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Inverse solutions to the inelastic scattering problem
Volume 110A, n u m b e r 4
PHYSICS LETTERS
22 July 1985
INVERSE SOLUTIONS TO THE INELASTIC SCATTERING PROBLEM J.M. BLACKLEDGE Physics Department, King's College (KQC), London University, Strand, London WC2R 2LS, UK Received 22 April 1985; accepted for publication 7 May 1985
Exact inverse solutions to the integral equation q~(r,l~, k) = f~3f(r, o~)g(rlro, k)g(rlr ,, k)d3r, where g(rl~, k); j = 0 or s is the free space Green function, are derived in plane and cylindrical coordinates for fixed w. These solutions allow an inelastic scattering potential f ( r , ~ ) which is of compact support r E 9 3 to be recovered from scattering data collected over the surfaces of a plane and cylinder respectively.
1. Introduction. The integral equation ¢(rslr 0' k) = f d3rf(r, ~)g(rlr O, k)g(rlrs, k), ~3 where
(1)
g(rlr/, k) = exp(ilkllr - r/I)/47rlr - r/I,
(2)
/ = 0 or s,
is the free space Green function and k (=27r/X) = w/c Ois the free space wave number, is common to a number of inverse problems which model scattering data in the Born and Rytov approximations. Two examples of current interest are: (i) the "two parameter problem" for electromagnetic fields where, given that
f(r, w) = e(r) + io(r)/w,
(3)
we are required to find solutions that map e(r) (the permittivity) and o(r) (the conductivity) to the scattered field q~(rs Ir 0' k) and (ii) the "four parameter problem" for acoustic fields where, given that
f(r, w) = 7~(r) + (1 + ito)Tp(r)h "m + ieo [Tx(r) + 27u(r)] (ti .m)2,
(4)
we are required to find inverse solutions for the dimensionalized compressibility, density and first and second Lam6 parameters [Tn(r), 7p(r) and 7x(r) and 7u(r ) respectively]. Since the scattering potentialf(r, ~ ) is frequency dependent, this is an inelastic scattering problem where the wave field experiences dispersive scattering by the "dispersive scatter generating functions" a(r) or 7x(r) and 7u(r ). Solutions to these problems can be found in refs. [1] and [2] respectively where eq. (1) is inverted using a spherical polar system of coordinates r = (r, 0, ~b).This solution if of the form
f(r, ~) = k---~-3 f f d~2h, dfZh" la' + a"lT(ka'l--k~") exp[ik(ti' + n").r] , 47r3 ~,i' S~h"
(5)
where
oo
N=0 M=0
f f a n o d. s [¢k(rslro,k)(-i)N+m(2N+ 1)(2M+ 1) I20 ~ s
X PN(ti" tio)PM(~ ".tis)/HN(kro)HM(krs) ] 188
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and H N and PN are the Hankel functions and Legendre polynomials respectively. By generating scattering data at two fixed frequencies ~b(rs Iro, kn); n = 1, 2, k 14= k 2 and recovering f(r, co) via eq. (5), e(r) and o(r) can be found from eq. (3). In a similar way, by generating data at four different frequencies q~(rs[r 0, kn); n = 1, 2, 3, 4, k 14= k 24= k 3 :~ k4, the acoustic parameters 'yK(r), ~,p(r), 7x(r) and ~/u(r) can be found from eq. (4). This solution is valid for scattering data collected on the inner surface of a sphere with a constant radius a = r 0 = r s which completely encloses a scattering domain of compact support c-/)3. In this paper, we extend the solution o f eq. (1) for data collected on the surfaces of a plane and circular cylinder. The solutions for these surfaces are: (i) complimentary to the solutions for spherical detection and (ii) designed to be of equal or greater practical significance - particularly the solution for a plane.
2. Solution for a plane: r = (x, y, z}. Let us write eq. (1) in the form ¢(rslr 0, k) = f
(7)
d3rf(r, ~)K(rlro,rlr s, k),
~3 where
(8)
K (rlr 0 , r lr s , k) = g(rlr 0 , k)g(rlr s, k).
The solution for a plane is obtained by writing the kernel K(r Ir 0, r Irs, k) as a decomposition of plane waves. Let us fix this plane at z s = z 0 = 0 so that the kernel can be written as f** d3K exp Igl 2ILK" (r0___~-r)] _**JTd3K , exp[iK"(rs-r)]lKi 2 _ k2 K(rlro, rlrs, k ) - (27061 _o.
