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Principles of Terahertz Science and Technology

Yun-Shik Lee

Principles of Terahertz Science and Technology

Yun-Shik Lee Physics Department Oregon State University Corvallis, Oregon 97331 USA

ISBN 978-0-387-09539-4 e-ISBN 978-0-387-09540-0 Library of Congress Control Number: 2008935382 2009 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com

To my parents Su-Ho Lee and Soon-Im Shin

Preface

Over the last two decades, THz technology has ripened enough that a thor-ough summary and review of the relevant topics is in order. Many differentdisciplines such as ultrafast spectroscopy, semiconductor device fabrication,and bio-medical imaging involve the recent development of THz technology.It is an important task to lay down a common ground among the researchers,so that they can communicate smoothly with one another. Besides, the THzcommunity is growing fast and the THz technology is in a transitional period.The THz research activities have mainly focused on generation and detectionuntil lately, but the focal point has shifted to the practical applications suchas high-speed communication, molecular spectroscopy, security imaging, andmedical diagnosis, among many others.

This book covers a broad range of topics and fundamental issues. Individu-als from distinct disciplines have helped developing new THz technologies, andin order to reach the next level, i.e., practical applications, the technology re-quires its researchers to understand and communicate with one another. Thisbook serves this general purpose by providing the researchers with a commonreference, thus bridging “the THz gap”.

I have tried to elucidate the fundamentals of THz technology and sciencefor their potential users. This book surveys major techniques of generating,detecting, and manipulating THz waves. It also discusses a number of essentialprocesses where THz waves interact with physical, chemical, and biologicalsystems. Scientists and engineers of various disciplines realize that the THzgap in the electromagnetic spectrum is now accessible thanks to the recentadvances in THz source and detection technologies. Many are seeking waysby which they can incorporate the new technologies into their expertise andresearch agenda. Younger researchers, who wish or are to join THz researchgroups, would also find this new field challenging due to many barriers, thelack of comprehensive introduction and/or instruction among them. Potentialusers of THz technology should be prepared in the essential concepts andtechniques of THz science and technology; I hope this book be an introductoryguide for the new comers.

viii Preface

During the process of writing this book, many colleagues, friends, and stu-dents gave me worthy criticism and introduction. Although it is impossible toacknowledge all scientific contributions, I am deeply grateful of those whoseworks I use in this book. I am much obliged to Joe Tomaino, Andy Jameson,and Jeremy Danielson for their invaluable advice. I am indebted to the Na-tional Science Foundation and the Alexander von Humboldt-Foundation fortheir generous support. Finally, I thank my wife, JungHwa, for her supportin every possible ways.

CorvallisSeptember 2008 Yun-Shik Lee

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Terahertz Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Terahertz Generation and Detection . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Terahertz Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Terahertz Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Terahertz Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Basic Theories of Terahertz Interaction with Matter . . . . . . . 112.1 Electromagnetic Waves in Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Reflection and Transmission . . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Coherent Transmission Spectroscopy . . . . . . . . . . . . . . . . . 172.1.4 Absorption and Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.5 Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.6 Electric Dipole Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.7 Quasi-Optical Propagation in Free Space . . . . . . . . . . . . . 24

2.2 Terahertz Radiation and Elementary Excitations . . . . . . . . . . . . 282.2.1 Quantum Theory of Electric Dipole Interaction . . . . . . . 282.2.2 Energy Levels of Hydrogen-like Atoms . . . . . . . . . . . . . . . 342.2.3 Rotational and Vibrational Modes of Molecules . . . . . . . 362.2.4 Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3 Laser Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Generation and Detection of Broadband Terahertz Pulses . 513.1 Ultrafast Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.1 Optical Pulse Propagation in Linear and DispersiveMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.2 Femtosecond Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1.3 Time-resolved Pump-Probe Technique . . . . . . . . . . . . . . . 583.1.4 Terahertz Time-Domain Spectroscopy . . . . . . . . . . . . . . . . 59

x Contents

3.2 Terahertz Emitters and Detectors Based on PhotoconductiveAntennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.1 Photoconductive Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.2 Generation of Terahertz Pulses from Biased

Photoconductive Antennas . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.3 Substrate Lenses: Collimating Lens and Hyper-

Hemispherical Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2.4 Terahertz Radiation from Large-Aperture

Photoconductive Emitters . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2.5 Time-Resolved Terahertz Field Measurements with

Photoconductive Antennas . . . . . . . . . . . . . . . . . . . . . . . . . 743.3 Optical Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3.1 Nonlinear Optical Interactions withNoncentrosymmetric Media . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.2 Second-Order Nonlinear Polarization andSusceptibility Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3.3 Wave Equation for Optical Rectification . . . . . . . . . . . . . . 843.3.4 Dispersion at Optical and Terahertz Frequencies . . . . . . 873.3.5 Absorption of Electro-Optic Crystals at the Terahertz

Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.4 Free-Space Electro-Optic Sampling . . . . . . . . . . . . . . . . . . . . . . . . 923.5 Ultrabroadband Terahertz Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.5.1 Optical Rectification and Electro-Optic Sampling . . . . . . 983.5.2 Photoconductive Antennas . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.6 Terahertz Radiation from Electron Accelerators . . . . . . . . . . . . . 1033.7 Novel Techniques for Generating Terahertz Pulses . . . . . . . . . . . 106

