Abstract

Frequency conversion in second-order nonlinear materials is sensitive to the phase velocities of interacting optical waves. Accurate modeling of such problems with the finite-difference time-domain method requires extremely fine grid resolutions to minimize numerical dispersion errors. We propose an alternative approach based on a pseudospectral time-domain (PSTD) method for solving the nonlinear Maxwell’s equations. Low-dispersion PSTD schemes with second- and fourth-order time stepping are developed and investigated. Benchmark simulations of second-harmonic generation (SHG) demonstrate that the PSTD schemes offer significant improvements in computational efficiency and accuracy. We demonstrate use of these schemes by modeling SHG in a nonlinear grating illuminated at an oblique angle, where phase matching is achieved in two dimensions.

© 2004 Optical Society of America

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    [CrossRef] [PubMed]

2001 (1)

T. W. Lee, S. C. Hagness, D. L. Zhou, and L. J. Mawst, “Modal characteristics of ARROW-type vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 13, 770–772 (2001).
[CrossRef]

2000 (1)

1999 (7)

D. W. Prather and S. Y. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
[CrossRef]

Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudo-spectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Remote Sens. 37, 917–926 (1999).
[CrossRef]

S. T. Yang and S. P. Velsko, “Frequency-agile kilohertz repetition-rate optical parametric oscillator based on periodically poled lithium niobate,” Opt. Lett. 24, 133–135 (1999).
[CrossRef]

H. F. Chou, C. F. Lin, and S. Mou, “Comparisons of finite difference beam propagation methods for modeling second-order nonlinear effects,” J. Lightwave Technol. 17, 1481–1486 (1999).
[CrossRef]

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75, 1036–1038 (1999).
[CrossRef]

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31, 843–855 (1999).
[CrossRef]

1998 (5)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

B. D’Urso, O. Painter, J. O’Brien, T. Tombrello, A. Yariv, and A. Scherer, “Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities,” J. Opt. Soc. Am. B 15, 1155–1159 (1998).
[CrossRef]

G. W. Ross, M. Pollnau, P. G. R. Smith, W. A. Clarkson, P. E. Britton, and D. C. Hanna, “Generation of high-power blue light in periodically poled LiNbO3,” Opt. Lett. 23, 171–173 (1998).
[CrossRef]

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transform (NUFFT),” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

1997 (5)

R. W. Ziolkowski, “The incorporation of microscopic mate-rial models into the FDTD approach for ultrashort opticalpulse simulation,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
[CrossRef]

Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158–165 (1997).
[CrossRef]

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

G. Y. Wang, J. Zhao, Q. Chen, and M. Cronin-Golomb, “Widely tunable efficient intracavity quasiphase-matched midinfrared generation,” Appl. Phys. Lett. 70, 2218–2220 (1997).
[CrossRef]

1996 (2)

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach-Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

1995 (3)

H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quantum Electron. 31, 2107–2113 (1995).
[CrossRef]

R. X. Bian, R. C. Dunn, and X. S. Xie, “Single molecule emission characteristics in near-field microscopy,” Phys. Lett. A 75, 4772–4775 (1995).
[CrossRef]

D. M. Sullivan, “Nonlinear FDTD formulation using Z transforms,” IEEE Trans. Microwave Theory Tech. 43, 676–682 (1995).
[CrossRef]

1993 (2)

R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186–198 (1993).
[CrossRef]

P. S. Weitzman and U. Osterberg, “A modified beam propagation method to model second harmonic generation in optical fibers,” IEEE J. Quantum Electron. 29, 1437–1443 (1993).
[CrossRef]

1992 (1)

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

1991 (2)

1984 (1)

B. Hermansson, D. Yevick, and L. Thylen, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Arnold, J. M.

H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quantum Electron. 31, 2107–2113 (1995).
[CrossRef]

Babic, D. I.

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Baek, Y.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach-Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

Barness, D. C.

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Baumann, I.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach-Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

Bhat, R.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75, 1036–1038 (1999).
[CrossRef]

Bian, R. X.

