Library of Congress Cataloging-in-Publication Data

O’Sullivan, Créidhe M. M.

Field guide to terahertz sources, detectors, and optics / Créidhe O’Sullivan, J. Anthony Murphy.

p. cm. – (The ﬁeld guide series ; FG28)

Includes bibliographical references and index.

ISBN 978-0-8194-9167-1

1. Millimeter wave devices–Handbooks, manuals, etc.

2. Terahertz technology–Handbooks, manuals, etc.

3. Infrared equipment–Handbooks, manuals, etc.

4. Submillimeter waves–Handbooks, manuals, etc.

I. Murphy, J. Anthony. II. Title.

TK7876.5.O88 2012

621.381′33--dc23

2012012372

Published by

SPIE

P.O. Box 10

Bellingham, Washington 98227-0010 USA

Phone: +1.360.676.3290

Fax: +1.360.647.1445

Email: books@spie.org

Web: http://spie.org

The content of this book reflects the work and thought of the author. Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon.

Printed in the United States of America.

First printing

## Introduction to the Series

Welcome to the * SPIE Field Guides*—a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the

*is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.*

**SPIE Field Guides**Each * SPIE Field Guide* addresses a major field of optical science and technology. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights, and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships, and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.

The * SPIE Field Guides* are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at

**fieldguides@SPIE.org.**

John E. Greivenkamp, **Series Editor**

College of Optical Sciences

The University of Arizona

## The Field Guide Series

Keep information at your fingertips with all of the titles in the Field Guide Series:

Field Guide to

* Adaptive Optics*, Robert Tyson & Benjamin Frazier

* Atmospheric Optics*, Larry Andrews

* Binoculars and Scopes*, Paul Yoder, Jr. & Daniel Vukobratovich

* Diffractive Optics*, Yakov Soskind

* Geometrical Optics*, John Greivenkamp

* Illumination*, Angelo Arecchi, Tahar Messadi, & John Koshel

* Image Processing*, Khan M. Iftekharuddin & Abdul Awwal

* Infrared Systems, Detectors, and FPAs*,

*, Arnold Daniels*

**Second Edition*** Interferometric Optical Testing*, Eric Goodwin & Jim Wyant

* Laser Pulse Generation*, Rüdiger Paschotta

* Lasers*, Rüdiger Paschotta

* Microscopy*, Tomasz Tkaczyk

* Optical Fabrication*, Ray Williamson

* Optical Fiber Technology*, Rüdiger Paschotta

* Optical Lithography*, Chris Mack

* Optical Thin Films*, Ronald Willey

* Polarization*, Edward Collett

* Probability, Random Processes, and Random Data Analysis*, Larry C. Andrews & Ronald L. Phillips

* Radiometry*, Barbara Grant

* Special Functions for Engineers*, Larry Andrews

* Spectroscopy*, David Ball

* Visual and Ophthalmic Optics*, Jim Schwiegerling

## Field Guide to THz

The region of the electromagnetic spectrum between microwaves and infrared radiation has come to be known as the “THz gap,” mainly due to the lack of readily available laboratory sources and detectors. For many years technology development was driven by astronomers and planetary scientists, but other potential uses, particularly in medical and security applications, have led to increased activity by the mainstream physics and engineering community in recent times. Because diffraction is important at these frequencies, THz systems cannot be successfully designed using traditional optical techniques alone.

The primary objective of this Field Guide is to provide the reader with a concise description of the quasi-optical techniques used at THz frequencies, as well as the basic principles of operation of the most common THz system components in use today. More detailed accounts of specific devices can be found in the bibliography and references therein.

We would like to thank our families and our colleagues at NUI Maynooth, in particular Neil Trappe, Marcin L. Gradziel, and Ian McAuley of the THz Optics group and also Stafford Withington of the Cavendish Laboratory at Cambridge.

