Theoretical and Computational Aspects
of Scattering from Periodic Surfaces:
One-dimensional Transmission Interface
J. DeSanto, G. Erdmann,
W. Hereman, and M. Misra
MCS-00-04R
May 2001
Revised version submitted to: Waves in Random Media (2001)
Research supported by AFOSR Grant # F49620-96-1-0039
Department of Mathematical and Computer Sciences
Colorado School of Mines
Golden, CO 80401-1887, USA
Phone: (303) 273-3860
Fax: (303) 273-3875
Email: jdesanto@mines.edu
Abstract
We consider the scattering from and transmission through a one-dimensional
periodic surface. For this problem, the electromagnetic cases of TE and TMpolarization reduce to the scalar acoustic examples. Three different theoretical and
computational methods are described, all involving the solution of integral equations
and their resulting discrete matrix system of equations for the boundary unknowns.
They are characterized by two sample spaces for their discrete solution, coordinate (C) space and spectral (S) space, and labelled by the sampling of the rows
and columns of the discretized matrices. They are coordinate-coordinate (CC), the
usual coordinate-space method, spectral-coordinate (SC) where the matrix rows are
discretized or sampled in spectral space, and spectral-spectral (SS) where both rows
and columns are sampled in spectral space. The SS method uses a new topological
basis expansion for the boundary unknowns.
Equations are derived for infinite surfaces, then specialized and solved for periodic surfaces. Computational results are presented for the transmission problem as
a function of roughness, near grazing incidence as well as many other angles, density
and wavenumber ratios. Matrix condition numbers and different sampling method
are considered. An error criterion is used to gauge the validity of the results.
The computational results indicated that the SC method was by far the fastest
(by several orders of magnitude), but that it became ill-conditioned for very rough
surfaces. The CC method was most reliable, but often required very large matrices
and was consequently extremely slow. It is shown that the SS method is computationally efficient and accurate at near grazing incidence and can be used to fill a
gap in the literature. Extensive computational results indicate that both SC and
SS are highly robust computational methods. Spectral based methods thus provide
viable computational schemes to study periodic surface scattering.
1
Introduction
In this paper we consider the scattering from and transmission through a one-dimensional
periodic surface. For this problem, the electromagnetic cases of TE and TM-polarization
reduce to the scalar acoustic examples. Very many methods have been employed to study
this type of scattering problem. A summary can be found in the book by Voronovich [22]
and in a forthcoming review paper by DeSanto [9]. The variety of different methods used
to study the problem can also be found in the book by Petit [21]. Recent developments in
differential-based methods [19], coupled-mode methods [20], the C-method [16], and the
method of variation of boundaries [2] are also available.
Here we present computational results for coordinate (C) methods as well as two
methods either partly or wholly based on spectral (S) space. The three methods are all
formulated using integral equations for the boundary unknowns, which are discretized and
solved as linear systems. The methods are characterized by the sampling of the rows and
columns of the discretized matrix. They are referred to as CC (both rows and columns
formed by sampling in coordinate space), SC (rows sampled in spectral space and columns
in coordinate space) and SS (both rows and columns sampled in spectral space). The SC
method was first derived in [8]. The three methods were previously used to study the
scattering from perfectly reflecting surfaces [11, 12], and the SC method was applied to
1
sinusoidal surfaces with a Dirichlet boundary condition where the objective was to study
the coordinate sampling [18].
The purpose of this paper is to present results for all three methods for the full
transmission problem. A much more extensive collection of computational results for this
problem can be found in our recent report [13].
2
CC Equations
We consider the scattering from and transmission through a one-dimensional surface specified by z = s(x) (see Figure 1).
sc
+
HR
sc
in
sc
+
DR
s(x)
R
−
DR
−
HR
Figure 1: Geometry for the one-dimensional rough interface z = s(x) separating media
+
−
z > s(x) (DR
in the limit as R → ∞) and z < s(x) (DR
in the limit as R → ∞) with
constant (j = 1, 2) ρj (density) and kj (wavenumber). Density and wave number ratios
ρ = ρ2 /ρ1 and κ = k2 /k1 are used throughout. A plane wave field is incident at angle θi .
HR+ and HR− are semicircles at radius R.
In this section we derive the equations for an infinite surface and specify it to be
periodic later. Some of the development is similar to that in a previous report [11] and
paper [12], which treated perfectly reflecting surfaces with a Dirichlet boundary condition
but we include the remainder of the full transmission development here for completeness.
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 = ∂/∂xi (i = 1, 2) and the normal
derivative ∂n = ni ∂i where ni = δi2 − δi1 s (x) is the (non-unit) surface normal (δij is
the Kronecker delta) and repeated subscripts are summed (here from 1 to 2). Fields are
represented by ψ and correspond to a velocity potential (acoustics), the y-component of
the electric vector for TE-polarization, or the y-component of the magnetic vector for
TM-polarization. A brief discussion of these relations is given in Appendix B of [13],
with particular attention to the limiting cases of Dirichlet and Neumann boundary value
problems. Here, since the surface generator is parallel to the y-axis, no polarization
2
change occurs, and the problem can be treated as a scalar transmission problem. All
fields are time-harmonic so that a factor exp(−iωt) is suppressed throughout (ω is circular
frequency, and t is time). The two regions of the problem (see Figure 1) are defined by
+
z > s(x) (region 1, DR
in the limit as R → ∞) with constant parameters ρ1 (density or
electromagnetic parameters, see Appendix B in [13]) and wave number k1 = 2π/λ where
−
in the limit as R → ∞) with ρ2 and k2 . We
λ is wavelength, and z < s(x) (region 2, DR
use density and wave number ratios ρ = ρ2 /ρ1 and κ = k2 /k1 .
Fields in the jth region (j = 1, 2) satisfy the scalar Helmholtz equation
(∂i ∂i + kj2 )φj (x) = 0,
where
φj (x) =
ψ1SC (x),
ψ2 (x),
1
x ∈ DR
(j = 1),
2
x ∈ DR (j = 2).
(2.1)
(2.2)
Here ψ1SC is the scattered field in region 1 and ψ2 the total field in region 2. The appropriate
free-space Green’s functions Gj satisfy the equations
(∂i ∂i + kj2 )Gj (x, x ) = −δ(x − x ),
(2.3)
i (1)
Gj (x, x ) = H0 (kj |x − x |) ,
4
(2.4)
and are explicitly given by
the Hankel function of zeroth order and first kind.
+
using ψ1SC and G1 yields in the limit as R → ∞ an integral
Green’s theorem in DR
SC
representation for ψ1 as an integral on the full surface s∞ (x). It is convenient to introduce
acoustic single and double layer potentials to express this. The single (S) layer potential
with density u is given by the single integral (j = 1, 2)
(Sj u)(x) =
S∞
Gj (x, xs )u(xs ) dx ,
(2.5)
and the double (D) layer potential with density v is given by
(Dj v)(x) =
S∞
∂n Gj (x, xs )v(xs ) dx .
(2.6)
+
as R → ∞ is then written as
The result of Green’s theorem in DR
θ1 (x)ψ1SC (x) = (D1 ψ1SC )(x) − (S1 N1SC )(x),
(2.7)
+
and N1SC is the normal derivwhere θ1 (x) is the characteristic function of the region D∞
ative
N1SC (x) = ∂n ψ1SC (x).
(2.8)
+
A second equation can be formed from (2.7) by taking its normal derivative for x ∈ D∞
.
It is
θ1 (x)N1SC (x) = ∂n (D1 ψ1SC )(x) − ∂n (S1 N1SC )(x).
