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Optimal channel spacing of wavelength division multiplexing optical soliton communication systems
I5 August 1995
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
I I9 ( 1995) 4 I45
Optimal channel spacing of wavelength division multiplexing optical soliton communication systems Xiong-Yan Tang, Mee-Koy Chin .Sclzool ofElectricu1
and Electronic Engineering. Nunyang Technological University. Nanyang Avenue, Singyzpore 2263. Singupore Received 23 January
1995; revised version received 1 I April 1995
Abstract
For soliton-based WDM systems with lumped amplifiers, the variance of the timing jitter resulting from soliton collisions is derived. The BER expression of the systems is given by considering soliton collisions and the Gordon-Haus effect. The dependence of the BER on the channel spacing is discussed, and the optimal channel spacing is determined.
1. Introduction
Optical soliton communication has attracted great attention in recent years. This technique has matured considerably since the use of erbium-doped fiber amplifiers (EDFA). The transmission capacity of solitonbased systems can be enhanced through wavelength division multiplexing (WDM) [ 11. For a soliton WDM system with lumped amplifiers (such as EDFAs), there are two main sources of bit error resulting from fluctuations in the arrival times of solitons due to changes in their central frequencies and the group velocity dispersion of the optical fiber. One frequency shift is caused by the addition of ASE noise of the amplifiers to the soliton, which is known as the Gordon-Haus effect [ 31. The other source of frequency fluctuation is soliton collisions, first studied by Mollenauer et al. [ 21. In ultralong-distance systems, a soliton in one channel suffers many collisions with solitons in other channels at random positions. In a uniform lossless fiber, there is no net frequency shift after each collision. However, in a fiber where amplification is provided, or dispersion is varied, periodically. there will be a net shift in the soliton central frequency after 0030.4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(9S)OO319-3
each collision due to the variation in the pulse energy or in the dispersion parameter. The behavior of the frequency shift can be described in terms of two characteristic lengths. One is a collision length between a pair of solitons with angular frequency -t 0 (in soliton units), defined as Lcoll= ~T/DA& or in normalized units, L, =LcO,,lzc = 1.763/n. The second is a perturbation length, Lpert,which is characteristic of the amplification or the dispersion variation. In the above expression, AA is the channel spacing, T is the FWHM pulse width, D is the fiber dispersion, z, = (2/7r)zo, where 3, is the soliton period, and 0 is related to AA through 0= (c/A ‘) H~AA, where t, = r/l .763. The main result of Mollenauer et al. [2] is that the net frequency shift is negligible so long as the collision length is more than twice the perturbation period. However, if the collision length is too long, the central position displacement after each collision is also very large. The adjacent channel spacing stands out as the major parameter on which the collision-induced shift in the soliton center frequency and the soliton center position depend. Clearly, it is also a very important system design parameter for a soliton-based WDM system.
42
X.-Y. Tang, M.-K. Chin /Optics Communications 119 (1995) 4145
However, to our knowledge, the problem of optimal design of channel spacing has not been investigated quantitatively. The aim of this paper is to study the influence of channel spacing on the bit error rate (BER) of soliton-based WDM systems. We first derive the variance of timing jitter resulting from soliton collisions. Then we give the bit error rate expression which takes into account the timing jitter induced by the soliton collisions and by the Gordon-Haus effect. Based on the BER expression, the dependence of the BER on the channel spacing is investigated. Finally, the safe regions and the optimum values for channel spacing are discussed.
where pn = Re v;1>, qn = ImV;I), Z is the normalized transmission distance, and Z, is the distance from the input to the ith collision position. For ultralong-distance transmission systems, a soliton in one channel will suffer hundreds of collisions with the solitons in other channels. Therefore it is reasonable to assume that & is a random variable with uniform distribution between 0 and 27r, so (St;) = 0, where ( ) is the averaging operator. The variance of the time jitter induced by the ith collision then becomes
X 2. Theoretical
(5)
model
When two solitons of different WDM channels collide in a lossy fiber with lumped amplifiers, the net frequency shift has been given in Ref. [ 21, which can be rewritten in the following form:
(1) where C= 16L,lrr’, x=22.8LclL,, L, is the normalized collision length given by L, = 1.763/f&20 is the normalized angle frequency difference of the two solitons, Lp = LFrt/zc is the normalized fiber perturbation period, & = 2rr( @“‘d( z) dz) lL,, Zco,, is the distance from the collision center to the nearest amplifier, cf(z) is the normalized fiber dispersion, and
Since the time jitter contributed by each collision is statistically independent, the total variance of timing jitter is a; = c
((&r’) .
