, 17)), the combined difficulties being (1) the polynomial growth of the solution at infinity (2) the half-infinite periodic part of the strip. We use DtN operators (see Chapter 1) to restrict the computation on the interface between the homogeneous half-strip and the periodic half-strip
Electromagnetic waves in periodic media, Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, p.34, 1993. ,
Mechanics of composite materials, Studies in applied mechanics, vol.29, issue.6, 1991. ,
Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators, of Mathematical Notes, vol.29, p.20, 1982. ,
Homogenization and two-scale convergence, SIAM J. Math. Anal, vol.23, issue.6, pp.1482-1518, 1992. ,
URL : https://hal.archives-ouvertes.fr/hal-01111805
Boundary layer tails in periodic homogenization. ESAIM: Control, Optimisation and Calculus of Variations, vol.4, pp.209-243, 1999. ,
A comparison between two-scale asymptotic expansions and bloch wave expansions for the homogenization of periodic structures, SEMA journal, vol.73, issue.3, p.113, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01215580
Discretization of the poisson equation with non-smooth data and emphasis on non-convex domains, Numer. Methods Partial Differential Equations, vol.32, issue.5, pp.1433-1454, 2016. ,
On integral equation and least squares methods for scaterring by diffraction gratings, Commun. Comput. Phys, vol.1, issue.6, pp.1010-1042, 2006. ,
Quantitative analysis of boundary layers in periodic homogenization, Arch. Ration. Mech. Anal, vol.226, issue.2, p.113, 2017. ,
Quantum percolation and ballistic conductance on a lattice of wires, Phys. Rev. B, vol.45, issue.3, 1973. ,
Homogenisation averaging processes in periodic media: mathematical problems in the mechanics of composite materials, p.94, 1989. ,
Variational approximation of Maxwell's equations in biperiodic structures, SIAM J. Appl. Math, vol.57, issue.2, pp.364-381, 1997. ,
Couplage des méthodes modale et éléments finis pour la diffraction des ondes élastiques guidées, vol.35, p.42, 2009. ,
Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide, J. Comput. Appl. Math, vol.234, issue.6, p.42, 2010. ,
URL : https://hal.archives-ouvertes.fr/hal-00975075
Stability of perfectly matched layers, group velocities and anisotropic waves, J. Comput. Phys, vol.188, issue.2, pp.399-433, 2003. ,
Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol.5, p.95, 1978. ,
A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys, vol.114, issue.2, pp.185-200, 1994. ,
Homogenization with corrector term for periodic elliptic differential operators, St. Petersburg Math. J, vol.17, issue.6, pp.897-973, 2006. ,
Homogenization with corrector for periodic differential operators. approximation of solutions in the sobolev class h1(rd). Algebra i Analiz, vol.18, 2006. ,
On correctors for linear elliptic homogenization in the presence of local defects, p.94, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01697104
On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves, Wave Motion, vol.51, issue.2, pp.266-283, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-00816895
Mathematical analysis of the junction of two acoustic open waveguides, SIAM J. Appl. Math, vol.71, issue.6, pp.2048-2071, 2011. ,
URL : https://hal.archives-ouvertes.fr/hal-00849568
Diffraction by an acoustic grating perturbed by a bounded obstacle, Adv. Comput. Math, vol.16, issue.2-3, pp.113-138, 2002. ,
URL : https://hal.archives-ouvertes.fr/hal-00990202
Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem, Bonnet-Ben Dhia and Starling, vol.331, pp.305-338, 1994. ,
URL : https://hal.archives-ouvertes.fr/hal-01010197
A generalized mode matching method for scattering problems with unbounded obstacles, J. Comput. Acoust, vol.9, issue.4, p.63, 2001. ,
URL : https://hal.archives-ouvertes.fr/hal-01008838
A limiting absorption principle for scattering problems with unbounded obstacles, Math. Methods Appl. Sci, vol.24, issue.14, p.63, 2001. ,
URL : https://hal.archives-ouvertes.fr/hal-01008796
Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math, vol.78, pp.1-96, 1946. ,
Multiscale nanorod metamaterials and realizable permittivity tensors, Commun. Comput. Phys, vol.11, issue.2, pp.489-507, 2012. ,
Homogenization of the 3d maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci, vol.347, issue.9, pp.571-576, 2009. ,
Homogenization near resonances and artificial magnetism in three dimensional dielectric metamaterials, Arch. Ration. Mech. Anal, vol.225, issue.3, pp.1233-1277, 2017. ,
Homogenization near resonances and artificial magnetism from dielectrics, C. R. Math. Acad. Sci, vol.339, issue.5, pp.377-382, 2004. ,
Etude mathématique et numérique de cristaux photoniques fortement contrastés, p.93, 2010. ,
Spectrum created by line defects in periodic structures, Math. Nachr, vol.287, pp.1972-1985, 2014. ,
On the spectrum of waveguides in planar photonic bandgap structures, Proc. A, vol.471, p.76, 2015. ,
On the homogenization of a scalar scattering problem for highly oscillating anisotropic media, SIAM J. Math. Anal, vol.48, issue.4, p.99, 2016. ,
Periodic Schrödinger operators with local defects and spectral pollution, Numer. Math, vol.28, pp.663-706, 2014. ,
A gap in the essential spectrum of a cylindrical waveguide with a periodic perturbation of the surface, Math. Nachr, vol.283, issue.9, pp.1222-1244, 2010. ,
Adjoint and self-adjoint differential operators on graphs, Electron. J. Differential Equations, vol.6, pp.1-10, 1998. ,
Riccati equation for the impedance of waveguides, Mathematical and numerical aspects of wave propagation-WAVES 2003, p.25, 2003. ,
Variational approach in weighted sobolev spaces to scattering by unbounded rough surfaces, SIAM J. Math. Anal, vol.42, issue.6, pp.2554-2580, 2010. ,
A well-posed integral equation formulation for three-dimensional rough surface scattering, Proc. A., volume, vol.462, p.34, 2006. ,
Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal, vol.37, issue.2, pp.598-618, 2005. ,
Scattering by infinite one-dimensional rough surfaces, Proc. A., volume, vol.455, p.34, 1990. ,
Mechanics of composite materials, 1979. ,
of oxford lecture series in mathematics and its applications, vol.17, p.95, 1999. ,
High order asymptotics for wave propagation across thin periodic interfaces, Asymptot. Anal, vol.83, issue.1-2, p.103, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00682386
Helmholtz equation in periodic media with a line defect, J. Comput. Phys, vol.231, issue.4, pp.1675-1704, 2012. ,
A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl, vol.106, issue.2, pp.367-413, 1985. ,
, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions, vol.1341, p.55, 2006.
On the well-posedness, stability and accuracy of an asymptotic model for thin periodic interfaces in electromagnetic scattering problems, Math. Models Methods Appl. Sci, vol.98, issue.1, p.103, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-00682357
, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simul, vol.12, issue.2, pp.488-513, 2014.
