UTMS 2003–5

January 29, 2003

An inverse problem in periodic diﬀractive optics:
Global nuiqueness with a single wave number
byJ. Elschner, G. Schmidt and M. Yamamoto

T

UNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANAn inverse problem in periodic diractive optics:
Global uniqueness with a single wave numberyy and M. Yamamotoz

J. Elschner , G. Schmidt

y Weierstrass Institute for Applied Analysis and Stochastics,

Mohrenstrasse 39, D  10117 Berlin, Germanyz Graduate School of Mathematical Sciences, The University of Tokyo,

3-8-1 Komaba Meguro Tokyo 153-8914, Japan

E-mail: elschner@wias-berlin.de, schmidt@wias-berlin.de,myama@ms.u-tokyo.ac.jp
We consider the problem of recovering a perfectly reecting twodimensional diraction grating from the knowledge of one wave number, one incident
direction and the total eld measured above the grating. We prove a global uniquenessresult within the class of polygonal grating proles. The proof relies on the analyticity
of solutions to the Helmholtz equation and the Rayleigh expansion of the scatteredeld.

Abstract.

AMS classication scheme numbers: 78A46, 35R30, 35Q60
1. Introduction

The problem of recovering a periodic structure from knowledge of the scattered eldoccurs in many applications, e.g., in diractive optics, radar imaging and nondestructive
testing. In this paper we consider the scattering of monochromatic plane waves by aperfectly reecting diraction grating and restrict ourselves to the transverse electric
polarization (the TE mode) in an isotropic lossless medium. Our goal is to prove global
uniqueness in determining polygonal periodic grating proles by near eld observations
with a single wave number.
Let the prole of the diraction grating be given by the curvef := f(x1; x2) 2 R2 : x2 = f (x1 )g
where f is a 2 -periodic Lipschitz function. Suppose that a plane wave given byuin := exp(i x1 i x2); ( ; ) = k (sin ; cos )
is incident on f from the top, where the wave number k is a positive constant and 2 ( =2; =2) is the incident angle. The domain above the curve is denoted by2
f := fx 2 R : x2 > f (x1 )g:2

An inverse problem in periodic diractive optics

Then the total eld u = u(x1; x2), which is the sum of uin and the scattered eld,satises the Dirichlet problem

u + k 2 u = 0; u = 0 on f ;and is assumed to be -quasiperiodic in x1:
u(x1 + 2; x2) = exp(i2  )u(x1; x2 ) :Moreover, u is required to satisfy the radiation conditionXu(x) = uin + An exp(i(n + )x1 + i n x2) ; x2 > max(f )
in

Z

n2

with the Rayleigh coecients An

(

n

:=

f

2 C andjn + j  k ;jn + j > k :(k 2 (n + )2 )1=2 ifi((n + )2 k 2)1=2 if

(1.1)

(1.2)
(1.3)

(1.4)

Since n is real for at most a nite number of indices, we notice that only a nite
number of plane waves in the sum (1.4) propagate into the far eld, with the remaining
evanescent waves decaying exponentially as x2.
1 ( ) of the direct
It is known that there exists exactly one solution u
Hlocfdiraction problem (1.1)(1.3); see [9] for suciently smooth (fC 2 ) and [7] for
Lipschitz proles.
The inverse problem or the prole reconstruction problem can now be formulated
as follows.
(IP): Determine the prole function f from the knowledge of one wave number k ,one incident direction  and the total eld u x2=b on a straight line x R2 : x2 = b
with b > max(f ).
Note that this problem also involves near eld measurements since the evanescentmodes cannot be measured far away from the grating prole.
The global uniqueness in problem (IP) is known if the wave number or the amplitudeof the grating are suciently small [8]; see also [1] in the case of a lossy medium (i.e.,
Im k > 0). For related stability results we refer to [4], [5]. A recent review on uniquenessresults in scattering theory for periodic structures can be found in [2].
In general, global uniqueness may not be true when k is real. This can be seen fromthe simple counterexample of the scattering of uin = exp( ikx2) when one moves the atgrating in certain multiples of the wavelength. Even though we exclude this at case, theglobal uniqueness is not known with a single general k > 0. The purpose of this paper
is to solve this open problem in some case, that is, to establish the global uniquenesswithin piecewise linear proles for any xed k . Notice that a class of piecewise linear
proles is acceptable from a practical viewpoint (e.g., [10]). Now we state our main
result.