Créidhe O’Sullivan

J. Anthony Murphy

Department of Experimental Physics

National University of Ireland,

Maynooth

## Table of Contents

**Glossary of Symbols and Acronyms** xi

**Introduction** 1

The THz Band 1

The THz Gap 2

THz Absorption in Air 3

Reflection and Transmission 4

**THz Sources** 5

Natural Sources of THz Radiation 5

THz Generation Techniques 6

Gunn Diodes 8

IMPATT Diodes 9

TUNNETT Diodes 10

Resonant Tunnel Diodes 11

Difference Frequency Generation 12

Electro-optic Crystals (Optical Rectification) 13

Optical Parametric Oscillators 15

Frequency Multipliers 16

Photoconductive Antennas 17

Photomixing 18

Optically Pumped Far-IR Gas Lasers 19

p-Type Germanium Lasers 20

Quantum Cascade Lasers 21

Gyrotrons 22

Synchrotrons 23

Free-Electron Lasers 24

Backward Wave Oscillators 25

Smith–Purcell Emitters 26

**THz Detectors** 27

THz Detection Techniques 27

Responsivity and Signal-to-Noise Ratio 29

Noise Equivalent Power 30

Shot and Thermal Noise 31

Extrinsic Semiconductor Detectors 32

Photoconductive Detectors 33

Photomixers 34

Schottky Diodes 35

Schottky Diode Mixers 37

SIS Mixers 38

Electro-optic Sampling 39

Semiconductor Bolometers 41

Microbolometer Arrays 43

Transition-Edge Sensors 44

SQUIDs 45

Hot-Electron Bolometers 47

Hot-Electron Bolometer Mixers 48

Pyroelectric Detectors 49

Golay Cells 50

**THz Optics** 51

Gaussian Beam Propagation 51

Complex Radius of Curvature 52

Gaussian Beam Parameters 53

Beamwidth 55

Edge Taper 56

Truncation and Spillover (Gaussian Beams) 57

Truncation and Spillover (Non-Gaussian Beams) 58

Confocal Distance 59

Far-Field Divergence 60

Ray Matrices and Gaussian Beam Transformation 61

Higher-Order Modes (Cylindrical Coordinates) 64

Higher-Order Modes (Rectangular Coordinates) 65

Mode Coefficients 66

Power Coupling Efficiency 69

Gaussicity 70

Mismatched Beams and Tolerancing 71

Scattering Matrix Formulism 72

Transmission Matrices 73

Linear Scattering Operators 74

Truncation at an Aperture 75

Perfect Lenses and Pure Propagation 76

Reflections at Dielectric Interfaces (Lens Surfaces) 77

Standing Waves in Horn-Fed Systems 78

Cascading Scattering and Transmission Matrices 79

Off-Axis Mirrors (Ellipsoidal) 80

Off-Axis Mirrors (Parabolic) 81

Off-Axis Mirrors (Distortion and Cross-Polarization) 82

Polarizing Grids 83

Roof Mirrors as Polarization Rotators 84

Dual-Beam Interferometers (Tunable Filters) 85

Diffraction Losses in Interferometers 86

Diplexers and Multiplexers 87

Four-Port Dual-Beam Interferometer (Diplexer) 88

Horn Antenna Feeds 89

Corrugated Conical Horns (Scalar Feed) 90

Smooth-Walled Horns (Pyramidal and Diagonal) 91

Conical Smooth-Walled Horns (Single and Dual Mode) 92

Shaped Horns and Multimode Feeds 93

Lens Antennas 94

System Design 95

Modeling Techniques 96

**THz Applications** 97

THz Imaging 97

THz Spectroscopy 99

THz Time-Domain Spectroscopy 100

**Equation Summary** 101

**Bibliography** 114

**Index** 117

#### Glossary of Symbols and Acronyms

**A**

absorbance

**A**

detector area

**A**

profiled horn parameter

**a**

aperture radius

**a**

horn aperture side length

**a**

semi-major axis of an ellipse

**a**

wire radius

${A}_{n},{A}_{n}^{h},{A}_{n}^{\upsilon}$

**n**^{th} mode coefficient of beam * a*,

**n**^{th}mode coefficient of orthogonal components

**a**_{n}

mode coefficients for beam * n* at the input port

