## Non-reciprocal transmission and Schmitt trigger operation in strongly modulated asymmetric WBGs

Optics Express, Vol. 14, Issue 26, pp. 12782-12793 (2006)

http://dx.doi.org/10.1364/OE.14.012782

Acrobat PDF (463 KB)

### 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

## 1. Introduction

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]

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]

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]

## 2. Numerical Experiment

### 2.1. Asymmetric waveguide Bragg grating

*z*as

*µ*m for this paper, and

*L*

_{0}is the total length of the grating. The averaged width of the waveguide was chosen to be

*W*=0.4

*µ*m such that it supports only the dominant propagation mode at the operating frequency. The maximum amplitude of the sidewall modulation is determined by

*W*

_{g}/2. The nearly linear variation of the band edge frequency is described by a 3rd-order polynomial in terms of the normalized distance

*z*′=

*z*/

*L*

_{0}as

*g*(

*z*′)=

*az*′

^{3}+

*bz*′

^{2}+

*cz*′+

*d*, whose coefficients have been found by fitting the polynomial to some FDTD results of stopbands for uniform waveguide gratings as shown in Fig. 2; they are chosen to be

*a*=-0.4856,

*b*=-0.0009,

*c*=0, and

*d*=1, and the resulting profile of the upper band-edge frequency for

*W*

_{g}=0.1

*µ*m is shown in Fig. 1(c). With this profile the sidewall modulation amplitude varies nonlinearly from

*W*

_{g}/2 at the left-hand-side of the waveguide to

*W*

_{g}/4 at the right-hand-side of the waveguide.

*x*=0.025

*µ*m and Δ

*z*=Λ/10=0.0241

*µ*m. Note that Δ

*x*is comparable to the side wall modulation, and it may appear too coarse to resolve the depth of the modulation. However, Δ

*z*is sufficiently small to resolve the periodic variation of the grating. We have therefore checked in preliminary analyses that the present cell size can resolve the side-wall modulation sufficiently for the purpose to realize the nearly linear profile of the band edge frequency.

*n*=3.17 at wavelength

*λ*=1.55

*µ*m) for the upper and the lower cladding, InGaAsP (

*n*=3.42 at the same wavelength) for the core, and surrounding air. For the numerical representation, the structure is approximated by a dielectric slab of

*n*=3.34 to be analyzed by a 2D FDTD method. The polarization of the incident light is quasi-TM where the dominant electric field component is perpendicular to the substrate. We employ the same fabrication and design procedures used in [14

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]

^{-18}m

^{2}/V

^{2}by our extensive numerical and experimental investigations [14

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]

### 2.2. FDTD analysis of WBG

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]

*E*and the light power P has been derived by numerically evaluating the effective area of the waveguide

*A*=5.3×10

^{-13}m

^{2}, thus we obtain

*P*=

*A*|

*E*|

^{2}/(2

*η*), where

*η*=377 Ω is the intrinsic wave impedance of vacuum.

*W*

_{g}=0.1

*µ*m and

*W*

_{g}=0.2

*µ*m, the transmission spectra for weak incidence have been analyzed as shown in Fig. 3. It is clearly seen in this figure that each spectrum is the combined one of those found in Fig. 2 for uniform gratings; due to the asymmetric variation of the grating modulation, the total transmission is also asymmetric with respect to frequency. In the previous 1D model e.g. in [2

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]

*λ*

_{0}=1442 nm) for the case of

*W*

_{g}=0.1

*µ*m, i.e. the frequency difference from the upper band edge was Δ

*f*=0.3 THz, and launched a sinusoidal wave from either the left-hand-side or the right-hand-side port of the asymmetric WBG. The incident light power is gradually increased, followed by a flat region such that the field in the grating establishes a stationary state. This staircase-like input is repeatedly cumulated until the incident light becomes strong enough to cause switching, and then the incident power is reduced in a similar manner until it vanishes. This allows the detection of the switch-on and -off thresholds of the WBG. The incident and the output time signals are plotted in Fig. 4 for the WBG of

*L*

_{0}=235Λ; due to the rapidly oscillating sinusoidal carrier, only the envelopes of the incident and the output signals for the RTL and the LTR configurations are visible in the figure. It is observed that the switching threshold is lower for the RTL configuration than for the LTR configuration; the switch-on threshold of the electric field is

^{7}V/m (equivalent light power

^{7}V/m (

^{7}V/m (

^{7}V/m (

13. O.H. Schmitt, “A thermionic trigger,” J. Scientific Instruments **15**, 24 (1938). [CrossRef]

^{-4}, which is much smaller than those used in literature, e.g. [4

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]

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]

