## Parallel-coupled dual racetrack silicon micro-resonators for quadrature amplitude modulation |

Optics Express, Vol. 19, Issue 16, pp. 14892-14902 (2011)

http://dx.doi.org/10.1364/OE.19.014892

Acrobat PDF (1082 KB)

### Abstract

A parallel-coupled dual racetrack silicon micro-resonator structure is proposed and analyzed for *M*-ary quadrature amplitude modulation. The over-coupled, critically coupled, and under-coupled scenarios are systematically studied. Simulations indicate that only the over-coupled structures can generate arbitrary *M*-ary quadrature signals. Analytic study shows that the large dynamic range of amplitude and phase of a modulated over-coupled structure stems from the strong cross-coupling between two resonators, which can be understood through a delicate balance between the direct sum and the “interaction” terms. Potential asymmetries in the coupling constants and quality factors of the resonators are systematically studied. Compensations for these asymmetries by phase adjustment are shown feasible.

© 2011 OSA

## 1. Introduction

1. P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE **94**(5), 952–985 (2006). [CrossRef]

_{3}) modulators can be used for such modulation. However, LiNbO

_{3}modulators are relatively large in size. For a general

*M*-ary modulation format that requires a large number of optical modulator components along with their driving signal circuitries, the overall size of the entire modulator is rather cumbersome. Recent breakthroughs in silicon photonics [3

3. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. **12**(6), 1678–1687 (2006). [CrossRef]

4. B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. **24**(12), 4600–4615 (2006). [CrossRef]

5. A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature **427**(6975), 615–618 (2004). [CrossRef] [PubMed]

6. Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature **435**(7040), 325–327 (2005). [CrossRef] [PubMed]

6. Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature **435**(7040), 325–327 (2005). [CrossRef] [PubMed]

10. D. M. Gill, S. S. Patel, M. Rasras, K. Y. Tu, A. E. White, Y. K. Chen, A. Pomerene, D. Carothers, R. L. Kamocsai, C. M. Hill, and J. Beattie, “CMOS-compatible si-ring-assisted Mach-Zehnder interferometer with internal bandwidth equalization,” IEEE J. Sel. Top. Quantum Electron. **16**(1), 45–52 (2010). [CrossRef]

11. 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**(18), 11564–11569 (2007). [CrossRef] [PubMed]

12. L. Zhang, J. Y. Yang, Y. C. Li, M. P. Song, R. G. Beausoleil, and A. E. Willner, “Monolithic modulator and demodulator of differential quadrature phase-shift keying signals based on silicon microrings,” Opt. Lett. **33**(13), 1428–1430 (2008). [CrossRef] [PubMed]

13. 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 **15**(25), 17264–17272 (2007). [CrossRef] [PubMed]

*Q*microring quadrature modulator incorporating dual 2 × 2 Mach-Zehnder interferometers has also been recently proposed with beneficial performance [14

14. W. D. Sacher and J. K. S. Poon, “Microring quadrature modulators,” Opt. Lett. **34**(24), 3878–3880 (2009). [CrossRef] [PubMed]

*M*-ary quadrature amplitude modulation (QAM). Two identical racetrack resonators are symmetrically side-coupled in parallel to a through waveguide in the center. The modulator can be fabricated on a silicon-on-insulator (SOI) wafer. The carriers can be injected or depleted from the racetrack resonators using a

*pin*diode [15

15. L. L. Gu, W. Jiang, X. N. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett. **90**(7), 071105 (2007). [CrossRef]

16. X. N. Chen, Y. S. Chen, Y. Zhao, W. Jiang, and R. T. Chen, “Capacitor-embedded 0.54 pJ/bit silicon-slot photonic crystal waveguide modulator,” Opt. Lett. **34**(5), 602–604 (2009). [CrossRef] [PubMed]

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

*n*

_{1}, Δ

*n*

_{3}

*in each racetrack resonator, which modifies the cross-coupled resonances of the two racetrack resonators. By carefully choosing the voltage signals applied to each resonator, the amplitude and phase of output optical signal can be controlled to generate arbitrary*

_{,}*M*-ary quadrature signals.

*M*-ary quadrature signal generations including quadrature phase shift keying (QPSK). The outcome of the cross-coupling of the resonances is fairly complex. However, our analysis shows that it can be understood through the direct sum and coherent “interaction” of the optical characteristics of two individual resonators as presented in Sec. 2.4. The structure of this paper is organized as follows. First the cross-coupling between the racetrack resonators is analyzed and the output transfer function of the proposed structure is presented. The critical coupling condition is obtained. Systematic studies of the over-coupled, critically coupled, and under-coupled scenarios for the parallel-coupled racetrack resonator structure indicate that strong over-coupling case is desired for arbitrary

*M*-ary quadrature signal generation. The interaction between the resonances of two racetracks is analyzed, and its critical role in

*M*-ary quadrature signal generation is presented. The effects of asymmetries in the coupling strengths and quality factors of resonators are systematically studied, and phase compensations for such asymmetries are presented. Lastly, the electrical aspects of the proposed modulators are briefly discussed, followed by a conclusion.

