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\begin{document}
% Memory Effect section
\section{Expanding the optical field control using the Memory Effect and free-space propagation} \label{sec:mem_effect}

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\subsection{Memory Effect} \label{sec:ME}

%%%%
This chapter describes the memory effect and its use in expanding the area of control from the reflector location to an isoplanatic patch around it.

The term ``memory effect'' describes in the optical field, a correlation in terms of tip and tilt in coherent fields. This correlation allows for a some amount of control even after the light undergoes limited scattering, as any tip or tilt modulated into the input light will result in a similar tip and tilt in the output scattered light.

In the area of fibre optics, the general memory effect is less apparent, due to cylindrical boundaries the geometry of an optical fibre introduces. As shown in \cite{mem_eff} while limited azimuthal control can be achieved, it is not the case radially, without negatively affecting the output pattern. A different approach, also named memory effect (ME), described in the same publication \cite{mem_eff}, works by assuming that the TM of a slightly bent fibre remains diagonal, though the phase of the diagonal elements can change. This could be considered a TM memory effect, as the diagonal form of the TM remains with phase-only diagonal elements, while the bending is assumed to only introduce small phase changes in the diagonal elements.

In order to describe the memory effect mathematically and make use of it for bending compensation, the derivation starts with the relationship $\mathbf{y} = \mathbf{T}_{bent} \mathbf{x}$ being expanded into a set of equations for each respective element in the column vectors:

\begin{equation}
    y_k = e^{i\phi_k} x_k
\end{equation}

The reason the diagonal elements of the assumed TM have an amplitude of 1 is due to the requirement of the TM being unitary, such that the total energy of the optical field is maintained. The useful property of a diagonal TM is that the diagonal elements can be calculated while only knowing a single pair of $\mathbf{x}$ and $\mathbf{y}$. This allows for the entire assumed TM to be calculated after generating a single focus.

A perfect focus can be described as the projection of a delta function into the modal space:
\begin{equation} \label{eq:focus_field}
    \mathbf{y}_{focus} = \mathbf{\Psi}^\dag \mathbf{\Delta}_h
\end{equation}
where $\dag$ represents the complex conjugate and
\begin{equation}
    \mathbf{\Delta}_h = 
    \begin{pmatrix}
        \delta_1 \\
        \delta_2 \\
        \vdots \\
        \delta_P
    \end{pmatrix} \nonumber
\end{equation}
\begin{equation}
    \delta_j =
    \begin{cases}
        1, & \mathrm{if}\; j=h \\
        0, & \mathrm{if}\; j\neq h
    \end{cases}
\end{equation}

Thus, by expanding the matrix multiplication in \ref{eq:focus_field} and, using the definition of $\mathbf{\Delta}_h$ above, the coefficient of a specific fibre mode of index $k$ in the output field can be computed, where $\ast$ represents the complex conjugate:
\begin{equation}
    y_k = \psi_{k,h}^\dag = \psi_{h,k}^\ast
\end{equation}

If a focus has been generated at the DF, then the input optical field $\mathbf{x}_{c,focus}$ which generates said focus can be projected into the fibre mode space to obtain $\mathbf{x}_{focus}$. Then the phase diagonal elements of the assumed TM can be determined:

\begin{align}
    \phi_k & = \mathrm{Arg}(y_{focus,k}) - \mathrm{Arg}(x_{focus,k}) \nonumber\\
    & = \mathrm{Arg}(\psi_{h,k}^\ast) - \mathrm{Arg}(x_{focus,k}) \nonumber\\
    & = -\mathrm{Arg}(\psi_{h,k}) - \mathrm{Arg}(x_{focus,k})
\end{align}

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\subsection{Simulating the memory effect} \label{ssec:me_res}

%%%
\subsubsection{Methodology}

As the memory effect approach is based on the assumption that the bent fibre TM is diagonal, it is expected that memory effect's correction ability is dependent on how close the actual bent fibre TM is to a diagonal matrix, a property which can be characterised through the measurement of diagonal energy:
\begin{equation}
    D_{energy} = \frac{\sum_{j=1}^{j=N} |t_{j,j}|^2}{\sum_{j=1}^{j=N} \sum_{k=1}^{k=N} |t_{j,k}|^2}
\end{equation}

This parameter characterises the amount of energy which is kept within modes rather than being coupled out. In the case of the straight fibre TM, which is diagonal, the diagonal energy is $1$ (or $100\%$).

