Aso'ophysics, Vol. 42, No. 3, 1999

SOME DIFFERENTIAL EQUATIONS FOR PROBLEMS OF WAVE SCATTERING IN A ONE-DIMENSIONAL MEDIUM

D. M. Sedrakian and A. Zh. Khachatrian UDC 52+530.145+519.222

A system of linear differential equations determining the amplitude of reflection R and the amplitude of trans- mission T for a plane wave (or an electron) and for an arbitral:v medium (or a one-dimensional potential of an arbitrary type) is obtained. It is shown that the problem of determining the scattering parameters R and T reduces, in general, to a Cauchy problem for a stational:v wave equation (or for the Schr6dinger equation).

1. Introduction

It is well known that one of the important problems in astrophysics is the problem of studying the phenomenon of

the transfer of light energy in various randomly inhomogeneous media and, in particular, in a medium with an arbitrary

nonuniform index of refraction [ 1 ]. In the present paper we suggest a new method of solving the problem of the propagation

of a plane electromagnetic wave in a one-dimensional linear medium with an arbitrary index of refraction. In obtaining

differential equations describing wave propagation in a one-dimensional medium, we use the method of V. A. Ambartsumian

of "the addition of a layer to a medium." The equations obtained exactly allow for all interference effects.

Let us consider a stationary wave equation that is satisfied by the amplitude of a wave in a one-dimensional medium,

d 2 = 0, ( 1 ) dx 2

where V(x) is an arbitrary function, finite along the entire x axis, that approaches zero as x ~ _ oo. In the case of an

e consists of to:/c 2, while V ( x ) = , o V c 2 ( 1 - n 2 ( x ) ) , where n(x) is the index of refraction of the electromagnetic wave,

medium, to is the frequency of the light, and c is the speed of light. Note that the wave equation written in the form (1)

coincides with the stationary Schrtidinger equation (h 2 = 2m = 1 ), and in this case E is the energy of an electron and V(x)

is the scattering potential. It is clear that problems of scattering of an electromagnetic wave or an electron are equivalent

from a mathematical standpoint.

Let the asymptotic form of the solution of Eq. (1) have the form

~(x) =eit~ R(ko)e -ik~ as x ~ - o o ,

~(X) "= T(ko)e ik~ a s x--, o0, (2)

where k 0 = , r E . The quantities R(k 0) and T(ko) are the amplitudes of wave reflection and transmission.

The problem consists in calculating R(k o) and T(k o) from the given form of the function V(x).

Yerevan State University, Armenia; State Engineering University of Armenia. Translated from Astrofizika, Vol. 42,

No. 3, pp. 419-426, July-September, 1999. Original article submitted June 1, 1999.

316 0571-7256/99/4203-0316522.00 �9 1999 Kluwer Academic/Plenum Publishers

2. Recursive Equations for T N and RN.

Before considering the problem (1), (2), let us consider the problem of wave scattering from a one-dimensional

system consisting of a finite number of uniform layers. It is well known that in this case the problem of determining R and

T reduces to the problem of calculating a product of second-order matrices [2, 3],

(.I.; - . ; I . ; I - fir.I.: -'.:1';;I. - R N I T N lIT N , 1/t,, } (3)

where T N and R N are scattering parameters of the system, N is the number of uniform media, and r,, and t,, are scattering

parameters of the nth uniform layer [4];

t,~l = e2"~ [ c~ k'~ + k,,k o sin2 k,fl,, ], (4)

r,,= i e2ik,,.~.(k~_ko)sin2k,,d,," t,, 2k, k o (5)

In (4) and (5) k 0 = 4E" and k = ~ . The quantities V characterize the value of the function V(x) in the nth

layer and x is the coordinate of the middle of the nth layer.

As shown in [5], the problem (3) in general form reduces to the problem of solving some system of finite-difference

equations. We introduce the notation

1 * * * H (ll--tr: -rn/tnl_ ( ) = 2 . , ,

k--RN_,/TN_, l/To_ , ) / t lit, f (6) n=N_l \ n n

It is clear from (6) that TN. ) and RN. ~ are the amplitudes of wave reflection and transmission for the system in which

the first and Nth uniform layers are absent.

For the quantities S u= 1/T u and S,, = R N/T N as functions of the discrete variable N we can obtain the following

system of difference equations directly from (3) and (6):

SN 1 "u'~ * '7 - "N * = - SN-)+ --u S N - j + ~ SN-l+ SN-j, (7) tNtl tNt I tNtl tNtj

1 = rNq - , r 1 _ r N , = , . . . . . + ~ S u i . (8) SN tNtl HN-I+ tNtl SN-I+ tNtl SN-I tNtl -

Equations (7) and (8) can be used to solve the problem (1), (2), since an arbitrary function V(x) of (1) can be

approximated with any given accuracy using a system of uniform layers.

