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The evolution of ASE noise and the generation of nonlinear phase shift are analyzed based on the travelling wave solution of ASE noise and its probability density function by solving the Fokker-Planck equation including dispersion effect. Nonlinear effect has strong impact on ASE noise. After the transmission in non-zero dispersion shift fibers + dispersion compensation fibers, due to the nonlinear effect, ASE noise is enhanced. Detailedly, the real part of ASE decreases but the image part increases greatly compared to that with dispersion effect only. Nonlinear phase shift, related to the image part of ASE noise, occurs in this kind of link. The impact of signal intensity on ASE noise induces fluctuations to the both curves of ASE noise and nonlinear phase shift as functions of time, respectively. Furthermore, it results in the non-Gaussian distribution of ASE noise probability density function (side-bands occurring) and brings more than 1 dB additive BER.

Nonlinear phase noise [

From the power spectral density, ASE noise probability density function (p.d.f.) is the Fourier transform of its characteristic function which is determined by the power distribution [

In addition, by the power spectral density, the interplay among Kerr nonlinearity, chromatic dispersion and ASE noise in the frequency domain can also be analyzed by the transfer function method [

On the other hand, the p.d.f. of ASE noise can also be calculated by the Fokker-Planck equation [

To avoid the complex noise probability evolution in fibers, in [

So, in this paper, we attempt to give an inclusive and explicit description about the change of ASE noise and the generation of nonlinear phase noise in dispersion and nonlinear fibers. By my knowledge, this is the first time that in the estimation related to ASE noise, the nonlinear effect is taken into account and it is also the first time that in the p.d.f. of ASE noise the dispersion effect is taken into account.

We start with the transmissions of ASE noise and signals, which satisfy the nonlinear partial differential equations (NLPDE) [

In Section 2, the analytical solution and probability density function of ASE noise are achieved by the travelling wave solution of NLPDE. Based on these solutions, the change of ASE noise originated from dispersion and nonlinear effects is detected for the link of non-zero dispersion shift fibers + dispersion compensation fibers(NZDSFs + DCFs), and nonlinear phase shift caused by nonlinear effect on ASE noise, is analyzed in Section 3. Section 4 presents the conclusion.

In this section, we will derive the nonlinear partial differential equation describing ASE noise in transmission fibers and solve it with the travelling wave method. Subsequently, its probability can been gained based on this solution. Both are the base for the analyses of nonlinear phase noise.

The envelope of electric field satisfies the nonlinear Schrodinger equation:

i ∂ U ( z , t ) ∂ z − β 2 2 ∂ 2 U ( z , t ) ∂ t 2 = − γ ( z ) exp ( − 2 α z ) | U ( z , t ) | 2 U ( z , t ) (1)

where β 2 is the group velocity dispersion, γ is the nonlinear coefficient and α is the fiber loss. The field is:

U ( z , t ) = ∑ l = 1 N [ u l ( z , t ) + A l ( z , t ) ] exp ( − i ω l t ) (2)

where u l ( z , t ) and A l ( z , t ) are the complex envelopes of signal and ASE noise, respectively. N is the channel number. ASE noise generated in erbium-doped fiber amplifiers (EDFAs) is: A l ( 0 , t ) = A l R ( 0 , t ) + i A l I ( 0 , t ) .

A l R ( 0 , t ) and A l I ( 0 , t ) are statistically real independent stationary white Gaussian processes and 〈 A l R ( 0 , t + τ ) A l R ∗ ( 0 , t ) 〉 = 〈 A l I ( 0 , t + τ ) A l I ∗ ( 0 , t ) 〉 = n s p h v l ( G l − 1 ) Δ v l δ ( τ ) . In the complete inversion case, n s p = 1 . h is the Planck constant. G l is the gain for channel l.

Substituting (2) into (1), we can get the equation that A l ( z , t ) satisfies:

i ∂ A l ( z , t ) ∂ z = β 2 2 ( − ω l 2 + ∂ 2 ∂ t 2 − i 2 ω l ∂ ∂ t ) A l ( z , t ) − γ ( z ) exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 A l ( z , t ) (3)

So, the in-phase and quadrature components of ASE noise obey:

∂ A l R ( z , t ) ∂ z = − β 2 ω l ∂ A l R ( z , t ) ∂ t + 1 2 β 2 ∂ 2 A l I ( z , t ) ∂ t 2 − 1 2 β 2 ω l 2 A l I − γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 A l I (4)

