## Modeling of the dynamic transmission properties of chalcogenide ring resonators in the presence of fast and slow nonlinearities |

Optics Express, Vol. 19, Issue 2, pp. 649-659 (2011)

http://dx.doi.org/10.1364/OE.19.000649

Acrobat PDF (1219 KB)

### Abstract

We propose a simple iterative method for calculating the dynamic behavior of ring resonators with fast and slow (cumulative) optical nonlinearities when an optical pulse with an arbitrary-shaped envelope is incident into them. In the case of a slow nonlinearity, the nonlinear phase shift and nonlinear absorption are temporally-integrated over the incident pulse. In this paper, we consider two types of single-ring resonators made out of As_{2}Se_{3} chalcogenide glass with high nonlinearity and investigate the dynamic properties (especially the effect of the cumulative nonlinearity on optical bistability) using known nonlinear material parameters. It is found that the cumulative nonlinearity suppresses overshoot and ringing after switching, decreases the width of the hysteresis loop between the input and output powers, and shifts its center corresponding to the operating point. The obtained results are useful in developing chalcogenide-based bistable optical devices and the proposed approach is applicable to modeling of a variety of nonlinear optical devices.

© 2011 OSA

## 1. Introduction

17. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. **26**(4), 760–769 (1990). [CrossRef]

19. K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 microm,” Opt. Express **16**(19), 14780–14791 (2008). [CrossRef] [PubMed]

20. K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As_{2}Se_{3} chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express **16**(22), 18230–18240 (2008). [CrossRef] [PubMed]

22. K. Ogusu and Y. Oda, “Transient absorption in As_{2}Se_{3} and Ag(Cu)-doped As_{2}Se_{3} glasses photoinduced at 1.06 μm,” Jpn. J. Appl. Phys. **48**(11), 110204 (2009). [CrossRef]

_{2}Se

_{3}chalcogenide glass for its realization. Two types of single-ring resonators with a single coupler and double couplers are investigated in this work. Although the existing iterative method under the assumption of an ideal Kerr nonlinearity is described by a set of simple difference equations [13

13. H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. **157**(1-6), 27–32 (1998). [CrossRef]

23. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. **30**(2), 257–261 (1979). [CrossRef]

24. A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. **137**(1-3), 136–142 (1997). [CrossRef]

## 2. Numerical analysis of nonlinear ring resonators

### 2.1 Single-coupler ring resonator

#### A. Iterative method considering n_{2} and α only

*L*(=2π

*r*,

*r*being the ring radius). In order to facilitate understanding of a novel approach proposed in this work, we briefly summarize the conventional iterative method for this configuration [13

13. H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. **157**(1-6), 27–32 (1998). [CrossRef]

23. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. **30**(2), 257–261 (1979). [CrossRef]

24. A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. **137**(1-3), 136–142 (1997). [CrossRef]

*n*

_{0}is the linear refractive index,

*n*

_{2}is the nonlinear refractive index with an ultrafast (instantaneous) response time, and

*η*

_{0}is the wave impedance in vacuum.

*I*is the optical intensity inside the waveguide,

*E*is the corresponding optical electric field, and

*P*is the optical power.

*S*

_{eff}is the effective mode area of the waveguide. Moreover we define the slowly-varying complex electric field at each position as shown in Fig. 1(a). At the coupling point between the bus waveguide and the ring waveguide, the field amplitudes of incident wave

*E*

_{in}(

*t*), transmitted wave

*E*

_{out}(

*t*), and circulating cavity wave

*E*

_{c}(

*z*,

*t*) satisfy the following equations: where

*γ*and

*κ*are the fractional intensity loss and intensity coupling coefficient of the coupler, respectively.

*E*

_{c}(

*L*,

*t*) at the end of the ring can be expressed by the prior cavity field

*E*

_{c}(0,

*t*-

*τ*

_{R}) at the entrance as follows:where

*α*is the intensity attenuation coefficient of the ring waveguide and

*τ*

_{R}( =

*n*

_{0}

*L*/

*c*) is the round-trip time of the cavity.

