Resonant scattering by open periodic waveguides 1

This article concerns the interaction between guided electromagnetic or acoustic modes of a penetrable periodic planar waveguide and plane waves originating from sources exterior to the waveguide. The interaction causes resonant enhancement of fields in the waveguide and anomalous transmission of energy across it. A guided mode is an eigenfunction of a member of the family of operators in the Floquet- Bloch decomposition of the periodic differential operator underlying the waveguide structure. The theory of existence or nonexistence of modes in ideal lossless waveguides is founded on variational principles. The mechanism for resonant scattering behavior is the dissolution of an embedded eigenvalue into the continuous spectrum, which corresponds to the destruction of a guided mode of a waveguide, upon pertur- bation of the wavevector or the material properties or geometry of the structure. Analytic perturbation of functions that unify the guided modes and the extended scattering states gives rise to asymptotic formulas for transmission anomalies.

[1]  R. Kress,et al.  Integral equation methods in scattering theory , 1983 .

[2]  Peter Kuchment,et al.  On guided waves in photonic crystal waveguides , 2004 .

[3]  C. M. Linton,et al.  Resonant effects in scattering by periodic arrays , 2007 .

[4]  Thomas W. Ebbesen,et al.  Crucial role of metal surface in enhanced transmission through subwavelength apertures , 2000 .

[5]  P. Lalanne,et al.  Microscopic theory of the extraordinary optical transmission , 2008, Nature.

[6]  D. V. Evans,et al.  Existence theorems for trapped modes , 1994, Journal of Fluid Mechanics.

[7]  T. Ebbesen,et al.  Light in tiny holes , 2007, Nature.

[8]  S. Shipman,et al.  Resonant transmission near nonrobust periodic slab modes. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Claus Müller,et al.  Foundations of the mathematical theory of electromagnetic waves , 1969 .

[10]  Henri Lezec,et al.  Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays. , 2004, Optics express.

[11]  D. Manda,et al.  Spectral and Scattering-Theory for Wave-Propagation in Perturbed Stratified Media , 1995 .

[12]  J. Pendry,et al.  Evanescently coupled resonance in surface plasmon enhanced transmission , 2001 .

[13]  P. Lalanne,et al.  Surface plasmons of metallic surfaces perforated by nanohole arrays , 2005 .

[14]  C. M. Linton,et al.  Construction of trapped modes for wave guides and diffraction gratings , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  M. Groves.  Examples of embedded eigenvalues for problems in acoustic waveguides , 1998 .

[16]  C. Goldstein.  Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries , 2011 .

[17]  H. Lezec,et al.  Control of optical transmission through metals perforated with subwavelength hole arrays. , 1999, Optics letters.

[18]  C. Linton,et al.  Trapped modes for off-centre structures in guides , 2002 .

[19]  M. Nevière,et al.  The Homogeneous Problem , 1980 .

[20]  C. Linton,et al.  Embedded trapped modes for obstacles in two-dimensional waveguides , 2001 .

[21]  S. Kukhlevsky.  Enhanced transmission of light through subwavelength nanoapertures by far-field multiple-beam interference , 2007, 0712.3156.

[22]  C. Linton,et al.  Trapped modes in cylindrical waveguides , 1998 .

[23]  N. Ptitsyna.  A discrete model of guided modes and anomalous scattering in periodic structures , 2009 .

[24]  C. Goldstein.  Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. II. Applications to scattering theory , 2011 .

[25]  A. Rau.  Perspectives on the Fano Resonance Formula , 2004 .

[26]  Martin Costabel,et al.  A direct boundary integral equation method for transmission problems , 1985 .

[27]  D. V. Evans,et al.  Trapped acoustic modes , 1992 .

[29]  H. Lezec,et al.  Extraordinary optical transmission through sub-wavelength hole arrays , 1998, Nature.

[30]  M. J. Lighthill,et al.  The eigenvalues of ∇2u + λu=0 when the boundary conditions are given on semi-infinite domains , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[31]  V. P. Shestopalov,et al.  Spectral Theory and Excitation of Open Structures , 1996 .

[32]  L. Parnovski,et al.  Trapped modes in a waveguide with a long obstacle , 2000, Journal of Fluid Mechanics.

[33]  Interference-induced enhancement of intensity and energy of a quantum optical field by a subwavelength array of coherent light sources , 2007, quant-ph/0701095.

[34]  Resonant frequencies for a narrow strip grating , 1999 .

[35]  Jeff F. Young,et al.  Two-dimensional vector-coupled-mode theory for textured planar waveguides , 2000 .

[36]  F. G. D. Abajo.  Colloquium: Light scattering by particle and hole arrays , 2007, 0903.1671.

[37]  M. Bertolotti.  Scattering Theory for Diffraction Gratings , 1984 .

[38]  J. Pendry,et al.  Theory of extraordinary optical transmission through subwavelength hole arrays. , 2000, Physical review letters.

[39]  C. Elachi,et al.  Waves in active and passive periodic structures: A review , 1976, Proceedings of the IEEE.

[40]  T. Ishihara,et al.  Quasiguided modes and optical properties of photonic crystal slabs , 2002 .

