## Energy-bandwidth trade-off in all-optical photonic crystal microcavity switches |

Optics Express, Vol. 19, Issue 19, pp. 18410-18422 (2011)

http://dx.doi.org/10.1364/OE.19.018410

Acrobat PDF (1934 KB)

### Abstract

The performance of all-optical switches is a compromise between the achievable bandwidth of the switched signal and the energy requirement of the switching operation. In this work we consider a system consisting of a photonic crystal cavity coupled to two input and two output waveguides. As a specific example of a switching application, we investigate the demultiplexing of an optical time division multiplexed signal. To quantify the energy-bandwidth trade-off, we introduce a figure of merit for the detection of the demultiplexed signal. In such investigations it is crucial to consider patterning effects, which occur on time scales that are longer than the bit period. Our analysis is based on a coupled mode theory, which allows for an extensive investigation of the influence of the system parameters on the switching dynamics. The analysis is shown to provide new insights into the ultrafast dynamics of the switching operation, and the results show optimum parameter ranges that may serve as design guidelines in device fabrication.

© 2011 OSA

## 1. Introduction

*Q*cavities with ultra small mode volumes and waveguides with highly tailorable dispersion characteristics [1

1. M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature Mater. **3**, 211–219 (2004). [CrossRef]

*Q*-values it is possible to achieve a large field enhancement, which reduces the energy requirement. However, it also limits the transmission of the cavity, and the increased photon lifetime causes patterning effects, which restrict the signal bandwidth. So again, there is a trade-off between energy consumption and bandwidth.

2. J. Y. Lee, L. H. Yin, G. P. Agraval, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express **18**, 11514–11523 (2010). [CrossRef] [PubMed]

3. M. Waldow, T. Plotzing, M. Gottheil, M. Forst, and J. Bolten, “25 ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator,” Opt. Express **16**, 7693–7702 (2008). [CrossRef] [PubMed]

4. C. Husko, A. De Rossi, S. Combré, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett. **94**, 021111 (2009). [CrossRef]

5. K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics **4**, 477–483 (2010). [CrossRef]

6. L. O’Faolain, D. M. Beggs, T. P. White, T. Kampfrath, K. Kuipers, and T. F. Krauss, “Compact optical switches and modulators based on dispersion engineered photonic crystals,” IEEE Photon. J. **2**, 404–414 (2010). [CrossRef]

7. O. Wada, “Recent progress in semiconductor-based photonic signal-processing devices,” IEEE J. Sel. Top. Quantum Electron. **17**, 309–319 (2011). [CrossRef]

8. P. A. Andrekson, H. Sunnerud, S. Oda, T. Nishitani, and J. Yang, “Ultrafast, atto-Joule switch using fiber parametric amplifier operated in saturation,” Opt. Express **16**, 10956–10961 (2008). [CrossRef] [PubMed]

9. J. Xu, X. Zhang, and J. Mørk, “Investigation of patterning effects in ultrafast SOA-based optical switches,” IEEE J. Quantum Electron. **46**, 87–94 (2010). [CrossRef]

10. J. B. Khurgin, “Performance of nonlinear photonic crystal devices at high bit rates,” Opt. Lett. **30**, 643–645 (2005). [CrossRef] [PubMed]

## 2. Model

*ε*

_{r}= 12, surrounded by air and has two input waveguides and two output waveguides coupled to a cavity at the center, similar to Ref. [11

11. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. **28**, 2506–2508 (2003). [CrossRef] [PubMed]

*ω*i = 2/

*τ*[13]. Fig. 2 illustrates these transmission spectra as well as the spectra of the signal and control pulses and indicates the corresponding parameters. The FWHM of the pulses are denoted Ω

_{i}*. In a structure like the one shown in Fig. 1(a), where the waveguides for the signal and control are spatially separated, the*

_{i}*Q*-values of the modes can be controlled independently by placing a different number of extra rods adjacent to the cavity in the two waveguides. This turns out to be important, since the results in Sec. 5 will show that an optimum switching performance is achievable by using very different linewidths for the two modes. In Ref. [11

11. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. **28**, 2506–2508 (2003). [CrossRef] [PubMed]

*Q*by placing more rods next to the cavity does not significantly affect

*κ*, if the corresponding change in the field distributions in the center of the cavity is negligible. This justifies using Δ

_{ij}*ω*

_{S}and Δ

*ω*

_{C}as independent parameters, which may be varied significantly in value.

