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Mid-span compensation of nonlinear phase noise
Optics Communications 245 (2005) 391–398 www.elsevier.com/locate/optcom
Mid-span compensation of nonlinear phase noise Keang-Po Ho
*
Department of Electrical Engineering, Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan Received 2 May 2004; received in revised form 25 August 2004; accepted 6 September 2004
Abstract The nonlinear phase noise, often called the Gordon–Mollenauer effect, is correlated with the intensity of the optical signal. With a single linear compensator, the optimal location is not preceding the receiver but located at about 2/3 of the fiber link. When N compensators are approximately optimally placed, the standard deviation of nonlinear phase noise is reduced by a factor of 2N + 1. The optimal compensation factors are all approximately equal to each other. 2004 Elsevier B.V. All rights reserved. PACS: 42.65.k; 05.40.a; 42.81.Uv Keywords: Nonlinear phase noise; Fiber nonlinearities; Noise compensation
1. Introduction Gordon and Mollenauer [1] showed that when optical amplifiers are used in lightwave communication systems to compensate for fiber loss, the interaction of amplifier noises and the fiber Kerr effect causes nonlinear phase noise, often called the Gordon–Mollenauer effect, or more precisely, self-phase modulation induced nonlinear phase noise. Directly added to the signal phase, nonlinear phase noise degrades the performance of both phase-shifted keying (PSK) and differential phaseshift keying (DPSK) signals [2–8]. This class of *
Tel.: +886223635251; fax: +886223683824. E-mail address: [email protected].
constant-intensity modulation has received renewed attention recently for long-haul [9–13] or spectrally efficiency transmissions [14–18]. Because the nonlinear phase noise is correlated with the optical intensity, the optical intensity can be used to correct part of the nonlinear phase noise [7,19–22]. Compensated preceding or together with the receiver, the transmission distance can be approximately doubled if the nonlinear phase noise is the dominant impairment [19,22]. The simplest nonlinear phase noise compensator is a linear compensator in which the correction term is proportional to the received intensity [7,19–22]. In term of variance, linear compensator performs almost the same as the optimal nonlinear compensator designed by the minimum mean-
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.10.009
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K.-P. Ho / Optics Communications 245 (2005) 391–398
squared error (MMSE) criterion without linearity constraint [23]. In all previous literatures, the linear compensator precedes the receiver and implemented using nonlinear all-optical device [19], phase modulator [20,21], or electronic circuits [22,23]. As shown later, even with a single compensator, mid-span compensation performs better than compensation preceding the receiver. With a single compensator, the optimal position is about 2/3 from the beginning of the fiber link. With N compensators, the optimal positions are about 2k/(2N + 1), k = 1,2, . . . ,N. Fig. 1 shows a linear compensator of nonlinear phase noise using a phase modulator [20,21]. Part of the signal is received by a photodetector and amplified by a trans-impedance amplifier (TIA), and applied to a phase modulator followed a driver. The phase modulator applies a correction to the signal phase. The amplification factor of the received signal is determined by the optimal compensation factor derived in [22] or derived later in this communication. Nonlinear compensator can be implemented using a nonlinear driver amplifier [23]. While most external modulators are polarization sensitive, polarization independent modulator is also available [24,25]. If polarization dependent modulator is used in Fig. 1, polarization control is required. Unlike the electronic compensator of [22,23], the phase-modulator based compensator of Fig. 1 can be used in the middle of the fiber link. When more than one compensator are used, the performance of the system can be further improved. With a compensator preceding the receiver, the variance of the residual nonlinear phase noise can be reduced to about a quarter of the variance of nonPhase Mod. tap
TIA
Driver
Fig. 1. The compensation of nonlinear phase noise using a phase modulator.
linear phase noise without compensation [19,22]. Here in this communication, the optimal linear compensator with mid-span compensation is derived analytically, to our knowledge, the first time. The compensator can be applied to both return-tozero (RZ) and non-return-to-zero (NRZ) signals. The same approach can also be used to reduce the phase variance of a soliton signal [26–30].
2. The optimal linear compensator The optimal linear compensator is derived analytical here. The model of nonlinear phase noise is first provided according to the distributed model of [5,31]. The optimal linear compensator is derived using the covariance between optical intensity and nonlinear phase noise. 2.1. Model for nonlinear phase noise The linear electric field at the kth fiber span is equal to E k ¼ E 0 þ n1 þ þ nk ;
ð1Þ
where E0 is the transmitted signal, nk, k = 1, . . . ,N, are the optical amplifier noise introduced into the system at the kth fiber span, nk are independent identically distributed (i.i.d.) complex zero-mean circular Gaussian random variables with E{|nk|2} = 2r2, where r2 is the noise variance per dimension per span. For an N-span system, the total nonlinear phase noise is [1,19,22,23] h 2 2 UNL ¼ cLeff jE0 þ n1 j þ jE0 þ n1 þ n2 j i 2 þ þ jE0 þ n1 þ þ nN j ; ð2Þ where c is the fiber nonlinear coefficient and Leff is the effective length per span. Similar to the approach of [5,31,30], the distributed model of nonlinear phase noise is assumed for a fiber link having infinite number of spans. Valid for systems having more than 32 fiber spans [5,6,31], the distributed model distributes the amplifier noise and nonlinear phase noise over the fiber link. Mathematically, the distributed model replaces all summation in (1) and (2) with integration. When the amplifier noise is approxi-
K.-P. Ho / Optics Communications 245 (2005) 391–398
mated by a complex Wiener variance with unit pffiffiffiffiffi variance of b(t), the received signal is qs þ bð1Þ 2 with the same SNR as EN of qs = |E0| /(2Nr2). Valid for either a constant intensity signal or soliton, the normalized nonlinear phase noise is Z 1 pffiffiffiffiffi U¼ j qs þ bðtÞj2 dt: ð3Þ 0
In (3), the transmitted signal is assumed to be a pffiffiffiffiffi pffiffiffiffiffi real signal of qs . Because EN and qs þ bð1Þ have the same SNR, the actual received signal pffiffiffiffiffi ENejUNL is proportional to ½ qs þbð1ÞejbU ; 1 where b ¼ hUNL i=ðqs þ 2Þ is the normalization factor to scale the normalized nonlinear phase noise to the actual nonlinear phase noise with mean nonlinear phase shift of ÆUNLæ [5,31]. From (A.6), hUi ¼ qs þ 12. The normalization factor is the ratio of the mean nonlinear phase shift of ÆUNLæ to ÆUæ. The mean nonlinear phase shift of ÆUNLæ has a simple definition for both NRZ and flat-top RZ signal [22]. In soliton system, as shown in [30], the mean nonlinear phase shift is half of an equivalent NRZ system having a mean power equal to the peak power of the soliton.