[41]  Shanhui Fan,et al.  Analysis of guided resonances in photonic crystal slabs , 2002 .

[42]  P. Neittaanmäki,et al.  On surface waves in diffraction gratings , 2000 .

[43]  Habib Ammari,et al.  Guided Waves in a Photonic Bandgap Structure with a Line Defect , 2004, SIAM J. Appl. Math..

[44]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[45]  R. G. Cooke.  Functional Analysis and Semi-Groups , 1949, Nature.

[46]  W. Burnside.  Theory of Functions of a Complex Variable , 1893, Nature.

[47]  I. M. Glazman,et al.  Theory of linear operators in Hilbert space , 1961 .

[48]  S. Shipman,et al.  Fano resonance of waves in periodic slabs , 2008, 2008 12th International Conference on Mathematical Methods in Electromagnetic Theory.

[49]  H. Ammari,et al.  Resonances for Maxwell’s equations in a periodic structure , 2000 .

[50]  A. Soffer,et al.  Time Dependent Resonance Theory , 1998 .

[51]  D. V. Evans,et al.  TRAPPED MODE FREQUENCIES EMBEDDED IN THE CONTINUOUS SPECTRUM , 1993 .

[52]  Theodor Tamir,et al.  Theory of Periodic Dielect Waveguides , 1975 .

[53]  Peter Monk,et al.  Existence, Uniqueness, and Variational Methods for Scattering by Unbounded Rough Surfaces , 2005, SIAM J. Math. Anal..

[54]  Thomas W. Ebbesen,et al.  Surface plasmons enhance optical transmission through subwavelength holes , 1998 .

[55]  D. V. Evans,et al.  Embedded Rayleigh–Bloch surface waves along periodic rectangular arrays , 2005 .

[56]  V. Weisskopf,et al.  Effects of Configuration Interaction on Intensities and Phase Shifts , 2001 .

[57]  C. Hill,et al.  On the Principle of Limiting Absorption , 2011 .

[58]  Tosio Kato.  A Short Introduction to Perturbation Theory for Linear Operators , 1982 .

[59]  A. Nosich.  Radiation conditions, limiting absorption principle, and general relations in open waveguide scattering , 1994 .

[60]  R. Newton.  Scattering theory of waves and particles , 1966 .

[61]  F. Ursell,et al.  Trapping modes in the theory of surface waves , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.

[62]  W. A. Murray,et al.  Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film. , 2004, Physical review letters.

[63]  Darko Volkov,et al.  Guided Modes in Periodic Slabs: Existence and Nonexistence , 2007, SIAM J. Appl. Math..

[64]  Hermann A. Haus,et al.  Attenuation of cutoff modes and leaky modes of dielectric slab structures , 1986 .

[65]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[66]  P. Kuchment.  The mathematics of photonic crystals , 2001 .

[67]  V. Edwards.  Scattering Theory , 1973, Nature.

[68]  Israel Michael Sigal,et al.  Introduction to Spectral Theory: With Applications to Schrödinger Operators , 1995 .

[69]  Raj Mittra,et al.  Analytical techniques in the theory of guided waves , 1971 .

[70]  Shanhui Fan,et al.  Mechanism for designing metallic metamaterials with a high index of refraction. , 2005, Physical review letters.

[71]  R. Kress.  Linear Integral Equations , 1989 .

[72]  R. A. Silverman,et al.  Theory of Functions of a Complex Variable, Volume I , 1965 .

[73]  D. V. Evans,et al.  Trapped modes in two-dimensional waveguides , 1991, Journal of Fluid Mechanics.

[74]  F. Medina,et al.  Extraordinary Transmission Through Arrays of Electrically Small Holes From a Circuit Theory Perspective , 2008, IEEE Transactions on Microwave Theory and Techniques.

[75]  J. Nédélec.  Acoustic and Electromagnetic Equations : Integral Representations for Harmonic Problems , 2001 .

[76]  C. M. Linton,et al.  The existence of Rayleigh–Bloch surface waves , 2002, Journal of Fluid Mechanics.

[77]  Gregory A. Kriegsmann,et al.  Complete Transmission Through a Two-Dimensional Diffraction Grating , 2004, SIAM J. Appl. Math..

[78]  A. Bonnet-Bendhia,et al.  Guided waves by electromagnetic gratings and non‐uniqueness examples for the diffraction problem , 1994 .

[79]  Stanly Steinberg,et al.  Meromorphic families of compact operators , 1968 .

[80]  L. Zalcman,et al.  Fourier Series and Integrals , 2007 .

[81]  Stephanos Venakides,et al.  Boundary-Integral Calculations of Two-Dimensional Electromagnetic Scattering in Infinite Photonic Crystal Slabs: Channel Defects and Resonances , 2002, SIAM J. Appl. Math..

[82]  A. Weinstein,et al.  Methods of intermediate problems for eigenvalues: theory and ramifications , 1972 .

[83]  Stephanos Venakides,et al.  Resonance and Bound States in Photonic Crystal Slabs , 2003, SIAM J. Appl. Math..

[84]  Ricardo Weder,et al.  Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media , 1990 .