Abstract

We investigate numerically a non-reciprocal switching behavior in strongly modulated waveguide Bragg gratings (WBGs) having a longitudinally asymmetric stopband configuration. The minimum power predicted for a stable switching operation is found to be approximately 77 mW for a realistic waveguide structure made of prospective materials; we assume in this paper a nano-strip InGaAsP/InP waveguide having longitudinally asymmetric modulation of the waveguide width. The analysis has been performed with our in-house nonlinear finite-difference time-domain (FDTD) code adapted to parallel computing. The numerical results clearly show low-threshold Schmitt trigger operation, as well as non-reciprocal transmission property where the switching threshold for one propagation direction is lower than that for the other direction. In addition, we discuss the modulation-like instability phenomena in such nonlinear periodic devices by employing both an instantaneous Kerr nonlinearity and a more involved saturable nonlinearity model.

© 2006 Optical Society of America

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  1. M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, and C.M. Bowden, "Thin-film nonlinear optical diode," Appl. Phys. Lett. 66, 2324-2326 (1995).
    [CrossRef]
  2. A. Maitra, C.G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
    [CrossRef]
  3. W. Freude, A. Maitra, J. Wang, C. Koos, C. Poulton, M. Fujii, and J. Leuthold, "All-optical signal processing with nonlinear resonant devices," in Proc. 8th Intern. Conf. on Transparent Optical Networks (ICTON’06), Vol. , (Nottingham, UK, 2006), paper We.D2.1, pp. 215-219.
  4. W. Chen and D.L. Mills, "Gap solitons and the nonlinear optical response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).
    [CrossRef] [PubMed]
  5. C. de Sterke and J.E. Sipe, "Switching dynamics of finite periodic nonlinear media: A numerical study," Phys. Rev. A 42, 2858-2869 (1990).
    [CrossRef] [PubMed]
  6. C. de Sterke and J.E. Sipe, "Gap solitons," in Progress in Optics, vol.XXXIII, pp.203-260, North-Holland, Amsterdam (1994).
  7. M. Scalora, J.P. Dowling, C.M. Bowden, and M.J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
    [CrossRef] [PubMed]
  8. M.W. Feise, I.V. Shdrivov, and Y.S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037,602 (2005).
    [CrossRef]
  9. X-H. Jia, Z-M. Wu, and G-Q. Xia, "Analysis of bistable steady characteristics and dynamic stability of linearly tapered nonlinear Bragg gratings," Opt. Express 12, 2945-2953 (2004).
    [CrossRef] [PubMed]
  10. E. Lidorikis and C.M. Soukoulis, "Pulse-driven switching in one-dimensional nonlinear photonic band gap materials: a numerical study," Phys. Rev. E 61, 5825-5829 (2000).
    [CrossRef]
  11. X.-S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
    [CrossRef]
  12. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
    [CrossRef] [PubMed]
  13. O.H. Schmitt, "A thermionic trigger," J. Scientific Instruments 15, 24 (1938).
    [CrossRef]
  14. M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP/InP racetrack micro-resonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
    [CrossRef]
  15. C. Koos, M. Fujii, C. Poulton, R. Steingrueber, J. Leuthold, andW.Freude, "FDTD-modeling of dispersive nonlinear ring resonators: Accuracy studies and experiments," IEEE J. Quantum Electron. In print.
  16. J. Koga, "Simulation model for the effects of nonlinear polarization on the propagation of intense pulse lasers," Optics Lett. 24, 408-410 (1999).
    [CrossRef]
  17. K.S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media," IEEE Trans. Antennas Prop. 14, 302-307 (1966).
    [CrossRef]
  18. M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, "High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2D Kerr and Raman nonlinear dispersive media," IEEE J. Quantum Electron. 40(2), 175-182 (2004).
    [CrossRef]
  19. A. Taflove and S.C. Hagness, Computational electrodynamics: The finite-difference time-domain method, 3rd ed., chap. 9 (Artech House, 2005).

