Theoretical and Computational Aspects
of Scattering from Periodic Surfaces:
Two-dimensional Perfectly Reflecting Surfaces
Using the Spectral-Coordinate Method
J. DeSanto, G. Erdmann, W. Hereman,
B. Krause, M. Misra, and E. Swim
MCS-00-06R
June 2001
submitted to: Waves in Random Media (2001)
Research supported by AFOSR Grant # F49620-96-1-0039
Department of Mathematical and Computer Sciences
Colorado School of Mines
Golden, CO 80401-1887, USA
Phone: (303) 273-3860
Fax: (303) 273-3875
Email: jdesanto@mines.edu
Abstract
We consider the scattering from a two-dimensional periodic surface. From our
previous work on scattering from one-dimensional surfaces (Waves in Random Media
8, 385(1998)) we have learned that the spectral-coordinate (SC) method was the
fastest method we have available. Most computational studies of scattering from
two-dimensional surfaces require a large memory and a long calculation time unless
some approximations are used in the theoretical development. Here by using the
SC method we are able to solve exact theoretical equations with a minimum of
calculation time.
We first derive (in PART I) in detail the SC equations for scattering from twodimensional infinite surfaces. Equations for the boundary unknowns (surface field
and/or its normal derivative) result as well as an equation to evaluate the scattered
field once we have solved for the boundary unknowns. Special cases for the perfectly
reflecting Dirichlet and Neumann boundary value problems are presented as is the
flux-conservation relation.
The equations are reduced to those for a two-dimensional periodic surface in
PART II and we discuss the numerical methods for their solution. The twodimensional coordinate and spectral samples are arranged in one-dimensional strings
in order to define the matrix system to be solved.
The SC equations for the two-dimensional periodic surfaces are solved in PART
III. Computations are done for both Dirichlet and Neumann problems for various
periodic sinusoidal surface examples. The surfaces vary in roughness as well as
period and are investigated when the incident field is far from grazing incidence
(“no grazing”) and when it is near-grazing. Extensive computations are included in
terms of the maximum roughness slope which can be computed using the method
with a fixed maximum error as a function of azimuthal angle of incidence, polar
angle of incidence, and wavelength to period ratio.
The results show that the SC method is highly robust. This is demonstrated
with extensive computations. Further the SC method is found to be computationally
efficient and accurate for near-grazing incidence. Computations are presented for
grazing angles as low as 0.01o . In general we conclude that the SC method is a
very fast, reliable and robust computational method to describe scattering from
two-dimensional periodic surfaces. Its major limiting factor is high slope and we
quantify this limitation.
PART I: THEORETICAL DEVELOPMENT FOR AN INFINITE SURFACE
1
Introduction
In previous reports [7, 8] and a paper [10] we presented the theoretical and computational
results of scattering from one-dimensional rough surfaces. Many different methods were
studied. They were all exact theoretical developments which were discretized and solved
as matrix systems. They were characterized by the space in which the rows and columns
of the matrices were sampled, either coordinate space (C) or spectral space (S). The
coordinate-coordinate (CC) method was the usual method of moments. The spectralcoordinate (SC) method used the Weyl representation of the free-space Green’s function
1
to generate equations in a mixed spectral and coordinate representation, and the spectralspectral (SS) method had the equations fully in spectral space. The results of our onedimensional scattering confirmed that the SC method was the fastest method (usually by
several orders of magnitude) in terms of fill time, i. e. the time necessary to create the
matrix [10]. This is not a minor point for the rapid solution of matrix systems. Once the
linear systems were created, the solution time for each of the methods was comparable.
The reason that the fill time of the SC method was so short is easily explained. The
matrix is created by sampling a simple exponential function. The function can be highly
oscillatory but this presents no difficulty for SC. The CC method requires an infinite
sum over these functions (to form the periodic Green’s function for example) and the
SS method a surface integral over these same functions. Both of the latter can be very
time consuming computations. The SC method was designed to avoid this complexity
[5]. The price of the SC method comes in the fact that for large slopes the method
becomes ill-conditioned. As yet we have not found a consistent way to overcome this illconditioning so that the dramatic time advantage we get in extending the SC method to
computations from two-dimensional surfaces must be tempered to stay within parametric
surface variability to ensure convergence.
The geometry and notation are presented in this section. In Sec. 2 we derive the SC
equations for an infinite surface. We keep both boundary unknowns in Sec. 2 and then
specify the Dirichlet and Neumann problems in Sec. 3 and the flux conservation in Sec. 4.
The equations are reduced to those for a periodic surface in Sec. 5 with the energy balance
criteria in Sec. 6. The numerical solutions of the periodic surface equations is discussed in
Sec. 7. Section 8 is a brief outline of the computational results for the Dirichlet problem
(Sec. 9) and the Neumann problem (Sec. 10). The summary and conclusions are presented
in Sec. 11.
Our computations are limited by maximum roughness and we quantify these limitations. On the other hand, extensive computations reveal that the SC method is quite
robust in terms of its ability to produce highly accurate results over a wide range of
incident angles (including near-grazing incidence and reflection) and surface period.
Other methods have been developed to treat the scattering from two-dimensional surfaces. Kim et al. [12] based their approach on previous work of ours [4, 5] involving
spectral methods. They differed from our work by expanding the boundary unknowns
in a two-dimensional Fourier series, producing a spectral-Fourier method. For this expansion, the matrix elements were again integrals over rapidly oscillating functions and
they limited their work to Dirichlet and Neumann problems and further limited their
results to shallow surfaces. Our surface heights are much larger and we have focused the
results on maximum roughness which we define here to be essentially maximum slope
divided by π. Kinsman [13] refers to this as steepness for one-dimensional waves. Tran
and Maradudin [15], Tran, Celli and Maradudin [16] and Macaskill and Kachoyan [14]
use the Liouville-Neumann expansion on integral equations to treat random surface scattering. The iterative expansion cuts down on solution time but has limitations on the
size involved in the expansion parameter. Our method uses exact inversion. A method
of Boag et al. [2] sets up fictitious patch sources displaced away from the surface. The
amplitudes of these sources are adjusted to match the boundary conditions at selected
points on the surface. Our method computes the surface sources directly. Wagner et al.
2
[17] developed the fast multipole method as an approximate analytical tool to speed up
the computations. A special case of normal incidence is studied in terms of approximately
computing the Mueller matrix for the scattering process [3]. All these authors point out
the difficulties of computing any fully two-dimensional scattering problem, and all are
faced with fill-time questions which our SC method enables us to resolve. An example of
a direct approximation scheme was presented by Jin and Lax [11] who used the doubleKirchhoff scattering approximation. Although these authors were primarily interested
in computing the backscatter enhancement effect, it is known that Kirchhoff scattering
fails near grazing incidence. We show that the SC method works very well near grazing
incidence. In Figure 1 we illustrate the geometry and definitions for the derivation of
the spectral-coordinate method in 2 dimensions. In what follows x = (x, y, z) = (xt , z)
denotes a vector in the 3 dimensional Cartesian space. Obviously, xt = (x, y). The z−axis
points vertically upwards, the x−axis points towards the right, and the y−axis is such that
(x, y, z) form a right-handed coordinate system. The two-dimensional surface is specified
by z = h(xt ). So, xh = (xt , h(xt )) is a position vector on the surface.
− L2
z
y
S1
L
2
ρ1 , k1
V1
x
h(xt )
Figure 1: V1 is the region between the rough surface and the highest surface excursion.
This region is bounded by h(xt ) and S1 vertically and ± L2 horizontally. In the derivation
of the equations we let L → ∞ for the results through Section 4.
There are 2 regions of interest: R and V1 . They are defined as follows. For region R :
z ≥ S1 , where the value S1 is assumed to be above the highest surface excursion. For
region V1 : h(xt ) < z ≤ S1 . In addition, ψ is the symbol for the acoustic velocity potential
which satisfies the appropriate Helmholtz equation. The derivation is analogous to that
of the electromagnetic case discussed previously in [6]. Further, the relation of acoustic
and electromagnetic boundary value problems can be found in [8, Appendix B] for onedimensional problems.
2
Derivation of Spectral-Coordinate (SC) Equations
in 2D: Region V1
Figure 1 illustrates region V1 , where the density is denoted by ρ1 , and the wave number
of the incident sound wave by k1 = ω/c1. The shaded area is defined by h(xt ) < z ≤ S1
and − L2 ≤ x, y ≤ L2 . V1 is specified by the characteristic function
θ1 (x) =
1,
0,
3
x ∈ V1 ,
x∈ V1 ,
(2.1)
which is given by
θ1 (x) = θ(z − h(xt ))θ(S1 − z)θ+ (xt ),
(2.2)
where θ is the Heaviside function and
θ+ (xt ) = θ(x +
L L
L L
)θ( − x)θ(y + )θ( − y).
2
2
2
2
(2.3)
The vector derivative of θ1 is
∂l θ1 (x) = nl (xt )δ(z − h)θ+ − δl3 δ(z − S1 )θ+ + θ(z − h)δ(S1 − z)∂l θ+ ,
(2.4)
with
L
L
L L
L L
)θ(y + )θ( − y) − δl1 δ(x − )θ(y + )θ( − y)
2
2
2
2
2
2
L L
L L
L
L
+δl2 δ(y + )θ(x + )θ( − x) − δl2 δ(y − )θ(x + )θ( − x).
2
2
2
2
2
2
∂l θ+ = δl1 δ(x +
(2.5)
Here
nl (xt ) = δl3 − hx δl1 − hy δl2
(2.6)
is a non-unit normal. Furthermore, the field ψ is given by ψ (1) (x) ∈ V1 .
Assume that ψ (1) (x) is source free in V1 and therefore satisfies the Helmholtz equation
(∂l ∂l + k12 )ψ (1) (x) = 0.
(2.7)
Planar wave states are defined in V1 as
±
φ±
1 (x) = exp[ik1 M · x],
(2.8)
where M± = (−Mt , ±m1 (Mt )), Mt = (Mx , My ), and
⎧
⎨
m1 (Mt ) = ⎩
+ 1 − Mt2 ,
+i Mt2 − 1,
Mt2 < 1,
Mt2 > 1.
They satisfy the same Helmholtz equation
(∂l ∂l + k12 )φ±
1 (x) = 0.
