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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 13342–13348
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Coupled-ring-resonator-based silicon modulator for enhanced performance

Yunchu Li, Lin Zhang, Muping Song, Bo Zhang, Jeng-Yuan Yang, Raymond G. Beausoleil, Alan E. Willner, and P. Daniel Dapkus  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 13342-13348 (2008)
http://dx.doi.org/10.1364/OE.16.013342


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Abstract

A compact silicon coupled-ring modulator structure is proposed. Two rings are coupled to each other, and only one of these rings is actively driven and over-coupled to a waveguide, which enables high modulation speed. The resultant notch filter profile is steeper than that of the single ring and has exhibited a smaller resonance shift and lower driving electrical power. Simulations show that: (i) potentially 60-Gb/s non-return-to-zero (NRZ) data modulation with over 20-dB extinction ratio can be achieved by shifting the active ring by a 20 GHz resonance shift, (ii) the frequency chirp of the modulated signals can be adjusted by varying the coupling coefficient between the two rings, and (iii) dispersion tolerance at 0.5-dB power penalty is extended from 18 to 85 ps/nm, for a 40-Gb/s NRZ signal.

© 2008 Optical Society of America

1. Introduction

Electro-optic modulators are key elements of any optical communication system, and there has been much interest in using ring resonators in electro-optic modulators. When compared to Mach-Zehnder modulators, ring resonators tend to be smaller in size and have the potential for lower power consumption and array integration [1

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

, 2

2. T. Sadagopan, S. J. Choi, S. J. Choi, P. D. Dapkus, and A. E. Bond, “High-speed, low-voltage modulation in circular WGM micro-resonators,” 2004 IEEE/LEOS Summer Topical Meetings (LEOS, San Diego, Calif, 2004) MC2-3.

]. This topic has taken on new excitement due to the potential for using compact rings in silicon-based photonic circuits [3

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

]. Therefore, any advance in the basic functionality of a ring-based silicon modulator would be of benefit, and the key characteristics tend to be: (i) operating bandwidth, (ii) driving power, (iii) extinction ratio, and (iv) induced frequency chirp [4

4. 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 15, 11564–11569, (2007). [CrossRef] [PubMed]

].

There have been several demonstrations of silicon ring resonator based modulators [5

5. 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 15, 5069–5076, (2007). [CrossRef] [PubMed]

, 6

6. S. Manipatruni, Q. Xu, and M. Lipson, “PINIP based high-speed high-extinction ratio micron-size silicon electro-optic modulator,” Opt. Express 15, 13035–13042 (2007). [CrossRef] [PubMed]

]. The typical design is a waveguide with a single-ring resonator coupled to it, in which the driving voltage applied to the ring shifts the resonance wavelength and produces a modulation of the pass-through optical field at a given wavelength. A laudable goal would be to develop novel structures to enhance the bandwidth, efficiency, and flexibility in electro-optic modulation based on silicon. We note that novel electrical structures and driving schemes have been reported to increase the modulator speed [6

6. S. Manipatruni, Q. Xu, and M. Lipson, “PINIP based high-speed high-extinction ratio micron-size silicon electro-optic modulator,” Opt. Express 15, 13035–13042 (2007). [CrossRef] [PubMed]

], and the fundamental limitation to the modulation speed comes mainly from the optical domain. For a single ring modulator, the photon lifetime limits the achievable modulation bandwidth to the resonance linewidth when the resonance shift is typically comparable to the resonance linewidth. Multiple-ring structures can also be used as modulator [7

7. L. Zhang, M. Song, T. Wu, L. Zou, R. G. Beausoleil, and A. E. Willner, “Embedded ring resonators for micro-photonic applications,” Opt. Letters 33, to be published in issue no. 17, 2008.

], which opens the possibility of further increasing modulation speed. Coupled-ring-resonator structures composed of a sequence of micro-resonators coupled to an optical waveguide have been reported [8

8. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713, (1999). [CrossRef]

, 9

9. J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. of Modern Opt. 49, 2629–2636 (2002). [CrossRef]

]. Previously, these structures are used for slow light or nonlinear optics.

