## Dynamic response of modulators based on cascaded-ring-resonator |

Optics Express, Vol. 20, Issue 20, pp. 21847-21859 (2012)

http://dx.doi.org/10.1364/OE.20.021847

Acrobat PDF (1282 KB)

### Abstract

We investigated the dynamic response of a cascaded-ring-resonator-loaded Mach–Zehnder modulator (CRR-MZM), in which a number of cascaded ring resonators (RRs) are loaded in the interferometer as phase modulators. The analytical form is derived for the small-signal response of CRR-MZM using temporal-coupled-mode (TCM) theory, and its validity is confirmed by numerical calculations. It is revealed that the bandwidth of the CRR-MZM is maximized by setting proper delays in driving signals between neighboring RRs; the optimized delay is twice the photon lifetime of each RR. The calculated performances of CRR-MZMs are compared with those of standard modulators based on a single-ring-resonator (SRR) without interferometer, in terms of the modulation depth and bandwidth. For a given degree of the refractive index change in a waveguide, CRR-MZM can provide a larger modulation depth than a SRR-type modulator in frequency ranges exceeding 25 GHz.

© 2012 OSA

## 1. Introduction

1. D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE **97**(7), 1166–1185 (2009). [CrossRef]

1. D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE **97**(7), 1166–1185 (2009). [CrossRef]

2. Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express **15**(2), 430–436 (2007). [CrossRef] [PubMed]

17. M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B **19**(9), 2052–2059 (2002). [CrossRef]

16. H. F. Taylor, “Enhanced electrooptic modulation efficiency utilizing slow-wave optical propagation,” J. Lightwave Technol. **17**(10), 1875–1883 (1999). [CrossRef]

17. M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B **19**(9), 2052–2059 (2002). [CrossRef]

2. Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express **15**(2), 430–436 (2007). [CrossRef] [PubMed]

13. A. Brimont, D. J. Thomson, P. Sanchis, J. Herrera, F. Y. Gardes, J. M. Fedeli, G. T. Reed, and J. Martí, “High speed silicon electro-optical modulators enhanced via slow light propagation,” Opt. Express **19**(21), 20876–20885 (2011). [CrossRef] [PubMed]

18. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. **23**(1), 123–129 (1987). [CrossRef]

2. Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express **15**(2), 430–436 (2007). [CrossRef] [PubMed]

13. A. Brimont, D. J. Thomson, P. Sanchis, J. Herrera, F. Y. Gardes, J. M. Fedeli, G. T. Reed, and J. Martí, “High speed silicon electro-optical modulators enhanced via slow light propagation,” Opt. Express **19**(21), 20876–20885 (2011). [CrossRef] [PubMed]

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

21. F. Xia, L. Sekaric, and Y. A. Vlasov, “Ultra-compact optical buffers on a silicon chip,” Nat. Photonics **1**(1), 65–71 (2007). [CrossRef]

**15**(2), 430–436 (2007). [CrossRef] [PubMed]

7. G. Li, X. Zheng, J. Yao, H. Thacker, I. Shubin, Y. Luo, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “25Gb/s 1V-driving CMOS ring modulator with integrated thermal tuning,” Opt. Express **19**(21), 20435–20443 (2011). [CrossRef] [PubMed]

7. G. Li, X. Zheng, J. Yao, H. Thacker, I. Shubin, Y. Luo, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “25Gb/s 1V-driving CMOS ring modulator with integrated thermal tuning,” Opt. Express **19**(21), 20435–20443 (2011). [CrossRef] [PubMed]

22. I. L. Gheorma and R. M. Osgood, “Fundamental limitations of optical resonator based on high-speed EO modulators,” IEEE Photon. Technol. Lett. **14**(6), 795–797 (2002). [CrossRef]

25. T. Ye and X. Cai, “On power consumption of silicon-microring-based optical modulators,” J. Lightwave Technol. **28**(11), 1615–1623 (2010). [CrossRef]

8. S. Akiyama, T. Kurahashi, T. Baba, N. Hatori, T. Usuki, and T. Yamamoto, “A 1V peak-to-peak driven 10-Gbps slow-light silicon Mach-Zehnder modulator using cascaded ring resonators,” Appl. Phys. Express **3**(7), 072202 (2010). [CrossRef]