(2.9)
From (2.7) and (2.9) we obtain the following vector identity:
(1)
±
(1)
∂l [∂l φ±
1 (x)ψ (x) − φ1 (x)∂l ψ (x)] = 0.
(2.10)
Multiplying (2.10) by θ1 (x) and integrating over all space in x we get
∞
−∞
(1)
±
(1)
θ1 (x) ∂l [∂l φ±
1 (x)ψ (x) − φ1 (x)∂l ψ (x)] dx = 0.
(2.11)
Integrating (2.11) by parts results in
∞
−∞
(1)
±
(1)
[∂l φ±
1 (x)ψ (x) − φ1 (x)∂l ψ (x)] ∂l θ1 (x) dx = 0.
4
(2.12)
Note that the partially integrated term drops because θ1 (x) defines a bounded region.
Next evaluate the integrals in (2.12). Using ∂l θ1 (x) from (2.4) we have to consider
three types of integrals: An integral over h; an integral over z = S1 ; and ‘Edge’ or side
integrals involving ∂l θ+ . Also, we will normalize the integrals to a per unit length by
dividing them all by L2 . (Equivalently, we could put a factor 1/L2 in the plane wave
definition). The result is
U ± (L) = S1± (L) − E1± (L),
(2.13)
where U ± (L) are integrals on the upper side of h. Hence,
U ± (L) =
1 L2
L
2
−L
2
(1)
±
(1)
[nl ∂l φ±
1 (xh )ψ (xh ) − φ1 (xh )nl ∂l ψ (xh )] dxt ,
(2.14)
±
(1)
(1)
±
where φ±
1 (xh ) = lim+ φ1 (x) and nl ∂l ψ (xh ) = lim+ nl ∂l ψ (x). S1 (L) are the integrals
z→h
z→h
over z = S1 and are given by
S1± (L)
1 = 2
L
L
2
−L
2
[
∂ ±
∂ (1)
φ1 (x1 )ψ (1) (x1 ) − φ±
ψ (x1 )] dxt ,
1 (x1 )
∂z
∂z
(2.15)
where x1 = (xt , S1 ). Finally, E1± (L) are the integrals on the ‘edge’, given by
E1± (L) =
1
L2
S1
h(xt )
dz
L
2
−L
2
(1)
±
(1)
[∂l φ±
1 (x)ψ (x) − φ1 (x)∂l ψ (x)]∂l θ+ dxt .
(2.16)
Note that the limits for the integration with respect to z come from the ∂l θ+ part of ∂l θ1 .
For the edge integrals we note the following:
1. For E1± in (2.16), ∂l θ+ yields at least one delta function in x and y and so one
horizontal integral can be carried out. One integral of order L remains. Then,
±
±
1
(1)
2. For bounded functions φ±
1 , ψ , ... an estimate of E1 (L) yields E1 (L) ∼ O( L ).
3. The other integrals behave like U ± (L) ∼ O(1) and S1± (L) ∼ O(1) for L large.
4. In the limit of large L, E1± (L) → 0 relative to U ± (L), andS1± (L), and thus edge
integrals are dropped. For any large finite value of L we assume that these integrals
can be as small as we want. For periodic surfaces (which we introduce later) the
edge integrals cancel exactly using Floquet boundary conditions.
If we define the remaining limits as
U ± = lim L2 U ± (L),
(2.17)
S1± = lim L2 S1± (L),
(2.18)
U ± = S1± ,
(2.19)
L→∞
and
L→∞
then the equations reduce to
5
where
U± =
and
S1±
=
∞
±
[nl ∂l φ±
1 (xh )ψ(xh ) − ik1 φ1 (xh )N(xh )] dxt ,
(2.20)
∂ ±
∂ (1)
φ1 (x1 )ψ (1) (x1 ) − φ±
ψ (x1 )] dxt .
1 (x1 )
∂z
∂z
(2.21)
−∞
∞
−∞
[
The boundary unknowns are defined by ψ(xh ) = ψ (1) (xh ) and N(xh ) = nl ∂l ψ (1) (xh )/ik1 ,
i.e. the function and (scaled) normal derivative evaluated on the boundary. We note that:
1. Equation (2.19) relates integrals on z = h to those on z = S1 , a kind of analytic
continuation.
2. U ± and S1± are functions of the spectral parameter M± through φ±
1 from (2.8). We
have suppressed it. Indeed, using (2.8),
± ±
nl ∂l φ±
1 (xh ) = ik1 nl Ml φ1 (xh ),
(2.22)
and
∂ ±
φ (x1 ) = ik1 (±m1 (Mt ))φ±
1 (x1 ),
∂z 1
so that (2.20) and (2.21) are (after ik1 is factored out)
U ± = ik1
∞
−∞
±
φ±
1 (xh ) [nl Ml ψ(xh ) − N(xh )] dxt ,
(2.23)
(2.24)
and
S1± = ik1
∞
−∞
(1)
φ±
1 (x1 ) [±m1 (Mt )ψ (x1 ) −
1 ∂ (1)
ψ (x1 )] dxt ,
ik1 ∂z
(2.25)
which we use along with (2.19).
We next evaluate S1± where ψ (1) is the sum of the incident and scattered fields. Above
the highest surface excursion in region R we can write the total field ψ (1) (x) as the sum
of incident and scattered fields, the latter of which is written as a superposition of purely
upgoing waves using a spectral expansion
ψ (1) → ψR (x) = ψ (0) (x) + ψ SC (x),
where
SC
ψ (x) =
R(αt )eik1 α·x dαt ,
(2.26)
(2.27)
with
α = (αt , αz ),
⎧
⎨
αz =
⎩
(2.28)
+ 1 − αt2 ,
+i αt2 − 1,
6
αt2 < 1,
αt2 > 1.
(2.29)
Note that we only consider upgoing waves. ψ (0) (x) is the incident field which we will
define later in this section. Computing the z−derivative of (2.26), i.e.
∂
∂ (0)
ψR (x) =
ψ (x) +
∂z
∂z
R(αt )ik1 αz eik1 α·x dαt ,
(2.30)
and using (2.26), we can write S1± as the sum of two terms
±(0)
±(SC)
S1± = S1
+ S1
,
(2.31)
with
±(0)
S1
= ik1
∞
−∞
(0)
φ±
1 (x1 ) [±m1 (Mt )ψ (x1 ) −
1 ∂ (0)
ψ (x1 )] dxt ,
ik1 ∂z
(2.32)
1 ∂ SC
ψ (x1 )] dxt .
ik1 ∂z
(2.33)
and
±(SC)
S1
= ik1
∞
−∞
SC
φ±
1 (x1 ) [±m1 (Mt )ψ (x1 ) −
Next, using
1 ∂ SC
ψ (x1 ) =
±m1 (Mt )ψ (x1 ) −
ik1 ∂z
SC
R(αt )eik1 α·x1 [±m1 (Mt ) − αz ] dαt ,
(2.34)
equation (2.33) can be rewritten as
±(SC)
S1
= ik1
∞
R(αt ) (±m1 (Mt ) − αz ) [
−∞
ik1 α·x1
φ±
dxt ] dαt .
1 (x1 )e
(2.35)
With (2.8), the second double integral in (2.35) can be recast as
∞
−∞
ik1 α·x1
φ±
1 (x1 )e
dxt =
∞
−∞
eik1 [M
±+
α]·x1 dx
= eik1 (±m1 (Mt )+αz )S1
∞
t
−∞
eik1 [αt −Mt ]·xt dxt
(2π)2 ik1 (±m1 (Mt )+αz )S1
=
e
δ(αt − Mt ).
k12
(2.36)
Inserting this result in (2.35), we obtain
±(SC)
S1
=
ik1 (2π)2
k12
R(αt ) (±m1 (Mt ) − αz ) eik1 (±m1 (Mt )+αz )S1 δ(αt − Mt ) dαt . (2.37)
To achieve further simplification we note that when αt = Mt :
Mt2
− 1 − αt2 |αt =Mt
±m1 (Mt ) − αz = ± 1 −
0
=
−2m1 (Mt ),
and
±m1 (Mt ) + αz |αt =Mt =
7
2m1 (Mt )
0.
(2.38)
(2.39)
Thus,
±(SC)
S1
−2(2π)2 m1 (Mt )
= ik1 [
]
k12
0
R(Mt ).
(2.40)
Note that the upgoing wave expansion for the scattered field and φ±
1 projects out 0 for
+
the φ1 , which itself is upgoing and projects out the reflection coefficients R for φ−
1 which
is downgoing.
±(0)
We continue with the evaluation of the incident field S1 . An arbitrary incident field
can be spectrally expanded as
(0)
ψ (x) =
I(β t )eik1 β ·x dβ t ,
where
(2.41)
βz = + 1 − βt2 .
β = (β t , −βz ),
(2.42)
The sign in −βz indicates a downgoing wave superposition (i.e. in the negative z direction).
If in (2.32) on the surface S1 , we use
1 ∂ (0)
±m1 (Mt )ψ (x1 ) −
ψ (x1 ) =
ik1 ∂z
(0)
I(β t )eik1 β ·x1 [±m1 (Mt ) + βz ] dβ t ,
(2.43)
then equation (2.32) can be rewritten as
±(0)
S1
= ik1
∞
I(β t ) (±m1 (Mt ) + βz ) [
−∞
ik1 β ·x1
φ±
dxt ] dβ t .
1 (x1 )e
(2.44)
The second double integral in (2.44) can be recast as
∞
−∞
ik1 β ·x1
φ±
1 (x1 )e
ik1 (±m1 (Mt )−βz )S1
dxt = e
∞
−∞
eik1 [βt −Mt ]·xt dxt
(2π)2 ik1 (±m1 (Mt )−βz )S1
=
e
δ(β t − Mt ),
k12
(2.45)
which leads to
±(0)
S1
ik1 (2π)2
=
k12
I(β t ) (±m1 (Mt ) + βz ) eik1 (±m1 (Mt )−βz )S1 δ(β t − Mt ) dβ t .
When β t = Mt :
±m1 (Mt ) + βz =
and
±m1 (Mt ) − βz =
Thus,
±(0)
S1
= ik1 [
(2.46)
2m1 (Mt )
0,
(2.47)
0
−2m1 (Mt ).