*ϕ*

_{0}( =

*n*

_{0}

*k*

_{0}

*L*)is the linear phase shift due to propagation around the ring and

*ϕ*

_{Ν}(

*t*-

*τ*

_{R}) is the nonlinear phase shift, which is given bywhere

*z*is the propagation distance along the ring from the coupling point and

*k*

_{0}is the propagation constant of free space. In the actual computation, it is convenient to replace

*ϕ*

_{0}by the detuning from resonance Δ

*ϕ*

_{0}=

*ϕ*

_{0}-2

*q*π, since exp(-

*jϕ*

_{0}) has period 2π. Substituting Eq. (4) into Eqs. (2) and (3), we have

*E*

_{out}(

*t*) is simply an iteration of the cavity field

*E*

_{c}(0,

*t*), with respect to the cavity round-trip time

*τ*

_{R}. Thus, we can calculate the dynamic properties when an optical pulse with an arbitrary temporal profile is incident into the nonlinear ring using Eqs. (6) and (7).

#### B. Iterative method considering ultrafast and cumulative nonlinearities

*E*

_{c}(0,

*t-τ*

_{R}) and

*E*

_{c}(

*L*,

*t*). Since these two cavity fields are complex, we decide to express them in polar form as follows: where

*θ*

_{0}(

*t-τ*

_{R}) is the phase with respect to the input field

*E*

_{in}(

*t-τ*

_{R}), i.e., the phase difference between

*E*

_{c}(0,

*t-τ*

_{R}) and

*E*

_{in}(

*t-τ*

_{R}), which changes with time as seen from Eq. (3). In the general case where there are a variety of nonlinearities, we must determine the relation between

*E*

_{c}(0,

*t-τ*

_{R}) and

*E*

_{c}(

*L*,

*t*) by directly solving the wave equation that governs the propagation through the nonlinear medium.

*I*(∝|

*E*|

^{2}) and the phase

*ϕ*

_{N}are usually employed for the purpose and the material parameters associated with their optical nonlinearities are defined. Using the slowly-varying envelope approximation, we can fully determine the light propagation within the nonlinear medium by [18

18. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. **9**(3), 405–414 (1992). [CrossRef]

20. K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As_{2}Se_{3} chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express **16**(22), 18230–18240 (2008). [CrossRef] [PubMed]

*β*is the two-photon absorption coefficient and Δ

*n*is the change in the refractive index.

*N*(

*z*,

*t*) is the density (henceforth called the carrier density) of the excited electronic state (for example, a free-carrier state or a higher-lying bound state) induced by one-photon absorption. σ

_{ab}and σ

_{r}are the changes in absorption coefficient and refractive index per unit carrier density (or absorbed photon density), respectively. The photogenerated carrier density is governed by the following rate equation:where

*τ*is the decay time of the excited state. If

*τ*is much longer than the incident pulse width

*τ*

_{p}, we can neglect the loss term

*N*(

*z*,

*t*)/τ in Eq. (12) and then have

*I*(0,

*t*), which changes with time. However we cannot expect its analytical solutions since that is an integro-differential equation. Therefore we must develop a simple numerical approach for solving Eq. (10).

*I*(

*z*,

*t*) in Eq. (10) can be assumed to be a constant within it, giving a solution of exponential function. Figure 2 shows a schematic drawing for explaining the concept of a proposed approach, where a computation scheme for the light propagation along the ring over one round-trip is shown using the space (

*z*,

*t*). We subdivide the ring length

*L*into

*M*equally spaced segments so that the step size Δ

*z*=

*L*/

*M*and the time step size Δ

*t*=

*τ*

_{R}/

*M*. The time region from

*t*=

*t-τ*

_{R}to

*t*=

*t*is shown in Fig. 2. We can numerically solve Eq. (10) along the straight line

*t*=

*t-τ*

_{R}+(

*τ*

_{R}/

*L*)

*z*by successively connecting the solutions of exponential function as follows:

*I*(Δ

*z,t-τ*

_{R}+Δ

*t*) from the given initial value

*I*(0

*,t*-

*τ*

_{R}) at

*z*=0. Next, using the obtained

*I*(Δ

*z,t-τ*

_{R}

*+*Δ

*t*) as an initial value for the next step, we calculate

*I*(Δz

*,t-τ*

_{R}+2Δ

*t*) at

*z*=Δ

*z*. Such a process is repeated so as to obtain the value of

*I*(

*L,t*). In connection with these computations, the carrier density

*N*(

*z,t*) is numerically integrated at every breakpoint (

*z*=

*i*Δ

*z*)) by adding the density increment

*N*(

*z,t-*Δ

*t*). On the other hand, from Eq. (11), the nonlinear phase shift

*ϕ*

_{Ν}(

*t*-

*τ*

_{R}) is given by

*E*

_{c}(

*L,t*) at

*z=L*from the known complex value of

*E*

_{c}(0

*,t-τ*

_{R}) at

*z=*0. Moreover we can determine the transmitted field

*E*

_{out}(

*t*) and the cavity field

*E*

_{c}(0

*,t*) for the given incident field

*E*

_{in}(

*t*) by substituting the obtained

*E*

_{c}(

*L,t*) into Eqs. (2) and (3). The obtained cavity field

*E*

_{c}(0,

*t*) will be used as an initial value for the computation after

*τ*

_{R}. We can entirely calculate the transmission changes when an optical pulse with an arbitrary temporal profile is incident into the nonlinear ring using repeating such a process at a fixed time interval of Δ

*t*.

### 2.2 Double-coupler ring resonator

*L*is coupled to two identical waveguides and is divided into a right and a left half. The naming of each port is based on the analogy between the ring resonator and the Fabry-Perot resonator.

*E*

_{r}(

*t*) and

*E*

_{t}(

*t*) are the transmitted fields at the reflection and transmission ports, respectively.

*E*

_{c1}(

*z,t*) and

*E*

_{c2}(

*z,t*) are the right-half (0<

*z*<

*L*/2) and left-half (

*L*/2<

*z*<

*L*) cavity fields, respectively. The complex electric fields are connected as follows: for the first coupler at the input port (or reflection port) and for the second coupler at the transmission port. We must calculate a unknown complex value of

*E*

_{c2}(

*L,t*) at

*z*=

*L*from the known complex value of

*E*

_{c1}(0,

*t-τ*

_{R}) at

*z*=0 as done in the preceding subsection. The presence of the second coupler at

*z*=

*L*/2 forces the amplitude of the circulating cavity field to decrease by [(1-

*γ*)(1-

*κ*)]

^{1/2}with no phase change.

## 3. Numerical results and discussion

_{2}Se

_{3}glass, the practical computation should be performed using as realistic device and material parameters as possible. We also assume a Nd:YAG laser as a light source. All numerical results presented here were calculated for the following values: free-space wavelength

*λ*=1.064 μm, ring radius

*r*=100 μm, effective mode area

*S*

_{eff}=2.0 μm

^{2}, linear refractive index

*n*

_{0}=2.818, linear absorption coefficient

*α*=0.621 cm

^{−1}, nonlinear refractive index

*n*

_{2}=3.0×10

^{−17}m

^{2}/W, two-photon absorption coefficient

*β*=5.0×10

^{−11}m/W, refractive index change per unit photo density

*σ*

_{r}=0.89×10

^{−22}cm

^{3}, absorption change per unit photo density

*σ*

_{ab}=4.46×10

^{−18}cm

^{2}. These linear and nonlinear material parameters are the experimental data obtained using the Brewster-angle technique [25

25. K. Ogusu, K. Suzuki, and H. Nishio, “Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle,” Opt. Lett. **31**(7), 909–911 (2006). [CrossRef] [PubMed]

20. K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As_{2}Se_{3} chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express **16**(22), 18230–18240 (2008). [CrossRef] [PubMed]

21. K. Ogusu and K. Shinkawa, “Optical nonlinearities in As_{2}Se_{3} chalcogenide glasses doped with Cu and Ag for pulse durations on the order of nanoseconds,” Opt. Express **17**(10), 8165–8172 (2009). [CrossRef] [PubMed]

*τ*

_{R}given by

*n*

_{0}

*L*/

*c*is 5.90 ps for

*r*=100 μm and the linear absorption coefficient

*α*=0.621 cm

^{−1}corresponds to a transmission loss of 0.17 dB per round-trip. In order to clarify the effect of the linear loss and the cumulative nonlinearity on optical bistable behavior, the numerical simulations will be performed for the following three cases:

- Case (i):
*n*_{2}≠0,*α*=0, and the other nonlinear parameters=0. - Case (ii):
*n*_{2}≠0,*α*≠0, and the other nonlinear parameters=0. - Case (iii): All linear and nonlinear parameters≠0.

