Abstract

The conventional phase-conjugate twin wave (PCTW) scheme can effectively mitigate the fiber nonlinear distortions but reduce the spectral efficiency (SE) by half. In this paper, we show that by modulating one of the conjugated signals with additional bits, the information encoded in the conjugated variants can still suppress the nonlinear distortions in a similar manner as in the conventional PCTW method, while that encoded in the additional bits, representing the improved SE, is the dominant source limiting the system performance. We further introduce redundancy to the additional bits via error correction coding to overcome this performance bottleneck. The overall system performance can thus approach that of the conventional PCTW method while the SE is significantly enhanced. Simulations of a 25-Gbaud polarization-division-multiplexed quadrature phase shift keying system over 15,200 km show that the proposed scheme can increase the normalized SE from 50% to 80% for a similar performance as the conventional PCTW method. We also show that the proposed concept can be applied to higher-level formats.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (4)

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

X. Liu, “Twin-wave-based optical transmission with enhanced linear and nonlinear performances,” J. Lightwave Technol. 33(5), 1037–1043 (2015).
[Crossref]

S. T. Le, M. E. McCarthy, N. M. Suibhne, M. A. Z. A. Khateeb, E. Giacoumidis, N. Doran, A. D. Ellis, and S. K. Turitsyn, “Demonstration of phase-conjugated subcarrier coding for fiber nonlinearity compensation in CO-OFDM transmission,” J. Lightwave Technol. 33(11), 2206–2212 (2015).
[Crossref]

2014 (5)

2013 (2)

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7(7), 560–568 (2013).
[Crossref]

Y. Tian, Y. K. Huang, S. Zhang, P. R. Prucnal, and T. Wang, “Demonstration of digital phase-sensitive boosting to extend signal reach for long-haul WDM systems using optical phase-conjugated copy,” Opt. Express 21(4), 5099–5106 (2013).
[Crossref] [PubMed]

2012 (5)

2011 (2)

2010 (2)

2008 (1)

2006 (2)

Aldaya, I.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

Asif, R.

Borowiec, A.

Bosco, G.

Calabro, S.

Carena, A.

Cartledge, J. C.

Chandrasekhar, S.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7(7), 560–568 (2013).
[Crossref]

Chen, X.

Chraplyvy, A. R.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7(7), 560–568 (2013).
[Crossref]

Cotter, D.

Curri, V.

de Waardt, H.

Dong, Z.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Doran, N.

Doran, N. J.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

Dou, L.

Du, L. B.

Ellis, A. D.

Forghieri, F.

Gao, Y.

Ghassemlooy, Z.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

Giacoumidis, E.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

S. T. Le, M. E. McCarthy, N. M. Suibhne, M. A. Z. A. Khateeb, E. Giacoumidis, N. Doran, A. D. Ellis, and S. K. Turitsyn, “Demonstration of phase-conjugated subcarrier coding for fiber nonlinearity compensation in CO-OFDM transmission,” J. Lightwave Technol. 33(11), 2206–2212 (2015).
[Crossref]

Gui, T.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Guiomar, F. P.

Hari, S.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Holtmannspoetter, M.

Hoshida, T.

Huang, Y. K.

Ip, E.

Jansen, S. L.

Jarajreh, M. A.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

Kahn, J. M.

Karar, A. S.

Ke, J. H.

Khateeb, M. A. Z. A.

Khoe, G.-D.

Krummrich, P. M.

Kschischang, F. K.

M. I. Yousefi and F. K. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

Kschischang, F. R.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Kumar, S.

Laperle, C.

Lau, A. P. T.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Le, S. T.

Li, C.

Li, G.

Li, L.

Li, Z.

Liang, X.

Lin, C. Y.

Liu, X.

X. Liu, “Twin-wave-based optical transmission with enhanced linear and nonlinear performances,” J. Lightwave Technol. 33(5), 1037–1043 (2015).
[Crossref]

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7(7), 560–568 (2013).
[Crossref]

Lowery, A. J.

Lu, C.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Luo, M.

McCarthy, M. E.

Morshed, M. M.

O’Sullivan, M.

Pinto, A. N.

Poggiolini, P.

Prilepsky, J. E.

Prucnal, P. R.