(2.48)
2(2π)2 m1 (Mt )
]
k12
I(Mt )
0.
(2.49)
Note that this time the projection goes the other way. The incident field which is down−
going is projected out by upgoing φ+
1 ; and 0 is projected out by downgoing φ1 .
8
Now we combine the results. First, substitute (2.40) and (2.49) into the right hand
±(0)
±(SC)
side of U ± = S1± = S1 + S1
, which yields
2(2π)2 m1 (Mt )
]
U = ik1 [
k12
±
I(Mt )
−R(Mt ).
(2.50)
Second, use the expression for U ± in (2.24). The final result is the two equations:
∞
−∞
±
φ±
1 (xh ) [nl Ml ψ(xh ) − N(xh )] dxt =
2(2π)2 m1 (Mt )
k12
I(Mt )
−R(Mt ).
(2.51)
The top equation will be used in the solution of the appropriate boundary unknown and
the bottom equation in the evaluation of the reflection coefficient R and thus the reflected
field by (2.27). The plane wave states are defined as
ik1 M
φ±
1 (xh ) = e
where
± ·x
h
,
M± · xh = −Mt · xt ± m1 (Mt )h(xt ).
(2.52)
(2.53)
The term nl Ml± occurs frequently in the subsequent analysis and it is given by
nl Ml± = ±m1 (Mt ) + hx Mx + hy My .
(2.54)
Further, we confine many of our results to single plane wave incidence. The spectral
amplitude in (2.41) can be written as
(0)
I(Mt ) = δ(Mt − αt ),
where
αx(0) = cos φi sin θi ,
and
αy(0) = sin φi sin θi ,
(0)
m1 (αt ) = cos θi ,
(2.55)
(2.56)
(2.57)
in terms of incident polar (θi ) and azimuthal (φi ) angles.
3
Dirichlet and Neumann Problem
For the Dirichlet problem we have
ψ(xh ) = 0,
(3.1)
so that (2.51) becomes
∞
±
kD
(Mt , xt ) N(xh ) dxt = D ± (Mt ),
(3.2)
±
kD
(Mt , xt ) = φ± (xh )
= e−ik1 Mt ·xt e±ik1 m1 (Mt )h(xt ) ,
(3.3)
−∞
±
where kD
is defined by
9
and the terms on the right-hand side are defined as
2m1 (Mt )(2π)2
D (Mt ) =
k12
±
−I(Mt )
R(D) (Mt ).
(3.4)
We solve the equation with the “+” sign for N(xh ), and then evaluate the equation with
the “–” sign for R(D) (Mt ). For the Dirichlet problem the solution was discussed in [7, 10]
±
are functions of a spectral
for the one dimensional surface. Note that the kernels kD
argument Mt and a coordinate argument xt , the former leading to row discretization
and the latter to column discretization to form the resulting matrix inversion problem
we solve, as well as the designation of the method as spectral-coordinate (SC). Note also
that N is the normal derivative scaled by ik1 (see the remark following (2.21)). For the
Neumann problem we have
N(xh ) = 0,
(3.5)
so that (2.51) can be written as
∞
−∞
where
or
±
kN
(Mt , xt ) ψ(xh ) dxt = N ± (Mt ),
(3.6)
±
±
(Mt , xt ) = φ±
kN
1 (xh ) nl Ml ,
(3.7)
±
±
kN
(Mt , xt ) = kD
(Mt , xh ) nl Ml± ,
(3.8)
and
2m1 (Mt )(2π)2 I(Mt )
(3.9)
N (Mt ) =
−R(N) (Mt ).
k12
Here N + = −D + . We solve (3.6) with the “+” sign for ψ(xh ), and evaluate the equation
with the “–” sign for R(N) (Mt ). Solution methods are similar to those described in [7, 10]
±
for the Dirichlet problem in one dimension. The kernels kN
are also functions of spectral
and coordinate variables and this is the SC method for the Neumann problem.
±
4
Flux Conservation
For any complex field φ(x) the z-component of flux is defined as
∂
∂
(4.1)
φ(x) − φ(x) φ(x)],
∂z
∂z
where ρB is the density of the medium and the bar indicates complex conjugation.
A spectral representation of φ is
Jz (x) = ρB [φ(x)
φ(x) =
B(t ) eik·x dt ,
(4.2)
where
= (t , z ),
⎧
⎨
z =
⎩
(4.3)
+ 1 − 2t ,
+i 2t − 1,
10
2t < 1,
2t > 1.
(4.4)
We can thus write
Jz (x) = ρB
dt dt B(t ) B(t ) {−ikz − ikz } eik(t −t )·xt eik(z −z )z .
(4.5)
The flux through an area L2 is
Jˆz =
L
2
−L
2
Jz (x) dxt .
(4.6)
In the limit as L → ∞ this becomes
2
8iπ
Jˆz = −
ρB
k
|B(t )|2 (Re z ) dt ,
(4.7)
which is independent of z and where only the real part of the z-component z appears in
the integrals. The latter represents real propagating orders.
For the scattered field replace z by αz = m1 (αt ) from (2.27) and (2.38), and k by k1
in (4.7), so that
8iπ 2
SC
ˆ
Jz = −
ρ1
|R(αt )|2 (Re m1 (αt )) dαt ,
(4.8)
k1
and for the incident field replace z by −βz = −m1 (β t ) from (2.41) and (2.47), and k by
k1 in (4.7), so that
2
8iπ
Jˆzi = −
ρ1
k1
|I(β t )|2 (−Re m1 (β t )) dβt .
(4.9)
The overall energy flux conservation is
Jˆzi = −JˆzSC ,
(4.10)
which yields
2
|I(βt )| (Re m1 (β t )) dβt =
|R(αt )|2 (Re m1 (αt )) dαt .
(4.11)
This is used as a check in our computations. For a single plane wave incident on a periodic
surface see Section 6.
11
PART II: THEORETICAL DEVELOPMENT FOR A PERIODIC
SURFACE
5
Equations for a Periodic Surface
In this section we reduce the equations in Section 2 to those for a periodic surface of
period L1 in x and period L2 in y. We illustrate the derivation keeping both boundary
unknowns for the moment. Equation (2.51) is
∞
−∞
−ik1 Mt ·xt ik1 m1 (Mt )h(xt )
[n+
e
dxt = F in (Mt ),
1 (Mt , xt )ψ(xh ) − N(xh )] e
where
(5.1)
n+
1 (Mt , xt ) = m1 (Mt ) + Mt · ht ,
(5.2)
Mt = (Mx , My ),
(5.3)
and
2(2π)2
m1 (Mt )I(Mt ).
k12
F in (Mt ) =
(5.4)
Floquet conditions on the surface fields
(0)
ψ
ik1 αt ·Lt ψ
(x
+
L
,
y
+
L
)
=
e
1
2
N
N (x, y),
(5.5)
where
(0)
αt = (αx(0) , αy(0) ),
Lt = (L1 , L2 ),
(5.6)
yield a sum over an infinite number of finite cells (p, q run from −∞ to ∞) from (5.1):
L1 L2
p
q
Ip(1)q (Mt ) = F in(Mt ),
(5.7)
where
Ip(1)q (Mt )
1
=
L1 L2
(2p+1)L1 /2
(2p−1)L1 /2
dx
(2q+1)L2 /2
(2q−1)L2 /2
dy [n+
1 (Mt , xt )ψ(xh ) − N(xh )] ·
· e−ik1Mt ·xt eik1 m1 (Mt )h(xt ) .
(5.8)
If we change variables to x = x − pL1 and y = y − qL2 , and note that n+
1 (Mt , xt ) is
periodic, we can then use the Floquet conditions (5.5) on ψ and N to write
(0)
Ip(1)q (Mt ) = eik1 αx
pL1
(0)
eik1 αy
qL2
(1)
e−ik1 Mt ·(pL1 ,qL2 ) I0 0 (Mt ).
(5.9)
Then (5.7) becomes
(1)
L1 L2 I0 0 (Mt )
p
(0)
eik1 (αx
−Mx )pL1
q
12
(0)
eik1 (αy
−My )qL2
= F in(Mt ).
(5.10)
These are just Poisson sums:
(1)
L1 L2 I0 0 (Mt )
∞
∞
λ1 λ1 δ(Mx − αjx )
δ(My − αj y ) = F in (Mt ),
L1 j=−∞
L2 j =−∞
(5.11)
where
λ1
λ1
, and αj y = αy(0) + j ,
L1
L2
are the Bragg equations in the 2 dimensions.
The result of (5.10) is thus
αjx = αx(0) + j
(1)
I0 0 (Mt )
∞
j=−∞
δ(Mx − αjx )
∞
j =−∞
(5.12)
1 in
F (Mt ).
λ21
(5.13)
j = 1, 2,
(5.14)
dMy F in(Mt ).
(5.15)
δ(My − αj y ) =
Integrating (5.13) using the following integration scheme,
lim
j →0
αpx +1 λ1
L1
λ
αpx −1 L1
αqy +2 λ1
L2
dMx
λ
αqy −2 L1
1
dMy
0 < j < 1,
2
it becomes
(1)
I0 0 (αjx , αj y )
1
= 2
λ1
αpx +1 λ1
L1
λ
αpx −1 L1
dMx
αqy +2 λ1
L2
1
λ
αqy −2 L1
2
For a single plane wave incidence, from Section 2,
2(2π)2
m1 (Mt ) I(Mt )
k12
2(2π)2
(0)
m1 (Mt ) δ(Mt − αt )
=
2
k1
2(2π)2
=
m1 (Mt ) δ(Mx − αx(0) ) δ(My − αy(0) ),
2
k1
F in(Mt ) =
(5.16)
so that equation (5.15) becomes
(1)
(0)
I0 0 (αjx , αj y ) = 2m1 (αt ) δj0 δj 0 .
(5.17)
The equation to solve is (5.17), which is explicitly,
L2 /2
1 L1 /2
dx
dy [(m1 (αjx , αj y ) + αjx hx + αj y hy ) ψ(xh ) − N(xh )] ·
L1 L2 −L1 /2
−L2 /2
(0)
· e−ik1(αjx x+αj y y) eik1 m1 (αjx ,αj y )h(xt ) = 2m1 (αt )δj0 δj 0 ,
Note that
(5.18)
m1 (αjx , αj y ) =
2
1 − αjx
− αj2 y .