### 3.1 Single-coupler ring resonator

*κ*=0.1and Δ

*ϕ*

_{0}=−0.1π for three cases (i) to (iii). In this numerical example, two values of the coupler loss

*γ*=0 (Fig. (a)) and

*γ*=0.1(Fig. (b)) are assumed. Note that the nonlinear ring resonator is completely lossless (

*γ*=0 and

*α*=0) for the case (i) in Fig. 3(a)). For the cases of (i) and (ii), we can calculate the nonlinear pulse response using Eqs. (6) and (7), as described in the preceding section. In these two cases, the numerical results obtained by the proposed approach coincide with those obtained using Eqs. (6) and (7). The number of segments used for these numerical simulations is

*M*=500 and the time required for computing one pulse response is within a few seconds on a standard personal computer. For a proper understanding of these pulse responses, we show the corresponding input-output characteristics in Fig. 4 . In the simulations for the cases (i) and (ii), optical bistability never occurs although clear switch-off, i.e., switching from high to low transmissions and subsequent overshoot and ringing take place, as already pointed out in [12

12. K. Ogusu, “Dynamic behavior of reflection optical bistability in a nonlinear fiber ring resonator,” IEEE J. Quantum Electron. **32**(9), 1537–1543 (1996). [CrossRef]

12. K. Ogusu, “Dynamic behavior of reflection optical bistability in a nonlinear fiber ring resonator,” IEEE J. Quantum Electron. **32**(9), 1537–1543 (1996). [CrossRef]

*α*(case (iii)), we can have optical bistability in which the output power describes a clockwise hysteresis loop as the input power is increased and then decreased. As seen from these examples, the losses in the ring resonator are absolutely essential for the appearance of optical bistability. However the extinction ratio, i.e., the ratio of the high and low power levels is not high and the switch-off response is not sharp. Hence, let us calculate the cavity response (build-up or decay) time and discuss the steady-state operation of the ring resonator. From the analogy between the ring resonator and the Fabry-Perot resonator [26], the cavity decay time

*τ*

_{c}(of the intensity) is given byIn the case of

*γ*<<1 and

*α*<<1,

*τ*

_{c}=59 ps. Since the pulse width

*τ*

_{P}=500 ps corresponds to 8.5

*τ*

_{c}, and 85

*τ*

_{R}, transient behavior remains in the response (see Fig. 6 ). According to our previous work [12

12. K. Ogusu, “Dynamic behavior of reflection optical bistability in a nonlinear fiber ring resonator,” IEEE J. Quantum Electron. **32**(9), 1537–1543 (1996). [CrossRef]

*ϕ*

_{0}. In this numerical example, a Gaussian pulse with

*τ*

_{P}=500 ps is assumed to be incident into the ring resonator with

*γ*=0.1 and

*κ*=0.1. It is found that the width of the hysteresis loop and the switch-off power increase as the magnitude of initial detuning is increased. If the input power does not reach the switching threshold, the ring resonator exhibits linear behavior.

*τ*

_{P}of an incident Gaussian pulse. The ring resonator with

*γ*=0.1,

*κ*=0.1, and Δ

*ϕ*

_{0}=−0.1π is assumed in this numerical example. For purpose of comparison, the numerical result for case (ii) and

*τ*

_{P}=5.0 ns is also presented in the figure. It is confirmed that the width of the hysteresis loop decreases with increasing pulse width. In the absence of the cumulative nonlinearity, the transient hysteresis loop approaches steady-state solution as the pulse width is increased. However such a situation does not occur in the presence of the cumulative nonlinearity. Cumulative nonlinear refraction mainly shifts the operating point (initial detuning) of the device, moving the center of the hysteresis loop on the higher input-power side. Moreover cumulative nonlinear absorption increases the switch-on threshold, decreasing the width of the hysteresis loop. It can be expected that the width of the hysteresis loop decreases with increasing incident pulse width.