3.7.1 Phase-Matching with Tilted Optical Pulses in LithiumNiobate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.7.2 Terahertz Generation in Air . . . . . . . . . . . . . . . . . . . . . . . . 1083.7.3 Narrowband Terahertz Generation in Quasi-Phase-

Matching Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.7.4 Terahertz Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4 Continuous-Wave Terahertz Sources and Detectors . . . . . . . . 1174.1 Photomixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.2 Difference Frequency Generation and Parametric Amplification 122

4.2.1 Principles of Difference Frequency Generation . . . . . . . . . 1234.2.2 Difference Frequency Generation with Two Pump

Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2.3 Optical Parametric Amplification . . . . . . . . . . . . . . . . . . . . 129

4.3 Far-Infrared Gas Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.4 P-Type Germanium Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.5 Frequency Multiplication of Microwaves . . . . . . . . . . . . . . . . . . . . 1364.6 Quantum Cascade Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.6.1 Lasing and Cascading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Contents xi

4.6.2 Prospective Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.7 Backward Wave Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.8 Free-Electron Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.8.1 Operational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444.8.2 Free Electron Laser Facilities . . . . . . . . . . . . . . . . . . . . . . . 146

4.9 Thermal Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.9.1 Bolometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1474.9.2 Pyroelectric Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1514.9.3 Golay Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.10 Heterodyne Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5 Terahertz Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.1 Dielectric Properties of Solids in the Terahertz Region . . . . . . . 1595.2 Materials for Terahertz Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.2.1 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.2.2 Dielectrics and Semiconductors . . . . . . . . . . . . . . . . . . . . . . 1645.2.3 Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.3 Optical Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1705.3.1 Focusing Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1705.3.2 Antireflection Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.3.3 Bandpass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1725.3.4 Polarizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.3.5 Wave Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

5.4 Terahertz Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.4.1 Theory of Rectangular Waveguides . . . . . . . . . . . . . . . . . . 1775.4.2 Hollow Metallic Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1795.4.3 Dielectric Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815.4.4 Parallel Metal Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.4.5 Metal Wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5.5 Artificial Materials at Terahertz Frequencies . . . . . . . . . . . . . . . . 1895.5.1 Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1895.5.2 Photonic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1945.5.3 Plasmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

5.6 Terahertz Phonon-Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

6 Terahertz Spectroscopy of Atoms and Molecules . . . . . . . . . . . 2156.1 Manipulation of Rydberg Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . 2156.2 Rotational Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.2.1 Basics of Rotational Transitions . . . . . . . . . . . . . . . . . . . . . 2206.2.2 High-Resolution Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 2226.2.3 Atmospheric and Astronomical Spectroscopy . . . . . . . . . . 224

6.3 Biological Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2326.3.1 Liquid Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2336.3.2 Normal Modes of Small Biomolecules . . . . . . . . . . . . . . . . 2366.3.3 Dynamics of Large Molecules . . . . . . . . . . . . . . . . . . . . . . . 248

xii Contents

7 T-Ray Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.2 Imaging with Broadband THz Pulses . . . . . . . . . . . . . . . . . . . . . . 261

7.2.1 Amplitude and Phase Imaging . . . . . . . . . . . . . . . . . . . . . . 2617.2.2 Real-Time 2-D Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2647.2.3 T-Ray Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

7.3 Imaging with Continuous-Wave THz Radiation . . . . . . . . . . . . . . 2737.3.1 Raster-Scan Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2747.3.2 Real-Time Imaging with a Microbolometer Camera . . . . 278

7.4 Millimeter-Wave Imaging for Security . . . . . . . . . . . . . . . . . . . . . . 2817.4.1 Active Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2827.4.2 Passive Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

7.5 Medical Applications of T-Ray Imaging . . . . . . . . . . . . . . . . . . . . 2887.5.1 Optical Properties of Human Tissue . . . . . . . . . . . . . . . . . 2887.5.2 Cancer Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2897.5.3 Reflective Imaging of Skin Burns . . . . . . . . . . . . . . . . . . . . 2927.5.4 Detection of Dental Caries . . . . . . . . . . . . . . . . . . . . . . . . . . 294

8 Terahertz Spectroscopy of Condensed Matter . . . . . . . . . . . . . . 2958.1 Intraband Transitions in Semiconductors . . . . . . . . . . . . . . . . . . . 295

8.1.1 Band Structure of Intrinsic Semiconductors . . . . . . . . . . . 2968.1.2 Photocarrier Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2978.1.3 Impurity States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2998.1.4 Semiconductor Nanostructures: Quantum Wells,

Quantum Wires, and Quantum Dots . . . . . . . . . . . . . . . . . 3028.2 Strongly Correlated Electron Systems . . . . . . . . . . . . . . . . . . . . . . 311

8.2.1 Quasiparticle Dynamics in ConventionalSuperconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.2.2 Low Energy Excitations in High TemperatureSuperconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

1

Introduction

Terahertz (THz) radiation is electromagnetic radiation whose frequency liesbetween the microwave and infrared regions of the spectrum. We cannot seeTHz radiation, but we can feel its warmth as it shares its spectrum withfar-infrared radiation. Naturally occurring THz radiation fills up the spaceof our everyday life, yet this part of the electromagnetic spectrum remainsthe least explored region mainly due to the technical difficulties involved inmaking efficient and compact THz sources and detectors. The lack of suitabletechnologies led to the THz band being called the “THz gap”. This techno-logical gap has been rapidly diminishing for the last two decades. Opticaltechnologies have made tremendous advances from the high frequency side,while microwave technologies encroach up from the low frequency side. Thischapter gives a brief perspective on the basic properties of THz radiation andits interaction with materials, which lays down the foundation to discuss theprogress of THz science and technology in subsequent chapters.

