. The THz spectral region accessible via ultrafast optical pulse generation and detection extends from less than 0.1 THz to more than 30 THz. Some source and detector techniques have enhanced low-frequency response, whereas other techniques are used to achieve high frequencies. Both the high- and low-frequency limits have associated experimental difficulties. The low-frequency end of this spectral region extends to frequencies that are easily achievable using high-precision electronic instruments. In addition, long temporal scans are required. For the high-frequency limit above about 10 THz, highly mature optical spectroscopy techniques such as Fourier transform spectroscopy exist. In brief, the choice to use time-resolved THz spectroscopy techniques depends on specific applications. The advantages of time-resolved THz spectroscopy are: coverage of more than a decade frequency range with a single measurement, peak spectral response in the 0.3–3.0 THz spectral range, and the ability to measure phenomena with subpicosecond time resolution. The THz spectral region is rich in spectroscopic features. Chapters later in this book provide much more complete reviews of the types of systems that can be probed with such pulses. From the perspective of trying to set up a system to perform measurements, THz spectroscopy can be broadly separated into measurement of gases and that of liquids and solids. At the low number density characteristic of gases, the frequency of collisions between molecules determines dephasing times that occur on time scales generally much longer than 1 ps. Spectral features are then less than 1 THz in width and can be less than 1 MHz at low pressures with Q (Dn/no) of 106 or more. In this limit, time-resolved THz spectroscopy resolves individual transitions over a broad range of energies, but does not have the frequency resolution to measure extremely narrow line widths. Limitations on frequency resolution are discussed in detail later. For liquids and solids resonances are broadened on femtosecond timescales resulting in broad, featureless spectra with corresponding to Q < 1 in most cases. For such systems, the spectral width of the short pulse-based measurement techniques can prove extremely valuable. Free carrier absorption occurs

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14


in semiconductor materials in the THz spectral range and the broad spectral coverage is extremely useful for determining material properties. One other difference between THz spectroscopy and spectroscopic techniques at optical frequencies is that the energies of THz photons correspond to thermally accessible energy states. At room temperature the thermal energy, kT, corresponds to 25 meV, 208 cm–1, or a frequency of 6.25 THz. Thus THz spectroscopy using photoconductive antennas typically covers the region kT > hn where thermal energies are higher than photon energies and can probe states that are populated at room temperature. Using short excitation pulses to achieve high frequencies, it is also possible to simultaneously measure the region hn > kT. The technique of THz time domain spectroscopy measures the change between time-resolved electric fields of THz pulses propagating through a sample and an equivalent length of free space. Depending on the sample geometry, the change in the THz pulse shape permits either analytic or numeric calculation of the complex sample index, permittivity, or conductivity. One assumption in the following treatment of THz time domain spectroscopy is that the interaction of an electric field with a sample is linear. Maxwell’s equation, and Equation 1.1, which describes propagation through a material with constituent parameters given by the permittivity and permeability, is linear for small electric field when higher order terms of the susceptibility can be ignored. THz pulses generated in the laboratory typically have peak powers on the order of 1 µWcm–2 corresponding to an electric field on the order of 1 Vcm–1. Because the assumption of linear interactions in most materials is a good one, linear dispersion theory8 is used to represent the time domain pulse in the frequency domain as a superposition of plane waves of frequency w, all with a k vector along the THz system optical axis, z. The relative amplitude and phase of each plane wave component of the THz pulse are obtained through a Fourier transform of the timeresolved electric field:

E (z , w ) =

1 2p

∫

∞

E (z , t )e − iwt ∂t

−∞

(1.8)

Here E(z,w) is the complex field amplitude and E(z,t) is the experimentally measured THz pulse electric field in the time domain. A typical measurement of two THz pulses is shown in Figure 1.6. The three frames on the left correspond to the reference pulse that has propagated through free space—in this case, nearly dry air. The three frames on the right correspond to the pulse that has propagated through the sample—humid air containing approximately 1 percent water vapor. The experimentally measured pulses are in the top two frames. The effect of water vapor on the THz pulse in the measurements of Figure 1.6a are visible as oscillations after the pulse. Figure 1.6b shows the magnitude of the complex spectral amplitude obtained through a numeric Fourier transform. Comparing the two spectral amplitudes, the discrete rotational transitions of H2O can be seen as distinct absorption lines. The relative phase of the plane wave components that make up the two THz pulses is displayed as modulo 2p in Figure 1.6c. As can be seen from

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15


E(t) (nA)

4 2

(a)

Esamp(t)

Eref(t)

0 –2 0

10

20

0

10

Time (ps)

20

30

Time (ps)

|E(ω)|

1.00

(b)

0.75 0.50

|Esamp(ω)|

|Eref(ω)|

0.25

∠E(ω)

0.00 π

(c)

0 –π 0.0

0.5

1.0

1.5

2.0

0.5

Frequency (THz)

1.0

1.5

2.0

2.5

Frequency (THz)

Figure 1.6 The reference and sample pulses discussed in the text are shown in (a) for dry and humid air. Numerical Fourier transforms of the amplitude and phase, (b) and (c), respectively, illustrate the effect of the sample on the plane wave components that comprise the THz pulse.

the figure, the phase changes linearly except near the resonance of the water vapor rotational frequencies. The gray regions on the plot correspond to invalid phase data caused by the reduced spectral amplitudes beyond 2 THz. Obtaining the complex electric field amplitude spectrum via a Fourier transform permits determination of the complex amplitude after propagating a distance Dz through a sample. Given the measured spectrum at the input of the sample, z = 0, the field exiting the sample is given by:

E ( Dz , w ) = E (0, w )e − ik (w ) Dz

(1.9)

The complex k vector, k(w), expresses complete information about the THz pulse interaction with the sample material. The wave vector k(w) can be expressed several ways, most commonly through the complex index, n(w) = n′(w)-jn″(w) with k(w) = wn(w)/c. The complex index is related to the permittivity through er(w) = n(w)2. The imaginary term is related to the power absorption coefficient by ap(w) = 2wn″(w)/c.

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16


The choice of sign for the imaginary term of n(w) is made so that plane waves are attenuated and not amplified. The complex, frequency-dependent k vector is then:

k (w ) =

w w[n ′(w ) − 1] i α p (w ) + − 2 c c

(1.10)

where the first term, k0 = w/c, corresponds to the phase change of the THz pulse through a length of free space equivalent to the sample thickness. The reason for expressing the wave vector in this form will be made clear later. To measure the complex refractive index of the sample, a measurement is first done without the sample in place and with the spectrometer purged with dry air to ensure n = 1 along the THz pulse path. The measured THz pulse, Eref (t) in Figure 1.6a, is numerically Fourier transformed to obtain Eref (ω). The measured THz pulse in a general sense can be written as:

E ref (w ) = E gen (w )T (w )H (w ) = E gen (w )T (w )e − ik 0 Dz

(1.11)

where Egen(ω) is the THz pulse generated by the source, T(ω) is the frequency response of the THz spectrometer including the THz source and detector, and H(ω) is the response function of the sample. For the reference sample of length ∆z of free space, H(w) = exp(-ikoDz). The concept of a transfer function is drawn from engineering and describes the effect of a system on a complex input signal. The terms Egen(ω) and T(ω) are not directly measured and depend on numerous experimental parameters and the THz source and detector used. After the sample is placed into the system, the measured pulse that has propagated through the sample is

E samp (w ) = E gen (w )T (w )H (w ) = E gen (w )T (w )e − ik (w ) Dz

(1.12)

The material response is determined by dividing the complex amplitude spectrum of the sample pulse by the free space pulse

E samp (w ) E gen (w )T (w )e − ik (w ) Dz = = e − i ( k (w )− k 0 ) Dz E ref (w ) E gen (w )T (w )e − ik 0 Dz

(1.13)

As can be seen from the k(ω)-ko term of Equation 13, division of the sample spectrum by the reference spectrum effectively removes the effect of an equivalent length of free space. The absorption coefficient and index of refraction can be determined from the magnitude and phase of the spectral ratio using Equation 1.10. The refractive index and absorption coefficient of the time domain measurements of humid air, Figure 1.6, are shown in Figure 1.7. The complex index of the rotational transitions have the form of the quasi-Lorentzian line shape with phase and magnitude similar to that of a harmonic oscillator.20 Besides permitting direct measurement of the

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas 2

17

(b)

n(ω)z

1 0 –1 –2 6

(a)

α(ω)z

4 2 0 0.0

0.5

1.0 Frequency (THz)

1.5

2.0

Figure 1.7 Refractive index and absorption coefficient extracted from the data of Figure 1.6.

index and absorption, dividing the Fourier transforms of the measured pulses in the frequency domain is mathematically equivalent to a time domain deconvolution to remove the frequency response of the generated THz pulse and optical system from the sample response function. The effect of this deconvolution is one reason that the response time of the semiconductor does not need to be much shorter than the rate of change of the THz field to perform high-resolution THz spectroscopy. The formalism of the transfer function in the derivation above is useful because most samples measured with THz time domain spectroscopy do not have simple transfer functions. The derivation previously considered propagation in which the Fresnel reflection and transmission coefficients can be ignored. A more general case is the measurement of a finite thickness of material with index n2(w) surrounded by a material of index n1(w), usually air, so n1 = 1. In this case the Fresnel coefficients need to be taken into account and the transfer function H(ω) is then defined as:21

H (w ) =

t12t 21e ib (1 − r12r21e i 2b

)

(1.14)

where t12(w), t21(w), r11(w), and r 21(w) are the Fresnel transmission and reflection coefficients21 and depend on the index of the sample, n2(w). b = n2 kodcos q is the phase delay associated with multiple reflections inside the slab where q is the angle of propagation inside the slab determined from Snell’s law. Although there are no

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18


exact analytical solutions for the index and absorption coefficient, the equation can be solved numerically either using an iterative guessing approach or piecewise in frequency trying to minimize phase and magnitude errors.22 Analytical approximations also serve in many cases. Although Equation 1.14 describes propagation through a dielectric slab, any other sample geometry can be represented equally well with the transfer function formalism. Assuming that the system being measured is one for which linear dispersion theory applies, after the system transfer function is determined, the THz measurements can compared with theory with equal ease in either the frequency or time domain. For a known H(ω) the time domain THz pulse is calculated after propagating through the sample from the inverse Fourier transform:

E samp (t ) =

∫

∞

E ref (w )H (w )e − i (wt )∂w

−∞

(1.15)

To summarize the process for extracting the index and absorption from measurements of the THz pulse, the THz pulse is Fourier transformed to express it in terms of a plane wave expansion. Taking the ratio of a THz pulse transmitted through a sample to one through a reference sample with known index deconvolves the response of the THz antenna and optical system and returns the sample transfer function. The index and absorption data are extracted from an analytic expression for the sample transfer function H(w). The complex index is measured over the system bandwidth without resorting to Kramers-Kronig analysis. As with any experimental technique, there are several assumptions inherent to the treatment discussed previously that can affect the results. The most common source of error arises from fluctuations of the THz pulse shape over the course of the measurement. Because two THz pulses—a reference pulse and a sample pulse—are measured to extract the complex index, any changes in the THz pulse shape over the duration of the measurement result in errors of the measured index. Relatively long-term experimental system drifts can be difficult to control so due experimental diligence and careful experimental design are required to avoid long-term drifts from thermal or mechanical fluctuations. Linear dispersion theory requires that the THz pulse can be expanded into a superposition of plane waves propagating along the z axis. Unlike laser beams at optical frequencies, this is not always true for THz pulses. Typical THz beam diameters are approximately 1 cm or less with a wavelength from millimeters to tens of microns. The beam diameter:wavelength ratio then ranges from 3 to 100 compared with values of several thousand for optical beams. Thus THz beams are more correctly represented as a summation of plane waves over a range of angles to the z axis. The angular spread of the k vectors can be estimated from Dq = 2Dk x/ko with ko = w/c, Dk x ≅ (2Dx)–1 and ∆x ≅ 1 cm. The angular spread is then on the order 0.3° at 1 THz and 0.9° at 0.3 THz. The angular spread of the beam can affect the measured transmission of samples angled to the beam.23 If the spatial profile of the THz pulse is measured, a spatial Fourier transform permits the angular-dependent transmission of the sample to be resolved.

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A final assumption is that the shape of the THz beam is not changed by passing through the sample. Referring to Equation 1.13, it was assumed that transmission through the sample does not affect T(w). Again, this is usually not an issue with optical beams, and is generally true for thin, low index samples that do not act as optical elements. The larger effect of diffraction at THz frequencies can result in beam reshaping for thick samples of high index. Techniques to calculate the effect of the sample on the THz beam will be discussed in the following section on experimental considerations of THz spectroscopy.

1.5 Experimental Considerations of TerahertZ Spectroscopy The basic technique of THz time domain spectroscopy using photoconductive switches has been outlined previously from the viewpoints of the fundamental physics of the THz generation and detection process and also from the mathematical treatment of extracting complex index from measurements. This section introduces the third element required for successful THz spectroscopic measurement: the experimental measurement system. This section is divided into three parts that attempt to provide practical considerations for performing THz spectroscopy: the optical pulses used to generate and detect the THz pulse, the propagation and coupling of the THz beam from source to detector, and miscellaneous experimental considerations required for successful measurements. A simplified schematic representation of a simple THz spectroscopy setup is shown in Figure 1.8. On the left side of the figure, a femtosecond laser provides a pulse train of femtosecond pulses. A beamsplitter separates the pulse train into beams that go to the THz source and detectors with nominally equivalent power. In Figure 1.8, the optical pulse going to the THz source (a) passes through an adjustable delay. The THz pulse created by the THz source (b) goes through several beamforming optics, passes through the sample, which changes the pulse shape (c), and is incident on the THz detector. The second optical pulse (d) arrives at nominally the

fs Laser Beamsplitter

Optical Delay (a)

THz Source

THz Detector

Sample

(b)

(c) (d)

Figure 1.8 Schematic illustration of a THz spectrometer showing the laser excitation source, optical delay line, THz source and detector, and THz coupling optics.

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same time as the THz pulse and gates the THz detector. Changes in the delay permit the THz pulse shape to be mapped out as discussed in previous sections. A key experimental issue is ensuring the time of arrival of the THz pulse on the detecting dipole antenna coincides with the arrival of the optical pulse. The laser source and associated optics are shown on the left side of Figure 1.8. THz spectroscopy using photoconductive switches can be done successfully using a wide range of laser excitation sources. There are a large number of ways to specify or characterize laser sources, some of which have a strong impact on THz spectroscopic measurements and others that are much less important. The characteristics of the laser source are discussed briefly in decreasing order of importance. The ranking in importance is rather arbitrary and other experimental data can certainly change the required laser source characteristics. Arguably the most important characteristic of the laser source is stability, defined as constant power output and beam pointing over periods longer than several seconds. The importance of this for THz spectroscopic measurements arises from the need to measure two data scans for the reference and sample. Because the complex index is determined by the ratio of the two data scans, any change in the laser power or beam position can introduce large errors in measurements. Experimentally, measurements usually are performed by taking a reference scan before and after a scan with the sample in the beam in order to determine if there has been any drift of the system during the measurement period. Temperature fluctuations in the laboratory or problems with optical beam pointing stability can also degrade measurements. Figure 1.9a illustrates differences between two commercial laser sources determined by measuring the peak of the THz signal—point tB in Figure 1.5—over 30 minutes. As can be seen from the figure, there is a significant difference in stability between different sources. Of nearly equal importance to medium-term stability is the optical noise of the laser source. The primary noise source on THz experiments is noise from the laser coupled onto the THz beam.24 The noise specification of interest depends on the data acquisition technique. For the majority of THz experiments, phase-sensitive or lock-in detection is used and the noise figure of interest is the noise at the modulation frequency of the THz time domain spectroscopy system. For data acquisition systems that use rapid scanning and signal averaging, the noise at lower frequencies is of more importance. Figure 1.9b shows the effect of the stability and noise of the laser sources on the spectral representation of data used to determine the complex index through THz spectroscopy. The figures show ratios of the magnitude of the spectral amplitudes of five sequential reference scans, |EN|/|EM| where M ≠ N, performed in dry air. The variation between separate ratios indicates approximately the accuracy of the transfer function which is used to determine the complex index. Laser 1, with less noise and drift, results in a transfer function accurate to about ±1% over a large bandwidth. The noisier laser 2 has a factor of three less accuracy. The experimental artifact near 1.6 THz is due to small changes in the water vapor content in the experimental system on the order of less than 2%. The increasing error of the ratios at low and high frequencies is due to the finite bandwidth of the THz system.25 A third important, but not necessarily vital, specification of the laser source is the optical pulse width, which under some experimental circumstances, has a critical effect on the THz signal bandwidth.

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Normalized Peak Signal (%)

+1

Commercial Laser #1

0 –1 +3

Commercial Laser #2

+2 +1 0 –1 –2

0

10

20

Time (minutes)

30

Ratio of Spectral Amplitudes (%)

(a) 105 100 Commercial Laser #1

95 110

Commercial Laser #2 100

90

|EN| |EM| 0

0.4

0.8

1.2

1.6

2

Frequency (THz) (b)

Figure 1.9 The upper figure, (a), compares the peak THz signals measured for two commercial laser sources over a 30-minute period to illustrate the impact of the excitation source on system signal-to-noise ratio. The lower figure, (b), compares five consecutive scans with the two laser sources illustrating how short-term stability impacts measurement accuracy.

For experiments using photoconductive switches over the spectral range 0.1–3 THz, the laser pulse width is not generally the limiting factor to the system bandwidth. Rather, the optical system and dipole response limit the bandwidth. However, to reach higher THz frequencies using relatively short dipole antennas on semiconductor substrates with fast response time, the bandwidth is critically dependent on the laser pulse width because the rise time of the conductivity limits the system spectral response. As a rule of thumb, pulse widths lower than 100 fs are more than adequate for most THz experiments and noise considerations can have a much larger effect on the effective system bandwidth than the laser pulse width. This is illustrated in Figure 1.10, in which the Fourier transforms of the five consecutive scans of Figure 1.9 are shown on a logarithmic scale for the two commercial lasers. The pulse width of the first laser, on the upper half of the figure, is specified by the manufacturer as > zo, the divergence full angle scales approximately as θ(n) ≅ .06/n (rad), whereas the beam waist scales as w(z) ≅ 0.03z/n (cm). This is illustrated graphically in Figure 1.13 that shows propagation of 0.1, 0.3, 1.0, 3.0, and 10.0 THz components of a THz pulse propagating a distance of 1 m from a lenscoupled THz source as calculated from Equation 1.16. At 100 cm distance, the beam at 0.3 THz has a radius of 10 cm; in practice, freely propagating THz beams cover much smaller distances and diffraction is controlled using optics. Because of the differing propagation characteristics of different frequency components of the THz beam, the propagation of THz beams through optical systems

100

Waist Radius, w(z) (cm)

0.1 THz 0.3 THz

10

1 THz 3 THz

1.0

10 THz 0.1

0

20

40

60

80

100

Distance from Source, z (cm)

Figure 1.13 Variation of the THz beam waist radius (1/e amplitude point) with distance from the lens-coupled, photoconductive THz source.

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exhibits effects that are not commonly seen at optical frequencies. A good first-order approximation of THz beam propagation through optical systems that assumes paraxial propagation and ignores aberrations and aperturing effects of mirrors can be made using Gaussian beam formalism combined with the ABCD matrix method for describing optical systems.31 To calculate propagation through optical systems, both the amplitude and phase terms, w(w,z) and R(w,z), respectively, are represented by the complex q parameter: 2c / w 1 1 ≡ −j 2 q ( w , z ) R( w , z ) w (w, z )

(1.19)

As the THz beam propagates through an optical system with a given ABCD matrix, the phase and amplitude profiles are changed, which transforms the q parameter according to:

1 = qout

  C +D 1   q in    A + B 1   q in 

(1.20)

where qin is the q parameter at the input plane of the optical system, which is usually defined as the THz source coupling lens and qout is the value at the system output, usually the coupling lens of the THz detector. To achieve optimal power transfer, the beam that is detected at the receiver must have an identical phase and amplitude profile to the emitted THz beam at all component frequencies.32 It can be shown that this occurs if the phase and amplitude profiles of the beam from the THz source and a hypothetical beam from the THz detector fully overlap at some point in the system. The coupling efficiency is determined by integrating the normalized field profiles of the beam from the source and the hypothetical detector beam over a plane at some point in the system.32 The normalized power coupling is then given by:

Cp =

 

∫∫

∫∫

E s (w , z )E D (w , z ) * dS

E s (w , z )ES (w , z ) * dS  ×   

∫∫

2

E D (w , z )E D (w , z ) * dS  

(1.21)

where for TEM0,0 modes the source and detector beams are given by ES (w,z) and ED (w,z), respectively. For ideal coupling, CP must be in unity for all the frequencies comprising the THz pulse. Assuming the THz source and THz detector are identical, unity power coupling occurs when the beam at the output of the optical system has identical phase and amplitude profiles to that at the input so that qout = qin. From Equation 1.20, this condition requires that the elements along the main diagonal of the ABCD matrix,

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Terahertz Spectroscopy: Principles and Applications 0.1 THz ƒ1

ƒ1

0.3 THz 1 THz

3 THz 10 THz ƒ1

ƒ1

2ƒ1 (a)

ƒ2

ƒ1

ƒ1

ƒ 1 + ƒ2

ƒ2

2ƒ2

ƒ1

ƒ1 + ƒ2

ƒ1

(b)

Figure 1.14 Two common configurations of THz spectrometers. The configuration shown in (a) is a confocal system with a frequency-dependent beam waist at the midpoint of the system. The configuration of (b) is commonly used in THz imaging systems and creates a frequency-independent waist between the short focal length lenses, ƒ2.

A and D, have values ±1 and the off-diagonal elements; B and C are zero—in other words, a unity transfer function. T(w), with ideal coupling for all THz frequencies is achieved for optical systems whose ABCD matrix is the identity matrix, I. Two separate optical systems with unity coupling and an identity transfer matrix are shown in Figure 1.14 along with the calculated beam diameters (1/e amplitude point) at 0.1 THz, 0.3 THz, 1.0 THz, 3.0 THz, and 10 THz. In both systems, lenses are spaced from each other by the sum of their focal lengths. Figure 1.14a illustrates the most common system for THz spectroscopy. Because the field pattern at the back focal plane of a lens is the spatial Fourier transform of the pattern at the front focal plane, the frequency-independent Gaussian beam waist at the coupling lens forms overlapping frequency dependent waists midway between the two lenses in Figure 1.14a. This frequency-dependent waist pattern is then transformed back to a frequency independent waist at the THz detector. The fact that most samples are placed midway between the two lenses for THz spectroscopy needs to be accounted for in data analysis because experimental artifacts from the frequency-dependent waist size can be introduced if the sample is not uniform over large areas. One method of

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∆z = 10%

1 Power Coupling Efficiency, CP

29

∆z = 20%

0.8 0.6

∆z = 10%

0.4 Figure 14(a) Figure 14(b)

0.2 0

0

2

∆z = 20%

6 4 Frequency (THz)

8

10

Figure 1.15 Effect of changing the spacing of focusing optics by 10% and 20% on the frequency-dependent THz power coupling efficiency for the optical configurations shown in Figure 1.14. The amplitude coupling coefficient can be determined by the square root of the plotted values.

measuring nonuniform samples or those with small area is shown in Figure 1.14b. Here, ƒ2 = ¼ƒ1 and the THz beam midway between the two short focal length lenses has a frequency-independent beam waist that is four times smaller than that created by the THz source. This configuration is often used in THz imaging applications33 in which a sample is raster-scanned at the beam waist. THz beam waist diameters smaller than 1 mm can be routinely achieved in this configuration. For both configurations shown in Figure 1.14, the spacing between the lenses affects the frequency-dependent coupling efficiency. The coupling efficiency is shown in Figure 1.15 as a function of frequency at lens spacings corresponding to 80% and 90% of the ideal distances in Figure 1.14, 2ƒ1 or ƒ1 + ƒ2, respectively. As can be seen from the figure, the optical transfer function is an important parameter for the design of any terahertz system. For nonideal optical configurations, it is possible to choose corrective optics to ensure the system has a unity transfer by simply finding the matrix inverse of the transfer function. Because T(w)T(w)–1 = ±I where I is the identity matrix, for nonideal system with transfer matrix T(w), the matrix T(w)–1 gives the transfer function of the corrective optics required to achieve unity coupling. For complex THz optical systems, it is good practice to build THz beam optics in blocks so that the transfer function of each block is the identity matrix. The general treatment outlined previously in this chapter can also be used when coupling THz pulses into waveguides or other photonic devices.34 In this case, the overlap integral is done between the waveguide or device modes and the free-space THz beam. Two final considerations are important when designing THz spectroscopy systems. First, as mentioned previously, the TEM0,0 Gaussian beam formalism is a useful approximation, but it suffers from the fact that the antenna pattern at the lens surface is not a true Gaussian. The propagation can be more precisely, but not

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uniquely, described in terms of a summation higher order Laguerre-Gauss modes.30 The coupling theory still applies in this case for each of the orthogonal modes. For most situations involving beam propagation, the TEM0,0 approximation has acceptable limits of error and the increased complexity of a full modal expansion is unwarranted. There are, however, several situations where the more complex description of the beam profile may need to be accounted for. The first of these is in the near-field region of the coupling lens, where z 10 –3 cm can be accurately measured. To improve the accuracy for thin film or low absorbance samples several techniques have been developed. The most widely used is known as differential time domain spectroscopy.36 In differential time domain spectroscopy, the sample position is spatially modulated as shown in Figure 1.16a so that the sample moves in and out of the THz beam, which is focused to a small spot size. The film on a substrate is attached to a shaker operating at frequency fs. The difference between the sample and reference signals, Eref (t)-Esamp(t), is measured by a phase-sensitive detector that detects only the modulated signal. Because the THz pulse is not modulated, the noise on the THz background spectrum is effectively eliminated. The measurement of a difference between the reference and sample scans rather than a ratio complicates data analysis. Approximate analytic expressions for the index and absorption have been developed,35 and spectroscopic information can also be extracted numerically.

