Inverse Scattering for Gratings and Wave Guides
aa r X i v : . [ m a t h . A P ] A p r Inverse Scattering for Gratings and Wave Guides
Gregory Eskin, Department of Mathematics, UCLA, E-mail: [email protected] Ralston, Department of Mathematics, UCLA, E-mail: [email protected] Yamamoto, Graduate School of Mathematical Sciences,University of Tokyo, E-mail: [email protected]
Abstract:
We consider the problem of unique identification of dielectric coefficientsfor gratings and sound speeds for wave guides from scattering data. We provethat the “propagating modes” given for all frequencies uniquely determine thesecoefficients. The gratings may contain conductors as well as dielectrics and theboundaries of the conductors are also determined by the propagating modes. § Introduction
Consider Maxwell’s equations for time-harmonic electric and magnetic fields,exp( − iωt ) E ( x , x , x ) and exp( − iωt ) H ( x , x , x ), in the absence of currents andcharges ∇ × E − iωµ H = 0 , ∇ × H + iωǫ ( x ) E = 0 . In this paper we study the inverse problem of determining the electric permittivity, ǫ , and hence the dielectric coefficient, ǫ/ǫ , from scattering data for these equations.The fundamental assumptions are that ǫ is independent of x , 2 π -periodic in x andconstant (= ǫ ) for | x | > T. These conditions are designed to model a dielectric“grating” extending throughout the region | x | < T (c.f. [P], [BDC]). We also allowfor conducting bodies embedded in the dielectric as long as they satisfy conditionsanalogous to our conditions on ǫ : their boundaries should be invariant with respectto all translations in x and translations by 2 π in x , and they should be containedin | x | < T .To define data sets for this inverse problem it is customary to consider thescattering problem for fields with either the transverse electric (TE) or transversemagnetic (TM) polarizations, respectively E ( x ) = (0 , , u ( x , x )) and H ( x ) =(0 , , v ( x , x )). These polarizations reduce Maxwell’s equations, respectively, to1 ǫ ( x ) ∆ u + ω µ u = 0 ( T E )and ∇ · ǫ ( x ) ∇ v + ω µ v = 0 . ( T M )In the case of embedded conductors we consider the TE polarization in the exteriorof the conductors with the Dirichlet condition, u = 0, on the boundary, since thiscorresponds to E = 0 and H · ˆ n = 0 on the surfaces of the conductors. For ourpurposes it is convenient to write both (TE) and (TM) as Lu − k u = 0 , where L = − ∆ for | x | > T and k = ω µ ǫ .We will also present the analogous inverse problem for acoustic wave guides.This requires only small modifications of the arguments for gratings. The waveguides that we consider are simply slabs, { < x < B } , in which the sound speed c is a function of ( x , x ). We assume that c ( x , x ) = c ( x ) for | x | > T , andimpose Dirichlet condition on x = 0, and the Neumann condition, ∂ x u = 0, on x = B . These boundary conditions correspond to an acoustically soft reflectingsurface at x = 0 and an acoustically hard reflecting surface at x = B , modellingunderwater sound propagation with x as depth (c.f. [BGWX]). We will show thatscattering data from propagating modes for the operator L = − c ( x )∆ with theseboundary conditions determine c ( x ).In both these settings we will apply recent results on inverse coefficient problemsfor hyperbolic equations (Belishev [B], Kachalov-Kurylev-Lassas [KKL] and Eskin[E1],[E2]). In those papers the data for the inverse problem is the Dirichlet-to-Neumann map. Hence the objective here will be to show that the scattering datadetermine the Dirichlet-to-Neumann map on a line x = T .Inverse scattering problems for dielectric gratings have been studied previouslyin [BDC], [BF], [K], [HK] and [EY]. These articles consider primarily the inverseproblem of finding the boundaries of conductors embedded in a dielectric of constantpermitivity from scattering data. To the best of our knowledge the present paperis the first to show that a variable dielectric is uniquely determined by scatteringdata.Inverse coefficient problems for wave guides were studied in [BGMX], [GMX],[M], [X] and [DM]. These papers give methods for recovering the sound speed. Weonly consider the uniqueness problem and prove that the sound speed is uniquelydetermined by the propagating modes. Our approach was influenced by the workof S.Dediu and J. McLaughlin, [DM], which also uses propagating modes. § Statement of results
