Electromagnetic Waves and Antennas combined - Chapter 23 potx

Multilayer Dielectric Structures brewster - calculates Brewster and critical anglesfresnel - Fresnel reflection coefficients for isotropic or birefringent median2r - refractive indices t

23 Appendices

A Physical Constants

We use SI units throughout this text Simple ways to convert between SI and other

popular units, such as Gaussian, may be found in Refs [123–126].

The Committee on Data for Science and Technology (CODATA) of NIST maintains

the values of many physical constants [112] The most current values can be obtained

from the CODATA web site [1330] Some commonly used constants are listed below:

speed of light in vacuum c0, c 299 792 458 m s−1

In the table, the constants c, μ0are taken to be exact, whereas 0, η0 are derived

from the relationships:

in moving across a voltage of one volt, that is, 1 eV = 1 602 176 462 × 10−19C · 1 V, or

1 eV = 1 602 176 462 × 10−19J

In units of eV/Hz, Planck’s constant h is:

h = 4 135 667 27 × 10−15eV / Hz = 1 eV / 241 8 THz that is, 1 eV corresponds to a frequency of 241.8 THz, or a wavelength of 1.24 μ m.

B Electromagnetic Frequency Bands

The ITU†divides the radio frequency (RF) spectrum into the following frequency and wavelength bands in the range from 30 Hz to 3000 GHz:

RF Spectrum

ELF Extremely Low Frequency 30–300 Hz 1–10 Mm

VF Voice Frequency 300–3000 Hz 100–1000 km VLF Very Low Frequency 3–30 kHz 10–100 km

An alternative subdivision of the low-frequency bands is to designate the bands 3–30 Hz, 30–300 Hz, and 300–3000 Hz as extremely low frequency (ELF), super low frequency (SLF), and ultra low frequency (ULF), respectively.

Microwaves span the 300 MHz–300 GHz quency range Typical microwave and satellite com- munication systems and radar use the 1–30 GHz band The 30–300 GHz EHF band is also referred to

fre-as the millimeter band.

The 1–100 GHz range is subdivided further into the subbands shown on the right.

Cell phones, personal communication systems (PCS), pagers, cordless phones, global positioning systems (GPS), RF identification systems (RFID), UHF-TV channels, microwave ovens, and long-range surveillance radar fall within the UHF band.

†International Telecommunication Union.

B Electromagnetic Frequency Bands 951

The SHF microwave band is used in radar (traffic control, surveillance, tracking,

mis-sile guidance, mapping, weather), satellite communications, direct-broadcast satellite

(DBS), and microwave relay systems Multipoint multichannel (MMDS) and local

multi-point (LMDS) distribution services, fall within UHF and SHF at 2.5 GHz and 30 GHz.

Industrial, scientific, and medical (ISM) bands are within the UHF and low SHF, at 900

MHz, 2.4 GHz, and 5.8 GHz Radio astronomy occupies several bands, from UHF to L–W

microwave bands.

Beyond RF, come the infrared (IR), visible, ultraviolet (UV), X-ray, and γ -ray bands.

The IR range extends over 3–300 THz, or 1–100 μ m Many IR applications fall in the

1–20 μ m band For example, optical fiber communications typically use laser light at

1.55 μ m or 193 THz because of the low fiber losses at that frequency The UV range lies

beyond the visible band, extending typically over 10–400 nm.

infrared 100–1 μ m 3–300 THz

ultraviolet 400–10 nm 750 THz–30 PHz

X-Ray 10 nm–100 pm 30 PHz–3 EHz 0.124–124 keV

γ -ray < 100 pm > 3 EHz > 124 keV

The CIE†defines the visible spectrum to be the wavelength range 380–780 nm, or

385–789 THz Colors fall within the following typical wavelength/frequency ranges:

Visible Spectrum

red 780–620 nm 385–484 THz orange 620–600 nm 484–500 THz yellow 600–580 nm 500–517 THz green 580–490 nm 517–612 THz blue 490–450 nm 612–667 THz violet 450–380 nm 667–789 THz X-ray frequencies fall in the PHz (petahertz) range and γ -ray frequencies in the EHz

(exahertz) range.‡X-rays and γ -rays are best described in terms of their energy, which is

related to frequency through Planck’s relationship, E = hf X-rays have typical energies

of the order of keV, and γ -rays, of the order of MeV and beyond By comparison, photons

in the visible spectrum have energies of a couple of eV.

The earth’s atmosphere is mostly opaque to electromagnetic radiation, except for

three significant “windows”, the visible, the infrared, and the radio windows These

three bands span the wavelength ranges of 380-780 nm, 1-12 μ m, and 5 mm–20 m,

respectively.

Within the 1-10 μ m infrared band there are some narrow transparent windows For

the rest of the IR range (1–1000 μ m), water and carbon dioxide molecules absorb infrared

radiation—this is responsible for the Greenhouse effect There are also some minor

transparent windows for 17–40 and 330–370 μ m.

†Commission Internationale de l’Eclairage (International Commission on Illumination.)

A × ( B × C ) = B ( A · C )− C ( A · B ) (BAC-CAB rule) (C.3) ( A × B ) ·( C × D ) = ( A · C )( B · D ) −( A · D )( B · C ) (C.4) ( A × B ) ×( C × D ) =  ( A × B ) · D 

C −  ( A × B ) · C 

A = n ˆ × ( A × n ˆ ) +( n ˆ · A ) ˆ n = A⊥+ A (C.6) where ˆ n is any unit vector, and A⊥, A are the components of A perpendicular and

parallel to ˆ n Note also that ˆ n × ( A × ˆ n ) = ( n ˆ × A ) × n A three-dimensional vector can ˆ equally well be represented as a column vector:

I = ˆ nˆ nT− N ˆ2, where ˆ n =

⎡

⎢ n n ˆ ˆx

yˆ

This corresponds to the matrix form of the parallel/transverse decomposition (C.6).

Indeed, we have a= ˆ n ( ˆ nTa ) and a⊥= ( ˆ n × a ) × ˆ n = − ˆ n × ( ˆ n × a ) = − N( ˆ N ˆ a ) = − N ˆ2a

Therefore, a = I a = ( nˆ ˆ nT− N ˆ2) a = a+ a⊥.

C Vector Identities and Integral Theorems 953

With r = x ˆ x + y ˆ y + z ˆ z, r = | r | = x2+ y2+ z2, and the unit vector ˆ r = r /r , we have:

Integral Theorems for Closed Surfaces

The theorems involve a volume V surrounded by a closed surface S The divergence or

Gauss’ theorem is:

n · ( A × ∇ ∇ ∇ × B − B × ∇ ∇ ∇ × A ) dS (C.35)

Integral Theorems for Open Surfaces

Stokes’ theorem involves an open surface S and its boundary contour C :



Sˆ

n · ∇ ∇ ∇ × A dS =



C

A · d l (Stokes’ theorem) (C.36) where d l is the tangential path length around C Some related theorems are:



C( ∇ ∇ψ) A · d l (C.38)

 ˆ

n × ∇ ∇ ∇ψ dS =



n dS = 1 2

The Green’s functions for the Laplace, Helmholtz, and one-dimensional Helmholtz

equa-tions are listed below:

where r = | r | Eqs (D.2) and (D.3) are appropriate for describing outgoing waves We

considered other versions of (D.3) in Sec 21.3 A more general identity satisfied by the

Green’s function g( r ) of Eq (D.1) is as follows (for a proof, see Refs [134,135]):

∂i∂jg( r )= − 1

3 δijδ(3)( r )+ 3 xixj− r2δij

r4 g( r ) i, j = 1 , 2 , 3 (D.4) where ∂i= ∂/∂xiand xistands for any of x, y, z By summing the i, j indices, Eq (D.4)

reduces to (D.1) Using this identity, we find for the Green’s function G( r ) = e−jkr/ 4 πr :

∂i∂jG( r )= − 1

3 δijδ(3)( r )+



jk + 1 r

3 xixj− r2δij

r3 − k2xixj

r2

 G( r ) (D.5)

This reduces to Eq (D.2) upon summing the indices For any fixed vector p, Eq (D.5)

is equivalent to the vectorial identity:

∇

∇ × ∇ ∇ ∇ ×  p G( r ) 

= 2 3 p δ(3)( r ) + 

jk + 1 r

3ˆ r ( ˆ r · p )− p

r2 + k2ˆ r × ( p × ˆ r )

 G( r ) (D.6) The second term on the right is simply the left-hand side evaluated at points away

from the origin, thus, we may write:

δ → 0 of the integrals over V −Vδ( r ) , where Vδ( r ) is an excluded small sphere of radius

δ centered about r The 2P ( r )/ 3 term has a different form if the excluded volume Vδ( r ) has shape other than a sphere or a cube See Refs [1179,483,495,621] and [129–133] for the definitions and properties of such principal value integrals.

