Abstract

We present modeling and simulation results on a new family of waveguided interferometric 2×2 optical routing switches actuated by electro-optic or thermo-optic or all-optical control. Two pairs of coupled microring resonators provide two 3dB coupling regions within a compact Mach-Zehnder geometry. An index perturbation Δn of 2×10-3 is sufficient to produce 100% 2×2 switching. This perturbation can be applied to one arm of the MZI or to the four rings in the device or to an additional ring that is coupled to one arm. We find that push-pull control is effective for switching: for example, when carriers are injected in one region and depleted in a corresponding second region. An optical transfer-matrix technique is employed to determine the electromagnetic response (the 1550-nm switching characteristics) of the three device-types. Microdisks can be employed instead of microrings, if desired.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  23. V. Van, W. N. Herman, and P. T. Ho, "Linearized microring-loaded Mach-Zehnder modulator with RF gain," IEEE J. Lightwave Technol. 24, 1850-1854 (2006).
    [CrossRef]
  24. A. Leinse, M. B. J. Diemeer, A. Rousseau, and A. Driessen, "A Novel High-Speed Polymeric EO Modulator Based on a Combination of a Microring Resonator and an MZI," IEEE Photon. Technol. Lett. 17, 2074-2076 (2005).
    [CrossRef]
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2008 (3)

K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nature Photon. 2, 242-246 (2008).
[CrossRef]

S. Darmawan, Y. M. Landobasa, P. Dumon, R. Baets, and M. K. Chin, "Nested-Ring Mach-Zehnder Interferometer in Silicon-on-Insulator," IEEE Photon. Technol. Lett. 20, 9-11(2008).
[CrossRef]

2007 (6)

2006 (2)

S. Y. Cho and N. M. Jokerst, "A Polymer Microdisk Photonic Sensor Integrated Onto Silicon," IEEE Photon. Technol. Lett. 18, 2096-2098 (2006).
[CrossRef]

V. Van, W. N. Herman, and P. T. Ho, "Linearized microring-loaded Mach-Zehnder modulator with RF gain," IEEE J. Lightwave Technol. 24, 1850-1854 (2006).
[CrossRef]

2005 (4)

A. Leinse, M. B. J. Diemeer, A. Rousseau, and A. Driessen, "A Novel High-Speed Polymeric EO Modulator Based on a Combination of a Microring Resonator and an MZI," IEEE Photon. Technol. Lett. 17, 2074-2076 (2005).
[CrossRef]

S. J. Emelett and R. A. Soref, "Synthesis of dual-microring-resonator cross-connect filters," Opt. Express 13, 4439-4456 (2005).
[CrossRef] [PubMed]

S. J. Emelett and R. A. Soref, "Analysis of dual-microring-resonator cross-connect switches and modulators," Opt. Express 13, 7840-7853 (2005).
[CrossRef] [PubMed]

S. J. Emelett and R. A. Soref, "Design and simulation of silicon microring optical routing switches," IEEE J. Lightwave Technol. 23, 1800-1807 (2005).
[CrossRef]

2004 (3)

2003 (1)

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P. T. Ho, and C. H. Lee, "Lightwave Switching in Semiconductor Microring Devices by Free Carrier Injection," IEEE J. Lightwave Technol. 21, 2997-3003 (2003).
[CrossRef]

2000 (2)

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

P. P. Absil, J. V. Hryniewicz, B. E. Little, R. A. Wilson, L. G. Joneckis, and P. T. Ho, "Compact Microring Notch Filters," IEEE Photon. Technol. Lett. 12, 398-400 (2000).
[CrossRef]

1999 (1)

Absil, P. P.

P. P. Absil, J. V. Hryniewicz, B. E. Little, R. A. Wilson, L. G. Joneckis, and P. T. Ho, "Compact Microring Notch Filters," IEEE Photon. Technol. Lett. 12, 398-400 (2000).
[CrossRef]

Baets, R.

S. Darmawan, Y. M. Landobasa, P. Dumon, R. Baets, and M. K. Chin, "Nested-Ring Mach-Zehnder Interferometer in Silicon-on-Insulator," IEEE Photon. Technol. Lett. 20, 9-11(2008).
[CrossRef]

Boyd, R. W.

Cao, W.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P. T. Ho, and C. H. Lee, "Lightwave Switching in Semiconductor Microring Devices by Free Carrier Injection," IEEE J. Lightwave Technol. 21, 2997-3003 (2003).
[CrossRef]

Chak, P.

Chin, M. K.

