REFLECTION OF ELECTROMAGNETIC WAVES FROM A MOVING DIFFUSE INTERFACE OF TWO MEDIA UDC 538.574.2

V. A. Davydov and V. E. Rok

Taking Maxwell's equations as the basis, a general expression is derived for t h e angular and spectral distributions of the energy radiated by arbitrary source located in a medium, whose dielectric constant contains a small correction as a function of coordinates and time. A general expression is also obtained for the characteristics of waves reflected from an arbitrarily moving boundary. The spectral energy density, the total power, and the reflection coefficient are obtained for waves reflected from a smooth interface of two media having the form of Epshtein transition layer and executing harmonic oscillations.

A quantum-electrodynamic perturbation theory was developed in [1] for the radiation of electromagnetic waves in inhomogeneous and nonstationary media. For media with dielectric constant of the form

e=~o+~l(r,t)

(1~11 ~ ~o),

(1)

the following expression was obtained for the angular distribution o f the energy in the radiated spectrum for a wave with frequency co and wave vector k:

Wk,x d3le = (2~)' to2/4% ] ~ dkxdtots, (k -- k,, to -- ~,) e~e,

(t~,, to,)]~ a,~k.

(2)

Here e x is the unit polarization vector and E (k,, %) and Sl (k,, to,) are, respectively, the Fourier transforms of the electric field o f the source located in the nonstationary and inhomogeneous medium (charge, dipole, etc.) and the variable part of the dielectric constant (1). Formula (2) can be derived also by the classical approach. We shall seek the solution of Maxwell equations in the form E = E0 -? E,,

/ - / = H0 § H, , where E o and H o are the solutions of Maxwell equations with dielectric constant

eo and E 1 and H I are small corrections to the field intensities caused by the variable part of the dielectric constant. Discarding terms of higher order than first, from Maxwell equations we obtain a system of equations for E 1 and H, : %dlvE 1 =--div(s,

E0) ,

divH1 ~---0, (3)

rot E1

1 0 H,, c ot

rot H,

% 0 Ei + 1 O cot -2 ~ (~eo).

System (3) formally coincides with Maxwell equations if we put 9 = (-- 1/4 ~) div (el E0),

j = (1/4 ~) (0/0 t) (s 1 E0).

We note that the "charge and current densities" thus defined satisfy the continuity equation. We shall use the method discussed in [21 for computing the radiated energy. At large distances the Fourier components of the vector potential of the radiation field have the following form: e tke~

(4)

where R o is the distance from the coordinate origin to the observation point and k ----(~o/c) ]/e~

9 Expanding e , E o into

Fourier integral, from (4) we get A~ =

ietkt~~ 170

to

2c

(2~) ~ (~,E0)k.~ =

ietk~o

to

Ro

2c

(2~)~ ~ dU, ato, E0 (U,, to1)~, (t~ -- U,, to --~l).

(5)

Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiof'~zika, Vol. 25, No. 3, pp. 334-338, March, 1982. Original article submitted April 27, 1981.

236

0033-8443/82/2503-0236507.50

9 1982 P l e n u m Publishing C o r p o r a t i o n

From (5) we obtain the Fourier component of the magnetic field intensity H~ =

e ~ o ,o~ (2~) ~ Ro

VTo in, (e, E),,~],

(61

2d

where n is the unit vector in the direction of k, i.e., n =

[e* e q ,

(7)

with e ~'2 are the polafization unit vectors. The energy radiated into a solid angle d~2 in the frequency interval dw is given by the square modulus of (6). Considering (7) we get vr do~ d a =

(2a)~ co' 1/~ ~ [e) (qe)~,~[ ~d~ d a , 4ca x~l

(8)

which coincides with (2) summed over polarizations, if d a k is replaced by (m=/ca) ,a/~ de df2. If an electromagnetic wave is taken as the source of the external field E in formula (2), then in this particular case (2) will describe the reflection and refraction of this wave in the inhomogeneous and nonstationary medium. Let the dielectric constant o f the medium be of the form e x p [ a ( z - - g(t))] e(z, t) = e 0 + A e l + e x p [ a ( z _ _ g ( t ) ) ] ,

lh~[ << co.

(9)

Th2s expression represents a diffuse interface o f two media moving in accordance with the law z = ~(t), where L = 1/~ is the characteristic width of the diffuse zone. We compute the Fourier transform of e x (k, co):

2 exp[i(o#--kr)l

I +e~e-~(~e-~(t) drdt.

(10)

Making the substitution u ~ e~z e-~g(t) we get s~ (k, o~)= A~g(k~)6 (k~)_(2ma

where F(c0,

Jf dudt (tt)-ikz/~-il--+-uexp[i(tot--kz ~(t))] = - - ihe4~_..~6(kx)6(kV)-sh(ak~/~)F(ro, kz),

(11)

k~) = 5dtexp[i(ojt--k~(t))].