Another useful result is the so-called Weyl representation or plane-wave-spectrum representation [22,26,1179,27,538] of the outgoing Helmholtz Green’s function G( r ) :

imag-To prove (D.9), we consider the two-dimensional spatial Fourier transform of G( r ) and its inverse Indicating explicitly the dependence on the coordinates x, y, z , we have:

whose outgoing/evanescent solution is g(kx, ky, z) = e−jkz|z|/ 2 jkz.

A more direct proof of (D.9) is to use cylindrical coordinates, kx= k⊥cos ψ , ky =

k sin ψ , x = ρ cos φ , y = ρ sin φ , where k2

⊥= k2

x+ k2

yand ρ2= x2+ y2 It follows that

D Green’s Functions 957

kxx + kyy = k⊥ρ cos (φ − ψ) Setting dx dy = ρ dρ dφ = r dr dφ , the latter following

from r2= ρ2+ z2, we obtain from Eq (D.11) after replacing ρ = √ r2− z2:

k r2− z2 where we used the integral representation (17.9.2) of the Bessel function J0(x) Looking

up the last integral in the table of integrals [1299], we find:

where kz must be defined exactly as in Eq (D.10) A direct consequence of Eq (D.11)

and the even-ness of G( r ) in r and of g(kx, ky, z) in kx, ky, is the following result:

The proof is obtained by splitting the integral over the sub-intervals [ 0 , z] and

[z, ∞) To handle the limits at infinity, kzmust be assumed to be slightly lossy, that is,

kz= βz− jαz, with αz> 0 Eqs (D.14) and (D.15) can be combined into:

Oseen extinction theorem in Sec 14.6.

A related Weyl-type representation is obtained by differentiating Eq (D.9) with spect to z Assuming that z ≥ 0 and interchanging differentiation and integration (and multiplying by − 2), we obtain the identity:

 1 sin θ

∂Ar

∂φ − ∂(rAφ)

∂r

 (E.3d)

+ φ ˆ 1 r

Transformations Between Coordinate Systems

A vector A can be expressed component-wise in the three coordinate systems as:

A = ˆ x Ax+ ˆ y Ay+ ˆ z Az

= ρ ˆ ρ Aρ+ φ φ A ˆ φ+ ˆ z Az

= ˆ r Ar+ θ θ A ˆ θ+ φ φ A ˆ φ

(E.4)

The components in one coordinate system can be expressed in terms of the

compo-nents of another by using the following relationships between the unit vectors, which

φ = − ˆ x sin φ + y cos ˆ φ

ˆ

x = ρ ˆ cos φ − φ ˆ sin φ ˆ

y = ρ ˆ sin φ + φ ˆ cos φ (E.5)

ρ = r sin θ

z = r cos θ

ˆ r = ˆ z cos θ + ρ ˆ sin θ ˆ

θ = − ˆ z sin θ + ρ ˆ cos θ

ˆ

z = ˆ r cos θ − θ ˆ sin θ ˆ

ρ = ˆ r sin θ + θ ˆ cos θ (E.6)

Aθ= cos φ cos θAx+ sin φ cos θAy− sin θAz

Aφ= − sin φAx+ cos φAy

(E.9)

Similarly, using Eq (E.6) the cylindrical components Aρ, Azcan be expressed in terms

of spherical components as:

Aρ= ρ ˆ · A = ρ ˆ · ( ˆ r Ar+ θ θ A ˆ θ+ φ φ A ˆ φ) = sin θAr+ cos θAθ

Az= ˆ z · A = ˆ z · ( ˆ r Ar+ θ θ A ˆ θ+ φ φ A ˆ φ) = cos θAr− cos θAθ

(E.10)

F Fresnel, Exponential, Sine, and Cosine Integrals

The Fresnel functions C(x) and S(x) are defined by [1298]:

C(x) =

x0cos

F Fresnel, Exponential, Sine, and Cosine Integrals 961

At x = 0, we have F( 0 )= 0 and F( 0 )= 1, so that the Taylor series approximation

is F(x) x , for small x The asymptotic expansions of C(x) , S(x) , and F(x) are for

 π

2 x2

 , F(x)= F2

The Fresnel function F2(x) can be evaluated numerically using Boersma’s

approx-imation [1156], which achieves a maximum error of 10−9 over all x The algorithm

approximates the function F2(x) as follows:

n

, if x > 4

(F.8)

where the coefficients an, bn, cn, dnare given in [1156] Consistency with the small- and

large- x expansions of F(x) requires that a0+ jb0= √ 8 /π and c0+ jd0= j/ √ 8 π We

have implemented Eq (F.8) with the MATLAB function fcs2:

F2 = fcs2(x); % Fresnel integralsF2(x)= C2(x)−jS2(x)

The ordinary Fresnel integral F(x) can be computed with the help of Eq (F.7) The

MATLAB function fcs calculates F(x) for any vector of values x by calling fcs2:

F = fcs(x); % Fresnel integralsF(x) = C(x)−jS(x)

In calculating the radiation patterns of pyramidal horns, it is desired to calculate a

Fresnel diffraction integral of the type:

F0(v, σ) =

1

−1ejπvξe−j(π/2)σ2ξ2

 v

σ + σ



− F

 v

F1(v, σ) = 1 2  F0(v + 0 5 , σ) +F0(v − 0 5 , σ) 

(F.14)

It can be verified easily that F0( 0 5 , σ) = F0( − 0 5 , σ) , therefore, the value of F1(v, σ)

at v = 0 will be given by:



ejπvξdξ = 1 2  F0(v + 0 5 , 0 ) +F0(v − 0 5 , 0 ) 

= sin

π(v + 0 5 ) π(v + 0 5 ) + sin

π(v − 0 5 ) π(v − 0 5 ) = 4

F Fresnel, Exponential, Sine, and Cosine Integrals 963

F0 = diffint(v,sigma,0); % diffraction integralF0(v, σ), Eq (F.9)

F1 = diffint(v,sigma,1); % diffraction integralF1(v, σ), Eq (F.13)

The vectors v,sigma can be entered either as rows or columns, but the result will

be a matrix of size length(v) x length(sigma) The integral F0(v, σ) can also be

calculated by the simplified call:

F0 = diffint(v,sigma); % diffraction integralF0(v, σ), Eq (F.9)

Actually, the most general syntax of diffint is as follows:

F = diffint(v,sigma,a,c1,c2); % diffraction integralF(v, σ, a), Eq (F.18)

It evaluates the more general integral:

 v

σ − σc1



− F

 v

Stationary Phase Approximation

The Fresnel integrals find also application in the stationary-phase approximation for

evaluating integrals The approximation can be stated as follows:

where x0is a stationary point of the phase φ(x) , that is, the solution of φ(x0) = 0,

where for simplicity we assume that there is only one such point (otherwise, one has a

sum of terms like (F.22), one for each solution of φ(x) = 0) Eq (F.22) is obtained by

expanding φ(x) in Taylor series about the stationary point x = x0and keeping only up

to the quadratic term:

φ(x)  φ(x0) +φ(x

0)(x − x0) + 1 2 φ(x0)(x − x0)2= φ(x0) + 1 2 φ(x0)(x − x0)2

Making this approximation in the integral and assuming that f (x) is slowly varying

in the neighborhood of x0, we may replace f (x) by its value at x0:

