Analysis of Time-Domain Scattering by Periodic Structures
aa r X i v : . [ m a t h . A P ] A p r ANALYSIS OF TIME-DOMAIN SCATTERING BY PERIODIC STRUCTURES
YIXIAN GAO AND PEIJUN LIA
BSTRACT . This paper is devoted to the mathematical analysis of a time-domain electromagnetic scattering byperiodic structures which are known as diffraction gratings. The scattering problem is reduced equivalently intoan initial-boundary value problem in a bounded domain by using an exact transparent boundary condition. Thewell-posedness and stability of the solution are established for the reduced problem. Moreover, a priori energyestimates are obtained with minimum regularity requirement for the data and explicit dependence on the time.
1. I
NTRODUCTION
This paper is concerned with the mathematical analysis of an electromagnetic scattering problem in peri-odic structures, where the wave propagation is governed by the time-domain Maxwell equations. The scat-tering theory in periodic diffractive structures, also known as diffraction gratings, has applications in manycutting-edge scientific areas including ultra-fast and high-energy lasers, space flight instruments, astronomy,and synchrotron spectrometers. A good introduction can be found in [10] to diffraction grating problemsand various numerical approaches. The book [19] contains descriptions of several mathematical problemsthat arise in diffractive optics modeling in industry. Some more recent developments are addressed in [6] ontheory, analysis, and computational techniques of diffractive optics.The time-harmonic grating problems have been extensively studied by many researchers via either the inte-gral equation methods or the variational methods [2–4,12,17]. A survey may be found in [7] for mathematicalstudies in rigorous grating theory. The general result may be stated as follows: The diffraction problem has aunique solution for all but a countable sequence of singular frequencies. Unique solvability for all frequen-cies can be obtained for gratings which have absorbing media or perfectly electrically conducting surfaceswith Lipschitz profiles. Numerical methods are developed for both the two-dimensional Helmholtz equa-tion (one-dimensional gratings) and the three-dimensional Maxwell equations (crossed or two-dimensionalgratings) [5, 8, 9, 15, 30].The time-domain scattering problems have attracted considerable attention due to their capability of cap-turing wide-band signals and modeling more general material and nonlinearity [11,22,23,25,29]. Comparingwith the time-harmonic problems, the time-domain problems are much less studied due to the additional chal-lenge of the temporal dependence. Rigorous mathematical analysis is very rare. The analysis can be foundin [14,28] for the time-domain acoustic and electromagnetic obstacle scattering problems. We refer to [24] forthe analysis of the time-dependent electromagnetic scattering from a three-dimensional open cavity. Numer-ical solutions can be found in [18, 27] for the time-dependent wave scattering by periodic structures/surfaces.The theoretical analysis is still lacking for the time-domain scattering by periodic structures.The goal of this work is to analyze mathematically the time-domain scattering problem which arises fromthe electromagnetic wave propagation in a periodic structure. Specifically, we consider an electromagneticplane wave which is incident on a one-dimensional grating in R . So the structure is assumed to be invariantin the y -direction and periodic in the x -direction. The three-dimensional Maxwell equations can be de-composed into two fundamental polarizations: transverse electric (TE) polarization and transverse magnetic(TM) polarization, where Maxwell’s equations are reduced to the two-dimensional wave equation. We shallstudy the wave equation in two dimensions for both polarizations. The structure can also be characterizedby the medium parameters: the electric permittivity and the magnetic permeability. They are periodic in x Key words and phrases.
Time-domain Maxwell’s equations, diffraction gratings, well-posedness and stability, a priori estimates.The research of YG was partially supported by NSFC grant 11571065 and Jilin Science and Technology Development Project.The research of PL was supported in part by the NSF grant DMS-1151308. and assumed only to be bounded measurable functions. Hence our method works for very general gratingswhose surfaces/interfaces are allowed to be Lipschitz profiles or even graphs of some Lipschitz continuousfunctions.There are two challenges of the problem: time dependence and unbounded domain. In the frequencydomain, various approaches have been developed to truncate unbounded domains into bounded ones, such asabsorbing boundary conditions (ABCs), transparent boundary conditions (TBCs), and perfectly matched layer(PML) techniques. These effective boundary conditions are being extended to handle time-domain problems[1, 13, 20, 21]. Utilizing the Laplace transform as a bridge between the time-domain and the frequencydomain, we develop an exact time-domain TBC and reduce the problem equivalently into an initial boundaryvalue problem in a bounded domain. Using the energy method with new energy functions, we show thewell-posedness and stability of the time-dependent problem. The proofs are based on examining the well-posedness of the time-harmonic Helmholtz equations with complex wavenumbers and applying the abstractinversion theorem of the Laplace transform. Moreover, a priori estimates, featuring an explicit dependenceon time and a minimum regularity requirement of the data, are established for the wave field by studyingdirectly the time-domain wave equation.The paper is organized as follows. In section 2, we introduce the model problem and develop a TBC toreduce it into an initial boundary value problem. Section 3 is devoted to the analysis of the reduced problem,where the well-posdeness and stability are addressed and a priori estimates are provided. We conclude thepaper with some remarks and directions for future work in section 4.2. P
ROBLEM FORMULATION
In this section, we introduce the mathematical model of interest and develop an exact TBC to reduce thescattering problem from an unbounded domain into a bounded domain.2.1.
A model problem.
