Theoretical and Computational Aspects
of Scattering from Rough Surfaces:
One-dimensional Transmission Interface
J. DeSanto, G. Erdmann,
W. Hereman, and M. Misra
MCS-00-02
March 2000
Technical Report # 3 MURI Project
AFOSR Grant # F49620-96-1-0039
Report of Work in Progress
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
Contents
1 Derivation of CC Equations for an Infinite One-Dimensional Transmission Interface
1
2 CC Surface Integral Equations
4
3 CC Equations for a Periodic Surface
5
4 Derivation of SC Equations for an Infinite One-Dimensional Transmission Interface
7
5 SC Equations for a Periodic Surface
8
6 SS Equations for a Periodic Surface: Topological Basis
6.1 The first equation: defining Dj . . . . . . . . . . . . . . .
6.2 The second equation: an equation for Dj . . . . . . . . .
6.3 The third equation: an equation for Aj . . . . . . . . . .
6.4 The fourth equation: an equation for Bj . . . . . . . . . .
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7 Energy
9
10
10
11
12
13
8 Reduced Equations for the Dirichlet Problem: Perfect Electric Conductor, TE Polarization
13
9 Reduced Equations for the Neumann Problem: Perfect Electric Conductor, TM Polarization
14
10 Computational Results: Dirichlet Problem (Additional Surfaces)
10.1 Gaussian-cosine surface . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.1 Trial 1.1a, λ L, θi = 20◦ . . . . . . . . . . . . . . . . . . .
10.1.2 Trial 1.1b, λ L, θi = 75◦ . . . . . . . . . . . . . . . . . . .
10.1.3 Trial 1.2a, λ ≈ L, θi = 20◦ . . . . . . . . . . . . . . . . . . . .
10.1.4 Trial 1.2b, λ ≈ L, θi = 75◦ . . . . . . . . . . . . . . . . . . .
10.2 Wave-superposition surface . . . . . . . . . . . . . . . . . . . . . . .
10.2.1 Trial 2.1a, λ L, θi = 20◦ . . . . . . . . . . . . . . . . . . .
10.2.2 Trial 2.1b, λ L, θi = 75◦ . . . . . . . . . . . . . . . . . . .
10.2.3 Trial 2.2a, λ ≈ L, θi = 20◦ . . . . . . . . . . . . . . . . . . . .
10.2.4 Trial 2.2b, λ ≈ L, θi = 75◦ . . . . . . . . . . . . . . . . . . .
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15
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18
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24
26
28
30
11 Computational Results: Neumann Problem
11.1 Trial 1.1a, λ L, θi = 20◦ . . . . . . . . . .
11.2 Trial 1.1b, λ L, θi = 75◦ . . . . . . . . . .
11.3 Trial 1.2a, λ ≈ L, θi = 20◦ . . . . . . . . . . .
11.4 Trial 1.2b, λ ≈ L, θi = 75◦ . . . . . . . . . . .
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32
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35
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12 Discussion of Computational Results: Transmission Problem
13 Transmission Interface Results for the SS Method
13.1 Case 1: Fairly Rough Surface . . . . . . . . . . . . .
13.2 Case 2: Less Rough Surface . . . . . . . . . . . . . .
13.3 Case 3: Fairly Rough Surface, Near Grazing . . . . .
13.4 Case 4: Very Smooth Surface, Near Grazing . . . . .
13.5 Case 5: Flat Surface, No Physical Interface . . . . .
13.6 Case 6: Flat Surface, Reflection and Refraction . . .
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36
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37
37
38
40
41
43
44
14 Comparison of Transmission Results
45
15 A Near-Null Field from Nonzero Surface Conditions
48
i
16 Comparison of Transmission Interface Results: Surface Currents and Fields
51
17 Comparison of Transmission Interface Results: Near Grazing Incidence
59
18 Roughness Graphs, Transmission Interface,
18.1 First coordinate sampling . . . . . . . . . .
18.2 Second coordinate sampling . . . . . . . . .
18.3 Third coordinate sampling . . . . . . . . . .
SC Formalism, Coordinate
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
Sampling
. . . . . . . .
. . . . . . . .
. . . . . . . .
19 Condition Number Graphs, Transmission Interface, SC Formalism, Coordinate
pling
19.1 First coordinate sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.2 Second coordinate sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.3 Third coordinate sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20 Roughness Graphs, Transmission Interface,
tions
20.1 First spectral sampling . . . . . . . . . . . .
20.2 Second spectral sampling . . . . . . . . . .
20.3 Third spectral sampling . . . . . . . . . . .
66
66
70
73
Sam76
76
79
82
. . .
. . .
. . .
SC Formalism, Spectral Sampling Varia85
85
88
91
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
21 Condition Number Graphs, Transmission Interface,
Variations
21.1 First spectral sampling . . . . . . . . . . . . . . . . .
21.2 Second spectral sampling . . . . . . . . . . . . . . .
21.3 Third spectral sampling . . . . . . . . . . . . . . . .
SC Formalism, Spectral Sampling
94
. . . . . . . . . . . . . . . . . . . . . 94
. . . . . . . . . . . . . . . . . . . . . 97
. . . . . . . . . . . . . . . . . . . . . 100
22 Roughness and Condition Number Graphs, Transmission Interface, SS Formalism
103
23 Variation of the Transmission Interface SS Formalism with κ, the Wave Number
Ratio
105
24 Variation of the Transmission Interface SS Formalism with ρ, the Density Ratio
107
25 Variation of the Transmission Interface SS Formalism with ρ, the Density Ratio,
Rougher Surface
111
26 Summary and Conclusions
113
A Brief Remarks on the Numerical Solution of the Coupled Equations
114
B Acoustic and Electromagnetic Boundary Conditions
115
B.1 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.2 Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
ii
Abstract
We consider the scattering from and transmission through a one-dimensional rough
surface. For this problem, the electromagnetic cases of TE and TM-polarization
reduce to the scalar acoustic examples. Three different theoretical and computational
methods are 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 coordinate-coordinate (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 are derived for infinite surfaces, then specialized and solved for several
periodic surfaces including a simple cosine surface, a Gaussian tapered cosine, and a
wave-superposition surface. An extensive suite of computational results is 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.
PART I: THEORETICAL DEVELOPMENT
1
Derivation of CC Equations for an Infinite OneDimensional Transmission Interface
We consider the scattering from and transmission through a one-dimensional surface specified by z = s(x)
(see Figure 1).
In this section we derive the equations for an infinite surface and specify it to be periodic later
in Section 3. Some of the development is similar to that in a previous report [8] and paper [9], which
treated perfectly reflecting surfaces with a Dirichlet boundary condition. 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, 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 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,
+
in the
and t is time). The two regions of the problem (see Figure 1) are defined by z > s(x) (region 1, DR
limit as R → ∞) with constant parameters ρ1 (density or electromagnetic parameters, see Appendix B)
−
and wave number k1 = 2π/λ where λ is wavelength, and z < s(x) (region 2, DR
in the limit as R → ∞)
with ρ2 and k2 . We use density and wave number ratios ρ = ρ2 /ρ1 and κ = k2 /k1 .
1
sc
+
HR
sc
in
sc
+
DR
R
s(x)
-
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 → inf) and z < s(x) (DR in the limit as R → inf) 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.
2
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
2
x ∈ DR
(1.1)
(j = 1),
(j = 2).
(1.2)
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 ),
(1.3)
and are explicitly given by
i (1)
H (kj |x − x |) ,
(1.4)
4 0
the Hankel function of zeroth order and first kind.
+
using ψ1SC and G1 yields in the limit as R → ∞ an integral representation for
Green’s theorem in DR
SC
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)
Gj (x, x ) =
(Sj u)(x) =
S∞
Gj (x, xs )u(xs ) dx ,
and the double (D) layer potential with density v is given by
(Dj v)(x) =
∂n Gj (x, xs )v(xs ) dx .
(1.5)
(1.6)
S∞
+
The result of Green’s theorem in DR
as R → ∞ is then written as
θ1 (x)ψ1SC (x) = (D1 ψ1SC )(x) − (S1 N1SC )(x),
(1.7)
+
where θ1 (x) is the characteristic function of the region D∞
and N1SC is the normal derivative
N1SC (x) = ∂n ψ1SC (x).
(1.8)
+
A second equation can be formed from (1.7) by taking its normal derivative for x ∈ D∞
. It is
θ1 (x)N1SC (x) = ∂n (D1 ψ1SC )(x) − ∂n (S1 N1SC )(x).
(1.9)
+
since ψ1SC and G1 each satisfy the Sommerfeld
There are no contributions from the semicircle at H∞
radiation condition. In fact it has been shown by us [10] that there is no contribution so long as ψ1SC
contains no horizontal plane waves.
−
as R → ∞ yields analogous equations (with the same normal now pointing
Green’s theorem in DR
outward from the domain). They are
−θ2 (x)ψ2 (x) = (D2 ψ2 )(x) − (S2 N2 )(x),
(1.10)
−θ2 (x)N2 (x) = ∂n (Dn ψ2 )(x) − ∂n (S2 N2 )(x),
(1.11)
and
−
where θ2 is the characteristic function of D∞
and N2 is the normal derivative of ψ2 .
−
i
The incident field ψ satisfies (1.1) for j = 1 but in DR
(where there are no sources 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 (1.10) and (1.11)
−θ2 (x)ψ i (x) = (D1 ψ i )(x) − (S1 N i )(x),
3
(1.12)
and
−θ2 (x)N i (x) = ∂n (D1 ψ i )(x) − ∂n (S1 N i )(x),
i
(1.13)
i
where N is the normal derivative of ψ .
Combining (1.7) and (1.12) to form the total field ψ1
we get the equation
ψ1 = ψ i + ψ1SC ,
(1.14)
θ1 (x)ψ1 (x) = ψ i (x) + (D1 ψ1 )(x) − (S1 N1 )(x),
(1.15)
and combining (1.9) and (1.13) yields
θ1 (x)N1 (x) = N i (x) + ∂n (D1 ψ1 )(x) − ∂n (S1 N1 )(x).
(1.16)
+
Equations (1.15) and (1.16) are used to form surface integral equations from region D∞
, and (1.10) and
−
(1.11) from region D∞ . Equations (1.10) and (1.15) are integral representations for the total fields in
each region.
2
CC Surface Integral Equations
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 [4]
lim (Sj u)(x) = (Sj u)(xs ),
x→x±
s
(2.1)
since it is a continuous function. The double layer potential has a jump discontinuity
lim (Dj v)(x) = (Dj v)± (xs ),
x→x±
s
(2.2)
where
1
(Dj v)± (xs ) = (Dj v)(xs ) ± v(xs ).
2
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
(2.3)
(2.4)
The integrals on the right hand sides of (2.3) and (2.4) 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) [3]
lim ∂n (Dv)(x) = FP∂n (Dv)(xs ).
(2.5)
x→x±
s
The limits of (1.15) and (1.16) are thus
1
ψ1 (xs ) = ψ i (xs ) + (D1 ψ1 )(xs ) − (S1 N1 )(xs ),
2
(2.6)
and
1
N1 (xs ) = N i (xs ) + FP∂n (D1 ψ1 )(xs ) − ∂n (S1 N1 )(xs )
2
and the limits (from below) of (1.10) and (1.11) are given by
and
(2.7)
1
− ψ2 (xs ) = (D2 ψ2 )(xs ) − (S2 N2 )(xs ),
2
(2.8)
1
− N2 (xs ) = FP∂n (D2 ψ2 )(xs ) − ∂n (S2 N2 )(xs ).
2
(2.9)
4
The continuity conditions at the rough interface are
ψ1 (xs ) = ρψ2 (xs ),
(2.10)
(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 (see Appendix A for details). Secondly,
we have
(2.11)
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 ),
(2.12)
N T (x) ≡ N1 (xs ).
(2.13)
1
F (x) = ψ i (xs ) + (D1 F )(xs ) − (S1 N T )(xs ),
2
(2.14)
and normal derivative of the total field
Then (2.6)–(2.9) can be written as
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
and
1 T
1
N (x) = − FP∂n (D2 F )(xs ) + ∂n (S2 N T )(xs ).
2
ρ
(2.15)
(2.16)
(2.17)
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 later. 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.
3
CC Equations for a Periodic Surface
The reduction of (2.14)–(2.17) to integral equations over a single period (−L/2 to L/2) of a periodic
surface follows procedures outlined in [8, 9]. 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)
L2
Gpj (x, x )N T (x ) dx ,
(3.1)
(Sj N T )(x) =
−L
2
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 )
(3.2)
where αn = sin(θn ) and θn is the angle of the nth scattered or transmitted Bragg wave, where the Bragg
equation is
λ
(3.3)
αn = α0 + n ,
L
5
with α0 = sin θi . Here, θi is the plane wave angle of incidence, measured from the positive z-direction,
and
1
1 − α2n 2
|αn | ≤ 1
(3.4)
m1 (αn ) =
2
12
i αn − 1
|αn | > 1,
1
κ2 − α2n 2
m2 (αn ) =
1
i α2n − κ2 2
and
|αn | ≤ κ
|αn | > κ.
(3.5)
For the other terms we have first
T
∂n (Sj N )(x) =
L
2
Gpj (x, x )N T (x ) dx ,
−L
2
(3.6)
where Gpj is the exterior normal derivative of Gpj (with respect to the x-variable),
Gpj (x, x ) = ∂n Gpj (x, x ),
second,
(Dj F )(x) =
L
2
−L
2
G̃pj (x, x )F (x ) dx ,
(3.7)
(3.8)
using the interior normal derivative
G̃pj (x, x ) = ∂n Gpj (x, x ),
and finally
∂n (Dj F )(x) =
with
L
2
−L
2
Gpj (x, x )F (x ) dx ,
Gpj (x, x ) = ∂n ∂n Gpj (x, x ),
the second normal derivative.