Qiu, K.

Rasmussen, J. C.

Reis, J. D.

Roberts, K.

Schmauss, B.

Sharma, D.

Sohler, W.

Spinnler, B.

Suche, H.

Suibhne, N. M.

Tao, Z.

Teixeira, A. L.

Tian, Y.

Tkach, R. W.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7(7), 560–568 (2013).
[Crossref]

Tsokanos, A.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

Turitsyn, S. K.

van den Borne, D.

Wai, P. K. A.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Wang, T.

Wei, X.

Winzer, P. J.

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7(7), 560–568 (2013).
[Crossref]

Yam, S. S.-H.

Yan, W.

Yang, Q.

Yi, X.

Yousefi, M. I.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

M. I. Yousefi and F. K. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

Zhang, S.

Zhao, J.

Zhong, K.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

Zhong, K. P.

Zhu, L.

IEEE Photonics Technol. Lett. (2)

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmission based on NFT,” IEEE Photonics Technol. Lett. 27(15), 1621–1623 (2015).
[Crossref]

IEEE Trans. Inf. Theory (1)

M. I. Yousefi and F. K. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

J. Lightwave Technol. (8)

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, W. Sohler, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra-long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24(1), 54–64 (2006).
[Crossref]

Y. Gao, J. H. Ke, K. P. Zhong, J. C. Cartledge, and S. S.-H. Yam, “Assessment of intrachannel nonlinear compensation for 112 Gb/s dual polarization 16QAM systems,” J. Lightwave Technol. 30(24), 3902–3910 (2012).
[Crossref]

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. 28(4), 423–433 (2010).
[Crossref]

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairment using digital back propagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[Crossref]

X. Liu, “Twin-wave-based optical transmission with enhanced linear and nonlinear performances,” J. Lightwave Technol. 33(5), 1037–1043 (2015).
[Crossref]

S. T. Le, M. E. McCarthy, N. M. Suibhne, M. A. Z. A. Khateeb, E. Giacoumidis, N. Doran, A. D. Ellis, and S. K. Turitsyn, “Demonstration of phase-conjugated subcarrier coding for fiber nonlinearity compensation in CO-OFDM transmission,” J. Lightwave Technol. 33(11), 2206–2212 (2015).
[Crossref]

G. Bosco, V. Curri, A. Carena, P. Poggiolini, and F. Forghieri, “On the performance of Nyquist-WDM Terabit superchannels based on PM-BPSK, PM-QPSK, PM-8QAM or PM-16QAM subcarriers,” J. Lightwave Technol. 29(1), 53–61 (2011).
[Crossref]

Nat. Photonics (1)

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7(7), 560–568 (2013).
[Crossref]

Opt. Express (10)

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express 20(2), 1360–1369 (2012).
[Crossref] [PubMed]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22(22), 26720–26741 (2014).
[Crossref] [PubMed]

X. Yi, X. Chen, D. Sharma, C. Li, M. Luo, Q. Yang, Z. Li, and K. Qiu, “Digital coherent superposition of optical OFDM subcarrier pairs with Hermitian symmetry for phase noise mitigation,” Opt. Express 22(11), 13454–13459 (2014).
[Crossref] [PubMed]

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010).
[Crossref] [PubMed]

L. Zhu and G. Li, “Nonlinearity compensation using dispersion-folded digital backward propagation,” Opt. Express 20(13), 14362–14370 (2012).
[Crossref] [PubMed]

C. Y. Lin, R. Asif, M. Holtmannspoetter, and B. Schmauss, “Nonlinear mitigation using carrier phase estimation and digital backward propagation in coherent QAM transmission,” Opt. Express 20(26), B405–B412 (2012).
[Crossref] [PubMed]

Y. Gao, J. C. Cartledge, A. S. Karar, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping,” Opt. Express 22(2), 1209–1219 (2014).
[Crossref] [PubMed]

L. B. Du, M. M. Morshed, and A. J. Lowery, “Fiber nonlinearity compensation for OFDM super-channels using optical phase conjugation,” Opt. Express 20(18), 19921–19927 (2012).
[Crossref] [PubMed]

