## Dynamics of microring resonator modulators

Optics Express, Vol. 16, Issue 20, pp. 15741-15753 (2008)

http://dx.doi.org/10.1364/OE.16.015741

Acrobat PDF (195 KB)

### Abstract

A dynamic model for the transmission of a microring modulator based on changes in the refractive index, loss, or waveguide-ring coupling strength is derived to investigate the limitations to the intensity modulation bandwidth. Modulation bandwidths approaching the free spectral range frequency are possible if the waveguide-ring coupling strength is varied, rather than the refractive index or loss of the ring. The results illustrate that via controlled coupling, resonant modulators with high quality factors can be designed to operate at frequencies much larger than the resonator linewidth.

© 2008 Optical Society of America

## 1. Introduction

1. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightwave Technol. **20**, 1968–1975 (2002). [CrossRef]

2. A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Gunter, “Electrooptically tunable microring resonators in lithium niobate,” Nature Photonics **1**, 407–410 (2007). [CrossRef]

3. Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature **435**, 325–327 (2005). [CrossRef] [PubMed]

4. Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nature Photonics **2**, 242–246 (2008). [CrossRef]

5. T. A. Ibrahim, V. Van, and P.-T. Ho, “All-optical time-division demultiplexing and spatial pulse routing with a GaAs/AlGaAs microring resonator,” Opt. Lett. **27**, 803–805 (2002). [CrossRef]

6. D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel-dropping filters using ring resonators with integrated SOAs,” IEEE Photon. Technol. Lett. **14**, 1442–1444 (2002). [CrossRef]

7. T. Sadagopan, S. J. Choi, S. J. Choi, K. Djordjev, and P. D. Dapkus, “Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation,” IEEE Photon. Technol. Lett. **17**, 414–416 (2005). [CrossRef]

1. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightwave Technol. **20**, 1968–1975 (2002). [CrossRef]

10. K. Djordjev, S.-J. Choi, S.-J. Choi, and P. D. Dapkus, “Active semiconductor microdisk devices,” J. Lightwave Technol. **20**, 105–113 (2002). [CrossRef]

1. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightwave Technol. **20**, 1968–1975 (2002). [CrossRef]

2. A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Gunter, “Electrooptically tunable microring resonators in lithium niobate,” Nature Photonics **1**, 407–410 (2007). [CrossRef]

3. Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature **435**, 325–327 (2005). [CrossRef] [PubMed]

7. T. Sadagopan, S. J. Choi, S. J. Choi, K. Djordjev, and P. D. Dapkus, “Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation,” IEEE Photon. Technol. Lett. **17**, 414–416 (2005). [CrossRef]

*Q*, the coupling coefficient, not the refractive index or loss of the ring resonator waveguide, should be modulated.

## 2. Time-dependent microring transmission

*E*

*(*

_{ξ}*t*)=

*ξ*(

*t*)exp(

*iω*

_{0}

*t*), where

*ξ*=

*B*,

*C*,

*D*and is a slowly varying amplitude, and ω0 is the frequency of the input optical wave. The input amplitude is constant, such that

*E*

*(*

_{A}*t*)=

*A*exp(

*iω*

_{0}

*t*).

*ϕ*, and attenuation,

*a*, experienced by a circulating wave at a frequency

*ω*after each round-trip in the resonator can be expressed by

*τ*=

*n*

*L*/

*c*is the resonator round-trip time,

*n*is the effective index,

*L*is the ring circumference, and

*C*(

*t*) propagates around the ring and experiences a different phase-shift, such that

*ω*-

*ω*

_{0}and

*C*̃(Ω) is the Fourier transform of

*C*(

*t*). To simplify Eq. (3), we assume that

*ϕ*(

*t*,

*ω*)≈

*ϕ*(

*t*,

*ω*

_{0})+Ω

*τ*, which is equivalent to approximating that the change in the phase-shift of each frequency component circulating in the resonator due to the index modulation, the

*η*(

*t*) term in Eq. (1a), is the same or is negligible compared to Ω

*τ*. This assumption is reasonable since typical index changes are on the order of ~10

^{-3}. With this approximation, Eq. (3) simplifies to

*ϕ*(

*t*)=

*ϕ*(

*t*,

*ω*

_{0}) and

*κ*(

*t*) and

*σ*(

*t*) are the resonator-waveguide coupling and transmission coefficients, and

*σ*

^{2}(

*t*)+

*κ*

^{2}(

*t*)|=1 for a lossless coupler.

11. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**, 321–322 (2000). [CrossRef]

*a*and

*σ*are interchangeable in |

*T*

*|*

_{ss}^{2}. The situation when

*σ*=

*a*is referred to as

*critical coupling*. At critical coupling, the wave in the bus waveguide destructively interferes with the wave coupled out of the ring to result in zero transmission [11

11. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**, 321–322 (2000). [CrossRef]

12. J. M. Choi, R. K. Lee, and A. Yariv, “Control of critical coupling in a ring resonator-fiber configuration: application to wavelength-selective switching, modulation, amplification, and oscillation,” Opt. Lett. **26**, 1236–1238 (2001). [CrossRef]

*Q*of the resonator must be high (

*a*,

*σ*≈1), so that a circulating wave can, in essence, experience any small changes in device parameters many times before being dissipated.

*T*(

*t*), we eliminate

*C*(

*t*) in Eq. (5), to arrive at

*a*(

*t*),

*κ*(

*t*),

*σ*(

*t*), and

*ϕ*(

*t*) are periodic with a period equal to

*τ*, then

*T*(

*t*) is equal to

*T*

*but with the static parameters replaced by their time-dependent counterparts. Sinusoidally periodic modulation of the refractive index of ring resonators at the free spectral range (FSR) has been recently demonstrated in electro-optic polymers [13*

_{ss}13. B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, “Electrooptic polymer ring resonator modulation up to 165 GHz,” IEEE J. Sel. Top. Quantum Electron. **13**, 104–110 (2007). [CrossRef]

14. B. Crosignani and A. Yariv, “Time-dependent analysis of a fiber-optic passive-loop resonator,” Opt. Lett. **11**, 251–253 (1986). [CrossRef] [PubMed]

## 3. Loss modulation

*a*(

*t*) varies in time, but

*ϕ*,

*σ*, and

*κ*are constant. The solution of Eq. (8) for the transmission with loss modulation,

*T*

*(*

_{a}*t*), is

*t*. Each prior round-trip is weighted by

*σ*e

^{-}

*, so that for high*

^{iϕ}*Q*resonators, a large number of terms in the summation will be significant to

*T*

*(*

_{a}*t*).

## 3.1. Small-signal approximation

*a*(

*t*)=

*a*

_{0}+

*a*

^{′}cos(Ω

_{m}*t*), where Ω

*is the modulation frequency, and*

_{m}*a*

^{′}/

*a*

_{0}≪ 1. The Fourier transform of

*a*(

*t*) is

*T*̃

*(Ω) is the Fourier transform of*

_{a}*T*

*(*

_{a}*t*). Since we consider a sinusoidal modulation,

*T*̃

*(Ω) consists only of the Ω=0 component and the harmonics of Ω*

_{a}*.*

_{m}*T*̃

*(Ω) order by order in*

_{a}*a*

^{′}. We obtain an approximate solution by keeping only the terms up to

*O*(

*a*

^{′}) to find that

*a*

^{′}. Eqs. (12) and (13) show that when the input wave is near resonance so that exp(-

*iϕ*) ≈1 and the modulation amplitude is small, the output intensity of the ring resonator is sinusoidal with the same frequency as the loss modulation but with a constant offset determined by the static response of the resonator. Eqs. (12) and (13) can also be used to study the distortion of a signal and the linearity of the modulator by evaluating the relative magnitudes and phases of

*T*̃

*(±Ω*

_{a}*) and*

_{m}*T*̃

*(0).*

_{a}*. The modulation depth of a signal is defined as*

_{m}*f*(

*t*)

*and*

_{max}*f*(

*t*)

*are the maximum and minimum amplitudes of the signal. Comparing the Fourier transform of a sinusoidally modulated signal with Eq. (14), we find, after some algebra, that*

_{min}*T*̃(Ω) is the Fourier transform of

*T*(

*t*). For loss modulation, substituting Eqs. (12) and (13) into Eq. (15), the modulation depth, Δ

*, is*

_{a}

_{m}*τ*≪ 1, Eq. (16) shows that the modulation depth decreases with increasing modulation frequency. By taking the derivative of Δ

*with respect to Ω*

_{a}

_{m}*τ*, we find that for

*a*,

*σ*≈1, there exists a special condition when the modulation depth is maximum:

*ϕ*+Ω

*τ ≈2*

_{m}*pπ*, where

*p*is an integer, i.e. when one of the sideband frequencies is on resonance. We shall refer to this situation as a

*modulation resonance*. The output distortion when the modulator operates close to a modulation resonance is dictated by the relative amplitudes and phase of the resonant and non-resonant sidebands.

