Abstract

We demonstrate that an optical regenerator architecture providing re-amplification, re-shaping, and re-timing based on the principle of spectral shift followed by filtering can lead to bit error ratio improvement of the signal passing through it. This is in contrast with typical regenerators based on the usual principle of power conversion from a transfer function, which are unable to improve the bit error ratio. At first, we provide the theoretical basis that explains this improvement. Then we present the regenerator architecture based on spectral shift followed by filtering and provide experimental evidence of bit error ratio improvement of a noisy signal from 3×10-6 without regenerator to 2×10-10 with regenerator.

© 2006 Optical Society of America

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References

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  1. P. V. Mamyshev, "All-optical data regeneration based on self-phase modulation effect," in Proc. of 24th European Conference on Optical Communication, 1998 (IEE, UK, 1998), pp. 475-476.
  2. T. Her, T. Leng, G. Raybon, J. C. Bouteiller, C. Jorgensen, K. Feder, K. Brar, P. Steinvurzel, D. Patel, N. M. Litchinitser, P. S. Westbrook, L. E. Nelson, C. Headley, and B. J. Eggleton, "Enhanced 40 Gbit/s OTDM receiver sensitivity with all-fiber optical 2R regenerator," in Technical Digest of the Conference on Lasers and Electro-Optics, 2002, (Optical Society of America, Washington DC, 2002), pp. 534-535.
  3. G. Raybon, Y. Su, J. Leuthold, R. J. Essiambre, T. Her, C. Joergensen, P. Steinvurzel, and K. D. K. Feder, "40 Gbit/s Pseudo-linear Transmission Over One Million Kilometers," in Proceeding of the Optical Fiber Conference and Exhibit, 2002, (Optical Society of America and the Laser and Electro-Optics Society, Washington DC, 2002), pp. FD10-1 - FD10-3.
    [CrossRef]
  4. M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
    [CrossRef]
  5. P. Z. Huang, A. Gray, I. Khrushchev, and I. Bennion, "10-Gb/s transmission over 100 Mm of standard fiber using 2R regeneration in an optical loop mirror," IEEE Photon. Technol. Lett. 16, 2526-2528 (2004).
    [CrossRef]
  6. D. Rouvillain, F. Seguineau, L. Pierre, P. Brindel, H. Choumane, G. Aubin, J. L. Oudar, and O. Leclerc, "40 Gbit/s optical 2R regenerator based on passive saturable absorber for WDM long-haul transmission," in Proceeding of the Optical Fiber Conference and Exhibit, 2002, (Optical Society of America and the Laser and Electro-Optics Society, Washington DC, 2002), p. FD11-1.
    [CrossRef]
  7. F. Ohman, S. Bischoff, B. Tromborg, and J. Mork, "Semiconductor devices for all-optical regeneration," in Proceedings of the 5th International Conference on Transparent Optical Networks, 2003, (2003), pp. 41-46.
  8. J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, "All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber," IEEE Photon. Technol. Lett. 17, 423-425 (2005).
    [CrossRef]
  9. M. Rochette, J. N. Kutz, J. L. Blows, D. Moss, J. T. Mok, and B. J. Eggleton, "Bit error ratio improvement with 2R optical regenerators," IEEE Photon. Technol. Lett. 17, 908-910 (2005).
    [CrossRef]
  10. M. Rochette, J. L. Blows, and B. J. Eggleton, "An All-Optical Regenerator that Discriminates Noise from Signal," in Proceedings of the 31st European Conference on Optical Communication, 2005, (IEE, UK, 2005), We2.4.1.
  11. M. Rochette, L. B. Fu, V. Ta’eed, D. J. Moss, and B. J. Eggleton, " 2R Optical Regeneration: An All-Optical Solution for BER Improvement," to be published in J. Sel. Topics in Quant.Electron.  12(6), (2006).
  12. A. Papoulis, and S. U. Pillai, Probability, random variables, and stochastic processes, 4th ed (McGraw Hill science, New York, 2001).

2006 (1)

M. Rochette, L. B. Fu, V. Ta’eed, D. J. Moss, and B. J. Eggleton, " 2R Optical Regeneration: An All-Optical Solution for BER Improvement," to be published in J. Sel. Topics in Quant.Electron.  12(6), (2006).

