Waves and Antennas

Sophocles J. Orfanidis

ECE Department
Rutgers University
94 Brett Road
Piscataway, NJ 08854-8058

Tel: 848-445-5017
e-mail: orfanidi@rci.rutgers.edu

Any feedback from readers is welcome.

This book provides a broad and applications-oriented introduction to electromagnetic waves and antennas, with MATLAB examples. Current interest in these areas is driven by the growth in wireless and fiber-optic communications, information technology, and materials science.

Communications, antenna, radar, and microwave engineers must deal with the generation, transmission, and reception of electromagnetic waves. Device engineers working on ever-smaller integrated circuits and at ever higher frequencies must take into account wave propagation effects at the chip and circuit-board levels. Communication and computer network engineers routinely use waveguiding systems, such as transmission lines and optical fibers. Novel recent developments in materials, such as photonic bandgap structures, omnidirectional dielectric mirrors, birefringent multilayer films, surface plasmons, negative-index metamaterials, slow and fast light, promise a revolution in the control and manipulation of light and other applications. These are just some examples of topics discussed in this book.

The book is organized around three main topic areas:

The text emphasizes connections to other subjects. For example, the mathematical techniques for analyzing wave propagation in multilayer structures, multisegment transmission lines, and the design of multilayer optical filters are the same as those used in DSP, such as the lattice structures of linear prediction, the analysis and synthesis of speech, and geophysical signal processing. Similarly, antenna array design is related to the problem of spectral analysis of sinusoids and to digital filter design, and Butler beams are equivalent to the FFT.

Initially posted online in November 2002. Latest revision date - August 1, 2016.

Please note that the book is now completed and, except for corrections, the August 1, 2016 revision will be the last one.

The entire book is freely available in PDF 2-up format, and in PDF 1-up format. The MATLAB toolbox is available here . The book is also available in printed form. Individual chapters are available below in PDF in 2-up format.

An errata file (updated December 10, 2020) is also available here. Many thanks to all who have sent me errata and, especially, to Lee Byeong-Yoon who found most of them.

Front Matter and Preface
Table of Contents

Ch.1: Maxwell's Equations
Review of Maxwell's equations, Lorentz force, constitutive relations, boundary conditions, charge and energy conservation, Poynting's theorem, simple models of dielectrics, conductors, and plasmas, relaxation time in conductors.

Ch.2: Uniform Plane Waves
Uniform plane waves in lossless media, monochromatic waves, wave impedance, polarization, waves in lossy media, waves in weakly lossy dielectrics, propagation in good conductors, propagation in oblique directions, complex waves, propagation in negative-index media, Doppler effect.

Ch.3: Pulse Propagation in Dispersive Media
Propagation filter, front velocity and causality, exact medium response examples, transient and steady-state behavior, pulse propagation and group velocity, group velocity dispersion and pulse spreading, propagation and chirping, dispersion compensation, slow, fast, and negative group velocities, chirp radar and pulse compression.

Ch.4: Propagation in Birefringent Media
Linear and circular birefringence, uniaxial and biaxial media, chiral media, natural vs. Faraday rotation, gyrotropic media, linear and circular dichroism, oblique propagation in birefringent media.

Ch.5: Reflection and Transmission
Reflection and transmission at normal incidence, propagation and matching matrices, reflected and transmitted power, single and double dielectric slabs, reflectionless slab, time-domain reflection response, lattice diagrams, reflection by a moving boundary, such as a moving mirror.

Ch.6: Multilayer Structures
Multiple dielectric slabs at normal incidence, antireflection coatings, dielectric mirrors, propagation bandgaps, narrow-band transmission filters, quarter-wave phase-shifted Fabry-Perot resonators, fiber Bragg gratings, equal travel-time multilayer structures, applications of layered structures, Chebyshev design of reflectionless multilayers.

Ch.7: Oblique Incidence
Oblique incidence and Snel's laws, transverse impedance, propagation and matching of transverse fields, Fresnel reflection coefficients, total internal reflection, Brewster angle, complex waves, lossy media, Zenneck surface wave, surface plasmons, oblique reflection by a moving interface, geometrical optics, Fermat's principle of least time, ray tracing techniques in geometrical optics illustrated by several exactly solvable examples drawn from several applications, such as atmospheric refraction, mirages, ionospheric refraction, propagation in a standard atmosphere and the effect of Earth's curvature, and propagation in graded-index optical fibers, Snel's law in negative-index media.

