- Email: [email protected]
Analysis of Timing-Jitter Statistics and Bit-Error Rate in Regenerated Soliton Systems
Optical Fiber Technology 5, 301᎐304 Ž1999. Article ID ofte.1999.0305, available online at http:rrwww.idealibrary.com on
Analysis of Timing-Jitter Statistics and Bit-Error Rate in Regenerated Soliton Systems O. Leclerc, E. Desurvire, P. Brindel, and E. Maunand Alcatel Corporate Research Center, Route de Nozay, 91460 Marcoussis, France E-mail: [email protected] Received January 11, 1999
Non-Gaussian soliton timing-jitter statistics after synchronous phaserintensity modulation are analyzed through a simple theory. Corresponding deviations in bit-error rates from Gaussian approximation are derived. 䊚 1999 Academic Press
Regeneration of soliton signals through synchronous phase or intensity modulation ŽPM or IM. causes dramatic reduction of jitter and amplitude noise, therefore enabling unlimited propagation distances w1, 2x. Through perturbation theory and extensive numerical simulations, it was shown that after regeneration, the output statistics of the jitter are non-Gaussian w3, 4x. In this paper, we derive through a simpler analytical method the non-Gaussian probability density functions ŽPDF. for PM or IM. We use them to evaluate the respective deviations in bit-error rates ŽBER. with respect to the Gaussian PDF approximation. We consider soliton signals at rate B reaching the modulator with arrival times t 0 with respect to the center of the modulation Žstandard deviation 0 .. In-line PM with depth ⌬⌽ at synchronous rate B results in a uniform frequency change, hence a velocity shift of the soliton pulses w5x. Propagation over ZR km of fiber with dispersion D induces then a temporal delay ⌬T PM : ⌬T PM Ž t 0 . s
2 c
ZR D⌬⌽max B sin Ž 2 Bt 0 . .
Ž 1.
In-line IM induces asymmetrical loss wtransmittance ⌬ I Ž t .x on the jittered soliton envelopes I Ž t .. Upon propagation, the pulse’s temporal position moves toward a new center of energy delayed by ⌬T IM , which depends on IM depth and is 301 1068-5200r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
302
LECLERC ET AL.
expressed by ⌬T IM Ž t 0 . s
H t⌬ I Ž t . I Ž t y t 0 . dt H I Ž t y t 0 . dt
,
Ž 2.
where ⌬ I Ž t . s 1 q Ž1 y ⌬ I .cosŽ2 Bt .r2. The analytical values of ⌬T IM and ⌬T PM in Eqs. Ž1. and Ž2. are used to determine the output timing-jitter statistics after modulation and propagation. Indeed, we can derive from a basic probability conservation principle a simple relation between input and output PDFs, i.e., X Pout Ž t q ⌬TX . s Pin Ž t .
1 1 q d⌬TX rdt
,
Ž 3.
where ⌬TX Ž X s PM or IM. are the delays defined in Eqs. Ž1. and Ž2.. The output X . PDF Ž Pout is then determined through Eqs. Ž1. ᎐ Ž3. using I Ž t . s sech 2 Ž1.73tr⌬t ., with a soliton width ⌬ t s 1r5B and assuming a Gaussian input PDF PinX Ž t . with deviation 0 . The exact output PDFs are plotted in Fig. 1 for both PM and IM cases. For comparison, Gaussian PDFs with the same as the exact Žnon-Gaussian . PDFs are also shown. In both PMrIM cases, the output deviations are smaller than the input deviations, thus showing a reduction of the timing jitter. The exact output PDFs are non-Gaussian in both cases, confirming results in w3, 4x. For PM, the Gaussian PDF approximation has tails dropping significantly faster than the exact PDF with a fitting over less than 1 decade from the peak. For IM, fitting occurs over more than 3 decades. Timing-jitter reduction is thus significantly overestimated in the PM case, which could make BER predictions overly optimistic.
FIG. 1. Timing-jitter PDF of s 1.55 m, 20 Gbitrs modulated solitons with Ža. ⌬ s 10⬚ PM, ZR s 400 km, D s 0.25 psrnm ⭈ km, Žb. ⌬ I s 10 dB IM for Žc. a Gaussian input-jitter statistics with s 5.0 ps. Dashed lines show results from the Gaussian approximation and plain lines those from our calculations.
ANALYSIS OF TIMING-JITTER STATISTICS
303
Using the exact PDFs for PM and IM in Eq. Ž3., we calculate the corresponding BER through BER X s
yTwr2
Hy⬁
X Pout Ž t . dt q
q⬁
X out
HT r2 P
Ž t . dt.