Equations (2.14)–(2.17) can thus be written in operator notation as
1
I − G̃p1 F = ψ i − Gp1 N T ,
2
1
I + Gp1 N T = N i + Gp1 F,
2
1
I + G̃p2 F = ρGp2 N T ,
2
and
ρ
1
I − Gp2
2
N T = −Gp2 F.
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
We can then combine these equations in various ways. We choose to avoid all equations where the
Hadamard Finite Part plays a part, namely (3.13) or (3.15) when F = 0. For the Dirichlet case (ρ = 0
and F = 0), we can use (3.12) or (3.13), or a linear combination of them. We call these options CC1, CC2,
and CFIE, refering 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
(3.12). For a 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 (3.12) and (3.14), which we again term CC1. Another option
is to add (3.15) to (3.13), removing the hyper singularity and use of Finite Part. Equations (3.12), (3.14),
or a linear combination thereof, can then be used for the second equation. This combination, termed
6
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 discretizaton 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, 5]. The pulse basis does not have the requisite differentiability to use the FP integrals
alone, and, as mentioned, we avoid them.
4
Derivation of SC Equations for an Infinite OneDimensional Transmission Interface
In the previous sections 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 [6].
First, use (1.15) for the scattered field with the boundary unknowns defined as in (2.12) and (2.13).
The result is
+
x ∈ D∞
,
ψ1SC (x) = (D1 F )(x) − (S1 N T )(x),
(4.1)
where the single and double layer functions are defined in (1.5) and (1.6). Similarly the transmitted field
can be found from (1.10)
ψ2 (x) = − ρ1 (D2 F )(x) + (S2 N T )(x),
−
x ∈ D∞
.
Next, define the Weyl representations for the Green’s functions [7]
∞
1
πi
eik1 [μ(x−x )+mj (μ)|z−s(x )|] dμ,
Gj (x, xs ) =
2
(2π) −∞ mj (μ)
where
mj (μ) =
1
(1 − μ2 ) 2
1
(κ2 − μ2 ) 2
j=1
j = 2,
(4.2)
(4.3)
(4.4)
with appropriate pure positive imaginary extensions when |μ| exceeds 1 or κ. For z > max s(x ) and
j = 1, drop the absolute value sign in the phase and use the result in (4.1). The scattered field can then
be represented as
ψ1SC (x) =
where
A(μ) =
and
A(μ, x ) =
∞
−∞
∞
−∞
A(μ)eik1 [μx+m1 (μ)z] dμ,
A(μ, x )e−ik1 [μx +m1 (μ)s(x )] dx ,
1
[(m1 (μ) − μs (x )) F (x ) + N (x )] ,
2λm1 (μ)
(4.5)
(4.6)
(4.7)
where we have scaled the boundary unknown N T as
N (x) = (ik1 )−1 N T (x).
(4.8)
Given the two boundary unknowns F and N we can thus find the scattered field.
In a similar way for z < min s(x ) and j = 2, drop the absolute value sign in (4.3) and use the result
in (4.2). The transmitted field is then
∞
ψ2 (x) =
B(μ)eik1 [μx−m2 (μ)z] dμ,
(4.9)
−∞
7
where
B(μ) =
and
B(μ, x ) =
∞
−∞
B(μ, x )e−ik1 [μx −m2 (μ)s(x )] dx ,
1
[(m2 (μ) + μs (x )) F (x ) − ρN (x )] .
2ρλm2 (μ)
(4.10)
(4.11)
Given these same boundary unknowns, we can thus find the transmitted field.
Next, we need two equations to solve for the boundary unknowns. From (1.15) we get
−ψ i (x) = (D1 F )(x) − (S1 N T )(x),
and from (1.10) we get
0 = ρ1 (D2 F )(x) − (S2 N T )(x),
−
x ∈ D∞
,
+
x ∈ D∞
.
In (4.12) we use (4.3) with j = 1 and z < min s(x ) to get
∞
I1 (μ)eik1 [μx−m1 (μ)z] dμ,
ψ i (x) =
(4.12)
(4.13)
(4.14)
−∞
where
I1 (μ) =
and
I1 (μ, x ) =
∞
−∞
I1 (μ, x )e−ik1 [μx −m1 (μ)s(x )] dx ,
1
{[m1 (μ) + μs (x )] F (x ) − N (x )} .
2λm1 (μ)
In (4.13) we use (4.3) with j = 2 and z > max s(x ) to get
∞
I2 (μ, x )e−ik1 [μx +m2 (μ)s(x )] dx ,
0=
(4.15)
(4.16)
(4.17)
−∞
where
I2 (μ, x ) =
1
{[m2 (μ) − μs (x )] F (x ) + N (x )} .
2ρλm2 (μ)
(4.18)
Equations (4.15) and (4.17) are the coupled equations to solve for the two boundary unknowns. This is
the general formulation for an infinite surface [6]. 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 ),
(4.19)
in (4.15). 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.
In the previous sections we treated the case where both rows and columns of the matrix to be inverted
were sampled in coordinate space. Here we have derived (from the CC equations) 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 [6].
5
SC Equations for a Periodic Surface
The derivation of the SC equations for a perfectly reflecting periodic surface was presented in [8, 9]. The
derivation here is a straightforward generalization of this. We omit the details and merely summarize the
results. The scattered and transmitted fields from (4.5) and (4.9) reduce to discrete infinite sums given
by
∞
ψ1SC (x) =
An eik1 (αn x+m1 (αn )z) ,
(5.1)
n=−∞
8
and
ψ2 (x) =
∞
Bn eik1 (αn x−m2 (αn )z) .
(5.2)
n=−∞
If we define the four phase functions (j = 1, 2)
φ±
j (μ, x) = μx ± mj (μ)s(x)
(5.3)
and the four terms resulting from taking normal derivatives
n±
j (μ, x) = mj (μ) ± μs (x),
(5.4)
then the two coupled equations (4.15) and (4.17) reduce for a single plane wave incident to the coupled
system
L2
+
−
1
n1 (αj , x)F (x) − N (x) e−ik1 φ1 (αj ,x) dx = 2m1 (α0 )Dδj0 ,
(5.5)
L
L −2
and
1
L
L
2
−L
2
−ik1 φ+ (αj ,x)
2
n−
dx = 0.
2 (αj , x)F (x) + ρN (x) e
(5.6)
Once these are solved for F and N , the scattered and transmitted amplitudes can be evaluated using the
periodic reduction of (4.6) and (4.10) as
1
L
and
1
L
6
L
2
−
+
n1 (αj , x)F (x) + N (x) e−ik1 φ1 (αj ,x) dx = 2m1 (αj )Aj ,
(5.7)
+
−
n2 (αj , x)F (x) − ρN (x) e−ik1 φ2 (αj ,x) dx = 2ρm2 (αj )Bj .
(5.8)
−L
2
L
2
−L
2
SS Equations for a Periodic Surface: Topological
Basis
In this section we derive the SS equations, which are found by choosing topological expansions for the
unknowns in the SC equations, namely
∞
F (x) =
−
Fj eik1 φ1 (αj ,x) ,
(6.1)
j =−∞
and
N (x) =
∞
−
ik1 φ1 (αj ,x)
N j n+
.
1 (αj , x)e
(6.2)
j =−∞
For a discussion on the completeness of the basis see [2, 12, 13]. We choose these bases to reduce the size
of the linear system from that of the SC equations, and using numerical trials and an energy check show
the results are accurate within certain slope limitations.
9
6.1
The first equation: defining Dj
Using (6.1) and (6.2), (5.5) can be written
∞
(1)
(1)
Kjj Fj − Mjj Nj = 2m1 (α0 )Dδj0 ,
(6.3)
j =−∞
where
(1)
Kjj =
1
L
and
(1)
Mjj =
1
L
L
2
−
−L
2
L
2
−
ik1 [φ1 (αj ,x)−φ1 (αj ,x)]
dx,
n+
1 (αj , x)e
−
−L
2
−
ik1 [φ1 (αj ,x)−φ1 (αj ,x)]
dx.
n+
1 (αj , x)e
(6.4)
(6.5)
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 ).
(6.6)
If m1 (αj ) = m1 (αj ) and αj = αj , then j = j , αj = −αj , and using integration by parts
Ly
iαj (j − j ) π
(1)
(1)
s
Kjj = −Mjj =
e−iy(j−j ) dy.
L
2π
−π
(6.7)
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
−π
e−i(j−j
and
(1)
Vjj =
(6.8)
)y ik1 s( Ly
2π )[m1 (αj )−m1 (αj )]
e
dy,
1 − m1 (αj )m1 (αj ) − αj αj .
m1 (αj ) − m1 (αj )
(1)
(6.9)
(6.10)
(1)
Using the relationship between M(1) and K(1) (i.e. Mjj = (2δjj − 1)Kjj ), we can rewrite (6.3) as
Nj = −Dδj0 +
∞
1
(1)
Kjj Dj ,
2m1 (αj ) (6.11)
j =−∞
where we have defined
Dj = Fj + Nj .
6.2
(6.12)
The second equation: an equation for Dj
Using (6.1) and (6.2), (5.6) can be written
∞
(2)
(2)
Kjj Fj + ρMjj Nj = 0,
(6.13)
j =−∞
where
(2)
Kjj 1
=
L
L
2
−L
2
−
+
ik1 [φ1 (αj ,x)−φ2 (αj ,x)]
n−
dx,
2 (αj , x)e
10
(6.14)
and
(2)
Mjj =
1
L
L
2
−
−L
2
+
ik1 [φ1 (αj ,x)−φ2 (αj ,x)]
n+
dx.
1 (αj , x)e
(6.15)
Using integration by parts, these can be rewritten
(2)
(2)
(2)
Kjj = Vjj Φjj ,
and
(2)
(2)
(6.16)
(2)
Mjj = Wjj Φjj ,
where
(2)
Φjj =
1
2π
π
e−i(j−j
−π
(2)
Vjj =
(6.17)
)y ik1 s( Ly
2π )[−m2 (αj )−m1 (αj )]
e
dy,
(6.18)
κ2 + m2 (αj )m1 (αj ) − αj αj ,
m2 (αj ) + m1 (αj )
(6.19)
1 + m2 (αj )m1 (αj ) − αj αj .
m2 (αj ) + m1 (αj )
(6.20)
and
(2)
Wjj =
Using (6.11) and (6.12), (6.13) can be rewritten as
(2)
(2)
Kjj Dj = D ρMj0 − Kj0 ,
∞
(6.21)
j =−∞
where the matrix K is defined by
K
=
jj ∞
(2)
(2)
+
ρMjj − Kjj (2)
Kjj j =−∞
1
(1)
K .
2m1 (αj ) j j
(6.22)
Equation (6.21) is a single equation for {Dj }. Its solution is used to evaluate {Nj } from (6.11) and
subsequently {Fj } from (6.12).
6.3
The third equation: an equation for Aj
Using (6.1) and (6.2), (5.7) can be written
∞
(3)
(3)
Kjj Fj + Mjj Nj = 2m1 (αj )Aj ,
(6.23)
j =−∞
where
(3)
Kjj =
1
L
and
(3)
Mjj =
1
L
L
2
−L
2
L
2
−L
2
−
+
ik1 [φ1 (αj ,x)−φ1 (αj ,x)]
n−
dx,
1 (αj , x)e
−
+
ik1 [φ1 (αj ,x)−φ1 (αj ,x)]
dx.
n+
1 (αj , x)e
(6.24)
(6.25)
Using integration by parts, these can be rewritten
(3)
(3)
(3)
(3)
Kjj = Mjj = Vjj Φjj ,
where
(3)
Φjj 1
=
2π
π
−π
e−i(j−j
Ly
)y ik1 s( 2π )[−m1 (αj )−m1 (αj )]
e
11
(6.26)
dy,
(6.27)
and
1 + m1 (αj )m1 (αj ) − αj αj .
m1 (αj ) + m1 (αj )
(3)
Vjj =
(6.28)
Using (6.26) and (6.12), (6.23) can be written
∞
1
(3)
Aj =
Kjj Dj .
2m1 (αj ) (6.29)
j =−∞
Thus the {Aj } can be directly evaluated once the {Dj } are known from (6.21).
6.4
The fourth equation: an equation for Bj
Using (6.1) and (6.2), (5.8) can be written
∞
j =−∞
where
(4)
Kjj 1
=
L
(4)
Mjj 1
=
L
and
(4)
(4)
Kjj Fj − ρMjj Nj = 2ρm2 (αj )Bj ,
L
2
−
−L
2
L
2
(6.30)
−
ik1 [φ1 (αj ,x)−φ2 (αj ,x)]
dx,
n+
2 (αj , x)e
−
−
ik1 [φ1 (αj ,x)−φ2 (αj ,x)]
dx.
n+
1 (αj , x)e
−L
2
(6.31)
(6.32)
If m1 (αj ) = m2 (αj ) and αj = αj , then j = j and
(4)
(4)
Kjj = Mjj = m1 (αj ).
(6.33)
If m1 (αj ) = m2 (αj ) and αj = αj , then, using integration by parts,
Ly
iαj (j − j ) π
(4)
Kjj =
s
e−iy(j−j ) dy,
L
2π
−π
and
(4)
Mjj =
iαj (j − j )
L
π
s
−π
Ly
2π
(6.34)
e−iy(j−j ) dy.