*S*

_{outS,C}, where Γ would be the rate of energy loss due to out of plane scattering or absorption in the cavity. We neglect such a term here to keep the description as simple as possible. The second terms in Eqs. (1) and (2) describe the nonlinear shift in the resonance frequency of the cavity modes due to self- and cross phase modulation. By combining the first and second terms in Eq. (1), we may define a power dependent effective detuning The operation principle of the switch is easily understood from Fig. 2 and Eqs. (1) and (4). The injected control pulse redshifts the transmission spectrum of cavity mode S, which increases its overlap with the signal spectrum and opens the switch.

11. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. **28**, 2506–2508 (2003). [CrossRef] [PubMed]

*ω*

_{S}= 2.344

*c*/

*a*,

*ω*

_{C}= 2.231

*c*/

*a*,

*κ*

_{SS}= 0.0943,

*κ*

_{CC}= 0.105,

*κ*

_{SC}= 0.0312, and

*κ*

_{CS}= 0.0343, where

*a*is the lattice constant of the PhC structure.

## 3. Transmission of a Single Signal Pulse

*t*

_{S}is the FWHM pulse width.

_{S}is larger than the cavity linewidth Δ

*ω*

_{S}. Although we will primarily focus on Gaussian pulses, it is more illustrative to use square pulses for this purpose, because the steady state appears as the limit of an infinitely long square pulse. While keeping the cavity linewidth constant, we have varied the peak power of the input pulse for different values of the pulse width. Fig 3(a) shows the ratio of output energy and pulse width as a function of the input power for different values of Δ

*ω*

_{S}/Ω

_{S}for a square pulse. The steady state solution of Eq. (1) is also plotted (dashed red) and agrees well with the results in [11

**28**, 2506–2508 (2003). [CrossRef] [PubMed]

*δ*

_{S}. The physical origin of the oscillations is a transient interference beating between the incoming pulse oscillating at

*ω*

_{LS}and the excited cavity mode oscillating at

*ω*

_{S}. In the top left graph of Fig. 3(c) the oscillation period is ∼4 times the pulse width, and the output energy is at a local maximum. In the bottom left graph, the period is ∼2 times the pulse width and the output energy is at a local minimum. In the upper (lower) right graph, the oscillation period is ∼4/3 (∼1) times the pulse width and again the output energy is at a maximum (minimum). Thus, it makes a big difference whether the output pulse has reached a maximum or a minimum of the oscillation in a time determined by the pulse width. As the input power is varied, the effective detuning changes due to the nonlinear frequency shift of the cavity. This, in turn, changes the effective oscillation period of the output pulse, causing the local extrema of the curves in Figs. 3(a) and 3(b).

*ω*

_{S}/Ω

_{S}< 1, is thus seen to be qualitatively different from the quasi steady state regime, Δ

*ω*

_{S}/Ω

_{S}> 1. For short pulses, the nonlinear change of the resonance frequency relative to the

*pulse*bandwidth has a significant effect on the transmission properties of the switch. For long pulses, the switching mechanism can be understood from the bistability curve in Fig. 3(a). By increasing the input power, the transmission jumps from a small to a large value. The criterion for the bistability to occur is

**28**, 2506–2508 (2003). [CrossRef] [PubMed]

*cavity*linewidth, which is important for the transmission in the quasi steady state regime.

15. J. Mørk, F. Öhmann, and S. Bischoff, “Analytical expression for the bit error rate of cascaded all-optical regenerators,” Photon. Technol. Lett. **15**, 1479–1481 (2003). [CrossRef]

## 4. Demultiplexing of a Data Signal

9. J. Xu, X. Zhang, and J. Mørk, “Investigation of patterning effects in ultrafast SOA-based optical switches,” IEEE J. Quantum Electron. **46**, 87–94 (2010). [CrossRef]

^{15}– 1 bits as the input in Eqs. (1) and (2). Fig. 4(c) shows the output energy registered by the detector as a function of the bit number. The figure also shows the probability distribution function (pdf) of the detected energies. The dashed black lines in Fig. 4(c) indicate

*t*

_{S}= Δ

*t*

_{C}= 10

^{3}

*a*/

*c*, constant phases,

*ϕ*

_{inS}(

*t*) =

*ϕ*

_{inC}(

*t*) = 0, the delay between them is zero, and the bit rate

*B*is 1/(4Δ

*t*

_{S}). The varied parameters are the detunings, the cavity linewidths, and the peak power of the control pulse

*t*

_{S}) and the number of OTDM channels fixed at 10. This corresponds to a variation of the signal bandwidth, and the energy-bandwidth trade-off is investigated by evaluating how much energy is required to obtain a certain FoM for each bandwidth.

