4.2 Probability density function of ASE noise

Because AlR and AlI have been solved, the time differentials of (49) and (50) can be calculated. Thus, the stochastic differential equations (ITO forms) around AlR and AlI are

Nonlinear Schrödinger Equation DOI: http://dx.doi.org/10.5772/intechopen.81093

$$\frac{\partial A\_{lR}(\mathbf{z},t)}{\partial \mathbf{z}} = f(A\_{lR}(\mathbf{z},t)) + \mathbf{g}(A\_{lR}(\mathbf{z},t))A\_{lR,\mathbf{z}=\mathbf{0}} \tag{60}$$

$$\frac{\partial A\_{ll}(\mathbf{z},t)}{\partial \mathbf{z}} = \mathbf{f}'(A\_{ll}(\mathbf{z},t)) + \mathbf{g}'(A\_{ll}(\mathbf{z},t))A\_{ll,\mathbf{z}=\mathbf{0}}\tag{61}$$

Here,

φ0

ð Þ¼ β2ω<sup>l</sup> � c

φ00ð Þ¼ β2ω<sup>l</sup> � c

ϕ00ð Þ β2ω<sup>l</sup> � c

<sup>φ</sup> <sup>¼</sup> <sup>B</sup> <sup>β</sup>2ω<sup>2</sup>

and

B ¼ AlRð Þ 0; t ð Þ β2ω<sup>l</sup> � c k=

β2ω<sup>2</sup>

2k2

<sup>=</sup><sup>4</sup> <sup>þ</sup> <sup>β</sup>2ω<sup>2</sup>

4

4

8 < :

8 < :

<sup>c</sup> ¼ � <sup>β</sup><sup>2</sup>

and AlI are

28

8 < :

4

1 2 β2ω<sup>2</sup>

Nonlinear Optics ‐ Novel Results in Theory and Applications

1 2 β2ω<sup>2</sup>

<sup>2</sup> ¼ � <sup>1</sup>

β2 1 2 β2ω<sup>2</sup>

4

From (51) and (54), we can easily obtain

<sup>l</sup> =2 þ γ exp ð Þ �2αz ∑

γβ<sup>2</sup> exp ð Þ �2αz ∑

4.2 Probability density function of ASE noise

AlR, AlI are the functions of the solo variable ξ, respectively.

<sup>l</sup> =2 þ γ exp ð Þ �2αz ∑

4

<sup>2</sup> β2ω<sup>2</sup>

N j¼1

> N j¼1

<sup>l</sup> =2 þ γ exp ð Þ �2αz

N j¼1

� � � � �

� � � � �

<sup>2</sup> 2

� � � � �

<sup>2</sup> 2

4

4

(52) is differentiated to ξ

<sup>l</sup> þ γ exp ð Þ �2αz ∑

<sup>l</sup> þ γ exp ð Þ �2αz ∑

N j¼1

N j¼1

� � � � �

Replacing ϕ<sup>0</sup> and ϕ<sup>000</sup> in (53) with (51) and the differential of (51), there are

<sup>l</sup> þ γ exp ð Þ �2αz ∑

<sup>l</sup> þ γ exp ð Þ �2αz ∑

ujð Þþ z; t Ajð Þ z; t

2 2

� � � � �

2 2

ujð Þþ z; t Ajð Þ z; t

ujð Þþ z; t Ajð Þ z; t

N j¼1

N j¼1

� � � � �

� � � � �

ujð Þþ z; t Ajðz; tÞ

<sup>2</sup> ∑ N j¼1

� � � � �

ujð Þþ z; t Ajð Þ z; t

In the above calculation process, B, c, and k should be regarded as constants, and

Because AlR and AlI have been solved, the time differentials of (49) and (50) can be calculated. Thus, the stochastic differential equations (ITO forms) around AlR

<sup>2</sup> 2

3

<sup>5</sup> cos <sup>k</sup><sup>ξ</sup> <sup>þ</sup> <sup>β</sup>2k<sup>2</sup>

� � � � � 3

ujð Þþ z; t Ajð Þ z; t

k ¼ arcsinð Þ AlIð Þ 0; t =B =t (59)

1=2

� � � � �

2 9 = ;

φ ¼ B sin kξ (56)

<sup>5</sup> cos kt <sup>þ</sup> <sup>β</sup>2k<sup>2</sup>

� � � � � 3

<sup>5</sup>=k<sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup>

þ β2ω<sup>l</sup> (58)

2ω<sup>2</sup> <sup>l</sup> =2þ

<sup>2</sup> 2

� � � � �

2 2

� � � � �

� � � � �

ujð Þþ z; t Ajðz; tÞ

ujð Þþ z; t Ajðz; tÞ

3 <sup>5</sup>ϕ<sup>0</sup> � <sup>1</sup> 2

> � � � � �

� � � � �

=2 � cos kξ

3

<sup>5</sup>ϕ″ <sup>þ</sup>

9 = ;