φ(x0)

∞

−∞ejπu2/2du =

 π

φ(x0)



Using  F(∞)−F(−∞) ∗= 2 F∗( ∞)= 1 + j = 2 j , we obtain

Exponential, Sine, and Cosine Integrals

Several antenna calculations, such as mutual impedances and directivities, can be duced to the exponential integral, which is defined as follows [1298]:

Si(z) =

z0

sin u

Ci(z)= γ + ln z +

z0

F Fresnel, Exponential, Sine, and Cosine Integrals 965

while for z ≤ 0, we have Si(z)= −Si(−z) and Ci(z)= Ci(−z)+jπ Conversely, we have

vector of z ’s by using the relations (F.26) and the built-in function expint:

y = Si(z); % sine integral, Eq (F.24)

y = Ci(z); % sine integral, Eq (F.24)

y = Cin(z); % sine integral, Eq (F.25)

A related integral that appears in calculating mutual and self impedances is what

may be called a “Green’s function integral”:

Gi (d, z0, h, s) =

h0

e−jkR

−jkszdz , R =  d2+ (z − z0)2, s = ± 1 (F.28) This integral can be reduced to the exponential integral by the change of variables:

v = jk  R + s(z − z0)

v = dz R which gives

h0

The function Gi evaluates Eq (F.29), where z0, s , and the resulting integral J , can be

vectors of the same dimension Its usage is:

J = Gi(d,z0,h,s); % Green’s function integral, Eq (F.29)

Another integral that appears commonly in antenna work is:

π

0

cos (α cos θ) − cos α

sin θ dθ = Si( 2 α) sin α − Cin( 2 α) cos α (F.30) Its proof is straightforward by first changing variables to z = cos θ , then using

partial fraction expansion, and finally changing variables to u = α( 1 + z) , and using

the definitions (F.24) and (F.25):

where wi, xiare appropriate weights and evaluation points (nodes) This can be written

in the vectorial form:

ba

The function quadr returns the column vectors of weights w and nodes x, with usage:

[w,x] = quadr(a,b,N); Gauss-Legendre quadratureThe function quadrs allows the splitting of the interval [a, b] into subintervals, computes N weights and nodes in each subinterval, and concatenates them to form the

overall weight and node vectors w , x:

[w,x] = quadrs(ab,N); Gauss-Legendre quadrature over subintervalswhere ab is an array of endpoints that define the subintervals, for example,

ab = [a, b] , single interval

ab = [a, c, b] , two subintervals, [a, c] and [c, b]

ab = [a, c, d, b] , three subintervals, [a, c] , [c, d] , and [d, b]

ab = a : c : b , subintervals, [a, a +c, a+ 2 c, , a +Mc] , with a + Mc = b

As an example, consider the following function and its exact integral:

f (x)= ex+ 1

2

1f (x) dx = e2− e1+ ln 2 = 5 36392145 This integral can be evaluated numerically by the MATLAB code:

N = 5; % number of weights and nodes[w,x] = quadr(1,2,N); % calculate weights and nodes for the interval[1, 2]

f = exp(x) + 1./x; % evaluatef (x) at the node vector

J = w’*f % approximate integralThis produces the exact value with a 4 23 × 10−7percentage error If the integration interval is split in two, say, [ 1 , 1 5 ] and [ 1 5 , 2 ] , then the second line above can be replaced by

†J Stoer and R Burlisch, Introduction to Numerical Analysis, Springer, NY, (1980); and, G H Golub and

G Gauss-Legendre Quadrature 967

[w,x] = quadrs([1,1.5,2],N); % or by, [w,x] = quadrs(1:0.5:2, N);

which has a percentage error of 1 28 × 10−9 Next, we discuss the theoretical basis of



z +



b + a 2



(G.3)

If wiand ziare the weights and nodes with respect to the interval [− 1 , 1 ] , then those

with respect to [a, b] can be constructed simply as follows, for i = 1 , 2 , , N :

xi=



b − a 2



zi+



b + a 2



wxi =

 b − a 2



wi

(G.4)

where the scaling of the weights follows from the scaling of the differentials dx =

dz(b − a)/ 2, so the value of the integral (G.1) is preserved by the transformation.

Gauss-Legendre quadrature is nicely tied with the theory of orthogonal polynomials

over the interval [ − 1 , 1 ] , which are the Legendre polynomials For N -point quadrature,

the nodes zi, i = 1 , 2 , , N are the N roots of the Legendre polynomial PN(z) , which

all lie in the interval [ − 1 , 1 ] The method is justified by the following theorem:

For any polynomial P(z) of degree at most 2 N − 1, the quadrature formula (G.1) is

satisfied exactly, that is,

provided that the ziare the N roots of the Legendre polynomial PN(z)

The Legendre polynomials Pn(z) are obtained via the process of Gram-Schmidt

or-thogonalization of the non-orthogonal monomial basis { 1 , z, z2, , zn .}

Orthogo-nality is defined with respect to the following inner product over the interval [ − 1 , 1 ] :

, n = 0 , 1 , 2 , (G.7) The first few of them are listed below:

They are normalized such that Pn( 1 )= 1 and are mutually orthogonal with respect

to (G.6), but do not have unit norm:

(Pn, Pm) =

1

−1Pn(z)Pm(z)dz = 2

2 n + 1 δnm (G.9) Moreover, they satisfy the three-term recurrence relation:

Pn(z) = fn(z) −

n−1

k=0

(fn, Pk) (Pk, Pk) Pk(z)

A few steps of the construction will clarify it:

P1(z) = f1(z) − (f1, P0)

(P0, P0) P0(z) = z where (f1, P0)= (z, 1 )=

1

−1z

3dz = 0, and (f2, P0) = (z2, 1 ) =

P2(z)= z2− 2 / 3

2 = z2− 1

3 Then, normalize it such that P2( 1 ) = 1, and so on For our discussion, we are going

to renormalize the Legendre polynomials to unit norm Because of (G.9), this amounts

to multiplying the standard Pn(z) by the factor ( 2 n + 1 )/ 2 Thus, we re-define:

Pn(z) =



2 n + 1 2

, n = 0 , 1 , 2 , (G.11) Thus, (G.9) becomes (Pn, Pm) = δnm In particular, we note that now

P0(z) = √ 1

G Gauss-Legendre Quadrature 969

By introducing the same scaling factors into each term of the recurrence (G.10), we

find that the renormalized Pn(z) satisfy:

zPn(z) = αnPn−1(z) +αn+1Pn+1(z) , αn= √ n

4 n2− 1 (G.13) This relationship can be assumed to be valid also at n = 0, provided we define

P−1(z)= 0 For each order n , the Gram-Schmidt procedure replaces the non-orthogonal

monomial basis by the orthonormalized Legendre basis:

1 , z, z2, , zn!

 P0(z), P1(z), P2(z), , Pn(z) ! Thus, any polynomial Q(z) of degree n can be expanded uniquely in either basis:

with the expansion coefficients calculated from ck= (Q, Pk) This also implies that if

Q(z) has order n − 1 then, it will be orthogonal to Pn(z)

Next, we turn to the proof of the basic Gauss-Legendre result (G.5) Given a

polyno-mial P(z) of order 2 N − 1, we can expand it uniquely in the form:

where Q(z) and R(z) are the quotient and remainder of the division by the Legendre

polynomial PN(z) , and both will have order N − 1 Then, the integral of P(z) can be

written in inner-product notation as follows:

1

−1P(z)dz = (P, 1 ) = (PNQ + R, 1 ) = (PNQ, 1 ) +(R, 1 ) = (Q, PN) +(R, 1 )

But (Q, PN)= 0 because Q(z) has order N − 1 and PN(z) is orthogonal to all such

polynomials Thus, the integral of P(z) can be expressed only in terms of the integral

of the remainder polynomial R(z) , which has order N − 1:

we had not assumed initially that the ziwere the zeros of PN(z) , and took them to be

an arbitrary set of N distinct points in [ − 1 , 1 ] , then (G.18) would read as

Inserting, for example, the monomial basis into (G.18) and matching the coefficients

of rkon either side, we obtain the system of N equations for the weights:

i and the vector uk=  1 + (− 1 )k

/(k + 1 ) , we may write (G.20) in the compact matrix form:

ck(PK, P0) = √ 2

N−1k=0

ckδk0= √ 2 c0

The right-hand side of (G.18) may be written as follows Defining the N ×N trix Pki = Pk(zi) , i = 1 , 2 , N , and k = 0 , 1 , , N − 1, and the row vector cT = [c0, c1, , cN−1] of expansion coefficients, we have,

G Gauss-Legendre Quadrature 971

Because the vector c is arbitrary, we must have the condition:

P w = √ 2 u0 ⇒ w = √ 2 P−1u0 (G.22) The matrix P has some rather interesting properties First, it has mutually orthogonal

columns Second, these columns are the eigenvectors of a Hermitian tridiagonal matrix

whose eigenvalues are the zeros zi Thus, the problem of finding both zi and wi is

reduced to an eigenvalue problem.

