4.1 The in-phase and quadrature components of ASE noise

The field including the complex envelopes of signal and ASE noise is:

$$U(z,t) = \sum\_{l=1}^{N} \left[ u\_l(z,t) + A\_l(z,t) \right] \exp\left(-i\alpha\_l t\right) \tag{47}$$

where ulð Þ z; t and Alð Þ z; t are the complex envelopes of signal and ASE noise, respectively [37, 38]. N is the channel number. ASE noise generated in erbium-doped fiber amplifiers (EDFAs) is Alð Þ¼ 0; t AlRð Þþ 0; t iAlIð Þ 0; t , AlRð Þ 0; t and AlIð Þ 0; t are statistically real independent stationary white Gaussian processes, and AlRð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>A</sup><sup>∗</sup> lRð Þ <sup>0</sup>; <sup>t</sup> � � <sup>¼</sup> AlIð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>A</sup><sup>∗</sup> lIð Þ <sup>0</sup>; <sup>t</sup> � � <sup>¼</sup> nsphvlð Þ Gl � <sup>1</sup> <sup>Δ</sup>vlδ τð Þ. In the complete inversion case, nsp ¼ 1. h is the Planck constant. Gl is the gain for channel l.

Substituting Eq. (47) into (1), we can get the equation that Alð Þ z; t satisfies:

$$\begin{split} i \frac{\partial A\_l(z,t)}{\partial z} &= \frac{\beta\_2}{2} \left( -\alpha\_l^2 + \frac{\partial^2}{\partial t^2} - i2\alpha \eta \frac{\partial}{\partial t} \right) A\_l(z,t) - \\ &\quad \gamma(z) \exp\left( -2\alpha z \right) \left| \sum\_{j=1}^N u\_j(z,t) + A\_j(z,t) \right|^2 A\_l(z,t) \end{split} \tag{48}$$

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

$$\begin{split} \frac{\partial A\_{IR}(z,t)}{\partial z} &= -\beta\_2 \boldsymbol{\alpha} \boldsymbol{\eta} \frac{\partial A\_{IR}(z,t)}{\partial t} + \frac{1}{2} \beta\_2 \frac{\partial^2 A\_{II}(z,t)}{\partial t^2} - \\ &\frac{1}{2} \beta\_2 \boldsymbol{\alpha}\_l^2 A\_{ll} - \boldsymbol{\gamma} \exp\left(-2\boldsymbol{\alpha}z\right) \left| \sum\_{j=1}^N \boldsymbol{u}\_j(\boldsymbol{z},t) + A\_j(\boldsymbol{z},t) \right|^2 A\_{ll} \end{split} \tag{49}$$
 
$$\begin{split} \frac{\partial A\_{II}(z,t)}{\partial z} &= -\beta\_2 \boldsymbol{\alpha}\_l \frac{\partial A\_{II}(z,t)}{\partial t} - \frac{1}{2} \beta\_2 \frac{\partial^2 A\_{IR}(z,t)}{\partial t^2} + \frac{1}{2} \beta\_2 \boldsymbol{\alpha}\_l^2 A\_{IR}(\boldsymbol{z},t) + \\ &\boldsymbol{\gamma} \exp\left(-2\boldsymbol{\alpha}z\right) \left| \sum\_{j=1}^N \boldsymbol{u}\_j(\boldsymbol{z},t) + A\_j(\boldsymbol{z},t) \right|^2 A\_{ll} \end{split} \tag{50}$$

We now seek their traveling wave solution by taking [37] AlR ¼ ϕ ξð Þ, AlI ¼ φ ξð Þ, and ξ ¼ t � cz.

Then, (49) and (50) are converted into

$$\phi'(\beta\_2 \alpha \eta - c) = -\left[\frac{1}{2}\beta\_2 \alpha\_l^2 + \chi \exp\left(-2\alpha z\right) \left|\sum\_{j=1}^N u\_j(z, t) + A\_j(z, t)\right|^2\right] \rho + \frac{1}{2}\beta\_2 \rho'' \tag{51}$$

Figure 6.

Table 1. Fiber parameters.

Figure 7.

Units (ps<sup>4</sup>

Table 2. The average.

26

The process of calculating β4.

/km).

are the same as Figure 5.