=i2n) 6 X
dK x dKy dK z exp{i[Kx(xo
fFf JJJ -**
dK x
dK z
x)+Ky('Vo - y ) - K z Z ] }
exp(i[Kx(X s - x ) + K y 0 ~ s - Y ) +Kzz])
K xp2 + K 'y2 +K;2 _ k 2
(9)
This decomposition is easily derived from the equation (V 2 +kZ)g(rlQ,k) = - 6 3 ( r - r / ) ,
/'=0ors,
(lo)
where upon taking the Fourier transform and using the relation f
exp ( - i K . r) 63(r - r/.) d3r = exp ( - i K . r/)
(11)
we have (IKI 2 - k 2)
f
d3r e x p ( - i K . r ) g ( r l ~ , k) = e x p ( - i K - r j ) .
(12)
Thus, taking the inverse Fourier transform, eq. (11) can be written as
g(rl~, k) =
1 ~ exp[iK.(rj-r)l (21r) ----~ _L IKI 2 - k 2
d3K"
(13)
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Note that the evaluation of this integral yields the expression for the Green function given by eq. (2). If we substitute eq. (9) into eq. (7) and multiply both sides by the function (21r)-4exp [ - i ( u x 0 + oy0) ] exp [ - i ( u ' x s + o'.Ys)] , then using the relation 1
J J exp{it(K x (2.) 4 . .
- u)SG
U)Xo+ (Ky. .
-o,(q
v)Y0] } dx 0 dy 0
-
JJexp(i[(K x -
U)Xs+
(K~ - v ' ) Y s ] } dx s dy s (14)
-o')
allows us to write eq. (7) in the form
qJ(u,v,u',v',k) - 1(2rr) 2 f f f a x dy dz f(x, y, z, co)exp [ - l ( u
+ u ')x] exp [-flu +
v')y]
q)3
~ dKz;~ p(-iKzz) ~ "
X
_
+ p2 _ k 2 _
z
.
exp(-iKzz) . . . . k2
(15)
K;2 + p'2 _
where
~b(u,v,u',v',k)= f f f f dxodYo~sdYs ¢(Xo,Yo,Xs, Ys, k)exp[-i(UXo+VYo)]exp[-i(UXs+OYs)],
(16)
and p2 = u2 + e2 and O' 2 = u,2 + e,2. For k > 0, k > 0 and z > O, the integration over K (and therefore K') yields (ref. [3], p. 312, 3.354 (5) f o r a -= (02 + k 2) ; i(k 2 - 02)1/2)
** exp(-iKzz ) f-** dKzK2+o2 k 2-
exp [ - i z ( k 2 - p2)1/2] rri
(17)
(k 2 _ p2)1/2
Thus, eq. (15) becomes
~(u, o, u', o', k) = f f f dx dy dzf(x, y, z, ¢o)exp[-i(u + u')x] e x p [ - i ( v + v')y] Cl)3 X e x p ( - i [ ( k 2 - p2)1/2 + (k 2 _ p,2)l/2]z},
(18)
where ~(u, o, u', o', k) = -4~b(u, o, u', o', k)[(k 2 - p2)(k2 - p,2)] 1/2
(19)
We can now use the Beylkin identity [4] which in the third dimension gives
k3 f f
f(r, ~o) = 4-~ s~ s]
as21, 1 +e21T(kall-kti2)exp[ik(n
I +ri2)'rl
,
(20)
where T(k~ll-ka2)
= f d3rf(r, ¢o)exp[-ik(~
1 + ti2)-r ] .
(21)
c-/)3 Here, ri I and ri 2 are unit vectors normal to the surfaces S 2 and S 2 and T(k~ll - k t i 2 ) is known only for Ik(ti 1 190
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+ ti 2) 1< 2k. Thus, on comparing eq. (18) with eq. (21) we can express the scattering potential f(x, y, z, w) in the form
f(x,y,z, co)--~ 3k3 / f d u d v
/fdu'
do"[(u + u') 2 + (o + v,)2 + [(k 2 _ 92)1/2 + (k 2 _ ,o,2)1/2] 2}1/2
X ~(u, v, u', v', k) exp [i(u + u')x] exp [i(v + v')y] exp(i[(k 2 + p2) 1/2 + (k 2 - p'2) 1/2] z ) .