These eigenvalue properties follow from the recursion (G.13) of the normalized

Leg-endre polynomials For n = 0 , 1 , 2 , 3, the recursion reads explicitly:

zP0(z) = α1P1(z)

zP1(z) = α1P0(z) +α2P2(z)

zP2(z) = α2P1(z) +α3P3(z)

zP3(z) = α3P2(z)+α4P4(z) which can be written in matrix form:

0

Now, if z is replaced by the i th zero ziof PN(z) , the last column will vanish and we

obtain the eigenvalue equation:

Denoting the above tridiagonal matrix by A and the column of Pk(zi) ’s by pi, we

may write compactly:

A pi= zipi, i = 1 , 2 , , N (G.24)

Thus, the eigenvalues of A are the zeros ziand the corresponding eigenvectors are

the columns piof the matrix P that we introduced in (G.22) Because the zeros ziare distinct and A is a Hermitian matrix, its eigenvectors will be mutually orthogonal:

eigen-H Lorentz Transformations

According to Einstein’s special theory of relativity [458], Lorentz transformations scribe the transformation between the space-time coordinates of two coordinate sys- tems moving relative to each other at constant velocity Maxwell’s equations remain invariant under Lorentz transformations This is demonstrated below.

de-Let the two coordinate frames be S and S By convention, we may think of S as the “fixed” laboratory frame with respect to which the frame Sis moving at a constant

velocity v For example, if v is in the z -direction, the space-time coordinates {t, x, y, z}

of S are related to the coordinates {t, x, y, z} of Sby the Lorentz transformation:

H Lorentz Transformations 973

where c is the speed of light in vacuum Defining the scaled quantities τ = ct and

β = v/c , the above transformation and its inverse, obtained by replacing β by −β , may

These transformations are also referred to as Lorentz boosts to indicate the fact that

one frame is boosted to move relative to the other Interchanging the roles of z and x , or

z and y , one obtains the Lorentz transformations for motion along the x or y directions,

respectively Eqs (H.1) may be expressed more compactly in matrix form:

quadratic form as follows, where xT denotes the transposed vector, that is, the row

vector xT= [τ, x, y, z] :

xTG x = τ2− x2− y2− z2= c2t2− x2− y2− z2 (H.4) More generally, a Lorentz transformation is defined as any linear transformation x=

L x that leaves the quadratic form xTG x invariant The invariance condition requires

that: xTG x= xTLTGL x = xTG x, or

In addition to the Lorentz boosts of Eq (H.1), the more general transformations

satisfying (H.5) include rotations of the three spatial coordinates, as well as time or

space reflections For example, a rotation has the form:

where R is a 3 × 3 orthogonal rotation matrix, that is, RTR = I , where I is the 3 × 3

identity matrix The most general Lorentz boost corresponding to arbitrary velocity

v1/c and β2= v2/c leads to the combined boost L(β) = L(β1)L(β2) , where:

β = β1+ β2

1 + β1β2  v = v1+ v2

1 + v1v2/c2 (H.8) with β = v/c Eq (H.8) is Einstein’s relativistic velocity addition theorem The same group property implies also that L−1(β)= L(−β) The proof of Eq (H.8) follows from the following condition, where γ1= 1 /

ay= ay

(H.10)

Four-vectors transforming according to Eq (H.9) are referred to as contravariant Under the general Lorentz boost of Eq (H.6), the spatial components of a that are trans-

verse to the direction of the velocity vector v remain unchanged, whereas the parallel

component transforms as in Eq (H.10), that is, the most general Lorentz boost mation for a four-vector takes the form:

H Lorentz Transformations 975

where a= β ˆTa and a = [ax, ay, az]Tis the spatial part of a Then,

a= β ˆ βa= β β( ˆ β ˆTa ) and a⊥= a − a= a − β βa ˆ Setting β = β β ˆ and using Eq (H.7), the Lorentz transformation (H.6) gives:

a − β βa ˆ + β βγ(a ˆ − βa0)



from which Eq (H.11) follows.

For any two four-vectors a, b , the quadratic form aTGb remains invariant under

Lorentz transformations, that is, aTGb= aTGb , or,

frequency and wavenumber ω/c kx ky kz

energy and momentum E/c px py pz

charge and current densities cρ Jx Jy Jz

scalar and vector potentials ϕ cAx cAy cAz

(H.13)

For example, under the z -directed boost of Eq (H.1), the frequency-wavenumber

transformation will be as follows:

ω= γ(ω − βckz)

kz= γ  kz− β

c ω

ky= ky

, βc = v , β c = v

c2 (H.14)

where we rewrote the first equations in terms of ω instead of ω/c The change in

frequency due to motion is the basis of the Doppler effect The invariance property

(H.12) applied to the space-time and frequency-wavenumber four-vectors reads:

This implies that a uniform plane wave remains a uniform plane wave in all reference

frames moving at a constant velocity relative to each other Similarly, the charge and

current densities transform as follows:

j

Lji ∂

∂ xj

⇒ ∂x= LT∂x ⇒ ∂x= L−T∂xFor the z -directed boost of Eq (H.1), we have L−T= L−1, which gives:

Under a Lorentz transformation, this remains invariant, and therefore, if it is zero

in one frame it will remain zero in all frames Using ∂T

do not Rather, they transform like six-vectors or rank-2 antisymmetric tensors.

H Lorentz Transformations 977

A rank-2 tensor is represented by a 4 × 4 matrix, say F Its Lorentz transformation

properties are the same as the transformation of the product of a column and a row

four-vector, that is, F transforms like the quantity abT, where a, b are column

four-vectors This product transforms like abT= L(abT)LT Thus, a general second-rank

tensor transforms as follows:

An antisymmetric rank-2 tensor F defines, and is completely defined by, two

three-dimensional vectors, say a = [ax, ay, az]Tand b = [bx, by, bz]T Its matrix form is:

Given the tensor F , one may define its covariant version through ¯ F = GFG , and its

dual, denoted by ˜ F and obtained by the replacements a → b and b → − a, that is,

Thus, ¯ F corresponds to the pair (− a , b ) , and ˜ F to ( b ,− a ) Their Lorentz

transfor-mation properties are:

¯

F= L−TFL ¯ −1, F ˜= L FL ˜ T (H.26) Thus, the dual ˜ F transforms like F itself For the z -directed boost of Eq (H.1), it

follows from (H.23) that the two vectors a , b transform as follows:

More generally, under the boost transformation (H.6), it can be verified that the

components of a , b parallel and perpendicular to v transform as follows:

un-It is evident also that Eqs (H.28) remain invariant under the duality transformation

a → b and b → − a, which justifies Eq (H.26) Some examples of ( a , b ) six-vector pairs defining an antisymmetric rank-2 tensor are as follows:

where P , M are the polarization and magnetization densities defined through the

rela-tionships D = 0E + P and B = μ0( H + M ) Thus, the ( E , B ) and ( D , H ) fields have the following Lorentz transformation properties:

where we may replace cβ β = v and β β/c = v /c2 Note that the two groups of equations

transform into each other under the usual duality transformations: E → H, H → − E,

D → B, B → − D For the z -directed boost of Eq (H.1), we have from Eq (H.30):

H Lorentz Transformations 979

Associated with a six-vector ( a , b ) , there are two scalar invariants: the quantities

( a · b ) and ( a · a − b · b ) Their invariance follows from Eq (H.28) Thus, the scalars

( E · B ) , ( E · E − c2B · B ) , ( D · H ) , (c2D · D − H · H ) remain invariant under Lorentz

transformations In addition, it follows from (H.30) that the quantity ( E · D − B · H ) is

invariant.