(6.35)
If m1 (αj ) = m2 (αj ), then the matrix elements can be rewritten
(4)
(4)
(4)
Kjj = Vjj Φjj ,
and
(4)
(4)
(6.36)
(4)
Mjj = Wjj Φjj ,
where
(4)
Φjj =
1
2π
π
−π
(4)
Vjj =
e−i(j−j
)y ik1 s( Ly
2π )[m2 (αj )−m1 (αj )]
e
dy,
(6.38)
κ2 − m2 (αj )m1 (αj ) − αj αj ,
m2 (αj ) − m1 (αj )
(6.39)
1 − m2 (αj )m1 (αj ) − αj αj .
m1 (αj ) − m2 (αj )
(6.40)
and
(4)
(6.37)
Wjj =
Then, (6.30) can be written
Bj =
∞
1
(4)
(4)
Kjj Fj − ρMjj Nj ,
2ρm2 (αj ) j =−∞
using the above simplifications for the matrices K and M.
12
(6.41)
7
Energy
The energy constraint is
∞
2
|Aj |
j=−∞
∞
e(m1 (αj ))
2 e(m2 (αn ))
+ρ
= D2 .
|Bn |
m1 (α0 )
m
(α
)
1
0
n=−∞
(7.1)
This can be derived directly using Green’s theorem on combinations of field representations in both
regions. The details are in [6]. We generally choose D = 1 and the difference between 1 and the
left-hand-side of (7.1) is generally quoted as the error in the calculation, i.e.,
Error = log10 |1 − LHS|,
(7.2)
where LHS is the left hand side of (7.1).
For the computational results we define
e(m1 (αj ))
|Aj |2 ,
m1 (α0 )
(7.3)
ρe(m2 (αn ))
|Bn |2 ,
m1 (α0 )
(7.4)
Rj =
and
Tn =
as the (scaled) reflected and transmitted energy in the repective modes.
8
Reduced Equations for the Dirichlet Problem: Perfect Electric Conductor, TE Polarization
For the Dirichlet problem (ρ = 0 and F = 0) the CC1 equation is from (3.12)
ψ i = Gp1 N T ,
and the CC2 equation from (3.13)
1
I + Gp1
2
N T = N i.
(8.1)
(8.2)
The SC equation from (5.5) is
1
L
L
2
−L
2
−
N (x) e−ik1 φ1 (αj ,x) dx = −2m1 (α0 )Dδj0 ,
(8.3)
and the SS equation from (6.3) (with Fj = 0) is
∞
j =−∞
(1)
Mjj Nj = −2m1 (α0 )Dδj0 ,
(8.4)
These are the equations we solved in [8, 9]. We repeat them here for completeness since we present
computational results for new surfaces in Section 10.
13
9
Reduced Equations for the Neumann Problem: Perfect Electric Conductor, TM Polarization
For the Neumann problem (ρ = ∞ and N T = 0 or N = 0) the CC2 equation is from (3.12)
1
I − G̃p1 F = ψ i ,
2
(9.1)
and this is the only equation we use.
The SC equation from (5.5)
1
L
L
2
−L
2
−
−ik1 φ1 (αj ,x)
n+
dx = 2m1 (α0 )Dδj0 ,
1 (αj , x)F (x) e
(9.2)
and the SS equation from (6.3) (with Nj = 0)
∞
j =−∞
(1)
Kjj Fj = 2m1 (α0 )Dδj0 ,
Computational results for this boundary value problem are presented in Section 11.
14
(9.3)
PART II: COMPUTATIONAL RESULTS
10
Computational Results: Dirichlet Problem (Additional Surfaces)
We solve the equations from Section 8. Details of the solution can be found in [8, 9]. Here we treat two
new classes of complicated surfaces listed below. For these two classes, the CC equations were often quite
large before a good energy check was found. The SC equations gave the fastest solutions but not always
a reliable energy check particularly in the case of a large height and/or large slope. The tables refer to
SS, SC, CC1 (first kind), CC2 (second kind defined in Section 3) and CG1, a coordinate-space Galerkin
method whose details are in [8, 9].
The first surface is a Gaussian-tapered cosine surface
s(x) = −(d/2) exp(−2 (x/L)2 ) cos(2π1 x/L),
(10.1)
presented in Section 10.1. Here 2 is chosen to taper the surface to near zero at the periodic end points.
Results are presented for λ L, λ ≈ L and incident angles of 20◦ and 75◦ . For λ L, the coordinate
based methods worked best provided the matrix size was large enough. These resulted in reliable but slow
solutions. For λ ≈ L, the SC method was highly reliable and very fast. Surface currents were extremely
variable and required extensive sampling.
The second class of surfaces are referred to as wave-superposition surfaces of the form
s(x) = −(d/2) cos(2π1 x/L) cos(2πx/L).
(10.2)
They are also referred to as modulated wave trains and arise from the interaction of gravity-capillary
waves with longer waves and can lead to polarization (HH/VV) ratios greater than one. For details see
[14, 15]. The surface can also be written as the sum of two sinusoids. The scattering results are presented
in Section 10.2. The conclusions are analogous to those for the Gaussian-cosine surface.
The scattered energy components Rj are defined in (7.3).
15
10.1
Gaussian-cosine surface
10.1.1
Trial 1.1a, λ L, θi = 20◦
S(x)
d/λ
d/L
L/λ
θi
ε1
ε2
Formalism
SS
SS
SS
SC
SC
SC
CC1
CC1
CC1
CC1
CG1
CG1
CG1
CC2
CC2
CC2
CC2
Matrix
Size
128
138
148
128
138
148
64
128
256
512
65
129
257
64
128
256
512
−(d/2) exp(−ε2 (x/L)2 ) cos(2πε1 x/L)
4.8
0.075
64
20◦
10
18.42
Q = λ/Δ
2.0
2.2
2.3
1.0
2.0
4.0
8.0
1.0
2.0
4.0
1.0
2.0
4.0
8.0
Fill Time
8123
9639
1.1 · 104
0.7
0.8
0.9
745
2984
1.2 · 104
4.8 · 104
690
2755
1.1 · 104
1195
4876
1.9 · 104
7.6 · 104
log10 |1−Energy Check|
-0.7
1.4
1.8
0.7
0.8
0.8
-0.2
-0.7
-1.1
-2.9
-0.2
-0.6
-1.2
1.7
2.3
0.4
-4.4
Table 1. Comparison of five different solution methods (SS, SC, CC1, CG1 and CC2) for the Gaussiantapered cosine defined in (10.1) which, near its center, is very rough (see Fig. 2(a)). The incidence angle
θi is 20o from vertical. Here L/λ (λ is wavelength) yields 128 real Bragg modes (see Fig. 2(b)) and a
rapidly varying surface current (see Fig. 3). The fill time is in seconds necessary to compute the matrix
elements and Δ is the coordinate sampling length referred to wavelength. Solution time was small for
all cases. There is a dramatic difference in fill time for SC versus other methods, but for this very rough
surface good convergence was achieved only with full coordinate-based methods.
16
0.4
0.08
0.3
0.07
0.2
0.06
0.1
0.05
0
0.04
−0.1
0.03
0.02
−0.2
0.01
−0.3
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.06
0.5
−0.04
−0.02
(a)
0
0.02
0.04
(b)
Figure 2: Trial 1.1a. (a) Surface and incoming plane wave and (b) Scattered Energy (CC2, 512 by 512),
polar plot of Rj for the Bragg directions.
8
8
6
7
4
6
2
5
0
4
−2
3
−4
2
−6
1
−8
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.5
0.5
(a)
Figure 3: Trial 1.1a.
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b)
Surface current N (x), (a) Real part and (b) Magnitude (CC2, 512 by 512) vs. x.
17
10.1.2
Trial 1.1b, λ L, θi = 75◦
S(x)
d/λ
d/L
L/λ
θi
ε1
ε2
Formalism
SS
SS
SS
SC
SC
SC
CC1
CC1
CC1
CG1
CG1
CG1
CC2
CC2
CC2
CC2
Matrix
Size
128
138
148
128
138
148
64
128
256
65
129
257
64
128
256
512
−(d/2) exp(−ε2 (x/L)2 ) cos(2πε1 x/L)
4.8
0.075
64
75◦
10
18.42
Q = λ/Δ
2.0
2.2
2.3
1.0
2.0
4.0
1.0
2.0
4.0
1.0
2.0
4.0
8.0
Fill Time
8139
9767
1.1 · 104
0.7
0.8
0.9
790
3145
1.3 · 104
697
2783
1.1 · 104
1270
5162
2.1 · 104
8.2 · 104
log10 |1−Energy Check|
-0.4
1.3
2.1
1.8
0.8
0.1
-0.4
-0.2
-2.5
-0.5
-0.5
-2.2
1.1
1.0
-2.9
-3.3
Table 2. Comparison of five different solution methods (SS, SC, CC1, CG1 and CC2) for the Gaussiantapered cosine defined in (10.1) which, near its center, is very rough (see Fig. 4(a)). The incidence angle
θi is 75o from vertical, close to grazing. Here L/λ (λ is wavelength) yields 128 real Bragg modes (see Fig.
4(b)) and a rapidly varying surface current (see Fig. 5). The fill time is in seconds necessary to compute
the matrix elements and Δ is the coordinate sampling length referred to wavelength. Solution time was
small for all cases. There is a dramatic difference in fill time for SC versus other methods, but for this
very rough surface good convergence was achieved only with full coordinate-based methods.
18
0.4
0.15
0.3
0.1
0.2
0.05
0.1
0
0
−0.1
−0.05
−0.2
−0.1
−0.3
−0.5
−0.2
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−0.15
−0.1
−0.05
0
0.05
0.1
0.5
(a)
(b)
Figure 4: Trial 1.1b. (a) Surface and incoming plane wave and (b) Scattered Energy (CC2, 512 by 512),
polar plot of Rj for the Bragg directions.
4
4
3.5
3
3
2
2.5
1
2
0
1.5
−1
1
−2
−3
−0.5
0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.5
0.5
(a)
Figure 5: Trial 1.1b.
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b)
Surface current N (x), (a) Real part and (b) Magnitude (CC2, 512 by 512) vs. x.
19
10.1.3
Trial 1.2a, λ ≈ L, θi = 20◦
S(x)
d/λ
d/L
L/λ
θi
ε1
ε2
Formalism
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
CC1
CC1
CC1
CC1
CG1
CG1
CG1
CC2
CC2
CC2
CC2
Matrix
Size
2
6
10
14
18
22
2
6
10
14
18
64
128
256
512
65
129
257
64
128
256
512
−(d/2) exp(−ε2 (x/L)2 ) cos(2πε1 x/L)
0.26
0.25
1.05
20◦
10
18.42
Q = λ/Δ
1.9
5.7
9.5
13.3
17.1
61
122
243
486
62
123
244
61
122
243
486
Fill Time
0.5
3.6
9.9
19.8
34
51
0.02
0.02
0.03
0.05
0.05
522
5088
8381
3.3 · 104
529
2100
8533
831
3322
1.3 · 104
5.4 · 104
log10 |1−Energy Check|
-0.6
-0.5
-0.5
-0.8
-1.0
-1.1
-15.4
-3.2
-5.0
-2.8
-4.1
-5.4
-6.1
-6.7
-7.3
-8.1
-9.2
-10.4
0.0
-1.5
-2.6
-4.7
Table 3. Comparison of five different solution methods (SS, SC, CC1, CG1 and CC2) for the Gaussiantapered cosine defined in (10.1). The surface is much less rough in height (see Fig. 6(a)) than the one
described in Tables 1 and 2 but has much larger slopes (πd/L). Only two real Bragg modes are present.
The best and fastest result was for the SC method with only the two real modes considered (see Fig.
6(b)). The surface current (Fig. 7) was considerably simpler than those in Figs. 3 and 5.
20
0.5
0.9
0.4
0.8
0.7
0.3
0.6
0.2
0.5
0.1
0.4
0
0.3
−0.1
0.2
0.1
−0.2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.6
0.5
−0.4
−0.2
(a)
0
0.2
0.4
(b)
Figure 6: Trial 1.2a. (a) Surface and incoming plane wave and (b) Scattered Energy (CG1, 257 by 257),
polar plot or Rj for the two real Bragg directions.
1
12
0
10
−1
−2
8
−3
−4
6
−5
4
−6
−7
2
−8
−9
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.5
0.5
(a)
Figure 7: Trial 1.2a.
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b)
Surface current N (x), (a) Real part and (b) Magnitude (CG1, 257 by 257) vs. x.
21
10.1.4
Trial 1.2b, λ ≈ L, θi = 75◦
S(x)
d/λ
d/L
L/λ
θi
ε1
ε2
Formalism
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
SC
SC
CC1
CC1
CC1
CC1
CG1
CG1
CG1
CC2
CC2
CC2
CC2
Matrix
Size
3
7
11
15
19
23
3
7
11
15
19
23
27
64
128
256
512
65
129
257
64
128
256
512
−(d/2) exp(−ε2 (x/L)2 ) cos(2πε1 x/L)
0.26
0.25
1.05
75◦
10
18.42
Q = λ/Δ
2.9
6.7
10.5
14.3
18.1
21.9
25.7
61
122
243
486
62
123
244
61
122
243
486
Fill Time
0.9
4.4
11.0
21
34.5
52
0.01
0.02
0.03
0.04
0.06
0.08
0.09
501
2012
81171
3.2 · 104
525
2097
8430
828
3318
1.3 · 104
5.4 · 104
log10 |1−Energy Check|
-1.6
-1.4
-1.3
-2.4
-2.7
-1.5
-2.0
-2.4
-5.3
-4.2
-2.5
-2.8
-2.9
-6.2
-6.9
-7.5
-8.1
-9.5
-9.9
-9.9
-0.1
-2.0
-3.1
-5.4
Table 4. Comparison of five different solution methods (SS, SC, CC1, CG1 and CC2) for the Gaussiantapered cosine defined in (10.1). The surface is much less rough in height (see Fig. 8(a)) than the one
described in Tables 1 and 2 but has much larger slopes (πd/L). Only two real Bragg modes are present.
