Abstract

Modulation characteristics of a novel InGaAs/InAlAs multiple quantum well (MQW) microring-enhanced Mach-Zehnder modulator (MRE-MZM) is investigated in detail and its low-voltage operation with high extinction ratio is demonstrated. The MZM has a single microring resonator in one arm and is driven by the change in electrorefractive index induced by the quantum-confined Stark effect in the MQW core layer. As the MQW, a multiple five-layer asymmetric coupled quantum well (FACQW) is used to obtain a large electrorefractive index change. The driving voltage of the proposed MZM is significantly reduced owing to the enhanced phase shift in the microring resonator. High-mesa waveguide structures are grown by solid-source molecular beam epitaxy and fabricated by inductively coupled plasma etching. A directional coupler with an asymmetric branching ratio is used as an input coupler to prevent the degradation of the extinction ratio of the MZM. The extinction ratio of the fabricated MRE-MZM is approximately 27 dB. The product of the half-wave voltage and phase shifter length, Vπ·L, is 1.7 Vmm in static modulation. This value is one-quarter that of a conventional MZM with the same waveguide structure.

© 2013 OSA

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2012 (3)

T. Arakawa, T. Hariki, Y. Amma, M. Fukuoka, M. Ushigome, and K. Tada, “Low-voltage Mach-Zehnder modulator with InGaAs/InAlAs five-layer asymmetric coupled quantum well,” Jpn. J. Appl. Phys.51, 042203 (2012).
[CrossRef]

H. Kaneshige, Y. Ueyama, H. Yamada, H. Yajima, T. Arakawa, and Y. Kokubun, “InGaAs/InAlAs multiple quantum well Mach-Zehnder modulator with single microring resonator,” Jpn. J. Appl. Phys.51(2), 02BG01 (2012).
[CrossRef]

S. Akiyama and S. Nomura, “Dynamic response of modulators based on cascaded-ring-resonator,” Opt. Express20(20), 21847–21859 (2012).
[CrossRef] [PubMed]

2011 (2)

T. Makino, T. Gotoh, R. Hasegawa, T. Arakawa, and Y. Kokubun, “Microring resonator wavelength tunable filter using five-layer asymmetric coupled quantum well,” J. Lightwave Technol.29(16), 2387–2393 (2011).
[CrossRef]

T. Arakawa, T. Toya, M. Ushigome, K. Yamaguchi, T. Ide, and K. Tada, “InGaAs/InAlAs five-layer asymmetric coupled quantum well exhibiting giant electrorefractive index change,” Jpn. J. Appl. Phys.50, 032204 (2011).
[CrossRef]

2010 (3)

2009 (2)

P. Dong, S. Liao, D. Feng, H. Liang, D. Zheng, R. Shafiiha, C. C. Kung, W. Qian, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Low Vpp, ultralow-energy, compact, high-speed silicon electro-optic modulator,” Opt. Express17(25), 22484–22490 (2009).
[CrossRef] [PubMed]

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

2008 (3)

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

S. Akiyama, H. Itoh, S. Sekiguchi, S. Hirose, T. Takeuchi, A. Kuramata, and T. Yamamoto, “InP-based Mach-Zehnder modulator with capacitively loaded traveling-wave electrodes,” J. Lightwave Technol.26(5), 608–615 (2008).
[CrossRef]

Y. Lu, L. Xu, M. Shu, P. Wang, and J. Yao, “Proposal to produce coupled resonator-induced transparency and bistability using microresonator enhanced Mach–Zehnder interferometer,” IEEE Photon. Technol. Lett.20(7), 529–531 (2008).
[CrossRef]

2007 (4)

W.-S. Choi, W. Zhao, J.-W. Bae, I. Adesida, B.-A. Yu, Y. L. Lee, and J.-H. Jang, “Micro-racetrack notch filters based on InGaAsP/InP high mesa optical waveguides,” Jpn. J. Appl. Phys.46(4B), 2434–2439 (2007).
[CrossRef]

Y. Kokubun, “High Index contrast optical waveguides and their applications to microring filter circuit and wavelength selective switch,” IEICE Trans. Electron.E90-C(5), 1037–1045 (2007).
[CrossRef]

T. Kawanishi, S. Sakamoto, and M. Izutsu, “High-speed control of lightwave amplitude, phase, and frequency by use of electrooptic effect,” IEEE J. Sel. Top. Quantum Electron.13(1), 79–91 (2007).
[CrossRef]

C. R. Doerr, L. Zhang, P. J. Winzer, J. H. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact high- speed InP DQPSK modulator,” IEEE Photon. Technol. Lett.19(15), 1184–1186 (2007).
[CrossRef]

2006 (1)

A. Rostami, “Low threshold and tunable all-optical switch using two-photon absorption in array of nonlinear ring resonators coupled to MZI,” Microelectron. J.37(9), 976–981 (2006).
[CrossRef]

2005 (4)

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

Y. Kokubun, “Vertically coupled micro-ring resonator filter for integrated add/drop node,” IEICE Trans. Electron.E-88C, 349–362 (2005).