(2.9)
3
+
There are no contributions from the semicircle at H∞
since ψ1SC and G1 each satisfy the
Sommerfeld radiation condition. In fact it has been shown by us [14] that there is no
contribution so long as ψ1SC contains no horizontal plane waves.
−
as R → ∞ yields analogous equations (with the same normal
Green’s theorem in DR
now pointing outward from the domain). They are
−θ2 (x)ψ2 (x) = (D2 ψ2 )(x) − (S2 N2 )(x),
(2.10)
−θ2 (x)N2 (x) = ∂n (Dn ψ2 )(x) − ∂n (S2 N2 )(x),
(2.11)
and
−
and N2 is the normal derivative of ψ2 .
where θ2 is the characteristic function of D∞
−
i
(where there are no sources
The incident field ψ satisfies (2.1) for j = 1 but in DR
−
or if it is a plane wave) and with Green’s theorem in DR (using ψ i and G1 and excluding
horizontal plane waves) yields relations like (2.10) and (2.11)
and
−θ2 (x)ψ i (x) = (D1 ψ i )(x) − (S1 N i )(x),
(2.12)
−θ2 (x)N i (x) = ∂n (D1 ψ i )(x) − ∂n (S1 N i )(x),
(2.13)
where N i is the normal derivative of ψ i .
Combining (2.7) and (2.12) to form the total field ψ1
ψ1 = ψ i + ψ1SC ,
(2.14)
θ1 (x)ψ1 (x) = ψ i (x) + (D1 ψ1 )(x) − (S1 N1 )(x),
(2.15)
we get the equation
and combining (2.9) and (2.13) yields
θ1 (x)N1 (x) = N i (x) + ∂n (D1 ψ1 )(x) − ∂n (S1 N1 )(x).
(2.16)
+
,
Equations (2.15) and (2.16) are used to form surface integral equations from region D∞
−
and (2.10) and (2.11) from region D∞ . Equations (2.10) and (2.15) are integral representations for the total fields in each region.
To form surface integral equations, let x → xs . That is, the field point approaches
the surface from above (+) it or from below (−). The limiting behavior of the single layer
potential is [5]
lim (Sj u)(x) = (Sj u)(xs ),
(2.17)
x→x±s
since it is a continuous function. The double layer potential has a jump discontinuity
where
lim ± (Dj v)(x) = (Dj v)± (xs ),
x→xs
(2.18)
1
(Dj v)± (xs ) = (Dj v)(xs ) ± v(xs ).
2
(2.19)
4
The normal derivative of the single layer potential also has a jump discontinuity
1
∂n (Sj u)± (xs ) = ∂n (Sj u)(xs ) ∓ u(xs ).
2
(2.20)
The integrals on the right hand sides of (2.19) and (2.20) are improper. The normal
derivative of the double layer potential has the same limit from both directions but is
singular, and we take its Hadamard Finite Part (FP) [4]
lim ∂n (Dv)(x) = FP∂n (Dv)(xs ).
x→x±s
(2.21)
The limits of (2.15) and (2.16) are thus
1
ψ1 (xs ) = ψ i (xs ) + (D1 ψ1 )(xs ) − (S1 N1 )(xs ),
2
(2.22)
and
1
N1 (xs ) = N i (xs ) + FP∂n (D1 ψ1 )(xs ) − ∂n (S1 N1 )(xs )
2
and the limits (from below) of (2.10) and (2.11) are given by
1
− ψ2 (xs ) = (D2 ψ2 )(xs ) − (S2 N2 )(xs ),
2
(2.23)
(2.24)
and
1
− N2 (xs ) = FP∂n (D2 ψ2 )(xs ) − ∂n (S2 N2 )(xs ).
2
The continuity conditions at the rough interface are
ψ1 (xs ) = ρψ2 (xs ),
(2.25)
(2.26)
(which is continuity of pressure for the acoustic case, continuity of the tangential electric
field for TE polarization, and continuity of the tangential H-field for TM polarization).
Here ρ is the density ratio for acoustics or the appropriate ratio of electromagnetic parameters. Secondly, we have
(2.27)
N1 (xs ) = N2 (xs ),
the continuity of velocity or the appropriate tangential magnetic or electric fields. We
define the surface field (which is a function of a single variable)
F (x) ≡ ψ1 (xs ),
(2.28)
and normal derivative of the total field
N T (x) ≡ N1 (xs ).
(2.29)
Then (2.22)–(2.25) can be written as
1
F (x) = ψ i (xs ) + (D1 F )(xs ) − (S1 N T )(xs ),
2
5
(2.30)
1 T
N (x) = N i (xs ) + FP∂n (D1 F )(xs ) − ∂n (S1 N T )(xs ),
2
1
1
F (x) = − (D2 F )(xs ) + (S2 N T )(xs ),
2ρ
ρ
(2.31)
(2.32)
and
1
1 T
N (x) = − FP∂n (D2 F )(xs ) + ∂n (S2 N T )(xs ).
(2.33)
2
ρ
We use various combinations of these integral equations to solve for the two boundary
unknowns F and N T . We discuss this in more detail below. The kernels of each of these
equations are functions of two variables, both in coordinate space and we thus refer to
them as coordinate-coordinate or CC methods to distinguish them from other equations
we derive which involve a functional dependence on the spectral variable.
The reduction of (2.30)–(2.33) to integral equations over a single period (−L/2 to L/2)
of a periodic surface follows procedures outlined in [11, 12]. Briefly, the integration over
−∞ to ∞ is written as an infinite sum on integrals over periodic cells [(2n − 1)L/2, (2n +
1)L/2] where n runs from −∞ to ∞. The Floquet periodicity of the fields collapses the
integration to a single period cell and replaces the Green’s function with its periodic
extension. We can write the explicit representations as one-dimensional integrals. For the
single layer (j = 1, 2)
T
(Sj N )(x) =
L
2
−L
2
Gpj (x, x )N T (x ) dx ,
(2.34)
and the periodic Green’s functions are
∞
1
i
eik1 [αn (x−x )+mj (αn )|s(x)−s(x )|] ,
Gpj (x, x ) =
2k1 L n=−∞ mj (αn )
(2.35)
where αn = sin(θn ) and θn is the angle of the nth scattered or transmitted Bragg wave,
where the Bragg equation is
λ
αn = α0 + n ,
(2.36)
L
with α0 = sin θi . Here, θi is the plane wave angle of incidence, measured from the positive
z-direction, and
⎧
⎨ (1 − α2 ) 12
|αn | ≤ 1
n
(2.37)
m1 (αn ) = ⎩
1
i (αn2 − 1) 2 |αn | > 1,
⎧
⎨
and
1
(κ2 − αn2 ) 2 |αn | ≤ κ
m2 (αn ) =
1
⎩ i (α2 − κ2 ) 2 |α | > κ.
n
n
(2.38)
For the other terms we have first
∂n (Sj N T )(x) =
L
2
−L
2
Gpj (x, x )N T (x ) dx ,
(2.39)
where Gpj is the exterior normal derivative of Gpj (with respect to the x-variable),
Gpj (x, x ) = ∂n Gpj (x, x ),
6
(2.40)
second,
(Dj F )(x) =
L
2
−L
2
G̃pj (x, x )F (x ) dx ,
(2.41)
using the interior normal derivative (on the integration variable)
G̃pj (x, x ) = ∂n Gpj (x, x ),
and finally
∂n (Dj F )(x) =
with
L
2
Gpj (x, x )F (x ) dx ,
−L
2
Gpj (x, x ) = ∂n ∂n Gpj (x, x ),
(2.42)
(2.43)
(2.44)
the second normal derivative.