The pulse shapes with and without δγ. <sup>β</sup><sup>2</sup> ¼ �21:7 ps<sup>2</sup> ð Þ <sup>=</sup>km , <sup>s</sup> <sup>¼</sup> <sup>0</sup>:01, <sup>γ</sup> <sup>¼</sup> <sup>1</sup>:<sup>3</sup> ð Þ <sup>=</sup>km=<sup>W</sup> . Other parameters

DCF 0.59 5.5 0.01 110 0.1381 NZDSF 0.21 2.2 0.01 �5.6 0.115 SMF 0.21 1.3 0.01 �21.7 �0.5

Z = 1.5LD Z = 5LD Z = 50LD

DCF 0.0003 0.00035 0.00032 NZDSF 0.0022 0.003 0.0032 SMF 0.0012 0.002 0.0025

/km) β<sup>3</sup> (ps<sup>3</sup>

/km)

a (dB/km) γ (/km/W) s β<sup>2</sup> (ps<sup>2</sup>

Nonlinear Optics ‐ Novel Results in Theory and Applications

Nonlinear Optics ‐ Novel Results in Theory and Applications

$$\rho'(\beta\_2 \alpha \eta - \varepsilon) = \left[\frac{1}{2}\beta\_2 \alpha\_l^2 + \gamma \exp\left(-2\alpha z\right) \left|\sum\_{j=1}^N u\_j(z, t) + A\_j(z, t)\right|^2\right] \phi - \frac{1}{2}\beta\_2 \phi'' \tag{52}$$

<sup>∂</sup>AlRð Þ <sup>z</sup>; <sup>t</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81093

<sup>∂</sup>AlIð Þ <sup>z</sup>; <sup>t</sup> <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>f</sup>

B β2ω<sup>2</sup>

> B β2ω<sup>2</sup>

> > f 0

β2ω<sup>2</sup>

ð Þ β2ω<sup>l</sup> � c k BAlI, <sup>z</sup>¼<sup>0</sup>

plR <sup>¼</sup> <sup>C</sup>

plI <sup>¼</sup> <sup>C</sup><sup>0</sup> g0

½ � g Að Þ lR

ð Þ¼ AlIð Þ z; t B

C, C<sup>0</sup> are determined by Ð <sup>þ</sup><sup>∞</sup>

vuuuuuuut

Nonlinear Schrödinger Equation

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>

vuuuuuuut

g0

noise are

29

Here,

0

<sup>l</sup> <sup>=</sup>2þ<sup>γ</sup> exp ð Þ �2α<sup>z</sup> ∑

<sup>l</sup> <sup>=</sup>2þ<sup>γ</sup> exp ð Þ �2α<sup>z</sup> ∑

ð Þþ AlIð Þ z; t g

f Að Þ¼ lRð Þ <sup>z</sup>; <sup>t</sup> <sup>β</sup>2kω<sup>l</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

N j¼1

ð Þ β2ωl�c k

N j¼1

Now, they can be regarded as the stationary equations, and we can gain their

corresponding Fokker-Planck equations of (60) and (61), the probabilities of ASE

ð Þ AlI ½ �<sup>2</sup> exp 2 <sup>ð</sup>AlI

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

<sup>2</sup> exp 2 <sup>ð</sup>AlR

� � � � �

� � � � �

ð Þ¼� AlIð Þ z; t β2kω<sup>l</sup>

<sup>l</sup> <sup>=</sup>2þ<sup>γ</sup> exp ð Þ �2α<sup>z</sup> ∑

q

probabilities according to Sections (7.3) and (7.4) in [39]. By solving the

ð Þ β2ωl�c k

N j¼1

� � � � �

<sup>∂</sup><sup>z</sup> <sup>¼</sup> f Að Þþ lRð Þ <sup>z</sup>; <sup>t</sup> g Að Þ lRð Þ <sup>z</sup>; <sup>t</sup> AlR, <sup>z</sup>¼<sup>0</sup> (60)

� � � � �

2 <sup>þ</sup>β2k<sup>2</sup> =2

� � � � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>B</sup><sup>2</sup> � <sup>A</sup><sup>2</sup>

ujð Þþ z; t Ajðz; tÞ

lIð Þ z; t

� � � � �

2 <sup>þ</sup>β2k<sup>2</sup> =2

2 <sup>þ</sup>β2k<sup>2</sup> =2

ð Þ AlIð Þ z; t AlI, <sup>z</sup>¼<sup>0</sup> (61)