3. Differential Equations for T ( x ) and R ( x )

We introduce the functions S(x , ,x2)= 1 T(x, ,x2) and S(xl ,x2) = R(x,,x2) T(xl ,x2) , where T (XI~X2) and R (Xl~2) are the amplitudes of wave transmission and reflection for the part of the "function" V(x) lying between the points x~ and

x 2. Thus, for x~ < x 2 we have

V(x ,x , ,x~)= V(x)O(x-x,)O(x2- x), (9)

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where q(x) is the Heaviside function.

Then the function V(x, x l - A x j ,x2WAx2) , i.e., the part of V(x) lying between the points x t -Ax ~ and x 2 + m ~ . 2 , where

Av~ and zXx 2 are small quantities, will look like the function V(X, Xl,X2) of (9) with the addition to it of uniform layers.

And the layer added on the left will be characterized by the value of the function V(x~) and the width ~ '~, while the layer

added on the right will be characterized by the parameters V(x 2) and Av 2.

For the amplitudes of reflection and transmission of an infinitely narrow layer, from (4) and (5) we have

t,, 1+ iV,, rn iV,, �9 . = , = - e2'k"'~" �9 (10) 2 k o 2 k o

Substituting SIv=S(x , -Ax , , x2+Ax2) , S N = S ( x t - A x , , x 2 + A x 2 ) , SN_I-~-S(xI,x2), and SN_I..~.S(xI,X2) into

(7) and (8) and expanding the resulting equations with respect to the small quantities Ar m and Ar 2, with allowance for (10)

we can obtain the system of equations

os =_iy(xQ s _ i V ( x ! ) e_2i,o.,. S ' (11) 0 x I 2 k o 2 k o

aS i V ( x l ) s + i V ( x , ) . . . = ........ e2&~aLS,

Oxl 2/% 2k0 (12)

OS = iV(x2) s iV(x2.)e2i,,,x,_S *, Ox2 2ko 2ko

(13)

a S iV(x., ) S + iv(x., ) e2itox" S* ax 2 2k0 2ko

(14)

Uniqueness of solutions of the system (11)-(14) is provided by means of the corresponding initial conditions:

s( ) = l, s ( x , , x 2 ) =o. Xl'X2 1.1__ x t ' l - - ~" " - - 2 " - - ' 2

We now show that the requirement of conservation of flux density,

S(Xl,X2) 2- S(xl,x2)i 2,~- T(Xl,X2)2+ R(XI,X2) :2 = 1 ,

(15)

(16)

for any x, and x 2, follows from the equations defining S(x,,x2) and S(Xl,X2).

In fact, differentiating (16) with respect to x I and x 2, we obtain

sOS:+s * a-S - s ~ =0, OX 1 OX 1 OX l OX I

(17)

s OS* + s , o_S _s OS*_ a. _~ =0. (18) Ox z bx2 Ox 2 Ox 2

Substituting (11)-(14) into Eqs. (17) and (18), we easily ascertain that the latter are satisfied.

We now consider Eqs. (11) and (12) for the case of a variable x~ and a fixed x 2 . Then, introducing the definitions

S(Xl,X2) = - P(x) and ~ ( x , , x 2 ) = P ( x ) , for the functions P(x) and P (x ) we obtain the system of equations

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d P = _ iV(x) P - iV(x) e_Zit0.,~, dr 2 k 0 2 k 0 (19)

dP = iV(x ) 1:'+ iV(x) e2ik~ dx 2 k o 2 k o (20)

The functions P(x) and P(x) describe the variation of the scattering parameters T(x) and R(x) for the function

u(x) = v(x)o(x- b),

as a function of x.

In the case of a variable :r and a fixed x,, the functions S(Xl,X2) = D(x) and S*(Xl,X2)= D(x) describe the

variation of T(x) and R(x) for the function

U(x) = V(x)O(a- x).

From (13) and (14) we obtain the following system of equations for the functions D(x) and D(x) :

dO iV(x) iV(x) 2 i t , ; = - D - - e t D ,

dx 2 k o 2 k o (21)

d D = i V ( x ) D+ !V(x) e.2ito.,.D. dx 2 k o 2 k o (22)

4. The Cauchy Problem for Scattering

We now show that from the systems of equations (19), (20) and (21), (22) one can obtain linear equations for the

combinations of functions P(x), P(x) and D(x), s one of which coincides with the wave equation (with the Schr6dinger

equation if one is considering a problem of electron scattering) while the other coincides with a simple first-order equation. In fact, let us write Eqs. (19), (20) in the form

We introduce the notation

and then Eqs. (23) and (24) take the form

eiko.rdP= iV(x)(peito.r+e-ikoX~]" dx 2 k o J

e-ikox dP _ iV(x) (eeiko.r..l_~e-ikox ~ dx 2k 0 /"