Consider the system of time-domain Maxwell equations in R for t > : ( ∇ × E ( x, y, z, t ) + µ∂ t H ( x, y, z, t ) = 0 , ∇ × H ( x, y, z, t ) − ε∂ t E ( x, y, z, t ) = 0 , (2.1)where E is the electric field, H is the magnetic field, ε and µ are the dielectric permittivity and magneticpermeability, respectively, and satisfy < ε min ≤ ε ≤ ε max < ∞ , < µ min ≤ µ ≤ µ max < ∞ . Here ε min , ε max , µ min , µ max are constants. We assume that the structure is invariant in the y -direction andthus focus on the one-dimensional grating. The more complicated problem in biperiodic structures will beconsidered in a separate work. There are two fundamental polarizations for the one-dimensional structure:(i) TE polarization. The electric and magnetic fields are E ( x, y, z, t ) = [0 , E ( x, z, t ) , ⊤ , H ( x, z, t ) = [ H ( x, z, t ) , , H ( x, z, t )] ⊤ . Eliminating the magnetic field from (2.1), we get the wave equation for the electric field: ε∂ t E ( x, z, t ) = ∇ · ( µ − ∇ E ( x, z, t )) . (2.2)(ii) TM polarization. The electric and magnetic fields are E ( x, y, z, t ) = [ E ( x, z, t ) , , E ( x, z, t )] ⊤ , H ( x, y, z, t ) = [0 , H ( x, z, t ) , ⊤ . We may eliminate the electric field from (2.1) and obtain the wave equation for the magnetic field: µ∂ t H ( x, z, t ) = ∇ · ( ε − ∇ H ( x, z, t )) . (2.3)It is clear to note from (2.2) and (2.3) that the TE and TM polarizations can be handled in a unified way byformally exchanging the roles of ε and µ . We will just present the results by using (2.2) as the model equationin the rest of the paper. NALYSIS OF TIME-DOMAIN SCATTERING BY PERIODIC STRUCTURES 3 ΩΩ Ω Γ Γ z = h z = h F IGURE
1. Problem geometry of the time-domain scattering by a periodic structureNow let us specify the problem geometry, which is shown in Figure 1. Since the structure and medium areassumed to be periodic in the x direction, there exists a period Λ > such that ε ( x + n Λ , z ) = ε ( x, z ) , µ ( x + n Λ , z ) = µ ( x, z ) , ∀ ( x, z ) ∈ R , n ∈ Z . We assume that ε and µ are constants away from the region Ω = { ( x, z ) : 0 ≤ x ≤ Λ , h ≤ z ≤ h } , where h j are constants. Denote Ω := { ( x, z ) : 0 ≤ x ≤ Λ , z > h } and Ω := { ( x, z ) : 0 ≤ x ≤ Λ , z < h } .There exist constants ε j and µ j such that ε ( x, z ) = ε j , µ ( x, z ) = µ j in Ω j . Throughout we also assume that εµ ≥ ε µ , which is usually satisfied since ε and µ are the electricpermittivity and magnetic permeability in the free space Ω . Finally we define Γ = { ( x, z ) : 0 ≤ x ≤ Λ , z = h } and Γ := { ( x, z ) : 0 ≤ x ≤ Λ , z = h } .Consider an incoming plane wave E inc which is incident on the structure from above. Explicitly we have E inc ( x, z, t ) = f ( t − c x − c z ) , where f is a smooth function and its regularity will be specified later, and c = cos θ/c, c = sin θ/c . Here θ , satisfying < θ < π , is the incident angle, and c = 1 / √ ε µ > is the light speed in the free space.Clearly, the incident field E inc ( x, z, t ) satisfies the wave equation (2.2) when ε = ε , µ = µ .Although the incident field E inc may not be a periodic function in the x -direction, we can verify that E inc ( x + Λ , z, t ) = E inc ( x, z, t − c Λ) , ∀ ( x, z ) ∈ R , t > . Motivated by the uniqueness of the solution, we assume that the total field satisfies the same translationproperty, i.e., E ( x + Λ , z, t ) = E ( x, z, t − c Λ) , ( x, z ) ∈ R , t > . We define U ( x, z, t ) = E ( x, z, t + c ( x − Λ)) , U inc ( x, z, t ) = E inc ( x, z, t + c ( x − Λ)) . (2.4)It follows from (2.4) that we get U ( x + Λ , z, t ) = E ( x + Λ , z, t + c x ) = E ( x, z, t + c x − c Λ) = U ( x, z, t ) , which shows that U is a periodic function in the x -direction with period Λ . Similarly, we can verify that theincident field U inc is a trivially periodic function of x (independent of x ) since U inc ( x, z, t ) = E inc ( x, z, t + c ( x − Λ)) = f ( t − c z − c Λ) . Using the change of variables, we have ∂ t E = ∂ t U, ∂ x E = ∂ x U − c ∂ t U. The equation (2.2) becomes ( ε − c µ − ) ∂ t U = ∇ · ( µ − ∇ U ) − c ( µ − ∂ tx U + ∂ x ( µ − ∂ t U )) . (2.5) YIXIAN GAO AND PEIJUN LI
A simple calculation yields that ε − c µ − = ( εµ − ε µ cos θ ) µ − ≥ ε µ (1 − cos θ ) µ − = ε µ µ − sin θ > , ∀ θ ∈ (0 , π ) , which shows that the equation (2.5) is a well-defined wave equation.It is easy to verify that the incident field U inc satisfies (2.5) with ε = ε , µ = µ . To impose the initialconditions, we assume that the total field and the incident field vanish for t < so that the incident field U inc = 0 and the scattered field V = U − U inc = 0 for t < . The initial conditions are U | t =0 = ∂ t U | t =0 = 0 . (2.6)In addition U is Λ -periodic in the x -direction. This paper aims to study the well-posedness and stability ofthe scattering problem (2.5)–(2.6).We introduce some notation. For any s = s + i s with s , s ∈ R , s > , define by ˘ u ( s ) the Laplacetransform of the function u ( t ) , i.e., ˘ u ( s ) = L ( u )( s ) = Z ∞ e − st u ( t )d t. Define a weighted periodic function space H s, p (Ω) = { u ∈ H (Ω) : u (0 , z ) = u (Λ , z ) } , which is Sobolev space with the norm characterized by k u k H s, p (Ω) = Z Ω (cid:0) |∇ u | + | s | | u | (cid:1) d x d z. Given u ∈ H s, p (Ω) , it has a Fourier expansion with respect to x : u ( x, z ) = X n ∈ Z u n ( z ) e i α n x , α n = 2 nπ Λ − . A simple calculation yields an equivalent norm in H s, p (Ω) via Fourier coefficients: k u k H s, p (Ω) = X n ∈ Z (cid:0) | s | + α n (cid:1) Z h h | u n ( z ) | d z + X n ∈ Z Z h h | u ′ n ( z ) | d z. (2.7)For a periodic function u defined on Γ j with Fourier coefficients u n , we define a weighted trace functionalspace H λs (Γ j ) = { u ∈ L (Γ j ) : k u k H λ (Γ j ) = X n ∈ Z (cid:0) | s | + α n (cid:1) λ | u n | < ∞} , (2.8)where λ ∈ R . It is clear to note that the dual space of H / s (Γ j ) is H − / s (Γ j ) under the L (Γ j ) inner product h u, v i Γ j = Z Γ j u ¯ v d γ j . The weighted Sobolev spaces H s, p (Ω) and H νs (Γ j ) are equivalent to the standard Sobolev spaces H (Ω) and H λ (Γ j ) since | s | 6 = 0 . Hereafter, the expression ‘ a . b ” stands for “ a ≤ Cb ” , where C is a positiveconstant and its specific value is not required but should be always clear from the context. NALYSIS OF TIME-DOMAIN SCATTERING BY PERIODIC STRUCTURES 5
Transparent boundary condition.
