Abstract

A wavelength selective switch (WSS) can route optical signals into any of output ports by wavelength, and is a key component of the reconfigurable optical add/drop multiplexer. We propose a wavefront control type WSS using silicon photonics technology. This consists of several arrayed waveguide gratings sharing a large slab waveguide, wavefront control waveguides and distributed Bragg reflectors. The structure, design method, operating principle, and scalability of the WSS are described and discussed. We designed and fabricated a 1 × 2 wavefront control type WSS using silicon waveguides. This has 16 channels with a channel spacing of 200 GHz. The chip size is 5 mm × 10 mm. The switching operation was achieved by shifting the phase of the light propagating in each wavefront control waveguide, and by controlling the propagation direction in the shared large slab waveguide. Our WSS has no crossing waveguide, so the loss and the variation in loss between channels were small compared to conventional waveguide type WSSs. The heater power required for switching was 183 mW per channel, and the average extinction ratios routed to Output#1 and Output#2 were 9.8 dB and 10.2 dB, respectively.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

2015 (1)

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

2014 (2)

T. Yoshida, H. Asakura, H. Tsuda, T. Mizuno, and H. Takahashi, “Switching Characteristics of a 100-GHz-Spacing Integrated 40-λ 1× 4 Wavelength Selective Switch,” IEEE Photonics Technol. Lett. 26(5), 451–453 (2014).
[Crossref]

B. Robertson, H. Yang, M. M. Redmond, N. Collings, J. R. Moore, J. Liu, A. M. Jeziorska-Chapman, M. Pivnenko, S. Lee, A. Wonfor, I. H. White, W. A. Crossland, and D. P. Chu, “Demonstration of multi-casting in a 1× 9 LCOS wavelength selective switch,” J. Lightwave Technol. 32(3), 402–410 (2014).
[Crossref]

2013 (2)

K. Sorimoto, K. Tanizawa, H. Uetsuka, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, and H. Tsuda, “Compact and phase-error-robust multilayered AWG-based wavelength selective switch driven by a single LCOS,” Opt. Express 21(14), 17131–17149 (2013).
[Crossref] [PubMed]

K. Sorimoto, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, H. Tsuda, and H. Uetsuka, “MEMS mirror with slot structures suitable for flexible-grid WSS,” IEICE Electron. Express 10(3), 20120924 (2013).
[Crossref]

2012 (1)

2010 (1)

T. A. Strasser and J. L. Wagener, “Wavelength-selective switches for ROADM applications,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1150–1157 (2010).
[Crossref]

2006 (1)

D. T. Neilson, C. R. Doerr, D. M. Marom, R. Ryf, and M. P. Earnshaw, “Wavelength selective switching for optical bandwidth management,” Bell Labs Tech. J. 11(2), 105–128 (2006).
[Crossref]

2002 (1)

1994 (1)

D. H. Bailey and P. N. Swarztrauber, “A fast method for the numerical evaluation of continuous Fourier and Laplace transforms,” SIAM J. Sci. Comput. 15(5), 1105–1110 (1994).
[Crossref]

Asakura, H.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

T. Yoshida, H. Asakura, H. Tsuda, T. Mizuno, and H. Takahashi, “Switching Characteristics of a 100-GHz-Spacing Integrated 40-λ 1× 4 Wavelength Selective Switch,” IEEE Photonics Technol. Lett. 26(5), 451–453 (2014).
[Crossref]

Bailey, D. H.

D. H. Bailey and P. N. Swarztrauber, “A fast method for the numerical evaluation of continuous Fourier and Laplace transforms,” SIAM J. Sci. Comput. 15(5), 1105–1110 (1994).
[Crossref]

Buhl, L. L.

Capmany, J.

Chen, L.

Chu, D.

Chu, D. P.

Collings, N.

Crossland, W. A.

Devarajan, A.

A. Devarajan, K. Sandesha, R. Gowrishankar, B. S. Kishore, G. Prasanna, R. Johnson, and P. Voruganti, “Colorless, directionless and contentionless multi-degree ROADM architecture for mesh optical networks,” in Proceedings of 2010 Second International Conference on Communication Systems and Networks (IEEE, 2010), pp. 1–10.
[Crossref]

Doerr, C. R.

C. R. Doerr, L. L. Buhl, L. Chen, and N. Dupuis, “Monolithic flexible-grid 1× 2 wavelength-selective switch in silicon photonics,” J. Lightwave Technol. 30(4), 473–478 (2012).
[Crossref]

D. T. Neilson, C. R. Doerr, D. M. Marom, R. Ryf, and M. P. Earnshaw, “Wavelength selective switching for optical bandwidth management,” Bell Labs Tech. J. 11(2), 105–128 (2006).
[Crossref]

Dupuis, N.