R. X. Bian, R. C. Dunn, and X. S. Xie, “Single molecule emission characteristics in near-field microscopy,” Phys. Lett. A 75, 4772–4775 (1995).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Boroditsky, M.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75, 1036–1038 (1999).
[CrossRef]

Bourgeade, A.

Brener, I.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

Britton, P. E.

Chaban, E. E.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Chen, Q.

G. Y. Wang, J. Zhao, Q. Chen, and M. Cronin-Golomb, “Widely tunable efficient intracavity quasiphase-matched midinfrared generation,” Appl. Phys. Lett. 70, 2218–2220 (1997).
[CrossRef]

Chou, H. F.

Chou, M. H.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

Christman, S. B.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Chuang, S. L.

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Clarkson, W. A.

Coccioli, R.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75, 1036–1038 (1999).
[CrossRef]

Corzine, S. W.

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Cronin-Golomb, M.

G. Y. Wang, J. Zhao, Q. Chen, and M. Cronin-Golomb, “Widely tunable efficient intracavity quasiphase-matched midinfrared generation,” Appl. Phys. Lett. 70, 2218–2220 (1997).
[CrossRef]

D’Urso, B.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Dunn, R. C.

R. X. Bian, R. C. Dunn, and X. S. Xie, “Single molecule emission characteristics in near-field microscopy,” Phys. Lett. A 75, 4772–4775 (1995).
[CrossRef]

Fan, S. H.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Fejer, M. M.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Freysz, E.

Goorjian, P. M.

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Hagness, S. C.

T. W. Lee, S. C. Hagness, D. L. Zhou, and L. J. Mawst, “Modal characteristics of ARROW-type vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 13, 770–772 (2001).
[CrossRef]

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
[CrossRef] [PubMed]

Hanna, D. C.

Haus, H. A.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Hayata, K.

Hermansson, B.

B. Hermansson, D. Yevick, and L. Thylen, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

Ho, S. T.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Joseph, R. M.

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
[CrossRef]

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
[CrossRef] [PubMed]

Judkins, J. B.

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Koshiba, M.

Krauss, T. F.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75, 1036–1038 (1999).
[CrossRef]

Krijnen, G.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach-Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Lee, T. W.

T. W. Lee, S. C. Hagness, D. L. Zhou, and L. J. Mawst, “Modal characteristics of ARROW-type vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 13, 770–772 (2001).
[CrossRef]

Lin, C. F.

Little, B. E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Liu, G.

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Liu, Q. H.

Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudo-spectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Remote Sens. 37, 917–926 (1999).
[CrossRef]

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transform (NUFFT),” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158–165 (1997).
[CrossRef]

Masoudi, H. M.

H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quantum Electron. 31, 2107–2113 (1995).
[CrossRef]

Mawst, L. J.

T. W. Lee, S. C. Hagness, D. L. Zhou, and L. J. Mawst, “Modal characteristics of ARROW-type vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 13, 770–772 (2001).
[CrossRef]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Mou, S.

Nguyen, N.

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transform (NUFFT),” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

O’Brien, J.

Osterberg, U.

P. S. Weitzman and U. Osterberg, “A modified beam propagation method to model second harmonic generation in optical fibers,” IEEE J. Quantum Electron. 29, 1437–1443 (1993).
[CrossRef]

Painter, O.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Pollnau, M.

Prather, D. W.

Rafizadeh, D.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

Ross, G. W.

Scherer, A.

Schiek, R.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach-Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

Seurin, J. F.

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Shi, S. Y.

Smith, P. G. R.

Sohler, W.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach-Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

Stegeman, G. I.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach-Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Sullivan, D. M.

D. M. Sullivan, “Nonlinear FDTD formulation using Z transforms,” IEEE Trans. Microwave Theory Tech. 43, 676–682 (1995).
[CrossRef]

Taflove, A.

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
[CrossRef]

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
[CrossRef] [PubMed]

Tan, M.

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Tanaka, M.

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31, 843–855 (1999).
[CrossRef]

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Thylen, L.

B. Hermansson, D. Yevick, and L. Thylen, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

Tiouririne, T. N.