Our results for gratings hold under mild conditions on the operator on L in (1).We will assume that L is a second order elliptic operator on D ⊂ R which issymmetric in the inner product( f, g ) = Z D f ( x ) g ( x ) a ( x ) dx. The weight a ( x ) is continuous and strictly positive on D . The coefficients of L + ∆are supported in | x | < T , and L commutes with translation by 2 π in x . Likewise a ( x ) − | x | < T and a ( x + 2 π, x ) = a ( x , x ). We will alsoassume that the region D is invariant under translation by 2 π in x , and boundary D is smooth. There are two cases that we wish to consider.Case 1: D is connected and contains {| x | > T } . In other words, while there maybe some holes in D , they do not disconnect D , and they are contained in | x | < T .Case 2: D is connected and we have the inclusions { x > T } ⊂ D ⊂ { x > − R } for some R > The domain of L will be H α ( D ) ∩ H α, ( D ). By H kα ( D ) (resp. H kα, ( D )) we meanfunctions satisfying u ( x + 2 π, x ) = e πiα u ( x , x ) (1)such that φ ( x ) u ( x , x ) ∈ H k ( D ) (resp. H k ( D )) for all φ ∈ C ∞ c ( R ). Note thatthis domain for L corresponds to the Dirichlet boundary condition on ∂D .For wave guides we simply take L = − c ( x ) ∆ on D = { x ∈ R : 0 < x T .For both gratings and wave guides scattering data at fixed energy k are obtainedfrom the “propagating modes”. In the case of gratings the scattered wave v + isobtained by solving ( L − k ) u = 0 in D with u = exp( l · x ) + v + , where l · l = k and v + is the “outgoing” solution to( L − k ) v = − ( L − k ) e il · x obtained as the limit as Im { k } → + (see below). For gratings we will only usescattering data from incident waves exp( il · x ) which satisfy the condition (2), i.e.those with l = n + α, n ∈ Z , l = − p k − ( m + α ) . For x > T the scatteredwave v + has the form v + ( x, n, k ) = X m ∈ Z e i [( m + α ) x + x √ k − ( m + α ) ] a m ( n, k ) (2)= X ( m + α )
0, provided that k does not belong to the set of “thresholds ”, { k : k = ( p + α ) , p ∈ Z } . We call { a m ( n, k ) , n, m ∈ Z : ( m + α ) < k , ( n + α ) < k } the scattering data at energy k from “propagating modes”.For wave guides, since we are taking L = − c ( x ) ∆ as the unperturbedoperator, the scattered wave v + ( x, k, n ) is obtained by solving ( L − k ) u = 0with u = Φ( x, k, n ) + v + , where Φ is a generalized eigenfunction for L , i.e.Φ( x, k, n ) = exp( − ix p µ n ( k )) φ n ( x , k ), where φ ′′ n + k c − φ n = µ n ( k ) φ n , φ n (0) = 0 , φ ′ n ( B ) = 0 , and µ n ( k ) > . For x > T the scattered wave v + has the form v + ( x, k, n ) = X m ∈ N b m ( k, n ) e ix √ µ m ( k ) φ m ( x , k ) X µ m ( k ) > b m ( k, n ) e ix √ µ m ( k ) φ m ( x , k ) + O( e − δx ) (2 ′ )for some δ >
0, provided that k does not belong to the set of thresholds, { k : µ p ( k ) = 0 , p ∈ N } . Here we call { b m ( k, n ) , n, m ∈ N : µ n ( k ) > , µ m ( k ) > } thescattering data at energy k from propagating modes.Note that in these definitions the functions p k − ( m + α ) and p µ m ( k ) will bechosen so that they extend into Im { k } > z ( k ) stand for either k − ( m + α ) or µ m ( k ), this choice amounts to choosing p z ( k ) > z ( k ) > k > p z ( k ) < z ( k ) > k <
0, and p z ( k ) = i p | z ( k ) | when z ( k ) <
0. We will follow these conventions in the rest ofthe paper.With the preceding definitions we have:
Theorem 1:
The scattering data from propagating modes in x > T given for all k determine D and ǫ ( x ) for gratings with either the (TE) or (TM) polarizations,and c ( x ) for wave guides.The proof of theorem will proceed as follows. We will consider “generalizeddistorted plane waves” u + ( x, k, n ) = exp( i ( n + α ) x − ix p k − ( n + α ) ) + v + ( x, k, n )and u + ( x, k, n ) = exp( − ix p µ n ( k )) φ n ( x, k ) + v + ( x, k, n ) , which are defined without the restrictions k > ( n + α ) and µ n ( k ) >