If only one compensator is used, the compensator can be located anywhere in the fiber link to optimize the system performance. The performance of the compensator in arbitrary location is analyzed here. Using the compensation factor of a, the optimal compensator minimizes the variance of the residual nonlinear phase noise of pffiffiffiffiffi 2 Ur ¼ U aj qs þ bðsÞj ; ð4Þ where s is the location of the compensator. The variance of the residual nonlinear phase noise is r2Ur ¼ r2U þ a2 jðs; sÞ 2afðsÞ;
The variance of (5) is a quadratic function of a, the optimal compensation factor is given by aopt ¼
ð5Þ
where j( Æ , Æ ) and f( Æ ) are defined by (A.7) and (A.9), respectively, and r2U is the variance of nonlinear phase noise given by (A.10). The covariance of optical intensity at s and t is given by j(s,t) and the covariance of optical intensity at s with the normalized nonlinear phase noise of (3) is given by f(s).
fðsÞ q sð2 sÞ þ s2 ð1 2s=3Þ ¼ s : jðs; sÞ 2qs s þ s2
ð6Þ
If s = 1, the optical compensation factor of aopt is the same as that of [5] when the compensator precedes the receiver. If s = 0, there is no nonlinear phase noise compensation. For high SNR of qs, the optimal compensation factor is approximately equal to aopt ! 1 12s. When the compensator precedes the receiver with s = 1, the approximation of aopt ! 12 was found in both [19,22] in high SNR. When the optimal compensation factor of (6) is substituted to the variance of (5), we obtain r2Ur ;min ¼ r2U
f2 ðsÞ : jðs; sÞ
ð7Þ
The optimal location of the compensator is given by solving dr2Ur ;min =ds ¼ 0 to obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 3 9qs 9 þ 42qs sopt ¼ þ 1 1þ 8 8 81q2s
2.2. Single compensator
393
2 2 þ : 3 91qs
ð8Þ
In most value of SNR, the optimal location is about 2/3 from the beginning of the fiber link. At high SNR of qs, the ratio of the variance of residual nonlinear phase noise to that of nonlinear phase noise of (A.10) is equal to r2Ur ;min 3 ! 1 3s þ 3s2 s3 4 r2U
ð9Þ
with a minimum value of 19 when sopt ! 23. Fig. 2 shows the ratio of standard deviation (STD) of residual nonlinear phase noise rRES to the mean nonlinear phase shift ÆUNLæ. The STD of residual nonlinear phase noise is given by rRES ¼ brUr without normalization. Fig. 2 shows rRES/ÆUNLæ when the compensator is located at s = 1/4, 1/2 and 3/4 from the beginning of fiber link, or preceding the receiver of s = 1. Fig. 2 also shows rRES/ÆUNLæ without compensation and without optimal location of s = sopt of (8). From Fig. 2, the same as that in [19,22], a compensator preceding the receiver about halves the
394
K.-P. Ho / Optics Communications 245 (2005) 391–398 0.4
w/o
RES
STD σ
com p
s=
0.2 5 1 s= 0.5
NL
/<Φ >
0.2
s=
0.1 0.08
s=
0.7
5 s
0.06
opt
0.04
0.02 5
10
15
20
SNR ρ (dB) s
Fig. 2. The ratio of STD to mean of nonlinear phase noise as a function of SNR qs when the compensator located at various locations.
STD of the nonlinear phase noise. From Fig. 2, a compensator at the optimal location of s = sopt of (8) reduces the STD of nonlinear phase noise to about 1/3 of the case without compensation. To illustrate the effects of compensator location to the performance of the nonlinear phase noise compensation scheme, Fig. 3 shows the ratio of rRES/ÆUNLæ as a function of compensator location for several values of SNR qs. The residual nonlinear phase noise reduces when a compensator is used in the fiber link. For a location before about
0.2
ρ = 10 dB
STD σRES/<ΘNL>
s
0.1 0.08
20
16 18
14
12
0.06 0.04
0.02 0
0.2
0.4
0.6
0.8
1
Normalized Location, s
Fig. 3. The ratio of STD to mean of nonlinear phase noise as a function of compensator location.
2/3 of the fiber link, the STD of residual nonlinear phase noise decreases with the location of the compensator and reaches a value of about 1/3 of the value of STD without compensation when s = 0. For a location after about 2/3 of the fiber link, the STD of residual nonlinear phase noise increases with the location of the compensator and reaches a value of about 1/2 of the value of STD without compensation when the compensator precedes the receiver. Mid-span compensation is obviously better than compensation preceding the receiver. The nonlinear phase noise of (3) is an integration of pffiffiffiffiffi the noisy power of j qs þ bðtÞj2 over distance of t. As an example, the mid-span intensity of pffiffiffiffiffi j qs þ bð1=2Þj2 has less noise than the received pffiffiffiffiffi 2 intensity of j qs þ bð1Þj . However, the nonlinear phase R 1 pffiffiffiffiffi noise 2 after the compensator of j q þ bðtÞj dt can still be compensated using 1=2 ffiffiffiffiffi s p 2 j qs þ bð1=2Þj . Because the intensity of pffiffiffiffiffi 2 j qs þ bð1=2Þj is not suitable to compensate the nonlinear phase noise at the end of the fiber link, R 1 pffiffiffiffi ffi i.e., 0:9 j qs þ bðtÞj2 dt, the optimal location needs to balance the usage of less noisy optical intensity and the ability to compensate the nonlinear phase noise at the end of the fiber link. The optimal location of about 2/3 is the balance between those two factors. The optimal compensation factor is aopt ! 2/3 at high SNR when s = 2/3. From the same length fiber interval, the nonlinear phase R 1 of pffiffiffiffiffi noiseRof 0:9 j qs þ bðtÞj2 dt is significantly larger 0:1 pffiffiffiffiffi than 0:0 j qs þ bðtÞj2 dt, the optimal compensator location is not at the middle of s = 1/2 but s = 2/3 toward the end of the fiber link. The compensation factor of (6) minimizes the variance of residual nonlinear phase noise at the end of the fiber link. The compensator may increase the phase noise immediately at the output of the compensator. However, at the end of the fiber link, the residual nonlinear phase noise is definitely minimized. Fig. 4 shows the ratio of rRES/ ÆUNLæ as a function of normalized distance when a compensator is located at s = 1/2, 2/3 and 3/4. Without compensation, the STD of nonlinear phase noise as a function of distance is given by (A.11). With compensation, the STD of residual nonlinear phase noise as a function of distance is given by (A.13). When the amplifier noise is accu-
K.-P. Ho / Optics Communications 245 (2005) 391–398
where the second term is the covariance of the pffiffiffiffiffi 2 intensity at sk of j qs þ bðsk Þj with the nonlinear phase noise of (3) and the third term is the covariance of the intensity at sk and sl. The functions of j( Æ , Æ ) and f( Æ ) can be found in Appendix A. The optimal compensator factors can be found by solving all equations of or2Ur =oak ¼ 0, or
0.18
w/o comp
0.16
0.08
RES
NL
/<Φ >
STD σ
0.14 0.12
s = 1/2 3/4
0.1
N X
2/3
0.06
395
al jðsk ; sl Þ ¼ fðsk Þ;
k ¼ 1; . . . ; N :
ð12Þ
0.04
l¼1
0.02
In the case of N = 1, the solution is the same as that of (6). There are N linear equations in (12) to determine the N optimal compensation factors of ak, k = 1, . . . ,N. The optimal compensation factors are equal to
0 0
0.2
0.4
0.6
0.8
1
Normalized Distance
Fig. 4. The ratio of STD to mean of nonlinear phase noise as a function of normalized distance when a compensator is located at s = 1/2, 2/3 and 3/4.