S. Darmawan, Y. M. Landobasa, P. Dumon, R. Baets, and M. K. Chin, "Nested-Ring Mach-Zehnder Interferometer in Silicon-on-Insulator," IEEE Photon. Technol. Lett. 20, 9-11(2008).
[CrossRef]

Cho, S. Y.

S. Y. Cho and N. M. Jokerst, "Integrated thin film photodetectors with vertically coupled microring resonators for chip scale spectral analysis," Appl. Phys. Lett. 90, 101105 (2007).
[CrossRef]

S. Y. Cho and N. M. Jokerst, "A Polymer Microdisk Photonic Sensor Integrated Onto Silicon," IEEE Photon. Technol. Lett. 18, 2096-2098 (2006).
[CrossRef]

Choi, S. J.

A. Stapleton, S. Farrell, Z. Peng, S. J. Choi, L. Christen, J. O�??Brien, P. D. Dapkus, and A. Willner, "Low V�? modulators containing InGaAsP/InP microdisk phase modulators," Appl. Phys. Lett. 90, 161121 (2007).
[CrossRef]

Christen, L.

A. Stapleton, S. Farrell, Z. Peng, S. J. Choi, L. Christen, J. O�??Brien, P. D. Dapkus, and A. Willner, "Low V�? modulators containing InGaAsP/InP microdisk phase modulators," Appl. Phys. Lett. 90, 161121 (2007).
[CrossRef]

Dapkus, P. D.

A. Stapleton, S. Farrell, Z. Peng, S. J. Choi, L. Christen, J. O�??Brien, P. D. Dapkus, and A. Willner, "Low V�? modulators containing InGaAsP/InP microdisk phase modulators," Appl. Phys. Lett. 90, 161121 (2007).
[CrossRef]

Darmawan, S.

S. Darmawan, Y. M. Landobasa, P. Dumon, R. Baets, and M. K. Chin, "Nested-Ring Mach-Zehnder Interferometer in Silicon-on-Insulator," IEEE Photon. Technol. Lett. 20, 9-11(2008).
[CrossRef]

Diemeer, M. B. J.

A. Leinse, M. B. J. Diemeer, A. Rousseau, and A. Driessen, "A Novel High-Speed Polymeric EO Modulator Based on a Combination of a Microring Resonator and an MZI," IEEE Photon. Technol. Lett. 17, 2074-2076 (2005).
[CrossRef]

Dong, P.

K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

P. Dong, S. F. Preble, and M. Lipson, "All-optical compact silicon comb switch," Opt. Express 15, 9600-9605 (2007).
[CrossRef] [PubMed]

Driessen, A.

A. Leinse, M. B. J. Diemeer, A. Rousseau, and A. Driessen, "A Novel High-Speed Polymeric EO Modulator Based on a Combination of a Microring Resonator and an MZI," IEEE Photon. Technol. Lett. 17, 2074-2076 (2005).
[CrossRef]

Dumon, P.

S. Darmawan, Y. M. Landobasa, P. Dumon, R. Baets, and M. K. Chin, "Nested-Ring Mach-Zehnder Interferometer in Silicon-on-Insulator," IEEE Photon. Technol. Lett. 20, 9-11(2008).
[CrossRef]

Emelett, S. J.

Farrell, S.

A. Stapleton, S. Farrell, Z. Peng, S. J. Choi, L. Christen, J. O�??Brien, P. D. Dapkus, and A. Willner, "Low V�? modulators containing InGaAsP/InP microdisk phase modulators," Appl. Phys. Lett. 90, 161121 (2007).
[CrossRef]

Goldhar, J.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P. T. Ho, and C. H. Lee, "Lightwave Switching in Semiconductor Microring Devices by Free Carrier Injection," IEEE J. Lightwave Technol. 21, 2997-3003 (2003).
[CrossRef]

Green, W. M. J.

Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nature Photon. 2, 242-246 (2008).
[CrossRef]

Grover, R.

Heebner, J. E.

Herman, W. N.

V. Van, W. N. Herman, and P. T. Ho, "Linearized microring-loaded Mach-Zehnder modulator with RF gain," IEEE J. Lightwave Technol. 24, 1850-1854 (2006).
[CrossRef]

Ho, P. T.

