Theoretical and computational aspects of scattering
from rough surfaces: one-dimensional perfectly
reflecting surfaces
J DeSanto, G Erdmann, W Hereman and M Misra
Department of Mathematical and Computer Sciences, Colorado School of Mines,
Golden, CO 80401-1887, USA, Phone: (303) 273-3036, Fax: (303) 273-3875, email:
jdesanto/gerdmann/whereman/mmisra@mines.edu
Abstract. We discuss the scattering of acoustic or electromagnetic waves from one
dimensional rough surfaces. We restrict the discussion in this report to perfectly
reflecting Dirichlet surfaces (TE-polarization). The theoretical development is for both
infinite surfaces and periodic surfaces, the latter equations derived from the former. We
include both derivations for completeness of notation. Several theoretical developments
are presented. They are characterized by integral equation solutions for the surface
current or normal derivative of the total field. All the equations are discretized to a
matrix system and further characterized by the sampling of the rows and columns of
the matrix which is accomplished in either coordinate space (C) or spectral space (S).
The standard equations are referred to here as CC equations of either first kind (CC1)
or second kind (CC2). Mixed representation equations or SC type are solved as well
as SS equations fully in spectral space.
Computational results are presented for scattering from various periodic surfaces.
The results include examples with grazing incidence, a very rough surface and a
highly oscillatory surface. The examples vary over a parameter set which includes the
geometrical optics regime, physical optics or resonance regime, and a renormalization
regime.
The objective of this study was to determine the best computational method for
these problems. Briefly, the SC method was the fastest but did not converge for large
slopes or very rough surfaces for reasons we explain. The SS method was slower and
had the same convergence difficulties as SC. The CC methods were extremely slow but
always converged. The simplest approach is to try the SC method first. Convergence,
when the method works, is very fast. If convergence doesn’t occur then try SS and
finally CC.
2
1. Derivation of CC Equations For an Infinite One-Dimensional Perfectly
Reflecting Rough Surface
We consider the scattering from an infinite one-dimensional rough surface specified by
z = s(x) (see Figure 1). For the examples we consider in this report the surface is
perfectly reflecting and periodic. In this section we first consider the surface to be
infinite and specify it to be periodic later in Section 4. Our computational results
are restricted to periodic surface cases. Notationally we have a spatial 2-vector
x = (x, z) = (x1 , x2 ) and its restriction to the surface xs = (x, s(x)). The gradient
operator is ∂i = ∂x∂ i (i = 1, 2) and the normal derivative ∂n = ni ∂i where ni is
the normal to the surface and repeated subscripts are summed (here from 1 to 2).
Fields are represented by ψ and correspond to a velocity potential (acoustics) [6] or the
y−component of the electric (Dirichlet boundary value problem) or magnetic (Neumann
boundary value problem) fields. Since the surface is one-dimensional its generator is
parallel to the y−axis and no polarization change occurs during scattering from such a
surface. The electromagnetic problem thus reduces to a scalar one, which is what we
treat. All fields are time-harmonic so that a factor exp(−iωt) is suppressed throughout
(ω is circular frequency and t is time).
is the
The scattered field satisfies the scalar Helmholtz equation (k1 = 2π
λ
wavenumber and λ is wavelength)
(∂i ∂i + k12 )ψ sc (x) = 0,
+
x ∈ DR
.
(1.1)
The free-space two-dimensional Green’s function G(2) for this problem satisfies the nonhomogeneous Helmholtz equation
(∂i ∂i + k12 )G(2) (x, x ) = −δ(x − x ),
(1.2)
where the right hand side is the Dirac delta function in two-dimensions. G(2) is explicitly
given by [6, pg. 54]
i (1)
G(2) (x, x ) = H0 (k1 |x − x |),
4
the Hankel function of zeroth-order, first kind. Its Fourier transform relation is
1
α,
eiα . (x−x ) G̃(α)d
G(2) (x, x ) =
2
(2π)
(1.3)
(1.4)
where
G̃(α) =
α2
1
2
− k1+
(1.5)
and we have chosen k1+ = lim→0+ (k1 + i) to indicate that we have an outgoing wave.
+
is specified by the characteristic function
The region DR
θ+ (x) = θ(z − s(x))θ(R − r),
(1.6)
3
sc
+
HR
sc
in
sc
+
DR
R
s(x)
-
DR
-
HR
Figure 1. Infinite perfectly reflecting one-dimensional rough surface z = s(x). Incident
+
(in) and scattered (sc) wave are indicated. The region DR
is specified by z > s(x)
+
and r = |x| < R and is bounded by the rough surface and the semicircle HR
at radius
−
−
R. The complementary region (DR ) and semi-circle (HR ) below the surface are also
illustrated.
in the limit as R → ∞. Here θ is the usual Heaviside function
θ(x) =
1,
0,
x > 0,
x < 0,
(1.7)
and r = |x|. The function θ+ thus represents the region bounded by the rough surface
s(x) truncated at R and the upper semicircle at radius R denoted by HR+ (see Figure
1). Vertical segments joining the surface and the semicircle can also be included [8] but
are omitted in the interests of brevity.
To form equations for the scattered field we use Green’s theorem. Multiply Eq. (1.1)
by G(2) and Eq. (1.2) by ψ sc and subtract the resulting equations. We get
∂i ∂i G(2) (x, x )ψ sc (x ) − G(2) (x, x )∂i ψ sc (x ) = −δ(x − x )ψ sc (x ). (1.8)
Next, multiply Eq. (1.8) by θ+ (x ), integrate over all space (in x ) and then integrate
4
by parts using the vector derivative of the characteristic function
∂j θ+ (x ) = nj (x )δ(z − s(x ))θ(R − r ) − nj (R)δ(r − R)θ(z − s(x )), (1.9)
where
nj (x ) = δj2 − δj1 s (x ),
(1.10)
is the non-unit normal to the surface s and
nj (R) = ∂j r |r=R ,
(1.11)
the radial normal to the semicircle HR+ . Two surface integrals result. The integral over
the semicircle is
+
HR
∂r G(2) (x, x )ψ sc (x ) − G(2) (x, x )∂r ψ sc (x )
r =R
Rdθ ,
(1.12)
where θ is the integration angle in [− π2 , π2 ]. If ψ sc satisfies a Sommerfeld radiation
condition this integral vanishes as R → ∞. More generally, so long as ψ sc does not
contain any horizontally propagating plane waves, the integral vanishes as R → ∞
[8]. We include the latter restriction since we treat the case of plane wave incidence in
our periodic surface examples later. If there is an incident plane wave we must admit
scattered plane waves in order to balance the total energy on this far away semicircle
[7]. For horizontal plane wave incidence and scattering other equations result than the
ones we quote below [8].
We thus assume that Eq. (1.12) vanishes as R → ∞. The result using Eq. (1.9) is
a single integral over the infinite surface s∞ (x). To write this result in convenient form,
define single (S) and double (D) layer acoustic potentials with respective densities u
and v as
(Su)(x) =
and
s∞
(Dv)(x) =
s∞
G(2) (x, xs )u(xs )dx ,
(1.13)
∂n G(2) (x, xs )v(xs )dx ,
(1.14)
as well as the normal derivative of ψ sc
N sc (x) = ∂n ψ sc (x).
(1.15)
The resulting equations can then be written as
(Dψ sc )(x) − (SN sc )(x) = θ+ (x)ψ sc (x),
(1.16)
+
−
which for x ∈ D∞
gives the representation of the scattered field, and for x ∈ D∞
(the
lower region below the surface) the equation is referred to as an extinction theorem [13].
5
To form surface integral equations use the limiting properties of single and double
layer potentials [3]. Define the limits from above (+) and below (−) as
lim (Su)(x) = (Su)± (xs ),
(1.17)
lim± (Dv)(x) = (Dv)± (xs ).
(1.18)
x→x±
s
and
x→xs
The single layer is continuous
(Su)+ (xs ) = (Su)− (xs ),
(1.19)
and the double layer has a jump discontinuity
(Dv)+ (xs ) − (Dv)− (xs ) = v(xs ),
(1.20)
with each limit defined as
1
(1.21)
(Dv)± (xs ) = PV(Dv)(xs ) ± v(xs ),
2
where PV stands for Cauchy Principal Value. Using these limiting values the surface
integral equation which follows from Eq. (1.16) is
1
(1.22)
PV(Dψ sc )(xs ) − (SN sc )(xs ) = ψ sc (xs ).
