Abstract

A novel Si four-wavelength multiplexer (MUX) for 100/400GbE composed of (a)symmetric directional couplers and a rib-waveguide TE1-TM0 mode converter is proposed and experimentally demonstrated. Two-lane signals are multiplexed as TE0 modes via symmetric directional coupler (DC) and other two-lane signals are multiplexed as TE1 modes via asymmetric directional couplers (ADCs). Finally, TE1 modes are converted to TM0 modes and the device acts as a 4-lane MUX. The proposed device is fabrication-tolerant due to the removal of the 1600-GHz filter used in conventional two-stage Mach-Zehnder MUXs, and the insertion loss is smaller than those of previously reported Si-based MUXs.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (1)

2018 (3)

2017 (4)

S. Makino, T. Fujisawa, and K. Saitoh, “Wavefront matching method based on full-vector finite-element beam propagation method for polarization control devices,” J. Lightwave Technol. 35(14), 2840–2845 (2017).
[Crossref]

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

L. Chang, Y. Gong, L. Liu, Z. Li, and Y. Yu, “Low-loss broadband Silicon-on-insulator demultiplexers in the O-band,” IEEE Photonics Technol. Lett. 29(15), 1237–1240 (2017).
[Crossref]

G. Gao, D. Chen, S. Tao, Y. Zhang, S. Zhu, X. Xiao, and J. Xia, “Silicon nitride O-band (de)multiplexers with low thermal sensitivity,” Opt. Express 25(11), 12260–12267 (2017).
[Crossref]

2016 (1)

2015 (2)

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

K. Hassan, C. Sciancakepore, J. Harduin, T. Ferrotti, S. Menezo, and B. B. Bakir, “Toward athermal silicon-on-insulator (de)multiplexers in the O-band,” Opt. Lett. 40(11), 2641–2644 (2015).
[Crossref]

2014 (3)

L. Zhang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Nonlinear Group IV photonics based on silicon and germanium: From near-infrared to mid-infrared,” Nanophotonics 3(4-5), 247–268 (2014).
[Crossref]

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Y. Ueda, T. Fujisawa, K. Takahata, M. Kohtoku, and H. Ishii, “InP-based compact transversal filter for monolithically integrated light source array,” Opt. Express 22(7), 7844–7851 (2014).
[Crossref]

2013 (1)

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

2012 (2)

2011 (3)

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

T. Fujisawa, K. Takahata, W. Kobayashi, T. Tadokoro, N. Fujiwara, S. Kanazawa, and F. Kano, “1.3-µm, 50-Gbit/s electroabsorption modulators integrated with a DFB laser for beyond 100G parallel LAN applications,” Electron. Lett. 47(12), 708–710 (2011).
[Crossref]

M. Shinkawa, N. Ishikura, Y. Hama, K. Suzuki, and T. Baba, “Nonlinear enhancement in photonic crystal slow light waveguides fabricated using CMOS-compatible process,” Opt. Express 19(22), 22208–22218 (2011).
[Crossref]

2009 (1)

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Agarwal, A. M.

L. Zhang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Nonlinear Group IV photonics based on silicon and germanium: From near-infrared to mid-infrared,” Nanophotonics 3(4-5), 247–268 (2014).
[Crossref]

Akage, Y.

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Arai, M.

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Aruga, H.

T. Murao, K. Mochizuki, K. Ikeda, K. Hasegawa, and H. Aruga, “Efficient optical alignment technique for compact 100 and 400 GbE TOSAs with integrated multiplexing with free space optics,” J. Lightwave Technol. 37(3), 799–807 (2019).
[Crossref]

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Baba, T.

Bakir, B. B.

Beaudoin, G.

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

Bouchoule, S.

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

Bowers, J. E.

Chang, L.

L. Chang, Y. Gong, L. Liu, Z. Li, and Y. Yu, “Low-loss broadband Silicon-on-insulator demultiplexers in the O-band,” IEEE Photonics Technol. Lett. 29(15), 1237–1240 (2017).
[Crossref]

Chen, D.

