probability density function and moments of the field in a slab of one-dimensional random medium
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Probability density function and moments of the field in a slab of onedimensionalrandom mediumR. H. Lang Citation: Journal of Mathematical Physics 14, 1921 (1973); doi: 10.1063/1.1666270 View online: http://dx.doi.org/10.1063/1.1666270 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/14/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Density functional theory of one-dimensional two-particle systems Am. J. Phys. 66, 512 (1998); 10.1119/1.18892 Stability of a onedimensional plasma in slab geometry Phys. Fluids 30, 3502 (1987); 10.1063/1.866431 Mean power reflection from a onedimensional nonlinear random medium J. Math. Phys. 27, 1760 (1986); 10.1063/1.527042 Backscattering and localization of highfrequency waves in a onedimensional random medium J. Math. Phys. 25, 1378 (1984); 10.1063/1.526279 Propagation in a onedimensional random medium J. Acoust. Soc. Am. 74, S96 (1983); 10.1121/1.2021242
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Probability density function and moments of the field in a . slab of one-dimensional random medium
R. H. Lang
Department of Electrical Engineering and Computer Science, George Washington University, Washington, D. C (Received 15 December 1972)
The problem of a plane wave normally incident on a slab of one-dimensional random medium is studied. The refr/lctive index variations of the random medium are taken to be a stationary Gaussian-Markov process. By employing an invariant imbedding technique and by using the Markov property of the refractive index variations, two cascaded diffusion equations are obtained for the probability density function of the reflection coefficient and the field in the slab. These equations are then solved approximately for small refractive index fluctuations and an expression is obtained for mean intensity in the slab interior.
1. INTRODUCTION
This paper is a study of wave propagation in a slab of one-dimensional random medium. The slab has width L; the medium inside the slab has a refr~ctive index variation n(x) = [1 + E~(x)]l/2, O:::s X:::S L .. We assume that ~(x) is a stationary Gaussian-Markov process with zero mean and exponential correlation l (OrnsteinUhlenbeck proce~) and that E is a small parameter. The regions x > L and x < 0 are homogeneous regions with unit refractive indexes; and a plane wave is normally incident on the slab from the region x > L.
This problem has been the subject of a number of investigations which have appeared in the literature. The investigators have directed their efforts toward finding approximate expressions for the moments of various statistical quantities associated with the slab such as reflection coefficient, transmission coefficient, and the field. Gertsenshtein and Vasil'iev2 considered a medium composed of discrete random inhomogeneities. They found an expression for the mean square reflection coefficient in the limit as their discrete medium approached a continuum. Gazaryan,3 again using a discrete model, found expressions for the mean field and intensity of the field in the continuum limit. The problem was then treated by a number of investigators who used a continuum model of the random medium directly instead of first considering a discretized medium. Kupiec, et aZ., 4 found an expression for the mean field by applying the method of smooth perturbations to the Dyson equation. Papanicolaou5 and Morrison, Papanicolaou, and Keller6 have found expressions for the probability density function (p.d.f.) of the transmission coefficient and from this have calculated the mean squared transmission and reflection coefficients.
In this paper we employ the medium model used by Morrison-Papanicolaou-Keller and find the p.d.f. of the reflection coefficient and the field inside the slab. From these p.d.f.'s we obtain an expression for the mean intensity in the slab and, in addition, recover the results of Refs. 4,5, and 6. Our expression for the mean intensity is similar to Gazaryan's, however, his result has been obtained by using different methods and a different model.
In Sec. 2 we start by reformulating the original problem as a boundary value problem over the interval [0, L]. The boundary value problem is then imbedded in two cascaded initial value problems by employing an invariant imbedding technique. 7 ,8 The solution to the first initial value problem will be called the generalized reflection coefficient r. This solution provides initial conditions for the second initial valu~ problem whose solution yields the desired field u.
1921 J. Math. Phys., Vol. 14, No. 12, December 1973
The probabilistic nature of the problem is introduced by first representing r in terms of its amplitude and phase, i.e., r = p expi¢ and then recognizing that X = (p, cp, ~) is a vector Markov process. As a result, a forward Kolmogorov equation can be written for the p.d.f. of X. In a similar manner, one finds that the second initiai value problem generates a vector Markov process Y = (v, e,p, cp, ~). Here v and e are the log amplitude and phase of the field u, respectively. A second forward Kolmogorov equation can then be written for the p.d.f. of Y. The initial distribution of the random variable Y is obtained from the solution of the first diffusion equation. Thus the original stochastic problem for the field u has been replaced by the deterministic problem of solving two cascaded diffusion equations.