(5.19)
For a periodic surface the reflected fields are discrete infinite sums of Bragg waves. These
can be written using (2.27)
SC
ψ (x) =
R(Mt ) eik1 Mt ·xt eik1 m1 (Mt )z dMt ,
13
(5.20)
where here
∞
R(Mt ) =
∞
Ajj δ(Mx − αjx ) δ(My − αj y ),
(5.21)
Ajj eik1 (αjx x+αj y y) eik1 m1 (αjx ,αj y )z .
(5.22)
j=−∞ j =−∞
so that
∞
ψ SC (x) =
∞
j=−∞ j =−∞
The “-” equation (2.51) reduces to
(1)
L1 L2 J0 0 (Mt )
p
(0)
eik1 (αx
−Mx )pL1
q
(0)
eik1 (αy
−My )qL2
=
2(2π)2
m1 (Mt ) R(Mt ),
k12
(5.23)
where
(1)
J0 0 (Mt )
(L2 /2
1 L1 /2
=
dx
dy [(m1 (Mt ) − Mt · ht )ψ(xh ) + N(xh )]
L1 L2 −L1 /2
−L2 /2
e−ik1 Mt ·xt e−ik1 m2 (Mt )h(xt ) .
(5.24)
Next, use the Poisson sum evaluation and integration as above, to get the explicit equation
that must be evaluated
(1)
J0 0 (αjx , αj y ) = 2m1 (αjx , αj y ) Ajj ,
(5.25)
where
(1)
J0 0 (αjx , αj y ) =
1
L1 L2
L1 /2
−L1 /2
L2 /2
dx
−L2 /2
dy [(m1 (αjx , αj y ) − αjx hx − αj y hy ) ψ(xh ) + N(xh )] ·
· e−ik1 (αjx x+αj y y) e−ik1 m1 (αjx ,αj y )h(xt ) .
(5.26)
The procedure is to compute the boundary unknowns ψ(xh ) or N(xh ) using (5.18), and
then use them in (5.26) to compute the scattered amplitudes by (5.25). The scattered
field is then found from (5.22).
14
6
Energy Balance
The flux conservation or energy balance follows from the results in Section 4. The major
difference is that the reflection coefficient is a discrete sum as in (5.21). For a single plane
wave as defined in (2.55)-(2.57) with amplitude D it can easily be shown that the energy
balance result is analogous to (4.11) and given by
D 2 m1 (αx(0) , αy(0) ) =
j,j |Ajj |2 (Re m1 (αjx , αj y )),
(6.1)
where the summations extend over all j, j values such that m1 (defined in (5.19) ) is real,
i.e. over all real scattered Bragg orders. This is used as a check in our calculations as
follows: Set D = 1 in (6.1) and divide the equation by m1 (αx(0) , αy(0) ) so the left hand
side of (6.1) is 1 and the resulting right hand side is called the normalized energy. The
resulting error is
Error = log10 |1 − Normalized Energy|.
(6.2)
We have effectively scaled the incident energy to 1, and the normalized energy is the total
energy in the scattered field.
15
7
Numerical Methods
In this section we summarize the equations and the computational methodology.
For a periodic transmission interface, the integral equation which we solve is from
(5.18):
1
L1 L2
L1 /2
−L1 /2
dx
L2 /2
−L2 /2
dy[(m1 (αjx , αj y ) + αjx hx (xt ) + αj y hy (xt )) ψ(xh ) − N(xh )]
(0)
e−ik1 (αjx x+αj y y) eik1 m1 (αjx ,αj y )h(xt ) = 2m1 (αt )δj0δj 0 ,
(7.1)
where (to summarize all the notations),
xt = (x, y),
(7.2)
xh = (xt , h(xt )) ,
(7.3)
∂h
(xt ),
∂x
∂h
(xt ),
hy (xt ) =
∂y
hx (xt ) =
(7.4)
(7.5)
αx(0) = cosφi sinθi ,
(7.6)
αy(0) = sinφi sinθi ,
(7.7)
(0)
αt = (αx(0) , αy(0) ),
λ
,
L1
λ
= αy(0) + j ,
L2
αjx = αx(0) + j
αj y
and
(7.8)
(7.9)
(7.10)
m1 (αjx , αj y ) =
2
1 − αjx
− αj2 y .
(7.11)
The equations are thus already discrete in spectral space.
The integral equation is then discretized over the rough surface in coordinate space to
give (wp are weight functions)
N
M q=1 p=1
[(m1 (αjx , αj y ) + αjx hx (xp , yq ) + αj y hy (xp , yq )) ψ(xp , yq , h(xp , yq ))
−N(xp , yq , h(xp , yq ))] e−ik1 αjx xp e−ik1 αj y yq eik1 m1 (αjx ,αj y )h(xp ,yq ) wp wq
(0)
= 2L1 L2 m1 (αt ) δj0 δj 0 .
(7.12)
This integral equation can be written as a matrix equation by defining the following
matrices:
[M1]jj , pq = e−ik1 αjx xp e−ik1 αj y yq eik1 m1 (αjx ,αj y )h(xp ,yq ) wp wq ,
[K1]jj , pq = [(m1 (αjx , αj y ) + αjx hx (xp , yq ) + αj y hy (xp , yq )] [M1]jj , pq .
16
(7.13)
(7.14)
The coordinate indices p and q and spectral indices j and j are formed as products for
the matrix indices. These products are defined as one-dimensional using the schematic
representations in Tables 1 and 2.
The vectors b, Ψand N are defined as
(0)
bjj = 2L1 L2 m1 (αt )δj0 δj 0 ,
ψpq = ψ(xp , yq , h(xp , yq )),
Npq = N(xp , yq , h(xp , yq )),
(7.15)
(7.16)
(7.17)
so the whole system can be expressed as
[K1] Ψ − [M1] N = b.
(7.18)
For the Dirichlet problem, Ψ = 0 and N is given by
N = −[M1]−1 b.
For the Neumann problem, N = 0 and Ψ is given by
Ψ = [K1]−1 b.
Coordinate Index
1
2
3
.
.
.
M
M+1
M+2
M+3
.
.
.
M ·N
p
1
1
1
.
.
.
1
2
2
2
.
.
.
N·
q
1
2
3
.
.
.
M
1
2
3
.
.
.
M
Table 1: A schematic representation of the labelling scheme to write the two dimensional
coordinate sampling indices p (N samples in x) and q (M samples in y). The total of M ·N
coordinate samples are strung out in a coordinate sample line running from 1 to M · N.
17
Spectral Index
1
2
3
.
.
.
m
m+1
m+2
m+3
.
.
.
n
j
jmin
jmin
jmin
.
.
.
jmin
jmin+1
jmin+1
jmin+1
.
.
.
jmax
j
j min
j min+1
j min+2
.
.
.
j max
j min
j min+1
j min+2
.
.
.
j max
Table 2: A schematic representation of the labelling scheme to write the total numbers
of real Bragg modes as n. The indices j and j each run from a minimum to a maximum
value for which the right hand sides of (7.9) and (7.10) do not exceed a magnitude of one
for m1 in (7.11). The mode j refers to the x−direction and j to the y−direction.
18
PART III: COMPUTATIONAL RESULTS FOR PERIODIC SURFACES
8
Outline of the Computational Results
In Sections 9 and 10 we present the computational results for scattering from many
examples of surfaces periodic in two dimensions. In Section 9 the results for the perfectly
reflecting Dirichlet boundary value problem are presented and in Section 10 the results for
the perfectly reflecting Neumann boundary value problem are presented. All the results
were generated using the two-dimensional spectral-coordinate (SC) method developed in
PART II. Computations were performed using MATLAB 5.2 on a Sun SPARC station
20.
The objective of this study was to provide results for a broad suite of parameter
values in order to assess the computational speed and accuracy of the SC method for
two-dimensional scattering. Various surface examples and cases were presented where the
surface periods, heights and slopes were varied as were the incident polar and azimuthal
angles. In addition, for fixed error, the maximum roughness (related to largest slope)
was studied as the incident polar angle was varied from 0◦ (vertical incidence) to 89.99◦
(near grazing incidence) and as the ratio of wavelength to surface period (both x−and
y−periods the same) varied over three orders of magnitude.
The number of spectral samples was generally fixed by the number of real Bragg
modes although we also present many examples where we increased the spectral sampling
by adding non-radiating (surface wave or evanescent) modes. To form a square matrix
in our SC development requires the same number of coordinate samples and we studied
the convergence behavior of increasingly larger square systems. Some non-square systems
where more coordinate than spectral samples were chosen are also presented. These latter
were extensively studied in [9].
19
9
Dirichlet Computations
In this section we present results for the perfectly reflecting Dirichlet boundary value
problem based on the theoretical development in Section 3. Four sinusoidal surface examples are presented in Secs. 9.1-9.4, the first three under near-grazing conditions and with
varying wavelength to period ratio. In Example 1 the wavelength was much less than the
surface periods. In Example 2 the wavelength was approximately equal to the surface
periods, and in Example 3 the wavelength was much greater than the surface periods.
Example 4 was similar to Example 1 except one period was twenty times the other.
In Sec. 9.5 we present an example where we vary the incident azimuthal angle from
0◦ to 90◦ . In Secs. 9.6 and 9.7 we fix the error in the computations to be less than −2
and plot the maximum value of roughness d/L with respect to incident polar angle θi
(Sec. 9.6) and wavelength to period ratio λ/L (Sec. 9.7). Some non-square systems were
included.
We present both tabular and visual representations of the computations. In the tables
the number of spectral and coordinate samples is indicated as well as the overall matrix
size. Tabular results were for square systems. The results in several figures were for nonsquare systems which were studied extensively in [9]. The number of real Bragg modes
fixes the number of real spectral samples. Occasionally we explore the effect of adding
additional non-radiating modes to the spectral sampling. Basically, for a square system,
the number of coordinate samples equals the number of spectral samples. If the system is
not square then the number of coordinate samples has been increased. The computational
procedure is outlined in Section 7. In addition, the tables contain sampling distances in
the x− and y− directions as a function of wavelength as well as the fill time (times in
seconds necessary to compute each term in the matrix), linear solution time (times in
seconds necessary to invert the matrix, compute the boundary unknown functions and
the resulting scattered field), the condition number of the matrix [1] and the error in the
computations (see Section 6 for the definition of error). The graphical presentation for
the examples consists in plotting the real part and magnitude of the surface boundary
unknown (here the normal derivative or surface current N) as well as the real part and
magnitude of the scattered field on hemispheres of radii R above the surface where R is
related to the surface period and on planes a distance H above the surface where H is
also related to the surface period. All are ploted with respect to a gray scale magnitude.