Given a six-vector ( a , b ) and its dual ( b , − a ) , we may define the following

four-dimensional “current” vectors that are dual to each other:

It can be shown that both J and ˜ J transform as four-vectors under Lorentz

trans-formations, that is, J= LJ and ˜ J= L J ˜ , where J J ˜are defined with respect to the

coordinates of the Sframe:

The calculation is straightforward but tedious For example, for the z -directed boost

(H.1), we may use Eqs (H.20) and (H.27) and the identity γ2( 1 − β2) = 1 to show:

J0= ∇ ∇· a= ∂xax+ ∂yay+ ∂zaz

= γ∂x(ax− βby) +γ∂y(ay+ βbx) +γ(∂z+ β∂τ)az

= γ  (∂xax+ ∂yay+ ∂zaz) −β(∂xby− ∂ybx− ∂τaz) 

= γ(J0− βJz)

In this fashion, one can show that J and ˜ J satisfy the Lorentz transformation

equa-tions (H.10) for a four-vector To see the significance of this result, we rewrite Maxwell’s

equations, with magnetic charge and current densities ρm, Jmincluded, in the

J

 ,

Thus, applying the above result to the six-vector (c D , H ) and to the dual of ( E , c B )

and assuming that the electric and magnetic current densities transform like

four-vectors, it follows that Maxwell’s equations remain invariant under Lorentz

transfor-mations, that is, they retain their form in the moving system:

 (H.35)

The Lorentz transformation properties of the electromagnetic fields allow one to solve problems involving moving media, such as the Doppler effect, reflection and trans- mission from moving boundaries, and so on The main technique for solving such prob- lems is to transform to the frame (here, S) in which the boundary is at rest, solve the reflection problem in that frame, and transform the results back to the laboratory frame

by using the inverse of Eq (H.30).

This procedure was discussed by Einstein in his 1905 paper on special relativity in connection to the Doppler effect from a moving mirror To quote [458]: “All problems

in the optics of moving bodies can be solved by the method here employed What is essential is that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of co-ordinates at rest relatively to the body By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies.”

I MATLAB Functions

The MATLAB functions are grouped by category They are available from the web page: www.ece.rutgers.edu/~orfanidi/ewa.

Multilayer Dielectric Structures

brewster - calculates Brewster and critical anglesfresnel - Fresnel reflection coefficients for isotropic or birefringent median2r - refractive indices to reflection coefficients of M-layer structurer2n - reflection coefficients to refractive indices of M-layer structuremultidiel - reflection response of isotropic or birefringent multilayer structuresmultidiel1 - simplified version of multidiel for isotropic layers

multidiel2 - reflection response of lossy isotropic multilayer dielectric structuresomniband - bandwidth of omnidirectional mirrors and Brewster polarizers

omniband2 - bandwidth of birefringent multilayer mirrorssnel - calculates refraction angles from Snel’s law for birefringent media

Quarter-Wavelength Transformers

bkwrec - order-decreasing backward layer recursion - from a,b to rfrwrec - order-increasing forward layer recursion - from r to A,Bchebtr - Chebyshev broadband reflectionless quarter-wave transformerchebtr2 - Chebyshev broadband reflectionless quarter-wave transformerchebtr3 - Chebyshev broadband reflectionless quarter-wave transformer

Dielectric Waveguides

dguide - TE modes in dielectric slab waveguidedslab - solves for the TE-mode cutoff wavenumbers in a dielectric slab

I MATLAB Functions 981

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 - wave impedance 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 impedance transformer

dualbw - two-section dual-band transformer bandwidths

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 - Pi to T transformation

lmatch - L-section reactive conjugate matching network

pmatch - Pi-section reactive conjugate matching network

S-Parameters

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 directivitydmax - computes directivity and beam solid angle of g(th) gaindipole - gain of center-fed linear dipole of length L

traveling - gain of traveling-wave antenna of length Lvee - gain of traveling-wave vee antennarhombic - gain of traveling-wave rhombic antennaking - King’s 3-term sinusoidal approximationkingeval - evaluate King’s 3-term sinusoidal current approximationkingfit - fits a sampled current to King’s 2-term sinusoidal approximationkingprime - converts King’s 3-term coefficients from unprimed to primed formhbasis - basis functions for Hallen equation

hdelta - solve Hallen’s equation with delta-gap inputhfield - solve Hallen’s equation with arbitrary incident E-fieldhmat - Hallen impedance matrix with method of moments and point-matchinghwrap - wraps a Toeplitz impedance matrix to half its size

kernel - thin-wire kernel computation for Hallen equationpfield - solve Pocklington’s equation with arbitrary incident E-fieldpmat - Pocklington impedance matrix with method of moments and point-matchinghcoupled - solve Hallen’s equation for 2D array of non-identical parallel dipoleshcoupled2 - solve Hallen’s equation for 2D array of identical parallel dipolesgain2d - normalized gain of 2D array of parallel dipoles with Hallen currentsgain2s - normalized gain of 2D array of parallel dipoles with sinusoidal currentsimped - mutual impedance between two parallel standing-wave dipoles

imped2 - mutual impedance between two parallel standing-wave dipolesimpedmat - mutual impedance matrix of array of parallel dipole antennasresonant - calculates the length of a resonant dipole antenna

yagi - simplified Yagi-Uda array design

Aperture Antenna Functions

diffint - generalized Fresnel diffraction integraldiffr - knife-edge diffraction coefficientdsinc - 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)hband - horn antenna 3-dB width

heff - aperture efficiency of horn antennahgain - horn antenna H-plane and E-plane gainshopt - optimum horn antenna design

hsigma - optimum sigma parametes for horn antenna

Antenna Array Functions

gain1d - normalized gain computation for 1D equally-spaced isotropic arraybwidth - beamwidth mapping from psi-space to phi-space

binomial - binomial array weightsdolph - Dolph-Chebyshev array weightsdolph2 - Riblet-Pritchard version of Dolph-Chebyshevdolph3 - DuHamel version of endfire Dolph-Chebyshev

I MATLAB Functions 983

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)

Gain Plotting Functions

abp - polar gain plot in absolute units

abz - azimuthal gain plot in absolute units

ab2p - 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

Si - sine integral Si(z)

Gi - Green’s function integralsinhc - hyperbolic sinc functionasinhc - inverse hyperbolic sinc functionsqrte - evanescent SQRT for waves problemsflip - flip a column, a row, or bothblockmat - manipulate block matricesupulse - generates trapezoidal, rectangular, triangular pulses, or a unit-stepustep - unit-step or rising unit-step function

dnv - dn elliptic function at a vector of modulisnv - sn elliptic function at a vector of moduliellipK - complete elliptic integral of first kind at a vector of moduliellipE - complete elliptic integral of second kind at a vector of modulilandenv - Landen transformations of a vector of elliptic moduli

MATLAB 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 linespulse2movie - step propagation on two cascaded lines

RLCmovie - step getting reflected off a reactive terminationTDRmovie - fault location by time-domain reflectometryxtalkmovie - crosstalk signals on coupled transmission linesdipmovie - electric field pattern of radiating Hertzian dipole

3-dB width, 167

9-dB per delta rule, 55

acoustic tube models, 185

Yagi-Uda, 934antireflection coatings, 169, 185, 187

at oblique incidence, 329aperture antennas, 726aperture-field method, 754current-distribution method, 754directivity of waveguide apertures, 729dual-reflector, 766

horn design, 740horn directivity, 737horn radiation fields, 732horn radiation patterns, 734horns, 730

lens, 769microstrip, 743open-ended waveguides, 726parabolic reflector beamwidth, 751parabolic reflector gain, 751parabolic reflectors, 749reflector radiation patterns, 757aperture efficiency, 609, 672aperture-field method for reflector antennas, 754apertures

