Journal of Mathematical Sciences, Vol. 111, No. 4, 2002

DIFFRACTION OF ELECTROMAGNETIC WAVES ON AN IMPEDANCE INTERFACE BETWEEN TWO MEDIA WITH DIFFERENT PERTURBATIONS IN IMPEDANCE N. Ya. Kirpichnikova, V. B. Philippov, and A. S. Kirpichnikova

UDC 517.9

The problem on the diffraction of the electromagnetic plane wave on the impedance interface between two media is investigated. The impedance is assumed to be different from a constant on a segment of the interface, where the impedance is described by piecewise-linear, quadratic, or step functions. Bibliography: 4 titles.

In [1], the problem on the diffraction of the electromagnetic plane wave on a jump of the impedance on the interface between two media was considered. The impedance was assumed to be constant everywhere, except for a small part (strip) of the interface. On that small part, the impedance had another constant value, which was rather close to that of the remaining part. In the present paper, we analyze problems on the diffraction of the electromagnetic plane wave on an interface with impedance weakly varying in a segment |x| ≤ l. It is proved that the behavior of the scattered wave essentially depends both on the length of the segment and on the character of variation in impedance. We consider two cases: (1) where the length of the segment is significantly less than the wavelength: kl  1; (2) where the length of the interval is of the same order as the wavelength: kl ∼ O(1).

Fig. 1 The following types of behavior of the impedance (see Fig. 1) are investigated: I. the impedance g of the first type is described by  g=

g0 g0 + f(x)

for |x| > l/2, for |x| ≤ l/2.

(0.1)

Here, the absolute value of the function f(x) of impedance perturbation is small enough, and Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 264, 2000, pp. 101–121. Original article submitted October 12, 1999. 3678

c 2002 Plenum Publishing Corporation 1072-3374/02/1114-3678 $27.00

(i) the function f(x) depends piecewise linearly on x, varying from 0 to g1 − g0 on the segment [−l/2, 0] and from g1 − g0 to 0 on the segment [0, l/2]:   2x    (g1 − g0 ) 1 + for x ∈ [−l/2, 0], l (0.10 ) f(x) =     (g1 − g0 ) 1 − 2x for x ∈ [0, l/2]; l (ii) the function f(x) is smooth; it increases monotonically from 0 to g1 − g0 on the segment [−l/2, 0] and decreases monotonically from g1 − g0 to 0 on the segment [0, l/2]. It should be noted here that max |f(x)| ≤ (g1 − g0 ) as in the previous case. Here, we assume that the function f(x) is quadratic:   2  2x f(x) = (g1 − g0 ) 1 − for x ∈ [−l/2, l/2]; l

(0.1 00 )

(iii) the function f(x) is a nonzero constant: f(x) = g1 − g0

for x ∈ [−l/2, l/2],

so that in this case the impedance g is a step function. II. The impedance g of the second type is described by the following conditions: (j) the impedance g varies linearly from a constant value g0 to another value g1 :  for x < −l/2, g0     g1 for x > l/2, g=    g1 − g0 2x   +1 for |x| ≤ l/2;  g0 + 2 l (jj) the impedance has a jump from a constant value g0 to another value g1 at the point x = 0:  g0 for x < 0, g= g1 for x > 0.

(0.1 000 )

(0.2)

(0.20 )

In both cases (I and II), we assume that the impedance perturbation is small enough, i.e., |g1 − g0 |  1. For this reason, (1) both media are assumed homogeneous, isotropic electromagnetic media, and the Maxwell equations are converted to the Helmholtz equations; (2) the method of iterations is used to obtain solutions of integral equations. The electric and magnetic components E and H of the electromagnetic field satisfy the Helmholtz equation, the impedance conditions on the interface between the two media, and the Sommerfeld conditions of radiation. To solve these problems, we use a fruitful idea (see [2, 3]) concerning the transformation of these problems to some integral equations. Then, we use the iteration method to solve the integral equations. We introduce a Cartesian system of coordinates, so that the plane xy coincides with the interface between the media, and the direction vector of the incident wave lies in the plane xz. Let θ0 be the angle between the direction vector and the z axis, and let θ1 be the angle of refraction. We denote by ϕ0 (ϕ1 ) the angle between the directions of the incident wave and the reflected (respectively, refracted) wave in the upper and lower media. Then, we denote the magnetic and electric permeabilities of the upper medium (z > 0) and the lower medium (z < 0) by µ = µ1 and ε = ε1 , respectively. The interfaces are usually assumed to be ideally conducting. Thus, if the plane z = 0 is assumed to be ideally conducting, then we have the following boundary conditions on it: Ey = 0,

Ex = 0

for z = 0.