Shaker fs Beamsplitter

Sample

Al Film

nfilm nair

nsub Iamp

(a)

LIA Silicon Prisms fs

(b)

Figure 1.16 Two experimental configurations used for measuring thin film samples with thickness much less than one wavelength. The differential time domain spectroscopy (DTDS) system is shown in (a) while a system for interferometric THz measurements is shown in (b).

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Using combinations of lock-in amplifiers, it is possible to measure samples with thicknesses of less than 100 nm and simultaneously extract the THz reference pulse using a single measurement.37,38 Extremely high-power dynamic ranges down to –80 dB have also been reported using this technique.39 An alternative technique to measure thin films is to create destructive interference between two THz pulses by splitting the THz pulse train into two equal parts and providing a near-π, frequency-independent phase shift to one of the pulses.40,41 Experimentally, this can be accomplished using a Michelson or Mach-Zender interferometer. Destructive interference between the pulses eliminates the background signal, drift, and some sources. Because of the broad spectral bandwidth, adjusting the phase through path length changes is not possible so the phase shift is accomplished from a Gouy phase shift,10 total internal reflection,9 or geometrical rotation of the electric field vector with a roof mirror.42 An experimental configuration for a THz interferometer is shown in Figure 1.16b. The Michaelson configuration achieves a near-π phase shift through total internal reflection in two silicon prisms, of which one face has been aluminized. The film thickness resolution of THz interferometric measurements is determined by the Sparrow criterion43; for typical THz photoconductive sources and detectors, time resolution of 1 fs is resolvable corresponding to measurements of films on the order of hundreds of nanometers of thickness. A second modification of THz spectroscopy systems is to measure the THz signal reflected or scattered from a sample rather than the transmitted THz pulse. A typical configuration for reflection measurements is shown in Figure 1.17. The THz pulse from the source can be scattered from the sample as shown in Figure 1.17a or reflected specularly as shown in Figure 1.17b. The detector used to measure the scattered pulse is fiber-coupled so the scattering angle β can be varied. Fiber-coupling photoconductive antennas are accomplished by prechirping the optical pulse so that the pulse is recompressed at the output of the fiber.

θ θ

β Optical Fiber Scattered THz Pulse

Fiber Coupled Detector (a)

(b)

Figure 1.17 THz scattering (a) and reflection (b) experimental configurations.

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It is more difficult to extract spectroscopic information from scattered or reflected pulses than it is from THz transmission measurements. For the specularly reflected pulse, the reflection coefficient is primarily determined by the nonresonant index. Rarely are there large enough changes in the real part of the index to result in large changes in the reflection coefficient. Additionally, there is no temporal shift of the THz pulse as there is in transmission measurements. As a rough estimate, the normalized change in reflectivity, DR/R, is approximately equal to the normalized change in index, Dn/n, with larger reflectivity changes for lower index materials. The complex index n″ is related to the absorption coefficient by n″ = a/2ko, where ko is the wave vector in free space with a value of approximately 2 × 10 4 m–1 at 1 THz. Except for extremely lossy materials, the real part of the index dominates the imaginary component and contributes negligibly to the reflection coefficient. It is possible to characterize a sample using THz ellipsometry44 and also to make spectroscopic measurements near Brewster’s angle 45 where the reflection coefficient has large changes. For nonspecular reflection, the scattered THz pulse depends to a much larger extent on the sample geometry than the sample composition. Various wave front reconstruction techniques can be used to reconstruct the sample from the scattered wave front; these are discussed in subsequent chapters. Besides adapting the basic technique of photoconductive THz spectroscopy through sample modulation or to measure reflection, it is possible to use photoconductive THz spectroscopy to measure the time-dependent change of a sample after external optical perturbation. Two common ways to accomplish this are optical pump THz probe spectroscopy and THz emission spectroscopy. This chapter concludes with an overview of these techniques with selected experimental results. Optical pump THz probe spectroscopy optically excites a sample with an ultrafast pulse and measures the dynamic processes that occur in the sample by probing with a THz pulse. The experimental modification required for optical pump THz probe spectroscopy shown in Figure 1.18 permits measurements of an optically excited

THz Source τTHz THz Detector fs Laser τopt

Figure 1.18 Schematic experimental configuration of optical pump THz probe spectroscopy described in the text.

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sample in the far infrared spectral region with picosecond time resolution. To minimize timing jitter between the optical excitation of the sample and the THz probe, the excitation pulse typically comes from the same THz source used to generate and detect the THz pulses. For this reason, an amplified laser system is often required to ensure there is sufficient excitation pulse energy to cause a measurable change in the sample. The timing between optical and THz pulses is controlled using two delay lines as shown in the figure. The delay line labeled τTHz is used to delay the optical gating pulse for the THz detector, so that the THz electric field can be mapped out in time. The delay line labeled τOPT changes the delay between the arrival of the optical and THz pulses on the sample. The THz optical system has also been modified by adding additional focusing optics to create a small, frequency-independent beam waist to maximize overlap of the THz and optical pulses. Ultrafast optical beams with a TEM0,0 Gaussian profile and a center wavelength of λ ideally have spot size of approximately wf = λƒ/πd where wf is the radius of the focused beam waist, ƒ is the lens focal length, and d is the laser spot diameter. Because the optical wavelengths are submicron size, whereas THz wavelengths are submillimeter size, without careful selection of optics there is poor overlap which can result in data with experimental artifacts. The optical pump THz probe experiment measures a time-resolved THz pulse for different delays between the THz probe and optical pump pulses, E(tTHz,topt). When the optical delay is zero, topt = 0, the THz and optical pulses arrive at the sample at the same time. As in simple THz spectroscopy (Figure 1.8), a reference pulse is needed, and the THz pulse measured before the arrival of the optical pulse is used as the reference pulse, Eref (t, topt < 0). The THz pulse measured after arrival of the optical pulse constitutes the sample pulse, Esamp(t, topt > 0). The dynamic response of the sample to optical excitation, H(w,topt), can then be mapped out as a function of time. Inherent in the analysis given previously was the assumption that the properties of the sample are time invariant during transmission of the THz pulse. Because the THz pulse may be of longer temporal duration than the sample properties being measured this assumption may not be true in optical pump THz probe measurements. In this case, determining the dynamic response of the sample is done by deconvolving the THz pulse from the time-dependent complex refractive index. There are many reports in the literature of optical pump THz probe spectroscopy that have been used to make measurements that are difficult using other spectroscopic techniques; for example, the optical pump pulse is able to create charge carriers in normally insulating materials in order to measure time-resolved carrier dynamics. The acceleration of the created charge carriers by the THz electric field attenuates the THz probe pulse allowing the complex conductivity as a function of THz frequency and as a function of time after optical excitation, topt, to be experimentally mapped. The wide range photon energies available from current ultrashort pulse sources can excite carriers from the valence band to the conduction band of semiconductors as well as materials that are insulators at room temperature. A good example of these measurements is optical pump THz probe spectroscopy of sapphire,46 a material typically considered an outstanding insulator with a dielectric strength of nearly 5 × 106 Vm–1, resistivity over 1014 Ω-cm and a bandgap of 8.7 eV. To excite carriers, an amplified Ti:Al2O3 laser pulse with photon energy of 1.5 eV was frequency tripled

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to 4.5 eV and two photon absorption moved electrons from the valence to the conduction band. The transmitted THz pulse showed the carriers behaved according to the Drude theory and had strongly temperature-dependent mobility that could be as high as µ = 3 × 104 cm2v–1s–1 at low temperatures when the scattering of the carriers by phonons (lattice vibrations) was reduced. In chemistry, optical pump THz probe measurements have been used to measure solvation dynamics in liquids, critical to many aspects of liquid-phase chemistry including the growth and self-assembly of nanostructures. Because the reorientation of the solvent occurs on time scales of 0.1–10 ps, THz frequencies are useful for probing these systems. In one set of measurements,47 the optical pulse was used to excite a solvated dye molecule and the reorientation of the solvent molecule dipole moments surrounding the optically excited dye molecule was measured with the THz pulse. Another example of optical pump THz probe spectroscopy is measurement in superconducting materials. In the previous example, the optical pulse caused conductivity changes by creating charge carriers; in the prototypical high Tc superconductor, YBCO short-pulse optical excitation breaks superconducting pairs.48 The pair breaking causes transient changes in the complex conductivity and the recovery time of the superconducting pairs can be determined by using a THz probe to measure the conductivity as a function of delay between the optical excitation pulse and the THz pulse. These measurements are only a representative fraction of the physical systems that have been examined to date with optical pump THz probe spectroscopy; more detailed discussion is found in later chapters. It is not necessary to use photoconductively generated THz pulses in reflection or transmission for spectroscopic measurements. Because the dynamics of the photogenerated carriers in the THz source determine the temporal profile of THz pulse, the generated pulse itself can provide information on carrier dynamics on a subpicosecond time scale and can be used for spectroscopic measurements. Using the generated pulse itself for spectroscopic measurement is known as THz emission spectroscopy and a simplified experimental configuration is illustrated in Figure 1.19. THz emission spectroscopy provides fundamentally different information than optical pump-probe spectroscopy. Optical techniques generally give insight into the dynamics of how carriers evolve in energy, the vertical axis on an E versus k diagram.

Delay

Sample

fs Laser

Figure 1.19 Experimental configuration of THz emission spectroscopy.

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THz emission spectroscopy, in contrast, probes the velocity, or momentum, evolution of carriers in time. Photoconductively generated THz pulses have electric fields proportional to the rate of change of the current, E(t) ∝ δJ/δt. Because the current is proportional to carrier density and velocity, J = N(t)ev(t), the THz field strength is proportional to the dynamic velocity ensemble of the carriers. For example, the rising edge of the THz pulse arises from carrier acceleration in conjunction with an increasing number of carriers because of optical excitation. The peak of the THz pulse occurs when the carrier ensemble is undergoing the maximum acceleration. The THz field then decreases and has a zero crossing when the ensemble velocity is a maximum (δv/δt = 0). Accurately extracting relevant information on carrier dynamics using THz emission spectroscopy requires the ability to accurately measure the electric field profile. Because the pulse shape is affected by the detector bandwidth, dispersion effects, and propagation, it is important to account for these effects of the measurement system to avoid experimental artifacts. Because THz emission spectroscopy is sensitive to the motion carriers with excess energy from photoexcitation in accelerating fields, it provides information on the dynamics of hot carriers. The ability to study hot carrier transport processes in high fields has impact on the development of high speed semiconductor devices such as high-electron mobility transistors. The THz pulse shape depends on both acceleration and collisions of carriers, so the pulse profile provides insight into carrier scattering processes including collisions with impurities and dopants, interaction with phonons (quantized lattice vibrations), and scattering from other carriers. It is possible to generate THz radiation from most semiconductors so that THz emission spectroscopy can measure the hot carrier dynamics in semiconductor materials without requiring direct electrical contact. It is straightforward to create heterostructures or lithographically define structures so that conditions such as temperature, the electrical bias, or the optical excitation density can be probed to map out carrier dynamics over a large parameter space. Similarly, by tuning the optical excitation wavelength, the dynamics of carriers created at different energies in the conduction band can be experimentally probed by looking at how the shape of the generated THz pulse changes. Even states below the conduction band, the so-called Urbach tail, can be investigated. THz emission spectroscopy directly measure Bloch oscillations in semiconductors. Bloch oscillations are changes in electron energy along with an oscillatory motion in real space that result from motion in the periodic crystal potential and were first predicted by Zener in 1934 based on Bloch’s 1928 theory of electrons in a periodic band structure.49 Much of the interest in Bloch oscillations arises from their potential to produce powerful, tunable sources of radiation from the oscillatory motion of the electrons. Because electron scattering processes are usually much faster than the period of the Bloch oscillation, dephasing processes need to be controlled. The development of controllable band structures in semiconductor superlattices allowed materials to be created in which Bloch oscillations could be observed. THz emission spectroscopy directly observes Bloch oscillations as oscillations following the initial THz pulse. As the bias on the sample increases, so does the frequency of the Bloch oscillations, as does the spectrum of the emitted THz radiation.

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37

Any other process that gives emits a THz pulse is a candidate for THz emission spectroscopy; examples are carrier acceleration in the surface depletion field of a semiconductor50 and emission from metal-air surfaces from optical rectification.51 THz emission spectroscopy is not limited to crystalline semiconductors because many materials produce a transient polarization after optical excitation. Emission from semiconducting polymer materials has helped to clarify the process of charge generation in these materials. Work has also looked at magnetic nonlinearities as well as electronic nonlinearities.48 Some of the more novel work on THz emission spectroscopy has been done on polar molecules in liquids. In these experiments, a static electric field is placed across the liquid, and the polar molecules orient in the field. Orientation of the ensemble is required to generate THz radiation because the net dipole moment of an ensemble of randomly aligned molecules will be zero. When the molecules are photoexcited, they undergo intermolecular charge transfer on a picosecond time scale, which causes a change in the dipole moment and a transient polarization of the ensemble of molecules that emits a THz pulse. These measurements provide information on the time scale and direction of intermolecular charge transfer.47

References 1. Harnessing light: optical science and engineering in the 21st century. Washington, D. C.: National Academy Press, 1998. 2. M. Inguscio et al., A review of frequency measurements of optically pumped lasers from 0.1 to 8 Thz, J. Appl. Phys., 60, R161, 1986. 3. D. S. Kurtz, et al., Submillimeter-wave sideband generation using Varactor Schottky diodes, IEEE Trans. Microwave Theory Tech., 50, 2610, 2002. 4. A. Tredicucci, et al., Terahertz quantum cascade lasers—first demonstration and novel concepts, Semicond Sci Technol, 20, S222, 2005. 5. S. D. Ganichev and W. Prettle, Intense THz excitation of semiconductors. New York: Oxford University Press, 2006. 6. C. Meystre and M. Sargent III, Elements of quantum optics. Berlin: Springer-Verlag, 1991. 7. Y. R. Shen, The principles of nonlinear optics. Hoboken, N. J.: John Wiley & Sons, 2003. 8. L. Allen and J. H. Eberly, Optical resonance and two-level atoms. New York: John Wiley & Sons, 1975. 9. S. R. Keiding and D. Grischkowsky, Measurements of the phase shift and reshaping of terahertz pulses due to total internal reflection, Opt. Lett., 15, 48, 1990. 10. S. Feng, H. G. Winful, and R. W. Hellwarth, Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses, Opt. Lett., 23, 385, 1998. 11. P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, Generation and detection of terahertz pulses from biased semiconductor antennas, J. Opt. Soc. Am. B, 13, 2424, 1996. 12. K. Bock, H. L. Hartnagel, and J. Dumas, Surface induced electromigration in GaAs devices, in Reliability of gallium arsenide MMICs, A. Christou, ed. New York: John Wiley & Sons, 1992, 192. 13. C. A. Balanis, Advanced engineering electromagnetics. New York: John Wiley & Sons, 1989. 14. R. W. P. King and G. S. Smith, Antennas in matter. Cambridge: MIT Press, 1981. 15. K. Liu, et al., Terahertz radiation from InAs induced by carrier diffusion and drift, Phys. Rev. B, 73, 2006.

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Terahertz Spectroscopy: Principles and Applications 16. S. D. Benjamin, H. S. Loka, and P. W. E. Smith, Tailoring of low-temperature MBE-grown GaAs for ultrafast photonic devices, Can. J. Phys., 74, S68, 1996. 17. F. E. Doany, D. Grischkowsky, and C.-C. Chi, Carrier lifetime versus ion-implantation dose in silicon on sapphire, Appl. Phys. Lett., 50, 462, 1987. 18. D. R. Dykaar, et al., Log-periodic antennas for pulsed terahertz radiation, Appl. Phys. Lett., 59, 262, 1991. 19. H. Murakami, et al., Tetrahertz pulse radiation properties of a Bi2Sr2CaCu2O8 + delta bowtie antenna by optical pulse illumination, Jap. J. Appl. Phys. Part 1, 41, 1992, 2002. 20. H. Harde, R. A. Cheville, and D. Grischkowsky, Terahertz studies of collision-broadened rotational lines, J. Phys. Chem. A, 101, 3646, 1997. 21. M. Born and E. Wolf, Principles of optics, ed 7. New York: Cambridge University Press, 1999. 22. L. Duvillaret, F. Garet, and J. L. Coutaz, A reliable method for extraction of material parameters in terahertz time-domain spectroscopy, IEEE J. Selected Topics Quantum Electr., 2, 739, 1996. 23. M. T. Reiten, et al. Incidence-angle selection and spatial reshaping of terahertz pulses in optical tunneling, Opt. Lett., 26, 1900, 2001. 24. L. Duvillaret, F. Garet, and J. L. Coutaz, Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy, Appl. Opt., 38, 409, 1999. 25. P. U. Jepsen and B. M. Fischer, Dynamic range in terahertz time-domain transmission and reflection spectroscopy, Opt. Lett., 30, 29, 2005. 26. S. C. Howells, S. D. Herrera, and L. A. Schlie, Infrared wavelength and temperature dependence of optically induced terahertz radiation from InSb, Appl. Phys. Lett., 65, 2946, 1994. 27. G. Zhao, et al., Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter, Rev. Sci. Instr., 73, 1715, 2002. 28. W. H. Press, et al., Numerical recipes in Fortran, 2nd ed., New York: Cambridge University Press, 1993. 29. W. Lukosz and Z. S. Eth, Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation, J. Opt. Soc. Am., 69, 1495, 1979. 30. M. T. Reiten, S. A. Harmon, and R. A. Cheville, Terahertz beam propagation measured through three-dimensional amplitude profile determination, J. Opt. Soc. Am. B, 20, 2003. 31. A. E. Siegman, Lasers. Mill Valley: University Science Books, 1986. 32. J. C. G. LeSurf, Millimetre-wave optics, devices, and systems. Bristol: Adam Hilger, 1990. 33. D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, T-ray imaging, IEEE J. Sel. Topics Quantum Electr, 2, 679, 1996. 34. G. Gallot, et al., Terahertz waveguides, J. Opt. Soc. Am. B, 17, 851, 2000. 35. D. Grischkowsky, et al., Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors, J. Opt. Soc. Am. B, 7, 2006, 1990. 36. Z. P. Jiang, M. Li, and X. C. Zhang, Dielectric constant measurement of thin films by differential time-domain spectroscopy, Appl. Phys. Lett., 76, 3221, 2000. 37. K. S. Lee, T. M. Lu, and X. C. Zhang, The measurement of the dielectric and optical properties of nano thin films by THz differential time-domain spectroscopy, Microelectr J, 34, 63, 2003. 38. S. P. Mickan, et al., Double modulated differential THz-TDS for thin film dielectric characterization, Microelectr J, 33, 1033, 2002.

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39. M. Brucherseifer, P. H. Bolivar, and H. Kurz, Combined optical and spatial modulation THz-spectroscopy for the analysis of thin-layered systems, Appl. Phys. Lett., 81, 1791, 2002. 40. J. L. Johnson, T. D. Dorney, and D. M. Mittleman, Enhanced depth resolution is terahertz imaging using phase shift interferometry, Appl. Phys. Lett., 78, 835, 2001. 41. S. Krishnamurthy, et al., Characterization of thin polymer films using terahertz time-domain interferometry, Appl. Phys. Lett., 79, 875, 2001. 42. J. A. Small and R. A. Cheville, Measurement and noise characterization of optically induced index changes using terahertz interferometry, Appl. Phys. Lett., 84, 4328, 2004. 43. D. Guenther, Modern optics, 2nd ed., New York: John Wiley & Sons, 1990. 44. T. Nagashima and M. Hangyo, Measurement of complex optical constants of a highly doped Si wafer using terahertz ellipsometry, Appl. Phys. Lett., 79, 3917, 2001. 45. M. Li, et al., Time-domain dielectric constant measurement of thin film in GHz-THz frequency range near the Brewster angle, Appl. Phys. Lett., 74, 2113, 1999. 46. J. Shan, et al., Measurement of the frequency-dependent conductivity in sapphire, Phys. Rev. Lett., 90, 247401, 2003. 47. M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, Terahertz spectroscopy, J. Phys. Chem. B, 106, 7146, 2002. 48. D. J. Hilton, e. al., On photo-induced phenomena in complex materials: probing quasiparticle dynamics using infrared and far-infrared pulses, J. Phys. Soc. Japan, 75, 011006, 2006. 49. K. Leo, Interband optical investigation of Bloch oscillations in semiconductor superlattices, Semicond Sci Technol, 13, 249, 1998. 50. J. Darmo, et al., Surface-modified GaAs terahertz plasmon emitter, Appl. Phys. Lett., 81, 871, 2002. 51. F. Kadlec, P. Kuzel, and J. L. Coutaz, Study of terahertz radiation generated by optical rectification on thin gold films, Opt. Lett., 30, 1402, 2005.

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2

Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection—Optical Rectification and Electrooptic Sampling Ingrid Wilke and Suranjana Sengupta Rensselaer Polytechnic Institute

Contents 2.1

Introduction.................................................................................................... 42 2.1.1 Terahertz Time-Domain Spectroscopy: An Overview....................... 42 2.2 Optical Rectification and Linear Electrooptic Effect....................................44 2.3 Experimental Results of Terahertz-Frequency Radiation Generation by Optical Rectification of Femtosecond Laser Pulses................................. 49 2.3.1 Materials............................................................................................. 50 2.3.1.1 Semiconductors..................................................................... 50 2.3.1.2 Inorganic Electrooptic Crystals............................................ 52 2.3.1.3 Organic Electrooptic Crystals............................................... 52 2.3.2 Recent Developments.......................................................................... 56 2.4 Experimental Results of Terahertz Electrooptic Detection........................... 57 2.4.1 Materials............................................................................................. 57 2.4.1.1 Semiconductors and Inorganic Crystals................................ 57 2.4.1.2 Organic Crystals....................................................................64 2.5 Application of Electrooptic Sampling of Terahertz Electric Field Transients....................................................................................................... 65 References................................................................................................................. 69

41

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2.1 Introduction The generation and detection of short pulses of terahertz (THz) frequency electromagnetic radiation are attracting great interest. Research activities in this area are driven by applications of THz-frequency radiation pulses in time-domain THz spectroscopy and time-domain THz imaging.1,2 Recent examples of major scientific advancements in THz wave research include the detection of single-base pair differences in femtomolar concentrations of DNA,3 the observation of the temporal evolution of exciton formation in semiconductors4 and understanding of carrier dynamics in high-temperature superconductors.5 Real-world applications of time-domain THz spectroscopy and imaging address important problems such as nondestructive testing,6 as well as new approaches to medical diagnostics and rapid screening in drug development.7 One of the research directions in THz science and technology is to increase the bandwidth of short THz-frequency radiation pulses. Nonlinear optical phenomena such as optical rectification and the linear electrooptic effect (Pockels’ effect) are attractive options for broadband generation and detection of THz radiation. This chapter provides an overview of the generation and detection of freely propagating THz pulses based on nonlinear optical techniques. In Section 2.1.1 of this chapter, the principles of time-domain THz spectroscopy and time-domain THz imaging using nonlinear optical techniques for THz pulse generation and detection are briefly discussed. Section 2.2 describes the principles of generation and detection of THz-frequency radiation by optical rectification of femtosecond laser pulses and femtosecond electrooptic sampling. THz radiation emission has been reported from a variety of nonlinear materials such as lithium niobate (LiNbO3),8,9 lithium tantalate (LiTaO3),9,10 zinc telluride (ZnTe),11,12 indium phosphide (InP),13 gallium arsenide (GaAs),14 gallium selenide (GaSe),15,16 cadmium telluride (CdTe),17 cadmium zinc telluride (CdZnTe),18 DAST,10,19 and metals.20,21 Section 2.3 summarizes the performance of THz emitters based on these materials in terms of bandwidth and signal strength of the emitted THz radiation. Materials for the detection of THz radiation by femtosecond electrooptic sampling are the subjects of Section 2.4. In Section 2.5, an application of electrooptic detection of THz-frequency electromagnetic transients is discussed.

2.1.1 Terahertz Time-Domain Spectroscopy: An Overview The basic ideas of time-domain THz spectroscopy and a time-domain THz imaging system is illustrated in Figure 2.1. A subpicosecond pulse of THz-frequency electromagnetic radiation passes through a sample placed in the THz beam and its time profile is compared to a reference pulse. The latter can be a freely propagating pulse or a pulse propagating through a medium with known properties. The frequency spectra of the transmitted and reference THz radiation pulses are obtained by Fourier transformation. Analysis of the frequency spectra yields spectroscopic information on the material under investigation. In the case of time-domain THz imaging the THz radiation focal spot is scanned across the sample and a THz image of the object under investigation is obtained. The time-domain THz spectroscopy system shown in Figure 2.1 is powered by a laser, which emits a train of femtosecond duration pulses at near-infrared frequencies.

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 43 Delay Stage M1

Laser

M2

Probe Beam Beam Splitter

M5

PM I

PM II

Pump Beam

THz Beam PBS

Chopper

Emitter

L2

Detector

L1 M3

M6

Quarter Wave Plate

M4

M7 Data Acquisition Software

Lock-in Amplifier

Diodes

Wollaston Prism

Figure 2.1 Experimental arrangements for time-domain THz spectroscopy measurements using nonlinear optical techniques for the generation and detection of freely propagating THz-radiation pulses. (M1–M7 optical beam steering mirrors, PMI, PMII parabolical THzbeam steering mirrors, L1, L2 lenses).