2
The kernel terms in this integral equation have both arguments in coordinate space (on
the surface) so that a discretized version of them will yield (kernel) matrices whose rows
and columns result from coordinate-space sampling. We thus refer to this equation as
a coordinate-coordinate (CC) equation.
A second equation can be formed by taking the normal derivative of Eq. (1.16) for
+
. The normal derivative of the single layer potential is discontinuous with limits
x ∈ D∞
1
∂n (Su)± (xs ) = PV∂n (Su)(xs ) ∓ u(xs ),
(1.23)
2
and the normal derivative of the double layer has the same limit from above and below
but is singular and we take its Hadamard Finite Part (FP) [9]
lim ∂n (Dv)(x) = FP∂n (Dv)(xs ).
x→x±
s
(1.24)
[Note: For now the use of PV and FP notation is purely formal.] The result is another
CC integral equation
1
(1.25)
FP∂n (Dψ sc )(xs ) − PV∂n (SN sc )(xs ) = N sc (xs ).
2
The usual boundary value problems we wish to discuss involve total field quantities. We
treat the incident field next in Section 2 and combine the results into integral equations
on the total field.
6
2.
Incident and Total Fields
We form integral equations of the total field (ψ T ) and normal derivative (N T ). These
are defined by
ψ T (x) = ψ i (x) + ψ sc (x),
(2.1)
N T (x) = N i (x) + N sc (x),
(2.2)
and
in terms of the incident (i) field and its normal derivative.
homogeneous Helmholtz equation
ψ i (x) satisfies the
(∂j ∂j + k12 )ψ i (x) = 0.
(2.3)
Our examples later are for a single plane wave
ψ i (x) = φp (x) = Deik1 (α0 x−β0 z) ,
(2.4)
where α0 = sin(θi ), β0 = cos(θi ), and θi is the angle of incidence defined from the
z−direction. We choose D = 1 for computations. We also omit horizontally incident
waves, so β0 > 0. More generally, we could have a continuous superposition (or spectral
decomposition) of plane waves
ψ i (x) = φc (x) =
where
⎧
⎪
⎨
m(µ) =
⎪
⎩
∞
−∞
I(µ)eik1 [µx−m(µ)z] dµ,
1
(1 − µ2 ) 2 ,
µ ≤ 1,
i(µ2 −
µ > 1.
1
1) 2 ,
(2.5)
(2.6)
The density I(µ) is a continuous function. Eq. (2.4) is a special case if we let I(µ) be
the distribution Dδ(µ − α0 ).
1
Asymptotically for r = (x2 +z 2 ) 2 large, Eqs. (2.4) and (2.5) behave quite differently.
Eq. (2.4) has no limit and Eq. (2.5) (for continuous I(µ)) behaves like a cylindrical
+
and an outgoing
wave and satisfies the incoming cylindrical radiation condition in D∞
−
cylindrical wave radiation condition in D∞ , the domain below the surface.
The results of Green’s theorem can be summarized by defining the bracket integral
of Green’s theorem on a surface P for any field φ
(2)
[G , φ; x, P ] =
P
G(2) (x, xp )∂n φ(xp ) − ∂n G(2) (x, xp )φ(xp ) dx .
(2.7)
The result is [8]
lim [G(2) , φp ; x, HR+ ] = φp (x),
R→∞
(2.8)
7
and
lim [G(2) , φp ; x, HR− ] = 0.
R→∞
(2.9)
For a single plane wave the equation corresponding to Eq. (1.16) is
(Dφp)(x) − (SN p )(x) = θ+ (x)φp (x) − φp (x),
(2.10)
where the last term follows from Eq. (2.8) and N p is the normal derivative of the plane
wave. Combining Eqs. (2.10) and (1.16) we have a representation for the total field at
x (off the surface)
θ+ (x)ψ T (x) = φp (x) + (Dψ T )(x) − (SN T )(x),
(2.11)
and in the limit as x approaches the surface s(x) the surface integral equation
1 T
ψ (xs ) = φp (xs ) + PV (Dψ T )(xs ) − (SN T )(xs ).
2
(2.12)
+
A second equation can be derived using the normal derivative of Eq. (2.11) for x ∈ D∞
and subsequently taking the limit as x approaches the surface. It is
1 T
N (xs ) = N p (xs ) + FP ∂n (Dψ T )(xs ) − PV ∂n (SN T )(xs ).
2
(2.13)
−
Further, using Green’s theorem in D∞
on G(2) and φc and combining the result with
the scattered field results in Section 1, an analogous set of equations to Eqs. (2.12)
and (2.13) results (with the modification that the plane wave term is replaced by the
continuous superposition).
Thus, for an incident field of either form (excluding horizontal plane waves) we can
write Eqs. (2.12) and (2.13) as
1 T
ψ (xs ) = ψ i (xs ) + PV (Dψ T )(xs ) − (SN T )(xs ),
2
(2.14)
1 T
N (xs ) = N i (xs ) + FP ∂n (Dψ T )(xs ) − PV ∂n (SN T )(xs ).
2
(2.15)
and
These are both coordinate-coordinate (CC) integral equations on the boundary
unknowns ψ T and N T . Both are valid for an infinite surface with the restriction that
no horizontal plane waves occur. For completeness and later use we include the field
representation which follows from Eq. (2.11) for an incident field satisfying the above
restrictions. It is
θ+ (x)ψ T (x) = ψ i (x) + (Dψ T )(x) − (SN T )(x).
(2.16)
8
3.
Dirichlet Problem
For the Dirichlet (D) boundary value problem
ψ T (xs ) = 0.
(3.1)
Acoustically this describes a soft surface and electromagnetically it is the case of TEpolarization. With this condition Eq. (2.14) becomes
ψ i (xs ) = (SN T )(xs ),
(3.2)
which is referred to as CC1, a first-kind integral equation [15] for the remaining boundary
unknown N T . Eq. (2.15) becomes
1 T
N (xs ) = N i (xs ) − PV ∂n (SN T )(xs ),
(3.3)
2
which is an integral equation of the second kind, and we refer to it as the CC2 equation.
Often the two equations are linearly combined as follows. Choose real constants α and
β, multiply Eq. (3.2) by α and Eq. (3.1) by β, and add the resulting equations. Since all
the functions are evaluated on the surface they are functions of a single variable. Define
the incident field function by
F i (x) = βψ i (xs ) + αN i (xs ).
(3.4)
The resulting added equations can be put in the impedance form
i
F (x) =
∞
−∞
Z D (x, x )N T (x )dx ,
(3.5)
where the “impedance” kernel is defined symbolically as
1
Z D (x, x ) = αδ(x − x ) + α PV ∂n G(2) (xs , xs ) + βG(2) (xs , xs ).
(3.6)
2
For α = 0 we have CC1. For β = 0, CC2, and for β = 1 and α arbitrary we have
what is referred to as the Combined Field Integral Equation (CFIE) (see [12]). It
becomes particularly important for scattering from bounded bodies where the CC1 and
CC2 solutions contain different resonances but the CFIE solutions remain finite at all
frequencies.
4.
Periodic Surface
For a periodic surface with period L, s(x + L) = s(x). We can then reduce Eq. (3.5) to
an integration over a single period cell, say from − L2 to L2 . The single layer potential
term in Eq. (3.2) can be written as
(SN T )(xs ) =
∞
−∞
G(2) (xs , xs )N T (x )dx
(4.1)
9
∞
=
n=−∞
In (x),
(4.2)
where
In (x) =
(2n+1) L
2
(2n−1) L
2
G(2) (xs , xs )N T (x )dx .
(4.3)
In Eq. (4.3) use the Weyl representation for the Green’s function [6, pg. 63]
πi ∞ 1 ik1 [µ(x−x )+m(µ)|s(x)−s(x )|]
dµ,
e
(2π)2 −∞ m(µ)
G(2) (xs , xs ) =
(4.4)
(with m(µ) defined by Eq. (2.6)), the Floquet (pseudo-)periodicity of the boundary
unknown
N T (x + nL) = eik1 α0 nL N T (x),
(4.5)
and the change of variables x = x − nL.
