Abstract

The split-field finite-difference time-domain (SF-FDTD) method for one-dimensionally periodic structures is extended to include the coefficient-tensor description of second- and third-order nonlinear-optical media. A set of nonlinear equations related to the split-field values of the electric field is established. An iterative fixed-point approach for solving the coupled nonlinear system of equations needed to update the electric field components in the SF-FDTD is then developed. The third-order nonlinear susceptibility dispersion is also considered by means of the Raman effect and its implementation in the SF-FDTD scheme. Different scenarios are considered in order to verify the reliability of the method for simulating second- and third-order nonlinear-optical media. First, second-harmonic generation and its efficiency are investigated in a homogeneous layer with and without the quasi-phase-matching technique. Second, the nonlinear dispersion is analyzed by means of the generation of solitons in Kerr media due to the Raman effect. Last, a set of binary phase gratings with nonlinear pillars is considered under oblique incidence. Here the nonlinear refractive index is generated by different physical mechanisms modeled with the nonscalar third-order susceptibility.

© 2013 Optical Society of America

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  2. G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012)457–492.
  3. M. Qasymeh, M. Cada, and S. A. Ponomarenko, “Quadratic electro-optic Kerr effect: applications to photonic devices,” IEEE J. Quantum Electron. 44, 740–746 (2008).
    [CrossRef]
  4. P. S. Balourdos, D. J. Frantzeskakis, M. C. Tsilis, and I. G. Tigelis, “Reflectivity of a nonlinear discontinuity in optical waveguides,” Pure Appl. Opt. 7, 1–11 (1998).
    [CrossRef]
  5. W. D. Deering and M. I. Molina, “Power switching in hybrid coherent couplers,” IEEE J. Quantum Electron. 33, 336–340 (1997).
    [CrossRef]
  6. Y. Wang, “Nonlinear optical limiter and digital optical switch by cascaded nonlinear couplers: analysis,” J. Lightwave Technol. 17, 292–297 (1999).
    [CrossRef]
  7. L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
    [CrossRef]
  8. S. I. Bozhevolny, K. Pedersen, T. Skettrup, X. Zhang, and M. Belmonte, “Far- and near-field second-harmonic imaging of ferroelectric domain walls,” Opt. Commun. 152, 221–224 (1998).
    [CrossRef]
  9. I. I. Smolyaninov, H. Y. Liang, C. H. Lee, C. C. Davis, V. Nagarajan, and R. Ramesh, “Near-field second harmonic imaging of the c/a/c/a polydomain structure of epitaxial pbzrxti1−xo3 thin films,” J. Microsc. 202, 250–254 (2001).
    [CrossRef]
  10. A. V. Zayats, T. Kalkbrenner, V. Sandoghdar, and J. Mlynek, “Second harmonic generation from individual surface defects under local excitation,” Phys. Rev. B 61, 4545–4548 (2000).
    [CrossRef]
  11. S. Takahashi and A. V. Zayats, “Near-field second-harmonic generation at a metal tip apex,” Appl. Phys. Lett 80, 3479–3481 (2002).
    [CrossRef]
  12. F. Zhou, Y. Liu, Z.-Y. Li, and Y. Xia, “Analytical model for optical bistability in nonlinear metal nano-antennae involving kerr materials,” Opt. Express 18, 13337–13344 (2010).
    [CrossRef]
  13. S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
    [CrossRef]
  14. S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
    [CrossRef]
  15. D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97, 041903 (2010).
    [CrossRef]
  16. W. T. Dong, L. Gao, and C. W. Qiu, “Goos–Hänchen shift at the surface of chiral negative refractive media,” Progress Electromagn. Res. 90, 255–268 (2009).
    [CrossRef]
  17. J. Yuan and J. Yang, “Computational design for efficient second-harmonic generation in nonlinear photonic crystals,” J. Opt. Soc. Am. B 30, 205–210 (2013).
    [CrossRef]
  18. L.-C. Zhao and J. Liu, “Localized nonlinear waves in a two-mode nonlinear fiber,” J. Opt. Soc. Am. B 29, 3119–3127 (2012).
    [CrossRef]
  19. K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
  20. J. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Microwave Theor. Tech. 46, 420–427 (1998).
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  21. J. G. Maloney and M. P. Kesler, “Analysis of antenna arrays using the split-field update FDTD method,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 1998), Vol. 4, pp. 420–427.
  22. S. M. Amjadi and M. Soleimani, “Design of band-pass waveguide filter using frequency selective surfaces loaded with surface mount capacitors based on split-field update FDTD method,” Progress Electromagn. Res. B 43, 271–281 (2008).
    [CrossRef]
  23. A. Belkhir and F. I. Baida, “Three-dimensional finite-difference time-domain algorithm for oblique incidence with adaptation of perfectly matched layers and nonuniform meshing: application to the study of a radar dome,” Phys. Rev. E 77, 056701 (2008).
    [CrossRef]
  24. C. Oh and M. J. Escuti, “Time-domain analysis of periodic anisotropic media at oblique incidence: An efficient FDTD implementation,” Opt. Express 14, 11870–11884 (2006).
    [CrossRef]
  25. F. I. Baida and A. Belkhir, “Split-field FDTD method for oblique incidence study of periodic dispersive metallic structures,” Opt. Lett. 34, 2453–2455 (2009).
    [CrossRef]
  26. A. Belkhir, O. Arar, S. S. Benabbes, O. Lamrous, and F. I. Baida, “Implementation of dispersion models in the split-field-finite-difference- time-domain algorithm for the study of metallic periodic structures at oblique incidence,” Phys. Rev. E 81, 046705 (2010).
    [CrossRef]
  27. A. Shahmansouri and B. Rashidian, “Comprehensive three-dimensional split-field finite-difference time-domain method for analysis of periodic plasmonic nanostructures: near- and far-field formulation,” J. Opt. Soc. Am. B 28, 2690–2700 (2011).
    [CrossRef]
  28. A. Shahmansouri and B. Rashidian, “GPU implementation of split-field finite-difference time-domain method for Drude-Lorentz dispersive media,” Progress Electromagn. Res. 125, 55–77 (2012).
    [CrossRef]
  29. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005), pp. 353–406.
  30. 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]
  31. 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]
  32. P. M. Goorjian and A. Taflove, “Direct time integration of maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).
    [CrossRef]
  33. K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Progress Electromagn. Res. 116, 441–456 (2011).
    [CrossRef]
  34. Y.-Q. Zhang and D. B. Ge, “A unified FDTD approach for electromagnetic analysis of dispersive objects,” Progress Electromagn. Res. 96, 155–172 (2009).
    [CrossRef]
  35. M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation,” IEEE J. Quantum Electron. 40, 175–182 (2004).
    [CrossRef]
  36. J. Francés, J. Tervo, and C. Neipp, “Split-field finite-difference time-domain scheme for Kerr-type nonlinear periodic media,” Progress Electromagn. Res. 134, 559–579 (2013).
  37. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
    [CrossRef]
  38. G. Zheng, A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, “Implementation of Mur’s absorbing boundaries with periodic structures to speed up the design process using finite-difference time-domain method,” Progress Electromagn. Res. 58, 101–114 (2006).
    [CrossRef]
  39. P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1990), pp. 12–149.
  40. H. M. Al-Mudhaffar, M. A. Alsunaidi, and H. M. Masoudi, “Full-wave solution of the second harmonic generation problem using a nonlinear FDTD algorithm,” in Progress in Electromagnics Research Symposium Proceedings, Prague, Czech Repulic (2007), pp. 479–482.
  41. M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photon. Technol. Lett. 5, 310–312 (2009).
    [CrossRef]
  42. M. A. Alsunaidi and F. S. Al-Hajiri, “Efficient NL-FDTD solution schemes for the phase-sensitive second harmonic generation problem,” J. Lightwave Technol. 27, 4964–4969 (2009).
    [CrossRef]
  43. C. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express 18, 21427–21448 (2010).
    [CrossRef]
  44. M. Ammann, “Non-trivial materials in EM-FDTD,” Master’s thesis (Department of Physics, Swiss Federal Institute of Technology, 2007).
  45. J. H. Greene and A. Taflove, “General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics,” Opt. Express 14, 8305–8310 (2006).
    [CrossRef]
  46. S. Malaguti, A. Armaroli, G. Bellanca, S. Trillo, S. Combrié, P. Colman, and A. De Rossi, “Temporal dynamics of nonlinear switching in GaAs photonoic-crystal-based devices,” in 36th European Conference and Exhibition on Optical Communication (ECOC) (IEEE, 2010), pp. 1–3.

