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\begin{document}
% General Correction section
\section{Combining the described techniques into a correction procedure}

%%%
\subsection{Introduction}

The target of the correction procedure is to recover the focusing capability in the plane of excitation, during imaging.

The previous chapters have described the procedure to recover the focus at a specific location using sensorless adaptive optics, then expand the area of focusing through the use of the memory effect. This chapter will describe the procedure which combines these techniques to achieve control of the entire optical field at the distal facet by optimising for focusing at multiple reflectors and applying the memory effect on these optimisations then combining the information acquired. Furthermore, the ability to control the optical field over the entire distal facet allows to further expand the focusing capability to a different focus plane than the distal facet.

%%%
\subsection{Free-space propagation}

During imaging, the desired excitation plane can be at a specific axial depth with respect to the distal facet. In the case of a straight fibre, this could be achieved by calibrating the fibre while setting the focal plane of the calibration module at a specific distance away from the distal facet. 

The TM measured will describe the relationship between the proximal facet and the excitation plane. However, as the recovered TM using the memory effect approach only describes the transmission up to the reflector plane, a method is needed to relate the optical field in the reflector plane to the optical field in the excitation plane. 

Ideally, the reflector plane and the distal facet are identical, but this might not be the case. However, if the reflector is very close to the distal facet, the fibre between the reflector and distal facet can be assumed straight and the change in the optical field between the two planes can be calculated using the ideal straight fibre case.

However, the change in the optical field between the distal facet and the excitation plane needs to be calculated using a different method. A diagram of the planes is shown in figure \ref{fig:planes_diagram}.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{images/planes_diagram.png}
    \caption{Diagram of the different planes showing the reflector plane, distal facet and excitation plane.}
    \label{fig:planes_diagram}
\end{figure}

Fourier optics allows for the description of free-space propagation \cite{fourier_optics}. The change in the optical field between two planes is described by the transfer function in equation \ref{eq:TF_freespace}, where $A$ represents the angular spectrum of the optical field, $f_x$ and $f_y$ are the frequency components of the angular spectrum, $z$ is the axial depth of the plane, as shown in the diagram in figure \ref{fig:planes_diagram}, and $\lambda$ is the wavelength of light.

\begin{equation} \label{eq:TF_freespace}
    A(f_x, f_y, z_2) = A(f_x, f_y, z_1) \exp \left( i2\pi(z_2-z_1)\sqrt{\frac{1}{\lambda^2} - f_x^2 - f_y^2} \right)
\end{equation}

\begin{sloppypar}
The angular spectrum can be calculated using a Fourier Transform: $A(f_x, f_y, z) = \mathcal{F}[ u(x, y, z) ]$.    
\end{sloppypar}

It can be seen that in the case of $f_x^2 + f_y^2 > \frac{1}{\lambda^2}$, the square root term becomes imaginary. Thus, some high frequency components of the optical field decay exponentially as an evanescent wave. This phenomenon leads to a diffraction limit to the achievable resolution in an imaging system.

A perfect focus can be modelled by a delta function. The Fourier spectrum of a delta function is a constant function over the entire frequency domain. However, as higher frequency components are evanescent and decay, only the frequency components within a disc centred at (0,0) and radius $\frac{1}{\lambda}$ in the frequency domain propagate. Thus, the achievable focus is in the diffraction limit an airy disc (Fourier transform of a disc), whose radial size up to the first null is $0.61 \frac{\lambda}{NA}$, with the numerical aperture of the system also being an additional constraint on the radial size of the disc in the frequency domain.

\begin{equation} 
    A_{desired}(f_x,f_y,z=z_{exc}) = \mathrm{circ} \left[ \mathcal{F}  (\delta_{x_{focus}, y_{focus}}), \frac{1}{\lambda^2} \right] 
\end{equation}

where

\begin{equation}
    \delta_{x_{focus}, y_{focus}} = 
    \begin{cases}
        1, & \mathrm{if}\; x=x_{focus} \;\mathrm{and}\; y = y_{focus} \\
        0, & \mathrm{if}\; x \neq x_{focus} \;\mathrm{or}\; y \neq y_{focus} \\
    \end{cases} \nonumber
\end{equation}

\begin{equation}
    \iint_{-\infty}^\infty \delta_{x_{focus}, y_{focus}} \,dx \,dy = 1\nonumber
\end{equation}

\begin{equation}
    \mathrm{circ} \left[ A(f_x, f_y),  \frac{1}{\lambda^2} \right] = 
    \begin{cases}
        A(f_x, f_y), & \mathrm{if}\;  f_x^2 + f_y^2 \leq \frac{1}{\lambda^2}\\
        0, & \mathrm{if}\; f_x^2 + f_y^2 > \frac{1}{\lambda^2} \\
    \end{cases} \nonumber
\end{equation}

The optical field at the distal facet which generates this focus can be found by inverting the free-space propagation transfer function.

\begin{equation}
    A_{desired}(f_x, f_y, z_{DF}) = A(f_x, f_y, z_{exc}) \exp \left( -i2\pi(z_{exc}-z_{DF})\sqrt{\frac{1}{\lambda^2} - f_x^2 - f_y^2} \right)
\end{equation}

It can be observed that filtering out the evanescent components prior to inversion is important, as, if they were included, these components would exponentially increase in value.

