# A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
## 3 The Inverse Nonlinear Fourier Transform Method
### The Numerical Method Algorithm for the Marchenko System  
Given the above considerations, the points on which we will decide to approximate the function \(u(t,z)\) will be equally spaced at a distance \(\Delta T\). The points where we will find \(u(t,z)\) are collected in the following vector: \(T=[t_{1}=a,\ldots,t_{n}=a+(n-1)\Delta T,\ldots,t_{N}=b]\), \(n=1,\ldots,N\), where \(a<0\) and \(b>0\) are extreme points of this vector. To solve the Marchenko system in Equation (45), we are going to use the Nystrom method, approximating the integrals by means of a quadrature formula to derive a discrete system to be solved through an appropriate method. In particular, we want to find \(B_{2}(t,\beta)\) for \(\beta=0\). The system in (45) with \(\beta=0\) will be:  
\[\left\{ \begin{array}{c}B_{1}(t_{n},0)=\int_{0}^{\infty}B_{2}(t,r)F(r+2t_{n} )dr\\ B_{2}(t_{n},0)=\int_{0}^{\infty}B_{1}(t,r)F^{*}(r+2t_{n})dr-F^{*}(2t)\end{array} \right. \tag{49}\]  
Equation (49) should be solved numerically. We observe that inside the integral, we need to evaluate \(B_{2}(t,r)\) and \(B_{1}(t,r)\) for values of higher than zero. As a result, we need to solve for the values of \(\beta\
eq 0\). In particular, we are going to take \(\beta\) and \(r\) from the same domain \(D=\left[2m\Delta t\right]_{m=0}^{l}\) where \(l=ceil\left[\frac{N}{2}-n\right]\) is the first value of \(m\), such that we are outside the domain of \(F\). The reason for this choice is that such a domain should take into account the fact that we know the support of \(F\), which is evaluated in both integrals as \(r+2t_{n}=r+2n\Delta t\). The sampling step of variable \(r\) where we will integrate is: \(\Delta\beta=2\Delta t\)Now, we can finally approximate both integrals in Equation (49) using the Simpson rule to find the system of equations:  
\[\left\{ \begin{array}{c}B_{1}\big{(}t_{n},r_{j}\big{)}=\sum_{m=0}^{l}d_{m}B_{2}( t_{n},r_{m})F\big{(}r_{n+j+m}\big{)}\\ B_{2}\big{(}t_{n},r_{j}\big{)}=\sum_{m=0}^{l}d_{m}B_{1}(t_{n},r_{m})F^{*}\big{(} r_{n+j+m}\big{)}-F^{*}\big{(}r_{n+j}\big{)}\end{array} \right. \tag{50}\]  
where \(r_{n+j+m}=(k+j+l)\Delta\beta\), \(d_{m}=\frac{\Delta\beta}{3}\{1,4,2,4,2.\}\), \(j=0,\ldots,l\). The system in Equation (50) can be represented using a matrix notation such as:  
\[\left| \begin{array}{cc}I&-\mathcal{F}D\\ \mathcal{F}^{*}D&I\end{array} \right|\left| \begin{array}{c}\mathbf{B}_{1}\\ \mathbf{B}_{2}\end{array} \right|=\left| \begin{array}{c}0\\ \mathcal{F}^{*}\end{array} \right| \tag{51}\]  
where \(\mathcal{F}=\big{(}F\big{(}r_{n+j+m}\big{)}\big{)}_{m,j=0}^{l}\), \(\mathcal{F}^{*}=\big{(}F\big{(}r_{n+j}\big{)}\big{)}_{j=0}^{l}\), \(D=diag\{d_{m}\}_{m=0}^{l}\), are known variables. The unknown variables are:  
\[\mathbf{B}_{1}=\left(B_{1}(t_{n},r_{m})\right)_{m=0}^{l}\mathbf{B}_{2}=\left( B_{2}(t_{n},r_{m})\right)_{m=0}^{l} \tag{52}\]  
In Equation (51), we are interested in the first component of \(\mathbf{B}_{2}\). After finding \(\mathbf{B}_{1}=\mathcal{F}D\mathbf{B}_{2}\) from the first part of the system, we substitute it in the second part, and then solve the following system:  
\[(\mathcal{F}^{*}D\mathcal{F}D+I)\mathbf{B}_{2}=\mathcal{F}^{*} \tag{53}\]  
Once we have solved it, we are able to find out the signal propagated at a desired distance \(z\):  
\[u(t,z)=\mathbf{B}_{2}(r_{0}) \tag{54}\]  
In summary, to find out the solution \(u(t,z)\) of Equation (2) through the implementation of the INFT (in this case, when \(a(\lambda)\
eq 0\) for every \(\lambda\in\mathbb{C}\)), we need to follow the steps below:  
1. Based on the above algorithm, we first need to find the scattering data \(a(\lambda)\) and \(b(\lambda)\), and their ratio \(\frac{b(\lambda)}{a(\lambda)}\). 2. We need to find the spatial evolution of continuous spectral amplitudes \(\frac{b(\lambda)}{a(\lambda)}\) by multiplying them with \(e^{i\lambda^{2}z}\) (step 2 should be applied only in case we will transmit the signal in an optical fiber described by Equation (2)). 3.