(6)
It is easy to show that n3 C
(p,
sin
nk
+qn
cos
nhA
smh2(nx) (7)
Furthermore, if the average distance between collision centers is f!+ we can write
(2)
G(z) =
2IZ., exp( - 2rz) 1 -exp(
-2rz)
(3)
’
where L, is the normalized amplifier spacing, and r is the normalized fiber loss. The time shift induced by the net frequency shift of the ith collision can be expressed as 6t; = sn,(z-Z;)
XC (p, n
=C(Z-Z;) n3x4
. sinnA +qncosn&,> smh*(nr)
’
(4)
where K is the number of collisions, K=ZIL,,. If the channel spacing is not very small, then K is very large, and we can replace the summation Et=, with the integral 1: dk, obtaining E
(Z-kL,)2=
g
.
(9)
k=l
If the occurrences of ONE and ZERO have the same probability, & is given by & = T/ii& where T is the normalized bit period. In the case of very small channel spacing, the above-mentioned approximation is not appropriate any more, but in this case, the frequency
X.-Y. Tang, M.-K. Chin/Optics Communications 119 (1995) 41-45
shift after collision is also very small because of a large collision length, and the influence of frequency jitter on the system performance will be insignificant. MakinguseofEqs. (5), (7)-(9),Eq. (6) becomes
(10) In addition to the shift in the arrival time induced by the net frequency shift, there is a pure time shift or, in other words, a central position displacement of the soliton induced by the temporary frequency shift during the collision. The pure time shift per collision is given by 1 I.@, so in the worst case where the adjacent channel consists of a sequence of ONE’s, the maximum pure time shift will be &,,,, = 221 To .
(11)
For multi-channel systems, we consider only soliton collisions between every two channels, and neglect three-channel and higher-order interactions. Under this approximation, Eqs. ( 10) and ( 11) may be generalized as
(13) where the subscript lm in Lb, f1 and x indicate that the variables are for channels 1 and m. Similar to single-channel soliton systems, the accumulated ASE noise of the amplifiers will also cause timing jitter, which is well known as the Gordon-Haus effect. This is another source of bit error for solitonbased WDM systems. For a soliton system with lumped amplifiers, the Gordon-Haus jitter can be expressed in real world quantities as [ 31 a&, = 0.1959n,ha~Z’F(G)D/rAeff,
(14)
where n2 is the fiber nonlinear Kerr coefficient, h is the Planck constant, CYis the fiber loss coefficient, /3 is the amplifier noise parameter, D is the fiber dispersion, r is the soliton FWHM, A,, is the fiber effective core area, and F(G) is the noise penalty factor, F(G) = [ (G - 1) /In G] ‘/G, where G is the amplifier power gain. Treating the soliton collision and the Gordon-Haus effect as independent random processes, the total var-
43
iance of the timing jitter is the sum of (T: and a&. bit error rate is then given by
BER=
$iexp(--
The
(15)
$ds,
Q
where
Q=
w- bt,,,/2
(16) ’
2J_
and W is the detection time window.