We introduce a TBC to reformulate the scattering problem into anequivalent initial-boundary value problem in a bounded domain. The idea is to design a Dirichlet-to-Neumann(DtN) operator which maps the Dirichlet data to the Neumann data of the wave field.Subtracting the incident field U inc from the total field U in (2.5) and (2.6), we obtain the equation for thescattered field ( ε − c µ − ) ∂ t V = ∇ · ( µ − ∇ V ) − c ( µ − ∂ tx V + ∂ x ( µ − ∂ t V )) in Ω , t > , (2.9)and the initial conditions V | t =0 = ∂ t V | t =0 = 0 in Ω . (2.10)Let ˘ V ( x, z, s ) = L ( V ) be the Laplace transforms of V ( x, z, t ) with respect to t . Recall that L ( ∂ t V ) = s ˘ V ( x, z, s ) − V ( x, z, , L ( ∂ t V ) = s ˘ V ( x, z, s ) − sV ( x, z, − ∂ t V ( x, z, . Taking the Laplace transform of (2 . and using the initial conditions (2.10), we have ( ε − c µ − ) s ˘ V = ∇ · ( µ − ∇ ˘ V ) − c ( µ − s∂ x ˘ V + s∂ x ( µ − ˘ V )) , which reduces to ( ε µ − c ) s ˘ V = ∆ ˘ V − c s∂ x ˘ V in Ω . (2.11)Since ˘ V is a periodic function in x , it has the Fourier expansion ˘ V ( x, z ) = X n ∈ Z ˘ V n ( z ) e i α n x , z > h . Substituting the Fourier expansion of ˘ V into (2.11), we obtain an ordinary differential equation for the Fouriercoefficients: ( ∂ z ˘ V n ( z ) − ( β ( n )1 ) ˘ V n ( z ) = 0 , z > h , ˘ V n ( z ) = ˘ V n ( h ) where β ( n )1 = ( ε µ s + ( α n + i c s ) ) / , Re β ( n )1 < . Using the outgoing radiation condition, we have ˘ V n ( z ) = ˘ V n ( h ) e β ( n )1 ( z − h ) , Thus we get the Rayleigh expansion for the scattered field in Ω : ˘ V ( x, z ) = X n ∈ Z ˘ V n ( h ) e i α n x e β ( n )1 ( z − h ) . Taking the normal derivative of the above equation on Γ yields ∂ ν ˘ V ( x, h ) = X n ∈ Z β ( n )1 ˘ V n ( h ) e i α n x , where ν = [0 , ⊤ is the unit normal vector on Γ .Similarly, we can obtain the Rayleigh expansion for the total field in Ω : ˘ U ( x, z ) = X n ∈ Z ˘ U n ( h ) e i α n x e − β ( n )2 ( z − h ) , where β ( n )2 = ( ε µ s + ( α n + i c s ) ) / , Re β ( n )2 < . Taking the normal derivative of ˘ U on Γ gives ∂ ν ˘ U ( x, h ) = X n ∈ Z β ( n )2 ˘ U n ( h ) e i α n x , YIXIAN GAO AND PEIJUN LI where ν = [0 , − ⊤ is the normal vector on Γ . For any function u ( x, h j ) defined on Γ j , we define the DtNoperators ( B j u )( x, h j ) = X n ∈ Z β ( n ) j u n ( h j ) e i α n x , u ( x, h j ) = X n ∈ Z u n ( h j ) e i α n x . (2.12) Lemma 2.1.
There exists a positive constant C such that k u k H / s (Γ j ) ≤ C k u k H s, p (Ω) , ∀ u ∈ H s, p (Ω) . Proof.
First we have ( h − h ) | ζ ( h j ) | = Z h h | ζ ( z ) | d z + Z h h Z h j z dd t | ζ ( t ) | d t d z ≤ Z h h | ζ ( z ) | d z + ( h − h ) Z h h | ζ ( z ) || ζ ′ ( z ) | d z, which gives (cid:0) | s | + α n (cid:1) / | ζ ( h j ) | ≤ ( h − h ) − (cid:0) | s | + α n (cid:1) / Z h h | ζ ( z ) | d z + Z h h (cid:0) | s | + α n (cid:1) / | ζ ( z ) || ζ ′ ( z ) | d z. It follows from the Cauchy–Schwarz inequality that (cid:0) | s | + α n (cid:1) / | ζ ( h j ) | ≤ ( h − h ) − (cid:0) | s | + | α n | (cid:1) Z h h | ζ ( z ) | d z + (cid:0) | s | + α n (cid:1) Z h h | ζ ( z ) | d z + Z h h | ζ ′ ( z ) | d z. Using the fact that s = s + i s with s > , we have | s | ≤ s − | s | , | α n | ≤ (2 π ) − Λ α n . Letting C = max { h − h ) − s − , π ) − ( h − h ) − Λ } , we can show that (cid:0) | s | + α n (cid:1) / | ζ ( h j ) | ≤ C (cid:16)(cid:0) | s | + α n (cid:1) Z h h | ζ ( z ) | d z + Z h h | ζ ′ ( z ) | d z (cid:17) . The proof is completed by combing the above estimates and the definition (2.7). (cid:3)
Lemma 2.2.
The DtN operator B j : H / s, p (Γ j ) → H − / s, p (Γ j ) is continuous, i.e., k B j u k H − / s, p (Γ j ) ≤ C k u k H / s, p (Γ j ) , where C > is a constant.Proof. For any u ∈ H / s, p (Γ j ) , it follow form (2.8) that k B j u k H − / s, p (Ω) = X n ∈ Z (cid:0) | s | + α n (cid:1) − / | β ( n ) j | | u n ( h j ) | = X n ∈ Z (cid:0) | s | + α n (cid:1) / (cid:0) | s | + α n (cid:1) − | β ( n ) j | | u n ( h j ) | ≤ C k u k H / s, p (Γ j ) , NALYSIS OF TIME-DOMAIN SCATTERING BY PERIODIC STRUCTURES 7 where we have used | β ( n ) j | = | ε j µ j s + ( α n + i c s ) | ≤ ε j µ j | s | + 2( α n + c | s | ) ≤ C (cid:0) | s | + α n (cid:1) . Here C = max { , c + ε max µ max } , which completes the proof. (cid:3) Lemma 2.3.
