Solving the Nonlinear Schrödinger Equation using the Split-Step Fourier Method to model pulses through an optical fiber

\( \frac{\partial A}{\partial z} = -\frac{\alpha}{2} A - i\frac{\beta}{2} \frac{\partial^2 A}{\partial T^2} + i\gamma |A|^2 A \)

This nonlinear partial differential equation models how the envelope and phase of light pulse changes when propagating through an optical fiber, when taking power attenuation α, Group Velocity Dispersion β and waveguide nonlinearity γ causing Self-Phase Modulation (SPM) into account.

Define Simulation Parameters

Number of Points :
Time Resolution:	\( dt =\)		e		(s)

Create a Pulse to be Simulated

Gaussian pulse

\( A \cdot \exp\left[-\left(\frac{1+j \cdot C_h}{2}\right)\left(\frac{t}{\tau}\right)^{2 \cdot m} - j \cdot 2 \pi f_c \cdot t\right] \)

Sinc Pulse

\( A \cdot \text{sinc}\left(\frac{t }{\tau}\right) \cdot \exp\left[-\left(\frac{1+j \cdot C_h}{2}\right)\left(\frac{t}{\tau}\right)^{2} - j \cdot 2 \pi f_c \cdot t\right] \)

Sech pulse (Soliton)

\( \frac{A}{\text{cosh}\left(\frac{t }{\tau}\right)} \cdot \exp\left[\frac{-j \cdot C_h}{2} \cdot \left(\frac{t}{\tau}\right)^{2} - j \cdot 2 \pi f_c \cdot t\right] \)

Amplitude:	\( A =\)	(W^1/2)
Carrier frequency:	\( f_c =\)	(Hz)
Chirp	\( C_h =\)
Order	\( m =\)
Duration: \(\tau =k \cdot dt; \)	\( k =\)

• Choose a Pulse :

Add Noise: Noise Amplitude \( = A / \)

• Plot :

Set Up Properties of the Fiber

Pulses launched into an optical fiber will evolve according to the NLSE When \( \beta \) is negative (anormalous dispersion), the term representing Group Velocity Dispersion (GVD) : \( -i\frac{\beta}{2} \frac{\partial^2 A}{\partial T^2} \) will cause a positive (blue) chirp in the front and a negative (red) chirp in the back. Similarly, the term representing self-phase modulation (SPM) : \( i\gamma |A|^2 A \) will cause a red chirp in the front and a blue chirp in the back. Since the chirp caused by GVD depends on the 2nd derivative (curvature) of the pulse, while the oppositely signed chirp of SPM depends on the squared amplitude of the pulse, we can ask the following question: "Is there some pulse envelope, where the effects of GVD and SPM exactly cancel out, causing the pulse to retain its shape as it propagates?" Such a pulse is called a "soliton". The Sech Pulse is the pulse shape that corresponds to a Fundamental soliton.

Number of Steps:
Length:		e	(m)
Gamma:	\( \gamma =\)	e	(W^-1.m^-1)
Beta :	\( \beta =\)	e	(fs².m^-1)
Alpha :	\( \alpha =\)	e	(dB.m^-1)

• If Server Overload, try to reduce the Number of Steps of the Fiber,Number of Points of the Simulation And use Characteristic Fiber Length and Pulse Amplitude:

	Use Characteristic Length:	\( Z \cdot \frac{\pi}{2 \cdot \tau^2 \cdot \|\beta\|} \)	;	\(Z =\)
	Use Characteristic Amplitude:	\( A \cdot \left( \frac{\|\beta\|}{\gamma \cdot \tau^2} \right)^{1/2} \)	;	\( A =\)

• Plot :