Abstract

A polarization independent reconfigurable optical demultiplexer with low crosstalk between adjacent channels and high number of potential allocated channels is designed on silicon on insulator technology. On to off state transitions can be implemented by changing the coupling factor or the ring length. Wavelength selective switch units are cascaded to form the demultiplexer. Crosstalks below −30dB with 50GHz channel spacing and losses below 1.5dB in the off state are obtained from simulations. Designs using carrier dispersion effect and power consumption estimations are included.

© 2014 Optical Society of America

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    [CrossRef]

2013

C. Vázquez, S. Vargas, and P. Contreras, “Low power consumption in silicon photonics tuning filters based on compound ring resonators,” in Silicon Photonics VIII, Photonics West, Proc. SPIE 8629, 44–50 (2013).
[CrossRef]

S. Ghosh, S. Keyvaninia, W. Van Roy, T. Mizumoto, G. Roelkens, and R. Baets, “Adhesively bonded Ce:YIG/SOI integrated optical circulator,” Opt. Lett. 38(6), 965–967 (2013).
[CrossRef] [PubMed]

2012

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), 1–14 (2012).
[CrossRef]

2011

L. Shuai, W. Yuanda, Y. Xiaojie, A. Junming, L. Jianguang, W. Hongjie, and H. Xiongwei, “Tunable filters based on an SOI nano-wire waveguide micro ring resonator,” J. Semiconduc. 32, 084007 (2011).

2010

2007

2006

I. Kiyat, A. Aydinli, and N. Dagli, “Low-Power Thermooptical Tuning of SOI Resonator Switch,” IEEE Photon. Technol. Lett. 18(2), 364–366 (2006).
[CrossRef]

2005

S. J. Emelett and R. A. Soref, “Analysis of dual-microring-resonator cross-connect switches and modulators,” Opt. Express 13(20), 7840–7853 (2005).
[CrossRef] [PubMed]

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

S. P. Chang, C. E. Png, S. T. Lim, V. M. N. Passaro, and G. T. Reed, “Single mode and polarization independent SOI waveguides with small cross section,” J. Lightwave Technol. 23, 1573–1582 (2005).

2004

W. Headley, G. Reed, S. Howe, A. Liu, and M. Paniccia, “Polarization-independent optical racetrack resonators using rib waveguides on silicon-on-insulator,” Appl. Phys. Lett. 85(23), 5523–5526 (2004).
[CrossRef]

Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004).
[CrossRef] [PubMed]

2003

C. Vázquez, S. Vargas, J. M. S. Pena, and P. Corredera, “Tunable Optical Filters Using Compound Ring Resonators for DWDM,” IEEE Photon. Technol. Lett. 15(8), 1085–1087 (2003).
[CrossRef]

2000

H. Zang, J. P. Jue, and B. Mukherjeea, “Review of Routing and Wavelength Assignment Approaches for Wavelength-Routed Optical WDM Networks,” Opt. Netw. Mag. 1, 47–60 (2000).

1987

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

1977

J. Dziewior and W. Schmid, “Auger coefficients for highly doped and highly excited silicon,” Appl. Phys. Lett. 31(5), 346–348 (1977).
[CrossRef]

Asghari, M.

Aydinli, A.

I. Kiyat, A. Aydinli, and N. Dagli, “Low-Power Thermooptical Tuning of SOI Resonator Switch,” IEEE Photon. Technol. Lett. 18(2), 364–366 (2006).
[CrossRef]

Baets, R.

Bauters, J.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), 1–14 (2012).
[CrossRef]

Bennett, B.

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

Boeck, R.

Bowers, J. E.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), 1–14 (2012).
[CrossRef]

Chang, S. P.

S. P. Chang, C. E. Png, S. T. Lim, V. M. N. Passaro, and G. T. Reed, “Single mode and polarization independent SOI waveguides with small cross section,” J. Lightwave Technol. 23, 1573–1582 (2005).

Chen, S.

F. Sun, J. Yu, and S. Chen, “Directional-coupler-based Mach-Zehnder interferometer in silicon-on-insulator technology for optical intensity modulation,” Opt. Eng. 42, 25601–25605 (2007).