10. S. Akiyama, T. Kurahashi, K. Morito, T. Yamamoto, T. Usuki, and S. Nomura, “Cascaded-ring-resonator-loaded Mach-Zehnder modulator for enhanced modulation efficiency in wide optical bandwidth,” Opt. Express **20**(15), 16321–16338 (2012). [CrossRef]

10. S. Akiyama, T. Kurahashi, K. Morito, T. Yamamoto, T. Usuki, and S. Nomura, “Cascaded-ring-resonator-loaded Mach-Zehnder modulator for enhanced modulation efficiency in wide optical bandwidth,” Opt. Express **20**(15), 16321–16338 (2012). [CrossRef]

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

## 2. Analytical modeling

23. W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator modulators,” Opt. Express **16**(20), 15741–15753 (2008). [CrossRef] [PubMed]

25. T. Ye and X. Cai, “On power consumption of silicon-microring-based optical modulators,” J. Lightwave Technol. **28**(11), 1615–1623 (2010). [CrossRef]

*ω*is a time-dependent function

_{r}*ω*(

_{r}*t*). When a voltage is applied to the ring resonator, the resonant frequency

*ω*is modulated at a frequency of Ω around its static resonant frequency

_{r}*ω*with an amplitude of

_{0}*δ*.where

_{ω}*d*is an arbitrary time delay set in the modulation signal. When we deal with CRRs in the later part of this section, multiple values of

*d*are assigned to RRs. With an input light of continuous wave with angular frequency

*ω*and amplitude

*S*

_{0}, i.e.,

*S*(

_{in}*t*) =

*S*

_{0}⋅e

*, (1) becomesWe defined a frequency detuning of Δ*

^{iωt}*=*

_{ω}*ω*-

_{0}*ω*and a slowly-varying amplitude

*a*(

*t*) when

*A*(

*t*) =

*a*(

*t*) e

*. Hereafter, we used a net photon lifetime of the RR,*

^{iωt}*τ*, which is the principal parameter that governs the dynamic response of the RR.Equation (5) is a linear ordinary differential equation. The solution of Eq. (5) is analytically given in integral form as follows: In the above formula, we used the initial condition of

*a*(0) = 0. Throughout the analysis, we used a small signal approximation, which means that we only consider the first order of

*δ*. Therefore, to perform the integral in Eq. (7), we apply:By using the approximation of Eq. (9), the integral in Eq. (7) is calculated for a sufficiently large

_{ω}*t*, as follows:

*h*= 1 –

*i*Δ

*. Equation (10) expressed the frequency response of RR for small signal modulation with an arbitrary choice of the parameter sets of*

_{ω}τ*τ*,

*μ*, Δ

*, and Ω. Note that*

_{ω}*μ*, Δ

*,*

_{ω}*δ*, and Ω are all normalized by

_{ω}*τ*in the above equation, with the exception of exponential terms. The right hand side of Eq. (10) contains three terms. The first term is the static solution of RR, while the second and third terms are frequency components with angular frequencies of

*ω*± Ω, respectively. These two components are created as side-band components around

*ω*due to the modulation of Ω. Because of the small signal approximation, the side bands of

*ω*± 2Ω,

*ω*± 3Ω, ..., which have higher orders of

*δ*, do not appear in Eq. (10). We introduced three functions,

_{ω}*F*

^{(0)}(

*μ*

^{2}

*τ*, Δ

*),*

_{ω}τ*F*

^{+}(

*μ*

^{2}

*τ*, Δ

*, Ω*

_{ω}τ*τ*), and

*F*

^{-}(

*μ*

^{2}

*τ*, Δ

*, Ω*

_{ω}τ*τ*), which correspond to the three terms, as in the second line in Eq. (10).

### 2.1 SRR-SSM

22. I. L. Gheorma and R. M. Osgood, “Fundamental limitations of optical resonator based on high-speed EO modulators,” IEEE Photon. Technol. Lett. **14**(6), 795–797 (2002). [CrossRef]

25. T. Ye and X. Cai, “On power consumption of silicon-microring-based optical modulators,” J. Lightwave Technol. **28**(11), 1615–1623 (2010). [CrossRef]

*μ*

^{2}

*τ*= 1 and Δ

*= 1. The former indicates a condition of the critical coupling, i.e.,*

_{ω}τ*τ*=

_{c}*τ*[27

_{l}27. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**(4), 321–322 (2000). [CrossRef]

*= 1, the output in a static solution from the RR is equal to one half of the input power. By substituting these formulas in Eq. (10), for*

_{ω}τ*S*

_{0}= 1, we obtained|

*S*|

_{out}^{2}varies with the angular frequency of Ω, and its amplitude decreases with Ω, which is inversely proportional to Ω

*τ*for large Ω

*τ*.

