3.3 Estimation of the fourth-order dispersion coefficient β<sup>4</sup>

The NLSE governing the wave's transmission in fibers is

$$\frac{\partial u}{\partial \mathbf{z}} + \frac{i}{2} \rho\_2 \frac{\partial^2 u}{\partial t^2} - \frac{1}{6} \rho\_3 \frac{\partial^3 u}{\partial t^3} - i\gamma \exp\left(-2a\mathbf{z}\right) \left[ \left| u \right|^2 u + i\epsilon \frac{\partial \left| u \right|^2}{\partial t} u + i\epsilon \left| u \right|^2 \frac{\partial u}{\partial t} \right] = \mathbf{0} \tag{41}$$

where s is the self-steepening parameter. In the frequency domain, its solution is

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

## Figure 3.

where ϕ<sup>m</sup> ¼ exp ð Þ irmt , m ¼ 1, 2, 3 and c1, c2, c<sup>3</sup> are determined by the initial

<sup>0</sup> ð Þ¼ <sup>a</sup>1φ<sup>1</sup> <sup>þ</sup> <sup>a</sup>2φ<sup>2</sup> <sup>þ</sup> <sup>a</sup>3φ3, t><sup>t</sup>

1 t <sup>0</sup> ð Þþ b2ϕ<sup>0</sup>

<sup>0</sup> ð Þ� b2ϕ″

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

<sup>2</sup> <sup>ϕ</sup>ð Þ<sup>1</sup> 3

<sup>2</sup> <sup>ϕ</sup>ð Þ<sup>2</sup> 3

<sup>0</sup> ð Þ ; E V t<sup>0</sup> ð Þ

<sup>0</sup> ð Þ ; E V t<sup>0</sup> ð Þ

ð G<sup>0</sup> t l ; t

⋯

The accuracy can be estimated by the last item of (40). The algorithm is plotted

2

where s is the self-steepening parameter. In the frequency domain, its solution is

<sup>u</sup> <sup>þ</sup> is <sup>∂</sup>j j <sup>u</sup>

2 ∂t

" #

<sup>u</sup> <sup>þ</sup> is uj j<sup>2</sup> <sup>∂</sup><sup>u</sup>

∂t

<sup>0</sup> ð Þ¼ b1ϕ<sup>1</sup> t

<sup>0</sup> ð Þ¼ b1ϕ<sup>0</sup>

1 t

ϕð Þ<sup>1</sup> <sup>1</sup> <sup>ϕ</sup>ð Þ<sup>1</sup>

ϕð Þ<sup>2</sup> <sup>1</sup> <sup>ϕ</sup>ð Þ<sup>2</sup>

Finally, the solution of (27) can be written with the eigen function and Green

<sup>0</sup> ð Þ¼ δ t � t

b1φ<sup>1</sup> þ b2φ<sup>2</sup> þ b3φ3, t < t

<sup>0</sup> ð Þþ b2ϕ<sup>2</sup> t

2 t

2 t <sup>0</sup> ð Þþ b3ϕ<sup>0</sup>

<sup>0</sup> ð Þ� b3ϕ″

W t<sup>0</sup> ð Þ , a<sup>3</sup> <sup>¼</sup> <sup>φ</sup>1φ\_ <sup>2</sup> � <sup>φ</sup>\_ <sup>1</sup>φ<sup>2</sup>

� � � � � � �

> ð G<sup>0</sup> t 0 ; t ″ ; E � �V t″ � �φ t

> ð G<sup>0</sup> t 0 ; t ″ ; E � �V t″ � �ϕ t

<sup>l</sup>þ<sup>1</sup> � �V t<sup>l</sup>þ<sup>1</sup> � �φ t

<sup>0</sup> ð Þ (33)

(34)

(39)

″ � �dt″

<sup>l</sup>þ<sup>1</sup> � �dt<sup>l</sup>þ<sup>1</sup>

″ � �dt″ <sup>þ</sup> <sup>⋯</sup>

(40)

¼ 0 (41)

<sup>0</sup> ð Þ (35)

<sup>0</sup> ð Þ (36)

<sup>0</sup> ð Þ¼�6i=β<sup>3</sup> (37)

W t<sup>0</sup> ð Þ (38)

