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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7663–7671
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Chirp-free optical return-to-zero modulation based on a single microring resonator

Lili Sun, Tong Ye, Xiaowen Wang, Linjie Zhou, and Jianping Chen  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7663-7671 (2012)
http://dx.doi.org/10.1364/OE.20.007663


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Abstract

This paper proposes a chirp-free optical return-to-zero (RZ) modulator using a double coupled microring resonator. Optical RZ modulation is achieved by applying a clock (CLK) driving signal to the input coupling region and a non-return-to-zero (NRZ) driving signal to the output coupling region. Static and time-domain coupled-mode theory (CMT) based dynamic analyse are performed to theoretically investigate its performance in RZ modulation. The criteria to realize RZ modulation are deduced. Various RZ modulation formats, including RZ phase-shift-keying (RZ-PSK), carrier-suppressed RZ (CSRZ), and RZ intensity modulation formats, can be implemented by using CLK and NRZ signals with different combinations of polarities. Numerical simulations are performed and the feasibility of our modulator at 10 Gbit/s for the multiple RZ modulation formats is verified.

© 2012 OSA

1. Introduction

Non-return-to-zero (NRZ) and return-to-zero (RZ) are important modulation formats in optical transmission systems. The NRZ modulation format, widely used in short-reach networks, requires less bandwidth and has higher timing tolerance. The RZ modulation format, though requiring a more complex modulation scheme, is more robust to intersymbol interferences (ISI) and nonlinear propagation distortions [1

1. P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94, 952–985 (2006). [CrossRef]

], and thus has better receiver sensitivity. Therefore, optical RZ modulators receive sustained attention from the optical communication society.

In this paper, we propose a method to perform chirp-free RZ modulations using a single microring resonator side-coupled to two straight waveguides. To implement the RZ modulation, the coupling coefficients of the couplers at the input and output sides are independently controlled by an electrical clock (CLK) signal and an NRZ signal. In this case, if continuous wave (CW) light at the resonance wavelength is injected to the input waveguide, one can obtain an optical RZ signal at the output waveguide. Moreover, our dynamic analysis and simulation demonstrate that the proposed microring-based modulator is able to generate multiple RZ modulation formats, including RZ, carrier-suppressed RZ (CSRZ), and RZ-PSK, if the polarities of the driving signals are properly adjusted.

2. Principle

2.1. Static analysis

The schematic structure of the proposed modulator is given in Fig. 1(a). The modulator is composed by a microring resonator side-coupled to a pair of straight waveguides. The CW light from the input port successively passes through the input and output coupling regions before being detected at the drop port. The optical field transfer matrix at the input and output coupling regions is given as
Mi=ejθi(tijkijkiti),
(1)
where the subscript i = 1, 2 represents the input and output couplers, θi is the propagation phase-shift for the coupler, ti is the self-coupling coefficient, and ki is the cross-coupling coefficient. In the lossless case, ti2+ki2=1.

Fig. 1 Schematics of a coupling modulated microring resonator for (a) the general case and (b) two MZI couplers.

Let a1 and a2 be the propagation losses of ring waveguides, φ1 and φ2 be the phase-shifts of the ring waveguides. The microring resonator transfer function can be expressed as follows:
Tdrop=k1k2a1ej(θ1+θ2+φ1)1t1t2a1a2ej(θ1+θ2+φ1+φ2).
(2)
At resonance wavelengths, ej(θ1+θ2+φ1+φ2)= 1, and Eq. (2) becomes
Tdrop=k1k2a1ejφ21t1t2a1a2.
(3)

If k1, k2 ≪ 1, Eq. (3) can reduce to
Tdropa1ejφ21a1a2k1k2.
(4)
It clearly indicates that one can obtain an optical RZ signal, if the two coupling coefficients are respectively modulated by an electrical NRZ and a CLK signal with the assumption k1, k2 ≪ 1. We will show by simulations in Section 3 that the analytical results are still valid even when the small signal assumptions are relaxed.