In Sec. 3- 5, we find approximate solutions to the diffusion equations for small E. Our perturbation method follows the one that Morrison- Papanicolaou- Keller 6 used. Once approximate expressions for the p.d.f.' s of X and Y have been developed, the moments of the reflection coefficient and field are calculated.
2. FORMULATION
We consider a plane wave which is normally incident on a slab of one-dimensional random medium. The field u obeys the one- dimensional wave equation
d2u - + K 2n2 (x)u = 0 - 00 < x < 00, dx2 '
(2. 1)
where we require that du/ dx be continuous. Here we take K to be the free space wavenumber and n(x) to be
11 ,x < 0,
n(x) = [1 + E~(x)]1/2, O:::s x:::s L,
1 ,x> L. (2.2)
If we assume that the process Hx) has a correlation length Z and if we define
x = x/Z, L = L/Z, k = ZK, )leX) = ~(x), (2.3)
then (2.1) can be put in the following normalized form:
d2u - + k 2 [1 + EJ.L(X)]U = 0, dx2
O:::s X:::S L,
in the slab region. Outside the slab, the normalized solution can be obtained explicitly. It is
{
e-ik(x-L) + Re +ik(x-L),
u(x) = Te- ikx ,
x~ L,
x:::s 0,
Copyright © 1973 by the American Institute·of Physics
(2.5)
1921
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1922 Roger H. Lang: Probability density function
Here R and T are called the reflection and transmission coefficients of the slab.
The problem on the infinite interval can be reformulated in terms of the following problem on the finite interval:
d2u(x) d;2 + k 2[1 + EI-L(X)]U(x) ::::: 0, 0 :$ X:$ L, (2.6a)
du(O) -- + iku(O) ::::: 0, dx
du(L) -- - iku(L) ::::: - 2ik.
dx
(2.6b)
(2.6c)
The boundary conditions have been obtained by using (2.5) along with the continuity of u and du/dx at x::::: L. We also find that
R ::::: u(L) - 1, T ::::: u(O). (2.7)
Next we convert the boundary value problem (2.6) to an initial value problem. This is accomplished by using an invariant imbedding procedure. 7 ,8 To apply this procedure, we exhibit the dependence of the field u on the slab width L explicitly, i.e., u ::::: u(x, L). Next we replace L by the variable t, 0:$ t:$ L, and riote that u(x, t) satisfies (2.6) with L replaced by t. Thus we have
(
d2 - + k 2[1 + EI-L(X)] u(x, t) ::::: 0, dx2
0:$ x, t :s L,
~ u(O, t) + iku(O, t) ::::: 0, dx
~ u(t, t) - iku(t, t) ::::: - 2ik. dx
(2. Sa)
(2. Sb)
(2. Sc)
In Appendix A, we convert the above differential equation in x with t as a parameter to two cascaded initial value problems in t with x as a parameter. These initial value problems are:
Problem 1:
dr(t) -- ::::: 2ikr(t) + tiEkl-L(t)[l + r(t)]2,
dt
r(o) ::::: 0, 0 :s t :$ Xj
Problem 2:
du(x, t) 1
-- ::::: iku(x, t) + "2iEkI-L(t)[l + r(t)]u(x, t), dt
u(x, x) ::::: 1 + r(x),
dr(t) -- ::::: 2ikr(t) + ~EkI-L(t)[l + r(t)]2,
dt
r(t) I t~x ::::: r(x), x :s t :s L,
(2.9a)
(2.9b)
(2. lOa)
(2. lOb)
(2. 11a)
(2.l1b)
The solution to the first problem obeys a Ricatti equation. We will call this solution, r(t), the generalized reflection coefficient since r(I,) ::::: R as is shown in (A7) of Appendix A. Once r(t), O:s t:s x, is determined, it provides the initial conditions for the second problem.
J. Math. Phys., Vol. 14, No. 12, December 1973
1922
The solution to the second problem provides us with u(x, t), .. x :$ t :s L, and thus we have the solution to the original boundary vaIue problem (2.6) by setting t ::::: lin u(x, t).