!1

2

j

2

f 2

g

Theorem. Exclude the Rayleigh frequencies by assumingn6= 0 ;i.e.

; k 2 6= (n + )2

for alln 2 Z:(1.5)

3

An inverse problem in periodic diractive optics

f1 and f2 be continuous 2 -periodic piecewise linear prole functions consisting ofnitely many segments, where the case f1  const , f2  const is excluded. Let u1
and u2 solve the corresponding direct diraction problem (1.1)(1.3) in
f1 andf2 ,respectively, and let b > max(f1; f2 ). Then the relation
u1(x1 ; b) = u2(x1 ; b) for all x1 2 R
implies f1 = f2 .
Let

Remark 1. If condition (1.5) is violated, one obtains further non-uniqueness examples

for (IP) (in addition to the case of parallel half-planes).
(i) Let k = 1 and = 0 (orthogonal incidence). Then we havethe nite Rayleigh expansion= 1; 1 = 0, andu(x) = exp( ix2) + exp(ix2) exp( ix1) exp(ix1)fgsatises the Helmholtz equation in the whole plane and vanishes on the lines x2 = x1 ,x2 = x1 and on the (quadratic) grids obtained by the 2 -periodic extensions of
x2 = x1 , x2 = x1 .
(ii) For k = 2 and  = =6 we have = 1; = 3; 1 = 3 = 0. Then the nite
sum of plane waves

f
f

g
gf

g

u(x) = exp(ix1

p

p

p

i 3x2 ) + exp( ix1 + i 3x2) exp(2ix1) exp( 2ix1 )

f

p

gf

satises the Helmholtz equation and vanishes on the lines x2 = 3x1 , x2 =and on the grids generated by the 2 -periodic extensions of these lines.

px1= 3g

The proof of the theorem is based on the analyticity of solutions to the Helmholtz
equation (e.g., [6]) and the Rayleigh expansion (1.3). The following section is devoted
to this proof, which is divided into several steps.
2. Proof of the theorem

2.1. Reection argument

We need the following auxiliary lemma. Consider a triangular or quadrilateral domain
R2 which is symmetric with respect to a line L. Furthermore let 0 and 1 be the1
two segments of @ 1 , which are not perpendicular to L and where 0 is the reectionof 1 with respect to L. Finally, set 2 = 1 L and let 2 be the subdomain of 1
lying between 0 and 2 ; see Fig. 1.

\

uj 2 H 2 ( j ) satisfy the Helmholtz equation ( + k 2 )u = 0 inthat uj j j = 0 (j = 1; 2), and assume that u1 = u2 in
2 . Then u1 j 0 = 0.

Lemma 1. Letj suchSince the Laplace operator is invariant under translation and rotation, we may
assume that 1 is symmetric with respect to the x1 axis. Setting v = u1 in 02 (theProof.

4

An inverse problem in periodic diractive optics

1

1
2

L

L
2
0

0
Figure 1.

Symmetric domain 1

reection of 2 with respect to the x1 axis) and v (x1; x2) =obviously have( + k 2)v = ( + k 2)u1 = 0 inj2:

u1(x1; x2) in

2 , wejMoreover, since u1 = u2 in 2 and u2 2 = 0, we obtain v 2 = 0, and the jump across
1=2
2 of the normal derivative @2 v = @v=@x2 (which is dened in the sense of H ( 2 )) iszero. Hence v is a distributional solution of the Helmholtz equation in 1 = 0222,20which belongs to H ( 1 ) by the elliptic regularity. Since v = u1 in 2 , we also havev = u1 in 1 by unique continuation. This implies u1 0 = u1 1 = 0 by the denitionof v .

