Appendix A: Derivation of the Generalized Nonlinear Schrödinger Equation
A.1 Wave Equation in Nonlinear Optics
In order to derive the wave equation for the propagation of light in a nonlinear optical medium, we begin from the Maxwell’s equations for dielectric medium written as follows:
where, are the electric and magnetic fields, respectively. The electric and magnetic flux densities and, respectively, are related to the electric and magnetic fields via
where ε0 is the permittivity of free space, μ0 is the permeability of free space, and is the induced electric polarization.
By taking curl of (A.4) and using (A.3) we obtain
Substitution of (A.2) into (A.7) yields the generic wave equation:
where c is the speed of light in a vacuum and is given by,
the induced polarization consists of linear and nonlinear components as
which are defined as
where 
and 
are the first- and third-order susceptibility tensors.
A.2 Generalized Nonlinear Schrödinger Equation (NSE)
The starting point for the derivation of the NSE is the wave equation (A.8). In order to cover a larger number of nonlinear effects, the general form of nonlinear polarization in (A.11) must be used, and an approximation of the is given by
where R(t) is the nonlinear response function normalized in a manner similar to the delta function (i.e., ). By introducing (A.12) into (A.11) with the slowly varying approximation, the nonlinear polarization in scalar form is given by
The assumptions in (A.13) are also applied for simplification of the NSE derivation. It will be clearer to describe numerous effects in frequency domain with using the following Fourier transforms:
In the frequency domain, the convolution of Equation (A.15) becomes a simple multiplication and the time derivatives can be replaced by and Hence, a modified Helmholtz equation can be derived from (A.9) by using (A.10) and (A.13) and taking the Fourier transform to give
whereare Fourier transforms ofin the time domain. Equation A.16 can be solved by using the method of separation of variables. The slowly varying part of the electric fieldis approximated by
where A(z,ω) is the slowly varying function of z, β0 is the wave number, and F(x,y) is the function of the transverse field distribution that is assumed to be independent of ω. Then substituting (A.17) into (A.16), the Helmholtz equation is split into two equations:
Equation (A.18) is an eigenvalue equation that needs to be solved for the wave number β and the fiber modes. In (A.18), can be approximated by, where Δn is a small perturbation and can be determined from.
In the case of a single mode, (B.19) can be solved using the first-order perturbation theory in which does not affect F(x,y), but only the eigenvalues. Hence, β in (A.19) becomes, where accounts for the effect of the perturbation term (referred to ) to change the propagation constant for the fundamental mode. Using (A.17) and (A.15), the electric field can be approximated by
where A(z,t) is the slowly varying complex envelope propagating along z in the optical fiber. From (A.19) after integrating over x and y, the following equation in the frequency domain is obtained:
where R(ω) is the Fourier transform of R(t), the nonlinear coefficient γ that has been introduced is given by
and where Aeff is the effective area of the optical fiber and given by
Equation (A.21) is the NSE that describes generally the pulse propagation in the frequency domain. It is useful to take into account the frequency dependence of the effects of the propagation constant β, the loss α, and the nonlinear coefficient γ by expanding them in the Taylor series as
However, the pulse spectrum in most cases of practical interest is narrow enough such that γ and α are constant over the pulse spectrum. Therefore, the NSE in the time domain can be obtained by using the inverse Fourier transform:
Equation (A.27) is the basic propagation equation, commonly known as the generalized nonlinear Schrödinger equation (NLSE) that is very useful for studying the evolution of the amplitude of the optical signal and the phase of the lightwave carrier under most effects of third-order nonlinearity in optical waveguides as well as optical fibers.