Pe ik~ = F 1 and Pe -ik~ = FI,

(23)

(24)

(25)

d-Fl = - ~ k o (Fx + FI )' (26)

[+ikoFl = / V (Fl+ Fl}. ZK o

(27)

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From these equations it is easy to show that for the quantities L, = (Fl + ~ )

equations

d x " + ( E - V ( x ) L , (x )=O, Q , = - k o ~/x"

and Q1 = (F1-FI) one can obtain the

(28)

Introducing the quantities De-ik~ F 2 and De ik~ ~ and performing analogous operations to the system (21),

(22), for the quantities L 2 = ( F 2 - F 2 ) and Q2 =(F2 + F2) we obtain the equations

2 +(E-v(x ) L,(x)=0, 02 i dL, = . . . . . ' ( 2 9 )

" k o dx

Since the additions of a layer on the right and on the left are equivalent, it is convenient to consider the case in which

a layer is added on the right. We designate L 2 = L,F 2 = F , and F 2 = F , and then the first equation of (29) takes the form

[jx2 (30)

while F and ff are expressed in terms of L by the equations

F=I~( - i -dL+L t F = l ( i dL L~ 2 iix ,ii dx - j (31)

Let us assume that we have a layer starting at the point x = a and with a thickness d = b - a. Then, in accordance

with (15), the boundary conditions for the function L(x) at the point x = a will be

L(a) = e -ik~176 _dL_ = _ikoe_ikoa" dx

We seek the solution of Eq. (30) in the form

L(x) = e-ikaa( H I - ikon 2),

and then the real functions n t and H 2 satisfy Eq. (30) with boundary conditions that follow from (32):

H , ( a ) = 1, H 2 ( a ) = 0 , dill all2] dX lx_a =0' dx l.,.=a =1"

(32)

(33)

(34)

Substituting the solution (33) into (31) and using the relationship between F and

-1- = l~ei~od {H, + dH~-ikoH2+ i_ dH'-} T 2 dx k o '

and D, i ) we finally obtain

(35)

- - e2ik~176 -- H1 + - ikoH2 , (36) T 2 dx k o dx

where x o = (a + b)/2 is the coordinate of the center of the layer.

We have thus shown that the solution of the problem (1), (2) reduces to the Cauchy problem for the wave equation

(Schr6dinger equation) (30).

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5. Conclusion

In conclusion, we make two important, in our opinion, remarks. First, we note that the explicit form of the functional

T IV(x)] can be given. Substituting (10) into (7), we obtain

where

N N �9 n - I

T N ' = I + E E tVi'djl ""iVjvdiP I-I(l-expi2/`(xJ,. ,-xJ,)" p = l l<--jl<...<jp 2ko 2ko /=1

Taking N --, o0 and max d,, ~ 0, we can obtain the explicit form of T [V(x)]:

o0

T -1 = I + E W , , , n = l

(37)

in [6] by the method of phase functions. In fact, substituting P(x) = 1/T(x) and e(x) = R(x) T(x)

R(x), in particular, we obtain the usual Riccati equation:

(38)

.", iv(.,., ) x,, ) w,,= J " Y 2/,,, "'" 2/, l l ( I - e x p 2 i / , o ( x , + , - x , l ) a x , . . , dx,,.

an_ I -oo 0 I=1

Second, from Eqs. (19) and (20) it is easy to obtain the well-known equations for the functions T(x) and R(x) found

into (19) and (20), for

dR(_x) = iV (eito.,+R(x)e_ikox)2 ' R(oo)=0. dx 2k 0 (39)

It is important to note that using equations of the type (38) to solve the scattering problem (1), (2) encounters serious

mathematical difficulties. Equation (30) proposed here for the solution of this problem is not only linear but an equation

coinciding with the usual wave equation (the Schr6dinger equation).

The authors wish to thank D. Badalian, A. Nikoghossian, and A. Sahakyan for a discussion of the results obtained.

REFERENCES

1. I. V. Klyatskin, Embedding Method in the Theolw. of Wave Propagation [in Russian], Nauka, Moscow (1986). 2. P. Erdos and R. C. Hemdon, Adv. Phys., 31, 65 (1982). 3. M. Ya. Azbel, Phys. Rev., B22, 4106 (1983).

4. N. D. Blokhintsev, Fundamentals of Quantum Mechanics [in Russian], Nauka, Moscow (1983) [D. I. Blokhintsev, Quantum Mechanics, translated by J. B. Sykes and M. J. Kearsley, Kluwer Academic, Hingham, Mass. (1964)].

5. D. M. Sedrakian and A. Zh. Khachatrian, lzv. Nats. Akad. Nauk Armenii, 34, 138 (1999). 6. V. V. Babikov, Method of Phase Ftmctions in Quantum Mechanics [in Russian], Nauka~ Moscow (1976).

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