∂ A l I ( z , t ) ∂ z = − β 2 ω l ∂ A l I ( z , t ) ∂ t − 1 2 β 2 ∂ 2 A l R ( z , t ) ∂ t 2 + 1 2 β 2 ω l 2 A l R ( z , t ) + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 A l R (5)

We now seek their travelling wave solution by taking [

ϕ ′ ( β 2 ω l − c ) = − [ 1 2 β 2 ω l 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 ] φ + 1 2 β 2 φ ″ (6)

φ ′ ( β 2 ω l − c ) = [ 1 2 β 2 ω l 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 ] ϕ − 1 2 β 2 ϕ ″ (7)

(7) is differentiated to ξ :

φ ″ ( β 2 ω l − c ) = [ 1 2 β 2 ω l 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 ] ϕ ′ − 1 2 β 2 ϕ ‴ (8)

Replace ϕ ′ and ϕ ‴ in (8) with (6) and the differential of (6), there are:

φ ″ ( β 2 ω l − c ) 2 = − [ 1 2 β 2 ω l 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 ] 2 φ + β 2 [ 1 2 β 2 ω l 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 ] φ ″ + 1 4 β 2 2 φ ( 4 ) (9)

From (6) and (9), We can easily obtain:

ϕ = B { [ β 2 ω l 2 / 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 ] cos k ξ + β 2 k 2 / 2 ⋅ cos k ξ } / ( β 2 ω l − c ) / k (10)

φ = B sin k ξ (11)

and

B = A l R ( 0 , t ) ( β 2 ω l − c ) k / { [ β 2 ω l 2 / 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 ] cos k t + β 2 k 2 / 2 ⋅ cos k t } (12)

c = ± { β 2 2 k 2 / 4 + [ β 2 ω l 2 / 2 + γ exp ( − 2 α z ) 2 | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 ] / k 2 + β 2 2 ω l 2 / 2 + γ β 2 exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 } 1 / 2 + β 2 ω l (13)

k = arcsin ( A l I ( 0 , t ) / B ) / t (14)

In the above calculation process, B, c and k should be regarded as constants and A l R , A l I are the functions of the solo variable ξ , respectively.

Because A l R and A l I have been solved, the time differentials of (4) and (5) can be calculated. Thus, the stochastic differential equations (Ito forms) around A l R and A l I are:

∂ A l R ( z , t ) ∂ z = f ( A l R ( z , t ) ) + g ( A l R ( z , t ) ) A l R , z = 0 (15)

∂ A l I ( z , t ) ∂ z = f ′ ( A l I ( z , t ) ) + g ′ ( A l I ( z , t ) ) A l I , z = 0 (16)

where

f ( A l R ( z , t ) ) = β 2 k ω l [ B β 2 ω l 2 / 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 + β 2 k 2 / 2 ( β 2 ω l − c ) k ] 2 − A l R 2 ( z , t ) (17)

g ( A l R ( z , t ) ) = − ( β 2 ω l − c ) k A l R , z = 0 × [ B β 2 ω l 2 / 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 + β 2 k 2 / 2 ( β 2 ω l − c ) k ] 2 − A l R 2 ( z , t ) (18)

f ′ ( A l I ( z , t ) ) = − β 2 k ω l B 2 − A l I 2 ( z , t ) (19)

g ′ ( A l I ( z , t ) ) = [ B β 2 ω l 2 / 2 + γ exp ( − 2 α z ) | ∑ j = 1 N u j ( z , t ) + A j ( z , t ) | 2 + β 2 k 2 / 2 ( β 2 ω l − c ) k ] 2 × ( β 2 ω l − c ) k B A l I , z = 0 B 2 − A l I 2 ( z , t ) (20)

Now, they can be regarded as the stationary equations and we can gain their probabilities according to Section (7.3) and (7.4) in [

p l R = C [ g ( A l R ) ] 2 exp [ 2 ∫ − ∞ A l R f ( s ) [ g ( s ) ] 2 d s ] (21)

p l I = C ′ [ g ′ ( A l I ) ] 2 exp [ 2 ∫ − ∞ A l I f ′ ( s ) [ g ′ ( s ) ] 2 d s ] (22)

C , C ′ are determined by ∫ − ∞ + ∞ p d p = 1 . Compared with [

of ASE noise take dispersion effect into account. This is the first time that the p.d.f. of ASE noise simultaneously including dispersion and nonlinear effects is presented.

(21) and (22) are efficient in the models of Gaussian and correlated non-Gaussian processes as our (4) (5).