1.1 Terahertz Band

“Terahertz radiation” is the most common term used to refer to this frequencyband, analogous to microwaves, infrared radiation, and x-rays. It is ratherawkward to use a frequency unit for naming a spectral band. Nevertheless, as“terahertz” has become a symbolic word designating the entire field, we willuniversally use this term throughout this book. An alternative and seeminglybetter terminology is “T-rays”, where “T” stands for terahertz. It was initiallycoined for an imaging technique, and will be used in Chapter 7, which isdedicated to T-ray imaging technologies and applications.

Until quite recently, THz technologies had been independently developedby researchers from several different disciplines. In practice, different com-munities use different units to describe the spectrum of THz radiation. Wewill use THz (1012 Hz) as the universal unit in this book, but other units

2 1 Introduction

will also be used when they are appropriate. Frequently used units and theirconversions at 1 THz are as follows:

• Frequency: ν = 1 THz = 1000 GHz• Angular frequency: ω = 2πν = 6.28 THz• Period: τ = 1/ν = 1 ps• Wavelength: λ = c/ν = 0.3 mm = 300 µm• Wavenumber: k = k/2π = 1/λ = 33.3 cm−1

• Photon energy: hν = hω = 4.14 meV• Temperature: T = hν/kB = 48 K

where c is the speed of light in vacuum, h is Plank’s constant, and kB isBoltzmann’s constant. Physicists tend to use µm and meV as units of photonwavelength and energy, respectively; chemists use cm−1 as a unit of wavenum-ber; engineers use mm and GHz as units of wavelength and frequency, respec-tively. In physics, angular wavenumber (k = 2π/λ) is usually abbreviated aswavenumber. In this book, we will use the abbreviated notation when it isclearly defined.

Fig. 1.1. Terahertz band in the electromagnetic spectrum

The THz band does not have a standard definition yet. Commonly useddefinitions are included in the spectral region between 0.1 and 30 THz. Therange of 10-30 THz, however, exceeds the far-IR band and intrudes on the mid-IR band, where well established optical technologies exist. Unless we deal with

1.2 Terahertz Generation and Detection 3

ultrabroadband THz pulses, we will use 0.1-10 THz as a universal definitionof the THz band. Figure 1.1 illustrates the THz band in the electromagneticspectrum. The THz band merges into neighboring spectral bands such asthe millimeter-wave band, which is the highest radio frequency band knownas Extremely High Frequency (EHF), the submillimeter-wave band, and thefar-IR band. The definitions of these bands are as follows:

• Millimeter wave (MMW): 1-10 mm, 30-300 GHz, 0.03-0.3 THz• Submillimeter wave (SMMW): 0.1-1 mm, 0.3-0.3 THz• Far infrared radiation (Far-IR): (25-40) to (200-350) µm, (0.86-1.5) to (7.5

to 12) THz• Sub-THz radiation: 0.1-1 THz

These bands are also distinguished by their characteristic technologies. Mil-limeter wave emitters and sensors are solid-state devices based on microwavetechnologies. Traditionally, far-IR applications rely on optical and thermaldevices.

1.2 Terahertz Generation and Detection

Technological advances in optics and electronics have resulted in many dif-ferent types of THz sources and sensors. Chapters 3 and 4 are devoted tobroadband and continuous-wave (CW) THz technologies, which are classifiedby similarities in radiation characteristics. In this section, we make brief de-scriptions of the schemes used for THz generation and detection, grouped byoperational concepts.

1.2.1 Terahertz Sources

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Fig. 1.2. Terahertz generation in nonlinear media

4 1 Introduction

One way of generating THz radiation is to exploit a nonlinear mediumin which incident electromagnetic waves undergo nonlinear frequency con-version (Fig. 1.2). Optical rectification and difference frequency generation(DFG) are second order nonlinear optical processes in which a THz photon atfrequency ωT is created by interaction of two optical photons at frequenciesω1 and ω2 with a nonlinear crystal, such that ωT = ω1 − ω2. Femtosecondlaser pulses with a broad spectrum (bandwidth∼10 THz) generate broadbandTHz pulses, whose shape resembles the optical pulse envelope, via opticalrectification. Two CW optical beams produce CW THz radiation by DFG.Solid-state THz sources based on microwave technology convert incoming mi-crowaves into their harmonic waves utilizing diodes with strongly nonlinearI-V characteristics.

Fig. 1.3. THz radiation from accelerating electrons

Accelerating charges, and time-varying currents, radiate electromagneticwaves (Fig. 1.3). THz radiation can be generated from a biased photocon-ductive (PC) antenna excited by laser beams. A PC antenna consists of twometal electrodes deposited on a semiconductor substrate. An optical beam,illuminating the gap between the electrodes, generates photocarriers, and astatic bias field accelerates the free carriers. This photocurrent varies in timecorresponding to the incident laser beam intensity. Consequently, femtosecondlaser pulses produce broadband THz pulses. Mixing two laser beams with dif-ferent frequencies forms an optical beat, which generates CW THz radiationat the beat frequency. This technique is called photomixing.