We have the estimate Re h ( sµ j ) − B j u, u i Γ j ≤ , ∀ u ∈ H / s, p (Γ j ) . Proof.
It follows from the definitions of (2.12) and (2.8) that we have h ( sµ j ) − B j u, u i Γ j = X n ∈ Z ¯ sβ ( n ) j | s | µ j | u n ( h j ) | . Let β ( n ) j = a j + i b j , s = s + i s with s > , a j < . Taking the real part of the above equation gives Re h ( sµ j ) − B j u, u i Γ j = X n ∈ Z ( s a j + s b j ) | s | µ j | u n ( h j ) | . (2.13)Recalling ( β ( n ) j ) = ε j µ j s + ( α n + i c s ) , we have a j − b j = ( ε j µ j − c )( s − s ) + α n − α n c s (2.14)and a j b j = ( ε j µ j − c ) s s + α n c s . (2.15)Using (2.15), we get s a j + s b j = s a j (cid:2) a j + ( ε j µ j − c ) s + α n c s (cid:3) . (2.16)Plugging (2.14) into (2.16) gives s a j + s b j = s a j (cid:2) b j + ( ε j µ j − c ) s + α n − α n c s (cid:3) . (2.17)Adding (2.16) and (2.17), we obtain s a j + s b j = s a j (cid:2) a j + b j + ( ε j µ j − c )( s + s ) + α n (cid:3) . (2.18)Substituting (2.18) into (2.13) yields Re h ( sµ j ) − B j u, u i Γ j = X n ∈ Z s a j | s | µ j (cid:2) a j + b j + ( ε j µ j − c )( s + s ) + α n (cid:3) | u n ( h j ) | ≤ , which completes the proof. (cid:3) Using the DtN operators (2.12), we obtain the following TBC in the s -domain: ( ∂ ν ˘ U = B ˘ U + ˘ ρ on Γ ,∂ ν ˘ U = B ˘ U on Γ , (2.19)where ˘ ρ = ∂ z ˘ U inc − B ˘ U inc . Taking the inverse Laplace transform of (2.19) yields the TBC in the timedomain: ( ∂ ν U = T U + ρ on Γ ,∂ ν U = T U on Γ , (2.20)where ρ is the inverse Laplace transform of ˘ ρ , i.e., ρ = L − (˘ ρ ) , and T j = L − ◦ B j ◦ L . YIXIAN GAO AND PEIJUN LI
3. T HE R EDUCED P ROBLEM
In this section, we present the main results of this work, which include the well-posedness and stability ofthe scattering problem and related a priori estimates.3.1.
Well-posedness in the s -domain. Taking the Laplace transform of (2.5) and using the TBC (2.19), wemay consider the following reduced boundary value problem: ( ε − c µ − ) s ˘ U = ∇ · (( sµ ) − ∇ ˘ U ) − c ( µ − ∂ x ˘ U + ∂ x ( µ − ˘ U )) in Ω ,∂ ν ˘ U = B ˘ U + ˘ ρ on Γ ,∂ ν ˘ U = B ˘ U on Γ . (3.1)Next we introduce a variational formulation of the boundary value problem (3.1) and give a proof of itswell-posedness in the space H s, p (Ω) .Multiplying (3.1) by the complex conjugate of a test function v ∈ H s, p (Ω) , using the integration by partsand TBCs, we arrive at the variational problem: To find ˘ U ∈ H s, p (Ω) such that a ( ˘ U , v ) = h ( sµ ) − ˘ ρ, v i Γ , ∀ v ∈ H s, p (Ω) , (3.2)where the sesquilinear form a ( ˘ U , v ) = Z Ω (cid:2) ( sµ ) − ∇ ˘ U · ∇ ¯ v + ( ε − c µ − ) s ˘ U ¯ v + c ( µ − ∂ x ˘ U + ∂ x ( µ − ˘ U ))¯ v (cid:3) d x d z − X j =1 h ( sµ j ) − B j ˘ U , v i Γ j . (3.3) Theorem 3.1.
The variational problem (3.2) has a unique solution ˘ U ∈ H s, p (Ω) , which satisfies k∇ ˘ U k L (Ω) + k s ˘ U k L (Ω) . s − | s |k ˘ ρ k H − / s (Γ ) . Proof.
It suffices to show the coercivity of the sesquilinear form of a , since the continuity follows directlyfrom the Cauchy–Schwarz inequality, Lemma 2.1, and Lemma 2.2.Letting v = ˘ U in (3.3), we get a ( ˘ U , ˘ U ) = Z Ω (cid:2) ( sµ ) − |∇ ˘ U | + ( ε − c µ − ) s | ˘ U | + c ( µ − ∂ x ˘ U + ∂ x ( µ − ˘ U )) ¯˘ U (cid:3) d x d z − X j =1 h ( sµ j ) − B j ˘ U , ˘ U i Γ j . Taking the real part of the above equation yields Re a ( ˘ U , ˘ U ) = Z Ω (cid:0) s | s | µ |∇ ˘ U | + ( ε − c µ − ) s | ˘ U | (cid:1) d x d z − Re X j =1 h ( sµ j ) − B j ˘ U , ˘ U i Γ j + c Re Z Ω (cid:0) µ − ∂ x ˘ U ¯˘ U + ∂ x ( µ − ˘ U ) ¯˘ U (cid:1) d x d z. Since µ and ˘ U are periodic in x , we have from the integration by part that Z Ω (cid:0) µ − ∂ x ˘ U ¯˘ U + ∂ x ( µ − ˘ U ) ¯˘ U (cid:1) d x d z + Z Ω (cid:0) ˘ U ∂ x ( µ − ¯˘ U ) + µ − ˘ U ∂ x ¯˘ U (cid:1) d x d z = 0 , which gives Re Z Ω (cid:0) µ − ∂ x ˘ U ¯˘ U + ∂ x ( µ − ˘ U ) ¯˘ U (cid:1) d x d z = 0 . NALYSIS OF TIME-DOMAIN SCATTERING BY PERIODIC STRUCTURES 9
Combining the above estimate and Lemma 2.3, we obtain Re a ( ˘ U , ˘ U ) ≥ C s | s | Z Ω (cid:0) |∇ ˘ U | + | s ˘ U | (cid:1) d x d z, (3.4)where C = µ − min { , ε µ sin θ } .It follows from the Lax–Milgram lemma that the variational problem (3.2) has a unique solution ˘ U ∈ H s, p (Ω) . Moreover, we have from (3.2) and Lemma 2.1 that | a ( ˘ U , ˘ U ) | ≤ ( | s | µ ) − k ˘ ρ k H − / s (Γ ) k ˘ U k H / s (Γ ) ≤ C ( | s | µ ) − k ˘ ρ k H − / s (Γ ) k ˘ U k H s, p (Ω) . (3.5)Combing (3.4) and (3.5) leads to k∇ ˘ U k L (Ω) + k s ˘ U k L (Ω) . s − | s |k ˘ ρ k H − / s (Γ ) k ˘ U k H s, p (Ω) , which completes the proof after applying the Cauchy–Schwarz inequality. (cid:3) Well-posedness in the time-domain.