2005 (2)

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, "All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber," IEEE Photon. Technol. Lett. 17, 423-425 (2005).
[CrossRef]

M. Rochette, J. N. Kutz, J. L. Blows, D. Moss, J. T. Mok, and B. J. Eggleton, "Bit error ratio improvement with 2R optical regenerators," IEEE Photon. Technol. Lett. 17, 908-910 (2005).
[CrossRef]

2004 (2)

M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
[CrossRef]

P. Z. Huang, A. Gray, I. Khrushchev, and I. Bennion, "10-Gb/s transmission over 100 Mm of standard fiber using 2R regeneration in an optical loop mirror," IEEE Photon. Technol. Lett. 16, 2526-2528 (2004).
[CrossRef]

Bennion, I.

P. Z. Huang, A. Gray, I. Khrushchev, and I. Bennion, "10-Gb/s transmission over 100 Mm of standard fiber using 2R regeneration in an optical loop mirror," IEEE Photon. Technol. Lett. 16, 2526-2528 (2004).
[CrossRef]

Benz, A.

M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
[CrossRef]

Blows, J. L.

M. Rochette, J. N. Kutz, J. L. Blows, D. Moss, J. T. Mok, and B. J. Eggleton, "Bit error ratio improvement with 2R optical regenerators," IEEE Photon. Technol. Lett. 17, 908-910 (2005).
[CrossRef]

Cvecek, K.

M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
[CrossRef]

Eggleton, B. J.

M. Rochette, L. B. Fu, V. Ta’eed, D. J. Moss, and B. J. Eggleton, " 2R Optical Regeneration: An All-Optical Solution for BER Improvement," to be published in J. Sel. Topics in Quant.Electron.  12(6), (2006).

M. Rochette, J. N. Kutz, J. L. Blows, D. Moss, J. T. Mok, and B. J. Eggleton, "Bit error ratio improvement with 2R optical regenerators," IEEE Photon. Technol. Lett. 17, 908-910 (2005).
[CrossRef]

Fu, L. B.

M. Rochette, L. B. Fu, V. Ta’eed, D. J. Moss, and B. J. Eggleton, " 2R Optical Regeneration: An All-Optical Solution for BER Improvement," to be published in J. Sel. Topics in Quant.Electron.  12(6), (2006).

Gray, A.

P. Z. Huang, A. Gray, I. Khrushchev, and I. Bennion, "10-Gb/s transmission over 100 Mm of standard fiber using 2R regeneration in an optical loop mirror," IEEE Photon. Technol. Lett. 16, 2526-2528 (2004).
[CrossRef]

Huang, P. Z.

P. Z. Huang, A. Gray, I. Khrushchev, and I. Bennion, "10-Gb/s transmission over 100 Mm of standard fiber using 2R regeneration in an optical loop mirror," IEEE Photon. Technol. Lett. 16, 2526-2528 (2004).
[CrossRef]

Khrushchev, I.

P. Z. Huang, A. Gray, I. Khrushchev, and I. Bennion, "10-Gb/s transmission over 100 Mm of standard fiber using 2R regeneration in an optical loop mirror," IEEE Photon. Technol. Lett. 16, 2526-2528 (2004).
[CrossRef]

Kikuchi, K.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, "All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber," IEEE Photon. Technol. Lett. 17, 423-425 (2005).
[CrossRef]

Kutz, J. N.

M. Rochette, J. N. Kutz, J. L. Blows, D. Moss, J. T. Mok, and B. J. Eggleton, "Bit error ratio improvement with 2R optical regenerators," IEEE Photon. Technol. Lett. 17, 908-910 (2005).
[CrossRef]

Leuchs, G.

M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
[CrossRef]

Meissner, M.

M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
[CrossRef]

Mok, J. T.

M. Rochette, J. N. Kutz, J. L. Blows, D. Moss, J. T. Mok, and B. J. Eggleton, "Bit error ratio improvement with 2R optical regenerators," IEEE Photon. Technol. Lett. 17, 908-910 (2005).
[CrossRef]

Moss, D.

M. Rochette, J. N. Kutz, J. L. Blows, D. Moss, J. T. Mok, and B. J. Eggleton, "Bit error ratio improvement with 2R optical regenerators," IEEE Photon. Technol. Lett. 17, 908-910 (2005).
[CrossRef]

Moss, D. J.