We can then write Eq. (4.1) on the domain [− L2 , L2 ] as
T
(SN )(xs ) =
L
2
−L
2
Gp1 (x, x )N T (x )dx ,
(4.6)
where Gp1 is the periodic Green’s function with wavenumber k1 given by
πi
Gp1 (x, x ) =
(2π)2
∞
−∞
∞
1 ik1 [µ(x−x )+m(µ)|s(x)−s(x )|] e
.
eink1 L(α0 −µ) dµ.
m(µ)
n=−∞
(4.7)
For scalar arguments x and x , Gp1 is confined to the surface.
Next, use the Poisson sum [17]
∞
n=−∞
eint = 2π
∞
δ(t + 2πj),
(4.8)
j=−∞
where t = k1 L(α0 − µ). The delta function in Eq. (4.8) reduces to
δ(t + 2πj) =
1
δ(µ − αj ),
k1 L
(4.9)
where
λ
αj = α0 + j .
L
(4.10)
This is the Bragg equation with αj = sin(θj ) and θj the angle of the j th outgoing Bragg
wave. Using Eqs. (4.8) to (4.10) to evaluate Eq. (4.7) we get
Gp1 (x, x ) =
∞
1 ik1 [αj (x−x )+βj |s(x)−s(x )|]
i λ e
,
4π L j=−∞ βj
(4.11)
10
with
⎧
⎪
⎨
βj = ⎪
⎩
1
(1 − αj ) 2 ,
2
i(αj −
1
1) 2 ,
|αj | ≤ 1,
(4.12)
|αj | > 1.
Other representations for this periodic Green’s function, useful for its evaluation in
computations, are discussed in Appendix A. Further, it is also convenient to have a
representation for this function off the surface, and it is obviously given by
Gp1 (x, x ) =
∞
1 ik1 [αj (x−x )+βj |z−z |]
i λ e
.
4π L j=−∞ βj
(4.13)
Similarly, the normal derivative of the single layer potential term in Eq. (3.3) can be
reduced to a single period cell. The result is
PV
∞
−∞
L
2
∂n G(2) (xs , xs )N T (x )dx = − L Gp1 (x, x )N T (x )dx ,
−2
(4.14)
where
Gp1 (x, x ) = [nj ∂j Gp1 (x, x )]|z=s(x)
|z =s(x )
(4.15)
follows from Eq. (4.13). The slash on the integral in Eq. (4.14) represents Cauchy
principal value (if appropriate). Using Eqs. (4.6) and (4.14), Eq. (3.4) reduces to
F i (x) =
L
2
−L
2
ZpD1 (x, x )N T (x )dx ,
(4.16)
where the impedance kernel is given by
1
(4.17)
ZpD1 (x, x ) = αδ(x − x ) + α PV Gp1 (x, x ) + βGp1 (x, x ).
2
Again, for α = 0 we have the CC1 equation, for β = 0 the CC2 equation, and for β = 1
and α arbitrary the CFIE equation. These are the equations which are solved for our
examples. The numerical solution is discussed in Appendix B. Discretization of Eq.
(4.16) yields an impedance matrix whose rows and columns result from sampling both
in coordinate space, thus the acronym coordinate-coordinate or CC.
5. Derivation of SC Equations For an Infinite One-Dimensional Perfectly
Reflecting Rough Surface
In the previous four sections we confined our attention to problems where both rows and
columns of the matrix to be inverted were sampled in coordinate space. Here we derive
a mixed representation where the rows are sampled in the conjugate spectral (S) space
and the columns still sampled in the coordinate space. These are the SC equations.
A straightforward derivation without using any of the results in the first four sections
11
can be found in the literature [5]. A different derivation which however yields the same
results proceeds as follows.
Use the representation for the total field given by Eq. (2.16) for the Dirichlet problem
T
(ψ (xs ) = 0). We have
ψ sc (x) = −(SN T )(x),
+
x ∈ D∞
,
(5.1)
and
ψ i (x) = (SN T )(x),
−
x ∈ D∞
.
(5.2)
The Weyl representation Eq. (4.4) written off the surface in the x−variable is
πi
(2π)2
G(2) (x, xs ) =
∞
−∞
1 ik1 [µ(x−x )+m(µ)|z−s(x )|]
e
dµ.
m(µ)
(5.3)
For z > max[s(x)] we can remove the absolute value in the phase of Eq. (5.3) and for
these values of z write Eq. (5.1) using Eqs. (1.13) and (5.3) as
sc
ψ (x) =
where
A(µ) =
∞
−∞
A(µ)eik1 [µx+m(µ)z] dµ,
(5.4)
−i ∞ −ik1 [µx +m(µ)s(x )] T e
N (x )dx .
4πm(µ) −∞
(5.5)
Once we know N T we can thus evaluate A(µ) and the scattered field. Eq. (5.4) is a
1
spectral representation of the scattered field. For large r = (x2 + z 2 ) 2 (where x = r sin θ
and z = r cos θ), a stationary phase evaluation of Eq. (5.4) can be written in terms of
the scattering amplitude T (θ) as
eik1 r
ψ sc (x) ∼ T (θ) √ ,
r
(5.6)
which is an outgoing cylindrical wave where
1
T (θ) = (−2πi) 2 A(µsp ),
(5.7)
and the stationary phase point is µsp = sin θ. The amplitude A(µ) is thus directly related
to the scattering amplitude.
To solve for N T use Eq. (5.2) and the representation Eq. (5.3) now for z < min[s(x)].
The result is
i
ψ (x) =
∞
−∞
I(µ)eik1 [µx−m(µ)z] dµ,
(5.8)
which is just Eq. (2.5) and where
i
I(µ) =
4πm(µ)
∞
−∞
e−ik1 [µx−m(µ)s(x)] N T (x)dx.
(5.9)
12
Given the properties of the incident field, i.e. I(µ), we solve the first kind integral
equation Eq. (5.9) for N T and use this to evaluate the scattered field. The kernel of
the integral equation is now a function of µ (spectral, S) and x (coordinate, C) and the
method is referred to as spectral-coordinate (SC).
We can write Eqs. (5.5) and (5.9) in the symmetric representation
±
I (µ) =
∞
−∞
e−ik1 [µx∓m(µ)s(x)] N T (x)dx,
where
±
I (µ) = 4πi m(µ)
6.
−I(µ)
A(µ) .
(5.10)
(5.11)
SC Equations For a Periodic Surface
For a periodic surface s(x + L) = s(x), and we can write Eqs. (5.10) as
∞
n=−∞
±
Q±
n (µ) = I (µ),
(6.1)
where
Q±
n (µ) =
(2n+1) L
2
(2n−1) L
2
e−ik1 [µx∓m(µ)s(x)] N T (x)dx.
(6.2)
Again, change variables to x = x − nL, and use the Floquet periodicity of N T given by
Eq. (4.5). The result is
ink1 L(α0 −µ) ±
Q0 (µ).
Q±
n (µ) = e
(6.3)
Use of the Poisson sum, Eq. (4.8), and Eq. (4.9) yield for Eq. (6.1)
∞
λ ±
Q0 (µ)
δ(µ − αj ) = I ± (µ).
L
j=−∞
(6.4)
Integration of both sides of Eq. (6.4) over the µ−domain [αn − Lλ , αn + Lλ ] where
0 < < 1 yields
αn +
λ ±
L ±
Q0 (αn ) =
λ I (µ)dµ.
L
αn −
L
λ
(6.5)
For a single incident plane wave (see Eq. (5.8))
I(µ) = Dδ(µ − α0 ),
(6.6)
and, for a periodic surface, the scattered field spectra are discrete
A(µ) =
∞
n=−∞
An δ(µ − αn ).
(6.7)
13
(This can be seen by reducing Eq. (5.5) to the integration over a single periodic cell).
Using these results in Eq. (6.5) and the definitions Eq. (5.11) we get
λ ±
(6.8)
Q (αn ) = I ± (αn ),
L 0
where
−Dδn0
±
(6.9)
I (αn ) = 4πi βn
An .
T
T
The integrals Q±
0 have dimensions of length times the dimensions of N . Also, N has
dimensions of inverse length times the dimensions of the field. It is convenient to scale
out this inverse length by defining the function N(x) as
N T (x) = ik1 N(x),
(6.10)
so that N(x) has the same dimensions as the field ψ T . (The scaling Eq. (6.10) obviously
relates to the fact that differentiation of a wave-like field quantity produces a factor
ik1 .) The result can be written as
P0± (αn ) = F ± (αn ),
where
P0± (αn )
1
=
L
−L
2
−L
2
and
±
F (αn ) = 2βn
e−ik1 [αn x∓βn s(x)] N(x)dx,
−Dδn0
An .
