## Critical conditions of control light for a silicon-based photonic crystal bistable switching |

Optics Express, Vol. 18, Issue 16, pp. 17313-17321 (2010)

http://dx.doi.org/10.1364/OE.18.017313

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### Abstract

In this paper, we investigate the critical conditions of control light for a silicon-based photonic crystal bistable switching. By establishing a time-dependent evolution equation, the critical pump power and pump time of the control light are derived, respectively. It is found that with the increase of the frequency detuning of the incident light, the critical power of the control light will rise, while the corresponding critical time will be shortened. It is also revealed that under the same conditions, the critical total power of the multiple-beam control light is less than the one of the single-beam control light. The theoretical predictions show perfect agreement with the simulation results.

© 2010 OSA

## 1. Introduction

1. K. Ogusu and K. Takayama, “Optical bistability in photonic crystal microrings with nonlinear dielectric materials,” Opt. Express **16**(10), 7525–7539 (2008). [CrossRef] [PubMed]

3. M. K. Kim, I. K. Hwang, S. H. Kim, H. J. Chang, and Y. H. Lee, “All-optical bistable switching in curved microfiber-coupled photonic crystal resonators,” Appl. Phys. Lett. **90**(16), 161118 (2007). [CrossRef]

3. M. K. Kim, I. K. Hwang, S. H. Kim, H. J. Chang, and Y. H. Lee, “All-optical bistable switching in curved microfiber-coupled photonic crystal resonators,” Appl. Phys. Lett. **90**(16), 161118 (2007). [CrossRef]

5. M. Belotti, J. F. Galisteo Lòpez, S. De Angelis, M. Galli, I. Maksymov, L. C. Andreani, D. Peyrade, and Y. Chen, “All-optical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities,” Opt. Express **16**(15), 11624–11636 (2008). [PubMed]

3. M. K. Kim, I. K. Hwang, S. H. Kim, H. J. Chang, and Y. H. Lee, “All-optical bistable switching in curved microfiber-coupled photonic crystal resonators,” Appl. Phys. Lett. **90**(16), 161118 (2007). [CrossRef]

6. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. **87**(15), 151112 (2005). [CrossRef]

## 2. Time-dependent evolution equation for a PC bistable switching: using a single-beam-CW control light

7. G. Priem, P. Bienstman, G. Morthier, and R. Baets, “Impact of absorption mechanisms on Kerr-nonlinear resonator behavior,” J. Appl. Phys. **99**(6), 063103 (2006). [CrossRef]

8. D. Brissinger, B. Cluzel, A. Coillet, C. Dumas, P. Grelu, and F. de Fornel, “Near-field control of optical bistability in a nanocavity,” Phys. Rev. B **80**(3), 033103 (2009). [CrossRef]

9. C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. **283**(14), 2957–2960 (2010). [CrossRef]

*ω*and initial phase (set to be zero in our case) with the CW signal light. When the signal light superposed with the control light is launched into the nonlinear cavity along WG1, according to the coupled mode theory (CMT) [10], the mode amplitude

*A*can be written aswhere

*s*

_{s},

*s*

_{x},

*s*

_{T}and

*s*

_{R}are the field amplitudes of the incident signal light, control light, transmitted light and reflective light, respectively,

*ω*

_{0}is the resonant frequency of the PC cavity, and

*p*

_{0}is the characteristic power [4

4. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **66**(5), 055601 (2002). [CrossRef]

*γ*

_{TPA}is proportional to the intensity of energy stored in the PC cavity,

*γ*

_{TPA}can be expressed as [9

9. C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. **283**(14), 2957–2960 (2010). [CrossRef]

*k*is defined as TPA coefficient which denotes the strength of the TPA process, and can be calculated by theory or simulation [9

9. C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. **283**(14), 2957–2960 (2010). [CrossRef]

*ω*is the frequency of the incident light. Thus, the instant values of the energy stored inside the cavity and the transmission rate can be easily calculated by using Eqs. (3) and (4) where

*t*

_{0}, then the instant photon energy stored in the cavity

## 3. Critical pump-power condition of the control light

### 3.1 Theory

*p*

_{1}; B is the up-jump point (B→C), and the corresponding incident power of the signal light is

*p*

_{2}. Alternatively, we can also fix the power of the signal light at a certain value within the bistable region, and trigger the switching to the “on” state by a control light. Here we should keep in mind that the jump from the “off” state to “on” state is indirect, e.g., the jump from A to D must pass through B and C points, although this process may be very quick.

*p*

_{2}. Thus we havei.e.,

*p*

_{1}. That is to say, if

*p*

_{x}is less than the critical value, one will not be able to trigger the bistable switching successfully even if the pump time of the control light is long enough.