M. Rochette, L. B. Fu, V. Ta’eed, D. J. Moss, and B. J. Eggleton, " 2R Optical Regeneration: An All-Optical Solution for BER Improvement," to be published in J. Sel. Topics in Quant.Electron.  12(6), (2006).

Ozeki, Y.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, "All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber," IEEE Photon. Technol. Lett. 17, 423-425 (2005).
[CrossRef]

Rochette, M.

M. Rochette, L. B. Fu, V. Ta’eed, D. J. Moss, and B. J. Eggleton, " 2R Optical Regeneration: An All-Optical Solution for BER Improvement," to be published in J. Sel. Topics in Quant.Electron.  12(6), (2006).

M. Rochette, J. N. Kutz, J. L. Blows, D. Moss, J. T. Mok, and B. J. Eggleton, "Bit error ratio improvement with 2R optical regenerators," IEEE Photon. Technol. Lett. 17, 908-910 (2005).
[CrossRef]

Schmauss, B.

M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
[CrossRef]

Sponsel, K.

M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
[CrossRef]

Suzuki, J.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, "All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber," IEEE Photon. Technol. Lett. 17, 423-425 (2005).
[CrossRef]

Ta’eed, V.

M. Rochette, L. B. Fu, V. Ta’eed, D. J. Moss, and B. J. Eggleton, " 2R Optical Regeneration: An All-Optical Solution for BER Improvement," to be published in J. Sel. Topics in Quant.Electron.  12(6), (2006).

Taira, K.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, "All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber," IEEE Photon. Technol. Lett. 17, 423-425 (2005).
[CrossRef]

Tanemura, T.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, "All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber," IEEE Photon. Technol. Lett. 17, 423-425 (2005).
[CrossRef]

Weisser, S.

M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
[CrossRef]

Electron (1)

M. Rochette, L. B. Fu, V. Ta’eed, D. J. Moss, and B. J. Eggleton, " 2R Optical Regeneration: An All-Optical Solution for BER Improvement," to be published in J. Sel. Topics in Quant.Electron.  12(6), (2006).

IEEE Photon. Technol. Lett. (4)

M. Meissner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauss, and G. Leuchs, "3.9-dB OSNR gain by an NOLM-based 2-R regenerator," IEEE Photon. Technol. Lett. 16,2105-2107 (2004).
[CrossRef]

P. Z. Huang, A. Gray, I. Khrushchev, and I. Bennion, "10-Gb/s transmission over 100 Mm of standard fiber using 2R regeneration in an optical loop mirror," IEEE Photon. Technol. Lett. 16, 2526-2528 (2004).
[CrossRef]

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, "All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber," IEEE Photon. Technol. Lett. 17, 423-425 (2005).
[CrossRef]

M. Rochette, J. N. Kutz, J. L. Blows, D. Moss, J. T. Mok, and B. J. Eggleton, "Bit error ratio improvement with 2R optical regenerators," IEEE Photon. Technol. Lett. 17, 908-910 (2005).
[CrossRef]

Other (7)

M. Rochette, J. L. Blows, and B. J. Eggleton, "An All-Optical Regenerator that Discriminates Noise from Signal," in Proceedings of the 31st European Conference on Optical Communication, 2005, (IEE, UK, 2005), We2.4.1.

P. V. Mamyshev, "All-optical data regeneration based on self-phase modulation effect," in Proc. of 24th European Conference on Optical Communication, 1998 (IEE, UK, 1998), pp. 475-476.

T. Her, T. Leng, G. Raybon, J. C. Bouteiller, C. Jorgensen, K. Feder, K. Brar, P. Steinvurzel, D. Patel, N. M. Litchinitser, P. S. Westbrook, L. E. Nelson, C. Headley, and B. J. Eggleton, "Enhanced 40 Gbit/s OTDM receiver sensitivity with all-fiber optical 2R regenerator," in Technical Digest of the Conference on Lasers and Electro-Optics, 2002, (Optical Society of America, Washington DC, 2002), pp. 534-535.