_{m}*τ*≪ 1, the modulation depth simplifies to

*,*

_{a}_{3}

_{dB},

*is higher for lower*

_{res}*Q*resonators with smaller values of

*a*

_{0}and

*σ*. Therefore, for loss modulated microrings, the modulation bandwidth is limited by the resonator

*Q*.

## 3.2. Numerical results

*µ*m and the waveguide index to be

*n*=3, resulting in a round-trip time of 0.628 ps. For the summation in the Neumann series [Eq. (9)], we include terms up to

*O*(10

^{-5}).

*calculated using the small signal approximation, Eq. (17), and the exact expression, Eq. (9), when the loss of the microring is modulated between 2 dB/cm and 5 dB/cm (*

_{m}*a*

_{0}=0.9975,

*a*

^{′}=0.0011) while

*σ*=0.9928. The 3 dB roll-off frequency is 4.3 GHz, in good agreement with Eq. (18). Figure 2(b) shows the presence of the modulation resonance when the input wavelength is detuned from resonance by

*f*

*. The ratio between the modulation resonance and Δ*

_{m}*(Ω*

_{a}*=0) is larger for an input wavelength that is greater detuned from resonance. As evidenced by the figures, there is good agreement between the small signal approximation and the exact equation at frequencies below the modulation resonance. At higher frequencies, higher order (harmonic) terms become more significant.*

_{m}## 4. Index modulation

*ϕ*(

*t*) varies in time and

*a*and

*σ*are constant. The Neumann series solution of Eq. (8) for the transmission coefficient,

*T*

_{ϕ}(

*t*), is

*T*

*(*

_{ϕ}*t*) consists of an instantaneous response, which is given by the first two terms, and a summation of memory terms where each preceding round-trip is weighted by

*σa*.

## 4.1. Small-signal approximation

*ϕ*

^{′}≪ 1, only the

*J*

_{0}and

*J*

_{1}terms dominate and

*J*

_{0}(

*ϕ*

^{′}) ≈ 1 and

*J*

_{1}(

*ϕ*

^{′}) ≈

*ϕ*

^{′}/2. Therefore,

*ϕ*

^{′}.

*T*̃ϕ

_{(0)}=

*T*̃

*(0)=*

_{a}*T*

*, which is the static response of the resonator, and*

_{ss}*, can be found using Eq. (15) to be*

_{ϕ}*=0. Intuitively, this is because the resonance wavelength is at the minimum of the static transmission spectrum. Thus, to first order in*

_{ϕ}*ϕ*

^{′}, there is no modulation in the transmission amplitude, and the microring operates as a phase modulator rather than an intensity modulator. As

*a*,

*σ*→ 1, the 3 dB roll-off frequency decreases, so the modulation bandwidth is again

*Q*limited. By taking the derivative of Eq. (25), we find that for high

*Q*resonators where

*a*,

*σ*≈ 1, a modulation resonance also exists for index modulation, with the modulation depth reaching a maximum at

*ϕ*

_{0}+Ω

_{m}*τ*≈ 2

*pπ*, where

*p*is an integer.

## 4.2. Numerical results

*versus the modulation frequency for a sinusoidally index modulated microring resonator. The figure compares the results of the small signal modulation depth from Eq. (25) and the exact solution from Eq. (19). For the calculations,*

_{ϕ}*ϕ*

_{0}=0.039477 and

*ϕ*

^{′}=0.005, which corresponds to an index change of 2×10

^{-5}at a wavelength of 1.55 µm. The two sets of calculations in Fig. 3 are identical except the values of

*a*and

*σ*are interchanged. The low frequency modulation depth is identical between the two cases and is therefore symmetric in

*a*and

*σ*, as can be seen in Eq. (25). However, at higher frequencies, the modulation depth is slightly larger for the over-coupled (

*σ*<

*a*) ring.

## 5. Coupling modulation

*a*and

*ϕ*are constant in time. The solution of Eq. (8) for

*T*

*(*

_{σ}*t*) is

*κ*(

*t*), the instantaneous value of the coupling coefficient – such a term is absent in Eq. (9) and Eq. (19). This, as we shall further demonstrate, implies that coupling modulation does not suffer from the same limitations as loss and index modulation.