S. Matsuo, Y. Ohiso, and T. Segawa, “A high-speed tunable optical filter using a semiconductor ring resonator for photonic packet switch,” IEICE Trans. Electron.E88-C(3), 295–302 (2005).
[CrossRef]

W. Green, R. Lee, G. Derose, A. Scherer, and A. Yariv, “Hybrid InGaAsP-InP Mach-Zehnder racetrack resonator for thermooptic switching and coupling control,” Opt. Express13(5), 1651–1659 (2005).
[CrossRef] [PubMed]

2004 (2)

J. E. Heebner, N. N. Lepeshkin, A. Schweinsberg, G. W. Wicks, R. W. Boyd, R. Grover, and P.-T. Ho, “Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators,” Opt. Lett.29(7), 769–771 (2004).
[CrossRef] [PubMed]

S. J. Choi, K. Djordjev, Z. Peng, Q. Yang, S. J. Choi, and P. D. Dapkus, “Laterally coupled buried heterostructure high-Q ring resonators,” IEEE Photon. Technol. Lett.16(10), 2266–2268 (2004).
[CrossRef]

2003 (2)

2002 (1)

M. Hamacher, H. Heidrich, D. G. Rabus, and U. Troppenz, Proc. SPIE4640, 37–45 (2002).
[CrossRef]

1999 (1)

1998 (1)

H. Feng, J. P. Pang, M. Sugiyama, K. Tada, and Y. Nakano, “Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential,” IEEE J. Quantum Electron.34(7), 1197–1208 (1998).
[CrossRef]

1991 (1)

R. G. Walker, “High-speed III-V semiconductor intensity modulators,” IEEE J. Quantum Electron.27(3), 654–667 (1991).
[CrossRef]

Adamiecki, A.

C. R. Doerr, L. Zhang, P. J. Winzer, J. H. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact high- speed InP DQPSK modulator,” IEEE Photon. Technol. Lett.19(15), 1184–1186 (2007).
[CrossRef]

Adesida, I.

W.-S. Choi, W. Zhao, J.-W. Bae, I. Adesida, B.-A. Yu, Y. L. Lee, and J.-H. Jang, “Micro-racetrack notch filters based on InGaAsP/InP high mesa optical waveguides,” Jpn. J. Appl. Phys.46(4B), 2434–2439 (2007).
[CrossRef]

Akiyama, S.

Amma, Y.

T. Arakawa, T. Hariki, Y. Amma, M. Fukuoka, M. Ushigome, and K. Tada, “Low-voltage Mach-Zehnder modulator with InGaAs/InAlAs five-layer asymmetric coupled quantum well,” Jpn. J. Appl. Phys.51, 042203 (2012).
[CrossRef]

Arai, M.

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

Arakawa, T.

H. Kaneshige, Y. Ueyama, H. Yamada, H. Yajima, T. Arakawa, and Y. Kokubun, “InGaAs/InAlAs multiple quantum well Mach-Zehnder modulator with single microring resonator,” Jpn. J. Appl. Phys.51(2), 02BG01 (2012).
[CrossRef]

T. Arakawa, T. Hariki, Y. Amma, M. Fukuoka, M. Ushigome, and K. Tada, “Low-voltage Mach-Zehnder modulator with InGaAs/InAlAs five-layer asymmetric coupled quantum well,” Jpn. J. Appl. Phys.51, 042203 (2012).
[CrossRef]

T. Arakawa, T. Toya, M. Ushigome, K. Yamaguchi, T. Ide, and K. Tada, “InGaAs/InAlAs five-layer asymmetric coupled quantum well exhibiting giant electrorefractive index change,” Jpn. J. Appl. Phys.50, 032204 (2011).
[CrossRef]

T. Makino, T. Gotoh, R. Hasegawa, T. Arakawa, and Y. Kokubun, “Microring resonator wavelength tunable filter using five-layer asymmetric coupled quantum well,” J. Lightwave Technol.29(16), 2387–2393 (2011).
[CrossRef]

Asghari, M.

Baba, T.

S. Akiyama, T. Kurahashi, T. Baba, N. Hatori, T. Usuki, and T. Yamamoto, “A 1 V peak-to-peak driven 10-Gbps slow-light silicon Mach–Zehnder modulator using cascaded ring resonators,” Appl. Phys. Express3(7), 072202 (2010).
[CrossRef]

Bae, J.-W.

W.-S. Choi, W. Zhao, J.-W. Bae, I. Adesida, B.-A. Yu, Y. L. Lee, and J.-H. Jang, “Micro-racetrack notch filters based on InGaAsP/InP high mesa optical waveguides,” Jpn. J. Appl. Phys.46(4B), 2434–2439 (2007).
[CrossRef]

Boyd, R. W.

Cao, W.

Choi, S. J.