### 2.2 CRR-MZM

*ω*is inputted, and three frequency components are created, as indicated by black bold arrows in Fig. 2 and Eq. (10). We can consider these three components as the input to the second RR, and as continuous waves with frequencies

*ω*and

*ω*± Ω, respectively. Therefore, Eq. (10) can be re-applied to each of these three components to calculate their respective outputs. At the second RR, the inputs with a frequency of

*ω*± Ω may create frequency components of

*ω*and

*ω*± 2Ω, respectively. However, these four components are all for second-order terms about

*δ*,, and are therefore neglected in small-signal analysis. Consequently, the second RR also outputs continuous lights with frequencies of

_{ω}*ω*and

*ω*± Ω. By making this speculation regarding the following RRs, it is noted that even for CRR with

*N*> 2, the frequency components with

*ω*and

*ω*± Ω are enough to be considered for our small signal analysis; the input and output of the

*j*-th RR both have three frequency components of

*ω*and

*ω*± Ω. As shown in Fig. 2, we defined the amplitude of the outputs from the

*j*-th RR with frequencies

*ω*,

*ω*+ Ω, and

*ω*− Ω as

*a*

_{j}^{(0)},

*a*

_{j}^{+}, and

*a*

_{j}^{-}, respectively. Next, we derive a recurrence formula for

*a*

_{j}^{(0)},

*a*

_{j}^{+}, and

*a*

_{j}^{-}. The red bold arrows correspond to the dependencies of

*a*

_{j}^{(0)},

*a*

_{j}^{+}and

*a*

_{j}^{-}on those in the previous step. Obviously,

*a*

_{j-1}^{+}and

*a*

_{j-1}^{-}are at least with the first order of

*δ*, and therefore, they only affect

_{ω}*a*

_{j-1}^{+}and

*a*, and

_{j-1}^{-}*a*

_{j}^{(0)}is therefore determined only by

*a*

_{j-1}^{(0)}. The red arrows in Fig. 2 indicate the remaining dependencies which we have to consider. The coefficients are determined by Eq. (10). As a result, we obtained with initial values of

*a*

_{0}

^{(0)}= 1 and

*a*

_{0}

^{±}= 0. As shown in Fig. 1(c), we assumed a constant time delay of

*d*between all pairs of neighboring RRs, which adds a phase factor of e

^{±}

^{i}^{Ω(}

^{j}^{-1)}

*to*

^{d}*α*

_{j}^{±}as in Eq. (12a). In addition, CRR-MZM uses each RR as a phase modulator. The phase change is most enhanced at the resonant frequencies of the RRs, and we therefore set the detuning as Δ

*= 0 [8*

_{ω}8. S. Akiyama, T. Kurahashi, T. Baba, N. Hatori, T. Usuki, and T. Yamamoto, “A 1V peak-to-peak driven 10-Gbps slow-light silicon Mach-Zehnder modulator using cascaded ring resonators,” Appl. Phys. Express **3**(7), 072202 (2010). [CrossRef]

10. S. Akiyama, T. Kurahashi, K. Morito, T. Yamamoto, T. Usuki, and S. Nomura, “Cascaded-ring-resonator-loaded Mach-Zehnder modulator for enhanced modulation efficiency in wide optical bandwidth,” Opt. Express **20**(15), 16321–16338 (2012). [CrossRef]

*δ*and the frequency components of

_{ω}*ω*and

*ω*± Ω.

*a*is obtained from Eq. (12a) and (12b) as follows: As shown in Fig. 1(c), CRRs are loaded as phase modulators on each arm of the Mach-Zehner interferometer. The two CRRs are operated by the signals having opposite polarities. When the static phase difference of π/2 is given between two arms, the output

_{N}*S*from the CRR-MZM in Fig. 1(c) becomesUnlike SRR-SSM, CRR-MZMs do not need the condition of critical coupling, and the propagation loss of the waveguide is ideally zero for the maximum output from the modulator. When we consider the loss-less case, 1/

_{out}*τ*= 0,

_{l}*μ*

^{2}

*τ*= 2, and

*F*

^{(0)}(

*μ*

^{2}

*τ*,0) = 1. With all of these simplifications, we obtainedas the response of CRR-MZM. For the single RR case, i.e.,

*N*= 1, the summation becomes unity in the above equation. In this case, the amplitude of sin(Ω

*t*+

*φ′*) decreases with Ω, and is inversely proportional to Ω

*τ*for large Ω

*τ*, which is similar to the case with SRR-SSM. In the case of CRRs, i.e.,

*N*> 1, the summation part is not equal to unity, and represents the distortion of the frequency response of CRR-MZM due to the cascading of RRs.