Ch.8: Multilayer Film Applications
Multilayer dielectric structures at oblique incidence, lossy multilayers, frustrated total internal reflection, surface plasmon resonance, perfect lenses in negative-index media, antireflection coatings at oblique incidence, omnidirectional dielectric mirrors, polarizing beam splitters, reflection and refraction in birefringent media, Brewster and critical angles in birefringent media, multilayer birefringent structures, giant birefringent optics.

Ch.9: Waveguides
Longitudinal-transverse decompositions of Maxwell's equations, power transfer and attenuation in guiding systems, TEM, TE, TM modes, rectangular waveguides, higher TE and TM modes, operating bandwidth, power transfer, energy density, and group velocity in waveguides, power attenuation, reflection model of waveguide propagation, dielectric slab guides.

Ch.10: Surface Waveguides
Plasmonic waveguides, plasmonic and oscillatory modes, MDM and DMD configurations, gap surface plasmons, anomalous complex modes, Sommerfeld wires for THz applications, skin effect for round wire, Goubau lines and their attenuation properties.

Ch.11: Transmission Lines
General properties of TEM transmission lines, parallel-plate, microstrip, coaxial, and two-wire lines, distributed circuit model of a transmission line, wave impedance and reflection response, two-port equivalent circuits, terminated lines, power transfer from generator to load, open- and short-circuited lines, Thevenin and Norton equivalent circuits, standing wave ratio, determination of unknown load impedance, Smith chart. Transient Response.

Ch.12: Coupled Lines
Coupled transmission lines, even-odd mode decomposition for identical matched or unmatched lines, crosstalk between lines, weakly coupled lines with arbitrary terminations, coupled-mode theory, co-directional couplers, fiber Bragg gratings as examples of contra-directional couplers, quarter-wave phase-shifted fiber Bragg gratings as narrow-band transmission filters, and the Schuster-Kubelka-Munk theory of diffuse reflection and transmission as an example of contra-directional coupling.

Ch.13: Impedance Matching
Conjugate and reflectionless matching, multisection transmission lines, quarter-wavelength impedance transformers, two-section dual-band Chebyshev transformers, quarter-wavelength transformers with series sections and shunt stubs, two-section series impedance transformers, single-stub matching, balanced stubs, double- and triple-stub matching, L-, T-, and Pi-section lumped reactive matching networks and their Q-factors.

Ch.14: S-Parameters
Scattering parameters, power flow, parameter conversions, input and output reflection coefficients, stability circles, transducer, operating, and available power gains, generalized S-parameters and power waves, simultaneous conjugate matching, power gain circles, unilateral gain circles, operating and available power gain circles, noise figure circles, design examples of low-noise high-gain microwave amplifiers and their microstrip matching circuits.

Ch.15: Radiation Fields
Currents and charges as sources of fields, retarded potentials, fields of a linear wire antenna, near and far fields of electric and magnetic dipoles, Ewald-Oseen extinction theorem of molecular optics, radiation fields, radiation field approximation, computing the radiation fields, radiation vector.

Ch.16: Transmitting and Receiving Antennas
Energy flux and radiation intensity from a radiating system, directivity, gain, and beamwidth of an antenna, effective area, gain-beamwidth product, antenna equivalent circuits, effective length and polarization and load mismatches, communicating antennas, Friis formula, antenna noise temperature, system noise temperature, limits on bit rates, satellite links, radar equation.

Ch.17: Linear and Loop Antennas
Linear antennas, Hertzian dipole, standing-wave antennas, half-wave dipole, monopole antennas, traveling wave antennas, vee and rhombic antennas, loop antennas, circular and square loops, dipole and quadrupole radiation.

Ch.18: Radiation from Apertures
Radiation from apertures and diffraction. Field equivalence principle, magnetic currents and duality, radiation fields from magnetic currents, radiation fields from apertures, Kottler's formula, Huygens sources, directivity and effective area of apertures, uniform, rectangular, and circular apertures and their gain-beamwidth products, Rayleigh diffraction limit, vector diffraction theory, Stratton-Chu, Kottler, Franz, and Kirchhoff diffraction integral formulas, extinction theorem, vector diffraction from apertures, Fresnel diffraction, Knife-edge diffraction, Fresnel zones, geometrical theory of diffraction and Sommerfeld's solution for a conducting half-plane.