**28**, 2506–2508 (2003). [CrossRef] [PubMed]

*a*if the structure is a slab with a thickness of some fraction of

*a*. By choosing

*a*= 0.6

*μ*m and

*χ*

^{(3)}= 6.5 × 10

^{−19}m

^{2}/V

^{2}, which corresponds to a nonlinear refractive index of

*n*

_{2}= 1.5 × 10

^{−17}m

^{2}/W [16], we have an energy unit of

*ε*

_{0}

*a*

^{2}/

*χ*

^{(3)}

*a*= 3 pJ, and a time unit of 10

^{3}

*a*/

*c*= 2 ps, giving a bit rate of 125 Gbit/s. The value of

*n*

_{2}we are using is achievable in AlGaAs below half the electronic bandgap [11

**28**, 2506–2508 (2003). [CrossRef] [PubMed]

## 5. Switching Dynamics with Signal and Control Pulse

*ω*

_{C}= 2Ω

_{C}. The control energy

*U*

_{inC}and the linewidth of cavity mode S Δ

*ω*

_{S}is varied, while the pulse widths of the signal and control are fixed. For each (

*U*

_{inC}, Δ

*ω*

_{S}), both detunings are varied in order to find the maximum FoM. Fig. 5 shows how the FoM depends on

*δ*

_{S}and

*δ*

_{C}for 4 different values of (

*U*

_{inC}, Δ

*ω*

_{S}). In Fig. 5(a), the quasi steady state limit with Δ

*ω*

_{S}/Ω

_{S}= 10 is shown, while Fig. 5(b) gives the dependence in the short pulse regime with Δ

*ω*

_{S}/Ω

_{S}= 0.06. Generally, the maximum FoM occurs at a larger

*δ*

_{S}when the control energy is increased. A large control energy provides a large nonlinear frequency shift of cavity mode S, which in combination with a large signal detuning results in a large change in the signal transmission. This is the reason why the FoM may be increased by using a larger control energy. Fig. 5 also shows that there is a large qualitative difference in the dependence of the FoM on the detunings in the quasi steady state and short pulse regimes. This is expected from the results in Sec. 3. The presence of multiple extrema of the FoM as a function of

*δ*

_{S}in Fig. 5(b) is caused by the same effect as the one discussed in relation to the appearance of local extrema in Fig. 3(a). Here, we have varied the detuning instead of the power as in Fig. 3(a), but it is still the oscillations in the output power that causes the extrema. A local extremum occurs for values of

*δ*

_{S}, where the fixed control power causes the effective oscillation period to change by ∼4 times the pulse width.

*U*

_{inC}and Δ

*ω*

_{S}. Notice that FoM < 1 is possible, although not practically acceptable, since the energy in the denominator of Eq. (6) is an integral over 9 pulses, whereas the numerator results from an integration over 1 pulse. The price to pay for the increase in the FoM is a reduction in transmission,

*δ*

_{S},

*δ*

_{C})-plane, cf. Fig. 5(b). The value of the signal detuning, where the maximum in the FoM occurs,

*U*

_{inC}than Δ

*ω*

_{S}. Since the signal power is too small to shift the resonance frequency of cavity mode C, it makes sense that Δ

*ω*

_{S}does not have a significant influence on

*ω*

_{C}and in order to get a maximum amount of power in the cavity

*δ*

_{S}, cf. Eq. (4), it seems reasonable to expect that the optimum value of

*δ*

_{S}/Ω

_{S}is found close to a maximum of the relative change of

*U*

_{outS}with respect to

*δ*

_{S}/Ω

_{S}In Fig. 7(a) we have plotted the maximum of

*ω*

_{S}/Ω

_{S}. The observed increase in max

*ω*

_{S}is decreased thus explains why the FoM also increases when Δ

*ω*

_{S}is decreased, as it is seen in Fig. 6(a). A more intuitive way to understand this behavior in the short pulse regime, is to consider the transient oscillations resulting from a step function input given in Eq. (5). As Δ

*ω*

_{S}is decreased, the ratio between the maxima and minima of the oscillations as well as the cavity lifetime increase, which causes a larger difference in the output energy between the upper and lower pulses in Fig. 3(c). This means that the change in output energy due to the presence of the control pulse increases as Δ

*ω*

_{S}is decreased.