While the desired field at the distal facet is obtained through inverse Fourier transform, this field would most likely contain components which are not part of the fibre mode space. The actual field achievable at the distal facet is the desired field projected into the modal space, through the use of the mode profile matrix $\mathbf{\Psi}$.

\begin{equation}
    \mathbf{y}_{c,desired} = \mathcal{F}^{-1} \left[ A_{desired}(f_x,f_y,z=z_{DF}) \right] \nonumber
\end{equation}
\begin{equation}
    \mathbf{y} = \mathbf{\Psi}^{\dag} \mathbf{y}_{c,desired} \nonumber
\end{equation}
\begin{equation}
    \mathbf{y}_c = \mathbf{\Psi} \mathbf{\Psi}^{\dag} \mathbf{y}_{c,desired}
\end{equation}

Finding the field at the proximal facet can then be achieved through inverting the TM of the fibre.

\subsection{Free-space propagation results}

The first simulations undertaken to showcase the ability to generate foci at a desired excitation plane using free-space propagation were done using a straight fibre. The parameters are identical to the ones described previously in section \ref{ssec:me_res}. The plane of excitation was chosen at a depth of $100 \;\mathrm{\mu m}$.

\begin{figure}[!htbp] 
    \centering
    \begin{subfigure}[t]{0.25\textwidth}
        \includegraphics[width = \textwidth]{images/prop_with_padding_desired_with_outline.png}
        \caption{The desired optical field}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.25\textwidth}
        \includegraphics[width = \textwidth]{images/prop_with_padding_desired_phase.png}
        \caption{The phase of the desired optical field}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.25\textwidth}
        \includegraphics[width = \textwidth]{images/prop_with_padding_fibre.png}
        \caption{The mode filtered optical field}
        \label{sfig:DF_field_prop}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.073\textwidth}
        \includegraphics[width = \textwidth]{images/colorbar.png}
    \end{subfigure}
    
    \caption{Optical fields at the distal facet: the desired field determined by back-propagation with the edge of the fibre core outlined in red (a), its phase component (b) and its mode space projection (c). The hue represents phase, as indicated by the colorbar and the value represents the absolute value of the field normalised to its maximum (HSV representation). The maximum absolute value of the filtered field is $\approx 1\%$ of the desired field.}
    \label{fig:fields_prop}
\end{figure}

As observed in figure \ref{fig:fields_prop}, where a focus at a radius position of $\frac{0.9 \cdot d}{2}$ was targeted, the free-space propagation approach generates a quadratic phase desired field at the distal facet. This is to be expected, as the back-propagation results in a defocus being applied to the original desired focus. 

However, such a field cannot be the output of the fibre, one of the main reasons being that the field extends outside the core facet area. Projecting the desired field into the modal space results in the achievable field at the distal facet. In this case, the maximum absolute value is $\approx 1\%$ of the desired field, the filtered version only containing $\approx 6.3\%$ of the desired total intensity. 

The next step is to propagate the obtained filtered field back to the excitation plane. The result is shown in figure \ref{fig:exc_focus}. While a focus is generated, it does not have a circular shape, due to the effect of the edge of the fibre core edge affecting the amplitude of the input field. The phase inside the fibre core seems not to be affected.

\begin{figure}[!htbp]  
    \centering
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/focus_at_exc.png}
        \caption{The focus obtained at the excitation plane}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/focus_at_exc_zoomed.png}
        \caption{Zoomed-in focus at the excitation plane}
    \end{subfigure}
    
    \caption{The resulting focus at the excitation field by propagating the filtered optical field shown in figure \ref{fig:fields_prop}. The figure shows the intensity of the focus, normalised to the maximum intensity.}
    \label{fig:exc_focus}
\end{figure}

\begin{figure}[!htbp] 
    \centering
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/composite_foci_only.png}
        \caption{A composite image of foci generated in the excitation plane at different locations. The red star shows the location of the focus generated in figure \ref{fig:exc_focus}.}
        \label{sfig:comp_foci}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/focus_shape_excitation_plane.png}
        \caption{The axial shape of the focus at the location of the red star in figure \ref{sfig:comp_foci}.}
        \label{sfig:one_focus}
    \end{subfigure}
    
    \caption{A showcase of the foci generated at different positions in the excitation plane. An in-depth look at one of the foci is also included.}
    \label{fig:comp_foci}
\end{figure}

Figure \ref{sfig:comp_foci} shows a composite image of different foci generated in the excitation plane, at different positions. The foci were generated individually and their vicinity was cropped. Combining the cropped images generated the composite image.