2013

J. Francés, J. Tervo, and C. Neipp, “Split-field finite-difference time-domain scheme for Kerr-type nonlinear periodic media,” Progress Electromagn. Res. 134, 559–579 (2013).

J. Yuan and J. Yang, “Computational design for efficient second-harmonic generation in nonlinear photonic crystals,” J. Opt. Soc. Am. B 30, 205–210 (2013).
[CrossRef]

2012

L.-C. Zhao and J. Liu, “Localized nonlinear waves in a two-mode nonlinear fiber,” J. Opt. Soc. Am. B 29, 3119–3127 (2012).
[CrossRef]

A. Shahmansouri and B. Rashidian, “GPU implementation of split-field finite-difference time-domain method for Drude-Lorentz dispersive media,” Progress Electromagn. Res. 125, 55–77 (2012).
[CrossRef]

2011

A. Shahmansouri and B. Rashidian, “Comprehensive three-dimensional split-field finite-difference time-domain method for analysis of periodic plasmonic nanostructures: near- and far-field formulation,” J. Opt. Soc. Am. B 28, 2690–2700 (2011).
[CrossRef]

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Progress Electromagn. Res. 116, 441–456 (2011).
[CrossRef]

2010

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97, 041903 (2010).
[CrossRef]

F. Zhou, Y. Liu, Z.-Y. Li, and Y. Xia, “Analytical model for optical bistability in nonlinear metal nano-antennae involving kerr materials,” Opt. Express 18, 13337–13344 (2010).
[CrossRef]

C. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express 18, 21427–21448 (2010).
[CrossRef]

A. Belkhir, O. Arar, S. S. Benabbes, O. Lamrous, and F. I. Baida, “Implementation of dispersion models in the split-field-finite-difference- time-domain algorithm for the study of metallic periodic structures at oblique incidence,” Phys. Rev. E 81, 046705 (2010).
[CrossRef]

2009

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photon. Technol. Lett. 5, 310–312 (2009).
[CrossRef]

F. I. Baida and A. Belkhir, “Split-field FDTD method for oblique incidence study of periodic dispersive metallic structures,” Opt. Lett. 34, 2453–2455 (2009).
[CrossRef]

M. A. Alsunaidi and F. S. Al-Hajiri, “Efficient NL-FDTD solution schemes for the phase-sensitive second harmonic generation problem,” J. Lightwave Technol. 27, 4964–4969 (2009).
[CrossRef]

W. T. Dong, L. Gao, and C. W. Qiu, “Goos–Hänchen shift at the surface of chiral negative refractive media,” Progress Electromagn. Res. 90, 255–268 (2009).
[CrossRef]

Y.-Q. Zhang and D. B. Ge, “A unified FDTD approach for electromagnetic analysis of dispersive objects,” Progress Electromagn. Res. 96, 155–172 (2009).
[CrossRef]

2008

S. M. Amjadi and M. Soleimani, “Design of band-pass waveguide filter using frequency selective surfaces loaded with surface mount capacitors based on split-field update FDTD method,” Progress Electromagn. Res. B 43, 271–281 (2008).
[CrossRef]

A. Belkhir and F. I. Baida, “Three-dimensional finite-difference time-domain algorithm for oblique incidence with adaptation of perfectly matched layers and nonuniform meshing: application to the study of a radar dome,” Phys. Rev. E 77, 056701 (2008).
[CrossRef]

M. Qasymeh, M. Cada, and S. A. Ponomarenko, “Quadratic electro-optic Kerr effect: applications to photonic devices,” IEEE J. Quantum Electron. 44, 740–746 (2008).
[CrossRef]

2006

S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
[CrossRef]

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

J. H. Greene and A. Taflove, “General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics,” Opt. Express 14, 8305–8310 (2006).
[CrossRef]