### 3.2 FDTD simulation: verify the theoretical predictions

11. C. Li, J. F. Wu, and W. C. Xu, “Precise control of light frequency via a linear photonic crystal microcavity,” Appl. Opt. **49**(14), 2597–2600 (2010). [CrossRef]

^{−5}μm

^{2}/W, while the imaginary part is set to be 2.5 x 10

^{−6}μm

^{2}/W, denoting TPA effect. To measure the transmission rate, one monitor is set at the output port of WG2. The grid sizes in the horizontal and vertical directions are chosen to be

*a*/20 (where

*a*= 1μm is the lattice constant), and a perfectly matched layer (PML) of 1μm is employed as boundary. Thus, we can begin numerical simulation and focus on the TM modes. By employing nonlinear FDTD technique [12], it is not difficult to obtain the values of

*p*

_{1}and

*p*

_{2}(these values can also be obtained by theoretical calculations, as presented in Ref [9

**283**(14), 2957–2960 (2010). [CrossRef]

*δ*= 4.3478,

*p*

_{1}and

*p*

_{2}are calculated to be 0.120 kW/μm and 0.215 kW/μm, respectively. Thus, the critical pump power of the control light at the down-jump point (

*p*

_{s}=

*p*

_{1}) can be obtained immediately by using Eq. (8):

*p*

_{x}= 0.0136 kW/μm. To examine the precision of the critical value predicted by theory, we begin simulation by using

*p*

_{x}= 0.0136 kW/μm and

*p*

_{x}= 0.0134 kW/μm as the power of the CW control light, respectively, for comparison. When a CW signal light (

*δ*= 4.3478,

*p*

_{s}=

*p*

_{1}= 0.120 kW/μm) superposed with the control light (which has the same frequency and initial phase with the signal light) is launched into the PC switching along WG1, the time-dependent evolution process of the transmission rate recorded by a monitor is shown in Fig. 3 (since the critical pump time keeps unknown at present,to ensure a long enough pump time in this simulation, the pump time of the control light is selected to be 4000(

*a*/

*c*), about 13.33ps). One can see when

*p*

_{x}= 0.0136 kW/μm, the bistable switching is successfully triggered to the “on” state with

*T*= 0.6; while when

*p*

_{x}is slightly weakened to 0.0134 kW/μm, the switching can only works on the “off” state with

*T*= 0.054.

## 4. Critical pump-time condition of the control light

*T*

_{1}is the steady-state transmission rate at D point after the control light is shut off. Thus, we obtain the critical condition of the energy stored inside the cavity for the incident signal power

*p*

_{1}Obviously, Eqs. (8) and (9) can be readily generalized to the arbitrary signal power in the bistable region by substituting

*p*

_{1}with

*p*

_{s}.

*p*is the incident signal power within the bistable region and

_{s}*T*

_{s}is the corresponding steady transmission rate of “on” state.

*p*

_{s}=

*p*

_{1},

*T*

_{s}=

*T*

_{1}). In this case, we haveSubstituting formula (11) into Eq. (10), we obtainFrom Eq. (12) the desired critical pump time of the control light can be calculated. Noticing that the left part of Eq. (12) is actually the time-dependent transmission rate under the pump of the signal light and control light, Eq. (12) can be rewritten as a more brief form

*γ*are time dependent. Therefore, we have to fall back on a numerical method: plot the evolution curve of

*T*(

*t*) and a horizon line of

*T = T*

_{1}, and the first intersection point of them reads the critical value, as shown in Fig. 4(a) (

*δ*= 4.3478). One can see clearly that the critical pump time of the control light is

*t*

_{s}= 3522(

*a*/

*c*) [before we know this critical value, the pump time of the control light should be long enough, e.g., 4000(

*a*/

*c*) in our case]. To verify it, in Fig. 4(b) we make a FDTD simulation on the bistable switching when the control light is shut off at the critical value of

*t*

_{s}= 3522(

*a*/

*c*) (about 11.74ps). One can see that the bistable switching is successfully triggered to the “on” state with transmission rate of 0.6.

## 5. Critical condition of multiple-beam-CW control light and pulsed control light

*N*beams of CW control light, for convenience, we suppose these beams of control light have the same frequency and initial phase with the signal light. Therefore, the total power of the incident light iswhere

*i*th beam of control light.

*N*. Enlightened by this find, if the power of the control-light source in lab is unfortunate to be lower than the critical value, we still have opportunity to trigger the bistable switching by dividing the control light into several beams. Obviously, this method is promising in actual applications.

*p*

_{1}with

*p*

_{s}in Eqs. (12) and (15).

