## Asymmetric Fano resonance and bistability for high extinction ratio, large modulation depth, and low power switching

Optics Express, Vol. 14, Issue 26, pp. 12770-12781 (2006)

http://dx.doi.org/10.1364/OE.14.012770

Acrobat PDF (256 KB)

### Abstract

We propose a two-ring resonator configuration that can provide optical switching with high extinction ratio (ER), large modulation depth (MD) and low switching threshold, and compare it with two other conventional one-ring configurations. The achievable input threshold is n_{2}I_{IN} ~10^{-5}, while maintaining a large ER (> 10dB) and MD (~ 1) over a 10-GHz (0.1 nm) optical bandwidth. This performance can also be achieved by the ring-enhanced Mach-Zehnder interferometer, and is one to two orders of magnitude better than the simple bus-coupled one-ring structures, because of the use of asymmetric Fano resonance as opposed to the usual symmetric resonance of a single ring. The sharpness and the asymmetricity of the Fano resonance are linked to the low switching threshold and the high extinction ratio, respectively, and also accounts for the different dependence on ring dimensions between the one- and two-ring structures.

© 2006 Optical Society of America

## 1. Introduction

^{4}) and low loss [1–3

1. B.E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. **16**, 2263–2265 (2004). [CrossRef]

4. D. A. B. Miller, “Refractive Fabry-Perot Bistability with Linear Absorption: Theory of Operation and Cavity optimization,” IEEE J. Quantum Electron , **17**, 306–311 (1981). [CrossRef]

5. F. Sanchez, “Optical bistability in a 2×2 coupler fiber ring resonator: parametric formulation,” Opt. Commun. **142**, 211 (1997). [CrossRef]

7. Y. Dumeige and P. Feron, “Dispersive tristability in microring resonator,” Phys. Rev. E, 72 066609 (2005). [CrossRef]

8. J. Danckaert, K. Fobelets, and I. Veretennicoff, “Dispersive optical bistability in stratified structures,” Phys. Rev. B , **44**, 15, 8214 (1991). [CrossRef]

9. B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic crystal resonators,” J. Opt. Soc. Am. B **22**(8), 1778–1784 (2005). [CrossRef]

10. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. **29**, 2387–2389 (2004). [CrossRef] [PubMed]

11. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A , **20**(3), 569–572 (2003). [CrossRef]

12. Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavityresonator-coupled Mach Zehnder Interferometer,” Opt. Lett. **30**, 3069–3071 (2005). [CrossRef] [PubMed]

13. L. B. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, “Tunable delay line with interacting whispering-gallery-mode resonator,” Opt. Lett. **29**, 626 (2004). [CrossRef] [PubMed]

_{2}I

_{IN}~10

^{-6}), high extinction ratio (>30dB) and large modulation depth (~1). We also consider their sensitivity to wavelength and show that a larger optical bandwidth can be obtained but at the expense of switching threshold.

12. Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavityresonator-coupled Mach Zehnder Interferometer,” Opt. Lett. **30**, 3069–3071 (2005). [CrossRef] [PubMed]

## 2. Bistability in a one-ring configuration

15. Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix Analysis of 2-D Micro-resonator Lattice Optical Filters,” IEEE J. Quantum Electronics **41**, 1410–1418 (2005). [CrossRef]

*k*

_{0}=2π/λ,

*n*

_{eff}is the waveguide effective index,

_{R}/I

_{IN}is given by

_{IN}is the input intensity to the bus waveguide. We have assumed that the nonlinear absorption is small compared with the nonlinear refraction, as is usually done [4

4. D. A. B. Miller, “Refractive Fabry-Perot Bistability with Linear Absorption: Theory of Operation and Cavity optimization,” IEEE J. Quantum Electron , **17**, 306–311 (1981). [CrossRef]

_{IN}, and consequently with Eq. (1), the transmission can be plotted as a function of I

_{IN}. This approach is analogous to the parametric formulation [6

6. Y. Dumeige, D. Arnaud, and P. Feron, “Combining FDTD with coupled mode theories for bistability in microring resonators,” Opt. Commun. 250 (2005) 376–383. [CrossRef]

^{2}, Eq. (3) is a cubic equation in I

_{R}, and hence, for a suitable value of I

_{IN}, there can exist three real solutions.