Using the time-domain TBC (2.20), we consider the reducedinitial-boundary value problem: ( ε − c µ − ) ∂ t U = ∇ · ( µ − ∇ U ) − c ( µ − ∂ tx U + ∂ x ( µ − ∂ t U )) in Ω , t > ,U | t =0 = ∂ t U | t =0 = 0 in Ω ,∂ ν U = T U + ρ on Γ , t > ,∂ ν U = T U on Γ , t > . (3.6)The following lemma (cf. [26, Theorem 43.1]) is an analogue of Paley–Wiener–Schwarz theorem forFourier transform of the distributions with compact support in the case of Laplace transform. Lemma 3.2.
Let ˘ h ( s ) denote a holomorphic function in the half-plane s > σ , valued in the Banach space E . The two following conditions are equivalent: (1) there is a distribution h ∈ D ′ + ( E ) whose Laplace transform is equal to ˘ h ( s ) ; (2) there is a real σ with σ ≤ σ < ∞ and an integer m ≥ such that for all complex numbers s with Re s = s > σ , it holds that k ˘ h ( s ) k E . (1 + | s | ) m ,where D ′ + ( E ) is the space of distributions on the real line which vanish identically in the open negative halfline. Theorem 3.3.
The initial-boundary value problem (3.6) has a unique solution U ( x, z, t ) , which satisfies U ( x, z, t ) ∈ L (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)) and the stability estimate max t ∈ [ t,T ] (cid:0) k ∂ t U k L (Ω) + k ∂ t ( ∇ U ) k L (Ω) (cid:1) . (cid:0) k ρ k L (0 ,T ; H − / (Γ )) + max t ∈ [0 ,T ] k ∂ t ρ k H − / (Γ ) + k ∂ t ρ k L (0 ,T ; H − / (Γ )) (cid:1) . (3.7) Proof.
First we have Z T (cid:0) k∇ U k L (Ω) + k ∂ t U k L (Ω) (cid:1) d t ≤ Z T e − s ( t − T ) (cid:0) k∇ U k L (Ω) + k ∂ t U k L (Ω) (cid:1) d t = e s T Z T e − s t (cid:0) k∇ U k L (Ω) + k ∂ t U k L (Ω) (cid:1) d t . Z ∞ e − s t (cid:0) k∇ U k L (Ω) + k ∂ t U k L (Ω) (cid:1) d t. Hence it suffices to estimate the integral Z ∞ e − s t (cid:0) k∇ U k L (Ω) + k ∂ t U k L (Ω) (cid:1) d t. Taking the Laplace transform of (3.6) yields ( ε − c µ − ) s ˘ U = ∇ · (( sµ ) − ∇ ˘ U ) − c ( µ − ∂ x ˘ U + ∂ x ( µ − ˘ U )) in Ω ,∂ ν ˘ U = B ˘ U + ˘ ρ on Γ ,∂ ν ˘ U = B ˘ U on Γ . The well-posedness of ˘ U ∈ H s, p (Ω) follows directly from Theorem 3.1. By the trace theorem in Lemma 2.1,we get k∇ ˘ U k L (Ω) + k s ˘ U k L (Ω) . s − | s | k ˘ ρ k H − / (Γ ) . s − | s | k ˘ U inc k H (Ω) . It follows from [26, Lemma 44.1] that ˘ U is a holomorphic function of s on the half plane s > ¯ γ > , where ¯ γ is any positive constant. Hence we have from Lemma 3.2 that the inverse Laplace transform of ˘ U existsand is supported in [0 , ∞ ] . One may verify from the inverse Laplace transform that ˘ U = L ( U ) = F ( e − s t U ) , where F is the Fourier transform with respect to s . Recall the Plancherel or Parseval identity for the Laplacetransform (cf. [16, (2.46)]) π Z ∞−∞ ˘ u ( s )˘ v ( s )d s = Z ∞ e − s t u ( t ) v ( t )d t, ∀ s > λ, (3.8)where ˘ u = L ( u ) , ˘ v = L ( v ) and λ is abscissa of convergence for the Laplace transform of u and v. Using (3.8), we have Z ∞ e − s t (cid:0) k∇ U k L (Ω) + k ∂ t U k L (Ω) (cid:1) d t = 12 π Z ∞−∞ (cid:0) k∇ ˘ U k L (Ω) + k s ˘ U k L (Ω) (cid:1) d s . s − Z ∞−∞ | s | (cid:0) k ˘ U inc k L (Ω) + k∇ ˘ U inc k L (Ω) (cid:1) d s . Since U inc | t =0 = ∂ t U inc | t =0 = 0 in Ω , we have L ( ∂ t U inc ) = s ˘ U inc in Ω . It is easy to note that | s | ˘ U inc = (2 s − s ) s ˘ U inc = 2 s L ( ∂ t U inc ) − L ( ∂ t U inc ) , | s | ∇ ˘ U inc = (2 s − s ) s ∇ ˘ U inc = 2 s L ( ∂ t ∇ U inc ) − L ( ∂ t ∇ U inc ) . Hence we have Z ∞ e − s t (cid:0) k∇ U k L (Ω) + k ∂ t U k L (Ω) (cid:1) d t . Z ∞−∞ k L ( ∂ t U inc ) k L (Ω) d s + s − Z ∞−∞ k L ( ∂ t U inc ) k L (Ω) d s + Z ∞−∞ k L ( ∂ t ∇ U inc ) k L (Ω) d s + s − Z ∞−∞ k L ( ∂ t ∇ U inc ) k L (Ω) d s . Using the Parseval identity (3.8) again gives Z ∞ e − s t (cid:0) k∇ U k L (Ω) + k ∂ t U k L (Ω) (cid:1) d t . Z ∞ e − s t k ∂ t U inc k H (Ω) d t + s − Z ∞ e − s t k ∂ t U inc k H (Ω) d t, which shows that U ( x, z, t ) ∈ L (0 , T ; H (Ω) ∩ H (0 , T ; L (Ω)) . NALYSIS OF TIME-DOMAIN SCATTERING BY PERIODIC STRUCTURES 11