Equations (2.30)–(2.33) can thus be written in operator notation as
and
1
I − G̃p1 F = ψ i − Gp1 N T ,
2
(2.45)
1
I + Gp1 N T = N i + Gp1 F,
2
1
I + G̃p2 F = ρGp2 N T ,
2
(2.46)
(2.47)
1
(2.48)
ρ I − Gp2 N T = −Gp2 F.
2
We can then combine these equations in various ways. We choose to avoid all equations
where the Hadamard Finite Part plays a part, namely (2.46) or (2.48) when F = 0. For
the Dirichlet case (ρ = 0 and F = 0), we can use (2.45) or (2.46), or a linear combination
of them. In [11] and [12] we called these options CC1, CC2, and CFIE, referring to the
coordinate-coordinate integral equation of the first or second kind, and the combined field
integral equation. For the Neumann case (ρ = ∞ and N T = 0), our only option is to use
(2.45). A detailed discussion of these cases and their computational solution is presented
in [11] and [12]. For the transmission case (0 < ρ < ∞), we must use a coupled system of
two equations, since both F and N are unknown. One option is to use (2.45) and (2.47),
which is the transmission generalization of CC1 in [11] and [12]. Another option is to add
(2.48) to (2.46), removing the hyper singularity and use of Finite Part. Equations (2.45),
(2.47), or a linear combination thereof, can then be used for the second equation. This
combination (termed CC3) requires many evaluations of complicated functions, and we
used it only when it is necessary to verify the accuracy of programs.
The standard coordinate discretization method we use is called a pulse basis, in which
the surface is divided into many small pieces, and the unknowns are assumed to be
constant on each piece. With all of the above methods, one can perform a change of basis
on the equations to introduce Fourier or wavelet transforms [1, 6, 11]. The pulse basis
however does not have the requisite differentiability to use the FP integrals alone, and,
as mentioned, we avoid them.
7
3
SC Equations
In the previous section we treated the case where both rows and columns of the matrix to
be inverted were sampled in coordinate space. Here we derive (from the previous) a set
of equations in a mixed representation where the rows of the matrix are sampled in the
conjugate spectral (S) variable, and the columns still in the coordinate (C) space. These
are the SC equations. A direct derivation without using the CC equations can be found
in the literature [8], where they were first derived.
First, use (2.15) for the scattered field with the boundary unknowns defined as in
(2.28) and (2.29). The result is
ψ1SC (x) = (D1 F )(x) − (S1 N T )(x),
+
x ∈ D∞
,
(3.1)
where the single and double layer functions are defined in (2.5) and (2.6). Similarly the
transmitted field can be found from (2.10)
ψ2 (x) = − 1ρ (D2 F )(x) + (S2 N T )(x),
−
x ∈ D∞
.
(3.2)
Next, define the Weyl representations for the Green’s functions [10]
Gj (x, xs )
πi
=
(2π)2
∞
−∞
1
eik1 [µ(x−x )+mj (µ)|z−s(x )|] dµ,
mj (µ)
(3.3)
where the mj are defined in (2.37) and (2.38). For z > max s(x ) and j = 1, drop the
absolute value sign in the phase and use the result in (3.1). The scattered field can then
be represented as
∞
SC
ψ1 (x) =
A(µ)eik1 [µx+m1 (µ)z] dµ,
(3.4)
−∞
where
A(µ) =
and
A(µ, x ) =
∞
−∞
A(µ, x )e−ik1 [µx +m1 (µ)s(x )] dx ,
1
[(m1 (µ) − µs (x )) F (x ) + N(x )] ,
2λm1 (µ)
(3.5)
(3.6)
where we have scaled the boundary unknown N T as
N(x) = (ik1 )−1 N T (x).
(3.7)
Given the two boundary unknowns F and N we can thus find the scattered field using
(3.4)-(3.6).
In a similar way for z < min s(x ) and j = 2, drop the absolute value sign in (3.3) and
use the result in (3.2). The transmitted field is then
ψ2 (x) =
where
B(µ) =
∞
−∞
∞
−∞
B(µ)eik1 [µx−m2 (µ)z] dµ,
B(µ, x )e−ik1 [µx −m2 (µ)s(x )] dx ,
8
(3.8)
(3.9)
and
B(µ, x ) =
1
[(m2 (µ) + µs (x )) F (x ) − ρN(x )] .
2ρλm2 (µ)
(3.10)
Given these same boundary unknowns, we can thus find the transmitted field using (3.8)(3.10).
Next, we need two equations to solve for the boundary unknowns. From (2.15) we get
−ψ i (x) = (D1 F )(x) − (S1 N T )(x),
−
x ∈ D∞
,
(3.11)
and from (2.10) we get
0 = ρ1 (D2 F )(x) − (S2 N T )(x),
+
x ∈ D∞
.
(3.12)
In (3.11) we use (3.3) with j = 1 and z < min s(x ) to get
i
ψ (x) =
where
I1 (µ) =
and
I1 (µ, x ) =
∞
−∞
∞
−∞
I1 (µ)eik1 [µx−m1 (µ)z] dµ,
I1 (µ, x )e−ik1 [µx −m1 (µ)s(x )] dx ,
1
{[m1 (µ) + µs (x )] F (x ) − N(x )} .
2λm1 (µ)
(3.13)
(3.14)
(3.15)
In (3.12) we use (3.3) with j = 2 and z > max s(x ) to get
0=
where
I2 (µ, x ) =
∞
−∞
I2 (µ, x )e−ik1 [µx +m2 (µ)s(x )] dx ,
1
{[m2 (µ) − µs (x )] F (x ) + N(x )} .
2ρλm2 (µ)
(3.16)
(3.17)
Equations (3.14) and (3.16) are the coupled equations to solve for the two boundary
unknowns. This is the general formulation for an infinite surface [8]. We use the case of
a single plane wave incident on the surface in the subsequent development. This is given
by
I1 (µ) = Dδ(µ − α0 ),
(3.18)
in (3.14). Here α0 = sin θi and m1 (α0 ) = cos θi , where θi is the angle of incidence measured
from the positive z-direction. D is the arbitrary amplitude and we generally set D = 1
in the calculations.
The derivation of the SC equations for a perfectly reflecting periodic surface was presented in [11, 12]. The derivation here for the transmission problem is a straightforward
generalization of this. We omit the details and merely summarize the results. The scattered and transmitted fields from (3.4) and (3.8) reduce to discrete infinite sums given
by
ψ1SC (x) =
∞
n=−∞
An eik1 (αn x+m1 (αn )z) ,
9
(3.19)
and
ψ2 (x) =
∞
n=−∞
Bn eik1 (αn x−m2 (αn )z) .
(3.20)
If we define the four phase functions (j = 1, 2)
φ±
j (µ, x) = µx ± mj (µ)s(x)
(3.21)
and the four terms resulting from taking normal derivatives
n±
j (µ, x) = mj (µ) ± µs (x),
(3.22)
then the two coupled equations (3.14) and (3.16) reduce for a single plane wave incident
to the coupled system
1
L
L
2
−
−ik1 φ1 (αj ,x)
n+
dx = 2m1 (α0 )Dδj0 ,
1 (αj , x)F (x) − N(x) e
−L
2
and
1
L
L
2
−L
2
+
−ik1 φ2 (αj ,x)
n−
dx = 0.
2 (αj , x)F (x) + ρN(x) e
(3.23)
(3.24)
Once these are solved for F and N, the scattered and transmitted amplitudes can be
evaluated using the periodic reduction of (3.5) and (3.9) as
1
L
and
1
L
4
L
2
−L
2
L
2
−L
2
+
−
−ik1 φ1 (αj ,x)
n−
dx = 2m1 (αj )Aj ,
1 (αj , x)F (x) + N(x) e
−ik1 φ2 (αj ,x)
n+
dx = 2ρm2 (αj )Bj .