Earnshaw, M. P.

D. T. Neilson, C. R. Doerr, D. M. Marom, R. Ryf, and M. P. Earnshaw, “Wavelength selective switching for optical bandwidth management,” Bell Labs Tech. J. 11(2), 105–128 (2006).
[Crossref]

Gowrishankar, R.

A. Devarajan, K. Sandesha, R. Gowrishankar, B. S. Kishore, G. Prasanna, R. Johnson, and P. Voruganti, “Colorless, directionless and contentionless multi-degree ROADM architecture for mesh optical networks,” in Proceedings of 2010 Second International Conference on Communication Systems and Networks (IEEE, 2010), pp. 1–10.
[Crossref]

Hasama, T.

K. Sorimoto, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, H. Tsuda, and H. Uetsuka, “MEMS mirror with slot structures suitable for flexible-grid WSS,” IEICE Electron. Express 10(3), 20120924 (2013).
[Crossref]

K. Sorimoto, K. Tanizawa, H. Uetsuka, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, and H. Tsuda, “Compact and phase-error-robust multilayered AWG-based wavelength selective switch driven by a single LCOS,” Opt. Express 21(14), 17131–17149 (2013).
[Crossref] [PubMed]

Ikeda, K.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

Ishikawa, H.

K. Sorimoto, K. Tanizawa, H. Uetsuka, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, and H. Tsuda, “Compact and phase-error-robust multilayered AWG-based wavelength selective switch driven by a single LCOS,” Opt. Express 21(14), 17131–17149 (2013).
[Crossref] [PubMed]

K. Sorimoto, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, H. Tsuda, and H. Uetsuka, “MEMS mirror with slot structures suitable for flexible-grid WSS,” IEICE Electron. Express 10(3), 20120924 (2013).
[Crossref]

Jeziorska-Chapman, A. M.

Johnson, R.

A. Devarajan, K. Sandesha, R. Gowrishankar, B. S. Kishore, G. Prasanna, R. Johnson, and P. Voruganti, “Colorless, directionless and contentionless multi-degree ROADM architecture for mesh optical networks,” in Proceedings of 2010 Second International Conference on Communication Systems and Networks (IEEE, 2010), pp. 1–10.
[Crossref]

Kawashima, H.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

K. Sorimoto, K. Tanizawa, H. Uetsuka, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, and H. Tsuda, “Compact and phase-error-robust multilayered AWG-based wavelength selective switch driven by a single LCOS,” Opt. Express 21(14), 17131–17149 (2013).
[Crossref] [PubMed]

K. Sorimoto, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, H. Tsuda, and H. Uetsuka, “MEMS mirror with slot structures suitable for flexible-grid WSS,” IEICE Electron. Express 10(3), 20120924 (2013).
[Crossref]

Kishore, B. S.

A. Devarajan, K. Sandesha, R. Gowrishankar, B. S. Kishore, G. Prasanna, R. Johnson, and P. Voruganti, “Colorless, directionless and contentionless multi-degree ROADM architecture for mesh optical networks,” in Proceedings of 2010 Second International Conference on Communication Systems and Networks (IEEE, 2010), pp. 1–10.
[Crossref]

Koshino, K.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

Lee, S.

Liu, J.

Marom, D. M.

D. T. Neilson, C. R. Doerr, D. M. Marom, R. Ryf, and M. P. Earnshaw, “Wavelength selective switching for optical bandwidth management,” Bell Labs Tech. J. 11(2), 105–128 (2006).
[Crossref]

Matsumaro, K.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

Mizuno, T.

T. Yoshida, H. Asakura, H. Tsuda, T. Mizuno, and H. Takahashi, “Switching Characteristics of a 100-GHz-Spacing Integrated 40-λ 1× 4 Wavelength Selective Switch,” IEEE Photonics Technol. Lett. 26(5), 451–453 (2014).
[Crossref]

Moore, J. R.

Mori, M.

K. Sorimoto, K. Tanizawa, H. Uetsuka, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, and H. Tsuda, “Compact and phase-error-robust multilayered AWG-based wavelength selective switch driven by a single LCOS,” Opt. Express 21(14), 17131–17149 (2013).
[Crossref] [PubMed]

K. Sorimoto, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, H. Tsuda, and H. Uetsuka, “MEMS mirror with slot structures suitable for flexible-grid WSS,” IEICE Electron. Express 10(3), 20120924 (2013).
[Crossref]

Munoz, P.