To realize the coupling modulation, MZI couplers operated in a push-pull mode [7

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

] can be used for these two coupling regions, as illustrated in Fig. 1(b). The coupling coefficients can then be expressed as ki = jei sinΔθi, where θi is the phase-shift induced by the MZI coupler.

2.2. Dynamic analysis

Suppose two electrical driving signals are applied to the input and output coupling regions, the refractive indices are modulated as ni(t) = n ± (mbi + micosωit), where mbi is the bias, mi is the modulation amplitude, and ωi is the angular frequency of the applied electrical signal. Thereby, the input and output cross-coupling coefficients are modulated as
ki(t)=sin[(mbi+micosωit)βLe/n]kbi+rbicosωit,i=1,2
(7)
where kbi = sin (mbiβLe/n), rbi = mi(βLe/n) cos (mbiβLe/n), i = 1, 2, β is the propagation constant in the waveguide, and Le is the MZI arm length.

Substituting k1(t) and k2(t) into Eq. (5) and solving Eqs. (5) and (6) with the small signal assumption rb1 ≪ 1 and rb2 ≪ 1, we have:
sd(t)vgL[kb1kb2Ω+kb2rb1ω12+Ω2cos(ω1tϕ1)+kb1rb2Ωcos(ω2t)]+vg2L2kb1kb2Ω[i=12kbirbiωi2+Ω2cos(ωitϕi)]
(8)
where ϕi=arcsin[ωi2/(ωi2+Ω2)] and Ω=1/τl+(vg/2L)(kb12+kb22). It shows that the proposed modulation scheme is chirp-free since sd(t) is a real number. If kb1kb2 = 0 (or 1/τl(vg/2L)(kb12+kb22)) and ω1 ≪ Ω, Eq. (8) will immediately reduce to
sd(t)vgLk1(tϕ1/ω1)k2(t)Ω.
(9)

Equation (9) indicates that it is feasible to use a single ring to realize RZ modulation, as long as the following criteria are satisfied:
  1. ω1 cannot be very high;
  2. kb1kb2 = 0 or (1/τl)(vg/2L)(kb12+kb22), i.e., the unloaded quality factor (Q-factor) of the microring resonator cannot be very high;
  3. Signal applied to the input coupling region needs to be shifted with a phase ϕ1 in advance.

Condition 1 suggests that the modulation is inherently bandwidth-limited. Such characteristic is manifested by the Optical Modulation Amplitude (OMA), which is defined as:
OMA=f(t)maxf(t)min2,
(10)
where f(t)max and f(t)min are the maximum and minimum amplitudes of the signal. In the case of our scheme,
OMA=vgL|kb2rb1Ω2kb12vg2LΩω12+Ω2|+vgL[(kb1rb2Ω)2+(vg2L(2kb1kb2rb2)2kb22vg2LΩΩ2(ω22+Ω2))]1/2.
(11)

When ω2 = 0,
OMA|ω2=0=vgL|kb2rb1Ω2kb12vg2LΩω12+Ω2|.
(12)
If ω1 → ∞, OMA|ω2=0 rapidly reduces to zero, as illustrated in Fig. 2(a). The roll-off frequency ω3dB defined as the range of frequency where the OMA is reduced to 1/2 of the maximum frequency response. Clearly, we have ω3dB1 = Ω according to Eq. (12). Also, when ω1 = 0,
OMA|ω1=0=vgL[(kb1rb2Ω)2+(vg2L(2kb1kb2rb2)2kb22vg2LΩΩ2(ω22+Ω2))]1/2.
(13)
OMA|ω1=0 almost keeps unchanged even when ω2 → ∞, as shown in Fig. 2(b). This implies that: 1) The RZ modulation is bandwidth-limited at the input coupling region; 2) A square-wave NRZ signal, which has a broad frequency spectrum, may incur severe distortion if it is applied to the input coupling region. To minimize the signal distortion, we put the NRZ signal at the output coupling region and the CLK signal at the input coupling region, as illustrated in Fig. 3(a).