Electron accelerators produce extremely bright THz radiation using rel-ativistic electrons. A femtosecond laser pulse triggers an electron source to

1.2 Terahertz Generation and Detection 5

generate a ultrashort pulse of electrons. After being accelerated to a relativis-tic speed, the electrons are smashed into a metal target, or are forced intocircular motion by a magnetic field. Coherent THz radiation is generated bythis transient electron acceleration.

Backward wave oscillators (BWOs) are laboratory-size equipment, andfree-electron lasers (FELs), small scale electron accelerators, are large facili-ties. In spite of the huge difference in size, there is a similarity in their THzgeneration mechanism. In both, an electron beam is undulated by a periodicstructure: a BWO has a metal grating and a FEL consists of a magnet array.CW THz radiation is produced by the periodic acceleration of electrons.

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• ��

Fig. 1.4. THz emission from laser

Laser action requires a population-inverted two-level quantum system(Fig. 1.4). Far-IR gas lasers utilize molecular rotation energy levels, whosetransition frequencies fall into the THz region. P-type germanium lasers areelectrically pumped solid-state lasers. Their lasing action relies on the pop-ulation inversion of two Landau levels formed by hot-carriers submerged incrossed electric and magnetic fields. Quantum cascade lasers (QCLs) are semi-conductor heterostructure lasers consisting of periodically alternating layersof dissimilar semiconductors. Transitions between subbands of these semi-conductor nanostructures involve THz photons. In a QCL, electrons undergosuccessive intersubband transitions to generate coherent THz radiation.

1.2.2 Terahertz Detectors

THz detection schemes are largely classified as either coherent or incoherenttechniques. The fundamental difference is that coherent detection measuresboth the amplitude and phase of the field, whereas incoherent detection mea-sures the intensity. Coherent detection techniques are closely associated withgeneration techniques in that they share underlying mechanisms and key com-ponents. In particular, optical techniques utilize the same light source for bothgeneration and detection.

Figure 1.5 illustrates the commonly used coherent detection schemes. Free-space electro-optic (EO) sampling measures the actual electric field of broad-band THz pulses in the time domain by utilizing the Pockels effect, which isclosely related to optical rectification. A THz field induces birefringence ina nonlinear optical crystal which is proportional to the field amplitude. The

6 1 Introduction

entire waveform is determined by a weak optical probe measuring the field-induced birefringence as a function of the relative time delay between the THzand optical pulses.

Sensing with a PC antenna also measures broadband THz pulses in thetime domain. In the absence of a bias field, a THz field induces a current in thephotoconductive gap when an optical probe pulse injects photocarriers. Theinduced photocurrent is proportional to the THz field amplitude. The THzpulse shape is mapped out in the time domain by measuring the photocurrentwhile varying the time delay between the THz pulse and the optical probe.

A combined setup of broadband THz generation and detection measureschanges in both the amplitude and phase of THz pulses induced by a sample,which provides enough information to simultaneously determine the absorp-tion and dispersion of the sample. This technique is named THz time-domainspectroscopy, or, in short, THz-TDS.

χ(2)��

��

��

� ��

��

� ��

��

��

��

��

��

�

∆t

ETHz(t))(tETHz∝∆Φ

)()( tEtI THz∝

��

��

��

��

��

��

��

��

��

!��

��

��

�

( ) ( )φφ ∆∝∆ THzEI

��

��

∆φ

THzO EV ∝

Fig. 1.5. Coherent detection of THz radiation

Photomixing measures CW THz radiation by exploiting photoconductiveswitching. In this case, the photocurrent shows sinusoidal dependence on therelative phase between the optical beat and the THz radiation.

Heterodyne detection utilizes a nonlinear device called a “mixer”. Schottkydiodes are commonly used as mixers. The key process in a mixer is frequencydownconversion, which is carried out by mixing a THz signal ωs with reference

1.3 Terahertz Applications 7

radiation at a fixed frequency ωLO. The mixer produces an output signal at thedifference frequency called the “intermediate frequency”, ωD = |ωS − ωLO|.The amplitude of the output signal is proportional to the THz amplitude.Unlike the optical techniques, heterodyne detection is usually used to detectincoherent radiation.

Commonly used incoherent detectors are thermal sensors such as bolome-ters, Golay cells, and pyroelectric devices. A common element of all thermaldetectors is a radiation absorber attached to a heat sink. Radiation energyis recorded by a thermometer measuring the temperature increase in the ab-sorber. Each type of thermal detector is distinguished by its specific schemeused to measure the temperature increase. Bolometers are equipped with anelectrical resistance thermometer made of a heavily doped semiconductor suchas Si or Ge. In general, bolometers operate at cryogenic temperature. Pyro-electric detectors employ a pyroelectric material in which temperature changegives rise to spontaneous electric polarization. In a Golay cell, heat is trans-ferred to a small volume of gas in a sealed chamber behind the absorber. Anoptical reflectivity measurement detects the membrane deformation inducedby the pressure increase. These thermal detectors respond to radiation over avery broad spectral range. Because a radiation absorber must reach to thermalequilibrium for a temperature measurement, detection response is relativelyslow compared with typical light detectors.