2 (αj , x)F (x) − ρN(x) e
(3.25)
(3.26)
SS Equations for a Periodic Surface: Topological
Basis
In this section we derive the SS equations for the transmission problem. They are found
by choosing topological basis expansions for the unknowns in the SC equations, namely
∞
F (x) =
−
j =−∞
and
N(x) =
∞
j =−∞
Fj eik1 φ1 (αj ,x) ,
−
ik1 φ1 (αj ,x)
Nj n+
.
1 (αj , x)e
(4.1)
(4.2)
For a discussion on the completeness of the basis see [3, 17, 21]. We choose these bases to
reduce the size of the linear system from that of the SC equations. Using numerical trials
and an energy check we show that the results are accurate within certain slope limitations.
10
Using (4.1) and (4.2), (3.23) can be written
∞ j =−∞
where
(1)
Kjj and
(1)
Mjj (1)
(1)
Kjj Fj − Mjj Nj = 2m1 (α0 )Dδj0 ,
1
=
L
L
2
−
−L
2
(4.3)
−
ik1 [φ1 (αj ,x)−φ1 (αj ,x)]
n+
dx,
1 (αj , x)e
(4.4)
−
1 2 +
ik1 [φ−
1 (αj ,x)−φ1 (αj ,x)] dx.
, x)e
=
n
(α
j
1
L − L2
L
(4.5)
These matrix elements are related as follows:
If m1 (αj ) = m1 (αj ) and αj = αj , then j = j and
(1)
(1)
Kjj = Mjj = m1 (αj ).
(4.6)
If m1 (αj ) = m1 (αj ) and αj = αj , then j = j , αj = −αj , and using integration by parts
(1)
Kjj =
(1)
−Mjj iαj (j − j )
=
L
π
Ly −iy(j−j )
s
e
dy.
2π
−π
(4.7)
If m1 (αj ) = m1 (αj ), then αj = αj , j = j , and
(1)
(1)
(1)
(1)
Kjj = −Mjj = Vjj Φjj ,
where
(1)
Φjj =
1
2π
π
(1)
Vjj =
Ly
e−i(j−j )y eik1 s( 2π )[m1 (αj )−m1 (αj )] dy,
−π
and
(4.8)
(4.9)
1 − m1 (αj )m1 (αj ) − αj αj .
m1 (αj ) − m1 (αj )
(1)
(4.10)
(1)
Using the relationship between M(1) and K(1) (i.e. Mjj = (2δjj − 1)Kjj ), we can rewrite
(4.3) as
∞
1
(1)
Kjj Dj ,
(4.11)
Nj = −Dδj0 +
2m1 (αj ) j =−∞
where we have defined
Dj = Fj + Nj .
(4.12)
Using (4.1) and (4.2), (3.24) can be written
∞ j =−∞
where
(2)
Kjj 1
=
L
L
2
−L
2
(2)
(2)
Kjj Fj + ρMjj Nj = 0,
−
+
ik1 [φ1 (αj ,x)−φ2 (αj ,x)]
dx,
n−
2 (αj , x)e
11
(4.13)
(4.14)
and
(2)
Mjj =
1
L
L
2
−
+
ik1 [φ1 (αj ,x)−φ2 (αj ,x)]
n+
dx.
1 (αj , x)e
L
−2
(4.15)
Using integration by parts, these can be rewritten
(2)
(2)
(2)
Kjj = Vjj Φjj ,
and
(2)
(2)
(4.16)
(2)
Mjj = Wjj Φjj ,
where
(2)
Φjj 1
=
2π
π
Ly
−π
(2)
Vjj =
(4.17)
e−i(j−j )y eik1 s( 2π )[−m2 (αj )−m1 (αj )] dy,
(4.18)
κ2 + m2 (αj )m1 (αj ) − αj αj ,
m2 (αj ) + m1 (αj )
(4.19)
1 + m2 (αj )m1 (αj ) − αj αj .
m2 (αj ) + m1 (αj )
(4.20)
and
(2)
Wjj =
Using (4.11) and (4.12), (4.13) can be rewritten as
∞
j =−∞
(2)
(2)
Kjj Dj = D ρMj0 − Kj0 ,
(4.21)
where the matrix K is defined by
∞
(2)
Kjj = Kjj +
j =−∞
(2)
(2)
ρMjj − Kjj 1
(1)
K .
2m1 (αj ) j j
(4.22)
Equation (4.21) is a single equation for {Dj }. Its solution is used to evaluate {Nj } from
(4.11) and subsequently {Fj } from (4.12). The use of the topological basis expansions
thus reduces the problem to the solution of a single equation.
To solve for the scattered and transmitted amplitudes again use (4.1) and (4.2). Then
(3.25) for the scattered amplitudes Aj becomes
∞ j =−∞
where
(3)
Kjj 1
=
L
(3)
Mjj 1
=
L
and
(3)
(3)
Kjj Fj + Mjj Nj = 2m1 (αj )Aj ,
L
2
−L
2
−
+
ik1 [φ1 (αj ,x)−φ1 (αj ,x)]
n−
dx,
1 (αj , x)e
L
2
−
+
ik1 [φ1 (αj ,x)−φ1 (αj ,x)]
dx.
n+
1 (αj , x)e
L
−2
(4.23)
(4.24)
(4.25)
Using integration by parts, these can be rewritten
(3)
(3)
(3)
(3)
Kjj = Mjj = Vjj Φjj ,
12
(4.26)
where
(3)
Φjj 1
=
2π
π
Ly
−π
e−i(j−j )y eik1 s( 2π )[−m1 (αj )−m1 (αj )] dy,
(4.27)
1 + m1 (αj )m1 (αj ) − αj αj .
m1 (αj ) + m1 (αj )
(4.28)
and
(3)
Vjj =
Using (4.26) and (4.12), (4.23) can be written
Aj =
∞
1
(3)
Kjj Dj .
2m1 (αj ) j =−∞
(4.29)
Thus the {Aj } can be directly evaluated once the {Dj } are known from (4.21).
Finally, (3.26) for the transmitted amplitudes Bj becomes
∞ j =−∞
where
(4)
Kjj (4)
−
−
1 2 +
=
n2 (αj , x)eik1 [φ1 (αj ,x)−φ2 (αj ,x)] dx,
L
L −2
(4.30)
L
and
(4)
Mjj (4)
Kjj Fj − ρMjj Nj = 2ρm2 (αj )Bj ,
1
=
L
L
2
−L
2
−
−
ik1 [φ1 (αj ,x)−φ2 (αj ,x)]
n+
dx.
1 (αj , x)e
(4.31)
(4.32)
If m1 (αj ) = m2 (αj ) and αj = αj , then j = j and
(4)
(4)
Kjj = Mjj = m1 (αj ).
(4.33)
If m1 (αj ) = m2 (αj ) and αj = αj , then, using integration by parts,
(4)
Kjj iαj (j − j )
=
L
and
π
Ly −iy(j−j )
s
e
dy,
2π
−π
Ly −iy(j−j )
iαj (j − j ) π
=
s
e
dy.