All methods worked well and the SC method was the fastest. The scattered energy distribution and the
surface currents are illustrated in Figs. 8 and 9.
22
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
0
−0.2
−0.1
−0.4
−0.2
−0.6
−1
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−0.5
0
0.5
0.5
(a)
(b)
Figure 8: Trial 1.2b. (a) Surface and incoming plane wave and (b) Scattered Energy (CG1, 257 by 257),
polar plot of Rj for the two real Bragg directions.
3.5
1
0.5
3
0
2.5
−0.5
2
−1
1.5
−1.5
1
−2
0.5
−2.5
−3
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.5
0.5
(a)
Figure 9: Trial 1.2b.
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b)
Surface current N (x), (a) Real part and (b) Magnitude (CG1, 257 by 257) vs. x.
23
10.2
10.2.1
Wave-superposition surface
Trial 2.1a, λ L, θi = 20◦
S(x)
d/λ
d/L
L/λ
θi
ε1
Formalism
SS
SS
SS
SC
SC
CC1
CC1
CC1
CC1
CG1
CG1
CG1
CC2
CC2
CC2
CC2
Matrix
Size
128
138
148
128
138
64
128
256
512
65
129
257
64
128
256
512
−(d/2) cos(2πε1 x/L) cos(2πx/L)
4.8
0.075
64
20◦
10
Q = λ/Δ
2.0
2.2
1.0
2.0
4.0
8.0
1.0
2.0
4.0
1.0
2.0
4.0
8.0
Fill Time
7835
9289
1.1 · 104
0.7
0.8
649
2606
1.0 · 104
4.2 · 104
634
2529
1.0 · 104
952
3883
1.6 · 104
6.2 · 104
log10 |1−Energy Check|
-0.5
1.2
1.1
0.9
1.2
-0.2
-0.4
-0.9
-2.8
-0.2
-0.4
-0.9
1.7
1.5
0.4
-4.2
Table 5. Comparison of five different solution methods (SS, SC, CC1, CG1 and CC2) for the wavesuperposition surface defined in (10.2), which in parts is very rough (see Fig. 10(a)). The incidence angle
θi is 20o from vertical. Here L/λ (λ is wavelength) yields 128 real Bragg modes (see Fig. 10(b)) and a
very rapidly oscillating surface current (see Fig. 11). Fill time for the SC method was dramatically less
than for other methods but good convergence was achieved only with the full coordinate-based methods.
24
0.4
0.09
0.08
0.3
0.07
0.2
0.06
0.1
0.05
0
0.04
−0.1
0.03
−0.2
0.02
0.01
−0.3
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
0.5
−0.06
−0.04
−0.02
(a)
0
0.02
0.04
(b)
Figure 10: Trial 2.1a. (a) Surface and incident plane wave and (b) Scattered Energy (CC2, 512 by 512),
polar plot of Rj for the Bragg directions.
8
8
6
7
4
6
2
5
0
4
−2
3
−4
2
−6
1
−8
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.5
0.5
(a)
Figure 11: Trial 2.1a.
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b)
Surface current N (x), (a) Real part and (b) Magnitude (CC2, 512 by 512) vs. x.
25
10.2.2
Trial 2.1b, λ L, θi = 75◦
S(x)
d/λ
d/L
L/λ
θi
ε1
Formalism
SS
SS
SS
SC
SC
SC
CC1
CC1
CC1
CG1
CG1
CG1
CC2
CC2
CC2
CC2
Matrix
Size
128
138
148
128
138
148
64
128
256
65
129
257
64
128
256
512
−(d/2) cos(2πε1 x/L) cos(2πx/L)
4.8
0.075
64
75◦
10
Q = λ/Δ
2.0
2.2
2.3
1.0
2.0
4.0
1.0
2.0
4.0
1.0
2.0
4.0
8.0
Fill Time
7820
9411
1.1 · 104
0.7
0.8
0.9
687
2769
1.1 · 104
654
2599
1.0 · 104
1033
4194
1.7 · 104
6.8 · 104
log10 |1−Energy Check|
-0.2
0.8
1.1
1.5
1.4
1.6
-0.3
-0.4
-2.3
-0.3
-0.4
-1.9
2.2
1.3
-2.6
-3.1
Table 6. Comparison of five different solution methods (SS, SC, CC1, CG1 and CC2) for the wavesuperposition surface defined in (10.2), which in parts is very rough (see Fig. 12(a)). The incidence angle
θi is 75o from vertical, near grazing. Here L/λ (λ is wavelength) yields 128 real Bragg modes (see Fig.
12(b)) and a surface current (see Fig. 13) much less rapidly oscillating than Fig. 11. Fill time for the SC
method was dramatically less than for other methods but good convergence was achieved only with the
full coordinate-based methods.
26
0.4
0.06
0.3
0.04
0.2
0.1
0.02
0
0
−0.1
−0.02
−0.2
−0.04
−0.3
−0.1
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.5
(a)
(b)
Figure 12: Trial 2.1b. (a) Surface and incident field and (b) Scattered Energy (CC2, 512 by 512), polar
plot of Rj for the Bragg directions.
5
5
4.5
4
4
3
3.5
2
3
2.5
1
2
0
1.5
−1
1
−2
−3
−0.5
0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.5
0.5
(a)
Figure 13: Trial 2.1b.
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b)
Surface current N (x), (a) Real part and (b) Magnitude (CC2, 512 by 512) vs. x.
27
10.2.3
Trial 2.2a, λ ≈ L, θi = 20◦
S(x)
d/λ
d/L
L/λ
θi
ε1
Formalism
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
CC1
CC1
CC1
CC1
CG1
CG1
CG1
CC2
CC2
CC2
CC2
Matrix
Size
2
6
10
14
18
22
2
6
10
14
18
64
128
256
512
65
129
257
64
128
256
512
−(d/2) cos(2πε1 x/L) cos(2πx/L)
0.26
0.25
1.05
20◦
10
Q = λ/Δ
1.9
5.7
9.5
13.3
17.1
61
122
243
486
62
123
244
61
122
243
486
Fill Time
0.5
2.5
9.7
19
33
50
0.02
0.02
0.03
0.05
0.06
377
1512
6035
2.5 · 104
386
1531
6092
561
2210
9015
3.6 · 104
log10 |1−Energy Check|
-0.3
-0.2
-0.2
-0.4
-1.5
-1.2
−∞
-4.6
-2.7
-2.9
-5.6
-7.4
-8.1
-8.8
-9.4
-8.3
-12.7
-14.8
-0.9
-1.0
-3.1
-6.7
Table 7. Comparison of five different solution methods (SS, SC, CC1, CG1 and CC2) for the wavesuperposition surface defined in (10.2), which is less rough in height than the surfaces in Tables 5 and 6
but has much larger slope (πd/L). All methods worked well with SC being the fastest. The surface is
illustrated in Fig. 14(a) and nearly all the energy is in the specular mode (see Fig. 14(b)). The surface
currents are illustrated in Fig. 15. The case was for near normal incidence θi = 20o.
28
0.5
1
0.4
0.8
0.3
0.2
0.6
0.1
0.4
0
−0.1
0.2
−0.2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
0.5
−0.6
−0.4
−0.2
(a)
0
0.2
0.4
0.6
(b)
Figure 14: Trial 2.2a. (a) Surface and incident field and (b) Scattered Energy (CG1, 257 by 257), polar
plot of Rj for the two real Bragg directions.
2
12
0
10
−2
8
−4
6
−6
4
−8
2
−10
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.5
0.5
(a)
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b)
Figure 15: Trial 2.2a. Surface current N (x), (a) Real part and (b) Magnitude (CG1, 257 by 257) vs. x.
29
10.2.4
Trial 2.2b, λ ≈ L, θi = 75◦
S(x)
d/λ
d/L
L/λ
θi
ε1
Formalism
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
SC
SC
CC1
CC1
CC1
CC1
CG1
CG1
CG1
CC2
CC2
CC2
CC2
Matrix
Size
3
7
11
15
19
23
3
7
11
15
19
23
27
64
128
256
512
65
129
257
64
128
256
512
−(d/2) cos(2πε1 x/L) cos(2πx/L)
0.26
0.25
1.05
75◦
10
Q = λ/Δ
2.9
6.7
10.5
14.3
18.1
21.9
25.7
61
122
243
486
62
123
244
61
122
243
486
Fill Time
0.9
4.6
11
22
36
53
0.01
0.02
0.03
0.05
0.07
0.08
0.10
356
1433
5713
2.3 · 104
377
1493
6003
560
2188
8889
3.6 · 104
log10 |1−Energy Check|
-1.3
-1.0
-0.8
-1.7
-2.6
-2.7
-2.6
-2.4
-5.0
-3.1
-4.3
-3.4
-4.5
-8.0
-8.7
-9.3
-9.7
-9.3
-9.9
-9.9
-1.3
-0.4
-3.8
-7.9
Table 8. Comparison of five different solution methods (SS, SC, CC1, CG1 and CC2) for the wavesuperposition surface defined in (10.2), which is less rough in height than the surfaces in Tables 5 and 6
but has much larger slope (πd/L). All methods worked well for near-grazing incidence at θi = 75o . The
surface is illustrated in Fig. 16(a) and nearly all the energy is in the specular direction (see Fig. 16(b)).
Surface current is illustrated in Fig. 17.
30
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
0
−0.2
−0.1
−0.4
−0.2
−0.6
−1
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−0.5
0
0.5
0.5
(a)
(b)
Figure 16: Trial 2.2b. (a) Surface and incident field and (b) Scattered Energy (CG1, 257 by 257), polar
plot of Rj for the two real Bragg directions.
3
3
2
2.5
1
2
0
1.5
−1
1
−2
0.5
−3
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.5
0.5
(a)
Figure 17: Trial 2.2b.
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b)
Surface current N (x), (a) Real part and (b) Magnitude (CG1, 257 by 257) vs. x.
31
11
Computational Results: Neumann Problem
We present some brief results for the Neumann boundary value problem in this section. The theoretical
results are in Section 9. Only Tables are included to compare the energy checks with the Dirichlet results
in [8, 9] for a simple cosine surface. It is seen that the SC and SS Methods give a different level of
performance than for the Dirichlet cases included in [8, 9]. They work quite well away from grazing
incidence even for very large heights, but for large slopes deteriorate near grazing.
11.1
Trial 1.1a, λ L, θi = 20◦
S(x)
d/λ
d/L
L/λ
θi
Formalism
SS
SS
SS
SC
SC
SC
CC2
CC2
CC2
Matrix
Size
128
138
148
128
138
148
64
128
256
−(d/2) cos(2πx/L)
4.8
0.075
64
20◦
Q = λ/Δ
2.0
2.2
2.3
1.0
2.0
4.0
Fill Time
2321
2964
3715
0.9
1.0
1.1
814
3301
1.3 · 104
log10 |1−Energy Check|
-5.0
-5.0
-5.0
-10.2
-10.2
-10.2
-0.1
-4.5
-5.3
Table 9. Comparison of three different solution methods (SS, SC and CC2) for the pure cosine surface
which is very rough (d/λ = 4.8). There are 128 real Bragg modes. All methods worked well with the
SC methods several orders of magnitude faster in terms of fill time. The incidence angle θi was 20o from
vertical.
32
11.2
Trial 1.1b, λ L, θi = 75◦
S(x)
d/λ
d/L
L/λ
θi
Formalism
SS
SS
SS
SC
SC
SC
CC2
CC2
CC2
Matrix
Size
128
138
148
128
138
148
64
128
256
−(d/2) cos(2πx/L)
4.8
0.075
64
75◦
Q = λ/Δ
2.0
2.2
2.3
1.0
2.0
4.0
Fill Time
2328
2960
3687
0.9
1.0
1.1
853
3463
1.4 · 104
log10 |1−Energy Check|
-0.3
-0.1
-0.3
-2.0
-0.3
0.0
0.8
-0.1
-3.3
Table 10. Comparison of three different solution methods (SS, SC and CC2) for the pure cosine surface
which is very rough. There are 128 real Bragg modes. The incidence angle θi is 75o so it is near grazing.
The SC method deteriorated when evanescent modes were included, whereas CC2 continued to improve
as the matrix size (and fill time) increased.
33
11.3
Trial 1.2a, λ ≈ L, θi = 20◦
S(x)
d/λ
d/L
L/λ
θi
Formalism
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
CC2
CC2
CC2
Matrix
Size
2
6
10
14
18
22
2
6
10
14
18
64
128
256
−(d/2) cos(2πx/L)
0.26
0.25
1.05
20◦
Q = λ/Δ
1.9
5.7
9.5
13.3
17.1
61
122
243
Fill Time
0.2
2.6
7.7
15.8
27
42
0.01
0.02
0.05
0.07
0.09
480
1934
7981
log10 |1−Energy Check|
-1.1
-1.1
-1.1
-1.2
-2.1
-1.1
-0.4
-0.7
-0.6
-0.6
-0.6
-1.8
-2.1
-2.4
Table 11. Comparison of three different solution methods (SS, SC and CC2) for the pure cosine surface
with very large slopes (πd/L). There are only two real Bragg modes. The spectral related methods
generally converged poorly, and CC2 improved as the matrix size increased. The incidence angle θi was
20o from vertical.
34
11.4
Trial 1.2b, λ ≈ L, θi = 75◦
S(x)
d/λ
d/L
L/λ
θi
Formalism
SS
SS
SS
SS
SS
SS
SC
SC
SC
SC
SC
SC
SC
CC2
CC2
CC2
Matrix
Size
3
7
11
15
19
23
3
7
11
15
19
23
27
64
128
256
−(d/2) cos(2πx/L)
0.26
0.25
1.05
75◦
Q = λ/Δ
2.9
6.7
10.5
14.3
18.1
21.9
25.7
61
122
243
Fill Time
0.5
3.5
9.4
18.2
30
46
0.01
0.03
0.05
0.07
0.09
0.12
0.14
469
1926
7802
log10 |1−Energy Check|
0.7
0.7
0.9
1.2
1.5
1.6
-0.8
-1.2
-0.7
-0.1
0.2
-1.3
-0.9
-2.3
-2.6
-2.9
Table 12. Comparison of three different solution methods (SS, SC and CC2) for the pure cosine surface
with very large slopes (πd/L). There are only two real Bragg modes. For this near grazing incidence
angle of 75o , the SS method did not converge, the SC method converged poorly if at all, and the CC2
method converged better as matrix size increased.