S. J. Choi, K. Djordjev, Z. Peng, Q. Yang, S. J. Choi, and P. D. Dapkus, “Laterally coupled buried heterostructure high-Q ring resonators,” IEEE Photon. Technol. Lett.16(10), 2266–2268 (2004).
[CrossRef]

S. J. Choi, K. Djordjev, Z. Peng, Q. Yang, S. J. Choi, and P. D. Dapkus, “Laterally coupled buried heterostructure high-Q ring resonators,” IEEE Photon. Technol. Lett.16(10), 2266–2268 (2004).
[CrossRef]

Choi, W.-S.

W.-S. Choi, W. Zhao, J.-W. Bae, I. Adesida, B.-A. Yu, Y. L. Lee, and J.-H. Jang, “Micro-racetrack notch filters based on InGaAsP/InP high mesa optical waveguides,” Jpn. J. Appl. Phys.46(4B), 2434–2439 (2007).
[CrossRef]

Chow, F.-S.

X. Xie, J. Khurgin, J. Kang, and F.-S. Chow, “Linearized Mach–Zehnder intensity modulator,” IEEE Photon. Technol. Lett.15(4), 531–533 (2003).
[CrossRef]

Dapkus, P. D.

S. J. Choi, K. Djordjev, Z. Peng, Q. Yang, S. J. Choi, and P. D. Dapkus, “Laterally coupled buried heterostructure high-Q ring resonators,” IEEE Photon. Technol. Lett.16(10), 2266–2268 (2004).
[CrossRef]

Derose, G.

Ding, Y.

Djordjev, K.

S. J. Choi, K. Djordjev, Z. Peng, Q. Yang, S. J. Choi, and P. D. Dapkus, “Laterally coupled buried heterostructure high-Q ring resonators,” IEEE Photon. Technol. Lett.16(10), 2266–2268 (2004).
[CrossRef]

Doerr, C. R.

C. R. Doerr, L. Zhang, P. J. Winzer, J. H. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact high- speed InP DQPSK modulator,” IEEE Photon. Technol. Lett.19(15), 1184–1186 (2007).
[CrossRef]

Dong, P.

Feng, D.

Feng, H.

H. Feng, J. P. Pang, M. Sugiyama, K. Tada, and Y. Nakano, “Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential,” IEEE J. Quantum Electron.34(7), 1197–1208 (1998).
[CrossRef]

Fukuoka, M.

T. Arakawa, T. Hariki, Y. Amma, M. Fukuoka, M. Ushigome, and K. Tada, “Low-voltage Mach-Zehnder modulator with InGaAs/InAlAs five-layer asymmetric coupled quantum well,” Jpn. J. Appl. Phys.51, 042203 (2012).
[CrossRef]

Goldhar, J.

Gotoh, T.

Green, W.

Grover, R.

Hamacher, M.

M. Hamacher, H. Heidrich, D. G. Rabus, and U. Troppenz, Proc. SPIE4640, 37–45 (2002).
[CrossRef]

Hariki, T.

T. Arakawa, T. Hariki, Y. Amma, M. Fukuoka, M. Ushigome, and K. Tada, “Low-voltage Mach-Zehnder modulator with InGaAs/InAlAs five-layer asymmetric coupled quantum well,” Jpn. J. Appl. Phys.51, 042203 (2012).
[CrossRef]

Hasegawa, R.

Hatori, N.

S. Akiyama, T. Kurahashi, T. Baba, N. Hatori, T. Usuki, and T. Yamamoto, “A 1 V peak-to-peak driven 10-Gbps slow-light silicon Mach–Zehnder modulator using cascaded ring resonators,” Appl. Phys. Express3(7), 072202 (2010).
[CrossRef]

Heebner, J. E.

Heidrich, H.

M. Hamacher, H. Heidrich, D. G. Rabus, and U. Troppenz, Proc. SPIE4640, 37–45 (2002).
[CrossRef]

Hirose, S.

Ho, P.-T.

Hvam, J. M.

Ibrahim, T. A.

Ide, T.

T. Arakawa, T. Toya, M. Ushigome, K. Yamaguchi, T. Ide, and K. Tada, “InGaAs/InAlAs five-layer asymmetric coupled quantum well exhibiting giant electrorefractive index change,” Jpn. J. Appl. Phys.50, 032204 (2011).
[CrossRef]

Iga, R.

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

Ishibashi, T.

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

Ishikawa, M.

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

Ito, T.

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

Itoh, H.

Izutsu, M.

T. Kawanishi, S. Sakamoto, and M. Izutsu, “High-speed control of lightwave amplitude, phase, and frequency by use of electrooptic effect,” IEEE J. Sel. Top. Quantum Electron.13(1), 79–91 (2007).
[CrossRef]

Jang, J.-H.

W.-S. Choi, W. Zhao, J.-W. Bae, I. Adesida, B.-A. Yu, Y. L. Lee, and J.-H. Jang, “Micro-racetrack notch filters based on InGaAsP/InP high mesa optical waveguides,” Jpn. J. Appl. Phys.46(4B), 2434–2439 (2007).
[CrossRef]

Kaneshige, H.

