## Low-threshold bistability in nonlinear microring tower resonator |

Optics Express, Vol. 18, Issue 25, pp. 25509-25518 (2010)

http://dx.doi.org/10.1364/OE.18.025509

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### Abstract

Microring tower resonators, which are a chain of microring resonators stacked on top of each other, are of great interest for nonlinear optics due to their unique features such as very high compactness, coupling efficiency and quality factor. In this research, we investigate the optical bistability in microring tower (MRT) with Kerr nonlinearity by using the coupled mode theory, and demonstrate how a proper defect into the structure can lead to low threshold bistability. In particular, we observed optical bistability in nonlinear defect modes with switching power as low as

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## 1. Introduction

5. 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), 055601–055604 (2002). [CrossRef]

6. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

8. M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express **13**(17), 6354–6375 (2005). [CrossRef] [PubMed]

9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. **48**(31), G148–G155 (2009). [CrossRef] [PubMed]

9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. **48**(31), G148–G155 (2009). [CrossRef] [PubMed]

*Q*factor. In passing, we derive the nonlinear coupled wave equations for light propagation along the structure. Finally, the resulting equations are solved numerically. Calculations show that optical bistability can be formed in the structure with a very low input power of

## 2. Linear response

### 2.1. Simple model for fast calculation of cavity- modes

10. D. N. Christodoulides and E. D. Eugenieva, “Minimizing bending losses in two-dimensional discrete soliton networks,” Opt. Lett. **26**(23), 1876–1878 (2001). [CrossRef]

*R*. Making use of coupled mode theory, the fields can be described by following equations for the amplitudes

^{th}ring [9

9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. **48**(31), G148–G155 (2009). [CrossRef] [PubMed]

^{th}ring,

^{th}ring satisfies the condition of closure of rings as follows:where

*ξ*is effective propagation constant along the vertical direction,

*d*is the period of the structure [9

**48**(31), G148–G155 (2009). [CrossRef] [PubMed]

*t*and

*r*, to be determined, are transmission and reflection coefficients associated with a forward and reflected wave, respectively. Substituting Eq. (4) into Eqs. (3) and applying some straightforward manipulations, one arrives at the following equations for the coefficients

*r*and

*q*:whereIn steady state, we can also expect that the defect modes assure the conditions of ring closure and therefore satisfy the following dispersion relation [9

**48**(31), G148–G155 (2009). [CrossRef] [PubMed]

*l*is an integer number. Note that for defect modes placed in the bandgap, the propagation constant

*ξ*is imaginary. Inserting the expression for

*t*and

*r*are infinitely large, so Eq. (4) shows that the field is localized at the defect position and decays exponentially away with the rate of

**48**(31), G148–G155 (2009). [CrossRef] [PubMed]

**48**(31), G148–G155 (2009). [CrossRef] [PubMed]

### 2.1. Influence of cavity parameters on the Q factor

*Q*factor to the cavity geometry, we first study the influence of

*Q*factor. The results are shown in Fig. 3(a) where we consider a uniform MRT with 9 rings and coupling parameters

*Q*factor is higher. This can readily explain by considering the position of the defect modes in the bandgap. By increasing the defect size, the position of the defect modes goes far from the band edges. As the defect mode approaches the band edges, the linewidth broadens so that the defect mode has a smaller

*Q*. Next, we study the influence of the coupling parameters. Calculation show that if the coupling parameter between adjacent microrings is equal to the coupling parameter between input/output waveguides and adjacent microring, i. e.

## 3. Nonlinear response

### 3.1. Coupled wave equations

**48**(31), G148–G155 (2009). [CrossRef] [PubMed]

10. D. N. Christodoulides and E. D. Eugenieva, “Minimizing bending losses in two-dimensional discrete soliton networks,” Opt. Lett. **26**(23), 1876–1878 (2001). [CrossRef]

*α*is the loss (or gain) per unit length in the ring. Therefore, using conventional nonlinear coupled mode theory in the presence of the optical Kerr effect, the following nonlinear coupled equations are obtained [11, 12

12. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. **18**(10), 1580–1583 (1982). [CrossRef]

*ρ*denote the intensity-insertion-loss coefficient [11]. In the following, for simplicity we assume that the ingoing amplitude

13. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**(9), 794–796 (1988). [CrossRef] [PubMed]

14. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. **81**(16), 3383–3386 (1998). [CrossRef]

### 3. 2. Nonlinear transmission properties

**48**(31), G148–G155 (2009). [CrossRef] [PubMed]

**A**collect the constant coefficient as well as terms proportional to

**b**is a vector which depends on the boundary conditions. Because the matrix elements of

**A**are not constant and depend on the solution as well, this is not an ordinary linear system and therefore cannot be solved directly. It is, however, possible to find a solution iteratively. Substituting an initial guess for

**A**are known, therefore, Eq. (13) transforms to an ordinary system that can be solved for x. An appropriate initial guess

15. A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. **62**(20), 2465–2467 (1993). [CrossRef]

*Q*-factor defect modes. An example is shown in Fig. 6 , where we consider an NMRT consisting of 9 rings with coupling parameters

*Q*factor. As mentioned above, the Q factor is very sensitive to the number of the rings, defect size,

16. A. Shinya, S. Matsuo, T. Yosia, E. Tanabe, T. Kuramochi, T. Sato, Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express **16**(23), 19382–19387 (2008). [CrossRef]

18. E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. **90**(10), 101–118 (2007). [CrossRef]

19. G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express **13**(23), 9623–9628 (2005). [CrossRef] [PubMed]

20. N. G. R. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express **16**(20), 16247–16254 (2008). [CrossRef] [PubMed]