0. Thesegeneralized distorted plane waves exist for k ∈ R \ S , where S is a discrete set. Notethat when k < ( n + α ) or µ n ( k ) < x → ∞ . In § § k and all n , uniquely determines the Dirichlet-to-Neumann map on a suitable line x = T for all choices of k outside a discreteset. We also show that, making use of the analytic continuation to Im { k } > v + ( x, k, n )’s, these generalized distorted plane waves are determined by thescattering data from propagating modes. Thus, under the hypotheses of Theorem 1,the Dirichlet-to-Neumann map is known on x = T for all k ∈ R outside a discreteset. Since this is equivalent to knowing the hyperbolic Dirichlet-to-Neumann mapfor the wave equations u tt = Lu on x = T , the proof of Theorem 1 will be reducedto the results on hyperbolic inverse coefficient problems cited above. Since analyticcontinuation plays a big role here, there are many variations on the set of k forwhich the propagating modes are known which lead to the same results. § Determination of the Dirichlet-to-Neumann Map for Gratings
In this section we will show that the scattering data from propagating modesdetermine the Dirichlet-to-Neumann map on a line x = T for the case of gratings.To do this we will first show that the traces of an appropriate family of distortedplane waves on x = T are dense in L (0 < x < π ).To begin we need the incoming and outgoing fundamental solutions for − ∆ − k on R in a form compatible with (1). Using Fourier series in x to reduce this toan ODE in x , one computes that for Im { k } > − ∆ − k ) − f ]( x ) = ∞ X m = −∞ π Z ∞−∞ ( Z π e i [( m + α )( x − y )+ λ m ( k ) | x − y | ] f ( y )2 iλ m ( k ) dy ) dy , (3)where λ m ( k ) = p k − ( m + α ) with the branch chosen ∼ k near infinity and the cut on ( −| m + α | , | m + α | ). Notethat Im { λ m ( k ) } > { k } >
0, and hence ( − ∆ − k ) − maps H α ( R ) into H α ( R ). The continuous extension of λ m ( k ) from Im { k } > λ m ( k ) = k p − ( m + α ) /k when ( m + α ) < k and λ m ( k ) = i p ( m + α ) − k when ( m + α ) > k . The corresponding extension of ( − ∆ − k ) − to R \{± ( m + α ) , m ∈ Z } gives the outgoing fundamental solution, G + ( k ). For the incomingfundamental solution we take λ m ( k ) = − p k − ( m + α ) , i.e. the branch chosen ∼ − k near infinity. Substituting this in the formula for ( − ∆ − k ) − f to get ( − ∆ − k ) − in Im { k } <
0, and define the incoming fundamentalsolution, G − ( k ), by continuous extension from Im { k } < G + ( k ) extends analytically to ( − ∆ − k ) − in Im { k } >
0, and G − ( k ) extends analytically to ( − ∆ − k ) − in Im { k } < L .Choose ψ ∈ C ∞ ( R ) such that ψ ≡ {| x | ≥ T } , and thesupport of ψ ( x ) is contained in the set where L = − ∆ and a = 1. An (outgoing)generalized distorted plane wave for L is a solution of ( L − k ) u = 0 in D of theform u + = ψ ( x ) exp( il · x ) + v + , where l ≡ α mod 1 with 0 ≤ α < , l = − p k − ( n + α ) , and v + is defined by limiting absorption, i.e. v + = − lim ǫ → + ( L − ( k + iǫ ) ) − ( L − k )( ψ ( x ) e il · x ) . These are generalized distorted plane waves in the sense used in §
1, since the secondcomponent of l is not necessarily real. Ordinarily, outgoing distorted plane wavesare defined as solutions of the form u + = exp( il · x ) + v + where v + is outgoing.However, we have u + = ψ ( x ) e il · x + v + = e il · x + [( ψ ( x ) − e il · x − lim ǫ → + ( L − ( k + iǫ ) ) − ( L − k )( ψ ( x ) e il · x )] . Since ( ψ ( x ) −
1) exp( il · x ) is outgoing, the term in brackets is outgoing.The limit defining v + will exist unlessi) k is one of the “thresholds”, k = ( n + α ) , where G ± ( k ) are undefined, orii) there is a solution to the homogeneous equation ( L − k ) u = 0 in D which issquare-integrable on D ∩ { < x < π } .We denote the set of exceptional k ’s defined by i) and ii) as S . Note that case ii) can occur. Choose V with compact support so that − ∂ x + V ( x ), consideredas a Schr¨odinger operator on R , has a bound state, u ∈ L ( R ), i.e. ( − ∂ x + V ( x )) u = Eu.