Conditions of the impedance type (see [2–4]) are a natural generalization of these conditions. If the electromagnetic field is considered in the half-space z > 0 and there is a large complex index of refraction in the half-space z < 0, then the Leontovich boundary conditions must be laid down on the interface: Ey = Z1 Hx ,

Ex = −Z1 Hy for z = 0. 3679

These conditions yield two relations between the tangential components of the electric and magnetic fields. The parameter Z1 is independent of z. It is called the wave drag of the lower medium or the impedance. Two conditions on the interface between the media can be replaced by a single one if the impedance Z1 is approximately taken as independent of the angle of refraction θ1 . Then, for the component E which lies in the plane of the incident wave we have r ∂Hy ωε µ1 z = 0, + γk Hy = 0, γk = i Z1 , Z1 = , ∂z c ε1 and for the component E orthogonal to the incident plane we have z = 0,

∂Ey + γ⊥ Ey = 0, ∂z

γ⊥ = i

µω . cZ1

Here, we have obtained “a boundary condition of the third kind” or “an impedance boundary condition.” Using this boundary condition, we can completely solve the problem in the upper medium without considering the lower medium. Thus, let a plane electromagnetic wave be incident on the interface between the media (the xy plane or z = 0) such that the direction vector lies in the plane xz. The wave is assumed to be represented as the superposition of two waves. Moreover, the vector E of the first wave is orthogonal to the plane xz and the vector E of the second wave lies in this plane (see Fig. 2). We denote by u the components Ey and Hy of the electromagnetic field in the upper medium for √ two polarizations. The function u satisfies the Helmholtz equation, the standard radiation conditions as r = x2 + z2 → ∞, and “impedance conditions” on the interface between the media. The Helmholtz equation is of the form ∆u + k 2 u = 0,

(0.3)

where

ω√ εµ. c The Sommerfeld condition of radiation for the field u is as follows:   √ ∂u lim r − iku = 0, Im k > 0, r→∞ ∂r k=

(0.4)

and finally the impedance condition on the interface between the media can be written as ∂u + ikgu = 0, z = 0. (0.5) ∂z For the impedance which is constant on the whole interface plane the function ikg = ikg0 is determined by the expressions r ε0 µ1 ε0 ε0 ikg0k = ik = ik1 = ik1 κε , κε = , (0.6k) ε1 µ0 ε1 ε1 r ε1 µ0 µ0 µ0 ikg0⊥ = ik0 = ik1 = ik1 κµ , κµ = (0.6⊥) ε0 µ1 µ1 µ1 (the modulus of ε1 may be large enough if the conductivity of the lower medium σ1 is large). We represent the solution of problem (0.3)–(0.5) with impedance (0.6) as the sum of incident and reflected waves with different reflection coefficients corresponding to two polarizations (we use the signs ⊥ and k for the reflection coefficients and the other parameters).

Fig. 2 3680

In the case of impedance boundary condition (0.5) with a constant impedance g0 , the solution is given by the relation   cos θ0 − g0⊥ ikz cos θ0 e . (0.7) u0⊥ = E0 eikx sin θ0 e−ikz cos θ0 + cos θ0 + g0⊥ For the other polarization, we have  u0k

ikx sin θ0

= H0 e

−ikz cos θ 0

e

 cos θ0 − g0k ikz cos θ0 . + e cos θ0 + g0k

(0.8)

1. Reflection from a small interface segment with perturbed impedance I. Consider the problem on the reflection of an incident electromagnetic plane wave from the interface between media, on which impedance conditions (0.1) are perturbed on a strip |x| ≤ l. Assume that the width of the strip is small in comparison with the wavelength: l  λ (λ = 2π k is the length of the electromagnetic wave). Let a field u satisfy problem (0.3)–(0.5) for z > 0. The impedance g in (0.5) is determined by condition (0.1):  g=

g0

for |x| > l/2,

g0 + f(x)

for |x| ≤ l/2.

Here, the function f(x), which describes the impedance perturbation, is of the form (i), (ii), or (iii). The solution u of problem (0.3)–(0.5) is represented as the sum of an unknown term U and the known solution u0 of the problem with constant impedance g0 : u = U + u0 .

(1.1)

In (1.1), u0 is either u0⊥ or u0k (see (0.7)–(0.8)), depending on the polarization of the incident field. For the function U , we obtain the problem ∆U + k2 U = 0, ∂U + ikgU = −iku0 ∂z



k ≡ k0 ,

z > 0,

0

for |x| > l/2, z = 0,

f(x)

for |x| ≤ l/2, z = 0.