ac

alternating current

**A**_{12}, **A**_{21}

scattering matrices for an absorbing stop

B

magnetic field

**B**

magnetic field strength

**b**

horn aperture side width

**b**

semi-minor axis of an ellipse

${B}_{n},{B}_{n}^{h},{B}_{n}^{\upsilon}$

**n**^{th} mode coefficient of beam * b*,

**n**^{th}mode coefficient of orthogonal components

**b**_{n}

mode coefficients for beam * n* at the output port

BWO

backward wave oscillator

**C**

capacitance

**C**

heat capacity

**c**

speed of light

**C**_{e} , **C**_{ph}

electron heat capacity, phonon heat capacity

c.c.

complex conjugate

CMOS

complementary metal-oxide- semiconductor

CW

continuous wave

**D**

detectivity

**d**

diameter

**d**

distance between the focal point and the apex of a lens antenna

**d**

propagation distance

**D**^{*}

specific detectivity

**D**_{12}, **D**_{21}

scattering matrices for pure propagation

dc

direct current

DFG

difference frequency generation

**E**, **E**_{a}

electric field

**E**(* t*),

**E**(

*)*

**r**electric field

* E*,

**E**_{0}

electric field amplitude

* E*(

*,*

**x***,*

**y***),*

**z***(*

**E****r**)

electric field amplitude

**e**

elementary charge

**E**_{a} , **E**_{b}

electric field amplitude

**E**_{ap}

electric field across an aperture

**e**_{F}

vector field

**e**_{G}

vector Gaussian field

**E**_{g} , **E**_{C} , **E**_{V} , **E**_{F}

energy gap, conduction band energy, valence band energy, Fermi energy

${\widehat{\mathbf{\text{e}}}}_{h},{\widehat{\mathbf{\text{e}}}}_{\upsilon}$

unit orthogonal vectors

**E**_{i}, **E**_{j}, **E**_{k}

electric field amplitude

**E**_{inc}, **E**_{refl}, **E**_{trans}

incident, reflected, transmitted electric field

**E**_{L}, **E**_{H}

light-, heavy-hole energy

**E**_{OP}

optical phonon energy

**E**_{out}

output electric field

**E**_{p, t}, **E**_{p, r}, **E**_{p, i}

transmitted, reflected, incident electric field amplitude for p-polarization

**E**_{s, t}, **E**_{s, r}, **E**_{s, i}

transmitted, reflected, incident electric field amplitude for s-polarization

**E**_{THz}

terahertz electric field amplitude

${E}_{p}^{+},{E}_{p}^{-}$

forward and reverse traveling **p**^{th} symmetric Laguerre mode

EO

electro-optic

**F**

Fresnel number

**f**

focal length

FEL

free-electron laser

FS

free space

FWHM

full width half maximum

**G**

thermal conductance

**g**

wire spacing

GBMA

Gaussian beam mode analysis

GBT

Gaussian beam telescope

**H**, **H**_{a}, **H**_{b}

magnetic field

**h**

Planck’s constant

**H**_{m} (* x*)

**m**^{th} Hermite polynomial

HEB

hot electron bolometer

HIFI

Heterodyne instrument for the far-infrared

HPBW

half-power beamwidth

**I**

current

**I**

intensity, transmitted intensity

**I**

identity matrix

**i**

$\sqrt{-1}$

**i**

angle of incidence

**I**_{c}

critical current

**I**_{x}, **I**_{y}

* x*,

*component of intensity*

**y****I**_{0}

incident intensity

IF

intermediate frequency

IMPATT

impact ionization avalanche transit time

IR

infrared

**J**

free current density

**J**

rotational state number

**J**

screening current

**K**

power coupling coefficient

**k**

wavenumber (2π/λ)

**K**_{ab}

power coupling between two beams

**k**_{B}

Boltzmann constant

**K**_{G}

power coupling to a pure Gaussian (Gaussicity)