Next we prove the stability. Let ˜ U ( x, z, t ) be the extension of U ( x, z, t ) with respect to t in R such that ˜ U ( x, z, t ) = 0 outside the interval [0 , t ] . By the Parseval identity (3.8), we follow the proof of Lemma 2.3and get Re Z t e − s t h T j U, ∂ t U i Γ j d t = Re Z Γ j Z ∞ e − s t h T j ˜ U , ∂ t ˜ U i Γ j d t = 12 π Z ∞−∞ Re h T j ˘˜ U, s ˘˜ U i Γ j d s ≤ , which yields after taking s → that Re Z t h T j U, ∂ t U i Γ j d t ≤ . (3.9)For any < t < T, consider the energy function e ( t ) = k ( ε − c µ − ) / ∂ t U ( · , t ) k L (Ω) + k µ − / ∇ U ( · , t ) k L (Ω) . It follows from (3.6) that we have Z t e ′ ( t ) dt =2Re Z t Z Ω (cid:0) ( ε − c µ − ) ∂ t U ∂ t ¯ U + µ − ∂ t ( ∇ U ) · ∇ ¯ U (cid:1) d x d z d t =2Re Z t Z Ω (cid:0) ∇ · ( µ − ∇ U ) ∂ t ¯ U + µ − ∂ t ( ∇ U ) · ∇ ¯ U (cid:1) d x d z d t − Z t Z Ω (cid:0) c ( µ − ∂ tx U + ∂ x ( µ − ∂ t U )) ∂ t ¯ U (cid:1) d x d z d t. Since µ and U are periodic functions in x , integrating by parts yields Z t Z Ω (cid:0) µ − ∂ tx U ∂ t ¯ U + µ − ∂ tx ¯ U ∂ t U + ∂ x ( µ − ∂ t U ) ∂ t ¯ U + ∂ x ( µ − ∂ t ¯ U ) ∂ t U (cid:1) d x d z d t = 0 , which gives Re Z t Z Ω (cid:0) c ( µ − ∂ tx U + ∂ x ( µ − ∂ t U )) ∂ t ¯ U (cid:1) d x d z d t = 0 . Since e (0) = 0 , we obtain from (3.9) that e ( t ) = Z t e ′ ( t )d t = 2Re Z t Z Ω (cid:0) − µ − ∇ U · ∂ t ( ∇ ¯ U ) + µ − ∂ t ( ∇ U ) · ∇ ¯ U (cid:1) d x d z d t + 2Re Z t X j =1 Z Γ j µ − j ∂ ν U ∂ t ¯ U d γ j d t =2Re Z t X j =1 µ − j h T j U, ∂ t U i Γ j d t + 2Re Z t h ρ, ∂ t U i Γ d t ≤ Z t (cid:0) k ρ k H − / (Γ ) k ∂ t U k H / (Γ ) (cid:1) d t . Z t (cid:0) k ρ k H − / (Γ ) k ∂ t U k H (Ω) (cid:1) d t ≤ (cid:0) max t ∈ [0 ,T ] k ∂ t U k H (Ω) (cid:1) k ρ k L (0 ,T ; H − / (Γ )) . Taking the derivative of (3.6) with respect to t, we know that ∂ t U also satisfies the same equations with ρ replaced by ∂ t ρ . Hence, we may consider the similar energy function e ( t ) = k ( ε − c µ − ) / ∂ t U ( · , t ) k L (Ω) + k µ − / ∂ t ( ∇ U ( · , t )) k L (Ω) and get the estimate e ( t ) ≤ Z t Z Γ ∂ t ρ ∂ t ¯ U d γ d t =2Re Z Γ ∂ t ρ ∂ t ¯ U | t d γ − Z t Z Γ ∂ t ρ ∂ t ¯ U d γ d t ≤ (cid:0) max t ∈ [0 ,T ] k ∂ t U k H (Ω) (cid:1)(cid:0) max t ∈ [0 ,T ] k ∂ t ρ k H − / (Γ ) + k ∂ t ρ k L (0 ,T ; H − / (Γ )) (cid:1) . Combing the above estimates, we can obtain max t ∈ [0 ,T ] k ∂ t U k H (Ω) . max t ∈ [0 ,T ] e ( t ) + e ( t ) . (cid:0) k ρ k L (0 ,T ; H − / (Γ )) + max t ∈ [0 ,T ] k ∂ t ρ k H − / (Γ ) + k ∂ t ρ k L (0 ,T ; H − / (Γ )) (cid:1) k ∂ t U k H p (Ω) , which give the estimate (3.7) after applying the Cauchy–Schwarz inequality. (cid:3) A priori estimates.
In this section, we derive a priori estimates for the total field with a minimumregularity requirement for the data and an explicit dependence on the time.The variation problem of (3.6) in time domain is to find U ∈ H (Ω) for all t > such that Z Ω ( ε − c µ − ) ∂ t U ¯ w d x d z = − Z Ω µ − ∇ U · ∇ ¯ w d x d z + X j =1 Z Γ j µ − j T j U ¯ w d γ j + Z Γ ρ ¯ w d γ − c Z Ω ( µ − ∂ tx U + ∂ x ( µ − ∂ t U )) ¯ w d x d z, ∀ w ∈ H (Ω) . (3.10)To show the stability of its solution, we follow the argument in [26] but with a careful study of the TBC. Theorem 3.4.
