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

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2010 (4)

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]

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]

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]

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

2009 (4)

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

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

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]

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. 21(4), 200–202 (2009).
[CrossRef]

2008 (2)

2007 (5)

2006 (3)

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

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

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

2005 (1)

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

2004 (1)

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]

2003 (1)

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]

2002 (1)

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

2000 (1)

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]

1998 (1)

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]

1987 (1)

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

1986 (1)

A. Hardy and W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. 4(1), 90–99 (1986).
[CrossRef]

Adibi, A.

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]

Beattie, J.

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]

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. 21(4), 200–202 (2009).
[CrossRef]

Beausoleil, R. G.

Bennett, B. R.

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

Carothers, D.

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]

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. 21(4), 200–202 (2009).
[CrossRef]

Chen, R. T.

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]

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]

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]

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]

Chen, X.

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]

Chen, X. N.

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]

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]

Chen, Y. K.

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]

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. 21(4), 200–202 (2009).
[CrossRef]

Chen, Y. S.

Cohen, O.

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]

Dapkus, P. D.

Essiambre, R. J.

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

Fathpour, S.

Gill, D. M.

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]

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. 21(4), 200–202 (2009).
[CrossRef]

Green, W. M. J.

Gu, L.

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]

Gu, L. L.

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]

Hardy, A.

A. Hardy and W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. 4(1), 90–99 (1986).
[CrossRef]

Hill, C.

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. 21(4), 200–202 (2009).
[CrossRef]

Hill, C. M.

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]

Im, Y.-J.

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]

Jalali, B.

Jiang, W.

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]

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]

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]

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]

Jones, R.

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]

Kamocsai, R.

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. 21(4), 200–202 (2009).
[CrossRef]

Kamocsai, R. L.

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]

Khan, M. H.

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]

Kim, C.-M.

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]

Leaird, D. E.

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]

Li, C.

Li, Q.

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]

Li, Y.

Li, Y. C.

Liao, L.

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]

Lipson, M.

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

Liu, A. S.

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]

Mashkov, V. A.

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]

Miller, D. A. B.

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

Nicolaescu, R.

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]

Paniccia, M.

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]

Patel, S. S.

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]

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. 21(4), 200–202 (2009).
[CrossRef]

Pomerene, A.

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]

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. 21(4), 200–202 (2009).
[CrossRef]

Poon, A. W.

Poon, J. K. S.

Pradhan, S.

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

Qi, M. H.

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]

Rasras, M.

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]

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. 21(4), 200–202 (2009).
[CrossRef]

Rooks, M. J.

Rubin, D.

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]

Sacher, W. D.

Samara-Rubio, D.

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]

Schmidt, B.

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

Sekaric, L.

Shen, H.

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]

Soltani, M.

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]

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Song, M. P.

Soref, R.

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

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[CrossRef]

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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]

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Tu, K. Y.

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]

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. 21(4), 200–202 (2009).
[CrossRef]

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Wang, L.

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]

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Weiner, A. M.

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]

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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]

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. 21(4), 200–202 (2009).
[CrossRef]

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Winzer, P. J.

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

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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]

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

Xuan, Y.

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]

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Yariv, A.

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

Yegnanarayanan, S.

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]

Zhang, B.

Zhang, L.

Zhao, L.

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]

Zhao, Y.

Zhou, L. J.

Zortman, W. A.

Appl. Phys. Lett. (1)

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]

IEEE J. Quantum Electron. (3)

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

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]

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]

IEEE J. Sel. Top. Quantum Electron. (3)

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]

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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[CrossRef]

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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. 21(4), 200–202 (2009).
[CrossRef]

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

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

A. Hardy and W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. 4(1), 90–99 (1986).
[CrossRef]

Nat. Photonics (1)

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]

Nature (2)

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]

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

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

W. Jiang and R. T. Chen, “Multichannel optical add-drop processes in symmetrical waveguide-resonator systems,” Phys. Rev. Lett. 91(21), 213901 (2003).
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P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94(5), 952–985 (2006).
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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).
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N. H. E. Weste and D. M. Harris, CMOS VLSI Design: A Circuit and Systems Perspective (Addison-Wesley, 2010).

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Figures (6)

Fig. 1
Fig. 1

Parallel-coupled dual racetrack resonators. (a) schematic of the structure, and typical spectra for an over-coupled structure: (b) output intensity and (c) phase in radians.

Fig. 2
Fig. 2

Intensity (a) and phase (b) variations under refractive index modulation for the parallel-coupled dual racetrack resonators. r 1 = r 3 = 3μm, L = 3μm, η 1 = η 3 = 0.994, c 1 = 0.4243. The intensity vanishes at two points (Δn 1n 3) = ( ± 3.5 × 10−4, ∓3.5 × 10−4). The color code for the phase is in radians in (b).

Fig. 3
Fig. 3

Mapping of the normalized complex output field amplitude E out on the complex plane for refractive index Δn 1, Δn 3 varying in the range of −0.002 ~0.002. (a)-(c) for parallel-coupled dual racetrack resonators; (d)-(f) for two uncoupled racetrack resonators in series. Evidently, only case (a) is suitable for arbitrary M-ary quadrature signal generation. Constellations for QPSK (brown circles) and 16-QAM (red squares) modulation formats are illustrated in (a).