Crosnier, G.

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

Dai, D.

Ferrotti, T.

Fujisawa, T.

S. Ohta, T. Fujisawa, S. Makino, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Si-based Mach-Zehnder wavelength/mode multi/demultiplexer for a WDM/MDM transmission system,” Opt. Express 26(12), 15211–15220 (2018).
[Crossref]

S. Makino, T. Fujisawa, and K. Saitoh, “Wavefront matching method based on full-vector finite-element beam propagation method for polarization control devices,” J. Lightwave Technol. 35(14), 2840–2845 (2017).
[Crossref]

Y. Ueda, T. Fujisawa, and M. Kohtoku, “8-channel Reflection-type Transversal Filter for Compact and Easy-to-Fabricate InP-based Optical Multiplexer,” J. Lightwave Technol. 34(11), 2684–2691 (2016).
[Crossref]

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

Y. Ueda, T. Fujisawa, K. Takahata, M. Kohtoku, and H. Ishii, “InP-based compact transversal filter for monolithically integrated light source array,” Opt. Express 22(7), 7844–7851 (2014).
[Crossref]

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

T. Fujisawa, S. Kanazawa, K. Takahata, W. Kobayashi, T. Tadokoro, H. Ishii, and F. Kano, “1.3-µm, 4×25G, EADFB laser array module with large-output-power and low-driving-voltage for energy-efficient 100GbE transmitter,” Opt. Express 20(1), 614–620 (2012).
[Crossref]

T. Fujisawa, K. Takahata, W. Kobayashi, T. Tadokoro, N. Fujiwara, S. Kanazawa, and F. Kano, “1.3-µm, 50-Gbit/s electroabsorption modulators integrated with a DFB laser for beyond 100G parallel LAN applications,” Electron. Lett. 47(12), 708–710 (2011).
[Crossref]

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

J. Takano, T. Fujisawa, T. Sato, Y. Sawada, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Low-loss and fabrication-tolerant Si four-wavelength multiplexer using higher-order mode for 100/400GbE,” in Proc of OFC, Tu2J.2 (2019).

Fujiwara, N.

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

T. Fujisawa, K. Takahata, W. Kobayashi, T. Tadokoro, N. Fujiwara, S. Kanazawa, and F. Kano, “1.3-µm, 50-Gbit/s electroabsorption modulators integrated with a DFB laser for beyond 100G parallel LAN applications,” Electron. Lett. 47(12), 708–710 (2011).
[Crossref]

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Gao, G.

Gong, Y.

L. Chang, Y. Gong, L. Liu, Z. Li, and Y. Yu, “Low-loss broadband Silicon-on-insulator demultiplexers in the O-band,” IEEE Photonics Technol. Lett. 29(15), 1237–1240 (2017).
[Crossref]

Hama, Y.

Harduin, J.

Hasegawa, K.

Hassan, K.

Iga, R.

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Ikeda, K.

Imai, Y.

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Ishii, H.

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

Y. Ueda, T. Fujisawa, K. Takahata, M. Kohtoku, and H. Ishii, “InP-based compact transversal filter for monolithically integrated light source array,” Opt. Express 22(7), 7844–7851 (2014).
[Crossref]

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

T. Fujisawa, S. Kanazawa, K. Takahata, W. Kobayashi, T. Tadokoro, H. Ishii, and F. Kano, “1.3-µm, 4×25G, EADFB laser array module with large-output-power and low-driving-voltage for energy-efficient 100GbE transmitter,” Opt. Express 20(1), 614–620 (2012).
[Crossref]

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

Ishikura, N.

Itoh, T.

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

Kanazawa, S.

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

T. Fujisawa, S. Kanazawa, K. Takahata, W. Kobayashi, T. Tadokoro, H. Ishii, and F. Kano, “1.3-µm, 4×25G, EADFB laser array module with large-output-power and low-driving-voltage for energy-efficient 100GbE transmitter,” Opt. Express 20(1), 614–620 (2012).
[Crossref]

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

T. Fujisawa, K. Takahata, W. Kobayashi, T. Tadokoro, N. Fujiwara, S. Kanazawa, and F. Kano, “1.3-µm, 50-Gbit/s electroabsorption modulators integrated with a DFB laser for beyond 100G parallel LAN applications,” Electron. Lett. 47(12), 708–710 (2011).
[Crossref]

Kano, F.