mulated with distance, the nonlinear phase noise is increased faster at the end of the fiber link. From Fig. 4, the STD of nonlinear phase noise is the largest at the compensator output for both s = 1/2 and 2/3. When the compensator is located at s = 1/2, the compensator actually increases its output phase noise. When the compensator is located at s = 3/4, the compensator reduces its output phase noise. However, in all cases of Fig. 4, the compensator minimizes the nonlinear phase noise at the end of the fiber link. 2.3. Multiple compensators If there are N compensators located at s1, s2, . . . ,sN, the optimal compensation factors of a1,a2, . . . ,aN minimize the variance of the residual nonlinear phase noise of Ur ¼ U
N X
pffiffiffiffiffi 2 ak j qs þ bðsk Þj :
ð10Þ
k¼1
The variance of the residual nonlinear phase noise is equal to r2Ur ¼ r2U 2
N X
ak fðsk Þ
k¼1
þ
N X N X k¼1
l¼1
ak al jðsk ; sl Þ;
ð11Þ
~ Z; aopt ¼ K1~
ð13Þ
where the covariance matrix has elements of Kkl ¼ jðsk ; sl Þ and the correlation vector is T ~ Z ¼ ½fðs1 Þ; fðs2 Þ; . . . ; fðsN Þ . The solution of (13) is the same as that for most MMSE algorithms [32, p. 626]. Fig. 5 shows the ratio of rRES/ÆUNLæ as a function of SNR qs when multiple compensators are used. In Fig. 5(a), the N compensators are evenly located at sk = k/N, k = 1,2, . . . ,N, including one compensator preceding the receiver of sN = 1. In Fig. 5(b), the N compensators are evenly located at sk = 2k/(2N + 1), k = 1,2, . . . N. The compensator spacings in Fig. 5(a) and (b) are 1/N and 2/ (2N + 1), respectively. The two curves in Fig. 2 without compensation and with one compensator of s = 1 are also shown in Fig. 5 for comparison. From Fig. 5(a), the STD of nonlinear phase noise is reduced by a factor of about 2N. From Fig. 5(b), the STD of nonlinear phase noise is reduced by a factor of about 2N + 1. Because the intensity at sN = 1 has larger noise, better performance is achieved when sN = 1 1/(2N + 1). In Fig. 3, the STD of the residual nonlinear phase noise is very close to it minimum value for a large range of locations from s = 0.6 to s = 0.75. Because the compensation scheme is optimized over a large range of locations, the locations of sk = 2k/ (2N + 1) are very close to the optimal locations. The optimal compensation factor of ~ aopt can also be evaluated numerically. With the compen-
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K.-P. Ho / Optics Communications 245 (2005) 391–398
3. Discussion
STD σRES/< Θ NL>
w/o c
omp
N=1
–1
10
2 3 4 6 8 –2
10
5
10
(a)
SNR ρs (dB)
15
20
15
20
w/o c
omp
STD σRES/<Θ NL>
–1
N=1
10
2 3 4 6 8 –2
10
5
(b)
10
SNR ρ (dB) s
Fig. 5. The ratio of STD to mean of nonlinear phase noise as a function of SNR qs when the multiple compensators are used: (a) sk = k/N, k = 1, . . . ,N; (b) sk = 2k/(2N + 1), k = 1, . . . ,N.
sator location of sk = k/N, the optimal compensation factors are aopt,k ! 1/N, k = 1, . . . ,N 1, and aopt,N ! 1/2N. With the compensator locations of sk = 2k/(2N + 1), the optimal compensation factors all approach aopt,k ! 2/(2N + 1), k = 1, . . . ,N, that is the same as the compensator spacing. With the compensator locations of sk = 2k/(2N + 1), all compensators have approximately the same compensation factor. From Fig. 5, when sufficient number of compensators are used, nonlinear phase noise can be completely eliminated. We may infer that if one compensator is used per fiber span in the system of [22], nonlinear phase noise can be completely eliminated.
When N compensators are used in optimal locations, the STD of nonlinear phase noise is reduced by a factor of about 2N + 1. If nonlinear phase noise is the dominant noise source, the transmission distance can be increased by a factor of 2N + 1. However, the amplifier noises modeled by the Wiener process of b(t) also limits the transmission distance by adding noise into the signal for system with more fiber spans. With higher tolerance of nonlinear phase noise, the launched power to the system can be increased accordingly. Assumed the same amplifier noises, the system SNR is increased with the launched power. If the operation point is estimated when the variances of linear and nonlinear phase noise are equal [1], the overall transmission pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distance can be increased by a factor of 2N þ 1. The results here are very similar to that in [33] using phase conjugation. With a single compensator, the optimal location is both at 2/3 the fiber link and the STD is both reduced by a factor of 1/3. However, the techniques and methods for analysis are different. Instead of phase conjugator in [33], a phase modulator is used here instead. For example, while a phase conjugator is usually located in the mid-span, a phase-modulator based compensator is usually preceding the receiver. While the analysis of [33] is only for soliton, the method here is applicable to more general signal. The inclusion of compensators in the fiber link requires additional amplification to compensate for the compensator loss. If the compensator is properly located, for example, in the mid-stage of an amplifier, compensator loss gives minimal increase to the amplifier noise figure and overall amplifier noise to the system. The compensator of Fig. 1 operates on only one single channel. In a wavelength-divisionmultiplexed (WDM) system, each WDM channel requires separated compensator. A compensator with the receiver or preceding the receiver requires no additional WDM demultiplexer and multiplexer. Compared with [33], only one phase conjugator is required for all WDM channels.
K.-P. Ho / Optics Communications 245 (2005) 391–398
The model of nonlinear phase noise of (2) is valid for systems in which the pulse along the fiber does not have large distortion. For highly dispersive systems with large pulse broadening, the systems may be limited by pulse-to-pulse interaction instead of nonlinear phase noise.
