Computational Mathematics and Modeling, Vol. 12, No. 3, 2001

MATHEMATICAL MODELING INVESTIGATING THE REFLECTION OF AN ELECTROMAGNETIC FIELD FROM A WAVY PERIODIC BOUNDARY BETWEEN TRANSPARENT MEDIA A. S. Il’inskii, T. N. Galishnikova, and I. V. Berezhnaya

UDC 517.946

The article investigates the mathematical model for the reflection of a plane H-polarized electromagnetic wave from a transparent medium with a periodic wavy boundary. Numerical results are presented for the reflection and distribution of the fields on the boundary.

The present article is a continuation of our ongoing study of the reflection of the field of a plane electromagnetic wave from an irregular boundary. We have developed various mathematical models that take into account boundary geometries close to those occurring in natural experiments [1, 2]. Numerical algorithms have been designed for solving the problems of reflection of the field of a plane two-dimensional wave from a periodic wavy boundary between two transparent media [3 – 5] and from a sufficiently extended finite section of a wavy surface subject to impedance boundary conditions [4]. Numerical experiments have been conducted for the reflection of a plane E-polarized electromagnetic wave from a transparent wavy interface [3]. The numerical algorithms are based on the integral equation method. In the present article, we investigate the reflection and distribution of the magnetic field on a wavy transparent periodic interface of two media for the case when the incident field is H-polarized. Statement of the Problem We will investigate the diffraction of the field of a two-dimensional plane H-polarized electromagnetic field on a periodic transparent interface of two media. The mathematical model is constructed in the Cartesian coordinate system x y z so that the z axis is parallel to the cylindrical generator of the interface and the interface is periodic in the x-direction. Assume that the medium above the boundary (region D1 ) and the medium below the boundary (region D2 ) are dielectric with permittivity ε 1 , 2 and permeability µ 1 , 2 , respectively. The wavenumbers k 1 , 2 are defined by the formula k1, 2 = ω ε1, 2 µ1, 2 , where ω is the circular frequency. We assume that ε1 is a real number. The time dependence is taken in the form exp (− i ω t ) . Denote by v0 ( x, y) the normalized field of the incident wave v0 ( x, y) = exp (− ik1 sin θ 0 x − ik1 cos θ 0 y) ,

(1)

where θ 0 is the angle between the negative y-direction and the direction of propagation of the incident wave projected onto the plane z = 0 . Let v1, 2 ( x, y) = Hz(1, 2) ( x, y) be the z-projections of the magnetic fields in the regions D1, 2 , respectively. It can be shown [6] that the sought functions v1, 2 ( x, y) in the regions D1, 2 satisfy the two-dimensional Helmholtz equations  ∂2 ∂2 2   2 + 2 + k1, 2  v1, 2 ( x, y) = 0 .  ∂x  ∂y

(2)

Translated from Prikladnaya Matematika i Informatika, No. 6, pp. 5–11, 2000. 1046–283X/01/1203–0187 $25.00

© 2001 Plenum Publishing Corporation

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A. S. I L’INSKII, T. N. G ALISHNIKOVA,

AND

I. V. BEREZHNAYA

The continuity conditions of the tangential electromagnetic field components across the interface (we denote it by S ) are written in the form v1( x, y) = v2 ( x, y) ,

1 ∂v1( x, y) 1 ∂v2 ( x, y) = ; ε1 ε2 ∂n ∂n

(3)

here ∂ /∂n is the derivative with respect to the outer normal n to the region D1 . The scattered field should satisfy the radiation conditions, which in the region D1 are written in the form

{ t + b2πm x } exp  i

∞

∑

v1( M ) = v0 ( M ) +

m = −∞

Rm exp i

k12 −

M

( t + b2πm ) y

 , 

2

M

(4)

where t = k1b sin θ 0 , b is the interface period, Rm are the reflection coefficients. Allowing for the surface geometry and the character of the incident field, the diffraction problem is reduced from the entire space R2 to solution within one period, where the interface is denoted by S0 . To obtain the integral equations, we use Green’s formulas and quasiperiodic fundamental solutions g1, 2 ( M, P) for the nonhomogeneous Helmholtz equation in the regions D1 , 2 :  ∂2 ∂2 2   2 + 2 + k1, 2  g1, 2 ( M, P) = − 2π δ ( M, P) .  ∂x  ∂y

(5)

The functions g1, 2 ( M, P) are defined in the entire space R2, satisfy the radiation conditions, and have the form [1, 2, 6] g1, 2 ( M, P) =

(

(1, 2)

i π ∞ exp (i λ n ∆x ) exp i γ n 2 n =∑ γ (1, 2) –∞

∆y

),

n

λn =

t + 2π n , b

γ (n1, 2) =

Im γ (n1, 2) = 0 , M = ( x M , yM ) ,

P = ( x P , yP ) ,

k12, 2 − λ2n ,

Im γ (n1, 2) > 0 ,

(6)

Re γ (n1, 2) > 0 , ∆x = x M − x P ,

∆y = yM − yP .