(1.2) (1.3)

The condition of radiation is determined by Eq. (0.4), where the change u 7→ U is made. The solution of problem (1.2)–(1.3) is found by using the Green’s function for the problem with a constant impedance. The Green’s function G(M, M0 ) satisfies the problem (∆ + k2 )G(M, M0 ) = −δ(M − M0 ),

z > 0,

∂G + ikg0 G = 0, z = 0, ∂z

(1.4) (1.5)

and the condition of radiation (0.4). Using the Green’s function, the solution of problem (1.2)–(1.3) can be represented as Zl/2 U (x, z) = −ik

0

0

0

Zl/2

0

f(x )U (x , 0)G(x, z; x , 0)dx − ik

−l/2

f(x0 )u0 (x0 , 0)G(x, z; x0, 0)dx0 .

(1.6)

−l/2

First we consider (1.6) on the interface between two media, i.e., on z = 0: Zl/2 U (x, 0) = −ik −l/2

0

0

0

0

Zl/2

f(x )U (x , 0)G(x, 0; x , 0)dx − ik

f(x0 )u0 (x0 , 0)G(x, 0; x 0, 0)dx0.

(1.7)

−l/2

3681

Substituting the Green’s function G(M, M0 ) into (1.7), we obtain the Fredholm integral equation of the second kind for determining the solution of problem (1.2)–(1.3) for z = 0. Then, using (1.6) with known U (x, 0), we obtain U (x, z). Finally, substituting U (x, z) into relation (1.1), we obtain the solution of (0.3)–(0.5). It should be noted that the solution u0 corresponding to the problem with a constant impedance g0 is determined by (0.7)–(0.8). Now we turn to finding the Green’s function. The Green’s function G(M, M0 ) can be constructed by the standard method based on the Fourier transform: 1 G(M, M ) = 2π

Z∞

0

0

eik(x−x )ξ kG1 (z, z 0 , ξ)dξ.

−∞

Upon some calculations, for the Green’s function G1 (z, z 0 , ξ) we obtain the expression √ 2 p √ 2 p  0 0  eik(z−z ) 1−ξ ( 1−ξ 2 + g0 )+eik(z+z ) 1−ξ ( 1−ξ 2 − g0 )   p p , z > z0 ,   2 ( 1 − ξ2 + g ) −2ik 1 − ξ 0 G1 (z, z 0 , ξ) = √ √ p p 0 0 2 2  ik(z − z) 1 − ξ ik(z + z) 1−ξ 2  e ( 1−ξ + g0 )+e ( 1−ξ 2 − g0 )   p p , z < z0 .  −2ik 1 − ξ 2 ( 1 − ξ 2 + g0 )

(1.8)

Taking into account that in integral equation (1.6) only the Green’s function at z0 = 0 is used, we can find the following Green’s function: i G(x, z; x0, z 0 = 0) = 2π

Z∞ −∞

√ 0 2 eik[(x−x )ξ+z 1−ξ ] p dξ. 1 − ξ 2 + g0

(1.9)

Since max |f(x)| ≤ |g1 − g0 |  1 by the assumption of the problem, it follows that on a small interval of integration kl  1, the kernel of the integral equation is either small enough or is bounded, and we can find the solution of integral equation (1.7) approximately by the method of iterations. As the initial approximation, we choose the second (known) term on the right-hand side in (1.7): Zl/2 U

(0)

(x, 0) = −ik

f(x0 )u0 (x0 , 0)G(x, 0; x 0, 0)dx0 ,

(1.10)

−l/2

where G(x, 0; x0, 0) is given by expression (1.9). Then, the first iteration of the required function U(0) (x, z) has the form Zl/2 U

(0)

(x, z) = −ik

  Zl/2 f(x ) − ik f(x00 )u0 (x00 , 0)G(x0, 0; x00 , 0)dx00 G(x, z; x0, 0)dx0 0

−l/2

−l/2

Zl/2 − ik

f(x0 )u0 (x0 , 0)G(x, z; x0, 0)dx0.

(1.11)

−l/2

Now we find the wave field scattered on the interface with a perturbed impedance and investigate its behavior at large distances (kr  1) p from the part of the interface where impedance is perturbed. In a region kr  1, r = (x − x0 )2 + z2 , the Green’s function (1.9) is equal to i sin ϕ G(x, z; x , 0) = 2 (sin ϕ + g0 ) 0

r

   2 i(kr− π ) 1 4 e 1+O . πkr kr

Here the angle ϕ is the angular coordinate of the observation point. 3682

(1.12)

In the case where both the source point and the observation point are situated on the interface between the media (on z = 0), the Green’s function is also represented by (1.12) but with r = |x − x0 |. At the next step, we investigate the first approximation to the solution of integral equation (1.7), obtained above. Let the observation point be situated far from the strip of perturbed impedance (k|x|  1), and, moreover, let the width of the strip be small, kl  1. Taking into account the asymptotic expansion of the function G for k|x − x0 |  1, we obtain the following expression for the iteration U 0 (x, 0): l/2  s  1  Z 2 i(k|x|− π )  k 0 sin ϕ 4 U (x, 0) ∼ A e 1+O f(x0 )dx0 . 2 sin ϕ + g0 πk|x| k|x| 0