0

0

<sup>0</sup> ð Þþ b3ϕ<sup>3</sup> t

3 t

3 t

<sup>E</sup> � <sup>H</sup>^ <sup>0</sup>ð Þ<sup>t</sup> � �G<sup>0</sup> <sup>t</sup>; <sup>t</sup>

�

3 t

<sup>0</sup> ð Þ� b1ϕ″

W t<sup>0</sup> ð Þ¼

<sup>0</sup> ð Þdt<sup>0</sup>

ð G<sup>0</sup> t 0 ; t ″ � �V t″ � �dt″


3.3 Estimation of the fourth-order dispersion coefficient β<sup>4</sup>

The NLSE governing the wave's transmission in fibers is

<sup>∂</sup>t<sup>3</sup> � <sup>i</sup><sup>γ</sup> exp ð Þ �2α<sup>z</sup> j j <sup>u</sup>

<sup>0</sup> ð Þdt<sup>0</sup> þ

<sup>0</sup> ð Þdt<sup>0</sup> þ

ð dt<sup>0</sup> G<sup>0</sup> t; t

ð dt<sup>0</sup> G<sup>0</sup> t; t

W t<sup>0</sup> ð Þ , a<sup>2</sup> <sup>¼</sup> <sup>φ</sup>3φ\_ <sup>1</sup> � <sup>φ</sup>\_ <sup>3</sup>φ<sup>1</sup>

� � � � � � �

pulse. The Green function of (30) is

By the construction method, it is

0

<sup>0</sup> ð Þþ a2ϕ<sup>2</sup> t

2 t

Let b<sup>1</sup> ¼ b<sup>2</sup> ¼ b<sup>3</sup> ¼ 0, then

At the point t ¼ t

a1ϕ<sup>1</sup> t

<sup>0</sup> ð Þþ a2ϕ″

a1ϕ<sup>0</sup> 1 t <sup>0</sup> ð Þþ a2ϕ<sup>0</sup>

a1ϕ″ 1 t

function:

φðÞ ¼ t ϕðÞþt

in Figure 3.

∂u ∂z þ i 2 β2 ∂2 u <sup>∂</sup>t<sup>2</sup> � <sup>1</sup> 6 β3 ∂3 u

22

¼ ϕðÞþt

¼ ϕðÞþt

ð G<sup>0</sup> t; t

ð G<sup>0</sup> t; t

ð G<sup>0</sup> t; t

þ ð dt<sup>0</sup> G<sup>0</sup> t; t <sup>0</sup> ð ÞV t<sup>0</sup> ð Þ

G<sup>0</sup> t; t

Nonlinear Optics ‐ Novel Results in Theory and Applications

, there are

2 t <sup>0</sup> ð Þþ a3ϕ<sup>0</sup>

<sup>0</sup> ð Þþ a3ϕ″

<sup>a</sup><sup>1</sup> <sup>¼</sup> <sup>φ</sup>2φ\_ <sup>3</sup> � <sup>φ</sup>\_ <sup>2</sup>φ<sup>3</sup>

<sup>0</sup> ð ÞV t<sup>0</sup> ð Þφ t

<sup>0</sup> ð Þ ; E V t<sup>0</sup> ð Þϕ t

<sup>0</sup> ð Þ ; E V t<sup>0</sup> ð Þϕ t

<sup>0</sup> ð Þþ a3ϕ<sup>3</sup> t

3 t

The Green algorithm for solving NLSE.

$$u(z+dz,\alpha) = \exp\left(dz\hat{D}\right)\exp\left(dz\hat{N}\right)u(z,\alpha)\tag{42}$$

where <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>β3, <sup>N</sup>^ <sup>¼</sup> <sup>Γ</sup> <sup>i</sup><sup>γ</sup> exp ð Þ �2α<sup>z</sup> j j <sup>u</sup> <sup>2</sup> <sup>þ</sup> is <sup>∂</sup>j j <sup>u</sup> <sup>2</sup> <sup>∂</sup><sup>t</sup> <sup>þ</sup> is uj j<sup>2</sup> <sup>∂</sup> ∂t n o h i , and <sup>Γ</sup> represents the Fourier transform [32]. Let <sup>L</sup>^ <sup>¼</sup> <sup>∂</sup> <sup>∂</sup><sup>z</sup> � <sup>D</sup>^ � <sup>N</sup>^ and LG z ^ ; z 0 ;<sup>ω</sup> � � <sup>¼</sup> <sup>δ</sup> <sup>z</sup> � <sup>z</sup> � �<sup>0</sup> ; we obtain the Green function

$$G\left(z, z', \alpha\right) = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \frac{\exp\left[-ik\left(z - z'\right)\right]}{ik - \hat{D} - \hat{N}} dk \tag{43}$$