35
12
Discussion of Computational Results: Transmission Problem
In Sections 13-25, we present an extensive suite of computational results for the transmission problem
using the three methods CC, SC, and SS from Sections 3,5 and 6, respectively. We outline and summarize
the results here.
In Section 13, we study the SS method for the transmission interface (TISS) for six cases of the pure
cosine profile
s(x) = −(d/2) cos(2πx/L).
(12.1)
The cases include a fairly rough surface (Case 1), a less rough surface (Case 2), fairly rough and very
smooth surfaces near grazing incidence (Cases 3 and 4) where both cases have fewer Bragg transmitted
than reflected modes, a flat surface with no interface (Case 5) as a computational check and a flat surface
with interface (Case 6) again as a computational check. The matrix K and energy check are presented
for all cases. TISS worked well except for Case 3.
In Sections 14-16, we compare the results of the three formalisms, CC, SC, and SS. There are
small differences in the computed surface currents and field for the different methods, but only negligible
changes in the resulting energies. These surface current and field differences can be viewed as non-null
surface current and field values producing near-null scattered field values. Note in Section 16 the very
small number of topological basis modes necessary to describe the problem with very small error.
In Section 17, we demonstrate that all the codes worked for an extreme grazing incidence case where
θi = 89.995◦. For the SS example, the number of topological modes was again quite small. (Pure
Fourier simulations done for perfectly reflecting examples indicate that upwards of ten times the number
of 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 18-21, we present an extensive collection of computational results for the SC method.
These include three different coordinate sampling methods (Sections 18 and 19) and three different
spectral sampling methods (Sections 20 and 21). The 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 in Sections 18
and 20, and the maximum condition number in Sections 19 and 21. The coordinate sampling depends on
the number of spectral orders above and below the surface as explained in the sections. Reliable results
were found for large d/L ratios and very large condition numbers.
Finally, Sections 22-25 present results on the SS formalism for roughness (d/L) values and condition
number for 1% error (Section 22), as a function of κ (Section 23) and ρ (Sections 24 and 25) again over
an extensive parameter domain.
36
13
Transmission Interface Results for the SS Method
In this section, we briefly demonstrate that the SS method for the transmission interface (TISS) works
for a variety of surfaces and incidence angles. The equations can be found in Section 6. The matrix K
is defined in (6.22). For a discussion of the method of solution, see Appendix A. Except for Case 3, the
energy check was quite good. For all cases, we use the cosine profile
s(x) = −(d/2)cos(2πx/L)
(13.1)
The cases include a fairly rough surface (Case 1), a less rough surface (Case 2), fairly rough and very
smooth surfaces near grazing incidence (Cases 3 and 4) where both cases have fewer Bragg transmitted
than reflected modes, a flat surface with no interface (Case 5) as a computational check and a flat surface
for all cases. TISS worked well except for Case 3.
13.1
Case 1: Fairly Rough Surface
Matrix Size
64
74
84
Condition Number
1.8 × 107
1.3 × 108
1.7 × 109
Energy Check
1.000065
1.000080
1.000065
Table 13. Condition numbers and energy check for different matrix sizes for the SS method. The surface
is the cosine profile in (13.1), θi = 25o , d/L = 0.1, λ/L = 0.0625 (64 real scattered orders in the lower
region, 32 in the upper region), κ = 2 and ρ = 3. The matrix K is illustrated in Fig. 18 and the energy
in the modal orders in Fig. 19. Matrix size 64 means 64 × 64 matrix consisting of 64 real modes in the
lower region times 32 real plus 32 evanescent modes in the upper region. For matrix sizes 74 and 84 we
add more evanescent modes from each region.
5
4
x 10
x 10
1
3
2
0.5
1
0
0
−0.5
−1
−1
−60
−2
−60
−40
−40
40
−20
−20
20
−40
40
20
0
0
−20
20
40
−20
20
0
0
−40
40
−60
(a)
−60
(b)
Figure 18: (Case 1) Matrix K, size 84 by 84: (a) Real part (×104 ). (b) Imaginary part (×105 ).
37
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
−0.12
−0.14
−0.16
−0.18
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
Figure 19: (Case 1) Polar plot of reflected Rj and transmitted Tn modal energy components defined in
Section 7. There is very little energy in the reflected field.
13.2
Case 2: Less Rough Surface
Matrix Size
64
74
84
Condition Number
5.7
9.3
13.6
Energy Check
1.0000000014
1.0000000014
1.0000000014
Table 14. Condition numbers and energy check for different matrix sizes for the SS method. The surface
is the cosine profile in (13.1), θi = 25o , d/L = 0.01, λ/L = 0.0625 (64 real scattered orders in the lower
region, 32 in the upper region), κ = 2 and ρ = 3. The matrix K is illustrated in Fig. 20 and the energy in
the modal orders in Fig. 21. The roughness is much less than that in Table 13. Matrix size is explained
in Table 13.
38
3
6
2
4
1
0
2
−1
0
−2
−2
−3
−4
−60
−4
−60
−40
−40
40
−20
40
−20
20
0
0
−20
20
20
−20
20
−40
40
0
0
−40
40
−60
(a)
−60
(b)
Figure 20: (Case 2) Matrix K, size 84 by 84: (a) Real part and (b) Imaginary part.
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
−0.9
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Figure 21: (Case 2) Polar plot of reflected Rj and transmitted Tn modal energy components defined in
Section 7. Most of the energy is transmitted.
39
13.3
Case 3: Fairly Rough Surface, Near Grazing
Cosine profile, θ = 85◦ , d/L = 0.1, λ/L = 0.0314 (64 real scattered orders in upper region, 13 in lower
region), κ = 0.2, ρ = 3.
Matrix Size
64
74
84
Condition Number
3.7 × 108
1.7 × 109
2.3 × 109
Energy Check
314
0.56
0.875
Table 15. Condition numbers and energy check (which is poor) for different matrix sizes for the SS
method. The surface is the cosine profile in (13.1), θi = 85o (near grazing), d/L = 0.1, λ/L = 0.0314 (64
real scattered orders in the upper region, 13 in the lower region), κ = 2 (inverted from our usual choices)
and ρ = 3. The matrix K is illustrated in Fig. 22 and the energy in the modal orders in Fig. 23. Matrix
size is explained in Table 13.
4
4
x 10
x 10
1.5
3
2
1
1
0.5
0
0
−1
−0.5
−2
−1
−3
−80
−1.5
−80
−60
−60
20
−40
−20
−20
−40
0
−60
20
0
−20
−20
−40
0
20
−40
0
−60
20
−80
(a)
−80
(b)
Figure 22: (Case 3) Matrix K, size 84 by 84: (a) Real part (×104 ) and (b) Imaginary part (×104 ). There
are smooth trends here, so wavelets might be applicable.
40
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
−0.005
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 23: (Case 3) Polar plot of reflected Rj and transmitted Tn modal energy components defined in
Section 7. There is very little energy in the transmitted waves.
13.4
Case 4: Very Smooth Surface, Near Grazing
Cosine profile, θ = 85◦ , d/L = 0.001, λ/L = 0.0314 (64 real scattered orders in upper region, 13 in lower
region), κ = 0.2, ρ = 3.
Matrix Size
64
74
84
Condition Number
3.5
3.5
3.9
Energy Check
0.99982
1.00071
1.00071
Table 16. Condition numbers and energy check for different matrix sizes for the SS method. The surface
is the (very smooth) cosine profile in (13.1), θi = 85o (near grazing), d/L = 0.001, λ/L = 0.0314 (64 real
scattered orders in the upper region, 13 in the lower region), κ = 0.2 (inverted from our usual choices)
and ρ = 3. The condition numbers were all small and the energy check nearly perfect. The matrix K
is illustrated in Fig. 24 and the energy in the modal orders in Fig. 25. This is a case of total internal
reflection. Matrix size is explained in Table 13.
41
2
2
1.5
1.5
1
1
0.5
0.5
0
−80
0
−80
−60
−60
20
−40
20
−40
0
−20
−20
−40
0
0
−40
0
−60
20
−20
−20
−60
20
−80
−80
(a)
(b)
Figure 24: (Case 4) Matrix K, size 84 by 84: (a) Real part and (b) Imaginary part.
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
−0.01
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 25: (Case 4) Polar plot of reflected Rj and transmitted Tn modal energy components defined in
Section 7. This is essentially total internal reflection.
42
13.5
Case 5: Flat Surface, No Physical Interface
This is the (degenerate) case of the cosine profile in (13.1) with θi = 45◦ , d/L = 0, λ/L = 0.0313, κ = 1,
ρ = 1 so there are 64 real scattered orders in both upper and lower regions. The matrix size is 64,
condition number 6.8, and energy check of 1. There is no interface and we get pure transmission. The
matrix K is illustrated in Fig. 26 and the modal energy results in Fig. 27.
−15
x 10
1
4
0.8
2
0.6
0
0.4
−2
0.2
0
−60
−4
−60
20
−40
20
−40
0
−20
−20
0
−20
0
−40
20
0
−20
−40
20
−60
(a)
−60
(b)
Figure 26: (Case 5) Matrix K, size 64 by 64: (a) Real part and (b) Imaginary part (×10−15 ).
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 27: (Case 5) Polar plot of reflected Rj and transmitted Tn modal energy components defined in
Section 7. The energy goes through unhindered, as expected.
43
13.6
Case 6: Flat Surface, Reflection and Refraction
Cosine profile, θi = 45◦ , d = 0, λ/L = 0.0417 (48 real scattered orders in upper region, 64 in lower
region), κ = 1.333, ρ = 1.
Matrix Size
64
Condition Number
2.6
1.2
Energy Check
1
0.5
1
0.4
0.8
0.3
0.6
0.2
0.4
0.1
0.2
0
0
−0.2
−60
−0.1
−60
20
−40
20
−40
0
−20
0
0
−20
−20
0
−40
20
−20
−40
20
−60
(a)
−60
(b)
Figure 28: (Case 6) Matrix K, size 64 by 64: (a) Real part and (b) Imaginary part.
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
−0.9
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 29: (Case 6) Polar plot of reflected Rj and transmitted Tn modal energy components defined in
Section 7. There are only two nonzero modes, as expected.
44
14
Comparison of Transmission Results
We have three different formalisms for the transmission interface from Sections 3, 5, and 6. We present
here a comparison of the results for one system. The system is S(x) = −(d/2) cos(2πx/L), with d/L = 0.1,
λ/L = 0.0625, θ = 25◦ , ρ = 8, and κ = 2. This yields 32 real orders above the interface and 64 real orders
below the interface. The scattered fields are nearly indistinguishable but the surface currents and fields
generated by the three methods contain some relatively large differences. These differences are discussed
further in Sections 15 and 16.
SS Formalism
0.175
2.6
0.17
2.55
0.165
2.5
0.16
2.45
0.155
2.4
0.15
2.35
0.145
2.3
0.14
2.25
0.135
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
2.2
−0.5
−0.4
−0.3
−0.2
−0.1
(a)
0
0.1
0.2
0.3
0.4
(b)
0.06
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
−0.12
−0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
(c)
Figure 30: SS formalism, matrix size 64 by 64, (a) |N (x)|2 , magnitude of surface current squared, vs. x,
(b) |F (x)|2 , magnitude of surface field squared, vs. x, and (c) polar plot of reflected Rj and transmitted
Tn modal energy components defined in Section 7. The energy check error is 10−4.3 .
45
0.5
SC Formalism
0.175
2.6
0.17
2.55
0.165
2.5
0.16
2.45
0.155
2.4
0.15
2.35
0.145
2.3
0.14
2.25
0.135
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
2.2
−0.5
−0.4
−0.3
−0.2
−0.1
(a)
0
0.1
0.2
0.3
0.4
(b)
0.06
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
−0.12
−0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
(c)
Figure 31: SC formalism (uniform sampling in x), matrix size 256 by 256, (a) |N (x)|2 , magnitude of
surface current squared, vs. x, (b) |F (x)|2 , magnitude of surface field squared, vs. x, and (c) polar plot
of reflected Rj and transmitted Tn modal energy components defined in Section 7. The energy check
error is 10−10.2 .
46
0.5
CC1 Formalism
2.6
0.175
2.55
0.17
2.5
0.165
2.45
0.16
0.155
2.4
0.15
2.35
0.145
2.3
0.14
2.25
0.135
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
2.2
−0.5
0.5
−0.4
−0.3
−0.2
(a)
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b)
0.06
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
−0.12
−0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
(c)
Figure 32: CC1 formalism (uniform sampling in x), matrix size 256 by 256, (a) |N (x)|2 , magnitude of
surface current squared, vs. x, (b) |F (x)|2 , magnitude of surface field squared, vs. x, and (c) polar plot
of reflected Rj and transmitted Tn modal energy components defined in Section 7. The energy check
error is 10−6.0 .
Matrix Size
64
128
256
log10 |1 − Energy Check|
-2.5
-4.1
-5.7
Table 17. Verification of CC1 convergence (same system). Matrix sizes and convergence for the CC1
method for the parameters listed in the text of Section 14.