Y. Tian, Y. K. Huang, S. Zhang, P. R. Prucnal, and T. Wang, “Demonstration of digital phase-sensitive boosting to extend signal reach for long-haul WDM systems using optical phase-conjugated copy,” Opt. Express 21(4), 5099–5106 (2013).
[Crossref] [PubMed]

X. Liang and S. Kumar, “Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links,” Opt. Express 22(24), 29733–29745 (2014).
[Crossref] [PubMed]

Opt. Lett. (1)

Other (7)

T. Yoshida, T. Sugihara, K. Ishida, and T. Mizuochi, “Spectrally-efficient dual phase-conjugate twin waves with orthogonally multiplexed quadrature pulse-shaped signals,” in Proc. Optical Fiber Communication Conference (2014), paper M.3.C.6.
[Crossref]

M. Koga, A. Mizutori, T. Ohata, and H. Takara, “Optical diversity transmission with signal and its phase-conjugate lights through multi-core fiber,” in Proc. Optical Fiber Communication Conference (2015), paper Th1D.4.
[Crossref]

W. R. Peng, Z. Li, F. Zhu, and Y. Bai, “Training based determination of perturbation coefficients got fiber nonlinearity mitigation,” in Proc. Optical Fiber Communication Conference (2015), paper Th3D.2.

H. Bulow, “Experimental assessment of nonlinear Fourier transformation based detection under fiber nonlinearity,” in Proc. European Conference on Optical Communication (2014), paper We.2.3.2.
[Crossref]

H. Hu, R. M. Jopson, A. Gnauck, M. Dinu, S. Chandrasekhar, X. Liu, C. Xie, M. Montoliu, S. Randel, and C. McKinstrie, “Fiber nonlinearity compensation of an 8-channel WDM PDM-QPSK signal using multiple phase conjugations, ” in Proc. Optical Fiber Communication Conference (2014), paper M3C.2.
[Crossref]

Q. Zhuge, M. Reimer, A. Borowiec, M. O’Sullivan, and D. V. Plant, “Aggressive quantization on perturbation coefficients for nonlinear pre-distortion,” in Proc. Optical Fiber Communication Conference (2014), paper Th4D.7.
[Crossref]

D. Rafique and B. Spinnler, “Fiber nonlinearity compensation: practical use cases and complexity analysis,” in Proc. Optical Fiber Communication Conference (2015), paper Th3D.5.
[Crossref]

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Figures (16)

Fig. 1
Fig. 1 Principle of the conventional PCTW scheme.
Fig. 2
Fig. 2 The principle of the G-PCTW scheme.
Fig. 3
Fig. 3 Principle of the M-PCTW-I scheme.
Fig. 4
Fig. 4 An example to illustrate the encoding of Ey(0,t).
Fig. 5
Fig. 5 Constellation diagram of a path at the receiver, Er,i, when Ex is a QPSK signal and Ea is encoded as (a) 1 or −1; (b) 1, −1, j or –j. Solid dots are the constellation points and the marked dots represent that when these points occur at the output of a path, this path is selected.
Fig. 6
Fig. 6 Principle of the M-PCTW-II scheme.
Fig. 7
Fig. 7 (a) Q factor versus the signal launch power at 15,200 km for 25-, 18.75- and 12.5-Gbaud conventional PDM QPSK, 25-Gbaud PCTW, and 25-Gbaud G-PCTW. (b) Optimal Q factor of G-PCTW versus φ. In (a) and (b), Ea is a constant field exp(jφ) in G-PCTW.
Fig. 8
Fig. 8 Constellation diagrams of (a) 25-Gbaud conventional PDM QPSK (b) 25-Gbaud PCTW and (c) 25-Gbaud G-PCTW. The signal launch power is −1 dBm. Rectangular points represent the incorrectly decoded data.
Fig. 9
Fig. 9 Decoding setup of M-PCTW-I when (a) one bit and (b) two bits are modulated in Ea.
Fig. 10
Fig. 10 Q factor versus the signal launch power when (a) one bit and (b) two bits are modulated in Ea. Squares: the BER is obtained by dividing the errors only in Ex by the total transmitted bits in Ex (the number of symbols × 2).
Fig. 11
Fig. 11 Constellation diagrams of (a) the output of a path; (b) the output of the selection module; (c) the output of the selection module when the decoded Ea is correct (i.e. the selected paths are correct). One bit is modulated in Ea and the signal launch power is −1 dBm.
Fig. 12
Fig. 12 Constellation diagrams of (a) the output of a path; (b) the output of the selection module; (c) the output of the selection module when the decoded Ea is correct (i.e. the selected paths are correct). Two bits are modulated in Ea and the signal launch power is −1 dBm.
Fig. 13
Fig. 13 (a) The optimal Q factor versus the rotated phase for PCTW and Ex in M-PCTW-I when Ea is correctly decoded. (b) The optimal Q factor versus the percentage of dispersion pre-compensation for PCTW and Ex in M-PCTW-I when Ea is correctly decoded.
Fig. 14
Fig. 14 Q factor versus the signal launch power for different schemes when (a) one bit and (b) two bits are modulated in Ea. Diamonds and pluses: the BER is obtained by dividing all errors by the total net transmitted bits excluding the BCH overhead.
Fig. 15
Fig. 15 Q-factor improvement versus the NSE when (a) one bit and (b) two bits are modulated in Ea. Different NSEs are obtained by using the BCH code in Ea with different overhead.
Fig. 16
Fig. 16 Q-factor improvement versus the NSE when (a) one bit and (b) two bits are modulated in Ea. Ex is an 8-QAM signal and the fiber length is 6,400 km