Fig. 2 OMA changes with signal frequency. (a) mb1 = m1 = 10−4, mb2 = m2 = 10−3 and ω2 = 0 (b) mb1 = m1 = 10−3, mb2 = m2 = 10−4 and ω1 = 0.
Fig. 3 (a) Signal application schemes for the proposed modulator. (b)–(d) Illustration of the advanced phase-shift of the CLK signal for (b) RZ-PSK, (c) CSRZ, and (d) pure RZ intensity modulations.

As for condition (2), we consider three cases. First, kb1 ≠ 0 and kb2 = 0, i.e., the driving signal applied to the input coupling region is unipolar while the signal applied at the output coupling region is bipolar. In our scheme illustrated in Fig. 3(a), kb1 ≠ 0 means a unipolar CLK signal and kb2 = 0 implies a bipolar NRZ signal with constant amplitude. Following Eq. (7), the unipolar CLK signal will mix with the bipolar NRZ signal in the optical domain, which yields an optical signal with the waveform the same as that of the CLK signal and the phase alternating as the NRZ signal. In other words, the obtained optical signal is an optical RZ-PSK signal, as illustrated in Fig. 3(b). Second, kb1 = 0 and kb2 ≠ 0, which suggests the CLK signal is bipolar and the NRZ signal is unipolar. Similarly, the mixing of the bipolar CLK and the unipolar NRZ in the optical domain will produce an optical CSRZ signal in Fig. 3(c). Third, if kb1 ≠ 0 and kb2 ≠ 0, both NRZ and CLK signals are unipolar, and pure RZ intensity modulation in Fig. 3(d) can be implemented when (1/τl)(vg/2L)(kb12+kb22) is satisfied. Again, (1/τl)(vg/2L)(kb12+kb22) implies that the unloaded Q-factor of the modulator should satisfy the condition that Q01/[(2/ω0)(vg/2L)(kb12+kb22)], which can be deduced from the relation 1/Q0 = 2/(ω0τl) [10

10. H. A. Haus, Waves and fields in optoelectronics (Prentice-HallEnglewood Cliffs, 1985).

]. For example, if kb1 = 0.162, kb2 = 0, λ0 = 1551nm, L = 420μm, the modulator should have an unloaded Q-factor of Q0 ≪ 4 × 105. As indicated in Fig. 3(b)–(d), the signal applied to the input coupling region needs to be shifted with a phase-shift ϕ1 in advance.

In summary, multiple optical chirp-free RZ modulation formats can be implemented by applying a CLK signal to the input coupling region and an NRZ signal to the output coupling region. Note that a phase-shift in advance is necessary for the signal applied to the input coupling region.

3. Simulation

Our proposed modulation scheme is essentially independent on the material platform that it can be built on. In particular, silicon-on-insulator (SOI) can be used due to its potential monolithic integration of photonic and electronic devices [11

11. H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE Sel. Top. Quantum Electron. 17, 671–688 (2011). [CrossRef]

]. On the other hand, polymer materials, which have a much larger electro-optic coefficient and thus a lower driving voltage, can also be employed [12

12. M. Hochberg, T. Baehr-Jones, G. Wang, J. Huang, P. Sullivan, L. Dalton, and A. Scherer, “Towards a millivolt optical modulator with nano-slot waveguides,” Opt. Express 15, 8401–8410 (2007). [CrossRef] [PubMed]

]. In our simulation, we assume the ring perimeter is 420 μm, and the length of the MZI arm is 200 μm. The ring waveguide propagation loss is α = 21 dB/cm, and the waveguide effective refractive index is ne ≈ 2.5. Here we assume the refractive index change is linearly proportional to driving voltage after the diode is turned on (Vth = 0.7V) and the effective index will have a change of 0.006 per volt according to the free carrier dispersion effect mentioned in [13