In this paper, we propose and analyze a coupled-ring-resonator-based modulator, in which only one ring resonator is driven, which is heavily over-coupled to the waveguide to increase operation speed. A passive ring is coupled with it to obtain a high extinction ratio. It is shown that up to 60-Gb/s NRZ modulation with over 20-dB extinction ratio can be achieved by only 20-GHz resonance shift of the active ring. The frequency chirp of the modulated signals can be adjusted by varying the coupling coefficient between the two rings. The coupled-ring modulator enhances dispersion tolerance of 40 Gb/s NRZ from 18 to 85 ps/nm at 0.5-dB power penalty in data transmission over single mode fiber (SMF) without dispersion compensation, as compared to the signals generated by a single-ring modulator.

2. Principle

However, simply increasing the coupling will decrease the cavity Q and enlarge the resonator linewidth, which may either cause a lower extinction ratio in the modulated signals or require a larger resonance shift [1

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

]. Inevitably, there is a tradeoff between the bandwidth and resonance shift for single ring modulator when keeping a certain extinction ratio.

Instead, as illustrated in Fig. 1(b), we describe a coupled-ring-based modulator, which utilizes two coupled rings. We designate the ring immediately adjacent to the waveguide as the “inner” ring and the other one as the “outer” ring. The inner ring is heavily over-coupled (i.e., coupling » loss) to the bus waveguide, and only this ring is actively driven to produce Large resonance shift Electrode Single-ring modulation data modulation. Given a fixed cavity Q, and considering that the loss is less than the coupling, the light energy in the inner ring can be accumulated much faster compared to the critical coupling case (coupling=loss) that is desired in the single-ring modulator. Since the output signal in the waveguide is determined by the interference of the input CW light with the light coupled from only the inner ring, the over-coupled ring can potentially respond to a higher-speed electrical signal due to the faster energy accumulation speed. Moreover, compared to the single-ring modulator, the proposed coupled-ring modulator features several potential advantages: (1) as shown in Fig. 1(b), the coupled-ring structure has a deeper notch profile, and thus enables relatively high extinction ratio, (2) the transmission profile of the coupled-ring structure becomes steeper, which may allow a smaller resonance shift and lower driving electrical power, and (3) increased design degrees-of-freedom provide us with better design flexibility to optimize the performance of the modulated signals in different communication scenarios, such as when chirp adjustment is desired.

Fig. 1. (a) Single ring modulator scheme. (b) The proposed coupled-ring modulator scheme: only the ring adjacent to waveguide is driven, which requires smaller resonance shift and enables higher modulation speed and higher extinction ratio.

Referring to [10

10. C. A. Barrios, “Electrooptic modulation of multisilicon-on-insulator photonic wires,” J. Lightwave Technol. 24, 2146–2155 (2006). [CrossRef]

], to model the coupled-ring modulator, we utilized an ideal 60 Gb/s NRZ drive voltage sent through a five-pole Bessel electrical filter as the electrical input signal. The variation in carrier density is simulated as a charging process following the applied voltage. According to [11

11. R. D. Kekatpure and M. L. Brongersma, “CMOS compatible high-speed electro-optical modulator,” Proc. SPIE 5926, paper G1 (2005) [CrossRef]

], this is believed to be a good fit to real behavior in MOS capacitors. The carrier transit time, defined as the duration for carrier density to increase from 10% to 90% of its peak value when a voltage step is applied, is considered to be 10 ps [10

10. C. A. Barrios, “Electrooptic modulation of multisilicon-on-insulator photonic wires,” J. Lightwave Technol. 24, 2146–2155 (2006). [CrossRef]

, 12

12. C. A. Barrios and M. Lipson, “Modeling and analysis of high-speed electro-optic modulation in high confinement silicon waveguides using metal-oxide-semiconductor configuration,” J. Appl. Phys. 96, 6008–6015 (2004). [CrossRef]