T. Fujisawa, S. Kanazawa, K. Takahata, W. Kobayashi, T. Tadokoro, H. Ishii, and F. Kano, “1.3-µm, 4×25G, EADFB laser array module with large-output-power and low-driving-voltage for energy-efficient 100GbE transmitter,” Opt. Express 20(1), 614–620 (2012).
[Crossref]

T. Fujisawa, K. Takahata, W. Kobayashi, T. Tadokoro, N. Fujiwara, S. Kanazawa, and F. Kano, “1.3-µm, 50-Gbit/s electroabsorption modulators integrated with a DFB laser for beyond 100G parallel LAN applications,” Electron. Lett. 47(12), 708–710 (2011).
[Crossref]

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Kawaguchi, Y.

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

Kimerling, L. C.

L. Zhang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Nonlinear Group IV photonics based on silicon and germanium: From near-infrared to mid-infrared,” Nanophotonics 3(4-5), 247–268 (2014).
[Crossref]

Kobayashi, W.

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

T. Fujisawa, S. Kanazawa, K. Takahata, W. Kobayashi, T. Tadokoro, H. Ishii, and F. Kano, “1.3-µm, 4×25G, EADFB laser array module with large-output-power and low-driving-voltage for energy-efficient 100GbE transmitter,” Opt. Express 20(1), 614–620 (2012).
[Crossref]

T. Fujisawa, K. Takahata, W. Kobayashi, T. Tadokoro, N. Fujiwara, S. Kanazawa, and F. Kano, “1.3-µm, 50-Gbit/s electroabsorption modulators integrated with a DFB laser for beyond 100G parallel LAN applications,” Electron. Lett. 47(12), 708–710 (2011).
[Crossref]

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Kodera, H.

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Kohtoku, M.

Kotoku, M.

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

Koyanagi, H.

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Li, Z.

L. Chang, Y. Gong, L. Liu, Z. Li, and Y. Yu, “Low-loss broadband Silicon-on-insulator demultiplexers in the O-band,” IEEE Photonics Technol. Lett. 29(15), 1237–1240 (2017).
[Crossref]

Liu, L.

H. Xu, L. Liu, and Y. Shi, “Polarization-insensitive four-channel coarse wavelength-division (de)multiplexer based on Mach-Zehnder interferometers with bent directional couplers and polarization rotators,” Opt. Lett. 43(7), 1483–1486 (2018).
[Crossref]

L. Chang, Y. Gong, L. Liu, Z. Li, and Y. Yu, “Low-loss broadband Silicon-on-insulator demultiplexers in the O-band,” IEEE Photonics Technol. Lett. 29(15), 1237–1240 (2017).
[Crossref]

Makino, S.

Matsui, T.

S. Ohta, T. Fujisawa, S. Makino, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Si-based Mach-Zehnder wavelength/mode multi/demultiplexer for a WDM/MDM transmission system,” Opt. Express 26(12), 15211–15220 (2018).
[Crossref]

J. Takano, T. Fujisawa, T. Sato, Y. Sawada, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Low-loss and fabrication-tolerant Si four-wavelength multiplexer using higher-order mode for 100/400GbE,” in Proc of OFC, Tu2J.2 (2019).

Menezo, S.

Michel, J.

L. Zhang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Nonlinear Group IV photonics based on silicon and germanium: From near-infrared to mid-infrared,” Nanophotonics 3(4-5), 247–268 (2014).
[Crossref]

Mochizuki, K.

Monnier, P.

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

Morita, Y.

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Murao, T.

T. Murao, K. Mochizuki, K. Ikeda, K. Hasegawa, and H. Aruga, “Efficient optical alignment technique for compact 100 and 400 GbE TOSAs with integrated multiplexing with free space optics,” J. Lightwave Technol. 37(3), 799–807 (2019).
[Crossref]

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Nakajima, K.