397
where pffiffiffiffiffi pffiffiffiffiffi T ~ ~ ¼ ð qs ; 0; qs ; 0ÞT , x ¼ ðxs1 ; xs2 ; xt1 ; xt2 Þ ; m and 2 3 s 0 minðs; tÞ 0 0 s 0 minðs; tÞ 7 16 6 7 C¼ 6 7: 5 2 4 minðs; tÞ 0 t 0 0
minðs; tÞ
0
t
4. Conclusion
ðA:3Þ
With a single compensator, the optimal placement of the compensator is not preceding the receiver but at about 2/3 of the fiber link. With this optimal location, a single compensator reduces the STD of nonlinear phase noise by a factor of 3 instead of a factor of 2 for a compensator preceding the receiver. The optimal compensation factor is equal 2/3. When N compensators are approximately optimally placed at 2k/(2N + 1) of the fiber link, the STD of nonlinear phase noise is reduced by a factor of 2N + 1. The optimal compensation factors for all compensators are all approximately equal to 2/(2N + 1).
where 12 minðs; tÞ is the covariance between Xs and Xt, Ys and Yt. pffiffiffiffiffi The mean of j qs þ bðsÞj2 is equal to 2 2 mjpffiffiffi qs þbðsÞj2 ¼ EfX s þ Y s g
o2 W ðx ; 0; 0; 0Þ X s ;Y s ;X t ;Y t s1 2 oxs1 xs1 ¼0 o2 2 WX s ;Y s ;X t ;Y t ð0; xs2 ; 0; 0Þ oxs2 xs2 ¼0
¼
¼ qs þ s:
ðA:4Þ
Similarly, we can find mjAþbðtÞj2 ¼ qs þ t;
ðA:5Þ
and Appendix A. Covariance between the intensity of Wiener process
hUi ¼
Z 0
Here is this appendix, we will derive the autocovariance between the intensity of a complex Wiener process. For a complex Wiener process of b(s) with unity variance, we will find n pffiffiffiffiffi 2 jðs; tÞ ¼ E j qs þ bðsÞj mjpffiffiffi qs þbðsÞj2 pffiffiffiffiffi o 2 j qs þ bðtÞj mjpffiffiffi ; ðA:1Þ 2 qs þbðtÞj where mjpffiffiffi 2 and m pffiffiffi 2 are the mean valqp j qs þbðtÞj s þbðsÞj ffiffiffiffiffi pffiffiffiffiffi 2 2 ues of j qs þ bðsÞj and j qs þ bðtÞj , respecpffiffiffiffiffi pffiffiffiffiffi tively. If qs þ bðsÞ ¼ X s þ jY s and qs þ bðtÞ ¼ X t þ jY t , Xs, Ys, Xt, Yt are multivariate Gaussian random variables with characteristic function of
ðA:2Þ
1 mjAþbðtÞj2 dt ¼ qs þ : 2
ðA:6Þ
The covariance of (A.1) is jðs; tÞ ¼ EfX 2s X 2t þ Y 2s Y 2t g EfX 2s gEfX 2t g EfY 2s gEfY 2t g o4 W ðx ; 0; x ; 0Þ ¼ 2 X ;Y ;X ;Y s1 t1 s s t t oxs1 ox2t1 xs1 ¼0;xt1 ¼0 4 o þ 2 W ð0; x ; 0; x Þ X ;Y ;X ;Y s2 t2 s s t t oxs2 ox2t2 xs2 ¼0;xt2 ¼0 t s ts qs þ qs þ 2 2 4 ðA:7Þ ¼ 2qs minðs; tÞ þ minðs; tÞ2 :
pffiffiffiffiffi 2 Using (A.7), the variance of j qs þ bðsÞj is r2jpffiffiffi ¼ jðs; sÞ ¼ 2qs s þ s2 : q þbðsÞj2 s
WX s ;Y s ;X t ;Y t ðxs1 ; xs2 ; xt1 ; xt2 Þ 1 T T ~ C~ ¼ exp j~ xT ~ x ; m x 2
1
ðA:8Þ
For the normalized nonlinear phase noise U defined in (3), the covariance between U and the pffiffiffiffiffi intensity of j qs þ bðsÞj2 is
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K.-P. Ho / Optics Communications 245 (2005) 391–398
n pffiffiffiffiffi 2 fðsÞ ¼ E j qs þ bðsÞj qs s Z 1 1 pffiffiffiffiffi 2 j qs þ bðtÞj dt qs 2 0 Z 1 2s 2 ¼ jðs; tÞ dt ¼ qs sð2 sÞ þ s 1 : 3 0 ðA:9Þ The variance of the normalized nonlinear phase noise is Z 1 2q 1 2 ðA:10Þ rU ¼ fðsÞ dt ¼ s þ 6 3 0 the same as that in [5,34]. For Fig. 4, for a compensator located at s with a compensator factor of a, the variance of the residual nonlinear phase noise as a function of distance is equal to r2U ðtÞ þ a2 jðs; sÞ 2aft ðsÞ;
t P s;
ðA:11Þ
where ft ðsÞ ¼
Z
t
jðs; t1 Þ dt1
ðA:12Þ
0
is the correlation of the nonlinear phase noise at t with the intensity at s, and r2U ðtÞ is the variance of the normalized nonlinear phase noise as a function of distance that is equal to Z t 2q t4 r2U ðtÞ ¼ ðA:13Þ ft ðsÞ ds ¼ s t3 þ : 3 6 0
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[8] T. Mizuochi, et al., J. Lightwave Technol. 21 (2003) 1933. [9] A.H. Gnauck, et al., Proceedings of the OFC02, Optical Society of America, Washington, DC, 2002 (postdeadline paper FC2). [10] C. Rasmussen, et al., Proceedings of the OFC03, Optical Society of America, Washington, DC, 2003 (postdeadline paper PD18). [11] B. Zhu, et al., J. Lightwave Technol. 22 (2004) 209. [12] J.-X. Cai, et al., Proceedings of the OFC04, Optical Society of America, Washington, DC, 2004 (postdeadline paper PDP34). [13] A.H. Gnauck, et al., Proceedings of the OFC04, Optical Society of America, Washington, DC, 2004 (postdeadline paper PDP35). [14] H. Kim, R.-J. Essiambre, Photon. Technol. Lett. 15 (2003) 769. [15] P.S. Cho, et al., IEEE Photon. Technol. Lett. 15 (2003) 473. [16] C. Wree, et al., IEEE Photon. Technol. Lett. 15 (2003) 1303. [17] Y. Zhu, et al., Proceedings of the OFC04, Optical Society of America, Washington, DC, 2004 (paper TuF1). [18] N. Yoshikane, I. Morita, Proceedings of the OFC04, Optical Society of America, Washington, DC, 2004 (postdeadline paper PDP38). [19] X. Liu, X. Wei, R.E. Slusher, C.J. McKinstrie, Opt. Lett. 27 (2002) 1616. [20] C. Xu, X. Liu, Opt. Lett. 27 (2002) 1619. [21] C. Xu, L.F. Mollenauer, X. Liu, Electron. Lett. 38 (2002) 1578. [22] K.-P. Ho, J.M. Kahn, J. Lightwave Technol. 22 (2004) 779. [23] K.-P. Ho, Opt. Commun. 211 (2003) 419. [24] Y. Chen, J.E. Zucker, N.J. Sauer, T.Y. Chang, IEEE Photon. Technol. Lett. 4 (1992) 1120. [25] M. Sugiyama, Proceedings of the OFC03, Optical Society of America, Washington, DC, 2003 (paper FP1). [26] K. Blow, N. Doran, S. Phoenix, Opt. Commun. 88 (1992) 137. [27] M. Hanna, H. Porte, J.-P. Goedgebuer, W.T. Rhodes, IEEE J. Quantum Electron. 36 (2000) 1333. [28] M. Hanna, H. Porte, J.-P. Goedgebuer, W.T. Rhodes, Electron. Lett. 37 (2001) 644. [29] C.J. McKinstrie, C. Xie, IEEE J. Sel. Top. Quantum Electron. 8 (2002) 616, erratum 956. [30] K.-P. Ho, J. Opt. Soc. Am. B 21 (2004) 266. [31] K.-P. Ho, Opt. Lett. 28 (2003) 1350. [32] J.G. Proakis, Digital Communications, fourth ed., McGraw-Hill, New York, 2000. [33] C.J. McKinstrie, S. Radic, C. Xie, Opt. Lett. 2 (2003) 1519. [34] K.-P. Ho, IEEE Photon. Technol. Lett. 16 (2004) 1403.