Tables (1)

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Table 1 Outputs of the two receiver paths when Ex is a QPSK signal and Ea is 1 or −1

Equations (17)

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δ E x (or y) (L,ω)=i 9 8 γ P 0 L eff + d ω 1 + d ω 2 η( ω 1 ω 2 ) ×[ E x (or y) (0,ω+ ω 1 ) E x (or y) (0,ω+ ω 2 ) E x (or y) * (0,ω+ ω 1 + ω 2 ) + E y (or x) (0,ω+ ω 1 ) E x (or y) (0,ω+ ω 2 ) E y (or x) * (0,ω+ ω 1 + ω 2 )]
E x (or y) (0,ω)= + E x (or y) (0,t) e iωt dt / 2π
δ E y (L,ω)=δ E x (L,ω) * , δ E y (L,t)=δ E x (L,t) *
E r (t)= E x (L,t)+ E y (L,t) * =2 E x (0,t)
δ E y (L,ω)=δ E x (L,ω) * × E a , δ E y (L,t)=δ E x (L,t) * × E a
E r (t)= E x (L,t)+ ( E y (L,t)/ E a ) * =[ E x (0,t)+δ E x (L,t)]+[ E x (0,t)δ E x (L,t)]=2 E x (0,t)
δ E x (L,t)× E y * (0,t)=δ E y * (L,t)× E x (0,t)
E r (t)= E x (L,t)+ ( E y (L,t)/ E a ) * =2 E x (0,t)+δ E x (L,t)+δ E y * (L,t)/ E a * =2 E x (0,t)
E r,i (t)= E x (L,t)+ ( E y (L,t)/ E a,i ) * , 1iN
Normalized SE (NSE) =( m a /(1+r)+ m x )/(2 m x )
δ E x (or y) (L,t)= + δ E x(or y) (L,ω)exp(2πiωt)dω =i 9 8 γ P 0 L eff + + + η( ω 1 ω 2 )×[ E x(or y) (0,ω+ ω 1 ) E x (or y) (0,ω+ ω 2 ) E x (or y) * (0,ω+ ω 1 + ω 2 ) + E y(or x) (0,ω+ ω 1 ) E x (or y) (0,ω+ ω 2 ) E y (or x) * (0,ω+ ω 1 + ω 2 )]exp(2πiωt)dωd ω 1 d ω 2
δ E x (L,t)× E y * (0,t)= E y * (0,t)×i 9 8 γ P 0 L eff + + + η( ω 1 ω 2 ) exp(2πiωt)dωd ω 1 d ω 2 ×[ E x (0,ω+ ω 1 ) E x (0,ω+ ω 2 ) E x * (0,ω+ ω 1 + ω 2 ) + E y (0,ω+ ω 1 ) E x (0,ω+ ω 2 ) E y * (0,ω+ ω 1 + ω 2 )]
δ E y * (L,t)× E x (0,t)= E x (0,t)×i 9 8 γ P 0 L eff + + + η * ( ω 1 ω 2 )exp(2πiωt)dωd ω 1 d ω 2 ×[ E y * (0,ω+ ω 1 ) E y * (0,ω+ ω 2 ) E y (0,ω+ ω 1 + ω 2 ) + E x * (0,ω+ ω 1 ) E y * (0,ω+ ω 2 ) E x (0,ω+ ω 1 + ω 2 )]