Efficient numerical modelling of absorbing regions for boundaries of guided waves problems, AIP conference proceedings, vol.820, pp.126-140, 2006. ,
Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput, vol.15, issue.3, 1994. ,
Numerical simulation of waves in periodic structures, Commun. Comput. Phys, p.69, 2008. ,
Exact artificial boundary conditions for problems with periodic structures, J. Comput. Phys, vol.227, issue.14, pp.6877-6894, 2008. ,
The limiting absorption and amplitude principles for the diffraction problem with two unbounded media, Comm. Math. Phys, vol.107, issue.1, pp.29-38, 1986. ,
On the principle of limiting absorption, 1963. ,
Lattice kronig-penney models, Phys. Rev. Lett, vol.74, issue.18, p.3503, 1995. ,
Contact interactions on graph superlattices, J. Phys. A, vol.29, issue.1, 1996. ,
Honeycomb lattice potentials and Dirac points, J. Amer. Math. Soc, vol.25, issue.4, 2012. ,
Topologically protected states in one-dimensional continuous systems and Dirac points, Proc. Natl. Acad. Sci. USA, vol.111, issue.24, pp.8759-8763, 2014. ,
Bifurcations of edge states-topologically protected and non-protected-in continuous 2D honeycomb structures, 2D Materials, vol.3, issue.1, 2016. ,
Edge States in Honeycomb Structures, Ann. PDE, vol.2, issue.2, 2016. ,
Wave Packets in Honeycomb Structures and Two-Dimensional Dirac Equations, Waves Random Complex Media, vol.326, issue.1, pp.245-256, 1997. ,
Left-handed media and homogenization of photonic crystals, Optics letters, vol.30, issue.10, pp.1189-1191, 2005. ,
URL : https://hal.archives-ouvertes.fr/hal-00437977
Theory of mesoscopic magnetism in photonic crystals, Phys. Rev. Lett, vol.94, issue.18, 2005. ,
URL : https://hal.archives-ouvertes.fr/hal-00437978
The computation of spectra of some 2D photonic crystals, J. Comput. Phys, vol.136, issue.2, p.76, 1997. ,
Localized electromagnetic waves in a periodic-layered dielctric medium with a defect, Phys. Rev. B, vol.58, issue.1, pp.180-188, 1998. ,
Localized classical waves created by defects, J. Statist. Phys, vol.86, issue.1-2, pp.165-177, 1997. ,
Localization of light in lossless inhomogeneous dielectrics, JOSA A, vol.15, issue.5, pp.1423-1435, 1998. ,
Midgap defect modes in dielectric and acoustic media, SIAM J. Appl. Math, vol.58, issue.6, p.76, 1998. ,
Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model, SIAM J. Appl. Math, vol.56, issue.1, p.84, 1996. ,
Band-gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals, SIAM J. Appl. Math, vol.56, issue.6, 1996. ,
Orthogonality relation for the rayleigh-lamb modes of vibration of a plate, JASA, 1976. ,
Cross-points in domain decomposition methods with a finite element discretization, Electron. Transac. Numer. Anal, vol.45, pp.219-240, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-00980386
The Mourre theory for analytically fibered operators, J. Funct. Anal, vol.152, issue.1, pp.202-219, 1998. ,
Homogenization in polygonal domains, J. Eur. Math. Soc, vol.13, issue.5, p.113, 2011. ,
Homogenization and boundary layers, Acta Math, vol.209, issue.1, p.113, 2012. ,
Numerical Methods for Problems in Infinite Domains, p.49, 1992. ,
Non-reflecting boundary conditions for elastic waves, Wave motion, vol.12, issue.3, pp.261-279, 1990. ,
Étude mathématique et numérique de guides d'ondes ouverts non uniformes, par approche modale, vol.46, 2010. ,
A note on bi-orthogonality relations for elastic cylinders of general cross section, J. Elasticity, vol.13, issue.3, pp.351-355, 1983. ,
Radiation boundary conditions for the numerical simulation of waves, Acta numerica, vol.8, p.15, 1999. ,
Hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs, Numer. Math, vol.133, issue.1, pp.103-139, 2016. ,
Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems, Wave Motion, vol.59, pp.94-110, 2015. ,
Dirichlet-to-Neumann maps for unbounded wave guides, J. Comput. Phys, vol.143, issue.1, pp.200-223, 1998. ,
Factorization of second-order elliptic boundary value problems by dynamic programming, Nonlinear Anal, vol.59, issue.5, pp.629-647, 2004. ,
The limiting absorption principle for a periodic semi-infinite waveguide, SIAM J. Appl. Math, vol.71, issue.3, p.41, 2011. ,
Solving timeharmonic scattering problems based on the pole condition, I. Theory. SIAM J. Math. Anal, vol.35, issue.1, p.15, 2003. ,
Solving timeharmonic scattering problems based on the pole condition. II. Convergence of the PML method, SIAM J. Math. Anal, vol.35, issue.3, p.15, 2003. ,
Riesz bases and jordan form of the translation operator in semi-infinite periodic waveguides, J. Math. Pures Appl, vol.100, issue.1, pp.113-135, 2013. ,
Principe d'absorption limite pour l'opérateur de Laplace-Dirichlet sur un ouvert au bord périodique, Math. Rep. (Bucur.), vol.5, issue.55, pp.239-249, 2003. ,
Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs, vol.102, p.101, 1992. ,
Multiple commutator estimates and resolvent smoothness in quantum scattering theory, Ann. Inst. H. Poincaré Phys. Théor, vol.41, issue.2, pp.207-225, 1984. ,
Homogenization of differential operators and integral functionals, vol.93, p.95, 2012. ,
Photonic Crystal -Molding the Flow of Light, vol.6, p.75, 1995. ,
Photonic Crystal -The road from theory to practice, vol.6, p.75, 2002. ,
A numerical method for the computation of electromagnetic modes in optical fibres, Math. Methods Appl. Sci, vol.22, issue.5, pp.389-447, 1999. ,
Perturbation theory for the Schrödinger operator with a periodic potential, Lecture Notes in Mathematics, vol.1663, p.76, 1997. ,
Perturbation theory for linear operators, Classics in Mathematics, vol.22, p.23, 1995. ,
Mechanics of composite materials, 2005. ,
The limiting absorption principle and a radiation condition for the scattering by a periodic layer, SIAM J. Math. Anal, p.34, 2017. ,
A radiation condition arizing from the limiting absorption principle for a closed full or half waveguide problem, Kirsch and Lechleiter, p.25, 2018. ,
Elliptic boundary value problems in domains with point singularities, vol.52, p.104, 1997. ,
, Floquet theory for partial differential equations, vol.70, p.79, 1993.
The mathematics of photonic crystals, Mathematical modeling in optical science, vol.22, p.75, 2001. ,
On some spectral problems of mathematical physics, Partial differential equations and inverse problems, vol.362, p.75, 2004. ,
On guided waves in photonic crystal waveguides, Waves in periodic and random media, vol.339, p.76, 2002. ,
Convergence of spectra of mesoscopic systems collapsing onto a graph, J. Math. Anal. Appl, vol.258, issue.2, p.87, 2001. ,
The floquet-bloch transform and scattering from locally perturbed periodic surfaces, J. Math. Anal. Appl, vol.446, issue.1, p.35, 2017. ,
Elliptic operators with honeycomb symmetry: Dirac points, edge states and applications to photonic graphene, 2017. ,
The localized finite element method and its application to the two-dimensional sea-keeping problem, SIAM J. Numer. Anal, vol.25, issue.4, pp.729-752, 1988. ,
URL : https://hal.archives-ouvertes.fr/hal-00974512
Acoustic waves in perturbed periodic layer: a limiting absorption principle, Asymptot. Anal, vol.16, issue.1, p.22, 1998. ,
A literature survey of mathematical study of metamaterials, Int. J. Numer. Anal. Model, vol.13, issue.2, pp.230-243, 2016. ,
Bw media-media with negative parameters, capable of supporting backward waves, Microw. Opt. Techn. Let, vol.31, issue.2, pp.129-133, 2001. ,
, Problèmes aux limites non homogènes et applications, vol.1, p.56, 1968.