(6.11)
(6.12)
(6.13)
The method of solution is to solve the first kind equation for N(x) (the “ + ” equation)
then evaluate the “ − ” equation for An . The scattered field from Eqs. (5.4) and (6.7) is
then
ψ sc (x, z) =
∞
n=−∞
An eik1 (αn x+βnz) .
(6.14)
The kernels in Eq. (6.12) are functions of µ = αn and x, i.e. a discrete spectral parameter
n and a coordinate variable, thus again the spectral-coordinate (SC) acronym. An
alternative derivation of these results can be found in the literature [4].
7.
SS Equations For a Periodic Surface
We derive the spectral-spectral (SS) equations from the SC equations in Section 6. The
method is to expand the boundary unknown in the topological or surface wave basis
[10] in Eq. (6.12)
N(x) =
∞
j =−∞
Ñj eik1 (αj x−βj s(x)) .
(7.1)
14
Note that this “basis” is not a complete set. The justification for its choice rests on the
fact that in many cases it produces an extremely fast and highly accurate result. The
result is first a system of linear equations for the vector of expansion coefficients in the
discrete spectral domain Ñ = {Ñj }
K̃Ñ = F+ ,
(7.2)
with the components
Fj+ = −2β0 Dδj0 ,
and the matrix whose entries are
1 π −i(j−j )y ik1 (βj −βj )s( L y)
2π
e
e
dy,
K̃jj =
2π −π
(7.3)
(7.4)
where the integrals have been scaled to [π, π], and second, the set of equations to evaluate
for the Aj coefficients is
j
M̃jj Ñj = 2βj Aj ,
where the matrix elements are
1 π −i(j−j )y −ik1 (βj +βj )s( L )y
2π
e
e
dy.
M̃jj =
2π −π
(7.5)
(7.6)
The scattered field is then given by Eq. (6.14). Rows and columns of both K̃ and
M̃ are indexed in the (discrete) spectral integer j and the method is referred to as
spectral-spectral (SS).
8.
Energy
We know that the sum of the scattered energy must equal the energy in the incident
wave
j
|Aj |2 Re(βj ) = β0 D 2 .
(8.1)
Only real (Re) orders carry energy away from the surface. We set D = 1 for our trials
and quote the condition as
j
|Aj |2
Re(βj )
= 1.
β0
(8.2)
The left hand side of Eq. (8.2) is referred to as the normalized energy. The Aj are
computed and then we determine how well the energy check Eq. (8.2) is satisfied. It is
a necessary but not sufficient condition of accuracy.
15
9.
Computational Results
This section presents timing and reliability results for several formalisms on several
surfaces. λ/∆x is a measure of pulse width, with higher numbers corresponding to
faster sampling. λ/∆x = 10 is often quoted, but it can be either oversampling or
undersampling. The surface graphs use equal scales on the horizontal and vertical axes,
so apparent tangency is true tangency. The best CC method is used for the surface
current and scattered amplitude plots, since spectral methods often have incorrect
currents. An additional result is referred to as CG, a CC Galerkin approach with
Fourier basis functions for a first kind equation. This is very similar to CC1, but the
self-cell integral is performed exactly, not using the first few terms in an expansion.
A discussion of the numerical methods for CC, SC, and CG methods can be found in
Appendices B through D respectively. The SS numerical technique is treated in Section
7. For our calculations with CG, we set α = 0 and β = 1 (see Appendix D).
In Sections 9.1 to 9.5 we treat a variety of rough surface examples. In Sections 9.1
to 9.3 respectively we consider the cases when λ/L 1 (geometrical optics regime),
λ/L ≈ 1 (resonance or physical optics regime), and λ/L 1 (sometimes referred to as
a renormalization regime) all for a cosine surface. In Section 9.4 we treat a very rough
surface with a maximum slope of about 25. In Section 9.5 we present results of a case
with a highly oscillatory surface with the oscillations increasing as the end points of the
period are approached.
The results of these computations can be summarized as follows. The CC methods
always worked well in the sense that the error was small for a sufficiently large matrix.
This is illustrated in Figures 2-9 and the accompanying tables contained in Sections 9.1
to 9.5 (Examples 1-5). For the CC methods, fill time refers to the time necessary to
compute the matrix elements, here the time to compute the periodic Green’s function
and its normal derivative. This took a great deal of time (see the discussion in Appendix
B), and the result was that the CC methods were extremely slow.
The fill time for the SC method was several orders of magnitude faster than CC
(this is because we were only evaluating a function, as shown in Appendix C). SC was
clearly the solution method of preference when it worked. It failed to work for very rough
(Example 4) and highly oscillatory (Example 5) surfaces. The fill time for the SS method
was between that for CC and SC and consisted of the evaluation of matrix elements
of the form given in Eq. 7.4. The SS method is based on the same type of topological
basis expansion as the SC method and it had the same convergence difficulties as the
SC method.
16
9.1. Example 1, λ/L 1
Case A, No Grazing
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
0.075
0.01563553622559
20◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
λ/∆x Fill Time
SS
128 by 128
4788
SS
138 by 138
6272
SS
148 by 148
7930
SC
128 by 128
2.0
0.64
SC
138 by 138
2.2
0.74
SC
148 by 148
2.3
0.82
CC1
64 by 64
1.0
501
CC1
128 by 128
2.0
2039
CC1
256 by 256
4.0
8165
CC2
64 by 64
1.0
808
CC2
128 by 128
2.0
3283
CC2
256 by 256
4.0
13201
CG
65 by 65
1.0
515
CG
129 by 129
2.0
2133
CG
257 by 257
4.0
8401
Linear Solution
Time
Error
0.24
-14.8
0.74
-15.3
0.88
-15.4
0.62
-15.7
0.72
-15.7
0.91
-15.1
0.11
-0.2
0.61
-1.6
4.18
-2.7
0.10
0.1
0.62
-5.7
4.04
-8.0
0.10
-0.2
0.61
-1.3
4.15
-3.4
Table 1. Example 1, Case A, no grazing incidence and 128 real Bragg modes. The
error for SS and SC was extremely small and the fill time for the SC matrix elements
nearly negligible. The only way a reasonably small error could be attained for the
coordinate based methods was to increase the matrix size. Fill time plus solution time
for the SC method was negligible compared to the other methods.
17
Surface and Incoming Waves
Scattered Energy Distribution
0.1
0.4
0.3
0.08
0.2
0.06
0.1
0
0.04
−0.1
−0.2
0.02
−0.3
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0
−0.06
−0.04
−0.02
0
(a)
0.02
0.04
0.06
(b)
|N(x)| vs x
Real(N(x)) vs x
2.5
2.05
2
2
1.5
1.95
1
0.5
1.9
0
1.85
−0.5
−1
1.8
−1.5
1.75
−2
−2.5
−0.5
−0.4
−0.3
−0.2
−0.1
0
(c)
0.1
0.2
0.3
0.4
0.5
1.7
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 2. Example 1, Case A: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC2 formalism,
matrix size 256 by 256.
18
Case B, Near-Grazing Incidence/Reflection
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
0.075
0.01566499626662
75◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
λ/∆x Fill Time
SS
128 by 128
1305
SS
138 by 138
2058
SS
148 by 148
2901
SC
128 by 128
2.0
0.76
SC
138 by 138
2.2
0.74
SC
148 by 148
2.3
0.83
CC1
64 by 64
1.0
541
CC1
128 by 128
2.0
2194
CC1
256 by 256
4.0
8781
CC2
64 by 64
1.0
848
CC2
128 by 128
2.0
3450
CC2
256 by 256
4.0
13851
CG
65 by 65
1.0
574
CG
129 by 129
2.0
2296
CG
257 by 257
4.0
9070
Linear Solution
Time
Error
0.24
-0.9
0.76
0.1
0.86
-0.6
0.59
-0.5
0.73
-0.9
0.88
-3.1
0.10
-0.5
0.60
-0.4
4.11
-4.7
0.11
-0.3
0.59
-0.8
3.98
-4.9
0.11
-0.5
0.62
-0.5
4.12
-3.9
Table 2. Example 1, Case B, near-grazing incidence and reflection with 128 real Bragg
modes. The error for all methods (especially SS and SC) increased considerably. Errors
for the coordinate based methods were adequate only for large matrix size. Again, fill
time plus solution time for SC was negligible compared to all the other methods.
