Revista de Matema´tica: Teor´ıa y Aplicaciones 2005 12(1 & 2) : 121–128
cimpa – ucr – ccss issn: 1409-2433
scattering of e polarized whispering-gallery
mode from concave boundary
Alexander P. Anyutin∗ V.I. Stasevich†
Received/Recibido: 17 Feb 2004
Abstract
In this work we present numerical results for the 2D problem of scattering E po-
larised whispering-gallery mode from concave convex perfectly conducting boundary.
The results were obtained by applying the developed method of currants integral
equations (CIE) [6,7] for high frequency domain when the size of the scatterer match
is greater than the wave length. We have applied the described procedure in order
to find numerical solutions of scattering whispering-gallery mode by concave finite
convex boundary as a part of a circular cylinder or part of parabolic cylinder. As
incident wave we have considered cylindrical waves from line source and Gauss beam
[6] with different effective width. It is shown that we have a complicated process of
focusing and oscillating of the beam’s reflected field, both cylindrical and Gauss beam
incident fields. The distortions of reflected field depend on shape of the boundary and
parameters of the incident fields.
Keywords: Scattering problem, E polarized whispering-gallery mode, concave boundary,
integral equation of the first kind, numerical solution.
Resumen
En este trabajo presentamos resultados nume´ricos para el problema 2D de modo
de galer´ıa susurrante E polarizada, de una frontera perfectamente conducente co´ncava
y convexa. Los resultados fueron obtenidos aplicando el me´todo desarrollado de ecua-
ciones integrales pasas [6,7] para dominios de alta frecuencia cuando el taman˜o de
la pareja dispersora es mayor que la longitud de onda. Hemos aplicado el proced-
imiento descrito con tal de encontrar soluciones nume´ricas de modo galer´ıa susurrante
de dispersio´n por frontera convexa finita co´ncava, como parte de un cilindro circular
∗Russian New University, Radio Street 22, 107005 Moscow, Russia. E-Mail: anioutine@mtu-net.ru.
†Russian New University 60-2-94, Novocheremushkinskaya UI 117420 Moscow, Russia. E-Mail:
walter@robis.ru.
121
122 A.P. Anyutin – V.I. Stasevich Rev.Mate.Teor.Aplic. (2005) 12(1 & 2)
o parte de un cilindro parabo´lico. Como onda de incidencia hemos considerado on-
das cil´ındricas de fuente de l´ınea y rayo de Gauss [6] con diferentes anchos efectivos.
Se muestra que tenemos un proceso complicado de enfoque y oscilacio´n del campo
reflajado del rayo, tanto en el campo de incidencia cil´ındrico como con el rayo de
Gauss. Las distorsiones del campo reflejado depende en la forma de la frontera y los
para´metros de los campos de incidencia.
Palabras clave: Problema de dispersio´n, modo de galer´ıa susurrante E polarizada, fron-
tera co´ncava, ecuacio´n integral de primer tipo, solucio´n nume´rica.
Mathematics Subject Classification: 34L25, 74S15, 65R20.
1 Introduction
The problem of scattering whispering-gallery mode from concave convex boundary is
known to be under wide scientific discussion within middle part of 20th. It is impor-
tant to underline that all achieved results were obtained by asymptotic methods only: the
method of geometrical theory of diffraction (GTD), its uniform or local modifications, the
method of physical optics, the method of parabolic equation or Kirhhoff approximation [1-
5] and deal with describing of the field nearby scatterer’s surface. All these methods have
restrictions and all attempts to improve the results run against huge difficulties. Actually,
it is impossible.
In this work we present a strict numerical solution for 2D problem of scattering
whispering-gallery mode from concave smooth boundary with finite size. The results
were obtained by applying developed method of prolonged boundary conditions for cur-
rants integral equations (CIE) [6,7] in high frequency domain ( kD  1, k = 2pi/λ–wave
number,λ–wave length, D–maximum size of the scatterer surface).
