October  2012, 5(5): 989-1020. doi: 10.3934/dcdss.2012.5.989

A lattice model for resonance in open periodic waveguides

1. 

Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, United States

2. 

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, United States

Received  April 2011 Revised  September 2011 Published  January 2012

We present a discrete model of resonant scattering of waves by an open periodic waveguide. The model elucidates a phenomenon common in electromagnetics, in which the interaction of plane waves with embedded guided modes of the waveguide causes sharp transmission anomalies and field amplification. The ambient space is modeled by a planar lattice and the waveguide by a linear periodic lattice coupled to the planar one along a line. We show the existence of standing and traveling guided modes and analyze a tangent bifurcation, in which resonance is initiated at a critical coupling strength where a guided mode appears, beginning with a single standing wave and splitting into a pair of waves traveling in opposing directions. Complex perturbation analysis of the scattering problem in the complex frequency and wavenumber domain reveals the complex structure of the transmission coefficient at resonance.
Citation: Natalia Ptitsyna, Stephen P. Shipman. A lattice model for resonance in open periodic waveguides. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 989-1020. doi: 10.3934/dcdss.2012.5.989
References:
[1]

Anne-Sophie Bonnet-Bendhia and Felipe Starling, Guided waves by electromagnetic gratings and nonuniqueness examples for the diffraction problem,, Math. Meth. Appl. Sci., 17 (1994), 305.   Google Scholar

[2]

Shanhui Fan and J. D. Joannopoulos, Analysis of guided resonances in photonic crystal slabs,, Phys. Rev. B, 65 (2002).   Google Scholar

[3]

U. Fano, Effects of configuration interaction on intensities and phase shifts,, Phys. Rev., 124 (1961), 1866.  doi: 10.1103/PhysRev.124.1866.  Google Scholar

[4]

S. Fan, P. R. Villeneuve and J. D. Joannopoulos, Rate-equation analysis of output efficiency and modulation rate of photonic-crystal light-emitting diodes,, IEEE Quantum. Elec., 36 (2000), 1123.   Google Scholar

[5]

Alexander Figotin and Jeffrey H. Schenker, Spectral theory of time dispersive and dissipative systems,, J. Stat. Phys., 118 (2005), 199.  doi: 10.1007/s10955-004-8783-7.  Google Scholar

[6]

Alexander Figotin and Stephen P. Shipman, Open systems viewed through their conservative extensions,, J. Stat. Phys., 125 (2006), 363.   Google Scholar

[7]

Hermann A. Haus and David A. B. Miller, Attenuation of cutoff modes and leaky modes of dielectric slab structures,, IEEE J. Quantum Elec., 22 (1986), 310.  doi: 10.1109/JQE.1986.1072956.  Google Scholar

[8]

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, Jeff F. Young, S. R. Johnson, Jim MacKenzie and T. Tiedje, Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,, Appl. Phys. Lett., 70 (1997), 1438.  doi: 10.1063/1.118570.  Google Scholar

[9]

Haitao Liu and Philippe Lalanne, Microscopic theory of the extraordinary optical transmission,, Lett. Nature, 452 (2008), 728.  doi: 10.1038/nature06762.  Google Scholar

[10]

S. Longhi, Bound states in the continuum in a single-level Fano-Anderson model,, Eur. Phys. J. B, 57 (2007), 45.  doi: 10.1140/epjb/e2007-00143-2.  Google Scholar

[11]

G. D. Mahan, "Many-Particle Physics,", Second edition, (1990).   Google Scholar

[12]

Andrey E. Miroshnichenko, Sergei F. Mingaleev, Sergej Flach and Yuri S. Kivshar, Nonlinear Fano resonance and bistable wave transmission,, Phys. Rev. E (3), 71 (2005).   Google Scholar

[13]

P. Paddon and Jeff F. Young, Two-dimensional vector-coupled-mode theory for textured planar waveguides,, Phys. Rev. B, 61 (2000), 2090.  doi: 10.1103/PhysRevB.61.2090.  Google Scholar

[14]

S. T. Peng, T. Tamir and H. L. Bertoni, Theory of periodic dielectric waveguides,, IEEE Trans. Microwave Th. and Tech., 23 (1975), 123.  doi: 10.1109/TMTT.1975.1128513.  Google Scholar

[15]

Michael Reed and Barry Simon, "Methods of Mathematical Physics. IV: Analysis of Operators,", Academic Press, (1980).   Google Scholar

[16]

Stephen P. Shipman, Resonant scattering by open periodic waveguides,, in, (2010).   Google Scholar

[17]

Stephen P. Shipman, Jennifer Ribbeck, Katherine H. Smith and Clayton Weeks, A discrete model for resonance near embedded bound states,, IEEE Photonics J., 2 (2010), 911.   Google Scholar

[18]

Stephen P. Shipman and Stephanos Venakides, Resonance and bound states in photonic crystal slabs,, SIAM J. Appl. Math., 64 (2003), 322.  doi: 10.1137/S0036139902411120.  Google Scholar

[19]

Stephen P. Shipman and Stephanos Venakides, Resonant transmission near non-robust periodic slab modes,, Phys. Rev. E, 71 (2005).   Google Scholar

[20]