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Tombrello, T.

Velsko, S. P.

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Vrijen, R.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75, 1036–1038 (1999).
[CrossRef]

Wang, G. Y.

G. Y. Wang, J. Zhao, Q. Chen, and M. Cronin-Golomb, “Widely tunable efficient intracavity quasiphase-matched midinfrared generation,” Appl. Phys. Lett. 70, 2218–2220 (1997).
[CrossRef]

Weitzman, P. S.

P. S. Weitzman and U. Osterberg, “A modified beam propagation method to model second harmonic generation in optical fibers,” IEEE J. Quantum Electron. 29, 1437–1443 (1993).
[CrossRef]

Xie, X. S.

R. X. Bian, R. C. Dunn, and X. S. Xie, “Single molecule emission characteristics in near-field microscopy,” Phys. Lett. A 75, 4772–4775 (1995).
[CrossRef]

Yablonovitch, E.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75, 1036–1038 (1999).
[CrossRef]

Yang, S. T.

Yariv, A.

Yevick, D.

B. Hermansson, D. Yevick, and L. Thylen, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

Zhao, J.

G. Y. Wang, J. Zhao, Q. Chen, and M. Cronin-Golomb, “Widely tunable efficient intracavity quasiphase-matched midinfrared generation,” Appl. Phys. Lett. 70, 2218–2220 (1997).
[CrossRef]

Zhou, D. L.

T. W. Lee, S. C. Hagness, D. L. Zhou, and L. J. Mawst, “Modal characteristics of ARROW-type vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 13, 770–772 (2001).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31, 843–855 (1999).
[CrossRef]

R. W. Ziolkowski, “The incorporation of microscopic mate-rial models into the FDTD approach for ultrashort opticalpulse simulation,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186–198 (1993).
[CrossRef]

Appl. Phys. Lett. (4)

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach-Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

G. Y. Wang, J. Zhao, Q. Chen, and M. Cronin-Golomb, “Widely tunable efficient intracavity quasiphase-matched midinfrared generation,” Appl. Phys. Lett. 70, 2218–2220 (1997).
[CrossRef]

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals,” Appl. Phys. Lett. 75, 1036–1038 (1999).
[CrossRef]

G. Liu, J. F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barness, and T. N. Tiouririne, “Mode selectivity study of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

IEEE J. Quantum Electron. (3)

P. S. Weitzman and U. Osterberg, “A modified beam propagation method to model second harmonic generation in optical fibers,” IEEE J. Quantum Electron. 29, 1437–1443 (1993).
[CrossRef]

H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quantum Electron. 31, 2107–2113 (1995).
[CrossRef]

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

IEEE Microwave Guid. Wave Lett. (1)

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transform (NUFFT),” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 11, 653–655 (1999).
[CrossRef]

T. W. Lee, S. C. Hagness, D. L. Zhou, and L. J. Mawst, “Modal characteristics of ARROW-type vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 13, 770–772 (2001).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filiters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
[CrossRef]

R. W. Ziolkowski, “The incorporation of microscopic mate-rial models into the FDTD approach for ultrashort opticalpulse simulation,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudo-spectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Remote Sens. 37, 917–926 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

D. M. Sullivan, “Nonlinear FDTD formulation using Z transforms,” IEEE Trans. Microwave Theory Tech. 43, 676–682 (1995).
[CrossRef]

J. Lightwave Technol. (2)

H. F. Chou, C. F. Lin, and S. Mou, “Comparisons of finite difference beam propagation methods for modeling second-order nonlinear effects,” J. Lightwave Technol. 17, 1481–1486 (1999).
[CrossRef]

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (4)

Microwave Opt. Technol. Lett. (1)

Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158–165 (1997).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (2)

B. Hermansson, D. Yevick, and L. Thylen, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31, 843–855 (1999).
[CrossRef]

Phys. Lett. A (1)

R. X. Bian, R. C. Dunn, and X. S. Xie, “Single molecule emission characteristics in near-field microscopy,” Phys. Lett. A 75, 4772–4775 (1995).
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Phys. Rev. Lett. (1)

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Other (5)

W. K. Leung, Y. Chen, and R. Mittra, “Transformed-space non-uniform pseudo-spectral time-domain (NU-PSTD) algorithm without use of non-uniform FFT,” in IEEE AP-S International Symposium (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2001), pp. 498–501.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 2000).