]. The continuous wave from laser source is then modulated. The relationship between the resonance frequency and the index is given by the resonance condition ω=mc/neffR, where ω is the resonance frequency, m an integar, c the light speeding vacuum, neff the effective index and R the ring radius. Change in voltage from 0 to 5 V causes a resonance peak shift of 0.16 nm towards shorter wavelength. To obtain the modulated optical signals, a set of differential equations are solved. The following derivations are essentially based on Refs. [3

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

, 13

13. H. A. Haus, Waves and fields in optoelectronics (Prentic-Hall, Inc. Englewood Cliffs New Jersey 07632, 1984) chap. 7.

]. The time rate equations of the energy amplitude change in the coupled-ring resonator with the inner ring coupled to a single waveguide are:

ddta1=(jω11τo11τe)a1+jμ2a2+jμ1Ein
(1-a)
ddta2=(jω21τo2)a2+jμ2a1
(1-b)
Eout=Ein+jμ1a1
(1-c)

where a 1 and a 2 are the energy amplitudes and ω 1 and ω 2 are the resonance angular frequencies in the inner ring and outer ring, respectively; E in and E out are the incident wave field and transmitted wave field; 1/τ e is the amplitude decay rate due to the power coupling into the waveguide; 1/τ o1 and 1/τ o2 are amplitude decay rates due to the intrinsic loss in the inner ring and outer ring; µ 11(vg1/2πR1)1/2 and µ 22(vg1vg2/2πR12πR2)1/2; κ1 is the power coupling between the ring and the bus waveguide; κ2 is the mutual power coupling between the rings; R1 and R2 are the inner ring radius and outer ring radius respectively. vg1 and vg2 are the group velocities in the inner ring and outer ring; and ω 0 is the carrier wave frequency. Since the inner ring is active while the outer ring is passive, ω 1 is modulated around the carrier frequency ω 0, whereas ω 2 is actually fixed at ω 0.

The operation principle of the coupled-ring modulator is as follows. First we focus on the inner ring, which is over-coupled to the waveguide. Assuming µ2=0 in equations (1-a, b, c) and ref, the energy in the inner ring resonator evolves as:

ddta12=2τea122τo1a12+2jμ1·Im(Eina1*)
(2)

Since the cavity Q is fixed, the sum of 1/τe and 1/τ01 is also fixed. As shown in Eq. (2) for an over-coupled ring resonator, the ring energy |a1|2 can increase faster compared to the critically coupled scenario with same cavity Q, due to a higher µ1, thereby resulting in a potentially higher response speed. Over-coupling, however, will result in the energy amplitude inside the ring being stronger than that in the critical-coupling condition, which produces a high signal ‘0’ level and a low extinction ratio.

In order to remove extra energy in the inner ring, the outer ring is needed. Given µ2≠0 and using equations (1-a) and (1-b), the time rate change of the energy in the rings evolves as:

ddta12=2τea122τo1a122jμ2·Im(a1a2*)+2jμ1·Im(Eina1*)
(3-a)
ddta22=2τo2a22+2jμ2·Im(a1a2*)
(3-b)

As compared to equation (2), the mutual energy coupling (2 2Im(a1a2*)) from the inner ring to the outer ring in Eq.(3-a) can decrease the inner-ring energy |a1|2, resulting in a low signal ‘0’ level and a high extinction ratio. Adding an outer ring produces mutual coupling loss in the inner ring while it won’t decrease the modulation bandwidth significantly: the mutual coupling loss (2 2Im(a1a2*)) in inner ring is proportional to the energy amplitudes in both inner and outer ring (a1 and a2); according to Eq. (3-b), the accumulation rate of a2 is proportional to a1, so the substantial growth of a2, as well as the mutual coupling loss, only happens as a1 accumulates, at which time the signal ‘0’ level is already generated by interference of the input CW light with the light coupled from the inner ring.