Since I-L(t) is a Gaussian process, it can be specified completely by its first two moments. They are
(2.12)
where the brackets indicate the expected values of the quantities enclosed. It is also noted that the process p.(t) can be generated from the Ita stochastic differential equation9 , 10
(2.13)
where (3(t) is the standard Brownian motion process. The distribution of I-L at t ::::: 0 is taken to be (21T)-1/2 x exp(- 1-L2/2).
Equations (2.13) and (2.9) can be solved simultaneously. Before doing this, however, we rewrite r in terms of its amplitude and phase, i.e.,
r ::::: pe i¢, 0 :s p :$ 1, - 1T < rp :s 1T. (2.14)
Upon substituting (2.14) into (2.9a) and equating real and imaginary parts to zero, we obtain two equations involving p and rp. When these two equations are considered along with (2.13), we have the following system of stochastic differential equations:
d(~ {~::'~:k~~:)+~¢+ p-') COS¢~dt + fi~D' o :s t :s x. (2.15)
The initial conditions for the above system are: p ::::: 0 with probability one; rp is distributed uniformily; 11
and u has the initial distribution associated with (2.13). Noting that the initial data is independent of the Brownian increments, d{3, 0:$ t :$ x, one can show that (2.15) generates a three-dimensional Markov process, X::::: (p, rp, I-L) and that the P.d.f.P1(X' t) obeys the forward Kolmogorov equation10
(2.16)
where
(2.17)
and
L<p::::: kl-L {(p2 -1) sinrp ~ - [2 + (p + p-1)COSrp] ~ 2 3p arp
+ (3p + p-1) sin¢}. (2. IS)
The initial data for (2. 16) is given by
(2. 19)
where 6(p) is tpe Dirac delta function. By using p 1 evaluated at t ::::: L one can calculate the moments of R, but to obtain the moments of u(x, L), one must consider the diffusion equation associated with Problem 2.
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1923 Roger H. Lang: Probability density function
With this aim in mind, we introduce (2.14) and
u=eik(t-x)ev+i6, -~< v<~, -1T<e::S1T, (2.20)
- 2-1ke/J.P sinq,
2-1kef..L(1 + p cosq,)
= 2-1kef..L(1 - p2) sinq, dt + ...fi d
2k + 2-1kef..L[2 + (p + p-l) cosq,]
-f..L
The solution to (2.21) generates a five-dimensional Markov process, Y = (v, e, p, <p, f..L), since the initial distribution of Y at t = x is independent of d{3, x::s t ::s L. Therefore the p.d.f'P2(Y' x, t) satisfies the following forward diffusion equation:
ap 2 at = L 2P2, L 2 :::= L(R) + EL(~), (2.22)
where L(~) = L(~), which is defined in (2.17), and where
L(P = L(y> + kf..L (p sinq, l... - (1 + p cosq,) l...) 2 \ av ae
(2.23)
with L (}> being given by (2.18).
The initial distribution of Y at t = x is obtained by using PI (X, x) along with (2. lOb) and (2.20). The result is
P 2 (Y, x, x) = PI (X, x)o[ V - f(p, q, )]o[ e - g(p, q, )],
where
f(P, q,) = t In[1 + p2 + 2p cosq,],
g(p, q,) = tan-1[p sinq,/ (1 + P cosq, )].
3. PERTURBATION ANALYSIS OF PROBLEM 1
(2.24)
(2.25)
(2.26)
We will now find solutions to (2.16) for small E. We start by representing P 1 (X, t) in terms of the eigenfunctions of the operator L 1 , Le.,
P1(X,t) = ~ aqe>..qtVq(X), q
where the eigenfunctions satisfy the equation
(3.1)
(3.2)
Since Ll is a three-dimensional operator, it should be noted that index q is also three dimensional.
Because the eigenfunctions Vq are difficult to obtain, we take advantage of the fact that Ll is composed of a sum of an operator L (~), whose eigenfunctions are readily obtainable, plus a small operator EL <i). Expanding
(3.3)
in a power series in E, then plugging (3.3) into (3.2) and equating equal powers of E, one obtains an infinite set of perturbation equations. The solution to the first of these equations yields ~(o) and its corresponding eigenvalues A ~o). Higher order v~n) are then found by successively solving the higher order perturbation equations. The A (~), n = 1,2, ... , are obtained by requiring
J. Math. Phys., Vol. 14, No. 12, December 1973
1923
into (2.10) and (2.11). This results in four real equations for v,e,p,and q, to which we append (2.13). The resulting system of stochastic differential equations is
x::s t::s L. (2.21)
that the Vq(n) , n = 1,2"", by periodic in q, over [0, 21T]. This procedure for determining the A (q) eliminates secular terms in t from occurring in the perturbation expansion ofPl()(,t).
