2.1 Theory for continuous wave

The NLSE governing the field in nonlinear and dispersion medium is

$$\frac{\partial u}{\partial z} + \beta\_1 \frac{\partial u}{\partial t} + \frac{i}{2} \beta\_2 \frac{\partial^2 u}{\partial t^2} + \frac{a}{2} u = i\gamma \left[ |u|^2 + 2|u'|^2 \right] u \tag{1}$$

where β<sup>1</sup> and β<sup>2</sup> are the dispersions, γ is the nonlinear coefficient, and α is the fiber loss. In the frequency domain, the solution is

$$
\mu(z+dz,\alpha) = \exp\left(dz\hat{D}\right)\exp\left(dz\hat{N}\right)\mu(z,\alpha)\tag{2}
$$

where <sup>D</sup>^ <sup>¼</sup> <sup>i</sup> <sup>2</sup>ω<sup>2</sup>β<sup>2</sup> <sup>þ</sup> <sup>i</sup>ωβ<sup>1</sup> � <sup>a</sup> <sup>2</sup> and <sup>N</sup>^ <sup>¼</sup> <sup>i</sup><sup>γ</sup> j j <sup>u</sup> <sup>2</sup> <sup>þ</sup> <sup>i</sup><sup>2</sup> <sup>u</sup><sup>0</sup> j j<sup>2</sup> h i [19] (Figure 1). Usually, the field amplitudes can be written as

$$\mu(z,\alpha) = \sqrt{P(z,\alpha)} \exp\left[i\phi(z,\alpha)\right] \tag{3}$$

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

Figure 1.

the algebraic modification projected the extended NLSE as a frictional problem and

Since the numerical computation of solving NLSE is a huge time-consuming process, the fast algorithms and efficient implementations, focusing on (i) an accurate numerical integration scheme and (ii) an intelligent control of the longitudinal

The finite differential method [10] and the pseudo-spectral method [11] were adopted to increase accuracy and efficiency and suppress numerically induced spurious effects. The adaptive spatial step size-controlling method [12] and the predictor-corrector method [13] were proposed to speed up the implementation of split-step Fourier method (SSFM). The cubic (or higher order) B-splines were used to handle nonuniformly sampled optical pulse profiles in the time domain [14]. The Runge-Kutta method in the interaction picture was applied to calculate the effective

Recently, the generalized NLSE, taking into account the dispersion of the transverse field distribution, is derived [16]. By an inhomogeneous quasi-linear firstorder hyperbolic system, the accurate simulations of the intensity and phase for the Schrödinger-type pulse propagation were obtained [17]. It has been demonstrated that modulation instability (MI) can exist in the normal GVD regime in the higher-

In this chapter, several methods to solve the NLSE will be presented: (1) The small-signal analysis theory and split-step Fourier method to solve the coupled NLSE problem, the MI intensity fluctuation caused by SPM and XPM, can be derived. Furthermore, this procedure is also adapted to NLSE with high-order dispersion terms. The impacts of fiber loss on MI gain spectrum can be discussed. The initial stage of MI can be described, and then the whole evolution of MI can also be discussed in this way; (2) the Green function to solve NLSE in the time domain. By this solution, the second-, third-, and fourth-order dispersion coefficients is discussed; and (3) the traveling wave solution to solve NLSE for ASE noise and its

refractive index, effective area, dispersion, and nonlinear coefficients [15].

order NLSE in the presence of non-Kerr quintic nonlinearities [18].

2. Small-signal analysis solution of NLSE for MI generation

The NLSE governing the field in nonlinear and dispersion medium is

where β<sup>1</sup> and β<sup>2</sup> are the dispersions, γ is the nonlinear coefficient, and α is the

<sup>2</sup> and <sup>N</sup>^ <sup>¼</sup> <sup>i</sup><sup>γ</sup> j j <sup>u</sup>

u zð Þ¼ ;<sup>ω</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u ¼ iγ j j u

u zð Þ¼ <sup>þ</sup> dz;<sup>ω</sup> exp dzD^ � � exp dzN^ � �u zð Þ ;<sup>ω</sup> (2)

<sup>2</sup> <sup>þ</sup> <sup>i</sup><sup>2</sup> <sup>u</sup><sup>0</sup> j j<sup>2</sup> h i

<sup>2</sup> <sup>þ</sup> <sup>2</sup> <sup>u</sup><sup>0</sup> j j<sup>2</sup> h i

P zð Þ ;<sup>ω</sup> <sup>p</sup> exp ½ � <sup>i</sup>ϕð Þ <sup>z</sup>;<sup>ω</sup> (3)

u (1)

[19] (Figure 1).

successfully solved the soliton transmission problems [9].