T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Wireless Propag. Lett. (to be published).

J. Fang, “Time-domain finite-difference computations for Maxwell’s equations,” Ph.D. dissertation (Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, Calif., 1989).

B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge U. Press, Cambridge, UK, 1996).

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Figures (10)

Fig. 1
Fig. 1

Comparison of the free-space numerical phase velocity errors associated with the PSTD-2 and PSTD-4 schemes.

Fig. 2
Fig. 2

Pulse propagation in a 1-D free-space grid with periodic boundary conditions. The electric field data are plotted after 10,000 time steps. The arrow indicates the direction of propagation.

Fig. 3
Fig. 3

Percent error in the numerical propagation constant computed with 1-D PSTD simulations of wave propagation in free space. (a) In the PSTD-2 simulation, a decrease in the time step by 2:1 reduces the error by 4:1. (b) In the PSTD-4 simulation, a decrease in the time step by 2:1 reduces the error by 16:1.

Fig. 4
Fig. 4

Power density of the fundamental and higher harmonics as a function of propagation distance in a 1-D nondispersive second-order nonlinear material simulated with PSTD-4.

Fig. 5
Fig. 5

Power density of the second harmonic as a function of propagation distance in a nondispersive second-order nonlinear material. (a) PSTD-4 results with various Δt, (b) PSTD-2 results with various Δt, (c) FDTD results with various grid-sampling densities.

Fig. 6
Fig. 6

Cross-sectional profile of the dielectric constant of the 2-D waveguide. The solid line represents the profile of the actual waveguide modeled in the test simulation.

Fig. 7
Fig. 7

Convergence tests for (a) PSTD-2 and (b) PSTD-4. The power density of the second-harmonic wave is computed from PSTD simulations with different time steps.

Fig. 8
Fig. 8

Convergence test for FDTD. The power density of the second-harmonic wave is computed from FDTD simulations with different grid resolutions. The FDTD grid cell size is expressed as a fraction of the PSTD-4 grid cell size.

Fig. 9
Fig. 9

Schematic diagram of the phase-matching condition for second-harmonic generation in a QPM grating structure oriented at an oblique angle with respect to the direction of propagation of the incident fundamental wave.

Fig. 10
Fig. 10

(a) Power density and (b) direction of power flow of the second harmonic generated in three different QPM gratings, as a function of distance propagated by the fundamental wave.

Tables (1)