Ž 4.
w
Figures 2 and 3 show the BER X as a function of PM and IM depths, respectively, for different initial deviations 0 . In both cases, BERs are seen to decrease with
FIG. 2. BER as a function of PM depth in a 20 Gbitrs system of ZR s 400 km with D s 0.25 psrnm ⭈ km for different initial standard deviations 0 . Full lines stand for exact BERs and dashed lines for BERs with Gaussian PDF approximation.
FIG. 3. BER as a function of IM depth in a 20 Gbitrs system for different initial standard deviations 0 . Full lines stand for exact BERs and dashed lines for BERs with Gaussian PDF approximation.
304
LECLERC ET AL.
modulation depths, thus showing the regenerative effect. In the PM case, the Gaussian BER is overly optimistic; for typical system operation Ž ⌬⌽ s 10⬚., we find 2 = 10y1 0 ŽGaussian . vs 2 = 10y6 Žexact. with 0 s 4.0, showing that the Gaussian approximation is no longer usable. For IM, the Gaussian approximation also systematically underestimates the BER, although with error smaller than 1 decade. Through a simple theory, we have determined the timing-jitter PDFs of synchronously modulated solitons. We have shown that Gaussian PDF approximation leads to systematically optimistic predictions of the bit-error rate, although acceptable in the IM case. The result is of great importance for regenerated soliton system design.
REFERENCES w1x H. Kubota and M. Nakazawa, ‘‘Soliton transmission control in time and frequency domains,’’ J. Quantum Electron., vol. 29, no. 7, 218 Ž1993.. w2x M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, ‘‘Experimental demonstration of soliton data transmission over unlimited distances with soliton control in time and frequency domains,’’ Electron. Lett., vol. 29, no. 9, 729 Ž1993.. w3x T. Georges, ‘‘Non-Gaussian timing jitter statistics of controlled solitons,’’ in Proc. Opt. Fiber Commun., San Jose, CA, pp. 232᎐233, paper ThH1, 1996. w4x T. Georges, ‘‘Bit error rate degradation of interacting solitons owing to non-Gaussian statistics,’’ Electron. Lett., vol. 31, no. 14, 1174 Ž1995.. w5x N. J. Smith, N. J. Doran, K. J. Blow, and W. J. Firth, ‘‘Grodon᎐Haus jitter supression using a single phase modulator,’’ Electron. Lett., vol. 30, no. 12, 987 Ž1994..
Analysis of Timing-Jitter Statistics and Bit-Error Rate in Regenerated Soliton Systems O. Leclerc, E. Desurvire, P. Brindel, and E. Maunand Alcatel Corporate Research Center, Route de Nozay, 91460 Marcoussis, France E-mail: [email protected] Received January 11, 1999
Non-Gaussian soliton timing-jitter statistics after synchronous phaserintensity modulation are analyzed through a simple theory. Corresponding deviations in bit-error rates from Gaussian approximation are derived. 䊚 1999 Academic Press
Regeneration of soliton signals through synchronous phase or intensity modulation ŽPM or IM. causes dramatic reduction of jitter and amplitude noise, therefore enabling unlimited propagation distances w1, 2x. Through perturbation theory and extensive numerical simulations, it was shown that after regeneration, the output statistics of the jitter are non-Gaussian w3, 4x. In this paper, we derive through a simpler analytical method the non-Gaussian probability density functions ŽPDF. for PM or IM. We use them to evaluate the respective deviations in bit-error rates ŽBER. with respect to the Gaussian PDF approximation. We consider soliton signals at rate B reaching the modulator with arrival times t 0 with respect to the center of the modulation Žstandard deviation 0 .. In-line PM with depth ⌬⌽ at synchronous rate B results in a uniform frequency change, hence a velocity shift of the soliton pulses w5x. Propagation over ZR km of fiber with dispersion D induces then a temporal delay ⌬T PM : ⌬T PM Ž t 0 . s
2 c
ZR D⌬⌽max B sin Ž 2 Bt 0 . .
Ž 1.
In-line IM induces asymmetrical loss wtransmittance ⌬ I Ž t .x on the jittered soliton envelopes I Ž t .. Upon propagation, the pulse’s temporal position moves toward a new center of energy delayed by ⌬T IM , which depends on IM depth and is 301 1068-5200r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
302
LECLERC ET AL.
expressed by ⌬T IM Ž t 0 . s
H t⌬ I Ž t . I Ž t y t 0 . dt H I Ž t y t 0 . dt
,
Ž 2.
where ⌬ I Ž t . s 1 q Ž1 y ⌬ I .cosŽ2 Bt .r2. The analytical values of ⌬T IM and ⌬T PM in Eqs. Ž1. and Ž2. are used to determine the output timing-jitter statistics after modulation and propagation. Indeed, we can derive from a basic probability conservation principle a simple relation between input and output PDFs, i.e., X Pout Ž t q ⌬TX . s Pin Ž t .