Let an electromagnetic wave E = Eocos

(kor -- mot)

(12)

be incident on the moving interface; for the sake of being specific we shall take the direction of koa to be along the negative z axis. ,The Fourier transform of the field intensity of the wave (12) has the form E (t~,, ~) = 0/2) Eo (~ (t~ - - &) ~ (,~ _,oo) + ,~ (th + t~o)~(o~ + o~o)).

(13)

For the subsequent discussion it is necessary to note that the boundary conditions at the moving interface [3] imply that if the plane of the interface is perpendicular to the z axis and it is moving along the z axis, then the projections of the wave vector on x and y axes remain continuous during reflection and refraction of the electromagnetic wave. The integral in formula (2) is easily computed and is equal to (14) 8~

sh (~ (k~ - - ko~) /a)

Let us now compute the radiated energw. It follows from (2) and (14) that the radiated energy is proportional to the square of product of two 6-functions; therefore we must be concerned with radiation from a unit st~rface area o f the interface. Let us introduce spherical coordinates 0, 9, 0 o ; then

k~ -~- (o/c) ]/To sin 0 cos % ko.~= (too/c) ]/-~ sin 0o, kv ~- (r

1/-~ sin 0 sin %

k~ = (o~/c) ~7o cos 0,

(15)

kou = O,

ko, -----(o~olc) ~'~ cos 0o.

23 7

Then for tim energy radiated from

W,,~d~le

a

u n i t surface area of the interface we obtain the following expression:

i ~" ((o - - coo, (1/E/c) (~ cos 0 - - coo cos 0o)) I" • o ~(a~) ~(Eoi e:~)-"~o' ~'~'~64~ 2 c~ s h ~ ( ( ~ ' 7 o l c ~ ) (o~ cos 0 - - ~o~cos 0o) )

(16)

:< ~ ((yTo/c) (~o sin 0 cos ~p- - ~0 sin %) ) 6( (lI e~/c) a sin 0 sin tp) sin 0 dO dq9 din. The integration over the angles 0 and ~ is elementary using the formula 6 ( f ( x ) ) =

2

6(x--xd lf'(x01

[4], where x i

are the roots of the equation f(x) = 0. Finally we have

Wo do) ----- r176(Ae):l F (~o - - o)o, ( ( f ~ d c ) (o) cos 9' - - ~o~cos 0o) ) I ~(Eoi ei x) ~ do), 64c "1/ eo a 2 sh 2 ( (,n I/E/ca) (o) cos 0' - - oo cos %) ) cos 0'

(17)

where cos 0' = l/1 - - (0~%ho2) sin ~"%.

(18)

The q u a n t i t y (EoieiX)2 for the wave polarized in the plane of incidence (X = 1) is equal to E~ cos2(0o + 0')

, while for

the wave polarized perpendicular to the plane of incidence (X = 2) it equals - E ~ . Although the spectral energy distribution of the reflected waves is given b y the squared m o d u l u s o f f u n c t i o n F (or the law o f m o t i o n o f the diffuse interface), the general formula (17) derived for an arbitrary law of m o t i o n exhibits m a n y i m p o r t a n t characteristics o f reflection from a moving boundary. Specifically these are: when a m o n o c h r o m a t i c wave is incident on a moving boundary, a whole spectrum of waves is reflected from the b o u n d a r y at different angles; the angle of reflection is related to the angle o f incidence o f the wave through formula (18); the energy o f the reflected short waves

(c/~-~ ~ << L) is exponentially small; the first-order correction to the expression of the energy of waves reflected from a sharp b o u n d a r y due to the diffuse n a t u r e is proportional to L 2 . Let us n o w use these formulas in the case o f reflection of wave (12) from the diffuse interface of two media undergoing h a r m o n i c oscillations. In this case ~(t) = A cos a t . The q u a n t i t y F(co, kz), introduced in formula (11), is equal to

F(o~, kz) -~- S dtexp[i(~ot--kzAeosf~t)]

=2r~

~ t2 =

Here we have made use of the fact that exp (iz cos

q)) =

~

n~-r

i'~]n(--Ate~)f(o~+na).

(19)

--O0

i '~1~ (z)e~n~

[5].