Simulations have been undertaken for an MMF using the following parameters:
\begin{itemize}
    \item Wavelength: $\lambda = 488 \;\mathrm{nm}$
    \item Core diameter: $d = 50 \;\mathrm{\mu m}$
    \item Core refractive index: $n_{core} = 1.45$
    \item Numerical aperture: $NA = 0.22$
    \item Fibre length: $L = 1.5 \;\mathrm{cm}$
\end{itemize}

These parameters are in line with the fibre that was used in the experimental system.
Bending has been modelled as a constant curvature, parameterised through the use of the inverse bending radius $\rho$. The reason for using the inverse radius is such that a straight fibre is represented by a value of $0 \;\mathrm{m^{-1}}$ and not $\infty\;\mathrm{m}$, allowing for better numerical precision in simulating small bends.

While the desired input field at the proximal facet was computed by inverting the modal space TM, the actual input optical field was computed by back-propagating the desired field into the Fourier plane (the SLM plane) and extracting only the phase information (division with the absolute value) before propagating back into the proximal facet plane. This approach strives to better simulate the real-case scenario, where the input optical field is controlled through the use of a phase-only modulator (SLM). 

%%%
\subsubsection{Results}

Figure \ref{fig:d_energy_plot} shows the plot of the diagonal energy as a function of inverse bending radius. As expected, as the bending increases, the amount of coupling between the modes increases, leading to a lower diagonal energy parameter.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{images/diag_energy_vs_invr.png} 
    \caption{Diagonal energy as a function of inverse bending radius}
    \label{fig:d_energy_plot}
\end{figure}

The isoplanatic patch represents the area in which focus correction using memory effect approaches the ideal case achievable by recalibrating the entire TM. More specifically, a good threshold would be the area in which through the memory effect the focus generated using the assumed TM reaches $70\%$ of the intensity of a focus generated using the actual TM of the bent fibre. In order to observe the effect of bending on the size of the isoplanatic patch, six inverse radii were chosen to be simulated. The results are shown in figure \ref{fig:isopatch}. It is clear from figure \ref{fig:isopatch} that as bending increases, the isoplanatic patch decreases in size as expected.

The location of the reflector was chosen at a radius of $\frac{0.9 \cdot d}{2}$. The reason for the location being close to the edge of the facet is due to the requirement that for a good correction, the reflector needs to be in a position where it samples most, if not all, of the fibre modes. The field distributions of higher order fibre modes will have the largest amplitude close to the facet edge, while almost none close to the centre. In the case of lower order modes, while the amplitude is larger closer to the centre, they will also have a number of lobes at a larger radial position. Thus, they will also be sampled by a reflector closer to the edge.  

The reason for this mode profile field distributions is due to higher order modes having a profile defined by a higher order Bessel function of first kind ($J_n$). The field distribution of a fibre mode inside the fibre core follows the function: $A J_l(hr) e^{il\phi}$, where $A$ is a normalisation parameter, $h^2 = n_{core}^2 k_0^2 - \beta^2$ is a radial magnification parameter dependent on the propagation constant of the mode, $l$ is the azimuthal index of the mode and $r$ and $\phi$ are the polar coordinates. The higher the order of a Bessel function of first kind, the further from 0 (the centre) the first peak occurs, thus, as the order $l$ of a mode increases, the resulting energy is being concentrated closer to the core facet edge. 

\begin{sidewaysfigure}
    \centering
    \includegraphics[width=\textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_128.png} 
    \caption{Intensity of the foci generated using memory effect normalised to the intensity of foci generated by inverting the actual bent fibre TM. Results shown for six different inverse bending radii, each with specific diagonal energy associated. The isoplanatic patch is the region where the correction achieves a factor of 0.7.}
    \label{fig:isopatch}
\end{sidewaysfigure}

\end{document}