The diffraction problem (1) – (4) reduces to solving a system of two integral equations for the unknown magnetic ∂v ( P) ∂v1( P) fields v( P) = v1( P) , = on S0 in the region D1 . These equations have the form [1, 2] ∂n ∂n v (M) =

1 2π

ε ∂v ( P)   ∂2  ( ) v P ∫  ∂nP (g2 ( M, P) − g1( M, P)) + ∂nP  g1( M, P) − ε21 g2 ( M, P)  dsP + v0 ( M ) , S0

M ∈ S0 , (7)

ε ∂v ( M ) 1 1 1 + 2 = 2 2π ε1  ∂n



∂

2

∫ v (P) ∂nM ∂nP (g2 ( M, P) − g1( M, P))

S0

–

∂v ( M ) ∂v ( P) ∂  ε 2 , g ( M, P) − g1( M, P)  dsP + 0   ∂nP ∂nM  ε1 2 ∂nM

M ∈ S0 ,

(8)

INVESTIGATING THE REFLECTION OF AN E LECTROMAGNETIC FIELD FROM A WAVY PERIODIC BOUNDARY

189

where v0 ( M ) is the incident field (1). The system of integral equations (7) – (8) is solved by reduction to a system ∂v( P) are approximated by first-order of linear algebraic equations. To this end, the unknown functions v ( P ) and ∂n splines on a uniform grid [3]. Numerical Results We have investigated the reflection and distribution of the magnetic field on the interface of two transparent media. The numerical calculations have been carried for various frequencies, various incidence angles of the plane H-polarized wave, and various dielectric permittivities characterizing the reflecting medium. Cases with both single-wave and many-wave reflected fields have been considered. We have investigated the dependence of the electromagnetic field reflection coefficients on the incidence angle of the plane wave in the range 0o ≤ θ 0 ≤ 89o . We have studied in detail the field distribution on the interface and the behavior of the reflection coefficients in the neighborhood of critical angles (the reflected field harmonics are redistributed when we pass through these critical angles). The numerical calculations reported below have been carried out for the case of a lattice with a period b = 2π ; the contour S0 is defined by the formula f ( x ) = 1 − cos x , 0 ≤ x ≤ 2π, µ1 = µ2 = 1, ε1 = 1. For the reflection of an E-polarized plane wave from a transparent interface of two media such calculations are reported in [3]. Figure 1 shows the convergence of the sought solution v( x ) S0 as a function of the order of the system of linear algebraic equations. The wavenumber for the medium D1 is k 1 = 0.75, the dielectric permittivity of the medium D2 is ε 2 = 50 + 4i. The plane wave is incident at an angle θ0 = 20˚, which immediately follows the critical angle θcr = arcsin (1/3) . The reflected field contains two propagating harmonics ( n = 0, – 1 ). The curves 1 and 2 obtained by solving systems of linear algebraic equations of order 80 and 100 are virtually identical, which indicates that the numerical results are quite accurate.

Fig. 1. The dependence of v( x )

S0

on the order of the system for an obliquely incident plane wave (k1 = 0.75, θ0 = 20˚).

Figure 2 shows the dependence of the reflection coefficients on the angle of incidence of the plane wave for k1 = 0.75 and ε2 = 50 + 4i. For θ 0 ∈ 0o ; θcr the reflected field contains one propagating harmonic ( n = 0 ) ; for

(

o

)

[

)

θ 0 ∈ θcr ; 90 there are two propagating harmonics ( n = 0, – 1 ) . The angle θcr is between 19o < θcr < 20o , where the amplitude of the zeroth field harmonic changes abruptly. This is associated with a significant change in the distribution of the function v( x ) S0 when the incidence angle of the plane wave crosses θcr .

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A. S. I L’INSKII, T. N. G ALISHNIKOVA,

AND

I. V. BEREZHNAYA

Fig. 2. The dependence of reflection coefficients on the incidence angle of a plane wave (k1 = 0.75).