19
Surface and Incoming Waves
Scattered Energy Distribution
0.4
0.3
0.1
0.2
0.05
0.1
0
0
−0.1
−0.2
−0.05
−0.3
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−0.15
0.5
−0.1
−0.05
(a)
1.2
1
1
0.5
0.8
0
0.6
−0.5
0.4
−1
0.2
−0.3
−0.2
−0.1
0
(c)
0.1
|N(x)| vs x
1.5
−0.4
0.05
(b)
Real(N(x)) vs x
−1.5
−0.5
0
0.1
0.2
0.3
0.4
0.5
0
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 3. Example 1, Case B: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC1 formalism,
matrix size 256 by 256.
20
9.2. Example 2, λ/L ≈ 1
Case A, No Grazing
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
0.25
0.95
20◦
Error = log10 |1 − Normalized Energy|
Formalism
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
CC1
CC1
CC1
CC2
CC2
CC2
CG
CG
CG
A−1
A0
Matrix
Size
2 by 2
6 by 6
10 by 10
14 by 14
18 by 18
22 by 22
2 by 2
6 by 6
10 by 10
14 by 14
18 by 18
64 by 64
128 by 128
256 by 256
64 by 64
128 by 128
256 by 256
65 by 65
129 by 129
257 by 257
λ/∆x
1.9
5.7
9.5
13.3
17.1
60.8
122
243
60.8
122
243
61.8
123
244
Fill Time
0.29
1.63
4.7
10.1
18.2
29
0.11
0.02
0.02
0.05
0.06
298
1224
4993
478
1959
8126
310
1231
4898
Linear Solution
Time
0.01
0
0
0
0
0.01
0
0
0.01
0.01
0.01
0.11
0.63
4.2
0.10
0.60
4.1
0.11
0.61
4.05
Error
-0.4
-1.7
-2.2
-2.3
-2.4
-2.3
−∞
-2.5
-2.7
-2.7
-2.7
-4.7
-5.3
-5.9
-5.5
-6.4
-7.3
-8.3
-9.5
-10.7
= −0.17963135247142 − 0.65697150178152i
= −0.75962914626508 + 0.17615097897977i
Table 3. Example 2, Case A, no grazing incidence or reflection. Only two real Bragg
modes are present. Again, fill time plus solution time for SC were negligible compared
to other methods. Accuracy for the coordinate methods increased as the matrix size
increased.
21
Surface and Incoming Waves
Scattered Energy Distribution
0.5
0.6
0.4
0.5
0.3
0.4
0.2
0.3
0.1
0
0.2
−0.1
0.1
−0.2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0
−0.4
−0.3
−0.2
−0.1
0
0.1
(a)
(b)
Real(N(x)) vs x
|N(x)| vs x
0.2
0.3
3.5
0.5
0
3
−0.5
2.5
−1
−1.5
2
−2
1.5
−2.5
−3
−0.5
−0.4
−0.3
−0.2
−0.1
0
(c)
0.1
0.2
0.3
0.4
0.5
1
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 4. Example 2, Case A: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC1 formalism,
matrix size 256 by 256.
22
Case B, Near-Grazing Incidence/Reflection
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
0.25
0.95
75◦
Error = log10 |1 − Normalized Energy|
Formalism
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
SC
SC
CC1
CC1
CC1
CC2
CC2
CC2
CG
CG
CG
A−2
A−1
A0
Matrix
Size
3 by 3
7 by 7
11 by 11
15 by 15
19 by 19
23 by 23
3 by 3
7 by 7
11 by 11
15 by 15
19 by 19
23 by 23
27 by 27
64 by 64
128 by 128
256 by 256
64 by 64
128 by 128
256 by 256
65 by 65
129 by 129
257 by 257
=
=
=
λ/∆x
2.9
6.7
10.5
14.3
18.1
21.9
25.7
60.8
122
243
60.8
122
243
61.8
123
244
Fill Time
0.38
2.1
5.7
11.9
20.2
31.4
0.01
0.03
0.04
0.05
0.07
0.07
0.09
303
1182
4799
469
1928
7787
305
1216
4824
Linear Solution
Time
0
0
0
0
0.01
0.01
0
0
0.01
0
0.01
0.02
0.02
0.11
0.58
4.27
0.12
0.61
4.06
0.11
0.62
4.14
Error
-1.1
-1.9
-2.0
-2.0
-1.9
-1.8
-1.4
-2.8
-3.4
-3.6
-3.7
-3.7
-3.6
-5.7
-6.3
-6.9
-6.1
-7.0
-7.9
-9.7
-9.8
-9.7
0.07250263029849 + 0.00497683319193i
−0.01402209947690 − 0.18205512555385i
−0.91996242949398 + 0.13259130973127i
Table 4. Example 2, Case B, near-grazing incidence and reflection with three modes.
Note a general increase in accuracy from Case A. The SC method was the fastest.
23
Surface and Incoming Waves
Scattered Energy Distribution
0.5
0.6
0.4
0.3
0.4
0.2
0.2
0.1
0
0
−0.2
−0.1
−0.4
−0.2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
−0.8
−0.6
−0.4
−0.2
(a)
0
0.2
0.4
0.6
0.8
(b)
|N(x)| vs x
Real(N(x)) vs x
1.1
1
1
0.8
0.9
0.6
0.8
0.4
0.7
0.6
0.2
0.5
0
0.4
−0.2
−0.4
−0.5
0.3
−0.4
−0.3
−0.2
−0.1
0
(c)
0.1
0.2
0.3
0.4
0.5
0.2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 5. Example 2, Case B: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC2 formalism,
matrix size 256 by 256.
24
9.3. Example 3, λ/L 1
Case A, No Grazing
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
2.5
100
20◦
Error = log10 |1 − Normalized Energy|
Formalism
SS
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
SC
CC1
CC1
CC1
CC2
CC2
CC2
CG
CG
CG
Matrix
Size
1 by 1
5 by 5
9 by 9
13 by 13
17 by 17
21 by 21
29 by 29
1 by 1
5 by 5
9 by 9
13 by 13
17 by 17
21 by 21
64 by 64
128 by 128
256 by 256
64 by 64
128 by 128
256 by 256
65 by 65
129 by 129
257 by 257
A0 =
λ/∆x
100
500
900
1300
1700
2100
6400
12800
25600
6400
12800
25600
6500
12900
25700
Fill Time
0.36
1.27
3.9
8.5
15.6
25.5
55.4
0
0.02
0.02
0.04
0.05
0.07
149
526
2042
267
893
3317
159
542
2091
Linear Solution
Time
0
0
0.01
0.01
0.01
0.01
0.01
0
0.01
0.01
0.01
0.01
0
0.1
0.6
4.1
0.1
0.6
3.9
0.11
0.63
4.16
Error
-2.0
-4.7
-7.5
-6.2
-7.7
-8.2
-5.3
-15.7
−∞
-15.4
−∞
-13.1
-10.3
-15.4
-15.4
-15.1
-6.7
-8.7
-9.0
-15.1
−∞
-14.8
−0.99185964722787 + 0.12733593444511i
Table 5. Example 3, Case A, no grazing incidence or reflection with only the specular
mode. All the methods were highly accurate with the SC as the fastest. The surface
has a very large maximum slope (πd/L).
25
Surface and Incoming Waves
Scattered Energy Distribution
5
0.9
4
0.8
3
0.7
0.6
2
0.5
1
0.4
0
0.3
0.2
−1
0.1
−2
−4
−3
−2
−1
0
1
2
3
0
−0.6
4
−0.4
−0.2
0
(a)
9
−1
8
−2
7
−3
6
−4
5
−5
4
−6
3
−7
2
−8
1
−0.3
−0.2
−0.1
0
(c)
0.6
|N(x)| vs x
0
−0.4
0.4
(b)
Real(N(x)) vs x
−9
−0.5
0.2
0.1
0.2
0.3
0.4
0.5
0
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 6. Example 3, Case A: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC1 formalism,
matrix size 256 by 256.