V. Van, W. N. Herman, and P. T. Ho, "Linearized microring-loaded Mach-Zehnder modulator with RF gain," IEEE J. Lightwave Technol. 24, 1850-1854 (2006).
[CrossRef]

J. E. Heebner, N. N. Lepeshkin, A. Schweinsberg, G. W. Wicksm, R. W. Boyd, R. Grover, and P. T. Ho, "Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators," Opt. Lett. 29, 769-771 (2004).
[CrossRef] [PubMed]

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P. T. Ho, and C. H. Lee, "Lightwave Switching in Semiconductor Microring Devices by Free Carrier Injection," IEEE J. Lightwave Technol. 21, 2997-3003 (2003).
[CrossRef]

P. P. Absil, J. V. Hryniewicz, B. E. Little, R. A. Wilson, L. G. Joneckis, and P. T. Ho, "Compact Microring Notch Filters," IEEE Photon. Technol. Lett. 12, 398-400 (2000).
[CrossRef]

Hryniewicz, J. V.

P. P. Absil, J. V. Hryniewicz, B. E. Little, R. A. Wilson, L. G. Joneckis, and P. T. Ho, "Compact Microring Notch Filters," IEEE Photon. Technol. Lett. 12, 398-400 (2000).
[CrossRef]

Huang, Y.

Ibrahim, T. A.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P. T. Ho, and C. H. Lee, "Lightwave Switching in Semiconductor Microring Devices by Free Carrier Injection," IEEE J. Lightwave Technol. 21, 2997-3003 (2003).
[CrossRef]

Jokerst, N. M.

S. Y. Cho and N. M. Jokerst, "Integrated thin film photodetectors with vertically coupled microring resonators for chip scale spectral analysis," Appl. Phys. Lett. 90, 101105 (2007).
[CrossRef]

S. Y. Cho and N. M. Jokerst, "A Polymer Microdisk Photonic Sensor Integrated Onto Silicon," IEEE Photon. Technol. Lett. 18, 2096-2098 (2006).
[CrossRef]

Joneckis, L. G.

P. P. Absil, J. V. Hryniewicz, B. E. Little, R. A. Wilson, L. G. Joneckis, and P. T. Ho, "Compact Microring Notch Filters," IEEE Photon. Technol. Lett. 12, 398-400 (2000).
[CrossRef]

Kim, Y.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P. T. Ho, and C. H. Lee, "Lightwave Switching in Semiconductor Microring Devices by Free Carrier Injection," IEEE J. Lightwave Technol. 21, 2997-3003 (2003).
[CrossRef]

Landobasa, Y. M.

S. Darmawan, Y. M. Landobasa, P. Dumon, R. Baets, and M. K. Chin, "Nested-Ring Mach-Zehnder Interferometer in Silicon-on-Insulator," IEEE Photon. Technol. Lett. 20, 9-11(2008).
[CrossRef]

Lee, C. H.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P. T. Ho, and C. H. Lee, "Lightwave Switching in Semiconductor Microring Devices by Free Carrier Injection," IEEE J. Lightwave Technol. 21, 2997-3003 (2003).
[CrossRef]

Leinse, A.

A. Leinse, M. B. J. Diemeer, A. Rousseau, and A. Driessen, "A Novel High-Speed Polymeric EO Modulator Based on a Combination of a Microring Resonator and an MZI," IEEE Photon. Technol. Lett. 17, 2074-2076 (2005).
[CrossRef]

Lepeshkin, N. N.

Li, C.

Li, J.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P. T. Ho, and C. H. Lee, "Lightwave Switching in Semiconductor Microring Devices by Free Carrier Injection," IEEE J. Lightwave Technol. 21, 2997-3003 (2003).
[CrossRef]

Lipson, M.

Little, B. E.

P. P. Absil, J. V. Hryniewicz, B. E. Little, R. A. Wilson, L. G. Joneckis, and P. T. Ho, "Compact Microring Notch Filters," IEEE Photon. Technol. Lett. 12, 398-400 (2000).
[CrossRef]

Manipatruni, S.

Mookherjea, S.

O???Brien, J.

A. Stapleton, S. Farrell, Z. Peng, S. J. Choi, L. Christen, J. O�??Brien, P. D. Dapkus, and A. Willner, "Low V�? modulators containing InGaAsP/InP microdisk phase modulators," Appl. Phys. Lett. 90, 161121 (2007).
[CrossRef]

Paloczi, G. T.

Peng, Z.

A. Stapleton, S. Farrell, Z. Peng, S. J. Choi, L. Christen, J. O�??Brien, P. D. Dapkus, and A. Willner, "Low V�? modulators containing InGaAsP/InP microdisk phase modulators," Appl. Phys. Lett. 90, 161121 (2007).
[CrossRef]

Pereira, S.