Muramatsu, K.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

Namiki, S.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

Neilson, D. T.

D. T. Neilson, C. R. Doerr, D. M. Marom, R. Ryf, and M. P. Earnshaw, “Wavelength selective switching for optical bandwidth management,” Bell Labs Tech. J. 11(2), 105–128 (2006).
[Crossref]

Ohtsuka, M.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

Pastor, D.

Pivnenko, M.

Prasanna, G.

A. Devarajan, K. Sandesha, R. Gowrishankar, B. S. Kishore, G. Prasanna, R. Johnson, and P. Voruganti, “Colorless, directionless and contentionless multi-degree ROADM architecture for mesh optical networks,” in Proceedings of 2010 Second International Conference on Communication Systems and Networks (IEEE, 2010), pp. 1–10.
[Crossref]

Redmond, M. M.

Robertson, B.

Ryf, R.

D. T. Neilson, C. R. Doerr, D. M. Marom, R. Ryf, and M. P. Earnshaw, “Wavelength selective switching for optical bandwidth management,” Bell Labs Tech. J. 11(2), 105–128 (2006).
[Crossref]

Sandesha, K.

A. Devarajan, K. Sandesha, R. Gowrishankar, B. S. Kishore, G. Prasanna, R. Johnson, and P. Voruganti, “Colorless, directionless and contentionless multi-degree ROADM architecture for mesh optical networks,” in Proceedings of 2010 Second International Conference on Communication Systems and Networks (IEEE, 2010), pp. 1–10.
[Crossref]

Seki, M.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

Sorimoto, K.

K. Sorimoto, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, H. Tsuda, and H. Uetsuka, “MEMS mirror with slot structures suitable for flexible-grid WSS,” IEICE Electron. Express 10(3), 20120924 (2013).
[Crossref]

K. Sorimoto, K. Tanizawa, H. Uetsuka, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, and H. Tsuda, “Compact and phase-error-robust multilayered AWG-based wavelength selective switch driven by a single LCOS,” Opt. Express 21(14), 17131–17149 (2013).
[Crossref] [PubMed]

Strasser, T. A.

T. A. Strasser and J. L. Wagener, “Wavelength-selective switches for ROADM applications,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1150–1157 (2010).
[Crossref]

Suzuki, K.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

Swarztrauber, P. N.

D. H. Bailey and P. N. Swarztrauber, “A fast method for the numerical evaluation of continuous Fourier and Laplace transforms,” SIAM J. Sci. Comput. 15(5), 1105–1110 (1994).
[Crossref]

Takahashi, H.

T. Yoshida, H. Asakura, H. Tsuda, T. Mizuno, and H. Takahashi, “Switching Characteristics of a 100-GHz-Spacing Integrated 40-λ 1× 4 Wavelength Selective Switch,” IEEE Photonics Technol. Lett. 26(5), 451–453 (2014).
[Crossref]

Tanizawa, K.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

K. Sorimoto, K. Tanizawa, H. Uetsuka, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, and H. Tsuda, “Compact and phase-error-robust multilayered AWG-based wavelength selective switch driven by a single LCOS,” Opt. Express 21(14), 17131–17149 (2013).
[Crossref] [PubMed]

Toyama, M.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

Tsuda, H.

K. Muramatsu, H. Asakura, K. Suzuki, K. Tanizawa, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, K. Ikeda, S. Namiki, H. Kawashima, and H. Tsuda, “Evaluation of the phase error in Si-wire arrayed-waveguide gratings fabricated by ArF-immersion photolithography,” IEICE Electron. Express 12(7), 1–6 (2015).
[Crossref]

T. Yoshida, H. Asakura, H. Tsuda, T. Mizuno, and H. Takahashi, “Switching Characteristics of a 100-GHz-Spacing Integrated 40-λ 1× 4 Wavelength Selective Switch,” IEEE Photonics Technol. Lett. 26(5), 451–453 (2014).
[Crossref]

K. Sorimoto, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, H. Tsuda, and H. Uetsuka, “MEMS mirror with slot structures suitable for flexible-grid WSS,” IEICE Electron. Express 10(3), 20120924 (2013).
[Crossref]

K. Sorimoto, K. Tanizawa, H. Uetsuka, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, and H. Tsuda, “Compact and phase-error-robust multilayered AWG-based wavelength selective switch driven by a single LCOS,” Opt. Express 21(14), 17131–17149 (2013).
[Crossref] [PubMed]

Uetsuka, H.