Chowdhury, P.

Y. Zhang, P. Chowdhury, M. Tornatore, and B. Mukherjee, “Energy Efficiency in Telecom Optical Networks,” IEEE Commun. Surveys Tuts. 12(4), 441–458 (2010).
[CrossRef]

Chrostowski, L.

Contreras, P.

C. Vázquez, S. Vargas, and P. Contreras, “Low power consumption in silicon photonics tuning filters based on compound ring resonators,” in Silicon Photonics VIII, Photonics West, Proc. SPIE 8629, 44–50 (2013).
[CrossRef]

Corredera, P.

C. Vázquez, S. Vargas, J. M. S. Pena, and P. Corredera, “Tunable Optical Filters Using Compound Ring Resonators for DWDM,” IEEE Photon. Technol. Lett. 15(8), 1085–1087 (2003).
[CrossRef]

Cunningham, J. E.

Dagli, N.

I. Kiyat, A. Aydinli, and N. Dagli, “Low-Power Thermooptical Tuning of SOI Resonator Switch,” IEEE Photon. Technol. Lett. 18(2), 364–366 (2006).
[CrossRef]

Dai, D.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), 1–14 (2012).
[CrossRef]

Dong, P.

Driessen, A.

Dziewior, J.

J. Dziewior and W. Schmid, “Auger coefficients for highly doped and highly excited silicon,” Appl. Phys. Lett. 31(5), 346–348 (1977).
[CrossRef]

Emelett, S. J.

Feng, D.

Ghosh, S.

Headley, W.

W. Headley, G. Reed, S. Howe, A. Liu, and M. Paniccia, “Polarization-independent optical racetrack resonators using rib waveguides on silicon-on-insulator,” Appl. Phys. Lett. 85(23), 5523–5526 (2004).
[CrossRef]

Hilderink, L. T. H.

Hoekman, M.

Hongjie, W.

L. Shuai, W. Yuanda, Y. Xiaojie, A. Junming, L. Jianguang, W. Hongjie, and H. Xiongwei, “Tunable filters based on an SOI nano-wire waveguide micro ring resonator,” J. Semiconduc. 32, 084007 (2011).

Hossein-Zadeh, M.

M. Hossein-Zadeh and K. J. Vahala, “Optomechanical Oscillator on a Silicon Chip,” IEEE J. Sel. Top. Quantum Electron. 16(1), 276–287 (2010).
[CrossRef]

Howe, S.

W. Headley, G. Reed, S. Howe, A. Liu, and M. Paniccia, “Polarization-independent optical racetrack resonators using rib waveguides on silicon-on-insulator,” Appl. Phys. Lett. 85(23), 5523–5526 (2004).
[CrossRef]

Jaeger, N. A. F.

Jianguang, L.

L. Shuai, W. Yuanda, Y. Xiaojie, A. Junming, L. Jianguang, W. Hongjie, and H. Xiongwei, “Tunable filters based on an SOI nano-wire waveguide micro ring resonator,” J. Semiconduc. 32, 084007 (2011).

Jue, J. P.

H. Zang, J. P. Jue, and B. Mukherjeea, “Review of Routing and Wavelength Assignment Approaches for Wavelength-Routed Optical WDM Networks,” Opt. Netw. Mag. 1, 47–60 (2000).

Junming, A.

L. Shuai, W. Yuanda, Y. Xiaojie, A. Junming, L. Jianguang, W. Hongjie, and H. Xiongwei, “Tunable filters based on an SOI nano-wire waveguide micro ring resonator,” J. Semiconduc. 32, 084007 (2011).

Keyvaninia, S.

Kiyat, I.

I. Kiyat, A. Aydinli, and N. Dagli, “Low-Power Thermooptical Tuning of SOI Resonator Switch,” IEEE Photon. Technol. Lett. 18(2), 364–366 (2006).
[CrossRef]

Klein, E. J.

Krishnamoorthy, A. V.

Li, C.

Li, G.