*τ*<< 1, to obtain a sufficiently large modulator bandwidth for a given Ω. In this case, the response of each RR may be small, as was observed in Eq. (15) for

*N*= 1. Therefore, we should cascade RRs to accumulate the phase change and to obtain a sufficient modulator response. The important thing here is to choose a proper delay

*d*to minimize the effect of the summation in Eq. (15). Because we chose

*τ*so that Ω

*τ*<< 1, then

*θ*≈Ω

*τ*in Eq. (15). Consequently, if the delay

*d*= 2

*τ*, the summation in Eq. (15) becomes

*N*regardless of the frequency Ω. In this case, the impact of cascading RR on the response becomes a simple multiplication with a factor of

*N*. This choice of delay is considered to be an optimized value required for CRR-MZMs to minimize the distortion in the frequency response. This result is reasonable because the group delay of each RR is also equal to 2

*τ*.

*S*|

_{out}^{2}divided by

*δ*. SRR-SSM and SRR-MZM (CRR-MZM for

_{ω}τ*N*= 1) have a similar dependence on Ω

*τ*. For large Ω

*τ*, both decrease at a rate of −10 dB/decade, as shown by the black and green curves in Fig. 3. With respect to 10-CRR-MZMs, their responses are 10 times larger than that of SRR-MZM for the smallest Ω

*τ*, regardless of the delay

*d*, as shown by the blue and red curves in Fig. 3. However, the bandwidth of 10-CRR-MZM is much smaller than that of SRR-MZM with

*d*= 0, as shown by the blue curve. As expected, the bandwidth is much improved for 10-CRR-MZM with

*d*= 2

*τ*, and is comparable to that of SRR-MZM. If we compare 10-CRR-MZM with

*d*= 2

*τ*to standard SRR-SSM, the former shows a much larger response at small Ω

*τ*with a smaller modulation bandwidth, as shown in Fig. 3. In the next section, we compared both configurations with respect to response at low frequencies and the modulation bandwidth, using a wide range of parameter sets, including the propagation loss of the waveguide and

*τ*and

*N*.

## 3. Numerical calculations

### 3.1 Numerical method

*U*(

*t*,

*z*) =

*u*(

*t, z*) e

^{i}^{(}

^{ωt}^{-}

^{kz}^{)}.

*u*(

*t*,

*z*) is a slowly varying function when compared with the angular frequency of light

*ω*, and satisfies the following partial differential equation [28]:In the coupling region, we considered the boundary condition as follows: The term –

*α*(2

*πR*) +

*φ*(

*t*) means the attenuation and phase shift that are experienced by the light in the RR.

*τ*

_{0}= (2

*πR n*)/

_{g}*c*as

*t*=

*t*=

_{m}*mτ*

_{0}. Note that in these calculations, we neglected variations of

*u*(

*t*) in the RR that were faster than

*τ*

_{0}. As in the analysis in the previous section, we also neglected the modulation of the group index in the waveguide. With these assumptions from Eq. (16),We numerically solved the above equations for the first RR with a constant input

*u*

_{1}(

*t*) = 1. The outputs obtained from the first RRs were used as the inputs of the successive RR. The modulation of the

*j*-th RR was defined in the

*φ*(

*t*) aswhere

*f*refers to the free-spectral range of the RR in frequency and Δ

_{FSR}*and*

_{ω}*δ*are the same as in the previous section. For these modulation signals, we calculated the optical output power from CRR-MZMs. For sufficiently large

_{ω}*t*, the optical output power varied sinusoidally with Ω regardless of the modulator configuration or the initial state, and we extracted its amplitude as the response Δ

*P*(Ω) of the modulator.