*r*=

*a*). These examples are for the case where

*n*

_{2}is negative, as is the case for polymers [19

19. B. L. Lawrence, M. Cha, W. E. Torruellas, G. I. Stegeman, S. Etemad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64 (1994), 2773. [CrossRef]

4. D. A. B. Miller, “Refractive Fabry-Perot Bistability with Linear Absorption: Theory of Operation and Cavity optimization,” IEEE J. Quantum Electron , **17**, 306–311 (1981). [CrossRef]

*ω*-

*ω*

_{o}=Δ

*ω*

_{FWHM}√3/2, where

*ω*

_{0}is the resonance frequency and Δω

_{FWHM}is the FWHM linewidth of the resonator. The critical detuning is the point where the build-up resonant function (Eq. (3)) has the maximum slope. If this requirement is met, then bistability occurs as the incident power is increased because of the nonlinear shift of the resonance towards λ, causing the intra-cavity power to build up, which in turn hastens the shift, leading to an unstable situation. This positive feedback increases the slope of the leading edge of the transmission spectrum relative to the linear case [17

17. A.R. Cowan and J.F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E **68**046606 (2003) [CrossRef]

_{IN}reduces the build-up factor). This is the

*turn-off*point indicated by the down arrow.

*turn-on*point. Note that the minimum transmission is zero only under critical coupling. The ON/OFF ratio at the turn-off point is generally smaller than that at turn-on, and determines the extinction ratio (ER) of the switch if the power is held here in the OFF state. Hence, to maximize the ER and modulation depth, it is important to minimize the off-state transmission

*at*the turn-off point, while maximizing the on-state transmission at the initial detuning. To align the minimum transmission point closer to the turn-off point, one way is to operate very close to the critical detuning, but the problem with this is that the ON transmission amplitude is also substantially reduced (hence the modulation depth is small), and the difference between the turn-on and turn-off powers becomes very small. Similar effect may be achieved by increasing the round-trip loss, as shown in Fig. 2(b). This is because increasing the loss broadens the resonance and increases the critical detuning, which has the same effect as having the operating wavelength closer to critical detuning. However, the broadened linewidth increases the switching threshold, as expected because of the smaller build-up factor. To maximize the build-up factor at the given loss, one must further satisfy the critical coupling condition

*r*=

*a*. Under this condition the achievable input threshold is in the order

*n*

_{2}I

_{IN}~10

^{-4}.

## 3. Ring-enhanced Mach-Zehnder Interferometer

*asymmetric*resonance such as the Fano resonance. Fano resonance is a result of interference between two pathways [11

11. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A , **20**(3), 569–572 (2003). [CrossRef]

*r*=

*a*). Under this condition, the outputs at the two arms are given approximately by

*T*

_{BAR}=sin

^{2}(

*φ*-Δ

*ϕ*)/2,

*T*

_{CROSS}=cos

^{2}(

*φ*-Δ

*ϕ*)2, where

*T*

_{cross}, near resonance. The resonance is very sharp as we have assumed

*r*=0.95. A large

*r*is desirable as a sharp resonance minimizes the switching threshold. When the phase bias is zero, the output is a symmetric function of

*δ*. As Δϕ is increased, the asymmetricity increases and the Fano resonance shifts slightly to the left. In the lossless case (

*a*=1), by expanding Eq. (4) it can be shown that the shift of the minimum transmission point (where

*T*

_{CROSS}=0) is approximately given by

*δ*

_{MIN}=-Δ

*ϕ*(1-

*r*)/(1+

*r*), i.e., the shift is

*reduced*, relative to the phase bias, by the maximum build-up factor in the ring. Far away from resonance, the transmission approaches asymptotically the value given by cos

^{2}(Δ

*ϕ*/2).