C. Oh and M. J. Escuti, “Time-domain analysis of periodic anisotropic media at oblique incidence: An efficient FDTD implementation,” Opt. Express 14, 11870–11884 (2006).
[CrossRef]

G. Zheng, A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, “Implementation of Mur’s absorbing boundaries with periodic structures to speed up the design process using finite-difference time-domain method,” Progress Electromagn. Res. 58, 101–114 (2006).
[CrossRef]

2004

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[CrossRef]

2002

S. Takahashi and A. V. Zayats, “Near-field second-harmonic generation at a metal tip apex,” Appl. Phys. Lett 80, 3479–3481 (2002).
[CrossRef]

2001

I. I. Smolyaninov, H. Y. Liang, C. H. Lee, C. C. Davis, V. Nagarajan, and R. Ramesh, “Near-field second harmonic imaging of the c/a/c/a polydomain structure of epitaxial pbzrxti1−xo3 thin films,” J. Microsc. 202, 250–254 (2001).
[CrossRef]

2000

A. V. Zayats, T. Kalkbrenner, V. Sandoghdar, and J. Mlynek, “Second harmonic generation from individual surface defects under local excitation,” Phys. Rev. B 61, 4545–4548 (2000).
[CrossRef]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

1999

1998

S. I. Bozhevolny, K. Pedersen, T. Skettrup, X. Zhang, and M. Belmonte, “Far- and near-field second-harmonic imaging of ferroelectric domain walls,” Opt. Commun. 152, 221–224 (1998).
[CrossRef]

P. S. Balourdos, D. J. Frantzeskakis, M. C. Tsilis, and I. G. Tigelis, “Reflectivity of a nonlinear discontinuity in optical waveguides,” Pure Appl. Opt. 7, 1–11 (1998).
[CrossRef]

J. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Microwave Theor. Tech. 46, 420–427 (1998).
[CrossRef]

1997

W. D. Deering and M. I. Molina, “Power switching in hybrid coherent couplers,” IEEE J. Quantum Electron. 33, 336–340 (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]

1996

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

1992

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]

P. M. Goorjian and A. Taflove, “Direct time integration of maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).
[CrossRef]

1966

K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).

Agrawal, G. P.

Ahmed, I.

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Progress Electromagn. Res. 116, 441–456 (2011).
[CrossRef]

Al-Hajiri, F. S.

Al-Mudhaffar, H. M.

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photon. Technol. Lett. 5, 310–312 (2009).
[CrossRef]

H. M. Al-Mudhaffar, M. A. Alsunaidi, and H. M. Masoudi, “Full-wave solution of the second harmonic generation problem using a nonlinear FDTD algorithm,” in Progress in Electromagnics Research Symposium Proceedings, Prague, Czech Repulic (2007), pp. 479–482.

Alsunaidi, M. A.

M. A. Alsunaidi and F. S. Al-Hajiri, “Efficient NL-FDTD solution schemes for the phase-sensitive second harmonic generation problem,” J. Lightwave Technol. 27, 4964–4969 (2009).
[CrossRef]

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photon. Technol. Lett. 5, 310–312 (2009).
[CrossRef]

H. M. Al-Mudhaffar, M. A. Alsunaidi, and H. M. Masoudi, “Full-wave solution of the second harmonic generation problem using a nonlinear FDTD algorithm,” in Progress in Electromagnics Research Symposium Proceedings, Prague, Czech Repulic (2007), pp. 479–482.

Amjadi, S. M.

S. M. Amjadi and M. Soleimani, “Design of band-pass waveguide filter using frequency selective surfaces loaded with surface mount capacitors based on split-field update FDTD method,” Progress Electromagn. Res. B 43, 271–281 (2008).
[CrossRef]

Ammann, M.

M. Ammann, “Non-trivial materials in EM-FDTD,” Master’s thesis (Department of Physics, Swiss Federal Institute of Technology, 2007).

Arar, O.

A. Belkhir, O. Arar, S. S. Benabbes, O. Lamrous, and F. I. Baida, “Implementation of dispersion models in the split-field-finite-difference- time-domain algorithm for the study of metallic periodic structures at oblique incidence,” Phys. Rev. E 81, 046705 (2010).
[CrossRef]

Armaroli, A.

S. Malaguti, A. Armaroli, G. Bellanca, S. Trillo, S. Combrié, P. Colman, and A. De Rossi, “Temporal dynamics of nonlinear switching in GaAs photonoic-crystal-based devices,” in 36th European Conference and Exhibition on Optical Communication (ECOC) (IEEE, 2010), pp. 1–3.

Baida, F. I.

A. Belkhir, O. Arar, S. S. Benabbes, O. Lamrous, and F. I. Baida, “Implementation of dispersion models in the split-field-finite-difference- time-domain algorithm for the study of metallic periodic structures at oblique incidence,” Phys. Rev. E 81, 046705 (2010).
[CrossRef]

F. I. Baida and A. Belkhir, “Split-field FDTD method for oblique incidence study of periodic dispersive metallic structures,” Opt. Lett. 34, 2453–2455 (2009).
[CrossRef]

A. Belkhir and F. I. Baida, “Three-dimensional finite-difference time-domain algorithm for oblique incidence with adaptation of perfectly matched layers and nonuniform meshing: application to the study of a radar dome,” Phys. Rev. E 77, 056701 (2008).
[CrossRef]

Balourdos, P. S.

P. S. Balourdos, D. J. Frantzeskakis, M. C. Tsilis, and I. G. Tigelis, “Reflectivity of a nonlinear discontinuity in optical waveguides,” Pure Appl. Opt. 7, 1–11 (1998).
[CrossRef]

Belkhir, A.

A. Belkhir, O. Arar, S. S. Benabbes, O. Lamrous, and F. I. Baida, “Implementation of dispersion models in the split-field-finite-difference- time-domain algorithm for the study of metallic periodic structures at oblique incidence,” Phys. Rev. E 81, 046705 (2010).
[CrossRef]

F. I. Baida and A. Belkhir, “Split-field FDTD method for oblique incidence study of periodic dispersive metallic structures,” Opt. Lett. 34, 2453–2455 (2009).
[CrossRef]

A. Belkhir and F. I. Baida, “Three-dimensional finite-difference time-domain algorithm for oblique incidence with adaptation of perfectly matched layers and nonuniform meshing: application to the study of a radar dome,” Phys. Rev. E 77, 056701 (2008).
[CrossRef]

Bellanca, G.