Fig. 3. Changes in the distribution of induced currents when passing through critical angles (k1 = 0.75). Figure 3 plots the distribution of v( x ) S0 calculated for θ 0 = 19˚, 20˚, 21˚, 25˚ (curves 1 – 4) and beyond the angle θcr . Curve 1 corresponds to the current on S0 before θcr , and curves 2 – 4 represent the situation after θcr . The numerical results presented in Fig. 3 show that a slight change of the incidence angle of the plane wave near the critical angles leads to a substantial change in the behavior of the function v( x ) S0 . Figure 4 analyzes for k1 = 1.75 and ε 2 = 36 + 4i the frequency dependence of the reflection coefficients on the incidence angle of the plane wave. Unlike the results in Fig. 2, the distribution of the reflected field by the harmonics changes on passing through three critical angles: θcr,1 = arcsin (1/ 7) ≈ 8o 12 ′′ , θcr,2 = arcsin (3/ 7) ≈ 25o 21′′ , and θcr,3 = arcsin (5/ 7) ≈ 45o 35′′ . In the range θ 0 ∈ 0o ; θcr,1 the reflected field contains three propa-

[

)

gating harmonics ( n = 0, ± 1 ) ; in the range θ 0 ∈(θcr,1, θcr, 2 ) it contains four propagating harmonics ( n = 0,

± 1, – 2 ) ; in the range θ 0 ∈(θcr, 2 , θcr, 3 ) it contains three propagating harmonics ( n = 0, – 1, – 2 ) ; and finally in the

(

)

range θ 0 ∈ θcr, 3, 90o it contains four propagating harmonics ( n = 0, – 1, – 2, – 3 ).

INVESTIGATING THE REFLECTION OF AN E LECTROMAGNETIC FIELD FROM A WAVY PERIODIC BOUNDARY

191

Fig. 4. The dependence of the reflection coefficients on the incidence angle of the plane wave (k1 = 1.75).

Fig. 5. Changes in the distribution of induced currents on passing through the critical angles (k1 = 1.75). Curves 1 – 5 in Fig. 4 correspond to Ri , i = 0, 1, – 1, – 2, – 3, where R i are the amplitudes of the reflection coefficients. As expected, the smooth behavior of the curves is disrupted near the critical angles. The zeroth harmonic attains its maximum for angles close to 90˚. Figure 5 shows the changes in current behavior on passing through θcr for the same parameter values as in Fig. 4. Curves 1 and 2 correspond to currents when the plane wave is incident at an angle before and after θcr,1, i.e., θ 0 = 8˚ and 9˚; curves 3 and 4, 5 and 6 correspond to incidence angles of the plane wave before and after θcr,2 and θcr,3 , respectively, i.e., for θ0 = 25˚, 26˚ and θ0 = 45˚, 46˚. A change of one degree in the incidence angle near the critical angles produces a marked change in currents. The numerical results reported in this paper demonstrate the high accuracy of the proposed numerical method and the possibility of conducting numerical experiments in the resonance frequency range for the computation of

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I. V. BEREZHNAYA

field reflection and distribution characteristics on the interface of two transparent periodic media for various incidence angles of the plane wave. The study has been supported by the Russian Foundation for Basic Research (grant No. 99-02-16972) and the interuniversity scientific program “Russia’s universities: basic research” (grant No. 990894). REFERENCES 1. A. S. Il’inskii and T. N. Galishnikova, “Mathematical modeling of the process of reflection of a plane electromagnetic wave from a wavy surface,” Radiotekhnika Élektronika, 44, No. 7, 773–786 (1999). 2. T. N. Galishnikova and A. S. Il’inskii, “Scattering of a plane wave on a wavy surface,” in: Mathematical Models in Natural Sciences [in Russian], Izd. MGU, Moscow (1995), pp. 86–111. 3. A. S. Il’inskii, T. N. Galishnikova, and I. V. Berezhnaya, “Reflection from a wavy boundary between transparent media,” in: Applied Mathematics and Informatics [in Russian], No. 3, pp. 33–42, Izd. MGU, Moscow (1999). 4. A. S. Il’inskii, T. N. Galishnikova, and I. V. Berezhnaya, “Comparison of two mathematical models in the diffraction problem for an H-polarized wave on an irregular interface,” Vestnik MGU. Ser. 15: Computational Mathematics and Cybernetics (in press). 5. A. S. Ilinski and T. N. Galishnikova, “Mathematical models in EM wave scattering by wavy surfaces,” Proc. 1 st Workshop on Electromagnetic and Light Scattering: Theory and Applications [in English], Moscow Lomonosov State University, Moscow (1997), pp. 84–88. 6. T. N. Galishnikova and A. S. Il’inskii, Numerical Methods in Diffraction Problems [in Russian], Izd. MGU, Moscow (1987).