[ [jj

2.2. Construction of an exit direction

6

To prove the theorem, let f1 = f2 be 2 -periodic continuous piecewise linear prolefunctions, with the case f1 = const , f2 = const excluded. Consider the solutions
1 (uj Hlocfj ) of the corresponding direct diraction problems, and dene2f (t) := max(f1(t); f2 (t)); t 2 R:

Note that by the elliptic regularity for the Dirichlet problem (1.1), each function uj isinnitely smooth up to the boundary, with the exception of the corner points of fj .
Moreover, since uj satises the Helmholtz equation, uj is real-analytic in fj (e.g., [6]).
Since u1 = u2 and then also @2u1 = @2 u2 on x2 = b , we havef

u1 = u2 =: u

in

f

g

;

(2.6)

see, e.g., [1]. Henceforth by a ray we mean a straight line starting from one point andextended to the point at innity.
Lemma 2. There exists a ray

S such that

S is not parallel to the coordinate axes; S 
Proof. We have to consider the following cases.

2

f

and

ujS = 0 :(2.7)

1) f2(t) > f1(t) ; t R ;a) f2 = const : Consider a segment of f2 with an endpoint of maximal x2 coordinate,6

5

An inverse problem in periodic diractive optics

which is not parallel to the x1 direction, and extend it to a ray S
f1 . Thenu S = u1 S = 0 since u1 vanishes on an open subset of S and is real-analytic in f1(and thus real-analytic as a function of a single independent variable on S
f1 ).
b) f2 = const : Consider the reection of f1 with respect to f2 , which is denotedby . Then u  = 0 by Lemma 1, and as in a) we can choose a ray S
f2 such thatu S = 0, because f1 = const by the assumption.
2) The graphs of f1 and f2 do intersect:Henceforth P1 P2 denotes the open segment connecting points P1 and P2 . By theperiodicity of f1 and f2 , we can choose two intersection points. Let Q1 and Q2 beintersection points of f1 and f2 such that f2 (t) > f1 (t) between Q1 and Q2. Choosea directed segment P1 Q of f2 between Q1 and Q2 with origin P1 and an endpoint Q
of the maximal x2 coordinate. Then the ray S extending P1 Q either satises (2.7) (cf.
the case 1a)), or intersects f at some point P2 . In the latter case, we have u P1 P2 = 0,
where the real analyticity of u1 was used again.
Assume, for example, that the segment P1 P2 has non-negative slope with respect
to the positive x1 direction. Here P1 may coincide with Q1 . If P2 is dierent from Q2 ,then there exists a segment  of f lying below P1 P2 and intersecting P1 P2 at P2 fromthe left, since f is the graph of a piecewise linear function. If P2 coincides with Q2, we
can choose a segment  f1 with the same properties. In each case we have u1  = 0.
By Lemma 1, we obtain u = 0 on the reection of  with respect to P1 P2 . A possibly
repeated application of the reection argument and the above analyticity argument yield
either a ray S satisfying (2.7), or a segment of angle 2 > 0 with the negative x1 direction
such that u P2 P3 = 0, P3 f and P3 lies above P2 and on the left from P1 ; see Fig. 2.

j

j

\

j

j



6

j



j

j

2

P3
2

P1
Figure 2.

P2
Reection step at P2

Then we repeat the construction with respect to the point P3 and apply reection
at P2 P3 etc., which gives either a ray S satisfying (2.7), or a segment P3 P4 of angle
f and P4 is lying above
3 > 0 (with positive x1 direction) such that u P3 P4 = 0; P4
P3 and on the right from P2 .
Thus we obtain a sequence of segments Pn Pn+1 with slopes n such that u Pn Pn+1 = 0
and Pn+1 Pn+2 lies above Pn Pn+1 for n = 1; 2; : : :. This process must terminate with

j

2

j

6

An inverse problem in periodic diractive optics!