Let U ∈ H (Ω) be the solution of (3.6). Given ρ ∈ L (0 , T ; H − / (Γ )) , we have for any T > that k U k L ∞ (0 ,T ; L (Ω)) + k∇ U k L ∞ (0 ,T ; L (Ω)) . T k ρ k L (0 ,T ; H − / (Γ )) + k ∂ t ρ k L (0 ,T ; H − / (Γ )) . (3.11) and k U k L (0 ,T ; L (Ω)) + k∇ U k L (0 ,T ; L (Ω)) . T / k ρ k L (0 ,T ; H − / (Γ )) + T / k ∂ t ρ k L (0 ,T ; H − / (Γ )) . (3.12) Proof.
Let < ξ < T and define an auxiliary function ψ ( x, z, t ) = Z ξt U ( x, z, τ )d τ, ( x, z ) ∈ Ω , ≤ t ≤ ξ. It is clear that ψ ( x, z, ξ ) = 0 , ∂ t ψ ( x, z, t ) = − U ( x, z, t ) . (3.13)For any φ ( x, z, t ) ∈ L (0 , ξ ; L (Ω)) , we have Z ξ φ ( x, z, t ) ¯ ψ ( x, z, t )d t = Z ξ (cid:0) Z t φ ( x, z, τ ) dτ (cid:1) ¯ U ( x, z, t )d t. (3.14) NALYSIS OF TIME-DOMAIN SCATTERING BY PERIODIC STRUCTURES 13
Indeed, using integration by parts and (3.13), we have Z ξ φ ( x, z, t ) ¯ ψ ( x, z, t )d t = Z ξ (cid:0) φ ( x, z, t ) Z ξt ¯ U ( x, z, τ )d τ (cid:1) d t = Z ξ Z ξt ¯ U ( x, z, τ )d τ d (cid:0) Z t φ ( x, z, ς )d ς (cid:1) = Z ξt ¯ U ( x, z, τ )d τ Z t φ ( x, z, ς )d ς | ξ + Z ξ (cid:0) Z t φ ( x, z, ς )d ς (cid:1) ¯ U ( x, z, t )d t = Z ξ (cid:0) Z t φ ( x, z, τ )d τ (cid:1) ¯ U ( x, z, t )d t. Next, we take the test function w = ψ in (3.10) and get Z Ω ( ε − c µ − ) ∂ t U ¯ ψ d x d z = − Z Ω µ − ∇ U · ∇ ¯ ψ d x d z + X j =1 Z Γ j µ − j T j U ¯ ψ d γ j + Z Γ ρ ¯ ψ d γ − c Z Ω ( µ − ∂ tx U + ∂ x ( µ − ∂ t U )) ¯ ψ d x d z. (3.15)It follows from (3.13) and the initial conditions in (3.6) that Re Z ξ Z Ω ( ε − c µ − ) ∂ t U ¯ ψ d x d z d t = Re Z Ω Z ξ (cid:0) ∂ t (( ε − c µ − ) ∂ t U ¯ ψ ) + ( ε − c µ − ) ∂ t U ¯ U (cid:1) d t d x d z = Re Z Ω (cid:0) ( ε − c µ − ) ∂ t U ¯ ψ ) | ξ + 12 ( ε − c µ − ) | U | | ξ (cid:1) d x d z = 12 k ( ε − c µ − ) / U ( · , ξ ) k L (Ω) . Integrating (3.15) from t = 0 to t = ξ and taking the real part yield k ( ε − c µ − ) / U ( · , ξ ) k L (Ω) + Re Z ξ Z Ω µ − ∇ U · ∇ ¯ ψ d x d z d t = 12 k ( ε − c µ − ) / U ( · , ξ ) k L (Ω) + 12 Z Ω µ − | Z ξ ∇ U ( · , t )d t | d x d z =Re Z ξ X j =1 Z Γ j µ − j T j U ¯ ψ d γ j d t + Re Z ξ Z Γ ρ ¯ ψ d γ d t − c Re Z ξ Z Ω ( µ − ∂ tx U + ∂ x ( µ − ∂ t U )) ¯ ψ d x d z d t. (3.16)In what follows, we estimate the three terms of the right-hand side of (3.16) separately.By the property (3.14), we have Re Z ξ Z Γ j µ − j T j U ¯ ψ d γ j d t = Re Z ξ Z t (cid:0) Z Γ j µ − j T j U ( · , τ )d γ j (cid:1) d τ ¯ U ( · , t )d t. Let ˜ U be the extension of U with respect to t in R such that ˜ U = 0 outside the interval [0 , ξ ] . We obtain fromthe Parseval identity and Lemma 2.3 that Re Z ξ e − s t Z t (cid:0) Z Γ j µ − j T j U ( · , τ )d γ j (cid:1) d τ ¯ U ( · , t )d t = Re Z Γ j Z ∞ e − s t (cid:0) Z t µ − j T j ˜ U ( · , τ )d τ (cid:1) ¯˜ U ( · , t )d tdγ j = Re Z Γ j Z ∞ e − s t (cid:0) Z t L − ◦ µ − j B j ◦ L ˜ U ( · , τ )d τ (cid:1) ¯˜ U ( · , t )d γ j d t = Re Z Γ j Z ∞ e − s t (cid:0) L − ◦ ( sµ j ) − B j ◦ L ˜ U ( · , t ) (cid:1) ¯˜ U ( · , t )d γ j d t = 12 π Z ∞−∞ Re h ( sµ j ) − B j ˘˜ U, ˘˜ U i Γ j d s ≤ , where we have used the fact that Z t u ( τ )d τ = L − ( s − ˘ u ( s )) . After taking s → , we obtain that Re Z ξ X j =1 Z Γ j µ − j T j U ¯ ψ d γ j d t ≤ . (3.17)For ≤ t ≤ ξ ≤ T, we have from (3.14) that Re Z ξ Z Γ ρ ¯ ψ d γ d t = Z ξ (cid:0) Z t Z Γ ρ ( τ )d γ d τ (cid:1) ¯ U d t ≤ Z ξ Z t k ρ ( · , τ ) k H − / (Γ ) k U ( · , t ) k H / (Γ ) d τ d t . Z ξ Z t k ρ ( · , τ ) k H − / (Γ ) k U ( · , t ) k H (Ω) d τ d t ≤ (cid:0) Z ξ k ρ ( · , t ) k H − / (Γ ) d t (cid:1)(cid:0) Z ξ k U ( · , t ) k H (Ω) d t (cid:1) . (3.18)Using integration by parts and (3.13), we have Z ξ Z Ω µ − ∂ t ( ∂ x U ) ¯ ψ d x d z d t + Z ξ Z Ω ∂ x ( µ − ∂ t U ) ¯ ψ d x d z d t = Z Ω (cid:0) µ − ∂ x U ¯ ψ (cid:1) | ξ d x d z − Z ξ µ − ∂ x U ∂ t ¯ ψ d t d x d z + Z Ω ∂ x ( µ − U ) · ¯ ψ | ξ d x d z − Z ξ ∂ x ( µ − U ) · ∂ t ¯ ψ d x d z d t = Z ξ Z Ω (cid:0) µ − ∂ x U + ∂ x ( µ − U ) (cid:1) · ¯ U d x d z d t. By the periodicity of µ and U in x , it yields that Z ξ Z Ω (cid:0) µ − ∂ x U + ∂ x ( µ − U ) (cid:1) ¯ U d x d z d t + Z ξ Z Ω (cid:0) µ − ∂ x ¯ U + ∂ x ( µ − ¯ U ) (cid:1) U d x d z d t = 0 . NALYSIS OF TIME-DOMAIN SCATTERING BY PERIODIC STRUCTURES 15