26
Case B, Near-Grazing Incidence/Reflection
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
2.5
100
75◦
Error = log10 |1 − Normalized Energy|
Formalism
SS
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
SC
CC1
CC1
CC1
CC2
CC2
CC2
CG
CG
CG
Matrix
Size
1 by 1
5 by 5
9 by 9
13 by 13
17 by 17
21 by 21
29 by 29
1 by 1
5 by 5
9 by 9
13 by 13
17 by 17
21 by 21
64 by 64
128 by 128
256 by 256
64 by 64
128 by 128
256 by 256
65 by 65
129 by 129
257 by 257
A0 =
λ/∆x
100
500
900
1300
1700
2100
6400
12800
25600
6400
12800
25600
6500
12900
25700
Fill Time
0.24
1.25
3.9
8.5
15.5
25.2
54.7
0
0.01
0.03
0.05
0.05
0.07
154
539
2108
277
917
3442
165
557
2145
Linear Solution
Time
0
0
0.01
0.01
0.01
0
0.02
0
0.01
0.01
0.01
0
0.01
0.1
0.6
4.2
0.1
0.6
3.9
0.1
0.6
4.0
Error
-3.1
-5.8
-8.6
-8.0
-8.8
-8.2
-5.6
-15.2
-15.7
-15.2
-15.7
-13.7
-10.2
-12.7
-12.9
-13.2
-6.7
-8.7
-9.0
-12.7
-12.9
-13.2
−0.99938168553634 + 0.03516029884120i
Table 6. Example 3, Case B, near grazing incidence and reflection with one mode.
All methods highly accurate with SC the fastest. The maximum slope (πd/L) is quite
large.
27
Surface and Incoming Waves
Scattered Energy Distribution
5
0.8
4
0.6
3
0.4
2
0.2
1
0
0
−0.2
−1
−0.4
−2
−0.6
−1
−4
−3
−2
−1
0
1
2
3
−0.5
0
(a)
(b)
|N(x)| vs x
Real(N(x)) vs x
0
2.5
−0.5
2
−1
1.5
−1.5
1
−2
0.5
−2.5
−0.5
−0.4
−0.3
−0.2
−0.1
0
(c)
0.5
4
0.1
0.2
0.3
0.4
0.5
0
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 7. Example 3, Case B: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC1 formalism,
matrix size 256 by 256.
28
9.4. Example 4, Very Rough Surface
This section presents results for a surface with extremely large slopes. We show that
our code still converges for such cases. The maximum slope, πd/L, is about 25. λ
is relatively small here. It is expected that better results should be attainable with
larger values of λ. Times are reported in seconds of CPU time required by a SPARC 20
workstation with 32 MB of memory.
Interface
S(x)
d/L
λ/L
θi
Perfectly Reflecting
−(d/2) cos(2πx/L)
8
0.05
20◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
λ/∆x Fill Time
CC2
512
25.6
9870
CC2
1024
51.2
39488
CG
513
25.7
7288
CG
1025
51.3
28872
Linear Solution
Time
Error
28
1.9
250
-3.9
29
-0.2
250
-0.2
Table 7. Example 4 for a very rough surface with a maximum slope about 25 and
many modes. Only CC2 converged with small errors and then only for a very large
matrix size. The convergence was very slow.
29
Surface and Incoming Waves
Scattered Energy Distribution
300
0.25
250
200
0.2
150
0.15
100
50
0.1
0
−50
0.05
−100
−150
−10
−8
−6
−4
−2
0
2
4
6
8
10
0
−0.2
−0.15
−0.1
−0.05
0
(a)
(b)
Real(N(x)) vs x
|N(x)| vs x
0.05
0.1
250
100
80
200
60
40
150
20
0
100
−20
−40
50
−60
−80
−100
−10
−8
−6
−4
−2
0
(c)
2
4
6
8
10
0
−10
−8
−6
−4
−2
0
2
4
6
8
10
(d)
Figure 8. (a) Surface and incoming waves (carefully note the scale), (b) Scattered
energy distribution, and surface current ((c) real part (d) modulus) for the CC2
formalism, matrix size 1024 by 1024.
30
9.5. Example 5, Highly Oscillatory Surface with Continuous Derivative
Due to the roughness of the surface, it is sampled uniformly in arc length (instead of
x).
S(x)
d/L
λ/L
θi
−(d/2) cos (2πx/L + 10π(2x/L)3 ) , |x| ≤ L/2
S(x + nL) = S(x),
elsewhere
0.15
0.05
20◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
λ/∆x Fill Time
SS
40 by 40
1316
SS
48 by 48
1887
SS
56 by 56
2482
SC
40 by 40
2.0
0.26
SC
48 by 48
2.4
0.16
SC
56 by 56
2.8
0.23
CC1
512 by 512
25.6
22406
CC1
1024 by 1024 51.2
89674
CC2
512 by 512
25.6
36635
CC2
1024 by 1024 51.2
146970
CG
513 by 513
25.7
22111
CG
1025 by 1025 51.3
88367
Linear Solution
Time
Error
0.01
-0.2
0.05
1.2
0.08
1.7
0.05
1.4
0.05
1.5
0.10
0.7
28.0
-0.6
247
-0.8
28.3
-2.0
246
-3.1
30.2
-2.4
249
-4.4
Table 8. Example 5, highly oscillatory surface with continuous derivative at the end
points. The surface was sampled uniformly in arc length. The lack of convergence in
both spectral related methods is noted. The coordinate-related methods required a
very large size to get good convergence.
31
Surface and Incoming Waves
Scattered Energy Distribution
8
6
0.1
4
2
0
0.05
−2
−4
−6
−10
−8
−6
−4
−2
0
2
4
6
8
10
0
−0.05
0
(a)
0.05
0.1
(b)
|N(x)| vs x
Real(N(x)) vs x
45
30
40
20
35
30
10
25
0
20
15
−10
10
−20
5
−30
−10
−8
−6
−4
−2
0
(c)
2
4
6
8
10
0
−10
−8
−6
−4
−2
0
2
4
6
8
10
(d)
Figure 9. Example 5: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CG formalism,
matrix size 1025 by 1025.
10.
Conclusions
Computational results have been presented for scattering from various periodic surfaces.
The results include examples with grazing incidence, a very rough surface and a
highly oscillatory surface. The examples vary over a parameter set which includes the
geometrical optics regime, physical optics and resonance regime, and a renormalization
regime.
32
The main objective of this study was to determine the best computational method
for these problems. Briefly, the SC method was the fastest but did not converge
for large slopes or very rough surfaces. The topological basis used in the method
was not a complete set, and computationally, the dynamical range in the matrix
increased exponentially with surface height. The SS method was slower and had the
same convergence difficulties as SC. The CC methods were extremely slow but always
converged. The simplest approach is to try the SC method first. Convergence, when
the method works, is very fast. If convergence does not occur then try SS and finally
CC. Results for the remaining mixed representation (CS) can be found in the literature
[16].
Acknowledgments
Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials
Command, USAF, under the Multi-University Research Initiative (MURI) program
Grant #F49620-96-1-0039.
The US Government is authorized to reproduce and distribute reprints for
governmental purposes notwithstanding any copyright notation thereon. The views and
conclusions contained herein are those of the authors and should not be interpreted as
necessarily representing the official policies or endorsements, either expressed or implied
of the AFOSR or the US Government.
Erdmann’s research was supported in part by an Undergraduate Research Grant
from the Colorado Advanced Software Institute (CASI) and a Grant-in-Aid of Research
from Sigma Xi, The Scientific Research Society.
Discussions with Dr. Gary Brown and Capt. Jeff Boleng were very helpful.
References
[1] K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge
University Press, Cambridge (1997).
[2] J. Boleng, C. Craig, J. DeSanto, G. Erdmann, W. Hereman, M. Khebchareon, M. Misra, and A.
Sinex, “Computational Modeling of Rough Surface Scattering,” Department of Mathematical
and Computer Sciences, Colorado School of Mines, Golden, Colorado, Report No. MCS-96-09
(October 1996).
[3] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York (1983).
[4] J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: An exact theory,”
Radio Science 16, 1315–1326 (1981).
[5] J. A. DeSanto, “Exact spectral formalism for rough-surface scattering,” J. Opt. Soc. Am. A 2,
2202–2207 (1985).
33
[6] J. A. DeSanto, Scalar Wave Theory: Green’s Functions and Applications, Springer Verlag, New
York (1992).
[7] J. A. DeSanto and P. A. Martin, “On angular-spectrum representations for scattering by infinite
rough surfaces,” Wave Motion 24, 421-433 (1996).