By employing these approximate eigenfunctions and eigenvalues in (3.1), we have
. (>..(0) (1) 2,(2) P
1(X, t) :::= ~ aqe q +t>.. q +< " q)tV (~)(X) + O(e),
q
O::s E3t < 0(1). (3.4)
Since the calculation of ~(o) and A~n), n :::= 0,1,2, is similar to the calculation that has been performed in Ref. 6, we will not include it here. Once these quantities have been found the a q in (3.4) can be computed by using the initial data given by (2.19) and the orthogonality properties of the eigenfunctions. Next we integrate out the f..L dependence from PI (X, t) since it will not be required. We finally obtained
:::= (211)-1(z - 1)1/2(z + 1)3/2 (3.5)
O::s e3t < 0(1). (3.6) where
Z :::= (1 + p2)(1 - p2)-I,
AO(S) :::= - 8-1 (1 + 4k2)-lk2(4s2 + 1) (3.7)
with X :::= (p, q,). In the above Pn(z) is the Legendre polynomial of order n. The fact that the expressi0.!l for PI is correct to O( E2) instead of 0 (E) is because PI is an even function of E whereas P 1 is not.
The approximate p.d.f. given in (3.6) can now be used to find moments of the reflection coeffiCient by setting t :::= L. Because the approximate expreSSion for PI given by (3.6) is independent of q" one immediately finds (Rn) = O(e2), n :::= 1,2,···. In addition, the mean square reflection coefficient (I R2 I) and the mean square transmission coefficient (I T 12) :::= 1 - < 1 R 12) can be found by employing (3.6). The expressions obtained agree exactly with those given in Refs. 5 and 6.
4. PERTURBATION ANALYSIS OF PROBLEM 2
In order to calculate moments of u(x, L), the probability density functionP2(Y'x, t) will be needed. This p.d.f. satisfies the diffusion equation (2.22) with initial condition (2.24). The solution to this equation will now be found for small E.
Before doing this, we simplify (2.22) by expanding
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1924 Roger H. Lang: Probability density function
P2(Y'X, t) in terms of the eigenfunctions exp(in6 + iwv), n = 0, ± 1, ± 2, ..• , - co < w < co, i.e.,
+00 +00
Pz(Y,x, t) = L; L"" Qn(w,X,x, t)ei(wv+n6)dw. (4.1) n ::::-00
Using (4.1) in (2.22) and the orthogonality properties of the eigenfunctions, we have
aQn ~ -L Q at - 2 n'
where L(~) is defined in (2.17) and where
(4.2)
£<2) = L(1) + tklJ{iwp sin¢ - in(1 + p cos¢)] (4.3)
with L(}> being given by (2.18). The initial condition for (4.2) can be obtained by inverting (4.1) with t = x, using (2.24), (2.25), (2.26) and Simplifying. We have
Qn(w,X,x,x)
= (21T)-2(1 + pe+ i ¢)-(n+iw)/2(1 + pe-i¢)(n- iw)/ZP1(X,x).
(4.4) Since (4.2) is similar to the diffusion equation treated
in Sec. 3, we again employ the perturbation procedure used there. By using this method we Wid an approximate expression for Qn(w,X,x, t) which is similar to (3.4). Putting this in (4.1), we obtain an approximate expression for the p.d.f'P2(X,x, t) over the interval x ::s t ::s L. Then integrating out the iJ. dependence, we finally obtain
P2(Y'X, t) = j +00 P2(Y,x, t)diJ. -00
(4.5)
+00 +00
- "\' j 7\ (w X x t)e in6 + iwvdw - L.J \t"n" , , n =-00 _00 .
(4.6)
where
Q (w,X,x, t) = j +00 Qn(w,X,x, t)diJ., n -00
(4.7)
with Y = (v, e, p, ¢) and X = (p, ¢). The expression for Q n is given by
~ (~ ~. <I> 100 ( ) ( ) Qn w,X,x, t) = L1 e,m hmn(s)Mmn z P::'1/2+iS Z
n =-00 0
x e'l-2imk+Ymnt2 ] (t-x)ds + O(£Z), x::s t::s L, (4.8)
where
M (z) = (z - 1)1/2+m(z + 1)(3+iw-4m-4)/2 and mn
'Ymn = 8-1 (1 + 4k2)-lk2[(2is + 2m + n)2
- 1 - 2(2m + n)2 + 4ik(2m + n)
- 2(2m + n)2(1 + 4k2)].