(22)
3. Solution for a circular cylinder: r -- (r, O, z). The solution to eq. (7) for scattering data collected on the inner surface of a cylinder is obtained by expanding the kernel K(rlr O, r [rs, k) in cylindrical coordinates. If we let r = (r, 0, z) and K = (/~,/3, Kz), then the cylindrical coordinates of a point in three dimensional r-space are given by x = r cos 0, y --- r sin 0 and z and in K-space by K x =/a cos ~, Ky =/a sin/3 and K z. Thus, IKI = Kx2 + Ky2 + Kz2 = K 2 +/a 2, d3K = dK z d~/d/z #, K . r = lar cos(~ - 0) + zK z and K .rl = lar cos(l~ - 0/) + z/Kz,] = 0 or s, and it follows from eq. (13), that the Green function can now be written as 21r
g(r,rj,k)=(27r) -3 ; d12. i dKz f d~exp{ila[r/cos(fl-O/)-rcos([3-O)]} 0 _o. o × exp(ilz/. - z IKz)/(la 2 + K2z - k 2 ) '
] = 0 or s.
(23)
The exponentials containing cosines can also be expanded via the expression (ref. [3], p. 973, 8.511 (4)) oo
exp[i/~r cos(/3 - 0)] = ~
inexp[in cos(l~ - O)]Jn(#r ),
(24)
n=_o¢
where Jn are the Bessel functions and we can write 2~"
f
0
¢m
dl3 exp[i/ar] cos(fl - 0])1 e x p [ - i ~ r cosO - 0)] = ~
n=--~
00
i n exp(inO/)Jn(l~)
~
m=-~
( - i ) m exp(-imO)Jm(lar )
21t
x
f
dfl exp[iC/(n + m)] = 2n
0
~
n=_..
Jn(t~ri)Jn(l~r)exp[in(~ - 0)] ,
(25)
since 2ff
f 0
(26)
d/3 exp [i/3(n + m)] = 2Zr6nm,
where 6nm is the Kronecker delta,
6nm = 1
n = m,
6nm = 0
n #:m.
(27)
Substituting eq. (25) into eq. (23) we now have
g(rlr/, k)- (2if) 21 _~dgz exp [i(Zs _Z)Kz] 0;/~2 +~z2/2--k 2 X ~
n=-**
Jn(la~)JnOar)exp[in(O]-O)],
/=0ors,
(28)
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and this expression allows eq. (7) to be written in the form
¢(O0, Os, ZO,Zs, k)=(21r)-2
fff (-1)3
X ~ n = -
?
oo
Jn(#a)Jn(#r)exp[in(Oo-O)l *~
f
dKzexp[i(Zs-Z)K~] **
'
du'~u'
0
la' 2 -+-K'2 -- k2
--
X ~
dp p
drrdO dz f(r, O, z, ¢o) f dKz exp[i(Zo - Z)Kz] a u2 + K2z _ k 2 -** 0
7.
Jm(#'a)Jm(#'r)exp[im(Os-O)],
(29)
m=_oo
where a = r 0 = r s is the radius of the cylinder which remains fixed. Let us now multiply both sides of eq. (29) by exp(-ir/zo) exp(-i~7 'Zs) and use the relations 21rl y exp [izo(K z - 7/)] dz 0 = 6 (K z - 71),
(30)
_oo
and oo
1 21r
f
exp[izs(Kz - ~ ' ) ] dZs=6(K z -71')
(31)
,
to obtain oo
~O(0 0 , 0 s, rt, 'q', k ) =
f f f dr rdO dz f (r, O, z, w) exp [-i(r/+ ~?')z] / dis Is ~ Jn(Pa)ln(Ur) exp [in (0 0 - 0)] c-D3 0 n =- o*l't2 +r/2 -- k2 Jm (P 'a) Jm (P'r)
Xf 0
dp' #'
n=-** p'2 + r/'2 _ k 2
exp [im(O s - 0)] ,
(32)
where
4(00, Os, 71, rl', k) = / f ~(o O, 0 s, z O, z s, k) exp(-iT?z0) exp(-ir/'Zs).