L
2π
−π
If m1 (αj ) = m2 (αj ), then the matrix elements can be rewritten
(4)
Mjj (4)
(4)
(4)
Kjj = Vjj Φjj ,
and
(4)
(4)
(4)
Mjj = Wjj Φjj ,
where
(4)
Φjj =
1
2π
(4)
π
(4.35)
(4.36)
(4.37)
e−i(j−j )y eik1 s( 2π )[m2 (αj )−m1 (αj )] dy,
(4.38)
κ2 − m2 (αj )m1 (αj ) − αj αj ,
m2 (αj ) − m1 (αj )
(4.39)
−π
Vjj =
Ly
(4.34)
13
and
(4)
Wjj =
1 − m2 (αj )m1 (αj ) − αj αj .
m1 (αj ) − m2 (αj )
(4.40)
Then, (4.30) can be written
Bj =
∞ 1
(4)
(4)
Kjj Fj − ρMjj Nj ,
2ρm2 (αj ) j =−∞
(4.41)
using the above simplifications for the matrices K and M.
The procedure is thus to solve for {Dj } from (4.21), evaluate {Nj } from (4.11), solve
for {Fj } from (4.12), and use the latter two to evaluate the scattered amplitudes from
(4.29), and the transmitted amplitudes from (4.41).
5
Energy
The energy constraint is
∞
∞
e(m1 (αj ))
e(m2 (αn ))
+ρ
= D2 .
|Aj |
|Bn |2
m1 (α0 )
m1 (α0 )
n=−∞
j=−∞
2
(5.1)
This can be derived directly using Green’s theorem on combinations of field representations in both regions. The details are in [8]. We generally choose D = 1 and the difference
between 1 and the left-hand-side of (5.1) is quoted as the error in the calculation, i.e.,
Error = log10 |1 − LHS|,
(5.2)
where LHS is the left hand side of (5.1).
For the computational results we define
e(m1 (αj ))
|Aj |2 ,
m1 (α0 )
(5.3)
ρ e(m2 (αn ))
|Bn |2 ,
m1 (α0 )
(5.4)
Rj =
and
Tn =
as the (scaled) reflected and transmitted energy in the respective modes.
6
Computational Results
In Sections 7-10, we present an extensive suite of computational results for the transmission problem using the three methods CC, SC, and SS from Sections 2, 3 and 4,
respectively. We outline and summarize the results here.
In Section 7, we demonstrate that the SS method works for an extreme grazing incidence case where θi = 89.995◦ . For the SS example, the number of topological modes is
quite small for both non-grazing and near-grazing incidence. (Pure Fourier simulations
done for perfectly reflecting examples indicate that upwards of ten times the number of
14
Fourier modes would be necessary.) For these grazing incidence cases we do not include
scattered field plots since all the energy is in the specular direction.
In Sections 8 and 9, we present an extensive collection of computational results for
the SC method. These include two different coordinate sampling methods (Section 8)
and two different spectral sampling methods (Section 9). The maximum error is fixed
at Error = −2. With this fixed error, the maximum value of d/L (slope is πd/L)
as a function of these samplings is treated as well as the maximum condition number.
The coordinate sampling depends on the number of spectral orders above and below the
surface. Reliable results are found for large d/L ratios and very large condition numbers.
Finally, in Section 10 we present results on the SS formalism for roughness (d/L)
values and condition number for 1% error, as a function of κ, and as a function of ρ again
over an extensive parameter domain.
We present these computational results as representative of much more extensive results which can be found in our report [13]. Summary and conclusions are in Section 11.
7
SS Method — Near Grazing Incidence
In this section, we exemplify the behavior of the SS method for two cases only. Additional
cases are in [13]. The number of topological basis modes is quite small for both cases but
is also quite different for non-grazing and near-grazing incidence. The surface field and
normal derivative expansions are in Section 4.
Fig. 2 illustrates the case of non-grazing incidence (θi = 25◦ ) for a moderately rough
surface (d/L = 0.1). To compute the topological basis modes in Fig. 2 the surface parameters were S(x) = −(d/2) cos(2πx/L), where λ/L = 0.047, ρ = 0.5, and κ = 1.5. The
number of real propagating orders above and below the surface is 43 and 64, respectively.
As is apparent from Fig. 2, for this moderately rough surface very good convergence (Error
= −4.3) is achieved with a very small number of topological basis terms.
15
2
0.4
1.5
0.3
1
0.2
0.5
0.1
0
−40
−20
0
20
0
−40
40
(a)
−20
0
20
40
(b)
Figure 2: SS Formalism: 64 by 64 matrices, Error = −4.3 The values of the magnitudes
of the topological basis components for the normal derivative |Nj | (a) and the field |Fj |
(b) vs. order. The small number of non-zero values is characteristic of SS solutions for a
good energy check.
16
The SS methods are also illustrated in Fig. 3 for a weaker (d/L = 0.07) rough surface
but where the angle of incidence is nearly parallel to the mean surface (θi = 89.995o). For
Fig. 3 we used the surface parameters S(x) = −(d/2) cos(2πx/L) where λ/L = 0.0469,
ρ = 1.5, and κ = 1.5. The number of real orders above and below the surface again was 43
and 64. Convergence is very good but at near-grazing incidence many more topological
basis modes are necessary to achieve this than were necessary at non-grazing incidence
(see Fig. 2). In general, we have also noted (in [13]) that the surface field and normal
derivative values are considerably smaller than in the corresponding cases for non-grazing
incidence. Near grazing incidence does not excite strong surface fields. No plots of the
modal energy distribution are included since nearly all of the energy is in the specular
direction.
−3
3
−4
x 10
2
x 10
2.5
1.5
2
1.5
1
1
0.5
0.5
0
−60
−40
−20
0
0
−60
20
(a)
−40
−20
0
20
(b)
Figure 3: SS Formalism, Near Grazing Incidence with 64 by 64 matrices, Error = −4.3.
The values of the magnitudes of the topological basis components for the surface normal
derivative |Nj | (a) and the surface field |Fj | (b) vs. order. The small number of non-zero
values is characteristic of SS solutions for a good energy check.
17
8
SC Results using Different Coordinate Sampling
Methods
In this section, we present extensive results on the effect of two different coordinate
sampling methods on the SC method as a function of incidence angle and density and
wavenumber ratios. The maximum error is fixed at Error = −2. In particular, we study
the maximum value of d/L (slope is π times this value) with fixed error for a large suite
of parameters and the behavior of the condition number. The coordinate samplings are
related to the spectral orders as explained below.
The results below are for S(x) = −(d/2) cos(2πx/L) and λ/L = 0.0625. Uniform
coordinate sampling is used throughout. All real orders and only real orders are used.
The method stopped unconditionally at d/L = 1, which was only achieved when κ =
ρ = 1. Additional sampling methods can be found in [13]. The methods we present are
representative.
8.1
First coordinate sampling method
We define this method as follows. First, choose the spectral discretization or sampling
by choosing all real propagating orders both above and below the surface. There are n1
real propagating orders above the surface and n2 below. Only real orders are kept for a
total of n1 + n2 rows (related to the spectral sampling). The first coordinate sampling
method is then defined by the number of coordinate samples equal to (n1 + n2 )/2. Since
there are two boundary unknowns the result is a square matrix to invert. The results are
illustrated in Fig. 4, which shows the maximum value of d/L for different combinations
of wavenumber ratio, density ratio and incident angle at fixed maximum Error = −2.
In each of the twelve horizontal rectangular blocks of Fig. 4(a) there are five horizontal
bands corresponding to different wavenumber ratios (0.5, 1, 2, 4, 8) read from bottom to
top. The blocks themselves refer to incident angle from 1◦ to 89◦ in increments of 8◦ read
from top to bottom. Density ratio is plotted on the horizontal. It can be seen that very
large values of d/L can be computed at nearly all angles, density ratios and wavenumber
ratios. The (inverted media) wavenumber ratio 0.5 could only be computed for small
d/L values until near vertical incidence. Even for angles close to grazing there are wide
variations of density and wavenumber ratios for which appreciable d/L values could be
routinely and rapidly computed within the error criterion specified.