2 Scattering of E polarized wave from concave finite cylin-
drical structure
Let at first consider the scattering problem for E polarized incident cylindrical wave:
U0(−→r ) = H(2)0 (k | −→r −−→r 0 |) (1)
by perfectly conducting finite cylindrical surface S with cross-section function ρ0 = ρ(ϕ),
ϕ ∈ [ϕB , ϕE ] in cylindrical coordinate system {r, ϕ}. In (1), point −→r 0 = {R0, ϕ0} indicates
a position of the source.
The scattering field U1(−→r ) has to satisfy a wave equation outside of S, boundary
condition on S :
U0(−→r ) |S +U1(−→r ) |S= 0 (2)
and Sommerfield condition when | −→r |→ ∞:
∂Ui(−→r )
∂r
= ikUi(−→r ) = o(| −→r |−1/2), | −→r |→ ∞.
scattering of e polarized whispering-gallery mode 123
It is known that in this case the scattering problem could be reduced to Dirichlet value
boundary problem and following integral equation of the first kind with singular kernel
could be obtained:
U0(−→r )S =
∫
S
µ(−→r SH(2)0 (k | −→r S −−→r σ |)dσ, (3)
In (3) the unknown function µ(−→r ) is a current on surface −→r S ∈ S, −→rσ ∈ S,
H
(2)
0 (k | −→r −−→r Σ |), fundamental solution to Helmholts equation,
| r −−→r S |=
√
r2 + r2S − 2rrS cos(ϕ− ϕS)
is a distance between points −→r = {r, ϕ} and −→r s = {rS , ϕS) in a cylindrical coordinate
system {r, ϕ}.
An ordinary way to make a numerical solution of integral equation (3) is to extract a
singularity in the kernel and use a piece-wise system of basis functions [4] or some other
full system (for example, any kind of orthogonal polynoms, splines or Fourier sets). But it
is impossible to obtain stable and accurate results for high frequency domain (kD  1).
That is why we have made another approach, based on two ideas. First one, is that any
numerical solution makes with finite accuracy only. Second one, we have used analytical
properties for presentation of scattering field in simple potential layer form [7]. It allows
us to make displacement for points rS from S into Σ surface (Σ = S + i∆˜, ∆˜ 1, where
S is the original surface, and ∆˜ is a value of displacement from original surface). In other
words, we have a “small” displacement δ or continuation into complex region for surface’s
points −→r i and condition δ  1 guarantees us that we do not cross any singularities. So,
as a result we have an integral equation of the first kind with smooth kernel and Haar
wavelet functions (as a system of basis functions) could be effective applied for its stable
numerical solution in the region kD  1.
It was mentioned above that one of the main problems deals with applying the method
CIE for numerical solution of the scattering problems connected with applying the most
effective basis function. To solve this problem we have used a wavelet technique [8]. So
using the wavelet technique one can obtain the following presentation for current µ(θ) in
form of a set:
µ(θ) = c0φ0(θ) +
∞∑
j=0
2j−1∑
k=0
djkΨjk(θ) (4)
where
φ0(θ) =
{
1, 0 ≤ θ < 2pi
0, θ /∈ [0, 2pi[,
Ψjk(θ) = 2j/2ΨH(2jθ − 2kpi),
ΨH(t) =

1, 0 ≤ t < pi,
−1, pi ≤ t < 2pi,
0, t /∈ [0, 2pi[.
124 A.P. Anyutin – V.I. Stasevich Rev.Mate.Teor.Aplic. (2005) 12(1 & 2)
ΨH(t) is a Haar function [8] and d0,djk are to be found. We would like to emphasize
that Haar function are forming an orthogonal and unconditional basis on 0 ≤ θ < 2pi
interval and any function from L2(<) could be presented as a set of Haar functions.