The initial laser beam after passing through a beam splitter is split into two parts: the pump and the probe beam. The pump beam after being modulated by an optical chopper is focused on the THz emitter (a second-order nonlinear optical medium), which releases a subpicosecond pulse of THz radiation in response to the incident femtosecond near infrared laser pulse. The generated THz radiation is focused onto a detector using two off-axis parabolic mirrors. The detector is an electrooptic crystal. The probe beam gates the detector, whose response is proportional to the amplitude and sign of the electric field of the THz pulse. By changing the time delay between the pump and the probe beams by means of an optical delay stage, the entire time profile of the THz transient can be traced. Electrooptic detection of THz transients is possible when the THz radiation pulse and the probe beams coincide in a copropagating geometry inside the electrooptic crystal. A pellicle beam splitter is put in the THz beam line for this purpose. As the THz pulse and probe beam copropagate through the electrooptic crystal, a phase modulation is induced on the probe beam which depends on the electric field of the THz radiation. The phase modulation of the probe beam is analyzed by a quarter wave (l/4) plate and the beam is then split into two beams of orthogonal

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polarizations by a Wollaston prism. At this point, the phase modulation of the probe beam is converted to an intensity modulation of the two orthogonal polarizations of the probe beam, which are then steered into a pair of photodiodes. A lock-in amplifier subsequently detects the difference in probe laser light intensities measured by the photodiodes. Next, the key aspects of the nonlinear optical processes involved in the generation and detection of short pulses of THz-frequency radiation are discussed in depth.

2.2 Optical Rectification and Linear Electrooptic Effect Optical rectification and the linear electrooptic effect (Pockels’ effect) are nonlinear optical techniques for the generation and detection of freely propagating subpicosecond THz-frequency radiation pulses. Generally, optical rectification refers to the development of a DC or low- frequency polarization when intense laser beams propagate through a crystal. The linear electrooptic effect describes a change of polarization of a crystal from an applied electric field. Optical rectification and the linear electrooptic effect occur only in crystals that are not centrosymmetric. However, optical rectification of laser light by centrosymmetric crystals is possible if the symmetry is broken by a strong electric field. Furthermore, generation and detection of THz-radiation pulses by optical rectification and the Pockels’ effect require that the crystals are sufficiently transparent at THz and optical frequencies. Freely propagating subpicosecond THz-radiation pulses are generated by optical rectification of femtosecond (fs) near infrared laser pulses in crystals with appropriate nonlinear optical properties for this process. The detection of freely propagating THz-radiation pulses is performed by measuring the phase modulation of an fs near infrared laser pulse propagating through an electrooptic crystal simultaneously with a THz-radiation pulse. The electric field of the THz radiation induces a phase modulation of the fs laser pulse through the linear electrooptic effect. In the following section, the theory of optical rectification and the linear electrooptic effect is discussed. The discussion of theory is limited to material we consider useful for the reader to get a general idea of the underlying physics of THz pulse generation and detection by nonlinear optical techniques and to equations that are relevant to the implementation of the techniques in the laboratory. Our discussion mainly follows the original work by pioneers in the field. First, DC optical rectification of 694 nm continuous-wave laser radiation in potassium dihydrogen phosphate and potassium dideuterium phosphate was demonstrated experimentally by Bass et al. in 1962.22 After that, monochromatic THz- radiation generation at 3 THz by low-difference frequency mixing of near-infrared (nir) laser radiation (lnir = 1.059–1.073 Nm) in quartz was achieved in 1965 by Zernike and Berman.23 Yang et al. demonstrated generation of broadband (0.06–0.36 THz) freely propagating THz-radiation pulses by optical rectification of picosecond Nd: glass laser pulses in LiNbO3 in 1971.8 Eventually, Hu et al. produced free-space THz-frequency radiation with a bandwidth of 1 THz by optical rectification of femtosecond CPM dye laser pulses in LiNbO3 in 1990.10

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 45

In 1893, Friedrich Pockels discovered the linear electrooptic effect.24 Subsequently, the phase modulation of optical laser light at microwave frequencies using the linear electrooptic effect was demonstrated in 1962 by Harris et al.25 Then, Valdmanis and coworkers built the first electrooptic sampling system with picosecond resolution for the measurements of ultrafast electrical transients in 1982.26 Finally, electrooptic sampling of freely propagating THz-radiation pulses was demonstrated in 1995 by Wu and Zhang27 and in 1996 by Jepsen et al.28 and Nahata et al.29 An excellent description of laser light modulation based on the linear electrooptic effect is given by Yariv.30 Rigorous quantum mechanical calculations of optical rectification were carried out by Armstrong et al.31 Previous reviews of nonlinear generation and detection of sub-picosecond THz-radiation pulses were published by Bonvalent and Joffre,32 Shen et al.,33 and Jiang and Zhang.34 The discussions of optical rectification and the Pockels’ effect begin by considering scalar relationships of polarization P, electric susceptibility, and electric field E.35 P = c( E ) E

(2.1)

The electric polarization P of a material is proportional to the applied electric field E. Here, c(E) is the electric susceptibility. The nonlinear optical properties of the material are described by expanding c(E) in powers of the field E.

P = ( c1 + c 2 E + c3 E 2 + c 4 E 3 + ) E

(2.2)

Optical rectification and the linear electrooptic effect are second order nonlinear optical effects P2nl and described by the P2nl = c2 E2 term in the expansion. For example, consider an optical electric field E described by E = E 0 cos wt. In this case, the second-order nonlinear polarization P2nl consists of a dc polarization c2 E 02/2 and a polarization with a cos 2wt dependence.

P2nl = c 2 E 2 = c 2

E0 2 (1 + cos 2 w t ) 2

(2.3)

The DC polarization results from the rectification of the incident optical electric field by the second-order nonlinear electric susceptibility of the material. The polarization with the cos 2wt dependence describes second harmonic generation. This nonlinear optical process is not relevant to the generation and detection of THz radiation by nonlinear optical techniques and therefore not discussed further. Similarly, consider two optical fields oscillating at frequencies E1 = E 0cos w1t and E2 = E 0cos w2t. P2nl = c 2 E1 E2 = c 2

E0 2 [cos(w1 - w 2 )t + cos(w1 + w 2 )t ] 2

(2.4)

this case, the DC second-order nonlinear polarization P2nl consists of a term In P2w1-w2 proportional to the difference frequency w1 + w2 and a term P2w1+w2 proportional to the sum frequency w1 + w2. The production of THz radiation by optical rectification relies on low difference frequency generation and is described by the

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46


term P2w1 - w2. Again, sum frequency generation P2w1 + w2 is not relevant to generation of THz radiation by nonlinear optical techniques. In the same way, generation of THz-radiation pulses from optical rectification of fs laser pulses is based on difference frequency mixing of all frequencies within the bandwidth Dw of an fs near infrared laser pulse. Specifically, an fs pulse ∆t is characterized by a large frequency bandwidth Dw. The bandwidth of the laser pulse is described by a Gauss function of width 1/4G.36  ( w - w 0 )2  E (w ) ∝ exp   4G  

(2.5)

In the time domain, the fs laser pulse is described by an optical field oscillating at frequency w0 with a time dependence described by a Gauss function.

E (t ) = E0 exp (iw 0t ) exp( - Gt 2 )

(2.6)

The bandwidth of the THz-radiation pulse is determined by difference frequency generation by all frequencies within the bandwidth of the fs laser pulse. The time profile of the radiated THz pulse from optical rectification of the fs laser pulse is proportional to the second time derivative of the difference frequency term P2w1 - w2. The time dependence of P2w1 - w2 is determined by the Gaussian time profile of the optical laser pulse. The physics of the linear electrooptic effect is described by the same term P2n1 = 2 c2 E in the expansion of c(E) as optical rectification. To understand the relationship between optical rectification and the Pockels’ effect, it is now necessary to describe polarization P and electric field E by vectors and the susceptibility c by a third rank tensor. Quantum mechanical calculations of the polarization for the case of two laser beams with frequencies w2 , w1 present in a crystal demonstrate that the ith component of the second-order nonlinear polarization piw1-w2 is related to the components j and k of the electrical fields Ejw1 and Ekw2 via the susceptibility tensor component cijkw1 - w2 35 w1 - w 2

w1

w2

pi = c ijk E j Ek (2.7) DC optical rectification and the linear electrooptic effect result from Equation 2.7 considering the limits of w2 - > w1 and w2 - > 0, respectively.

w1 - w 2

w 2 → w1

pi0 = c ijk0 E wj 1 Ekw1

(2.8)

w2 → 0

piw1 = c ijkw1 E wj 1 Ek0

(2.9)

Equations 2.8 and 2.9 demonstrate that a strong electric field at frequency w1 gives rise to a dc polarization pi0 and a DC electric field Ek 0 changes the polarization piw1 at frequency w1.

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Further calculations demonstrate that c 0ijk is equal to c w1jik. Therefore, the third rank tensors are identical under simple interchange of indices as well as frequencies. This demonstrates the identity of optical rectification and the linear electrooptic effect. As a result of this identity, it is possible to estimate the magnitude of optical rectification from the linear electrooptic coefficients of a material. The most widely employed crystals for nonlinear optical generation and detection of THz radiation are ZnTe, gallium phosphide (GaP), and GaSe. ZnTe and GaP exhibit zincblende structure and 43m point group symmetry. GaSe is a hexagonal crystal with 62m point group symmetry. All three crystals are uniaxial. Uniaxial crystals have only one axis of rotational symmetry referred to as the c-axis or optical axis of the crystal. The efficacy of ZnTe, GaP, and GaSe for nonlinear optical THz generation and detection is rated by the linear electrooptic coefficient of the materials. The linear electroptic coefficients for ZnTe, GaP, and GaSe are listed in Table 2.1. The relationship between susceptibility tensor elements describing optical rectification cijk 0 and linear electrooptic effect c jik w is: cijk 0 + cikj 0 =

1 c jik w 2

(2.10)

and the relationship between the linear electrooptic coefficient rjki and the linear electrooptic effect tensor element c jik w is:  4p  rjki = -  2 2  c jik w  n0 ne 

(2.11)

Efficient generation and detection of THz radiation by optical rectification and the Pockels’ effect require single crystals with high second-order nonlinearity or large electrooptic coefficients, proper crystal thickness and proper crystal orientation with respect to the linear polarization of the THz radiation. The surfaces of the crystals should be optically flat at the laser excitation wavelengths and of high crystalline quality (e.g., low levels of impurities, structural defects, and intrinsic stress). The bandwidth of an electrooptic crystal for THz generation and detection is determined by the coherence lengths and optical phonon resonances in the material.

Table 2.1 Electrooptic Coefficients of Some Commonly Used Terahertz Emitters and Detectors Material ZnTe GaP GaSe

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Structure Zincblende(30) Zincblende(30) Hexagonal (53)

Point Group Symmetry ( 30 )

43m 43m( 30 ) 62 m( 30 )

Electrooptic Coefficient (pm/V) r41 = 3.9(30) r41 = 0.97(30) r41 = 14.4(76)

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48


Infrared active optical phonon resonances result in strong absorption of electromagnetic radiation at the absorption frequencies. The phonon absorption bands (optical phonons) of a material in the THz frequency range are not available for THz generation and detection. If the refractive indices of the near infrared laser excitation frequency and the THz radiation are identical, then the bandwidths of the THz radiation depend only on the pulse width of the incident femtosecond near infrared laser beam. Furthermore, the strength of emission and sensitivity of THz detection would increase similarly for all frequencies within the bandwidth with increasing crystal thickness. However, the refractive indices for the near infrared laser frequency and THz frequency are generally not identical. Consequently, electromagnetic waves at THz and near infrared frequencies travel at slightly different speeds through the crystal. The efficacy of nonlinear optical THz generation and detection decreases if the mismatch between the velocities becomes too large. The distance over which the slight velocity mismatch can be tolerated is called the coherence length. As a result, efficient THz generation and detection at a given frequency only occur for crystals that are equal in thickness or thinner than the coherence length for this frequency. The coherence length in case of pulsed near-infrared laser radiation is defined as:11 lc (w THz ) =

w THz

pc | nopt eff (w 0 ) - nTHz (w THz ) |

(2.12)

with  ∂nopt  nopt eff = nopt (w ) - l opt   ∂l  l

(2.13) opt

In Equations 2.12 and 2.13, c is the velocity of light, w = 2p/u the THz freTHz THz quency, w0 is the near infrared excitation frequency nTHz the refractive index at THz frequencies, nopt the index of refraction at near infrared wavelength lopt, and nopt eff the group velocity refractive indices of the femtosecond near infrared laser pulse. THz emission strength of a crystal and sensitivity of an electrooptic THz detector are proportional to the thickness of the crystal. However, the bandwidth of the THz emitter crystal and bandwidth of the THz detector crystal also depend on the crystal thickness. Generally THz emission strength and THz emission bandwidth for a crystal have a reciprocal relationship. The THz emission bandwidth increases when the crystal becomes thinner. The THz emission strength decreases when the crystal becomes thinner. The same rule applies to the sensitivity of an electrooptic THz detector and the detector bandwidth. Thinner crystals have a higher bandwidth but lower sensitivity than thicker crystals and vice versa. For efficient generation of THz radiation by optical rectification of femtosecond laser pulses from zincblende structure crystals, it is important to select the proper orientation of the crystal with respect to the linear polarization of the laser beam. The preferred crystal orientations of selected materials are discussed in Sections 2.3 and 2.4.

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In case of the longitudinal linear electrooptic effect, the electric field is applied parallel to the direction of propagation of the optical probe laser beam. In case of the transverse linear electrooptic effect, the electric field is applied perpendicular to the direction of propagation of the optical probe beam. Detection of THz radiation pulses is generally performed using the geometry of the transverse electrooptic effect. The amplitude and phase of the subpicosecond THz-radiation pulse are measured by recording the phase change of femtosecond near infrared laser pulses traveling collinearly and simultaneously with the THz radiation pulses through the electrooptic crystal. The electric field of the THz radiation induces a change of the index of refraction of the crystal via the linear electrooptic effect such that the material becomes birefringent. The phase retardation G between the ordinary and extraordinary ray after propagating through the birefringent crystal is proportional to the amplitude and phase of the THz electric field E, the crystal thickness l, the linear electrooptic coefficient r14, the index of refraction of the crystal at the near infrared laser frequency no, and the near-infrared wavelength l.30 G∝

1 3 ln0 r41ETHz l

(2.14)

Furthermore, the phase retardation also depends on the orientation of the crystal and directions of polarization of the THz radiation pulse and the near infrared laser pulse.37 For 43m zincblende structure crystals, preferred directions of the THz electric field ETHz, optical field Eopt, and crystal orientation are discussed in Sections 2.3 and 2.4. The temporal profile of the THz radiation is mapped out in time by delaying the much shorter femtosecond near infrared laser pulse with respect to the THz radiation pulse and measuring the phase retardation as a function of the delay between THz radiation pulse and optical probe pulse.

2.3 ExperimenTAL Results of Terahertz-Frequency Radiation Generation by Optical Rectification of Femtosecond Laser Pulses In this chapter, we review the experimental results for generation of THz radiation pulses based on optical rectification. Several nonlinear optical materials have been reported to generate THz radiation by optical rectification of a femtosecond near infrared laser pulse. Table 2.2 lists the various crystals investigated for generation of THz radiation by optical rectification. For the sake of organization, the experimental results will be divided into three broad material categories: semiconductors, inorganic electrooptic crystals and organic electrooptic crystals, and the characteristics of the emitted THz radiation will be discussed in brief under each material category. The performance of various emitters will be discussed on the basis of bandwidth and amplitude of the emitted THz radiation. We conclude our discussion by presenting recent experimental developments in THz emitter research.

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Table 2.2 Materials Used for Generation of Terahertz Optical Rectification Semiconductors

Inorganic Electrooptic Crystals

Organic Electrooptic Crystals

LiNbO3 LiTaO3

4-N-methylstilbazolium tosylate (DAST) N-benzyl-2-methyl-4-nitroaniline (BNA) (-) 2-(a-methylbenzyl-amino)- 5-nitropyridine (MBANP) electrooptic polymers

GaAs InP CdTe InAs InSb GaP ZnTe ZnCdTe GaSe

2.3.1 Materials 2.3.1.1 Semiconductors A significant amount of research dating back as early as the 1970s has explored the generation of THz radiation by optical rectification of pico- and femtosecond optical pulses in nonlinear optical crystals. Experimental work in the early 1990s reported that THz-frequency radiation could also be generated by femtosecond optical pulses incident on a semiconductor surface.38,39 This observation was initially explained as the dipole radiation of a time-varying current resulting from photo-excited charge carriers in the depletion field near the semiconductor surface.39–41 Although the ultrafast photocarrier transport model successfully explained the transverse magnetic polarization of the emitted THz radiation and the observed emission maxima for Brewster angle incidence, the intensity modulation of the emitted THz radiation observed when the crystal was rotated about its surface normal indicated that there was more than one mechanism responsible for the generation of THz radiation. In 1992, Chuang et al.42,43 proposed a theoretical model based on optical rectification of femtosecond laser pulses at semiconductor surfaces, which successfully explained all of the earlier experimental observations. The investigation of the physics of optical rectification indicates that depending on the optical fluence, the THz radiation generation by optical rectification is either a second-order nonlinear optical process governed by the bulk second-order susceptibility tensor c2, or a third-order nonlinear optical process whereby a second-order nonlinear susceptibility results from the mixing of a static surface depletion field and the third-order nonlinear susceptibility tensor c3.41,44 THz radiation generation from optical rectification of femtosecond laser pulses has been reported from , , and -oriented crystals with zincblende structure commonly displayed by most III–V (and some II–VI) semiconductors. These crystals have a cubic structure with 43m point group symmetry and only one independent nonlinear optical coefficient, namely r41 = r52 = r63.30 Theory predicts that for -oriented zincblende crystals, there is no optical rectification field at normal incidence (as the nonvanishing second-order optical coefficients r41, r52, and

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r63 are not involved), whereas a three-fold rotation symmetry about the surface normal may be observed for and orientations. Experimental studies conducted on , , and oriented GaAs, CdTe, and indium phosphide (InP) crystals13,17,41 and GaP crystal45 are in excellent agreement with the theoretical results providing evidence that second-order bulk optical rectification is the major nonlinear process under the condition of moderate optical fluence (~nJ/cm2) and normal incidence on unbiased semiconductors. A dramatic variation in the radiated THz signal along with polarity reversal is observed in and CdTe and GaAs crystals, as the incident photon energy is tuned near the bandgap.13,17,41 This phenomenon can be explained as the dispersion of the nonlinear susceptibility tensor near the electron resonance state.13,41 THz generation by optical rectification has recently also been reported from narrow bandgap semiconductors such as indium arsenide (InAs) and indium antimonide (InSb) at high optical fluence (1–2 mJ/cm2) and off-normal incidence.44,46–50 Experimental results of THz emission from InSb crystal and , , and InAs crystal for both n- and p-type doping were examined.44,49 A pronounced angular dependence of the emitted THz radiation on the crystallographic orientation of the samples as well as on the optical pump beam polarization was observed. The experimental results are in good agreement with the theory, which predicts that for high excitation fluence, THz emission from narrow band gap semiconductors results primarily from surface electric field-induced optical rectification with small bulk contribution.50 This observation is in contrast with the earlier results obtained for wide bandgap semiconductors such as GaAs, in which the THz emission is primarily dominated by strong bulk contribution. The anomaly can be explained in the light of the theoretical calculations by Ching and Huang51 whereby narrow bandgap III–V semiconductors (InAs and InSb) are predicted to have third-order nonlinear susceptibilities (c3) several orders of magnitude higher than those of larger bandgap semiconductors. Among the zincblende crystals, perhaps the most popular candidate for generation of THz radiation by optical rectification is ZnTe. Nahata et al.11 first reported THz generation in ZnTe in an experiment that used a pair of ZnTe crystals for both generation and detection of THz radiation. In their work, a ZnTe crystal was pumped by 130 femtosecond pulses at 800 nm from a mode-locked Ti:sapphire laser. The Fourier spectrum of the temporal waveform demonstrated a broad bandwidth with useful spectral information beyond 3 THz. Subsequently, Han et al.12 reported a higher frequency response of 17 THz limited only by a strong phonon absorption at 5.3 THz. Systematic studies have also been conducted on THz emitter application of cadmium zinc telluride ternary crystals with varying Cd composition. It has been reported18,52 that the optimum Cd composition x = 0.05 enhances THz radiation as well as crystal quality of ZnTe. Apart from the zincblende crystals described earlier, GaSe is a promising semiconductor crystal that has been exploited recently for THz generation with an extremely large bandwidth of up to 41 THz.15,53 GaSe is a negative uniaxial layered semiconductor with a hexagonal structure characteristic of 62m point group and a direct bandgap of 2.2 eV at 300 K. The crystal has a large nonlinear optical coefficient (54 pm/V),53 high damage threshold, suitable transparent range, and low absorption coefficient, which make it an attractive option for generation of

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broadband mid infrared electromagnetic waves. For broadband THz generation and detection using a sub-20 fs laser source, GaSe emitter-detector system performance is comparable to or better than the use of thin ZnTe crystals. Using GaSe crystals of appropriate thickness as emitter and detector it is also possible to obtain a frequencyselective THz wave generation and detection system. The disadvantage of GaSe lies in the softness of the material that makes it fragile during operation. 2.3.1.2 Inorganic Electrooptic Crystals In 1971 Yang et al. first demonstrated the generation of THz radiation by optical rectification of picosecond optical pulses in inorganic electrooptic materials. In their experiment they observed THz radiation from a LiNbO3 crystal illuminated by picosecond optical pulses from a mode-locked Nd:glass laser.8 In the late 1980s, Auston and coworkers reported THz radiation generation by optical rectification in LiTaO3 with spectral bandwidth as high as 5 THz.54 A major drawback in this experiment was the difficulty in extracting the THz pulses from the generating crystal because of total internal reflection owing to small critical angle (and high static dielectric constant) in LiTaO3. In their later publications Auston et al. reported an approach that minimized the reflection loss and permitted the extraction of the THz radiation into free space.10 Zhang et al.9 later measured and calculated optical rectification from LiNbO3 and y-cut LiTaO3 under normal incident optical excitation. The spatiotemporal shape of the radiated THz pulse was found to be similar for both materials with LiNbO3 emitting a stronger signal because of a smaller dielectric constant and hence smaller reflection losses at the crystal boundary. The maximum signal obtained from LiTaO3 was found to be 185 times smaller than that emitted by a 4-Nmethylstilbazolium tosylate (DAST) organic crystal.19 2.3.1.3 Organic Electrooptic Crystals Organic crystals have been a recent source of interest as THz emitters as they have been reported to generate stronger THz signals than commonly used semiconductors or inorganic electrooptic emitters owing to their large second-order nonlinear electric susceptibility. Zhang et al.19 first reported THz optical rectification from an organic crystal, dimethyl amino DAST, which is a member of the stilbazolium salt family. Electrooptic measurements at 820 nm have reported a high electrooptic coefficient (>400 pm/V).19 In their work on THz emission by optical rectification of femtosecond laser pulses in DAST, Zhang et al. observed a strong dependence of the radiated THz field amplitude on sample rotation about the surface normal, for both parallel and perpendicular orientations of the incident optical pump beam. The angular dependence was in excellent agreement with the predicted theory, with the maximum signal occurring for the angle at which the optical polarization, the crystal polar “a” axis and the detector dipole axis, were aligned in the same direction.19 Zhang and coworkers19 also reported that with a 180-mW optical pump beam focused to a 200-µm diameter spot, the best DAST sample provided a detected THz electric field that was 185 times larger than that obtained from an LiTaO3 crystal and 42 times larger than GaAs and InP crystals under the same experimental conditions. DAST has also been shown to

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perform well at higher frequencies with an observable bandwidth up to 20 THz from a 100-µm DAST crystal showing a six-fold increase over that reported from a 30-µm ZnTe crystal under similar experimental conditions.55 The frequency spectrum of DAST shows a characteristic strong absorption line at 1.1 THz (from TO phonon resonance) along with some additional absorption lines between 3 and 5 THz. The latter (with the exception of the line at 5 THz) are significantly weaker than the absorption line at 1.1 THz, such that the THz amplitude at these frequencies is still well above the noise level.56 Nahata et al.57 advocated the use of organic compounds in polymeric forms for THz-frequency radiation generation instead of single crystals as these compounds offer significant advantages over single crystals. In particular they are easily processable, can be poled to introduce noncentrosymmetry and have higher nonlinear coefficients than inorganic materials. In their experiment Nahata and coworkers used a copolymer of 4-N-ethyl-N- (2-methacryloxyethyl) amine-4′-nitro-azobenzene (MA1) and methyl methacrylate (MMA) (commonly known as MA1:MMA) for generation of THz radiation via optical rectification. The radiated field amplitude from a 16-mm thick sample (electrooptic coefficient ~11 pm/V) was found to be four times smaller than that observed from a 1-mm thick y-cut LiNbO3 crystal. However, the coherence length of the polymer was reported to be 20 times larger. Nahata and coworkers suggested that the conversion efficacy relative to LiNbO3 could be substantially improved using available polymers with six times the nonlinear coefficient and millimeter-thick films poled at high fields. This could be easily created by a dielectric stack of thin-poled films.58 After the success of DAST and MA1:MMA as THz emitters, Hashimoto et al. reported THz emission via optical rectification from organic single crystals of N-substituted (N-benzyl, N-diphenylmethyl, and N-2 naphthylmethyl) derivatives of 2-methyl-4-nitroaniline.59 While no substantial emission was observed from N-diphenylmethyl and N-2 naphthylmethyl derivatives (owing to the strong phonon modes existing in the range 0 to 2.5 THz), the integrated intensity of the THz radiation emitted by N-benzyl-2-methyl-4-nitroaniline (BNA) was as intense as that by the DAST crystal under similar experimental conditions. The dynamic range for BNA is limited to 2.1 THz because of a strong phonon mode that exists around 2.3 THz. Kuroyanagi and coworkers60 later reported improved signal intensity (about three times greater than that generated by ZnTe) and increased bandwidth (up to 4 THz) from highly purified BNA crystals. Although DAST and BNA have been reported to perform better than ZnTe when used for THz generation by optical rectification, the THz electric fields generated by these crystals are more complicated in both time and frequency domains. Phonon bands in these crystals also result in the production of smaller range of frequencies than ZnTe. Recent investigations of a range of organic molecular crystals revealed that the crystal (–) 2-(a-methylbenzyl-amino)-5-nitropyridine (MBANP) is superior to ZnTe when used for optical rectification at 800 nm.61 MBANP crystallizes in P21 monoclinic62 and has the advantage of being relatively easy to grow in single crystals, probably because of its L-shaped structure and strong H bonding in certain directions as well as a suitable habit.61 Optical rectification of 800-nm pulses in -oriented MBANP resulted in THz pulses that are very similar to those

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54


produced by ZnTe both in frequency and time domain. However, MBANP produces a signal that is an order of magnitude higher because of its larger electrooptic coefficient (18.2 pm/V at 632.8 nm).63 Several other crystals similar to MBANP have also been studied.61 It was found that 4-(N, N-dimethylamino)-3-acetamidonitrobenzene and 4‑nitro-4′-methylbenzylidene aniline (NMBA) produced THz pulses through optical rectification of 800-nm pulses of comparable strength to MBANP. While NMBA is easy to grow, 4-(N, N-dimethylamino)-3-acetamidonitrobenzene is much more difficult owing to its needle-like habit, and obtainable crystal size is limited. Strong phonon band absorption also limits the power in the range 1.6 to 3 THz, which makes these crystals unsuitable for high-frequency applications. We have thus reviewed THz optical rectification from a variety of materials including semiconductors and inorganic and organic electrooptic crystals. The performance of these emitters in terms of bandwidth has been summarized in Table 2.3. ZnTe is the most popular choice for THz generation by optical rectification because it can generate extremely short and high-quality THz pulses. Organic crystals have been of recent interest owing to their higher electrooptic coefficient and enhanced capacity to generate stronger signals than the commonly used ZnTe. However, the THz fields generated by organic crystals such as DAST and more recently BNA are more complicated in both time and frequency domain. Low-frequency phonon bands also limit the bandwidth in these crystals than compared with ZnTe. Organic crystals also have a large naturally occurring birefringence, which complicates their application.55 This has resulted in ZnTe remaining the THz generator of choice. The comparative performance of a ZnTe THz emitter pumped by an amplified titanium sapphire laser system is shown in Figure 2.2.64 The choice of a suitable THz emitter also largely depends on the operating conditions. As mentioned previously, to achieve reasonable efficacy for nonlinear optical processes, long interaction length and appropriate phase matching conditions must be met. The phase matching condition is satisfied when the phase of the THz wave travels at the group velocity of the optical pulse. In the case of ZnTe, the phase matching condition and the subsequent enhancement of coherence length are achieved at 800 nm, making it the most suitable electrooptic crystal for THz wave emission and detection using a Ti:sapphire laser system with a center wavelength of 800 nm.