## 2. Principles of parallel-coupled racetrack resonators

### 2.1 Cross-Coupling Analysis and Output Transfer Function

18. V. A. Mashkov and H. Temkin, “Propagation of eigenmodes and transfer amplitudes in optical waveguide structures,” IEEE J. Quantum Electron. **34**(10), 2036–2047 (1998). [CrossRef]

20. C.-M. Kim and Y.-J. Im, “Switching operations of three-waveguide optical switches,” IEEE J. Sel. Top. Quantum Electron. **6**(1), 170–174 (2000). [CrossRef]

*u*(

_{n}*z*)where

**M**

*(*

_{n}*x,y*) is the lateral mode profile,

*β*is the propagation constant along the waveguide axis

*z*for an isolated waveguide. For the parallel coupled racetrack resonator structure in Fig. 1, the input fields and output fields of the coupling segments are given bywhere

*a*and

_{n}*b*are the normalized input and output complex amplitudes, respectively. The solution of the coupled mode equations yields [20

_{n}20. C.-M. Kim and Y.-J. Im, “Switching operations of three-waveguide optical switches,” IEEE J. Sel. Top. Quantum Electron. **6**(1), 170–174 (2000). [CrossRef]

*c*

_{1}−1/2|. In addition, light propagation along a racetrack gives rise to the following relationswhere the amplitude attenuation along a racetrack is given by

*η*<1, and the phase shift is given by

_{n}*θ*. Assuming a unity input amplitude

_{n}*a*

_{2}= 1, the output amplitude

*b*can be solved from Eqs. (3) and (4) where

_{2}*ϕ = βL*, and

*u*

_{1}and Δ

*u*

_{3}. As such, the symmetry of the structure can be utilized to help simplify the understanding of the device principles, as noted in the study of other devices [21

21. W. Jiang and R. T. Chen, “Multichannel optical add-drop processes in symmetrical waveguide-resonator systems,” Phys. Rev. Lett. **91**(21), 213901 (2003). [CrossRef] [PubMed]

### 2.2 Critical Coupling Condition and Vanishing Amplitude for a Modulated Over-Coupled Structure

22. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**(4), 483–485 (2002). [CrossRef]

*b*= 0 in Eq. (5). For symmetric parallel-coupled racetracks without modulation (

_{2}*θ*in each ring will be a linear function of the refractive index changes, Δ

_{n}*n*, due to carrier injection or depletion in the respective racetrack resonator. Therefore the output amplitude

_{n}*b*

_{2}depends on Δ

*n*through the phase shift terms. To understand the modulation characteristics, it is helpful to rewrite the output amplitude in the following form

_{n}*c*

_{1}is a real number, for a modulated symmetric (

*η*

_{1}=

*η*

_{3}) dual-racetrack structure, the output amplitude can vanish only if

*modulated*amplitude can still vanish under the following modulation condition where

*m*

_{1}and

*m*

_{3}are two integers. For real nonzero Δ

*θ,*this requireswhich corresponds to over-coupling in comparison to Eq. (8). The spectra of an over-coupled dual racetrack structure (without modulation) are illustrated in Fig. 1(b) and 1(c).

### 2.3 Arbitrary M-*ary* Quadrature Signal Generation Capability

*n*can be achieved with carrier concentration changes Δ

*N*

_{e}, Δ

*N*

_{h}~3 × 10

^{17}cm

^{−3}according to the well-known plasma dispersion relation reported in [17

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

*r*

_{1}=

*r*

_{3}= 3μm,

*L*= 3μm,

*η*

_{1}=

*η*

_{3}= 0.994,

*c*

_{1}= 0.4243. Note that compact silicon racetrack resonators have been systematically characterized recently [23

23. M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. **46**(8), 1158–1169 (2010). [CrossRef]

23. M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. **46**(8), 1158–1169 (2010). [CrossRef]

*n*

_{1},Δ

*n*

_{3}) = ( ± 3.5 × 10

^{−4}, ∓3.5 × 10

^{−4}), in accordance with the analytic results given in Eq. (9b). In all phase plots starting from Fig. 2, the overall constant phase factor e

^{i}

*in*

^{ϕ}*b*

_{2}is omitted to better illustrate the symmetry of the modulated output. On a side note, if

*η*

_{1}and

*η*

_{3}decrease simultaneously (

*η*

_{1}=

*η*

_{3}), the two “eyes” on the diagonal of Fig. 2(a) widen and the phase contours in Fig. 2(b) expand accordingly.