!1nding a ray S satisfying (2.7) unless n
0 as n. We nally show that this caseis impossible.
Since there are only nitely many segments of f1 f2 , there exists a minimal
slope  > 0 (with respect to the x1 axis) among all segments not parallel to the x1 axis.
Therefore, if all n (n n ) are close to zero, then the reection step at Pn +1 yields a
segment Pn +1 Pn +2 of slope n +1 close to  , which is a contradiction.

[



2.3. Reduction to a nite sum of propagating waves

Consider the solution u of the Helmholtz equation in
the convergent Rayleigh series expansion

u(x) = A exp(i x1

i x2) +

X

f dened by (2.6). Then we have!

An exp(i( + n)x1 + i n x2)

n2P
0
1
X
+@
An exp(i(n + )x1 + i n x2 )A := v + w ; x2 > max(f ) ;

ZnP

(2.8)

n2

f 2

2 g



where A = 1 and P denotes the nite set n Z : n R . Note that i n
C>0
for all n Z P and nn i as n.
From Lemma 2 we have a ray S
f such that u S = 0. The next lemma showsthat then all Rayleigh coecients of the evanescent modes must vanish.

2 n

j j

j j!1


Lemma 3. Suppose there exists a rayj

S such that the conditions (2.7) hold. Then theu satises
An = 0 for all n 2 ZnP :Rayleigh expansion (2.8) of(2.9)

x coordinates only amounts to dierent coecients inthe expansion (2.8) with A 6= 0, we can assume that S = fx2 = ax1 : x1 > 0g for somea 2 R, a 6= 0. Consider the case a > 0, for example.
Then, for any " > 0, there exists N 2 N suciently large so thatjw(t; at)j  " ; t  N :From the relation ujS = 0 we then havejv(t; at)j  " ; t  N :(2.10)
The nite sum v1(t) := v (t; at) is a special case of an almost periodic function on R.
Proof. Since a translation of the

Therefore the relation

maxt2Rjv1(t)j = lim supt!+1 jv1 (t)jjholds (see, e.g., [3, p. 407]), which together with (2.10) implies v1 = 0, hence v S = 0.
From (2.7) and (2.8) we then obtain w S = 0.
It remains to verify that the last relation implies (2.9). Letj = minf i : n 2 ZnP g :n7

An inverse problem in periodic diractive optics

There are at most two indices n, say n0, n1 , such that
We obtain the convergent series

i n =  and n1 + = (n0 + ).

exp( at)w(t; at) = An0 exp(i(n0 + )t)X+ An1 exp( i(n0 + )t) +An exp(i(n + )t + (i n + )at) = 0

Z

n2 nP
n6=n0 ;n1(2.11)! 1

for t suciently large. Since the innite sum in (2.11) tends to zero as t
+ , thesum of the rst two terms on the right handside becomes arbitrarily small if t is large
enough. This implies An0 = An1 = 0, using the above argument for almost periodic
functions and the fact that the functions exp(i(n0 + )t) ; exp( i(n0 + )t) are linearly
independent. If there is only one index n = n0 with i n = , then An0 = 0, of course,
follows immediately. Repeating this reasoning, we obtain successively An = 0, n Z P .

2 n2.4. End of proof of the theorem

From (2.8) and Lemma 3 we obtain the nite sum of plane waves,v (x) = A exp(i x1

i x2) +X

n2P

An exp(i( + n)x1 + i n x2 ); A 6= 0;2(2.12)

jso that v (x) is analytically extended to any x R2 and satises the conditions v L = 0
by (2.6), where L is any straight line extending a segment of f1 f2 . Here we notethat v solves the Helmholtz equation (+ k 2 )v = 0 and is real-analytic in R2. Therefore,
since both f1 and f2 cannot be constant, we nd two lines L0 and L' ; ' (0; =2], so
that

[

2

v jL0 = v jL' = 0 :Here and in the following L denotes the line of polar angle
with respect to L0 .
Without loss of generality, we take the intersection point of L0 and L' as the origin O.
After repeated application of the reection argument of Lemma 1, only the following
two cases may occur:(i) Let ' =  , where  (0; 1=2) is irrational. Then we have2

v jLk' = 0 ; k 2 N
where the directions of Lk' with respect to L0 are dense in [0; 2 ).
(ii) The case of rational  leads to the relations
v jLk=N = 0 ; k = 0; 1; : : : ; N 1 ; for some N 2 N ; N  2
by the odd extensions by Lemma 1.
Proof in case (i).