1.3 Terahertz Applications

The THz region is crowded by innumerable spectral features associated withfundamental physical processes such as rotational transitions of molecules,large-amplitude vibrational motions of organic compounds, lattice vibrationsin solids, intraband transitions in semiconductors, and energy gaps in super-conductors. THz applications exploit these unique characteristics of materialresponses to THz radiation.

��

108 109 1010 1011 1012 1013 1014 1015 1016 1017

0

20

40

60

80

100

Tra

nsm

issi

on (

%)

��

10 10 10 10 10 10 10 10 10 10Frequency (Hz)

Fig. 1.6. Atmospheric transmission spectrum of electromagnetic waves

8 1 Introduction

Compared with the neighboring regions of radio waves and infrared radi-ation, the THz band shows exceedingly high atmospheric opacity due to therotational lines of constituent molecules (Fig. 1.6). In particular, absorptionby water vapor is the predominant process of atmospheric THz attenuation.Figure 1.7 shows a high-resolution transmission spectrum of water vapor. Inpractice, water absorption is an important factor to be considered when de-signing an operation scheme for a THz application.

1

0.5 1.0 1.5 2.0

0.1

1

Tran

smis

sion

Frequency (THz)

1 2 3 4 5 6

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

Tran

smis

sion

Frequency (THz)

Fig. 1.7. Water vapor transmission spectrum from 0.3 to 6 THz

Distinctive line structures of each molecular species can be utilized toidentify it in an unknown specimen. Furthermore, spectral line-shapes pro-vide crucial information concerning microscopic mechanisms of molecular col-lisions. High-resolution THz spectroscopy is being used to monitor the Earthsatmosphere and to observe molecules in the interstellar medium.

Spectral signatures of organic and biological molecules in the THz regionare associated with large amplitude vibrational motions and inter-molecularinteractions. THz spectroscopy is capable of analyzing these molecular dy-namics, it can therefore be applied to the detection of explosives and illicitdrugs, testing pharmaceutical products, investigating protein conformation,etc.

Based on optical properties at THz frequencies, condensed matter is largelygrouped into three types: water, metal, and dielectric. Water, a strongly po-lar liquid, is highly absorptive in the THz region. Because of high electrical

1.3 Terahertz Applications 9

conductivity, metals are highly reflective at THz frequencies. Nonpolar andnonmetallic materials, i.e., dielectrics such as paper, plastic, clothes, wood,and ceramics that are usually opaque at optical wavelengths, are transparentto THz radiation. A brief description of the optical properties of each materialtype is shown in Table 1.1.

Table 1.1. Optical Properties of Condensed Matter in the THz Band

Material Type Optical Property

liquid water high absorption (α ≈ 250 cm−1 at 1 THz)metal high reflectivity (>99.5% at 1 THz)plastic low absorption (α < 0.5 cm−1 at 1 THz)

low refractive index (n ≈ 1.5)semiconductor low absorption (α < 1 cm−1 at 1 THz)

high refractive index (n ∼ 3-4)

These stark contrasts of THz properties are useful for many imaging ap-plications. Since common packaging materials are dielectric, THz imaging isapplied to nondestructive testing to inspect sealed packages. Because of thehigh absorption of water in the THz region, hydrated substances are easilydifferentiated from dried ones. Metal objects also can be easily identified dueto their high reflectivity and complete opacity. The same concept is appliedto security applications. THz imaging is used to identify weapons, explosives,and illegal drugs concealed underneath typical wrapping and packaging ma-terials. The high sensitivity of THz radiation to water is useful for medicalapplications because, in a biological system, small changes in water contentcould indicate crucial defects emerging in the region.

In addition to the applications we briefly described in this section, THzspectroscopy and imaging techniques have been applied to many others rang-ing from basic scientific missions to commercial projects. We will look intothem in following chapters.

2

Basic Theories of Terahertz Interaction withMatter

This chapter is concerned with basic concepts and theories that form the foun-dation for understanding the unique characteristics of THz radiation and itsinteraction with materials. Classical electromagnetic theory provides a generaldescription of THz waves which propagate in and interact with macroscopi-cally uniform media. The basic framework of quantum theory is utilized todescribe elementary exciations at THz frequencies.

2.1 Electromagnetic Waves in Matter

We shall begin with Maxwell’s equations to describe THz waves as we would dofor any other spectral regions. The macroscopic form of Maxwell’s equationshave the form

∇ ·D = ρf , (2.1)∇ ·B = 0, (2.2)

∇×E = −∂B∂t

, (2.3)

∇×H = Jf +∂D∂t

, (2.4)

where ρf and Jf are the free charge density and the free current density.These deceptively simple equations, together with the Lorentz force law

F = q (E + v ×B) , (2.5)

constitute the entire theoretical basis of classical electrodynamics. The macro-scopic fields D and H are related to the fundamental fields E and B as

D ≡ ε0E + P = εE, (2.6)

H ≡ 1µ0

B−M =1µB, (2.7)

12 2 Basic Theories of Terahertz Interaction with Matter

where ε0 and µ0 are the permittivity and the permeability of free space. Thepolarization P and the magnetization M contain the information about themacroscopic-scale electromagnetic properties of the matter. The last termsin Eqs. 2.6 and 2.7, where ε and µ denote the the electric permittivity andthe magnetic permeability, are valid only if the media is isotropic and linear.Typical magnetic responses of matter are subtle, |µ− µ0| < 10−4µ0, compar-ing with their electric counter parts, largely because of the nonexistence ofmagnetic monopoles.