Table 2.3 Bandwidth Performances of Some Common Terahertz Emitters

Material GaAs GaSe ZnTe GaP DAST BNA

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Experimental Bandwidth (THz) 40(12) 41(15) 17(12) 3.5(45) 20(55) 2.1(59)

Lowest Optical Phonon Resonance (THz) at 300 K 8.02(71) 7.1(15) 5.4(12) 10.96(71) 1.1(56) 2.3(59)

Incident Pulse Wavelength (nm)

Incident Pulse Width (fs)

Detection Scheme

800(12) 780(15) 800(12) 1055(45) 800(55) 800(59)

12(12) 10(15) 12(12) 300(45) 15(55) 100(59)

Electrooptic Electrooptic Electrooptic PC antenna Electrooptic Electrooptic

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1000

Estimated based on measured saturation properties of ZnTe GaAs 10.7 kV/cm (You at al.) Fit based on (Darrow et al.)

(a)

G

0.1 1000

d

m a2 n

ae

sm

Pla

P

m

Ha

a(

m las

r ste

et

al.)

ETHz ∝ Jopt

Estimated based on measured saturation properties of ZnTe Fit based on (Darrow et al.) /cm 1kV s A Ga Te al.) Zn et m eld er t fi m s . 2 m xt Ha ae a( sm a m l s P Pla

10 1

on

ic

0.1

nd

ha

rm

0.01

JTHz ∝ Jopt2

as

m

a2

1E–3

Pl

1E–4 10–2 Pulse Energy Conversion (η)

eld

Fi xt.

GaAs 10.7 kV/cm (You et al.)

(b)

100

10

Te

Zn

as

m 2m

Pl

1

/cm

kV

ha

1 aAs

on ic

10

rm

100

GaAs 10.7 kV/cm (You et al.)

(c)

Estimated based on measured saturation properties of ZnTe

–3

Fit based on (Darrow et al.)

10–4 10–5

η∝

GaAs 1 kV/cm

10–6

m

10–7

2m

–8

1×10

1×10–9 1

10

J opt η∝

e ZnT Pla Ha sma 2 n rm d on ic

THz Pulse Energy (nJ)

Maximum Focused Field Amplitude (kV/cm)


100

d

. Fiel

a ext

Plasm

ma

Plas

ter ams

(H

1000

l.)

et a

J opt

η ∝ J op

t

10000

100000

Optical Pulse Energy (µJ)

Figure 2.2 Comparison of the THz emission efficacy of a 2-mm ZnTe crystal with other types of THz emitters: (a) the focused peak electric field, (b) the THz pulse energy, and (c) the energy conversion efficacy are displayed as a function of the energy per pulse of the femtosecond near infrared pump laser. (From Loeffler T., Semicond. Sci. Technol. 20, S140, 2005, IOP Publishing. With permission.)

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56


However, the large size of these lasers prevents them from being used in portable THz systems. To build a compact, integrated, and sensitive THz spectroscopy or THz imaging system, it has thus become crucial to develop a THz wave generation and detection technique coupled to an optical fiber cable. The erbium (Er)- and ytterbium (Yb)-doped fiber lasers operating at 1,550 nm and 1,000 nm, respectively, are especially attractive from this viewpoint. The large velocity mismatch and consequently a small coherent length, however, make ZnTe unsuitable for application at these wavelengths. Although GaP and CdTe have been suggested as potentially compatibles emitters for Yb-doped laser systems operating at 1,000 nm, GaAs is an excellent candidate for THz wave emission with erbium-doped fiber lasers. Calculations17 show that the phase matching condition is satisfied at 1,050 nm and 1 THz in CdTe crystals resulting in an enhancement of coherence length, whereas GaAs crystal phase matching occurs at 1,330 nm. This optical wavelength in GaAs is the longest among all binary semiconductors.14 Even at 1,550 nm, the coherence length still retains a large value, thus enabling GaAs to be used a suitable emitter in this wavelength range. GaAs also has the added advantage of having its lowest phonon resonance lying at higher THz frequencies (10 THz) and consequently increased bandwidth capability. The coherence length of some organic crystals such as BNA is also quite large in the long wavelength regime; however, because of the difficulties highlighted previously, organic crystals have not gained popularity as THz emitters.

2.3.2 Recent Developments The increasing interest in the development of novel THz sources has stimulated indepth studies of microscopic mechanisms of THz field generation in conventional semiconductors,38–53 electrooptic materials,55–63 and an extensive search for new materials to be employed in THz generation and detection. We conclude our discussion on THz emitters by highlighting some recent developments in THz emitter research. The THz emission from nonresonant optical rectification discussed in earlier sections resulted from dipolar excitations from such materials as GaAs, ZnTe, and LiNbO3. THz emission may also result from nondipolar excitations. Emission via optical rectification of femtosecond laser pulses in YBa2Cu3O7,65 single crystals of iron (Fe),66 and films consisting of nanosized graphite crystallites67 was observed and was found to have resulted from quadrupole magnetic dipole nonlinearities. For Fe, which crystallizes in a body-centered cubic lattice, optical rectification is forbidden by lattice symmetry. A nonvanishing second-order optical nonlinearity can, however, result from an electric quadrupole magnetic dipole contribution, surface nonlinearity or sample magnetization. Each of these contributions to the second-order nonlinear electric susceptibility has a characteristic dependence on sample azimuth and optical pump pulse polarization for a given crystal orientation. The THz emission reported from Fe was found to be approximately three orders of magnitude weaker than that from a 1-mm thick ZnTe crystal under the same optical excitation conditions. THz emission from metals such as gold and silver has also come to light recently.20,21 Although second harmonic generation had been reported from metal surfaces as early as the 1960s, optical rectification of laser light had not been reported until recently. Kadlec et al.20,21 generated intense THz emission by optical

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rectification of p‑polarized 810-nm laser pulses at a fluence of 6 mJ/cm2 in gold films. The THz amplitude was found to scale quadratically with optical fluence for values up to 2 mJ/cm2. The s-polarized optical pump beam resulted in THz radiation emission, which was approximately three times weaker than the signal obtained from a p-polarized optical pump beam. THz emission from silver showed a different behavior. At the highest optical pump fluence, the emitted THz radiation was 15 times weaker than for gold and showed a linear dependence on incident pump fluence.20 These recent experimental findings reveal the possibility of accessing a wealth of new information about nonlinear optical properties and electronic structures of metal surfaces via femtosecond optically excited THz emission measurements.

2.4 Experimental Results of Terahertz Electrooptic Detection Much research activity has been devoted to increasing the sensitivity and bandwidth of electrooptic detection of THz-frequency radiation pulses. Other than the duration of the optical probe pulse, the bandwidth of free-space electrooptic sampling is only limited by the dielectric properties of the electrooptic crystal. Electrooptic sampling of THz-frequency radiation is a powerful tool for detection of electromagnetic radiation pulses in and beyond the mid-infrared range. A variety of electrooptic materials, including semiconductors and inorganic and organic electrooptic crystals, have been tested for detector applications in THz spectroscopy. In this section, we present a brief overview of the experimental results of broadband THz electrooptic detection with various materials.

2.4.1 Materials 2.4.1.1 Semiconductors and Inorganic Crystals In 1995, Wu and Zhang27 first demonstrated free-space electrooptic sampling of short THz radiation pulses. In their experiment, a GaAs photoconductive emitter triggered by 150-fs optical pulses at 820 nm radiated electromagnetic waves of THz frequency. A 500-µm thick LiTaO3 crystal, with its c axis parallel to the electric field polarization, was used as the electrooptic sampling element. To improve detection efficacy, the THz radiation beam was focused onto the detector crystal by a highresistivity silicon lens. The temporal resolution of the detected THz transient was limited by the velocity mismatch between optical and THz frequencies. Later work by Wu and Zhang68,69 addressed this shortcoming by careful geometric design of the radiation emitter and a special cut detector crystal. A signal-to-noise ratio of about 170:1 was obtained with a 0.3-second lock-in integration time constant. Wu and coworkers also tested ZnTe for detector applications in a novel experimental geometry in which the optical and THz beams were made to propagate collinearly in the electrooptic crystal.68 In this setup, a 1.5-mm thick, -oriented ZnTe crystal was used as the detector with the optical probe and THz beam set parallel to the edge of the ZnTe crystal for optimal electrooptic phase modulation. A signal-to-noise ratio of about 10,000:1 was obtained for a single scan using again a lock-in amplifier with time constant of 0.3 seconds. Nahata et al.11 demonstrated a

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broadband time domain THz spectroscopy system using -oriented ZnTe crystals for generation of THz radiation by optical rectification and detection of THz radiation by electrooptic sampling. An important condition for generation and detection of THz radiation by nonlinear optical techniques is the appropriate phase matching between the optical and THz pulses in the nonlinear optical crystal. For a medium with dispersion at optical frequencies, phase matching is achieved when the phase velocity of THz wave is equal to the velocity of optical pulse envelope (or the optical group velocity).11 The data in Figure 2.8 show the effective index of refraction of ZnTe at optical wavelengths nopt eff and THz frequencies nTHz. The coherence length lc of ZnTe at THz frequencies for an optical wavelength λ = 780 nm decreases as illustrated in Figure 2.3. The temporal waveform of the measured THz electric field and its Fourier amplitude and Fourier phase are displayed in Figures 2.3, 2.4, and 2.5, respectively. The peak of the THz pulse amplitude shows a three-fold rotational symmetry when the ZnTe detector crystal is rotated by 360° about an axis normal to the surface. This result is inherent to all and zincblende crystals.40 The broadband detection capability of ZnTe is obvious from the Fourier amplitude spectrum (Figure 2.4), which extends well beyond 2 THz. The rolloff in sensitivity for higher frequencies from phase mismatch and the shorter coherence length are in agreement with data presented in Figures 2.6, 2.7, and 2.9. A thinner crystal detects higher frequencies better than a thicker crystal. However, the detector sensitivity of a thinner crystal is reduced compared to a thicker crystal. In addition to limitations imposed by

Electro-optic Signal (arb. units)

4 2 0 –2 –4 2

4

6

8

10

Time (ps)

Figure 2.3 Time domain measurement of a THz pulse by electrooptic sampling with a ZnTe crystal 10 mm × 10mm × 1 mm in size. The THz pulse and the optical probe pulse are incident onto the surface of the ZnTe crystal parallel to the plane. The optical axis of the crystal is in the direction. The linear polarization of the THz radiation pulse is perpendicular to the optical axis. The linear polarization of the incident optical probe pulse is parallel to the optical axis. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 7,2000. With permission.)

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Fourier-amplitude (arb. units)

104

103

102

101

100

0.5

1.0

1.5

2.0

2.5

3.0

Frequency f (THz)

Figure 2.4 Fourier amplitude of the time domain measurement displayed in Figure 2.3. The bandwidth of the signal is approximately 2.75 THz. The dynamic range of the measurement is 500:1. (Adapted from Selig, Hanns, Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 7, 2000. With permission.)

phase mismatch, the absorption caused by a transverse optical phonon resonance at 5.4 THz11 in ZnTe is expected to attenuate the high-frequency components of THz radiation. Later experiments by Han and Zhang70 reported a detection bandwidth up to 40 THz using a 27-µm ZnTe detector crystal. 2.0×104

Fourier-phase (arb. units)

1.9×104 1.8×104 1.7×104 1.6×104 1.5×104 1.4×104 1.3×104 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Frequency f (THz)

Figure 2.5 Fourier phase of the time domain measurement displayed in Figure 2.3. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 7, 2000. With permission.)

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60



6 4 2 0 –2 ZnTe 0.5 mm ZnTe 1.0 mm

–4 –6

0

1

2

3

4

5

6

7

8

Time (ps)

Figure 2.6 Time domain measurements of THz pulses by electrooptic sampling with ZnTe crystals of different thickness. For both measurements, the optical probe beam and the THz pulse both incident onto the surface of the crystals parallel to the plane. The surface of the crystals is 10 mm × 10 mm. The linear polarizations of the incident THz-pulse and probe pulse are perpendicular to each other. The polarization of the probe beam is parallel to the optical axis ( direction). The ratio of the peak-to-peak amplitude of the THz pulses measured by the 1-mm crystal and the 0.5-mm crystal is 2.3 and scales approximately with the thickness of the two crystals (1 mm/0.5 mm = 2). The thicker ZnTe crystal exhibits a larger signal than the thinner crystal because the phase retardation G ∝ l is linearly proportional to the thickness l of the electrooptic crystal. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 35,2000. With permission.)

Although ZnTe delivers excellent performance in the 800-nm wavelength regime, its efficacy is inherently limited by large group velocity mismatch (GVM) of 1 ps/mm68 between optical and THz frequencies. An ideal material would be one with larger electrooptic coefficient and lower GVM. At 886 nm, GaAs has been found to possess a moderate electrooptic coefficient (25 pm/V), but a low GVM of 15 fs/mm. The electrooptic coefficient of LiTaO3 was found to be comparable to that of ZnTe, but LiTaO3 has a large GVM (>14 ps/mm), whereas for organic crystals such as DAST, the electrooptic coefficient is very large but GVM is comparable to ZnTe.68 A substantial improvement in bandwidth detection and sensitivity may be obtained by using materials with phonon resonance occurring at higher THz frequencies. Among inorganic media with comparable nonlinear optical properties zinc selenide (lowest TO phonon resonance at 6 THz),11 GaAs (TO phonon resonance at 8 THz), and GaP (TO phonon resonance at 11 THz)71 are excellent candidates. To verify the performance of electrooptic sampling with GaAs, Vosseburger et al.72 performed time-resolved experiments of THz radiation detection using bulk GaAs as the electrooptic sampling element. An excitation wavelength of 900 nm was chosen to avoid photoexcitation of carriers by interband absorption in the GaAs. The measured THz radiation pulse had a complex

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 61 104

Fourier Amplitude (arb. units)

ZnTe 0.5 mm ZnTe 1.0 mm 103

102

101

100 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Frequency (THz)

Figure 2.7 Fourier amplitudes of the time domain measurements displayed in Figure 2.6. The thinner ZnTe crystal has a slightly larger bandwidth than the thicker ZnTe crystal. The calculated bandwidths of a 0.5-mm and a 1.0-mm ZnTe crystal are 2.86 THz and 2.62 THz, respectively. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 35,2000. With permission.)

Optical Wavelength (Nm) 6.0

0.5

0.6

0.7

0.8

0.9

1.0 6.0

Index of Refraction n

5.5

5.5 nTHz

nopt eff

5.0

5.0

4.5

4.5

4.0

4.0

3.5

3.5

3.0

3.0

2.5

2.5

2.0

0

1

2

3

4

5

2.0

Frequency f (THz)

Figure 2.8 Index of refraction of ZnTe at optical wavelength (nopt eff ) and THz frequencies (nTHz). Optical radiation at 760-µm wavelength propagates at the same speed as 2.6 THz- frequency radiation in ZnTe. The effective index of refraction at λ = 780 µm is nopt eff = 3.27. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 33, 2000. With permission.)

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62


Coherence Length lc (mm)

10

1

0.1

0.01

0

1

2

3

4

5

Frequency (THz)

Figure 2.9 Calculated coherence length lc of ZnTe as a function of THz frequency for λ = 780 µm. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 7, 2000. With permission.)

temporal structure and was smaller by a factor of approximately 20 compared with the same pulse measured by a ZnTe crystal. The reduction in amplitude had its origins in the smaller electrooptic coefficient of GaAs than that of ZnTe and poor phase matching of GaAs at 900 mm and 1 THz. Among zincblende crystals, GaP is an excellent alternative to ZnTe for electrooptic sampling of THz-frequency radiation because of its high-frequency phonon band at 11 THz,71 the highest available among all zincblende crystals. The electrooptic coefficient of GaP (r41 ≈ 0.97 pm/V30), however, is one fourth that of ZnTe. In 1997, Wu and Zhang73 first demonstrated electrooptic sampling of THz radiation with a GaP crystal. Using a GaAs emitter excited by 50-fs pulses and a 150-µm thick GaP crystal, they demonstrated a bandwidth resolution up to 7 THz. Broadband THz electrooptic sampling with GaP crystals was further developed by Leitenstorfer et al.74 They carefully evaluated the frequency dependence of the refractive index, the electrooptic coefficient and the response function of the GaP electrooptic THz detector. Using a 13-µm thick crystal and 12-fs titanium–sapphire laser pulses the detected spectral bandwidth of GaP was estimated to be ~70 THz. One of the recently reported materials used for free-space electrooptic sampling of THz radiation is ZnSe in crystalline and polycrystalline form.75 Although the electrooptic coefficient of ZnSe (r41 ≈ 2pm/V75) is only half of that of ZnTe, the TO phonon resonance frequency (6 THz)11 is higher than that of ZnTe (5.4 THz),11 thus promising a higher detection bandwidth potential for the former. The GVM for ZnSe (0.96 pm/V75) is comparable to that for ZnTe. Using a 10-fs titanium–sapphire laser pulse to excite a 100-µm GaAs photoconducting antenna and a 0.5-mm, -oriented ZnSe single crystal as detector, a spectral bandwidth of 3 THz was

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 63 90 120

1.0

φ (°)

60


0.8 30

150

0.6 0.4 0.2 0.0

180

0

0.2 0.4 0.6

330

210

0.8 1.0

240

300 270

Figure 2.10 Measurements of the peak electrooptic signal as a function of the angle Φ between the direction of a ZnTe crystal and the direction of linear polarization of the optical probe beam. The polarization of the THz pulse is perpendicular to the polarization of the optical probe pulse. During the measurements, the orientations of the polarization of the optical and the THz pulses are fixed and the crystal is turned in steps of 10°. An angle Θ = 90° corresponds to the situation in which the polarization of the optical probe beam is parallel to the direction of the ZnTe. Two major maxima at Θ = 90° and Θ = 270° and four minor maxima at Θ = 30°, Θ = 150°, Θ = 210°, and Θ = 330° are observed. The amplitude of the peak THz electric field is estimated to be 5 V/m. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 47, 2000. With permission.)

measured. In this experiment, the bandwidth detected by the ZnSe crystal was limited only by the generation of the THz radiation. It was found that for thicker (1 mm) polycrystalline electrooptic ZnSe samples, the random nature of crystallographic orientations within the interaction length distorted the phase of THz waveform. However, a reduction of the thickness to 0.15 mm minimized the distortion and extended the bandwidth up to 4 THz. Because polycrystalline semiconductors offer practical advantages in terms of ease of fabrication75 over their single crystal counterparts, the successful application of polycrystalline materials for free-space electrooptic sampling permits the possibility of utilizing non-lattice matched thin film integrated electrooptic detectors of THz radiation. As discussed previously, the mismatch between THz phase velocity and the group velocity of the optical probe pulse limits the detection bandwidth of zincblende electrooptic crystals such as ZnTe and GaP. However, this limitation can be eliminated by using a novel detection scheme, taking advantage of the type II phase matching in a naturally birefringent crystal such as GaSe.53,76 The phase matching condition can be satisfied by angle tuning whereby the electrooptic crystal

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64


Table 2.4 Properties of Typical Electrooptic Sensor Materials*

Material

Crystal Structure

Point Group

ZnTe GaP ZnSe LiTaO3

Zincblende(30) Zincblende(30) Zincblende(30) Trigonal(30)

43m( 30 ) 43m( 30 ) 43m( 30 ) 3m(30)

GaSe DAST Poled polymers

Hexagonal(53) — —

62 m( 53) — —

Electrooptic Coefficient (pm/V)

Group Velocity Mismatch (ps/mm)

Surface Orientation

1.1(68) — 0.96(75) 14.1(68)

110(11) 110(73) 111(75) —

2.5(11) 7(73) 3(75) —

0.10(76) 1.22(68) —

— — —

120(76) 6.7(77) 33(78)

r41 = 3.90(30) r41 = 0.97(30) r41 = 2.00(30) r33 = 30.3(30) r13 = 5.70 r22 = 14.4(76) r11 = 160(68) —

Experimental Detection Bandwidth (THz)†

* A more comprehensive listing may be found in reference 68. † The bandwidth reported here is subject to experimental conditions and not the absolute bandwidth for a particular emitter/sensor material.

(z-cut GaSe crystal) is tilted by an angle qdet (the phase matching angle) about a horizontal axis perpendicular to the direction of time-delayed probe beam.76 In the experiments conducted by Kubler and coworkers,76 a THz radiation pulse was generated by optical rectification of 10-fs titanium–sapphire laser pulses in a 20-µm thick GaSe crystal (qem = 57°) and detected with a GaSe sensor (qdet = 60°) of thickness 30 µm. qdet/em = 0 signifies normal incidence. The amplitude spectrum of the THz pulse peaked at 33.8 THz and extended from 7 THz to beyond 120 THz. Compared with a 12-µm thick ZnTe crystal that displayed a local minimum at 34 THz under the same excitation, the GaSe spectrum showed a nearly flat response from 10 THz up to 105 THz. The enhanced performance of GaSe crystals over ZnTe crystals is attributed mainly to a greater interaction length because of better phase matching and a larger electrooptic coefficient of GaSe (Table 2.4). 2.4.1.2 Organic Crystals In the quest for novel materials for electrooptic sampling of THz radiation pulses, organic electrooptic crystals have attracted attention owing to their very high electrooptic coefficients. Han and coworkers55 first demonstrated the application of the organic ionic salt crystal DAST as a free space electrooptic detector of THz radiation. The electrooptic coefficient for DAST is 160 pm/V at 820 nm68 and is almost two orders higher than that found in ZnTe (Table 2.4). However, DAST exhibits two confirmed phonon absorption peaks at 1.1 THz and 3.05 THz.77 Recently, nonlinear optical generation and detection of THz radiation pulses was investigated theoretically and experimentally by Schneider et al.77 They demonstrated that the THz radiation spectrum generated and detected using DAST crystals extended from 0.4 THz to 6.7 THz, depending on the laser excitation wavelength in the 700-nm

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to 1,600-nm wavelength range. Furthermore, they found that the technologically important wavelengths around 1,500 nm are velocity matched to THz frequencies between 1.5 THz and 2.7 THz. Among other organic materials, poled polymers have been reported to exhibit excellent broadband electrooptic detection capability.78 These materials demonstrate large electrooptic coefficients,79 low dispersion between the THz and optical refractive indices,57,80 ease of fabrication in large area thin films, and easily modifiable chemical structures for better THz and optical properties because of their organic nature. Using poled polymers for free-space electrooptic detection, Nahata and coworkers78 recently measured an amplitude spectrum up to frequencies of 33 Hz.