H. Kaneshige, Y. Ueyama, H. Yamada, H. Yajima, T. Arakawa, and Y. Kokubun, “InGaAs/InAlAs multiple quantum well Mach-Zehnder modulator with single microring resonator,” Jpn. J. Appl. Phys.51(2), 02BG01 (2012).
[CrossRef]

Kang, J.

X. Xie, J. Khurgin, J. Kang, and F.-S. Chow, “Linearized Mach–Zehnder intensity modulator,” IEEE Photon. Technol. Lett.15(4), 531–533 (2003).
[CrossRef]

Kawaguchi, Y.

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

Kawanishi, T.

T. Kawanishi, S. Sakamoto, and M. Izutsu, “High-speed control of lightwave amplitude, phase, and frequency by use of electrooptic effect,” IEEE J. Sel. Top. Quantum Electron.13(1), 79–91 (2007).
[CrossRef]

Khurgin, J.

X. Xie, J. Khurgin, J. Kang, and F.-S. Chow, “Linearized Mach–Zehnder intensity modulator,” IEEE Photon. Technol. Lett.15(4), 531–533 (2003).
[CrossRef]

Kikuchi, N.

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

Kim, Y.

Kokubun, Y.

H. Kaneshige, Y. Ueyama, H. Yamada, H. Yajima, T. Arakawa, and Y. Kokubun, “InGaAs/InAlAs multiple quantum well Mach-Zehnder modulator with single microring resonator,” Jpn. J. Appl. Phys.51(2), 02BG01 (2012).
[CrossRef]

T. Makino, T. Gotoh, R. Hasegawa, T. Arakawa, and Y. Kokubun, “Microring resonator wavelength tunable filter using five-layer asymmetric coupled quantum well,” J. Lightwave Technol.29(16), 2387–2393 (2011).
[CrossRef]

Y. Kokubun, “High Index contrast optical waveguides and their applications to microring filter circuit and wavelength selective switch,” IEICE Trans. Electron.E90-C(5), 1037–1045 (2007).
[CrossRef]

Y. Kokubun, “Vertically coupled micro-ring resonator filter for integrated add/drop node,” IEICE Trans. Electron.E-88C, 349–362 (2005).

Kondo, Y.

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

Krishnamoorthy, A. V.

Kung, C. C.

Kurahashi, T.

S. Akiyama, T. Kurahashi, T. Baba, N. Hatori, T. Usuki, and T. Yamamoto, “A 1 V peak-to-peak driven 10-Gbps slow-light silicon Mach–Zehnder modulator using cascaded ring resonators,” Appl. Phys. Express3(7), 072202 (2010).
[CrossRef]

Kuramata, A.

Lee, C. H.

Lee, R.

Lee, Y. L.

W.-S. Choi, W. Zhao, J.-W. Bae, I. Adesida, B.-A. Yu, Y. L. Lee, and J.-H. Jang, “Micro-racetrack notch filters based on InGaAsP/InP high mesa optical waveguides,” Jpn. J. Appl. Phys.46(4B), 2434–2439 (2007).
[CrossRef]

Lepeshkin, N. N.

Li, G.

Li, J.

Liang, H.

Liao, S.

Liu, L.

Lu, Y.

Y. Lu, L. Xu, M. Shu, P. Wang, and J. Yao, “Proposal to produce coupled resonator-induced transparency and bistability using microresonator enhanced Mach–Zehnder interferometer,” IEEE Photon. Technol. Lett.20(7), 529–531 (2008).
[CrossRef]

Makino, T.

Matsuo, S.

S. Matsuo, Y. Ohiso, and T. Segawa, “A high-speed tunable optical filter using a semiconductor ring resonator for photonic packet switch,” IEICE Trans. Electron.E88-C(3), 295–302 (2005).
[CrossRef]

Nakano, Y.

H. Feng, J. P. Pang, M. Sugiyama, K. Tada, and Y. Nakano, “Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential,” IEEE J. Quantum Electron.34(7), 1197–1208 (1998).
[CrossRef]

Nomura, S.

Ohiso, Y.

S. Matsuo, Y. Ohiso, and T. Segawa, “A high-speed tunable optical filter using a semiconductor ring resonator for photonic packet switch,” IEICE Trans. Electron.E88-C(3), 295–302 (2005).
[CrossRef]

Oku, S.

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

Ou, H.

Pang, J. P.

H. Feng, J. P. Pang, M. Sugiyama, K. Tada, and Y. Nakano, “Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential,” IEEE J. Quantum Electron.34(7), 1197–1208 (1998).
[CrossRef]

Peng, Z.

S. J. Choi, K. Djordjev, Z. Peng, Q. Yang, S. J. Choi, and P. D. Dapkus, “Laterally coupled buried heterostructure high-Q ring resonators,” IEEE Photon. Technol. Lett.16(10), 2266–2268 (2004).
[CrossRef]

Pu, M.

Qian, W.

Rabus, D. G.