On the schwarz alternating method. i, First international symposium on domain decomposition methods for partial differential equations, pp.1-42, 1988. ,
, , p.50
The riccati method for the helmholtz equation, JASA, vol.100, issue.3, pp.1432-1446, 1996. ,
Hill's equation, Interscience Tracts in Pure and Applied Mathematics, vol.25, issue.20, 1966. ,
Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Birkhäuser, vol.1, p.99, 2012. ,
A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces, J. Integral Equations Appl, vol.12, issue.3, pp.281-321, 2000. ,
First-order corrections to the homogenised eigenvalues of a periodic composite medium. a convergence proof, Proc. Roy. Soc. Edinburgh Sect. A, vol.127, issue.6, p.97, 1997. ,
First order corrections to the homogenized eigenvalues of a periodic composite medium. the case of neumann boundary conditions, Indiana Univ. Math. J, p.94, 1997. ,
Absence of singular continuous spectrum for certain selfadjoint operators, Commu. Math. Phys, vol.78, issue.3, pp.391-408, 1981. ,
Opening of a gap in the continuous spectrum of a periodically perturbed waveguide, Math. Notes, vol.87, issue.5-6, pp.738-756, 2010. ,
The asymptotic analysis of gaps in the spectrum of a waveguide perturbed with a periodic family of small voids, J. Math. Sci. (N. Y.), vol.186, issue.2, p.84, 2012. ,
On the spectrum of the laplace operator on the infinite dirichlet ladder, St. Petersburg Math. J, vol.23, issue.6, p.87, 2012. ,
The mandelshtam energy radiation conditions and the umov-poynting vector in elastic waveguides, Probl. Mat. anal, vol.72, p.43, 2013. ,
Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder, Comput. Math. Math. Phys, vol.54, issue.8, pp.1261-1279, 2014. ,
Umov-mandelshtam radiation conditions in elastic periodic waveguides, Sbornik: Mathematics, vol.205, issue.7, p.41, 2014. ,
On radiation conditions for selfadjoint elliptic problems, Dokl. Akad. Nauk SSSR, vol.311, issue.3, pp.532-536, 1990. ,
Radiation principles for selfadjoint elliptic problems, Probl. Mat. Fiz, vol.13, pp.192-244, 1991. ,
Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Expositions in Mathematics, vol.13, p.45, 1994. ,
Asymptotic behavior of solutions to the helmholtz equations with sign changing coefficients, Trans. Amer. Math. Soc, vol.367, issue.9, pp.6581-6595, 2015. ,
Limiting absorption principle and well-posedness for the helmholtz equation with sign changing coefficients, J. Math. Pures Appl, vol.106, issue.2, pp.342-374, 2016. ,
Remarks on a transmission problem, J. Math. Anal. Appl, vol.196, issue.2, pp.639-658, 1995. ,
Lamb wave propagation in inhomogeneous elastic waveguides, Proc. A, vol.458, 2002. ,
Scattering matrix properties with evanescent modes for waveguides in fluids and solids, JASA, vol.116, p.42, 1913. ,
Lamb wave propagation in elastic waveguides with variable thickness, Proc. A, vol.462, pp.1315-1339, 2006. ,
Asymptotic partial decomposition of domain for spectral problems in rod structures, J. Math. Pures Appl, vol.87, issue.1, pp.1-36, 2007. ,
A method for computing guided waves in integrated optics. I. Mathematical analysis, SIAM J. Numer. Anal, vol.39, issue.2, p.77, 2001. ,
URL : https://hal.archives-ouvertes.fr/hal-01010356
A method for computing guided waves in integrated optics. II. Numerical approximation and error analysis, SIAM J. Numer. Anal, vol.39, issue.5, p.77, 2002. ,
URL : https://hal.archives-ouvertes.fr/hal-01010356
Magnetism from conductors, and enhanced non-linear phenomena, IEEE T. Microw. Theory, vol.47, pp.2075-2084, 1999. ,
Electromagnetic theory of gratings, Topics in Current Physics, vol.22, p.33, 1980. ,
Remark on the structure of the spectrum of a two-dimensional Schrödinger operator with periodic potential, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol.109, pp.183-184, 1981. ,
Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincaré, vol.7, issue.5, p.87, 2006. ,
Energetic criterion for the radiation of floquet waves in infinite anisotropic periodically multilayered media, Acustica-Acta Acustica, vol.87, issue.3, pp.340-351, 2001. ,
, The principle of limit absorption and limit amplitude for periodic operators, p.21, 2009.