Placing (4) into (3) will we get the following expression for (3):
c0Lφ0 +
p∑
j=0
2j−1∑
k−0
djkLΨjk ≈ Ψ(α) (5)
where is an integral operator in (3), Ψ(α) = U0(−→r S). Then making a procedure of
description with functions in form of the delta functions: ξm = δ(α − αm) (where αm
are the points from the interval [0, 2pi] on the contour S, m = 1, . . . ,M) one could get a
system of linear equations. This system could be written in matrix form as
[Amn][dn] = [bm] (6)
where
Amn = 〈ξm, LΨik〉, m = 1, . . . ,M1; bm = 〈ξm,Ψ〉. (7)
j = 0, . . . , p; k = 0, . . . , 2j − 1; n = p+ 2j .
If the µ(−→r ) is known, then the scattering pattern can be calculated as follows
g(ϕ) =
∫
S
exp[ikρ0(θ) cos(θ − ϕ)]µ(θ)dθ (8)
We estimate the error in the solution of the problem as the residual ∆ of the boundary
condition on S′.
3 Numerical results
We have applied the described procedure for numerical solution of scattering whispering-
gallery mode by concave finite convex boundary as a part of circular cylinder with cross
section function ρ(θ):
ρ(θ) = a, ϕ ∈ [ϕB , ϕE ], (9)
or part of parabolic cylinder with cross section function (parabola):
ρ(θ) = 2p/[1 + cos(ϕ)])ϕ ∈ [ϕB , ϕE ] (10)
As incident wave we have considered a cylindrical wave (1) and a Gauss beam [6]
U0(x, y) = exp{−ikx− k2(y − Y0)2/k2σ2}, (11)
where σ determines an “effective width” of the beam and Y0 a position of its center.
The relative amplitude of scattering pattern g(ϕ)(g(ϕ) ≡| g(ϕ)/max{g(ϕ)} | for Gauss
beam (11) with parameters: kσ = 1, kY0 = −98 , or kσ = 5, kY0 = −95; the surface as a
part of cylinder (9) with parameters: ka = 120, ϕB,E = ±pi/2 are illustrated by Figures
scattering of e polarized whispering-gallery mode 125
1 and 2. It is seen that the process of beam’s interaction with finite reflector (9) is
accompanied by a complicate oscillating of scattering field’s amplitude.
The relative amplitude of scattering pattern g(ϕ) for cylindrical E polarized cylindrical
wave (1) and cylindrical surface (8) presents at Figures 3 and 4. Parameters of surface (9)
were KA = 80; ϕB,E = ±pi/2 and coordinates of the source were kR0 = 78; ϕ0 = −pi/2.
126 A.P. Anyutin – V.I. Stasevich Rev.Mate.Teor.Aplic. (2005) 12(1 & 2)
The relative amplitude of scattering pattern g(ϕ) for cylindrical E polarized cylindrical
wave (1) and cylindrical surface (8) presents at Figures 5 and 6. Parameters of surface
(9) were ka = 120, ϕB,E = ±pi/2., and coordinates of the source were kR0 = 115, ϕ0 = 0.
These figures show that structure of the scattering pattern depends strongly on source’s
location with respect to scatterer surface.
The relative amplitude of scattering pattern is a parabolic reflector (10) (see Figure
7) depicted in Figures 8–10. Parameters of parabola were: kp = 60, ϕB,E = ±pi/3,
parameters of the source: kR0 = 58, kR0 = 54, kR0 = 30, ϕ0 = 0 (a case of symmetrical
location of the sourcer) for Figures 8–10 accordingly.
scattering of e polarized whispering-gallery mode 127
The case of “nonsymmetrical” location of the source with regard to the scattererb
when its parameter were : kR0 = 80, ϕ0 = −55◦ or kR0 = 74.26, ϕ0 = −55◦ presented at
Figures 11 and 12 accordingly.
It is seen that structure of the scattering pattern in this case is closed to a case
of cylindrical surface and difference deal with curvature’s changing of the scatterer. It
is important to underline that distortions of the scattering pattern depend strongly on
location of the source with regard to scatterer’s surface.
Acknowledgements
This work was supported by the Russian Foundation for Basic Research, Project Number
03-02-16336.
128 A.P. Anyutin – V.I. Stasevich Rev.Mate.Teor.Aplic. (2005) 12(1 & 2)
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