## 4. The Two-ring Configuration

*r*

_{1}, and that between the two rings,

*r*

_{2}, may be different, and this variability can be used to optimize the asymmetricity of the Fano resonance to achieve maximum ER, just as Δϕ does in the case of REMZI. As a four-port device, there are two possible output ports, denoted as “drop” (D) and “through” (T), respectively, and the switch can function in a pump and probe configuration [18

18. V. Van, T. A. Ibrahim, P.P. Absil, F. G. Johnson, R. Grover, and P-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Quantum Electron. **8**, 705–713 (2002). [CrossRef]

14. I. Chremmos and N. Uzunoglu, “Reflective properties of double-ring resonator system coupled to a waveguide,” IEEE Photon. Technol. Lett, 17, 2110–2112, 2005 [CrossRef]

*t*|

^{2}, and D=|

*d*|

^{2}, where

*T*

_{2}=

*E*

_{3}/

*E*

_{3}=[

*r*

_{2}-

*a*

_{2}exp(-

*iδ*

_{2})]/[1-

*a*

_{2}

*r*

_{2}exp(-

*iδ*

_{2})] is the factor that incorporates the feedback and interference effect from the upper ring. Note that

*T*

_{2}is the same as the transmittivity for a single ring coupled to one bus [

*cf.*Eq. (1)]. The coupling between the two resonators causes a splitting of the ring resonance. If the rings are slightly different then the splitting will be asymmetric. This can be shown as follows.

*T*

_{2}=|

*T*

_{2}|exp(

*iϕ*), and Eq. (5) may be written as

*a*=

*a*

_{1}|

*T*

_{2}|, and

*δ*represent the loading effect of the upper ring, for which the linear round trip phase is written as δ

_{2}=γδ

_{1}, where γ is the ratio of the linear round-trip phases in the two rings. Eq. (8) is plotted in Fig. 8(a) as a function of δ

_{1}for two values of γ, and Fig. 8(b) shows the resulting resonance splitting in the T spectra. In the T spectra the resonances appear as minima. Similar to a single-bus ring, the upper ring exhibits a nonlinear 2π phase shift centered at resonance (δ

_{2}=2πm). When the upper ring is off-resonance, the loading phase is zero meaning that it has no effect on the lower ring, hence the lower ring behaves as a simple dual-bus ring, giving a maximum T when the ring is off-resonance (i.e., δ

_{1}is an odd multiple of π). However, when the upper ring is on resonance, the nonlinear phase loading kicks in and the loaded lower ring becomes resonant when δ is an even multiple of π. This occurs at two points, giving rise to the two minima in T. When γ=1, the two resonances occur on the linear part of the phase hence they are symmetric. When γ≠1 one of the resonances occurs near the nonlinear part of the phase and becomes the asymmetric Fano resonance. The two minima are shifted unequally when γ departs from 1.

*γ*=1 the field is confined equally in both rings. However, closer inspection shows that the lower frequency resonance has a symmetric field profile at the coupling point, while the higher frequency resonance has an antisymmetric profile. As

*γ*is increased (decreased), the split resonances shift asymmetrically to the right (left). The sharp, faster-shifting, asymmetric resonance originates primarily from the upper ring (as shown by the field distribution), while the broader resonance is associated mainly with the bottom ring. We shall refer to the sharp asymmetric resonance as the Fano resonance as it is induced by the strong effect of the upper ring, and the broader resonance as the main resonance associated with the lower ring.

*r*

_{1}. This is more evident in the D spectra, as shown in Fig. 10. When

*r*

_{1}is reduced the Fano resonance becomes more asymmetric. Note that while the right edges of the Fano resonances follow the envelope of the main resonance, the steepness on the left edges is relatively stable due to the proximity of the minimum in D, which location is independent of

*r*

_{1}. The amplitude at the dip changes somewhat for different

*r*

_{1}but remains small. By operating at a wavelength just before the “dip”, it is possible to achieve switching with high extinction ratio, large modulation depth and low threshold power, as shown below.