G. Raybon, Y. Su, J. Leuthold, R. J. Essiambre, T. Her, C. Joergensen, P. Steinvurzel, and K. D. K. Feder, "40 Gbit/s Pseudo-linear Transmission Over One Million Kilometers," in Proceeding of the Optical Fiber Conference and Exhibit, 2002, (Optical Society of America and the Laser and Electro-Optics Society, Washington DC, 2002), pp. FD10-1 - FD10-3.
[CrossRef]

D. Rouvillain, F. Seguineau, L. Pierre, P. Brindel, H. Choumane, G. Aubin, J. L. Oudar, and O. Leclerc, "40 Gbit/s optical 2R regenerator based on passive saturable absorber for WDM long-haul transmission," in Proceeding of the Optical Fiber Conference and Exhibit, 2002, (Optical Society of America and the Laser and Electro-Optics Society, Washington DC, 2002), p. FD11-1.
[CrossRef]

F. Ohman, S. Bischoff, B. Tromborg, and J. Mork, "Semiconductor devices for all-optical regeneration," in Proceedings of the 5th International Conference on Transparent Optical Networks, 2003, (2003), pp. 41-46.

A. Papoulis, and S. U. Pillai, Probability, random variables, and stochastic processes, 4th ed (McGraw Hill science, New York, 2001).

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of an optical communication link with typical (Class I) optical regenerators. (a) The transmitter sends a signal that travels through several kilometers of fiber and amplification stages before regeneration. Examples of signal in the time domain and their associated probability distribution functions (pdfs) are illustrated. For Class I regenerators, the bit error ratio is the same just before (b) and just after (c) regeneration, as quantified by the area under the pdfs common surface (brown-colored). R: Regenerator, A: amplifier.

Fig. 2.
Fig. 2.

A sample of 10 bits of a noisy signal (left graph) and the probability distribution functions for the logical ones and logical zeros at sampling time (right graph). The sampling time is represented by the red dotted line. The optimal threshold is TO =0.269 W, as seen from the crossing point between pdf1 and pdf0 . The BER of this signal is 4.7×10-9 as measured from the common surface under both pdfs (brown-colored).

Fig. 3.
Fig. 3.

Example of Class I power transfer function (left graph) and pdfs before and after conversion with the power transfer function (right graph). The profile of the pdfs and the optimal threshold have been modified in the conversion but the corresponding bit error ratio remains 4.7×10-9 in both cases.

Fig. 4.
Fig. 4.

Distinct power transfer functions for the logical ones and the logical zeros (upper left graph) and pdfs before and after conversion with the power transfer functions (PTFs) (upper right graph). The pdfs before conversion provide BERII -=4.7×10-9 whereas the pdfs after conversion provide BER+II=3.7×10-10. The origins of this BER improvement come from the power transfer function of the logical ones which is closer to the upper power level, with respect to the power transfer function of the logical zeros. Plot of pdf0, pdf1 and 11(ℑ0(pdf0- )) providing a clear comparison between BER before and BER after conversion with distinct power transfer functions (lower graph). The BER after regeneration is evaluated from the common surface under 11(ℑ0(pdf0- )) and pdf1, BERII +=3.7×10-10.

Fig. 5.
Fig. 5.

Distinct power transfer functions (PTFs) for the logical ones and the logical zeros (upper left graph) leading to bit inversion, and pdfs before and after conversion with the power transfer functions (upper right graph). The pdfs before conversion provide BERII -=4.7×10-9 whereas the pdfs after conversion provide BER+II=3.7×10-10. Plot of pdf0, pdf1 and 01(ℑ0(pdf0- )) providing a graphic comparison between BER before and BER after conversion with distinct power transfer functions (lower graph). The BER after regeneration is evaluated from the common surface under 01(ℑ0(pdf0- )) and pdf1, BER+II=3.7×10-10.

Fig. 6.
Fig. 6.

(a) Schematic of the 3R regenerator based on XPM-induced spectral shift followed by filtering. (b) The low power clock pulses are overlapping with the edge of the high power signal pulses, which induce a spectral shift on the clock. At the output of the regenerator, data have been imprinted at the clock wavelength, and inverted. (c) The clock spectrum is composed of two lobes after XPM (i.e. after HNLF). One lobe at λc is identical to the clock spectrum before XPM and a second lobe at λc’ is a spectrally shifted replica of the clock spectrum. The first lobe is formed by input logical zeros whereas the second lobe is formed by input logical ones. λs/c: Wavelength for signal/clock. P: Power. PSD: Power spectral density.

Fig. 7.
Fig. 7.