2006

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP/InP racetrack micro-resonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

2005

M.W. Feise, I.V. Shdrivov, and Y.S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037,602 (2005).
[CrossRef]

A. Maitra, C.G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

X.-S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
[CrossRef]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

2004

X-H. Jia, Z-M. Wu, and G-Q. Xia, "Analysis of bistable steady characteristics and dynamic stability of linearly tapered nonlinear Bragg gratings," Opt. Express 12, 2945-2953 (2004).
[CrossRef] [PubMed]

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, "High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2D Kerr and Raman nonlinear dispersive media," IEEE J. Quantum Electron. 40(2), 175-182 (2004).
[CrossRef]

2000

E. Lidorikis and C.M. Soukoulis, "Pulse-driven switching in one-dimensional nonlinear photonic band gap materials: a numerical study," Phys. Rev. E 61, 5825-5829 (2000).
[CrossRef]

1999

J. Koga, "Simulation model for the effects of nonlinear polarization on the propagation of intense pulse lasers," Optics Lett. 24, 408-410 (1999).
[CrossRef]

1995

M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, and C.M. Bowden, "Thin-film nonlinear optical diode," Appl. Phys. Lett. 66, 2324-2326 (1995).
[CrossRef]

1994

M. Scalora, J.P. Dowling, C.M. Bowden, and M.J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

1990

C. de Sterke and J.E. Sipe, "Switching dynamics of finite periodic nonlinear media: A numerical study," Phys. Rev. A 42, 2858-2869 (1990).
[CrossRef] [PubMed]

1987

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

1966

K.S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media," IEEE Trans. Antennas Prop. 14, 302-307 (1966).
[CrossRef]

1938

O.H. Schmitt, "A thermionic trigger," J. Scientific Instruments 15, 24 (1938).
[CrossRef]

Bloemer, M.J.

M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, and C.M. Bowden, "Thin-film nonlinear optical diode," Appl. Phys. Lett. 66, 2324-2326 (1995).
[CrossRef]

M. Scalora, J.P. Dowling, C.M. Bowden, and M.J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Bowden, C.M.

M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, and C.M. Bowden, "Thin-film nonlinear optical diode," Appl. Phys. Lett. 66, 2324-2326 (1995).
[CrossRef]

M. Scalora, J.P. Dowling, C.M. Bowden, and M.J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Chen, W.

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

de Sterke, C.

C. de Sterke and J.E. Sipe, "Switching dynamics of finite periodic nonlinear media: A numerical study," Phys. Rev. A 42, 2858-2869 (1990).
[CrossRef] [PubMed]

Dowling, J.P.

M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, and C.M. Bowden, "Thin-film nonlinear optical diode," Appl. Phys. Lett. 66, 2324-2326 (1995).
[CrossRef]

M. Scalora, J.P. Dowling, C.M. Bowden, and M.J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Feise, M.W.

M.W. Feise, I.V. Shdrivov, and Y.S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037,602 (2005).
[CrossRef]

Freude, W.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP/InP racetrack micro-resonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

A. Maitra, C.G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, "High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2D Kerr and Raman nonlinear dispersive media," IEEE J. Quantum Electron. 40(2), 175-182 (2004).
[CrossRef]

Fujii, M.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP/InP racetrack micro-resonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, "High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2D Kerr and Raman nonlinear dispersive media," IEEE J. Quantum Electron. 40(2), 175-182 (2004).
[CrossRef]

Jia, X-H.

Kira, G.

Kivshar, Y.S.

M.W. Feise, I.V. Shdrivov, and Y.S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037,602 (2005).
[CrossRef]

Koga, J.

J. Koga, "Simulation model for the effects of nonlinear polarization on the propagation of intense pulse lasers," Optics Lett. 24, 408-410 (1999).
[CrossRef]

Koos, C.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP/InP racetrack micro-resonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

Kuramochi, E.

Lan, S.

X.-S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
[CrossRef]

Leuthold, J.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP/InP racetrack micro-resonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

A. Maitra, C.G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

Lidorikis, E.

E. Lidorikis and C.M. Soukoulis, "Pulse-driven switching in one-dimensional nonlinear photonic band gap materials: a numerical study," Phys. Rev. E 61, 5825-5829 (2000).
[CrossRef]

Lin, X.-S.

X.-S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
[CrossRef]

Maitra, A.