[8] J. A. DeSanto and P. A. Martin, “On the derivation of boundary integral equations for scattering
by an infinite one-dimensional rough surface,” J. Acoust. Soc. Am. 102, 67-77 (1997).
[9] P. A. Martin and F. J. Rizzo, “On boundary integral equations for crack problems,” Proc. R. Soc.
Lond. A 421, 341-355 (1989).
[10] D. Maystre, “Rigorous vector theories of diffraction gratings,” in: Prog. in Optics XXI, ed. E.
Wolf, North Holland, Amsterdam, pp. 3-67 (1984).
[11] R. C. McNamara and J. A. DeSanto, “Numerical determination of scattered field amplitudes for
rough surfaces,” J. Acoust. Soc. Am. 100, 3519-3526 (1996).
[12] N. Morita, N. Kumagai and J. R. Mautz, Integral Equation Methods for Electromagnetics, Artech
House, Boston (1990).
[13] M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, Wiley, New York (1991).
[14] F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer Verlag, Berlin (1973).
[15] D. Porter and D. S. G. Stirling, Integral Equations, Cambridge University Press, Cambridge (1990).
[16] M. Saillard and J. A. DeSanto, “Coordinate-spectral method for rough surface scattering,” Waves
in Random Media 6, 135-150 (1996).
[17] L. Schwartz, Mathematics for the Physical Sciences, Addison-Wesley, Reading MA (1966).
[18] M. E. Veysoglu, H. A. Yueh, R. T. Shin and J. A. Kong, “Polarimetric passive remote sensing of
periodic surfaces,” J. Elect. Waves & Appls. 5, 267-280 (1991).
Appendix A.
Periodic Green’s Function
In Section 4 we derived the first (or spectral) representation of Gp1 given by
∞
1 ik1 [αj (x−x )+βj |s(x)−s(x )|]
i λ e
.
Gp1 (x, x ) =
4π L j=−∞ βj
(A1)
A second representation follows from using Eqs. (4.1) to (4.3). We have
∞
−∞
G(2) (xs , xs )N T (x )dx =
∞
n=−∞
In ,
(A2)
where now we use the Hankel function representation for G(2) (from Eq. (1.3) )
In =
(2n+1) L
2
(2n−1) L
2
1
i (1)
H0 (k1 [(x − x )2 + (s(x) − s(x ))2 ] 2 )N T (x )dx .
4
(A3)
Shift the integration variable (x = x − nL), and use the periodicity of s(x) and the
Floquet periodicity of the boundary unknown Eq. (4.5). The result is
∞
−∞
G(2) (xs , xs )N T (x )dx =
L
2
−L
2
Gp1 (x, x )N T (x )dx ,
(A4)
34
where now
Gp1 (x, x ) =
∞
1
i (1)
2
2
eik1 α0 nL H0 (k1 {[(x − (x + nL))] + [s(x) − s(x )] } 2 ), (A5)
4 n=−∞
which is the representation of Gp1 in terms of a phased periodic array of Hankel functions.
A third representation can also be derived using Eq. (A5) [18]. From tables [14] we
have the Laplace transform representation
1
∞
√
2i
cos[a(t2 − 2it) 2 ]
(1)
e−st
dt.
H0 ( s2 + a2 ) = − eis
1
π
0
(t2 − 2it) 2
Transforming this equation using t = u2 we have
∞
√
4i
2
(1)
H0 ( s2 + a2 ) = − eis
e−su D(a, u)du,
π
0
where
(A6)
(A7)
1
D(a, u) =
cos[au(u2 − 2i) 2 ]
1
(u2 − 2i) 2
.
(A8)
Rewrite the sum in Eq. (A5) in three parts, the n = 0 term, a sum from 1 to ∞, and a
sum from −1 to −∞. Let n → −n in the latter sum. If we define as the s−variable in
Eq. (A7)
s± = k1 (±(x − x) + nL),
(A9)
and use
a = k1 [s(x) − s(x )],
(A10)
then Eq. (A5) can be written using Eq. (A7) as
1
i (1)
Gp1 (x, x ) = H0 (k1 [(x − x )2 + (s(x) − s(x ))2 ] 2 )
4
∞
∞
1
2
+
eik1 α0 nL eis+
e−s+ u D(a, u)du
π n=1
0
∞
1 −ik1 α0 nL is− ∞ −s− u2
+
e
e
e
D(a, u)du.
π n=1
0
(A11)
The summations can be performed. Using Eq. (A9) define the coefficients of n in Eq.
(A11)
b± = k1 L(u2 − i[1 ± α0 ]),
(A12)
and then the sums are
∞
n=1
e−nb± =
e−b±
.
1 − e−b±
(A13)
35
If we further define
p± = eik1 L(1±α0 ) ,
(A14)
then Eq. (A11) can be written as
1
i (1)
Gp1 (x, x ) = H0 (k1 [(x − x )2 + (s(x) − s(x ))2 ] 2 )
4
2
1 + ik1 (x−x ) ∞ e−u k1 (x −x+L)
+ p e
D(a, u)du
π
1 − p+ e−k1 Lu2
0
2
1 − ik1 (x −x) ∞ e−u k1 (x−x +L)
D(a, u)du,
+ p e
π
1 − p− e−k1 Lu2
0
(A15)
which is the third representation for Gp1 on the surface.
The same analysis follows for the function off the surface. Extend a to b where
b = k1 (z − z ),
(A16)
and we have the general off-the-surface representation
1
i (1)
Gp1 (x, x ) = H0 (k1 [(x − x )2 + (z − z )2 ] 2 )
4
2
1 + ik1 (x−x ) ∞ e−u k1 (x −x+L)
D(b, u)du
+ p e
π
1 − p+ e−k1 Lu2
0
2
1 − ik1 (x −x) ∞ e−u k1 (x−x +L)
D(b, u)du,
+ p e
π
1 − p− e−k1 Lu2
0
(A17)
which is used to compute the normal derivative of Gp1 as in Eq. (4.14).
To compute the impedance kernel in Section 4 we use the Green’s function
representation in various ways. To compute Gp1 (x, x ) for example consider the following
cases:
(a) For x = x and s(x), s(x ) far apart we use the spectral sum Eq. (A1).
(b) For x = x but s(x) close to s(x ) use Eq. (A15) and directly evaluate it. We
approximate the integrals using piecewise Gaussian quadrature. The integrands
decay rapidly, and we determine where the integrand is negligible and approximate
the number of integration intervals to achieve good accuracy.
(c) For x = x we again use the representation Eq. (A15) as follows. In the two integral
terms set x = x and evaluate as in case (b). The Hankel function term in Eq. (A15)
must be treated as described in Appendix B by evaluating the self-cell integral.
Although Eq. (A5) is the canonical representation for the periodic Green’s function in
two-dimensions, we generally do not use it for evaluation purposes since the convergence
is slow and evaluation time per term is long.
To evaluate Gp1 we have corresponding cases:
(a) For x = x and s(x), s(x ) far apart take the normal derivative of Eq. (A1).
36
(b) For x = x but s(x) close to s(x ) take the normal derivative of Eq. (A15).
(1)
(c) For x = x the normal derivative of the Hankel function H0 is
(1)
np |xs − xs |p H1 (k1 |xs − xs |)
,
−
=−
|xs − xs |
which is finite in the limit as xs → xs
s (x )
i
(1)
lim np ∂p H0 (k1 |xs − xs |) = −
,
xs →x s
π 1 + [s (x )]2
so the self-cell evaluation is not a problem.
(1)
np ∂p H0 (k1 |xs
xs |)
(A18)
(A19)
Although we do not use it for our calculations, we can also define the Fourier transform
of the periodic Green’s function. It is given by
Ĝp1 (kx , kx ) =
≈
L
2
−L
2
L
2
−L
2
j
m
e−i(kx x + kx x ) Gp1 (xs , xs ) dx dx
e−i(kx xj + kx xm ) Gp1 (xj , xm )∆xj ∆xm
(A20)
Several examples are presented in Section 10.7.
Appendix B.
Numerical Solution of the CC Equations
In Section 4 we derived the CFIE integral equation given by
i
F (x) =
L
2
−L
2
ZpD1 (x, x )N T (x )dx ,
where the impedance kernel is given by
1
ZpD1 (x, x ) = αδ(x − x ) + α PV Gp1 (x, x ) + βGp1 (x, x ),
2
and
F i (x) = αN i (xs ) + βψ i (xs ).