Nonlinear Optics ‐ Novel Results in Theory and Applications

spatial step size, are required.

probability density function.

2.1 Theory for continuous wave

where <sup>D</sup>^ <sup>¼</sup> <sup>i</sup>

16

∂u ∂z þ β<sup>1</sup> ∂u ∂t þ i 2 β2 ∂2 u ∂t<sup>2</sup> þ a 2

fiber loss. In the frequency domain, the solution is

<sup>2</sup>ω<sup>2</sup>β<sup>2</sup> <sup>þ</sup> <sup>i</sup>ωβ<sup>1</sup> � <sup>a</sup>

Usually, the field amplitudes can be written as

Schematic illustration of medium. u(z, t) and u(z + dz, t) correspond to the field amplitudes at z and z + dz, respectively.

<sup>ϕ</sup>ð Þ <sup>z</sup>;<sup>ω</sup> is caused by the nonlinear effect, and <sup>ϕ</sup>ð Þ¼ <sup>z</sup>;<sup>ω</sup> <sup>Ð</sup> <sup>z</sup> <sup>0</sup> γ P zð Þþ ;ω 2P<sup>0</sup> ½ � ð Þ z;ω dz [3].

u zð Þ þ dz;ω (is)

$$u(z+dz,\omega) = \exp\left(dz\hat{D}\right)\sqrt{P(z,\omega)}\exp\left\{i\rho(z,\omega) + i\gamma[P+2P']dz\right\}$$

$$= e^{-adz/2}\exp\left(\beta\_1\alpha dz\right)\exp\left(\beta\_2/2\alpha^2 dz\right)\sqrt{P(z,\omega)}e^{i\rho(z+dz,\omega)}\tag{4}$$

$$=\sqrt{P(z+dz,\omega)}\exp\left[i\rho(z+dz,\omega)\right]$$

Assuming: P zð Þ¼ ;ω h i P zð Þ þ ΔP zð Þ ;ω

h i P zð Þ is the average signal intensity. ΔP zð Þ ;ω is the noise or modulation term. There is [20] h i P zð Þ ΔP zð Þ ;ω

The amplitude ffiffiffiffiffiffiffiffiffiffiffiffiffiffi P zð Þ ;<sup>ω</sup> <sup>p</sup> can be regarded as

$$\sqrt{P(z,\alpha)} \approx \sqrt{\langle P(z) \rangle} \left( 1 + \frac{\Delta P(z,\alpha)}{2P(z)} \right) \tag{5}$$

The small-signal theory implies that the frequency modulation or noise <sup>φ</sup>\_ð Þ¼ <sup>z</sup> <sup>þ</sup> dz;<sup>ω</sup> <sup>d</sup>φ\_ð Þ <sup>z</sup>þdz;<sup>ω</sup> dt is small enough. Finally ([21])

$$\begin{split} P(\mathbf{z} + d\mathbf{z}, \boldsymbol{\omega}) &= \langle P(\mathbf{z}) \rangle + \mathbf{2} e^{-ad\mathbf{z}/2} \times \\ &\quad \text{Re}\left\{ \langle P(\mathbf{z}) \rangle \exp\left( i\alpha\beta\_1 dz + i\alpha^2 \beta\_2 dz \right) \left[ \frac{\Delta P(\mathbf{z}, \boldsymbol{\omega})}{2\langle P(\mathbf{z}) \rangle} + i\rho(\mathbf{z} + d\mathbf{z}, \boldsymbol{\omega}) \right] \right\} \end{split} \tag{6}$$

The operation exp <sup>i</sup>ωβ1dz <sup>þ</sup> <sup>i</sup>ω<sup>2</sup> ð Þ <sup>β</sup>2dz can be split into its real and imaginary parts:

$$\exp\left(i\alpha\beta\_1d\mathbf{z} + i\alpha^2\beta\_2d\mathbf{z}\right) = \cos\left(\alpha\beta\_1d\mathbf{z} + \alpha^2\beta\_2d\mathbf{z}\right) + i\sin\left(\alpha\beta\_1d\mathbf{z} + \alpha^2\beta\_2d\mathbf{z}\right) \tag{7}$$