47
15
A Near-Null Field from Nonzero Surface Conditions
Different formalisms produce slightly different surface currents and surface fields, but they have nearly
the same scattering pattern. Thus, a near-null field is produced from the nonzero difference between
the surface conditions. The system used for comparison is S(x) = −(d/2) cos(2πx/L), with d/L = 0.1,
λ/L = 0.0625, θ = 25◦ , ρ = 8, and κ = 2. This yields 32 real orders above the interface and 64 real
orders below the interface. The SS matrix size is 64 by 64, and the CC1 matrix size is 256 by 256. The
N (x) and F (x) values are illustrated in Figure 33.
The difference between the N (x) and F (x) values are illustrated in Figure 34. They are defined as:
ΔN (x) = Nss (x) − NCC1 (x),
(15.1)
ΔF (x) = Fss (x) − FCC1 (x).
(15.2)
and
When scattered amplitudes are computed just using this difference, the normalized energy in the field is
1.88 × 10−4 . The scattering pattern is also in Figure 34. Note the scale in this latter figure.
48
SS
CC1
0.175
0.175
0.17
0.17
0.165
0.165
0.16
0.16
0.155
0.155
0.15
0.15
0.145
0.145
0.14
0.14
0.135
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.135
−0.5
0.5
−0.4
−0.3
−0.2
−0.1
(a)
2.6
2.55
2.55
2.5
2.5
2.45
2.45
2.4
2.4
2.35
2.35
2.3
2.3
2.25
2.25
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
2.2
−0.5
0.5
−0.4
−0.3
−0.2
(b)
0.2
0.3
0.4
0.5
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.035
0.04
(b)
0.06
0.06
0.04
0.04
0.02
0.02
0
0
−0.02
−0.02
−0.04
−0.04
−0.06
−0.06
−0.08
−0.08
−0.1
−0.12
−0.005
0.1
(a)
2.6
2.2
−0.5
0
−0.1
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
−0.12
−0.005
0.04
(c)
0
0.005
0.01
0.015
0.02
0.025
0.03
(c)
Figure 33: A comparison of SS and CC1 computational results: (a) |N (x)|2 , the magnitude of surface
current squared, vs. x, (b) |F (x)|2 , the magnitude of the surface field squared, vs. x, and (c) polar plots
of the reflected Rj and transmitted Tn modal energy components defined in Section 7. The SS matrix
size is 64 by 64, and the CC1 matrix size is 256 by 256. The SS energy check error is 10−4.3 , and the
CC1 energy check error is 10−6.0 .
49
−4
2.2
0.015
x 10
2
0.01
1.8
1.6
0.005
1.4
0
1.2
1
−0.005
0.8
−0.01
0.6
−0.015
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.4
−0.5
−0.4
−0.3
−0.2
−0.1
(a)
0
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
(b)
−4
0.02
4
0.015
3.5
0.01
3
0.005
2.5
0
2
−0.005
1.5
−0.01
1
−0.015
0.5
−0.02
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x 10
0
−0.5
−0.4
−0.3
−0.2
−0.1
(c)
0
(d)
−5
1
x 10
0.5
0
−0.5
−1
−1.5
−2
0
2
4
6
8
10
12
14
16
−6
x 10
(e)
Figure 34: Differences in surface currents and fields and the resulting scattering pattern calculated using
the differences: (a) Real(ΔN (x)) vs. x, (b) |ΔN (x)|2 vs. x, (c) Real(ΔF (x)) vs. x, (d) |ΔF (x)|2 vs. x,
and (e) polar plot of the reflected Rj and transmitted Tn modal energy components defined in Section 7
but using field and current differences. Note the scale.
50
16
Comparison of Transmission Interface Results: Surface Currents and Fields
In this section, we continue the comparisons of the three methods of solution. Differences in the surface
currents and fields computed using the different methods are illustrated. The SS method is illustrated
in Figures 35 and 36. Although this is a moderately rough surface (d/L = 0.1) very good convergence is
achieved with a small number of topological basis terms. The SC computations are illustrated in Figure
37. Compare the results to those in Figures 36 for SS, Figures 38 for CC1, and Figures 39 for CC3.
An expanded scale of some of the results are illustrated in Figures 40 and 41, and the differences and
their corresponding modal energy predictions in Figure 42. These computational differences are confined
mostly to additional oscillatory behavior in the surface field and normal derivative and produce negligible
energy differences in the scattered modes. See Figures 40-42.
−(d/2) cos(2πx/L)
0.1
0.047
25◦
0.5
1.5
43
64
S(x)
d/L
λ/L
θi
ρ
κ
Real Orders Above
Real Orders Below
Error = log10 |1 − Normalized Energy|
Table 18. Example of a pure cosine surface and parameters used for the comparison calculations in
Section 16.
2
0.4
1.5
0.3
1
0.2
0.5
0.1
0
−40
−20
0
(a)
20
0
−40
40
−20
0
(b)
20
40
Figure 35: 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.
51
1.7
2
1.6
1
1.5
0
1.4
−1
1.3
−2
−0.5
0
(a)
0.5
1.2
−0.5
0.4
0
(b)
0.5
0
0.5
0.4
0.38
0.2
0.36
0
0.34
−0.2
−0.4
−0.5
0.32
0
0.3
−0.5
0.5
(c)
(d)
0.1
0.05
0
−0.05
−0.1
−0.1
−0.05
0
0.05
0.1
(e)
Figure 36: SS Formalism: 64 by 64 matrices, Error = −4.3, N (x) = 1/(ik1 ) n · ∇F . Computations using
the SS formalism for the surface current (normal derivative) in (a) Real(N (x)) vs. x and (b) |N (x)| vs.
x, surface field in (c) Real(F (x)) vs. x and (d) |F (x)| vs. x, and the modal energy distribution in (e).
52
1.7
2
1.6
1
1.5
0
1.4
−1
−2
−0.5
1.3
0
(a)
0.5
1.2
−0.5
0
(b)
0.5
0.4
0.4
0.38
0.2
0.36
0
0.34
−0.2
−0.4
−0.5
0.32
0
(c)
0.3
−0.5
0.5
0
(d)
0.5
0.1
0.05
0
−0.05
−0.1
−0.1
−0.05
0
0.05
0.1
(e)
Figure 37: SC Formalism: 64 by 64 matrices, Error = −11.5, N (x) = 1/(ik1 ) n ·∇F . Computations using
the SC formalism for the surface current (normal derivative) in (a) Real(N (x)) vs. x and (b) |N (x)| vs.
x, surface field in (c) Real(F (x)) vs. x and (d) |F (x)| vs. x, and the modal energy distribution in (e).
53
1.7
2
1.6
1
1.5
0
1.4
−1
−2
−0.5
1.3
0
1.2
−0.5
0.5
0
(a)
0.5
(b)
0.4
0.4
0.38
0.2
0.36
0
0.34
−0.2
−0.4
−0.5
0.32
0
0.3
−0.5
0.5
0
(c)
0.5
(d)
0.1
0.05
0
−0.05
−0.1
−0.1
−0.05
0
0.05
0.1
(e)
Figure 38: CC1 Formalism: 256 by 256 matrices, Error = −3.9, N (x) = 1/(ik1 ) n · ∇F . Computations
using the CC1 formalism for the surface current (normal derivative) in (a) Real(N (x)) vs. x and (b)
|N (x)| vs. x, surface field in (c) Real(F (x)) vs. x and (d) |F (x)| vs. x, and the modal energy distribution
in (e).
54
2
1.7
1.5
1.65
1.6
1
1.55
0.5
1.5
0
1.45
−0.5
1.4
−1
1.35
−1.5
1.3
−2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
1.25
−0.5
0.5
−0.4
−0.3
−0.2
−0.1
(a)
0.4
0.1
0.2
0.1
0.2
0.3
0.4
0.5
0.4
0.3
0.39
0.2
0.38
0.1
0.37
0
0.36
−0.1
0.35
−0.2
0.34
−0.3
0.33
−0.4
−0.5
0
(b)
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.32
−0.5
0.5
−0.4
−0.3
−0.2
(c)
−0.1
0
0.3
0.4
0.5
(d)
0.1
0.05
0
−0.05
−0.1
−0.1
−0.05
0
0.05
0.1
(e)
Figure 39: CC3 Formalism: 128 by 128 matrices, Error = −3.5, N (x) = 1/(ik1 ) n · ∇F . Computations
using the CC3 formalism for the surface current (normal derivative) in (a) Real(N (x)) vs. x and (b)
|N (x)| vs. x, surface field in (c) Real(F (x)) vs. x and (d) |F (x)| vs. x, and the modal energy distribution
in (e).
55
0.4
0.38
0.36
0.34
0.32
0.3
−0.5
0
0.5
Figure 40: |F (x)| vs. x, SS Formalism: Expanded view of Fig. 36(d). Note the oscillations not present
in Fig. 41.
56
0.4
0.38
0.36
0.34
0.32
0.3
−0.5
0
0.5
Figure 41: |F (x)| vs. x, CC1 Formalism: Expanded view of Fig. 38(d). Compare with the oscillations in
Fig. 40.
57
0.04
0.04
0.02
0.03
0
0.02
−0.02
0.01
−0.04
−0.5
0
0
−0.5
0.5
0
(a)
(b)
0.02
0.02
0.01
0.015
0
0.01
−0.01
0.005
−0.02
−0.5
0.5
0
0
−0.5
0.5
0
(c)
0.5
(d)
−5
3
x 10
2
1
0
−1
−1
0
1
2
3
−5
x 10
(e)
Figure 42: Difference between CC1 and SS Results: ΔN (x) = NSS (x) − NCC1 (x), ΔF (x) = FSS (x) −
FCC1 (x). Surface current differences are in (a) Real(ΔN (x)) vs. x and (b) |ΔN (x)| vs. x, and surface
field differences in (c) Real(ΔF (x)) vs. x and (d) |ΔF (x)| vs. x. The corresponding modal energy
differences computed using these surface field and normal derivative differences are illustrated in (e).
58
17
Comparison of Transmission Interface Results: Near
Grazing Incidence
This section continues the comparison of the three methods for near grazing incidence. There are no field
pictures since for the parameters chosen, all the scattered energy is in the specular direction. All methods
worked well with differences in the surface fields and currents. The SS method is illustrated in Figures
43 and 44. This is a weakly rough surface. The angle of incidence is nearly parallel to the mean surface
(θi = 89.995o). Convergence is very good but many more topological basis modes are necessary to achieve
this than were necessary at 25o incidence (compare Figures 43 and 35). The SC results are illustrated
in Figure 45. Compare these to the SS results in Figure 44 and CC1 in Figure 46. The computational
differences are highlighted in Figures 47-49. The SS results contain more oscillations but on a scale here
which is much smaller than the differences in Section 16. The field and normal derivative values are also
considerably smaller than in Section 16. Near grazing incidence does not excite strong surface fields.
−(d/2) cos(2πx/L)
0.07
0.0469
89.995◦
1.5
1.5
43
64
S(x)
d/L
λ/L
θi
ρ
κ
Real Orders Above
Real Orders Below
Error = log10 |1 − Normalized Energy|
Table 19. Example of a pure cosine surface and parameters used for comparison calculations in Section 17.
−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
(a)
0
0
−60
20
−40
−20
(b)
0
20
Figure 43: 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.
59
−4
4
−4
x 10
x 10
4
2
3
0
2
−2
1
−4
−0.5
0
(a)
0.5
−0.5
−4
0.5
0
(d)
0.5
−4
x 10
x 10
6
7
4
6
2
5
0
4
3
−2
2
−4
1
−6
−0.5
0
(b)
0
(c)
0.5
0
−0.5
Figure 44: SS Formalism, Near Grazing Incidence with 64 by 64 matrices, Error = −4.3, N (x) =
1/(ik1 ) n · ∇F. Near-grazing incidence results for the SS formalism for the surface normal derivative (a)
Real(N (x)) vs. x and (b) |N (x)| vs. x and the surface field (c) Real(F (x)) vs. x and (d) |F (x)| vs. x.
60
−4
4
−4
x 10
x 10
4
2
3
0
2
−2
1
−4
−0.5
0
(a)
0.5
−0.5
−4
0.5
0
(d)
0.5
−4
x 10
x 10
6
7
4
6
2
5
0
4
3
−2
2
−4
1
−6
−0.5
0
(b)
0
(c)
0.5
0
−0.5
Figure 45: SC Formalism, Near Grazing Incidence with 64 by 64 matrices, Error = −8.3, N (x) =
1/(ik1 ) n · ∇F. Near-grazing incidence results for the SC formalism for the surface normal derivative (a)
Real(N (x)) vs. x and (b) |N (x)| vs. x and the surface field (c) Real(F (x)) vs. x and (d) |F (x)| vs. x.
61
−4
4
−4
x 10
x 10
4
2
3
0
2
−2
1
−4
−0.5
0
(a)
0.5
−0.5
−4
0.5
0
(d)
0.5
−4
x 10
x 10
6
7
4
6
2
5
0
4
3
−2
2
−4
1
−6
−0.5
0
(b)
0
(c)
0.5
0
−0.5
Figure 46: CC1 Formalism, Near Grazing Incidence with 128 by 128 matrices, Error = −5.7, N (x) =
1/(ik1 ) n · ∇F. Near-grazing incidence results for the CC1 formalism for the surface normal derivative (a)
Real(N (x)) vs. x and (b) |N (x)| vs. x and the surface field (c) Real(F (x)) vs. x and (d) |F (x)| vs. x.
62
−4
x 10
7
6
5
4
3
2
1
0
−0.5
0
0.5
Figure 47: |F (x)| vs. x, CC1 Formalism: Magnitude of the surface field for the CC1 formalism. This is
an expanded version of Fig. 46 (d).
63
−4
x 10
7
6
5
4
3
2
1
0
−0.5
0
0.5
Figure 48: |F (x)| vs. x, SS Formalism: Magnitude of the surface field for the SS formalism. This is an
expanded version of Fig. 44 (d). Note the difference between this and the previous figure, but also notice
the overall scale.