An example plot of spectral orders is also included.
20
9.1
Example 1: Near-Grazing Incidence/Reflection
The results in Table 3 and Figs. 2 and 3 are based on the following surface parameters:
S(x, y) = −(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = L2 = 1, d/L1 = d/L2 = 0.075,
λ/L1 = λ/L2 = 0.12. The number of radiating orders is 220, θi = 75◦ and φi = 15◦ . For
this example the wavelength was much less than the surface period. The scaled height
d/λ ≈ 0.625. All the examples in Table 3 contained evanescent waves. The matrix systems
were squared by the choice of the number of coordinate samples. For this example the
more evanescent modes the better the convergence. In Figure 2 well-defined peaks in the
magnitude of the normal derivative on the surface are clearly evident and the scattered
field in Figure 3 is characteristic of surfaces with large periods.
Matrix
Size
289 × 289
324 × 324
361 × 361
400 × 400
441 × 441
484 × 484
529 × 529
576 × 576
625 × 625
676 × 676
729 × 729
Number of Samples
Coord. Spectral
x y j
j
17 17 17
17
18 18 18
18
19 19 19
19
20 20 20
20
21 21 21
21
22 22 22
22
23 23 23
23
24 24 24
24
25 25 25
25
26 26 26
26
27 27 27
27
λ/∆x
2.0400
2.1600
2.2800
2.4000
2.5200
2.6400
2.7600
2.8800
3.0000
3.1200
3.2400
λ/∆y
2.0400
2.1600
2.2800
2.4000
2.5200
2.6400
2.7600
2.8800
3.0000
3.1200
3.2400
Fill
Time (s)
3.0300
3.6700
4.5600
5.8200
6.9900
8.7500
9.0700
11.4300
12.5100
14.9000
17.3100
Linear
Solution
Time (s)
5.4900
7.5800
10.2800
14.6800
19.7400
27.6500
31.1600
42.3800
53.5300
74.1900
88.7600
Condition
Number
12.7476
21.3189
31.3239
49.5966
70.9439
108.9295
153.3484
230.9881
321.8926
478.6422
662.4547
Error
-2.0498
-3.2254
-3.2300
-4.2416
-4.2266
-5.2112
-5.2294
-6.1944
-6.2196
-7.1775
-7.2064
Table 3: Parameters and computational results for the Dirichlet problem for Example
1. Angle parameters were chosen so that near-grazing incidence and reflection occurred.
Only square systems were included. Convergence was very good for all cases considered.
21
|N(x, y, S(x, y))|
Re[N(x, y, S(x, y)]
0.5
0.5
0.4
0.8
0.4
0.3
0.6
0.3
0.2
0.4
0.2
0.1
0.2
0.1
0
0
−0.1
−0.2
0.9
0.8
0.7
y
y
0
0.6
−0.1
0.5
−0.2
−0.4
−0.3
−0.6
−0.2
0.4
−0.3
−0.4
−0.4
−0.8
−0.5
−0.5
0
x
0.3
−0.5
−0.5
0.5
(a)
0
x
0.5
(b)
Figure 2: Example 1 for the Dirichlet problem and a matrix size of 729×729 with an error
of −7.2064. Real part (a) and magnitude (b) of the surface current or normal derivative
N.
22
√
SC
ψ
x, y, R2
√
Re ψ SC x, y, R2 − x2 − y 2 , R = 10L
10
1.5
− x2 − y 2 , R = 10L
10
1.5
8
8
1
6
6
4
4
0.5
y 0
0
y 0
−2
−2
−0.5
−4
−6
−1
0.5
−4
−6
−8
−8
−10
−10
1
2
2
−5
0
x
5
−10
−10
10
(a)
−5
0
x
5
10
0
(b)
Figure 3: Example 1 for the Dirichlet problem and a matrix size of 729 × 729. The real
part and magnitude of the scattered field plotted on a hemisphere of radius R = 10L in
(a) and (b). Here L = L1 = L2 and the resolution is 100 × 100.
23
9.2
Example 2: Near-Grazing Incidence/Reflection
The results in Table 4 and Figs. 4 and 5 are based on the following surface parameters:
S(x, y) = −(d/2) cos(2πx/L1 ) cos(2πy/L2 ), where L1 = L2 = 1, d/L1 = d/L2 = 0.25,
λ/L1 = λ/L2 = 0.95. The number of radiating orders is 4, θi = 75◦ and φi = 15◦ .
This example has much larger slopes (d/L1 ) than Example 1, but smaller scaled heights
(d/λ) and the wavelength was approximately equal to a surface period. All the examples
in Table 4 contained evanescent modes but, unlike Example 1, here adding many more
evanescent modes did not improve the convergence. The best convergence was obtained
by the first example in the Table, the one with the fewest evanescent modes. There were
again well-defined maxima for the surface normal derivative evident in Figure 4, and they
were much sharper than those in Figure 2. The scatter field pattern in Figure 5 was
characteristic of surfaces where the period was approximately equal to the wavelength.
Matrix
Size
6×6
16 × 16
36 × 36
64 × 64
100 × 100
144 × 144
196 × 196
256 × 256
324 × 324
400 × 400
484 × 484
Number of Samples
Coord. Spectral
x y j
j
3 2 3
2
4 4 4
4
6 6 6
6
8 8 8
8
10 10 10
10
12 12 12
12
14 14 14
14
16 16 16
16
18 18 18
18
20 20 20
20
22 22 22
22
λ/∆x
2.8500
3.8000
5.7000
7.6000
9.5000
11.4000
13.3000
15.2000
17.1000
19.0000
20.9000
λ/∆y
1.9000
3.8000
5.7000
7.6000
9.5000
11.4000
13.3000
15.2000
17.1000
19.0000
20.9000
Fill
Time (s)
0.0300
0.0700
0.1800
0.3600
0.6600
1.1100
1.7500
2.7700
4.0700
5.8500
8.2900
Linear
Solution
Time (s)
0.0100
0.0100
0.0300
0.1100
0.2900
0.8200
1.8200
3.9000
7.8900
13.7300
23.8900
Condition
Number
1.0000
4.3221
32.4522
258.9573
2.0444·103
1.6185·104
1.2956·105
1.0499·106
8.6189·106
7.1485·107
5.9908·108
Error
< −15.9
-2.8789
-3.5466
-3.7416
-3.8465
-3.8987
-3.8942
-3.8663
-3.7994
-3.7253
-3.6150
Table 4: Parameters and computational results for the Dirichlet problem for Example
2. Angle parameters were chosen so that near-grazing incidence and reflection occurred.
Only square systems were considered. Note that although in principle there are 6 possible
combinations of j and j values which radiate, only four modes in fact radiate. The
wavelength λ was approximately equal to the two (equal) periods. Convergence was still
very good.
24
|N(x, y, S(x, y))|
Re[N(x, y, S(x, y)]
0.5
0.5
0.9
0.6
0.4
0.4
0.3
0.4
0.3
0.2
0.2
0.2
0.8
0.7
0.6
0.1
0.1
0
y
y
0
0
0.5
−0.2
−0.1
−0.1
−0.2
−0.4
−0.2
−0.3
−0.6
−0.3
0.4
0.3
0.2
−0.4
−0.4
−0.8
−0.5
−0.5
−0.4
−0.3
−0.2
−0.1
0
x
0.1
0.2
0.3
0.4
−0.5
−0.5
0.5
−0.4
−0.3
−0.2
−0.1
0
x
(a)
0.1
0.2
0.3
0.4
0.5
0.1
(b)
Figure 4: Example 2 for the Dirichlet problem. Real part (a) and magnitude (b) of the
surface current or normal derivative N generated with a matrix of size 484 × 484. and an
error of −3.6150.
√
SC
ψ
x, y, R2
√
Re ψ SC x, y, R2 − x2 − y 2 , R = 10L
10
− x2 − y 2 , R = 10L
10
1.1
8
1.08
6
1.06
4
1.04
2
1.02
1
8
0.8
6
0.6
4
0.4
2
0.2
y 0
y 0
0
−0.2
−2
−0.4
−4
−0.6
−6
1
−2
0.98
−4
0.96
−6
0.94
−8
0.92
−0.8
−8
−1
−10
−10
−8
−6
−4
−2
0
x
2
4
6
8
−10
−10
10
(a)
−8
−6
−4
−2
0
x
2
4
6
8
10
0.9
(b)
Figure 5: Example 2 for the Dirichlet problem with a matrix size of 484 × 484. Real part
(a) and magnitude (b) of the scattered field plotted on hemispheres of radius R = 10L
where L = L1 = L2 . The resolution is 100 × 100.
25
9.3
Example 3: Near-Grazing Incidence/Reflection
The results in Table 5 and Fig. 6 are based on the following surface parameters: S(x, y) =
−(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = L2 = 1, d/L1 = d/L2 = 2.5, λ/L1 = λ/L2 =
100. There is one radiating order, θi = 75◦ and φi = 15◦ . This example has much larger
slopes (d/L1 ) than either Example 1 or Example 2, but smaller scaled heights (d/λ).
For this example the wavelength was much greater than the surface period. The
first example in Table 5 with no evanescent modes produced the best result. Adding a
few evanescent modes still produced good error results, but the error deteriorated as a
larger number of evanescent modes were added. The highly patterned surfaces normal
derivative is illustrated in Figure 6 (note the scale) as is the scattered field interference
pattern characteristic of surfaces with very small periods. For our choice of parameters it
is obvious that only one radiating order is present. The example was included to illustrate
the fact that the code produced this single order with extremely high accuracy.
