Abstract

We propose and numerically verify a novel scheme of all-optical format conversion from non-return-to-zero (NRZ) to binary phase-shift-keying (BPSK) at 160 Gb/s using cascaded microring resonators (CMRR) on a single silicon chip. The conversion is based on large phase shift and flattened intensity-response characteristics in the CMRR. A continuous-wave light experiences different phase shifts controlled by the power of an input NRZ signal with an ~8.8-dB extinction ratio, while maintaining approximately the constant intensity. All-optical format conversion from NRZ to quadrature phase-shift-keying (QPSK) is also demonstrated based on parallel NRZ/BPSK converters in a Mach-Zehnder interferometer structure.

© 2007 Optical Society of America

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2007 (2)

2006 (2)

C. Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K. Kitayama, "NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter," J. Lightwave Technol. 24, 3751-3758 (2006).
[CrossRef]

2005 (3)

2004 (3)

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

W. Li, M. Chen, Y. Dong, and S. Xie, "All-optical format conversion from NRZ to CSRZ and between RZ and CSRZ using SOA-based fiber loop mirror," IEEE Photon. Technol. Lett. 16, 203-205 (2004).
[CrossRef]

K. Croussore, C. Kim and G. Li, "All-optical regeneration of differential phase-shift keying signals based?on phase-sensitive amplification," Opt. Lett. 28, 2357-2359 (2004).
[CrossRef]

2003 (4)

Y. Chen and S. Blair, "Nonlinear phase shift of cascaded microring resonators," J. Opt. Soc. Am. B 20, 2125-2132 (2003).
[CrossRef]

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, "A comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise," J. Lightwave Technol. 21, 1933-1943 (2003).
[CrossRef]

L. Xu, B. C. Wang, V. Baby, I. Glesk, and P. R. Prucnal, "All-optical data format conversion between RZ and NRZ based on a Mach-Zehnder interferometric wavelength converter," IEEE Photon. Technol. Lett. 15, 308-310 (2003).
[CrossRef]

2002 (3)

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

S. Blair, J. E. Heebner, and R. W. Boyd, "Beyond the absorption-limited nonlinear phase shift with microring resonators," Opt. Lett. 27, 357-359 (2002).
[CrossRef]

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

1999 (1)

Abe, J.

Absil, P. P.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

Agrawal, G. P.

Asghari, M.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Baby, V.

L. Xu, B. C. Wang, V. Baby, I. Glesk, and P. R. Prucnal, "All-optical data format conversion between RZ and NRZ based on a Mach-Zehnder interferometric wavelength converter," IEEE Photon. Technol. Lett. 15, 308-310 (2003).
[CrossRef]

Barry, J.

J. Yu, G. K. Chang, J. Barry, and Y. Su, "40 Gbit/s signal format conversion from NRZ to RZ using a Mach-Zehnder delay interferometer," Opt. Commun. 248, 419-422 (2005).
[CrossRef]

Birner, A.

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

Blair, S.

Boyd, R. W.

Burrows, E.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Chandrasekhar, S.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Chang, G. K.

J. Yu, G. K. Chang, J. Barry, and Y. Su, "40 Gbit/s signal format conversion from NRZ to RZ using a Mach-Zehnder delay interferometer," Opt. Commun. 248, 419-422 (2005).
[CrossRef]

Chen, M.

W. Li, M. Chen, Y. Dong, and S. Xie, "All-optical format conversion from NRZ to CSRZ and between RZ and CSRZ using SOA-based fiber loop mirror," IEEE Photon. Technol. Lett. 16, 203-205 (2004).
[CrossRef]

Chen, Y.

Chul, Y. J.

Croussore, K.

Day, I. E.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Doerr, C.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Dong, Y.

W. Li, M. Chen, Y. Dong, and S. Xie, "All-optical format conversion from NRZ to CSRZ and between RZ and CSRZ using SOA-based fiber loop mirror," IEEE Photon. Technol. Lett. 16, 203-205 (2004).
[CrossRef]

Drake, J.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Feng, L.

C. Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Glesk, I.

L. Xu, B. C. Wang, V. Baby, I. Glesk, and P. R. Prucnal, "All-optical data format conversion between RZ and NRZ based on a Mach-Zehnder interferometric wavelength converter," IEEE Photon. Technol. Lett. 15, 308-310 (2003).
[CrossRef]

Gnauck, A. H.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Gosele, U.

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

Grover, R.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

Harpin, A.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Hatta, T.

Heebner, J. E.

Ho, P.-T.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

Ibrahim, T. A.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

Ishida, K.

Izutsu, M.

Johnson, F. G.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

Kasahara, K.

Kawanishi, T.

Kim, C.

Kim, Y. J.

Kinjo, K.

Kitayama, K.

Kobayashi, T.

Lee, C. G.

Leonard, S. W.

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

Leuthold, J.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Li, G.

Li, W.

W. Li, M. Chen, Y. Dong, and S. Xie, "All-optical format conversion from NRZ to CSRZ and between RZ and CSRZ using SOA-based fiber loop mirror," IEEE Photon. Technol. Lett. 16, 203-205 (2004).
[CrossRef]

Liang, T. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Maruta, A.

Mishina, K.

Mitani, S.

Miyahara, T.

Mizuochi, T.

Motoshima, K.

Nissanka, S. M.

Prucnal, P. R.

L. Xu, B. C. Wang, V. Baby, I. Glesk, and P. R. Prucnal, "All-optical data format conversion between RZ and NRZ based on a Mach-Zehnder interferometric wavelength converter," IEEE Photon. Technol. Lett. 15, 308-310 (2003).
[CrossRef]

Raybon, G.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Roberts, S. W.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Sakamoto, T.

Schilling, J.

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

Schweizer, S. L.

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

Shimizu, K.

Stulz, L.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Su, Y.

C. Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

J. Yu, G. K. Chang, J. Barry, and Y. Su, "40 Gbit/s signal format conversion from NRZ to RZ using a Mach-Zehnder delay interferometer," Opt. Commun. 248, 419-422 (2005).
[CrossRef]

Tan, H.-W.

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

Tian, X.

C. Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Tian, Y.

C. Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Tsang, H. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Van, V.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

van Driel, H. M.

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

Wang, B. C.

L. Xu, B. C. Wang, V. Baby, I. Glesk, and P. R. Prucnal, "All-optical data format conversion between RZ and NRZ based on a Mach-Zehnder interferometric wavelength converter," IEEE Photon. Technol. Lett. 15, 308-310 (2003).
[CrossRef]

Wehrspohn, R. B.

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

Wong, C. S.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Xie, S.

W. Li, M. Chen, Y. Dong, and S. Xie, "All-optical format conversion from NRZ to CSRZ and between RZ and CSRZ using SOA-based fiber loop mirror," IEEE Photon. Technol. Lett. 16, 203-205 (2004).
[CrossRef]

Xu, L.

L. Xu, B. C. Wang, V. Baby, I. Glesk, and P. R. Prucnal, "All-optical data format conversion between RZ and NRZ based on a Mach-Zehnder interferometric wavelength converter," IEEE Photon. Technol. Lett. 15, 308-310 (2003).
[CrossRef]

Xu, X.

C. Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Yan, C.

C. Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Yi, L.

C. Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Yin, L.

Yu, J.