Figure \ref{sfig:one_focus} takes a zoomed-in look at the axial shape and extent of the focus generated previously in figures \ref{fig:fields_prop} and \ref{fig:exc_focus}. It is clear that the focus is propagating at an angle away from the centre axis of the fibre facet. This phenomenon of a spatially variant point spread function (PSF) has been observed in practice \cite{deconv} and it is intrinsic to MMF imaging, as the MMF forms an inherently a non-telecentric system. The accord between the simulation and experimental observations gives confidence in the correctness of the simulation procedure.

%%%%
\subsection{Combining memory effect with free-space propagation} \label{ssec:ME_and_F}

%%
\subsubsection{Single correction target}
Generating a focus away from the distal facet requires the control of a larger area of the field at the distal facet, as illustrated by figure \ref{sfig:DF_field_prop}. Thus, it is expected that the area of the isoplanatic patch decreases with depth of the focus plane.

A set of simulations were done to investigate the effectiveness of the memory effect at different depths and the results for three different depths of 20 $\mu m$, 50 $\mu m$ and 100 $\mu m$ are shown in figure \ref{fig:iso_patch_depth}. As described above, the isoplanatic patch decreases in size with increasing depth. The simulations were done for the same set of bending inverse radii and fibre length as in chapter \ref{sec:mem_effect}, leading to the same values for TM diagonal energy.

\begin{sidewaysfigure} 
    \centering
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_20um_depth.png}
        \caption{The isoplanatic patch at a focus depth of 20 $\mu m$}
        \label{sfig:iso_patch_depth_20}
    \end{subfigure}
\end{sidewaysfigure}
\begin{sidewaysfigure}
    \ContinuedFloat
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_50um_depth.png}
        \caption{The isoplanatic patch at a focus depth of 50 $\mu m$}
        \label{sfig:iso_patch_depth_50}
    \end{subfigure}
\end{sidewaysfigure}
\begin{sidewaysfigure}
    \ContinuedFloat
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_100um_depth.png}
        \caption{The isoplanatic patch at a focus depth of 100 $\mu m$}
        \label{sfig:iso_patch_depth_100}
    \end{subfigure}
    
    \caption{Intensity of the foci generated using memory effect at different focus plane depths normalised to the intensity of foci generated at the same focus plane by inverting the actual bent fibre TM. Results shown for six different inverse bending radii, each with specific diagonal energy associated. The isoplanatic patch has been defined as the region where the correction achieves a factor of at least 0.7.}
    \label{fig:iso_patch_depth}
\end{sidewaysfigure}

%%
\subsubsection{Multiple correction targets}

The isoplanatic patch generated by a reflector using memory effect does not cover the entire distal facet. As during imaging foci need to be generated over the entire field of view, a single reflector would not suffice. Furthermore, as can be seen in figure \ref{sfig:DF_field_prop}, generating a focus further away from the distal facet requires the generation of a desired optical field at the distal facet which is not confined to a single focus. 

Based on the results in figure \ref{fig:isopatch}, if a limit to the bending during imaging is assumed, such as $\approx 10 \;\mathrm{m}^{-1}$, it would be possible to correct for the entire facet, while only using four reflectors. The value of $10 \;\mathrm{m}^{-1}$ of the inverse bending radius for a 1.5 cm of fibre represents a translation of the tip of the fibre by $\approx 1 \;\mathrm{mm}$. This would be within the expected deviations which would occur at the brain level during implantation and imaging.

Applying the memory effect to four reflectors results in the generation of four assumed diagonal TMs, each being applicable to to its associated isoplanatic patch. Thus, the desired optical field at the distal facet needs to be split into four regions, each with its own TM. Thus, any desired optical field was considered to be a superposition of the four regions.

In order to calculate the desired input field at the proximal facet, each region component was treated independently and the inverse of its associated diagonal TM was applied. The resulting proximal facet optical fields were then added together, making use of the linearity property of a matrix described transmission to generate the necessary input field to be used.