13. C. Angulo Barrios, V. R. Almeida, R. Panepucci, and M. Lipson, “Electrooptic modulation of silicon-on-insulator submicrometer-size waveguide devices,” J. Lightwave Technol. 21, 2332– (2003). [CrossRef]

]. We assume that the power of the input CW light is 0 dBm. The electrical NRZ signal is generated using a 27 – 1 pseudo-random binary sequence (PRBS), and filtered by a low-pass Butterworth filter to obtain a rising/falling time of 15 ps. To mimic the light wave propagation, we perform the simulation using the finite-difference method [14

14. L. Ghisa, Y. Dumeige, N. N. T. Kim, Y. G. Boucher, and P. Feron, “Performances of a fully integrated all-optical pulse reshaper based on cascaded coupled nonlinear microring resonators,” J. Lightwave Technol. 25, 2417–2426 (2007). [CrossRef]

].

Figure 4 shows the performance for 10-Gb/s RZ-PSK modulation. The refractive index modulation Δn1 and Δn2 on the MZI couplers are shown in Fig. 4(a) and 4(b). The Δn1 and Δn2 modulation amplitudes (defined as the peak-to-peak value) are 4 × 10−4 and 1.4 × 10−3, respectively. In this case, the microring operates at the over-coupling region. The phase-shift for the CLK signal is 0.25π. The driving voltages for the refractive index modulations are also indicated in the Fig. 4(a) and 4(b). Figure 4(c) shows the modulated optical signal with a power contrast ratio (CR) of larger than 17 dB and the phase of the output signal, which confirms the modulation is almost chirp-free. Our dynamic analysis in Section 2 shows that the CLK signal applied to the input coupling region suffers from a low-pass filtering effect. To suppress such an effect, we increase the amplitude of the NRZ signal to increase the output cross-coupling coefficient, and thus the Q-factor of the resonator is decreased. As visualized in Fig. 5, the CR of the output signal has a significant improvement from 7 dB to 23 dB when Δn2 increases from 2 × 10−4 to 1.8 × 10−3. Moreover, Fig. 4(d) shows the modulation performance in the scenario where the CLK signal isn’t phase-shifted in advance. Figure 4(d) shows that there is misalignment between the output power waveform and the output phase waveform, which will increase the bit error rate at the receiver. This is consistent with the prediction by our analytical results in Subsection 2.2.

Fig. 4 (a) 10 GHz CLK driving signal and (b) 10 Gb/s NRZ driving signal for the RZ-PSK modulation. (c) Optical output signal when the CLK signal is phase-shifted in advance. (d) Optical output signal when there is no phase shift for the CLK signal. Insets in (c) and (d) show the output optical signal eye diagrams.
Fig. 5 Output power and CR change as a function of Δn2 when the NRZ signal amplitude is varied.

Figure 6 shows the result for the CSRZ modulation at 10-Gb/s, where the amplitude of Δn1 is 8 × 10−4 and that of Δn2 is 4 × 10−4. The phase-shift for the CLK signal is 0.2π. In this case, the microring operates at the under-coupling region. Figure 6 illustrates that the resulting optical CSRZ signal has a large extinction ratio. We also find the optical CSRZ signal is nearly chirp-free except for a thorn-like phase peak at each 0–π transition. Such thorn-like phase peaks appear due to the fact that the propagation loss on each arm of the push-pull MZI couplers is slightly different. Fortunately, the thorn-like phase peaks correspond to a very low optical signal, and thus have a negligible effect.

Fig. 6 Output optical signal power and phase for the CSRZ modulation at 10-Gb/s. Inset shows the optical signal eye diagram.

Pure RZ intensity modulation can also be generated by applying a unipolar CLK signal and a unipolar NRZ signal to the input and output couplers, respectively. Figure 7(a) shows the modulated optical signals when the amplitudes of Δn1 and Δn2 are both 4 × 10−4, and the CLK signal has a phase-shift of 0.35π. In this case, the microring operates at the under-coupling region. The resulting optical signal suffers from a little distortion. To reduce the distortion, we increase to 45 dB/cm to degrade the unloaded Q-factor of the modulator, while keeping the driving NRZ and CLK signals unchanged. Note that a more practical way to decrease the unloaded Q-factor is to introduce another waveguide to couple with the microring. Figure 7(b) shows the improved signal quality, which is consistent with our analysis. Note that the improvement in signal quality is at the expense of lower output signal power.