3. Results and discussion Before discussing the dependence of the timing jitter and the system BER on the channel spacing, we must first determinef, as given by Eq. (2)) which depends on the special form of the fiber energy perturbation and dispersion variation in a particular system. In particular, we assume, as in Ref. [ 21, a system with periodically varying D, such that the normalized dispersion has two constant values between amplifiers, changing from d, to d2 at a distance L,
d,
(O
d2
(L,
, ’
(17)
Hence Eq. (2) becomes fn=
2na 1- exp( -2&J x l-exp( (
2n,
-212,+in+,) + i27mdi
+ exp( - 2I2,
+ in+,) - exp( - 212,) 2IL, + i27md2
where 4, = 27rdlL1/La. In the numerical examples discussed below, the other relevant real world parameters usedare:Z=9000km,L,=L,,,,=40km,D=1.0ps/ nm. km, CX=0.21 dB/km, p= 1.2, Aeff= 35 p,m2, the single-channel bit rate R=4.0 Gbit/s, r= 50 ps, T=250 ps, W=0.7T= 175 ps. Fig. 1 shows the standard variance of timing jitter induced by soliton collisions versus the adjacent channel spacing for the case of 2 and 5 channels, respectively (N is the number of channels). The solid curve
X.-Y. Tang, M.-K. Chin/Optics
44
-
t
I
I 40 aI 5 :;
*\ :
‘\
CT30 .s
.-E 75 20 ul &
10
1
091
<. JI...
Adjacent
,
IJ.
chdnnel
1
.
.
spacing
.
.
..I
1
[nom)
Fig. 1. The standard channel variance of timing jitter by soliton collisions versus the adjacent channel spacing (solid curve: without dispersion variation; dashed curve: with dispersion variation).
is for the case where d(z) is given by Eq. ( 17) with d, = 0.5, d2 = 1.5, L1 = 0.5L,, whereas the dashed curve is obtained by assuming that the fiber dispersion is constant. The difference between the two cases shows that the dispersion variation has significant effect on timing jitter. From Fig. 1, we can see that a, first increases with AA, reaches a maximum, and then decreases as Ah is increased. This result is similar to Fig. 8 of Ref. [ 221, which shows the dependence of the frequency shift on the collision length, where the change in collision length is obtained by changing A,h. When Ah is small, the collision length is large, and the number of collisions is fewer. In general, our results show that, in a multi-channel system, when the collision length (L,) between the two extreme channels (i.e. the two channels with greatest separation) is such that L,> 2L,, then the net frequency shift will be negligibly small. This is a more general result than that first noted in Ref. [ 21 for the case of two channels. This condition defines the “safe” region for AA to make a, negligibly small. Note that this means that the safe region is different for different numbers of channels (N). In the other extreme, where Ah is very large, the collision length is much smaller than Lp (or La).Since in this case the solitons do not sense the perturbation within a collision length, the frequency shift during soliton collision will be very small, and hence a, will also be small. In Fig. 2, we plot BER versus AA. When Ah is smaller, the BER is large due to the large pure time
Communications 119(1995) 4145
shift (At in Eq. ( 11) ). As AA increases, the influence of the pure time shift gets weaker and the effect of the net frequency shift becomes dominant. For sufficiently large Ah, the pattern of BER mainly depends on a,, and thus resembles the behavior in Fig. 1. We note that recently Midrio et al. have also investigated the performance degradation of a two-channel soliton-based WDM system due to soliton collisions through eyediagram simulation [4]. The results in Ref. [ 41 are similar with ours. In the absence of soliton collisions, the BER caused by the Gordon-Haus effect is 6 X lo-l4 in this example. If the BER of less than 10e9 is desired, we can see from Fig. 2 that there exists two safe regions for the adjacent channel spacing Ah. In the case ofN = 2, without dispersion variation, one safe region for Ah is from 0.13 to 2.30 nm. The other is the region beyond 14.3 nm. With dispersion variation, the first safe region shrinks, and ranges from 0.13 to 1.87 nm. The other region is from 3 1.O nm onward. For WDM systems, the maximum total wavelength span allowed for multiplexing is about 20 nm, being limited by the fiber characteristics and the amplifier bandwidth. In order to have more channels for a fixed maximum wavelength span, the adjacent channel spacing should be small. The value of 31.0 nm is thus too large for channel spacing, so the only appropriate region for Ah for N= 2 is the region from 0.13 to 2.30 nm. Similarly, for N = 5, the only safe region is from 0.27 to 0.50 nm without dispersion variation, and smaller with dispersion vari-
-b*2
Adjacent
channel
spacing
;irn)
Fig. 2. Bit error rate versus the adjacent channel spacing (solid curve: without dispersion variation; dashed curve: with dispersion variation).