Mid-span compensation of nonlinear phase noise Keang-Po Ho
*
Department of Electrical Engineering, Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan Received 2 May 2004; received in revised form 25 August 2004; accepted 6 September 2004
Abstract The nonlinear phase noise, often called the Gordon–Mollenauer effect, is correlated with the intensity of the optical signal. With a single linear compensator, the optimal location is not preceding the receiver but located at about 2/3 of the fiber link. When N compensators are approximately optimally placed, the standard deviation of nonlinear phase noise is reduced by a factor of 2N + 1. The optimal compensation factors are all approximately equal to each other. 2004 Elsevier B.V. All rights reserved. PACS: 42.65.k; 05.40.a; 42.81.Uv Keywords: Nonlinear phase noise; Fiber nonlinearities; Noise compensation
1. Introduction Gordon and Mollenauer [1] showed that when optical amplifiers are used in lightwave communication systems to compensate for fiber loss, the interaction of amplifier noises and the fiber Kerr effect causes nonlinear phase noise, often called the Gordon–Mollenauer effect, or more precisely, self-phase modulation induced nonlinear phase noise. Directly added to the signal phase, nonlinear phase noise degrades the performance of both phase-shifted keying (PSK) and differential phaseshift keying (DPSK) signals [2–8]. This class of *
Tel.: +886223635251; fax: +886223683824. E-mail address: [email protected].
constant-intensity modulation has received renewed attention recently for long-haul [9–13] or spectrally efficiency transmissions [14–18]. Because the nonlinear phase noise is correlated with the optical intensity, the optical intensity can be used to correct part of the nonlinear phase noise [7,19–22]. Compensated preceding or together with the receiver, the transmission distance can be approximately doubled if the nonlinear phase noise is the dominant impairment [19,22]. The simplest nonlinear phase noise compensator is a linear compensator in which the correction term is proportional to the received intensity [7,19–22]. In term of variance, linear compensator performs almost the same as the optimal nonlinear compensator designed by the minimum mean-
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.10.009
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squared error (MMSE) criterion without linearity constraint [23]. In all previous literatures, the linear compensator precedes the receiver and implemented using nonlinear all-optical device [19], phase modulator [20,21], or electronic circuits [22,23]. As shown later, even with a single compensator, mid-span compensation performs better than compensation preceding the receiver. With a single compensator, the optimal position is about 2/3 from the beginning of the fiber link. With N compensators, the optimal positions are about 2k/(2N + 1), k = 1,2, . . . ,N. Fig. 1 shows a linear compensator of nonlinear phase noise using a phase modulator [20,21]. Part of the signal is received by a photodetector and amplified by a trans-impedance amplifier (TIA), and applied to a phase modulator followed a driver. The phase modulator applies a correction to the signal phase. The amplification factor of the received signal is determined by the optimal compensation factor derived in [22] or derived later in this communication. Nonlinear compensator can be implemented using a nonlinear driver amplifier [23]. While most external modulators are polarization sensitive, polarization independent modulator is also available [24,25]. If polarization dependent modulator is used in Fig. 1, polarization control is required. Unlike the electronic compensator of [22,23], the phase-modulator based compensator of Fig. 1 can be used in the middle of the fiber link. When more than one compensator are used, the performance of the system can be further improved. With a compensator preceding the receiver, the variance of the residual nonlinear phase noise can be reduced to about a quarter of the variance of nonPhase Mod. tap
TIA
Driver
Fig. 1. The compensation of nonlinear phase noise using a phase modulator.
linear phase noise without compensation [19,22]. Here in this communication, the optimal linear compensator with mid-span compensation is derived analytically, to our knowledge, the first time. The compensator can be applied to both return-tozero (RZ) and non-return-to-zero (NRZ) signals. The same approach can also be used to reduce the phase variance of a soliton signal [26–30].