*δ*

_{S}/Ω

_{S}, where the maximum in

*U*

_{inC}= 0.105

*ε*

_{0}

*a*

^{2}/

*χ*

^{(3)}(dashed red curve). The agreement between the curves is seen to be good for large values of Δ

*ω*

_{S}, but there is a discrepancy when the cavity linewidth decreases. This is also the case for the transmission corresponding to the maximum of

*U*

_{inC}= 0.105

*ε*

_{0}

*a*

^{2}/

*χ*

^{(3)}from Fig. 6(b) (dashed red curve). The reason for this discrepancy is that the denominator in Eq. (6) is an integration over 9 bits. When Δ

*ω*

_{S}decreases, the signal energy in one bit does not escape the cavity before the next pulse arrives. This gives rise to a much more complicated behavior, which can not be accounted for by an analysis based on a single pulse input. The discontinuities in Fig. 6(a)–(c) are not described by the linear analysis either. These can only be understood as an interplay between the nonlinear frequency shift and the transient oscillations of the output pulses observed in Fig. 3(c). This conclusion is supported by the fact that the discontinuities appear in the short pulse regime, where the oscillations were observed in Fig. 3(c). Even though the linear analysis is not able to describe all the details of Fig. 6(a)–(c), it is still very useful for understanding the general trends.

*ω*

_{S}∼Ω

_{S}/3, or equivalently, a quality factor of,

*Q*

_{S}∼2500. Fig. 6(a) also shows that to achieve a figure of merit well above 1 with this cavity linewidth, we must use larger control energies.

*ω*

_{S}= Ω

_{S}/3, we now vary the control energy and the linewidth of cavity mode C in the same way as in Fig. 6. The results are given in Fig. 8(a)–(d). From Fig. 8(a) it is observed that there is an optimum value of Δ

*ω*

_{C}, which minimizes the required control energy to achieve a certain value of the figure of merit. If the cavity linewidth is large, the field enhancement inside the cavity is small, and thus it requires a larger input power to achieve a certain frequency shift. On the other hand, if the linewidth becomes smaller, the figure of merit is reduced by patterning effects as mentioned in Sec. 4. Fig. 8(b) shows the dependence of the signal transmission,

*δ*

_{S}. This makes sense because the difference between the numerator and denominator in Eq. (6) can be made larger by using a larger detuning if sufficient control power is available to deliver a correspondingly large frequency shift. From Fig. 8(d) it is observed that the behavior of

*δ*

_{C}, where a maximum amount of control power is available to shift the resonance frequency of cavity mode S. This is the reason for the increase in

*U*

_{inC}, because the shift of

*ω*

_{C}also increases with control power.

## 6. Energy-Bandwidth Trade-Off

*ω*

_{C}and control energy as in Fig. 8 for different values of the two pulse widths, Δ

*t*

_{S}= Δ

*t*

_{C}. The bit rate is still given by 1/(4Δ

*t*), so this allows us to investigate the dependence of the minimum required control energy on the bit rate of the signal. We keep the ratio Δ

*ω*

_{S}/Ω

_{S}fixed at 1/3 to get sufficient transmission, cf. Sec. 5. The result is shown in Fig. 9(a). The different curves correspond to the minimum energy required to achieve different values of the FoM. It is clearly observed how the energy requirement increases as the bit rate increases, and as the value of the FoM increases. Fig. 9(b) shows the optimum ratio, Δ

*ω*

_{C}/Ω

_{C}, corresponding to the minimum in

*U*

_{inC}. The different curves correspond to the same values of the FoM as in Fig. 9(a). The optimum ratio is seen to be independent of the bit rate, but increases as the value of the FoM is increased. Since the bit rate is varied by changing Ω

_{C}, we see that the optimum linewidth of cavity mode C changes, corresponding to a quality factor of

*Q*

_{C}≈ 900 for

*B*= 40 GHz and

*Q*

_{C}≈ 300 for

*B*= 125 GHz, when Δ

*ω*

_{C}/Ω

_{C}≈ 3. For a bit rate of 125 GHz, the quality factors of the two modes,

*Q*

_{S}and

*Q*

_{C}, differ by a factor of 9. As explained in Sec. 2, such a large difference is easy to achieve in a structure with spatially separated waveguides, like the one shown in Fig. 1(a), suggesting that such structures are advantageous when operating at high bit rates.

## 7. Conclusion

*Q*

_{C}at

*B*= 125 GHz shows that these effects are dominant in the energy-bandwidth trade-off compared to the field enhancement caused by the cavity at high bit rates. This conclusion underlines the importance of using realistic data signals consisting of many bit periods to estimate the performance of switching devices.