(1.13)

−l/2

In accordance with one of the assumptions of the problem, in (1.13) max |f(x)| ≤ |g1 −g0 | is small for any polarization of the electromagnetic field. Moreover, the values of the amplitude factor at the exponent exp{−ikx0 cos ϕ0 } for the field u0 (x0 , 0) are distinct for different polarizations:   2 sin ϕ0 0 , (1.14) A⊥ = E0 sin ϕ0 + g0⊥   2 sin ϕ0 A0k = H0 . (1.15) sin ϕ0 + g0k The first approximation U 0 (x, 0) of the unknown function U (x, z) is determined by the equation Zl/2 U (x, z) = −ik 0

f(x0 )(U 0 (x0 , 0) + u0 (x0 , 0))G(x, z; x0, 0)dx0 .

(1.16)

−l/2

Finally, from (1.13)–(1.16) we obtain the function U 0 (x, z), which should be added to the incident wave u0 , as  l/2  R the product of a cylindrical wave with scattering coefficient F (ϕ0 , ϕ) and a quantity of order k f(x0 )dx0 : −l/2

r U (x, z) ∼ E0(H0 ) 0

2 i(kr− π ) 4 F (ϕ , ϕ) k e 0 πkr

Zl/2

  f(x0 )dx0 1 + O(kl(g1 − g0 ) ,

(1.17)

−l/2

F (ϕ0 , ϕ) =

sin ϕ sin ϕ0 . (sin ϕ + g0⊥(k) ) (sin ϕ0 + g0⊥(k) )

(1.18)

Remark. It should be noted that (i) in the case where the function f(x) is a piecewise-linear function described by (0.10 ), we have k

l/2 R

f(x0 )dx0

−l/2

= (g1 − g0 ) kl/2; (ii) if the function f(x) is a quadratic function described by (0.100 ), we have k

l/2 R

−l/2

(iii) if the function f(x) is a constant function (g(x) is a step function), then k

f(x0 )dx0 = (g1 − g0 )kl2/3;

l/2 R

−l/2

f(x0 )dx0 = (g1 − g0 )kl.

Recall that, by assumption, the jump of the impedance (g1 − g0 ) is small and kl  1. By (1.17) and (1.18), taking into account |kl|  1, for the iteration U 0 (x, z) and for kr  1 we obtain the resultant expression for the wave field scattered on the interface with perturbed impedance (0.1): r sin ϕ sin ϕ0 2 i(kr− π ) 0 4 e U (x, z) ∼ E0 (H0 ) (sin ϕ + g0⊥(k) ) (sin ϕ0 + g0⊥(k) ) πkr 3683

  Zl/2 
0 0 1 + O kl(g1 − g0 ) . × k f(x )dx

(1.19)

−l/2

For the three cases (i), (ii), and (iii) of different perturbations of f(x) in the impedance, the expression in the h l/2 i R square brackets k f(x0 )dx0 can be estimated as −l/2

Zl/2 (g1 − g0 )kl/2 ≤ k

f(x0 )dx0 ≤ (g1 − g0 )kl,

−l/2

where the estimates depend both on the deviation of the impedance r r r  µ0 ε2 ε1 , (g1⊥ − g0⊥ ) = − ε0 µ2 µ1 r  r r ε0 µ2 µ1 (g1k − g0k ) = − µ0 ε2 ε1

(1.20) (1.21)

and on the length |kl|  1 of the interval of perturbation of the impedance. Diffraction on the interface with an impedance perturbed (smoothly, piecewise smoothly, or stepwise) on an interval |x| ≤ l with |kl|  1 in accordance with (0.1) (cases (i), (ii), and (iii)) is similar to the radiation in the xz plane of a point source, which is situated on the y axis. Compare the cases of piecewise-linear and quadratic variations in impedance (i) and (ii) of the present paper with the case of impedance described by step function (iii), investigated earlier in [1]. Here, we have f(x) = g1 −g0 , g0 ≈ g1 , on a small interval |kl|  1. The qualitative picture proves to be the same in all three cases because of the smallness of the interval of the impedance variation. The difference quantitatively depends on the area of the curvilinear figure bounded by the function f(x). In accordance with the remark, in the cases under consideration l/2 R the variation reduces to the factor χ ∈ [1/2, 1] in k f(x0 )dx0 = χ(g1 − g0 ) kl. −l/2