2. The optimal linear compensator The optimal linear compensator is derived analytical here. The model of nonlinear phase noise is first provided according to the distributed model of [5,31]. The optimal linear compensator is derived using the covariance between optical intensity and nonlinear phase noise. 2.1. Model for nonlinear phase noise The linear electric field at the kth fiber span is equal to E k ¼ E 0 þ n1 þ þ nk ;
ð1Þ
where E0 is the transmitted signal, nk, k = 1, . . . ,N, are the optical amplifier noise introduced into the system at the kth fiber span, nk are independent identically distributed (i.i.d.) complex zero-mean circular Gaussian random variables with E{|nk|2} = 2r2, where r2 is the noise variance per dimension per span. For an N-span system, the total nonlinear phase noise is [1,19,22,23] h 2 2 UNL ¼ cLeff jE0 þ n1 j þ jE0 þ n1 þ n2 j i 2 þ þ jE0 þ n1 þ þ nN j ; ð2Þ where c is the fiber nonlinear coefficient and Leff is the effective length per span. Similar to the approach of [5,31,30], the distributed model of nonlinear phase noise is assumed for a fiber link having infinite number of spans. Valid for systems having more than 32 fiber spans [5,6,31], the distributed model distributes the amplifier noise and nonlinear phase noise over the fiber link. Mathematically, the distributed model replaces all summation in (1) and (2) with integration. When the amplifier noise is approxi-
K.-P. Ho / Optics Communications 245 (2005) 391–398
mated by a complex Wiener variance with unit pffiffiffiffiffi variance of b(t), the received signal is qs þ bð1Þ 2 with the same SNR as EN of qs = |E0| /(2Nr2). Valid for either a constant intensity signal or soliton, the normalized nonlinear phase noise is Z 1 pffiffiffiffiffi U¼ j qs þ bðtÞj2 dt: ð3Þ 0
In (3), the transmitted signal is assumed to be a pffiffiffiffiffi pffiffiffiffiffi real signal of qs . Because EN and qs þ bð1Þ have the same SNR, the actual received signal pffiffiffiffiffi ENejUNL is proportional to ½ qs þbð1ÞejbU ; 1 where b ¼ hUNL i=ðqs þ 2Þ is the normalization factor to scale the normalized nonlinear phase noise to the actual nonlinear phase noise with mean nonlinear phase shift of ÆUNLæ [5,31]. From (A.6), hUi ¼ qs þ 12. The normalization factor is the ratio of the mean nonlinear phase shift of ÆUNLæ to ÆUæ. The mean nonlinear phase shift of ÆUNLæ has a simple definition for both NRZ and flat-top RZ signal [22]. In soliton system, as shown in [30], the mean nonlinear phase shift is half of an equivalent NRZ system having a mean power equal to the peak power of the soliton.
If only one compensator is used, the compensator can be located anywhere in the fiber link to optimize the system performance. The performance of the compensator in arbitrary location is analyzed here. Using the compensation factor of a, the optimal compensator minimizes the variance of the residual nonlinear phase noise of pffiffiffiffiffi 2 Ur ¼ U aj qs þ bðsÞj ; ð4Þ where s is the location of the compensator. The variance of the residual nonlinear phase noise is r2Ur ¼ r2U þ a2 jðs; sÞ 2afðsÞ;
The variance of (5) is a quadratic function of a, the optimal compensation factor is given by aopt ¼
ð5Þ
where j( Æ , Æ ) and f( Æ ) are defined by (A.7) and (A.9), respectively, and r2U is the variance of nonlinear phase noise given by (A.10). The covariance of optical intensity at s and t is given by j(s,t) and the covariance of optical intensity at s with the normalized nonlinear phase noise of (3) is given by f(s).
fðsÞ q sð2 sÞ þ s2 ð1 2s=3Þ ¼ s : jðs; sÞ 2qs s þ s2
ð6Þ
If s = 1, the optical compensation factor of aopt is the same as that of [5] when the compensator precedes the receiver. If s = 0, there is no nonlinear phase noise compensation. For high SNR of qs, the optimal compensation factor is approximately equal to aopt ! 1 12s. When the compensator precedes the receiver with s = 1, the approximation of aopt ! 12 was found in both [19,22] in high SNR. When the optimal compensation factor of (6) is substituted to the variance of (5), we obtain r2Ur ;min ¼ r2U
f2 ðsÞ : jðs; sÞ
ð7Þ
The optimal location of the compensator is given by solving dr2Ur ;min =ds ¼ 0 to obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 3 9qs 9 þ 42qs sopt ¼ þ 1 1þ 8 8 81q2s
2.2. Single compensator
393
2 2 þ : 3 91qs
ð8Þ
In most value of SNR, the optimal location is about 2/3 from the beginning of the fiber link. At high SNR of qs, the ratio of the variance of residual nonlinear phase noise to that of nonlinear phase noise of (A.10) is equal to r2Ur ;min 3 ! 1 3s þ 3s2 s3 4 r2U
ð9Þ
with a minimum value of 19 when sopt ! 23. Fig. 2 shows the ratio of standard deviation (STD) of residual nonlinear phase noise rRES to the mean nonlinear phase shift ÆUNLæ. The STD of residual nonlinear phase noise is given by rRES ¼ brUr without normalization. Fig. 2 shows rRES/ÆUNLæ when the compensator is located at s = 1/4, 1/2 and 3/4 from the beginning of fiber link, or preceding the receiver of s = 1. Fig. 2 also shows rRES/ÆUNLæ without compensation and without optimal location of s = sopt of (8). From Fig. 2, the same as that in [19,22], a compensator preceding the receiver about halves the
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K.-P. Ho / Optics Communications 245 (2005) 391–398 0.4
w/o
RES
STD σ
com p
s=
0.2 5 1 s= 0.5
NL
/<Φ >
0.2
s=
0.1 0.08
s=
0.7
5 s
0.06
opt
0.04
0.02 5
10
15
20
SNR ρ (dB) s
Fig. 2. The ratio of STD to mean of nonlinear phase noise as a function of SNR qs when the compensator located at various locations.
STD of the nonlinear phase noise. From Fig. 2, a compensator at the optimal location of s = sopt of (8) reduces the STD of nonlinear phase noise to about 1/3 of the case without compensation. To illustrate the effects of compensator location to the performance of the nonlinear phase noise compensation scheme, Fig. 3 shows the ratio of rRES/ÆUNLæ as a function of compensator location for several values of SNR qs. The residual nonlinear phase noise reduces when a compensator is used in the fiber link. For a location before about
0.2
ρ = 10 dB
STD σRES/<ΘNL>
s
0.1 0.08
20
16 18
14
12
0.06 0.04
0.02 0
0.2
0.4
0.6
0.8
1
Normalized Location, s
Fig. 3. The ratio of STD to mean of nonlinear phase noise as a function of compensator location.