In c o m p u t i n g the spectral energy distribu-

tion of the reflected waves from (19), we come across squares of ~-functions; therefore, we are concerned only with energy reflected from the interface i n u n i t time. Integrating over r in ( t 7 ) we o b t a i n the expression for the total power of the reflected waves: =

~~]j~ 3 2 c 1/-7o , ~

where ~0~ = coo- - nO,

=_

Q

-- A

V~

) (~o~ cos 0 - - ~ cos 0o)

012/',oXp "12 s h ~ @/4V~lc)(o~,c o s o~ - ,oo c o s Oo)) c o s o,<

(20)

cos 0~ = ]/1 --ttt0~/~2~01 n~ sin '2 00 , and N is the largest integer contained in w o f12,

(e}Eo32 = leg cos2(0o + 0~), ~ = 1.

t For c o m p u t i n g the reflection coefficient we have to find the ratio of (20) to the power incident at a u n i t area of the interface: I = P

c

)-1 V % E02cos 0o 8~

(21)

(we recall that since the direction o f koz is along the negative z axis, cos 0 o is a negative quantity). In the particular case ~2 = 0, L = 0 we easily o b t a i n Fresnel formulas (of course, u n d e r the c o n d i t i o n Ae << % ) keeping i n m i n d that d~ (x) = 1 . Thus, when a m o n o c h r o m a t i c wave is incident at an oscillating interface of two media, waves forming

238

a discrete spectrum are reflected from the interface with each wave having its own angle of reflection and the frequencies of the waves are separated by multiples of the oscillation frequency of the interface. The reflection from unifornfly moving transition layers had been investigated earlier by Stolyarov [6, 7]. The authors are ~ateful to him for valuable discussions of the present work. They also thank B. M. Bolotovskii for attention and interest. LITERATURE CITED 1. 2. 3. 4. 5. 6. 7.

V.A. Davydov, Zh. Eksp. Teor. Fiz., 8_00,No. 3, 859 (1981). L.D. Landau and E. M. Lifshits, Field Theory [in Russian], Nauka, Moscow (1973). B.M. Bolotovskii and S. N~ Stolyarov, Einshtein Collections [in Russian], Nauka, Moscow (1975). D.D. Ivanenko and A. A. Sokolov, Classical Field Theory [in Russian], Gosteldfizdat, Moscow-Leningrad (195t). I.S. Gradshtein and I. 1~ Ryzhik, Tables of Integrals, Series, and Products, Academic Press (1966). S.N. Stolyarov, Kvantovaya Elektron., 4, No. 4(58), 763 (1977). S.N. Stolyarov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 2_[1,No. 2, 174 (1978).

METHOD OF NONORTHOGONAL SERIES AND RAYLEIGH HYPOTHESIS IN EXTERNAL PROBLEMS OF DIFFRACTION V. F. Apel'tsin

UDC 535.12:535.42:517.947:517.52

Under consideration is the possibility of representing the solutions to external plane problems of diffraction of waves within regions bounded by smooth contours in the form of series of divergent cylindrical waves, which implies solving these problems by the method of nonorthogonal series. The validity of such a representation of a solution throughout a given region must be proved, since it is based on the Rayleigh hypothesis (assumption that such a representation of the solution is regular all the way to the boundary of the region). A method is proposed for obtaining effective sufficient conditions which will make the Rayleigh hypothesis applicable to the regular part of the Green function in an external problem in the class of contours p = p@) describable in polar coordinates by analytical functions p@) holomorphic within some range - ~ < Im ~0 < ~. Also considered is the possibility of applying the method of nonorthogonal series to the case of piecewise-analytical curves bounding a region.

The method of constructing the solutions to boundary-value problems in the theory of diffraction in the form of series of partJ.cular solutions to the corresponding problems with constant coefficients (these coefficients determined from limiting conditioias) is well known to be effective when the solution is sought in a region bounded by coordinate surfaces. Extending this method to the case of noncoordinate boundary surfaces often encounters certain difficulties. The series coefficients can be determined only approximately, through truncation of infinite systems of algebraic equations which satisfying the boundary conditions in one or another sense yields. The matrices of the corresponding systems are quite nondiagonal (particular solutions to the equations on noncoordinate surfaces already do not form orthogonal complete systems of functions but, as a rule, are only complete linearly independent systems). The truncation method is valid, in the strict sense, only in the case of basality of the selected system of particular solutions on the region's boundary. Furthermore, such a method of constructing the solution usually imposes on the diffracted field a representation in the fom~ of an infinite series of waves of a certain kind. In external problems of diffraction in open regions, for example, the field everywhere outside the body surface is represented as a series of divergent spherical (or cylindrical) waves with a common center inside the body surface. The proposition that the scattered field is representable in this form constitutes the essence of the Rayleigh hypothesis [ 1] (the abbreviation R.h. will be used henceforth), the validity of which has been questioned for a long time on account of unsuccessful attempts to numerically implement the method of nonorthogonal series for bodies of shapes very different than a sphere (circIe) [2]. In various analytical studies of this problem [3, 4] there have been established the applicability limits for the Rayleigh hypothesis (tLh.) in certain special cases. It has also been demonstrated [5] that an obstacle to such a representation of the scattered field will be the location of its singularities, as the field is analytically extended inside the surface (contour) Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 25, No. 3, pp. 339~347, March, 1982. Original article submitted March 3, 1981.

0033-8443/82/2503-0239$07.50

9 1982 Plenum Publishing C o r p o r a t i o n

239