(4.9)
(4.10)
In the above z is given in terms of p by (3.7) and ~~(z) is the associated Legendre function of degree II
and order iJ.. The coefficients hmn(s) appearing in (4.8) are found by setting t = x and inverting this expression. We find
hmn(s) = Jm(s) j+1f 100
[Mnm(z')]-l -1f 1
X pm . (z')e-im¢'Q~ (w X X x)dn-'dz' (4.11) -liz +, s n , " 'I-' ,
where
J (s) = (21T 3)-1/2s sinh1Tsr(t - m + is)r(~ - m - is). m (4.12)
J. Math. Phys., Vol. 14, No. 12, December 1973
1924
The initial data Q(w,X,x,x) is obtained by integrating both sides of (4.4) with respect to iJ., then using (4.7) (3.5) and (3.6).
Although the expression derived above for fi 2 (X, x, t) is fairly complex, we will see that expression for specific moments are much simpler.
5. MOMENTS OF u(x, L)
The mean (u(x, L» and intensity ( 1 u(x, LJ 122 will be calculated to O( £2) from our knowledge of P2 (X, x, L). We first consider the mean of u(x, L). Using (2.20), we have
(u) = e ik(L -x)(e v + W)
= eik(L-~) j e v+i6 P2&'X, L)ctY.
(5.1)
(5.2)
Substituting (4.6) with t = L into (5.2) and integrating with respect to v, e, and w gives
(5.3)
We now use (4.8) and (5.3) and then we perform the ¢ integration. After this we change integration variables from p to z with the aid of (3.7). The result is
00 100 (u) = (21T)3e ik(L-x) ~ 0 h O._1(s)
2 X P . . (z)e t Oo.-l(s)(L-X)dzds + 0(£2) (5.4) -1/2+.s ,
where
h _ (s) = s tanh1Ts 100 1"" s' tanh1Ts' o. 1 (21T)3 1 0
and
X P -1/2 +is (z ')P -1/2 + is ,(z ')
x eAo(S')t2Ldz'ds'
00._I(S) =- 8-1(1 + 4k2)-lk2[(2s + i)2
(5.5)
+ 3 + 4ik + 2(1 + 4kZ)]. (5.6)
The double integrals appearing in (5.5) and (5.4) are evaluated in Appendix B using properties of the Mehler transform. We use (B5) to evaluate the double integral in (5.5). This result is then used in (5.4) and that double integral is evaluated with the aid of (B6). We find
(u) = eik(L-X)+t2oo._1(-;/2)(L-x)+t2Ao(-i/2)t2L + 0(£2).
(5.7) Now using (5.6), (3.7) and introducing the unnorma
lized variables in (2.3), we obtain
(5.8)
where
This result has been obtained by Kupiec, et al. 4 when they applied the method of smooth perturbation to the Dyson equation. It should be noted that our K is the complex conjugate of their K. This is because 01,lr waves are incident from opposite sides of the slab.
The intensity (I u(x, L) 12) can be evaluated by using (2.20) and p.d.f.Pz' We have
(luI2)={e2v ) =jezvp2 {Y,x,L)dY. (5.10)
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Substituting (4.6) with t = L into (5.10) and integrating with respect to v, e, and w gives
(I U 12) = (21T)2 J Qo(2i, X,x, L)dX. (5.11)
We use (4.8) in (5.11) and then we evaluate the cf> integral. Following this, the p integration variable is transformed to z with the aid of (3.7). We obtain
(I u 12) = (21T)3 100 100 ho.o(s) p . (z)e Ao(S)E2(L-X) dzds
1 0 1 + Z -1/2 HS
+ O( £2), (5.12)
where
h () 2s tanh1Ts 100 100 I t h( ') Ip ( ') o 0 s = s an 1TS z -1/2 +ts Z . (21T)3 1 0
X P_1I2+iS.(zl)eAo(s.)<2Ldzlds' (5.13)
with AO(S) given in (3.7). The recurrence relation12
(5. 14)
for the Legendre function P v(z) can be used in (5.13). Two double integrals result which are given in (B7) and (B8) of Appendix B. We obtain
ho.o(s) = [tanh1Ts/(21T)3][(2s - i)e AO(S-i)E2X
+ (2s + i)e AO (S+i)E2X]. (5.15)
Now by putting (5.15) into (5.12) and evaluating the z integral with the aid Of13
100
(z + 1)-lp_1/2+ is (z)dz = 1T sech1Ts, (5.16)
the desired expression for (I u 12) is obtained. This result is then expressed in terms of the unnormalized variables by using (2.3). We find
L 100 tanh1Ts . - - 2-<I U 12) = 1Te 4i - --- [smSxs + 2s cos8xs]e- 4s Lds o cosh1Ts
o $ x $ 1, (5.17) where
x = dE2X, L = d E2I,
(5.18)
2.0
1.0 C = .05",
.1 .2 .3 .4 .5 .6
FIG. 1. Intensity vs xl L with l as a parameter.