Fig. 7 Output optical signal power and phase for the RZ-intensity modulation at 10-Gb/s with the ring waveguide losses of (a) α = 21 dB/cm and (b) α = 45 dB/cm. Insets show the optical signal eye diagrams.

In practice, it is hard for the input wavelength to exactly match the resonance wavelength. Therefore, we investigate the impact of the wavelength deviation on the modulation performance of the proposed RZ modulator. The deteriorated modulation performances when a 0.01 nm wavelength deviation is introduced are shown in Fig. 8. It suggests that the modulated signals only incur a small chirp if the deviation is not very large.

Fig. 8 Output modulated signals when input wavelength deviates from resonance by 0.01 nm for (a) RZ-PSK, (b) CSRZ, and (c) RZ-intensity modulations. Insets show the corresponding eye-diagrams of the modulated signals.

4. Conclusion

We proposed an optical chirp-free RZ modulator based on a double coupled microring resonator. The static analysis suggests the output optical signal is resulted from the multiplication of two electrical signals (CLK and NRZ signals) applied to the input and output coupling regions. The time-domain CMT based dynamic analysis further reveals the modulation mechanism and the criteria for achieving RZ modulation are deduced. Due to the modulation bandwidth limitation at the input coupler, the CLK signal can only be applied to the input coupler and the NRZ signal to the output coupler. Multiple RZ modulation formats, such as RZ-PSK, CSRZ, and RZ intensity modulation, can be achieved by using different polarities of the applied electrical signals. We also did numerical simulations to confirm the feasibility of our modulator in realizing the various RZ modulation formats. The simulation results at 10 Gbit/s verify the feasibility of our proposed modulator for RZ-PSK, CSRZ, and RZ intensity modulations.

Acknowledgments

This work was supported in part by 973 program (ID2011CB301700), NSFC ( 61001074, 61071011, 61007039, 61127016), Project of MoE Key Lab of Optical Fiber Sensing & Communication (UESTC), The state Key Laboratory of Optoelectronics Project (No. 2010KFB002), STCSM Project (10DJ1400402), State Key Lab Projects (GKZD030004/09/15/20/21).

References and links

1.

P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94, 952–985 (2006). [CrossRef]

2.

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

3.

S. Manipatruni, X. Qianfan, B. Schmidt, J. Shakya, and M. Lipson, “High speed carrier injection 18 Gb/s silicon micro-ring electro-optic modulator,” in Proceedings of Lasers and Electro-Optics Society (IEEE, 2007), 537–538.

4.

W. D. Sacher, W. M. Green, S. Assefa, T. Barwicz, S. M. Shank, Y. A. Vlasov, and J. Poon, “Controlled coupling in silicon microrings for high-speed, high extinction ratio, and low-chirp modulation,” in CLEO:2011 - Laser Applications to Photonic Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA8.

5.

K. Padmaraju, N. Ophir, S. Manipatruni, C. B. Poitras, M. Lipson, and K. Bergman, “DPSK modulation using a microring modulator,” in CLEO:2011 - Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuN4.

6.

L. Zhang, Y. Li, J. Yang, R. G. Beausoleil, and A. E. Willner, “Creating RZ data modulation formats using parallel silicon microring modulators for pulse carving in DPSK,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CWN4.

7.

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

8.

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]

9.

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. 16, 149–158 (2010). [CrossRef]

10.

H. A. Haus, Waves and fields in optoelectronics (Prentice-HallEnglewood Cliffs, 1985).

11.

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE Sel. Top. Quantum Electron. 17, 671–688 (2011). [CrossRef]

12.