2.1.1 The Wave Equation

The coupled electric and magnetic fields in Maxwell’s equations are disentan-gled by taking the curl of Eqs. 2.3 and 2.4 and using the linear relations ofEqs. 2.6 and 2.7:

∇× (∇×E) + εµ∂2E∂t2

= −µ∂Jf

∂t, (2.8)

∇× (∇×H) + εµ∂2H∂t2

= ∇× Jf . (2.9)

These are the most general wave equations for E and H. Using the vectoridentity

∇× (∇×A) = ∇(∇A)−∇2A (2.10)

with Eqs. 2.1 and 2.2 we can rewrite the wave equations as

∇2E− εµ∂2E∂t2

= µ∂Jf

∂t+

1ε∇ρf , (2.11)

∇2H− εµ∂2H∂t2

= −∇× Jf . (2.12)

Assuming Jf is linear with E,

Jf = σE, (2.13)

where σ is the electric conductivity, and neglecting charge density fluctuations,i.e., ∇ρf = 0, we simplify the wave equation for E as

∇2E = σµ∂E∂t

+ εµ∂2E∂t2

. (2.14)

Here σ and ε are real and independent. The wave equation for H takes anidentical form. These time-varying fields are closely intertwined by Maxwell’sequations: if one is known, the other is fully determined. The coupled entityof the two fields is called an electromagnetic wave.

If the material is a dielectric or an insulator, the wave equation takes theuniversal form

2.1 Electromagnetic Waves in Matter 13

∇2E = εµ∂2E∂t2

=1v2

∂2E∂t2

, (2.15)

which signifies that electromagnetic waves propagate in homogeneous mediaat a speed

v =1√εµ

=c

n, (2.16)

where c(= 1/

√ε0µ0

)is the speed of light in free space and n

(=

√ε/ε0

)is

the refractive index, assuming µ = µ0.General solutions of the wave equation are linearly-polarized monochro-

matic plane waves:

E(r, t) = E0ei(k·r−ωt) and H(r, t) = H0e

i(k·r−ωt), (2.17)

where k is the wave vector and ω is the angular frequency. From Maxwell’sequations we can draw the relations between E and H associated with k andω. Substituting the plane waves into ∇ ·E = 0 and ∇ ·B = 0 we obtain

k ·E = 0 and k ·H = 0. (2.18)

This means that E and H are both perpendicular to the wave vector, that is,electromagnetic waves are transverse. The curl equations give the relation

k×E = ωµH. (2.19)

Inserting Eq. 2.17 into Eq. 2.15 we obtain the dispersion relation

k2 = εµω2. (2.20)

As ε and µ quantify the electromagnetic properties of the material, the dis-persion relation governs how the wave propagates in the medium. For a non-magnetic medium, the wavenumber k is related to the wavelength λ by therelation

k =2π

λ= n

ω

c. (2.21)

The energy flux of an electromagnetic wave is the time-averaged Poyntingvector

〈S〉 =12E×H∗ =

12v ε|E0|2ek, (2.22)

where ek(=k/k) is a unit vector in the direction of the wave propagation. Themagnitude of the energy flux,

I = |〈S〉| = 12v ε|E0|2, (2.23)

is the radiation intensity, which is the measurement quantity of typical lightdetectors. A commonly used unit of light intensity is W/cm2.


0.4

0.6

0.8

1.0

E(z

)/E

(0)

0 1 2 3 4 5 6-0.2

0.0

0.2

E(z

)/E

(0)

z/δ

Fig. 2.1. Electric field decay in a conductor

In a conducting medium, general solutions of the wave equation, Eq. 2.14,also take the form of the plane waves in Eq. 2.17. Even Eqs. 2.18 and 2.19 arestill valid. Wave propagation in a conductor is, however, quite different fromthat in a dielectric medium. If the medium has a very large conductivity suchthat σ ¿ ωε, the wave equation leads to the dispersion relation

k2 ≈ iσµω. (2.24)

Evidently the amplitude of the wave vector is a complex number,

k = kR + ikI ≈√

ωµσ

2(1 + i). (2.25)

This means that when an electromagnetic wave is incident on a conductor thefield decays exponentially with an attenuation length δ, which is called thepenetration depth or the skin depth:

δ =√

2ωµσ

. (2.26)

Typical metals behave like an ideal conductor for THz waves. For example,the skin depth of copper is δ ≈ 0.07 µm for the frequency ν(= ω/2π) =1 THz, which is almost negligible when compared to the free-space wavelength,300 µm.

2.1.2 Reflection and Transmission

When an electromagnetic wave reflects from and transmits through an inter-face of two linear dielectric media, both E and H obey the boundary condition


that the parallel components of the vector fields are continuous across the in-terface. An apparent ramification of the boundary condition is Snell’s law

n1 sin θ1 = n2 sin θ2, (2.27)

where n1,2 are the refractive indices of the media and θ1,2 are the angle ofincidence and the angle of refraction.