Ch.19: Diffraction -- Plane-Wave Spectrum
Plane-wave spectrum point of view of diffraction and its equivalence to the Rayleigh-Sommerfeld diffraction theory both for scalar and vector fields, including Smythe diffraction integrals, apertures in conducting screens, Bethe-Bouwkamp theory of diffraction by small holes, and the Babinet principle for scalar and vector electromagnetic fields.

Ch.20: Diffraction -- Fourier Optics
Fourier optics concepts, Fresnel approximation, Talbot effect, Fourier transformation properties of lenses, one- and two-dimensional apodizer design and aperture synthesis for narrow beamwidths and low sidelobes including Fourier-Bessel and Dini series expansions, realization of apodizers using star-shaped masks, coronagraphs and starshade occulters, superresolving apertures, overview of superdirectivity, superresolution, and superoscillation concepts based on prolate spheroidal wavefunctions.

Ch.21: Aperture Antennas
Open-ended waveguides, horn antennas, horn radiation fields, horn directivity, optimum horn design, microstrip antennas, parabolic reflector antennas, gain and beamwidth of reflector antennas, aperture-field and current-distribution methods, radiation patterns of reflector antennas, dual-reflector antennas, lens antennas.

Ch.22: Antenna Arrays
Antenna arrays and translational phase shift, array pattern multiplication, one-dimensional arrays, visible region, grating lobes, uniform arrays, array directivity, steering, and beamwidth.

Ch.23: Array Design Methods
Schelkunoff's zero-placement method, Fourier series design method with windowing, sector beam array design, Woodward-Lawson frequency-sampling design, discretization of continuous line sources, narrow-beam low-sidelobe designs, binomial arrays, Dolph-Chebyshev arrays, Taylor one-parameter source, prolate arrays, Taylor n-bar distribution, Villeneuve arrays, multi-beam arrays, emphasis on the connections to DSP methods of digital filter design and spectral analysis of sinusoids.

Ch.24: Currents on Linear Antennas
Hallen and Pocklington integral equations, delta-gap, frill generators, and plane-wave sources, solving Hallen's equation, sinusoidal current approximation, reflecting and center-loaded receiving antennas, King's three-term approximation, evaluation of the exact kernel using elliptic functions, method of moments, pulse, triangular, NEC, and delta-function bases, Hallen's equation for arbitrary incident field, solving Pocklington's equation.

Ch.25: Coupled Antennas
Near fields of linear antennas, improved near-field calculation, self and mutual impedance, coupled two-element arrays, arrays of parallel dipoles, Yagi-Uda antennas, Hallen equations for coupled antennas.

Ch.26: Appendices
Physical constants, electromagnetic frequency bands, vector identities, integral theorems, Green's functions, coordinate systems, Fresnel integrals, sine and cosine integrals, stationary-phase approximation, Gauss-Legendre quadrature, tanh-sinh double-exponential quadrature, prolate spheroidal wavefunctions including MATLAB functions for their computation, Lorentz transformations, and a detailed list of the book's MATLAB functions.


Copyright Notice

Copyright (c) 1996-2016 by Sophocles J. Orfanidis, All Rights Reserved. The book exists in online form through the web page www.ece.rutgers.edu/~orfanidi/ewa. Links to this page may be placed on any web site. The book may cited as follows:

S. J. Orfanidis, "Electromagnetic Waves and Antennas", 2016 [Online]. Available: https://www.ece.rutgers.edu/~orfanidi/ewa/.

Any part of this book may be downloaded and printed for personal or educational use only, as long as the printed or photocopied pages are not altered in any way from the original PDF files posted on the book's web page.

No part of this book may be reproduced, altered in any way, or transmitted in any form for commercial, profit, sale, or marketing purposes.

MATLAB (R) is a registered trademark of The MathWorks, Inc.

