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While the signal field + ASE noise pass through a span of transmission fiber, a dispersion compensation grating and a fiber amplifier(with the generation of ASE noise), the nonlinear Fokker-Plank equations, describing the probability transforms of the field, are established and solved. Based on these statistical theories, the probability distributions of the signal + ASE noise field through 50km NZDSF, a dispersion compensation grating and a fiber amplifier link, are obtained. The dispersion and nonlinear effects in transmission fiber induce frequency offsets in the probability distribution of field and they cannot be dissipated by dispersion compensation. The generation of ASE noise in the amplifier will accelerate this frequency offset.

The statistical model of phase noise induced by the interplay between amplifier spontaneous emission (ASE) noise and fiber Kerr nonlinearity, is extensively studied during the last decade. The probability density function (p.d.f.) of ASE noise + signal field is necessary to analyze the noise properties and evaluate the system performances.

By converting a Cartesian into a polar description, the nonlinear phase noise (NPN) (the additive component of ASE noise) was identified to be nearly Gaussian [

But by the experimental observation and performance study of the differential phase-shift-keying transmission systems [

Taking the dispersion effect into account, the p.d.f. of signal pulse with nonlinear phase noise was broadened and the broadening was asymmetrical with respect to the mean nonlinear phase shift [

In this paper, the probability transform is studied when ASE noise +signal field pass through a span of transmission fiber and a fiber amplifier where ASE noise is generated and described by a delta (δ)-function. The dispersion effect of fiber is taken into account, so the statistical properties of the field are nonstationary. In the amplifier, ASE noise is generated, and thus based on the birth theory of stochastic, the field’s statistical transform is established and the probability distribution through 50 km nonzero-dispersion-shifted fiber (NZDSF), a dispersion compensation grating and an erbium-doped fiber amplifier (EDFA), are obtained.

The optical field envelope in an amplified transmission system is governed by the nonlinear Schrodinger(NLS) equation [

where

where

Disregard the stochastic item in (1), we can calculate the optical field without ASE noise item by the split-step method [

Now, (1) can be written as

Assume

where

According to the property of

we get the generalized Fokker-Plank equation describing the signal field with ASE noise

In particular, the isolated system (

We assume the transient solution (only

Taking the ASE noise as a perturbation item, we now try to solve the perturbed Fokker-Planck Equation (8) [

Its eigenvalue and eigenfunction can be expanded as

Finally, comparing the terms

(17) and (18) can be written as

(20)

The item of p in (23) is definite because of the orthogonality and normalization of (13). Also, the coefficients b_{p} and Y_{n} are determined by the input field of the amplifier. Note that, this model requires that the first order differential of

In this section, we will simulate the p.d.f. of the ASE noise+ signal field after transmitted in NZDSF, grating and an amplifier link such as

Fibers parameters are: a = 0.21 (dB/km), γ = 2.2 (/km/W), D = 4.4 (ps/nm/km). The pulse is:

There are frequency offsets in the p.d.f. when the dispersion and nonlinear effects in transmission fiber are taken into account and they cannot be dissipated by dispersion compensation.

where D is the compensation dispersion supplied by grating.

The sideband induced by dispersion and nonlinear effects is not transient and it exists at a certain probability. From the mathematical formula, the solution (11) is the Hermite polynomial and it is a series of vibration functions. When they add together several times, the side bands occur but still with a lot of chances, they are located neither in the field nor in the sideband regions. These are determined by the coefficient

+ signal field) and similar to the case of [

Phase shifts caused by the nonlinear effect are also consistent with [

It is most likely that this figure can’t clearly show the impact of local ASE noise (generated in EDFA). So, in

Therefore, in the fiber + dispersion grating + amplifier transmission systems, the evaluation about the statistical transform of signal + ASE noise field shows that the frequency offset of the field’s probability distribution induced by dispersion and nonlinear effects in transmission fiber cannot be dissipated by dispersion compensation and the ASE noise generated in the amplifier is a perturbation item and has weak impact on the field’s p.d.f., but it accelerates the field’s frequency offset.

Jing Huang,Jianquan Yao, (2016) Statistical Transform of Signal Field with ASE Noise through a Fiber Amplifier. Optics and Photonics Journal,06,69-74. doi: 10.4236/opj.2016.68B012