Simulations were undertaken by choosing four target foci at a radial coordinate of $\frac{0.9 \cdot d}{2}$ and azimuthal coordinates of $0$, $\frac{\pi}{2}$, $\pi$ and $\frac{3\pi}{2}$, where an azimuthal angle of $0$ is down as shown on the axis definition in figure \ref{sfig:comp_foci}. The associated regions were chosen by cutting the optical field diagonally, as shown in figure \ref{sfig:reg_ill}.

In order to consider an edge case scenario, the focus to be generated has been chosen at a position on the edge of two regions (1 and 2) at a radial coordinate of $\frac{0.9 \cdot d}{2}$ and azimuthal coordinate of $\frac{\pi}{4}$. The calculated desired distal facet field using free-space propagation is illustrated in figure \ref{sfig:des_field}.

Figure \ref{fig:me_regions_split} shows the splitting of the optical field. Even though the complete optical field is the superposition of the 4 components, it is important to note that these components lose the property of being entirely in the fibre mode space. Thus, even with an entirely known TM, generating the individual components can only be done approximately. This is similar to the previous phenomenon where the quadratic phase field was filtered by the fiber modes, though in this case, the desired field lies entirely within the core area, thus, the approximation will be more similar to the desired field. In figure \ref{fig:me_regions_mode_filter} the projection of the individual components into the fibre mode space is shown. While there is not a large amount of change in the output optical fields, there is some leakage present between the regions. However, this is negligible compared to the desired optical field itself.

\begin{figure} 
    \centering
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/isoplanatic_pathces_chosen_text.png}
        \caption{An illustration of the region split of the optical field.}
        \label{sfig:reg_ill}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/all_reg.png}
        \caption{The desired distal facet optical field}
        \label{sfig:des_field}
    \end{subfigure}

    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/reg_1_nf.png}
        \caption{Region 1 component}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/reg_2_nf.png}
        \caption{Region 2 component}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/reg_3_nf.png}
        \caption{Region 3 component}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/reg_4_nf.png}
        \caption{Region 4 component}
    \end{subfigure}
 
    \caption{The desired optical field at the distal facet for generating a focus at a depth of $100 \;\mathrm{\mu m}$ and its division into reflector region components. The location is at a radial coordinate of $\frac{0.9 \cdot d}{2}$ and azimuthal coordinate of $\frac{\pi}{4}$. }
    \label{fig:me_regions_split}
\end{figure}

The simulation was then performed to compute the isoplanatic patch at different depths with the use of four targets for the memory effect optimisation. As expected, overall, the system performs similarly to having four different copies of the isoplanatic patch resulted in the case with only one target, though there are some differences.

Firstly, it seems that the direction of bending can influence the amount of correction achieved. In this case, the bending was applied vertically with respect to the image. The two targets which lie along the bending direction have performed worse compared with the two targets which are on the direction perpendicular to bending. The reason for this effect is not completely understood. However, as the bending initially distorts the focus along the bending direction, the reason for the difference might have been due to the foci which lie on the bending direction having had their focus distorted radially, while the other two targets having had their focus distorted in an azimuthal direction. As mentioned in section \ref{sec:ME}, optical fibres are more robust to rotational changes that radial.

Secondly, the location which are at the borders of the regions and the centre of the facet perform worse. This could be due to an effect generated by the splitting into regions or it could be simply due to the locations being the furthest in the facet from any optimisation target location.

The final result of the simulation was that the approach is usable and could achieve an adequate correction over the entire focus plane.

\begin{figure}
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/reg_1_wf.png}
        \caption{Region 1 component, projected into modal space}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/reg_2_wf.png}
        \caption{Region 2 component, projected into modal space}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/reg_3_wf.png}
        \caption{Region 3 component, projected into modal space}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.4\textwidth}
        \includegraphics[width = \textwidth]{images/reg_4_wf.png}
        \caption{Region 4 component, projected into modal space}
    \end{subfigure}

    \caption{The optical fields generated by splitting the optical field into reflector regions projecting each region into fibre mode space.}
    \label{fig:me_regions_mode_filter}
\end{figure}

\begin{sidewaysfigure}
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_20um_depth_4targets.png}
        \caption{The isoplanatic patch at a focus depth of 20 $\mu m$.}
    \end{subfigure}
\end{sidewaysfigure}
\begin{sidewaysfigure}
    \ContinuedFloat
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_50um_depth_4targets.png}
        \caption{The isoplanatic patch at a focus depth of 50 $\mu m$.}
    \end{subfigure}

    \caption{Intensity of the foci generated using memory effect at different focus plane depths normalised to the intensity of foci generated at the same focus plane by inverting the actual bent fibre TM. Results shown for six different inverse bending radii, each with specific diagonal energy associated. The isoplanatic patch has been defined as the region where the correction achieves a factor of at least 0.7. Four optimisation targets were chosen spread uniformly over azimuthal range.}
    \label{fig:iso_patch_depth_4_targets}
\end{sidewaysfigure}

To further the understanding into the approach, a second set of simulation were done with six targets instead of four, the expectation being that increasing the number of targets will result in a better correction. The targets were uniformly distributed around the facet, while keeping the radial position the same. The regions associated which each target were also generated based on the azimuthal coordinate of each pixel location. The results are shown in figure \ref{fig:iso_patch_depth_6_targets}. 