M. Hamacher, H. Heidrich, D. G. Rabus, and U. Troppenz, Proc. SPIE4640, 37–45 (2002).
[CrossRef]

Raybon, G.

C. R. Doerr, L. Zhang, P. J. Winzer, J. H. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact high- speed InP DQPSK modulator,” IEEE Photon. Technol. Lett.19(15), 1184–1186 (2007).
[CrossRef]

Rostami, A.

A. Rostami, “Low threshold and tunable all-optical switch using two-photon absorption in array of nonlinear ring resonators coupled to MZI,” Microelectron. J.37(9), 976–981 (2006).
[CrossRef]

Sakamoto, S.

T. Kawanishi, S. Sakamoto, and M. Izutsu, “High-speed control of lightwave amplitude, phase, and frequency by use of electrooptic effect,” IEEE J. Sel. Top. Quantum Electron.13(1), 79–91 (2007).
[CrossRef]

Sanjoh, H.

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

Sato, T.

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

Sauer, N.

C. R. Doerr, L. Zhang, P. J. Winzer, J. H. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact high- speed InP DQPSK modulator,” IEEE Photon. Technol. Lett.19(15), 1184–1186 (2007).
[CrossRef]

Scherer, A.

Schweinsberg, A.

Segawa, T.

S. Matsuo, Y. Ohiso, and T. Segawa, “A high-speed tunable optical filter using a semiconductor ring resonator for photonic packet switch,” IEICE Trans. Electron.E88-C(3), 295–302 (2005).
[CrossRef]

Sekiguchi, S.

Shafiiha, R.

Shibata, Y.

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

Shu, M.

Y. Lu, L. Xu, M. Shu, P. Wang, and J. Yao, “Proposal to produce coupled resonator-induced transparency and bistability using microresonator enhanced Mach–Zehnder interferometer,” IEEE Photon. Technol. Lett.20(7), 529–531 (2008).
[CrossRef]

Sinsky, J. H.

C. R. Doerr, L. Zhang, P. J. Winzer, J. H. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact high- speed InP DQPSK modulator,” IEEE Photon. Technol. Lett.19(15), 1184–1186 (2007).
[CrossRef]

Sugiyama, M.

H. Feng, J. P. Pang, M. Sugiyama, K. Tada, and Y. Nakano, “Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential,” IEEE J. Quantum Electron.34(7), 1197–1208 (1998).
[CrossRef]

Tada, K.

T. Arakawa, T. Hariki, Y. Amma, M. Fukuoka, M. Ushigome, and K. Tada, “Low-voltage Mach-Zehnder modulator with InGaAs/InAlAs five-layer asymmetric coupled quantum well,” Jpn. J. Appl. Phys.51, 042203 (2012).
[CrossRef]

T. Arakawa, T. Toya, M. Ushigome, K. Yamaguchi, T. Ide, and K. Tada, “InGaAs/InAlAs five-layer asymmetric coupled quantum well exhibiting giant electrorefractive index change,” Jpn. J. Appl. Phys.50, 032204 (2011).
[CrossRef]

H. Feng, J. P. Pang, M. Sugiyama, K. Tada, and Y. Nakano, “Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential,” IEEE J. Quantum Electron.34(7), 1197–1208 (1998).
[CrossRef]

Takeuchi, T.

Tohmori, Y.

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

Toya, T.

T. Arakawa, T. Toya, M. Ushigome, K. Yamaguchi, T. Ide, and K. Tada, “InGaAs/InAlAs five-layer asymmetric coupled quantum well exhibiting giant electrorefractive index change,” Jpn. J. Appl. Phys.50, 032204 (2011).
[CrossRef]

Troppenz, U.

M. Hamacher, H. Heidrich, D. G. Rabus, and U. Troppenz, Proc. SPIE4640, 37–45 (2002).
[CrossRef]

Tsuzuki, K.

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

Ueyama, Y.

H. Kaneshige, Y. Ueyama, H. Yamada, H. Yajima, T. Arakawa, and Y. Kokubun, “InGaAs/InAlAs multiple quantum well Mach-Zehnder modulator with single microring resonator,” Jpn. J. Appl. Phys.51(2), 02BG01 (2012).
[CrossRef]

Ushigome, M.

T. Arakawa, T. Hariki, Y. Amma, M. Fukuoka, M. Ushigome, and K. Tada, “Low-voltage Mach-Zehnder modulator with InGaAs/InAlAs five-layer asymmetric coupled quantum well,” Jpn. J. Appl. Phys.51, 042203 (2012).
[CrossRef]

T. Arakawa, T. Toya, M. Ushigome, K. Yamaguchi, T. Ide, and K. Tada, “InGaAs/InAlAs five-layer asymmetric coupled quantum well exhibiting giant electrorefractive index change,” Jpn. J. Appl. Phys.50, 032204 (2011).
[CrossRef]

Usuki, T.