2/3 of the fiber link, the STD of residual nonlinear phase noise decreases with the location of the compensator and reaches a value of about 1/3 of the value of STD without compensation when s = 0. For a location after about 2/3 of the fiber link, the STD of residual nonlinear phase noise increases with the location of the compensator and reaches a value of about 1/2 of the value of STD without compensation when the compensator precedes the receiver. Mid-span compensation is obviously better than compensation preceding the receiver. The nonlinear phase noise of (3) is an integration of pffiffiffiffiffi the noisy power of j qs þ bðtÞj2 over distance of t. As an example, the mid-span intensity of pffiffiffiffiffi j qs þ bð1=2Þj2 has less noise than the received pffiffiffiffiffi 2 intensity of j qs þ bð1Þj . However, the nonlinear phase R 1 pffiffiffiffiffi noise 2 after the compensator of j q þ bðtÞj dt can still be compensated using 1=2 ffiffiffiffiffi s p 2 j qs þ bð1=2Þj . Because the intensity of pffiffiffiffiffi 2 j qs þ bð1=2Þj is not suitable to compensate the nonlinear phase noise at the end of the fiber link, R 1 pffiffiffiffi ffi i.e., 0:9 j qs þ bðtÞj2 dt, the optimal location needs to balance the usage of less noisy optical intensity and the ability to compensate the nonlinear phase noise at the end of the fiber link. The optimal location of about 2/3 is the balance between those two factors. The optimal compensation factor is aopt ! 2/3 at high SNR when s = 2/3. From the same length fiber interval, the nonlinear phase R 1 of pffiffiffiffiffi noiseRof 0:9 j qs þ bðtÞj2 dt is significantly larger 0:1 pffiffiffiffiffi than 0:0 j qs þ bðtÞj2 dt, the optimal compensator location is not at the middle of s = 1/2 but s = 2/3 toward the end of the fiber link. The compensation factor of (6) minimizes the variance of residual nonlinear phase noise at the end of the fiber link. The compensator may increase the phase noise immediately at the output of the compensator. However, at the end of the fiber link, the residual nonlinear phase noise is definitely minimized. Fig. 4 shows the ratio of rRES/ ÆUNLæ as a function of normalized distance when a compensator is located at s = 1/2, 2/3 and 3/4. Without compensation, the STD of nonlinear phase noise as a function of distance is given by (A.11). With compensation, the STD of residual nonlinear phase noise as a function of distance is given by (A.13). When the amplifier noise is accu-
K.-P. Ho / Optics Communications 245 (2005) 391–398
where the second term is the covariance of the pffiffiffiffiffi 2 intensity at sk of j qs þ bðsk Þj with the nonlinear phase noise of (3) and the third term is the covariance of the intensity at sk and sl. The functions of j( Æ , Æ ) and f( Æ ) can be found in Appendix A. The optimal compensator factors can be found by solving all equations of or2Ur =oak ¼ 0, or
0.18
w/o comp
0.16
0.08
RES
NL
/<Φ >
STD σ
0.14 0.12
s = 1/2 3/4
0.1
N X
2/3
0.06
395
al jðsk ; sl Þ ¼ fðsk Þ;
k ¼ 1; . . . ; N :
ð12Þ
0.04
l¼1
0.02
In the case of N = 1, the solution is the same as that of (6). There are N linear equations in (12) to determine the N optimal compensation factors of ak, k = 1, . . . ,N. The optimal compensation factors are equal to
0 0
0.2
0.4
0.6
0.8
1
Normalized Distance
Fig. 4. The ratio of STD to mean of nonlinear phase noise as a function of normalized distance when a compensator is located at s = 1/2, 2/3 and 3/4.
mulated with distance, the nonlinear phase noise is increased faster at the end of the fiber link. From Fig. 4, the STD of nonlinear phase noise is the largest at the compensator output for both s = 1/2 and 2/3. When the compensator is located at s = 1/2, the compensator actually increases its output phase noise. When the compensator is located at s = 3/4, the compensator reduces its output phase noise. However, in all cases of Fig. 4, the compensator minimizes the nonlinear phase noise at the end of the fiber link. 2.3. Multiple compensators If there are N compensators located at s1, s2, . . . ,sN, the optimal compensation factors of a1,a2, . . . ,aN minimize the variance of the residual nonlinear phase noise of Ur ¼ U
N X
pffiffiffiffiffi 2 ak j qs þ bðsk Þj :
ð10Þ
k¼1
The variance of the residual nonlinear phase noise is equal to r2Ur ¼ r2U 2
N X
ak fðsk Þ
k¼1
þ
N X N X k¼1
l¼1
ak al jðsk ; sl Þ;
ð11Þ
~ Z; aopt ¼ K1~
ð13Þ
where the covariance matrix has elements of Kkl ¼ jðsk ; sl Þ and the correlation vector is T ~ Z ¼ ½fðs1 Þ; fðs2 Þ; . . . ; fðsN Þ . The solution of (13) is the same as that for most MMSE algorithms [32, p. 626]. Fig. 5 shows the ratio of rRES/ÆUNLæ as a function of SNR qs when multiple compensators are used. In Fig. 5(a), the N compensators are evenly located at sk = k/N, k = 1,2, . . . ,N, including one compensator preceding the receiver of sN = 1. In Fig. 5(b), the N compensators are evenly located at sk = 2k/(2N + 1), k = 1,2, . . . N. The compensator spacings in Fig. 5(a) and (b) are 1/N and 2/ (2N + 1), respectively. The two curves in Fig. 2 without compensation and with one compensator of s = 1 are also shown in Fig. 5 for comparison. From Fig. 5(a), the STD of nonlinear phase noise is reduced by a factor of about 2N. From Fig. 5(b), the STD of nonlinear phase noise is reduced by a factor of about 2N + 1. Because the intensity at sN = 1 has larger noise, better performance is achieved when sN = 1 1/(2N + 1). In Fig. 3, the STD of the residual nonlinear phase noise is very close to it minimum value for a large range of locations from s = 0.6 to s = 0.75. Because the compensation scheme is optimized over a large range of locations, the locations of sk = 2k/ (2N + 1) are very close to the optimal locations. The optimal compensation factor of ~ aopt can also be evaluated numerically. With the compen-
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3. Discussion
STD σRES/< Θ NL>
w/o c
omp
N=1
–1
10
2 3 4 6 8 –2
10
5
10
(a)
SNR ρs (dB)
15
20
15
20
w/o c
omp
STD σRES/<Θ NL>
–1
N=1
10
2 3 4 6 8 –2
10
5
(b)
10
SNR ρ (dB) s
Fig. 5. The ratio of STD to mean of nonlinear phase noise as a function of SNR qs when the multiple compensators are used: (a) sk = k/N, k = 1, . . . ,N; (b) sk = 2k/(2N + 1), k = 1, . . . ,N.
sator location of sk = k/N, the optimal compensation factors are aopt,k ! 1/N, k = 1, . . . ,N 1, and aopt,N ! 1/2N. With the compensator locations of sk = 2k/(2N + 1), the optimal compensation factors all approach aopt,k ! 2/(2N + 1), k = 1, . . . ,N, that is the same as the compensator spacing. With the compensator locations of sk = 2k/(2N + 1), all compensators have approximately the same compensation factor. From Fig. 5, when sufficient number of compensators are used, nonlinear phase noise can be completely eliminated. We may infer that if one compensator is used per fiber span in the system of [22], nonlinear phase noise can be completely eliminated.