3-dB angles, 675aperture efficiency, 672

1033

circular, 675diffraction theory, 678directivity of, 671effective area of, 671extinction theorem, 682field equivalence principle, 661Fourier optics, 711

Franz diffraction formulas, 681Fresnel diffraction, 685, 711geometrical theory of diffraction, 697Huygens source, 669

Huygens-Fresnel principle, 661Kirchhoff diffraction formulas, 680knife-edge diffraction, 689Kottler’s formulas, 666, 681lenses, 719

plane wave spectrum, 706Poisson’s spot, 715radiation fields from, 666radiation from, 661radiation vectors of, 667Rayleigh-Sommerfeld theory, 703rectangular, 673

Sommerfeld’s solution, 697Stratton-Chu diffraction formulas, 680uniform, 673

vector diffraction for, 684apparent depth, 253Appleton-Hartree equations, 150array design methods, 802and FIR filters, 807binomial, 825Blass matrix, 852Butler beams, 851continuous line sources, 817continuous to discrete, 819Dolph-Chebyshev, 826DSP analogies, 807endfire DuHamel, 837Fourier series with windowing, 807frequency-sampling, 814multibeam, 850narrow-beam, low-sidelobe, 821prolate array, 843

Riblet method, 831sampled current sources, 818Schelkunoff’s zero-placement, 805secant array, 816

sector beam, 808spatial sampling theorem, 818Taylor line source, 845Taylor n-bar distribution, 845Taylor’s one-parameter, 839Villeneuve, 849

Woodward-Lawson, 812, 819zero mapping, 819array factor, 773array pattern multiplication, 773

array processing, 785array space factor, 773arrays, 771

array factor, 773array processing, 785beamwidth, 797beamwidth of uniform array, 791coupled two-element, 922directivity, 793discrete-space Fourier transform, 785DSP analogies, 785

grating lobes, 787ground effects, 779interferometry, 787one-dimensional, 783optimum directivity, 794parallel dipoles, 925parasitic, 931pattern multiplication, 773prolate matrix, 794rectangular window, 790sidelobes of uniform array, 792space factor, 773

spatialz-transform, 785steering and scanning, 794translational phase shift, 771uniform, 789

very large array, 788very long base line, 788visible region, 785Yagi-Uda, 934atmospheric refraction, 288attenuation constant, 55attenuation in waveguides, 367, 379attenuator noise temperature, 622available gain circles, 557available power gain, 541

Babinet principle, 715BAC-CAB rule, 31backward recursion, see layer recursionsbackward waves, 38

balanced stubs, 505bandwidth

in waveguides, 376noise, 617

of dielectric mirrors, 196beam efficiency, 620beam solid angle, 605beamwidth, 602biaxial media, 132bilinear transformation, 216binomial arrays, 825birefingent plasmas, 34birefringencecircular, 131linear, 131birefringent media, 131

INDEX 1035

Brewster and critical angles, 349

Brewster angle in, 350

critical angle of incidence in, 349

giant birefringent optics, 354, 355, 359

maximum angle of refraction in, 349

multilayer structures, 354

reflection and refraction, 345

bite-error rate (BER), 627

for birefringent media, 350

for lossy media, 260

complementary error function, 627

complex refractive index, 54

cutoff wavenumber and frequency, 364cylindrical coordinates, 366, 593

data rate limits, 627delta-gap generator, 858density

current, 1electric flux, 1Lorentz force, 3magnetic flux, 1momentum, 13, 33polarization, 8surface charge, 7surface current, 7volume charge, 1dichroism, linear and circular, 138dielectric constant, 4

dielectric mirrors, 192bandwidth of, 196Fabry-Perot resonators, 203multiband, 201

narrow-band transmission filters, 203omnidirectional, 198

shortpass/longpass filters, 202dielectric model, 17

dielectric polarization, 4dielectric slab, 162, 306half-wave, 165quarter-wave, 165reflectionless, 165dielectric waveguides, 386diffraction integrals, 960

diffuse reflection and transmissionKubelka-Munk model of, 472dipole moment density, 4dipole radiation, 658directive gain, 602directivity, 602directivity of apertures, 671discretization of continuous line sources, 817dish antennas, 609, 749

dispersionanomalous, 19intermodal, 6material, 6normal, 19waveguide, 6dispersion coefficient, 99dispersion compensation, 103dispersive materials, 5displacement current, 1Dolph-Chebyshev arrays, 826Dolph-Chebyshev-Riblet arrays, 831doppler ambiguity, 119

Doppler effect, 66Doppler radar, 180Doppler shift, 177, 180double-stub tuner, 507Drude model, 21DSFT, discrete-space Fourier transform, 785DTFT, discrete-time Fourier transform, 785dual-reflector antennas, 766

duality transformation, 663dynamic predictive deconvolution, 185

effective area of an antenna, 607effective area of apertures, 671effective length of an antenna, 613effective noise temperature, 563, 621efficiency factor, 603

EIRP, effective isotropic radiated power, 603electric and magnetic dipoles, 580electric field, 1

electric flux density, 1electromagnetic frequency bands, 950elliptic functions, 873

elliptic integrals, 873endfire DuHamel arrays, 837energy conservation, 12energy density, 12, 14, 25, 45energy flux, 10, 12, 45, 601energy velocity, 29, 40equivalent noise temperature, 623error function, 627

evanescent waves, 251penetration depth for, 251Ewald-Oseen extinction theorem, 585exponential integrals, 960

extinction theorem, 585, 682

Fabry-Perot interferometer, 203, 308far-field approximation, 590Faraday rotation, 131, 138Faraday’s law of induction, 1Fermat’s principle of least time, 281fiber Bragg gratings, 185, 203, 469quarter-wave phase-shifted gratings, 203fiber, single mode, 100

fiber, standard, 100field equivalence principle, 661field intensities, 1

fields of dipoles, 580fields of wire antennas, 578fingerprint identification, 257flux

charge, 9definition of, 9energy, 10, 12momentum, 10Poynting vector, 12flux densities, 1forward recursion, see layer recursionsforward waves, 38

Fourier optics, 711Fourier series method with windowing, 807Franz diffraction formulas, 681

Fraunhofer region, 592free-space loss, 616frequency bands, 950frequency-sampling array design, 814Fresnel coefficients, 246, 247Fresnel diffraction, 685, 711Fresnel drag, 179

Fresnel integrals, 960Fresnel region, 592Fresnel rhomb, 254Fresnel zones, 692Friis formula, 615frill generator, 858front delay, 84front velocity, 84frustrated total internal reflection, 308

gain, 602gain-beamwidth relationship, 605, 611Galerkin weighting, 878

gamma-ray bands, 951gauge transformation, 572Gauss’s laws, 1

geometrical optics, 278geometrical theory of diffraction, 697geosynchronous satellite, 606giant birefringent optics, 139, 354, 355, 359mirrors, 355, 356

reflective polarizers, 358glass prisms, 252

Goos-H¨anchen shift, 255

half-wave dipole antennas, 645

half-wave reflectionless slab, 165

Hall effect, 34, 148

Hall´en equation with arbitrary field, 894

Hall´en equations for coupled antennas, 939

Hall´en integral equation, 857, 860

Hertzian dipole antenna, 639

high resolution microscopy, 257

horn antennas, 730

horn design, 740

horn directivity, 737

horn radiation fields, 732

horn radiation patterns, 734

double- and triple-stub tuners, 507

dual-band Chebyshev transformer, 485

flat line, 477

L-section matching network, 509

matching networks, 477

one-section transformer, 501Pi-section matching network, 512quarter-wavelength transformer, 185, 479quarter-wavelength with series section, 491quarter-wavelength with shunt stub, 494reversed matching networks, 519single-stub tuner, 501

two-section transformer, 496impedance matrix, 525impedance transformers, 189infrared bands, 951inhomogeneous materials, 5inhomogeneous waves, 64, 260integral theorems, 952intermodal dispersion, 6internal reflection spectroscopy, 257inverse power iteration, 845ionospheric refraction, 285isotropic radiator, 602

