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 and 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), 305-338.

[2]

Shanhui Fan and J. D. Joannopoulos, Analysis of guided resonances in photonic crystal slabs, Phys. Rev. B, 65 (2002), 23511, 8 pp.

[3]

U. Fano, Effects of configuration interaction on intensities and phase shifts, Phys. Rev., 124 (1961), 1866-1878. doi: 10.1103/PhysRev.124.1866.

[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), 1123-1130.

[5]

Alexander Figotin and Jeffrey H. Schenker, Spectral theory of time dispersive and dissipative systems, J. Stat. Phys., 118 (2005), 199-263. doi: 10.1007/s10955-004-8783-7.

[6]

Alexander Figotin and Stephen P. Shipman, Open systems viewed through their conservative extensions, J. Stat. Phys., 125 (2006), 363-413.

[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), 310-318. doi: 10.1109/JQE.1986.1072956.

[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), 1438-1440. doi: 10.1063/1.118570.

[9]

Haitao Liu and Philippe Lalanne, Microscopic theory of the extraordinary optical transmission, Lett. Nature, 452 (2008), 728-731. doi: 10.1038/nature06762.

[10]

S. Longhi, Bound states in the continuum in a single-level Fano-Anderson model, Eur. Phys. J. B, 57 (2007), 45-51. doi: 10.1140/epjb/e2007-00143-2.

[11]

G. D. Mahan, "Many-Particle Physics," Second edition, Physics of Solids and Liquids, Plenum Press, New York, 1990.

[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), 036626, 8 pp.

[13]

P. Paddon and Jeff F. Young, Two-dimensional vector-coupled-mode theory for textured planar waveguides, Phys. Rev. B, 61 (2000), 2090-2101. doi: 10.1103/PhysRevB.61.2090.

[14]

S. T. Peng, T. Tamir and H. L. Bertoni, Theory of periodic dielectric waveguides, IEEE Trans. Microwave Th. and Tech., 23 (1975), 123-133. doi: 10.1109/TMTT.1975.1128513.

[15]

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

[16]

Stephen P. Shipman, Resonant scattering by open periodic waveguides, in "Progress in Computational Physics" (ed. M. Ehrhardt), Vol. I, Bentham Sci. Pub., 2010.

[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), 911-923.

[18]

Stephen P. Shipman and Stephanos Venakides, Resonance and bound states in photonic crystal slabs, SIAM J. Appl. Math., 64 (2003), 322-342. doi: 10.1137/S0036139902411120.

[19]

Stephen P. Shipman and Stephanos Venakides, Resonant transmission near non-robust periodic slab modes, Phys. Rev. E, 71 (2005), 026611, 10 pp.

[20]

Stephen P. Shipman and Darko Volkov, Guided modes in periodic slabs: Existence and nonexistence, SIAM J. Appl. Math., 67 (2007), 687-713. doi: 10.1137/050647189.

[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), 045102, 17 pp.

[22]

V. Weisskopf and E. Wigner, Berechnung der natürlichen linienbreite auf grund der diracschen lichttheorie, Zeitschrift für Physik, 63 (1930), 54-73. doi: 10.1007/BF01336768.

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), 305-338.

[2]

Shanhui Fan and J. D. Joannopoulos, Analysis of guided resonances in photonic crystal slabs, Phys. Rev. B, 65 (2002), 23511, 8 pp.

[3]

U. Fano, Effects of configuration interaction on intensities and phase shifts, Phys. Rev., 124 (1961), 1866-1878. doi: 10.1103/PhysRev.124.1866.

[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), 1123-1130.

[5]

Alexander Figotin and Jeffrey H. Schenker, Spectral theory of time dispersive and dissipative systems, J. Stat. Phys., 118 (2005), 199-263. doi: 10.1007/s10955-004-8783-7.

[6]

Alexander Figotin and Stephen P. Shipman, Open systems viewed through their conservative extensions, J. Stat. Phys., 125 (2006), 363-413.

[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), 310-318. doi: 10.1109/JQE.1986.1072956.

[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), 1438-1440. doi: 10.1063/1.118570.

[9]

Haitao Liu and Philippe Lalanne, Microscopic theory of the extraordinary optical transmission, Lett. Nature, 452 (2008), 728-731. doi: 10.1038/nature06762.

[10]

S. Longhi, Bound states in the continuum in a single-level Fano-Anderson model, Eur. Phys. J. B, 57 (2007), 45-51. doi: 10.1140/epjb/e2007-00143-2.

[11]

G. D. Mahan, "Many-Particle Physics," Second edition, Physics of Solids and Liquids, Plenum Press, New York, 1990.

[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), 036626, 8 pp.

[13]

P. Paddon and Jeff F. Young, Two-dimensional vector-coupled-mode theory for textured planar waveguides, Phys. Rev. B, 61 (2000), 2090-2101. doi: 10.1103/PhysRevB.61.2090.

[14]

S. T. Peng, T. Tamir and H. L. Bertoni, Theory of periodic dielectric waveguides, IEEE Trans. Microwave Th. and Tech., 23 (1975), 123-133. doi: 10.1109/TMTT.1975.1128513.

[15]

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

[16]

Stephen P. Shipman, Resonant scattering by open periodic waveguides, in "Progress in Computational Physics" (ed. M. Ehrhardt), Vol. I, Bentham Sci. Pub., 2010.

[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), 911-923.

[18]

Stephen P. Shipman and Stephanos Venakides, Resonance and bound states in photonic crystal slabs, SIAM J. Appl. Math., 64 (2003), 322-342. doi: 10.1137/S0036139902411120.

[19]

Stephen P. Shipman and Stephanos Venakides, Resonant transmission near non-robust periodic slab modes, Phys. Rev. E, 71 (2005), 026611, 10 pp.

[20]

Stephen P. Shipman and Darko Volkov, Guided modes in periodic slabs: Existence and nonexistence, SIAM J. Appl. Math., 67 (2007), 687-713. doi: 10.1137/050647189.

[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), 045102, 17 pp.

[22]

V. Weisskopf and E. Wigner, Berechnung der natürlichen linienbreite auf grund der diracschen lichttheorie, Zeitschrift für Physik, 63 (1930), 54-73. doi: 10.1007/BF01336768.

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