Poon, A. W.

Poon, J. K. S.

Preble, S. F.

Preston, K.

K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

Rousseau, A.

A. Leinse, M. B. J. Diemeer, A. Rousseau, and A. Driessen, "A Novel High-Speed Polymeric EO Modulator Based on a Combination of a Microring Resonator and an MZI," IEEE Photon. Technol. Lett. 17, 2074-2076 (2005).
[CrossRef]

Scheuer, J.

Schmidt, B.

K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

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

Schweinsberg, A.

Shaka, J.

Sipe, J. E.

Soref, R. A.

Stapleton, A.

A. Stapleton, S. Farrell, Z. Peng, S. J. Choi, L. Christen, J. O�??Brien, P. D. Dapkus, and A. Willner, "Low V�? modulators containing InGaAsP/InP microdisk phase modulators," Appl. Phys. Lett. 90, 161121 (2007).
[CrossRef]

Van, V.

V. Van, W. N. Herman, and P. T. Ho, "Linearized microring-loaded Mach-Zehnder modulator with RF gain," IEEE J. Lightwave Technol. 24, 1850-1854 (2006).
[CrossRef]

Vlasov, Y.

Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nature Photon. 2, 242-246 (2008).
[CrossRef]

Wicksm, G. W.

Willner, A.

A. Stapleton, S. Farrell, Z. Peng, S. J. Choi, L. Christen, J. O�??Brien, P. D. Dapkus, and A. Willner, "Low V�? modulators containing InGaAsP/InP microdisk phase modulators," Appl. Phys. Lett. 90, 161121 (2007).
[CrossRef]

Wilson, R. A.

P. P. Absil, J. V. Hryniewicz, B. E. Little, R. A. Wilson, L. G. Joneckis, and P. T. Ho, "Compact Microring Notch Filters," IEEE Photon. Technol. Lett. 12, 398-400 (2000).
[CrossRef]

Xia, F.

Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nature Photon. 2, 242-246 (2008).
[CrossRef]

Xu, Q.

Yariv, A.

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express 12, 90-103 (2004).
[CrossRef] [PubMed]

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

Zhou, L.

Appl. Phys. Lett. (3)

K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

A. Stapleton, S. Farrell, Z. Peng, S. J. Choi, L. Christen, J. O�??Brien, P. D. Dapkus, and A. Willner, "Low V�? modulators containing InGaAsP/InP microdisk phase modulators," Appl. Phys. Lett. 90, 161121 (2007).
[CrossRef]

S. Y. Cho and N. M. Jokerst, "Integrated thin film photodetectors with vertically coupled microring resonators for chip scale spectral analysis," Appl. Phys. Lett. 90, 101105 (2007).
[CrossRef]

Electron. Lett. (1)

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

IEEE J. Lightwave Technol. (3)

S. J. Emelett and R. A. Soref, "Design and simulation of silicon microring optical routing switches," IEEE J. Lightwave Technol. 23, 1800-1807 (2005).
[CrossRef]

V. Van, W. N. Herman, and P. T. Ho, "Linearized microring-loaded Mach-Zehnder modulator with RF gain," IEEE J. Lightwave Technol. 24, 1850-1854 (2006).
[CrossRef]

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P. T. Ho, and C. H. Lee, "Lightwave Switching in Semiconductor Microring Devices by Free Carrier Injection," IEEE J. Lightwave Technol. 21, 2997-3003 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

A. Leinse, M. B. J. Diemeer, A. Rousseau, and A. Driessen, "A Novel High-Speed Polymeric EO Modulator Based on a Combination of a Microring Resonator and an MZI," IEEE Photon. Technol. Lett. 17, 2074-2076 (2005).
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S. Y. Cho and N. M. Jokerst, "A Polymer Microdisk Photonic Sensor Integrated Onto Silicon," IEEE Photon. Technol. Lett. 18, 2096-2098 (2006).
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S. Darmawan, Y. M. Landobasa, P. Dumon, R. Baets, and M. K. Chin, "Nested-Ring Mach-Zehnder Interferometer in Silicon-on-Insulator," IEEE Photon. Technol. Lett. 20, 9-11(2008).
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P. P. Absil, J. V. Hryniewicz, B. E. Little, R. A. Wilson, L. G. Joneckis, and P. T. Ho, "Compact Microring Notch Filters," IEEE Photon. Technol. Lett. 12, 398-400 (2000).
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J. Opt. Soc. Am. B (1)

Nature Photon. (1)

Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nature Photon. 2, 242-246 (2008).
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Opt. Express (6)

Opt. Lett. (3)

Other (3)

A. Biberman, B. G. Lee, K. Bergman, P. Dong, and M. Lipson, "Demonstration of All-Optical Multi-Wavelength Message Routing for Silicon Photonic Networks," in Optical Fiber Communication Conference (OFC), San Diego, CA Feb. 24, 2008, Paper OTuF6.