S. Ohta, T. Fujisawa, S. Makino, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Si-based Mach-Zehnder wavelength/mode multi/demultiplexer for a WDM/MDM transmission system,” Opt. Express 26(12), 15211–15220 (2018).
[Crossref]

J. Takano, T. Fujisawa, T. Sato, Y. Sawada, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Low-loss and fabrication-tolerant Si four-wavelength multiplexer using higher-order mode for 100/400GbE,” in Proc of OFC, Tu2J.2 (2019).

Nakamura, K.

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Nunoya, N.

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

Ohki, A.

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

Ohta, S.

Oohashi, H.

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

Raineri, F.

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

Raj, R.

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

Sagnes, I.

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

Saitoh, K.

Sakamoto, T.

S. Ohta, T. Fujisawa, S. Makino, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Si-based Mach-Zehnder wavelength/mode multi/demultiplexer for a WDM/MDM transmission system,” Opt. Express 26(12), 15211–15220 (2018).
[Crossref]

J. Takano, T. Fujisawa, T. Sato, Y. Sawada, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Low-loss and fabrication-tolerant Si four-wavelength multiplexer using higher-order mode for 100/400GbE,” in Proc of OFC, Tu2J.2 (2019).

Sanchez, D.

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

Sanjoh, H.

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

Sato, T.

J. Takano, T. Fujisawa, T. Sato, Y. Sawada, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Low-loss and fabrication-tolerant Si four-wavelength multiplexer using higher-order mode for 100/400GbE,” in Proc of OFC, Tu2J.2 (2019).

Sawada, Y.

J. Takano, T. Fujisawa, T. Sato, Y. Sawada, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Low-loss and fabrication-tolerant Si four-wavelength multiplexer using higher-order mode for 100/400GbE,” in Proc of OFC, Tu2J.2 (2019).

Sciancakepore, C.

Shi, Y.

Shimono, M.

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Shinada, T.

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Shinkawa, M.

Suzuki, K.

Tadokoro, T.

T. Fujisawa, S. Kanazawa, K. Takahata, W. Kobayashi, T. Tadokoro, H. Ishii, and F. Kano, “1.3-µm, 4×25G, EADFB laser array module with large-output-power and low-driving-voltage for energy-efficient 100GbE transmitter,” Opt. Express 20(1), 614–620 (2012).
[Crossref]

T. Fujisawa, K. Takahata, W. Kobayashi, T. Tadokoro, N. Fujiwara, S. Kanazawa, and F. Kano, “1.3-µm, 50-Gbit/s electroabsorption modulators integrated with a DFB laser for beyond 100G parallel LAN applications,” Electron. Lett. 47(12), 708–710 (2011).
[Crossref]

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Takahata, K.

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

Y. Ueda, T. Fujisawa, K. Takahata, M. Kohtoku, and H. Ishii, “InP-based compact transversal filter for monolithically integrated light source array,” Opt. Express 22(7), 7844–7851 (2014).
[Crossref]

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

T. Fujisawa, S. Kanazawa, K. Takahata, W. Kobayashi, T. Tadokoro, H. Ishii, and F. Kano, “1.3-µm, 4×25G, EADFB laser array module with large-output-power and low-driving-voltage for energy-efficient 100GbE transmitter,” Opt. Express 20(1), 614–620 (2012).
[Crossref]

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

T. Fujisawa, K. Takahata, W. Kobayashi, T. Tadokoro, N. Fujiwara, S. Kanazawa, and F. Kano, “1.3-µm, 50-Gbit/s electroabsorption modulators integrated with a DFB laser for beyond 100G parallel LAN applications,” Electron. Lett. 47(12), 708–710 (2011).
[Crossref]

Takano, J.

J. Takano, T. Fujisawa, T. Sato, Y. Sawada, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Low-loss and fabrication-tolerant Si four-wavelength multiplexer using higher-order mode for 100/400GbE,” in Proc of OFC, Tu2J.2 (2019).