δ E x (L,t)× E y * (0,t)= E y * (0,t)×i 9 8 γ P 0 L eff η + + exp(2πiωt)dωd ω 2 ×[ E x (0,ω) E x (0,ω+ ω 2 ) E x * (0,ω+ ω 2 )+ E y (0,ω) E x (0,ω+ ω 2 ) E y * (0,ω+ ω 2 )] = E y * (0,t)×i 9 8 γ P 0 L eff η + + exp(2πi ω p t)d ω p d ω p,2 ×[ E x (0, ω p ) | E x (0, ω p,2 ) | 2 + E y (0, ω p ) E x (0, ω p,2 ) E y * (0, ω p2 )] =i 9 8 γ P 0 L eff η + d ω p,2 ×[ E x (0,t) E y * (0,t) | E x (0, ω p,2 ) | 2 + | E y (0,t) | 2 E x (0, ω p,2 ) E y * (0, ω p2 )]
δ E y * (L,t)× E x (0,t)= E x (0,t)×i 9 8 γ P 0 L eff η + + exp(2πiωt)dωd ω 2 ×[ E y * (0,ω) E y * (0,ω+ ω 2 ) E y (0,ω+ ω 2 )+ E x * (0,ω) E y * (0,ω+ ω 2 ) E x (0,ω+ ω 2 )] = E x (0,t)×i 9 8 γ P 0 L eff η + + exp(2πi ω p t)d ω p d ω p,2 ×[ E y * (0, ω p ) | E y (0, ω p,2 ) | 2 + E x * (0, ω p ) E y * (0, ω p,2 ) E x (0, ω p,2 )] =i 9 8 γ P 0 L eff η + d ω p,2 ×[ E x (0,t) E y * (0,t) | E y (0, ω p,2 ) | 2 + | E x (0,t) | 2 E x (0, ω p,2 ) E y * (0, ω p,2 )]
δ E x (L,t)× E y * (0,t)= E y * (0,t)×i 9 8 γ P 0 L eff η + + exp(2πiωt)dωd ω 1 ×[ E x (0,ω+ ω 1 ) E x (0,ω) E x * (0,ω+ ω 1 )+ E y (0,ω+ ω 1 ) E x (0,ω) E y * (0,ω+ ω 1 )] = E y * (0,t)×i 9 8 γ P 0 L eff η + + exp(2πi ω p t)d ω p d ω p,1 ×[ E x (0, ω p ) | E x (0, ω p,1 ) | 2 + E x (0, ω p ) | E y (0, ω p,1 ) | 2 ] =i 9 8 γ P 0 L eff η + d ω p,2 ×[ E x (0,t) E y * (0,t) | E x (0, ω p,1 ) | 2 + E x (0,t) E y * (0,t) | E y (0, ω p,1 ) | 2 ]
δ E y * (L,t)× E x (0,t)= E x (0,t)×i 9 8 γ P 0 L eff η + + exp(2πiωt)dωd ω 1 ×[ E y * (0,ω+ ω 1 ) E y * (0,ω) E y (0,ω+ ω 1 )+ E x * (0,ω+ ω 1 ) E y * (0,ω) E x (0,ω+ ω 1 )] = E x (0,t)×i 9 8 γ P 0 L eff η + + exp(2πi ω p t)d ω p d ω p,1 ×[ E y * (0, ω p ) | E y (0, ω p,1 ) | 2 + E y * (0, ω p ) | E x (0, ω p,1 ) | 2 ] =i 9 8 γ P 0 L eff η + d ω p,2 ×[ E x (0,t) E y * (0,t) | E y (0, ω p,1 ) | 2 + E x (0,t) E y * (0,t) | E x (0, ω p,1 ) | 2 ]

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