NLSE

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

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

Souce

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)

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

Parameters

Pulse Number of Points	Time Resolution \(dt\)(s)			Fiber Number of Steps
		e

Input Pulse :

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

Fiber Properties

Gamma \( \gamma \) (W^-1.m^-1)		Beta \( \beta \) (fs².m^-1)		Alpha \( \alpha \) (dB.m^-1)
	e		e		e
Length (m)		Use Characteristic Fiber Length : \( Z \cdot \frac{\pi}{2 \cdot \tau^2 \cdot \|\beta\|} \)		Use Characteristic Pulse Amplitude: \( A \cdot \left( \frac{\|\beta\|}{\gamma \cdot \tau^2} \right)^{1/2} \)
	e	\(Z =\)		\( A =\)

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