6. Y. Dumeige, D. Arnaud, and P. Feron, “Combining FDTD with coupled mode theories for bistability in microring resonators,” Opt. Commun. 250 (2005) 376–383. [CrossRef]

*E*

_{9}. The set of equations required to solve the drop and through amplitudes are

*k*

_{0}

*n*

_{eff}

*n*

_{2}

*cε*

_{o}[1-exp(

*αL*

_{mn})]/(2

*α*) and

*E*

_{i}|

^{2}is the cumulative nonlinear phase over the path

*L*

_{mn}from point

*m*to point

*n*. In the linear case,

*γ*

_{NL}is set to zero. It can be seen that by fixing E

_{9}, we can obtain E

_{2}and E

_{3}, and thus T

_{2}, which contains the phase perturbation due to the upper ring. From |E

_{2}|

^{2}we get |E

_{1}|

^{2}, which leads to the Through (T=|E

_{T}/E

_{IN}|

^{2}) and the Drop (D=|E

_{D}/E

_{IN}|

^{2}) transmission. To calculate the nonlinear response, the wavelength is fixed while

*E*

_{9}is varied from 0 to ∞. The nonlinear response is sensitive to wavelength, and bistability exists only for certain wavelengths which depend on the other design parameters (γ,

*r*

_{1}and

*r*

_{2}). In the following, we assume that the lower ring has a fixed radius of 15µm.

*n*

_{2}I

_{IN}for the asymmetric case (γ=1.05). In (a) we show the dependence on wavelength, with

*r*

_{1,2}=0.85 and

*a*

_{1,2}=0.99. Bistablity exists for the two curves on the right. It can be seen that the ER is greater than 10 dB even if the wavelength changes by 0.1 nm. With careful wavelength tuning, an ER of 20 dB can be achieved. In (b), we show the dependence on asymmetricity. We make

*r*

_{1}and

*r*

_{2}different so that the Fano resonance becomes more asymmetric (other parameters are λ=1555 nm and

*a*=0.999). It is evident that the more asymmetric Fano resonance (

*r*

_{1}≠

*r*

_{2}) gives a much larger extinction ratio compared with the case where

*r*

_{1}=

*r*

_{2}. In fact, the latter is similar to the one-ring case shown in Fig. 2(b). We further note that the extinction ratio generally decreases when

*r*approaches

*a*. In fact, the dotted curve shows that under the critical coupling condition (when

*r*

_{2}=

*a*), bistability disappears as power is entirely absorbed in the upper ring resonance and the Fano interaction between the two rings is quenched. Hence, critical coupling is undesirable for the two-ring configuration, unlike in the single-ring case.

*a*). This can be shown by taking the maximum and minimum values of Eqs. (6) and (7). In the simplest case where the loss in the upper ring is negligible (i.e., |T

_{2}|=1), this gives:

*T*

_{2}|=1, are plotted as a function of

*a*

_{1}in Fig. 12 for several values of

*r*

_{1}(while

*r*

_{2}is fixed). Note that the ER for T decreases, while that for D increases, with increasing

*r*

_{1}. The Drop port (D) generally has a higher ER, but is exceeded by the Through port (T) when the loss is sufficiently small. For

*a*=0.999, the ER can be as high as 40dB. Such a low loss is possible, as demonstrated by some recent reports., First, a very high order multi-ring filters with very high Q resonators have been realized using low-loss Hydex material [1

1. B.E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. **16**, 2263–2265 (2004). [CrossRef]

20. C. Y. Chao and L. J. Guo, “Reduction of surface scattering loss in polymer mirorings using thermal-reflow technique,” IEEE Photon. Technol. Lett. **16**, 1498–1500 (2004). [CrossRef]

3. J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-qualityfactor silicon-on-insulator microring resonator”, Opt. Lett. **29**, 2861–2863 (2004). [CrossRef]

*r*in Fig. 10), which means that the slope is relatively unchanged with cavity size. There are also more parameters (such as

*r*

_{1},

*r*

_{2}, and γ) that can be used to optimize the performance. A comparison between the two cases, for various ring sizes (where 1× refers to 15 µm radius), is shown in Fig. 13. Note that a two-ring structure with smaller rings still has lower threshold than a one-ring configuration with a larger ring. This clearly shows that the two-ring devices generally have much better performance than the onering configuration, and are not constrained by the conventional size dependence. The switching threshold can be one order of magnitude smaller, while the ON transmission is higher giving a better modulation depth. This winning edge is due to a tunable, narrow and highly asymmetric Fano resonance made possible by the field dynamics in a two-cavity structure.