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]

*L*

_{0}=200Λ,

*W*

_{g}=0.1

*µ*m at the operating frequency 207.85 THz for (a) an off-state and (b) an onstate. Similar fields are observed for the case of the LTR configuration. From these figures it is found that the fields have a few peaks, which indicates that the mode of resonance in the grating is of higher-order. Due to both the strong grating modulation and the structural asymmetry, and despite that the incident light power has been gradually increased, the fundamental mode is not generated but the higher-order mode takes over. It has been found in our results that the off-state fields are significantly different between the RTL and the LTR (not shown) configurations. For the RTL configuration the field penetrates into the waveguide, while for the LTR configuration the field is blocked at the entrance to the waveguide. When the waveguide grating switches on, the field distributions become similar for the RTL and the LTR configurations.

*f*

_{op}=207.85 THz, 207.9 THz and 207.95 THz for

*W*

_{g}=0.1

*µ*m, and

*f*

_{op}=214.03 THz for

*W*

_{g}=0.2

*µ*m as discussed later. The switching threshold no longer exhibits a linear dependence on the grating length when the grating is longer than approximately 210Λ. This behavior of the threshold value is considered to be due to the strong modulation of the grating. This tendency applies also to other cases of different

*W*

_{g}. The minimum threshold is obtained at length around 220Λ to 230Λ regardless of

*W*

_{g}. The WBGs longer than 235Λ are found unstable, causing a modulation-like field to grow rapidly, which is to be discussed in a later section. We have noted that for the minimum threshold condition at a length around 220Λ to 230Λ, the incident light and the light reflected from the grating become out of phase, and subsequently, the total field at the input port (right-hand-side port) becomes very small. Contrary, for shorter grating lengths the total field at the input port is observed significantly large (not shown). The total field near the input port will obviously affects the switching behavior. Therefore, the phase difference between the incident and the reflected lights, which is determined by the total length of the asymmetric grating, could be the main reason for the fact that the threshold is not linearly dependent on the grating length.

*W*

_{g}=0.20

*µ*m while maintaining the longitudinal profile of the modulation, i.e. the modulation amplitude

*W*

_{g}/2 of the grating varies according to the same polynomial, namely from 0.10

*µ*m to 0.05

*µ*m along the waveguide. The transmission spectra for the 200Λ-long asymmetric grating is compared with that of

*W*

_{g}=0.1

*µ*m in Fig. 3. The operating frequency for the nonlinear switching analysis was then chosen to be 214.03 THz, closer to the band edge (214.08 THz) than that of the previous case because the transmission spectrum is much steeper at the band edge, which allows the same level of extinction ratio for the nonlinear switching operation.

^{7}V/m (

^{7}V/m (

^{7}V/m for both RTL and LTR. The time evolution exhibits larger fluctuations than that of the previous case in Fig. 4 because the operating frequency is closer to the band edge and therefore the extinction ratio is worse than that in Fig. 4. The change in normalized permittivity for the switch-on threshold is 3.6×10

^{-5}. Note that the transmission spectrum for

*W*

_{g}=0.20

*µ*m has a very steep transition and a large dip near the band edge, in comparison to those for

*W*

_{g}=0.10

*µ*m. However, we observe that the stability of the switching operation is not largely affected by the dip. The steep transition reduces the switch-on power significantly. If the switch turns on, the bandgap shifts to the lower frequency side, and the device can maintain the on-state like an electronic Schmitt trigger, irrespective of the fine structure of the transmission curve near the band edge.

^{-16}m

^{2}/V

^{2}[1

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]

### 2.3. Stable state of nonlinear WBG

*L*=200Λ,

*W*

_{g}=0.15

*µ*m, uniform WBG at operating frequency 209.40 THz. The upper band-edge frequency of this structure is 209.55 THz. The time evolution of the field is shown in Fig. 8. In this case switching occurred at 40 ps with the incident threshold field

^{7}V/m (

*P*

_{th}=170 mW). For the time period between 40 ps and 50 ps, a relatively sharp pulse is observed. Further analyses showed that when the incident field is stronger the pulse becomes higher and narrower, leading to a number of soliton-like pulses generated and transmitted (not shown). After 50 ps, the incident power reduces gradually, and thereby the grating’s inner field switches to a stable state, which is retained until the incident light almost vanishes at time 100 ps. The corresponding field distributions are shown in Fig. 9(a) for an off-state at 20 ps, and in (b) for an on-state at 60 ps. These field distributions clearly indicate that the field is of a longitudinal fundamental mode, because only a single envelope peak is observed for the on-state, which is similar to those observed in previous literature for 1D solutions [5

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]

