Random Rough Surface Scattering
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Transcript of Random Rough Surface Scattering
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