Fig. 4
Fig. 4

Effect of asymmetric coupling constants. (a) Required phase compensation in each racetrack for up to 50% asymmetry in the coupling ratios. The characteristics of the asymmetric dual racetrack structure for the worst case scenario (κ2312 = 1.5) are illustrated in (b)-(d). (b) Output spectrum without modulation; (c) Intensity variation with index modulation; (d) Mapping of the output field on the complex plane. All parameters are the same as those used in Fig. 2 except κ 23/κ 12 is varied.

Fig. 5
Fig. 5

Effect of asymmetric quality factors. (a) Required phase compensation in each racetrack for asymmetry in the quality factors. The characteristics of the asymmetric dual racetrack structure for the worst case scenario (Q 3 = 0.5Q 1) are illustrated in (b)-(d). (b) Output spectrum; (c) Intensity variation with index modulations; (d) Mapping of the output field on the complex plane. All parameters are the same as those used in Fig. 2 except η 3 is varied to yield different Q 3.

Fig. 6
Fig. 6

Output intensity as a function of the driving power for parallel-coupled dual racetrack modulators with varying degrees of asymmetry. (a) For various κ2312 values and (b) For various Q 3 /Q 1 values. The output intensity is normalized by the input intensity. The driving power is normalized by the power level corresponding to the case that each racetrack is driven to Δn = 0.001.

Equations (19)

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E n ( x , y , z ) = M n ( x , y ) exp ( i β z ) u n ( z ) , n = 1, 2, 3,
E n ( i n ) ( x , y , 0 ) = M n ( x , y ) u n ( 0 ) M n ( x , y ) a n , E n ( o u t ) ( x , y , L ) = M n ( x , y ) exp ( i β L ) u n ( L ) M n ( x , y ) b n ,
[ b 1 b 2 b 3 ] = exp ( i β L ) [ c 1 + 1 / 2 c 2 c 1 1 / 2 c 2 2 c 1 c 2 c 1 1 / 2 c 2 c 1 + 1 / 2 ] [ a 1 a 2 a 3 ] ,
c 1 = 1 2 cos ( 2 κ L ) , c 2 = 1 2 i sin ( 2 κ L ) .
a 1 = η 1 exp ( i θ 1 ) b 1 , a 3 = η 3 exp ( i θ 3 ) b 3 ,
E o u t = b 2 = e i φ [ ( 1 / 2 c 1 ) ( Δ u 1 + Δ u 3 ) + 2 c 1 Δ u 1 Δ u 3 ] ( 1 / 2 c 1 ) ( Δ u 1 + Δ u 3 ) + Δ u 1 Δ u 3 ,
Δ u n 1 e i φ + i θ n η n 1 , n = 1, 3,
η 1 = 2 c 1 = cos 2 κ L .
b 2 = e i φ [ 1 + ( 2 c 1 + 1 ) ( 1 / 2 c 1 ) ( 1 / Δ u 1 + 1 / Δ u 3 ) + 1 ] .
φ + θ 1 = 2 m 1 π Δ θ , and φ + θ 3 = 2 m 3 π + Δ θ ,
cos Δ θ = η 1 [ 1 + c 1 ( 1 / η 1 2 1 ) c 1 + 1 / 2 ] ,
η 1 > 2 c 1 = cos 2 κ L ,
| Δ u 1 Δ u 3 | < < | ( 1 / 2 c 1 ) ( Δ u 1 + Δ u 3 ) |
| ( 1 / 2 c 1 ) ( Δ u 1 + Δ u 3 ) | < | Δ u 1 Δ u 3 | .
d d z [ u 1 ( z ) u 2 ( z ) u 3 ( z ) ] = i [ 0 κ 12 0 κ 12 0 κ 23 0 κ 23 0 ] [ u 1 ( z ) u 2 ( z ) u 3 ( z ) ] ,
[ u m ( z ) ] = exp ( i [ κ m n ] z ) [ u n ( 0 ) ] = X exp ( i Λ z ) X + [ u n ( 0 ) ] .
[ u 1 ( z ) u 2 ( z ) u 3 ( z ) ] = [ cos ( κ + z ) ρ 1 2 + ρ 3 2 i sin ( κ + z ) ρ 1 cos ( κ + z ) ρ 1 ρ 3 ρ 1 ρ 3 i sin ( κ + z ) ρ 1 cos ( κ + z ) i sin ( κ + z ) ρ 3 cos ( κ + z ) ρ 1 ρ 3 ρ 1 ρ 3 i sin ( κ + z ) ρ 3 cos ( κ + z ) ρ 3 2 + ρ 1 2 ] [ u 1 ( 0 ) u 2 ( 0 ) u 3 ( 0 ) ] ,
b 2 = e i φ [ 1 + 1 + cos ( κ + L ) [ 1 cos ( κ + L ) ] ( ρ 1 2 / Δ u 1 + ρ 3 2 / Δ u 3 ) + 1 ] ,
ρ 1 2 / Δ u 1 + ρ 3 2 / Δ u 3 = κ 12 2 κ + 2 e i φ + i θ 1 η 1 1 e i φ + i θ 1 η 1 + κ 23 2 κ + 2 e i φ + i θ 3 η 3 1 e i φ + i θ 3 η 3 .

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