S. Malaguti, A. Armaroli, G. Bellanca, S. Trillo, S. Combrié, P. Colman, and A. De Rossi, “Temporal dynamics of nonlinear switching in GaAs photonoic-crystal-based devices,” in 36th European Conference and Exhibition on Optical Communication (ECOC) (IEEE, 2010), pp. 1–3.

Belmonte, M.

S. I. Bozhevolny, K. Pedersen, T. Skettrup, X. Zhang, and M. Belmonte, “Far- and near-field second-harmonic imaging of ferroelectric domain walls,” Opt. Commun. 152, 221–224 (1998).
[CrossRef]

Benabbes, S. S.

A. Belkhir, O. Arar, S. S. Benabbes, O. Lamrous, and F. I. Baida, “Implementation of dispersion models in the split-field-finite-difference- time-domain algorithm for the study of metallic periodic structures at oblique incidence,” Phys. Rev. E 81, 046705 (2010).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), pp. 161–224.

Bozhevolny, S. I.

S. I. Bozhevolny, K. Pedersen, T. Skettrup, X. Zhang, and M. Belmonte, “Far- and near-field second-harmonic imaging of ferroelectric domain walls,” Opt. Commun. 152, 221–224 (1998).
[CrossRef]

Brzozowski, L.

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

Butcher, P.

P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1990), pp. 12–149.

Cada, M.

M. Qasymeh, M. Cada, and S. A. Ponomarenko, “Quadratic electro-optic Kerr effect: applications to photonic devices,” IEEE J. Quantum Electron. 44, 740–746 (2008).
[CrossRef]

Colman, P.

S. Malaguti, A. Armaroli, G. Bellanca, S. Trillo, S. Combrié, P. Colman, and A. De Rossi, “Temporal dynamics of nonlinear switching in GaAs photonoic-crystal-based devices,” in 36th European Conference and Exhibition on Optical Communication (ECOC) (IEEE, 2010), pp. 1–3.

Combrié, S.

S. Malaguti, A. Armaroli, G. Bellanca, S. Trillo, S. Combrié, P. Colman, and A. De Rossi, “Temporal dynamics of nonlinear switching in GaAs photonoic-crystal-based devices,” in 36th European Conference and Exhibition on Optical Communication (ECOC) (IEEE, 2010), pp. 1–3.

Cotter, D.

P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1990), pp. 12–149.

Davis, C. C.

I. I. Smolyaninov, H. Y. Liang, C. H. Lee, C. C. Davis, V. Nagarajan, and R. Ramesh, “Near-field second harmonic imaging of the c/a/c/a polydomain structure of epitaxial pbzrxti1−xo3 thin films,” J. Microsc. 202, 250–254 (2001).
[CrossRef]

De Rossi, A.

S. Malaguti, A. Armaroli, G. Bellanca, S. Trillo, S. Combrié, P. Colman, and A. De Rossi, “Temporal dynamics of nonlinear switching in GaAs photonoic-crystal-based devices,” in 36th European Conference and Exhibition on Optical Communication (ECOC) (IEEE, 2010), pp. 1–3.

Deering, W. D.

W. D. Deering and M. I. Molina, “Power switching in hybrid coherent couplers,” IEEE J. Quantum Electron. 33, 336–340 (1997).
[CrossRef]

Dissanayake, C.

Dong, W. T.

W. T. Dong, L. Gao, and C. W. Qiu, “Goos–Hänchen shift at the surface of chiral negative refractive media,” Progress Electromagn. Res. 90, 255–268 (2009).
[CrossRef]

Escuti, M. J.

Francés, J.

J. Francés, J. Tervo, and C. Neipp, “Split-field finite-difference time-domain scheme for Kerr-type nonlinear periodic media,” Progress Electromagn. Res. 134, 559–579 (2013).

Frantzeskakis, D. J.

P. S. Balourdos, D. J. Frantzeskakis, M. C. Tsilis, and I. G. Tigelis, “Reflectivity of a nonlinear discontinuity in optical waveguides,” Pure Appl. Opt. 7, 1–11 (1998).
[CrossRef]

Freude, W.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[CrossRef]

Fujii, M.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[CrossRef]

Gao, D.

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97, 041903 (2010).
[CrossRef]

Gao, L.

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97, 041903 (2010).
[CrossRef]

W. T. Dong, L. Gao, and C. W. Qiu, “Goos–Hänchen shift at the surface of chiral negative refractive media,” Progress Electromagn. Res. 90, 255–268 (2009).
[CrossRef]

S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
[CrossRef]

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

Ge, D. B.

Y.-Q. Zhang and D. B. Ge, “A unified FDTD approach for electromagnetic analysis of dispersive objects,” Progress Electromagn. Res. 96, 155–172 (2009).
[CrossRef]

Gedney, S. D.

J. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Microwave Theor. Tech. 46, 420–427 (1998).
[CrossRef]

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

Glisson, A. W.

G. Zheng, A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, “Implementation of Mur’s absorbing boundaries with periodic structures to speed up the design process using finite-difference time-domain method,” Progress Electromagn. Res. 58, 101–114 (2006).
[CrossRef]

Goh, R. S. M.

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Progress Electromagn. Res. 116, 441–456 (2011).
[CrossRef]

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]

P. M. Goorjian and A. Taflove, “Direct time integration of maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).
[CrossRef]

Greene, J. H.

Hagness, S. C.

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]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005), pp. 353–406.

Harms, P. H.

J. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Microwave Theor. Tech. 46, 420–427 (1998).
[CrossRef]

Hung, T. G. G.

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Progress Electromagn. Res. 116, 441–456 (2011).
[CrossRef]

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]

Kalkbrenner, T.

A. V. Zayats, T. Kalkbrenner, V. Sandoghdar, and J. Mlynek, “Second harmonic generation from individual surface defects under local excitation,” Phys. Rev. B 61, 4545–4548 (2000).
[CrossRef]

Kesler, M. P.

J. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Microwave Theor. Tech. 46, 420–427 (1998).
[CrossRef]

J. G. Maloney and M. P. Kesler, “Analysis of antenna arrays using the split-field update FDTD method,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 1998), Vol. 4, pp. 420–427.

Khoo, E. H.

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Progress Electromagn. Res. 116, 441–456 (2011).
[CrossRef]

Kishk, A. A.

G. Zheng, A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, “Implementation of Mur’s absorbing boundaries with periodic structures to speed up the design process using finite-difference time-domain method,” Progress Electromagn. Res. 58, 101–114 (2006).
[CrossRef]

Lamrous, O.

A. Belkhir, O. Arar, S. S. Benabbes, O. Lamrous, and F. I. Baida, “Implementation of dispersion models in the split-field-finite-difference- time-domain algorithm for the study of metallic periodic structures at oblique incidence,” Phys. Rev. E 81, 046705 (2010).
[CrossRef]

Lee, C. H.

I. I. Smolyaninov, H. Y. Liang, C. H. Lee, C. C. Davis, V. Nagarajan, and R. Ramesh, “Near-field second harmonic imaging of the c/a/c/a polydomain structure of epitaxial pbzrxti1−xo3 thin films,” J. Microsc. 202, 250–254 (2001).
[CrossRef]

Lee, K. H.

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Progress Electromagn. Res. 116, 441–456 (2011).
[CrossRef]

Li, E. P.

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Progress Electromagn. Res. 116, 441–456 (2011).
[CrossRef]

Li, Z.-Y.

Liang, H. Y.

I. I. Smolyaninov, H. Y. Liang, C. H. Lee, C. C. Davis, V. Nagarajan, and R. Ramesh, “Near-field second harmonic imaging of the c/a/c/a polydomain structure of epitaxial pbzrxti1−xo3 thin films,” J. Microsc. 202, 250–254 (2001).
[CrossRef]

Liu, J.

Liu, Y.

Malaguti, S.

S. Malaguti, A. Armaroli, G. Bellanca, S. Trillo, S. Combrié, P. Colman, and A. De Rossi, “Temporal dynamics of nonlinear switching in GaAs photonoic-crystal-based devices,” in 36th European Conference and Exhibition on Optical Communication (ECOC) (IEEE, 2010), pp. 1–3.

Maloney, J. G.

J. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Microwave Theor. Tech. 46, 420–427 (1998).
[CrossRef]

J. G. Maloney and M. P. Kesler, “Analysis of antenna arrays using the split-field update FDTD method,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 1998), Vol. 4, pp. 420–427.

Masoudi, H. M.

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photon. Technol. Lett. 5, 310–312 (2009).
[CrossRef]

H. M. Al-Mudhaffar, M. A. Alsunaidi, and H. M. Masoudi, “Full-wave solution of the second harmonic generation problem using a nonlinear FDTD algorithm,” in Progress in Electromagnics Research Symposium Proceedings, Prague, Czech Repulic (2007), pp. 479–482.

Mlynek, J.

A. V. Zayats, T. Kalkbrenner, V. Sandoghdar, and J. Mlynek, “Second harmonic generation from individual surface defects under local excitation,” Phys. Rev. B 61, 4545–4548 (2000).
[CrossRef]

Molina, M. I.

W. D. Deering and M. I. Molina, “Power switching in hybrid coherent couplers,” IEEE J. Quantum Electron. 33, 336–340 (1997).
[CrossRef]

Nagarajan, V.

I. I. Smolyaninov, H. Y. Liang, C. H. Lee, C. C. Davis, V. Nagarajan, and R. Ramesh, “Near-field second harmonic imaging of the c/a/c/a polydomain structure of epitaxial pbzrxti1−xo3 thin films,” J. Microsc. 202, 250–254 (2001).
[CrossRef]

Neipp, C.

J. Francés, J. Tervo, and C. Neipp, “Split-field finite-difference time-domain scheme for Kerr-type nonlinear periodic media,” Progress Electromagn. Res. 134, 559–579 (2013).

Oh, C.

Pan, T.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

Pedersen, K.

S. I. Bozhevolny, K. Pedersen, T. Skettrup, X. Zhang, and M. Belmonte, “Far- and near-field second-harmonic imaging of ferroelectric domain walls,” Opt. Commun. 152, 221–224 (1998).
[CrossRef]

Ponomarenko, S. A.

M. Qasymeh, M. Cada, and S. A. Ponomarenko, “Quadratic electro-optic Kerr effect: applications to photonic devices,” IEEE J. Quantum Electron. 44, 740–746 (2008).
[CrossRef]

Premaratne, M.

Qasymeh, M.

M. Qasymeh, M. Cada, and S. A. Ponomarenko, “Quadratic electro-optic Kerr effect: applications to photonic devices,” IEEE J. Quantum Electron. 44, 740–746 (2008).
[CrossRef]

Qiu, C. W.

W. T. Dong, L. Gao, and C. W. Qiu, “Goos–Hänchen shift at the surface of chiral negative refractive media,” Progress Electromagn. Res. 90, 255–268 (2009).
[CrossRef]

Ramesh, R.

I. I. Smolyaninov, H. Y. Liang, C. H. Lee, C. C. Davis, V. Nagarajan, and R. Ramesh, “Near-field second harmonic imaging of the c/a/c/a polydomain structure of epitaxial pbzrxti1−xo3 thin films,” J. Microsc. 202, 250–254 (2001).
[CrossRef]

Rashidian, B.

A. Shahmansouri and B. Rashidian, “GPU implementation of split-field finite-difference time-domain method for Drude-Lorentz dispersive media,” Progress Electromagn. Res. 125, 55–77 (2012).
[CrossRef]

A. Shahmansouri and B. Rashidian, “Comprehensive three-dimensional split-field finite-difference time-domain method for analysis of periodic plasmonic nanostructures: near- and far-field formulation,” J. Opt. Soc. Am. B 28, 2690–2700 (2011).
[CrossRef]

Roden, J.

J. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Microwave Theor. Tech. 46, 420–427 (1998).
[CrossRef]

Rukhlenko, I. D.

Russer, P.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[CrossRef]

Sakagami, I.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[CrossRef]

Sandoghdar, V.