(a) Temporal waveforms for ASE noise alone and (b-c) noisy pulses. (b) Pulses with OSNR=10 dB and in (c), pulses with OSNR=20 dB. (d) Pulsewidth fluctuation as a function of OSNR. Calculated from 100 noisy pulses at every OSNR value. The pulsewidth fluctuation is expressed as the standard deviation (σFWHM) divided by the average pulsewidth (FWHM). The spectra of ASE noise and pulses are identically Gaussian with 53 GHz spectral width in all simulations of this figure.

Fig. 8.
Fig. 8.

(a) Experimental setup for the power transfer function measurement. (b) Setup to demonstrate BER improvement. The 3R regenerator is identical to the one shown in Fig. 6. a. VA: Variable attenuator. PMIn/PMOut: Power meter at the input/output of the O3R. BPF: Band-pass filter. PRBS: Pseudo random bit sequence. BERT: Bit error ratio tester.

Fig. 9.
Fig. 9.

(a), power transfer functions for 8 ps pulses, 16 ps pulses, and continuous wave (cw) light. The input and output powers, Pin and Pout, are expressed in terms of peak power. (b) BER with and without the regenerator as a function of OSNR and constant received power. The 3R regenerator improves the BER for constant OSNR or equivalently, the 3R regenerator increases the OSNR margin for a given BER.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

Prob [ a < P E < b ] = a b pdf ( P ) dP
BER = 1 2 0 T pdf 1 ( P ) dP + 1 2 T pdf 0 ( P ) dP
BER = 1 2 0 T o pdf 1 ( P ) dP + 1 2 T o pdf 0 ( P ) dP .
P + = f ( P ) .
pdf + ( P + ) = pdf ( f 1 ( P + ) ) df 1 ( P + ) dP + [ pdf ( P ) ]
BER I + = 1 2 0 T + pdf 1 + ( P + ) dP + + 1 2 T + pdf 0 + ( P + ) dP + .
BER I + = 1 2 0 T o + pdf 1 + ( P + ) dP + + 1 2 T o + pdf 0 + ( P + ) dP + .
BER I + = 1 2 0 T o + [ pdf 1 ( P ) ] dP + + 1 2 0 T o + [ pdf 0 ( P ) ] dP + .
BER I + = 1 2 0 T o + [ pdf 1 ( f 1 [ P + ] ) ] dP + + 1 2 T o + [ pdf 0 ( f 1 [ P + ] ) ] dP + .
BER I + = 1 2 0 f 1 ( T o + ) 1 { [ pdf 1 ( f [ f 1 { P } ] ) ] } dP + 1 2 f 1 ( T o + ) 1 { [ pdf 0 ( f [ f 1 { P } ] ) ] } dP
BER I + = 1 2 0 T o pdf 1 ( P ) dP + 1 2 T o pdf 0 ( P ) dP
= BER I
BER I I + = 1 2 0 T + pdf 1 + ( P + ) dP + + 1 2 T + pdf 0 + ( P + ) dP + .
BER I I + = 1 2 0 T o + pdf 1 + ( P + ) dP + + 1 2 T o + pdf 0 + ( P + ) dP + .
BER I I + = 1 2 0 T o + 1 [ pdf 1 ( P ) ] dP + + 1 2 T o + 0 [ pdf 0 ( P ) ] dP + ,
BER II + = 1 2 0 T o + 1 [ p d f 1 ( f 1 1 [ P + ] ) ] d P + + 1 2 T o + 0 [ p d f 0 ( f 0 1 [ P + ] ) ] d P + .
BER II + = 1 2 0 T o + p d f 1 ( f 1 1 [ P + ] ) d P + + 1 2 T o + 0 1 { 0 [ p d f 0 ( f 0 1 [ P + ] ) ] } d P + .
BER II + = 1 2 0 f 1 1 ( T o + ) p d f 1 ( P ) d P + 1 2 f 1 1 ( T o + ) 1 1 { 0 [ p d f 0 ( f 0 1 [ f 1 { P } ] ) ] } d P .
BER II > BER II + ,
T o p d f 0 ( P ) d P > T o 1 1 { 0 [ p d f 0 ( f 0 1 [ f 1 { P } ] ) ] } d P .
δ ω XPM ( t ) = 2 γ L Eff dP ( t ) dt

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