Constructing the iteration <sup>β</sup><sup>3</sup> <sup>¼</sup> <sup>β</sup><sup>0</sup> <sup>3</sup> <sup>þ</sup> δβ3, u zð Þ¼ ;<sup>ω</sup> <sup>u</sup><sup>0</sup>ð Þþ <sup>z</sup>;<sup>ω</sup> <sup>δ</sup>u zð Þ ;<sup>ω</sup> , then there is

$$\delta u(z,\alpha) = \int G(z, z', \alpha) Z(z', \alpha, \delta \beta\_3 \left( z' \right), u^0 \left( z', \alpha \right)) dz' \tag{44}$$

where Z z<sup>0</sup> ;ω; δβ<sup>3</sup> z � �<sup>0</sup> ; u<sup>0</sup> z 0 ;<sup>ω</sup> � � � � ¼ � <sup>i</sup> <sup>6</sup> δβ<sup>3</sup> z � �<sup>0</sup> ω<sup>3</sup>u<sup>0</sup> z 0 ;ω � � and u<sup>0</sup> z 0 ;ω; β<sup>0</sup> 3 � � is determined by (42).

The minimum value of <sup>δ</sup>u zð Þ ;<sup>ω</sup> satisfies <sup>∂</sup>δu zð Þ ;<sup>ω</sup> <sup>=</sup>∂<sup>ω</sup> <sup>¼</sup> <sup>0</sup>, R <sup>∂</sup><sup>2</sup> <sup>δ</sup>u zð Þ ;<sup>ω</sup> <sup>=</sup>∂ω<sup>2</sup> � �>0, so

$$\delta\beta\_3 = \exp\left[\int\_{-\infty}^{+\infty} \left(-\frac{1}{G}\frac{\partial G}{\partial \alpha} - \frac{3}{\alpha} - \frac{1}{u^0}\frac{\partial u^0}{\partial \alpha}\right) d\alpha\right] \tag{45}$$

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 repeating the above process, we get

$$\delta\gamma \approx \exp\left[\int\_{-\infty}^{+\infty} \left(-\frac{\mathbf{1}}{G} \frac{\partial G}{\partial \nu} - \frac{\mathbf{3}\dot{\mathbf{s}}}{\mathbf{1} - \mathbf{3}\dot{\mathbf{s}}\alpha} - \frac{\mathbf{1}}{\mathbf{u}^{0}} \frac{\partial \boldsymbol{u}^{0}}{\partial \alpha}\right) d\alpha\right] \tag{46}$$

Now, we can simulate the pulse shape affected by high-order dispersive and nonlinear effects. Assume LD ¼ t 2 <sup>0</sup>=∣β2∣ and

<sup>u</sup>ð Þ¼ <sup>0</sup>; <sup>t</sup> <sup>Ð</sup> <sup>þ</sup><sup>∞</sup> �<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> <sup>2</sup>=t 2 <sup>0</sup>=<sup>2</sup> � �.

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 [3]. The self-steepening effect attributing to is <sup>∂</sup> j j <sup>u</sup> 2 u � �=∂<sup>t</sup> is also shown explicitly in the black line. When we reduce the s value to 0.0001 in (b), the split pulse's symmetry is improved.

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

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 pulse's redshift.

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

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

<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

The evolutions of pulse. The red line: without δγ; the black line: with δβ<sup>3</sup> and δγ. (a) s ¼ 0,

<sup>2</sup>ω<sup>2</sup>β<sup>2</sup> � <sup>i</sup>

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

<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.

dispersion operator including <sup>β</sup><sup>4</sup> is <sup>D</sup>^ <sup>¼</sup> <sup>i</sup>

parameters are the same as Figure 4.

Nonlinear Schrödinger Equation

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

Figure 5.

25
