Finite Element Analysis

The objective of this project is to study the Nonlinear Schrodinger equation (NLSE), which governs the propagation of optical pulses through a single mode optical fiber and use matlab to learn how to find the solution in the frequency domain and how to make use of the Fourier transform to solve the equation and represent in time domain solution.

Theory:-

The Nonlinear Schrodinger equation (NLSE), which governs the propagation of optical pulses through a single mode optical fiber [1-2], can be written as

A(z,t)/ t = (L+N(A(z,t)))A(z,t) -à (1)

Where operator L is equal to

Where L and N are the linear and nonlinear operators respectively, A (z, t) is the complex envelope of slowly time-varying optical pulse. Note that D denotes dispersion operator. Parameters β2, β3, α and γ represent fiber group velocity dispersion (GVD) GVD slope, attenuation, and nonlinearity, respectively. Please note that z is propagation distance and t = τ – z/vg is the retarded time, where τ is physical time and vg is the group velocity. Eq.1 is derived using an electrical engineering notation for complex envelop and Fourier transform, denoted by Equation 2 and equation 3 respectively.

E (t, z) =Re [A (t, z) e^iωct] à(2)

à (ω, z) = à (3)

Where in Eq. (2), E (t, z) is the normalized scalar electrical field with a unit of.

Answer to the theory question:-

Let’s consider the linear effect only, i.e., in the absence of nonlinearity, integration of equation (1) in the frequency domain yields equation (4)

A (ω, zf) =H (ω) A (ω, zt) à(4)

Where H(ω)=exp((-α/2-i/2*β2*w.^2-i/6*β3*w.^3)*t) and is fiber’s frequency response, A(ω,zf)=)and A(ω,zt) are the Fourier transforms of the fiber’s input and output, respectively. Hence, basically, first we integrate the linear part of NLSE or use Fourier transform in frequency domain which results in equation (4). Then we got simple form of equation in multiplication form. We can do either convert this to t-domain and do...