*E*

_{out}(Δ

*n*

_{1},Δ

*n*

_{3}), for

*M*-ary signal generations, the ensemble of complex

*b*values for all values of

_{2}*n*

_{1}, Δ

*n*

_{3}values in the aforementioned range. Evidently, the ensemble of blue points covers most part of the unit circle (the symbol space), therefore, allowing for the access of a wide range of amplitude and phase values. A close examination of Fig. 2 indicates that the intensity and phase varies widely in the second and fourth quadrants where Δ

*n*

_{1}and Δ

*n*

_{3}have opposite signs, which corresponds to a push-pull configuration. In contrast, the intensity and phase are much less sensitive to Δ

*n*

_{1}and Δ

*n*

_{3}when they have the same sign. Indeed, our simulations indicate that the push-pull configuration is usually responsible for over 80% of coverage on the complex

*E*plane. Hence a push-pull modulation configuration is preferred for such a parallel-coupled dual-racetrack structure.

### 2.4 The Cross-Coupling of Two Racetrack Resonances: Direct Sum and “Interaction”

*strong cross-coupling*between the two racetrack resonators mediated by the center waveguide. To illustrate this point, the simulated typical coverage of a critically coupled case and an under-coupled case is shown in Fig. 3(b) and 3(c), respectively, for parallel-coupled dual racetrack resonators. In addition, the simulated typical coverage for two

*uncoupled*racetrack resonator in series is plotted in Fig. 3(d)–3(f). None of the cases illustrated in Fig. 3(b)–3(f) has adequate coverage for arbitrary

*M*-ary quadrature signal generation.

*u*

_{1}+ Δ

*u*

_{3}and the “interaction” term Δ

*u*

_{1}Δ

*u*

_{3}on both the numerator and denominator in Eq. (5). Based on their definitions

*u*can be regarded as the

_{n}*normalized*change of the field amplitude after one round-trip propagation in a racetrack. Here the initial field amplitude is unity, and the amplitude change is

*normalized by the final field*amplitude

*n*

_{1}= Δ

*n*

_{3}= 0),

*η*

_{1}) near resonance, and Δ

*u*

_{1}and Δ

*u*

_{3}are in phase. Therefore, we findbecause 1−

*η*

_{1}<2(1/2−

*c*

_{1}) according to the strong coupling condition. The dominance of the direct sum term in Eq. (5) yields an output amplitude close to −1. With sufficient modulation in a push-pull configuration, Δ

*u*can gain large imaginary parts (Im(Δ

_{n}*u*)~Δ

_{n}*θ*, up to ± 0.09 at Δ

_{n}*n*= 0.001) with opposite signs whereas their real parts remain small. Therefore, the product term exceeds the sum by a large margin, |Δ

_{n}*u*

_{1}Δ

*u*

_{3}|>>|Δ

*u*

_{1}+ Δ

*u*

_{3}| such that

## 3. Asymmetry Effect in Parallel-Coupled Dual Racetrack Resonators

*L*>2μm. Optical path differences between the two racetracks can usually be compensated by a proper DC bias or by additional thermo-optic heaters [24

24. M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics **4**(2), 117–122 (2010). [CrossRef]

25. W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express **18**(23), 23598–23607 (2010). [CrossRef] [PubMed]

### 3.1 Asymmetric Coupling

*κ*

_{12}and

*κ*

_{23}, respectively. To solve such a set of differential equation,

*X*are the eigenvectors of

*XX*

^{+}= I. The original equation can then be integrated according to

*b*

_{2}can be solved in a procedure similar to that given for the symmetric case. After lengthy calculations, the final result is surprisingly simplewhere Δ

*u*are defined the same way as in the symmetric case. Comparing Eq. (14) and Eq. (8), it is evident that all asymmetry effects can be effectively factored into the term

_{n}*θ*

_{1}and Δ

*θ*

_{3}such that the output amplitude

*b*

_{2}vanishes. The required phase variations are plotted against the asymmetric coupling ratio, κ

_{23}/κ

_{12}, in Fig. 4(a) for up to 50% asymmetry. As Δ

*θ*

_{1}and Δ

*θ*

_{3}generally have opposite signs, we plot Δ

*θ*

_{1}and −Δ

*θ*

_{3}to better illustrate the deviation from symmetry. Note that Δ

*θ*

_{1}= −Δ

*θ*

_{3}is required for

*b*

_{2}= 0 in a symmetric structure (

*κ*

_{23}/

*κ*

_{12}= 1), according to Eq. (9). The difference between Δ

*θ*

_{1}and −Δ

*θ*

_{3}becomes larger as the asymmetry increases.