(33)
_oo
If we now multiply through by exp(-ip0 0) exp(-iq0 s) where p and q take on integer values from _co to +0% then integrating over 0 0 and 0 s from 0 to 2rr and using the relation f 0
27r 2n f d0 0 dOs exp[i(n - P)00] exp[i(m - q)0s] = 41r6npSmq, 0
(34)
we can then write eq. (32) as
¢pq(r~, rf, k) = f f f
dr rdO dz [(r, O, z, 6o) exp [-i(r/+ 7/')z] exp [-i(p + q)O]
~3
x of d where
192
Jp(ualSpO r)
of a ' u u,sqo'a)JqO',)
(35)
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27r 2~r
k)-- f f d00 dos (00.0s. r/.r/'.k) exp(-ip00)
exp(-iq0s).
(36)
0 o For 0 < r < a and Re(r/2 - k 2) > 0, the integration over p (and likewise over p') gives (ref. [3], p. 679, 6.541(1))
f ap/z . Jp(ua)Sp(Ur) -~-----:----z -/p 0
[r(r/2 - k2) 1/2 ] Kp [a(r/2 - k2) 1/2] ,
(37)
t~ +r/Z-k':
where Ip and Kp are the Bessel functions of an imaginary argument. For integral p we have (ref. [7], p. 952, 8.406(3))
Ip [r(r/2 - k2) 1/2] = Ip [ir(k 2 _r/2)I/2] = (_i)-Vjp [r(k 2 _ r/2)1/21 ,
(38)
where we have exploited the parity condition for a pth order Bessel function,
jp [r(k 2 _ r/2)1/2] = (_i)njp [_r(k 2 _ ,72)1/2].
(39)
From eqs. (37) and (38), eq. (35) becomes
%(?7, r/', k) =
f f f drrdO dz f(r, O, z, co) exp [-i(r/+
n')z] exp [-i(p + q)O] (-i) -(p+q)
(b3 X Jp [r(k 2 - r/2)1/2] jq [r(k 2 _ ~/,2)1/2] ,
(40)
where
Xltpq(r/, ~/', k) = ~pq0"/, r/', k){ G [a(r/2 - k2) 1/2 ] gq [a(~/2 - k2)1/21)-1.
(41)
Multiplying eq. (40) by (-i)2(p÷q) exp(ip/3) exp(iq3'), summing over p and q from _oo to +oo and using eq. (24) we can write
~O,{3',r/,r/',k)=
fff
drrdO dzf(r,O,z,k)exp[-i(r/+r/')z]
cD3
X exp{-ir[(k 2 - r/2) 1/2 cos(0 -/3) + (k 2 - r/,2)1/2 cos(0 - ~')]},
(42)
where oQ
• (fl, /3', r/, r/', k) = ~ ~ (-i)2(p+q)6xp[i(p[3+q[3')]~pq(r/,r/',k). p=-** q=-.~
(43)
Eq. (42) is eq. (21) in cylindrical coordinates [i.e. the Fourier transform o f f ( r , 0, z, co)]. Thus from eq. (20) we can express f (r, 0, z, w) in the form
f(r,O,z,¢o)= 4"3 k-~3 fO2~rf02~rd/3/3' -.0 f -~, dr/dr/' ((r/+ r/,)2 + [(k 2 _r/2)l/2cos( o _~)+(k 2 _r/,2)l/2cos( 0 _/3,)]2}1/2 X q ~ , / 3 ' , r/, r/', k) exp[i(r/+ r/')zl exp(ir[(k 2 - r/2)l/2cos(0 -/~) + (k 2 - r/'2)l/2cos(0 - ~')]}.
(44)
4. Summary. Eqs. (5), (22) and (44) are exact inverse solutions to the inelastic scattering problem as defined by eq. (1). They apply to scattering data which is obtained in the near field (i.e. when r "~ r 0 " rs), and is collected on the surfaces of a sphere, a plane and a circular cylinder respectively. This data is assumed to be generated by the interaction of an incident spherical wave g(rlro, k) with a frequency dependent potential f(r, co) where the scat193
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tered field CB(rslr0, k) under the Born approximation is related to the scattered field CR(rslr0, k) under the Rytov approximation by [2] CR(rs[r0, k) = k-2g(rslr o, k) ln[¢~(rslr0,
k)[g(rslro,
k)] .
References [1] [2] [3] [4]
194
J.M. Blackledge, Phys. Lett. 107A (1985) 59. J.M. Blackledge and L. Zapalowski, Inverse problems 1 (1985) 17. I.S. Gradshteyn and I.M. Ryshik, Tables of integrals, series and products (Academic Press, New York, 1980). G. Beylkin, J. Math. Phys~ 24 (1983) 1399.