*y*

^{2}(5%), which confirms the validity of the analysis in the previous section. The responses were slightly different when the resonance was broad with a relatively large 1-

*y*

^{2}(40%). This deviation at a large 1-

*y*

^{2}occurred because the TCM theory is only valid for the resonators which weakly couple to the outside [26]. Although the analysis in the previous section is useful to gain an insight into the dynamics in CRR-MZMs, it should be carefully applied to obtain quantitative results, especially for relatively low-Q RRs. In the rest of paper, we show the results calculated using the numerical model to include low-Q RRs. In fact, 2

*τ*, which was calculated using Eqs. (3) and (6), is equal to the group delay of RR at Δ

*= 0, for only high-Q RRs. A more rigorous expression of the group delay of RR was given in previous works [29*

_{ω}29. 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]

*d*that was calculated by this rigorous form, and inserted it in signals between two adjacent RRs.

### 3.2 Calculated performance comparison between SRR-SSM and CRR-MZM

*P*

_{DC}= Δ

*P*(0) and the 3-dB bandwidth of SRR-SSMs and CRR-MZMs. Table 1 summarizes the conditions of the calculations. As shown in Table 1, we directly assigned values to some of the parameters, whereas the others were calculated from those values indicated in the table. We used the same conditions of the frequency detuning for SRR-SSM and CRR-MZM as mentioned in sections 2.1 and 2.2, respectively. We determined the values of

*n*,

_{g}*R*, and

*α*

_{dB}by making reference to other studies on silicon-based modulators [2

**15**(2), 430–436 (2007). [CrossRef] [PubMed]

**20**(15), 16321–16338 (2012). [CrossRef]

*α*

_{dB}= 0 and different values of

*τ*and

*N*. The left axis is the response of the modulators to a DC input, which was analytically calculated by Eqs. (17) and (18) using static solutions. The right axis indicates the 3-dB bandwidth,

*f*

_{3dB}, at which the response Δ

*P*(Ω) becomes half of the DC response Δ

*P*

_{DC}. Figure 6(a) shows general dependencies in RR-based modulators, regardless of the structures of the modulators in which the DC responses increase with the photon lifetime of the RR, whereas the 3-dB bandwidths decrease. The slopes of the DC response are constant for an increasing photon lifetime, and are almost the same among different structures. For a constant lifetime, the differences between the red, green, and blue solid curves are about 6-dB, corresponding to

*N*= 1, 4, and 16 for CRR-MZMs. This means that an increase of the response by

*N*-times is obtained for CRR-MZMs with

*N*-CRRs. However, in this case, the 3-dB bandwidths decrease with increasing

*N*, as shown by the dashed curves. To compare the results on the same basis, we drew parametric plots between the DC response and the 3-dB bandwidth with the parameter of the photon lifetime, as shown in Fig. 6(b). As clearly shown in the graph, CRR-MZMs have a greater response with an increase in

*N*for a constant 3-dB bandwidth, which confirms our predictions for CRR-MZMs which were mentioned in the introduction of this paper. Without propagation loss of the RRs, these gains are preserved at any 3-dB bandwidth. In other words, they are independent of the photon lifetime of the RRs.

**20**(15), 16321–16338 (2012). [CrossRef]

*N*, as shown in Fig. 7(a). Figure 7(b) is the parametric plot of the photon lifetime, and corresponds to Fig. 6(b). For a 3-dB bandwidth smaller than 25 GHz, CRR-MZMs have a relatively smaller DC response when compared with the SRR-SSM, regardless of

*N*, and the strategy of cascading RRs does not work. This is due to the large photon lifetime and the enhancement of loss in RRs. However, this dependence is not the case for a large 3-dB bandwidth with a relatively small photon lifetime. For a 3-dB bandwidth larger than 25 GHz, the modulation of CRR-MZM accumulates with an increase of

*N*, and CRR-MZMs provide larger responses than those provided by SRR-SSM, as shown in Fig. 7(b). Therefore, cascading RRs are still more effective than SRR-SSM for obtaining larger responses at high frequencies, even with the propagation losses.