Tables Icon

Table 1 Number of FFT Subroutine Calls Required per Grid Cell per Time Step

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

ψxiFx-1[-jkxFx(ψ)]|i,
H|i,j,kn+1/2=H|i,j,kn-1/2-Δtμ(×E)|i,j,kn,
E|i,j,kn+1=E|i,j,kn+Δt(×H)|i,j,kn+1/2,
Hu|i,j,kn+1/2=Hu|i,j,kn-1/2-Δtμi,j,k{Fv-1[-jkvFv(Ew)]|i,j,kn-Fw-1[-jkwFw(Ev)]|i,j,kn},
Eu|i,j,kn+1=Eu|i,j,kn+Δti,j,k{Fv-1[-jkvFv(Hw)]|i,j,kn+1/2-Fw-1[-jkwFw(Hv)]|i,j,kn+1/2},
ψtn=ψ|n+1/2-ψ|n-1/2Δt-Δt2243ψt3n+O(Δt4).
H|i,j,kn+1/2=H|i,j,kn-1/2-Δtμ(×E)|i,j,kn+Δt3243Ht3i,j,kn,
E|i,j,kn+1=E|i,j,kn+Δt(×H)|i,j,kn+1/2+Δt3243Et3i,j,kn+1/2.
3Ht3i,j,kn=-1μ×1×-1μ×Ei,j,kn,
3Et3i,j,kn+1/2=1×-1μ×1×Hi,j,kn+1/2.
H|i,j,kn+1/2=H|i,j,kn-1/2-Δtμ(×E)|i,j,kn+Δt324v2μ×××Ei,j,kn,
E|i,j,kn+1=E|i,j,kn+Δt(×H)|i,j,kn+1/2-Δt324v2×××Hi,j,kn+1/2,
Hy|in+1/2=Hy|in-1/2+ΔtμiFx-1[-jkxFx(Ez)]|in+vi2Δt224Fx-1[jkx3Fx(Ez)]|in,
Ez|in+1=Ez|in+ΔtiFx-1[-jkxFx(Hy)]|in+1/2+vi2Δt224Fx-1[jkx3Fx(Hy)]|in+1/2.
Hx|i,jn+1/2=Hx|i,jn-1/2-Δtμi,jFy-1[-jkyFy(Ez)]|i,jn+vi,j2Δt224 [Fx-1(-kx2Fx{Fy-1×[-jkyFy(Ez)]})|i,jn+Fy-1[jky3Fy(Ez)]|i,jn],
Hy|i,jn+1/2=Hy|i,jn-1/2+Δtμi,jFx-1[-jkxFx(Ez)]|i,jn+vi,j2Δt224 [Fy-1(-ky2Fy{Fx-1×[-jkxFx(Ez)]})|i,jn+Fx-1[jkx3Fx(Ez)]|i,jn],
Ez|i,jn+1=Ez|i,jn+Δti,jFx-1[-jkxFx(Hy)]|i,jn+1/2-Fy-1[-jkyFy(Hx)]|i,jn+1/2+vi,j2Δt224 [2Fy-1(-ky2Fy{Fx-1×[-jkxFx(Hy)]})|i,jn+1/2+Fx-1[jkx3Fx(Hy)]|i,jn+1/2-Fy-1[jky3Fy(Hx)]|i,jn+1/2],
Hx|i,j,kn+1/2=Hx|i,j,kn-1/2-Δtμi,j,k×Ezy|i,j,kn-Eyz|i,j,kn+vi,j,k2Δt224{Fy-1[jky3Fy(Ez)]|i,j,kn-Fz-1[jkz3Fz(Ey)]|i,j,kn+Fx-1[-kx2Fx(Ezy)]|i,j,kn+Fz-1[-kz2Fz(Ezy)]|i,j,kn-Fx-1[-kx2Fx(Eyz)]|i,j,kn-Fy-1[-ky2Fy(Eyz)]|i,j,kn},
Ex|i,j,kn+1=Ex|i,j,kn-Δti,j,k×Hzy|i,j,kn+1/2-Hyz|i,j,kn+1/2+vi,j,k2Δt224{Fy-1[jky3Fy(Hz)]|i,j,kn+1/2-Fz-1[jkz3Fz(Hy)]|i,j,kn+1/2+Fx-1[-kx2Fx(Hzy)]|i,j,kn+1/2+Fz-1[-kz2Fz(Hzy)]|i,j,kn+1/2-Fx-1[-kx2Fx(Hyz)]|i,j,kn+1/2-Fy-1[-ky2Fy(Hyz)]|i,j,kn+1/2},
k˜=2cΔt sinωΔt2.
k˜1-c2Δt224k˜2=2cΔt sinωΔt2.
Δts=2Δxπc2.
D=0rE+0χ(2)E2,
E=D0[r+χ(2)E].
E=-0r+[(0r)2+40χ(2)D]1/220χ(2).
3Hyt3=v2μ3Ezx3-χ(2)μr3(Ez2)xt2,
3Dzt3=v2 3Hyx3-χ(2)μr3(Ez2)x2t.
Δts=2(rclad)1/2πc(Δx-2+Δy-2)1/2.
Lc=π|2βf-βs|=λ0f4|nefff-neffs|,
Δts=2(rclad)1/2πc(Δx-2+Δy-2)1/2,
2kf+kg=ks.
θ=2ϕ,
d=λ0f4n sin ϕ.

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