S. Akiyama, T. Kurahashi, T. Baba, N. Hatori, T. Usuki, and T. Yamamoto, “A 1 V peak-to-peak driven 10-Gbps slow-light silicon Mach–Zehnder modulator using cascaded ring resonators,” Appl. Phys. Express3(7), 072202 (2010).
[CrossRef]

Walker, R. G.

R. G. Walker, “High-speed III-V semiconductor intensity modulators,” IEEE J. Quantum Electron.27(3), 654–667 (1991).
[CrossRef]

Wang, P.

Y. Lu, L. Xu, M. Shu, P. Wang, and J. Yao, “Proposal to produce coupled resonator-induced transparency and bistability using microresonator enhanced Mach–Zehnder interferometer,” IEEE Photon. Technol. Lett.20(7), 529–531 (2008).
[CrossRef]

Wicks, G. W.

Winzer, P. J.

C. R. Doerr, L. Zhang, P. J. Winzer, J. H. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact high- speed InP DQPSK modulator,” IEEE Photon. Technol. Lett.19(15), 1184–1186 (2007).
[CrossRef]

Xie, X.

X. Xie, J. Khurgin, J. Kang, and F.-S. Chow, “Linearized Mach–Zehnder intensity modulator,” IEEE Photon. Technol. Lett.15(4), 531–533 (2003).
[CrossRef]

Xu, L.

Y. Lu, L. Xu, M. Shu, P. Wang, and J. Yao, “Proposal to produce coupled resonator-induced transparency and bistability using microresonator enhanced Mach–Zehnder interferometer,” IEEE Photon. Technol. Lett.20(7), 529–531 (2008).
[CrossRef]

Xue, W.

Yajima, H.

H. Kaneshige, Y. Ueyama, H. Yamada, H. Yajima, T. Arakawa, and Y. Kokubun, “InGaAs/InAlAs multiple quantum well Mach-Zehnder modulator with single microring resonator,” Jpn. J. Appl. Phys.51(2), 02BG01 (2012).
[CrossRef]

Yamada, E.

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

Yamada, H.

H. Kaneshige, Y. Ueyama, H. Yamada, H. Yajima, T. Arakawa, and Y. Kokubun, “InGaAs/InAlAs multiple quantum well Mach-Zehnder modulator with single microring resonator,” Jpn. J. Appl. Phys.51(2), 02BG01 (2012).
[CrossRef]

Yamaguchi, K.

T. Arakawa, T. Toya, M. Ushigome, K. Yamaguchi, T. Ide, and K. Tada, “InGaAs/InAlAs five-layer asymmetric coupled quantum well exhibiting giant electrorefractive index change,” Jpn. J. Appl. Phys.50, 032204 (2011).
[CrossRef]

Yamamoto, T.

S. Akiyama, T. Kurahashi, T. Baba, N. Hatori, T. Usuki, and T. Yamamoto, “A 1 V peak-to-peak driven 10-Gbps slow-light silicon Mach–Zehnder modulator using cascaded ring resonators,” Appl. Phys. Express3(7), 072202 (2010).
[CrossRef]

S. Akiyama, H. Itoh, S. Sekiguchi, S. Hirose, T. Takeuchi, A. Kuramata, and T. Yamamoto, “InP-based Mach-Zehnder modulator with capacitively loaded traveling-wave electrodes,” J. Lightwave Technol.26(5), 608–615 (2008).
[CrossRef]

Yang, Q.

S. J. Choi, K. Djordjev, Z. Peng, Q. Yang, S. J. Choi, and P. D. Dapkus, “Laterally coupled buried heterostructure high-Q ring resonators,” IEEE Photon. Technol. Lett.16(10), 2266–2268 (2004).
[CrossRef]

Yao, J.

Y. Lu, L. Xu, M. Shu, P. Wang, and J. Yao, “Proposal to produce coupled resonator-induced transparency and bistability using microresonator enhanced Mach–Zehnder interferometer,” IEEE Photon. Technol. Lett.20(7), 529–531 (2008).
[CrossRef]

Yariv, A.

Yasaka, H.

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

Yasui, T.

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

Yu, B.-A.

W.-S. Choi, W. Zhao, J.-W. Bae, I. Adesida, B.-A. Yu, Y. L. Lee, and J.-H. Jang, “Micro-racetrack notch filters based on InGaAsP/InP high mesa optical waveguides,” Jpn. J. Appl. Phys.46(4B), 2434–2439 (2007).
[CrossRef]

Yvind, K.

Zhang, L.

C. R. Doerr, L. Zhang, P. J. Winzer, J. H. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact high- speed InP DQPSK modulator,” IEEE Photon. Technol. Lett.19(15), 1184–1186 (2007).
[CrossRef]

Zhao, W.

W.-S. Choi, W. Zhao, J.-W. Bae, I. Adesida, B.-A. Yu, Y. L. Lee, and J.-H. Jang, “Micro-racetrack notch filters based on InGaAsP/InP high mesa optical waveguides,” Jpn. J. Appl. Phys.46(4B), 2434–2439 (2007).
[CrossRef]

Zheng, D.

Zheng, X.