*y*is less than 1, the presence of the RRs ensures that there is some enhancement of the phase modulation in the CRR-MZM with a factor of

*τ*/

*τ*

_{0}, when compared with the standard non-resonant Mach-Zehnder modulator [10

**20**(15), 16321–16338 (2012). [CrossRef]

^{7}-1. In the calculation, we used the same procedure as that described in section 3.1. The use of Eqs. (16) to (18) is not limited within small signal analysis and is valid for large signal modulation. Therefore, we simply applied the input signals of PRBS to those equations and the procedure to calculate eye diagrams. We set the rise and fall time to be 5.9 ps for the transition between 0 and 1 state with the thresholds of 10 and 90%, respectively. This rise/fall time was large enough for the round-trip time of 0.67 ps. By using Fig. 7(a), we determined the photon lifetimes of the RRs used in the two device configurations, respectively, such that both devices had the same 3-dB bandwidth of about 40 GHz. With this constant 3-dB bandwidth, it is expected that both devices equally show sufficient qualities of eye diagrams for 50-Gb/s operations. Thus, it is possible to evaluate the modulation depths of the two devices and compare them for 50-Gb/s operations. Choice of the smaller 3-dB bandwidth and the longer photon lifetimes would cause inadequate qualities of eye diagrams for 50-Gb/s operations. Figures 8(a) and (b) show the resulting eye diagrams, with the parameters used in the calculation shown in the graphs. By cascading 4-RRs with a relatively small photon lifetime, the CRR-MZM provided an eye opening that was wider than that of the SRR-SSM.

## 4. Conclusion

## Acknowledgments

## References and links

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12. | H. C. Nguyen, Y. Sakai, M. Shinkawa, N. Ishikura, and T. Baba, “10 Gb/s operation of photonic crystal silicon optical modulators,” Opt. Express |

13. | A. Brimont, D. J. Thomson, P. Sanchis, J. Herrera, F. Y. Gardes, J. M. Fedeli, G. T. Reed, and J. Martí, “High speed silicon electro-optical modulators enhanced via slow light propagation,” Opt. Express |

14. | H. Tazawa, Y. Kuo, I. Dunayevskiy, J. Luo, A. K. Y. Jen, H. Fetterman, and W. Steier, “Ring resonator based electrooptic polymer traveling-wave modulator,” J. Lightwave Technol. |

15. | H. Kaneshige, Y. Ueyama, H. Yamada, T. Arakawa, and Y. Kokubun, “Quantum well Mach-Zehnder modulator with single microring resonator and optimized arm length,” 17th Microoptics Conference (MOC' 11), Sendai, Japan, paper G-5, (2011). |

16. | H. F. Taylor, “Enhanced electrooptic modulation efficiency utilizing slow-wave optical propagation,” J. Lightwave Technol. |

17. | M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B |

18. | R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. |

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

20. | L. C. Kimerling, D. Ahn, A. B. Apsel, M. Beals, D. Carothers, Y.-K. Chen, T. Conway, D. M. Gill, M. Grove, C.-Y. Hong, M. Lipson, J. Liu, J. Michel, D. Pan, S. S. Patel, A. T. Pomerene, M. Rasras, D. K. Sparacin, K.-Y. Tu, A. E. White, and C. W. Wong, “Electronic-photonic integrated circuits on the CMOS platform,” Proc. SPIE |

21. | F. Xia, L. Sekaric, and Y. A. Vlasov, “Ultra-compact optical buffers on a silicon chip,” Nat. Photonics |

22. | I. L. Gheorma and R. M. Osgood, “Fundamental limitations of optical resonator based on high-speed EO modulators,” IEEE Photon. Technol. Lett. |

23. | W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator modulators,” Opt. Express |

24. | L. Zhang, Y. Li, J.-Y. Yang, M. Song, R. G. Beausoleil, and A. E. Willner, “Silicon-based microring resonator modulators for intensity modulation,” IEEE J. Sel. Top. Quantum Electron. |

25. | T. Ye and X. Cai, “On power consumption of silicon-microring-based optical modulators,” J. Lightwave Technol. |

26. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

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

28. | K. Okamoto, |

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

**OCIS Codes**

(230.5750) Optical devices : Resonators

(250.5300) Optoelectronics : Photonic integrated circuits

(250.7360) Optoelectronics : Waveguide modulators

(250.4110) Optoelectronics : Modulators

**ToC Category:**

Optoelectronics

**History**

Original Manuscript: July 5, 2012

Revised Manuscript: August 23, 2012

Manuscript Accepted: August 23, 2012

Published: September 10, 2012

**Citation**

Suguru Akiyama and Shintaro Nomura, "Dynamic response of modulators based on cascaded-ring-resonator," Opt. Express **20**, 21847-21859 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-21847

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