As shown in Fig. 2, the proposed modulator achieves both high response speed and high extinction ratio, exhibiting the same modulation performance with only 1/3 resonance shift, while the over-coupled single ring modulator produces heavily distorted signals.

Fig. 2. Comparison among the three examples of the modulation of a 60-Gb/s NRZ signal shows that the proposed modulator requires 1/3 resonance shift (R.S.) to achieve the modulation with negligible distortion compared to the critically coupled sing ring modulator. All figures are plotted in the same scale with the same output power.

3. Results

In Fig. 3, the 3-dB bandwidth and extinction ratio of the modulated signals are examined. The single ring modulator is compared to the coupled-ring modulator, where the power coupling coefficient κ1 between the ring and waveguide (WG) is varied. For the single ring modulator, the cavity Q of the ring cavity is maintained at 9500, corresponding to a 20-GHz resonance linewidth, and the ring radius is 2.7 µm. These parameters can be feasible according to the state-of-the-art fabrication [14

14. Q. Xu, D. Fattal, and R. G. Beausoleil, “1.5-µm-radius high-Q silicon microring resonators,” Opt. Express 16, 4309–4315 (2008). [CrossRef] [PubMed]

]. As shown in Fig. 3(a) and with a 20 GHz resonance shift, the 3-dB bandwidth of the single ring resonator modulator increases with the coupling coefficient κ1. Critical coupling occurs around κ1=0.095. As κ1>0.1, the modulation bandwidth has a trade-off with the extinction ratio and is limited to 30 GHz with extinction ratio >10 dB.

For the proposed structure, we would like to utilize an over-coupled inner ring with which the energy can be accumulated very fast at the resonance. Hence the probe wavelength is set equal to the inner ring resonance and the ring-ring coupling is set small enough to make the profile as a deep notch profile. The inner ring has a cavity Q-factor of 9500 for a fair comparison and its round-trip loss coefficient is 0.9991. The coupling coefficient κ2 is set to be 0.0094, and the round-trip loss coefficient of the outer ring is 0.9787. The two rings have the same radii of 2.7 µm. In Fig. 3(b), the critical coupling occurs at κ1=0.012. Given a resonance shift of 20 GHz, the simulated modulation bandwidth up to 50 GHz is observed with 12-dB extinction ratio.

Fig. 3. Modulation bandwidth and extinction ratio versus the ring-to-waveguide coupling coefficient κ1 for the (a) single ring and (b) coupled ring modulators.

The coupling coefficient κ2 between the two rings plays a critical role in optimizing the performance of the coupled-ring modulator. One can consider the coupled-ring structure as a single compound resonator and κ2 can be varied to adjust the energy distribution between the inner and outer rings. Noticing the loss in outer ring is larger than that in inner ring, people can modify overall loss coefficient of the compound resonator by changing κ2, which is different from a single ring resonator in which the loss is determined by the fabrication technique [3

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

]. As κ2 increases, the compound resonator becomes lossier and is switched from the over-coupled condition to the under-coupled one. Critical coupling is observed near κ2=0.013, in which the extinction ratio of the modulated signal is more than 20 dB (Fig. 4(a)). As a result, the frequency chirp is switched across the critical point (Fig. 4(b)). However, we note that large instantaneous frequency chirp occurs at the low-power region of pulse waveforms. Thus it is important to consider the chirp together with the instantaneous power. We define “effective chirp” as f(t)·p(t), where f(t) and p(t) represent the instantaneous frequency and power of the modulated pulses, respectively. With the input power of 0 dBm, we choose the peak value of the calculated effective chirp to evaluate the chirp effect of the modulated signals, as plotted in Fig. 4(b). As κ2 increases from 0.009 to 0.016, the peak value of effective chirp changes from -4.45 to -1.0 mW·GHz.