Then,taking m large enough that ( m + α ) + E − V is strictly positive, defining ǫ − ( x ) = (( m + α ) + E )(( m + α ) + E − V ) − and ψ = exp( i ( m + α ) x ) u ( x ), we have (1 /ǫ ( x ))∆ ψ +(( m + α ) + E ) ψ = 0 . Since l = n + α for a unique n ∈ Z , we use n and k to parametrize the generalizeddistorted plane waves, u = u ( x, n, k ). With these definitions we have outgoingdistorted plane waves for all ( n, k ) ∈ Z × R \ S . The analytic properties of G + ( k )discussed above carry over to the u + ( x, n, k )’s: they have analytic continuationsto Im { k } > R \ S . This leads directly to thefollowing conclusion which we state as a lemma. Lemma 0:
For each n the set { u + ( x, n, k ) , k ∈ I } , where I is an open interval in k > ( n + α ) determines u + ( x, n, k ) for k ∈ R \ S . Thus the true distorted planewaves determine the generalized distorted plane waves.The following observation is the main step in the proof. Lemma 1.
Letting ( L − ( k − i ) − g denote lim ǫ → − ( L − ( k + iǫ ) ) − g , the“incoming” solution, we have Z π f ( x ) u + ( x , T, m, k ) dx = Z D ∩{
The set of generalized distorted plane waves { u + ( x, k, m ) , m ∈ Z , x > T } for fixed k ∈ R \ ( S T ∪ S ), determine Λ( k ) on x = T .We want to recover Λ( k ) from the propagating modes. That follows easily atthis point. Lemma 3.
The scattering data from propagating modes, { a m ( n, k ) : ( m + α ) < k and ( n + α ) < k } given for all k ∈ R \ S , determine the distorted plane waves in x ≥ T .Proof: By (2) a m ( n, k ) = e − ix √ λ m ( k ) π Z π e − ix ( m + α ) v + ( x , T, k, n ) dx , it follows that a m ( n, k ) is analytic in k on the set where v + is analytic in k . For fixed m and n , a m ( n, k ) will be part of the scattering data from propagating modes when k is sufficiently large. Thus for each m and n the scattering data from propagatingmodes determine a m ( n, k ) on R \ S . Thus by (2) the propagating modes determinethe generalized distorted plane waves.Combining Corollary 1 and Lemma 3, we conclude the the propagating modesdetermine Λ( k ) for k ∈ \ ( S ∪ S T ). § Determination of the Dirichlet-to-Neumann Map for Wave Guides
The arguments of the preceding section apply to the wave guides with modifica-tions that we give here.Since now the unperturbed operator is − c ∆, we need to replace (3) with arepresentation for the outgoing fundamental solution for − c ∆. To obtain this representation we separate variables and use expansion in the eigenfunctions (chosento be real-valued) of the Sturm-Liouville problem φ ′′ m ( x , k ) + k c ( x ) φ m ( x , µ ) = µ m ( k ) φ m ( x , k )with φ m (0 , k ) = 0, φ ′ m ( B, k ) = 0. Using this basis and assuming that k is chosenso that µ m ( k ) = 0 , m ∈ N , one checks that for f with bounded support in x u ( x, k ) = ∞ X m =1 i p µ m ( k ) Z [0 ,B ] × R e i √ µ m ( k ) | x − y | φ m ( x , k ) φ m ( y , k ) f ( y ) c ( y ) dy, (9)is a solution to ( L − k ) u = f when k is real. To see that this is the outgoing solutionwe will show that u ( x, k ) continues to a square-integrable solution when k movesinto the upper half plane. Since the boundary conditions make d /dx + k /c ( x )self-adjoint when k is real, the functions φ m ( x , k ) and µ m ( k ) are analytic in k by Rellich’s theorem. This is an elementary result here, since µ m ( k ) is a simpleeigenvalue when k is real. Thus for ǫ >