## 5.1. Small-signal approximation

*Q*, such that

*κ*≪ 1 and

*σ*≈1. We take the coupling coefficient as

*κ*′/

*κ*

_{0}| ≪ 1. For |

*σ*(

*t*)|

^{2}+ |

*κ*(

*t*)|

^{2}= 1 to

*O*(

*κ*′), it follows that

*σ*

^{′}=-

*κ*

_{0}

*κ*

^{′}/

*σ*

_{0}and |

*σ*

^{′}/

*σ*

_{0}| ≈

*κ*

^{′}

*κ*

_{0}. Substituting Eq. (27) and (28) into Eq. (7) up to

*O*(

*κ*

^{′}), and taking the Fourier transform results in

*T*̃

*(0),*

_{σ}*T*̃

*(Ω*

_{σ}*), and*

_{m}*T*̃

*(-Ω*

_{σ}*) in the same fashion as was done for index and loss modulation. To*

_{m}*O*(

*κ*

^{′}),

*T*̃

^{σ}(0)=

*T*

*, the static response of resonator. Using Eq. (15) to solve for the modulation depth, we obtain*

_{ss}*iϕ*)=1, and Ω

_{m}*τ*≪ 1 to arrive at

*a*and

*σ*

_{0}. At low modulation frequencies,

_{σ,res}is approximately constant and equal to 2

*σ*

^{′}(1-

*a*

^{2})/[|

*σ*

_{0}-

*a*|(1-

*a*

*σ*

_{0})]. At high frequencies such that

## 5.2. Numerical results

*a*=0.9971. The series solutions, Eq. (26), closely follow the predictions of Eqs. (30) and (31). The low frequency modulation depth is smaller than the high frequency value for over-coupled ring resonators and vice-versa for under-coupled resonators. In addition, the results show that for both resonant and detuned inputs, the modulation depth is roughly constant at large frequencies. However, comparing Fig. 4 (a) with Fig. 4 (b), we can see that the input wavelength should be close to resonance to achieve large modulation depths. Fig. 4 (b) also shows the existence of modulation resonance for coupling modulation with the input detuned from resonance.

*B*(

*t*), in Fig. 1. The modulation of the coupling constant, similar to index or loss modulation, generates frequency sidebands to the input frequency,

*ω*

_{0}, that also circulate in the microring. The amplitude of these sidebands diminish with increasing modulation frequency or increasing

*Q*, which leads to the roll-off in the modulation depth for index and loss modulation. In contrast, for coupling modulation, as can be seen in Eq. (26), a factor of

*κ*(

*t*) is applied to any light that exits the cavity. Therefore, the output of the modulator is determined by the instantaneous modulation of the frequency components at

*ω*

_{0}and the sidebands.

*ω*

_{0}components are modulated simultaneously. However, there will be a modulation frequency range over which the sideband amplitudes diminish, leaving only the instantaneous modulation of

*ω*

_{0}, which is independent of modulation frequency. The flat high frequency modulation responses in Fig. 4 are due to this instantaneous modulation. The higher the

*Q*factor is, the lower the modulation frequency needs to be for the modulator to reach the flat high frequency response, i.e.

## 6. Discussion

*κ*≠0, which results in a certain transmission amplitude. If

*κ*is suddenly reduced to zero, immediately, no light can exit or enter the resonator. This leads to an instantaneous change in the transmission that is not limited by the resonator

*Q*, but only the response of the coupler. On the other hand, if the loss or index of the resonator is changed suddenly, light that was circulating inside the resonator can continue to escape from the resonator. The rate at whichthe steady-state intensity transmission, Eq. (6b),

*σ*and a are interchangeable. Therefore, a static description of the resonator would not distinguish between changes in a and σ. It is only through a dynamical description of the resonator that the differences in the modulation rate limits can be revealed.

*Q*of the resonator must be very large to produce pulses that closely resemble the coupling strength pulse shape. This is the opposite requirement compared to loss or index modulation which suffer from high frequency limitations and thus require low

*Q*resonators for undistorted output pulses.

*τ*in Eq. (7), the resonator output is identical to the low frequency response, neglecting any averaging of device parameters that occur as a result of Eq. (1). Therefore, the response of a coupling modulated microring resembles the low frequency response at modulation frequencies approaching the FSR of the resonator. However, for microring resonators, the FSR is on the order of ~1 THz, sufficient for most communication applications. Moreover, throughout this analysis, we have neglected the frequency, amplitude, and phase response of the coupler itself. The ultimate modulation rate would be determined by the modulation response of the coupler, which need not be a resonant device. For example, state-of-the-art electro-optic polymer Mach-Zehnder interferometric switches can operate at > 100 GHz [15

15. D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. **70**, 3335–3337 (1997). [CrossRef]

16. M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science **298**, 1401–1403 (2002). [CrossRef] [PubMed]

*κ*is only of the order of 10

^{-3}, i.e. the loaded

*Q*is almost constant. Therefore, a high

*Q*, microring modulator based on variable coupling can still be low power and compact in size. Recently, microrings integrated with a variable coupler have been proposed and demonstrated [17

17. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**, 483–485 (2002). [CrossRef]