A. Maitra, C.G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

Mills, D.L.

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

Mitsugi, S.

Notomi, M.

Poulton, C.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP/InP racetrack micro-resonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

Poulton, C.G.

A. Maitra, C.G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

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 in 2D Kerr and Raman nonlinear dispersive media," IEEE J. Quantum Electron. 40(2), 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 in 2D Kerr and Raman nonlinear dispersive media," IEEE J. Quantum Electron. 40(2), 175-182 (2004).
[CrossRef]

Scalora, M.

M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, and C.M. Bowden, "Thin-film nonlinear optical diode," Appl. Phys. Lett. 66, 2324-2326 (1995).
[CrossRef]

M. Scalora, J.P. Dowling, C.M. Bowden, and M.J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Schmitt, O.H.

O.H. Schmitt, "A thermionic trigger," J. Scientific Instruments 15, 24 (1938).
[CrossRef]

Shdrivov, I.V.

M.W. Feise, I.V. Shdrivov, and Y.S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037,602 (2005).
[CrossRef]

Shinya, A.

Sipe, J.E.

C. de Sterke and J.E. Sipe, "Switching dynamics of finite periodic nonlinear media: A numerical study," Phys. Rev. A 42, 2858-2869 (1990).
[CrossRef] [PubMed]

Soukoulis, C.M.

E. Lidorikis and C.M. Soukoulis, "Pulse-driven switching in one-dimensional nonlinear photonic band gap materials: a numerical study," Phys. Rev. E 61, 5825-5829 (2000).
[CrossRef]

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 in 2D Kerr and Raman nonlinear dispersive media," IEEE J. Quantum Electron. 40(2), 175-182 (2004).
[CrossRef]

Tanabe, T.

Tocci, M.D.

M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, and C.M. Bowden, "Thin-film nonlinear optical diode," Appl. Phys. Lett. 66, 2324-2326 (1995).
[CrossRef]

Wang, J.

A. Maitra, C.G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

Wu, Z-M.

Xia, G-Q.

Yee, K.S.

K.S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media," IEEE Trans. Antennas Prop. 14, 302-307 (1966).
[CrossRef]

Appl. Phys. Lett.

M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, and C.M. Bowden, "Thin-film nonlinear optical diode," Appl. Phys. Lett. 66, 2324-2326 (1995).
[CrossRef]

Chin. Phys. Lett.

X.-S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
[CrossRef]

IEEE J. Quantum Electron.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, "High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2D Kerr and Raman nonlinear dispersive media," IEEE J. Quantum Electron. 40(2), 175-182 (2004).
[CrossRef]

A. Maitra, C.G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP/InP racetrack micro-resonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

IEEE Trans. Antennas Prop.

K.S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media," IEEE Trans. Antennas Prop. 14, 302-307 (1966).
[CrossRef]

J. Scientific Instruments

O.H. Schmitt, "A thermionic trigger," J. Scientific Instruments 15, 24 (1938).
[CrossRef]

Opt. Express

Optics Lett.

J. Koga, "Simulation model for the effects of nonlinear polarization on the propagation of intense pulse lasers," Optics Lett. 24, 408-410 (1999).
[CrossRef]

Phys. Rev. A

C. de Sterke and J.E. Sipe, "Switching dynamics of finite periodic nonlinear media: A numerical study," Phys. Rev. A 42, 2858-2869 (1990).
[CrossRef] [PubMed]

Phys. Rev. E

M.W. Feise, I.V. Shdrivov, and Y.S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037,602 (2005).
[CrossRef]

E. Lidorikis and C.M. Soukoulis, "Pulse-driven switching in one-dimensional nonlinear photonic band gap materials: a numerical study," Phys. Rev. E 61, 5825-5829 (2000).
[CrossRef]

Phys. Rev. Lett.

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

M. Scalora, J.P. Dowling, C.M. Bowden, and M.J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Other

C. de Sterke and J.E. Sipe, "Gap solitons," in Progress in Optics, vol.XXXIII, pp.203-260, North-Holland, Amsterdam (1994).