M. Hochberg, T. Baehr-Jones, G. Wang, J. Huang, P. Sullivan, L. Dalton, and A. Scherer, “Towards a millivolt optical modulator with nano-slot waveguides,” Opt. Express 15, 8401–8410 (2007). [CrossRef] [PubMed]

13.

C. Angulo Barrios, V. R. Almeida, R. Panepucci, and M. Lipson, “Electrooptic modulation of silicon-on-insulator submicrometer-size waveguide devices,” J. Lightwave Technol. 21, 2332– (2003). [CrossRef]

14.

L. Ghisa, Y. Dumeige, N. N. T. Kim, Y. G. Boucher, and P. Feron, “Performances of a fully integrated all-optical pulse reshaper based on cascaded coupled nonlinear microring resonators,” J. Lightwave Technol. 25, 2417–2426 (2007). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.4080) Fiber optics and optical communications : Modulation
(130.0130) Integrated optics : Integrated optics

ToC Category:
Integrated Optics

History
Original Manuscript: December 15, 2011
Revised Manuscript: January 31, 2012
Manuscript Accepted: February 16, 2012
Published: March 20, 2012

Citation
Lili Sun, Tong Ye, Xiaowen Wang, Linjie Zhou, and Jianping Chen, "Chirp-free optical return-to-zero modulation based on a single microring resonator," Opt. Express 20, 7663-7671 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7663


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References

  1. P. J. Winzer, R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94, 952–985 (2006). [CrossRef]
  2. Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005). [CrossRef] [PubMed]
  3. S. Manipatruni, X. Qianfan, B. Schmidt, J. Shakya, M. Lipson, “High speed carrier injection 18 Gb/s silicon micro-ring electro-optic modulator,” in Proceedings of Lasers and Electro-Optics Society (IEEE, 2007), 537–538.
  4. W. D. Sacher, W. M. Green, S. Assefa, T. Barwicz, S. M. Shank, Y. A. Vlasov, J. Poon, “Controlled coupling in silicon microrings for high-speed, high extinction ratio, and low-chirp modulation,” in CLEO:2011 - Laser Applications to Photonic Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA8.
  5. K. Padmaraju, N. Ophir, S. Manipatruni, C. B. Poitras, M. Lipson, K. Bergman, “DPSK modulation using a microring modulator,” in CLEO:2011 - Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuN4.
  6. L. Zhang, Y. Li, J. Yang, R. G. Beausoleil, A. E. Willner, “Creating RZ data modulation formats using parallel silicon microring modulators for pulse carving in DPSK,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CWN4.
  7. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002). [CrossRef]
  8. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997). [CrossRef]
  9. L. Zhang, Y. Li, J.-Y. Yang, M. Song, R. G. Beausoleil, A. E. Willner, “Silicon-based microring resonator modulators for intensity modulation,” IEEE J. Sel. Top. Quantum Electron. 16, 149–158 (2010). [CrossRef]
  10. H. A. Haus, Waves and fields in optoelectronics (Prentice-HallEnglewood Cliffs, 1985).
  11. H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE Sel. Top. Quantum Electron. 17, 671–688 (2011). [CrossRef]
  12. M. Hochberg, T. Baehr-Jones, G. Wang, J. Huang, P. Sullivan, L. Dalton, A. Scherer, “Towards a millivolt optical modulator with nano-slot waveguides,” Opt. Express 15, 8401–8410 (2007). [CrossRef] [PubMed]
  13. C. Angulo Barrios, V. R. Almeida, R. Panepucci, M. Lipson, “Electrooptic modulation of silicon-on-insulator submicrometer-size waveguide devices,” J. Lightwave Technol. 21, 2332– (2003). [CrossRef]
  14. L. Ghisa, Y. Dumeige, N. N. T. Kim, Y. G. Boucher, P. Feron, “Performances of a fully integrated all-optical pulse reshaper based on cascaded coupled nonlinear microring resonators,” J. Lightwave Technol. 25, 2417–2426 (2007). [CrossRef]

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