## 5. Conclusion

*n*

_{2}I

_{IN}~10

^{-5}) over a signal bandwidth of 0.1 nm (10 GHz) can be simultaneously achieved by both the ring-enhanced MZI and the simple two-ring configuration. In contrast, for the one-ring case, both one-bus and two-bus, although they can achieve a similar ER, they inevitably fare poorer in terms of modulation depth, switching threshold and signal bandwidth.

*n*

_{2}=5×10

^{-12}cm

^{2}/W [19

19. B. L. Lawrence, M. Cha, W. E. Torruellas, G. I. Stegeman, S. Etemad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64 (1994), 2773. [CrossRef]

^{2}, n

_{2}I

_{IN}~ 10

^{-6}implies a threshold power of 2mW. This theoretical value is 30x smaller compared with the theoretical result of Dumeige which is based on the dual-bus one-ring structure [7

7. Y. Dumeige and P. Feron, “Dispersive tristability in microring resonator,” Phys. Rev. E, 72 066609 (2005). [CrossRef]

21. 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). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2678 [CrossRef] [PubMed]

## References and Links

1. | B.E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. |

2. | T. A. Ibrahim, R. Grover, L. -C. Kuo, S. Kanakaraju, L. C. Calhoun, and P. -T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. |

3. | J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-qualityfactor silicon-on-insulator microring resonator”, Opt. Lett. |

4. | D. A. B. Miller, “Refractive Fabry-Perot Bistability with Linear Absorption: Theory of Operation and Cavity optimization,” IEEE J. Quantum Electron , |

5. | F. Sanchez, “Optical bistability in a 2×2 coupler fiber ring resonator: parametric formulation,” Opt. Commun. |

6. | Y. Dumeige, D. Arnaud, and P. Feron, “Combining FDTD with coupled mode theories for bistability in microring resonators,” Opt. Commun. 250 (2005) 376–383. [CrossRef] |

7. | Y. Dumeige and P. Feron, “Dispersive tristability in microring resonator,” Phys. Rev. E, 72 066609 (2005). [CrossRef] |

8. | J. Danckaert, K. Fobelets, and I. Veretennicoff, “Dispersive optical bistability in stratified structures,” Phys. Rev. B , |

9. | B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic crystal resonators,” J. Opt. Soc. Am. B |

10. | V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. |

11. | S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A , |

12. | Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavityresonator-coupled Mach Zehnder Interferometer,” Opt. Lett. |

13. | L. B. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, “Tunable delay line with interacting whispering-gallery-mode resonator,” Opt. Lett. |

14. | I. Chremmos and N. Uzunoglu, “Reflective properties of double-ring resonator system coupled to a waveguide,” IEEE Photon. Technol. Lett, 17, 2110–2112, 2005 [CrossRef] |

15. | Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix Analysis of 2-D Micro-resonator Lattice Optical Filters,” IEEE J. Quantum Electronics |

16. | A. Yariv, “Critical coupling and its control in optical waveguide-resonator systems,” IEEE Photon. Technol. Lett. |

17. | A.R. Cowan and J.F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E |

18. | V. Van, T. A. Ibrahim, P.P. Absil, F. G. Johnson, R. Grover, and P-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Quantum Electron. |

19. | B. L. Lawrence, M. Cha, W. E. Torruellas, G. I. Stegeman, S. Etemad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64 (1994), 2773. [CrossRef] |