Now we turn to the investigation of the problem on diffraction on the interface with an impedance perturbed on a strip of finite width. Here the behavior of the scattered field is of a completely different type. 2. Scattering on the interface with impedance (0.1) perturbed on an interval of finite length II. Consider the problem on diffraction on the interface with an impedance of the type (0.1) perturbed on an interval of finite length kl ∼ O(1). We obtain and investigate the first approximation to the solution of integral equation (1.7) in the case of a finite segment [−l/2, l/2] with perturbation of the type (0.1). Let the observation point be situated at a large distance (kr  1) from the segment of perturbed impedance, and let the length of the segment be finite (have the same order as the length of the incident wave), i.e., kl ∼ O(1). The first iteration U 0 (x, z) of the function U (x, z) in solution (1.1) of the diffraction problem is represented by the integral Zl/2 0 f(x0 )u0 (x0 , 0)G(x, z; x0, 0)dx0. (2.1) U (x, z) = −ik −l/2

The application of the method of iterations to the solution of integral equation (1.16) on a finite interval of integration is justified by the fact that the function f(x) is small by assumption. For the absolute value of the function, we have |f(x)| ≤ |g1 − g0 |, g0 ≈ g1 . Substituting into (2.1) asymptotic expansion (1.12) of the Green’s function G(M, M 0 ) as kr  1 and the solution u0 (see (0.7)–(0.8)) for a constant impedance, we obtain the following expression for the principal approximation of U 0 (x, z): Zl/2 f(x0 ) ik(x−x0 ) cos ϕ0 eikr 0 0 0 √ dx . U (x, z) = C e (2.2) g1 − g0 kr −l/2

3684

Here, C 0 is a factor independent of the variable of integration x0 . It is equal to r C = kE0 (H0 )F (ϕ0 , ϕ)(g1 − g0 ) 0

2 −i π −ikx cos ϕ0 e 4e . π

(2.3)

In (2.2), we use the change of variables x − x0 = r cos ϕ,

−dx0 = dr cos ϕ

(2.4)

and obtain three different integrals corresponding to three different types (i), (ii), and (iii) of impedance (0.1): (i) First we consider the diffraction problem on the interface with piecewise-linear impedance (0.10 ) perturbed on a finite segment. We have

Ui0 (x, z)

rl/2   Zr0  Z 2x 2x dr = −C cos ϕ 1 + + 1− eikr[1+cos ϕ cos ϕ0 ] √ l l kr 0

r−l/2

2C 0 cos2 ϕ + kl

 Zr0

rl/2 Z √ eikr[1+cos ϕ0 cos ϕ] krdr, −

r−l/2

where rl/2 =

p (x − l/2)2 + z2 ,

r0

(2.5)

r0

r−l/2 =

p (x + l/2)2 + z2 ,

r0 =

p x2 + z2 .

To evaluate Ui0 (x, z), we use the change of variables kr[1 + cos ϕ0 cos ϕ] = t2 ,

k[1 + cos ϕ0 cos ϕ] dr = 2t dt.

(2.6)

Then, introducing the notation below for the Fresnel integral   Zz 2 z 2 Φ √ =√ eit dt, i iπ

(2.7)

0

with the help of (2.6) and (2.7) we can rewrite the integral Ui0 (x, z) in the form √        2C 0 iπ cos2 ϕ t∗∗ t∗ t0 √ r−l/2 Φ √ +rl/2 Φ √ −2r0 Φ √ kl 1+cos ϕ0 cos ϕ i i i  0 2 2 2 2 2C i cos ϕ + 2 [t∗∗eit∗∗ + t∗ eit∗ − 2t0 eit0 ] k l[1 + cos ϕ0 cos ϕ]3/2 √        πi t∗∗ t∗ t0 + Φ √ + Φ √ − 2Φ √ , 2 i i i

Ui0 (x, z)=

(2.8)

where the parameters t∗ , t∗∗ , and t0 are equal to q krl/2 [1 + cos ϕ0 cos ϕ], q t∗∗ = kr−l/2 [1 + cos ϕ0 cos ϕ], p t0 = kr0 [1 + cos ϕ0 cos ϕ]. t∗ =

(2.9)

Substituting expression (2.3) of C 0 into (2.8), we obtain the first approximation to the solution u0 of the diffraction problem on the interface with piecewise-linear perturbation (0.10 ) in impedance on a finite segment. It should be noted that the third group of terms, which include the Fresnel integrals and have the large parameter kr  1 in the denominators, can be omitted because of its smallness in comparison with the first and second 3685

groups. The second group of terms represents three cylindrical waves scattered on the points x = −l/2, 0, l/2, which are the points of discontinuity of impedance (0.1) and (0.10 ). Finally, we obtain the expression √ − 2E0 (H0 ) cos ϕF (ϕ0 , ϕ)(g1 − g0 )e−ikx cos ϕ0 √ Ui0 (x, z) = 1+cos ϕ0 cos ϕ        2 2 2x t0 t∗∗ ei(t0 +π/4) ei(t∗∗ +π/4) √ √ √ √ × 1+ Φ −Φ + − l πt0 πt∗∗ i i        2 2 t∗ 2x t0 ei(t∗+π/4) ei(t0+π/4) + 1− Φ √ −Φ √ + √ . − √ l πt∗ πt0 i i In (2.10), the expressions