X.-Y. Tang, M.-K. Chin /Optics Communications I19 (1995) 4145
1.4 0
_ 2 0 0.6
45
and the Gordon-Haus effect is studied in relation to the adjacent channel spacing. The dependence of the optimal channel spacing on the number of channels is also discussed. Finally, we point out that the simple analytical model presented in this paper only provides a guideline to the system design. Due to the complexity of soliton collisions in a real system, numerical simulation as in Ref. [5] is necessary to further understand the behavior of soliton collisions and to more accurately evaluate the timing jitter resulting from soliton collisions.
Acknowledgements Fig. 3. Optimal adjacent channel spacing versus channel number (circles: without dispersion variation; crosses: with dispersion variation )
ation. In general, the larger the number of channels, the narrower is the usable safe region for AA. Within the safe region, there exists an optimal value for channel spacing at which the BER is minimal. Fig. 3 shows the optimal adjacent channel spacing as a function of channel number. The optimal adjacent channel spacing decreases as the number of channels increases. With dispersion variation, the optimal adjacent channel spacing decreases slightly. In conclusion, we have derived the variance of the timing jitter induced by soliton collisions in solitonbased WDM systems with lumped amplifiers. The system bit error rate determined by the soliton collision
The authors would like to thank the reviewer for valuable comments and Prof. Ye Peida for useful discussion. This work was financially supported by the National Science and Technology Board of Singapore.
References [ 1] L.F. Mollenauer, J.P. Gordon and M.N. Islam, IEEE J. Quantum Electron. 22 (1986) 157. [2] L.F. Mollenauer, G. Evangelides and J.P. Gordon, IEEE J. Lightwave Technol. 9 (1991) 362. [ 31 J.P. Gordon and L.F. Mollenauer, IEEE J. Lightwave Technol. 9 (1991) 170. [4] M. Middo, P. France, F. Matera, M. Romagnoli and M. Settembre, Optics Comm. 112 (1994) 283. [ 51 Y. Kodama and S. Wabnitz, Optics Comm. 114 (1995) 395.
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
I I9 ( 1995) 4 I45
Optimal channel spacing of wavelength division multiplexing optical soliton communication systems Xiong-Yan Tang, Mee-Koy Chin .Sclzool ofElectricu1
and Electronic Engineering. Nunyang Technological University. Nanyang Avenue, Singyzpore 2263. Singupore Received 23 January
1995; revised version received 1 I April 1995
Abstract
For soliton-based WDM systems with lumped amplifiers, the variance of the timing jitter resulting from soliton collisions is derived. The BER expression of the systems is given by considering soliton collisions and the Gordon-Haus effect. The dependence of the BER on the channel spacing is discussed, and the optimal channel spacing is determined.
1. Introduction
Optical soliton communication has attracted great attention in recent years. This technique has matured considerably since the use of erbium-doped fiber amplifiers (EDFA). The transmission capacity of solitonbased systems can be enhanced through wavelength division multiplexing (WDM) [ 11. For a soliton WDM system with lumped amplifiers (such as EDFAs), there are two main sources of bit error resulting from fluctuations in the arrival times of solitons due to changes in their central frequencies and the group velocity dispersion of the optical fiber. One frequency shift is caused by the addition of ASE noise of the amplifiers to the soliton, which is known as the Gordon-Haus effect [ 31. The other source of frequency fluctuation is soliton collisions, first studied by Mollenauer et al. [ 21. In ultralong-distance systems, a soliton in one channel suffers many collisions with solitons in other channels at random positions. In a uniform lossless fiber, there is no net frequency shift after each collision. However, in a fiber where amplification is provided, or dispersion is varied, periodically. there will be a net shift in the soliton central frequency after 0030.4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(9S)OO319-3
each collision due to the variation in the pulse energy or in the dispersion parameter. The behavior of the frequency shift can be described in terms of two characteristic lengths. One is a collision length between a pair of solitons with angular frequency -t 0 (in soliton units), defined as Lcoll= ~T/DA& or in normalized units, L, =LcO,,lzc = 1.763/n. The second is a perturbation length, Lpert,which is characteristic of the amplification or the dispersion variation. In the above expression, AA is the channel spacing, T is the FWHM pulse width, D is the fiber dispersion, z, = (2/7r)zo, where 3, is the soliton period, and 0 is related to AA through 0= (c/A ‘) H~AA, where t, = r/l .763. The main result of Mollenauer et al. [2] is that the net frequency shift is negligible so long as the collision length is more than twice the perturbation period. However, if the collision length is too long, the central position displacement after each collision is also very large. The adjacent channel spacing stands out as the major parameter on which the collision-induced shift in the soliton center frequency and the soliton center position depend. Clearly, it is also a very important system design parameter for a soliton-based WDM system.