64
−4
4
−4
x 10
x 10
4
2
3
0
2
−2
1
−4
−0.5
0
(a)
0.5
0
−0.5
−4
0.5
0
(d)
0.5
−4
x 10
x 10
6
7
4
6
2
5
0
4
3
−2
2
−4
1
−6
−0.5
0
(b)
0
(c)
0.5
0
−0.5
−6
2
x 10
0
−2
−4
−6
−2
0
2
4
6
−6
x 10
(e)
Figure 49: Differences in the surface normal derivatives ((a) Real(ΔN (x)) vs. x and (b) |ΔN (x)| vs. x)
and the surface fields ((c) Real(ΔF (x)) vs. x and (d) |ΔF (x)| vs. x) between CC1 and SS formalisms.
Part (e) is the resulting energy difference computed using these difference fields (note scale). ΔN (x) =
NSS (x) − NCC1 (x), ΔF (x) = FSS (x) − FCC1 (x).
65
18
Roughness Graphs, Transmission Interface, SC
Formalism, Coordinate Sampling
In this section, we present extensive results on the effect of different coordinate sampling on the SC method
as a function of incidence angle and density and wavenumber ratios. The 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. The coordinate samplings are related to the spectral orders as explained below.
These results are for S(x) = −(d/2) cos(2πx/L) and λ = 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.
Recall that the maximum slope is πd/L. Convergence for much larger d/L values was achieved over
a much broader parameter domain for the third coordinate sampling which was the largest number of
coordinate samples taken for these examples. The pseudoinverse method is used extensively. Note that
even though we have cases where ρ = 0 (Dirichlet) or ρ = ∞ (Neumann), we may still have κ = 1, and
this formally affects the number of coordinate samples we choose.
18.1
First coordinate sampling
All real orders and only real orders are used. The number of coordinate samples is equal to the average
of the two numbers of spectral samples, (n1 + n2 )/2, where n1 , respectively n2 , is the number of real
spectral orders above (respectively below) the surface interface.
66
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)
67
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)
68
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 50: 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, and 8) read from the top rectangular box to the bottom box.
69
0
18.2
Second coordinate sampling
All real orders and only real orders are used. The number of coordinate samples is equal to the number
of spectral samples above the surface, n1 .
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)
70
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)
71
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 51: The figures 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 number 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 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 to the bottom 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, and 8) read from
the top rectangular box to the bottom box.
72
0
18.3
Third coordinate sampling
All real orders and only real orders are used. The number of coordinate samples is equal to the maximum
of the two numbers of spectral samples from above and below the surface.
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)
73
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)
74
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 52: The figures 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 maximum 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, and 8)
read from the top rectangular box to the bottom box.
75
0
19
Condition Number Graphs, Transmission Interface, SC Formalism, Coordinate Sampling
The results presented in this section are for S(x) = −(d/2) cos(2πx/L) and λ = 0.0625. Uniform
coordinate sampling is used throughout. All real orders and only real orders are used. Plotted is the
log10 of the condition number at the d/L value for which the Error = −2. The method stopped
unconditionally at d/L = 1, which was only achieved when κ = ρ = 1. The sampling methods are the
same as those in Section 18. The best conditioning occurred for the second coordinate sampling method
defined below.
19.1
First coordinate sampling
All real orders and only real orders are used. The number of coordinate samples is equal to the average
of the two numbers of spectral samples.
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)
76
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
0.5
8
1.5
0.5
8
1
0.5
8
0.5
0.5
20
40
Incident angle
(b)
77
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 53: 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 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) 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, and 8) read from the top rectangular box to the bottom box.
78
0
19.2
Second coordinate sampling
All real orders and only real orders are used. The number of coordinate samples is equal to the number
of spectral samples above the surface.
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)
79
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
0.5
8
1.5
0.5
8
1
0.5
8
0.5
0.5
20
40
Incident angle
(b)
80
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 54: 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, and 8) read from the top rectangular box to the bottom box.
81
0
19.3
Third coordinate sampling
All real orders and only real orders are used. The number of coordinate samples is equal to the maximum
of the two numbers of spectral samples.
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)
82
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
0.5
8
1.5
0.5
8
1
0.5
8
0.5
0.5
20
40
Incident angle
(b)
83
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 55: 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 maximum
of the two 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, and 8) read from the top
rectangular box to the bottom box.
84
0
20
Roughness Graphs, Transmission Interface, SC
Formalism, Spectral Sampling Variations
These results are for S(x) = −(d/2) cos(2πx/L) and λ = 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 question was to find the largest value of d/L for the different
sampling methods. Adding additional orders to the spectral sampling methods yielded better results.
20.1
First spectral sampling
The smaller number of real orders from above and below is used both above and below.
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)
85
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)
86
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 56: 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. 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, and 8) read from the top rectangular box to the bottom box.
87
20.2
Second spectral sampling
The smaller number of real 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.
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)
88
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)
89
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 57: 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,
and 8) read from the top rectangular box to the bottom box.
90
20.3
Third spectral sampling
The smaller number of real orders from above and below is used both above and below. In addition,
twenty positive spectral orders and twenty negative spectral orders are added both above and below.
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)
91
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)
92
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 58: 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 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, and 8) read from the top rectangular box to the bottom box.
93
21
Condition Number Graphs, Transmission Interface, SC Formalism, Spectral Sampling Variations
These results are for S(x) = −(d/2) cos(2πx/L) and λ = 0.0625. Uniform coordinate sampling is used
throughout. The number of coordinate samples, spectral samples above, and spectral samples below are
all equal. Plotted is the log10 of the condition number at the d/L value for which the Error = −2 The
method stopped unconditionally at d/L = 1, which was only achieved when κ = ρ = 1. It was possible
to handle very large condition numbers in a routine manner. The effect of adding evanescent orders is
treated in Figures 60 and 61. Adding a small number of evanescent orders often improved the results but
adding large numbers of evanescent orders increased the condition number.
21.1
First spectral sampling
The smaller number of real orders from above and below is used both above and below.
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)
94
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
0.5
8
1.5
0.5
8
1
0.5
8
0.5
0.5
20
40
Incident angle
(b)
95
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
0
(c)
Figure 59: 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. 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, and 8) read from the top rectangular
box to the bottom box.
96
21.2
Second spectral sampling
The smaller number of real orders from above and below is used both above and below. In addition, ten
positive orders and ten negative orders are added both above and below.
Incident angle=1 to 89, by 8
log(cond)
Wavenumber ratio=0.5,1,2,4,8
8
0.5
8
18
0.5
8
16
0.5
8
14
0.5
8
0.5
8
12
0.5
8
10
0.5
8
8
0.5
8
6
0.5
8
4
0.5
8
0.5
8
2
0.5
0
0.5
1
2
4
Density ratio
(a)
97
8
16
Inf
0
Density ratio=0,0.5,1,2,4,8,16,Inf
log(cond)
8
18
0.5
8
Wavenumber ratio=0.5,1,2,4,8
16
0.5
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)
98
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 60: 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 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, and 8) read from the top rectangular box to the bottom box.
99
21.3
Third spectral sampling
The smaller number of real orders from above and below is used both above and below. In addition,
twenty positive orders and twenty negative orders are added both above and below.
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)
100
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)
101
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 61: 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, and 8) read from the top rectangular box to the bottom box.
102
22
Roughness and Condition Number Graphs, Transmission Interface, SS Formalism
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.
Results are given approximately every ten degrees. d/L was varied in steps of 0.01. The graphed
results for d/L and the condition numbers utilize interpolation between tested values. The error was
fixed at 1%. The results indicate larger d/L values can be used as the incidence angle approaches zero
(normal incidence) but that the matrix condition number correspondingly increases. The effects of adding
evanescent orders is also included. Results are mixed but generally good results occur when (at least) all
real orders are included.
0.25
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53
0.20
43
+
64 ×
2
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+
×
0.15
74 d/L
0.10
0.05
+
2
◦
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0.00
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Incident angle
Figure 62: d/L values for 1% error in energy check for the SS formalism. Number of orders
labeled at left. ◦ – 43 orders included, 2 – 53, × – 64, and – 74. + – 43 above, 64 below.
103
15
53 ........
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64 ×
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43 ◦
×
74 +
9
log10 (cond(K))
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5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Incident angle
Figure 63: Condition numbers for 1 % error in energy check for the SS formalism. Number of
orders labeled at left. ◦ – 43 orders included, – 53, × – 64, and + – 74.
104
23
Variation of the Transmission Interface SS Formalism with κ, the Wave Number Ratio
In this section, 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 from Figure 64 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 banded structure of the matrix K is illustrated in Figure 65. The
matrix is not square and pseudoinverse methods are used throughout.
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), θ = 25◦ , ρ = 3. 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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× × ×
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+.......
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×
◦ + +
×
+
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×
+
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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 64: 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).
105
−150
−100
−50
0
50
100
150
−20
−15
−10
−5
0
5
Figure 65: Plot of the real part of K in the SS formalism. Matrix size is 320 × 32, parameters are listed
in Table 20.
S(x)
d/L
λ/L
θi
ρ
κ
Real Orders Above
Real Orders Below
Error
−(d/2) cos(2πx/L)
0.1
0.0625
25◦
3
10
32
320
-5.2
Error = log10 |1 − Normalized Energy|
Table 20. Parameters for the real part of the matrix K in Figure 65 showing banded structure of
overdetermined matrices from SS formalisms.
106
24
Variation of the Transmission Interface SS Formalism with ρ, the Density Ratio
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), θ = 25◦ . Error = − log10 |1 − Energy Check|. In Figure 66, we see that over a
decade of values of ρ the error is small and quite stable for various spatial sampling schemes. In the figures
that follow, Figures 67-71, we illustrate the real part of the matrix K and polar plots of the scattering
and transmission. These are small matrices, but they are not smooth at all. The energy distribution
varies greatly with ρ. There is little transmission for ρ very small or very large. However, there are some
values of ρ where there is almost no reflection.
−1.5
−2.0
−2.5
−3.0
−3.5
−4.0
−4.5
−5.0
Error
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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 66: Error vs. ρ for the SS formalism. Matrix sizes are: ◦ — 32 by 32, — 48 by 48, × — 64 by
64, + — 64 by 32.
107
0.12
0.1
0.3
0.2
0.08
0.1
0
0.06
−0.1
−0.2
0.04
−0.3
−30
10
−20
0.02
0
−10
−10
0
−20
10
−30
0
−0.02
0
0.02
0.04
(a)
0.06
0.08
0.1
0.12
(b)
Figure 67: SS Formalism, ρ = 0 (Energy check error ≈ 10−5.7 ): (a) Real part of matrix K, size 32 by 32.
(b) Polar plot of reflected Rj and transmitted Tn modal energy components defined in Section 7. All the
energy is reflected.
0.02
0
0.6
−0.02
0.4
−0.04
0.2
−0.06
0
−0.08
−0.2
−0.1
−0.4
−30
−0.12
10
−20
0
−10
−0.14
−10
0
−20
10
−0.16
−0.01
−30
(a)
0
0.01
0.02
0.03
0.04
0.05
(b)
Figure 68: SS formalism, ρ = 1 (Energy check error ≈ 10−4.0 ): (a) Real part of matrix K, size 32 by 32.
(b) Polar plot of reflected Rj and transmitted Tn modal energy components defined in Section 7. The
reflected energy arises solely from the wave number difference.
108
0.06
0.2
0
2
−0.2
1.5
−0.4
1
0.5
−0.6
0
−0.8
−0.5
−1
−1
−1.5
−30
−1.2
10
−20
0
−10
−1.4
−10
0
−20
10
−1.6
−0.1
−30
0
0.1
(a)
0.2
0.3
0.4
0.5
0.6
(b)
Figure 69: SS formalism, ρ = 10 (Energy check error ≈ 10−4.3 ): (a) Real part of matrix K, size 32 by 32.
(b) Polar plot of reflected Rj and transmitted Tn modal energy components defined in Section 7. Even
for this large density contrast, most of the energy is in the transmitted field.
0.12
0.1
15
10
0.08
5
0.06
0
−5
0.04
−10
0.02
−15
−30
10
−20
0
−10
0
−10
0
−20
10
−0.02
−0.02
−30
(a)
0
0.02
0.04
0.06
0.08
0.1
(b)
Figure 70: SS formalism, ρ = 100 (Energy check error ≈ 10−4.6 ): (a) Real part of matrix K, size 32 by
32. (b) Polar plot of reflected Rj and transmitted Tn modal energy components defined in Section 7.
Most of the energy is reflected.
109
0.12
0.12
0.1
30
20
0.08
10
0.06
0
−10
0.04
−20
0.02
−30
−30
10
−20
0
−10
0
−10
0
−20
10
−0.02
−0.02
−30
(a)
0
0.02
0.04
0.06
0.08
0.1
(b)
Figure 71: SS formalism, ρ = 200 (Energy check error ≈ 10−4.9 ): (a) Real part of matrix K, size 32 by
32. (b) Polar plot of reflected Rj and transmitted Tn modal energy components defined in Section 7.
Most of the energy is reflected.
110
0.12
25
Variation of the Transmission Interface SS Formalism with ρ, the Density Ratio, Rougher Surface
This section presents a second example of error versus density variation. The surface is slightly rougher
than the surface used to 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 Figure 72 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), θ = 0.1◦ (near normal incidence).
111
−1
× × ×
−2
−3
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Error −5
−6
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× × × × × ×
× ×
×
× × ×
× × × × ×
+ + + + + + +
× × ×
+ + +
+
+ + +
+ + +
+ + + +
+ + + +
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
×
−7 −8
◦
+
−9
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
ρ
Figure 72: 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).