Matrix
Size
1×1
4×4
9×9
16 × 16
25 × 25
36 × 36
49 × 49
64 × 64
81 × 81
100 × 100
121 × 121
144 × 144
169 × 169
196 × 196
225 × 225
256 × 256
Number of Samples
Coord. Spectral
x y j
j
1 1 1
1
2 2 2
2
3 3 3
3
4 4 4
4
5 5 5
5
6 6 6
6
7 7 7
7
8 8 8
8
9 9 9
9
10 10 10
10
11 11 11
11
12 12 12
12
13 13 13
13
14 14 14
14
15 15 15
15
16 16 16
16
λ/∆x
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
λ/∆y
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
Fill
Time (s)
0.0100
0.0200
0.0400
0.0800
0.1200
0.1800
0.2600
0.3600
0.5000
0.6500
0.8800
1.1200
1.4500
1.7600
2.2300
2.7200
Linear
Solution
Time (s)
< 10−3
< 10−3
0.0100
< 10−3
0.0100
0.0300
0.0500
0.1000
0.1700
0.2900
0.5200
0.8000
1.2700
1.9500
2.7300
4.0700
Condition
Number
1.0000
1.0000
3.9429·106
3.2356·108
4.6186·1013
1.5418·1018
1.8373·1020
1.3722·1023
8.8589·1027
2.7528·1031
4.5615·1033
3.9242·1037
6.4054·1039
8.3684·1043
3.3895·1046
1.1279·1050
Error
< −15.9
< −15.9
-15.3525
-15.6536
< −15.9
-14.5744
-15.3525
-15.3525
-15.6536
-15.6536
-13.9376
-14.2064
-11.4572
-10.5530
-12.6747
-9.0324
Table 5: Parameters and computational results for the Dirichlet problem for Example
3. Angle parameters were chosen to include near-grazing incidence and reflection. Only
square systems were considered. The wavelength λ is much greater than either of the
(equal) surface periods and only one radiating mode exists. The errors were extremely
small even for every large condition numbers.
26
|N(x, y, S(x, y))|
Re[N(x, y, S(x, y)]
0.5
0.5
0.4
0.4
0.4
0.5
0.3
0.3
0.2
0.2
0.2
y
0.4
0.1
0.1
0
0
y
0.3
0
−0.1
−0.1
−0.2
0.2
−0.2
−0.2
−0.3
−0.3
−0.4
0.1
−0.4
−0.4
−0.5
−0.5
0
x
0.5
−0.6
−0.5
−0.5
0
x
0.5
(a)
(b)
√
Re ψ SC x, y, R2 − x2 − y 2 , R = 100L
Re [ψ SC (x, y, H)] , H = 100L
100
1
1
150
80
0.8
60
0.6
40
0.4
20
0.2
0
0
−20
−0.2
−40
−0.4
−60
−0.6
−80
−0.8
0
0.8
0.6
100
0.4
50
y
0.2
y
0
0
−0.2
−50
−100
−100
−50
0
x
50
100
−1
(c)
−0.4
−100
−0.6
−0.8
−150
−150
−100
−50
0
x
50
100
150
−1
(d)
Figure 6: Example 3 for the Dirichlet problem. Real part (a) and magnitude (b) of the
surface current or normal derivative N generated with a matrix size of 25 × 625 and error
of −3.5373. Real part of the scattered field plotted on a hemisphere (c) of radius R = 100L
and on a plane (d) at height H = 100L above the surface. Here L = L1 = L2 . The matrix
size was 121 × 121 and the images have a resolution of 100 × 100.
27
9.4
Example 4: L1 >> L2
The results in Table 6 and Figs. 7 and 8 are based on the following surface parameters: S(x, y) = −(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = 1, L2 = 0.05, d/L1 =
0.075, d/L2 = 0.150, λ/L1 = 0.12 and λ/L2 = 2.40. The number of radiating orders
is 17, θi = 20◦ and φi = 15◦ .
For this example the period in the x-direction is twenty times the period in the ydirection. In Table 6 it can be seen that adding a few evanescent modes produced the
best results (second line in Table 6), but the addition of more evanescent modes caused
the solution to deteriorate. Figure 7 illustrates the fact that the non-zero values of the
normal derivative on the surface are confined to a very small surface area. The different
surface periods also produced a blurred scattered filed as is evident in Figure 8.
Matrix
Size
17 × 17
36 × 36
57 × 57
80 × 80
105 × 105
132 × 132
161 × 161
192 × 192
225 × 225
260 × 260
297 × 297
289 × 289
324 × 324
361 × 361
400 × 400
Number of Samples
Coord. Spectral
x y j
j
17 1 17
1
18 2 18
2
19 3 19
3
20 4 20
4
21 5 21
5
22 6 22
6
23 7 23
7
24 8 24
8
25 9 25
9
26 10 26
10
27 11 27
11
17 17 17
17
18 18 18
18
19 19 19
19
20 20 20
20
λ/∆x
2.0400
2.1600
2.2800
2.4000
2.5200
2.6400
2.7600
2.8800
3.0000
3.1200
3.2400
2.0400
2.1600
2.2800
2.4000
λ/∆y
0.1200
0.2400
0.3600
0.4800
0.6000
0.7200
0.8400
0.9600
1.0800
1.2000
1.3200
2.0400
2.1600
2.2800
2.4000
Fill
Time (s)
0.0800
0.1900
0.3200
0.5500
0.8000
1.0400
1.4000
1.8300
2.3900
3.0100
3.7200
3.7800
4.3700
5.4500
6.4100
Linear
Solution
Time (s)
0.0200
0.0300
0.0900
0.2000
0.3800
0.7300
1.2100
1.9600
3.0300
4.5800
6.4100
6.2800
8.3800
11.3300
14.8600
Condition
Number
1.6319
1.0000
786.0448
2.0248·104
7.2357·105
3.3771·108
4.3818·109
5.4034·1010
4.1184·1011
2.4100·1013
1.2410·1014
7.7568·1024
2.0658·1026
3.9084·1027
4.9564·1028
Error
-4.9707
< −15.9
-2.5743
-2.7686
-1.3685
-1.0049
-1.6574
-1.6080
-1.6731
-1.6480
-1.5578
-0.5850
-1.4543
1.2890
-0.8009
Table 6: Parameters and computational results for the Dirichlet problem for Example
4, where L1 = 20L2 . Only square systems were considered. Convergence was better for
smaller systems in this case of roughness values dramatically different in the two directions.
28
|N(x, y, S(x, y))|
Re[N(x, y, S(x, y)]
0.025
0.025
0.02
500
0.02
200
0.015
450
0.015
400
100
0.01
0.01
350
0.005
y
300
y
0
−100
−0.005
−0.01
−200
−0.015
−300
−0.02
−0.025
−0.5
0.005
0
0
x
0
(a)
200
−0.01
150
−0.015
100
−0.02
50
−0.025
−0.5
0.5
250
−0.005
0
x
0.5
0
(b)
Figure 7: Example 4 for the Dirichlet problem. Real part (a) and magnitude (b) of the
surface current or normal derivative N with added non-radiating orders of 3 rows in y,
above and below, and 1 column in x, left and right for a matrix size of 133 × 133 (where
133 = 19 × 7 = jj ).
29
√
SC
ψ
x, y, R2
√
Re ψ SC x, y, R2 − x2 − y 2 , R = 2L1
− x2 − y 2 , R = 2L1
2
2
1.8
1.5
1.5
1.5
1
1
1
1.4
0.5
0.5
0.5
1.2
0
0
1.6
y
y
1
0
0.8
−0.5
−0.5
−0.5
−1
−1
−1
0.6
−1.5
−2
−2
−1.5
−1.5
−1
−0.5
0
x
0.5
1
1.5
−2
−2
2
(a)
0.4
−1.5
0.2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
0
(b)
Figure 8: Example 4 for the Dirichlet problem and a matrix size of 57 × 57. Real part (a)
and magnitude (b) of the scattered field viewed on hemispheres of radius R = 2L1 with
a resolution of 100 × 100.
30
9.5
Suite of φi values (azimuthal angles of incidence)
The results in Table 7 and Fig. 9 are based on the following surface parameters: S(x, y) =
−(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = L2 = 1, d/L1 = d/L2 = 0.075, λ/L1 =
λ/L2 = 0.12 and θi = 75◦ . The number of coordinate samples in x and y directions is 19.
Here the periods were much greater than the wavelength and the polar angle of incidence
was near-grazing. In Table 7 the azimuthal angle of incidence was varied from 0◦ to 90◦
which produced different numbers of radiating orders (real Bragg modes). Some of the
resulting matrix systems were not square. Good results in terms of error occurred at
nearly all angles. In Figure 9 we illustrate the shift in the alignment of the real spectral
orders for the cases of φi = 0◦ and φi = 90◦ .
The SC method is seen to be a stable and robust computational method over the
entire range of incident azimuthal angles.
Matrix
Size
361 × 361
361 × 361
342 × 361
361 × 361
342 × 361
342 × 361
361 × 361
324 × 361
361 × 361
361 × 361
361 × 361
324 × 361
361 × 361
342 × 361
342 × 361
361 × 361
342 × 361
361 × 361
361 × 361
φi
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
Radiating
Orders
221
220
220
220
217
217
221
213
220
216
220
213
221
217
217
220
220
220
221
Spectral
Sampling
j
j
19 19
19 19
19 18
19 19
18 19
19 18
19 19
18 18
19 19
19 19
19 19
18 18
19 19
18 19
19 18
19 19
18 19
19 19
19 19
Fill
Time (s)
4.5200
4.5400
4.1900
4.5400
4.5700
4.2100
4.5200
4.0100
4.7800
4.6900
4.8400
4.2800
4.8500
4.5100
4.2200
4.4200
4.1900
4.4400
4.4200
Linear
Solution
Time (s)
10.1100
10.1400
449.8600
10.4000
481.1000
473.0100
10.8000
403.6700
11.2700
11.7900
11.0400
454.1000
11.3000
500.8500
440.8000
10.1700
440.3000
10.1400
10.1300
Condition
Number
31.0986
31.4446
16.7692
31.3239
16.9834
17.0979
31.0950
9.8004
31.3299
31.8821
31.3299
9.8004
31.0950
17.0979
16.9834
31.3239
16.7692
31.4446
31.0986
Error
-3.0615
-3.0616
-3.1014
-3.2300
-1.9265
-4.2997
-4.2530
-3.1581
-4.9944
-4.7223
-4.9944
-3.1581
-4.2530
-4.2997
-1.9265
-3.2300
-3.1014
-3.0616
-3.0615
Table 7: Parameters and computational results for the Dirichlet problem where the incident azimuthal angle φi varies. The spectral sampling includes one extra row or column
of non-radiating orders added to each side of the set of radiating orders. Convergence was
good for all cases. The polar angle of incidence θi was near grazing.