Solutions Manual

I am grateful to Mr. Davide Ramaccia, student of Prof. Alessandro Toscano, Department of Applied Electronics, University "Roma Tre", Rome, Italy, for producing very detailed solutions of the problems of Chapters 1 & 2 , and for allowing me to post them online. Solutions to the rest of the problems are not yet available.


Google Scholar citations.


The book is also available in print-on-demand form. Because of printer limitations, the book has been evenly split into two volumes, each with its own index, but with the list of references and appendices included only in volume two. The following versions are available:

MATLAB Toolbox

The text makes extensive use of MATLAB. We have developed an "Electromagnetic Waves & Antennas" toolbox containing about 200 MATLAB functions for carrying out all of the computations and simulation examples in the text. Code segments illustrating the usage of these functions are found throughout the book, and serve as a user manual. Our MATLAB-based numerical solutions are not meant to replace sophisticated commercial field solvers. The inclusion of numerical methods was motivated by the desire to provide the reader with some simple tools for self-study and experimentation. We felt that it would be useful and fun to be able to quickly carry out the computations illustrating various waves and antenna applications, and have included enough MATLAB code in each example (but skipping all figure annotations) that would enable the reader to reproduce the results. The functions may be grouped into the following categories:

Please read the license agreement before using the toolbox. The toolbox was developed under MATLAB v5.3, but runs also under v7.0 and v2013b. The zipped file ewa.zip (last revised on April 14, 2019) contains all the MATLAB functions. It should be uncompressed in a directory, say c:\antennas\ewa. To add this directory to the MATLAB path and to get initial help, use the following commands:
addpath c:\antennas\ewa;
help ewa;
Note that just typing the name of any function will produce a help/usage comment for that function. For gain plots that will eventually be exported into EPS and inserted in LaTeX files, we found it best to add the following lines to the MATLAB startup.m file:
set(0,'DefaultAxesLineWidth', 1);
set(0,'DefaultLineLineWidth', 1.5);
The detailed list of the MATLAB functions is as follows:
Multilayer Dielectric Structures
brewster - calculates Brewster and critical angles
fresnel - Fresnel reflection coefficients for isotropic or birefringent media
n2r - refractive indices to reflection coefficients of M-layer structure
r2n - reflection coefficients to refractive indices of M-layer structure
multidiel - reflection response of isotropic or birefringent multilayer structures
multidiel1 - simplified version of multidiel for isotropic layers
multidiel2 - reflection response of lossy isotropic multilayer dielectric structures
omniband - bandwidth of omnidirectional mirrors and Brewster polarizers
omniband2 - bandwidth of birefringent multilayer mirrors
snel - calculates refraction angles from Snel's law for birefringent media

Quarter-Wavelength Transformers
bkwrec - order-decreasing backward layer recursion - from a,b to r
frwrec - order-increasing forward layer recursion - from r to A,B
chebtr - Chebyshev design of broadband reflectionless quarter-wave transformer
chebtr2 - Chebyshev design of broadband reflectionless quarter-wave transformer
chebtr3 - Chebyshev design of broadband reflectionless quarter-wave transformer

Dielectric Waveguides
dguide - TE modes in dielectric slab waveguide
dslab - solves for the TE-mode cutoff wavenumbers in a dielectric slab
dguide3 - TE and TM modes in asymmetric 3-slab dielectric waveguide

Plasmonic Waveguides
drude - Drude-Lorentz model for Silver, Gold, Copper, Aluminum
dmda - asymmetric DMD plasmonic waveguide - iterative solution
dmds - symmetric DMD plasmonic waveguide - iterative solution
dmdcut - cutoff width for asymmetric DMD guides
pwg - plasmonic waveguide solution for symmetric guides
pwga - plasmonic waveguide solution for asymmetric guides
pwgpower - transmitted power in plasmonic waveguide

Sommerfeld and Goubau Wires
sommer - solve characteristic equation for Sommerfeld wire
goubau - solve characteristic equation of Goubau line
goubatt - Goubau line attenuation
gcut - cutoff function for Goubau line
attw - characteristic equation of Attwood surface waveguide
attwatt - attenuation of Attwood surface waveguide
J01 - J0(z)/J1(z) approximation for large imag(z)