θ

θ1 θ2

x

z θ

θ1 θ2

x

z

HR

ER

HT

ETkT

kR

HR

ER

HT

ET kT

kR

s-polarization p-polarization

medium 1 medium 2 medium 1 medium 2

θ1z θ1

z

HI

EI

kI

HI

EIkI

Fig. 2.2. Reflection and transmission of s- and p-polarized electromagnetic waves

Figure 2.2 illustrates the incident, reflected, and transmitted waves on aplane of incidence. S-polarization denotes the case when the polarization of theincident wave is perpendicular to the plane of incidence, and p-polarization iswhen the polarization of the incident wave is parallel to the plane of incidence.Eventually, the boundary conditions determine the ratios of the reflected andtransmitted field amplitudes to the incident field amplitude. These relationsare expressed in the Fresnel equations:

ER,s

EI,s=

n1 cos θ1 − n2 cos θ2

n1 cos θ1 + n2 cos θ2and

ET,s

EI,s=

2n1 cos θ1

n1 cos θ1 + n2 cos θ2(2.28)

for s-polarization and

ER,p

EI,p=

n2 cos θ1 − n1 cos θ2

n2 cos θ1 + n1 cos θ2and

ET,p

EI,p=

2n1 cos θ1

n2 cos θ1 + n1 cos θ2(2.29)

for p-polarization.Reflectivity R is defined as the fraction of incident radiation power that

reflects from the boundary, and transmittance T that transmits through. Since


the radiation intensity striking the interface is the normal component of thePoynting vector 〈S〉 · ek, the reflectivity and the transmittance are given as

R =|ER|2|EI |2 and T =

n2|ET |2n1|EI |2 . (2.30)

Figure 2.3 shows an example of the reflectivity of s- and p-polarized waves ver-sus the angle of incidence for n1=1 and n2=2. The reflectivity of p-polarizationis completely expunged at Brewster’s angle,

θB = tan−1

(n2

n1

). (2.31)

If medium 1 is optically denser than medium 2, i.e., n1 > n2, reflectivitybecomes unity for

θ1 > sin−1

(n2

n1

). (2.32)

This phenomenon is called total internal reflection.

0.4

0.6

0.8

1.0

Ref

lect

ivity

Rp

Rs

0 30 60 900.0

0.2

Angle of Incidence (degree)

θB

Rp

Fig. 2.3. Reflectivity of s- and p-polarization versus angle of incidence

The same boundary conditions are applied to a conducting surface. Con-sider an electromagnetic wave incident on an interface of air and a conductorwith normal angle. Eq. 2.28 is still valid if we substitute n2 with a complexindex of refraction,

n =ck

ω= c

√µσ

2ω(1 + i), (2.33)

using Eq. 2.25. Due to the large conductivity, |n| À 1,


ER =1− n

1 + nEI

∼= −EI , (2.34)

ET =2

1 + nEI

∼= 0. (2.35)

The reflectivity is close to unity, and very little energy is dissipated into theconducting medium.

2.1.3 Coherent Transmission Spectroscopy

Coherent THz spectroscopy in a transmission geometry is a commonly usedtechnique to measure the optical constants of materials. The coherent detec-tion scheme measuring both the amplitude and phase of THz fields warrantssimultaneous determination of the real and imaginary part of a dielectric func-tion εr(ω) or a conductivity σ(ω).

EI ET

r1 r2

)(ωn

t1 t2

……

d

Fig. 2.4. Transmission of an electromagnetic wave through a flat single-layer ofmaterial with a complex index of refraction n(ω) and a thickness d. r1 and r2 arethe reflection coefficients at the entrance and exit surfaces, respectively, and t1 andt2 are the transmission coefficients.

Figure 2.4 illustrates a normal-incident electromagnetic wave passingthrough a single-layer of material with a thickness d and a complex indexof refraction

n(ω) = n(ω) + iκ(ω). (2.36)

Inside the layer some parts of the wave undergo multiple reflections at theinterfaces before they transmit through. Ultimately the total transmission isthe superposition of all the parts having endured multiple reflections. Thus,the transmitted field ET can be expressed as


ET = EIt1t2eiφd + EIt1t2e

iφd · (r1r2e2iφd

)+ · · ·

= EIt1t2eiφd

∞∑m=0

(r1r2e

2iφd)m

=EIt1t2e

iφd

1− r1r2e2iφd, (2.37)

where

r1(ω) = r2(ω) =n(ω)− 1n(ω) + 1

, (2.38)

t1(ω) =2

n(ω) + 1, (2.39)

t2(ω) =2n(ω)

n(ω) + 1, (2.40)

obtained from Eq. 2.28 with the incident angle θ1 = 0, are the reflection andtransmission coefficients at the entrance and exit surfaces, and

φd(ω) = n(ω)ω

cd (2.41)

is the phase shift while the wave propagates a distance d within the material.What we actually measure is the complex transmission coefficient t(ω) withthe amplitude |t(ω)| and the phase Φ(ω), which is expressed as

t(ω) = |t(ω)|eΦ(ω)

=ET (ω)EI(ω)

=4n(ω)eiφd(ω)

[n(ω) + 1]2 − [n(ω)− 1]2 e2iφd(ω)(2.42)

in terms of n(ω). n(ω) is determined by fitting the transmission data toEq. 2.42.