31
Spectral Orders, φi = 0◦
j’
Spectral Orders, φi = 90◦
10
2
8
0
6
−2
4
−4
2
−6
0
j’
−8
−2
−10
−4
−12
−6
−14
−8
−16
−10
−20
−15
−10
j
−5
−18
0
(a)
−10
−5
0
j
5
10
(b)
Figure 9: Spectral orders for incident angles (a) φi = 0◦ and (b) φi = 90◦ illustrating the
shift of the radiating orders where stars refer to radiating modes, and dots to non-radiating
modes. There were 19 samples in both j and j .
9.6
Maximum Roughness with respect to incident polar angle
θi
The results in Table 8 and Fig. 10 are based on the following surface parameters: S(x, y) =
−(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = L2 = 1, λ/L1 = λ/L2 = 0.20, and φi = 50◦ .
Here the periods are five times the wavelength. In Table 8 we define four different spectral
sampling schemes to illustrate the different results of maximum roughness (d/L) as a
function of incident polar angle θi plotted in Figure 10. The best overall results were
achieved by adding a non-radiating row. The error was fixed at less than −2. The SC
method yielded very large increases in maximum roughness near grazing (also see Figs.
15 and 16).
Line Type
Dash Dot
Solid
Dashed
Dotted
Added Non-radiating Rows
Matrix Size and Columns on all Sides
100 × 100
0
144 × 144
1
196 × 196
2
256 × 256
3
Table 8: Parameters for the Dirichlet example chosen to illustrate the maximum roughness
d/L (L = L1 = L2 ) of the surface S(x, y) with respect to incident polar angle θi for error
fixed at less than −2. Four different sampling schemes are illustrated and the results are
plotted in Fig. 10. The different sampling schemes consist in adding non-radiating modes
to the spectral sampling. Sampling is symmetric in x and y, and j and j .
32
0.28
0.26
0.24
0.22
d/L
0.2
0.18
0.16
0.14
0.12
0.1
0
10
20
30
i
40
θ (deg)
50
60
70
80
90
Figure 10: Maximum roughness for fixed error less than −2 with respect to incident
polar angle θi for the Dirichlet problem. Data near θi = 0◦ is taken at .01◦ and data
near θi = 90◦ is taken at 89.99◦. The sampling schemes are explained in Table 8. The
best overall results were achieved by adding a non-radiating row in the spectral sampling
(solid curve). Large increases in maximum roughness were noted near grazing incidence
for several examples.
33
9.7
Maximum Roughness with respect to λ/L
The results in Table 9 and Fig. 11 are based on the following surface parameters: S(x, y) =
−(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = L2 = 1, θi = 20◦ , and φi = 15◦ . In Table 9
four different spectral sampling schemes are illustrated. The schemes are different from
others we have illustrated in that specific orders are added in either the forward or back
scatter directions or in both. The larger increase in maximum roughness illustrated in
Figure 11 occured as the number of real propagating Bragg modes decreased, i.e. as λ/L
approached one. The fewer the number of real propagating Bragg modes the larger the
value of the maximum roughness that could be successfully computed within a fixed error.
Line Type j left j right j Dash Dot 0
0
Solid 1
1
Dashed 2
2
Dotted 3
3
above j 0
1
2
3
below
0
1
2
3
Table 9: Parameters for the Dirichlet example chosen to illustrate the maximum roughness
d/L (L = L1 = L2 ) of the surface S(x, y) with respect to λ/L, the ratio of the wavelength
to surface period. Four different sampling schemes are illustrated and the results are
plotted in Fig. 11. The schemes refer to additional spectral orders added in the left (or
back scatter) direction or right (forward) scattered direction. Sampling is symmetric in x
and y, and j and j .
34
2
10
1
10
d/L
0
10
−1
10
−1
10
0
10
1
λ /L
10
2
10
Figure 11: Maximum roughness for fixed error less than −2 with respect to λ/L (L =
L1 = L2 ) for the Dirichlet problem. The spectral sampling schemes are explained in
Table 9. Dash-dot line extends above graph to unknown height. The maximum roughness
dramatically increased as λ/L approached 1 from below as the number of real Bragg modes
decreased.
35
newpage
10
Neumann Computations
In this section we present results for the perfectly reflecting Neumann boundary value
problem. The theoretical development is summarized in Section 3. Three sinusoidal
surface examples are presented in Secs. 10.1-10.3, two under near-grazing incidence and
reflection conditions. In Example 1, the wavelength was much less than the surface
periods. In Example 2 the wavelength was approximately equal to the surface periods,
and in Example 3 the wavelength was much greater than the surface periods. In addition,
in Secs. 10.4 and 10.5 we present studies of the maximum roughness (d/L) for error less
than −2 as a function of incident polar angle (Sec. 10.4) and wavelength to period ratio
(Sec. 10.5).
10.1
Example 1: Near-Grazing Incidence/Reflection
The results in Table 10 and Fig. 12 are based on the following surface parameters:
S(x, y) = −(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = L2 = 1, d/L1 = d/L2 = 0.075,
λ/L1 = λ/L2 = 0.12. The number of radiating orders is 220, θi = 75◦ and φi = 15◦ . For
this example the wavelength was much less than the surface period. The scaled height
d/L = 0.625. All the examples in Table 10 contained evanescent waves and the matrix
system was squared by the choice of coordinate samples. Convergence was poor for a
small number of evanescent modes but improved by adding more modes. The convergence was not as good as for the Dirichlet problem (compare with Table 3). The surface
field in Figure 12 was similar in appearance but not in magnitude to the surface normal
derivative in the Dirichlet problem (compare to Figure 2).
36
Matrix
Size
289 × 289
324 × 324
361 × 361
400 × 400
441 × 441
484 × 484
529 × 529
576 × 576
625 × 625
676 × 676
729 × 729
Number of Samples
Coord. Spectral
x y j
j
17 17 17
17
18 18 18
18
19 19 19
19
20 20 20
20
21 21 21
21
22 22 22
22
23 23 23
23
24 24 24
24
25 25 25
25
26 26 26
26
27 27 27
27
λ/∆x
2.0400
2.1600
2.2800
2.4000
2.5200
2.6400
2.7600
2.8800
3.0000
3.1200
3.2400
λ/∆y
2.0400
2.1600
2.2800
2.4000
2.5200
2.6400
2.7600
2.8800
3.0000
3.1200
3.2400
Fill
Time (s)
4.3300
5.1000
6.1700
7.3400
8.5800
10.3200
11.9700
14.0500
16.2600
18.7300
21.5900
Linear
Solution
Time (s)
5.5800
7.6100
10.3800
13.7800
18.2600
25.4000
31.3500
42.5200
53.4500
73.3500
86.3000
Condition
Number
86.0076
89.5040
87.4948
130.8250
167.8446
222.3018
276.9816
366.8100
452.7573
588.2078
801.8589
Error
-0.4139
-2.5973
-1.6433
-3.6024
-3.1785
-4.1685
-3.6104
-4.9838
-4.4744
-5.8107
-5.3146
Table 10: Parameters and computational results for the Neumann problem for Example
1. Angle parameters are chosen to include grazing incidence and reflection. Only square
systems were considered. The parameters are the same as for the Dirichlet problem in
Table 3. The results are comparable with both condition numbers and errors slightly
larger here. The wavelength λ is much less than either of the (equal) surface periods.
Convergence improved by adding evanescent waves.
37
|ψ(x, y, S(x, y))|
Re[ψ(x, y, S(x, y)]
0.5
0.5
2.5
0.4
2
0.3
1
0.2
0.1
0.5
0.1
0
0
−0.1
−0.5
−0.1
−0.2
−1
−0.2
−0.3
−1.5
−0.3
−2
−0.4
−0.4
−0.5
−0.5
−2.5
0
x
0.5
(a)
2.4
0.3
1.5
0.2
y
2.6
0.4
y
2.2
2
0
−0.5
−0.5
1.8
1.6
1.4
0
x
0.5
(b)
Figure 12: Example 1 for the Neumann problem with a matrix size of 729 × 729 and an
error of −5.3146. Real part (a) and magnitude (b) of the total field on the surface.
38
10.2
Example 2: No Grazing
The results in Table 11 and Fig. 13 are based on the following surface parameters:
S(x, y) = −(d/2) cos(2πx/L1 ) cos(2πy/L2 ), where L1 = L2 = 1, d/L1 = d/L2 = 0.25,
λ/L1 = λ/L2 = 0.95. The number of radiating orders is 3, θi = 20◦ and φi = 15◦ . For this
example the wavelength was approximately equal to the surface period. All the examples
in Table 12 contained evanescent waves. Although all the examples converged very well,
those with the smallest number of evanescent modes converged the best. All matrix systems were squared by the choice of coordinate samples. The three radiating orders are
evidenced in banding of the surface field in Fig. 13.
Matrix
Size
4×4
16 × 16
36 × 36
64 × 64
100 × 100
144 × 144
196 × 196
256 × 256
324 × 324
400 × 400
484 × 484
Number of Samples
Coord. Spectral
x y j
j
2 2 2
2
4 4 4
4
6 6 6
6
8 8 8
8
10 10 10
10
12 12 12
12
14 14 14
14
16 16 16
16
18 18 18
18
20 20 20
20
22 22 22
22
λ/∆x
1.9000
3.8000
5.7000
7.6000
9.5000
11.4000
13.3000
15.2000
17.1000
19.0000
20.9000
λ/∆y
1.9000
3.8000
5.7000
7.6000
9.5000
11.4000
13.3000
15.2000
17.1000
19.0000
20.9000
Fill
Time (s)
0.0200
0.1700
0.4100
0.8400
1.3400
2.2300
3.3900
3.8900
5.5900
8.0100
11.7000
Linear
Solution
Time (s)
0.0200
0.0100
0.0300
0.1300
0.3700
1.0100
2.5000
4.0100
7.8200
15.2300
26.8200
Condition
Number
2.6463
86.5420
191.9474
4.8087·103
2.5102·104
2.1760·105
5.9507·106
3.4245·107
5.9760·108
1.3403·1010
3.0515·1010
Error
< −15.9
-15.0003
-12.2171
-14.4775
-11.3225
-11.1530
-10.7172
-12.7071
-10.9631
-8.9165
-10.6510
Table 11: Parameters and computational results for the Neumann problem for Example
2. Angle parameters were chosen so that grazing incidence and reflection are avoided.
Only square systems were considered. The results are comparable with both condition
numbers and errors slightly larger here. The wavelength λ is much less than either of the
two equal surface periods. Note that the surface is much rougher than Example 1.