When N compensators are used in optimal locations, the STD of nonlinear phase noise is reduced by a factor of about 2N + 1. If nonlinear phase noise is the dominant noise source, the transmission distance can be increased by a factor of 2N + 1. However, the amplifier noises modeled by the Wiener process of b(t) also limits the transmission distance by adding noise into the signal for system with more fiber spans. With higher tolerance of nonlinear phase noise, the launched power to the system can be increased accordingly. Assumed the same amplifier noises, the system SNR is increased with the launched power. If the operation point is estimated when the variances of linear and nonlinear phase noise are equal [1], the overall transmission pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distance can be increased by a factor of 2N þ 1. The results here are very similar to that in [33] using phase conjugation. With a single compensator, the optimal location is both at 2/3 the fiber link and the STD is both reduced by a factor of 1/3. However, the techniques and methods for analysis are different. Instead of phase conjugator in [33], a phase modulator is used here instead. For example, while a phase conjugator is usually located in the mid-span, a phase-modulator based compensator is usually preceding the receiver. While the analysis of [33] is only for soliton, the method here is applicable to more general signal. The inclusion of compensators in the fiber link requires additional amplification to compensate for the compensator loss. If the compensator is properly located, for example, in the mid-stage of an amplifier, compensator loss gives minimal increase to the amplifier noise figure and overall amplifier noise to the system. The compensator of Fig. 1 operates on only one single channel. In a wavelength-divisionmultiplexed (WDM) system, each WDM channel requires separated compensator. A compensator with the receiver or preceding the receiver requires no additional WDM demultiplexer and multiplexer. Compared with [33], only one phase conjugator is required for all WDM channels.
K.-P. Ho / Optics Communications 245 (2005) 391–398
The model of nonlinear phase noise of (2) is valid for systems in which the pulse along the fiber does not have large distortion. For highly dispersive systems with large pulse broadening, the systems may be limited by pulse-to-pulse interaction instead of nonlinear phase noise.
397
where pffiffiffiffiffi pffiffiffiffiffi T ~ ~ ¼ ð qs ; 0; qs ; 0ÞT , x ¼ ðxs1 ; xs2 ; xt1 ; xt2 Þ ; m and 2 3 s 0 minðs; tÞ 0 0 s 0 minðs; tÞ 7 16 6 7 C¼ 6 7: 5 2 4 minðs; tÞ 0 t 0 0
minðs; tÞ
0
t
4. Conclusion
ðA:3Þ
With a single compensator, the optimal placement of the compensator is not preceding the receiver but at about 2/3 of the fiber link. With this optimal location, a single compensator reduces the STD of nonlinear phase noise by a factor of 3 instead of a factor of 2 for a compensator preceding the receiver. The optimal compensation factor is equal 2/3. When N compensators are approximately optimally placed at 2k/(2N + 1) of the fiber link, the STD of nonlinear phase noise is reduced by a factor of 2N + 1. The optimal compensation factors for all compensators are all approximately equal to 2/(2N + 1).
where 12 minðs; tÞ is the covariance between Xs and Xt, Ys and Yt. pffiffiffiffiffi The mean of j qs þ bðsÞj2 is equal to 2 2 mjpffiffiffi qs þbðsÞj2 ¼ EfX s þ Y s g
o2 W ðx ; 0; 0; 0Þ X s ;Y s ;X t ;Y t s1 2 oxs1 xs1 ¼0 o2 2 WX s ;Y s ;X t ;Y t ð0; xs2 ; 0; 0Þ oxs2 xs2 ¼0
¼
¼ qs þ s:
ðA:4Þ
Similarly, we can find mjAþbðtÞj2 ¼ qs þ t;
ðA:5Þ
and Appendix A. Covariance between the intensity of Wiener process
hUi ¼
Z 0
Here is this appendix, we will derive the autocovariance between the intensity of a complex Wiener process. For a complex Wiener process of b(s) with unity variance, we will find n pffiffiffiffiffi 2 jðs; tÞ ¼ E j qs þ bðsÞj mjpffiffiffi qs þbðsÞj2 pffiffiffiffiffi o 2 j qs þ bðtÞj mjpffiffiffi ; ðA:1Þ 2 qs þbðtÞj where mjpffiffiffi 2 and m pffiffiffi 2 are the mean valqp j qs þbðtÞj s þbðsÞj ffiffiffiffiffi pffiffiffiffiffi 2 2 ues of j qs þ bðsÞj and j qs þ bðtÞj , respecpffiffiffiffiffi pffiffiffiffiffi tively. If qs þ bðsÞ ¼ X s þ jY s and qs þ bðtÞ ¼ X t þ jY t , Xs, Ys, Xt, Yt are multivariate Gaussian random variables with characteristic function of
ðA:2Þ
1 mjAþbðtÞj2 dt ¼ qs þ : 2
ðA:6Þ
The covariance of (A.1) is jðs; tÞ ¼ EfX 2s X 2t þ Y 2s Y 2t g EfX 2s gEfX 2t g EfY 2s gEfY 2t g o4 W ðx ; 0; x ; 0Þ ¼ 2 X ;Y ;X ;Y s1 t1 s s t t oxs1 ox2t1 xs1 ¼0;xt1 ¼0 4 o þ 2 W ð0; x ; 0; x Þ X ;Y ;X ;Y s2 t2 s s t t oxs2 ox2t2 xs2 ¼0;xt2 ¼0 t s ts qs þ qs þ 2 2 4 ðA:7Þ ¼ 2qs minðs; tÞ þ minðs; tÞ2 :
pffiffiffiffiffi 2 Using (A.7), the variance of j qs þ bðsÞj is r2jpffiffiffi ¼ jðs; sÞ ¼ 2qs s þ s2 : q þbðsÞj2 s
WX s ;Y s ;X t ;Y t ðxs1 ; xs2 ; xt1 ; xt2 Þ 1 T T ~ C~ ¼ exp j~ xT ~ x ; m x 2
1
ðA:8Þ
For the normalized nonlinear phase noise U defined in (3), the covariance between U and the pffiffiffiffiffi intensity of j qs þ bðsÞj2 is
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K.-P. Ho / Optics Communications 245 (2005) 391–398
n pffiffiffiffiffi 2 fðsÞ ¼ E j qs þ bðsÞj qs s Z 1 1 pffiffiffiffiffi 2 j qs þ bðtÞj dt qs 2 0 Z 1 2s 2 ¼ jðs; tÞ dt ¼ qs sð2 sÞ þ s 1 : 3 0 ðA:9Þ The variance of the normalized nonlinear phase noise is Z 1 2q 1 2 ðA:10Þ rU ¼ fðsÞ dt ¼ s þ 6 3 0 the same as that in [5,34]. For Fig. 4, for a compensator located at s with a compensator factor of a, the variance of the residual nonlinear phase noise as a function of distance is equal to r2U ðtÞ þ a2 jðs; sÞ 2aft ðsÞ;
t P s;
ðA:11Þ
where ft ðsÞ ¼
Z
t
jðs; t1 Þ dt1
ðA:12Þ
0
is the correlation of the nonlinear phase noise at t with the intensity at s, and r2U ðtÞ is the variance of the normalized nonlinear phase noise as a function of distance that is equal to Z t 2q t4 r2U ðtÞ ¼ ðA:13Þ ft ðsÞ ds ¼ s t3 þ : 3 6 0
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