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PHYSICS LETTERS
22 July 1985
INVERSE SOLUTIONS TO THE INELASTIC SCATTERING PROBLEM J.M. BLACKLEDGE Physics Department, King's College (KQC), London University, Strand, London WC2R 2LS, UK Received 22 April 1985; accepted for publication 7 May 1985
Exact inverse solutions to the integral equation q~(r,l~, k) = f~3f(r, o~)g(rlro, k)g(rlr ,, k)d3r, where g(rl~, k); j = 0 or s is the free space Green function, are derived in plane and cylindrical coordinates for fixed w. These solutions allow an inelastic scattering potential f ( r , ~ ) which is of compact support r E 9 3 to be recovered from scattering data collected over the surfaces of a plane and cylinder respectively.
1. Introduction. The integral equation ¢(rslr 0' k) = f d3rf(r, ~)g(rlr O, k)g(rlrs, k), ~3 where
(1)
g(rlr/, k) = exp(ilkllr - r/I)/47rlr - r/I,
(2)
/ = 0 or s,
is the free space Green function and k (=27r/X) = w/c Ois the free space wave number, is common to a number of inverse problems which model scattering data in the Born and Rytov approximations. Two examples of current interest are: (i) the "two parameter problem" for electromagnetic fields where, given that
f(r, w) = e(r) + io(r)/w,
(3)
we are required to find solutions that map e(r) (the permittivity) and o(r) (the conductivity) to the scattered field q~(rs Ir 0' k) and (ii) the "four parameter problem" for acoustic fields where, given that
f(r, w) = 7~(r) + (1 + ito)Tp(r)h "m + ieo [Tx(r) + 27u(r)] (ti .m)2,
(4)
we are required to find inverse solutions for the dimensionalized compressibility, density and first and second Lam6 parameters [Tn(r), 7p(r) and 7x(r) and 7u(r ) respectively]. Since the scattering potentialf(r, ~ ) is frequency dependent, this is an inelastic scattering problem where the wave field experiences dispersive scattering by the "dispersive scatter generating functions" a(r) or 7x(r) and 7u(r ). Solutions to these problems can be found in refs. [1] and [2] respectively where eq. (1) is inverted using a spherical polar system of coordinates r = (r, 0, ~b).This solution if of the form
f(r, ~) = k---~-3 f f d~2h, dfZh" la' + a"lT(ka'l--k~") exp[ik(ti' + n").r] , 47r3 ~,i' S~h"
(5)
where
oo
N=0 M=0
f f a n o d. s [¢k(rslro,k)(-i)N+m(2N+ 1)(2M+ 1) I20 ~ s
X PN(ti" tio)PM(~ ".tis)/HN(kro)HM(krs) ] 188
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and H N and PN are the Hankel functions and Legendre polynomials respectively. By generating scattering data at two fixed frequencies ~b(rs Iro, kn); n = 1, 2, k 14= k 2 and recovering f(r, co) via eq. (5), e(r) and o(r) can be found from eq. (3). In a similar way, by generating data at four different frequencies q~(rs[r 0, kn); n = 1, 2, 3, 4, k 14= k 24= k 3 :~ k4, the acoustic parameters 'yK(r), ~,p(r), 7x(r) and ~/u(r) can be found from eq. (4). This solution is valid for scattering data collected on the inner surface of a sphere with a constant radius a = r 0 = r s which completely encloses a scattering domain of compact support c-/)3. In this paper, we extend the solution o f eq. (1) for data collected on the surfaces of a plane and circular cylinder. The solutions for these surfaces are: (i) complimentary to the solutions for spherical detection and (ii) designed to be of equal or greater practical significance - particularly the solution for a plane.
2. Solution for a plane: r = (x, y, z}. Let us write eq. (1) in the form ¢(rslr 0, k) = f
(7)
d3rf(r, ~)K(rlro,rlr s, k),
~3 where
(8)
K (rlr 0 , r lr s , k) = g(rlr 0 , k)g(rlr s, k).
The solution for a plane is obtained by writing the kernel K(r Ir 0, r Irs, k) as a decomposition of plane waves. Let us fix this plane at z s = z 0 = 0 so that the kernel can be written as f** d3K exp Igl 2ILK" (r0___~-r)] _**JTd3K , exp[iK"(rs-r)]lKi 2 _ k2 K(rlro, rlrs, k ) - (27061 _o.