(B1)
(B2)
(B3)
A standard discretization of Eq. (B1) is the method-of-moments approach, which utilizes
a pulse basis and collocation in x. There are three steps in using this approach. First,
partition the interval [− L2 , L2 ] into a certain number (say M) subintervals, with the pth
interval called ∆p . Then Eq. (B1) can be written as
i
F (x) =
M p=1 ∆p
ZpD1 (x, x )N T (x )dx .
(B4)
Second, choose a representative point (such as the midpoint) in each subinterval, with
the pth given by xp , and approximate N T (x ) for x ∈ ∆p as N T (xp ). Consequently,
i
F (x) ≈
M
p=1
T
N (xp )
∆p
ZpD1 (x, x )dx .
(B5)
37
Third, collocate this equation for x equal to each of the chosen points, yielding
M
i
F (xq ) ≈
p=1
T
N (xp )
∆p
ZpD1 (xq , x )dx .
(B6)
When the kernel is not singular (i.e. when p = q), approximate the integrals as
∆p
ZpD1 (xq , x )dx ≈ ∆p ZpD1 (xq , xp ).
(B7)
When the kernel is singular, we must analytically integrate any singular terms. As seen
in representation Eq. (A11) for Gp1 , the only singular term is the Hankel function. For
x ≈ x ,
(1)
H0
2
k1 [(x − x ) + (s(x) − s(x
1
2 2
)) ]
1
k1
2i
[(x − x )2 + (s(x) − s(x ))2 ] 2 ,
≈1+
γ + ln
(B8)
π
2
with Euler’s constant γ ≈ 0.5772. For small ∆q , we can therefore approximate the
integral as follows
∆q
(1)
H0
2
k1 [(xq − x ) + (s(xq ) − s(x
≈
∆q
2
−
∆q
2
1
2 2
)) ] dx
k1 || 1 + (s (xq ))2 2i
1+
γ + ln(
) d
π
2
k1 1 + (s (xq ))2 4i ∆2q
2i
) +
γ + ln(
ln d
= ∆q 1 +
π
2
π 0
k1 1 + (s (xq ))2 eγ ∆q 2i
) .
= ∆q 1 + ln(
(B9)
π
4e
This is the only nonrepairable singularity present in ZpD1 , so we can now form the matrix
equation we wish to solve
Fi = ZNT ,
(B10)
where Fni = F i (xn ), NnT = N T (xn ), and
⎧
α
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎨
Zmn = ∆n
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
+ α PV Gp1 (xm , xn ) + β U(xm , xn )
+ iβ4
1+
2i
π
ln
k1
√
1+(s (xn ))2 eγ ∆n
4e
α PV Gp1 (xm , xn ) + βGp1 (xm , xn ),
,
m = n,
(B11)
m = n,
where U(xm , xn ) is the nonsingular part of Gp1 (x, x ), namely, the two integral terms in
Eq. (A15) which are evaluated at x = x . The latter term in the m = n equation in Eq.
(B11) follows from Eq. (B9)
38
The scattered amplitudes are given by an approximation to the “−” version of Eqs.
(6.11)-(6.13)
2βn An =
Appendix C.
M
1
e−ik1 [αn xp +βn s(xp )] N(xp )∆p .
L p=1
(B12)
Numerical Solution of the SC Equations
In Section 6 we derived the SC integral representation:
−2Dβn δn0
1
=
L
−L
2
−L
2
e−ik1 [αn x−βn s(x)] N(x)dx.
(C1)
We approximate the integral with the discrete quadrature rule given by
−L
2
−L
2
f (x)dx ≈
M
p=1
wp f (xp ),
(C2)
where {wp } and {xp } are the weights and sampled points, respectively. Therefore, Eq.
(C1) becomes
−2Dβn δn0
M
1
=
wp e−ik1 [αn xp −βn s(xp )] N(xp ).
L p=1
(C3)
This can be written as the matrix equation
KN = F+ ,
where Fm+ = −2Dβ0 δm0 (see Eq. (7.3), Nn = N(xn ), and
wn −ik1 [αm xn −βm s(xn )]
e
.
Kmn =
L
(C4)
(C5)
The scattering amplitudes are obtained via the following approximation of Eq. (6.12)
2βn An =
M
1
wp e−ik1 [αn xp +βn s(xp )] N(xp ).
L p=1
(C6)
More details on a particular numerical method for solving this system of equations can
be found in [11].
Appendix D.
Method
Numerical Solution of the CC Equations: Discrete Galerkin
An alternative to the method-of-moments approach in Appendix B is set forth in [1].
This method allows one to integrate the logarithmic singularity in the CC1 equation
exactly, avoiding the series approximation used in the method-of-moments approach.
39
A rigorous derivation of this method, with error analysis, can be found in [1]. We
present a more intuitive derivation here. First, we parameterize the coordinate variable,
Lt
− L2 . Then Eq. (4.16) becomes
using x(t) = 2π
2
2π
0
ZpD1 (t, t )N(x(t ))dt =
4π i
F (x(t)),
L
(D1)
where
1
ZpD1 (t, t ) = αδ(x(t) − x(t )) + α PV Gp1 (x(t), x(t )) + βGp1 (x(t), x(t )).
2
Now we treat the singularity in Gp1 (x, x ) at x = x . We can write
Gp1 (x, x ) = Gp1 (x, x ) +
1
1
ln [(x − x )2 + (s(x) − s(x ))2 ] 2
2π
(D2)
1
1
ln [(x − x )2 + (s(x) − s(x ))2 ] 2 ,
−
(D3)
2π
where the singularity in the bracketed expression at x = x is repairable. In addition,
the logarithmic term can be written as
1
1
t − t
2
2 2
)| + B̃(t, t ), (D4)
ln [(x(t) − x(t )) + (s(x(t)) − s(x(t ))) ] = ln |2e− 2 sin(
2
where
B̃(t, t ) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1
e+ 21 [(x(t)−x(t ))2 +(s(x(t))−s(x(t )))2 ] 2 ln
,
|2 sin( t−t
)|
2
1
L
ln e+ 2 2π
1 + [s (x(t))]2 ,
t − t = 2mπ,
(D5)
t − t = 2mπ.
Since B̃(t, t ) is not singular, we will group it with the other nonsingular or repairable
terms of the kernel:
B(t, t ) = αδ(x(t) − x(t )) +
2α PV Gp1 (x(t), x(t ))
+ β 2Gp1 (x(t), x(t ))
1
1
1
ln [(x(t) − x(t ))2 + (s(x(t)) − s(x(t )))2 ] 2 − B̃(t, t ) . (D6)
π
π
Then Eq. (D1) becomes
2π
1
β 2π
t − t
4π i
B(t, t )N(x(t ))dt −
ln |2e− 2 sin(
(D7)
)|N(x(t ))dt =
F (x(t)).
π 0
2
L
0
Now we expand N(x) in a truncated Fourier series
+
N(x(t )) ≈
n
m=−n
Nm eimt ,
(D8)
and substitute this expression into Eq. (D7). The second integral can then be evaluated
exactly:
2π
1
t − t imt
βNm eimt
β
)|e dt =
.
(D9)
ln |2e− 2 sin(
− Nm
π
2
max{1, |m|}
0
40
Therefore, we can write Eq. (D7) as
n
m=−n
Nm
2π
0
imt
B(t, t )e
βeimt
4π i
dt +
=
F (x(t)).
max{1, |m|}
L
(D10)
We now approximate the remaining integral in a manner which avoids redundant
computation of the complicated function B(t, t ). We divide the interval [0, 2π] into
2π
(j + 12 ), j = 0, ..., 2n, is the midpoint of the jth
2n + 1 equal intervals, so that tj = 2n+1
subinterval. Use the integral approximation
2π
0
f (t )dt ≈
2n
2π f (tj )
2n + 1 j=0
(D11)
in Eq. (D10) and collocate the resulting equation at the tj points, yielding
n
m=−n
Nm
2n
βeimtj
4π i
2π B(tj , tk )eimtk +
=
F (x(tj )),
2n + 1 k=0
max{1, |m|}
L
j = 0, ..., 2n.(D12)
This linear system is used to determine the expansion coefficients of N(x(t)). This
method has good convergence properties [1]. The scattering amplitudes are computed
using the approximation
2βm Am =
2n
1 e−ik1 [αm x(tj )+βm s(x(tj ))] N(x(tj )).
2n + 1 j=0
(D13)