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]

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]

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]

### 2.4. Pulsative state of nonlinear WBG

*W*

_{g}=0.2

*µ*m except that the grating modulation is uniform, the stable switching is not observed clearly even when a relatively high-power incident light is launched. Instead, a pulsative state is suddenly observed as shown in Fig. 10. By comparison with the results for the asymmetric WBG in Fig. 7, it is found that the asymmetric grating has an effect of stabilizing the switch-on state. Similar stability characteristics have been investigated by Jia et.al. for a weakly-modulated-taper nonlinear Erbium-doped fiber Bragg grating [9

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]

### 2.5. Chaotic state of nonlinear WBG

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]

**12**, 2945–2953 (2004). [CrossRef] [PubMed]

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]

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]

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]

*x*is the displacement from equilibrium for a bound particle having an electric charge

*q*, and

*a*

_{0}is an equiliblium radius. These parameters can be determined from desired nonlinear properties such as linear sceptibility, third-order nonlinear sceptibility, and a characteristic resonance frequency of the charged particle. In its differential equation of electron motion it has a term for the Lorentz dispersion and a term for the 3rd-order nonlinearity, which can be efficiently implemented in the FDTD algorithm through the ADE formalism [18

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]

_{1}and at the output port V

_{2}, for a uniformly modulated WBG of 300Λ in length; (a) is for the instantaneous Kerr nonlinearity and the Lorentz dispersion with a damping factor

*δ*

_{p}=1.0×10

^{14}rad/s, and (b) for the SCP model with a damping factor

*δ*

_{p}=1.0×10

^{13}rad/s. From these results one can see that the SCP model allows the observation of the chaotic behavior, at least qualitatively, by virtue of the saturating nature of the model, while in (a) even with a larger damping factor the calculation stops due to a convergence problem of the nonlinear algorithm. Our extensive study shows that when the damping factor is increased for the Lorentz dispersion model, the modulation-like field builds up like the result in (a) at a slightly later time, and it results in the same convergence problem of the nonlinear algorithm.

## 3. Conclusions

## Acknowledgements

## References and links

1. | M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, and C.M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. |

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. |

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 |

4. | W. Chen and D.L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. |

5. | C. de Sterke and J.E. Sipe, “Switching dynamics of finite periodic nonlinear media: A numerical study,” Phys. Rev. A |

6. | C. Sterke and J.E. Sipe, “Gap solitons,” in |

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. |

8. | M.W. Feise, I.V. Shdrivov, and Y.S. Kivshar, “Bistable diode action in left-handed periodic structures,” Phys. Rev. E |

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 |

10. | E. Lidorikis and C.M. Soukoulis, “Pulse-driven switching in one-dimensional nonlinear photonic band gap materials: a numerical study,” Phys. Rev. E |

11. | X.-S. Lin and S. Lan, “Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity,” Chin. Phys. Lett. |

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. | O.H. Schmitt, “A thermionic trigger,” J. Scientific Instruments |

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. |

15. | C. Koos, M. Fujii, C. Poulton, R. Steingrueber, J. Leuthold, and W. 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. |

17. | K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Prop. |

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. |

19. | A. Taflove and S.C. Hagness, |

**OCIS Codes**

(130.4310) Integrated optics : Nonlinear

(190.1450) Nonlinear optics : Bistability

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: November 21, 2006

Revised Manuscript: December 12, 2006

Manuscript Accepted: December 13, 2006

Published: December 22, 2006

**Citation**

Masafumi Fujii, Ayan Maitra, Christopher Poulton, Juerg Leuthold, and Wolfgang Freude, "Non-reciprocal transmission and Schmitt trigger operation in strongly modulated asymmetric WBGs," Opt. Express **14**, 12782-12793 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-12782

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### References

- 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]
- 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]
- 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.
- W. Chen and D.L. Mills, "Gap solitons and the nonlinear optical response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987). [CrossRef] [PubMed]
- 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]
- C. de Sterke and J.E. Sipe, "Gap solitons," in Progress in Optics, vol.XXXIII, pp.203-260, North-Holland, Amsterdam (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]
- 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]
- 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]
- 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]
- 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]
- O.H. Schmitt, "A thermionic trigger," J. Scientific Instruments 15, 24 (1938). [CrossRef]
- 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]
- 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.
- J. Koga, "Simulation model for the effects of nonlinear polarization on the propagation of intense pulse lasers," Optics Lett. 24, 408-410 (1999). [CrossRef]
- 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]
- 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. Taflove and S.C. Hagness, Computational electrodynamics: The finite-difference time-domain method, 3rd ed., chap. 9 (Artech House, 2005).

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