A. V. Zayats, T. Kalkbrenner, V. Sandoghdar, and J. Mlynek, “Second harmonic generation from individual surface defects under local excitation,” Phys. Rev. B 61, 4545–4548 (2000).
[CrossRef]

Sargent, E. H.

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

Shahmansouri, A.

A. Shahmansouri and B. Rashidian, “GPU implementation of split-field finite-difference time-domain method for Drude-Lorentz dispersive media,” Progress Electromagn. Res. 125, 55–77 (2012).
[CrossRef]

A. Shahmansouri and B. Rashidian, “Comprehensive three-dimensional split-field finite-difference time-domain method for analysis of periodic plasmonic nanostructures: near- and far-field formulation,” J. Opt. Soc. Am. B 28, 2690–2700 (2011).
[CrossRef]

Skettrup, T.

S. I. Bozhevolny, K. Pedersen, T. Skettrup, X. Zhang, and M. Belmonte, “Far- and near-field second-harmonic imaging of ferroelectric domain walls,” Opt. Commun. 152, 221–224 (1998).
[CrossRef]

Smolyaninov, I. I.

I. I. Smolyaninov, H. Y. Liang, C. H. Lee, C. C. Davis, V. Nagarajan, and R. Ramesh, “Near-field second harmonic imaging of the c/a/c/a polydomain structure of epitaxial pbzrxti1−xo3 thin films,” J. Microsc. 202, 250–254 (2001).
[CrossRef]

Soleimani, M.

S. M. Amjadi and M. Soleimani, “Design of band-pass waveguide filter using frequency selective surfaces loaded with surface mount capacitors based on split-field update FDTD method,” Progress Electromagn. Res. B 43, 271–281 (2008).
[CrossRef]

Taflove, A.

J. H. Greene and A. Taflove, “General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics,” Opt. Express 14, 8305–8310 (2006).
[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]

P. M. Goorjian and A. Taflove, “Direct time integration of maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).
[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]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005), pp. 353–406.

Tahara, M.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[CrossRef]

Takahashi, S.

S. Takahashi and A. V. Zayats, “Near-field second-harmonic generation at a metal tip apex,” Appl. Phys. Lett 80, 3479–3481 (2002).
[CrossRef]

Tang, C. J.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

Tervo, J.

J. Francés, J. Tervo, and C. Neipp, “Split-field finite-difference time-domain scheme for Kerr-type nonlinear periodic media,” Progress Electromagn. Res. 134, 559–579 (2013).

Tigelis, I. G.

P. S. Balourdos, D. J. Frantzeskakis, M. C. Tsilis, and I. G. Tigelis, “Reflectivity of a nonlinear discontinuity in optical waveguides,” Pure Appl. Opt. 7, 1–11 (1998).
[CrossRef]

Trillo, S.

S. Malaguti, A. Armaroli, G. Bellanca, S. Trillo, S. Combrié, P. Colman, and A. De Rossi, “Temporal dynamics of nonlinear switching in GaAs photonoic-crystal-based devices,” in 36th European Conference and Exhibition on Optical Communication (ECOC) (IEEE, 2010), pp. 1–3.

Tsilis, M. C.

P. S. Balourdos, D. J. Frantzeskakis, M. C. Tsilis, and I. G. Tigelis, “Reflectivity of a nonlinear discontinuity in optical waveguides,” Pure Appl. Opt. 7, 1–11 (1998).
[CrossRef]

Wang, S. M.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
[CrossRef]

Wang, Y.

Xia, Y.

Yakovlev, A. B.

G. Zheng, A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, “Implementation of Mur’s absorbing boundaries with periodic structures to speed up the design process using finite-difference time-domain method,” Progress Electromagn. Res. 58, 101–114 (2006).
[CrossRef]

Yang, J.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).

Yuan, J.

Zayats, A. V.

S. Takahashi and A. V. Zayats, “Near-field second-harmonic generation at a metal tip apex,” Appl. Phys. Lett 80, 3479–3481 (2002).
[CrossRef]

A. V. Zayats, T. Kalkbrenner, V. Sandoghdar, and J. Mlynek, “Second harmonic generation from individual surface defects under local excitation,” Phys. Rev. B 61, 4545–4548 (2000).
[CrossRef]

Zhang, X.

S. I. Bozhevolny, K. Pedersen, T. Skettrup, X. Zhang, and M. Belmonte, “Far- and near-field second-harmonic imaging of ferroelectric domain walls,” Opt. Commun. 152, 221–224 (1998).
[CrossRef]

Zhang, Y.-Q.

Y.-Q. Zhang and D. B. Ge, “A unified FDTD approach for electromagnetic analysis of dispersive objects,” Progress Electromagn. Res. 96, 155–172 (2009).
[CrossRef]

Zhao, L.-C.

Zheng, G.

G. Zheng, A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, “Implementation of Mur’s absorbing boundaries with periodic structures to speed up the design process using finite-difference time-domain method,” Progress Electromagn. Res. 58, 101–114 (2006).
[CrossRef]

Zhou, F.

Appl. Phys. Lett

S. Takahashi and A. V. Zayats, “Near-field second-harmonic generation at a metal tip apex,” Appl. Phys. Lett 80, 3479–3481 (2002).
[CrossRef]

Appl. Phys. Lett.

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97, 041903 (2010).
[CrossRef]

IEEE J. Quantum Electron.

M. Qasymeh, M. Cada, and S. A. Ponomarenko, “Quadratic electro-optic Kerr effect: applications to photonic devices,” IEEE J. Quantum Electron. 44, 740–746 (2008).
[CrossRef]

W. D. Deering and M. I. Molina, “Power switching in hybrid coherent couplers,” IEEE J. Quantum Electron. 33, 336–340 (1997).
[CrossRef]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[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 Photon. Technol. Lett.

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photon. Technol. Lett. 5, 310–312 (2009).
[CrossRef]

IEEE Trans. Antennas Propag.

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).

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]

IEEE Trans. Microwave Theor. Tech.

J. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Microwave Theor. Tech. 46, 420–427 (1998).
[CrossRef]

J. Lightwave Technol.

J. Microsc.

I. I. Smolyaninov, H. Y. Liang, C. H. Lee, C. C. Davis, V. Nagarajan, and R. Ramesh, “Near-field second harmonic imaging of the c/a/c/a polydomain structure of epitaxial pbzrxti1−xo3 thin films,” J. Microsc. 202, 250–254 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

S. I. Bozhevolny, K. Pedersen, T. Skettrup, X. Zhang, and M. Belmonte, “Far- and near-field second-harmonic imaging of ferroelectric domain walls,” Opt. Commun. 152, 221–224 (1998).
[CrossRef]

S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

Phys. Rev. B

A. V. Zayats, T. Kalkbrenner, V. Sandoghdar, and J. Mlynek, “Second harmonic generation from individual surface defects under local excitation,” Phys. Rev. B 61, 4545–4548 (2000).
[CrossRef]

Phys. Rev. E

A. Belkhir and F. I. Baida, “Three-dimensional finite-difference time-domain algorithm for oblique incidence with adaptation of perfectly matched layers and nonuniform meshing: application to the study of a radar dome,” Phys. Rev. E 77, 056701 (2008).
[CrossRef]

A. Belkhir, O. Arar, S. S. Benabbes, O. Lamrous, and F. I. Baida, “Implementation of dispersion models in the split-field-finite-difference- time-domain algorithm for the study of metallic periodic structures at oblique incidence,” Phys. Rev. E 81, 046705 (2010).
[CrossRef]

Progress Electromagn. Res.

G. Zheng, A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, “Implementation of Mur’s absorbing boundaries with periodic structures to speed up the design process using finite-difference time-domain method,” Progress Electromagn. Res. 58, 101–114 (2006).
[CrossRef]

A. Shahmansouri and B. Rashidian, “GPU implementation of split-field finite-difference time-domain method for Drude-Lorentz dispersive media,” Progress Electromagn. Res. 125, 55–77 (2012).
[CrossRef]

J. Francés, J. Tervo, and C. Neipp, “Split-field finite-difference time-domain scheme for Kerr-type nonlinear periodic media,” Progress Electromagn. Res. 134, 559–579 (2013).

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Progress Electromagn. Res. 116, 441–456 (2011).
[CrossRef]

Y.-Q. Zhang and D. B. Ge, “A unified FDTD approach for electromagnetic analysis of dispersive objects,” Progress Electromagn. Res. 96, 155–172 (2009).
[CrossRef]

W. T. Dong, L. Gao, and C. W. Qiu, “Goos–Hänchen shift at the surface of chiral negative refractive media,” Progress Electromagn. Res. 90, 255–268 (2009).
[CrossRef]

Progress Electromagn. Res. B

S. M. Amjadi and M. Soleimani, “Design of band-pass waveguide filter using frequency selective surfaces loaded with surface mount capacitors based on split-field update FDTD method,” Progress Electromagn. Res. B 43, 271–281 (2008).
[CrossRef]

Pure Appl. Opt.

P. S. Balourdos, D. J. Frantzeskakis, M. C. Tsilis, and I. G. Tigelis, “Reflectivity of a nonlinear discontinuity in optical waveguides,” Pure Appl. Opt. 7, 1–11 (1998).
[CrossRef]

Other

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), pp. 161–224.

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012)457–492.

J. G. Maloney and M. P. Kesler, “Analysis of antenna arrays using the split-field update FDTD method,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 1998), Vol. 4, pp. 420–427.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005), pp. 353–406.

P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1990), pp. 12–149.

H. M. Al-Mudhaffar, M. A. Alsunaidi, and H. M. Masoudi, “Full-wave solution of the second harmonic generation problem using a nonlinear FDTD algorithm,” in Progress in Electromagnics Research Symposium Proceedings, Prague, Czech Repulic (2007), pp. 479–482.

M. Ammann, “Non-trivial materials in EM-FDTD,” Master’s thesis (Department of Physics, Swiss Federal Institute of Technology, 2007).

S. Malaguti, A. Armaroli, G. Bellanca, S. Trillo, S. Combrié, P. Colman, and A. De Rossi, “Temporal dynamics of nonlinear switching in GaAs photonoic-crystal-based devices,” in 36th European Conference and Exhibition on Optical Communication (ECOC) (IEEE, 2010), pp. 1–3.

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Figures (6)

Fig. 1.
Fig. 1.

(a) Scheme of a two-dimensional computational space. (b) Yee cell for a two-dimensional SF-FDTD scheme.

Fig. 2.
Fig. 2.

(a) Second-harmonic field amplitude as a function of space. (b) Time-domain results for the fundamental (thin blue line) and second-harmonic (thick green line) fields at a point along the device at the center of the simulation grid. (c) Second-harmonic intensity along the nonlinear media with (thick red line) and without QPM (thin blue line).

Fig. 3.
Fig. 3.

(a) Snapshots of the electric field R { E y } in different kinds of media detected from left to right at times t = 2000 Δ t (thin blue line) and 4000 Δ t (thick green line), respectively. (a) Linear dispersive Lorentz medium. (b) Nonlinear Lorentz and Kerr media. (c) Nonlinear Lorentz, Kerr, and Raman media.

Fig. 4.
Fig. 4.

Frequency spectrum for the case of a pulse propagating in nonlinear dispersive media. Sampling plane at z = 1500 Δ .

Fig. 5.
Fig. 5.

Transmission diffraction efficiencies as a function of the normalized depth of the pillars. The minus first, zeroth, and first-orders are represented in the first (a)–(c), second (d)–(f), and third (g)–(i) rows of the graphs, respectively. Each column of graphs belongs to a different value of the ratio B / A . (a), (d), (g) B / A = 1 . (b), (e), (h) B / A = 6 . (c), (f), (i) B / A = 0 .

Fig. 6.
Fig. 6.

Differences between electric field components for B / A = 1 and B / A = 0 in GV / cm . Parameters: Λ / λ = 20 and h / λ = 1 , and input intensity 79.5 GW / cm 2 . (a) Absolute difference between E x components. (b) Absolute difference between E z components.