Appl. Phys. Express (1)

S. Akiyama, T. Kurahashi, T. Baba, N. Hatori, T. Usuki, and T. Yamamoto, “A 1 V peak-to-peak driven 10-Gbps slow-light silicon Mach–Zehnder modulator using cascaded ring resonators,” Appl. Phys. Express3(7), 072202 (2010).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. G. Walker, “High-speed III-V semiconductor intensity modulators,” IEEE J. Quantum Electron.27(3), 654–667 (1991).
[CrossRef]

H. Feng, J. P. Pang, M. Sugiyama, K. Tada, and Y. Nakano, “Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential,” IEEE J. Quantum Electron.34(7), 1197–1208 (1998).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Kawanishi, S. Sakamoto, and M. Izutsu, “High-speed control of lightwave amplitude, phase, and frequency by use of electrooptic effect,” IEEE J. Sel. Top. Quantum Electron.13(1), 79–91 (2007).
[CrossRef]

IEEE Photon. Technol. Lett. (7)

K. Tsuzuki, T. Ishibashi, T. Ito, S. Oku, Y. Shibata, T. Ito, R. Iga, Y. Kondo, and Y. Tohmori, “A 40-Gb/s InGaAlAs-InAlAs MQW n-i-n Mach–Zehnder modulator with a drive voltage of 2.3 V,” IEEE Photon. Technol. Lett.17(1), 46–48 (2005).
[CrossRef]

T. Yasui, Y. Shibata, K. Tsuzuki, N. Kikuchi, M. Ishikawa, Y. Kawaguchi, M. Arai, and H. Yasaka, “10-Gb/s 100-km SMF transmission using InP Mach–Zehnder modulator monolithically integrated with semiconductor optical amplifier,” IEEE Photon. Technol. Lett.20(13), 1178–1180 (2008).
[CrossRef]

C. R. Doerr, L. Zhang, P. J. Winzer, J. H. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact high- speed InP DQPSK modulator,” IEEE Photon. Technol. Lett.19(15), 1184–1186 (2007).
[CrossRef]

N. Kikuchi, Y. Shibata, K. Tsuzuki, H. Sanjoh, T. Sato, E. Yamada, T. Ishibashi, and H. Yasaka, “80-Gb/s low-driving-voltage InP DQPSK modulator with an n-p-i-n Structure,” IEEE Photon. Technol. Lett.21(12), 787–789 (2009).
[CrossRef]

X. Xie, J. Khurgin, J. Kang, and F.-S. Chow, “Linearized Mach–Zehnder intensity modulator,” IEEE Photon. Technol. Lett.15(4), 531–533 (2003).
[CrossRef]

Y. Lu, L. Xu, M. Shu, P. Wang, and J. Yao, “Proposal to produce coupled resonator-induced transparency and bistability using microresonator enhanced Mach–Zehnder interferometer,” IEEE Photon. Technol. Lett.20(7), 529–531 (2008).
[CrossRef]

S. J. Choi, K. Djordjev, Z. Peng, Q. Yang, S. J. Choi, and P. D. Dapkus, “Laterally coupled buried heterostructure high-Q ring resonators,” IEEE Photon. Technol. Lett.16(10), 2266–2268 (2004).
[CrossRef]

IEICE Trans. Electron. (3)

Y. Kokubun, “Vertically coupled micro-ring resonator filter for integrated add/drop node,” IEICE Trans. Electron.E-88C, 349–362 (2005).

S. Matsuo, Y. Ohiso, and T. Segawa, “A high-speed tunable optical filter using a semiconductor ring resonator for photonic packet switch,” IEICE Trans. Electron.E88-C(3), 295–302 (2005).
[CrossRef]

Y. Kokubun, “High Index contrast optical waveguides and their applications to microring filter circuit and wavelength selective switch,” IEICE Trans. Electron.E90-C(5), 1037–1045 (2007).
[CrossRef]

J. Lightwave Technol. (3)

Jpn. J. Appl. Phys. (4)

W.-S. Choi, W. Zhao, J.-W. Bae, I. Adesida, B.-A. Yu, Y. L. Lee, and J.-H. Jang, “Micro-racetrack notch filters based on InGaAsP/InP high mesa optical waveguides,” Jpn. J. Appl. Phys.46(4B), 2434–2439 (2007).
[CrossRef]

H. Kaneshige, Y. Ueyama, H. Yamada, H. Yajima, T. Arakawa, and Y. Kokubun, “InGaAs/InAlAs multiple quantum well Mach-Zehnder modulator with single microring resonator,” Jpn. J. Appl. Phys.51(2), 02BG01 (2012).
[CrossRef]

T. Arakawa, T. Toya, M. Ushigome, K. Yamaguchi, T. Ide, and K. Tada, “InGaAs/InAlAs five-layer asymmetric coupled quantum well exhibiting giant electrorefractive index change,” Jpn. J. Appl. Phys.50, 032204 (2011).
[CrossRef]