22. Y. Li, L. Zhang, M. Song, B. Zhang, J. Y. Yang, R. G. Beausoleil, A. E. Willner, and P. D. Dapkus, “Coupledring-resonator-based silicon modulator for enhanced performance,” Opt. Express **16**, 13342–13348 (2008). [CrossRef] [PubMed]

17. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**, 483–485 (2002). [CrossRef]

20. L. Zhou and A. W. Poon, “Electrically reconfigurable silicon microring resonator-based filter with waveguidecoupled feedback,” Opt. Express **15**, 9194–9204 (2007). [CrossRef] [PubMed]

## 7. Conclusion

*Q*microresonators to realize low loss, low power, and compact modulators which also possess a high modulation bandwidth. Our model can be extended to incorporate the dynamic effects of the coupler and to analyze other properties of microring modulators, such as the chirp, linearity, and extinction ratio.

## Acknowledgments

## References and links

1. | P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightwave Technol. |

2. | A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Gunter, “Electrooptically tunable microring resonators in lithium niobate,” Nature Photonics |

3. | Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature |

4. | Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nature Photonics |

5. | T. A. Ibrahim, V. Van, and P.-T. Ho, “All-optical time-division demultiplexing and spatial pulse routing with a GaAs/AlGaAs microring resonator,” Opt. Lett. |

6. | D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel-dropping filters using ring resonators with integrated SOAs,” IEEE Photon. Technol. Lett. |

7. | T. Sadagopan, S. J. Choi, S. J. Choi, K. Djordjev, and P. D. Dapkus, “Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation,” IEEE Photon. Technol. Lett. |

8. | L. Zhang, J.-Y. Yang, M. Song, Y. Li, B. Zhang, R. G. Beausoleil, and A. E. Willner, “Microring-based modulation and demodulation of DPSK signal,” Opt. Express |

9. | I. L. Gheorma and R. M. Osgood, “Fundamental limitations of optical resonator based high-speed EO modulators,” J. Lightwave Technol. |

10. | K. Djordjev, S.-J. Choi, S.-J. Choi, and P. D. Dapkus, “Active semiconductor microdisk devices,” J. Lightwave Technol. |

11. | A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. |

12. | J. M. Choi, R. K. Lee, and A. Yariv, “Control of critical coupling in a ring resonator-fiber configuration: application to wavelength-selective switching, modulation, amplification, and oscillation,” Opt. Lett. |

13. | B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, “Electrooptic polymer ring resonator modulation up to 165 GHz,” IEEE J. Sel. Top. Quantum Electron. |

14. | B. Crosignani and A. Yariv, “Time-dependent analysis of a fiber-optic passive-loop resonator,” Opt. Lett. |

15. | D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. |

16. | M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science |

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

18. | W. M. J. Green, R. K. Lee, G. A. DeRose, A. Scherer, and A. Yariv, “Hybrid InGaAsP-InP Mach-Zehnder racetrack resonator for thermooptic switching and coupling control,” Opt. Express |

19. | C. Li, L. Zhou, and A. W. Poon, “Silicon microring carrier-injection-based modulators/switches with tunable extinction ratios and OR-logic switching by using waveguide cross-coupling,” Opt. Express |

20. | L. Zhou and A. W. Poon, “Electrically reconfigurable silicon microring resonator-based filter with waveguidecoupled feedback,” Opt. Express |

21. | W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Optical modulation using anti-crossing between paired amplitude and phase resonators,” Opt. Express |

22. | Y. Li, L. Zhang, M. Song, B. Zhang, J. Y. Yang, R. G. Beausoleil, A. E. Willner, and P. D. Dapkus, “Coupledring-resonator-based silicon modulator for enhanced performance,” Opt. Express |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.4110) Optical devices : Modulators

(230.5750) Optical devices : Resonators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: August 14, 2008

Revised Manuscript: September 11, 2008

Manuscript Accepted: September 16, 2008

Published: September 19, 2008

**Citation**

Wesley D. Sacher and Joyce K. S. Poon, "Dynamics of microring resonator modulators," Opt. Express **16**, 15741-15753 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15741

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### References

- P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002). [CrossRef]
- A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Gunter, "Electrooptically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407 - 410 (2007). [CrossRef]
- Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005). [CrossRef] [PubMed]
- Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nat. Photonics 2, 242-246 (2008). [CrossRef]
- T. A. Ibrahim, V. Van, and P.-T. Ho, "All-optical time-division demultiplexing and spatial pulse routing with a GaAs/AlGaAs microring resonator," Opt. Lett. 27, 803-805 (2002). [CrossRef]
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