J. Math. Phys .• Vol. 14, No. 12. December 1973
1925
Since we have not been able to evaluate this integral in terms of known functions, it has been evaluated numerically for various slab widths. These graphs are shown in Fig. I. For small and large L we find that (5.10) simplifies to
<luI2)=1+0(£2), o$;$L, L«l <I U 12) = erfc[L1I2(1- 2i/ L)] + 0(£2), (5.19)
o $ ; $ L/2, L » 1,
= 2 - erfc[];1/2(1 - 2x/L)] + 0(E2), L/2 <; < L (5.20)
where erfc(z) is the complementary error function of z. The approximate expression for large L was obtained by using the saddle point method to evaluate (5.17). The complementary error function arose because of the presence of a pole near a saddle point.
As a check of the consistency of the result (5.17), we find (I T 12) by evaluating < 1 u 12) at X = O. The expression for < 1 T 12) obtained agrees with that obtained in Ref. 6.
ACKNOWLEDGMENTS The author would like to thank Professor J. B. Keller
for bringing this problem to his attention and Professor R. Pickholtz for many helpful discussions. He also wishes to thank the National Science Foundation for supporting this work under Grant GK-2788L
APPENDIX A To derive (2.9)- (2.11) from (2.8), we first take the
derivative of (2.8) with respect to t and interchange orders of differentiation. We obtain
( '02 ~ au(x, t)
- + k 2 [1 + EfJ.(x)] -- = 0 ax2 at'
(~ + ik\ ~ u(O, t) = 0, ax J at
a2u(t, t) 02u(t, t) au(t, t) au(t, t) ---:c-- + --- - ik --- - ik --- = O.
ax2 axat ax at
(AI)
(A2)
(A3)
We denote alf,/ ox and auf of as the partial derivatives of u with respect to the first and second arguments of u respectively.
.7 .8
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1926 Roger H. Lang: Probability density function
An inspection of (AI), (A2), (2. 8a), and (2.8b) shows that u and ou/ at obey the same equation and the same boundary condition at x == O. Thus we conclude that
au (x , t) -- == C(t)u(x, t).
at (A4)
Now using (2. 8a) and (A4) with x == t in (A3), we obtain
C(t) == ik{l + h/J.(t)[l + r(t)J} (A5)
with iku(t, t) + (%x)u(t, t)
r(t) == • iku(t, t) - (o/ax)u(t, t)
(A6)
We shall call r(t) the generalized reflection coefficient since from (A6), (2. 6c), and (2.7), we see that
() iku(L, L) + (%x)u(L, L)
rL == ==u(L)-l==R. iku(L, L) - (%x)u(L, L) (A7)
An equation for r(t) can be obtained by taking the derivative of (A6) with respect to t and then using (2. 8a), (A4) with x == t and (A6) in this expression. We find
dr(t) 1 dt == 2ikr(t) + 2ik €/J.(t)[1 + r(t)]2, (A8)
where r(O) == O. The above equation is a Ricatti equation. An equation of this type was found for the reflection coefficient in Ref. 5. The initial condition is found by making use of (2. 8b) and (A6) with x == t == O.
The reflection coefficient of the slab can thus be found by solving (A8) over the interval O::s t::s L. The field u(x, L) can be found by first solving (A8) in the interval 0 ::s t ::s x and then solving (A4) and (A8) simultaneously in the interval x ::s t::s L. The initial conditions for the system of equations (A4) and (A8) are
r(t) I t~x == r(x),
u(x,x) == 1 + r(x).