42
X.-Y. Tang, M.-K. Chin /Optics Communications 119 (1995) 4145
However, to our knowledge, the problem of optimal design of channel spacing has not been investigated quantitatively. The aim of this paper is to study the influence of channel spacing on the bit error rate (BER) of soliton-based WDM systems. We first derive the variance of timing jitter resulting from soliton collisions. Then we give the bit error rate expression which takes into account the timing jitter induced by the soliton collisions and by the Gordon-Haus effect. Based on the BER expression, the dependence of the BER on the channel spacing is investigated. Finally, the safe regions and the optimum values for channel spacing are discussed.
where pn = Re v;1>, qn = ImV;I), Z is the normalized transmission distance, and Z, is the distance from the input to the ith collision position. For ultralong-distance transmission systems, a soliton in one channel will suffer hundreds of collisions with the solitons in other channels. Therefore it is reasonable to assume that & is a random variable with uniform distribution between 0 and 27r, so (St;) = 0, where ( ) is the averaging operator. The variance of the time jitter induced by the ith collision then becomes
X 2. Theoretical
(5)
model
When two solitons of different WDM channels collide in a lossy fiber with lumped amplifiers, the net frequency shift has been given in Ref. [ 21, which can be rewritten in the following form:
(1) where C= 16L,lrr’, x=22.8LclL,, L, is the normalized collision length given by L, = 1.763/f&20 is the normalized angle frequency difference of the two solitons, Lp = LFrt/zc is the normalized fiber perturbation period, & = 2rr( @“‘d( z) dz) lL,, Zco,, is the distance from the collision center to the nearest amplifier, cf(z) is the normalized fiber dispersion, and
Since the time jitter contributed by each collision is statistically independent, the total variance of timing jitter is a; = c
((&r’) .
(6)
It is easy to show that n3 C
(p,
sin
nk
+qn
cos
nhA
smh2(nx) (7)
Furthermore, if the average distance between collision centers is f!+ we can write
(2)
G(z) =
2IZ., exp( - 2rz) 1 -exp(
-2rz)
(3)
’
where L, is the normalized amplifier spacing, and r is the normalized fiber loss. The time shift induced by the net frequency shift of the ith collision can be expressed as 6t; = sn,(z-Z;)
XC (p, n
=C(Z-Z;) n3x4
. sinnA +qncosn&,> smh*(nr)
’
(4)
where K is the number of collisions, K=ZIL,,. If the channel spacing is not very small, then K is very large, and we can replace the summation Et=, with the integral 1: dk, obtaining E
(Z-kL,)2=
g
.
(9)
k=l
If the occurrences of ONE and ZERO have the same probability, & is given by & = T/ii& where T is the normalized bit period. In the case of very small channel spacing, the above-mentioned approximation is not appropriate any more, but in this case, the frequency
X.-Y. Tang, M.-K. Chin/Optics Communications 119 (1995) 41-45
shift after collision is also very small because of a large collision length, and the influence of frequency jitter on the system performance will be insignificant. MakinguseofEqs. (5), (7)-(9),Eq. (6) becomes
(10) In addition to the shift in the arrival time induced by the net frequency shift, there is a pure time shift or, in other words, a central position displacement of the soliton induced by the temporary frequency shift during the collision. The pure time shift per collision is given by 1 I.@, so in the worst case where the adjacent channel consists of a sequence of ONE’s, the maximum pure time shift will be &,,,, = 221 To .