1 1 q d⌬TX rdt
,
Ž 3.
where ⌬TX Ž X s PM or IM. are the delays defined in Eqs. Ž1. and Ž2.. The output X . PDF Ž Pout is then determined through Eqs. Ž1. ᎐ Ž3. using I Ž t . s sech 2 Ž1.73tr⌬t ., with a soliton width ⌬ t s 1r5B and assuming a Gaussian input PDF PinX Ž t . with deviation 0 . The exact output PDFs are plotted in Fig. 1 for both PM and IM cases. For comparison, Gaussian PDFs with the same as the exact Žnon-Gaussian . PDFs are also shown. In both PMrIM cases, the output deviations are smaller than the input deviations, thus showing a reduction of the timing jitter. The exact output PDFs are non-Gaussian in both cases, confirming results in w3, 4x. For PM, the Gaussian PDF approximation has tails dropping significantly faster than the exact PDF with a fitting over less than 1 decade from the peak. For IM, fitting occurs over more than 3 decades. Timing-jitter reduction is thus significantly overestimated in the PM case, which could make BER predictions overly optimistic.
FIG. 1. Timing-jitter PDF of s 1.55 m, 20 Gbitrs modulated solitons with Ža. ⌬ s 10⬚ PM, ZR s 400 km, D s 0.25 psrnm ⭈ km, Žb. ⌬ I s 10 dB IM for Žc. a Gaussian input-jitter statistics with s 5.0 ps. Dashed lines show results from the Gaussian approximation and plain lines those from our calculations.
ANALYSIS OF TIMING-JITTER STATISTICS
303
Using the exact PDFs for PM and IM in Eq. Ž3., we calculate the corresponding BER through BER X s
yTwr2
Hy⬁
X Pout Ž t . dt q
q⬁
X out
HT r2 P
Ž t . dt.
Ž 4.
w
Figures 2 and 3 show the BER X as a function of PM and IM depths, respectively, for different initial deviations 0 . In both cases, BERs are seen to decrease with
FIG. 2. BER as a function of PM depth in a 20 Gbitrs system of ZR s 400 km with D s 0.25 psrnm ⭈ km for different initial standard deviations 0 . Full lines stand for exact BERs and dashed lines for BERs with Gaussian PDF approximation.
FIG. 3. BER as a function of IM depth in a 20 Gbitrs system for different initial standard deviations 0 . Full lines stand for exact BERs and dashed lines for BERs with Gaussian PDF approximation.
304
LECLERC ET AL.
modulation depths, thus showing the regenerative effect. In the PM case, the Gaussian BER is overly optimistic; for typical system operation Ž ⌬⌽ s 10⬚., we find 2 = 10y1 0 ŽGaussian . vs 2 = 10y6 Žexact. with 0 s 4.0, showing that the Gaussian approximation is no longer usable. For IM, the Gaussian approximation also systematically underestimates the BER, although with error smaller than 1 decade. Through a simple theory, we have determined the timing-jitter PDFs of synchronously modulated solitons. We have shown that Gaussian PDF approximation leads to systematically optimistic predictions of the bit-error rate, although acceptable in the IM case. The result is of great importance for regenerated soliton system design.
REFERENCES w1x H. Kubota and M. Nakazawa, ‘‘Soliton transmission control in time and frequency domains,’’ J. Quantum Electron., vol. 29, no. 7, 218 Ž1993.. w2x M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, ‘‘Experimental demonstration of soliton data transmission over unlimited distances with soliton control in time and frequency domains,’’ Electron. Lett., vol. 29, no. 9, 729 Ž1993.. w3x T. Georges, ‘‘Non-Gaussian timing jitter statistics of controlled solitons,’’ in Proc. Opt. Fiber Commun., San Jose, CA, pp. 232᎐233, paper ThH1, 1996. w4x T. Georges, ‘‘Bit error rate degradation of interacting solitons owing to non-Gaussian statistics,’’ Electron. Lett., vol. 31, no. 14, 1174 Ž1995.. w5x N. J. Smith, N. J. Doran, K. J. Blow, and W. J. Firth, ‘‘Grodon᎐Haus jitter supression using a single phase modulator,’’ Electron. Lett., vol. 30, no. 12, 987 Ž1994..