, Methods of modern mathematical physics v. I-IV, vol.41, p.42, 1972.
Elastic waves in solids II: generation, acousto-optic interaction, applications, p.66, 1999. ,
Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum, Arch. Ration. Mech. Anal, vol.160, issue.4, pp.271-308, 2001. ,
Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Comput, vol.7, issue.3, pp.856-869, 1986. ,
The limiting equation for Neumann Laplacians on shrinking domains, Electron. J. Differential Equations, vol.25, issue.31, p.84, 2000. ,
A dispersive effective medium for wave propagation in periodic composites, 75 BIBLIOGRAPHY [Santosa and Symes, vol.6, pp.984-1005, 1991. ,
High-order numerical modelling of highly conductive thin sheets, Citeseer, vol.99, p.103, 2008. ,
Efficient computation of photonic crystal waveguide modes with dispersive material, Opt. Express, vol.18, issue.7, pp.7307-7322, 2010. ,
URL : https://hal.archives-ouvertes.fr/hal-00873062
Boundary layers in periodic homogenization of neumann problems, 2016. ,
Homogenization of 3-d-connected and nonconnected wire metamaterials, IEEE transactions on microwave theory and techniques, vol.53, 2005. ,
Guided elastic waves and perfectly matched layers, Wave motion, vol.44, issue.7, pp.573-592, 2007. ,
Proof of the Bethe-Sommerfeld conjecture in dimension 2, Dokl. Akad. Nauk SSSR, vol.248, issue.1, pp.39-42, 1979. ,
Multidimensional Schrödinger operator with periodic potential, Izv. Akad. Nauk SSSR Ser. Mat, vol.47, issue.3, pp.659-687, 1983. ,
The spectrum band structure of the threedimensional Schrödinger operator with periodic potential, Invent. Math, vol.80, issue.1, pp.107-121, 1985. ,
Convergence of the supercell method for defect modes calculations in photonic crystals, SIAM J. Numer. Anal, vol.43, issue.3, p.78, 2005. ,
Meromorphic families of compact operators. Arch. Ration, Mech. Anal, vol.31, issue.5, p.42, 1968. ,
The general theory of homogenization: a personalized introduction, vol.7, p.95, 2009. ,
Matching and multiscale expansions for a model singular perturbation problem, C. R. Math. Acad. Sci. Paris, vol.343, issue.10, p.99, 2006. ,
URL : https://hal.archives-ouvertes.fr/hal-00453366
Domain decomposition methods: algorithms and theory, vol.34, p.50, 2005. ,
Perturbation methods in fluid mechanics, vol.8, p.99, 1964. ,
, On the Bethe-Sommerfeld conjecture for certain periodic Maxwell operators, J. Math. Anal. Appl, vol.377, issue.1, pp.370-383, 2011.
Three-dimensional timeharmonic elastodynamic green's functions for anisotropic solids, Proc. A, volume, vol.449, p.50, 1937. ,
Spectral and scattering theory for wave propagation in perturbed stratfied media, p.20, 1990. ,
Steady-state wave propagation in homogeneous anisotropic media, Arch. Ration. Mech. Anal, vol.25, pp.201-242, 1967. ,
Sound propagation in stratified fluids, Appl. Math. Sci, vol.1, p.20, 1984. ,
An efficient bidirectional propagation method based on dirichlet-to-neumann maps, IEEE Photonics Techn. Lett, vol.18, pp.1967-1969, 2006. ,