=2 � cos kt

3 5

2

ϕþ

1 4 β2 2ϕð Þ <sup>4</sup>

(54)

=ð Þ β2ω<sup>l</sup> � c =k

(55)

9 = ; (57)

3 <sup>5</sup><sup>ϕ</sup> � <sup>1</sup> 2

β2ϕ<sup>00</sup> (52)

β2ϕ<sup>000</sup> (53)

f Að Þ¼ lRð Þ <sup>z</sup>; <sup>t</sup> <sup>β</sup>2kω<sup>l</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B β2ω<sup>2</sup> <sup>l</sup> <sup>=</sup>2þ<sup>γ</sup> exp ð Þ �2α<sup>z</sup> ∑ N j¼1 ujð Þþ z; t Ajðz; tÞ � � � � � � � � � � 2 <sup>þ</sup>β2k<sup>2</sup> =2 ð Þ β2ωl�c k 2 6 6 6 6 4 3 7 7 7 7 5 2 � <sup>A</sup><sup>2</sup> lRð Þ z; t vuuuuuuut (62) g Að Þ¼� lRð Þ <sup>z</sup>; <sup>t</sup> ð Þ <sup>β</sup>2ω<sup>l</sup> � <sup>c</sup> <sup>k</sup> AlR, <sup>z</sup>¼<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B β2ω<sup>2</sup> <sup>l</sup> <sup>=</sup>2þ<sup>γ</sup> exp ð Þ �2α<sup>z</sup> ∑ N j¼1 ujð Þþ z; t Ajðz; tÞ � � � � � � � � � � 2 <sup>þ</sup>β2k<sup>2</sup> =2 ð Þ β2ωl�c k 2 6 6 6 6 4 3 7 7 7 7 5 2 � <sup>A</sup><sup>2</sup> lRð Þ z; t vuuuuuuut (63) f 0 ð Þ¼� AlIð Þ z; t β2kω<sup>l</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>B</sup><sup>2</sup> � <sup>A</sup><sup>2</sup> lIð Þ z; t q (64) g0 ð Þ¼ AlIð Þ z; t B β2ω<sup>2</sup> <sup>l</sup> <sup>=</sup>2þ<sup>γ</sup> exp ð Þ �2α<sup>z</sup> ∑ N j¼1 ujð Þþ z; t Ajðz; tÞ � � � � � � � � � � 2 <sup>þ</sup>β2k<sup>2</sup> =2 ð Þ β2ωl�c k 2 6 6 6 6 4 3 7 7 7 7 5 2 ð Þ β2ω<sup>l</sup> � c k BAlI, <sup>z</sup>¼<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>B</sup><sup>2</sup> � <sup>A</sup><sup>2</sup> lIð Þ z; t q (65)

Now, they can be regarded as the stationary equations, and we can gain their probabilities according to Sections (7.3) and (7.4) in [39]. By solving the corresponding Fokker-Planck equations of (60) and (61), the probabilities of ASE noise are

$$p\_{IR} = \frac{C}{\left[\mathfrak{g}(A\_{IR})\right]^2} \exp\left[2\int\_{-\infty}^{A\_{IR}} \frac{f(s)}{\left[\mathfrak{g}(s)\right]^2} ds\right] \tag{66}$$

$$p\_{ll} = \frac{C'}{\left[\mathbf{g}'(A\_{ll})\right]^2} \exp\left[2\int\_{-\infty}^{A\_{ll}} \frac{f'(s)}{\left[\mathbf{g}'(s)\right]^2} ds\right] \tag{67}$$

C, C<sup>0</sup> are determined by Ð <sup>þ</sup><sup>∞</sup> �<sup>∞</sup> pdp <sup>¼</sup> 1. Compared with [40], these probabilities 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.

(66) and (67) are efficient in the models of Gaussian and correlated non-Gaussian processes as our (49) and (50). Obviously, the Gaussian distribution has been distorted. They are no longer symmetrical distributions, and both have phase shifts consistent with [40], and as its authors have expected that "if the dispersion

effect was taken into account, the asymmetric modulation side bands occur." The reasons are that item �iβ2ω<sup>l</sup> ∂ <sup>∂</sup><sup>t</sup> Alð Þ z; t in (48) brings the phase shift and item β2 2 ∂2 <sup>∂</sup>t<sup>2</sup> Alð Þ z; t brings the expansion and induces the side bands, the self-phase modulation effects, and the cross-phase modulation effects. Their synthesis impact is amplified by (66) and (67) and results in the complete non-Gaussian distributions.

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