(A9)
(AIO)
Here r(x) is the solution to the Ricatti equation in the interval O::s t ::s x With t == x. The initial condition (AIO) is obtained by making use of (2. 8c) and (A6) with t == x.
APPENDIX B The purpose of this appendix is to evaluate several
double integrals that appear in the text. We start by considering the identity
S tanh1Ts er(s) == S tanh1Ts ~"" P_ 1/2 +is (z)q(z)dz,
o ::s S ::s co, (Bl) where
q(z) == 1"" P 1/2+is'(Z)[S' tanh1Ts'er(S')]ds'. (B2) o -
Here r(s) is a second degree polynomial in s with complex coefficients, i.e.,
where we assume 0 > Rea2 > Ima 2 •
(B3)
The identity (Bl) follows immediately from the fact that we are taking the inverse Mehler transforms of the Mehler transform 14 of s tanh1Tse r (s).
J. Math. Phys., Vol. 14, No. 12, December 1973
1926
Simplifying (Bl), we have
er(s) = ~"" P_1/ 2 + is (z)q(z)dz. (B4)
One can now show that both sides of equality (B4) can be analytically continued into an infinite strip in the complex s plane bounded by the lines s == ± 2i. The analytic continuation is dependent on the fact that q(z) must decay rapidly enough as z -) co so that the integral in (B4) is uniformly convergent. We find q(z) has this property when 0 > Rea2 > Ima2 •
We now use (B4) to evaluate the integrals of interest. The functions AO(S) and 00,_l(S) used below are given in (3.7) and (5.6) respectively.
Case 1: Let r(s) == AO(s)€2L:
e Ao(-il2)€2L == 1"" 1"" s'tanh1TS'P . (z') 1 0 -1/2 HS
A (s,)€2L X P- 1/2 + is ,(z')e 0 dz'ds. (B5)
Case 2: Let r(s) == 00,-1 (s)€2(L - x) + Ao(s)e2L with s ==- i/2:
Here Po(z) == 1 was used.
Case 3: Let r(s) == AO(S) e2 L and replace s by S - i in (B4):
e AO(S-i)€2 L == 1"" 1"" s' tanh1TS'P . (z') 1 0 1/2 +IS
X P . (:Z')eAo(S,)€2Ldz'ds' (B7) -1/2 +lS" •
Case 4: Let r(s) == AO(S)€2L and replace s by S + i in (B4):
e AO(S+i)€2 L == 1"" 1"" s' tanh1Ts' P . (z') 1 0 -3/2 +IS
X P . (z ')e AO(S ,)€2 L dz 'ds' (B8) -1/2+,s' •
I L. Breiman, Probability (Addison-Wesley, New York, 1968), p. 347. 2 M. E. Gertsenshtein and V. B. Vasil'iev, Theor. Prob. Appl. 4, 392
(1959). 3 Yu. L. Gazaryan, Zh. Eksp. Teor. Fiz. 56, 1856 (1969) [SOy. Phys.
JETP 29, 996 (1969) J. 4 1. Kupiec, L. B. Felsen, S. Rosenbaum, J. B. Keller, and P. Chow,
Radio Sci. 4,1067 (1969). • G. C. Papanicolaou, SIAM J. Appl. Math. 21, 13 (1971). 6 1. A. Morrison, G. C. Papanicolaou, and J. B. Keller, Comm.
Pure Appl. Math. 24, 473 (1971). 7 R. Bellman and R. Kalaba, J. Math. Mech. 8,683 (1959). 8 R. Bellman and R. Kalaba, Electromagnetic Wave Propagation
(Academic, New York, 1960), p. 243. 9 E. Wong, Stochastic Processes in Information and Dynamical
Systems (McGraw-Hili, New York, 1972). IDA. Jazwinski, Stochastic Processes and Filtering Theory (Academic,
New York, 1970). II Since the transformation (2.14) is singular at p = 0, any initial dis
tribution of'" can be chosen. 12W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theo
rems for the Special Functions of Mathematical Physics (SpringerVerlag, Berlin, 1966), p. 171.
13Bateman Manuscript Project, Higher Transcedental Functions (McGraw-Hili, 1953), Vol. I, p. 174.
"See Ref. 10, p. 398.
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