The complex dielectric function

εr(ω) = [n(ω)]2 = εr1(ω) + iεr2(ω) (2.43)

and the complexity conductivity

σ(ω) = σ1(ω) + iσ2(ω) (2.44)

have the relation

σ1(ω) = −ε0ω εr2(ω), (2.45)σ2(ω) = ε0ω [εr1(ω)− εr1(∞)] , (2.46)

where εr1(∞) is the dielectric constant in the high-frequency limit.


2.1.4 Absorption and Dispersion

Frequency dispersion refers to the phenomenon in which waves of differentfrequencies propagate at different speeds. Dispersion, together with absorp-tion, characterizes how media respond to external electromagnetic fields. Allelectromagnetic phenomena involve the interaction of fields with charged par-ticles, electrons and nuclei at a microscopic scale, in matter. Electromagneticwaves force charged particles to move; their accelerated motion induces ra-diation. The effects of magnetic fields on naturally occurring materials aremostly negligible and the amplitude of the electron motions are usually verysmall. Consequently, electromagnetic properties of a medium are dominatedby electric dipoles induced by the applied electric fields. In the linear opticalregime, the electric dipole moments are proportional to the amplitude of theapplied electric fields.

The classical Lorentzian model provides a good qualitative description ofthis phenomenon. Assuming that a bound charge oscillates about its equilib-rium position with a very small amplitude, we model the system as a simpleharmonic oscillator. Figure 2.5 illustrates the Lorentzian harmonic oscillatormodel. The potential energy of the charged particle is quadratic for small dis-placements from equilibrium. The size of the oscillator is much smaller thanthe wavelength of the applied field, hence the electric field is constant nearthe equilibrium position for a given time t.

z

x

202

1)( xmxU ω=

Incident EM wave

)(0

tzkieE ω−

q

Fig. 2.5. The classical Lorentzian model accounts for the optical response of boundelectrons in dielectric media.

When the monochromatic wave

E(t) = E0e−iωt (2.47)

of angular frequency ω, polarized along the x axis, interacts with the chargeq, its equation of motion is expressed as

d2x

dt2+ γ

dx

dt+ ω2

0x =q

mE(t), (2.48)

where q and m are the charge and the mass. γ and ω0 are the dampingconstant and the resonant frequency, respectively. The solution of the equationof motion is


x(t) = x0e−iωt, where x0 =

q

m

E0

ω20 − ω2 − iωγ

. (2.49)

The electric dipole moment of the harmonic oscillator is

p(t) = qx(t). (2.50)

We suppose that a medium has N oscillators per unit volume, then theelectric polarization is given by

P (t) = Nqx(t) =Nq2

m

E0e−iωt

ω20 − ω2 − iωγ

≡ ε0χe(ω)E0e−iωt, (2.51)

where χe(ω) denotes the linear electric susceptibility of the medium. InsertingEq. 2.51 into Eq. 2.6 we obtain the dielectric constant of the medium

εr(ω) ≡ ε(ω)ε0

= 1 + χe(ω) = 1 +Nq2

mε0

1ω2

0 − ω2 − iωγ. (2.52)

The real and imaginary parts of the complex dielectric constant are writtenas

< [εr]− 1 =Nq2

mε0

ω20 − ω2

(ω20 − ω2)2 + ω2γ2

, (2.53)

= [εr] =Nq2

mε0

ωγ

(ω20 − ω2)2 + ω2γ2

. (2.54)

Figure 2.6 shows characteristic dielectric dispersion in the vicinity of the reso-nant frequency. This medium is dispersive because its response to an externalelectromagnetic wave depends on frequency. The imaginary part of the dielec-tric constant indicates that absorption is maximized at the resonant frequency,and the bandwidth is ∼ γ.

From the dispersion relation of Eq. 2.20 we get the complex amplitude ofthe wave vector

k(ω) = kR(ω) + ikI(ω) =√

εr(ω)ω

c, (2.55)

which governs how the wave propagates in the medium. The plane wave prop-agating along the z-axis is written as

E(z, t) = E0ei(kz−ωt) = E0e

−α2 ze−iω(t−n

c z). (2.56)

The radiation intensity exponentially decays with an absorption coefficient

α(ω) = 2=[k(ω)], (2.57)

and the wave propagates with the phase velocity

v =n(ω)

c=<[k(ω)]

ω. (2.58)


[ ] 1−ℜ rε

[ ]rεℑ

γ

0ω

Fig. 2.6. Dielectric dispersion at a resonance.

2.1.5 Plasma Frequency

Consider the interaction of electromagnetic waves with a system in whichcharges move freely, and scattering between the particles is negligible. Dopedsemiconductors or plasmas may behave in such a way upon incidence of THzradiation. Substituting ω0 = 0 and γ = 0 into Eq. 2.52 we get the dielectricconstant,

εr(ω) = 1− ω2p

ω2, (2.59)

where

ωp =

√Nq2

mε0(2.60)

is the plasma frequency. The dispersion relation,

ck =√

ω2 − ω2p, (2.61)

indicates that waves can propagate through the medium for ω > ωp, while itdecays with the absorption coefficient,

α(ω) =2c

√ω2

p − ω2, (2.62)

for ω < ωp. A typical electron density, 1016 cm−3, of laboratory plasmas anddoped semiconductors corresponds to ωp

∼= 6 THz.

2.1.6 Electric Dipole Radiation

Electromagnetic waves are generated by accelerating charges and time-varyingcurrents. In the present section, we discuss the emission of radiating fields from

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