Figure 4.

The pulse shapes with and without δβ<sup>3</sup> and δγ. The red line: without δβ<sup>3</sup> and δγ; the black line: with δβ<sup>3</sup> and δγ. <sup>ν</sup> <sup>¼</sup> <sup>ω</sup>=2=π, <sup>β</sup><sup>0</sup> <sup>3</sup> <sup>¼</sup> 0 ps<sup>3</sup> ð Þ <sup>=</sup>km , <sup>γ</sup> <sup>¼</sup> <sup>1</sup>:<sup>3</sup> � <sup>10</sup>�<sup>2</sup>ð Þ <sup>=</sup>km=<sup>W</sup> , <sup>t</sup><sup>0</sup> <sup>¼</sup> 80 fs ð Þ, <sup>z</sup> <sup>¼</sup> <sup>3</sup>:<sup>7</sup> � <sup>t</sup> 2 <sup>0</sup>= β<sup>2</sup> j j, <sup>β</sup><sup>2</sup> ¼ �21:7=150 ps2 ð Þ <sup>=</sup>km , <sup>u</sup><sup>0</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup> j j=γ=<sup>t</sup> 2 <sup>0</sup>. (a) s = 0.01 and (b) s = 0.0001.

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

Next, we take the higher-order nonlinear effect into account. Constructing another iteration related to δγ : <sup>γ</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup> <sup>þ</sup> δγ, u zð Þ¼ ;<sup>ω</sup> <sup>u</sup>0ð Þþ <sup>z</sup>;<sup>ω</sup> <sup>δ</sup>u zð Þ ;<sup>ω</sup> and

Now, we can simulate the pulse shape affected by high-order dispersive and

Firstly, we see what will be induced by the above items δβ<sup>3</sup> and δγ. To extrude their impact, we choose the other parameters to be small values in Figures 4 and 5. The deviation between the red and the black lines in Figure 4(a) indicates the impact of δβ<sup>3</sup> and δγ; that is, they induce the pulse's symmetrical split. This split does not belong to the SPM-induced broadening oscillation spectral or β3-induced oscillation in the tailing edge of the pulse, because here γ is very small and β<sup>3</sup> ¼ 0

the black line. When we reduce the s value to 0.0001 in (b), the split pulse's

Is the pulse split in Figure 4(a) caused by δβ<sup>3</sup> or δγ? The red lines in Figure 5 describe the evolution of pulse affected by the very small second-order dispersion and nonlinear (including self-steepening) coefficients. Here, δβ<sup>3</sup> induces the pulse's symmetrical split, and the maximum peaks of split pulse alter and vary from the spectral central to the edge and to the central again. Therefore, its effect is equal to

From the deviation between the red and black lines in Figure 5, we can also detect the impact of δγ. It only accelerates the pulse's split when the self-steepening effect is ignored (s = 0 in Figure 5(a)). This is similar to the self-phase modulationbroadening spectral and oscillation. The high nonlinear γ accelerating pulse's split is validated in [35, 36]. If s 6¼ 0 (Figure 5(b)), δγ simultaneously leads to the split

Generally, we do not take δγ into account, so we should clarify in which case it creates impact. Compared (c) with (b) in Figure 5, the red lines change little means that δβ<sup>3</sup> has a tiny relationship with γ. But with the increase of γ (Figure 5(c)), the

The pulse shapes with and without δβ<sup>3</sup> and δγ. The red line: without δβ<sup>3</sup> and δγ; the black line: with δβ<sup>3</sup> and

<sup>0</sup>. (a) s = 0.01 and (b) s = 0.0001.

<sup>3</sup> <sup>¼</sup> 0 ps<sup>3</sup> ð Þ <sup>=</sup>km , <sup>γ</sup> <sup>¼</sup> <sup>1</sup>:<sup>3</sup> � <sup>10</sup>�<sup>2</sup>ð Þ <sup>=</sup>km=<sup>W</sup> , <sup>t</sup><sup>0</sup> <sup>¼</sup> 80 fs ð Þ, <sup>z</sup> <sup>¼</sup> <sup>3</sup>:<sup>7</sup> � <sup>t</sup>

2

<sup>1</sup> � <sup>3</sup>is<sup>ω</sup> � <sup>1</sup>

<sup>2</sup>=t 2 <sup>0</sup>=<sup>2</sup> � �.

> 2 u � �

� �

� �

u0

∂u<sup>0</sup> ∂ω

dω

=∂t is also shown explicitly in

2 <sup>0</sup>= β<sup>2</sup> j j, (46)

repeating the above process, we get

nonlinear effects. Assume LD ¼ t

<sup>u</sup>ð Þ¼ <sup>0</sup>; <sup>t</sup> <sup>Ð</sup> <sup>þ</sup><sup>∞</sup>

symmetry is improved.

pulse's redshift.

Figure 4.

24

δγ. <sup>ν</sup> <sup>¼</sup> <sup>ω</sup>=2=π, <sup>β</sup><sup>0</sup>

<sup>β</sup><sup>2</sup> ¼ �21:7=150 ps2 ð Þ <sup>=</sup>km , <sup>u</sup><sup>0</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup> j j=γ=<sup>t</sup>

δγ≈exp

ðþ<sup>∞</sup> �∞

Nonlinear Optics ‐ Novel Results in Theory and Applications

�<sup>∞</sup> <sup>u</sup>ð Þ <sup>0</sup>;<sup>ω</sup> exp ð Þ �iω<sup>t</sup> <sup>d</sup><sup>ω</sup> <sup>¼</sup> <sup>u</sup><sup>0</sup> exp �<sup>t</sup>

[3]. The self-steepening effect attributing to is <sup>∂</sup> j j <sup>u</sup>

that of the fourth-order dispersion β<sup>4</sup> [33, 34, 3].