*b*

_{2}~0). The un-modulated output spectrum for the worst case (

*κ*

_{23}/

*κ*

_{12}= 1.5) is illustrated in Fig. 4(b) and shows no anomaly. However, the intensity variation upon refractive index modulation shows obvious distortion from the symmetric case. Nonetheless, two features remain: (1) there are two points with relatively small index changes ( ± 2.2 × 10

^{−4}, ∓5.4 × 10

^{−4}) where the intensity vanishes; (2) the intensity varies significantly in the push-pull configuration and much less otherwise. The coverage on the complex

*E*plane is slightly enhanced, although a small hole exists at a large amplitude value, which may limit the maximum accessible amplitude to 0.78 for a generic

*M*-ary modulation format.

### 3.2 Asymmetric Quality Factors

*θ*

_{1}and −Δ

*θ*

_{3}, for vanishing

*b*

_{2}, are plotted against the ratio of the quality factors in Fig. 5(a). The unloaded quality factor

*Q*

_{1}is fixed at its original value ~2.5 × 10

^{4}.

*θ*

_{1}= −Δ

*θ*

_{3}for the case of

*Q*

_{3}/

*Q*

_{1}= 1 in accordance to the symmetric case. The un-modulated output spectrum for the worst case (

*Q*

_{3}/

*Q*

_{1}= 0.5) is illustrated in Fig. 5(b). A small yet noticeable spike appears at the resonance due to the asymmetric quality factors of the two racetrack resonators. The modulated intensity variation upon refractive index modulation depicted in Fig. 5(c) shows less severe distortion compared to the distortion observed in the Fig. 4(c). Again, two features remain: (1) there are two points with relatively small index changes ( ± 4.4 × 10

^{−4}, ∓4.1 × 10

^{−4}) where the intensity vanishes; (2) the intensity varies significantly in the push-pull configuration and much less otherwise. The coverage on the complex

*E*plane slightly deteriorates. There exists a small hole, which may limit the maximum accessible amplitude to 0.74 for a generic

*M*-ary modulation format.

*n*

_{1}= Δ

*n*

_{3}= 0). As the asymmetry in the quality factors worsens, the “eye” centers do not narrow or rotate substantially although there are some deformations.

^{−4}, which can be readily provided with a low-power heater or a small change of the DC bias. Fundamentally, such compensations are possible because all these asymmetries enter the output amplitude, Eq. (14), through the term given in Eq. (15). For structures with asymmetric

*η*’s or

*Q’*s, asymmetric phase shifts can restore the value of the term given in Eq. (15) to a corresponding symmetric structure. Specifically, to achieve vanishing output intensity under modulation, a structure with 50% asymmetry in the coupling constant requires (Δ

*n*

_{1},Δ

*n*

_{3}) = ( ± 2.2 × 10

^{−4}, ∓5.4 × 10

^{−4}) whereas a symmetric structure requires (Δ

*n*

_{1},Δ

*n*

_{3}) = ( ± 3.5 × 10

^{−4}, ∓3.5 × 10

^{−4}). The difference between |Δ

*n*

_{1}| and |Δ

*n*

_{3}| in the asymmetric case is used to restore Eq. (15) to the value of the symmetric case such that

*b*= 0.

_{2}## 4. Discussion

*M*-ary digital signal into the driving signal for the modulator. Consider the case of a QPSK signal with four symbols shown in Fig. 3(a). The encoder will have a two-bit input and two output ports. Each output port has four output voltage levels. The design of such an encoder and its supporting circuitries has been well studied in the state-of-the-art high-speed data conversion systems [26] and CMOS VLSI [27]. Under the given specifications (resolution, signal-to-noise ratio, bandwidth, driving power, etc.), this encoder can be easily architected and implemented as a high-speed digital-to-analog data converter, which can be fabricated economically using the silicon-on-insulator technology together with the dual racetrack resonator modulator. Note that a conventional nested Mach-Zehnder QPSK [1

1. P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE **94**(5), 952–985 (2006). [CrossRef]

28. W. Jiang, L. Gu, X. Chen, and R. T. Chen, “Photonic crystal waveguide modulators for silicon photonics: device physics and some recent progress,” Solid-State Electron. **51**(10), 1278–1286 (2007). [CrossRef]

