Random Rough Surface Scattering

PEC

xy

z

consider the rough surface scattering problem depicted above

note that TM to z is equivalent to the TE case (transverse to the direction of propagation)

Integral Equation

PEC

xy

z

Fj

AkAE0

2s

Fj

AkAEEEE0

2si

0F

0n̂E

on the PEC surface

Integral Equation

dt)kR(H)'x(J4

kAjk)x(E 20zz

iz

dxdx

)x(dy1)dy()dx(dt2

22

'dx'dx

)'x(dy1)kR(H)'x(J4

k)x(E2

20z

iz

)'x('f1)'x(J z where f = y(x’)

Tapered Incident Field

the rough surface has a finite length, is truncated at x=±L/2

the incident field cannot be a uniform plane wave, otherwise, diffraction from the end points may be significant

the incident field is chosen as a tapered wave that reads

2

2)tan())(1(gyxrwrkj

i

e

22

2

cos

1)tan(2

)(i

i

kg

gyx

rw

g is the tapering parameter

Long Surface is Required for Near Grazing Incidence

2i

iz

2iz

2

coskg

1EkE

1cos kg i 4Lg

near grazing incidence, the RHS would not be close to zero

g and therefore L must be very large for near grazing incidence

TE-to-z or TM (to the Direction of Propagation)

yA

xA

tJH xyt

iz )(

'dt)kR(HR

)'t(yy)'t(cos

R)'t(xx

)'t(sinj4

k)nm(Z m21

m

m

m

mmn

)'t(cos'dt)kR(HR

))'t(yym())'t(xx)('t(tanj4

k)nm(Z m21

m

mmn

'dx)kR(HR

))'t(yym())'t(xx('fj4

k)nm(Z m21

m

mmn

Note that we can use the MFIE for the thin shell problem since a tapered wave is used, as the other side of the surface has zero fields.

Difference between TE and TM Case

the former one has a symmetric impedance matrix while that of the latter one is non-symmetric

when the surface is large, the number of unknowns will be large and the matrix solution time will be long

we have developed a banded matrix iterative approach to solve a large matrix for the one-dimensional rough surface (a two-dimensional scattering) problem 

Spatial Domain Methods

one disadvantage of the spectral domain is that it requires numerical integration of infinite extend

spatial domain Green’s functions are not readily available for layered media with applications in microstrip antennas and high-frequency circuits

methods have been developed to circumvent this difficulty

we will discuss the complex image method

in conjunction with the Rao-Wilton-Glisson triangular discretization

the mixed-potential integral equation (MPIE)

Microstrip Structures

Ѐr

W

h

ground plane

microstrip line

dielectriclayer

has arbitrary surface conductor geometry

ground plane and substrate extend to infinity in the transverse direction and the space above the dielectric is unbounded

the substrate is homogeneous and isotropic, but not necessarily lossless

the upper conductor and ground plane have zero resistivity, and the upper conductor is infinitely thin

our goal is to compute the surface current distribution from which other parameters can be extracted

Mixed-Potential Integral Equation (MPIE)

it has weaker singularities in its Green’s functions than the EFIE, rendering more quickly convergent solutions

AjEinc

'ds)'r,r(G)'r(JAS

a

'ds)'r,r(G)'r(qS

qs

n

N

1nns fI)r(J

ss qjJ

Rao-Wilton-Glisson (RWG) Triangular Basis Functions

T+

T-

A+

A-

V+

V-

'r

r

r

r

these functions overlap, and each plate can be part of up to three different basis functions

O

Rao-Wilton-Glisson (RWG) Triangular Basis Functions

nn

nn A2

)r(f r

nT

nn

nn A2

)r(f r

nT

0)r(fn

in

in

otherwise

Rao-Wilton-Glisson (RWG) Triangular Basis Functions

the current flows from plate to , with maximum current across the common edge, zero current at the isolated vertices and , and no currents with components normal to the other four sides

the weighting in the current representation is such that the current normal to the common edge is continuous across that edge, and hence, no fictitious charge singularities arise

when the continuity equation is applied to the basis function, the charge density is a constant equal to and on each plate and the total charge is zero

nT

nT

nV

nV

n

n

A

n

n

A

Method of Moments

S

bdsab,a

mmminc f,f,Ajf,E

mnmnm

mnm

mnmmn 2A

2AjZ

'ds)r,r(G)'r(fA cma

Snmn

'ds)r,r(G)'r(f' cmq

Snmn

Simplications

many identical integrations will be performed

considerable computational effort is saved by evaluating and storing the scalar potential integral for all plate combinations and recalling these results as needed in evaluating the matrix elements

the same cannot be said of the integral for the vector potential, whose integrand evaluated over a particular source triangle depends on the identity of the isolated vertex and, hence, the basis function to which the triangle is assigned

Simplications'ds)'r,r(G)'r(gI c

maS

ii

where is set alternately to x’, y’and 1

the basis function dependence is removed from the integrals and reintroduced in the evaluation of Zmn through a weighted sum of these three

for each plate combination, a total of four scalar integrals are evaluated and later recalled in constructing the elements in impedance matrix Z

the excitation vector is given by

)'r(g i

2)r(E

2)r(Ev

cmc

minc

cmc

minc

mm

Spectral-Domain Green’s Function

no closed-form expressions for Ga and Gq in the spatial domain, but they can be represented in closed-form in the spectral domain