Liang, H.

Liao, S.

Lim, S. T.

S. P. Chang, C. E. Png, S. T. Lim, V. M. N. Passaro, and G. T. Reed, “Single mode and polarization independent SOI waveguides with small cross section,” J. Lightwave Technol. 23, 1573–1582 (2005).

Lipson, M.

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

Liu, A.

W. Headley, G. Reed, S. Howe, A. Liu, and M. Paniccia, “Polarization-independent optical racetrack resonators using rib waveguides on silicon-on-insulator,” Appl. Phys. Lett. 85(23), 5523–5526 (2004).
[CrossRef]

McNab, S. J.

Mizumoto, T.

Mukherjee, B.

Y. Zhang, P. Chowdhury, M. Tornatore, and B. Mukherjee, “Energy Efficiency in Telecom Optical Networks,” IEEE Commun. Surveys Tuts. 12(4), 441–458 (2010).
[CrossRef]

Mukherjeea, B.

H. Zang, J. P. Jue, and B. Mukherjeea, “Review of Routing and Wavelength Assignment Approaches for Wavelength-Routed Optical WDM Networks,” Opt. Netw. Mag. 1, 47–60 (2000).

Paniccia, M.

W. Headley, G. Reed, S. Howe, A. Liu, and M. Paniccia, “Polarization-independent optical racetrack resonators using rib waveguides on silicon-on-insulator,” Appl. Phys. Lett. 85(23), 5523–5526 (2004).
[CrossRef]

Passaro, V. M. N.

S. P. Chang, C. E. Png, S. T. Lim, V. M. N. Passaro, and G. T. Reed, “Single mode and polarization independent SOI waveguides with small cross section,” J. Lightwave Technol. 23, 1573–1582 (2005).

Pellens, R.

Pena, J. M. S.

C. Vázquez, S. Vargas, J. M. S. Pena, and P. Corredera, “Tunable Optical Filters Using Compound Ring Resonators for DWDM,” IEEE Photon. Technol. Lett. 15(8), 1085–1087 (2003).
[CrossRef]

Png, C. E.

S. P. Chang, C. E. Png, S. T. Lim, V. M. N. Passaro, and G. T. Reed, “Single mode and polarization independent SOI waveguides with small cross section,” J. Lightwave Technol. 23, 1573–1582 (2005).

Poon, A. W.

Pradhan, S.

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

Qian, W.

Reed, G.

W. Headley, G. Reed, S. Howe, A. Liu, and M. Paniccia, “Polarization-independent optical racetrack resonators using rib waveguides on silicon-on-insulator,” Appl. Phys. Lett. 85(23), 5523–5526 (2004).
[CrossRef]

Reed, G. T.

S. P. Chang, C. E. Png, S. T. Lim, V. M. N. Passaro, and G. T. Reed, “Single mode and polarization independent SOI waveguides with small cross section,” J. Lightwave Technol. 23, 1573–1582 (2005).

Roelkens, G.

Rooks, M.

Rouger, N.

Schmid, W.

J. Dziewior and W. Schmid, “Auger coefficients for highly doped and highly excited silicon,” Appl. Phys. Lett. 31(5), 346–348 (1977).
[CrossRef]

Schmidt, B.

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

Sekaric, L.

Sengo, G.

Shafiiha, R.

Shuai, L.

L. Shuai, W. Yuanda, Y. Xiaojie, A. Junming, L. Jianguang, W. Hongjie, and H. Xiongwei, “Tunable filters based on an SOI nano-wire waveguide micro ring resonator,” J. Semiconduc. 32, 084007 (2011).

Soref, R.

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

Soref, R. A.

Sun, F.

F. Sun, J. Yu, and S. Chen, “Directional-coupler-based Mach-Zehnder interferometer in silicon-on-insulator technology for optical intensity modulation,” Opt. Eng. 42, 25601–25605 (2007).

Tornatore, M.

Y. Zhang, P. Chowdhury, M. Tornatore, and B. Mukherjee, “Energy Efficiency in Telecom Optical Networks,” IEEE Commun. Surveys Tuts. 12(4), 441–458 (2010).
[CrossRef]

Urban, P.