Tables (2)

Tables Icon

Table 1. Contracted Matrix Notation for d m ξ Indices

Tables Icon

Table 2. Setup Parameters of SF-FDTD for Results in Figs. 5 and 6

Equations (55)

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× E = μ 0 H t ,
× H = D t ,
D = ϵ 0 ϵ r E + F NL ,
× H = ϵ 0 ϵ r E t + J NL ,
= Ĕ e j k x x ,
= μ 0 c e j k x x ,
P x = P x a c 2 μ 0 κ F x NL e j k x x ,
P y = P y a + sin θ 0 κ Q z c 2 μ 0 κ F y NL e j k x x ,
P z = P z a sin θ 0 κ Q y c 2 μ 0 κ F z NL e j k x x ,
Q x = Q x a ,
Q y = Q y a sin θ 0 P z ,
Q z = Q z a + sin θ 0 P y ,
d = [ d 11 d 12 d 13 d 14 d 15 d 16 d 21 d 22 d 23 d 24 d 25 d 26 d 31 d 32 d 33 d 34 d 35 d 36 ] .
F m NL , ω f = 2 ϵ 0 σ , β d m ξ E σ ω f E β ω s ,
F m NL , ω s = ϵ 0 σ , β d m ξ E σ ω f E β ω f ,
[ F x NL , ω f F y NL , ω f F z NL , ω f ] e j ω f sin θ 0 x / c = 2 d ϵ 0 [ P x ω f E x ω s P y ω f E y ω s P z ω f E z ω s P z ω f E y ω s + P y ω f E z ω s P z ω f E x ω s + P x ω f E z ω s P x ω f E y ω s + P y ω f E x ω s ] ,
[ F x NL , ω s F y NL , ω s F z NL , ω s ] e j ω s sin θ 0 x / c = d ϵ 0 [ P x ω f P x ω f P y ω f P y ω f P z ω f P z ω f 2 P z ω f P y ω f 2 P z ω f P x ω f 2 P x ω f P y ω f ] .
P x ω f = P x a ω f 2 d 14 κ P z ω f E y ω s ,
P z ω f = P z a ω f sin θ 0 κ Q y ω f 2 d 14 κ P x ω f E y ω s ,
Q y ω f = Q y a ω f sin θ 0 P z ω f ,
P y ω s = P y a ω s + sin θ 0 κ Q z ω s 2 d 14 κ P x ω f P z ω f ,
Q x ω s = Q x a ω s ,
Q z ω s = Q z a ω s + sin θ 0 P y ω s ,
P ( p + 1 ) = U ( P ( p ) ) , p = 0 , 1 , 2 ,
F NL = F K + F R ,
F K ( t ) = α ϵ 0 χ 0 ( 3 ) | E | 2 E ,
F R ( t ) = ϵ 0 E [ χ Raman ( 3 ) * | E | 2 ] ,
S e + 1 / 2 = 2 ( ω R Δ t / 2 ) 2 1 + γ R Δ t / 2 S e + γ R Δ t / 2 1 1 + γ R Δ t / 2 S e 1 / 2 + ( 1 α ) χ 0 ( 3 ) ( Δ t ω R / 2 ) 2 1 + γ R Δ t / 2 | P e | 2 ,
ω R = τ 1 2 + τ 2 2 τ 1 2 τ 2 2 ; γ R = 1 τ 2 .
F R e + 1 e j k x x = ϵ 0 S e + 1 P e + 1 .
P x = P x a κ ( α χ 0 ( 3 ) | P x | 2 P x + S x P x ) ,
P y = P y a + sin θ inc κ Q z κ ( α χ 0 ( 3 ) | P y | 2 P y + S y P y ) ,
P z = P z a sin θ inc κ Q y κ ( α χ 0 ( 3 ) | P z | 2 P z + S z P z ) .
P z = P z a κ sin θ 0 Q y a 1 + κ ( χ 0 ( 3 ) | E z | 2 + S z sin 2 θ 0 ) = C z P ^ z a ,
Q y = Q y a sin θ 0 P z ,
P x = P x a 1 + κ ( χ 0 ( 3 ) | E x | 2 + S x ) = C x P x a ,
P y = P y a + κ sin θ 0 Q z a 1 + κ ( χ 0 ( 3 ) | E y | 2 + S y sin 2 θ 0 ) = C y P ^ y a ,
Q z = Q z a + sin θ 0 P y ,
C x = 1 1 + κ ( χ 0 ( 3 ) I x + S x ) ,
C y = 1 1 + κ ( χ 0 ( 3 ) I y + S y sin 2 θ 0 ) ,
C z = 1 1 + κ ( χ 0 ( 3 ) I z + S z sin 2 θ 0 ) ,
I = | E | 2 .
I x | p + 1 = C x | p ( C x | p ) * | P x a | 2 ,
I y | p + 1 = C y | p ( C y | p ) * | P ^ y a | 2 ,
I z | p + 1 = C z | p ( C z | p ) * | P ^ z a | 2 ,
C x | p = 1 1 + κ ( χ 0 ( 3 ) I x | p + S x ) ,
C y | p = 1 1 + κ ( χ 0 ( 3 ) I y | p + S y sin 2 θ 0 ) ,
C z | p = 1 1 + κ ( χ 0 ( 3 ) I z | p + S z sin 2 θ 0 )
F NL = ϵ 0 A ( E · E * ) E + 1 2 ϵ 0 B ( E · E ) E * .
F x NL = ϵ 0 [ ( A + 1 2 B ) | E x | 2 E x + A | E z | 2 E x + 1 2 B E z 2 E x * ] ,
F z NL = ϵ 0 [ ( A + 1 2 B ) | E z | 2 E z + A | E x | 2 E z + 1 2 B E x 2 E z * ] .
E z 2 E x * = | E z | 2 e j 2 Φ z E x E x E x * = | E z | 2 E x e j 2 Φ ,
E x 2 E z * = | E x | 2 e j 2 Φ x E z E z E z * = | E x | 2 E z e j 2 Φ
F x NL e j k x x = ϵ 0 [ ( A + 1 2 B ) | P x | 2 + ( A + 1 2 B e j 2 Φ ) | P z | 2 ] P x ,
F z NL e j k x x = ϵ 0 [ ( A + 1 2 B e j 2 Φ ) | P x | 2 + ( A + 1 2 B ) | P z | 2 ] P z ,

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