20. N. G. R. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express **16**(20), 16247–16254 (2008). [CrossRef] [PubMed]

*Q*defect modes. By modifying the structure, it is possible even more to reduce the bistability threshold. For example, to increase the Q factor and decrease the bistability threshold further, we change the dimensionless propagation constant of the first and last ring in the previous configuration by the amount of

## 4. Effects of the loss upon device performance

6. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

21. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B **21**(9), 1665–1673 (2004). [CrossRef]

23. F. Xu, P. Horak, and G. Brambilla, “Optical microfiber coil resonator refractometric sensor,” Opt. Express **15**(12), 7888–7893 (2007). [CrossRef] [PubMed]

*between the quality factor and the loss (especially for*

24. L. D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, “Extremely low power optical bistability in silicon demonstrated using 1D photonic crystal nanocavity,” Opt. Express **17**(23), 21108–21117 (2009). [CrossRef] [PubMed]

*α*in the example of the Fig. 7(a) is

22. M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express **12**(10), 2303–2316 (2004). [CrossRef] [PubMed]

23. F. Xu, P. Horak, and G. Brambilla, “Optical microfiber coil resonator refractometric sensor,” Opt. Express **15**(12), 7888–7893 (2007). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgment

## References and links

1. | H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Orlando, 1985). |

2. | 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 |

3. | A. Hurtado, A. Quirce, A. Valle, L. Pesquera, and M. J. Adams, “Power and wavelength polarization bistability with very wide hysteresis cycles in a 1550 nm-VCSEL subject to orthogonal optical injection,” Opt. Express |

4. | T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. |

5. | 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. |

6. | B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. |

7. | K. Sakoda, |

8. | M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express |

9. | M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. |

10. | D. N. Christodoulides and E. D. Eugenieva, “Minimizing bending losses in two-dimensional discrete soliton networks,” Opt. Lett. |

11. | K. Okamoto, |

12. | S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. |

13. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

14. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. |

15. | A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. |

16. | A. Shinya, S. Matsuo, T. Yosia, E. Tanabe, T. Kuramochi, T. Sato, Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express |

17. | H. Zhang, V. Gauss, P. Wen, and S. Esener, “Observation of wavelength and multiple bistabilities in 850nm Vertical-Cavity Semiconductor Optical Amplifiers (VCSOAs),” Opt. Express |

18. | E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. |

19. | G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express |

20. | N. G. R. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express |

21. | J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B |

22. | M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express |

23. | F. Xu, P. Horak, and G. Brambilla, “Optical microfiber coil resonator refractometric sensor,” Opt. Express |

24. | L. D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, “Extremely low power optical bistability in silicon demonstrated using 1D photonic crystal nanocavity,” Opt. Express |

**OCIS Codes**

(190.1450) Nonlinear optics : Bistability

(230.4320) Optical devices : Nonlinear optical devices

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 23, 2010

Revised Manuscript: November 11, 2010

Manuscript Accepted: November 12, 2010

Published: November 22, 2010

**Citation**

Mehdi Shafiei and Mohammad Khanzadeh, "Low-threshold bistability in nonlinear microring tower resonator," Opt. Express **18**, 25509-25518 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25509

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### References

- H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Orlando, 1985).
- 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(7), 2678–2687 (2005). [CrossRef] [PubMed]
- A. Hurtado, A. Quirce, A. Valle, L. Pesquera, and M. J. Adams, “Power and wavelength polarization bistability with very wide hysteresis cycles in a 1550 nm-VCSEL subject to orthogonal optical injection,” Opt. Express 17(26), 23637–23642 (2009). [CrossRef]
- T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. 30(19), 2575–2577 (2005). [CrossRef] [PubMed]
- 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), 055601–055604 (2002). [CrossRef]
- B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]
- K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
- M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005). [CrossRef] [PubMed]
- M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009). [CrossRef] [PubMed]
- D. N. Christodoulides and E. D. Eugenieva, “Minimizing bending losses in two-dimensional discrete soliton networks,” Opt. Lett. 26(23), 1876–1878 (2001). [CrossRef]
- K. Okamoto, Fundamentals of Optical Waveguides, (Elsevier, 2006), Chap. 4.
- S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. 18(10), 1580–1583 (1982). [CrossRef]
- D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13(9), 794–796 (1988). [CrossRef] [PubMed]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998). [CrossRef]
- A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993). [CrossRef]
- A. Shinya, S. Matsuo, T. Yosia, E. Tanabe, T. Kuramochi, T. Sato, Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express 16(23), 19382–19387 (2008). [CrossRef]
- H. Zhang, V. Gauss, P. Wen, and S. Esener, “Observation of wavelength and multiple bistabilities in 850nm Vertical-Cavity Semiconductor Optical Amplifiers (VCSOAs),” Opt. Express 15(18), 11723–11730 (2007). [CrossRef] [PubMed]
- E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007). [CrossRef]
- G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005). [CrossRef] [PubMed]
- N. G. R. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express 16(20), 16247–16254 (2008). [CrossRef] [PubMed]
- J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21(9), 1665–1673 (2004). [CrossRef]
- M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12(10), 2303–2316 (2004). [CrossRef] [PubMed]
- F. Xu, P. Horak, and G. Brambilla, “Optical microfiber coil resonator refractometric sensor,” Opt. Express 15(12), 7888–7893 (2007). [CrossRef] [PubMed]
- L. D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, “Extremely low power optical bistability in silicon demonstrated using 1D photonic crystal nanocavity,” Opt. Express 17(23), 21108–21117 (2009). [CrossRef] [PubMed]

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