Stephen P. Shipman and Darko Volkov, Guided modes in periodic slabs: Existence and nonexistence,, SIAM J. Appl. Math., 67 (2007), 687.  doi: 10.1137/050647189.  Google Scholar

[21]

Sergei G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius and Teruya Ishihara, Quasiguided modes and optical properties of photonic crystal slabs,, Phys. Rev. B, 66 (2002).   Google Scholar

[22]

V. Weisskopf and E. Wigner, Berechnung der natürlichen linienbreite auf grund der diracschen lichttheorie,, Zeitschrift für Physik, 63 (1930), 54.  doi: 10.1007/BF01336768.  Google Scholar

show all references

References:
[1]

Anne-Sophie Bonnet-Bendhia and Felipe Starling, Guided waves by electromagnetic gratings and nonuniqueness examples for the diffraction problem,, Math. Meth. Appl. Sci., 17 (1994), 305.   Google Scholar

[2]

Shanhui Fan and J. D. Joannopoulos, Analysis of guided resonances in photonic crystal slabs,, Phys. Rev. B, 65 (2002).   Google Scholar

[3]

U. Fano, Effects of configuration interaction on intensities and phase shifts,, Phys. Rev., 124 (1961), 1866.  doi: 10.1103/PhysRev.124.1866.  Google Scholar

[4]

S. Fan, P. R. Villeneuve and J. D. Joannopoulos, Rate-equation analysis of output efficiency and modulation rate of photonic-crystal light-emitting diodes,, IEEE Quantum. Elec., 36 (2000), 1123.   Google Scholar

[5]

Alexander Figotin and Jeffrey H. Schenker, Spectral theory of time dispersive and dissipative systems,, J. Stat. Phys., 118 (2005), 199.  doi: 10.1007/s10955-004-8783-7.  Google Scholar

[6]

Alexander Figotin and Stephen P. Shipman, Open systems viewed through their conservative extensions,, J. Stat. Phys., 125 (2006), 363.   Google Scholar

[7]

Hermann A. Haus and David A. B. Miller, Attenuation of cutoff modes and leaky modes of dielectric slab structures,, IEEE J. Quantum Elec., 22 (1986), 310.  doi: 10.1109/JQE.1986.1072956.  Google Scholar

[8]

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, Jeff F. Young, S. R. Johnson, Jim MacKenzie and T. Tiedje, Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,, Appl. Phys. Lett., 70 (1997), 1438.  doi: 10.1063/1.118570.  Google Scholar

[9]

Haitao Liu and Philippe Lalanne, Microscopic theory of the extraordinary optical transmission,, Lett. Nature, 452 (2008), 728.  doi: 10.1038/nature06762.  Google Scholar

[10]

S. Longhi, Bound states in the continuum in a single-level Fano-Anderson model,, Eur. Phys. J. B, 57 (2007), 45.  doi: 10.1140/epjb/e2007-00143-2.  Google Scholar

[11]

G. D. Mahan, "Many-Particle Physics,", Second edition, (1990).   Google Scholar

[12]

Andrey E. Miroshnichenko, Sergei F. Mingaleev, Sergej Flach and Yuri S. Kivshar, Nonlinear Fano resonance and bistable wave transmission,, Phys. Rev. E (3), 71 (2005).   Google Scholar

[13]

P. Paddon and Jeff F. Young, Two-dimensional vector-coupled-mode theory for textured planar waveguides,, Phys. Rev. B, 61 (2000), 2090.  doi: 10.1103/PhysRevB.61.2090.  Google Scholar

[14]

S. T. Peng, T. Tamir and H. L. Bertoni, Theory of periodic dielectric waveguides,, IEEE Trans. Microwave Th. and Tech., 23 (1975), 123.  doi: 10.1109/TMTT.1975.1128513.  Google Scholar

[15]

Michael Reed and Barry Simon, "Methods of Mathematical Physics. IV: Analysis of Operators,", Academic Press, (1980).   Google Scholar

[16]

Stephen P. Shipman, Resonant scattering by open periodic waveguides,, in, (2010).   Google Scholar

[17]

Stephen P. Shipman, Jennifer Ribbeck, Katherine H. Smith and Clayton Weeks, A discrete model for resonance near embedded bound states,, IEEE Photonics J., 2 (2010), 911.   Google Scholar

[18]

Stephen P. Shipman and Stephanos Venakides, Resonance and bound states in photonic crystal slabs,, SIAM J. Appl. Math., 64 (2003), 322.  doi: 10.1137/S0036139902411120.  Google Scholar

[19]

Stephen P. Shipman and Stephanos Venakides, Resonant transmission near non-robust periodic slab modes,, Phys. Rev. E, 71 (2005).   Google Scholar

[20]

Stephen P. Shipman and Darko Volkov, Guided modes in periodic slabs: Existence and nonexistence,, SIAM J. Appl. Math., 67 (2007), 687.  doi: 10.1137/050647189.  Google Scholar

[21]

Sergei G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius and Teruya Ishihara, Quasiguided modes and optical properties of photonic crystal slabs,, Phys. Rev. B, 66 (2002).   Google Scholar

[22]

V. Weisskopf and E. Wigner, Berechnung der natürlichen linienbreite auf grund der diracschen lichttheorie,, Zeitschrift für Physik, 63 (1930), 54.  doi: 10.1007/BF01336768.  Google Scholar

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