=i2n) 6 X
dK x dKy dK z exp{i[Kx(xo
fFf JJJ -**
dK x
dK z
x)+Ky('Vo - y ) - K z Z ] }
exp(i[Kx(X s - x ) + K y 0 ~ s - Y ) +Kzz])
K xp2 + K 'y2 +K;2 _ k 2
(9)
This decomposition is easily derived from the equation (V 2 +kZ)g(rlQ,k) = - 6 3 ( r - r / ) ,
/'=0ors,
(lo)
where upon taking the Fourier transform and using the relation f
exp ( - i K . r) 63(r - r/.) d3r = exp ( - i K . r/)
(11)
we have (IKI 2 - k 2)
f
d3r e x p ( - i K . r ) g ( r l ~ , k) = e x p ( - i K - r j ) .
(12)
Thus, taking the inverse Fourier transform, eq. (11) can be written as
g(rl~, k) =
1 ~ exp[iK.(rj-r)l (21r) ----~ _L IKI 2 - k 2
d3K"
(13)
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Note that the evaluation of this integral yields the expression for the Green function given by eq. (2). If we substitute eq. (9) into eq. (7) and multiply both sides by the function (21r)-4exp [ - i ( u x 0 + oy0) ] exp [ - i ( u ' x s + o'.Ys)] , then using the relation 1
J J exp{it(K x (2.) 4 . .
- u)SG
U)Xo+ (Ky. .
-o,(q
v)Y0] } dx 0 dy 0
-
JJexp(i[(K x -
U)Xs+
(K~ - v ' ) Y s ] } dx s dy s (14)
-o')
allows us to write eq. (7) in the form
qJ(u,v,u',v',k) - 1(2rr) 2 f f f a x dy dz f(x, y, z, co)exp [ - l ( u
+ u ')x] exp [-flu +
v')y]
q)3
~ dKz;~ p(-iKzz) ~ "
X
_
+ p2 _ k 2 _
z
.
exp(-iKzz) . . . . k2
(15)
K;2 + p'2 _
where
~b(u,v,u',v',k)= f f f f dxodYo~sdYs ¢(Xo,Yo,Xs, Ys, k)exp[-i(UXo+VYo)]exp[-i(UXs+OYs)],
(16)
and p2 = u2 + e2 and O' 2 = u,2 + e,2. For k > 0, k > 0 and z > O, the integration over K (and therefore K') yields (ref. [3], p. 312, 3.354 (5) f o r a -= (02 + k 2) ; i(k 2 - 02)1/2)
** exp(-iKzz ) f-** dKzK2+o2 k 2-
exp [ - i z ( k 2 - p2)1/2] rri
(17)
(k 2 _ p2)1/2
Thus, eq. (15) becomes
~(u, o, u', o', k) = f f f dx dy dzf(x, y, z, ¢o)exp[-i(u + u')x] e x p [ - i ( v + v')y] Cl)3 X e x p ( - i [ ( k 2 - p2)1/2 + (k 2 _ p,2)l/2]z},
(18)
where ~(u, o, u', o', k) = -4~b(u, o, u', o', k)[(k 2 - p2)(k2 - p,2)] 1/2
(19)
We can now use the Beylkin identity [4] which in the third dimension gives
k3 f f
f(r, ~o) = 4-~ s~ s]
as21, 1 +e21T(kall-kti2)exp[ik(n
I +ri2)'rl
,
(20)
where T(k~ll-ka2)
= f d3rf(r, ¢o)exp[-ik(~
1 + ti2)-r ] .
(21)
c-/)3 Here, ri I and ri 2 are unit vectors normal to the surfaces S 2 and S 2 and T(k~ll - k t i 2 ) is known only for Ik(ti 1 190
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+ ti 2) 1< 2k. Thus, on comparing eq. (18) with eq. (21) we can express the scattering potential f(x, y, z, w) in the form
f(x,y,z, co)--~ 3k3 / f d u d v
/fdu'
do"[(u + u') 2 + (o + v,)2 + [(k 2 _ 92)1/2 + (k 2 _ ,o,2)1/2] 2}1/2
X ~(u, v, u', v', k) exp [i(u + u')x] exp [i(v + v')y] exp(i[(k 2 + p2) 1/2 + (k 2 - p'2) 1/2] z ) .