2.5 Application of Electrooptic Sampling of THz Electric Field Transients Nonlinear techniques for the generation and detection of short pulses of THz- frequency radiation have found important applications such as time domain THz spectroscopy and time domain THz imaging.34 Here, we discuss briefly the application of electrooptic sampling of THz radiation pulses to longitudinal electron bunch length measurements. The electrooptic detection of the local nonradiative electric field that travels with a relativistic electron bunch has recently emerged as a powerful new technique for subpicosecond electron bunch length measurements.81–84 The method makes use of the fact that the local electric field of a highly relativistic electron bunch that moves in a straight line is almost entirely concentrated perpendicular to its direction of motion. Consequently, the Pockels’ effect induced by the electric field of the passing electron bunch can be used to produce birefringence in an electrooptic crystal placed in the vicinity of the beam. Specifically, the birefringence induced by a single electron bunch is measured by monitoring the change of polarization of the wavelength components of a chirped, synchronized titanium–sapphire laser pulse. When the electric field of an electron bunch and the chirped optical pulse copropagate in the electrooptic crystal, the polarizations of the various wavelength components of the chirped pulse that passes through the crystal are rotated by different amounts that correspond to different portions of the local electric field. The direction and degree of rotation are proportional to the amplitude and the phase of the electric field. Thus the time profile of the local electric field of the electron bunch is linearly encoded to the wavelength spectrum of the optical probe beam. An analyzer converts the modulation of the polarization of the chirped optical pulse into an amplitude modulation of its spectrum. The time profile of the electric field of the electron bunch is measured as the difference of the spectrum with and without a copropagating electron bunch. The width of the temporal profile corresponds directly to the electron bunch length, and the shape of the temporal profile is proportional to the longitudinal electron distribution within the electron bunch. The length and the shape of individual electron bunches are determined by measuring the spectra of single chirped laser pulses with an optical multichannel analyzer equipped with a nanosecond shutter. The method allows direct in situ electron bunch diagnostics with a high signal‑to‑noise ratio and subpicosecond time resolution.

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66


Grating f + ∆z

Lens

Mirror

f

Pulse Generator ccd Intensifier Spectrometer

Signal

(a) Optical Stretcher

Reference Ti: Sapphire Amplifier

Optical Stretcher (a)

Fiber Analyzer

Polarizer FEL Cavity

FEL

rf-clock ZnTe Accelerator

e-beam

Undulator Dump

Figure 2.11 Experimental setup for electron bunch length measurements by electrooptic sampling with chirped optical pulses. The electron bunch length is measured by using a ZnTe crystal placed inside the vacuum pipe at the entrance of the undulator. The femtosecond near infrared laser repetition rate is synchronized with the repetition rate of the electron bunches. The inset (a) exhibits a two-dimensional optical stretcher for laser pulse chirping. (Adapted figure with permission from I. Wilke, A. M. McLeod, W. A. Gillespie, G. Berden, G. M. H. Knippels, A. F. G. van der Meer, Phys. Rev. Lett. 88, 124801-1 (2002). Copyright 2002 by the American Physical Society.)

The experimental arrangements for the electrooptic electron bunch length measurements are schematically illustrated in Figure 2.11. The electron bunch source is the radiofrequency linear accelerator at the FELIX free electron laser facility in the Netherlands.85 The electron beam energy of FELIX was set at 46 MeV, and its charge per bunch at around 200 pC. The micropulse repetition rate was 25 MHz, and the macropulse duration was around 8 µ with a repetition rate of 5 Hz. A titanium– sapphire amplifier, producing 30-fs FWHM pulses at 800 nm with a repetition rate of 1 kHz, was used as a probe beam. The 30-fs optical laser pulses are chirped to pulses of up to 20 ps (FWHM) duration with an optical stretcher which consists of a grating, a lens, and a plane mirror.86 The duration of the chirped pulses has been measured with an optical autocorrelator based on second harmonic generation in a BBO crystal. The electron bunch length is measured inside the accelerator beam pipe at the entrance of the undulator. A 0.5-mm thick ZnTe crystal is used as an

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electrooptic sensor and is placed with its 4 × 4 mm 2 front face perpendicular to the propagation direction of the electron beam. The incoming chirped laser beam is linearly polarized. The outgoing chirped beam is split into a signal beam and a reference beam used to monitor possible laser power fluctuations. The signal beam passes through an analyzer (a second polarizer), which is crossed with respect to the first polarizer. Subsequently, the spectra of the chirped laser pulses are dispersed with a grating spectrometer and the line spectra are focused onto a charge-coupled device (CCD) camera. The CCD camera is equipped with an intensifier, which acts as a nanosecond shutter with a gate width of 100 ns. Single-shot measurements are performed by actively synchronizing the titanium–sapphire amplifier87 with both the repetition rate of the electron beam and the gate of the CCD camera, and therefore recording only the spectrum of one chirped laser pulse at a time. The temporal overlap between the electron bunch and chirped laser pulse is controlled by an electronic delay. Figure 2.12a shows measurements of the chirped laser pulse spectra with and without a copropagating electron bunch. The signal spectra are labeled by S and S′. The crossed polarizers exhibit a finite transmission of 1.7% if the electron bunch does not overlap with the chirped laser pulse. This is attributed to a small intrinsic stress birefringence of the ZnTe. The transmission of the crossed polarizers changes significantly when the electron bunch and the laser pulse copropagate: in these circumstances, a large peak, which corresponds to a strong enhancement of the transmission, is observed in the center of the spectrum. The strong change of the spectrum is attributed to the wave length-dependent change in polarization of the chirped laser pulse because of the electric field of the electron bunch. The length and shape of the electron bunch is obtained by subtracting the spectrum without copropagating electron bunch, which has been corrected for laser power fluctuations by multiplication with the ratio of the reference spectra R/R′ from the spectrum with copropagating electron bunch. This difference S-(S′R/R′) is corrected for the wavelength-dependent variations in intensity in the spectrum by dividing by the spectrum S′R/R′. The pixels are converted to time by measuring the length of the chirp t, the spectral resolution of the spectrometer and CCD setup, Dl/pixel, and the bandwidth of the chirped laser pulses Dlbw. Then, the time interval per pixel is given by (Dt/pixel) = (Dl/pixel)/(t/Dlbw). For the spectra of Figure 2.12a, the chirp was 4.48 ps FWHM, which results in an electron bunch measurement as displayed in Figure 2.12b. The width of the electron bunch is (1.72 ± 0.05) ps FWHM. The signal-to-noise ratio depends on the position in the spectrum; in the center of the spectrum, it is better than 200:1. The measured width and shape of a single electron bunch agree very well with the electron bunch measurements averaged over more than 8,000 electron bunches88 and CTR measurements.89 Linear accelerators used as drivers for new femtosecond x‑ray free‑electron lasers or employed in new teraelectron volt linear electronpositron colliders for high-energy physics, require dense, relativistic electron bunches with bunch lengths shorter than a picosecond. Precise measurements of the electron bunch length and its longitudinal charge distribution are important to monitor the preservation of the beam quality, whereas the electron bunch train travels through the beam pipe, as well as to tune and to operate a linear collider or a FEL.

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68

Terahertz Spectroscopy: Principles and Applications 5×104 4×104

S S´

Counts

3×104 2×104 1×104 0 100

200

300

400

Pixel (a)

Norm. Electro-optic Signal (a.u.)

1.0 0.8 0.6 0.4 0.2 0.0 –0.2

–4

–2

0

2

4

Time (ps) (b)

Figure 2.12 Fig. 2.12 (a) Measurements of the chirped laser pulse spectra with(s) and without (s′) a copropagating electron bunch. (b) single shot electron bunch length measurements. The width of the electron bunch is (1.72 ± 0.05) ps FWHM.

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3

Time-Resolved Terahertz Spectroscopy and Terahertz Emission Spectroscopy Jason B. Baxter and Charles A. Schmuttenmaer Yale University

Contents 3.1 3.2

3.3

Introduction.................................................................................................... 74 Time-Resolved Terahertz Spectroscopy........................................................ 75 3.2.1 Introduction 3.2.2 History and Examples......................................................................... 77 3.2.3 Experimental Setup and Data Collection........................................... 78 3.2.3.1 Basic Requirements............................................................... 78 3.2.3.2 Detailed Description of Spectrometer................................... 79 3.2.3.3 Data Collection...................................................................... 81 3.2.3.4 Importance of Spot Size........................................................ 83 3.2.4 Data Workup.......................................................................................84 3.2.4.1 Calculating Conductivity......................................................84 3.2.4.2 Use of Complex Transmission Coefficients.......................... 88 3.2.4.3 Special Treatment at Short Pump-Delay Times....................92 3.2.4.4 Treatment of Porous Media................................................... 95 Terahertz Emission Spectroscopy..................................................................96 3.3.1 Experimental.......................................................................................97 3.3.1.1 Far Field versus Near Field................................................... 98 3.3.1.2 Terahertz Focusing Optics....................................................99 3.3.2 Data Analysis......................................................................................99 3.3.2.1 Sample Orientation.............................................................. 100 3.3.2.2 Excitation Polarization........................................................ 100 3.3.2.3 Emitted Waveform.............................................................. 100 3.3.3 Specific Examples............................................................................. 103 3.3.3.1 Photoconductive Switches................................................... 103 3.3.3.2. Shift Currents and Optical Rectification............................. 103

73

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3.3.3.3 Intramolecular Charge Transfer in Orienting Field............ 107 3.3.3.4 Demagnetization Dynamics................................................ 112 3.3.4 General Formalism........................................................................... 113 3.4 Conclusions.................................................................................................. 115 References............................................................................................................... 115

3.1 Introduction This chapter explores time-resolved aspects of THz spectroscopy. THz spectroscopy involves generating and detecting subpicosecond far-infrared (IR) pulses with an ultrafast visible or near-IR laser, as has been discussed in previous chapters. THz spectroscopy covers the region from 0.3 to 20 THz (10–600 cm–1), with most of the work being done between 0.5 and 3 THz. When converted to other units, 1 THz is equivalent to 33.3 cm–1 (wave numbers) or 0.004 eV photon energy or 300 µm wave length. For example, the THz region is ideal for probing semiconductors because the frequency range closely matches typical carrier scattering rates of 1012 to 1014 s–1, allowing for more accurate modeling of the photoconductivity data. In conventional benchtop experiments, THz pulses are created and detected using short-pulsed Ti:sapphire lasers with pulsewidths ranging from ~100 fs to ~10 fs. The pulsed light allows time-resolved THz studies, in contrast to continuous sources of far-IR radiation such as arc lamps or globars. Pulsed sources such as free electron lasers or synchrotrons have also recently achieved subpicosecond pulse duration, but these are large, expensive user facilities and are less accessible than a Ti:sapphire laser in an individual laboratory. THz spectroscopy also has the advantage that it measures the transient electric field, not simply its intensity.1 Coherent detection allows direct determination of both the amplitude and the phase of each of the spectral components that make up the pulse. The absorption coefficient and refractive index of the sample are computed from the amplitude and phase. Thus the complex-valued permittivity of the sample is obtained without requiring a Kramers-Kronig analysis. The great majority of the results reported in the literature that use conventional far-IR sources and detectors present the frequency-dependent absorption coefficient, but not the refractive index. In this respect, THz spectroscopy provides a convenient method for determining the complex permittivity, even for studies that are not time-resolved. There are three categories of commonly performed experiments: THz time domain spectroscopy (THz-TDS), time-resolved THz spectroscopy (TRTS), and THz emission spectroscopy (TES). The information obtained in a THz-TDS experiment is equivalent to that obtained from a frequency domain linear absorption spectrometer, for example, an FTIR. THz-TDS made its debut in 1988–1989, and the majority of THz work done to date has been along these lines. There are distinct advantages of the THz-TDS method over frequency domain methods, as discussed in detail in Chapter 1. However, the present chapter concerns time-resolved experiments, and THz-TDS need not be considered further. Aside from a small number of initial studies,2–8 most of the development of TRTS has taken place since the year 2000. In contrast to a standard THz-TDS experiment,

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this type of experiment is somewhat more difficult to perform and analyze. In TRTS, the sample is typically photoexcited with a laser pulse (with wavelength ranging from near-IR to ultraviolet) and the time-dependent THz spectrum is obtained with the THz probe pulse. The THz spectrum can be obtained at times ranging from less than 100 fs to more than 1 ns after photoexcitation. This method is sometimes referred to as optical pump/THz probe spectroscopy, but that designation is too limiting. One need not pump optically to make a time-resolved measurement, although the examples given in this chapter all employ optical excitation. TRTS differs from THz-TDS at a very fundamental level. THz-TDS provides information about the static properties of the sample, but TRTS probes the dynamic, evolving properties of the material. For example, the THz transmission of a material depends strongly on how well it conducts electricity. In fact, TRTS is a noncontact electrical probe with subpicosecond temporal resolution, and the behavior of undoped semiconductors on photoexcitation has been studied in great detail as they switch from insulating to conducting. THz emission spectroscopy, TES, has been carried out for as long as THz-TDS has. To perform THz-TDS, it is necessary to generate THz pulses before sending them through the sample under study. However, in that capacity, one is not trying to learn about the material that generates the THz pulses, but is simply using it as a source of THz radiation. In contrast, TES refers to the situation where one is trying to understand the properties of the material being studied by analyzing the shape and amplitude of the THz waveform emitted. It is basically a THz-TDS setup in which the sample itself is the THz emitter. Over the years, semiconductors, superconductors, heterostructures, oriented molecules in solution, and magnetic films have all been studied with this method. All of these methods (THz-TDS, TRTS, and TES) take advantage of one or more of the unique characteristics of THz spectroscopy. The time-dependent electric field is measured (rather than its intensity). These are very bright far-IR sources, and the detectors employed have certain advantages over standard far-IR detectors as well. Most important, however, is the ability to carry out time-resolved experiments in the far-IR region of the spectrum with subpicosecond time resolution.

3.2 Time-Resolved Terahertz Spectroscopy 3.2.1 Introduction TRTS is a noncontact electrical probe capable of determining the complex-valued, frequency-dependent, far-IR permittivity with a temporal resolution of better than 200 fs. TRTS is used to study the changes in a material’s permittivity upon photoexcitation with an optical pump pulse. Changes in the permittivity arise from phenomena such as the photoinduced creation of mobile electrons and holes, polarizable excitons, and polarons. We will refer to “photoconductivity” as the change in sample permittivity upon photoexcitation, although conductivity does not necessarily arise from electronic conduction. Knowledge of a material’s frequency-dependent photoconductivity is important for its use in electronic and optoelectronic devices. Furthermore, the ability to characterize electrical properties in a noncontact fashion

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Terahertz Spectroscopy: Principles and Applications τpp

Sample

Figure 3.1 Schematic illustration of time-resolved THz spectroscopy experiment. The sample is photoexcited with an optical pump pulse and probed at a time, τpp, later with a THz pulse. The sample attenuates and delays the THz pulse according to its complex permittivity.

is critical in the study of nanomaterials, where it can be difficult or impossible to make electrical measurements using conventional probes. The intensity of the THz transmission change after photoexcitation is, in most cases, proportional to both the photoexcited carrier density and mobility. This is in contrast to methods such as transient absorption or time-resolved luminescence, which are sensitive to either the sum or the product of the electron and hole distribution functions, respectively. Additionally, luminescence methods are limited to direct gap semiconductors, whereas TRTS can be used to study carrier dynamics in both direct and indirect gap materials. Even methods that are sensitive to the diffusion of carriers, such as transient grating or four-wave mixing, are not able to determine the frequency-dependent conductivity. TRTS has high sensitivity to mobile electrons. For example, it can measure photoexcited carrier column densities as low as 5 × 1010 cm–2 in GaAs (equivalent to a carrier density of 5 × 1014 cm–3 with a skin depth of 1 µm). Because TRTS measures the sample permittivity, it is sensitive not only to mobile electrons and holes, but also to quasiparticles such as excitons and polarons. The different signatures of these particles in the frequency domain allow one to distinguish the source of permittivity changes using a combination of TRTS and appropriate models. Several different types of experiments make use of the subpicosecond pulses in the far-IR region of the spectrum. In each case, the sample is excited with an optical pulse and then probed with a terahertz pulse as shown in Figure 3.1. So-called “pump scans” monitor a single point on the THz waveform (or the integrated power using a bolometer) and vary the pump probe delay time to measure the average response of the material. In this way, it is possible to characterize the average far-IR response to an optical perturbation. One can measure the overall change in transmission as well as the time scales for the material to respond and to return to its equilibrium state. A second technique, often called a probe scan, is used to record the entire THz wave form at a fixed delay time after photoexcitation. This method obtains the frequencydependent, complex-valued photoconductivity. Finally, the most comprehensive type of study is to create a two-dimensional (2D) grid that maps out the THz waveform at many pump delay times. Although average response and fixed pump delay studies

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are quite useful, the maximum amount of information is obtained from a 2D study in which the photoconductivity can be determined as a function of both frequency and delay time. This method is often referred to as optical pump/THz probe spectroscopy or time-resolved THz spectroscopy. Many different materials systems have been investigated using TRTS. Carrier dynamics in bulk and nanostructured semiconductors, superconductors, and strongly correlated electron systems have been topics of recent research. A very brief accounting of some initial and more recent studies is given below to demonstrate some of the different phenomena that can be observed with TRTS, and other examples will be discussed in detail in subsequent chapters. The remainder of this section describes the experimental apparatus, collection of data, and analysis of data. It points out the common assumptions made during the interpretation of data as well as methods of dealing with possible pitfalls or artifacts.

3.2.2 History and Examples The methods of data collection and analysis used in TRTS were first extensively tested on the well-characterized GaAs system in the late 1990s. A gallium arsenide (GaAs) wafer was shown to have frequency-dependent photoconductivity that follows the Drude model and mobilities that match well with those determined by other experimental techniques.9 Carrier injection time was a function of optical pump wavelength. High-energy photons can excite electrons into low mobility valleys in the conduction band. Carriers scatter and relax into lower energy, high mobility regions of momentum space, resulting in an increase in THz absorbance. In low temperature-grown GaAs (LT-GaAs) thin films, the electron lifetime was found to be only 1.1 ps,10 compared with hundreds of picoseconds in GaAs,9 because of the increased recombination rates arising from higher defect densities. Mobility in lowtemperature-GaAs is also a factor of two lower than in GaAs single crystals. The mobilities determined in these studies could be compared with those from electrical measurements, with TRTS contributing previously unobtainable information at very short time scales. After the techniques of TRTS had been benchmarked using bulk samples with well-characterized electrical properties, its noncontact nature could be exploited to study semiconductor nanomaterials such as nanowire arrays and nanoparticle films. Although it is cumbersome to make contacts to and electrically characterize individual nanowires, TRTS allows characterization of a nanowire array without having to remove the nanowires from the substrate. In cases such as these, an appropriate effective medium theory must be taken into account. Conductivities of InP,11 CdSe,12,13 TiO2,14,15 and ZnO nanoparticle films16 and in ZnO nanowire arrays16 have been studied. In addition to studying carriers generated by bulk absorption of photons with energy larger than the semiconductor bandgap, TRTS is also sensitive to electrons injected into the semiconductor from adsorbed dye molecules.14,15 TRTS can be combined with transient absorption spectroscopy to gain detailed insight into interfacial electron transfer processes. In addition to detecting mobile electrons as in the above systems, TRTS is also sensitive to polarizable excitons. TRTS has given unprecedented information on the

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(a)

4

(b)

2 0

0 –4

4

–2 τ = 15 ps 0.5

τ = 200 ps 0.5

1.0

–4 1.0

σ0 (ω, τ = 200 ps) (10–1 S/m)

σ0 (ω, τ = 15 ps) (S/m)


ω (THz)

Figure 3.2 Conductivity of a ZnO wafer (a) 15 ps and (b) 200 ps after 400 nm photoexcitation. Open and closed circles are the real and imaginary components, respectively, of the complex conductivity. In (a), the response is dominated by free electrons and holes and is well-described by the Drude model (lines). In (b), excitons dominate the response, with lines modeling exciton polarizability. (Figure reprinted with permission from Hendry, E., Charge dynamics in novel semiconductors, Ph.D. thesis, 2005.)

time scales for exciton formation from free electrons and holes in systems such as single crystal ZnO wafers,17 carbon nanotubes,18 and semiconducting polymers.19–21 A dominant positive real conductivity indicates the presence of mobile free carriers, whereas a strong negative imaginary conductivity can be indicative of excitons, as shown in Figure 3.2. Simultaneous detection of both free carriers and bound excitons is another advantage of TRTS.

3.2.3 Experimental Setup and Data Collection 3.2.3.1 Basic Requirements The fundamental requirement of TRTS is the generation of synchronized optical pump and THz probe pulses. Figure 3.1 shows a sample being photoexcited with an optical pulse and subsequently probed with a THz pulse. Varying the delay time between the pump and probe pulses allows study of many different processes including carrier injection, cooling, decay, and trapping. Depending on the pulse energy and wavelength desired, these pulses could be provided from sources such as unamplified or amplified Ti:sapphire lasers or synchrotrons. Amplified Ti:sapphire laser systems are the most commonly used, as they can provide high energy, sub-100-fs pulses at kHz repetition rates with wavelengths tunable by nonlinear optics and optical parametric amplifiers. A detailed description of the amplified Ti:sapphire TRTS spectrometer used in the Schmuttenmaer lab is given in the following section. For samples that exhibit a large change in THz transmission per fundamental Ti:sapphire photon and whose longest time constants are on the order of nanoseconds, it is possible to perform TRTS with an 82-MHz Ti:sapphire oscillator alone. Samples requiring very high photon flux can be investigated using free electron lasers or synchrotron sources.

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Time-Resolved Terahertz Spectroscopy Ti: Sapphire Regenerative Amplifier 1 kHz, 100 fs, 1W Average Power

(a)

(c)

(b)

(f )

(p) (o) (l)

Chopper Pos. 1 (h)

(n)

(i)

Pump Beam

2

(m)

1

(q)

(k)

THz Beam

Lock-in Amplifier

(j) (e)

Chopper Pos. 2 (d)

(g)

Detector Beam

Oscilloscope

3

Computer

Figure 3.3 Time-resolved THz spectrometer. (a) Galilean telescope reduces the laser beam waist, (b) beam splitter, (c) optional frequency doubling crystal, (d) pump beam delay table, (e) optional lens, (f) dielectric mirror (99/1 beam splitter), (g) generator delay table, (h) 800-nm λ/2 waveplate, (i) 1-mm ZnTe crystal, (j) four off-axis parabolic mirrors, (k) sample position, (l) 0.5-mm ZnTe crystal, (m) detector delay table or galvanometer, (n) polarizer, (o) 800-nm λ/4 waveplate, (p) Wollaston polarizing beam splitter, and (q) balanced photodiodes. The balanced photodiodes are output to a lock-in amplifier.

3.2.3.2 Detailed Description of Spectrometer Figure 3.3 displays a detailed schematic of a typical TRTS spectrometer. This particular design has been employed in the Schmuttenmaer laboratory at Yale University, but many variations are possible. A Spectra Physics regenerative amplifier system (Millennia-Tsunami-Merlin-Spitfire) produces a 1-kHz pulse train of 800 µJ, 800 nm pulses of 100-fs duration (full width at half maximum). There are three arms of the spectrometer: THz generation, THz detection, and optical photoexcitation. Roughly two thirds of the light is used as the optical pump beam, whereas the other third generates and detects the THz radiation. The beam for the THz probe is split into a

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collimated 2-mm diameter near-IR beam of 250 µJ/pulse for THz generation and a focused 200-µm diameter near-IR beam of 5 nJ/pulse for THz detection. The intensity of the optical pump beam can be varied to a maximum of ~500 µJ/pulse for 800-nm excitation, ~200 µJ/pulse for 400-nm light, or lower energies for other wavelengths generated by an optical parametric amplifier. The pump spot size is usually between 5–10 mm diameter, but can be adjusted depending on the experimental requirements. The following list describes each component of the TRTS spectrometer shown in Figure 3.3. (a) A Galilean telescope (one focusing and one diverging lens) used to reduce the laser beam waist by a factor of two and collimate the beam. (b) 70:30 beam splitter (Melles Griot). Pump arm (c) 1-mm thick β-BaB2O4 crystal (Quantum Technology Inc.) for experiments requiring second harmonic generation to produce 400-nm excitation. (d) Optical delay table (Parker-Daedal) for pump arm of spectrometer. (e) Optional lens for expanding the pump-beam diameter. THz generation arm (f) A dielectric mirror that acts as a 99:1 beam splitter. The reflected beam generates the THz pulse and the transmitted beam is used to detect the THz pulse. (g) Optical delay table (Parker-Daedal) for THz generation arm of spectrometer. (h) A λ/2 waveplate (Melles Griot) that rotates the vertically polarized 800-nm beam before optical rectification; the output THz beam is polarized horizontally. (i) THz generator crystal, usually a 1-mm thick ZnTe single crystal (eV Products) mounted in a rotation stage. The crystal axis is rotated relative to the polarization axis of the visible pulse to maximize THz generation. (j) Four off-axis parabolic mirrors, (Melles Griot O2 POA 017). Mirror 1 collimates the diverging THz beam, Mirror 2 refocuses the beam onto the sample, Mirror 3 recollimates it, and Mirror 4 focuses it onto the detector crystal. Mirror 4 has a shorter focal length (38.1 mm) than the others (119 mm). Mirrors 2 and 4 have small holes drilled in them to allow transmission of the pump and detector beams, respectively. (k) Sample position. (l) A 0.5-mm thick (110) ZnTe crystal (eV Products) used to detect the THz pulses by free space electrooptic sampling. THz detection arm (m) Galvanometer (General Scanning) for detector arm of spectrometer. (n) 800-nm polarizer (Polacor). (o) A λ/4 waveplate (Melles Griot) to balance the detectors while THz is blocked.

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(p) Wollaston polarizing beam splitter (Karl Lambrecht) to send horizontal and vertical polarizations to different photodiodes. (q) Balanced large-area silicon photodiodes (ThorLabs) for signal detection. The voltages from the photodiodes are directed to a lock-in amplifier which measures the difference in their voltages. Coherent detection allows pulses below the blackbody radiation level to be measured without the use of specialized detectors. A computer interfaces with the experiment by controlling the delay tables and the lock-in amplifier. 3.2.3.3 Data Collection

Electric Field (a.u.)