There exists n0

2



Let
0. By (2.13) we nd a > 1 such that v (t;P such that(i + i a) = i(n0 + )i n0 a:(2.13)(2.14)

at) = 0 ; t 2 R.
(2.15)

8

An inverse problem in periodic diractive optics

Otherwise the rst term on the right hand side of (2.12) is linearly independent of
the second term and the equality (2.12) on x2 = ax1 and A = 0 lead us to acontradiction.
The relation (2.15) implies

f

g

6

a(k 2 2 )1=2 + a(k 2 (n0 + )2 )1=2 = n0 ;hence either n0 = 0, k = , or n0 > 0. The rst case is excluded by the conditionjj < =2. For n0 > 0, we obtainn0 a = a( 2 n20 2 n0 )1=2 :Since 2  n20 + 2 n20  n20 , we have
 jn0j so that n0 a  n0 an0 < 0,which is a contradiction. Analogously, for 0, we obtain a contradiction withv (t; at) = 0 ; t 2 R, for some a > 1. Thus the case (i) always leads to a contradiction,
proving uniqueness of (IP).

Let k ; k = 0; 1; : : : ; N , be the open two-sided sector of angle =N
lying between the lines Lk=N and L(k+1)=N , with the convention that N = 0 . Wenote that
Proof in case (ii).

[Nk=01 k = R2;

2

R 2

since we consider two-sided sectors. For a point x
k , letkx
k+1 denote its
reection with respect to the line L(k+1)=N . From the reection argument of Lemma 1(or rather its proof), we now deduce the relationv (R0x) = v (x) ; x 2 1 :In fact, the function v  dened byv (x) = v (x) ; x 2 0 ; v (x) = v (R0 1 x) ; x 2[(2.16)

1solves the homogeneous Helmholtz equation in 0
1 and vanishes on the line L=N , andthe jump of its normal derivative across L=N is zero. Hence v  solves the homogeneous
Helmholtz equation in 0 L=N
1 , and (2.16) follows by unique continuation.
Using the relations (2.14), noticing that[[

Rx := RN 1RN 2 : : : R0x = x ;x20

and applying the above reection argument N times, we then obtainv (x) = ( 1)N v (Rx) = ( 1)N v ( x) ; x 2 0 :By unique continuation, the last relation holds for all x 2 R2 and takes the form
A exp(i x1

i x2) +X

n2P

An exp(i(n + )x1 + i n x2)= ( 1) A exp( i x1 + i x2) + ( 1)N
N

X
n2P

An exp( i(n + )x1

i n x2) ;(2.17)

implying( 1)N A exp( i x1 + i x2) = An0 exp(i(n0 + )x1 + i n0 x2)(2.18)

9

An inverse problem in periodic diractive optics

2

2

6

for some n0 P and all x R2. Now A = 0 and (2.18) give the equalities n0 =and A( 1)N = An0 . Therefore (2.17) implies
n0 =X

n2P nfn0 g

An exp(i(n + )x1 + i n x2) =2

Xn2P nfn0 g

An exp( i(n + )x1

2 ,i n x2 )2 nf gfor (x1 ; x2) R2. Then, for n P n0 , we must have n = 0 (i.e., a Rayleigh frequency
would occur contradicting (1.5)) or An = 0 in (2.17). Therefore, v takes the formv (x) = A exp(i x1

i x2) + An0 exp( i x1 + i x2) ; A 6= 0

and vanishes on two dierent lines passing through the origin, which is also impossible.
This concludes the proof of the theorem.
The proof of Lemma 2 can be easily extended to the case that verticallines are present in the proles f1 and f2 . In that case we always nd a ray Sfwith u S = 0 which may be parallel to x1 or x2 direction. If S is parallel to the x2
axis, then a contradiction to the presence of the incoming wave can be directly obtainedusing the relation v S = 0. For S parallel to x1 direction, it follows easily by taking theFourier expansion on S that u (after a translation of the x coordinates) reduces to theexpression
Remark 2.