### 3.2 Double-coupler ring resonator

*κ*=0.1, Δ

*ϕ*

_{0}=−0.1π, and

*γ*=0 (a) and 0.1 (b). In order to clarify thecontribution of the cumulative nonlinearity, the output powers,

*P*

_{t}and

*P*

_{r}, were calculated for two cases (ii) and (iii). Regarding transmission bistability, the switch-on and switch-off occur at the leading edge and the trailing edge of the incident pulse, respectively. The cumulative nonlinearity can suppress overshoot and ringing after switching, but it brings an additional loss. The temporal variation of

*P*

_{t}shows that the induced loss increases with time since the nonlinear absorption continues to accumulate. Figure 8 shows the input-output characteristics at the transmission port of the double-coupler ring resonators with

*κ*=0.1, Δ

*ϕ*

_{0}=−0.1π, and

*γ*=0 (a) and 0.1(b). The numerical results were computed for two cases (ii) and (iii), and two pulse widths

*τ*

_{P}=500 ps and 5 ns. We can clearly confirm that the effect of the cumulative nonlinearity on the switch-off is greater than that on the switch-on. It is also confirmed that the width of the hysteresis loop decreases with increasing incident pulse width. Moreover it is worth pointing out that we cannot assume an infinite pulse width since the accumulated loss becomes infinite.

## 4. Conclusions

_{2}Se

_{3}chalcogenide glass with high nonlinearity were considered and the effect of the cumulative nonlinearity on optical bistability was investigated using the proposed numerical method and known nonlinear material parameters. It has been found that we can obtain optical bistability under suitable conditions although the cumulative nonlinearity is considerable intense. Moreover the cumulative nonlinearity can suppress overshoot and ringing after switching. But it decreases the width of the hysteresis loop between the input and output powers, and shifts its center corresponding to the operating point. The proposed approach is easily applicable to higher-order nonlinearities, finite carrier decay time, i.e., the case where the loss term

*Ν*(

*z,t*)/

*τ*in Eq. (12) is not negligible, and so on.

## Acknowledgement

## References and link

1. | O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters–A tutorial overview,” J. Lightwave Technol. |

2. | B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO |

3. | D. G. Rabus, |

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

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

6. | T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. |

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

8. | Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. |

9. | D. Sarid, “Analysis of bistability in a ring-channel waveguide,” Opt. Lett. |

10. | J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. |

11. | K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. |

12. | K. Ogusu, “Dynamic behavior of reflection optical bistability in a nonlinear fiber ring resonator,” IEEE J. Quantum Electron. |

13. | H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. |

14. | A. Zakery, and S. R. Elliott, |

15. | V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express |

16. | K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. |

17. | M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. |

18. | A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. |

19. | K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 microm,” Opt. Express |

20. | K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As |

21. | K. Ogusu and K. Shinkawa, “Optical nonlinearities in As |

22. | K. Ogusu and Y. Oda, “Transient absorption in As |

23. | K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. |

24. | A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. |

25. | K. Ogusu, K. Suzuki, and H. Nishio, “Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle,” Opt. Lett. |

26. | A. Yariv, |

**OCIS Codes**

(190.1450) Nonlinear optics : Bistability

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(230.1150) Optical devices : All-optical devices

(230.4320) Optical devices : Nonlinear optical devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 20, 2010

Revised Manuscript: December 13, 2010

Manuscript Accepted: December 14, 2010

Published: January 5, 2011

**Citation**

Kazuhiko Ogusu and Yosuke Oda, "Modeling of the dynamic transmission properties of chalcogenide ring resonators in the presence of fast and slow nonlinearities," Opt. Express **19**, 649-659 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-649

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### References

- O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters–A tutorial overview,” J. Lightwave Technol. 22(5), 1380–1394 (2004). [CrossRef]
- B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998). [CrossRef]
- D. G. Rabus, Integrated ring resonators (Springer, Berlin, 2007).
- V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002). [CrossRef]
- T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003). [CrossRef]
- T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004). [CrossRef] [PubMed]
- V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004). [CrossRef] [PubMed]
- Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006). [CrossRef] [PubMed]
- D. Sarid, “Analysis of bistability in a ring-channel waveguide,” Opt. Lett. 6(11), 552–553 (1981). [CrossRef] [PubMed]
- J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. 30(11), 2578–2588 (1994). [CrossRef]
- K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. 20(22), 2288–2290 (1995). [CrossRef] [PubMed]
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