Time domain scans are collected by scanning the probe delay line (delay line 2 in Figure 3.3) with the pump beam blocked and the chopper in position 1 in Figure 3.3. Figure 3.4 shows THz time domain scans with and without a sample in the THz beam path. Introducing a sample delays, disperses, and attenuates the THz wave form. Methods of extracting the absorbance and refractive index from the THz wave forms are described in the Section 3.2.4. Photoexcitation causes a change in transmission of the sample. For example, photoexcitation of many semiconductors generates free carriers that absorb THz radiation, resulting in a decrease in transmission. The transmission change is a function of both the delay time between the optical pump and THz probe and the time point on the THz waveform. Figure 3.5 shows a 2D grid of the transmission change as a function THz delay time and pump probe delay time.22 These data can be visualized either as a three-dimensional plot or as contour lines projected onto the pump delay–THz delay plane. Taking projections of the 2D grid allows us to measure the dynamics of an experiment. A cut parallel to the pump-delay axis is a 1D pump scan, and a cut parallel to the THz delay axis is a 1D probe scan. The 1D pump scans give the average THz absorption as a function of pump delay time. By Fourier transforming the 1D probe scans, we obtain the frequency-dependent optical constants of the photoexcited material at a given time after photoexcitation.

10 8 6 4 2 0 –2 –4 –6

Reference Sample

–4

–2

0

2

4

6

Time (ps)

Figure 3.4 Time domain scans of a terahertz pulse with and without a sample, in this case a ZnO wafer, at the focus of the terahertz beam. The sample delays, disperses, and attenuates the pulse.

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Terahertz Spectroscopy: Principles and Applications 1 D Pump Scan

Pump-Delay (ps)

3

1D Probe Scan

2 1 0 –1 1

2 3 THz Delay (ps)

4

Figure 3.5 Contour plot of THz difference scan of low-temperature-GaAs photoexcited at 400 nm solid lines represent negative values, whereas dashed lines correspond to positive values. Because the far-infrared response is slower when photoexcitation occurs at 400 nm compared with 800 nm, there is no need for detector deconvolution. One-dimensional (1D) cuts can be taken from the data: 1D pump scans are parallel to the pump delay axis and 1D probe scans are parallel the THz delay axis. (Figure reprinted with permission from Beard, M.C. et al., J. Appl. Phys., EPAPS, 90, 5915, 2001. Copyright 2000 by the American Institute of Physics.)

We can obtain these 1D cuts experimentally by fixing one delay line and scanning the other. We collect the 2D grid by collecting a series of 1D probe scans at a series of pump delay times. Pump scans are taken by fixing the THz delay (line 2) at the peak of the photoexcited THz waveform and scanning the pump delay (line 3) while chopping the optical pump beam (position 2 in Figure 3.3). By chopping the pump beam, we measure the difference between the photoexcited and non-photoexcited THz transmission through the sample. Figure 3.6a shows an 800-ps pump scan for a ZnO thin film, with the inset showing a 16-ps pump scan. Pump scans taken over the time immediately bracketing the optical pump are useful in determining the time constants of electron photoinjection, whereas long-time pump scans can be used to measure the decay dynamics of the photoexcited electron population. In this example, there is a fast injection component in the subpicosecond regime and a slower injection component on the 5-ps time scale. The long-time pump scan is very flat, indicating that the photoexcited electron population and mobility remain constant at higher than 800 ps. “Difference scans” are probe scans taken by fixing the pump delay (line 3) and scanning the THz delay (line 2) while chopping the optical pump beam. Figure 3.6b shows a difference scan for a ZnO wafer, the reference and sample THz scans of which are shown in Figure 3.4. The magnitude of the difference waveform is less than 10% of the sample THz waveform, allowing the photoexcitation to be treated as a small perturbation to the system. In many semiconductors, photoexcitation generates free carriers that absorb THz radiation. Therefore the transmission decreases on photoexcitation and the shape of the difference waveform resembles the negative THz waveform. Because the photoconductivity is determined by the difference between the photoexcited and nonphotoexcited waveforms, it is most accurate to measure this difference directly with the chopper in the pump beam path rather than

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Norm. Transmission

Time-Resolved Terahertz Spectroscopy 0.0

0.0 –0.2 –0.4 –0.6 –0.8 –1.0 –4

–0.2 –0.4 –0.6 –0.8

0

4

8

12

–1.0 0

200

400

800

600

Time (ps)

Electric Field (a.u.)

(a) 0.2 0.0 –0.2 –0.4 –0.6 –6

–4

–2

0

2

4

6

Time (ps) (b)

Figure 3.6 (a) Pump scans for a ZnO thin film. The inset shows injection at short pump– probe delay times and the main panel shows that the signal does not decay over 800 ps. (b) Difference scan, ∆ETHz (pump-on minus pump-off), taken 20 ps after photoexcitation of a ZnO wafer. Note the similar shape, but opposite sign and smaller magnitude, to the associated non-photoexcited THz waveform shown in Figure 3.4.

to measure the THz waveform with the chopper in position 1 both with and without pumping. 3.2.3.4 Importance of Spot Size To avoid frequency-dependent artifacts, it is important that the THz probe samples a region of uniformly photoexcited material. Therefore, the spot size of the visible (pump) beam should be least two times larger than that of the THz (probe) beam. The parabolic mirrors focus the THz probe at the sample to a spot size of approximately 3 mm, but higher THz frequencies are focused more tightly than lower frequencies. As a result, the extracted spectrum is skewed to higher frequencies when the spot size (the diameter at which the intensity of a Gaussian beam falls to 1/e2 of its value at the beam center) of the pump is the same size or smaller than the THz probe, as shown in Figure 3.7. If higher pump intensities are needed, then a smaller diameter pump beam can be used in conjunction with an iris at the sample. The iris blocks THz radiation that would have passed through nonphotoexcited regions of the sample. Additionally, the THz and visible beams should be copropagated to minimize temporal smearing.

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∆(OD) (arb. units)


0.3 0.2 0.1 0.0

0

1

2

Frequency (THz)

Figure 3.7 Change in optical density spectra of photoexcited GaAs as a function of visible (pump) spot size at a 20-ps pump–probe delay time. The visible spot sizes are 6.0 mm (solid line), 1.7 mm (dashed line), and 1.1 mm (dot dashed line), whereas the THz spot size is 2.3 mm. The spectrum becomes skewed toward higher frequencies as the visible spot size becomes smaller than the THz spot size. (Figure reprinted with permission from Beard, M.C. et al., Phys. Rev. B, 62, 15766, 2000. Copyright 2000 by the American Physical Society.)

3.2.4 Data Workup 3.2.4.1 Calculating Conductivity A THz waveform can be Fourier transformed to give the frequency-dependent, complex-valued permittivity. The frequency-dependent, complex-valued photoconductivity, σˆ , is obtained from

ˆ ˆ (w ) = eˆ (w ) + iσ(w ) , η e 0w

(3.1)

ˆ is the photoexcited permittivity, eˆ is the nonphotoexcited permittivity, w where η is the angular frequency, e0 is the free-space permittivity, and i is the unit imaginary. The photoconductivity refers to the response induced by the photoexcitation. This response could arise from a variety of sources such as mobile electrons (holes) excited into the conduction (valence) band of a semiconductor, polarizable excitons, polaritons, or other quasiparticles. Prior knowledge of the sample properties combined with careful experiments can determine the source of the conductivity. Determination of the conductivity from the data does not require any model, although models can later be fit to the data to extract relevant parameters about the system. Depending on the degree of accuracy required and the validity of certain assumptions, calculating permittivity and conductivity from THz waveforms can range from fairly simple to rather complex. If we assume that no THz radiation is reflected at any of the sample interfaces such that all THz is either transmitted or absorbed, the sample permittivity can be calculated in a straightforward way from the Fourier transform of the reference and sample scans shown in Figure 3.4. The power and phase, calculated from the Fourier transform of the time domain scans, are shown as a function of frequency in Figure 3.8. Because the phase is determined using the arctan, it can only be determined in the range (–π, π), resulting in a saw

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Power (a.u.)

Time-Resolved Terahertz Spectroscopy 10–3 10–4 10–5 10–6 10–7 10–8 10–9 10–10 10–11

Ref Sample

0

1

2

3

4

3

4

Frequency (THz) (a)

Phase (rad)

200

π

150

0 –π

100

Sample 1

2 Ref

50 0

0

1

2 Frequency (THz) (b)

Figure 3.8 (a) Power and (b) phase of a reference and sample pulse calculated from the time domain scans in Figure 3.4. The inset in (b) shows the reference phase before correcting for discontinuities.

tooth behavior with increasing frequency. The discontinuities should be removed to facilitate later calculations and for ease of visualization by adding 2π at each point of discontinuity. This process smooths the phase to a continuously increasing function without loss of information and allows ∆φ to be determined properly. Additional factors of 2π may also need to be added to the sample phase at all frequencies for thick or high index samples. The absorbance and refractive index of the sample are related to the power, P, and phase, f, of the Fourier transform of the sample and reference scans by a=

1  P ln d  P0 

(3.2)

and

n = 1+

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c (φ - φ 0 ). 2 pwd

(3.3)

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Absorbance (1/cm)

50 40 30 20 10

Refractive Index

0 0.2

0.4

0.6

0.8 1.0 Frequency (THz) (a)

0.4

0.6

1.2

1.4

1.2

1.4

3.4 3.2 3.0 2.8 2.6 0.2

0.8

1.0

Frequency (THz) (b)

Figure 3.9 (a) Absorbance and (b) refractive index of a ZnO wafer calculated using Equations 3.2 and 3.3 and the power and phase data in Figure 3.8. In this case, transmission losses are actually the result of reflection, but they appear as absorbance because of the assumptions in this simplistic procedure. Using complex transmission coefficients as described in Section 3.2.4.2 will remove this “absorbance offset.”

Figure 3.9 shows the absorbance and refractive index calculated using Equations 3.2 and 3.3 along with the power and phase data from Figure 3.8. The complex- valued permittivity and the complex-valued refractive index are related through eˆ = nˆ 2 where nˆ = n + ik , and k = λa / 4 p = c a / 2w. Thus the real and imaginary components of the permittivity are determined from a and n through

e′ = n2 - k 2

(3.4)

e ′′ = 2nk .

(3.5)

and

Photoconductivity can then be calculated from Equation 3.1 by using the photoexcited permittivity as the generalized permittivity, ηˆ , and the non-photoexcited permittivity as the static permittivity, eˆ . The real and imaginary parts of the

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photoconductivity are given by

σ ′ = e 0w ( η′′ - e ′′)

σ ′′ = e 0w (e ′ - η′).

(3.6) (3.7)

Example photoconductivities are plotted in Figure 3.2 for a ZnO wafer at 30 K at two pump delay times.17 At short times, the photoconductivity is well fit by the Drude model for free electrons, whereas at long times, the photoconductivity has the signature of excitons. The free electrons and holes become bound as excitons on the 50-ps timescale. ˆ is the photoIt is important to note that the measured change in conductivity, σ, conductivity and not the overall conductivity of the sample when using this method. In the case in which the change on photoexcitation is small, it is useful to measure the “difference scan” directly by placing the chopper in the pump beam as discussed in the experimental methods section. Hence, ∆E(t) = Ep* (t) – Enp(t) is measured and added to the measured nonphotoexcited THz waveform Enp(t) to yield the photoexcited waveform Ep* (t). An alternative method has been presented by Heinz et al. for samples with ∆E(t) > r 2 / ct0 , where d is the distance of the detector from the sample, r is the transverse radius of the visible pump beam, and t0 is the full width half maximum of the transient. For large bandwidth nearly signal cycle pulses, the pulse length, ct0, is a better metric for the near‑ and far‑field regimes than is the wavelength, λ. Typically, r 2 /ct0 ≈ 1 cm, and it is best to ensure that the far field has

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99

EΩ(arb. units)

Time-Resolved Terahertz Spectroscopy Near-Field d > r2/(ct0)

–1 –2

–2

0 THz Delay (ps)

d 2

Figure 3.15 THz generation is produced from ZnTe via optical rectification to illustrate near- versus far-field regimes. The open circles are the first derivatives of the near-field signal, whereas the lines are the measured near-field and far-field signals. The detector configuration is shown to the right. (Figure reprinted with permission from Beard, M.C. et al., J. Phys. Chem. A 106, 878, 2002. Copyright 2002 by the American Chemical Society.)

been achieved by increasing d until the pulse shape no longer changes shape with increasing distance, at which point d is usually about 3 cm or more. In rare cases, the signal can be collected in the near-field regime (that is, when it is possible to position the detector essentially in contact with the sample), which would allow the pulse shape to be measured directly. However, because of the relatively small excitation spot size and because it is often impossible to place the detector close enough to the photoexcited medium to be in the near field, the signal is typically collected in the far‑field regime. 3.3.1.2 Terahertz Focusing Optics Regardless of whether one collects data in the far field or near field, one of the most important considerations is that focusing optics not be used. The detector is placed adjacent to the sample as shown in Figure 3.14. Collecting the data in this manner allows one to obtain the true underlying dynamics, but at the cost of a lower signal-to-noise ratio compared with when THz focusing optics are used. However, one avoids difficulties of astigmatism, Guoy phase shift, and other diffraction‑induced pulse distortions that occur with large bandwidth pulses. The pitfall with using focusing optics is that variations in the THz waveform will be attributed to time-dependent changes in the polarization, when in fact it is nothing more than experimental misalignment. In addition, one must be absolutely certain that there are not other systematic artifacts. This is especially true when rotating the laser polarization, or if the sample is inside a cryostat. For extremely weak signals, it is imperative that every possible source of generation is ruled out; we have measured very weak THz generation from neutral density filters for example.

3.3.2 Data Analysis After the data are collected, useful information must be extracted. The data analysis begins with observing the dependence of the THz emission on sample rotation/

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orientation and excitation polarization. In addition, one can analyze the emitted waveforms for additional information. 3.3.2.1 Sample Orientation The sample may be rotated azimuthally (about the surface normal, qz) or obliquely (about the axes perpendicular to the surface normal, qy or qx). For example, several well-known generation mechanisms in semiconductors rely on a current surge along the surface normal. These are the surface depletion field and the photo-Dember effect. The surface depletion field occurs when the Fermi level is pinned to surface states or trap states thereby causing “band bending” at the surface. On photoexcitation, electrons are accelerated toward or way from the surface, depending on the sign of the field.42 The photo-Dember effect occurs even when band bending is not present because electrons and holes have different mobility. On photoexcitation, the carriers diffuse, but at different rates. Thus a net current normal to the surface results.43 If the sample is at or near normal incidence, THz pulses from either of these mechanisms will be very small. The only reason they are measurable at all is because either the sample is not perfectly normal to the excitation beam and detector, or more likely, because a focusing lens is being used.44 So, if either of these mechanisms is responsible, it will become apparent by a strong dependence on rotation of the sample about the x or y axis, but not on azimuthal rotation. If the current is along a well defined direction in the surface plane, then the THz emission will decrease as the sample is rotated about the axis perpendicular to the direction of the current surge, but not if it is rotated about the axis in the direction of the current surge (neglecting any effects due to differences in reflectivity as a function of angle). It will also depend (co)sinusoidally on the azimuthal angle. 3.3.2.2 Excitation Polarization Another important consideration is the dependence of the waveform on the polarization of the excitation beam (linear, either vertical or horizontal, or some of both, elliptical, or circular). For example, the THz emission in the x and y directions (lab-fixed) as a function of rotation of a linearly polarized excitation beam for a system of partially oriented dye molecules is shown in Figure 3.16. Conversely, the emission from a magnetic thin film is independent of excitation laser polarization since the fundamental mechanism is ultrafast thermal heating of the electrons: it only depends on sample orientation as seen in Figure 3.17.37 Similarly, the emission from a photoconductive switch is independent of the polarization direction (or ellipticity) of the excitation laser because the fundamental mechanism is simply generation of conduction band electrons, which are then accelerated by the applied bias voltage. Thus the underlying mechanism of the THz emission dictates whether it depends on the excitation laser polarization or not. 3.3.2.3 Emitted Waveform The shape of the emitted pulse reveals additional information about the underlying process. For example, in GaAs or other semiconductors, one can produce THz

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101

Time-Resolved Terahertz Spectroscopy 2e-5 EΩ(arb. units)

2e-5 1e-5 8e-6 4e-6 0 –4e-6 –8e-6

0

50

100

150

200

Polarization Angle (ζ) (b)

Figure 3.16 Dependence of the generated signal amplitude as a function of visible polarization angle, ζ, detecting the x component (filled circles) and y component (open circles) of the generated pulse amplitude. The static electric field is in the x direction. (Figure reprinted with permission from Beard, M.C. et al., J. Phys. Chem. A 106, 878, 2002. Copyright 2002 by the American Chemical Society.)

THz Amplitude (arb. units)

radiation (at normal incidence and without a bias voltage) from either optical rectification, if the photon energy is below the bandgap, or a shift current if the photon energy is above the bandgap.45 By analyzing the waveforms, it is possible to distinguish these two mechanisms, even though the amplitude for THz emission for both mechanisms depends on excitation polarization in the same fashion. As will be seen in Section 3.3.3.2, the two waveforms are related through a time derivative. With sample orientation and excitation polarization information in hand, the analysis of the waveform is guided by the putative mechanism of basis for THz pulse generation. That is, if one suspects that a current surge is the mechanism, either increasing or decreasing, he will then invoke a model that treats time-varying

1e-5 5e-6

M

0

y

–5e-6 –1e-5

0

90

180

270

360

z

x

Sample Rotation (Degrees)

Figure 3.17 (Left) THz emission from demagnetization as a function of sample orientation as it is rotated about z axis. (Right) Relevant coordinate system. A change in magnetization along the y axis produces a THz pulse polarized along the x axis. The THz emission is independent of excitation laser polarization.

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102

THz Amplitude (arb. units)

Terahertz Spectroscopy: Principles and Applications 1e-6 8e-7 6e-7 4e-7 2e-7 0

–2e-7

0

2

4 Time (ps)

6

8

(a)

0

1

2

3

4

5

6

7

8

6

7

8

Time (ps) (b)

0

1

2

3

4

5

Time (ps) (c)

Figure 3.18 Experimentally measured THz emission shown with solid line (a). The sample is a 40-nm thick in-plane magnetized Ni film. (b) The “basis set” of Gaussian functions whose amplitudes were then arbitrarily varied in a nonlinear least-squares loop to produce the change in magnetization shown with the thick line (c). One of the Gaussians was randomly chosen and shown with a thick line (b). The THz emission shown with the dashed line (a) is calculated by taking its first derivative and convoluting with the detector response function.

currents. Similarly, if a change in polarization or magnetization is suspected, then it is analyzed as such. It is also possible to extract the underlying change in current, polarization, or magnetization without assuming any model. For example, the solid line in Figure 3.18a is the emitted near-field THz pulse from a 40-nm thick magnetic Ni thin film. One generates (numerically) a series of Gaussians in the time domain as shown in Figure 3.18b, then uses a least-squares loop to vary their amplitudes, and convolute the underlying change in current, polarization, or magnetization with the

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103

detector response until agreement with the data is achieved. Figure 3.18c shows the underlying change in magnetization before convolution with the detector response function and before taking its first derivative. The dashed line in Figure 3.18a is the calculated THz pulse based on the time-dependent magnetization shown in Figure 3.18c (i.e., taking the first derivative of the magnetization shown in Figure 3.18c), and convoluting it with the detector response function. It might also be necessary to investigate both the first and second derivative of the underlying change if the source is not known.

3.3.3 Specific Examples In all cases, the goal is to determine Jx(t), Px(t), or Mx(t) from the measured THz waveform. Typically, a model is chosen to describe the underlying process, which is then convoluted it with an instrument response function. The parameters of the model are then varied using a nonlinear least squares fitting program until agreement between the calculated and measured waveforms is achieved, at which point one has obtained information about the underlying generation mechanism. 3.3.3.1 Photoconductive Switches Some of the earliest TES studies were to understand the mechanism behind THz generation from a photoconductive switch, the first type of source used for THzTDS. This required understanding the current surge because of the acceleration of photoexcited electrons by an applied bias field. That is, Jx(t) in Equation 3.26 is given by:

J x (t ) = -enf (t )v (t ),

(3.29)

where e is the magnitude of the electron charge, nf (t) is the time-dependent density of free (i.e., photogenerated) electrons, and v(t) is their time-dependent mean velocity. The time-dependent carrier density is related to the excitation pulsewidth as a generation mechanism, and any scattering, damping, or trapping that may be present. The time-dependent velocity depends on the effective mass of the electrons which depends on their location in the (nonparabolic) conduction band, as well as the local field, which results from the applied field as well as any screening effects. These effects can be treated either analytically,46–48 or numerically via a Monte Carlo approach.49 3.3.3.2 Shift Currents and Optical Rectification Although biased semiconductor photoconductive switches produce THz radiation because of a current surge, there exist other mechanisms as well. Perhaps most well known is optical rectification (the mechanism for THz generation in ZnTe emitters, which are popular sources of THz radiation for THz-TDS and TRTS experiments). Shift current and injection current are second-order mechanisms that do not require a bias field and can produce a pulse when the sample surface is normal to the direction of excitation.

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Sipe and coworkers have developed a detailed mechanistic treatment of secondorder responses in semiconductors by deriving an expression for the second-order susceptibility, χ(2) which is valid for below, above, and at bandgap conditions.45 The most salient results of their work are summarized below. For semiconductors, the second-order susceptibility may be written in the following form (in the frequency domain): χ( 2) (- W ;w , -w + W) = χ( 2)′ (- W; w , - w + W) +

σ ( 2) (- W; w , -w + W) iW

η( 2) (- W; w , -w + W) + (- i W) 2

(3.30)

The first term describes optical rectification, the second shift current, whereas the last term accounts for injection currents. W is a THz frequency, and W is an optical frequency. For zincblende materials such as GaAs, the injection current term vanishes because of symmetry. The current density has contributions due to rectification and shift currents:

J (t ) = J rect (t ) + J shift (t )

(3.31)

These current densities can be related to the applied electric field:

J rect (t ) = 2e o χ( 2)

∂ E (t )E * (t ) ∂t

J shift (t ) = 2e o σ ( 2)E (t )E * (t )

(3.32)

(3.33)

when detecting in the far field, where the signal appears as its first derivative with respect to time,41 the THz waveform should have the following profile with respect to the optical pulse:

rect (t ) = 2e χ ( 2 ) E THz o

∂2 E (t )E * (t ) ∂t 2

(3.34)

shift (t ) = 2e σ ( 2 ) ETHz o

∂ E (t ) E * (t ) ∂t

(3.35)

3.3.3.2.1 Laser Polarization Dependence of Second-Order Response The THz pulse generated from optical rectification or shift currents depends on the polarization state of the laser (linear, elliptical, circular) and the orientation of the sample. We employ the Jones matrix formalism to account for the various optics

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and sample orientation.50 In this formalism, the elements of the vectors and matrices below are complex quantities. The initial polarization of the optical beam is written as

 E x ,opt    Elab =  E y,opt   Ez,opt   

(3.36)

where x, y, and z are lab fixed coordinates. The z direction is the direction of propagation, and we can arbitrarily choose the x direction as horizontal and y vertical. Thus a horizontally, linearly polarized beam emerging from the laser is represented as

Elab

1    = 0  . 0   

(3.37)

When the beam passes through a waveplate, the polarization state of the emerging beam is expressed as:

 E x ′,opt   E x ,opt      Elab ′ =  E y′, opt  = W  E y, opt  = WElab E  E   z′, opt   z, opt 

(3.38)

where W is the product of the matrices:

W = R (-q) W0 R (q)

(3.39)

with:

e - i G / 2  W0 = e i φ  0  0 

0 e - iG / 2 0

0  0 1 

 cos q  R(q) =  - sin q  0 

sin q cos q 0

0  0. 1 

(3.40)

and

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(3.41)

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106


W0 is the matrix describing the phase retardation placed on the optical beam with Γ equal to p for a half-wave plate or p/2 for a quarter-wave plate. R(θ) is a rotation matrix that transforms the incident optical beam onto the fast and slow axes of the wave plate, and θ is the angle of the optical beam with respect to the fast axes. The emerging beam is then transformed back into lab coordinates using R(-θ) and carries the polarization induced by the waveplate. The optical field is then projected onto the crystallographic axis of the sample:

Extal

 Ei    = Ej  = T   Ek 

 Ex′    ′  E y ′  = TElab    Ez ′ 

(3.42)

where T is the transformation matrix projecting the electric field in lab coordinates onto the i, j, k crystallographic axes of the GaAs sample:    T =  

2 3 1

3

1

6 1 6

2

-

   1  3  1  3   1

0

1 2

(3.43)

For zincblende crystals such as GaAs, the induced polarization, or “current”, in the ith direction is proportional to the product of the optical fields polarized along the jth and kth direction: Pi ∝ E j E k* , where i, j, and k are mutually orthogonal Cartesian coordinates, and E* is the complex conjugate of E.51,52

0  P = Ek 0 

0 0

Ei

Ej   0 0 

 E i*   0  *   E j  = Ek  *  0 Ek  

0 0 Ei

Ej   * 0  Extal 0 

(3.44)

This time-dependent polarization leads to the observed THz emission, which is transformed back into lab coordinates:

ETHz

 E x ,THz    =  E y ,THz  = T -1P    E z ,THz 

(3.45)

where T–1 is the inverse of the transformation matrix T. The calculated signal is obtained by taking the real part of ETHz in Equation 3.45.

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Putting this together, we have:

ETHz = T-1 P TR ( - q)W 0 R(q) Elab .

(3.46)

On the other hand, if we were to rotate the sample instead of the waveplate, it would be

ETHz = R( - q) T -1 P TR (q)W 0 Elab

(3.47)

(with the optical axis of the waveplate along the polarization of the optical beam), or perhaps more simply, to eliminate the waveplate:

ETHz = R ( - q) T -1 P TR (q) Elab .