In Fig. 4(b) the eight horizontal rectangular blocks refer to density ratios (0, 0.5, 1,
2, 4, 8, 16, ∞). The bands in each block refer to wavenumber ratios all as a function of
incident angle. For the SC method the deterioration (in the sense of max d/L computable
within the stated error) is evident as the incident angle approaches grazing. For these
latter cases it was necessary to use the full spectral method SS.
In Fig. 4(c) the five horizontal rectangular blocks refer to wavenumber ratios (0.5, 1,
2, 4, 8, read from top to bottom) and the eight bands per block to density ratios all as
a function of incident angle. For the wavenumber ratio 0.5 (top block) reasonable d/L
values could be computed only closer to normal incidence. For other wavenumber values
much larger d/L values could be computed over a much wider parameter range.
The result of this section is to quantify the limiting values of d/L for which the SC
18
method can be computed within fixed error for the particular coordinate sampling method
employed.
Incident angle=1 to 89, by 8
Max. d/L
8
0.5
0.5
8
Wavenumber ratio=0.5,1,2,4,8
0.5
8
0.4
0.5
8
0.5
8
0.5
8
0.3
0.5
8
0.5
8
0.2
0.5
8
0.5
8
0.5
8
0.1
0.5
8
0.5
0
0.5
1
2
4
Density ratio
(a)
19
8
16
Inf
0
Density ratio=0,0.5,1,2,4,8,16,Inf
Max. d/L
0.5
8
Wavenumber ratio=0.5,1,2,4,8
0.5
8
0.4
0.5
8
0.5
8
0.3
0.5
8
0.5
0.2
8
0.5
8
0.1
0.5
8
0.5
20
40
Incident angle
(b)
20
60
80
0
Wavenumber ratio= 0.5,1,2,4,8
Max. d/L
0.5
Inf
0
0.4
Density ratio=0,0.5,1,2,4,8,16,Inf
Inf
0
0.3
Inf
0
0.2
Inf
0
0.1
Inf
0
20
40
Incident angle
60
80
(c)
Figure 4: The figures (a), (b) and (c) show the maximum value of d/L for convergence
with 1% error for the SC method. The number of coordinate samples is equal to the
average of the two numbers of spectral samples. The figures are: (a) wavenumber ratios
(0.5, 1, 2, 4, 8) as bands from bottom to top in each of the 12 rectangular boxes vs. density
as a function of incident angle (from 1◦ to 89◦ by 8◦ increments read from the top box to
the bottom box of the 12 rectangular boxes), (b) wave number ratios (read as in (a)) vs.
incident angle for eight density ratios (0, 0.5, 1, 2, 4, 8, 16, ∞) read from the top box to the
bottom box of the eight rectangular boxes, (c) eight density ratios (0, 0.5, 1, 2, 4, 8, 16, ∞)
read as bands from the bottom to the top of each of the five rectangular boxes vs. incident
angle for five wavenumber ratios (0.5, 1, 2, 4, 8) read from the top rectangular box to the
bottom box.
21
0
8.2
Second coordinate sampling method
All real propagating orders and only real propagating orders are used for the spectral
samples as in the first method. Thus the matrix has n1 + n2 rows. Here the number of
coordinate samples is chosen to be n1 , the number of spectral samples above the surface.
For two boundary unknowns this means the matrix has 2n1 columns and is not square. It
is squared by multiplying it by the adjoint. The condition number of this square matrix
is defined to be the product of the norm of the matrix times the norm of its inverse
or, alternatively, by the ratio of the magnitudes of its largest to smallest eigenvalue (see
[1, 7, 15] for details). Loosely speaking, a N by N matrix is ill-conditioned if the condition
number is much larger than the size (N) of the matrix. A problem is marginally illconditioned if the condition number is of the order of its size, and well-conditioned if it is
well below the size of the matrix. Of course, the larger the condition number, the harder
the matrix is to invert.
In Figs. 5(a)-(c) we plot the log of this condition number versus wavenumber ratio,
density ratio and incident angle. It can be seen that for the majority of cases the condition number is small. Large condition numbers occurred when (numerically) the density
became very large (Fig. 5(a) and (b)) or when the wavenumber ratio was equal to one
(Figs. 5(a), (b) and (c)). Although these condition numbers were sometimes larger, all
the systems were solved within Error = −2.
Generally speaking, although we often encountered large condition numbers of our
matrix systems, this did not present a significant difficulty to the computations.
22
Incident angle=1 to 89, by 8
log(cond)
8
0.5
8
3
Wavenumber ratio=0.5,1,2,4,8
0.5
8
0.5
8
2.5
0.5
8
0.5
8
2
0.5
8
0.5
8
1.5
0.5
8
1
0.5
8
0.5
8
0.5
0.5
8
0.5
0
0.5
1
2
4
Density ratio
(a)
23
8
16
Inf
0
Density ratio=0,0.5,1,2,4,8,16,Inf
log(cond)
8
Wavenumber ratio=0.5,1,2,4,8
0.5
8
3
0.5
8
2.5
0.5
8
2
0.5
8
1.5
0.5
8
0.5
8
1
0.5
8
0.5
0.5
20
40
Incident angle
(b)
24
60
80
0
Wavenumber ratio= 0.5,1,2,4,8
log(cond)
Inf
3
0
Density ratio=0,0.5,1,2,4,8,16,Inf
Inf
2.5
0
Inf
2
0
1.5
Inf
1
0
Inf
0.5
0
20
40
Incident angle
60
80
(c)
Figure 5: In the figures above, the value of the condition number is shown for convergent
examples within Error = −2 for the SC method. The number of coordinate samples
is equal to the numbers of spectral samples above the surface. The figures are: (a)
wavenumber ratios (0.5, 1, 2, 4, 8) as bands from bottom to top in each of the 12 rectangular
boxes versus density as a function of incident angle (from 1◦ to 89◦ by 8◦ increments read
from the top box to the bottom box of the 12 rectangular boxes), (b) wave number ratios
(read as in (a)) versus incident angle for eight density ratios (0, 0.5, 1, 2, 4, 8, 16, ∞) read
from the top box to the bottom box of the eight rectangular boxes, (c) eight density
ratios (0, 0.5, 1, 2, 4, 8, 16, ∞) read as bands from the bottom to the top of each of the
five rectangular boxes versus incident angle for five wavenumber ratios (0.5, 1, 2, 4, 8) read
from the top rectangular box to the bottom box.
25
0
9
SC Results using Different Spectral Sampling Methods
In this section we present results for the SC method with different spectral sampling methods. The results below are for S(x) = −(d/2) cos(2πx/L) and λ/L = 0.0625. Uniform
coordinate sampling is used throughout. The number of coordinate samples, spectral
samples above, and spectral samples below are all equal. The method stopped unconditionally at d/L = 1, which was only achieved when κ = ρ = 1. The error was fixed at
Error = −2. The purpose was to find the largest value of d/L and the best condition
number for the different sampling methods. Adding additional orders to the spectral
sampling methods yielded better results. Several other sampling methods are in [13].
9.1
First spectral sampling method
The smaller number of real propagating orders from above and below is used both above
and below. In addition, ten positive spectral orders and ten negative spectral orders are
added both above and below. For example, this means that for κ < 1, the number of real
orders chosen is n1 , the number of real propagating orders from above. Thus, adding ten
positive (negative) spectral orders above means adding evanescent orders which propagate
in the positive (negative) x−direction. Adding the same number of modes in the lower
region means adding a mixture of propagating and evanescent modes where the exact
mixture depends on the specific value of κ. For κ = 1, only evanescent modes are added
in both regions. The case of κ < 1 means that the number of real propagating orders
is n2 (< n1 here) and the explanation is the inverse of the κ > 1 case. The results are
presented in Fig. 6.