Transmission Lines
g2z - reflection coefficient to impedance transformation
z2g - impedance to reflection coefficient transformation
lmin - find locations of voltage minima and maxima
mstripa - microstrip analysis (calculates Z,eff from w/h)
mstripr - microstrip synthesis with refinement (calculates w/h from Z)
mstrips - microstrip synthesis (calculates w/h from Z)
multiline - reflection response of multi-segment transmission line
swr - standing wave ratio
tsection - T-section equivalent of a length-l transmission line segment
gprop - reflection coefficient propagation
vprop - voltage and current propagation
zprop - wave impedance propagation

Impedance Matching
qwt1 - quarter wavelength transformer with series segment
qwt2 - quarter wavelength transformer with 1/8-wavelength shunt stub
qwt3 - quarter wavelength transformer with shunt stub of adjustable length
dualband - two-section dual-band Chebyshev transformer
dualbw - bandwidth of dual-band transformer
stub1 - single-stub matching
stub2 - double-stub matching
stub3 - triple-stub matching
onesect - one-section impedance transformer
twosect - two-section impedance transformer
pi2t - Pi to T transformation
t2pi - T to Pi transformation
lmatch - L-section reactive conjugate matching network
pmatch - Pi-section reactive conjugate matching network

gin - input reflection coefficient in terms of S-parameters
gout - output reflection coefficient in terms of S-parameters
nfcirc - constant noise figure circle
nfig - noise figure of two-port
sgain - transducer, available, and operating power gains of two-port
sgcirc - stability and gain circles
smat - S-parameters to S-matrix
smatch - simultaneous conjugate match of a two-port
smith - draw basic Smith chart
smithcir - add stability and constant gain circles on Smith chart
sparam - stability parameters of two-port
circint - circle intersection on Gamma-plane
circtan - point of tangency between the two circles

Linear Antenna Functions
dipdir - dipole directivity
dmax - computes directivity and beam solid angle of g(th) gain
dipole - gain of center-fed linear dipole of length L
traveling - gain of traveling-wave antenna of length L
vee - gain of traveling-wave vee antenna
rhombic - gain of traveling-wave rhombic antenna

king - King's 3-term sinusoidal approximation
kingeval - evaluate King's 3-term sinusoidal current approximation
kingfit - fits a sampled current to King's 2-term sinusoidal approximation
kingprime - converts King's 3-term coefficients from unprimed to primed form

hbasis - basis functions for Hallen equation
hdelta - solve Hallen's equation with delta-gap input
hfield - solve Hallen's equation with arbitrary incident E-field
hmat - Hallen impedance matrix with method of moments and point-matching
hwrap - wraps a Toeplitz impedance matrix to half its size
kernel - thin-wire kernel computation for Hallen equation
pfield - solve Pocklington's equation with arbitrary incident E-field
pmat - Pocklington impedance matrix with method of moments and point-matching

hcoupled - solve Hallen's equation for 2D array of non-identical parallel dipoles
hcoupled2 - solve Hallen's equation for 2D array of identical parallel dipoles

gain2d - normalized gain of 2D array of parallel dipoles with Hallen currents
gain2s - normalized gain of 2D array of parallel dipoles with sinusoidal currents
imped - mutual impedance between two parallel standing-wave dipoles
imped2 - mutual impedance between two parallel standing-wave dipoles
impedmat - mutual impedance matrix of array of parallel dipole antennas
resonant - calculates the length of a resonant dipole antenna
yagi - simplified Yagi-Uda array design

Aperture Antenna Functions
BBnum - computation of fields in Bethe-Bouwkamp model
BBfar - far fields in Bethe-Bouwkamp model
BBnear - near fields in Bethe-Bouwkamp model

diffint - generalized Fresnel diffraction integral
diffr - knife-edge diffraction coefficient
dsinc - the double-sinc function cos(pi*x)/(1-4*x^2)
fcs - Fresnel integrals C(x) and S(x)
fcs2 - type-2 Fresnel integrals C2(x) and S2(x)

jinc - jinc and "shifted" jinc function

talbot - Gauss sums for fractional Talbot effect

tbw - Taylor's one-parameter window
tnb1 - Taylor's n-bar window (1-D)
tnb2 - Taylor's n-bar window (2-D)

hband - horn antenna 3-dB width
heff - aperture efficiency of horn antenna
hgain - horn antenna H-plane and E-plane gains
hopt - optimum horn antenna design
hsigma - optimum sigma parametes for horn antenna