*f*(

*t*) is the envelopment function in time domain, and

*ω*

_{0}is the central frequency. By using Fourier transformation, the pulse can be approximately regarded as a superposition of a series of CWs with frequencies near

*ω*

_{0}and appropriate weights:where

*ω*

_{i}and

*g*(

*ω*

_{i}) are the frequency and field amplitude of the

*i*th CW component, respectively, Δ

*ω*is the step width in frequency domain. Thus, we can use “|

*g*(

*ω*

_{i})|

^{2}Δ

*ω*” as the corresponding weight factor, and a smaller Δ

*ω*will lead to a more precise result.

*ω*

_{0}. In this case, the problem can be converted to the one in the multiple-beam-CW-control-light case, which has been above discussed. While for an ultrashort control pulse (e.g, several fs), the method presented above is not a good approximation anymore (due to the further broadening of the frequency spectrum), and a more precise model including multiple-beam-CW control light with deferent frequencies is required. However, we can still predict that the critical pump power of CWs with deferent frequencies must be greater than the one with same frequency, as has been testified by our FDTD simulations.

## 6. Conclusion

## Acknowledgments

## References and links

1. | K. Ogusu and K. Takayama, “Optical bistability in photonic crystal microrings with nonlinear dielectric materials,” Opt. Express |

2. | M. F. Yanik, S. H. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. |

3. | M. K. Kim, I. K. Hwang, S. H. Kim, H. J. Chang, and Y. H. Lee, “All-optical bistable switching in curved microfiber-coupled photonic crystal resonators,” Appl. Phys. Lett. |

4. | M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

5. | M. Belotti, J. F. Galisteo Lòpez, S. De Angelis, M. Galli, I. Maksymov, L. C. Andreani, D. Peyrade, and Y. Chen, “All-optical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities,” Opt. Express |

6. | T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. |

7. | G. Priem, P. Bienstman, G. Morthier, and R. Baets, “Impact of absorption mechanisms on Kerr-nonlinear resonator behavior,” J. Appl. Phys. |

8. | D. Brissinger, B. Cluzel, A. Coillet, C. Dumas, P. Grelu, and F. de Fornel, “Near-field control of optical bistability in a nanocavity,” Phys. Rev. B |

9. | C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. |

10. | H. A. Haus, |

11. | C. Li, J. F. Wu, and W. C. Xu, “Precise control of light frequency via a linear photonic crystal microcavity,” Appl. Opt. |

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

**OCIS Codes**

(190.1450) Nonlinear optics : Bistability

(230.1150) Optical devices : All-optical devices

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: July 12, 2010

Manuscript Accepted: July 22, 2010

Published: July 29, 2010

**Citation**

Chao Li, Hong Wang, Jun-Fang Wu, and Wen-Cheng Xu, "Critical conditions of control light for a silicon-based photonic crystal bistable switching," Opt. Express **18**, 17313-17321 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17313

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### References

- K. Ogusu and K. Takayama, “Optical bistability in photonic crystal microrings with nonlinear dielectric materials,” Opt. Express 16(10), 7525–7539 (2008). [CrossRef] [PubMed]
- M. F. Yanik, S. H. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83(14), 2739–2741 (2003). [CrossRef]
- M. K. Kim, I. K. Hwang, S. H. Kim, H. J. Chang, and Y. H. Lee, “All-optical bistable switching in curved microfiber-coupled photonic crystal resonators,” Appl. Phys. Lett. 90(16), 161118 (2007). [CrossRef]
- M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601 (2002). [CrossRef]
- M. Belotti, J. F. Galisteo Lòpez, S. De Angelis, M. Galli, I. Maksymov, L. C. Andreani, D. Peyrade, and Y. Chen, “All-optical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities,” Opt. Express 16(15), 11624–11636 (2008). [PubMed]
- T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. 87(15), 151112 (2005). [CrossRef]
- G. Priem, P. Bienstman, G. Morthier, and R. Baets, “Impact of absorption mechanisms on Kerr-nonlinear resonator behavior,” J. Appl. Phys. 99(6), 063103 (2006). [CrossRef]
- D. Brissinger, B. Cluzel, A. Coillet, C. Dumas, P. Grelu, and F. de Fornel, “Near-field control of optical bistability in a nanocavity,” Phys. Rev. B 80(3), 033103 (2009). [CrossRef]
- C. Li, J. F. Wu, and W. C. Xu, “Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity,” Opt. Commun. 283(14), 2957–2960 (2010). [CrossRef]
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).
- C. Li, J. F. Wu, and W. C. Xu, “Precise control of light frequency via a linear photonic crystal microcavity,” Appl. Opt. 49(14), 2597–2600 (2010). [CrossRef]
- A. Taflove, and S. C. Hagness, Computational Electrodynamics (Artech House, Norwood, MA, 2000)

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