112
26
Summary and Conclusions
We considered the scattering from and transmission through a one-dimensional rough 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 coordinate-coordinate (CC), the usual coordinatespace 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 several periodic surfaces
including a simple cosine surface, a Gaussian tapered cosine, and a wave-superposition surface. 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. For the Dirichlet
problem the application of our methods to the Gaussian tapered cosine and wave superposition surfaces
the coordinate based methods worked well when λ << L provide the matrix sizes were large enough.
When λ ≈ L the SC method was highly reliable and very fast. The surface currents produced by these
surfaces were highly oscillatory and required extensive sampling to achieve good results. Specialization
of the results to the Neumann boundary value problem for a pure cosine surface generally yielded a level
of performance of the SC method similar to the results for the Dirichlet case in [8, 9] although here the
results deteriorated near grazing incidence.
The computational results for the full transmission case were presented in Sections 13-25. The three
methods, CC, SC and SS, were all studied. In Section 13 the SS method was applied to a suite of six
surfaces varying from fairly rough to flat. The method worked well except for a fairly rough surface at
near grazing incidence. In Sections 14-16 the three methods were compared and for the cases considered
all worked with the SC method the fastest. Each method produced slightly different surface fields but
these differences led to negligible differences in the scattered fields and energy check. In Section 17,
all methods worked well at extreme grazing incidence. The SS example required only a small number
of topological modes to achieve a highly accurate solution. The SC method was extensively studied in
Sections 18-21 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. Finally, in Sections 22-25 the SS method was
studied with fixed error for largest slope values as a function of wavenumber and density ratios. Highly
stable results were found over an extensive suite of wavenumber, density and incidence angle variations.
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 for technical assistance in the production of this report.
113
A
Brief Remarks on the Numerical Solution of the
Coupled Equations
The numerical solution of the coupled equations follows the general discretization methods defined in
[8, 9]. The appendices of these papers treat the evaluation of the periodic Green’s function and general
techniques of coordinate discretization. Here we treat three coordinate sampling schemes in Sections 18
and 19. These latter are based on the number of spectral Bragg modes in scattering and transmission. Various spectral sampling techniques are treated in Sections 20 and 21. Computations involving a
truncation of the topological basis in the SS representation are treated in Sections 13-17 and 22-25.
To solve a Dirichlet or Neumann problem, one matrix equation must be solved. The method we use
to do so is a direct algorithm if the matrix is square and a pseudoinverse if it is non-square.
To solve the transmission case, two matrix equations must be solved. If all four matrices are square,
they become the four quadrants of a large square system, which is solved directly. If any of the four matrices is non-square, we do not form a large matrix, but rather solve smaller systems, using a pseudoinverse
for any non-square matrix.
114
B
B.1
Acoustic and Electromagnetic Boundary Conditions
Acoustics
The following symbols are used for acoustics: ψ is the velocity potential, v = −∇ψ is the velocity,
p = −ρ ∂ψ
∂t is the pressure, p = iωρψ if harmonic time dependence is assumed.
Figure 73 shows two media separated by a flat interface. Medium 1 has density ρ1 , and the wave
number of the incident sound wave is k1 = ω/c1 . Medium 2 has density ρ2 , and the wave number of
the transmitted sound wave is k2 = ω/c2 . As usual, ω = 2πf is the circular frequency, f is the natural
frequency, and c1 and c2 are the sound speeds.
The incident wave makes an angle θi with the vertical z−axis, and the scattered wave an angle θSC .
The transmitted waves makes an angle θt with the negative z−axis, as shown in Figure 73.
z
i
sc
θ
θ
ρ
R
1
k1
x
ρ
2
T
k2
t
θ
Figure 73: Flat interface z = 0 separating media of different (but constant) (j = 1, 2) ρj (density) and kj
(wavenumber). A plane wave is incident at angle θi from region 1. It produces a reflected (R, angle θsc )
and transmitted (T, angle θt ) plane wave. The relation of these scalar parameters to the corresponding
electromagnetic boundary value problems is discussed in Appendix B.
115
Continuity conditions on the velocity potential
The continuity conditions on the velocity potential follow from requiring that
(1) The pressure is continuous:
ρ1 ψ1 = ρ2 ψ2 ,
(B.1)
ψ1 = ρψ2 ,
(B.2)
or,
where ρ = ρ2 /ρ1 , and
(2) The normal (here z)-component of the velocity is continuous:
In region 1 the fields are
where
∂ψ1
∂ψ2
=
.
∂z
∂z
(B.3)
ψ1 (x, z) = ψ1i (x, z) + ψ1SC (x, z),
(B.4)
i
i
ψ1i (x, z) = Aeik1 (x sin θ −z cos θ ) ,
and
ψ1SC (x, z) = RAeik1 (x sin θ
SC
In region 2, the field is
ψ2 (x, z) = T Aeik2 (x sin θ
t
(B.5)
+z cos θ SC )
−z cos θ t )
.
(B.6)
.
(B.7)
where R and T are the reflection and transmission coefficients defined for the volocity potential. Substituting the fields into (B.2) and (B.3), and evaluating at z = 0 gives
i
eik1 x sin θ + R eik1 x sin θ
and
SC
i
= ρT eik2 x sin θ ,
i
−ik1 cos θi eik1 x sin θ + ik1 cos θSC R eik1 x sin θ
SC
(B.8)
t
= −ik2 cos θt eik2 x sin θ .
(B.9)
Since these equations must hold for all x we have
k1 sin θi = k1 sin θSC = k2 sin θt ,
(B.10)
which imply the law for specular reflection,
θi = θSC ,
(B.11)
and Snell’s law
k1 sin θi = k2 sin θt .
If we use κ = k2 /k1 = c1 /c2 and cos θt = 1 − κ12 sin2 θi , (B.8) and (B.9) are
1 + R = ρT,
and
cos θt
κ
R − 1 = −κ
T =−
i
cos θ
cos θi
Solution of (B.13) and (B.14) gives
1−
1
1 2
sin2 θi T = −
κ − sin2 θi T.
2
κ
cos θi
(B.12)
(B.13)
(B.14)
κ2 − sin2 θi
R=
,
ρ cos θi + κ2 − sin2 θi
(B.15)
2 cos θi
.
ρ cos θi + κ2 − sin2 θi
(B.16)
ρ cos θi −
and
T =
116
We now discuss two limits.
(1) For an acoustically soft surface one has ρ2 = 0, hence ρ = 0, and thus
R = −1.
(B.17)
To get the correct limit for T we have to use the soft surface ρ2 c2 → 0. We need c2 → 0 (or κ → ∞) to
get T = 0.
(2) For an acoustically hard surface ρ2 → ∞, hence ρ → ∞, and thus
R = +1,
T = 0.
(B.18)
Continuity conditions on the pressure
As an alternative, we can impose continuity conditions on the pressure. In that case the fields are
p1 (x, z) =
i
p (x, z) =
pSC
1 (x, z) =
p2 (x, z) =
pi1 (x, z) + pSC
1 (x, z),
Ae
ik1 (x sin θ i −z cos θ i )
Rp Aeik1 (x sin θ
Tp Ae
SC
(B.19)
,
(B.20)
+z cos θ
t
t
ik2 (x sin θ −z cos θ )
SC
)
,
.
(B.21)
(B.22)
Continuity of pressure, using Snell’s law etc., then leads to p1 = p2 and
1 + Rp = Tp .
(B.23)
Continuity of the normal components of the velocities leads to
1 ∂p1
1 ∂p2
=
.
ρ1 ∂z
ρ2 ∂z
which results in
− cos θi (1 − Rp ) =
(B.24)
κ
cos θt Tp .
ρ
(B.25)
The relations (B.23) and (B.25) are the same as (B.13) and (B.14) if we replace Tp by ρT and Rp by R.
The reflection and transmission coefficients on the pressure are
ρ cos θi − κ2 − sin2 θi
Rp =
,
(B.26)
ρ cos θi + κ2 − sin2 θi
2ρ cos θi
.
(B.27)
Tp =
ρ cos θi + κ2 − sin2 θi
B.2
Electromagnetics
The source-free boundary conditions are:
n × (E1 − E2 ) =
0,
(B.28)
n × (H1 − H2 ) =
n · (B1 − B2 ) =
0,
0,
(B.29)
(B.30)
n · (D1 − D2 ) =
0.
(B.31)
Here
Bj = μj Hj , and Dj = j Ej ,
(j = 1, 2).
(1) For symmetry we will keep all μj , although in general at optical frequencies μ1 = μ2 .
117
(B.32)
(2) Equations (B.30) and (B.31) can be shown to follow from Maxwell equations and equations (B.28)
and (B.29). We now discuss two cases.
Case 1: TE Polarization (H polarization, ⊥ polarization, E⊥ polarization)
The total electrical field E = (0, E(x, z), 0) has only a y−component, orthogonal to the plane of incidence.
Maxwell’s equations
∂B
∇×E = −
= iωB,
(B.33)
∂t
lead to
i
∂E
∂E
B = − (−
, 0,
).
(B.34)
ω
∂z
∂x
Here n = k̂.
Using (B.28) and (B.29), respectively, the boundary conditions become,
and
E1 = E2 ,
(B.35)
1 ∂E2
1 ∂E1
=
.
μ1 ∂z
μ2 ∂z
(B.36)
∂E1
∂E2
=
,
∂x
∂x
(B.37)
Furthermore, (B.30) gives
and (B.31) is satisfied identically.
Only the first two equations are independent. Indeed, (B.37) follows from (B.35) if the representations
of the fields are used.
Next, we write field expressions on the scalar components of the E field:
In medium 1,
E1 (x, z) = E i (x, z) + E1SC (x, z),
(B.38)
and in medium 2, using (B.7)
(with T → TH ).
E2 (x, z) = ψ2 (x, z)
(B.39)
Furthermore, from (B.5) and (B.6), we obtain
E i (x, z) = ψ1i (x, z),
and
E1SC (x, z) = ψ1SC (x, z)
(with R → RH ).
(B.40)
(B.41)
Substituting these field expansions into (B.35) and (B.36), allows one to solve for the reflection and
transmission coefficients, RH and TH , for H polarization.
Alternatively, these coefficients can be obtained by using the acoustic analogy. Indeed, if we define
ρ1 ψ1 = E1 ,
ρ2 ψ2 = E2 ,
(B.42)
then (B.35) agrees with (B.2), and (B.36) becomes
ρ1 ∂ψ1
ρ2 ∂ψ2
=
,
μ1 ∂z
μ2 ∂z
(B.43)
which agrees with (B.3) for acoustics if ρ = ρ2 /ρ1 = μ2 /μ1 = μ.
The corresponding reflection and transmission coefficients follow from the acoustic velocity potential
results by simply replacing ρ by μ. So,
μ cos θi − κ2 − sin2 θi
,
(B.44)
RH =
μ cos θi + κ2 − sin2 θi
118
and
TH =
2 cos θi
,
μ cos θi + κ2 − sin2 θi
where now for EM
κ=
√
2 μ2
k2
√
= √
= μ.
k1
1 μ1
(B.45)
(B.46)
Case 2: TM polarization (V polarization, polarization, H⊥ polarization)
The total magnetic field is H = (0, H(x, z), 0). Maxwell’s equations
∂D
= −iωD,
∂t
(B.47)
∂H
∂H
i
(−
, 0,
).
ω
∂z
∂x
(B.48)
∇×H=
lead to
D=
Again, n = k̂.
Using (B.28) and (B.29), respectively, the boundary conditions become,
1 ∂H1
1 ∂H2
=
,
1 ∂z
2 ∂z
(B.49)
H1 = H2 .
(B.50)
and
Furthermore, (B.30) is satisfied identically, and (B.31) gives
∂H2
∂H1
=
.
∂x
∂x
(B.51)
Here, (B.51) follows from (B.50), so only (B.49) and (B.50) are independent.
To compare (B.50) and (B.49) with (B.2) and (B.3), we define (by acoustic analogy)
H1 = ρ1 ψ1 ,
H2 = ρ2 ψ2 .
(B.52)
Then (B.50) agrees with (B.2), and (B.49) becomes
ρ2 ∂ψ2
ρ1 ∂ψ1
=
,
1 ∂z
2 ∂z
(B.53)
which agrees with (B.3) provided ρ = ρ2 /ρ1 = 2 /1 = .
The corresponding reflection and transmission coefficients defined on the magnetic field component
H1 (x, z) = H i (x, z) + H1SC (x, z),
(B.54)
and from (B.7)
H2 (x, z) = ψ2 (x, z)
From (B.5) and (B.6)
and
(with T → TV ).
(B.55)
H1i (x, z) = ψ1i (x, z),
(B.56)
H1SC (x, z) = ψ1SC (x, z),
(B.57)
with R → RV . The reflection and transmission coefficients can thus be found from the scalar values by
replacing ρ by . So,
cos θi − κ2 − sin2 θi
RV =
,
(B.58)
cos θi + κ2 − sin2 θi
2 cos θi
TV =
,
(B.59)
cos θi + κ2 − sin2 θi
119
where
κ=
√
k2
= μ.
k1
(B.60)
For both polarizations these are the reflection and transmission coefficients given in [11].
Perfect electric conductor
Now we consider the limit 2 → ∞ which corresponds to → ∞.
In that case RH = −1, TH = 0, RV = +1, and TV = 0.
This leads to the following result: Although the boundary conditions for the TE polarization contain
μ’s (see (B.50)), in the limit of a perfect electric conductor RH = −1, which corresponds to the acoustic
case of a soft boundary, whereas the limit of the TM polarization is RV = +1 which corresponds to a
hard boundary.
Conclusion:
TE or H polarization with a perfect electric conductor is identical to a soft boundary, and TM or V
polarization with a perfect electric conductor is identical to a hard boundary.
120
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121