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

## 5. Conclusion

*M*-ary quadrature signal generation. The critical coupling condition is obtained for such a structure. The intensity and phase modulations are obtained by varying the refractive indices of the silicon waveguides in the two parallel-coupled resonators. It is shown that a push-pull configuration effectively modulates the intensity and phase. The coverage of the complex plane of the output field

*E*

_{out}is systematically studied for over-coupling, critical-coupling, and under-coupling scenarios, and is compared to the corresponding scenarios of two uncoupled racetrack-resonators in series. It is found that only the over-coupling scenario of a parallel-coupled dual racetrack resonator structure results in adequate coverage for arbitrary

*M*-ary quadrature signal generation. The interaction between the parallel-coupled racetrack resonators is key to the coverage of the complex

*E*plane. In an over-coupled dual racetrack structure, a delicate balance is achieved between the direct sum and the interaction of the two racetrack resonances, which results in a large dynamic range of the output amplitude and phase. Particularly, the modulated intensity can reach zero in a push-pull configuration although the intensity of the un-modulated over-coupled racetrack resonators do not vanish at any wavelength. The effects of asymmetries in the coupling constants and quality factors are systematically studied. Despite the distortion of the intensity and phase mapping, small refractive index changes, which can be readily obtained with a reasonable thermal or electrical bias, can be used to compensate the asymmetry. The coverage of the complex

*E*plane remains sufficient despite asymmetries.

## Acknowledgments

## References and links

1. | P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE |

2. | G. P. Agrawal, |

3. | R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. |

4. | B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. |

5. | A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature |

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

7. | C. Li, L. J. 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 |

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

9. | D. M. Gill, M. Rasras, K. Y. Tu, Y. K. Chen, A. E. White, S. S. Patel, D. Carothers, A. Pomerene, R. Kamocsai, C. Hill, and J. Beattie, “Internal Bandwidth Equalization in a CMOS-Compatible Si-Ring Modulator,” IEEE Photon. Technol. Lett. |

10. | D. M. Gill, S. S. Patel, M. Rasras, K. Y. Tu, A. E. White, Y. K. Chen, A. Pomerene, D. Carothers, R. L. Kamocsai, C. M. Hill, and J. Beattie, “CMOS-compatible si-ring-assisted Mach-Zehnder interferometer with internal bandwidth equalization,” IEEE J. Sel. Top. Quantum Electron. |

11. | 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 |

12. | L. Zhang, J. Y. Yang, Y. C. Li, M. P. Song, R. G. Beausoleil, and A. E. Willner, “Monolithic modulator and demodulator of differential quadrature phase-shift keying signals based on silicon microrings,” Opt. Lett. |

13. | 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 |

14. | W. D. Sacher and J. K. S. Poon, “Microring quadrature modulators,” Opt. Lett. |

15. | L. L. Gu, W. Jiang, X. N. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett. |

16. | X. N. Chen, Y. S. Chen, Y. Zhao, W. Jiang, and R. T. Chen, “Capacitor-embedded 0.54 pJ/bit silicon-slot photonic crystal waveguide modulator,” Opt. Lett. |

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

18. | V. A. Mashkov and H. Temkin, “Propagation of eigenmodes and transfer amplitudes in optical waveguide structures,” IEEE J. Quantum Electron. |

19. | A. Hardy and W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. |

20. | C.-M. Kim and Y.-J. Im, “Switching operations of three-waveguide optical switches,” IEEE J. Sel. Top. Quantum Electron. |

21. | W. Jiang and R. T. Chen, “Multichannel optical add-drop processes in symmetrical waveguide-resonator systems,” Phys. Rev. Lett. |

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

23. | M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. |

24. | M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics |

25. | W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express |

26. | B. Razavi, |

27. | N. H. E. Weste and D. M. Harris, |

28. | W. Jiang, L. Gu, X. Chen, and R. T. Chen, “Photonic crystal waveguide modulators for silicon photonics: device physics and some recent progress,” Solid-State Electron. |

29. | D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE |

**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: March 28, 2011

Revised Manuscript: June 14, 2011

Manuscript Accepted: June 15, 2011

Published: July 19, 2011

**Citation**

Ryan A. Integlia, Lianghong Yin, Duo Ding, David Z. Pan, Douglas M. Gill, and Wei Jiang, "Parallel-coupled dual racetrack silicon micro-resonators for quadrature amplitude modulation," Opt. Express **19**, 14892-14902 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-14892

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### References

- P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94(5), 952–985 (2006). [CrossRef]
- G. P. Agrawal, Fiber-Optic Communication Systems (John Wiley & Sons, 1997).
- R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006). [CrossRef]
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