**K**_{lost}

power lost due to truncation

**k**_{P}, **k**_{S}, **k**_{I}

pump wavevector, signal wavevector, idler wavevector

**K**_{spillover}

power lost due to spillover

**K**_{Xpol}

power scattered into the cross-polar direction

**L**

inductance

* L*,

**L**_{x},

**L**_{y}

slant length, slant length along * x* direction, slant length along

*direction*

**y****l**

grating period

**L**_{mn}

operator representing a transfer function

**L**_{nn}

coupling to reflected modes at port **n**

${L}_{p}^{m}$

generalized Laguerre polynomials

**l**_{w}

walk-off length

**l**_{c}

coherence length

**l**_{1}, **l**_{2}

path length

LO

local oscillator

LT-GaAs

low-temperature-grown gallium arsenide

**M**

magnification

**M**_{n}

**n**^{th} matrix

**m**_{e}

electron mass

N

multiplication factor of a frequency multiplier

**n**

number density

* n*,

**n**_{signal},

**n**_{noise}

number of photons, signal photons, noise photons detected

* n*,

**n**_{1},

**n**_{2}

refractive index

**n**

rms noise level

**n**

spectral order

**n**_{optical}, **n**_{THz}

optical, terahertz refractive index

NDR

negative differential resistance

NEP

noise equivalent power

NTD

neutron-transmutation doped

OP

optical phonon

OPO

optical parametric oscillator

OPTL

optically pumped THz laser

P, **P**_{i}, **P**_{0}, **P**

polarization

* P*(

*)*

**r**power contained within a radius **r**

**P**

vertex-focus distance of a parabola

**P**_{noise}, **P**_{signal}

noise power, signal power

**P**_{∞}

total power

PCA

photoconductive antenna

p-Ge

p-type germanium laser

**q**

charge

* q*(

*),*

**z**

**q**_{0}

complex radius of curvature (also known as complex beam parameter or Gaussian beam parameter)