Thus Re Z ξ Z Ω ( µ − ∂ tx U + ∂ x ( µ − ∂ t U )) ¯ ψ d x d z d t = 0 . (3.19)Substituting (3.17)–(3.19) into (3.16), we have for any ξ ∈ [0 , T ] that k ( ε − c µ − ) / U ( · , ξ ) k L (Ω) + 12 Z Ω µ − | Z ξ ∇ U ( · , t )d t | d x d z ≤ (cid:0) Z ξ k ρ ( · , t ) k H − / (Γ ) d t (cid:1)(cid:0) Z ξ k U ( · , t ) k H (Ω) (cid:1) d t. (3.20)Taking the derivative of (3.6) with respect to t , we know that ∂ t U satisfies the same equation with ρ replaced by ∂ t ρ . Define ψ ( x, z, t ) = Z ξt ∂ t U ( x, z, τ )d τ, ( x, z ) ∈ Ω , ≤ t ≤ ξ. We may follow the same steps as those for ψ to obtain k ( ε − c µ − ) / ∂ t U ( · , ξ ) k L (Ω) + 12 Z Ω µ − | Z ξ ∂ t ( ∇ U ( · , t ))d t | d x d z =Re Z ξ X j =1 Z Γ j µ − j T j ∂ t U ¯ ψ d γ j d t + Re Z ξ Z Γ ∂ t ρ ¯ ψ d γ d t − c Re Z ξ Z Ω ( µ − ∂ ttx U + ∂ x ( µ − ∂ t U )) ¯ ψ d x d z d t. (3.21)Integrating by parts yields that Z Ω µ − | Z ξ ∂ t ( ∇ U ( · , t ))d t | d x d z = 12 k µ − / ∇ U ( · , ξ ) k L (Ω) . (3.22)The first and the third terms on the right-hand side of (3.21) are discussed as above. We only have to considerthe second term. By (3.13), Lemma 2.1, and Lemma 2.2, we get Z ξ Z Γ ∂ t ρ ¯ ψ d γ d t = Z ξ Z t ( Z Γ ∂ τ ρ ( · , τ )d γ )d τ ∂ t ¯ U ( · , t )d t = Z Γ (cid:0) Z t ∂ τ ρ ( · , τ )d τ (cid:1) ¯ U ( · , t ) | ξ d γ − Z ξ Z Γ ∂ t ρ ( · , t ) U ( · , t )d γ d t . Z ξ k ∂ t ρ ( · , t ) k H − / (Γ ) k U ( · , t ) k H / (Γ ) d t . Z ξ k ∂ t ρ ( · , t ) k H − / (Γ ) k U ( · , t ) k H (Ω) d t. (3.23)Substituting (3.22) and (3.23) into (3.21), we have for any ξ ∈ [0 , T ] that k ( ε − c µ − ) / ∂ t U ( · , ξ ) k L (Ω) + 12 k µ − / ∇ U ( · , ξ ) k L (Ω) . Z ξ k ∂ t ρ ( · , t ) k H − / (Γ ) k U ( · , t ) k H (Ω) d t. (3.24) Combing the estimates (3.20) and (3.24), we obtain k U ( · , ξ ) k L (Ω) + k∇ U ( · , ξ ) k L (Ω) . (cid:0) Z ξ k ρ ( · , t ) k H − / (Γ ) d t (cid:1)(cid:0) Z ξ k U ( · , t ) k H (Ω) d t (cid:1) + Z ξ k ∂ t ρ ( · , t ) k H − / (Γ ) k U ( · , t ) k H (Ω) d t. (3.25)Taking the L ∞ - norm with respect to ξ on both side of (3.25) yields k U k L ∞ (0 ,T ; L (Ω)) + k∇ U k L ∞ (0 ,T ; L (Ω) ) . T k ρ k L (0 ,T ; H − / (Γ )) k U k L ∞ (0 ,T ; H (Ω)) + k ∂ t ρ k L (0 ,T ; H − / (Γ )) k U k L ∞ (0 ,T ; H (Ω)) , which gives the estimate (3.11) after applying the Young inequality.Integrating (3.25) with respect to ξ from to T and using the Cauchy–Schwarz inequality, we obtain k U k L (0 ,T ; L (Ω)) + k∇ U k L (0 ,T ; L (Ω) ) . T / k ρ k L (0 ,T ; H − / (Γ )) k U k L (0 ,T ; H (Ω) )+ T / k ∂ t ρ k L (0 ,T ; H − / (Γ )) k U k L (0 ,T ; H (Ω) ) , which implies the estimate (3.12) by using the Young inequality again. (cid:3)
4. C
ONCLUSION
In this paper, we studied the time-domain scattering problem in a one-dimensional grating. The TE andTM cases were considered in a unified approach. The scattering problem was reduced equivalently into aninitial-boundary value problem in a bounded domain by using the exact time-domain DtN map. The reducedproblem was shown to have a unique solution by using the energy method. The stability was also presented.The main ingredients of the proofs were the Laplace transform, the Lax–Milgram lemma, and the Parsevalidentity. Moreover, by directly considering the variational problem of the time-domain wave equation, weobtained a priori estimates with explicit dependence on time. In the future, we plan to investigate the time-domain scattering by biperiodic structures where the full three-dimensional Maxwell’s equations should beconsidered. The progress will be reported elsewhere.R
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