(2.10)

    2x 2x (g1 − g0 ) 1 + and (g1 − g0 ) 1 − l l

are values of the function fi (−l/2 < x < 0) and fi (0 < x < l/2), which describe a piecewise-linear variation in impedance on the segment [−l/2, 0, l/2] between the constant values g0 and g1 (see Fig. 1). (ii) Performing calculations which are similar to the previous ones, for the first iteration of the solution u0 of the diffraction problem on the interface with a quadratic perturbation in impedance (0.1) and (0.100 ), we obtain √       −C 0 cos ϕ iπ 4x2 t∗ t∗∗ 0 Uii (x, z) = √ 1− 2 Φ √ −Φ √ k 1+cos ϕ0 cos ϕ l i i √       0 2 t∗∗ 8C ix cos ϕ πi t∗ it2∗ it2∗∗ √ t∗ e − t∗∗ e Φ −Φ √ + 2 − 2 k l[1 + cos ϕ0 cos ϕ]3/2 i i      2 2 4C 0 i cos3 ϕ 3i 3i − 32 t2∗∗ + t∗∗ eit∗∗ − t2∗ + t∗ eit∗ . (2.11) 2 2 k l [1 + cos ϕ0 cos ϕ]5/2 Using (2.9), we transform expression (2.11) to a more compact form: √        2 2  −C 0 iπ cos ϕ eit∗∗ 4x2 t∗ t∗∗ eiπ/4 eit∗ 0 − Uii (x, z) = √ 1− 2 Φ √ −Φ √ + √ l π t∗ t∗∗ k 1+cos ϕ0 cos ϕ i i  it2       2  3 e ∗ eit∗∗ ix 3 t∗ t∗∗ + √ . − − + 22 Φ √ −Φ √ 2 2 2 t∗∗ t∗ (x + l/2)t∗ 4t∗t∗∗ 2 iπ t∗ t∗∗ i i

(2.12)

In view of the fact that we investigate √ the scattered field for kr  1 and kl ∼ O(1), all the parameters t0 , t∗ , and t∗∗ in (2.9) are quantities of order kr  1. Therefore, the last two terms in (2.12) are small in comparison with the first term. The principal contribution is given by the difference between the Fresnel integrals that describe penumbra effects from the edges of the segment [−l/2, l/2]. Substituting the expression (2.3) of C0 into (2.12), we obtain the principal term of the first iteration of the solution u0 in the form √ − 2E0 (H0 )(g1 − g0 ) cos ϕF (ϕ0 , ϕ)e−ikx cos ϕ0 0 √ Uii (x, z) = 1 + cos ϕ0 cos ϕ        2 2  4x2 t∗ t∗∗ eiπ/4 eit∗ eit∗∗ × 1− 2 Φ √ −Φ √ + √ − . l π t∗ t∗∗ i i

(2.13)

Now, we consider case (iii) of a step function of impedance perturbation. Here the first iteration of the solution u0 is given by the formula √      −C 0 cos ϕ iπ t∗ t∗∗ 0 Uiii (x, z) = √ Φ √ −Φ √ . (2.14) k 1+cos ϕ0 cos ϕ i i Using (2.3) for C 0 , we obtain the final expression √      − 2E0 (H0 )(g1 − g0 ) cos ϕF (ϕ0 , ϕ)e−ikx cos ϕ0 t∗ t∗∗ 0 √ Uiii (x, z) = Φ √ −Φ √ . 1 + cos ϕ0 cos ϕ i i 3686

(2.15)

Compare the behavior of the scattered waves for three different types of small perturbations (2.10), (2.13), and (2.15) in impedance on a finite segment. The smallness of perturbation in impedance is determined by the factor (g1 − g0 ) occurring in all three formulas. The principal contribution from the points of singularities of the impedance on the segment in cases (ii) and (iii) and from the three points of singularities {−l/2, 0, l/2} of the impedance in case (i) is given by the linear combination of Fresnel integrals, which describe penumbra effects. It is important that these penumbra effects do not disappear at any position of the observation point for a discontinuous perturbation in impedance. The penumbra effects disappear at the observation points (±l/2, z) and (0, z) for a piecewise-linear or quadratic perturbation in impedance. 3. Diffraction on an interface where impedance has a jump from one value to another or changes linearly from one value to another on a segment of finite length III. Consider the problem on the scattering of an incident electromagnetic plane wave on the interface between two media with linear variation in impedance on a segment |x| ≤ l/2, kl ∼ O(1), or its stepwise variation at the point x = 0 from a given constant value to another one. Let, for z > 0, the field u satisfy the Helmholtz equation (0.3), radiation conditions (0.4) for z > 0, and impedance boundary condition (0.5): ∂u + ikgu = 0, z = 0. (3.1) ∂z The impedance g in (3.1) is determined by the following condition: (j) the impedance varies linearly from g0 to g1 on the segment [−l/2, l/2]:  g0 for x < −l/2    g for x > l/2 1 g=    g1 − g0 2x    g0 + + 1 for |x| ≤ l/2. 2 l

(3.2)

(jj) the impedance has a jump from g0 to g1 at the point x = 0:  g=

g0 g1

for x < 0, for x > 0.