W. Freude, A. Maitra, J. Wang, C. Koos, C. Poulton, M. Fujii, and J. Leuthold, "All-optical signal processing with nonlinear resonant devices," in Proc. 8th Intern. Conf. on Transparent Optical Networks (ICTON’06), Vol. , (Nottingham, UK, 2006), paper We.D2.1, pp. 215-219.

C. Koos, M. Fujii, C. Poulton, R. Steingrueber, J. Leuthold, andW.Freude, "FDTD-modeling of dispersive nonlinear ring resonators: Accuracy studies and experiments," IEEE J. Quantum Electron. In print.

A. Taflove and S.C. Hagness, Computational electrodynamics: The finite-difference time-domain method, 3rd ed., chap. 9 (Artech House, 2005).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic configuration of the waveguide Bragg grating having an asymmetric stopband; (a) the top view, and (b) the cross section of the waveguide. In the analysis the structure is approximated by a slab waveguide having an effective refractive index of the stacked structure. The ports for signal detection are located at a half hight of the InGaAsP core and at (x, z)=(0, 0) and (0, L 0). (c) An example of the upper band-edge frequency variation realized by the third-order polynomial g(z′) for Wg =0.1 µm.

Fig. 2.
Fig. 2.

Transmission spectra for the 100Λ-long waveguide Bragg gratings of uniform amplitudes of sidewall modulation. Wg in legend is in µm.

Fig. 3.
Fig. 3.

Comparison of the calculated transmission spectra for 200-Λ-long asymmetric gratings with the maximum modulation Wg =0.1 µm and Wg =0.20 µm. The upper frequency band edges are 208.20 THz for Wg =0.1 µm, and 214.08 THz for Wg =0.20 µm.

Fig. 4.
Fig. 4.

Switching for the LTR and RTL configuration of a 235Λ-long waveguide with Wg =0.1 µm at 207.9 THz operating frequency (Δf=0.3 THz). Maximum incident field value is 5.25×107 V/m at 100 ps. Right vertical axis shows the normalized permittivity change by the nonlinearity.

Fig. 5.
Fig. 5.

Electric field for 200Λ-long RTL configuration (positive variation) with Wg =0.1 µm at 207.85 THz operating frequency. (a) off-state at 90 ps, and (b) on-state at 130 ps.

Fig. 6.
Fig. 6.

Switch-on threshold electric field and power versus the length of the waveguide grating for the RTL (positive) configuration with Wg =0.1 µm and 0.2 µm. The error bars are the uncertainty due to the step increment of the incident field variation.

Fig. 7.
Fig. 7.

Switching for the LTR and RTL configuration of a 220Λ-long waveguide with Wg =0.20 µm at 214.03 THz operating frequency (Δf=0.05 THz). Maximum incident field is 2.6×107 V/m at 75 ps. Minimum switch-on threshold is Eth(on)=1.05×107 V/m (Pth(on)=77 mW) for RTL.

Fig. 8.
Fig. 8.

An example of the stable switching state for a uniformly modulated WBG of 200Λ in length, Wg =0.15µm at operating frequency 209.40 THz (Δf=0.15 THz). The maximum field is 1.75×107 V/m. Switch-on occurs at Eth(on)=1.6×107 V/m (Pth(on)=170 mW).

Fig. 9.
Fig. 9.

Stable state electric field for 200Λ-long uniform grating. Wg =0.15µm, 209.40 THz, 1.75×107 V/m. (a) off-state at 20 ps, and (b) on-state at 60 ps.

Fig. 10.
Fig. 10.

An example of the pulsative time signal for a uniformly modulated WBG of 220Λ in length, Wg =0.2µm at operating frequency 214.03 THz (Δf=0.05 THz). The maximum incident field is 5.25×107 V/m. Transition to the pulsative state occurs at incident field E=3.2×107 V/m (equivalent power P=720 mW).

Fig. 11.
Fig. 11.

Time signals that exhibit the modulation-like instability. For uniform WBGs of L 0=300 Λ. The incident light is a smoothly excited sinusoid with the maximum field 5.25×107 V/m.

Equations (2)

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x ( z ) = ± [ W 2 + W g 2 g ( z L 0 ) sin ( 2 π z Λ ) ] ,
U ( x ) = q 2 a 0 2 + x 2 ,

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