Vahala, K. J.

M. Hossein-Zadeh and K. J. Vahala, “Optomechanical Oscillator on a Silicon Chip,” IEEE J. Sel. Top. Quantum Electron. 16(1), 276–287 (2010).
[CrossRef]

van Dijk, P.

Van Roy, W.

Vargas, S.

C. Vázquez, S. Vargas, and P. Contreras, “Low power consumption in silicon photonics tuning filters based on compound ring resonators,” in Silicon Photonics VIII, Photonics West, Proc. SPIE 8629, 44–50 (2013).
[CrossRef]

S. Vargas and C. Vazquez, “Synthesis of optical filters using microring resonators with ultra-large FSR,” Opt. Express 18(25), 25936–25949 (2010).
[CrossRef] [PubMed]

C. Vázquez, S. Vargas, J. M. S. Pena, and P. Corredera, “Tunable Optical Filters Using Compound Ring Resonators for DWDM,” IEEE Photon. Technol. Lett. 15(8), 1085–1087 (2003).
[CrossRef]

Vazquez, C.

Vázquez, C.

C. Vázquez, S. Vargas, and P. Contreras, “Low power consumption in silicon photonics tuning filters based on compound ring resonators,” in Silicon Photonics VIII, Photonics West, Proc. SPIE 8629, 44–50 (2013).
[CrossRef]

C. Vázquez, S. Vargas, J. M. S. Pena, and P. Corredera, “Tunable Optical Filters Using Compound Ring Resonators for DWDM,” IEEE Photon. Technol. Lett. 15(8), 1085–1087 (2003).
[CrossRef]

Vlasov, Y.

Vlasov, Y. A.

Wang, X.

Xia, F.

Xiaojie, Y.

L. Shuai, W. Yuanda, Y. Xiaojie, A. Junming, L. Jianguang, W. Hongjie, and H. Xiongwei, “Tunable filters based on an SOI nano-wire waveguide micro ring resonator,” J. Semiconduc. 32, 084007 (2011).

Xiongwei, H.

L. Shuai, W. Yuanda, Y. Xiaojie, A. Junming, L. Jianguang, W. Hongjie, and H. Xiongwei, “Tunable filters based on an SOI nano-wire waveguide micro ring resonator,” J. Semiconduc. 32, 084007 (2011).

Xu, Q.

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

Yu, J.

F. Sun, J. Yu, and S. Chen, “Directional-coupler-based Mach-Zehnder interferometer in silicon-on-insulator technology for optical intensity modulation,” Opt. Eng. 42, 25601–25605 (2007).

Yuanda, W.

L. Shuai, W. Yuanda, Y. Xiaojie, A. Junming, L. Jianguang, W. Hongjie, and H. Xiongwei, “Tunable filters based on an SOI nano-wire waveguide micro ring resonator,” J. Semiconduc. 32, 084007 (2011).

Zang, H.

H. Zang, J. P. Jue, and B. Mukherjeea, “Review of Routing and Wavelength Assignment Approaches for Wavelength-Routed Optical WDM Networks,” Opt. Netw. Mag. 1, 47–60 (2000).

Zhang, Y.

Y. Zhang, P. Chowdhury, M. Tornatore, and B. Mukherjee, “Energy Efficiency in Telecom Optical Networks,” IEEE Commun. Surveys Tuts. 12(4), 441–458 (2010).
[CrossRef]

Zheng, X.

Zhou, L.

Appl. Phys. Lett.

W. Headley, G. Reed, S. Howe, A. Liu, and M. Paniccia, “Polarization-independent optical racetrack resonators using rib waveguides on silicon-on-insulator,” Appl. Phys. Lett. 85(23), 5523–5526 (2004).
[CrossRef]

J. Dziewior and W. Schmid, “Auger coefficients for highly doped and highly excited silicon,” Appl. Phys. Lett. 31(5), 346–348 (1977).
[CrossRef]

IEEE Commun. Surveys Tuts.