J. Yu, G. K. Chang, J. Barry, and Y. Su, "40 Gbit/s signal format conversion from NRZ to RZ using a Mach-Zehnder delay interferometer," Opt. Commun. 248, 419-422 (2005).
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H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5?m wavelength," App. Phys. Lett. 80, 416-418 (2002).
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C. Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

L. Xu, B. C. Wang, V. Baby, I. Glesk, and P. R. Prucnal, "All-optical data format conversion between RZ and NRZ based on a Mach-Zehnder interferometric wavelength converter," IEEE Photon. Technol. Lett. 15, 308-310 (2003).
[CrossRef]

W. Li, M. Chen, Y. Dong, and S. Xie, "All-optical format conversion from NRZ to CSRZ and between RZ and CSRZ using SOA-based fiber loop mirror," IEEE Photon. Technol. Lett. 16, 203-205 (2004).
[CrossRef]

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
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J. Yu, G. K. Chang, J. Barry, and Y. Su, "40 Gbit/s signal format conversion from NRZ to RZ using a Mach-Zehnder delay interferometer," Opt. Commun. 248, 419-422 (2005).
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Proc. SPIE (1)

H. M. van Driel, S. W. Leonard, H.-W. Tan, A. Birner, J. Schilling, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, "Tuning 2-D photonic crystals," Proc. SPIE 5511, 1-9 (2004).

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C. S. Langhorst, R. Ludwig,M. Galili, B. Huettl, F. Futami, S. Watanabe, and C. Schubert, "160 Gbit/s all-optical OOK to DPSK in-line format conversion," in Proceedings of the European Conference on Optical Communication (ECOC 2006), PD Th4.3.5.

S. H. Lee, K. K. Chow, C. Shu, and C. Lin, "All optical ASK to DPSK format conversion using cross-phase modulation in a nonlinear photonic crystal fiber," in Conference on Lasers and Electro-Optics (2005), paper CFJ2-5.

L. Zhou, H. Chen, and A.W. Poon, "NRZ-to-PRZ format conversion using silicon second-order coupled-microring resonator-based notch filters," in Conference on Lasers and Electro-Optics (Optical Society of America, 2007), paper CThP4.

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagrams of format conversions for (a) NRZ to BPSK and (b) NRZ to QPSK.

Fig. 2.
Fig. 2.

(a) Structure of a single ring resonator. (b) Nonlinear response versus incident power in a single ring resonator: intensity transmittance (solid) and output phase shift (dashed). (c) Operation principle of NRZ to BPSK format conversion in CMRR.

Fig. 3.
Fig. 3.

Simulation results of NRZ to BPSK format conversion. (a), (c) ,(e) Amplitude, eye diagram, and spectrum of the input NRZ signal. (b), (d), (f) Phase, phase eye diagram, and spectrum of the output BPSK data. (g), (h) Amplitude and signal eye diagram of the BPSK data. (i), (j) Waveform and eye diagram of the demodulated signal.

Fig. 4.
Fig. 4.

Simulation results of NRZ to QPSK format conversion. (a) Phase eye diagram of the converted QPSK, (b) Optical spectrum of the converted QPSK. (c), (d) Eye diagrams of the in-phase and quadrature branches of the demodulated QPSK.

Equations (11)

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[ E out E ( 0 ) ] = [ τ τ ] [ E in E ( L ) ] ,
E ( L ) = a exp ( + ) E ( 0 ) ,
E out E in = τ a exp ( + ) 1 τa exp ( + ) .
ΔΦ = ara tan τa sin ( ϕ ) 1 τa cos ( ϕ ) arctan a sin ( ϕ ) τ a cos ( ϕ ) ,
E 0 z + α 2 E 0 + β non deg 2 ( 2 E 1 2 ) E 0 = i n 2 ω 0 c ( 2 E 1 2 ) E 0 ,
I 0 z = β non deg ( 2 I 1 ) I 0 exp ( αz ) ,
ϕ NL z = n 2 ω 0 c ( 2 I 1 ) exp ( αz ) .
I 0 ( L ) = I 0 ( 0 ) exp ( αL ) 1 + 2 β non deg I 1 ( 0 ) L eff ,
ϕ NL = n 2 ω 0 c β non deg ln ( 1 + 2 β non deg I 1 ( 0 ) L eff ) ,
a 2 = exp ( αL ) 1 + 2 β non deg I 1 ( 0 ) L eff ,
ϕ = k 0 L + n 2 ω 0 c β non deg ln ( 1 + 2 β non deg I 1 ( 0 ) L eff ) ,

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