Tang, Y.

Tao, S.

Tsujikawa, K.

S. Ohta, T. Fujisawa, S. Makino, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Si-based Mach-Zehnder wavelength/mode multi/demultiplexer for a WDM/MDM transmission system,” Opt. Express 26(12), 15211–15220 (2018).
[Crossref]

J. Takano, T. Fujisawa, T. Sato, Y. Sawada, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Low-loss and fabrication-tolerant Si four-wavelength multiplexer using higher-order mode for 100/400GbE,” in Proc of OFC, Tu2J.2 (2019).

Tsuzuki, K.

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Uchiyama, A.

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Ueda, Y.

Y. Ueda, T. Fujisawa, and M. Kohtoku, “8-channel Reflection-type Transversal Filter for Compact and Easy-to-Fabricate InP-based Optical Multiplexer,” J. Lightwave Technol. 34(11), 2684–2691 (2016).
[Crossref]

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

Y. Ueda, T. Fujisawa, K. Takahata, M. Kohtoku, and H. Ishii, “InP-based compact transversal filter for monolithically integrated light source array,” Opt. Express 22(7), 7844–7851 (2014).
[Crossref]

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

Xia, J.

Xiao, X.

Xu, H.

Yamanaka, T.

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

Yasui, N.

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

Yu, Y.

L. Chang, Y. Gong, L. Liu, Z. Li, and Y. Yu, “Low-loss broadband Silicon-on-insulator demultiplexers in the O-band,” IEEE Photonics Technol. Lett. 29(15), 1237–1240 (2017).
[Crossref]

Zhang, L.

L. Zhang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Nonlinear Group IV photonics based on silicon and germanium: From near-infrared to mid-infrared,” Nanophotonics 3(4-5), 247–268 (2014).
[Crossref]

Zhang, Y.

Zhu, S.

Electron. Lett. (2)

T. Fujisawa, M. Arai, N. Fujiwara, W. Kobayashi, T. Tadokoro, K. Tsuzuki, Y. Akage, R. Iga, T. Yamanaka, and F. Kano, “25-Gbit/s 1.3-µm InGaAlAs-based electroabsorption modulator integrated with a DFB laser for metro-area (40 km) 100-Gbit/s Ethernet system,” Electron. Lett. 45(17), 900–901 (2009).
[Crossref]

T. Fujisawa, K. Takahata, W. Kobayashi, T. Tadokoro, N. Fujiwara, S. Kanazawa, and F. Kano, “1.3-µm, 50-Gbit/s electroabsorption modulators integrated with a DFB laser for beyond 100G parallel LAN applications,” Electron. Lett. 47(12), 708–710 (2011).
[Crossref]

IEEE J. Quantum Electron. (1)

T. Fujisawa, S. Kanazawa, Y. Ueda, W. Kobayashi, K. Takahata, A. Ohki, T. Itoh, M. Kotoku, and H. Ishii, “Low-loss cascaded Mach-Zehnder multiplexer integrated 25-Gbit/s×4-lane EADFB laser array for future CFP4 100GbE transmitter,” IEEE J. Quantum Electron. 49(12), 1001–1007 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (5)

L. Chang, Y. Gong, L. Liu, Z. Li, and Y. Yu, “Low-loss broadband Silicon-on-insulator demultiplexers in the O-band,” IEEE Photonics Technol. Lett. 29(15), 1237–1240 (2017).
[Crossref]

H. Xu and Y. Shi, “Flat-top CWDM (De)Multiplexer based on MZI with bent directional couplers,” IEEE Photonics Technol. Lett. 30(2), 169–172 (2018).
[Crossref]

S. Kanazawa, T. Fujisawa, K. Takahata, T. Itoh, Y. Ueda, W. Kobayashi, H. Ishii, and H. Sanjoh, “Flip-chip interconnection lumped-electrode EADFB laser for 100-Gb/s/λ transmitter,” IEEE Photonics Technol. Lett. 27(16), 1699–1701 (2015).
[Crossref]