(3.20 )

Recall that the absolute value of the difference of the impedances is assumed to be small enough: |g1 −g0 |  1. We represent the solution u of problem (0.3)–(0.5) as the sum of an unknown term U and the known solution u0 of the problem with a constant impedance g0 : u = U + u0 .

(3.3)

Here, u0 is either u0⊥ or u0k (see (0.7)–(0.8)), depending on the polarization of the incident field. For the function U we obtain the following problem (recall that the function U satisfies radiation condition (0.4)): ∆U + k2 U = 0, z > 0, k ≡ k0 ,  0 for x < −l/2, z = 0,    g − g for x > l/2, z = 0, ∂U 1 0 + ikU g = −iku0    ∂z g1 − g0 2x   + 1 for |x| ≤ l/2, z = 0.  2 l Using the Green’s function G(M, M 0 ) from (1.9) for the impedance g0 that is constant on the whole interface, we can represent the solution of the problem for U in the form Zl/2 U (x, z) = −ik(g1 − g0 ) −l/2

x0 1
+ (U (x0 , 0) + u0 (x0 , 0))G(x, z; x0, 0)dx0 l 2 3687

Z∞ −ik(g1 − g0 ) (U (x0 , 0) + u0 (x0 , 0))G(x, z; x0, 0)dx0.

(3.4)

l/2

We seek the solution of Eq. (3.4) by the iteration method, taking into account the smallness of the difference of the impedances |g1 − g0 |  1. p Consider the behavior of solution (3.4) in a distant zone for kr  1, where r = (x−x0 )2 +z2 . In such a zone, the Green’s function is given by the expression i sin ϕ G(x, z; x , 0) = 2 (sin ϕ + g0 ) 0

r

  1  2 i(kr− π ) 4 e 1+O . πkr kr

(3.5)

Now we find the first iteration of integral (3.4) in cases (3.2) and (3.20) of the variation in impedance. Upon calculations similar to those given in Secs. 1 and 2, we obtain the following expression for the integral over the infinite interval of integration: I

∞

√    C 0 cos ϕ iπ t∗ 1−Φ √ =− √ . k 1 + cos ϕ0 cos ϕ i

(3.6)

Using (2.3) for C 0 , we can represent the integral I∞ in the following ultimate form: √ I

∞

=−

   2E0 (H0 )e−ikx cos ϕ0 cos ϕF (ϕ0 , ϕ)(g1 − g0 ) t∗ √ 1−Φ √ . 1 + cos ϕ0 cos ϕ i

(3.7)

Finally, it remains to calculate the integral I(x, z) over the finite interval: Zl/2  I(x, z) = −ik(g1 − g0 ) −l/2

 x0 1 0 0 + u (x , 0)G(x, z; x0, 0)dx0 . l 2

The substitution of the Green’s function G(M, M0 ) and the solution u0 , which corresponds to a constant impedance on the interface, into the integral I(x, z) yields a sum of two different integrands and, therefore, a sum of two integrals for the integral I(x, z): Zl/2  I(x, z) = I1 + I2 = C

0 −l/2

x0 − x + l



x 1 + l 2



0 eikr eik(x−x ) cos ϕ0 √ dx0 . kr

Again, upon some calculations similar to those given in Sec. 1, for the integral I2 we obtain √       C 0 cos ϕ iπ x 1 t∗ t∗∗ + Φ √ −Φ √ , I2 = − √ 2 k 1 + cos ϕ0 cos ϕ l i i

(3.8)

where the parameters t∗ and t∗∗ are determined by (2.9). Using the change of variables (2.6), the integral I1 can be represented in the form 2C 0 cos2 ϕ I1 = 2 k l[1 + cos ϕ0 cos ϕ]3/2

Zt∗

2

t2 eit dt, t∗∗

where the lower and upper limits of integration t∗∗ and t∗ are determined by (2.9). Integrating this integral by parts, we obtain t ∗   Zt∗ C 0 cos2 ϕe−iπ/2 it2 it2 I1 = 2 te − e dt . k l[1 + cos ϕ0 cos ϕ]3/2 t∗∗ t∗∗