Kaiser window, 810, 839kernel

approximate, 856elliptic function representation, 873exact, 856

numerical evaluation, 872reduced, 856

King’s four-term approximation, 871, 915King’s three-term approximation, 865, 910Kirchhoff diffraction formulas, 680knife-edge diffraction, 689Kottler’s formulas, 666Kramers-Kronig dispersion relations, 6, 26, 34Kubelka-Munk model, 472

L-section matching network, 509layer recursions, 215, 303backward, 216forward, 212, 303left-handed media, 7, 70, 294lens antennas, 769lenses, 719Levinson recursion, see layer recursionslinear antennas, 637

linear prediction, 185link budget calculation, 631loop antennas, 653Lorentz dielectric, 18Lorentz force, 2Lorentz transformations, 177, 972Lorenz gauge condition, 572loss tangent, 24, 58lossless bounded real functions, 185lossy media, 53

lossy media, weakly, 59lossy multilayer structures, 304low-noise, high-gain, amplifier, 625

Macneille polarizers, see reflective polarizersmagnetic currents, 661, 663

magnetic field, 1magnetic flux density, 1magnetic induction, 1magnetic resonance, 138magnetization, 4magnetization current, 6matched filter, 121matching, see impedance matchingmatching matrices, 156, 186matching matrix, 246matching networks, 477material dispersion, 6MATLAB functions:

Cin, cosine integral Cin, 921, 965

Ci, cosine integral Ci, 921, 965

Gi, Green’s function integral, 920, 965RLCmovie, reactive termination, 445

Si, sine integral, 921, 965TDRmovie, time-domain reflectometry, 455abp2, polar gain in absolute units, 930abp, polar gain in absolute units, 605abz2, azimuthal gain in absolute units, 930abz, azimuthal gain in absolute units, 605asinhc, inverse hyperbolic sinh, 841binomial, binomial array, 826bkwrec, backward layer recursion, 221blockmat, manipulate block matrices, 944brewster, Brewster and critical angles, 350bwidth, array beamwidth, 798

c2p, cartesian to phasor form, 565chebarray, Dolph-Chebyshev array, 830chebtr2, Chebyshev transformer, 229, 480chebtr3, Chebyshev transformer, 229, 480chebtr, Chebyshev transformer, 229, 480dbp2, polar gain in dB, 930

dbp, polar gain in dB, 605dbz2, azimuthal gain in dB, 778dbz, azimuthal gain in dB, 605dguide, TE modes in dielectric slab, 393diffint, diffraction integrals, 733, 963dipdir, dipole directivity, 644dipmovie, radiating dipole movie, 585dipole, dipole gain, 646

dmax, dipole directivity, 646dnv, elliptic function dn, 874dolph2, Dolph-Chebyshev array, 834dolph3, Dolph-Chebyshev array, 838dolph, Dolph-Chebyshev array, 830dsinc, double sinc function, 728dslab, cutoff wavenumbers in slab, 393dualband, dual-band transformer, 489dualbw, dual-band bandwidth, 491ellipE, elliptic integral of 2nd kind, 874ellipK, elliptic integral of 1st kind, 874ellipse, polarization ellipse, 52fcs2, Fresnel integrals, 961

fcs, Fresnel integrals, 961fresnel, Fresnel coefficients, 249, 349frwrec, forward layer recursion, 221gain1d, one-dimensional array gain, 774gain2d, gain of 2D array of dipoles, 945gain2s, gain of sinusoidal dipole array, 928gin, input reflection coefficients, 537gout, output reflection coefficients, 537gprop, propagation ofΓ, 419grvmovie1, pulse propagation with vg<0, 112grvmovie2, pulse propagation with vg>c, 112hband, horn bandedges, 733

hbasis, Hall´en basis functions, 894hcoupled2, coupled Hall´en equations, 944hcoupled, coupled Hall´en equations, 944hdelta, Hall´en equation with delta-gap, 883heff, horn aperture efficiency, 733hfield, Hall´en with arbitrary field, 896hgain, horn gain patterns, 733hmat, Hall´en impedance matrix, 883hopt, optimum horn design, 733hsigma, hornσparameter, 733hwrap, wrapped impedance matrix, 883impedmat, mutual impedance matrix, 927imped, dipole impedance, 642

k2k, converts to King’s primed form, 866kernel, exact and approximate kernel, 874kingeval, King’s three-term evaluation, 870kingfit, King’s three-term fit, 870king, Kings three-term approximation, 868landev, vectorial Landen transformation, 874lmatch,L-section transformer, 511lmin, location of voltage min/max, 433mstripa, microstrip analysis, 407mstripr, microstrip synthesis, 407mstrips, microstrip synthesis, 407multbeam, multibeam array, 851multidiel1, multilayers, simplified, 304multidiel2, lossy multilayers, 305multidiel, multilayer structures, 304, 354multiline, response of multisection line, 479n2r, reflection coefficients, 221

nfcirc, noise figure circles, 537nfig, calculate noise figure, 537omniband2, birefringent bandwidth, 355omniband, omnidirectional bandwidth, 335onesect, one-section transformer, 501p2c, phasor to cartesian form, 565pfield, solves Pocklington equation, 901pi2t,ΠtoTtransformation, 513pmatch,Πmatching network, 515poly2, improved version of poly, 232, 830prolmat, prolate matrix, 845

prol, prolate array, 845pulse2movie, pulse on transmission line, 453pulsemovie, pulse on transmission line, 443quadr2, Gauss-Legendre quadrature, 972quadrs2, Gauss-Legendre quadrature, 972

INDEX 1039

quadrs Gauss-Legendre quadrature, 760

quadrs, Gauss-Legendre quadrature, 966

quadr, Gauss-Legendre quadrature, 966

qwt1, quarter-wavelength transformer, 494

qwt2, quarter-wavelength transformer, 495

qwt3, quarter-wavelength transformer, 495

r2n, refractive indices, 221

rhombic, rhombic antenna gain, 652

scan, array scanning, 796

sector, sector beam design, 812

sgain, calculate power gains, 537

sgcirc, stability and gain circles, 537

sinhc, hyperbolic sinc, 841

smatch, simultaneous conjugate match, 537

smithcir, draw stability or gain circles, 537

smith, draw a basic Smith chart, 537

snel, refraction angle, 348

snv, elliptic function sn, 874

sparam, calculate stability parameters, 537

sqrte, evanescent square root, 263

steer, array steering, 796

stub1, single-stub tuner, 504

stub2, double-stub tuner, 508

stub3, triple-stub tuner, 509

swr, standing wave ratio, 429

t2pi,TtoΠtransformation, 513

taylor1p, Taylor’s one-parameter, 841

taylorbw, Taylor’s B-parameter, 841

taylornb, Taylor’s n-bar method, 848

travel, traveling-wave antenna, 649

tsection, T-section equivalent, 420

twosect, two-section transformer, 189, 499

upulse, pulse generation, 452

ustep, unit-step generation, 453

vee, vee antenna gain, 652

ville, Villeneuve method, 850

vprop, propagationV, I, 419

woodward, Woodward-Lawson method, 814

yagi, Yagi-Uda array, 935

zprop, propagation ofZ, 419

maximum angle of refraction, 250, 349

maximum available gain (MAG), 542

maximum stable gain (MSB), 543

maximum usable frequency (MUF), 286

mirages, 287mobility, 22momentum conservation, 33momentum density, 13momentum flux, 10monopole antennas, 646moving boundaryFresnel drag, 179oblique reflection from, 274reflection and transmission from, 177moving media, 177

moving mirror, 180multibeam array design, 850multilayer optical film, 359multilayer structures, 185, 302

at oblique incidence, 302birefringent, 354dielectric mirrors, 192, 332energy conservation in, 214equal-travel time, 208lossy, 304

reflection frequency response of, 213scattering matrix of, 214

multiple dielectric slabs, 175multiple reflections, 173multisection transmission lines, 478mutual impedance, 916