(11)
For multi-channel systems, we consider only soliton collisions between every two channels, and neglect three-channel and higher-order interactions. Under this approximation, Eqs. ( 10) and ( 11) may be generalized as
(13) where the subscript lm in Lb, f1 and x indicate that the variables are for channels 1 and m. Similar to single-channel soliton systems, the accumulated ASE noise of the amplifiers will also cause timing jitter, which is well known as the Gordon-Haus effect. This is another source of bit error for solitonbased WDM systems. For a soliton system with lumped amplifiers, the Gordon-Haus jitter can be expressed in real world quantities as [ 31 a&, = 0.1959n,ha~Z’F(G)D/rAeff,
(14)
where n2 is the fiber nonlinear Kerr coefficient, h is the Planck constant, CYis the fiber loss coefficient, /3 is the amplifier noise parameter, D is the fiber dispersion, r is the soliton FWHM, A,, is the fiber effective core area, and F(G) is the noise penalty factor, F(G) = [ (G - 1) /In G] ‘/G, where G is the amplifier power gain. Treating the soliton collision and the Gordon-Haus effect as independent random processes, the total var-
43
iance of the timing jitter is the sum of (T: and a&. bit error rate is then given by
BER=
$iexp(--
The
(15)
$ds,
Q
where
Q=
w- bt,,,/2
(16) ’
2J_
and W is the detection time window.
3. Results and discussion Before discussing the dependence of the timing jitter and the system BER on the channel spacing, we must first determinef, as given by Eq. (2)) which depends on the special form of the fiber energy perturbation and dispersion variation in a particular system. In particular, we assume, as in Ref. [ 21, a system with periodically varying D, such that the normalized dispersion has two constant values between amplifiers, changing from d, to d2 at a distance L,
d,
(O
d2
(L,
, ’
(17)
Hence Eq. (2) becomes fn=
2na 1- exp( -2&J x l-exp( (
2n,
-212,+in+,) + i27mdi
+ exp( - 2I2,
+ in+,) - exp( - 212,) 2IL, + i27md2
where 4, = 27rdlL1/La. In the numerical examples discussed below, the other relevant real world parameters usedare:Z=9000km,L,=L,,,,=40km,D=1.0ps/ nm. km, CX=0.21 dB/km, p= 1.2, Aeff= 35 p,m2, the single-channel bit rate R=4.0 Gbit/s, r= 50 ps, T=250 ps, W=0.7T= 175 ps. Fig. 1 shows the standard variance of timing jitter induced by soliton collisions versus the adjacent channel spacing for the case of 2 and 5 channels, respectively (N is the number of channels). The solid curve
X.-Y. Tang, M.-K. Chin/Optics
44
-
t
I
I 40 aI 5 :;
*\ :
‘\
CT30 .s
.-E 75 20 ul &
10
1
091
<. JI...
Adjacent
,
IJ.
chdnnel
1
.
.
spacing
.
.
..I
1
[nom)
Fig. 1. The standard channel variance of timing jitter by soliton collisions versus the adjacent channel spacing (solid curve: without dispersion variation; dashed curve: with dispersion variation).