Antenna Array Functions
gain1d - normalized gain computation for 1D equally-spaced isotropic array

bwidth - beamwidth mapping from psi-space to phi-space
binomial - binomial array weights
dolph - Dolph-Chebyshev array weights
dolph2 - Riblet-Pritchard version of Dolph-Chebyshev
dolph3 - DuHamel version of endfire Dolph-Chebyshev
multibeam - multibeam array design
prol - prolate array design
prolmat - prolate matrix
scan - scan array with given scanning phase
sector - sector beam array design
steer - steer array towards given angle
taylornb - Taylor n-bar line source array design
taylor1p - Taylor 1-parameter array design
taylorbw - Taylor B-parameter and beamwidth
uniform - uniform array weights
woodward - Woodward-Lawson-Butler beams
ville - Villeneuve array design

chebarray - Bresler's Chebyshev array design method - written by P. Simon
I would like to thank Dr. Simon for premission to include
this function in this collection.

Gain Plotting Functions
abp - polar gain plot in absolute units
abz - azimuthal gain plot in absolute units
abp2 - polar gain plot in absolute units - 2*pi angle range
abz2 - azimuthal gain plot in absolute units - 2pi angle range

dbp - polar gain plot in dB
dbz - azimuthal gain plot in dB
dbp2 - polar gain plot in dB - 2*pi angle range
dbz2 - azimuthal gain plot in dB - 2pi angle range

abadd - add gain in absolute units
abadd2 - add gain in absolute units - 2pi angle range
dbadd - add gain in dB
dbadd2 - add gain in dB - 2pi angle range
addbwp - add 3-dB angle beamwidth in polar plots
addbwz - add 3-dB angle beamwidth in azimuthal plots
addcirc - add grid circle in polar or azimuthal plots
addline - add grid ray line in azimuthal or polar plots
addray - add ray in azimuthal or polar plots

Miscellaneous Utility Functions
ab - dB to absolute power units
db - absolute power to dB units

c2p - complex number to phasor form
p2c - phasor form to complex number

d2r - degrees to radians
r2d - radians to degrees

dtft - DTFT of a signal x at a frequency vector w
I0 - modified Bessel function of 1st kind and 0th order
ellipse - polarization ellipse parameters
etac - eta and c
wavenum - calculate wavenumber and characteristic impedance
poly2 - specialized version of poly with increased accuracy

quadr - Gauss-Legendre quadrature weights and evaluation points
quadrs - quadrature weights and evaluation points on subintervals
quadr2 - Gauss-Legendre quadrature weights and evaluation points
quadrs2 - quadrature weights and evaluation points on subintervals
quadts - tanh-sinh, double-exponential, quadrature

Ci - cosine integral Ci(z)
Cin - cosine integral Cin(z)
Gi - Green's function integral
Si - sine integral Si(z)

sinhc - hyperbolic sinc function
asinhc - inverse hyperbolic sinc function
sqrte - evanescent SQRT for waves problems

pswf - prolate spheroidal wave functions
spherj - spherical Bessel functions
legpol - evaluate Legendre polynomials

flip - flip a column, a row, or both
blockmat - manipulate block matrices
upulse - generates trapezoidal, rectangular, triangular pulses, or a unit-step
ustep - unit-step or rising unit-step function

Elliptic Function Computation
snv - sn elliptic function at a vector of moduli
dnv - dn elliptic function at a vector of moduli
ellipK - complete elliptic integral of first kind at a vector of moduli
ellipE - complete elliptic integral of second kind at a vector of moduli
landenv - Landen transformations of a vector of elliptic moduli

MATLAB Movies (in subdirectory ewa/movies)
grvmovie1 - pulse propagation with slow and negative group velocity (vg<0)
grvmovie2 - pulse propagation with slow and fast group velocity (vg>c)
pulsemovie - step and pulse propagation on terminated transmission lines
pulse2movie - step propagation on two cascaded lines
RLCmovie - step getting reflected off a reactive termination
TDRmovie - fault location by time-domain reflectometry
xtalkmovie - crosstalk signals on coupled transmission lines
dipmovie - electric field pattern of radiating Hertzian dipole