narrow-beam, low-sidelobe array design, 821natural rotation, 136

near fields of linear antennas, 905, 908negative index media, 7, 30, 70, 294Brewster angle in, 296equivalent conditions, 72propagation in, 70negative-index medialossy, 72perfect lens, 321Snel’s law, 294network analyzer, 526noise bandwidth, 617noise figure, 563, 622noise figure circles, 563noise model of a device, 621noise power, 617

noise temperature, 563, 617cellular base station, 618

of attenuator, 622

of cascaded devices, 623sky, 618

system, 621

normalized gain, 604numerical aperture, 254Nyquist frequency

in multilayer structures, 209

Ohm’s law, 6, 21ohmic power losses, 3, 12, 14, 24omnidirectional dielectric mirrors, 192, 332one-dimensional arrays, 783

operating gain circles, 557operating power gain, 541optical fibers, 253, 386graded index, 293numerical aperture, 254optical filters, 185antireflection coatings, 187dielectric mirrors, 192narrow-band transmission, 203shortpass and longpass, 202optical manhole, 252

optical rotation, 134, 136optically active media, 131optimum array directivity, 794

p-polarization, see polarizationparabolic reflector antennas, 749parasitic array, 931

PBG, see periodic bandgap structurespenetration depth, 55

for evanescent waves, 251perfect electric conductor, 662perfect lens, 297, 321perfect magnetic conductor, 662periodic bandgap structures, 203acoustic and vibration control, 203Bloch wavenumber, 194

fiber Bragg gratings, 203photonic crystals, 203transmission lines and waveguides, 203permeability, 3

permittivity, 3phase delay, 84phase thickness, 209, 303, 332phase velocity, 84

photonic crystals, 203physical constants, 949Pi-section matching network, 512plane wave incident on linear antenna, 858plane wave spectrum representation, 706plasma frequency, 18

plasmas, 25plasmonics, 274Pocklington equation solution, 899Pocklington integral equation, 857, 908Poisson spot, 715

Poisson’s spot, 715polarization, 4, 46, 241

charge density, 8linear, circular, 132

TE, perpendicular, s-polarization, 241

TM, parallel, p-polarization, 241polarization current and density, 6polarization ellipse, 47

polarizers, 133, 343beam splitters, 343reflective, 343, 358polarizers, dichroic, 138polaroid materials, 139positive real functions, 185power density, 601power gain circles, 554power gains, 539power losses, 3, 23power losses in transmission lines, 401power losses per unit volume, 3power transfer in transmission lines, 400, 423power transfer in waveguides, 367

power waves, 545Poynting vector, 12, 14, 40precursors, 91

principal-value integrals, 578prisms, 252

prolate array design, 843prolate matrix, 794, 843propagation

propagation impulse response examples, 87propagation matrices, 186

propagator frequency response, 83propagator impulse response, 83pulse compression, 112pulse compression filters, 114pulse compression, and chirping, 103pulse propagation, 82

pulse propagation, and group velocity, 94pulse spreading, 6, 97

QPSK modulation, 627

radiation fields from apertures, 666

radiation fields of magnetic currents, 665

radiation from apertures, 661

radiation from dipoles, 580

radiation from waveguide apertures, 726

RF spectrum, 950rhombic antennas, 650

S-parameters, 525available gain, 541available gain circles, 557generalized, 545input and output reflection coefficients, 531maximum available gain, 542

maximum stable gain, 543microwave amplifier design, 544network analyzers, 526noise figure circles, 563operating gain, 541operating gain circles, 557power flow, 529power gain circles, 554power gains, 539power waves, 545scattering matrix, 530simultaneous conjugate matching, 549stability circles, 533

stability criterion, 536transducer gain, 541traveling waves, 526unilateral gain circles, 555s-polarization, see polarizationsatellite links, 629

scattering matrix, 158, 214unitarity, 215scattering parameters, 525Schelkunoff’s zero-placement, 805Schur algorithm, 185

search radar, 633sector beam array design, 808Sellmeier equation, 20sensors, chemical and biological, 257, 320Shannon channel capacity, 627

SI units, 1, 949simultaneous conjugate matching, 549sine integrals, 960

single-stub tuner, 501sinusoidal current approximation, 861skin depth, 55

slef impedance, 916small dipole antenna, 641

Snel’s law, 242, 282Bouguer’s law, 290for lossy media, 248

in birefringent media, 353

in multilayer structures, 302

in negative-index media, 294solid angle, 595

solitons, 6Sommerfeld’s conducting half-space solution, 697spatial sampling theorem, 818

spherical coordinates, 593square loop antennas, 657stability circles, 533standard atmosphere, 291standing wave ratio, 428standing-wave antennas, 641stationary phase approximation, 963Stratton-Chu diffraction formulas, 680superluminal group velocity, 85, 105surface current, 60

surface impedance, 60surface plasmon resonance, 257, 312surface plasmons, 271, 312susceptibility, electric, magnetic, 4system noise temperature, 621system SNR, 622

Taylor line source array, 845Taylor one-parameter array design, 839Taylor’s ideal line source, 821Taylor’s one-parameter line source, 821Taylor-Kaiser arrays, 839

TE and TM impedance, 364

TE waves, 63

TE, TM, TEM modes, 363, 369telegrapher’s equations, 438, 457thick glasses, 170

thin films, 185, 191, 204thin-wire kernel, 856time averageenergy density, 14ohmic losses, 14Poynting vector, 14time-domain reflection response, 173time-domain reflectometry, 454time-domain response of transmission lines, 438

TM waves, 63total internal reflection, 249, 263critical angle of incidence, 250for birefringent media, 349frustrated, 308

maximum angle of refraction, 250transducer power gain, 541transfer matrix, 186, 194, 209, 212, 525transformers, see impedance matchingtransition matrix, see transfer matrixtranslational phase shift, 771

transmission lines, 397broadband terminations of, 185cascaded lines, 453

coaxial lines, 408coupled, 456coupled telegrapher’s equations, 457crosstalk, 462

determination of load impedance, 430distributed circuit model of, 415equivalent electrostatic problem, 397higher modes in, 412

impedance, inductance, capacitance, 398lattice timing diagrams, 441

microstrip lines, 404multisection lines, 478open and short circuited lines, 425parallel plate lines, 403

power losses, 401power transfer, 423reactive terminations, 443reflection response, 417rise time effects, 452Smith chart, 434standing wave ratio, 428telegrapher’s equations, 438terminated lines, 420Th´evenin equivalent circuit, 426, 476time-domain reflectometry, 454time-domain response, 438transient response, 439transmitted power, 400two-port equivalent circuit of, 419two-wire lines, 413

wave impedance, 417weakly coupled lines, 465transmittance, 165transmitted power, 159transverse

fields, 242Fresnel coefficients, 246, 247impedance, 244

propagation matrices, 245reflection coefficients, 245, 303refractive index, 244, 303wave impedance, 245transverse impedance, 64, 242, 364traveling wave antennas, 648traveling waves, 526triple-stub tuner, 507two-port network, 525two-section impedance transformer, 496

ultraviolet bands, 951uniaxial media, 132uniform apertures, 673uniform arrays, 789uniform plane waves, 36

INDEX 1043

units, 1

vector diffraction for apertures, 684

vector diffraction theory, 678

vector identities, 952

vector potential, 571

vee antennas, 650

velocity of light, 4

very large array (VLA), 788

very long base line array (VLBA), 788

cutoff wavenumber and frequency, 364

TE, TM, TEM modes, 363, 369

WMD, see wavelength division multiplexing

Woodward-Lawson array design, 812

Woodward-Lawson method, 819

Yagi-Uda antennas, 934

Zenneck surface wave, 269zero dispersion wavelength, 100zero-placement array design, 805

dualband - two-section dual-band Chebyshev impedance transformer

dualbw - two-section dual-band transformer bandwidths

stub1 - single-stub matching

stub2 - double-stub matching... integraldiffr - knife-edge diffraction coefficientdsinc - 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)hband - horn... 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