In Fig. 6(a) each of the twelve horizontal blocks represents a different angle of incidence
(from 1◦ to 89◦ in increments of 8◦ read from top to bottom block). Each band in a block
represents wavenumber ratios from 0.5 to 8 read from bottom to top. All are plotted
versus density ratio from zero (Dirichlet problem) to infinity (Neumann problem). Large
values of d/L could be computed for this extremely wide range of parameters. The method
was very robust. Only at κ = 0.5 and large angles of incidence was the maximum value
of d/L the smallest.
In Fig. 6(b) each of the eight horizontal blocks represents a different density ratio
(from zero to infinity read from top to bottom block). The bands in each block represent
wavenumber ratios as in Fig. 6(a). All are plotted versus incident angle. The same
conclusions can be reached as those of Fig. 6(a). The small values of d/L are highlighted
somewhat better than in Fig. 6(a). In Fig. 6(c) the five horizontal blocks represent
wavenumber ratios (0.5 to 8 read from top to bottom block), the bands in each block
are now density ratios (from zero to infinity read from bottom to top of each block), all
plotted versus incident angle. The smaller maximum values of d/L are highlighted in yet
another way in the top block.
This spectral sampling method yielded better results (in terms of large d/L values)
than the coordinate sampling method in Section 8.1.
26
Incident angle=1 to 89, by 8
Max. d/L
8
0.5
0.5
8
Wavenumber ratio=0.5,1,2,4,8
0.5
8
0.4
0.5
8
0.5
8
0.5
8
0.3
0.5
8
0.5
8
0.2
0.5
8
0.5
8
0.5
8
0.1
0.5
8
0.5
0
0.5
1
2
4
Density ratio
(a)
27
8
16
Inf
0
Density ratio=0,0.5,1,2,4,8,16,Inf
Max. d/L
0.5
8
Wavenumber ratio=0.5,1,2,4,8
0.5
8
0.4
0.5
8
0.5
8
0.3
0.5
8
0.5
8
0.2
0.5
8
0.1
0.5
8
0.5
20
40
Incident angle
(b)
28
60
80
0
Wavenumber ratio= 0.5,1,2,4,8
Max. d/L
0.5
Inf
0
0.4
Density ratio=0,0.5,1,2,4,8,16,Inf
Inf
0
0.3
Inf
0
0.2
Inf
0
0.1
Inf
0
20
40
Incident angle
60
80
0
(c)
Figure 6: The above figures show maximum d/L values for convergence within Error = −2
for the SC method. The smaller number of real orders from above and below is used both
above and below for the spectral sampling, with an additional ten positive and ten negative
orders added both above and below. The figures are: (a) wavenumber ratios (0.5, 1, 2, 4, 8)
as bands from bottom to top in each of the 12 rectangular boxes vs. density as a function
of incident angle (from 1◦ to 89◦ by 8◦ increments read from the top box to the bottom
box of the 12 rectangular boxes), (b) wave number ratios (read as in (a)) vs. incident
angle for eight density ratios (0, 0.5, 1, 2, 4, 8, 16, ∞) read from the top box to the bottom
box of the eight rectangular boxes, (c) eight density ratios (0, 0.5, 1, 2, 4, 8, 16, ∞) read as
bands from the bottom to the top of each of the five rectangular boxes vs. incident angle
for five wavenumber ratios (0.5, 1, 2, 4, 8) read from the top rectangular box to the bottom
box.
29
9.2
Second spectral sampling method
In Fig. 7(a)-(c) we plot the log10 of the condition number at the d/L value for which
Error = −2 for parameters in the same sequence as the results in Fig. 6(a)-(c). The
smaller number of real orders from above and below is used both above and below, but
here twenty positive orders and twenty negative orders are added both above and below.
Adding a small number of evanescent orders often improved the results (see Fig. 6) but
adding large numbers of evanescent orders increased the condition number as can be seen
in Figs. 7(a)-(c). It was possible to handle very large condition numbers in a routine
manner however.
Incident angle=1 to 89, by 8
log(cond)
8
0.5
8
18
Wavenumber ratio=0.5,1,2,4,8
0.5
8
16
0.5
8
0.5
8
14
0.5
8
12
0.5
8
10
0.5
8
8
0.5
8
0.5
8
6
0.5
8
4
0.5
8
2
0.5
0
0.5
1
2
4
Density ratio
(a)
30
8
16
Inf
0
Density ratio=0,0.5,1,2,4,8,16,Inf
log(cond)
Wavenumber ratio=0.5,1,2,4,8
8
0.5
8
18
0.5
16
8
14
0.5
8
12
0.5
8
10
0.5
8
8
6
0.5
8
4
0.5
8
2
0.5
20
40
Incident angle
(b)
31
60
80
0
Wavenumber ratio= 0.5,1,2,4,8
log(cond)
Inf
18
0
16
Density ratio=0,0.5,1,2,4,8,16,Inf
Inf
14
0
12
Inf
10
0
8
Inf
6
0
4
Inf
2
0
20
40
Incident angle
60
80
0
(c)
Figure 7: The figures above show the value of the condition number for convergent examples within Error = −2 for the SC method. The smaller number of real orders from above
and below is used both above and below for the spectral sampling, with an additional
twenty positive and twenty negative orders added both above and below. The figures are:
(a) wavenumber ratios (0.5, 1, 2, 4, 8) as bands from bottom to top in each of the 12 rectangular boxes vs. density as a function of incident angle (from 1◦ to 89◦ by 8◦ increments
read from the top box to the bottom box of the 12 rectangular boxes), (b) wave number
ratios (read as in (a)) vs. incident angle for eight density ratios (0, 0.5, 1, 2, 4, 8, 16, ∞)
read from the top box to the bottom box of the eight rectangular boxes, (c) eight density
ratios (0, 0.5, 1, 2, 4, 8, 16, ∞) read as bands from the bottom to the top of each of the five
rectangular boxes vs. incident angle for five wavenumber ratios (0.5, 1, 2, 4, 8) read from
the top rectangular box to the bottom box.
32
10
SS Results
In this section we present results using the SS method. These results are for S(x) =
−(d/2) cos(2πx/L), λ/L = 0.046875, 64 real scattered orders in lower (transmission)
region and 43 real scattered orders in upper (reflection) region, κ = 1.5, and ρ = 1.5.
In Fig. 8 we plot the d/L values (in steps of 0.01) versus incident angle (approximately
every ten degrees) for several different spectral sampling schemes all of which included
(at least) all real propagating orders in the reflected region. The plot labelled with circles
included only the 43 real orders above and only 43 of the 64 propagating orders below.
The square symbol plot included 53 orders above (10 of which were evanescent) and 53
of the 63 real orders below. The remaining schemes added more evanescent orders above
(64, 43 of which are real), added evanescent orders both above and below (74, diamond
plot) and finally the plot with the plus symbol kept only real orders both above and
below. The error was fixed at Error = −2 (percentage error of 1%). All of the schemes
performed credibly but generally good results occur when (at least) all real orders are
included. In Fig. 9 the log of the condition number was plotted versus incident angle
for the same spectral choices as those in Fig. 8. For near normal incidence the condition
number was very large but all calculations were done within the error constraint. The
condition number was smaller near grazing incidence which is exactly the opposite result
from coordinate based methods.
We can thus conclude that the full spectral method is an efficient way to compute
scattered fields at near grazing incidence.