39
|ψ(x, y, S(x, y))|
Re[ψ(x, y, S(x, y)]
0.5
0.5
5.5
0.4
5
0.4
0.3
4
0.3
5
4.5
0.2
0.2
4
3
0.1
0.1
y
2
0
−0.1
1
y
3.5
3
0
2.5
−0.1
2
−0.2
−0.2
1.5
0
−0.3
−0.3
1
−1
−0.4
−0.5
−0.5
0
x
−0.4
−0.5
−0.5
0.5
(a)
0.5
0
x
0.5
(b)
Figure 13: Example 2 for the Neumann problem with matrix size of 64 × 625 and an error
of −1.0708. Real part (a) and magnitude (b) of the total field on the surface.
40
10.3
Example 3: Near-Grazing Incidence/Reflection
The results in Table 12 and Fig. 14 are based on the following surface parameters:
S(x, y) = −(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = L2 = 1, d/L1 = d/L2 = 2.5,
λ/L1 = λ/L2 = 100. There is one radiating order, θi = 75◦ and φi = 15◦ . For this example the wavelength was one hundred times the surface period and the result was only
one radiating order. The slopes are very large as is the ratio of wavelength to surface
period. From Table 12 we observe that convergence was best with no evanescent orders
or with only a few added. It deteriorated as larger numbers of evanescent orders were
added although the convergence was still quite good. The surface field is plotted in Fig.
14.
Matrix
Size
1×1
4×4
9×9
16 × 16
25 × 25
36 × 36
49 × 49
64 × 64
81 × 81
100 × 100
121 × 121
Number of
Coordinate
x
y
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
Samples
Spectral
j j
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 10
11 11
λ/∆x
100
200
300
400
500
600
700
800
900
1000
1100
λ/∆y
100
200
300
400
500
600
700
800
900
1000
1100
Linear
Fill
Solution Condition
Time (sec) Time (sec) Number
0.0200
0.0100
1.0000
0.0300
0.0100
549.6318
0.0800
0.0100
2.4487·107
−3
0.1400
< 10
1.9106·1010
0.2200
0.0200 2.8038·1013
0.3300
0.0400 4.4693·1018
0.5000
0.0600 4.6066·1019
0.6500
0.1100 5.1819·1024
0.8600
0.2000 6.7276·1026
1.1000
0.3200 2.6052·1031
1.4300
0.5700 1.1520·1033
Error
< −15.9
< −15.9
-11.4872
-15.0515
-10.2382
-12.6217
-10.9173
-12.3248
-10.6541
-12.0901
-10.2995
Table 12: Parameters and computational results for the Neumann problem for Example
3. Angle parameters are chosen so that grazing incidence and reflection are included.
Only square systems were considered. The parameters are the same as for the Dirichlet
problem in Table 5. The results are comparable.
41
|ψ(x, y, S(x, y))|
Re[ψ(x, y, S(x, y)]
0.5
0.5
0.25
0.4
0.4
0.25
0.2
0.3
0.3
0.15
0.2
0.2
0.2
0.1
0.1
y
0.1
0.05
y
0.15
0
0
0
−0.1
−0.05
−0.1
−0.2
−0.1
−0.2
−0.3
−0.15
−0.3
0.1
−0.2
−0.4
0.05
−0.4
−0.25
−0.5
−0.5
0
x
−0.5
−0.5
0.5
(a)
0
x
0.5
0
(b)
Figure 14: Example 3 for the Neumann problem with matrix size of 25 × 729 and an error
of −1.7037. Real part (a) and magnitude (b) of the total field ψ on the surface.
42
10.4
Maximum Roughness with respect to incident polar angle
θi
The results in Table 13 and Figs. 15 and 16 are based on the following surface parameters:
S(x, y) = −(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = L2 = 1, λ/L1 = λ/L2 = 0.20, and
φi = 50◦ . Here the wavelength was one fifth the surface period. In Table 13 we illustrate
four different sampling schemes used to determine the maximum roughness (d/L) as a
function of incident polar angle θi . The latter is plotted in Figure 15 with an expanded
view near 90◦ plotted in Fig. 16. Large increase in maximum roughness were also observed
for the Dirichlet problem (see Fig. 10). Spectral based methods work efficiently and well
near grazing incidence.
Line Type
Dash Dot
Solid
Dashed
Dotted
Added Non-radiating Rows
Matrix Size and Columns on all Sides
100 × 100
0
144 × 144
1
196 × 196
2
256 × 256
3
Table 13: Parameters for the Neumann example chosen to illustrate the maximum roughness d/L (L = L1 = L2 ) of the surface S(x, y) with respect to incident polar angle θi for
fixed error less than −2. Four different sampling schemes are illustrated and the results
are plotted in Fig. 15. The different sampling schemes consist in adding non-radiating
modes to the spectral sampling. Sampling is symmetric in x and y, and j and j .
43
0.22
0.2
0.18
0.16
0.14
d/L
0.12
0.1
0.08
0.06
0.04
0
10
20
30
40
50
i
θ (deg)
60
70
80
90
Figure 15: Maximum roughness for error less than −2 with respect to the incident polar
angle θi for the Neumann problem. Data near θi = 0◦ is taken at .01◦ and data near
θi = 90◦ is taken at 89.99◦ . The sampling schemes are explained in Table 13. The best
overall results were achieved by adding at least one non-radiating row in the spectral
sampling. An expanded view near 90◦ is illustrated in Fig. 16. The roughness values were
not as large as those for the Dirichlet example (compare with Fig. 10). As in the Dirichlet
example, large increases in maximum roughness were noted near grazing incidence.
44
0.7
0.6
0.5
0.4
d/L
0.3
0.2
0.1
0
86
86.5
87
87.5
88
i
θ (deg)
88.5
89
89.5
90
Figure 16: Maximum roughness for error less than −2 with respect to incident polar
angle θi for the Neumann problem. Expanded view near θi = 90◦ of Fig. 15. Data
near θi = 90◦ is taken at 89.99◦. Large increases in maximum roughness were noted near
grazing incidence for the Dirichlet problem also (see Fig. 10).
45
10.5
Maximum Roughness with respect to λ/L
The results in Table 14 and Fig. 17 are based on the following surface parameters:
S(x, y) = −(d/2) cos(2πx/L1 ) cos(2πy/L2), where L1 = L2 = 1, θi = 20◦ and φi = 15◦ .
In Table 14 we illustrate different spectral sampling schemes similar to those discussed
for the Dirichlet problem in Table 9. Specific spectral orders are added in either forward
or backscatter directions or in both direction. In Figure 17 the increase in maximum
roughness (for fixed error= −2) as λ/L approaches one is due to the fact that fewer real
propagating modes exist. The fewer the number of real Bragg modes the larger the rough
surface whose scattering could be computed within fixed error.
Line Type j left j right j Dash Dot 0
0
Solid 1
1
Dashed 2
2
Dotted 3
3
above j 0
1
2
3
below
0
1
2
3
Table 14: Parameters for the Neumann example chosen to illustrate the maximum roughness d/L (L = L1 = L2 ) of the surface S(x, y) with respect λ/L, the ratio of the wavelength to surface period. Four different sampling schemes are illustrated and the results
are plotted in Fig. 17. The schemes refer to additional spectral orders added in the left
(or back scatter) direction or right (forward) scattered direction or added values in the
vertical direction (above and below). Sampling is symmetric in x and y, and j and j .
46
2
10
1
10
d/L
0
10
−1
10
−1
10
0
10
1
λ /L
10
2
10
Figure 17: Maximum roughness for error less than −2 with respect to λ/L (L = L1 = L2 )
for the Neumann problem. The spectral sampling schemes are explained in Table 14.
Dash-dot line extends above graph to unknown height. The maximum roughness dramatically increased as λ/L approached 1 from below due to the decrease in the number of
real Bragg modes.
47
11
Summary and Conclusions
We presented theoretical and computational results to describe the scattering from a twodimensional periodic surface. The equations used to describe the scattering process were
found using a reduction of the equations for an infinite surface. They were in a mixed
spectral-coordinate (SC) representation.
Calculations were presented for both perfectly reflecting Dirichlet and Neumann examples. The computations were extensive, involving not only surfaces of different roughness
under conditions of no grazing and near-grazing incidence and reflection as well as considerable variability in the incident angle and the wavelength to period ratio.
The general conclusions are straightforward. The method is very fast as evidenced
by the fill time of the matrix. Additional time savings can occur if different matrix
solution methods are employed. We only used Gaussian (row reduction) or pseudo-inverse
methods. The method is stable, robust, and accurate over (a) the entire range of azimuthal
angles of incidence, (b) wavelength to period ratios over three orders of magnitude, and (c)
polar incidence angles down to extreme near grazing. Our computations were limited only
by large values of surface slopes and we quantified these limitations. The computational
results presented were a representative selection of the extensive computational results in
[9].
Spectral methods can thus play a role in scattering from two-dimensional periodic
surfaces. In particular they are very efficient and accurate when angles of incidence and
reflection are near grazing.
Acknowledgements
Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials
Command, USAF, under the Multi-University Research Initiative (MURI) program Grant
# F49620-96-1-0039.
Erdmann’s research was supported in part by an Undergraduate Research Grant from
the Colorado Advanced Software Institute (CASI) and a Grant-in-Aid of Research from
Sigma Xi, The Scientific Research Society.
We are grateful to Mr. Guy Somberg and Mr. Douglas Baldwin for technical assistance
in the production of this paper.
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