T. Murao, N. Yasui, T. Shinada, Y. Imai, K. Nakamura, M. Shimono, H. Kodera, Y. Morita, A. Uchiyama, H. Koyanagi, and H. Aruga, “Integrated spatial optical system for compact 28-Gb/s×4-lane transmitter optical subassemblies,” IEEE Photonics Technol. Lett. 26(22), 2275–2278 (2014).
[Crossref]

T. Fujisawa, S. Kanazawa, H. Ishii, N. Nunoya, Y. Kawaguchi, A. Ohki, N. Fujiwara, K. Takahata, R. Iga, F. Kano, and H. Oohashi, “1.3-µm, 4×25-Gbit/s, monolithically integrated light source for metro area 100-Gbit/s Ethernet,” IEEE Photonics Technol. Lett. 23(6), 356–358 (2011).
[Crossref]

J. Lightwave Technol. (3)

Nanophotonics (1)

L. Zhang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Nonlinear Group IV photonics based on silicon and germanium: From near-infrared to mid-infrared,” Nanophotonics 3(4-5), 247–268 (2014).
[Crossref]

Nat. Photonics (1)

G. Crosnier, D. Sanchez, S. Bouchoule, P. Monnier, G. Beaudoin, I. Sagnes, R. Raj, and F. Raineri, “Hybrid indium phosphide-on-silicon nanolaser diode,” Nat. Photonics 11(5), 297–300 (2017).
[Crossref]

Opt. Express (6)

Opt. Lett. (2)

Other (2)

http://www.ieee802.org/3/

J. Takano, T. Fujisawa, T. Sato, Y. Sawada, T. Sakamoto, T. Matsui, K. Tsujikawa, K. Nakajima, and K. Saitoh, “Low-loss and fabrication-tolerant Si four-wavelength multiplexer using higher-order mode for 100/400GbE,” in Proc of OFC, Tu2J.2 (2019).

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Figures (14)

Fig. 1.
Fig. 1. Schematic of the conventional two-stage MZI 4λ MUX and four wavelengths used in LAN-WDM system.
Fig. 2.
Fig. 2. A configuration of the proposed 4λ MUX.
Fig. 3.
Fig. 3. (Top) A top-view schematic of the proposed 4λ MUX and (bottom) structures of each component.
Fig. 4.
Fig. 4. Cross sections of (left) Si-wire and (right) Si-rib waveguides.
Fig. 5.
Fig. 5. FSR as a function of n of (left) DC-A and (right) ADC-C.
Fig. 6.
Fig. 6. Calculated transmission spectra of (left) DC-A and (right) ADC-C.
Fig. 7.
Fig. 7. Calculated and measured transmission spectra of ADC-B.
Fig. 8.
Fig. 8. Calculated magnetic field distributions in tapered rib waveguide when TE1 mode (|Hy| is the major field) is launched from the rib waveguide.
Fig. 9.
Fig. 9. Calculated transmission spectra of tapered rib waveguide.
Fig. 10.
Fig. 10. Measured normalized transmission spectra of (left) DC-A and (right) ADC-C.
Fig. 11.
Fig. 11. (Top) A micrograph of fabricated TE1-TM0 mode converter followed by ADC. This component acts as polarization beam splitter when the light is launched to Port 1. (Bottom) Measured transmission spectra of (left) TE input and (right) TM input.
Fig. 12.
Fig. 12. A micrograph of the fabricated 4λ MUX.
Fig. 13.
Fig. 13. Measured transmission spectra of the fabricated 4λ MUX for Ports (left) 4 and (right) 1 (TM input) for different values of injection current.
Fig. 14.
Fig. 14. Measured transmission spectra of the fabricated 4λ MUX for (left) Ports 2 and 3 (TE input) and (right) Ports 1 and 4 (TM input).

Tables (1)

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Table 1. Waveguide parameters

Equations (2)

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d d z [ a ( z ) b ( z ) ] = j T [ a ( z ) b ( z ) ]
T = [ β 1 κ κ β 2 ]

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