The system performance of the coupled-ring modulator is simulated and compared to a single ring modulator. In the back-to-back case, a 60-Gb/s NRZ signal is modulated with a single ring modulator (both linewidth and resonance shift=60 GHz) and a coupled-ring modulator (both inner ring linewidth and resonance shift=20 GHz), respectively. As shown in Fig. 5(a), two bit-error-rate (BER) curves are quite close to each other, indicating that the coupled ring modulator can modulate 60 Gb/s NRZ signal by only 20 GHz resonance shift of the inner ring under negligible degradation. In contrast, a critical-coupled single ring modulator operated with both the linewidth and resonance shift=20 GHz will result in a low extinction ratio and cannot achieve a low BER<10-9.

Fig. 4. (a) Extinction ratio versus the coupling coefficient κ2 between the inner and outer rings. Critical coupling occurs around κ2=0.013 and the extinction ratio tends to be infinite. (b) Frequency chirp is switched from negative to positive as κ2 increases.

As stated above, the modulated signal exhibits the negative effective chirp, which can be controlled by κ2. As an example, a 40-Gb/s NRZ is modulated by the coupled-ring modulators with different κ2, compared to a single ring modulator. Fig. 5(b) shows the simulated power penalty of the 40-Gb/s signal in the SMF transmision without dispersion compensation. We note that less than 0.5 dB power penalty due to 85 ps/nm chromatic dispersion is achieved for the 40-Gb/s NRZ signal with the coupled ring modulator. This is consistent with the trend shown in Fig. 4(b). In contrast, a single-ring modulator shows power penalty as high as 3.5 dB under 35 ps/nm chromatic dispersion and exhibits less flexibility to modify the properties of the modulated signal due to the request to keep critically coupling. Dispersion tolerance at 0.5-dB power penalty is extended from 18 to 85 ps/nm, as shown in Fig. 5(b).

Fig. 5. (a) Back-to-back BER curves of the 60-Gb/s NRZ signals modulated with the single ring (linewidth=60 GHz, resonance shift=60 GHz and 20 GHz) and coupled-ring modulators (linewidth=20 GHz, resonance=20 GHz). (b) A 40-Gb/s NRZ signal under different chromatic dispersion values, where the coupled-ring modulators with different effective chirps are achieved by varying the ring–to- ring coupling coefficient κ2.

4. Conclusion

A silicon microring modulator with coupled-ring-resonator structure is proposed. A 60-Gb/s NRZ signal has been obtained from a 20 GHz resonance shift of the inner ring. This design is particular meaningful for the high speed signal modulation based on microring resonators.

Acknowledgments

This work was supported in part by funding from the Optical Code Division Multiple Access (OCDMA) program of DARPA administered by SPAWAR, Contract no. N66001-02-1-8939, by the Chip Scale WDM (CSWDM) program of DARPA monitored by AFOSR, Contract no. F49620-02-1-0403, and HP Labs and the DARPA University Photonics Research Program.

References

1.

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

2.

T. Sadagopan, S. J. Choi, S. J. Choi, P. D. Dapkus, and A. E. Bond, “High-speed, low-voltage modulation in circular WGM micro-resonators,” 2004 IEEE/LEOS Summer Topical Meetings (LEOS, San Diego, Calif, 2004) MC2-3.

3.

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

4.

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 15, 11564–11569, (2007). [CrossRef] [PubMed]

5.

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 15, 5069–5076, (2007). [CrossRef] [PubMed]

6.

S. Manipatruni, Q. Xu, and M. Lipson, “PINIP based high-speed high-extinction ratio micron-size silicon electro-optic modulator,” Opt. Express 15, 13035–13042 (2007). [CrossRef] [PubMed]

7.

L. Zhang, M. Song, T. Wu, L. Zou, R. G. Beausoleil, and A. E. Willner, “Embedded ring resonators for micro-photonic applications,” Opt. Letters 33, to be published in issue no. 17, 2008.

8.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713, (1999). [CrossRef]

9.

J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. of Modern Opt. 49, 2629–2636 (2002). [CrossRef]

10.

C. A. Barrios, “Electrooptic modulation of multisilicon-on-insulator photonic wires,” J. Lightwave Technol. 24, 2146–2155 (2006). [CrossRef]

11.