The modulation or noise ΔP zð Þ þ dz;ω is ΔP zð Þ þ dz;ω ≈P zð Þ� þ dz;ω h i P zð Þ So

$$\begin{aligned} P(\mathbf{z} + d\mathbf{z}, \boldsymbol{\omega}) &= e^{-ad\mathbf{z}/2 - i\alpha\beta\_1 d\mathbf{z}} \\ &\left[ \cos\left(\frac{1}{2}\beta\_2 \boldsymbol{\rho}^2 d\mathbf{z}\right) \Delta P(\mathbf{z}, \boldsymbol{\omega}) + \sin\left(\frac{1}{2}\beta\_2 \boldsymbol{\omega}^2 d\mathbf{z}\right) 2\langle P(\mathbf{z})\rangle \rho(\mathbf{z} + d\mathbf{z}, \boldsymbol{\omega}) \right] \end{aligned} \tag{8}$$

And

$$\begin{pmatrix} \frac{\Delta P(z+dz,\alpha)}{2\langle P(z)\rangle} \\\\ \varrho(z+dz,\alpha) \end{pmatrix} = e^{-adx/2-i\alpha\beta\_1 dx} e^{i\gamma(P(x))+2\langle P(z)\rangle)dx} $$
 
$$ \begin{pmatrix} \cos\left(\frac{1}{2}\beta\_2\alpha^2 dz\right) & -\sin\left(\frac{1}{2}\beta\_2\alpha^2 dz\right) \\\\ \sin\left(\frac{1}{2}\beta\_2\alpha^2 dz\right) & \cos\left(\frac{1}{2}\beta\_2\alpha^2 dz\right) \end{pmatrix} \begin{pmatrix} \Delta P(z,\alpha) \\\\ \Delta(P(z)) \\\\ \varrho(z,\alpha) \end{pmatrix} \tag{9} $$

Figure 2 shows a comparison of the gain spectra between Eq. (11) and [6]

<sup>2</sup> <sup>β</sup>2ω2dz <sup>¼</sup> <sup>π</sup>, which makes Eq. (11) have the same frequency regime as [26]. In Figure 2, the curves are different but have the same maximum value of gMI. In practice, researchers generally utilize the maximum value of gMI to estimate the amplified noises and SNR [3]. The result of small-signal analysis in Figure 2 has a phase delay of around ω0. Compared with the experiment result of [27], the reason is taking the fiber loss into account, the gain spectrum exhibits a phase delay close to ω0, and the curve descends a little [27]. Fiber loss results in the difference of gMI between the small-signal analysis method and the

2.2 The general theory on cross-phase modulation (XPM) intensity fluctuation

For the general case of two channels, the input optical powers are denoted by

resulting in intensity fluctuation. According to [4], the whole length L is separated

Lwo ¼ Δt=ð Þ DΔλ . Δt is the edge duration of the carrier wave, D is the dispersion coefficient, and Δλ is the wavelength spacing between the channels. By the smallsignal analysis, the phase modulation in channel 1 originating in dz at z can be

dϕXPMð Þ¼ z; t γ2P<sup>0</sup> z; t � zβ<sup>0</sup>

This phase shift is converted to an intensity fluctuation through the group velocity dispersion (GVD) from z to the receiver. So, at the fiber output, the intensity fluctuation originating in dz in the frequency domain is

�a Lð Þ �<sup>z</sup> � <sup>e</sup>

�a Lð Þ �<sup>z</sup> � <sup>e</sup>

�a Lð Þ �<sup>z</sup> � <sup>e</sup>

�az � <sup>e</sup> iωβ<sup>0</sup> <sup>1</sup><sup>z</sup> � <sup>e</sup>

of light. At the fiber output, the XPM-induced intensity fluctuation is the integral of

�az � <sup>e</sup> iωβ<sup>0</sup> <sup>1</sup><sup>z</sup> � <sup>e</sup>

The walk-off between co-propagating waves is regulated by the convolution

linearly along the fibers, and dispersion acts on the phase-modulated signal

into two parts 0 < z < Lwo and Lwo < z < L; Lwo is the walk-off length,

ð Þt , respectively [28]. Only in the first walk-off length, the nonlinear interaction (XPM) is taken into account; in the remaining fibers, signals are propagated

> 1 � �e

n o

<sup>i</sup>ωβ1ð Þ <sup>L</sup>�<sup>z</sup> sin ½ � b Lð Þ � <sup>z</sup> <sup>d</sup>φXPMð Þ <sup>z</sup>;<sup>ω</sup>