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Finally, we arrive at the following expression for the integral I1: √       C 0 cos2 ϕe−iπ/2 iπ t∗∗ t∗ it2∗ it2∗∗ [t∗e − t∗∗ e ] I1 = 2 Φ √ −Φ √ . 2 k l[1 + cos ϕ0 cos ϕ]3/2 i i

(3.9)

In the case of a linear variation in impedance from one constant value to another, the first approximation Uj0 (x, z) has the form √       C 0 cos ϕ iπ x 1 i t∗ Uj0 (x, z) = − √ × 1+ − 1+ 2 Φ √ k 1 + cos ϕ0 cos ϕ l 2 2t∗ i      √   it2   it2  i x 1 i t∗∗ x 1 e ∗ x 1 e ∗∗ + 1+ 2 Φ √ − √ − + . − − l 2 2t∗∗ π l 2 t∗ l 2 t∗∗ i

(3.10)

Substituting expression (2.6) of C 0 and the parameters q q t∗∗ = kr−l/2 [1 + cos ϕ0 cos ϕ], t∗ = krl/2 [1 + cos ϕ0 cos ϕ] into (2.6) and taking into account that the parameters t∗ and t∗∗ are large enough for kr  1, for the first approximation of the solution we have √ E0 (H0 )(g1 − g0 ) 2e−ikx cos ϕ0 cos ϕF (ϕ0 , ϕ) √ Uj0 (x, z) = 1 + cos ϕ0 cos ϕ   p   krl/2 [1 + cos ϕ0 cos ϕ] 2x 1 √ Φ × 1− 1− 2 l i p     kr−l/2 [1 + cos ϕ0 cos ϕ] 1 2x √ − 1+ Φ 2 l i   ikr [1+cos ϕ0 cos ϕ]−i π l/2 4 i 1 2x e p + √ 1− 2 π l krl/2 [1 + cos ϕ0 cos ϕ]   ikr  [1+cos ϕ0 cos ϕ]−i π 4 2x e −l/2 p + 1+ , (3.11) l kr−l/2 [1 + cos ϕ0 cos ϕ] where rl/2 =

p (x − l/2)2 + z2 ,

r−l/2 =

p (x + l/2)2 + z2 .

The last two terms in (3.11) determine cylindrical waves scattered on the endpoints of the segment [−l/2, l/2]. 0 Using previous formulas and substitutions, we can easily obtain the expression for the first iteration Ujj in the case of a discontinuous variation in impedance: √    C 0 cos ϕ iπ t0 0 Ujj (x, z) = − √ 1−Φ √ , (3.12) k 1 + cos ϕ0 cos ϕ i where r0 =

p x2 + z 2 ,

t0 =

p kr0 [1 + cos ϕ0 cos ϕ].

Finally, the substitution of expression (2.6) of C 0 into (3.12) yields √    E0 (H0 )(g1 − g0 ) 2e−ikx cos ϕ0 cos ϕF (ϕ0 , ϕ) t0 0 √ Ujj (x, z) = × 1−Φ √ . 1 + cos ϕ0 cos ϕ i

(3.13)

Now compare scattered fields (3.11) and (3.13) for two cases of impedance variation: (j) g varies linearly from g0 to g1 and (jj) g has a jump from g0 to g1 . In the case of a linear variation in impedance, we can see that the behavior of the scattered wave is determined by the surface wave e−ikx cos ϕ0 with a common factor for all problems. By (3.11), this is the interference of the surface wave and two other types of waves. First, there are waves given by linear combination of Fresnel 3689

integrals. They characterize penumbra effects. Second, there are cylindrical waves scattered on the endpoints of the segment [−l/2, l/2]. In the case where the impedance has a jump, we can see that the behavior of the scattered wave is determined by the same surface wave e−ikx cos ϕ0 with the same common factor. By (3.13), this surface wave interferes with the other types of waves, which are determined by Fresnel integrals. They arise as a result of the jump of the impedance and characterize penumbra effects. This paper was supported by the Russian Foundation for Basic Research under grant No. 99-01-00107. Translated by N. Ya. Kirpichnikova. REFERENCES 1. V. B. Philippov and N. Ya. Kirpichnikova, “Diffraction of electromagnetic waves on small inhomogeneities,” Zap. Nauchn. Semin. POMI, 257, 304–322 (1999). 2. B. A. Fock, Problems on the Diffraction and Propagation of Electromagnetic Waves [in Russian], Sov. Radio, Moscow (1970). 3. L. A. Vainshtein, Diffraction Theory and the Factorization Method [in Russian], Moscow (1966). 4. G. A. Grinberg and B. A. Fock, “On the theory of coast refraction of electromagnetic waves,” in: Investigations on the Propagation of Radiowaves, Vol. 2, Akad. Nauk USSR (1948), pp. 69–81.

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