(22)
3. Solution for a circular cylinder: r -- (r, O, z). The solution to eq. (7) for scattering data collected on the inner surface of a cylinder is obtained by expanding the kernel K(rlr O, r [rs, k) in cylindrical coordinates. If we let r = (r, 0, z) and K = (/~,/3, Kz), then the cylindrical coordinates of a point in three dimensional r-space are given by x = r cos 0, y --- r sin 0 and z and in K-space by K x =/a cos ~, Ky =/a sin/3 and K z. Thus, IKI = Kx2 + Ky2 + Kz2 = K 2 +/a 2, d3K = dK z d~/d/z #, K . r = lar cos(~ - 0) + zK z and K .rl = lar cos(l~ - 0/) + z/Kz,] = 0 or s, and it follows from eq. (13), that the Green function can now be written as 21r
g(r,rj,k)=(27r) -3 ; d12. i dKz f d~exp{ila[r/cos(fl-O/)-rcos([3-O)]} 0 _o. o × exp(ilz/. - z IKz)/(la 2 + K2z - k 2 ) '
] = 0 or s.
(23)
The exponentials containing cosines can also be expanded via the expression (ref. [3], p. 973, 8.511 (4)) oo
exp[i/~r cos(/3 - 0)] = ~
inexp[in cos(l~ - O)]Jn(#r ),
(24)
n=_o¢
where Jn are the Bessel functions and we can write 2~"
f
0
¢m
dl3 exp[i/ar] cos(fl - 0])1 e x p [ - i ~ r cosO - 0)] = ~
n=--~
00
i n exp(inO/)Jn(l~)
~
m=-~
( - i ) m exp(-imO)Jm(lar )
21t
x
f
dfl exp[iC/(n + m)] = 2n
0
~
n=_..
Jn(t~ri)Jn(l~r)exp[in(~ - 0)] ,
(25)
since 2ff
f 0
(26)
d/3 exp [i/3(n + m)] = 2Zr6nm,
where 6nm is the Kronecker delta,
6nm = 1
n = m,
6nm = 0
n #:m.
(27)
Substituting eq. (25) into eq. (23) we now have
g(rlr/, k)- (2if) 21 _~dgz exp [i(Zs _Z)Kz] 0;/~2 +~z2/2--k 2 X ~
n=-**
Jn(la~)JnOar)exp[in(O]-O)],
/=0ors,
(28)
191
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22 July 1985
and this expression allows eq. (7) to be written in the form
¢(O0, Os, ZO,Zs, k)=(21r)-2
fff (-1)3
X ~ n = -
?
oo
Jn(#a)Jn(#r)exp[in(Oo-O)l *~
f
dKzexp[i(Zs-Z)K~] **
'
du'~u'
0
la' 2 -+-K'2 -- k2
--
X ~
dp p
drrdO dz f(r, O, z, ¢o) f dKz exp[i(Zo - Z)Kz] a u2 + K2z _ k 2 -** 0
7.
Jm(#'a)Jm(#'r)exp[im(Os-O)],
(29)
m=_oo
where a = r 0 = r s is the radius of the cylinder which remains fixed. Let us now multiply both sides of eq. (29) by exp(-ir/zo) exp(-i~7 'Zs) and use the relations 21rl y exp [izo(K z - 7/)] dz 0 = 6 (K z - 71),
(30)
_oo
and oo
1 21r
f
exp[izs(Kz - ~ ' ) ] dZs=6(K z -71')
(31)
,
to obtain oo
~O(0 0 , 0 s, rt, 'q', k ) =
f f f dr rdO dz f (r, O, z, w) exp [-i(r/+ ~?')z] / dis Is ~ Jn(Pa)ln(Ur) exp [in (0 0 - 0)] c-D3 0 n =- o*l't2 +r/2 -- k2 Jm (P 'a) Jm (P'r)
Xf 0
dp' #'
n=-** p'2 + r/'2 _ k 2
exp [im(O s - 0)] ,
(32)
where
4(00, Os, 71, rl', k) = / f ~(o O, 0 s, z O, z s, k) exp(-iT?z0) exp(-ir/'Zs).