0, if we can show that Im { µ m ( k + iǫ ) } > µ m ( k ) >
0, the choice of p µ m ( k + iǫ ) that we use here (see the definitionspreceding Theorem 1 in §
1) will make Im { p µ m ( k + iǫ ) >
0. However, this followsimmediately from the observation that dµ m ( k ) /dk > k real. Thus we concludethat for all f for which (9) is a finite sum, u extends to a square-integrable solutionto ( L − k ) u = f as k moves into the upper half-plane. Thus, on the complementof the thresholds the operator G + ( k ), defined by G + ( k ) f = ∞ X m =1 φ m ( x , k ) Z [0 ,B ] × R e i √ µ m ( k ) | x − y | i p µ m ( k ) φ m ( y , k ) f ( y ) c ( y ) dy, coincides with the limit of ( − c ∆ − k I ) − as Im { k } → + on a dense set of f .Since an easy limiting absorption argument shows that lim ǫ → + ( − c ∆ − k I ) − f exists for f with bounded support, it follows that G + ( k ) is the outgoing fundamen-tal solution. The same construction, replacing the square roots in (9) with theircomplex conjugates, leads to the incoming fundamental solution G − ( k ).As stated in §
1, distorted plane waves for the wave guide are obtained by solving( L − k ) u = 0 with the given boundary conditions on x = 0 and x = B for u = Φ( x, k, m ) + v + , where Φ is a generalized eigenfunction for − c ( x )∆, i.e.Φ( x, m, k ) = exp( ix p µ m ( k )) φ m ( x , k )). Note that for this to be a true distortedplane wave µ m ( k ) should be positive. However, as in § µ m ( k ) <
0. As in § u + ( x, m, k ) = Φ( x, m, k ) + v ′ + ( x, m, k ) is done by limiting byabsorption. As in § u + has a representation u + = ψ ( x )Φ + v + where v + = − lim ǫ → + ( L − ( k + iǫ ) ) − ( L − k )( ψ ( x )Φ) , with L = − c ∆. Here the cutoff function ψ ∈ C ∞ ( R ) satisfies ψ ≡ | x | > T +1with support contained in | x | > T . As before, the functions u + do not depend on ψ . Moreover, the exceptional set S is again the union of the thresholds and the setof k for which there is a nontrivial, square-integrable solution to ( L − k ) u = 0 in [0 , B ] × R . The generalized distorted plane waves u + have analytic continuationsto Im { k } >
0, and hence as in the case of gratings, { u + ( x, m, k ) , k ∈ I } , where I is an open interval in { k : µ m ( k ) > } determines u + ( x, m, k ) for k ∈ R \ S (notethat µ m ( k ) > k sufficiently large for each m ). In other words the generalizeddistorted plane waves are again determined by the true distorted plane waves viaanalytic continuation.The analog of (4) for wave guides is Z B f ( x ) u + ( x , T, m, k ) dx = Z [0 ,B ] × R Φ( x, m, k )( L − k ) ψ ( L − ( k − i ) − ( f δ T ) dx, and this identity shows that Corollary 1 holds for wave guides. Likewise we havethe following analog of Lemma 3. Lemma 4.