jju(x) = A exp(i x1 i x2) + A0 exp(i x1 + i x2) ; A 6= 0
which must vanish on the x1 axis and another line passing through the origin, leading

to a contradiction again. This proves our global uniqueness result within a more generalclass of polygonal grating proles (which are not necessarily dened by the graph of apiecewise linear function).
Note that we can obtain the uniqueness in problem (IP) without excluding theRayleigh frequencies (condition (1.5)) if there exists a ray S parallel to one of thecoordinate axes on which the total eld vanishes.
The case of transverse magnetic polarization leads to the Neumann
problem, where the Dirichlet condition in (1.1) is replaced by the Neumann condition
Remark 3.

@ u = 0onfand @ denotes the normal derivative. Then the global uniqueness results of the theorem
and Remark 2 carry over to the inverse Neumann problem if we require the additional
condition that no segment of the proles f1 ; f2 is parallel to the direction  of theincident wave. If the latter condition is not satised, then one obtains further nonuniqueness examples for the reconstruction problem. The proof is essentially parallel tothat for the inverse Dirichlet problem, but needs some modications related with the
Neumann boundary condition. We do not present the details here and only note thatthe odd extension used in Subsections 2.1 and 2.4 has to be replaced by even extension.

An inverse problem in periodic diractive optics

10

Acknowledgments

The rst and the second authors gratefully acknowledge the support by the German
Research Foundation (DFG) and the Department of Mathematics of the University ofTokyo. The third author was partly supported by Grant-in-Aid for Scientic Research(14604005) from Japan Society for the Promotion of Science.
References[1] G. Bao, A uniqueness theorem for an inverse problem in periodic diractive optics, Inverse
Problems 10 (1994), 335340.
[2] G. Bao, Inverse and optimal design problems in diractive optics, in: Proceed. International Conf.
on Inverse Problems, City University of Hong Kong, World Sci. Press, to appear.
[3] A. Böttcher, B. Silbermann, Analysis of Toeplitz operators, Akademie-Verlag, Berlin, 1989.
[4] G. Bruckner, J. Cheng, M. Yamamoto, Conditional stability in determining a periodic structure ina lossy medium and the Tikhonov regularization, Trans. Japan Soc. Comput. Methods Engng.
1 (2001), 61-66.
[5] G. Bruckner, J. Cheng, M. Yamamoto, An inverse problem of diractive optics: conditional
stability, Inverse Problems 18 (2002), 415433.
[6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag,
Berlin, 1992.
[7] J. Elschner, M. Yamamoto, An inverse problem in periodic diractive optics: reconstruction of
Lipschitz grating proles, Applicable Analysis 81 (2002), 13071328.
[8] F. Hettlich, A. Kirsch, Schier's theorem in inverse scattering for periodic structures, Inverse
Problems 13 (1997), 351361.
[9] A. Kirsch, Diraction by periodic structures, in: Proc. Lapland Conf. Inverse Problems (eds.
Päivärinta et al), Springer, Berlin, 1993, pp. 87102.
[10] J. Turunen, F. Wyrowski (eds.), Diractive Optics for Industrial and Commercial Applications,Akademie-Verlag, Berlin, 1997.

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The Graduate School of Mathematical Sciences was established in the University ofTokyo in April, 1992. Formerly there were two departments of mathematics in the University of Tokyo: one in the Faculty of Science and the other in the College of Arts andSciences. All faculty members of these two departments have moved to the new graduate school, as well as several members of the Department of Pure and Applied Sciencesin the College of Arts and Sciences. In January, 1993, the preprint series of the former
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