University of Groningen
The Unique Solution of the Inverse Diffraction Problem
Hoenders, B.J.
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Optics Communications
DOI:
10.1016/0030-4018(79)90362-6
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Hoenders, B. J. (1979). The Unique Solution of the Inverse Diffraction Problem. Optics Communications,
30(3). DOI: 10.1016/0030-4018(79)90362-6
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Volume 30, number 3
THE
OPTICS COMMUNICATIONS
September 1979
UNIQUE SOLUTION OF THE INVERSE DIFFRACTION PROBLEM
B.J. HOENDERS
Department of Applied Physics, State University at Groningen,
Nijenborgh 18, 9747AG Groningen, The Netherlands
Received 30 May 1979
The problem of the determination of the values of a field on a surface from its values on a surface to which it has propagated is shown to have a unique solution if the field satisfies any linear elliptic partial differential equation.
Suppose that a scalar field ~ is the solution of a
linear second order elliptic partial differential equation
L~ =0,
(1)
in a domain D bounded by two closed surfaces S 1 and
S 2, (see fig. I). The equation (1) can for instance be
the Helmholtz equation (V 2 + k2n2)~b = 0, valid in a
medium with variable index of refraction, or the time
independent Schr6dinger equation in the presence of
an electromagnetic field, characterized by the vector
potential A and the scalar potential ¢:
~2 V 2 ~ + . e / i A . V ~
2m
~n
+
e(2--mmI A I 2 - e ~ ) ~b=E~b.(2)
lem then requires to determine the values of ~ at S 1
from its values at S 2 (or perhaps on some part of $2),
to which the field has propagated. For a review see
Hoenders [1].
For the solution of this problem we need the following corollary of a remarkable theorem derived by
Beckert [2]:
Theorem Let P denote a continuously differentiable
surface lying entirely within a n-dimensional domain E
with boundary aE. The dimension s of P satisfies 1 ~<
s ~< n - 1, and P does not separate E. Le.t u be a solution of the elliptic partial differential equation
n
i,k ~Xk ~ ik~ } OXi ]
The field outside A is supposed to be uniquely determined by its values on S 1 and Sommerfeld's radiation
condition at infinity, i.e. It is assumed that the Dirichlet
problem for the operator L in the domain D outside S 1
admits a unique solution. The inverse diffraction prob-
s~
S2
Fig. 1. The surfaces S1 ,S2 ,S'l, and S~ for n = 2.
+ ~bi(x)-~x(x)+ c(x)~,(x)=1Ix).
(3)
The functions aik are two times Hflder continuous differentiable, c and f are HOlder continuous, and b i are
one time HOlder continuous differentiabl.e.
Suppose that the interior Dirichlet problem L ~ = 0,
with ~ =/*(x) if xEaE, and if#(x) denotes an arbitrary
L 2 function is solvable. Then the set of fiJnctions
{~n(X)), xEl-', generated by a suitable set of boundary
values ~n(X)}, x E a E or x E any subset of aE, n = 1,2..
is together with its normal derivatives dense in the
Hilbert space of all L 2 functions on P. i.e. Any L 2 func.
tions h(x) and (a/an)h(x), x E v can be approximated
simultaneously in the mean arbitrarily closely by a set
of solutions {~kn } o f L ~0n = 0 generated by an appro327
Volume 30, number 3
OPTICS COMMUNICATIONS
September 1979
priate set 0an) of surface distributions on bE.
We will need Green's formula, which reads as
tion. The existence of such a function is ascertained
by the corollary to Beckert's theorem, stated above.
Eqs. (1) and (4) lead to:
f (~(L q,) - O(MCO)dr
f
J
T
{a(¢ b~b/an - tp a¢/an) + b~b¢)do
sl
= f(a(Vaq,/b.- qJa¢/an)+ b~b¢)do,
(4)
P
= - J { a ( ¢ b ~ / a n - ~bac~/an) + b~q~}dr,
s'2
t7
ifM denote the adjoint operator to L :
(9)
or, on using eqs. (7) and (8) we derive from (9):
n
M~ = ~
i,k
ik(X)
dP(X)
a(x)qJ(X)Ix~Si
P
(5)
- ~i a ~ i ( b i ( x ) ¢ ( x ) ) + e(x)¢(x),
o denotes the boundary of the domain r, and
n
n
a = ~i,k aikni'
b = ~i ein i,
&
aaik
e i = b i - Z.,t
k = 1 ax k '
(6)
- J (a(¢,aq, lan - qJa¢lan)+b~¢}do.
s~
(10)
v
Hence, the values of the field on S] can be determined
from the known values of the r.h.s, ofeq. (10) : ¢ and
a(~[an are known by construction, and ~k is assumed
to be known on S 2 . The knowledge of ~k on S 2 together with Sommerfeld's radiation condition enables
us to determine a~k[an, so that the rJa.s, ofeq. (10) is
a known quantity.
The proof of Beckert's theorem as well as the corollary used in this letter will be given in a forthcoming
paper, which also contains some explicit examples.
and n i denote the cartesian components of the normal
tl or o.
Let S'1 and S~ be the surfaces drawn in the figure.
Suppose that DCE and that ~ is a solution ofM(¢) = 0
if xEE, with
q~(x) ~- 0,
~b(x) ~Sr(X - y),
(7)
t
i f x and y E S 1,
(8)
and 5 r(X - y) denotes a regularisation of the 5-func-
328
The author is greatly endebted to dr. H.A. Ferwerda
and dr. A.M.J. Huiser for many discussions and a critical
reading of the manuscript.
References
[1] B.J. Hoenders, in: Inverse source problems in optics, ed.
H.P. BaRes (Springer, Heidelberg, 1978) Ch. 3.
[2] H. Beckert, Math. Ann. 139 (1960) 255;