is for the case where d(z) is given by Eq. ( 17) with d, = 0.5, d2 = 1.5, L1 = 0.5L,, whereas the dashed curve is obtained by assuming that the fiber dispersion is constant. The difference between the two cases shows that the dispersion variation has significant effect on timing jitter. From Fig. 1, we can see that a, first increases with AA, reaches a maximum, and then decreases as Ah is increased. This result is similar to Fig. 8 of Ref. [ 221, which shows the dependence of the frequency shift on the collision length, where the change in collision length is obtained by changing A,h. When Ah is small, the collision length is large, and the number of collisions is fewer. In general, our results show that, in a multi-channel system, when the collision length (L,) between the two extreme channels (i.e. the two channels with greatest separation) is such that L,> 2L,, then the net frequency shift will be negligibly small. This is a more general result than that first noted in Ref. [ 21 for the case of two channels. This condition defines the “safe” region for AA to make a, negligibly small. Note that this means that the safe region is different for different numbers of channels (N). In the other extreme, where Ah is very large, the collision length is much smaller than Lp (or La).Since in this case the solitons do not sense the perturbation within a collision length, the frequency shift during soliton collision will be very small, and hence a, will also be small. In Fig. 2, we plot BER versus AA. When Ah is smaller, the BER is large due to the large pure time
Communications 119(1995) 4145
shift (At in Eq. ( 11) ). As AA increases, the influence of the pure time shift gets weaker and the effect of the net frequency shift becomes dominant. For sufficiently large Ah, the pattern of BER mainly depends on a,, and thus resembles the behavior in Fig. 1. We note that recently Midrio et al. have also investigated the performance degradation of a two-channel soliton-based WDM system due to soliton collisions through eyediagram simulation [4]. The results in Ref. [ 41 are similar with ours. In the absence of soliton collisions, the BER caused by the Gordon-Haus effect is 6 X lo-l4 in this example. If the BER of less than 10e9 is desired, we can see from Fig. 2 that there exists two safe regions for the adjacent channel spacing Ah. In the case ofN = 2, without dispersion variation, one safe region for Ah is from 0.13 to 2.30 nm. The other is the region beyond 14.3 nm. With dispersion variation, the first safe region shrinks, and ranges from 0.13 to 1.87 nm. The other region is from 3 1.O nm onward. For WDM systems, the maximum total wavelength span allowed for multiplexing is about 20 nm, being limited by the fiber characteristics and the amplifier bandwidth. In order to have more channels for a fixed maximum wavelength span, the adjacent channel spacing should be small. The value of 31.0 nm is thus too large for channel spacing, so the only appropriate region for Ah for N= 2 is the region from 0.13 to 2.30 nm. Similarly, for N = 5, the only safe region is from 0.27 to 0.50 nm without dispersion variation, and smaller with dispersion vari-
-b*2
Adjacent
channel
spacing
;irn)
Fig. 2. Bit error rate versus the adjacent channel spacing (solid curve: without dispersion variation; dashed curve: with dispersion variation).
X.-Y. Tang, M.-K. Chin /Optics Communications I19 (1995) 4145
1.4 0
_ 2 0 0.6
45
and the Gordon-Haus effect is studied in relation to the adjacent channel spacing. The dependence of the optimal channel spacing on the number of channels is also discussed. Finally, we point out that the simple analytical model presented in this paper only provides a guideline to the system design. Due to the complexity of soliton collisions in a real system, numerical simulation as in Ref. [5] is necessary to further understand the behavior of soliton collisions and to more accurately evaluate the timing jitter resulting from soliton collisions.
Acknowledgements Fig. 3. Optimal adjacent channel spacing versus channel number (circles: without dispersion variation; crosses: with dispersion variation )
ation. In general, the larger the number of channels, the narrower is the usable safe region for AA. Within the safe region, there exists an optimal value for channel spacing at which the BER is minimal. Fig. 3 shows the optimal adjacent channel spacing as a function of channel number. The optimal adjacent channel spacing decreases as the number of channels increases. With dispersion variation, the optimal adjacent channel spacing decreases slightly. In conclusion, we have derived the variance of the timing jitter induced by soliton collisions in solitonbased WDM systems with lumped amplifiers. The system bit error rate determined by the soliton collision
The authors would like to thank the reviewer for valuable comments and Prof. Ye Peida for useful discussion. This work was financially supported by the National Science and Technology Board of Singapore.
References [ 1] L.F. Mollenauer, J.P. Gordon and M.N. Islam, IEEE J. Quantum Electron. 22 (1986) 157. [2] L.F. Mollenauer, G. Evangelides and J.P. Gordon, IEEE J. Lightwave Technol. 9 (1991) 362. [ 31 J.P. Gordon and L.F. Mollenauer, IEEE J. Lightwave Technol. 9 (1991) 170. [4] M. Middo, P. France, F. Matera, M. Romagnoli and M. Settembre, Optics Comm. 112 (1994) 283. [ 51 Y. Kodama and S. Wabnitz, Optics Comm. 114 (1995) 395.