� 1 G ∂G <sup>∂</sup><sup>ω</sup> � <sup>3</sup>is

2 <sup>0</sup>=∣β2∣ and

Figure 5.

The evolutions of pulse. The red line: without δγ; the black line: with δβ<sup>3</sup> and δγ. (a) s ¼ 0, <sup>γ</sup> <sup>¼</sup> <sup>1</sup>:<sup>3</sup> � <sup>10</sup>�<sup>4</sup>ð Þ <sup>=</sup>km=<sup>W</sup> ; (b) <sup>s</sup> <sup>¼</sup> <sup>0</sup>:01, <sup>γ</sup> <sup>¼</sup> <sup>1</sup>:<sup>3</sup> � <sup>10</sup>�<sup>4</sup>ð Þ <sup>=</sup>km=<sup>W</sup> ; (c) <sup>s</sup> <sup>¼</sup> <sup>0</sup>:01, <sup>γ</sup> <sup>¼</sup> <sup>1</sup>:3ð Þ <sup>=</sup>km=<sup>W</sup> . Other parameters are the same as Figure 4.

split pulse's redshift is strengthened, so δγ has a relationship with γ. In Figure 6, the pulse is not split until z = 9 LD, and the black line with δγ is completely overlapped by the red line without δγ, so the high second-order dispersion β<sup>2</sup> results in the impact of δγ covered and the impact of δβ<sup>3</sup> weakened. Therefore, only in the zerodispersion regime, δγ should be taken into account in the simulation of pulse shape.

So, we can utilize δβ<sup>3</sup> to determine the fourth-order dispersion coefficient β4. Fiber parameters are listed in Table 1. The process is shown in Figure 7, and the dispersion operator including <sup>β</sup><sup>4</sup> is <sup>D</sup>^ <sup>¼</sup> <sup>i</sup> <sup>2</sup>ω<sup>2</sup>β<sup>2</sup> � <sup>i</sup> <sup>6</sup>ω<sup>3</sup>β<sup>3</sup> <sup>þ</sup> <sup>i</sup> <sup>24</sup>ω<sup>4</sup>β4.

Table 2 is the average of β4. They are different from those determined by FWM or MI where β<sup>4</sup> is related to power and broadening frequency [35, 36]. By our method, the fourth-order dispersion is also a function of distance, and every type of

fibers has its special average β<sup>4</sup> which reveals the characteristic of fibers. These values are similar to those experiment results in highly nonlinear fibers [35, 36]. Although we take the higher-order nonlinear effect δγ into account which upgrades

u have a very tiny contribution to β4, only 10�<sup>26</sup> ps4

order for the typical SMF. Here, the impact of δγ is hidden by the relative strong β2.

2 u � �=∂<sup>t</sup> and

½ � ulð Þþ z; t Alð Þ z; t exp �iω<sup>l</sup> ð Þt (47)

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 com-

ujð Þþ z; t Ajðz; tÞ

ujð Þþ z; t Ajðz; tÞ

1 2 β2ω<sup>2</sup>

� � � � �

2 AlR

ujð Þþ z; t Ajð Þ z; t

� � � � �

2

Alð Þ z; t

� � � � �

<sup>l</sup> AlRð Þþ z; t

� � � � � 3 5φ þ 1 2

2 AlI (48)

(49)

(50)

β2φ<sup>00</sup> (51)

∂ ∂t � �Alð Þ� <sup>z</sup>; <sup>t</sup>

> AlIð Þ z; t ∂t<sup>2</sup> �

> > N j¼1

� � � � �

AlRð Þ z; t ∂t<sup>2</sup> þ

ujð Þþ z; t Ajðz; tÞ

N j¼1

� � � � �

<sup>2</sup> 2

N j¼1

� � � � � /km quantity

the pulse's symmetrical split and redshift, the items is <sup>∂</sup> j j <sup>u</sup>

4. Traveling wave solution of NLSE for ASE noise

U zð Þ¼ ; t ∑

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>

<sup>∂</sup>Alð Þ <sup>z</sup>; <sup>t</sup> <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup>

<sup>∂</sup><sup>z</sup> ¼ �β2ω<sup>l</sup>

<sup>∂</sup><sup>z</sup> ¼ �β2ω<sup>l</sup>

1 2 β2ω<sup>2</sup>

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

1 2 β2ω<sup>2</sup>

4

i

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

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

AlI ¼ φ ξð Þ, and ξ ¼ t � cz.

ð Þ¼� β2ω<sup>l</sup> � c

ϕ0

27

4.1 The in-phase and quadrature components of ASE noise

N l¼1

statistically real independent stationary white Gaussian processes, and

<sup>2</sup> �ω<sup>2</sup>

<sup>l</sup> þ ∂2 <sup>∂</sup>t<sup>2</sup> � <sup>i</sup>2ω<sup>l</sup>

γð Þz exp ð Þ �2αz ∑

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

þ 1 2 β2 ∂2

<sup>l</sup> AlI � γ exp ð Þ �2αz ∑

We now seek their traveling wave solution by taking [37] AlR ¼ ϕ ξð Þ,

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

<sup>∂</sup>AlIð Þ <sup>z</sup>; <sup>t</sup> <sup>∂</sup><sup>t</sup> � <sup>1</sup> 2 β2 ∂2

> N j¼1

<sup>l</sup> þ γ exp ð Þ �2αz ∑

� � � � �

γ exp ð Þ �2αz ∑

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

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

plete 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:

iδγ exp ð Þ �2αz j j u

AlRð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>A</sup><sup>∗</sup>

2

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

Nonlinear Schrödinger Equation