**q**_{in}, **q**_{out}

complex radius of curvature of input, output beam

**q**_{1}, **q**_{2}

complex radius of curvature of beam 1, 2

QCL

quantum cascade laser

* R*(

*)*

**z**Gaussian beam phase radius of curvature

**R**

radius

**R**

reflectivity

**R**

resistance

**R**

surface radius of curvature

**r**

complex reflection amplitude coefficient

**R**_{in,} **R**_{out}

phase radius of curvature

**r**_{in}, **r**_{out}

ray displacement

**R**_{surf}

surface radius of curvature

**r**_{T}

truncation radius

**R**_{V}

responsivity

**R**_{1}, **R**_{2}

surface radius of curvature

**R**_{1}, **R**_{2}

distances to foci of an ellipse

RF

radio frequency

rms

root-mean-square

RTD

resonant tunnel diode

**S,S**^{a}, **S**^{b}, **S**^{c}

full scattering matrix

**s**

phase error

**S**_{11}, **S**_{12},

scattering matrices

**S**_{21}, **S**_{22}

SIS

superconductor–insulator–superconductor

SNR

signal-to-noise ratio

SNR_{dB}

signal-to-noise ratio in dBs

S/N

signal-to-noise ratio

SPR

Smith–Purcell radiation

SQUID

superconducting quantum interference device

**T**

temperature

**T**

transmittance

**T**

fractional power transmission

**t**

complex transmission amplitude coefficient

**T**, **T**^{a}, **T**^{b}, **T**^{c}

full transmission matrix

**T**_{b}, **T**_{s}

bath, heat sink temperature

**T**_{c}

critical temperature

**T**_{e}

edge taper

**T**_{e} (dB)

edge taper expressed in dB

**T**_{e}, **T**_{ph}

electron temperature, phonon temperature

**T**_{11}, **T**_{12},

transmission matrices

**T**_{21}, **T**_{22}

TCR

thermal coefficient of resistance

TE, TM

transverse electric, magnetic

TES

transition edge sensor

THz-TDS

terahertz time-domain spectroscopy

TUNNETT

tunnel injection transit time

* u*(

*,*

**x***,*

**y***),*

**z**

**u**_{0}

non-plane-wave part of the electric field, constant

* V*,

**V**_{rms}

voltage, root-mean-square voltage

**υ**

vibrational state number

**V**_{bi}

built-in potential

**V**_{gap}

gap voltage

* w*,

*(*

**w***)*

**z**Gaussian beam radius

**w**_{m}

beam radius at a mirror

**w**_{0}, **w**_{0,a}, **w**_{0, b}

Gaussian beam waist radius

**w**_{0,in}, **w**_{0,out}

Gaussian beam waist radius

WG

waveguide

**x**

path length

**x**_{c}

point of intersection, * x* coordinate

**y**_{c}

point of intersection, * y* coordinate

**z**_{c}

confocal distance, Rayleigh range

α

absorption coefficient

α

diffraction parameter

α

incident beam polarization angle

α, α(* T*)

thermal coefficient of resistance

α

tilt of grid wires

β

mode balance constant

β_{trans}, β_{refl}

transmitted, reflected beam polarization direction

χ

electron affinity (volts)

${\text{\chi}}^{\text{(2)}},{\text{\chi}}_{ijk}^{(2)}$

second-order susceptibility

Δ

path length difference

Δ_{IF}

IF bandwidth

Δ_{max}

path length difference for maximum transmission

Δ_{min}

path length difference for minimum transmission

Δ_{SSB}

single-sideband path length difference

Δ_{1}, Δ_{2}

energy gap of a semiconductor

Δ**E**

energy gap

Δ**x**

lateral shift

Δ**z**

relative displacement

Δ**z**

position of beam waist behind aperture

Δθ

cone angle

Δν

bandwidth

δν

frequency of successive transmission bands

ε_{0}

permittivity of free space

Φ_{0}

flux quantum

ϕ

grid inclination angle

ϕ_{c}

critical emission angle

ϕ_{m}, ϕ_{sc}, ϕ_{B}

metal, semiconductor, barrier work function

ϕ_{0}(* z*), ϕ

_{in}, ϕ

_{out}

Gaussian beam phase slippage

γ

Lorentz factor $(1/\sqrt{\left(1-{\text{\beta}}^{2}\right),}\text{\hspace{0.17em}}\text{\beta}=\upsilon /c),$ * υ* is velocity

γ

reflection coefficient at an interface

η_{spillover}

Gaussian beam spillover efficiency

λ

wavelength

λ_{FEL}

wavelength of radiation from an FEL

λ_{IF}, λ_{LO}, λ_{s}

IF, LO, signal wavelength

λ_{w}

wiggler magnet spacing wavelength

ν

frequency

ν_{IF}, ν_{LO}, ν_{s}

IF, LO, signal frequency

ν_{THz}, ν_{pump}

terahertz, pump frequency

θ

angle from normal

θ

angle with respect to propagation axis

θ

incident field polarization angle

θ

grid tilt angle

θ′

projected grid tilt angle

θ_{c}

critical angle

θ_{in}, θ_{out}

ray angle with respect to normal

θ_{0}

asymptotic beam growth angle

θ_{1}, θ_{2}

angle of incidence, reflection

σ

absorption cross-section

τ

transmission

τ_{diff}

diffusion time

τ_{ep}, τ_{pe}, τ_{es}

electron-phonon, phonon-electron, electron-substrate energy transfer time

τ_{p}

pulse length

ω, ω_{1}, ω_{2}, ω_{3}

angular frequency

Ψ_{n}

**n**^{th} Gaussian beam mode

Ψ_{mn} (* x*,

*,*

**y***)*

**z**Gaussian–Hermite beam mode amplitude (rectangular coordinates)

Ψ_{pm} (* r*, ϕ,

*)*

**z**Gaussian–Laguerre beam mode amplitude (cylindrical coordinates)

${\text{\Psi}}_{n}^{SC}$

**n**^{th} scattered Gaussian beam mode

ψ_{a}, ψ_{b}, ψ_{F}

scalar field

ψ_{G}

scalar Gaussian field