2

2

� <sup>A</sup><sup>2</sup>

2

lRð Þ z; t

� <sup>A</sup><sup>2</sup>

lRð Þ z; t

(62)

(63)

(64)

(65)

0

ujð Þþ z; t Ajðz; tÞ

AlR, <sup>z</sup>¼<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ujð Þþ z; t Ajðz; tÞ

q

ð Þ β2ωl�c k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>B</sup><sup>2</sup> � <sup>A</sup><sup>2</sup>

lIð Þ z; t

�∞

�∞

f sð Þ

�<sup>∞</sup> pdp <sup>¼</sup> 1. Compared with [40], these probabilities of

f 0 ð Þs g<sup>0</sup> ½ � ð Þs

½ � g sð Þ <sup>2</sup> ds " # (66)

<sup>2</sup> ds " # (67)

(52) is differentiated to ξ

$$\rho''(\beta\_2\alpha\_l - c) = \left[\frac{1}{2}\beta\_2\alpha\_l^2 + \gamma \exp\left(-2\alpha z\right)\left|\sum\_{j=1}^N u\_j(z,t) + A\_j(z,t)\right|^2\right] \phi' - \frac{1}{2}\beta\_2\phi''' \tag{53}$$

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

$$\begin{aligned} \left(\phi''(\beta\_2\alpha\_l - c)\right)^2 &= -\left[\frac{1}{2}\beta\_2\alpha\_l^2 + \gamma \exp\left(-2c\text{z}t\right)\left|\sum\_{j=1}^N u\_j(z,t) + A\_j(z,t)\right|^2\right]^2 \phi + \\ \beta\_2 \left[\frac{1}{2}\beta\_2\alpha\_l^2 + \gamma \exp\left(-2c\text{z}t\right)\left|\sum\_{j=1}^N u\_j(z,t) + A\_j(z,t)\right|^2\right] \phi'' &+ \frac{1}{4}\beta\_2^2 \phi^{(4)} \end{aligned} \tag{54}$$

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

$$\rho = B \left\{ \left[ \beta\_2 \alpha\_l^2 / 2 + \gamma \exp \left( -2 \alpha \mathbf{z} \right) \left| \sum\_{j=1}^N u\_j(\mathbf{z}, t) + A\_j(\mathbf{z}, t) \right|^2 \right] \cos k\xi + \beta\_2 k^2 / 2 \cdot \cos k\xi \right\} / (\beta\_2 \alpha\_l - c) / k^2 \tag{55}$$

$$\rho = B \sin k\xi \tag{56}$$

and

$$\begin{aligned} B &= A\_{lR}(\mathbf{0}, t)(\beta\_2 \alpha \mathbf{\hat{o}} - c)k / \\ &\left\{ \left| \beta\_2 \alpha\_l^2 / 2 + \chi \exp\left( -2\alpha \mathbf{z} \right) \left| \sum\_{j=1}^N \mu\_j(\mathbf{z}, t) + A\_j(\mathbf{z}, t) \right|^2 \right] \cos kt + \beta\_2 k^2 / 2 \cdot \cos kt \right\} \end{aligned} \tag{57}$$

$$c = \pm \left\{ \theta\_2^2 k^2 / 4 + \left[ \beta\_2 \alpha\_l^2 / 2 + \gamma \exp\left( -2\alpha x \right)^2 \left| \sum\_{j=1}^N u\_j(x, t) + A\_j(x, t) \right|^2 \right] / k^2 + \beta\_2^2 \alpha\_l^2 / 2 + \gamma \right\} \tag{58}$$

$$\gamma \beta\_2 \exp\left( -2\alpha x \right) \left| \sum\_{j=1}^N u\_j(x, t) + A\_j(x, t) \right|^2 \Bigg\}^{1/2} + \beta\_2 \alpha\_l \tag{58}$$

$$k = \arcsin\left( A\_{ll}(0, t) / \mathcal{B} \right) / t \tag{59}$$

In the above calculation process, B, c, and k should be regarded as constants, and AlR, AlI are the functions of the solo variable ξ, respectively.