T. Arakawa, T. Hariki, Y. Amma, M. Fukuoka, M. Ushigome, and K. Tada, “Low-voltage Mach-Zehnder modulator with InGaAs/InAlAs five-layer asymmetric coupled quantum well,” Jpn. J. Appl. Phys.51, 042203 (2012).
[CrossRef]

Microelectron. J. (1)

A. Rostami, “Low threshold and tunable all-optical switch using two-photon absorption in array of nonlinear ring resonators coupled to MZI,” Microelectron. J.37(9), 976–981 (2006).
[CrossRef]

Opt. Express (5)

Opt. Lett. (2)

Proc. SPIE (1)

M. Hamacher, H. Heidrich, D. G. Rabus, and U. Troppenz, Proc. SPIE4640, 37–45 (2002).
[CrossRef]

Other (2)

A. Kaplan, A. Greenblatt, G. Harston, P. S. Cho, Y. Achiam, and I. Shpantzer: Proc. Co-herent Optical Technologies and Applications (COTA), 2006, CFC3.

H. Kaneshige, Y. Ueyama, H. Yamada, T. Arakawa, and Y. Kokubun, “Quantum well Mach-Zehnder modulator with single microring resonator and optimized arm length,” 17th Microoptics Conf. (MOC’11), G-5 (Nov. 1, 2011).

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Figures (15)

Fig. 1
Fig. 1

Schematic top view of microring resonator with busline waveguide.

Fig. 2
Fig. 2

Effective phase ϕeff and transmittance from Ports 1 to 3, T, as functions of single-pass phase ϕ for coupling efficiency K = 0.25, round-trip length of microring Lring = 480 μm, and power loss constant in microring α = 2.2 dB/mm.

Fig. 3
Fig. 3

Schematic top view of proposed InGaAs/InAlAs MQW microring-enhanced Mach-Zehnder modulator.

Fig. 4
Fig. 4

Schematic cross-sectional view of waveguide and coupling region.

Fig. 5
Fig. 5

(a) Calculated output port spectrum responses at various reverse bias voltages. (b) Calculated theoretical modulation characteristic of MRE-MZM with device parameters in Table 1 at wavelength of 1550.1 nm.

Fig. 6
Fig. 6

ON and OFF states for MRE-MZM in dependence of effective phase ϕeff on single-pass phase ϕ.

Fig. 7
Fig. 7

Calculated dependence of extinction ratio and effective phase-shift-enhancement factor of proposed MRE-MZM on branching ratio of input coupler.

Fig. 8
Fig. 8

Microscopic top view of fabricated MRE-MZM. (a) Entire image. (b) Magnified image of DC3.

Fig. 9
Fig. 9

Measured output port spectrum responses at various reverse bias voltages (a) from 0 to 10 V and (b) from 0 to 4 V.

Fig. 10
Fig. 10

Evaluated changes in refractive index in core layer of microring waveguide Δncore at various wavelengths in the vicinity of 1550 nm as functions of applied dc reverse voltage.

Fig. 11
Fig. 11

Static modulation characteristics of microring MZ modulator at wavelength of 1550.1 nm.

Fig. 12
Fig. 12

Static modulation characteristics of microring MZ modulator at various wavelengths around 1550.0 nm.

Fig. 13
Fig. 13

Calculated output port spectral responses at various applied reverse bias Va (a) from 0 to 10 V and (b) from 0 to 4 V.

Fig. 14
Fig. 14

(a) Dynamic modulation driven by sinusoidal reverse voltage with repetition rate of 3.5 GHz and amplitude of 4.0 V. (b) Dependence of electro-optic (EO) response on modulation frequency.

Fig. 15
Fig. 15

(a) Dependence of the 3-dB bandwidth of proposed MRE-MZM (Lring = 480 μm) determined by photon life time τp on coupling efficiency K. (b) 3-dB bandwidth of MRE-MZM with Lring = 480 and 280 μm as functions of effective phase-shift-enhancement factor Fpe(eff).

Tables (1)

Tables Icon

Table 1 Device parameters for MRE-MZM

Equations (11)

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ϕ eff =arg( E 3 E 1 ) =π+ϕ+ tan 1 ( rsinϕ arcosϕ )+ tan 1 ( arsinϕ 1arcosϕ ).
r= 1K ,
a=exp( α L ring 2 ),
F pe = 3π/2π/2 ϕ| ϕ eff =3π/2 ϕ| ϕ eff =π/2 = π φ| ϕ eff =3π/2 ϕ| ϕ eff =π/2 .
a> 1K .
F pe (eff) = Δ ϕ eff Δϕ = ϕ eff ( ϕ ON ) ϕ eff ( ϕ OFF ) ϕ ON ϕ OFF ,
1 f 3dB 2 = ( 2π τ p ) 2 + ( 2πRC ) 2 ,
τ p = λQ 2πc ,
Q=( Q loss 1 + Q couple 1 ),
Q loss = 2πc λ τ p = 2π L ring n eff λ{ 1exp(α L ring ) } ,
Q couple = 2π L ring n eff λK ,

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