## Acknowledgments

## References and links

1. | M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature Mater. |

2. | J. Y. Lee, L. H. Yin, G. P. Agraval, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express |

3. | M. Waldow, T. Plotzing, M. Gottheil, M. Forst, and J. Bolten, “25 ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator,” Opt. Express |

4. | C. Husko, A. De Rossi, S. Combré, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett. |

5. | K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics |

6. | L. O’Faolain, D. M. Beggs, T. P. White, T. Kampfrath, K. Kuipers, and T. F. Krauss, “Compact optical switches and modulators based on dispersion engineered photonic crystals,” IEEE Photon. J. |

7. | O. Wada, “Recent progress in semiconductor-based photonic signal-processing devices,” IEEE J. Sel. Top. Quantum Electron. |

8. | P. A. Andrekson, H. Sunnerud, S. Oda, T. Nishitani, and J. Yang, “Ultrafast, atto-Joule switch using fiber parametric amplifier operated in saturation,” Opt. Express |

9. | J. Xu, X. Zhang, and J. Mørk, “Investigation of patterning effects in ultrafast SOA-based optical switches,” IEEE J. Quantum Electron. |

10. | J. B. Khurgin, “Performance of nonlinear photonic crystal devices at high bit rates,” Opt. Lett. |

11. | M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. |

12. | J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightwave Technol. |

13. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

14. | H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A |

15. | J. Mørk, F. Öhmann, and S. Bischoff, “Analytical expression for the bit error rate of cascaded all-optical regenerators,” Photon. Technol. Lett. |

16. | R. W. Boyd, |

**OCIS Codes**

(200.4560) Optics in computing : Optical data processing

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: July 1, 2011

Revised Manuscript: August 8, 2011

Manuscript Accepted: August 22, 2011

Published: September 6, 2011

**Citation**

Mikkel Heuck, Philip Trøst Kristensen, and Jesper Mørk, "Energy-bandwidth trade-off in all-optical photonic crystal microcavity switches," Opt. Express **19**, 18410-18422 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18410

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### References

- M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature Mater.3, 211–219 (2004). [CrossRef]
- J. Y. Lee, L. H. Yin, G. P. Agraval, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express18, 11514–11523 (2010). [CrossRef] [PubMed]
- M. Waldow, T. Plotzing, M. Gottheil, M. Forst, and J. Bolten, “25 ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator,” Opt. Express16, 7693–7702 (2008). [CrossRef] [PubMed]
- C. Husko, A. De Rossi, S. Combré, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009). [CrossRef]
- K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics4, 477–483 (2010). [CrossRef]
- L. O’Faolain, D. M. Beggs, T. P. White, T. Kampfrath, K. Kuipers, and T. F. Krauss, “Compact optical switches and modulators based on dispersion engineered photonic crystals,” IEEE Photon. J.2, 404–414 (2010). [CrossRef]
- O. Wada, “Recent progress in semiconductor-based photonic signal-processing devices,” IEEE J. Sel. Top. Quantum Electron.17, 309–319 (2011). [CrossRef]
- P. A. Andrekson, H. Sunnerud, S. Oda, T. Nishitani, and J. Yang, “Ultrafast, atto-Joule switch using fiber parametric amplifier operated in saturation,” Opt. Express16, 10956–10961 (2008). [CrossRef] [PubMed]
- J. Xu, X. Zhang, and J. Mørk, “Investigation of patterning effects in ultrafast SOA-based optical switches,” IEEE J. Quantum Electron.46, 87–94 (2010). [CrossRef]
- J. B. Khurgin, “Performance of nonlinear photonic crystal devices at high bit rates,” Opt. Lett.30, 643–645 (2005). [CrossRef] [PubMed]
- M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett.28, 2506–2508 (2003). [CrossRef] [PubMed]
- J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightwave Technol.25, 2539–2546 (2007). [CrossRef]
- J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals, Molding the Flow of Light (Princeton University Press, 2008)
- H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A41, 5187–5198 (1990). [CrossRef] [PubMed]
- J. Mørk, F. Öhmann, and S. Bischoff, “Analytical expression for the bit error rate of cascaded all-optical regenerators,” Photon. Technol. Lett.15, 1479–1481 (2003). [CrossRef]
- R. W. Boyd, Nonlinear Optics (Academic Press, 2008)

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