Y. Zhang, P. Chowdhury, M. Tornatore, and B. Mukherjee, “Energy Efficiency in Telecom Optical Networks,” IEEE Commun. Surveys Tuts. 12(4), 441–458 (2010).
[CrossRef]

IEEE J. Quantum Electron.

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

IEEE J. Sel. Top. Quantum Electron.

M. Hossein-Zadeh and K. J. Vahala, “Optomechanical Oscillator on a Silicon Chip,” IEEE J. Sel. Top. Quantum Electron. 16(1), 276–287 (2010).
[CrossRef]

IEEE Photon. Technol. Lett.

C. Vázquez, S. Vargas, J. M. S. Pena, and P. Corredera, “Tunable Optical Filters Using Compound Ring Resonators for DWDM,” IEEE Photon. Technol. Lett. 15(8), 1085–1087 (2003).
[CrossRef]

I. Kiyat, A. Aydinli, and N. Dagli, “Low-Power Thermooptical Tuning of SOI Resonator Switch,” IEEE Photon. Technol. Lett. 18(2), 364–366 (2006).
[CrossRef]

in Silicon Photonics VIII, Photonics West, Proc. SPIE

C. Vázquez, S. Vargas, and P. Contreras, “Low power consumption in silicon photonics tuning filters based on compound ring resonators,” in Silicon Photonics VIII, Photonics West, Proc. SPIE 8629, 44–50 (2013).
[CrossRef]

J. Lightwave Technol.

S. P. Chang, C. E. Png, S. T. Lim, V. M. N. Passaro, and G. T. Reed, “Single mode and polarization independent SOI waveguides with small cross section,” J. Lightwave Technol. 23, 1573–1582 (2005).

J. Semiconduc.

L. Shuai, W. Yuanda, Y. Xiaojie, A. Junming, L. Jianguang, W. Hongjie, and H. Xiongwei, “Tunable filters based on an SOI nano-wire waveguide micro ring resonator,” J. Semiconduc. 32, 084007 (2011).

Light: Sci. Appl.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), 1–14 (2012).
[CrossRef]

Nature

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

Opt. Eng.

F. Sun, J. Yu, and S. Chen, “Directional-coupler-based Mach-Zehnder interferometer in silicon-on-insulator technology for optical intensity modulation,” Opt. Eng. 42, 25601–25605 (2007).

Opt. Express

P. Dong, S. Liao, H. Liang, R. Shafiiha, D. Feng, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Submilliwatt, ultrafast and broadband electro-optic silicon switches,” Opt. Express 18(24), 25225–25231 (2010).
[CrossRef] [PubMed]

Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004).
[CrossRef] [PubMed]

C. Li, L. Zhou, and A. W. Poon, “Silicon microring carrier-injection-based modulators/switches with tunable extinction ratios and OR-logic switching by using waveguide cross-coupling,” Opt. Express 15(8), 5069–5076 (2007).
[CrossRef] [PubMed]

P. Dong, W. Qian, H. Liang, R. Shafiiha, X. Wang, D. Feng, G. Li, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “1x4 reconfigurable demultiplexing filter based on free-standing silicon racetrack resonators,” Opt. Express 18(24), 24504–24509 (2010).
[CrossRef] [PubMed]

S. Vargas and C. Vazquez, “Synthesis of optical filters using microring resonators with ultra-large FSR,” Opt. Express 18(25), 25936–25949 (2010).
[CrossRef] [PubMed]

E. J. Klein, P. Urban, G. Sengo, L. T. H. Hilderink, M. Hoekman, R. Pellens, P. van Dijk, and A. Driessen, “Densely integrated microring resonator based photonic devices for use in Access networks,” Opt. Express 15(16), 10346–10355 (2007).
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Opt. Lett.

Opt. Netw. Mag.

H. Zang, J. P. Jue, and B. Mukherjeea, “Review of Routing and Wavelength Assignment Approaches for Wavelength-Routed Optical WDM Networks,” Opt. Netw. Mag. 1, 47–60 (2000).

