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% Conclusion section
\section{Conclusion and outlook}

\subsection{Conclusion}

This thesis had provided an approach to correcting for the change in light transmission through a bent multimode optical fibre for use in \textit{in vivo} microscopy.

The correction method approached the problem in steps, starting with recovering focusing capabilities for a single location at or near the output facet of the fibre. It made use of Walsh-Hadamard modes as an orthogonal basis to characterise the modulation applied by the SLM on the optical field being inputted into the fibre. The advantage of Walsh modes is that varying the coefficient of one of them lead to a sinusoidal variation in the output focus intensity. Such a predictable output allowed for a simple optimisation procedure which would compute the best coefficient with only three measurements at different Walsh mode coefficients. Repeating the optimisation process for different Walsh modes, lead to an iterative process through which the output focus intensity was mostly recovered. This type of process fits into the categorisation of sensorless adaptive optics. The thesis showed through simulations that a significant improvement in the quality of the focus can be achieved, however, this is at the cost of time as the process requires a large number of iterations.

In order to expand the correction from a single location to an area around it, the memory effect was utilised. The approach assumes that for small bends in the fibre, the TM does not change by a large amount and keeps its diagonal form, though with changes in the phase of the elements on the main diagonal. Restricting the assumed form of the TM allows for information about the entire TM to be obtained from the input optical field which generates a focus at the output of the fibre. Through simulations, it was shown that focusing capabilities are recovered in an area around the original target. This area where the intensity of the recovered focus is at least $70\%$ of the fully corrected focus was named the isoplanatic patch. The isoplanatic patch was shown to decrease when increasing the amount of bending applied to the fibre, as more coupling occurs between the fibre modes and the assumption about the diagonality of the TM becomes further from the actual TM. However, within some limits, the approach can correct for around a quarter of the output facet.

However, in order to use the fibre for imaging, the correction must be generated for the entire output facet. Thus, a method utilising multiple target correction foci was described in the thesis. As each correction target has an associated isoplanatic patch, a method of combining the information from different targets was required. The method split the desired output field into regions, each associated with a correction target, which were individually back-propagated through the fibre using their associated diagonal TMs. Then, the resulting fields at the input facet were summed to give the corrected input field.

The correction approach was not limited to focusing at the distal facet of the fibre. Free-space propagation was analytically described through the use of angular spectrum and was used to allow for the change in the axial position of the focusing plane. The methods described above were combined to show that adequate correction is achievable for axial distances up to 50 $\mu m$ and bending inverse radii of 12 $\mathrm{m^{-1}}$ with the use of six target corrected foci.

The second main contribution of the thesis was a novel approach in simulating Bragg gratings in multimode optical fibres. The method follows the literature approach of coupled-modes analysis, but continues the approach by directly solving the set of differential equations generated. The thesis presented simulated results of different gratings: their associated resulting optical fields at the Bragg wavelength and their spectrum. This approach can provide an important tool in optimising the parameters for a Bragg grating if it were to be used as a reflector in the correction approach described previously. However, it could also be used in other applications where Bragg gratings would be required inside multimode optical fibres.

In conclusion, the thesis provides a set of tools and approaches which will be useful in tackling the issue of changes in light transmission through an optical fibre due to bending being introduced. This can have an important impact in improving \textit{in vivo} imaging quality and allowing a better understanding of biological processes which can be observed with this method.

\subsection{Outlook}

The correction approach and the Bragg grating simulations were only tested in a simulated environment. An important step which needs to be taken is to validate the approaches with experimental data.

The sensorless adaptive optics was described in the thesis as a solution to recovering a focus at a desired reflector target. However, this process is lengthy in time and the results might not be able to reach an optimum correction. Thus, different approaches which improve upon these drawbacks might need to be implemented. However, as long as the result of recovering a specific focus is achieved, the rest of the correction procedure can be used.

The correction approach also has some drawbacks. Specifically, during the project, I have tried to implement the memory approach experimentally. However, I ran into issues relating to the aberrations present in the system. As the change in phase component of the diagonal elements is required to be measured, a precise evaluation of the optical field at the proximal facet is required. Thus, the experimental system must be robust to aberrations or have a very good understanding of them in order to eliminate their influence.

Furthermore, a real-world fibre is only an approximation to an ideal step-index fibre. Thus, to find the correct mode profiles, effects such as diffusion (index change at core-cladding interface is not exactly a step function) and manufacturing imperfections will affect the propagation mode basis.

There is also the question of the specific geometry of the Bragg grating if it were to be used as a reflector. For good localisation of the focus, the grating should be very small in both transversal and axial directions. However, a short grating leads to a very small amount of reflection, which can lead to issues in detecting the reflected signal, especially if there is a large amount of reflection at the proximal facet as well. It is also possible there are other geometries available which could solve this issue, though, this has not been investigated.

Another issue which can arise is that when using multiple reflectors and trying to optimise for the focus at one of them, light from the other reflectors could also be reflected, as even in the ideal case a perfect focus cannot be achieved due to the discrete nature of fibre modes, which would not allow for a good optimisation. Further investigation is necessary to find whether this is a large issue at small inverse bending radii. However, Bragg gratings could be the solution to this problem, as by changing the grating period and using light at multiple wavelengths, the reflection is discriminated between reflectors. But, even in this case, there is an issue in that the memory effect corrections at different wavelengths need to be correlated into an overall correction at the excitation wavelength.

In my opinion, the next steps in building upon the work in this project in terms of the bending correction procedure would be to find the right reflector, solve the aberrations issue and build as thorough as possible the mode basis of the real-world fibre.

Furthermore, I think that graded-index fibres should also be investigated in the context of this correction approach, as I think that due to their greater robustness to bending, they would take advantage even more of the correction approach to expand their range of bending robustness.    

Finally, as mentioned above, thorough experimental validation is also needed for the Bragg grating theory presented in this thesis.

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