As expected, the results are an improvement with respect to using four targets, as each location in the plane has a greater chance of being closer to an optimisation target. This result shows that there is a balancing act which need to be made in the choice of the number of target reflectors, as a larger number leads to better results but also implies more rounds of optimisation being needed to restore focus at each target location.

\begin{sidewaysfigure}
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_20um_depth_6targets.png}
        \caption{The isoplanatic patch at a focus depth of 20 $\mu m$.}
    \end{subfigure}
\end{sidewaysfigure}
\begin{sidewaysfigure}
    \ContinuedFloat
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_50um_depth_6targets.png}
        \caption{The isoplanatic patch at a focus depth of 50 $\mu m$.}
    \end{subfigure}

    \caption{Intensity of the foci generated using memory effect at different focus plane depths normalised to the intensity of foci generated at the same focus plane by inverting the actual bent fibre TM. Results shown for six different inverse bending radii, each with specific diagonal energy associated. The isoplanatic patch has been defined as the region where the correction achieves a factor of at least 0.7. Six optimisation targets were chosen spread uniformly over azimuthal range.}
    \label{fig:iso_patch_depth_6_targets}
\end{sidewaysfigure}

Finally, in order to showcase the improvement achieved by the correction method, figure \ref{fig:iso_patch_depth_no_corr} shows the resulting isoplanatic patch if no correction is applied. It is clear that even for a smaller amount of bending (inverse radius of 8.25 $m^{-1}$) the result falls under the 0.7 imposed limit over most of the facet plane.

\begin{sidewaysfigure}
    \centering
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_no_corr_20um_depth.png}
        \caption{The isoplanatic patch at a focus depth of 20 $\mu m$}
        \label{sfig:iso_patch_depth_20_no_corr}
    \end{subfigure}
\end{sidewaysfigure}
\begin{sidewaysfigure}
    \ContinuedFloat
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_no_corr_50um_depth.png}
        \caption{The isoplanatic patch at a focus depth of 50 $\mu m$}
        \label{sfig:iso_patch_depth_50_no_corr}
    \end{subfigure}
\end{sidewaysfigure}
\begin{sidewaysfigure}
    \ContinuedFloat
    \begin{subfigure}[t]{\textwidth}
        \includegraphics[width = \textwidth]{images/Memory_CP_full_TM_all_invr_one_pos_no_corr_100um_depth.png}
        \caption{The isoplanatic patch at a focus depth of 100 $\mu m$}
        \label{sfig:iso_patch_depth_100_no_corr}
    \end{subfigure}
    
    \caption{Intensity of the foci generated without applying any correction at different focus plane depths normalised to the intensity of foci generated at the same focus plane by inverting the actual bent fibre TM. Results shown for six different inverse bending radii, each with specific diagonal energy associated. The isoplanatic patch has been defined the region where the correction achieves a factor of at least 0.7.}
    \label{fig:iso_patch_depth_no_corr}
\end{sidewaysfigure}

\subsection{Conclusion}

In conclusion, the approach described in this chapter has been shown analytically to be feasible in the use for correcting the loss in focusing capabilities of an optical fibre setup generated by bending of said fibre. The approach makes use of the 
memory effect to infer information about focusing at any location in the excitation plane while only optimising for a small number of locations. Furthermore, it is also able not only to correct but also generate foci at different excitation plane depths without the need to calibrate the fibre numerous times, while changing the calibration arm CCD focus plane.

The number of reflectors used needs to be determined by the amount of bending expected. The more bending is expected, the smaller the isoplanatic patch of the memory effect approach is, thus, leading to a larger number of reflectors being required to cover the entirety of the fibre facet. However, each reflector reduces the amount of light behind it, or even blocks it entirely. Thus, if the focus plane is near the distal facet, there will be blind spots behind the reflectors and in the far-field, some of the spatial frequencies will be blocked, leading to a lower NA. In conclusion, the number of reflectors will need to be chosen in practice taking into consideration this disadvantage.

\end{document}