)k(R1k2j1

4G~ TE

0z

0a

TMTE2

20z

TE0z0

q RRkk

R1k2j1

41G~

Wave numbers kzo and k are the vertical and radial components of the free-space propagation constant ko in the cylindrical system

RTE and RTM are the reflection coefficients at the interface of TE and TM plane waves incident on the substrate with ground plane

Spatial Green’s Function through Transformation

the spatial-domain Green’s functions can then be expressed as an inverse Hankel-transform of the spectral counterparts, commonly referred to as Sommerfeld integrals

dkkkHG~G 2

0q,aq,a

Approximate analytic expressions exist for the evaluation of the Hankel-transform for in the near and far fields

a technique developed by Prof. Fang Dagang of the Nanjing University of Science and Technology and improved by Prof. Y. L. Chow allows efficient evaluation in all regions

Complex Image Method

the Sommerfeld integral is divided into three contributions: 1) quasi-dynamic images, 2) surface waves, and 3) complex images

the first two contributions, which dominate respectively in the near-and far-field, are extracted from RTE and RTM and handled analytically using the Sommerfeld Identity

what remains in RTE and RTM is relatively well behaved and exhibits exponential decay for sufficiently large values of k

the remainder can be accurately approximated with a short series of exponentials terms, which are interpreted as complex images

Complex Image Method

the exponents of the expansion are computed using Prony’s method or the matrix pencil method and the term weights then obtained through a least-square fit

the inverse Hankel-transform of the exponentials can be performed analytically, again using the Sommerfeld Identity

two to four expansion terms are appropriate, depending on the frequency

particular care should be taken in determining the number for expansion terms of Gq since its contribution in the scalar potential is a second-order difference arising from the source pulse-doublet and the testing procedure

Efficient Implementation

all elements in the impedance matrix can be computed from a linear combination of four scalar integrals evaluated for all source/test plate combinations

the surface integration over the source plate can be replaced by evaluation of the integrand at the plate centroid

cnma

cn

cnma

cn

n r,rGr,rG2

Amn

cnmq

cnmqn r,rGr,rG

mn

Efficient Implementation

the difference between the distances from the three vertices of the source plate to the test plate centroid is under some set maximum level, say 20%, then that approximation is viable

it makes sense to evaluate the Green’s functions with an interpolation table

both Ga and Gq exhibit 1/ and log singularities, so the table must begin at some minimum displacement governed by the interpolation scheme, the dominant 1/ singularity, and a maximum error criterion, say 1%

Efficient Implementation

the interactions can be catalogued by stepping through each plate combination

far-interactions are ignored, as they are too numerous to store and can be rapidly evaluated through the Ga and Gq

for near interactions, the four scalar integrals are evaluated and catalogued

subsequent plate combinations are then checked against the stored interactions and computed only if no equivalent interactions is available

Two plate interaction integrals are equivalent if the x- and y-displacements of the test plate centroid from the source plate vertices are identical

Expressions Needed

hk2jTE10

hk2jTE10

TE 1z

1z

er1er

R

qTMTE2

20z RRR

kk

)er1)(er1)(kk)(kk()e1)(1(k2

R hk2jTM10

hk2jTE100zr1z0z1z

hk4jr

20z

q 1z1z

1z

0z1z

0z1zTE10 kk

kkr

0zr1z

0zr1zTM10 kk

kkr

20

220z kkk

20r

221z kkk

Sommerfeld Identity

dkkkHkj

er

e

z

zjkrjk z)(

220

0

00

222 zr

Use of Sommerfeld Identity

)k(R1k2j1

4G~ TE

0z

0a

dkk)k(HRk2j1

re

4G 2

0TE0z0

rjk0

a

00

TMTE2

20z

TE0z0

q RRkk

R1k2j1

41G~

dkk)k(H)RR(k2j1

re

41G 2

0qTE0z0

rjk

0q

00

0r

Curve Fitting using Complex Exponentials

i

bjkiTE

izoeaR

i bi

r0jk

i0

rjk0

a rea

re

4G

bi00

2i

2bi br

it is unfortunate that brute force application of signal processing techniques would not yield satisfactory results in representing our functions

Quasi-Dynamic Contributions

at very low frequency, , RTE and Rq can be reduced to the quasi-dynamic form given by

1z0z kk

hjk20TETE

0zeRR

)Ke1)(e1(KKe1

)e1(KRRq hk2jhk4jhk2j

hk4j

0q0z0z

0z

0z

)1/()1(K rr

Quasi-Dynamic Contributions

1

rjk

0

rjk0

0a re

re

4G

1000

22n )nh2(r

3

rjk2

2

rjk

1

rjk2

0

rjk

0

rjk

00q r

eKr

eKr

eKr

eKr

e4

1G3020100000

surface-wave contribution dominates in the far field

Surface-Wave Contributions

)(212)(2/12/1

220

20

22

kHjdkkH

kkkk

TETEp

pzokk

Rkkkj

spp

)(

1 21

limRe

)(21

limRe),(

2 qTETMTEp

pzokk

RRkkkj

spp

ppaSW kkHsjG 2

010 Re2

4

ppqSW kkHsjG 2

020 Re2

4

Complete Expressions

dkkkH)k(Fk2j1

4GGG 2

01zo

0aSW0aa

dkkkH)k(F

k2j1

41GGG 2

02zo0

qSW0qq

0z2p

21p

0TETE1 k2jkk

sRek2RR)k(F

0z2p

22p

0q0TEqTE2 k2jkk

sRek2RRRR)k(F