The scattering data from propagating modes, { b m ( n, k ) : µ n ( k ) > ) and µ m ( k ) > } given for all k ∈ R \ S , determine the distorted plane waves in x ≥ T .Since (2’) gives, b m ( n, k ) = e − ix √ µ m ( k ) Z B φ m ( x , k ) v + ( x , T, k, n ) dx , it follows that a m ( n, k ) is analytic in k on the set where u + is analytic in k , theproof of Lemma 3 applies here, and again conclude that the propagating modesdetermine Λ( k ) for k ∈ R \ ( S ∪ S T ). § Reduction to the hyperbolic inverse problem
We will begin with the wave guide problem. Consider the hyperbolic equation v tt = c ( x )∆ v, t > { ( x , x ) ∈ [0 , B ] × ( −∞ , T ] } with zero initial conditions, v ( x,
0) = 0 , v t ( x,
0) = 0,and the boundary conditions v (0 , x , t ) = 0 , ∂v∂x ( B, x , t ) = 0 , and v ( x , T, t ) = g ( x , t ) . Let Λ H denote the hyperbolic Dirichlet-to-Neumann operator corresponding to thisinitial-boundary value problem:Λ H g = ∂v∂x for ( x , t ) ∈ [0 , B ] × [0 , ∞ ) . The following theorem is a particular case of results in [B] and [KKL] (see also [E1],[E2]).
Theorem 2.
The hyperbolic Dirichlet-to-Neumann map, Λ H on x = T , uniquelydetermines the sound speed c ( x ).To deduce Theorem 1 from Theorem 2 we proceed as follows. Let Λ( k ) be theelliptic Dirichlet-to-Neumann operator defined previously, for c ( x )∆, i.e. Λ( k ) h = ∂u/∂x on x = T , where u is the outgoing solution to the boundary value problem c ( x )∆ u + k u = 0 in [0 , B ] × ( −∞ , T ] , u = h on x = T with the zeroDirichlet boundary condition on x = 0 and the zero Neumann con-dition on x = B . As we observed earlier, Λ( k ) is analytic in k off the discrete set S ∪ S T . Hence, using the Fourier-Laplace transform in t ,we can recover Λ H fromΛ( k ), given for k − ǫ < k < k + ǫ . Since we showed in § k ), this completes the proof of Theorem 1 for wave guides.Since we have also shown for gratings that Λ( k ) for k ∈ R \ ( S ∪ S T ) is determinedby scattering data from propagating modes, the only change in the argument neededto prove Theorem 1 for gratings is in the citation of results on the hyperbolicDirichlet-to-Neumann map. Here the hyperbolic Dirichlet-to-Neumann operator,Λ H , is defined by Λ H g = ∂v∂x on x = T, where v is the solution to v tt = Lv in D ∩ { x < T } × { ≤ t < ∞} satisfying the periodicity condition (1), the initial-boundary conditions v ( x,
0) = v t ( x,
0) = 0 and v ( x , T, t ) = g . In this setting the uniqueness results of [B], [KKL]and [E1,2] imply that that Λ H given on x = T determine both the coefficientsof L , i.e. the permitivity ǫ ( x ), and the domain D . This completes the proof ofTheorem 1. References: [AK] Ahluwalia, D. and J. Keller, Exact and asymptotic representations of thesound field in a stratified ocean, Wave Propagation and Acoustics, Lecture Notesin Physics, (1977), 14-85.[B] Belishev, M., Boundary control in reconstruction of manifolds and metrics (theBC method), Inverse Problems (1997), R1-R45.[BDC] Bao, G., Dobson, D. and J. Cox, Mathematical studies in rigorous gratingtheory, J. Optical Soc. A (1995), 1029-1042.[BF] Bao, G. and A. Friedman, Inverse problems for scattering by periodic struc-tures, Arch. Rat. Mech. Anal. (1995), 49-72.[BGWX] Buchanan, J.L., Gilbert, R.P., Mawata, C., and Y. Xu, Marine Acoustics.Direct and Inverse Problems , SIAM, Philadelphia, 2004.[GMX] Gilbert, R.P., Mawata, C., and Y. Xu, Determination of a distributed inho-mogeneity in a two-layered wave guide from scattered sound,
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