(3.48)

In summary, the Jones matrix formalism50 is a well-known and efficient way to describe and understand the effects of essentially any optical element as well as the samples themselves. 3.3.3.3 Intramolecular Charge Transfer in Orienting Field It is possible to generate THz emission from a sample of oriented dye molecules that undergo intramolecular charge transfer on photoexcitation. This provides a way to understand this process without relying on secondary effects such as a change in absorption or fluorescence properties. The signal is simply because of the change in polarization of the sample. It is possible to orient the molecules several ways. If a single crystal can be grown, and if it has the appropriate symmetry, then that will lead to a net orientation. Also, in some cases it is possible to make monolayers or multilayers of oriented molecules. In the work discussed here, however, an applied electric field is used to partially orient the molecules. Any TES experiment can be described as a nonlinear process. However, it is sometimes useful to think more intuitively about the underlying physical process. For the purpose of illustration, we will compare and contrast TES from oriented dye molecules in terms of the considerations of intramolecular charge transfer in partially oriented molecules versus a description based on the third-order nonlinear susceptibility. In both cases, the x axis is defined as the direction of the applied electric field. The angle of a molecular ground state dipole moment relative to the x axis is denoted q. The angle of the linearly polarized photoexcitation beam relative to the x axis is denoted z. The angle of the polarization axis of the THz detector relative to the x axis is denoted f. 3.3.3.3.1 Intuitive Description The interaction of the static electric field with a dipolar molecule in solution results in a fractional orientation of the molecules along the field direction, which provides the underlying physical basis for this method. This fractional orientation can be calculated by considering the interaction energy of a dipole with an applied field, μgE 0 in comparison to kBT, where kB is Boltzmann’s constant and T is the temperature

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108


in Kelvin. It is expressed as a function of angle, θ, between the applied field and molecular dipole moment by53,54

f (q) =

1 e -V / k BT ≈ -V / k BT ∫ e d q 2p 2p 0

 mE 0 cos q  1+ , k T   B

(3.49)

where the interaction potential, V, is given by V = –mE 0cosq. Terms resulting from polarizability anisotropy have been neglected. Given the inherent azimuthal symmetry, we work in polar coordinates. Thus interaction with the static field results in a linear combination of an isotropic distribution and one that varies as cosθ. The visible pulse introduces a cos2(θ – ζ) distribution of excited molecules because of projection of the transition dipole moment onto its polarization. An additional factor of cos(θ – φ) results from a projection of the emitted polarization onto an axis at angle φ. Thus the emitted amplitude at a given polarization angle φ varies as a function of visible polarization angle, ζ, as

E φW (z) ∝

∫

2p

0

(

)

cos(q) cos2 (q - z) cos (q - φ) d q = A z, φ ,

(3.50)

where the integral evaluates to

A(z, φ) =

p [cos φ (3 cos2 z + sin 2 z) + 2 sin φ cos z sin z]. 4

(3.51)

3.3.3.3.2 Third-Order Nonlinear Susceptibility Description It is also possible to treat this process in terms of a third-order nonlinear susceptibility:55

3) w w 0 E iW ∝ χ(ijkl E j Ek El ,

(3.52)

where E W is the THz electric field (which arises from the time-dependent, third-order polarization, Pi(3)), E w is the optical electric field, E 0 is the static electric field, and i, j, k, and l are one of the( 3x, y, z Cartesian coordinates. The third-order nonlinear ) susceptibility is denoted χijkl . As defined previously, the applied field (E 0) is in the x direction (therefore, l = x). Because z is the direction of propagation, i, j, and k must be x or y. For isotropic media there are only three independent tensor elements and they are related by52

3) = χ ( 3) + χ ( 3) + χ ( 3) . χ(xxxx xyyx yxyx yyxx

(3.53)

3) = χ ( 3) = χ ( 3) Also, because Klienman symmetry52 holds in this case, χ(xyyx yyxx yxyx ( 3 ) ( 3 ) and χ xxxx = 3χ xyyx . From these relationships, we can determine the amplitude and

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109


3) . direction of the induced third‑order polarization, P(3), in terms of χ(xyyx The x and y components of P(3) are

3 ) E w E - w E 0 + χ ( 3 ) E w E - w E 0 = χ ( 3 ) I E 0 [3 cos 2 (z ) + sin 2 (z)], Px( 3) (z) = χ(xxx x x x xyyx y y x xyyx w x

(3.54)

and 3 ) E w E - w E 0 + χ ( 3 ) E w E - w E 0 = χ ( 3 ) I E 0 [2 cos(z )sin (z)] Py( 3) (z) = χ(yyxx y x x yxyx x y x xyyx w x

(3.55)

where I wjk is the intensity of the visible pulse with polarization angle ζ relative to the x-direction. The visible polarization vector, eˆ jk , is given by eˆ jk = eˆx cos z + eˆy sin z. The emitted polarization is at an angle φ with respect to the static field and is obtained from the x and y components P (φ3)= Px( 3) xˆ + Py( 3) yˆ,

(3.56)

where φ = arctan[2 cos(z)sin(z)/(3 cos 2 (z) + sin 2 (z))]. It can easily be verified that Equations 3.54 through 3.56 are equivalent to the angular dependence of Equation 3.51, and the two formalisms are complementary. The x and y components of the emitted field are shown in Figure 3.16 and compared with Equations 3.54 and 3.55 (shown with solid lines). 3.3.3.3.3 Propagation Effects and Solvent Response If propagation effects of the fields through the solvent can be neglected, then the emitted amplitude in the near-field regime is equal to the second derivative of the time-dependent polarization (see Equation 3.25). However, nonnegligible propagation effects such as group velocity mismatch between the visible and generated THz pulses, absorption of the visible pulse by the solution, and dispersion of the generated pulse by the solvent must be accounted for to correctly obtain the charge transfer dynamics. We do so by numerically solving Maxwell’s equations in the time domain56 coupled with the phenomenologic model below for the time-dependent polarization. We then perform a nonlinear least-squares fit of the model to the data to obtain the charge transfer dynamics. It should be noted that an alternative approach is described in Section 3.3.4 and in previous studies.57 Consider a delta function excitation pulse of a single dye molecule at z = 0 and t′ = 0. The pulse induces an electron transfer with rate kET, and subsequent back electron transfer with rate kBET, and the change in polarization is given by Dp (t ) =

7525_C003.indd 109

k ET [exp(- k ET t ] - exp(- k BET t )] D ′m, ( k BET - k ET )

(3.57)

11/15/07 10:52:01 AM

110


where ∆′µ is the change in dipole moment along the ground state dipole. We average the contributions from individual molecules over an anisotropic distribution that is created by the visible and static fields. Because reorientation of the excited molecules occurs on a longer timescale than these measurements, the orientational average is given by D ′m = m ′e

∫

2p 0

f (q) cos 2 (q - z) cos (q - φ ) dq

E0 = A (zz, φ ) (m ′e - m g ) m g , 8kBT

(3.58)

where µg is the ground state dipole moment and m e′ is the projection of the excited state dipole moment along the ground state dipole moment. Thus, the change in polarization is obtained by replacing D′m with 〈D′m〉 in Equation 3.57. The solvent molecules also affect the measured change in polarization, so we must account for their reaction field. The electrostatic potential of a dipole in solution is modified from its value in vacuum by the reaction potential of the polarized surrounding dielectric medium. On photoexcitation, the electrostatic potential changes abruptly and therefore a repolarization of the solvent occurs. If the change in dipole is fast compared with the solvent motions, then the measured change in polarization will reflect this solvent repolarization. We describe the solvent response as an impulse response function to a delta function change in the solute charge configuration. We treat the solvent response as a single exponential, and the combined solvent–solute polarization ps response to an impulse excitation pulse is given by

ps (t ) =

t

∫ dt ′ Dp (t ′) Φ (t - t ′),

(3.59)

0

where Φ(t) = exp(-kst) represents the response of the solvent with rate constant ks. If ks 100*

100*

Note: The corresponding time constants are included for convenience. The Gaussian width is given by ∆w. The asterisk denotes that the values were held fixed during the fit.

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3.3.3.4 Demagnetization Dynamics Finally, it has been shown recently that a sample with magnetization that changes on a picosecond to subpicosecond timescale (as achieved by heating a magnetic thin film with an ultrashort, ~100-fs, laser pulse) will emit THz radiation.37,38 Although initially surprising, it is completely understood through Maxwell’s equations (see Equation 3.28). This implies that the measurement of transient electric field emitted by the sample is related to the second time derivative of the magnetization. Figure 3.20a shows the THz emission from a 40-nm thick, in-plane magnetized Ni film on heating with an ultrafast laser pulse (~100-fs duration, 800-nm wavelength). The THz emission was calculated from a model of the temporal variation of the magnetization (Figure 3.20b) similar to the one previously used in optical pump–probe experiments of Kerr rotation in the same type of sample.58

DM (t ) = {-Θ (t )[k1 (1 - exp ( -t / τth )) exp ( -t / τ ep ) + k2 (1 - exp)( -t / τ ep ))]} ⊗ G (t )

(3.60)

where t is time, ∆M(t) is the time-dependent change in magnetization, Θ(t) is the Heaviside step function centered at t = 0, k1 and k2 are constants depicting the relative

THz Amplitude or Magnetization (arb. units)

1.0

0.5 (a) 0.0

–0.5 (b) –1.0

0

2

4

6

8

Time (ps)

Figure 3.20 The solid line (a) is the near-field THz pulse generated upon ultrafast laser heating of 40-nm thick in-plane magnetized Ni film (in “transmission” mode). (b) The timedependent magnetization as described by Equation 3.60. The dashed line in part (a) is the calculated emission obtained by convoluting the first derivative of the time-dependent magnetization of part (b) with the ZnTe detector response function.

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amount of transient response versus long-term values. In the present context, tth and tep may be viewed as phenomenologic constants describing the electron thermalization time and electron–phonon coupling time.58–60 ⊗G(t) represents convolution with a Gaussian instrument response function. The best fit of the second derivative of the magnetization after convolution with the detector response function is plotted as the dashed line in Figure 3.20a. Here we find that tth = 0.37 (0.01) ps, tep = 1.30 (0.04) ps, k1/k2 = 2.0 (held fixed), and the Gaussian instrument response function full width at half maximum is 640. (6.) fs.

3.3.4 General Formalism Recently, Wynne and Carey have developed an integrated description of THz generation from optical rectification, charge transfer, and current surge.61 Each of these three processes is described using an analytical formula in the frequency domain followed by a numerical Fourier transform into the time domain. They account for absorption and dispersion in the detector crystal, as well as the effect of propagation into the far field. Although their treatment avoids explicit FDTD calculations, it cannot describe a situation wherein the response of a sample changes on a timescale fast compared to the inverse of the THz spectral bandwidth. However, there are many scenarios where it is perfectly valid. They provide a recipe. First the THz signal in the near field is calculated and then converted to the far field.

f eikz ( n - nVIS ) - 1 ikzn VIS E THz, NF ( z, w ) = e 2 n nVIS - n E THz ( z, w ) = -iw E THz, NF ( z, w ).

(3.61) (3.62)

In the time domain, the far-field signal is simply the time derivative of the near-field signal,41 which in the frequency domain corresponds to multiplying it by –iw. Here z is the propagation distance, w is the THz frequency, ñ is the complex-valued refractive index of the sample, nVIS is the group refractive index of the material at the excitation wavelength, determined through vgroup = c/nVIS, where c is the speed of light in vacuum, and k is the free-space wavevector of the THz field, k = w/c. Response functions, f(w), are provided for four different scenarios: optical rectification, direct charge transfer, indirect charge transfer, and the current surge model. Optical rectification:

f (w ) = χ( 2) (w ; W, W - w ) Ipump (w ),

(3.63)

where χ(2) is the second-order nonlinear response of the medium, and W represents the range of frequencies contained in the ultrashort pump pulse. Note: Ipump (w ) is the Fourier transform of the intensity envelope of the excitation pulse rather than its spectrum.

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Direct charge transfer: 3e (w ) N Dm f (w ) = I (w ) 2e (w ) + 1 e 0 ( g - i w ) pump

(3.64)

where e (w ) is the continuum dielectric of the sample, N is the number density of photoexcited molecules, ∆m is the change in molecular dipole moment on photoexcitation, e0 is the permittivity of free space. In the case of direct charge transfer, g is the rate of decay back to the ground state with the original dipole moment. Indirect charge transfer: f (w ) =

3e (w ) N Dm  1 1   I pump (w ). 2 e (w ) + 1 e 0  G - iw g - iw 

(3.65)

In the case of indirect charge transfer, g is the rate of decay back to an intermediate state, and Γ is the rate of decay back to the ground state with the original dipole moment. Current surge model:

3e (w ) 1 -e 2Enf 1  1 1   I f (w ) = (w ) 2e (w ) + 1 e 0 g s m* - i w  G - i w g s - i w  pump

(3.66)

where E is a DC external electric field which accelerates the electrons, gs is the momentum relaxation rate, m* is the carrier effective mass, nf is the carrier density, and Γ is the carrier recombination rate (it is assumed that Γ 50 kV/cm) was compared to the transmission of lower field strength THz pulses (2∆) phonons, and τP describes the relaxation time of the phonons either by anharmonic decay to phonons with energies 2∆ phonons is greater than for K = 0. The dynamic signature for this regime is a pronounced rise time and appears to explain the pair breaking dynamics in MgB2 in Figure 4.14(d). That is, a significant fraction of the energy in the photoexcited quasiparticle distribution is initially transferred to high-energy phonons that subsequently break additional Cooper pairs before condensate recovery. This contrasts with a scenario in which the photoexcited quasiparticles reach quasi-equilibrium through direct pair breaking in an avalanche-like process not involving phonons. Fits to the rise time dynamics at various fluences (solid lines in Figure 4.14(d)) using Equations 4.10a and 4.10b permitted a determination of the phonon pair breaking and bare quasiparticle recombination rates,123 yielding β–1 = 15 ± 2 ps and R = 100 ± 30 unit cell/ps, respectively.

4.3.4 Quasiparticle Dynamics in High-TC Superconductors Knowledge of quasiparticle interactions is particularly relevant for the high-TC cuprate superconductors, where the mechanism for charge pairing at temperatures up to ≈100 K remains unresolved.127 These materials show a complex phase diagram as a function of chemical doping; with increased density of hole carriers in the Cu2O planes they transition from an insulating antiferromagnetic state, via a non-Fermi liquid state that shows superconductivity at relatively small TC (“underdoped”) to the state with highest TC (“optimally doped”). As the doping is further increased, the materials show Fermi liquid properties and a progressively smaller TC until they are no longer superconducting. The quest to understand the complex quasiparticle properties and interactions in cuprates has inspired numerous experiments that employ visible or near-IR fs pulses to observe optical reflectivity changes at E ≈ 1–2 eV probe energy.108,128–130 However, it should be understood that this energy scale far exceeds the fundamental excitations of a superconductor, such as the THz gap and collective quasiparticle and Cooper pair response discussed previously. Ultrafast THz studies of superconductors, in contrast, can provide a direct measure of quasiparticle and Cooper pair dynamics, but have remained scarce. In the following section, we discuss these first THz studies of high-TC superconductor ultrafast dynamics. From a simple extrapolation of the dynamics on conventional superconductors one might expect similar dynamics on the high-TC cuprates. For example, near TC it was shown that for conventional superconductors the condensate recovery time is proportional to 1/∆(T). This suggests that the condensate recovery in the cuprates might follow a similar trend, albeit with a shorter lifetime given the larger gaps in these materials. Although such behavior is observed and there are similarities between these two classes of superconductors, there are also important differences in the nonequilibrium dynamics that likely originate from their vastly different microscopic properties. The major microscopic differences are that in the cuprates (1) superconductivity is obtained through chemical doping of Mott-Hubbard insulators, (2) superconductivity is thought to occur in two-dimensional CuO2 planes, (3) the superconducting order parameter has d-wave symmetry resulting in nodes in the superconducting gap (along kX = kY), and (4) there is the distinct possibility

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that the boson mediating pairing is not a phonon but some other excitation such as antiferromagnetic spin fluctuations. The hope is that optical pump THz probe and time-resolved mid-IR studies can provide insight into the properties of the cuprates through, for example, extracting the quasiparticle recombination rate or unmasking other interactions that are hidden in various time-integrated spectroscopies. We will briefly describe the first optical pump THz probe experiments that were performed on near-optimally doped and underdoped samples of yttrium– barium–copper oxide.131,132 The measurements were made on the same epitaxial films in Figure 4.13 and others grown under similar conditions including YBa2Cu3O7-d with TC = 89 K (50 K) for δ = 0 (0.5) and Y0.7Pr0.3Ba2Cu3O7 with TC = 0.5. The Pr compounds provide an alternate approach to achieve underdoping which is easier than controlling the oxygen stoichiometry. Figure 4.15(a) shows σ2(ω) as a function of frequency at various delays after photoexcitation for YBa2Cu3O7 at 60 K, whereas Figure 4.15(b) shows the dynamics for Y0.7Pr0.3Ba2Cu3O7 at 25 K. In both cases, there is a rapid decrease in the condensate fraction followed by a fast picosecond recovery. The condensate recovery is nearly complete by 10 ps, in dramatic contrast to conventional superconductors. The extracted lifetimes of the conductivity dynamics, and thus the condensate recovery time, are shown in Figure 4.15(c) for all of the samples investigated. In the

σim/104 (Ω cm)–1

YBa2Cu3O7-δ 6

Unpumped

0

60 K

2.5

Unpumped

2

10 ps

4 2

Y0.7Pr0.3Ba2Cu3O7 25 K

10 ps

1.5 1

1 ps 0.5

1

1 ps

0.5

2 ps 1.5

2

0

0.4

0.8

2 ps 1.2

1.6

Frequency (THz)

Frequency (THz)

(a)

(b)

5 τσ(ps)

4 3 95 K

2 1

0

10

20

30

40 50 T(K)

60

70

80

90

(c)

Figure 4.15 Condensate dynamics on (a) YBa2Cu3O7 and (b) Y0.7Pr0.3Ba2Cu3O7. (c) Summary of lifetimes measured on near-optimally doped and underdoped films—the lines are to guide the eye. Triangles: YBa2Cu3O6.5, diamonds: Y0.7Pr0.3Ba2Cu3O7. Closed and open circles: two different YBa2Cu3O7 films taken at different fluences.

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Time-Resolved Terahertz Studies of Carrier Dynamics

1.3 ps

0 10 ps 35 ps 50 ps 4

6 8 10 4 6 8 10 Photon Energy (meV)

0

–2000

(c)

(d)

1

0 –2 –4

0.1

1/∆σ2 (10–3 Ω cm)

1000

(b) t = –1.2 ps

–∆σ2(1000 Ω–1 cm–1)

(a)

∆σ2(ω)(Ω–1 cm–1)

∆σ1 (ω)(Ω–1 cm–1)

optimally doped films (closed and open circles), the condensate recovery is approximately 1.7 ps and increases near TC, consistent with a decrease in the superconducting gap. Above TC at 95 K, the lifetime has decreased to 2 ps and is likely a measure of electron–phonon equilibration in the normal state. Furthermore, the lifetime is independent of fluence indicative of the absence of bimolecular kinetics although this was not pursued in detail in this work.131,132 In contrast, for the underdoped films (triangles and diamonds in Figure 4.15(c)), the lifetime was constant at ≈3 ps even above TC. The constant lifetime as a function of temperature is suggestive of a pseudogap. These results revealed that it is possible to sensitively probe the condensate dynamics in the cuprates using optical pump THz probe spectroscopy. A comprehensive study of the transient picosecond THz conductivity changes after ultrafast optical excitation in a different superconductor, Bi2Sr2CaCu2O8+d (Bi‑2212), has recently been reported.133 The pump-induced change ∆σ(ω) of the CuO2 plane THz conductivity was measured in 62-nm thick optimally doped Bi-2212 films with a transition temperature TC ≈ 88 K. As in YBCO, in the superconducting state a 1/ω component dominates the imaginary part of conductivity σ2(ω) at low frequencies, which provides a direct measure of the condensate density. Figures 4.16(a) and (b) show the conductivity change observed directly after photoexcitation. The imaginary part displays a strong decrease, ∆σ2 TC (T* ≈ 160 K in this sample), and results from ultrafast excitation of a second kind of correlated carrier that lies at the origin of the pseudogap phenomenon. The latter may be due to preformed pairs or antiferromagnetic correlations.135 Thus the femtosecond study provides a separation of different components of the ≈30 THz conductivity gap, which remain hidden in linear spectra. The temperature dependence of the transient signal amplitudes also closely follows the strength of an important, 41-meV antiferromagnetic resonance in cuprates, which supports theories in which spin fluctuations couple strongly to the carriers.136 The experiments discussed show the detailed insight into the dynamics of superconductors obtained by employing THz pulses as probes of the fundamental charge excitations. They motivate further work on a larger range of materials to elucidate the differences or similarities in the pseudogap correlations or quasiparticle recombination kinetics. Moreover, almost all previous time-resolved studies of superconductors have employed photoexcitation in the 1- to 2-eV range. This leads to a cascade of scattering processes as quasiparticles relax to the low-energy states, resulting in a fairly indirect and uncontrolled way to break Cooper pairs. We can envision that in the future resonant excitation of superconductors with intense THz pulses at the gap energy 2∆0 will be possible, leading to direct control of (and consequently even more detailed insight into) the nonequilibrium state of superconductors.

4.4 Half-Metallic Metals: Manganites and Pyrochlores 4.4.1 Overview Manganite perovskites such as R1-xDxMnO3 (where R = La, Nd and D = Ca, Sr) and related compounds are similar to the cuprates in that their parent materials (e.g., LaMnO3) are Mott-Hubbard insulators. As a function of doping, fascinating states occur ranging from charge-ordered insulator to ferromagnetic metal phases. The most studied materials of this class during the past decade are the optimally doped manganites around x ≈ 0.3, which includes La0.7Ca0.3MnO3 and La0.7Sr0.3MnO3. Here, “optimal” refers to the maximum ferromagnetic Curie temperature TC. Above TC, these materials are paramagnetic semiconductors, whereas below TC, they transition into ferromagnetic metals. In the vicinity of TC, an applied magnetic field yields a dramatic decrease in the resistivity. This large negative magnetoresistance has resulted in these materials being termed colossal magnetoresistive (CMR) manganites. They are also often called half-metallic manganites referring to the complete spin polarization that occurs below TC. Several excellent reviews of CMR materials

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are available.4,5,7,9,140 Recent research has focused on the structural and electronic properties away from optimal doping, with the nature and origin of intrinsic electronic inhomogeneities receiving considerable attention.5 Although the intense interest in manganites during the past decade141 derives from advances in cuprate superconductivity, these materials have been investigated since their discovery in the 1950s. This initial work led to fundamental theoretical insights related to double exchange and experimental explorations such as neutron scattering studies of their detailed magnetic structure.142–146 In the manganites, the structural units of interest are the MnO6 octahedra. The octahedral symmetry breaks the five-fold degeneracy of the Mn d-orbitals resulting in localized t2g spins (S = 3/2) and higher lying eg spins. It is in this eg‑derived band that transport occurs. Additionally, strong Hund’s rule coupling leads to slaving of the eg to the t2g spins, an essential ingredient to the so-called double exchange. In the double exchange model, electron transport between adjacent Mn ions via an intervening oxygen orbital is enhanced if the core spins on adjacent sites are parallel. However, it is now realized that the double exchange model alone cannot account for CMR behavior because the magnitude of the resistivity in the paramagnetic phase is larger than spin scattering predictions.147,148 Coherent lattice effects such as the Jahn-Teller distortion of the MnO6 octahedron also affect the charge transport. The simple picture is that in the paramagnetic phase lattice distortions can follow the carriers from site to site because disorder (of the t2g spins) reduces the carrier kinetic energy, resulting in polaronic transport. Ferromagnetic ordering increases the carrier kinetic energy. Hopping from site to site becomes too rapid for the lattice distortions to follow the carriers, resulting in a metallic state (though polaronic signatures persist below TC). Although the ultimate origin of CMR in the manganites has not been established, it is clear that the spins and polaronic behavior are basic ingredients to consider. In the quest to more fully understand the origin of CMR, many other materials have been synthesized and characterized.149 A particularly important class of materials in this regard are the pyrochlores such as Tl2Mn2O7, which also display a pronounced negative magnetoresistance.150,151 In Tl2Mn2O7, the transition temperature is TC = 120 K. Pyrochlores at first appear quite similar to perovskite manganites: both exhibit CMR, and MnO6 octahedrons are equally important structural subunits. However, experimental and theoretical studies have highlighted important differences. In particular, in pyrochlores double exchange and Jahn-Teller effects are negligible because of the low carrier density and the absence of the Jahn-Teller Mn3+ ion. In addition, separate subsystems are responsible for ferromagnetic ordering (Mn 3d electrons, t2g with S = 3/2 as in manganites) and conduction properties (Tl 6s electrons). The ferromagnetic ordering of the Mn4+ sublattice likely occurs through frustrated superexchange between Mn on adjacent sites. Below TC, indirect exchange between the Mn and Tl orbitals causes a band of primarily Tl 6s character to shift below the Fermi level. It thus appears that the microscopic origin of CMR in pyrochlores and manganites is quite different, though the possibility remains that CMR in both materials derives from mesoscopic effects related to intrinsic electronic inhomogeneities. Below, we briefly discuss time-integrated infrared spectra of La0.7Ca0.3MnO3 and Tl2Mn2O7, followed by a description of time-resolved studies.152–155 These THz studies provide insight into their fundamental

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properties and highlight important differences between these half-metallic correlated materials.