K. Sorimoto, K. Tanizawa, H. Uetsuka, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, and H. Tsuda, “Compact and phase-error-robust multilayered AWG-based wavelength selective switch driven by a single LCOS,” Opt. Express 21(14), 17131–17149 (2013).
[Crossref] [PubMed]

K. Sorimoto, H. Kawashima, M. Mori, T. Hasama, H. Ishikawa, H. Tsuda, and H. Uetsuka, “MEMS mirror with slot structures suitable for flexible-grid WSS,” IEICE Electron. Express 10(3), 20120924 (2013).
[Crossref]

Voruganti, P.

A. Devarajan, K. Sandesha, R. Gowrishankar, B. S. Kishore, G. Prasanna, R. Johnson, and P. Voruganti, “Colorless, directionless and contentionless multi-degree ROADM architecture for mesh optical networks,” in Proceedings of 2010 Second International Conference on Communication Systems and Networks (IEEE, 2010), pp. 1–10.
[Crossref]

Wagener, J. L.

T. A. Strasser and J. L. Wagener, “Wavelength-selective switches for ROADM applications,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1150–1157 (2010).
[Crossref]

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Figures (13)

Fig. 1
Fig. 1 Concept of 1 × N wavefront control type WSS.
Fig. 2
Fig. 2 Spatial distribution of the light field in the slab waveguide.
Fig. 3
Fig. 3 16-channel, 200-GHz-spacing WSSs with 2, 3, and 4 output ports and 8 or 16 wavefront control waveguides, Mwf, were designed. The transmittances of the WSSs when all the channels were switched to Output#1 are indicated.
Fig. 4
Fig. 4 The transmittances to Output#1, Output#2, Output#3, and Output#4 when channels #1, #5, #9, #13 were switched to Output #1, channels #2, #6, #10, #14 to Output #2, channels #3, #7, #11, #15 to Output #3, and channels #4, #8, #12, #16 to Output #4.
Fig. 5
Fig. 5 (a) Mask layout of the 1 × 2 wavefront control type WSS. (b)Enlarged view of the wavefront control waveguides array. (c)Enlarged view of the AWG.
Fig. 6
Fig. 6 Structure of low-loss joint between the slab and the array. (a) 3D view, (b) top view.
Fig. 7
Fig. 7 (a) Coupling efficiency from A to B as a function of L. (b) Coupling powers to the fundamental mode and to the 2nd mode around L = 5.0 µm for Wrib of 1.1, 1.2, and 1.3 μm.
Fig. 8
Fig. 8 Concept of wavefront control waveguides for one channel.
Fig. 9
Fig. 9 Photograph of the fabricated 1 × 2 WSS.
Fig. 10
Fig. 10 The transmission characteristics of AWG for input measured using the monitor tap waveguide.
Fig. 11
Fig. 11 (a) Transmittance from Input Port to Output Port#2 when one of the wavelength channels was selected. (b)Transmittance from the Input Port to Output Port#1.
Fig. 12
Fig. 12 Transmittance to each output port when the odd and even numbered channels (except for Ch.9,10, and 16) switched to Output #2. (a)Transmittance to Output Port#2 (b)Transmittance to Output Port#1
Fig. 13
Fig. 13 Simulated transmittance (a) from the Input Port to Output Port#2, and (b) from the Input Port to Output Port#1.The red curves show the transmittance when the odd numbered channels are switched to Output Port#2, and the blue curves shows the transmittance wheneven number channels switched to Output Port #2.

Tables (6)

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Table 1 Design parameter for the WSS.

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Table 2 Loss budget for the 1 × 2 wavefront control type WSS with Mwf of 8, 16, 32 when the 8th wavelength channel signal is routed to Output#2.

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Table 3 Characteristics of 200-GHz spacing, 16-channel WSS with various number of output ports

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Table 4 Characteristics of 1 × 2 WSS with various channel spacing when the number of wavefront control waveguides for each channel was 8.

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Table 5 Design parameter of AWG.

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Table 6 Transmittance to Output Port#1 when the heater on each channel was OFF, transmittance to Output Port#2 when the heaters were ON, extinction ratio for Output Port#1, extinction ratio for Output Port#2, largest crosstalk, and switching power for each channel.

Equations (7)

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Δl= m λ 0 n c ( λ 0 ) ,
m= c n c ν FSR n g λ 0 c n c Δν N ch n g λ 0 ,
n g = n c λ d n c dλ .
L f = n s d a λ 0 x FSR ,
x FSR = M wf d wf ν FSR Δν ,
δ ϕ ij = n s d wf Δ X ij L f 2π λ i ,
F(X)= n s λ L f F{f(x)}| u= X λ L f / n s = n s λ L f f(x) e ixu du| u= X λ L f / n s ,

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