Other

T. E1-Bawab, Optical Switching (Springer, 2010).

J. G. Proakis and D. G. Manolakis, Digital Signal Processing, (Pearson Prentice Hall, 2006).

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Figures (13)

Fig. 1
Fig. 1

WDM switch unit.

Fig. 2
Fig. 2

Poles and Zeroes Modulus of the RRMI through transfer function, with K = 0.25, γ = γc = 0.025 and losses in the RR less than 0.01 dB.

Fig. 3
Fig. 3

Schematic layout of the proposed rib waveguide with w = 0.67 μm, H = 1 μm and D = 0.62 μm.

Fig. 4
Fig. 4

Polarization independent directional coupler: a) TE polarization and b) TM polarization. Cross section of the waveguides is shown in Fig. 3.

Fig. 5
Fig. 5

Crosstalk for adjacent channels with separations of 50 GHz, and 25 GHz.

Fig. 6
Fig. 6

Dependence of Δβ with Δn for the proposed waveguides at 1550 nm.

Fig. 7
Fig. 7

WDM switch spectral response simulations at on state, of the drop a), and through b) outputs.

Fig. 8
Fig. 8

WDM switch spectral response, at off state on the drop a), and through b) outputs.

Fig. 9
Fig. 9

Transverse mode profile.

Fig. 10
Fig. 10

1x4 WDM demultiplexer, with each WDM Switch tuned to different wavelengths.

Fig. 11
Fig. 11

Spectral response of WDM demultiplexer with all the WDM switches at the off state. Through output a) and the four drop outputs b).

Fig. 12
Fig. 12

Spectral response of WDM demultiplexer with all the WDM switches at the on state. Through output a) and drop outputs b).

Fig. 13
Fig. 13

Spectral response of WDM demultiplexer: for one (fourth) channel extracting a) through output b) drop outputs. For two (second and fourth) channels extracting c) through output d) drop outputs. For three (first, second and third) channels extracting e) through output f) drop outputs.

Equations (16)

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A 2 A 1 = ( 1 γ c ) 1/2 ( 1 Z c1 z 1 )( 1 Z c2 z 1 ) ( 1 Z p1 z 1 )( 1 Z p2 z 1 )
A 1R A 1 = j( 1 γ c )( 1γ ) K c ( 12K )| r( Ω ) | e αL z 1 ( 1 Z p1 z 1 )( 1 Z p2 z 1 )
Z c1 = ( 1 γ c ) 1/2 ( 1γ )| r( Ω ) | e αL ( 1 K c ) 1/2 [ j ( ( 1 K c ) ( 12K ) 2 K c 2 ( K K 2 ) ) 1/2 +( 2 K c ) ( K K 2 ) 1/2 ]
Z c2 = ( 1 γ c ) 1/2 ( 1γ )| r( Ω ) | e αL ( 1 K c ) 1/2 [ j ( ( 1 K c ) ( 12K ) 2 K c 2 ( K K 2 ) ) 1/2 +( 2 K c ) ( K K 2 ) 1/2 ]
| Z p |=( 1γ )[ ( 1 γ c )( 1 K c ) ] 1/2 | r(Ω) | e αL
φ p =± tan 1 [ (12K) 2 ( K K 2 ) ]
K lim ={ 12K 1K 0K<0.5 2K1 K 0.5<K1
| Z c |= ( 1 γ c ) 1/2 ( 1γ )| r( Ω ) | e αL
φ c =± tan 1 ( ( ( 1 K c ) ( 12K ) 2 K c 2 ( K K 2 ) ) 1/2 ( 2 K c ) K K 2 )
K c = sin 2 ( δ 1+ ( ξ δ ) 2 )( 1 1+ ( ξ δ ) 2 )
K c = sin 2 ( κ· L C )
Δn=8.8× 10 22 Δ N e 8.5× 10 18 ( Δ N h ) 0.8
Δα=8.5× 10 18 Δ N e +6× 10 18 Δ N h
I= ΔNeS L c τ
R Aug =( C n n+ C p p )( np n i 2 )
τ= ΔN R Aug

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