R. D. Kekatpure and M. L. Brongersma, “CMOS compatible high-speed electro-optical modulator,” Proc. SPIE 5926, paper G1 (2005) [CrossRef]

12.

C. A. Barrios and M. Lipson, “Modeling and analysis of high-speed electro-optic modulation in high confinement silicon waveguides using metal-oxide-semiconductor configuration,” J. Appl. Phys. 96, 6008–6015 (2004). [CrossRef]

13.

H. A. Haus, Waves and fields in optoelectronics (Prentic-Hall, Inc. Englewood Cliffs New Jersey 07632, 1984) chap. 7.

14.

Q. Xu, D. Fattal, and R. G. Beausoleil, “1.5-µm-radius high-Q silicon microring resonators,” Opt. Express 16, 4309–4315 (2008). [CrossRef] [PubMed]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(130.0130) Integrated optics : Integrated optics
(200.4650) Optics in computing : Optical interconnects
(230.4110) Optical devices : Modulators

ToC Category:
Integrated Optics

History
Original Manuscript: July 7, 2008
Revised Manuscript: August 6, 2008
Manuscript Accepted: August 8, 2008
Published: August 14, 2008

Citation
Yunchu Li, Lin Zhang, Muping Song, Bo Zhang, Jeng-Yuan Yang, Raymond G. Beausoleil, Alan E. Willner, and P. Daniel Dapkus, "Coupled-ring-resonator-based silicon modulator for enhanced performance," Opt. Express 16, 13342-13348 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13342


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References

  1. T. Sadagopan, 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]
  2. T. Sadagopan, S. J. Choi, S. J. Choi, P. D. Dapkus, and A. E. Bond, "High-speed, low-voltage modulation in circular WGM micro-resonators," 2004 IEEE/LEOS Summer Topical Meetings (LEOS, San Diego, Calif, 2004) MC2-3.
  3. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997). [CrossRef]
  4. 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 15, 5069-5076 (2007). [CrossRef] [PubMed]
  5. 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 15, 11564-11569 (2007). [CrossRef] [PubMed]
  6. S. Manipatruni, Q. Xu, and M. Lipson, "PINIP based high-speed high-extinction ratio micron-size silicon electro-optic modulator," Opt. Express 15, 13035-13042 (2007). [CrossRef] [PubMed]
  7. L. Zhang, M. Song, T. Wu, L. Zou, R. G. Beausoleil and A. E. Willner, "Embedded ring resonators for micro-photonic applications," Opt. Lett.  33, to be published in issue no. 17, 2008.
  8. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999). [CrossRef]
  9. J. E. Heebner and R. W. Boyd, "'Slow' and 'fast' light in resonator-coupled waveguides," J. Mod. Opt. 49, 2629-2636 (2002). [CrossRef]
  10. C. A. Barrios, "Electrooptic modulation of multisilicon-on-insulator photonic wires," J. Lightwave Technol. 24, 2146-2155 (2006) [CrossRef]
  11. R. D. Kekatpure and M. L. Brongersma, "CMOS compatible high-speed electro-optical modulator," Proc. SPIE 5926, paper G1 (2005) [CrossRef]
  12. C. A. Barrios and M. Lipson, "Modeling and analysis of high-speed electro-optic modulation in high confinement silicon waveguides using metal-oxide-semiconductor configuration," J. Appl. Phys. 96, 6008-6015 (2004). [CrossRef]
  13. E. Hecht, Optics (3rd edition), (Addison Wesley Longman, Inc, 1998) Chap. 9.
  14. H. A. Haus, Waves and Fields in Optoelectronics (Prentic-Hall, Inc. Englewood Cliffs New Jersey 07632, 1984) Chap. 7. [CrossRef] [PubMed]
  15. Q. Xu, D. Fattal, and R. G. Beausoleil, "1.5-μm-radius high-Q silicon microring resonators," Opt. Express 16, 4309-4315 (2008).

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