(33)
_oo
If we now multiply through by exp(-ip0 0) exp(-iq0 s) where p and q take on integer values from _co to +0% then integrating over 0 0 and 0 s from 0 to 2rr and using the relation f 0
27r 2n f d0 0 dOs exp[i(n - P)00] exp[i(m - q)0s] = 41r6npSmq, 0
(34)
we can then write eq. (32) as
¢pq(r~, rf, k) = f f f
dr rdO dz [(r, O, z, 6o) exp [-i(r/+ 7/')z] exp [-i(p + q)O]
~3
x of d where
192
Jp(ualSpO r)
of a ' u u,sqo'a)JqO',)
(35)
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22 July 1985
27r 2~r
k)-- f f d00 dos (00.0s. r/.r/'.k) exp(-ip00)
exp(-iq0s).
(36)
0 o For 0 < r < a and Re(r/2 - k 2) > 0, the integration over p (and likewise over p') gives (ref. [3], p. 679, 6.541(1))
f ap/z . Jp(ua)Sp(Ur) -~-----:----z -/p 0
[r(r/2 - k2) 1/2 ] Kp [a(r/2 - k2) 1/2] ,
(37)
t~ +r/Z-k':
where Ip and Kp are the Bessel functions of an imaginary argument. For integral p we have (ref. [7], p. 952, 8.406(3))
Ip [r(r/2 - k2) 1/2] = Ip [ir(k 2 _r/2)I/2] = (_i)-Vjp [r(k 2 _ r/2)1/21 ,
(38)
where we have exploited the parity condition for a pth order Bessel function,
jp [r(k 2 _ r/2)1/2] = (_i)njp [_r(k 2 _ ,72)1/2].
(39)
From eqs. (37) and (38), eq. (35) becomes
%(?7, r/', k) =
f f f drrdO dz f(r, O, z, co) exp [-i(r/+
n')z] exp [-i(p + q)O] (-i) -(p+q)
(b3 X Jp [r(k 2 - r/2)1/2] jq [r(k 2 _ ~/,2)1/2] ,
(40)
where
Xltpq(r/, ~/', k) = ~pq0"/, r/', k){ G [a(r/2 - k2) 1/2 ] gq [a(~/2 - k2)1/21)-1.
(41)
Multiplying eq. (40) by (-i)2(p÷q) exp(ip/3) exp(iq3'), summing over p and q from _oo to +oo and using eq. (24) we can write
~O,{3',r/,r/',k)=
fff
drrdO dzf(r,O,z,k)exp[-i(r/+r/')z]
cD3
X exp{-ir[(k 2 - r/2) 1/2 cos(0 -/3) + (k 2 - r/,2)1/2 cos(0 - ~')]},
(42)
where oQ
• (fl, /3', r/, r/', k) = ~ ~ (-i)2(p+q)6xp[i(p[3+q[3')]~pq(r/,r/',k). p=-** q=-.~
(43)
Eq. (42) is eq. (21) in cylindrical coordinates [i.e. the Fourier transform o f f ( r , 0, z, co)]. Thus from eq. (20) we can express f (r, 0, z, w) in the form
f(r,O,z,¢o)= 4"3 k-~3 fO2~rf02~rd/3/3' -.0 f -~, dr/dr/' ((r/+ r/,)2 + [(k 2 _r/2)l/2cos( o _~)+(k 2 _r/,2)l/2cos( 0 _/3,)]2}1/2 X q ~ , / 3 ' , r/, r/', k) exp[i(r/+ r/')zl exp(ir[(k 2 - r/2)l/2cos(0 -/~) + (k 2 - r/'2)l/2cos(0 - ~')]}.
(44)
4. Summary. Eqs. (5), (22) and (44) are exact inverse solutions to the inelastic scattering problem as defined by eq. (1). They apply to scattering data which is obtained in the near field (i.e. when r "~ r 0 " rs), and is collected on the surfaces of a sphere, a plane and a circular cylinder respectively. This data is assumed to be generated by the interaction of an incident spherical wave g(rlro, k) with a frequency dependent potential f(r, co) where the scat193
Volume 110A, number 4
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22 July 1985
tered field CB(rslr0, k) under the Born approximation is related to the scattered field CR(rslr0, k) under the Rytov approximation by [2] CR(rs[r0, k) = k-2g(rslr o, k) ln[¢~(rslr0,
k)[g(rslro,
k)] .
References [1] [2] [3] [4]
194
J.M. Blackledge, Phys. Lett. 107A (1985) 59. J.M. Blackledge and L. Zapalowski, Inverse problems 1 (1985) 17. I.S. Gradshteyn and I.M. Ryshik, Tables of integrals, series and products (Academic Press, New York, 1980). G. Beylkin, J. Math. Phys~ 24 (1983) 1399.
(45)