33
0.25
2
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53
0.20
43
+
64 ×
2
◦
+
×
0.15
74 d/L
0.10
0.05
+
2
◦
×
◦
2
+
×
+
◦
2
×
◦
+
×
2
×
2
+
×
2
+
×
×
2
×
◦
◦
2
◦
◦
0.00
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Incident angle
Figure 8: d/L values for 1% error in energy check for the SS formalism. Number of orders
labeled at left. ◦ — 43 orders included, both above and below, 2 — 53, × — 64, and — 74.
+ — 43 above, 64 below.
34
15
53 ........
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.
64 ×
12
43 ◦
×
74 +
9
log10 (cond(K))
+
◦
×
+
◦
6
3
+
×
◦
×
+
×
+
×
+
◦
◦
×
◦
0
0
+
×
+
×
◦
◦
+
×
◦
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Incident angle
Figure 9: Condition numbers for 1 % error in energy check for the SS formalism. Number of
orders labeled at left. ◦ — 43 orders included, both above and below, — 53, × — 64, and +
— 74.
35
In Fig. 10, we present extensive results for the SS formalism on the error (defined
below) versus κ, the ratio of wavenumbers. For this case, it can be concluded that the
lowest error is maintained provided we choose a spectral sampling consisting of only real
orders. Smaller and larger matrices (fewer than all real orders or the addition of evanescent
orders) work well between κ = 1 to κ = 5 but very poorly after κ = 6. The matrix K is
banded but not always square and pseudo-inverse methods are used.
These results are for S(x) = −(d/2) cos(2πx/L), d/L = 0.1, λ/L = 0.0625 (32 real
orders above the surface, 320 real orders below), θi = 25◦ , ρ = 3, and κ = 10. Error =
log10 |1 − Energy Check|.
0.0
−0.5
−1.0
◦
−2.5
Error
−3.0
−3.5
−4.0
−4.5
−5.0
◦
◦
◦
◦
◦
◦
× ×
◦
× ×
×
◦
◦
◦
×
×
◦
×
◦
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×
◦
−1.5 ×
◦
−2.0
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........
+.......
×
◦
◦
×
+
◦
×
+
×
×
+
◦
×
+
× +
+
× +
× +
× ×
+ +
+ + +
+ + +
+ + +
−5.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
κ
Figure 10: Error versus κ for the SS formalism. Matrix size: ◦ — 32 by 32, 2 — 48 by
48, × — 64 by 64, and + — all real orders, (32 columns, approximately 32 κ rows). For
the + result Error = −5.2.
36
In Fig. 11, we see that over a decade of values of ρ the error is small and quite stable for
various sampling schemes. These results are for S(x) = −(d/2) cos(2πx/L), d/L = 0.1,
λ/L = 0.0625, κ = 2 (32 real orders above the surface, 64 below), and θi = 25◦ with Error
= log10 |1 − Energy Check|.
−1.5
−2.0
−2.5
−3.0
−3.5
−4.0
−4.5
−5.0
Error
◦.............
. .
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.....
......
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... ..
... ....
..
.....
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....
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.....
..
....
.
.....
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....
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.....
..
....
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◦
◦ ◦ ◦
◦ ◦ ◦
◦
◦
◦
+
+
◦
×
+
×
+
×
×
×
◦ +
×
+
+
+
× ×
×
+
+
◦ +
×
×
◦ +
×
◦ +
×
×
×
×
+
+
× +
×
+
×
+
×
×
+
+
+
×
+ +
× ×
×
+
◦
◦
◦
+
◦
◦
◦
◦ ◦ ◦
−5.5 +
◦
−6.0
−6.5 −7.0
−7.5
−8.0
−8.5
−9.0
−9.5
−10.0
−10.5 ×
−11.0
0
1
2
3
4
5
ρ
6
7
8
9
10
Figure 11: Error vs. ρ for the SS formalism. Matrix sizes are: ◦ — 32 by 32, — 48 by
48, × — 64 by 64, + — 64 by 32.
37
For the second example of error versus density variation in Fig. 12, the surface is
slightly rougher than the surface used to generate the results for Fig. 11 and the results
are different. The choice of only real orders works well but better results can be obtained
by taking away some real orders or by adding some evanescent orders. The results in
Fig. 12 are for S(x) = −(d/2) cos(2πx/L), d/L = 0.175, λ/L = 0.0406, κ = 1.3 (49
real orders above the surface, 64 below), θi = 0.1◦ (near normal incidence). Error =
log10 |1 − Energy Check|.
−1
−2
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ρ
Figure 12: Error vs. ρ for the SS formalism. Matrix sizes are: ◦ — 49 by 49, — 57 by
57, × — 64 by 64, + — 64 by 49 (real orders only).
38
11
Summary and Conclusions
We considered the scattering from and transmission of an acoustic or electromagnetic
wave through a one-dimensional periodic surface. For this problem, the electromagnetic
cases of TE and TM-polarization reduced to the scalar acoustic examples. Three different
theoretical and computational methods were described, all involving the solution of integral equations for the boundary unknowns. They are characterized by two sample spaces
for their discrete solution, coordinate (C) space and spectral (S) space, and labelled by
the sampling of the rows and columns of the discretized matrices. They are coordinatecoordinate (CC), the usual coordinate-space methods, spectral-coordinate (SC) where the
matrix rows are sampled in spectral space, and spectral-spectral (SS) where both rows and
columns are sampled in spectral space. The SS method uses a topological basis expansion
for the boundary unknowns.
Equations were derived for infinite surfaces, then specialized and solved for periodic
surfaces. An extensive suite of computational results was presented for the transmission
problem as a function of roughness, near grazing incidence as well as many other angles,
density and wavenumber ratios. Matrix condition numbers and different sampling method
were considered. An error criterion was used to gauge the validity of the results.
The computational results indicated that the SC method was by far the fastest (by
several orders of magnitude), but that it became ill-conditioned for very rough surfaces.
The CC method was most reliable, but often required very large matrices and was consequently extremely slow.
The computational results for the full transmission case were presented in Sections 610. The three methods, CC, SC and SS, were all studied. In Section 7, the SS example
required only a small number of topological modes to achieve a highly accurate solution at
extreme grazing incidence. The widely used Kirchoff approximation method is known to
fail near grazing incidence and coordinate-based numerical schemes have computational
difficulties near grazing. The SS method is computationally efficient and accurate at near
grazing incidence and thus we believe fills a gap in the literature in this regard.
The SC method was extensively studied in Sections 8 and 9 using different coordinate
and spectral sampling schemes and a fixed error. Maximum values of slope and condition
number of the matrix system were tabulated, and reliable results were found for both
large slopes and large condition numbers. The extensive parameter domain over which
the SC method was computed (κ = 0.5 − 8, ρ = 0 to infinity, all incident angles) clearly
indicates that it is a highly robust computational method.
Finally, in Section 10 the SS method was studied with fixed error for largest slope
values as a function of wavenumber and density ratios and large condition numbers.
Highly stable results were found over the same extensive suite of wavenumber, density
and incidence angle parameters as those of the SC method.
Spectral based methods thus provide efficient and robust computational alternatives
to pure coordinate based methods for periodic surface scattering.
As mentioned earlier, these computational results are a representative sample of the
results in our report [13]. This report also contains results for perfectly reflecting Dirichlet
and Neumann boundary conditions as well as applications to a Gaussian tapered cosine
surface and a wave-superposition surface. Additional coordinate and spectral sampling
schemes were also investigated.
39
Acknowledgements
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.
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.
We are grateful to Mr. Guy Somberg and Mr. Douglas Baldwin for technical assistance
in the production of this paper.
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