Abstract

In this paper, we review various numerical methods currently used to model nonlinear optical processes in nanophotonics. Among the different theoretical frameworks that have been used to study nonlinear photonic structures, we particularly focus on the application of both perturbation theory and coupled-mode theory to the analysis of complex nonlinear nanophotonic devices. This description is illustrated on several examples of how these techniques can be used to design photonic-crystal-based nonlinear devices. In addition, in all these examples, we show that the predictions made by the two mentioned techniques are in a good agreement with the numerical results obtained from a nonlinear finite-difference-time-domain approach to these problems.

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2005 (3)

2004 (2)

W. Suh, Z. Wang, S. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).

M. Soljačić, J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nat. Mater. 3, 211-219 (2004).

2003 (5)

M. Soljačić, C. Luo, J. D. Joannopoulos, S. Fan, "Nonlinear photonic crystal microdevices for optical integration," Opt. Lett. 28, 637-639 (2003).

M. Soljačić, M. Ibanescu, S. G. Johnson, J. D. Joannopoulos, Y. Fink, "Optical bistability in axially modulated OmniGuide fibers," Opt. Lett. 28, 516-518 (2003).

M. F. Yanik, S. Fan, M. Soljačić, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).

M. F. Yanik, S. Fan, M. Soljačić, J. D. Joannopoulos, "All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry," Opt. Lett. 28, 2506-2508 (2003).

A. R. Cowan, J. F. Young, "Optical bistability involving photonic crystal microcavities and Fano line shapes," Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 68, 046 606 (2003).

2002 (6)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 66, 055601R (2002).

C. M. Bowden, A. M. Zheltikov, "Nonlinear optics of photonic crystals," J. Opt. Soc. Amer. B, Opt. Phys. 19, 2046-2048 (2002).

A. R. Cowan, J. F. Young, "Mode matching for second-harmonic generation in photonic crystal waveguides," Phys. Rev. B, Condens. Matter 65, 085106 (2002).

S. F. Mingaleev, Y. S. Kivshar, "Nonlinear transmission and light localization in photonic-crystal waveguides," J. Opt. Soc. Amer. A, Opt. Image Sci. 19, 2241-2249 (2002).

M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Amer. B, Opt. Phys. 19, 2052-2059 (2002).

S. Mookherjea, A. Yariv, "Second-harmonic generation with pulses in a coupled-resonator optical waveguide," Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 65, 026607 (2002).

2001 (4)

E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, J. D. Joannopoulos, "Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at lambda = 1.55 mu m wavelengths," Opt. Lett. 26, 286-288 (2001).

M. L. Povinelli, S. G. Johnson, S. Fan, J. D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B, Condens. Matter 64, 0753131 (2001).

S. G. Johnson, J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis ," Opt. Express 8, 173-190 (2001).

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, M. Bertolotti, "Photonic band edge effects in finite structures and applications to chi(2) interactions ," Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 64, 16609 (2001).

2000 (4)

E. Centeno, D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B, Condens. Matter 62, R7683-R7686 (2000).

M. Bayindir, B. Temelkuran, E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals ," Phys. Rev. Lett. 84, 2140 (2000).

S. Noda, K. Tomoda, N. Yamamoto, A. Chutinan, "Full three-dimensional photonic bandgap crystals at near-infrared wavelengths," Science 289, 604-606 (2000).

Y. Xu, R. K. Lee, A. Yariv, "Quantum analysis and the classical analysis of spontaneous emission in a microcavity ," Phys. Rev. A, Gen. Phys. 61, 33 808 (2000).

1999 (1)

1998 (2)

S. Fan, P. R. Villenueve, J. D. Joannopoulos, H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).

1997 (1)

H. Karlsson, F. Laurell, "Electric field poling of flux grown KTiOPO4," Appl. Phys. Lett. 71, 3474-3476 (1997).

1996 (2)

S. J. B. Yoo, C. Caneau, R. Bhat, M. A. Koza, A. Rajhel, N. Antoniades, "Wavelength conversion by difference frequency generation in AlGaAs waveguides with periodic domain inversion achieved by wafer bonding," Appl. Phys. Lett. 68, 2609-2611 (1996).

K. Sakoda, K. Ohtaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B, Condens. Matter 54, 5742-5749 (1996).

1995 (2)

1992 (1)

M. M. Fejer, G. A. Magel, H. D. Jundt, R. L. Byer, "Quasi-phase-matched 2nd harmonic-generation—Tuning and tolerances," IEEE J. Quantum Electron. 28, 2631-2654 (1992).

1989 (3)

D. N. Christodoulides, R. I. Joseph, "Slow Bragg solitons in nonlinear periodic structures," Phys. Rev. Lett. 62, 1746-1749 (1989).

A. B. Aceves, S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media," Phys. Lett. A 141, 37-42 (1989).

C. M. de Sterke, J. E. Sipe, "Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures," Phys. Rev. A, Gen. Phys. 39, 5163-5178 (1989).

1988 (1)

1987 (4)

W. Chen, D. L. Mills, "Gap solitons and the nonlinear optical-response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).

D. L. Mills, S. E. Trullinger, "Gap solitons in nonlinear periodic structures," Phys. Rev. B, Condens. Matter 36, 947-952 (1987).

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).

1979 (1)

H. G. Winful, J. H. Marburger, E. Garmire, "Theory of bistability in non-linear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).