Obviously, the Gauss distribution has been distorted. They are no longer symmetrical distributions and both have phase shifts consistent with [

− i β 2 ω l ∂ ∂ t A l ( z , t ) in (3) brings the phase shift, and β 2 2 ∂ 2 ∂ t 2 A l ( z , t ) brings the

expanding and induces the side bands, the self- and cross-phase modulation effects. Their synthesis impact is amplified by (21), (22) and results in the complete non-Gauss distributions.

So, we have obtained the analytical solution of ASE noise:

a ( z , t ) = ∑ l = 1 N ( A l R p l R + i A l I p l I ) exp ( − i ω l t ) (23)

In this section, by calculating the signal and ASE noise fields point-to-point, we will simulate the above ASE noise statistical solution and attempt to get some extended results about nonlinear phase noise.

To simplify the problem, we only consider one signal channel. Fiber parameters

are listed in

is 33%. The bit rate is 2.5 Gb/s. In the calculation of ASE noise: Δ ν = 50 GHz , G = 16 .8 dB / km and ν = 3 × 10 8 / 1550 × 10 9 ( Hz ) . The link is shown in

We have derived the representation of ASE noise in transmission fibers in (23), so the noise evolution can be explicitly shown. We draws ASE noises after they have been transmitted in the link of NZDSFs + DCFs to disclose the nonlinear phase shift induced by nonlinear effect on ASE noise.

In

a ′ ( z , t ) = ( 〈 { h ( S z , t + τ ) ⊗ a ( 0 , t + τ ) } { h ( S z , t ) ⊗ a ( 0 , t ) } * 〉 ) 1 2 , S z = 4 β 2 0 z , h ( S z , t ) = 1 2 π S z [ sin ( t 2 S z ) + cos ( t 2 S z ) − i cos ( t 2 S z ) + i sin ( t 2 S z ) ] , and the solid lines are based on (23) ( 〈 a ( z , t + τ ) a ∗ ( z , t ) 〉 ) 1 / 2 .

a (dB/km) | γ (/km/W) | D (ps/nm/km) | |
---|---|---|---|

DCFs | 0.59 | 5.5 | −87 |

NZDSFs | 0.21 | 2.2 | 4.4 |

With the increase of distance (span number increases), the deviation between the dotted and the solid lines increases.

Even the dispersion is compensated, due to the nonlinear effect, ASE noise is enhanced. Detailedly, the real part of ASE decreases but the image part increases greatly compared to that with dispersion effect only. The term − i β 2 ω l ∂ A ( z , t ) / ∂ t in (3) brings a phase shift and partly distorts the odd and even symmetries of ASE noise compared with [

Only the Re ( 〈 a ( z , t + τ ) a ∗ ( z , t ) 〉 ) 1 / 2 plays roles in the estimation of nonlinear phase noise and according to the calculation formula of [

To explicitly exhibit the impact of nonlinear effect on ASE noise, we calculate the bit-error-rate (BER) of a PSK signal as a function of SNR in

SNR = P l N ASE + N NL

BER = p ( ϕ ) | ϕ = 2 rad

P l is the signal power, N ASE is the average power of ASE noise and N NL is the nonlinear fluctuations induced by SPM and XPM. p ( ϕ ) is the p.d.f. as a function of phase. With the decrease of SNR, for the worst case( 〈 ϕ 〉 = 2 rad , the side-band point such as the side band t = 4 × 10 − 11 ( s ) in

We also calculate the phase shift caused by ASE noise [

σ = arctan Im ( U ( z , t ) ) Re ( U ( z , t ) ) − arctan Im [ ∫ − ∞ + ∞ u ( z , ω ) exp ( − i ω t ) d ω ] Re [ ∫ − ∞ + ∞ u ( z , ω ) exp ( − i ω t ) d ω ]

This phase value is caused by the accumulated ASE noise ( ∑ 1 M ( 〈 a ( M z , t + τ ) a ∗ ( M z , t ) 〉 ) 1 / 2 , M is the span number). The values in

There is a trace that a symmetric two-peak spectrum occurs in

In the dispersion and nonlinear fibers, based on the transmission theory of ASE noise and the corresponding Fokker-Planck equations, the change of ASE noise

and the generation of nonlinear phase noise can be analyzed. ASE noise is enhanced due to the nonlinear effect. The strong signal not only changes ASE noise probability, statistical amplitude and phase, but also brings an oscillation in the statistical spectrum of nonlinear phase shift.

Gui, M.X. and Jing, H. (2017) Statistical Analyses of ASE Noise. Optics and Photonics Journal, 7, 160-169. https://doi.org/10.4236/opj.2017.710016