20. | C. Y. Chao and L. J. Guo, “Reduction of surface scattering loss in polymer mirorings using thermal-reflow technique,” IEEE Photon. Technol. Lett. |

21. | 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 |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(190.1450) Nonlinear optics : Bistability

(230.5750) Optical devices : Resonators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: October 18, 2006

Revised Manuscript: December 1, 2006

Manuscript Accepted: December 6, 2006

Published: December 22, 2006

**Citation**

Landobasa Y. Mario, S. Darmawan, and Mee K. Chin, "Asymmetric Fano resonance and bistability for high extinction ratio, large modulation depth, and low power switching," Opt. Express **14**, 12770-12781 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-12770

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### References

- B.E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004). [CrossRef]
- T. A. Ibrahim, R. Grover, L. -C. Kuo, S. Kanakaraju, L. C. Calhoun, P. -T. Ho, "All-optical AND/NAND logic gates using semiconductor microresonators," IEEE Photon. Technol. Lett. 15, 1422-1424 (2003). [CrossRef]
- J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, "Ultrahigh-quality-factor silicon-on-insulator microring resonator", Opt. Lett. 29, 2861-2863 (2004). [CrossRef]
- D. A. B. Miller, "Refractive Fabry-Perot Bistability with Linear Absorption: Theory of Operation and Cavity optimization," IEEE J. Quantum Electron, 17, 306-311 (1981). [CrossRef]
- F. Sanchez, "Optical bistability in a 2x2 coupler fiber ring resonator: parametric formulation," Opt. Commun. 142, 211 (1997). [CrossRef]
- Y. Dumeige, D. Arnaud, P. Feron, "Combining FDTD with coupled mode theories for bistability in micro-ring resonators," Opt. Commun. 250 (2005) 376-383. [CrossRef]
- Y. Dumeige, P. Feron, "Dispersive tristability in microring resonator," Phys. Rev. E, 72066609 (2005). [CrossRef]
- J. Danckaert, K. Fobelets, I. Veretennicoff, "Dispersive optical bistability in stratified structures," Phys. Rev. B, 44, 15, 8214 (1991). [CrossRef]
- B. Maes, P. Bienstman, R. Baets, "Switching in coupled nonlinear photonic crystal resonators," J. Opt. Soc. Am. B 22(8), 1778-1784 (2005). [CrossRef]
- V. R. Almeida and M. Lipson, "Optical bistability on a silicon chip," Opt. Lett. 29, 2387-2389 (2004). [CrossRef] [PubMed]
- S. Fan, W. Suh, and J. D. Joannopoulos, "Temporal coupled-mode theory for the Fano resonance in optical resonators, "J. Opt. Soc. Am. A, 20(3), 569-572 (2003). [CrossRef]
- Y. Lu, J. Yao, X. Li, and P. Wang, "Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach Zehnder Interferometer," Opt. Lett. 30, 3069-3071 (2005). [CrossRef] [PubMed]
- L. B. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonator," Opt. Lett. 29, 626 (2004). [CrossRef] [PubMed]
- I. Chremmos, and N. Uzunoglu, "Reflective properties of double-ring resonator system coupled to a waveguide," IEEE Photon. Technol. Lett, 17, 2110-2112, 2005 [CrossRef]
- Y. M. Landobasa, S. Darmawan, and M. K. Chin, "Matrix Analysis of 2-D Micro-resonator Lattice Optical Filters," IEEE J. Quantum Electronics 41, 1410-1418 (2005). [CrossRef]
- A. Yariv, "Critical coupling and its control in optical waveguide-resonator systems," IEEE Photon. Technol. Lett. 14, 483-485, 2002. [CrossRef]
- A.R. Cowan and J.F. Young, "Optical bistability involving photonic crystal microcavities and Fano line shapes," Phys. Rev. E 68046606 (2003) [CrossRef]
- V. Van, T. A. Ibrahim, P.P. Absil, F. G. Johnson, R. Grover, and P-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Quantum Electron. 8, 705-713 (2002). [CrossRef]
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