M. R. Watts, D. C. Trotter, and R. W. Young, "Maximally confined high-speed second-order silicon microdisk switches," in National Fiber Optic Engineers Conference (NFOEC), San Diego, CA Feb. 24, 2008, Postdeadline Session B, paper PDP-14.

S. Manipatruni, C. Poitras, Q. Xu, and M. Lipson, "Electro-optic Tuning of On-Chip Optical Transparency," in 20th Annual Meeting of the IEEE Lasers and Electro-Optic Society (LEOS), Lake Buena Vista, FL Oct. 21-25 2007, 539-540.

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Figures (9)

Fig. 1.
Fig. 1.

Internal and external field configurations in a double-microring coupler.

Fig. 2.
Fig. 2.

Normalized output power at through and drop ports of the double-microring coupler with different cross coupling coefficients (a) c 1=0.707, c 2=0.14; (b) c 1=0.707, c 2=0.34.

Fig. 3.
Fig. 3.

Mach-Zehnder Interferometer using two double-microring couplers.

Fig. 4.
Fig. 4.

Normalized output power at the through port of the double-microring based MZI for different phase bias conditions.

Fig. 5.
Fig. 5.

MRMZI switches with (a) double push and (b) push-pull perturbations.

Fig. 6.
Fig. 6.

Spectral responses and phase responses at the through port of (a, c) “double push” and (b, d) “push-pull” MRMZI switches for different index perturbations.

Fig. 7.
Fig. 7.

Schematic diagrams of (a) “single push” and (b) “push-pull” MRMZI with microring resonators coupled to the arms.

Fig. 8.
Fig. 8.

Spectral responses at the through port of the (a) “single push” and (b) “push-pull” MRMZI with microring resonators coupled to the waveguide arms.

Fig. 9.
Fig. 9.

Spectral responses at the through port of the lossy double push MRMZI switch for different index perturbations.

Equations (15)

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w = i = 1 N z i ,
[ e 1,1 e 2,1 ] = j c 1 [ t 1 1 1 t 1 ] [ E 1 E 2 ] K 1 [ E 1 E 2 ] ,
[ e 4,2 e 3,2 ] = j c 2 [ t 2 1 1 t 2 ] [ e 4,1 e 3,1 ] K 2 [ e 4,1 e 3,1 ] , [ E 3 E 4 ] = K 1 [ e 1,2 e 2,2 ] ,
t i 2 + c i 2 = 1 , i = 1 , 2 .
[ e 4,1 e 3,1 ] = [ 0 exp ( j β 1 π R ) exp ( j β 1 π R ) 0 ] [ e 1,1 e 2,1 ] P 1 [ e 1,1 e 2,1 ] ,
[ e 1,2 e 2,2 ] = P 2 [ e 4,2 e 3,2 ] ,
[ E 3 E 4 ] = ( K 1 P 2 K 2 P 1 K 1 ) [ E 1 E 2 ] M [ E 1 E 2 ] , M = [ m 11 m 12 m 21 m 22 ] .
P coupler , drop = Det [ M ] m 12 2 , P coupler , through = m 11 m 12 2 .
[ E 2 E 4 ] = 1 m 12 [ m 11 1 Det [ M ] m 22 ] [ E 1 E 3 ] T [ E 1 E 3 ] .
[ E 1 E 3 ] = [ exp ( j β L arm ) 0 0 exp ( j ( β L arm + Δ ϕ ) ) ] [ E 2 E 4 ] L [ E 2 E 4 ] ,
[ E 2 E 4 ] = TLT [ E 1 E 3 ] S [ E 1 E 3 ] , S = [ s 11 s 12 s 21 s 22 ]
P MRMZI , drop = s 21 2 , P MRMZI , Through = s 11 2 .
T MR = E out E in = 1 1 ( c 3 ) 2 exp ( j β 3 2 π R ) 1 ( c 3 ) 2 exp ( j β 3 2 π R ) ,
[ E 2 E 4 ] = TL T [ E 1 E 3 ] ,
L = { exp ( j β L arm 2 ) T MR exp ( j β L arm 2 ) } · I ,

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