1971 (1)

H. Kogelnik, C. V. Shank, "Stimulated emission in a periodic structure," Appl. Phys. Lett. 18, 152-154 (1971).

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, G. Weinreich, "Generation of optical harmonics," Phys. Rev. Lett. 7, 118-199 (1961).

Appl. Phys. Lett. (5)

S. J. B. Yoo, C. Caneau, R. Bhat, M. A. Koza, A. Rajhel, N. Antoniades, "Wavelength conversion by difference frequency generation in AlGaAs waveguides with periodic domain inversion achieved by wafer bonding," Appl. Phys. Lett. 68, 2609-2611 (1996).

H. Karlsson, F. Laurell, "Electric field poling of flux grown KTiOPO4," Appl. Phys. Lett. 71, 3474-3476 (1997).

H. Kogelnik, C. V. Shank, "Stimulated emission in a periodic structure," Appl. Phys. Lett. 18, 152-154 (1971).

H. G. Winful, J. H. Marburger, E. Garmire, "Theory of bistability in non-linear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).

M. F. Yanik, S. Fan, M. Soljačić, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).

IEEE J. Quantum Electron. (2)

W. Suh, Z. Wang, S. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).

M. M. Fejer, G. A. Magel, H. D. Jundt, R. L. Byer, "Quasi-phase-matched 2nd harmonic-generation—Tuning and tolerances," IEEE J. Quantum Electron. 28, 2631-2654 (1992).

J. Opt. Soc. Amer. A, Opt. Image Sci. (1)

S. F. Mingaleev, Y. S. Kivshar, "Nonlinear transmission and light localization in photonic-crystal waveguides," J. Opt. Soc. Amer. A, Opt. Image Sci. 19, 2241-2249 (2002).

J. Opt. Soc. Amer. B, Opt. Phys. (3)

B. Maes, P. Bienstman, R. Baets, "Modeling second-harmonic generation by use of mode expansion," J. Opt. Soc. Amer. B, Opt. Phys. 22, 1378-1383 (2005).

C. M. Bowden, A. M. Zheltikov, "Nonlinear optics of photonic crystals," J. Opt. Soc. Amer. B, Opt. Phys. 19, 2046-2048 (2002).

M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Amer. B, Opt. Phys. 19, 2052-2059 (2002).

Nat. Mater. (1)

M. Soljačić, J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nat. Mater. 3, 211-219 (2004).

Opt. Express (3)

Opt. Lett. (8)

Phys. Lett. A (1)

A. B. Aceves, S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media," Phys. Lett. A 141, 37-42 (1989).

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).

Phys. Rev. A, Gen. Phys. (2)

C. M. de Sterke, J. E. Sipe, "Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures," Phys. Rev. A, Gen. Phys. 39, 5163-5178 (1989).

Y. Xu, R. K. Lee, A. Yariv, "Quantum analysis and the classical analysis of spontaneous emission in a microcavity ," Phys. Rev. A, Gen. Phys. 61, 33 808 (2000).

Phys. Rev. B, Condens. Matter (5)

D. L. Mills, S. E. Trullinger, "Gap solitons in nonlinear periodic structures," Phys. Rev. B, Condens. Matter 36, 947-952 (1987).

M. L. Povinelli, S. G. Johnson, S. Fan, J. D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B, Condens. Matter 64, 0753131 (2001).

K. Sakoda, K. Ohtaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B, Condens. Matter 54, 5742-5749 (1996).

E. Centeno, D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B, Condens. Matter 62, R7683-R7686 (2000).

A. R. Cowan, J. F. Young, "Mode matching for second-harmonic generation in photonic crystal waveguides," Phys. Rev. B, Condens. Matter 65, 085106 (2002).

Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. (4)

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, M. Bertolotti, "Photonic band edge effects in finite structures and applications to chi(2) interactions ," Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 64, 16609 (2001).

S. Mookherjea, A. Yariv, "Second-harmonic generation with pulses in a coupled-resonator optical waveguide," Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 65, 026607 (2002).

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 66, 055601R (2002).

A. R. Cowan, J. F. Young, "Optical bistability involving photonic crystal microcavities and Fano line shapes," Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 68, 046 606 (2003).

Phys. Rev. Lett. (8)

S. Fan, P. R. Villenueve, J. D. Joannopoulos, H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).

M. Bayindir, B. Temelkuran, E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals ," Phys. Rev. Lett. 84, 2140 (2000).

D. N. Christodoulides, R. I. Joseph, "Slow Bragg solitons in nonlinear periodic structures," Phys. Rev. Lett. 62, 1746-1749 (1989).

W. Chen, D. L. Mills, "Gap solitons and the nonlinear optical-response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).

P. A. Franken, A. E. Hill, C. W. Peters, G. Weinreich, "Generation of optical harmonics," Phys. Rev. Lett. 7, 118-199 (1961).

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).

Science (1)

S. Noda, K. Tomoda, N. Yamamoto, A. Chutinan, "Full three-dimensional photonic bandgap crystals at near-infrared wavelengths," Science 289, 604-606 (2000).

Other (9)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

H. Gibbs, Optical Bistability: Controlling Light With Light (Academic, 1985).

R. W. Boyd, Nonlinear Optics (Academic, 1992).

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ., 1995).

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).

S. G. Johnson, Photonic Crystals: The Road From Theory to Practice (Kluwer, 2002).

R. E. Slusher, B. J. Eggleton, Nonlinear Photonic Crystals (Springer-Verlag, 2003).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

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