<sup>i</sup>ωβ1ð Þ <sup>L</sup>�<sup>z</sup> P<sup>0</sup>

n odz

<sup>i</sup>ωβ1ð Þ <sup>L</sup>�<sup>z</sup> P<sup>0</sup>

n odz

�azdz (12)

ð Þ z;ω sin ½ � b Lð Þ � z

=ð Þ 4πc , where c is the speed

ð Þ z;ω sin ½ � b Lð Þ � z

(13)

(14)

<sup>2</sup> <sup>β</sup>2ω2dz <sup>¼</sup> <sup>π</sup> [22, 23], and the maximum

<sup>4</sup>γh i P zð Þ <sup>=</sup> <sup>β</sup><sup>2</sup> j j <sup>p</sup> , so the choice of dz satisfies

for the case h i P zð Þ = P<sup>0</sup> h i ð Þz ¼ 1. The maximum frequency modulation index

caused by dispersion corresponds to <sup>1</sup>

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

Nonlinear Schrödinger Equation

perturbation approach.

1

P tð Þ, P<sup>0</sup>

expressed as

given by [29].

PXPM ¼

operation.

19

ðL 0

¼ ðL 0 4γ e

dPXPMð Þ¼ z;ω 2 e

¼ 4γ e

Eq. (13) with z ranging from 0 to L:

dPXPMð Þ z; ω dz

<sup>i</sup>ωzβ1P zð Þ ;<sup>ω</sup> � �<sup>⊗</sup> <sup>e</sup>

<sup>i</sup>ωzβ1P zð Þ ;<sup>ω</sup> � �<sup>⊗</sup> <sup>e</sup>

<sup>i</sup>ωzβ1P zð Þ ; <sup>ω</sup> � �<sup>⊗</sup> <sup>e</sup>

<sup>⊗</sup> representing the convolution operation <sup>b</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>Dλ<sup>2</sup>

value of the sideband is <sup>ω</sup><sup>c</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

When only intensity modulation is present and no phase modulation exists, the transfer function cos <sup>1</sup> <sup>2</sup> <sup>β</sup>2ω2dz � � is obtained. The 3 dB cutoff frequency corresponds to <sup>1</sup> <sup>2</sup> <sup>β</sup>2ω2dz <sup>¼</sup> <sup>π</sup>=4 in [22, 23]. This treatment is also adaptable to the case that only the nonlinear phase (frequency) modulation is present; then, the intensity modulation ΔP zð Þ þ dz;ω due to FM-IM conversion is given as

$$
\Delta P(z+dz,\omega) = 2\langle P(z)\rangle e^{-adz/2-i\alpha\beta\_1 dx} \sin\left(\frac{1}{2}\beta\_2 dz \alpha^2\right) \rho(z+dz,\omega) \tag{10}
$$

This is in very good agreement with [24] for small-phase modulation index. Even for large modulation index <sup>1</sup> <sup>2</sup> <sup>β</sup>2ω<sup>2</sup>dz <sup>¼</sup> <sup>π</sup>=2, the difference is within 0.5 dB. Eq. (10) does not include a Bessel function, so it is simpler than that in [24].

Obviously, the above process can be used to treat NLSE with higher-order dispersion (β3, β4) [25]. Similarly, the result in Eq. (10) will include ω<sup>3</sup> and ω<sup>4</sup> .

The corresponding MI gain gMI in the side bands of ω<sup>0</sup> (the frequency of signal) is given by

$$\begin{split} \mathbf{g}\_{MI}(\mathbf{z}, \boldsymbol{\omega}) &= \frac{|\Delta P(\mathbf{z} + d\mathbf{z}, \boldsymbol{\omega}) - \Delta P(\mathbf{z}, \boldsymbol{\omega})|}{\langle P(\mathbf{z}) \rangle d\mathbf{z}} \\ &= 2e^{-ad\mathbf{z}/2 - i\alpha\beta\_1 d\mathbf{z}} \sin\left(\frac{1}{2}\beta\_2 d\mathbf{z}\boldsymbol{\omega}^2\right) \left\{\gamma \int\_{\mathbf{z}}^{\mathbf{z} + d\mathbf{z}} [P(\mathbf{z}, \boldsymbol{\omega}) + 2\mathbf{P}'(\mathbf{z}, \boldsymbol{\omega})] d\mathbf{z}\right\} \Big/ \left\{d\mathbf{z}\right\} \end{split} \tag{11}$$