4.4.2 Optical Conductivity and Spectral Weight Transfer

σreal/103 (Ω cm)–1

For the cuprates, we discussed how, on entering the superconducting state, spectral weight transfers from the Drude-like response to the condensate response represented as a Dirac delta function at zero frequency. In the manganites, spectral weight transfer also occurs as a function of temperature.155 However, for these materials the spectral weight transfer occurs from higher energies (~1 eV) to a coherent Drude-like response with the crossover from paramagnetic semiconductor to ferromagnetic metal. This spectral weight transfer is intimately related to the CMR effect in DC conductivity measurements and involves the spin and lattice degrees of freedom. Figure 4.18 shows the temperature dependence of the optical conductivity from 0.2 to 5 eV as a function of temperature for La0.7Ca0.3MnO3 (TC ~ 260 K).155 This clearly shows spectral weight transfer with decreasing temperature and the development of a Drude peak centered at zero frequency. A detailed discussion of the various features of the optical conductivity has been presented.155 The broad feature at ~1.0 eV has been interpreted in terms of the photon-induced hopping of Jahn-Teller polarons. With decreasing temperature, it is this Jahn-Teller peak that partially evolves into a metallic response below TC. This is just as discussed previously, in that with increasing spin polarization the increased carrier kinetic energy is sufficient to overcome polaronic trapping. Further details regarding the higher energy peaks can be found elsewhere.155 In the pyrochlore Tl2Mn2O7 (TC = 120 K), there is also a strong transfer of spectral weight from higher to lower energies with decreasing temperature.153 However, in this material there is no high energy peak ascribable to polarons and the spectral weight transfer likely comes from higher energy (>2 eV) interband transitions consistent with the crossing of a single Tl 6s band as the ferromagnetic half-metallic state develops. Figure 4.19 shows the temperature dependence of the reflectivity for Tl2Mn2O7. At 295 and 160 K, the carrier density is very low such that the phonons

Mn3+

1.5

O eg

1.0

t2g

Mn4+ t

eg

10 K 100 K 125 K 150 K 200 K 225 K 250 K 300 K

t2g

0.5 La0.7Ca0.3MnO3 1

2

3

4

5

Photon Energy (eV)

Figure 4.18 Optical conductivity in La0.7Ca0.3MnO3 as a function of photon energy at various temperatures. The inset gives a schematic depiction of double exchange. (Reprinted with permission. Copyright 1998 by the American Physical Society.)

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Reflectivity


0.8 0.6 0.4 0.2

40 K

0.8

160 K

0.6 0.4 0.2

0.8 0.6 0.4 0.2

110 K

0.8

295 K

0.6 0.4 0.2

0.01

0.1

1

0.01 Energy (eV)

0.1

1

Figure 4.19 Infrared reflectivity in Tl2Mn2O7 at various temperatures.

are clearly observable. At 110 K, the development of a Drude peak occurs and it is fully developed at 40 K (though the phonons are still not fully screened). Comparing the 40 K data with Figure 4.1(b), we can see that the response is entirely consistent with Drude-like behavior including the dramatic decrease in the reflectivity at ωs which dips and then rises back up to a value consistent with ε∞. This behavior is quite different from that of the manganites, which are much more spectrally congested because of the distinct polaronic response. In Tl2Mn2O7, the carrier density below TC increases as the square of the magnetization to a maximum value of n ≈ 1019 cm-3, about two orders of magnitude smaller than in manganites.

4.4.3 Dynamic Spectral Weight Transfer in Manganites Figure 4.20 shows the results of THz-TDS and optical pump THz probe measurements on a 100-nm thick film of La0.7Ca0.3MnO3 grown on LaAlO3.152,156 In Figure 4.20(a), the real conductivity is plotted as a function of frequency at various temperatures. These conductivity measurements are in the regime ωτ |2>

ωTHz

ω12 ∆

ω*12 |1> |1'>

THz NI

R

Figure 6.28 (a) DFKE in an ideal two-dimensional system. In a strong terahertz field (dashed curve; solid curve is zero field), the “edge” shifts to higher frequency and subgap absorption increases. (b) The AC Stark effect: a strong field applied at frequency f ∼ f12 . causes f12 to shift. For f < (>) f12, the transition shifts to f12* > ( W), the component with the smaller frequency shift dominates state a. When this driven system is probed by a weaker beam, there can be a splitting or shift in the linear absorption because of these additional frequency components. The original states of the system are “dressed” by the strong coupling of the states in the presence of the driving field. In semiconductors, AC Stark shifts in interband absorption have been observed due to an interband pump, typically detuned below the band gap.67,68 Resonant pumping of interband and intersubband transitions has resulted in Autler-Townes splitting of the linear absorption.69,70 Autler-Townes splitting occurs when the strongly coupled states are probed by absorption to or from a remote third state. This splitting was first observed in a molecular system driven by a strong radio frequency field and probed with a microwave field.71 Autler-Townes splitting is often the precursor of electromagnetically induced transparency, in which a pump beam causes an otherwise absorbing resonance to become transparent.70,72 These AC Stark effects in semiconductors have primarily been induced using visible to mid-infrared pumps, in which there was significant absorption of the pump. Changes in the interband QW absorption due to the AC Stark effect at terahertz frequencies has been given a great deal of theoretical attention. For in-plane fields, the coupling between s and p exciton states has been studied,73,74 and for growth direction fields, intersubband coupling has been studied.75–77 Experimentally, the

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previously discussed measurements of the DFKE with in-plane terahertz fields show some evidence of the AC Stark effect, but the effects are small and competed with the DFKE.17 In this section, we will focus on experiments in which a growth direction terahertz field couples two heavy-hole exciton states together, and clearly manifests an Autler-Townes splitting in the interband absorption.20 These experiments are performed at low temperature (~10 K) on a sample with ten periods of undoped In0.06Ga0.94As QWs (each 143 Å) with Al0.3Ga0.7As barriers (300 Å). A 6% indium concentration is used to bring the QW exciton absorption below the bandgap of the semi-insulating GaAs substrate, allowing transmission of the NIR probe through the substrate. A 100-nm layer of aluminum is deposited on the epitaxial layer to improve the boundary conditions for the terahertz field. The aluminum layer eliminates in-plane terahertz fields and significantly enhances the growth direction field. The NIR probe consists of incoherent light from a 353-THz (1.46 eV) LED focused onto the sample backside. As illustrated in Figure 6.31 (left), the probe is transmitted through the substrate, interacts with the QWs, reflects off the aluminum layer, and is transmitted back through the QWs and substrate. The reflectivity of the sample (essentially double-pass transmission) without terahertz excitation is plotted as the lowest spectrum in Figure 6.31 (left). The strong absorption is due to the e1hh1X1s. The exciton e1hh1X1s is expected to be 3.34 THz (13.8 meV) above e1hh1X1s from calculations, but interband transitions to this state are optically forbidden. The light-hole exciton, e1lh1X1s, is shifted to much higher energies due to strain, and other higher exciton states (e.g., e2hh1X1s, e2hh2X1s) are expected to be sufficiently high in energy that the terahertz field only weakly couples them to e1hh1X1s. The states considered in this experiment are the ground state (no excitons), e1hh1X1s, and e1hh2X1s. These states will be referred to as the ground state, hh1X, and hh2X for simplicity in the remainder of this section. The different in-plane states (2s, 2p) are not considered as they have weaker oscillator strengths and 1s to 2p transitions will only occur for in-plane terahertz fields. The terahertz field couples hh1X and hh2X, and the NIR beam probes the transition from the ground state to e1hh1X. Schematics of these energy levels and fields are displayed to the right of the measured spectra in Figure 6.31 (left), with the terahertz frequency below, on, and above the e1hh1X – hh2X resonance. The terahertz beam is focused onto the edge of the sample (see top portion of Figure 6.31, left), with the maximum intensity inside the sample estimated at ~1 MW/cm2 (~15 kV/cm electric field amplitude). The effect of the terahertz field on the reflectivity is displayed in Figure 6.31 (left) for terahertz frequencies below, on, and above the hh1X – hh2X resonance. Below resonance in Figure 6.31A (left), at f THz = 2.52 THz (10.4 meV), the absorption line red shifts with increasing terahertz intensity, with a weaker absorption line appearing above the undriven exciton energy at the highest intensities. On resonance in Figure 6.31B (left), at f THz = 3.42 THz (14.1 meV), there is a symmetric splitting of the absorption that increases with intensity. Above resonance in Figure 6.31C (left), at f THz = 3.90 THz (16.1 meV), a weaker absorption line appears below the undriven exciton energy. These observations are clear manifestations of an Autler-Townes splitting because of terahertz-induced coupling of the hh1X and hh2X states. The two absorption lines observed represent the

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Optical Response of Semiconductor Nanostructures in Terahertz Fields NIR

259

Al In Al e1

Substrate

THz

1 0.8

h2X h1X 0

B

665 365 210 0

0.6

h2x (dark)

h1x

0.4 1.6

0

C

320 220 1.2 140 1 0 0.8 1.4

0.6

h2x (dark)

h1x

0.4 351

353

355

NIR Frequency (THz)

357

Reflectivity (theory)

1.2

h2x (dark)

h1x

h2X

1.4

930 710 460 175 0

h1X

Reflectivity

1.6

A

h1

h1X h2X

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4

QWs h2 Aluminum NIR Frequency (eV) 1.455 1.465 1.475

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 1.6 1.4 1.2 1 0.8

NIR Frequency (eV) 1.455 1.465 1.475 A

910 680 455 230 0

B

910 680 455 230 0

C

350 235 115 0

0.6 0

0.4 351

353

355

357

NIR Frequency (THz)

Figure 6.31 (Left). Reflectivity spectra for a series of terahertz intensities at f THz = (a) 2.52 THz (10.4 meV), (b) 3.42 THz (14.1 meV), and (c) 3.90 THz (16.1 meV). The spectra are offset and labeled according to the terahertz intensities (in kW/cm2). Level diagrams illustrating the detuning from the expected h1X-h2X resonance are shown to the right of each graph. Schematics of the experimental geometry and the band diagram of a QW along with the relevant subband energies are displayed above. The doublesided arrow represents the lowest excitation of the system, the h1X exciton. The AlGaAs barriers are labeled “Al” and the InGaAs layer is labeled “In.” (Right). Calculated reflectivity spectra for a series of terahertz intensities at f THz = (A) 2.52 THz (10.4 meV), (B) 3.42 THz (14.1 meV), and (C) 3.90 THz (16.1 meV). The spectra are offset and labeled according to the terahertz intensity (in kW/cm2). The absorption strength and energy position of the spectrum for zero terahertz field were set to best fit the measured reflectivity. (Reprinted with permission from Carter, S.G. et al., Science, 310, 651, 2005.)

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1.462

353.4

1.461

353.2 353.0

1.460

352.8

1.459

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

(b)

3 2 1

0

NIR frequency (eV)

(a)

353.6

200 400 Intensity (kW/cm2)

600

Splitting (meV)

Splitting (THz)

NIR Frequency (THz)

terahertz dressed states, each consisting of a linear combination of hh1X and hh2X. This Autler-Townes splitting can be observed at temperatures of up to 78 K. Calculated reflectivity spectra are displayed in Figure 6.31 (right) for the same terahertz frequencies and comparable terahertz intensities. These calculations are similar to the model discussed previously in this section in that they calculate how the terahertz field couples the states of the QW. The calculations are for an infinitely deep QW, and they include all of the quantized electron and hole states, the Coulomb interaction, and the effects of the terahertz field without resorting to perturbation theory.77 The model is equivalent to solving the semiconductor Bloch equations in the low-density limit.78 The agreement between the experimental and calculated spectra is quite good, although there are some differences. First, the observed splitting is more symmetric at f THz = 3.42 THz (14.1 meV) (Figure 6.31B, left) than in the calculations (Figure 6.31B, right), indicating the actual hh1X – hh2X resonance is closer to 3.42 THz (14.1 meV) than the calculated value of 3.34 THz (13.8 meV). More significantly, the experimental spectra show a red shift in both dressed states with increasing terahertz intensity that is not seen in calculations. This red shift is likely due to heating of the sample due to the intense terahertz field. The red shift can be seen more clearly in Figure 6.32(a), which plots the energies of the two absorption lines as a function of terahertz intensity. These energies are obtained by fitting the spectra to two Lorentzians. On resonance, the two dressed state energies are expected to

0

Figure 6.32 (a) Positions of the absorption lines and (b) the splitting as a function of terahertz power at f THz = 3.42 THz (14.1 meV). Error bars represent the uncertainty from the fits. The small squares in (b) give the calculated splitting, and the curve fits these points to a square root function, expected from theory. (Reprinted with permission from Carter, S.G. et al., Science, 310, 651, 2005.)

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symmetrically shift up and down in energy relative to the undriven exciton line, but terahertz-induced heating red shifts both of these lines as the intensity increases. The splitting between the absorption lines should be independent of this heating effect and is plotted in Figure 6.32(b), along with the calculated splitting. The splitting on resonance is expected to be proportional to the Rabi frequency, 2µE, as found in the simple Rabi model. The splitting should therefore be proportional to the square root of the terahertz intensity. Over the limited range of terahertz intensities in which the two absorption lines can be resolved, the measured splitting appears linear but is also not far from the expected square root behavior. Further measurements must be performed to better characterize the dependence of the splitting on terahertz intensity. Terahertz-induced changes in the QW reflectivity are measured over a wide range of terahertz frequencies from 1.53 to 3.90 THz (6.3 to 16.1 meV). The absorption spectra at several terahertz frequencies, with roughly equal intensities, are shown in Figure 6.33, along with the fits to two Lorentzians. The absorption spectra are obtained by correcting for the linear decrease in the reflectivity with energy (presumably due to the tail of absorption by the GaAs substrate) and taking the negative natural logarithm. Figure 6.34 displays the fitted absorption line positions as a function of terahertz frequency at a relatively low intensity of ~300 kW/cm2. This was the maximum intensity for f THz = 3.75 and 3.90 THz (15.5–16.1 meV). These positions represent the energies of the two dressed states, with the dark (light) circles representing strong (weak) NIR absorption. There is a clear anticrossing of the dressed states at the hh1X – hh2X resonance, as expected from theory. On resonance, the oscillator strength is equally shared between the dressed states because of the strong coupling of the two exciton states. Off resonance, the dressed state near the undriven hh1X energy (the dashed line in Figure 6.34) dominates the absorption strength, while the weaker dressed state approaches the hh2X energy with decreasing terahertz frequency. Clearly, in the zero-frequency limit, the electric field will produce an absorption line at hh2X since the field breaks the inversion symmetry of the QW, making hh2X optically allowed. The measured absorption positions follow the calculated dressed states quite well. These calculated positions from the model equivalent to the SBE are quite similar to positions obtained from the Rabi model: fdressed = fe1hh1X - ( D ± W 2 + D 2 ) / 2 . On resonance, the splitting is determined by the Rabi frequency, whereas far off resonance, the splitting is determined by the detuning. The deviations from calculations at 1.53 and 1.98 THz (6.3 and 8.2 meV) may be due to a higher terahertz intensity than estimated. The lack of a blue shift above resonance is probably due to heating. These experiments provide the first clear evidence of the terahertz AC Stark effect in the interband absorption spectrum, with a clear manifestation of AutlerTownes splitting. Although the AC Stark effect has previously been observed in semiconductors with MIR and optical pumps, this experiment at terahertz frequencies is unique for several reasons. First of all, there is negligible absorption of the terahertz pump because there are no free carriers in the QWs without optical excitation. This reduces heating of the sample, which tends to broaden the exciton line width, and likely makes the observation of this effect possible with a quasi-continuous wave pump. Second, the Rabi frequency, which is given by the splitting on resonance, is a significant fraction of the pump frequency because of the large intersubband dipole

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Terahertz Spectroscopy: Principles and Applications NIR Frequency (eV) 0.8 0.6

1.455

1.460

1.465

1.470

A

fTHz = 3.90 THz

B

fTHz = 3.75 THz

0.4 0.2 0.0 0.6 0.4 0.2

Absorbance

0.0 0.6

C

fTHz = 3.42 THz

0.4 0.2 0.0 0.6

D

fTHz = 3.09 THz

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E

fTHz = 2.82 THz

0.4 0.2 0.0

351

352

353

354

355

356

NIR Frequency (THz)

Figure 6.33 Absorption spectra and fits for f THz = 3.90 THz (16.1 meV) (A) to 2.82 THz (11.7 meV) (E). The thick black lines represent the absorption obtained from the reflectivity spectra by correcting for the downward slope due to continuum absorption and by taking the negative natural logarithm. The gray curves are fits to two Lorentzians; the other two curves are the individual Lorentzians. (Reprinted with permission from Carter, S.G. et al., Science, 310, 651, 2005.)

moment and the small pump frequency. On resonance at the highest terahertz intensity, the ratio Ω/f is ~0.2, and below resonance at 1.5 THz (6.2 meV), the ratio is close to 0.5. This system is therefore approaching the strong field regime, in which Ω/f ≥ 1, where the RWA breaks down and terahertz replicas can appear in the spectra. These large Rabi frequencies allow the observation of terahertz dressed states over a wide range of detunings that exceed half of the resonance frequency. Finally, the terahertz-driven QW is essentially a QW modulator driven at terahertz frequencies. As previously stated, growth-direction electric fields in QWs are currently used to modulate NIR light through the QCSE at frequencies up to approximately 50 GHz. At these frequencies which are lower than the carrier dephasing time, the QW absorption adiabatically follows the field. At terahertz frequencies,

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Drive Frequency (meV) 5

10

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20 1.468 1.466 1.464

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1.458

NIR Frequency (eV)

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355

1.456

352 1.5

2.5

3.5

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Figure 6.34 Measured absorption line positions (large circles) versus terahertz frequency for a terahertz intensity of approximately 300 kW/cm2. The absorption strength for each point is represented on a grayscale; darker circles indicate stronger absorption. The standard error of the positions from fitting is indicated by error bars except where smaller than the circle diameter. The data points and solid line represent the calculated absorption positions at a terahertz intensity of 375 kW/cm2. Near the resonance at f THz = 3.4 THz (14.1 meV), the calculated spectra were fit to two Lorentzians to determine absorption positions; away from resonance, the positions were taken at the minimum of the reflectivity for each peak. The horizontal dashed line marks the undriven exciton line position. (Reprinted with permission from Carter, S.G. et al., Science, 310, 651, 2005.)

quantum coherence between exciton states of the QW is preserved, leading to terahertz dressed states. This quantum coherence will be important to consider as the frequency of optoelectronic devices, such as QW modulators, continues to increase. One possible application of quantum coherence in a QW modulator is that two optical beams resonant with different dressed states of the QW interact very strongly, with sensitivity to the phase between the two beams. This effect allows for the intermodulation of two beams at arbitrarily low intensities.79

6.4.4 Conclusions The effect of a terahertz driving field on the interband absorption of QWs depends qualitatively on the polarization of the terahertz radiation. In-plane terahertz fields drive carriers back and forth, leading to the DFKE, which blue shifts excitons according the ponderomotive energy, EKE. The terahertz field also couples to inplane exciton transitions, leading to the AC Stark effect, which competes with the DFKE. Growth-direction terahertz fields couple to intersubband transitions in QWs, leading to a quantum-confined AC Stark effect. This effect is essentially the result of Rabi oscillations between two states, which dress the states. Perhaps the most dramatic evidence of these dressed states is the observation of an Autler-Townes splitting of the exciton absorption.

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6.5 Conclusion This chapter has described a number of experimental methods and results which arise from studying the optical properties of semiconductor heterostructures under illumination by essentially monochromatic, quasi-continuous wave radiation at frequencies ranging from 0.3 to 4 THz (1.2 to 16 meV). For some of the experiments, such as optically detected terahertz resonance, only modest terahertz power is necessary because this is essentially a linear spectroscopy tool. Others, such as the experiments in which terahertz radiation modifies the optical absorption (and hence the band structure) of semiconductor QWs, require terahertz electric fields on the order of 10 kV/cm, or intensities of several hundred kW/cm2. During the time that this chapter’s experiments were performed, terahertz source technology has made tremendous advances. From the higher frequency end, terahertz quantum cascade lasers have worked their way down to lower than 2 THz (8 meV) with average emitted powers exceeding 10 mW. From the lower frequency end, reliable sources based on multiplied microwave oscillators can now be purchased with powers ranging from 50 mW at 0.25 THz (1 meV) to approximately 10 µW at 1 THz (4 meV). Both of these advances, which are sure to continue, bode well for transitioning at least some of the experimental methods described in this chapter to experimental tabletops that include the terahertz source, and, perhaps, into useful devices. For example, the intensities inside a terahertz quantum cascade laser cavity can exceed 10 kW/cm2, enabling highly nonlinear experiments on structures embedded into the laser cavity. Sideband generation in a mid-infrared quantum cascade laser has already been demonstrated.80

kBΔT/IFIR (cm2/s)

10–21

6.3 meV

10–22

10–23

10–24 10

ω–2 Carrier Temperature = 15 K Carrier Temperature = 30 K Carrier Temperature = 50 K 100 FIR Frequency (cm–1)

Figure 6.35 kB∆T/ITHz versus far-infrared (FIR) frequency for the quantum dot sample at carrier temperatures of 15, 30, and 50 K. The lattice temperature is 7 K. Carrier temperature is determined from the PL lineshape, and the terahertz intensity required to heat carriers to 15, 30, and 50 K is measured at each terahertz frequency. For all three carrier temperatures, the heating efficiency reaches a maximum for FIR energy of 1.5 THz (6.3 meV). (From Cerne J. et al., Phys. Rev. Lett., 77, 1131, 1996. Copyright 1996, American Physical Society.)

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Many interesting theoretical predictions have been made on the response of undoped76,79 and doped81–83 QWs in strong terahertz fields, and these are worthy of pursuit. However, perhaps the biggest challenges and opportunities for optically detected terahertz science on semiconductor heterostructures lies in the investigation of lower dimensional structures, like quantum wires and dots. Some preliminary experimental results are shown in Figure 6.35. Here, terahertz-induced changes in PL from a sample containing quantum dots are investigated. The dependence on frequency is strikingly different from the f –2 observed generically for the same experiment in QWs (see Figure 6.10). Terahertz radiation is absorbed resonantly with a peak near 1.5 THz (6.3 meV).84 This likely indicates some sort of intraband transition within the quantum dots. The heterogeneity of most quantum dots presents new challenges, perhaps necessitating optically detected terahertz resonance spectroscopy under an optical microscope so that single quantum dots can be studied. The experiments described in this chapter are only the beginning. The future will continue to involve free-electron lasers, but will also be broadened to laboratories with ever-improving tabletop terahertz sources.

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Section III Applications in Chemistry and Biomedicine

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Terahertz Spectroscopy of Biomolecules Edwin J. Heilweil and David F. Plusquellic National Institute of Standards and Technology

Contents 7.1 7.2 7.3 7.4 7.5

Introduction.................................................................................................. 269 Experimental Procedures............................................................................. 270 Theoretical Spectral Modeling Methods..................................................... 272 Weakly Interacting Organic Model Compounds......................................... 274 Small Biomolecules as Crystalline Solids................................................... 276 7.5.1 Amino Acids..................................................................................... 277 7.5.2 Polypeptides...................................................................................... 278 7.5.3 Nucleic Acid Bases and Other Sugars..............................................280 7.5.4 Other Small Biomolecules................................................................ 282 7.6 Terahertz Studies of Large Biomolecules.................................................... 286 7.6.1 DNAs and RNAs.............................................................................. 286 7.6.2 Proteins............................................................................................. 288 7.6.3 Polysaccharides................................................................................. 292 7.7 Terahertz Studies of Biomolecules in Liquid Water.................................... 293 7.8 Conclusions and Future Investigations........................................................ 294 Acknowledgments................................................................................................... 295 References............................................................................................................... 295

7.1 Introduction A better comprehension of biomolecular function and activity in vivo requires a detailed picture of biopolymer secondary and tertiary structures and their dynamical motions on a range of timescales.1–3 At the cores of all proteins, for example, are the substituent amino acid building blocks that determine their structure and function. In an effort to develop methodologies to directly monitor complex macromolecule dynamics in real time, we4 and others5,6 suggested that an atomic level picture of the concerted motions of polypeptide chains and DNAs may be accessible through accurate measurement of low-frequency vibrational spectra. These vibrations are expected to occur in the terahertz (THz) frequency regime and may be observed using Raman, low-energy neutron7 and infrared absorption spectroscopies or related optical techniques. However, for even naturally occurring proteins and DNAs with >30 kDalton (kDa) molecular weights and large numbers of constituent peptide units or bases, one would expect the density 269

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of overlapping states to be so high in this frequency range that contributing absorption bands would “smear out” and yield essentially structureless spectra. It would clearly be of highest priority to obtain low-frequency vibrationally resolved spectra for biologic systems in aqueous-phase environments because this condition would most closely mimic their natural environment. However, this scenario has not been immediately feasible for most systems using far-infrared absorption spectroscopy because (1) absorption by the amino acids and most other biomolecules is masked by much stronger water absorption in the 1 to 3 THz spectral region8 and (2) spectral broadening arising from the full accessibility to conformational space and the rapid time scale for interconversion in these environments. Despite this limitation, recent studies of biomolecules in aqueous or high relative humidity environments reviewed in this chapter have revealed detailed information about the dynamics of these systems through careful broadband absorption measurements. Also reviewed here is the work performed to determine whether pure solid samples of organic model systems and protein fragments do indeed show sharp spectral features that are uniquely determined from their individual molecular symmetries and structures. Spectral THz absorption data and spectral modeling methods discussed in this review are the first of their kind and demonstrate that these expectations are indeed borne out.

7.2 Experimental Procedures As discussed in detail in our previous publications, solid organic species (e.g., di-substituted benzenes) and lyophilized samples of individual purified amino acids and short polypeptides were used as received from Sigma-Aldrich, Inc. or ICN Biomedicals, Inc. Powders requiring low-temperature storage were maintained at 273 K until use to prevent decomposition and exposure to atmospheric water. Matrix diluted samples for THz absorption measurements were rapidly prepared by first weighing 2 to 10 mg of solid and homogenizing the material in a mortar and pestle to reduce the solid particle size distribution. This procedure ensures particle sizes sufficiently smaller than THz wavelengths to reduce baseline offsets at higher frequencies arising from nonresonant light scattering. Each sample was thoroughly mixed with approximately 100 mg of spectrophotometric grade high density polyethylene powder (Sigma-Aldrich, Inc. with

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