\providecommand{\main}{..}

\documentclass[main.tex]{subfiles}
\graphicspath{ {../images/} }
\begin{document}
% Introduction section
\section{Introduction}

%%%
\subsection{Thesis motivation and brief description of achievements}

Being able to observe neurons at a cellular level, observing their activity and plasticity while still performing in their natural environment, allows for a better understanding of their structural and functional properties. While microscopy allows for imaging at a cellular level, such a system often comes with limitations in terms of optical capabilities.

Observing a neuron in its natural environment can be impeded by optical aberrations and scattering generated by the refractive index changes occurring due to the presence of surrounding biological tissue. Adaptive optics is the area of optical engineering which strives to correct such aberrations in optical systems through the use of measurement algorithms and sensors to enable the use of hardware able to control the optical field as desired. However, even the applicability of adaptive optics is limited by the amount of scattering produced in a thick tissue, as is the case for deep-brain imaging.

Multi-mode optical fibres have proven to be an important tool in combating optical aberrations present during \textit{in-vivo} imaging experiments, due to their ability to guide light to a desired location within an isolated propagation channel. Furthermore, their thin profile allows for imaging to be achieved while minimising the amount of damage to the surrounding tissue, limiting the observation of undesired malfunctioning neurons.

The light transmission though a multi-mode fibre can be characterised through the use of fibre modes. In an ideal fibre, the light coupled within these modes propagates unaffected by the other modes, nor influencing the propagation of other modes. However, due to differences in their propagation constants, the light is scrambled as it passes through the fibre. Thus, in order for a multi-mode fibre to be used in imaging, the light transmission needs to be characterised. This can be done through empirical interferometric measurements, the resulting transmission matrix describing the relationship between the input optical field and the resultant output optical field. Mathematically, this can be described as $\mathbf{y} = \mathbf{Tx}$, where $\mathbf{T}$ is the transmission matrix, $\mathbf{x}$ and $\mathbf{y}$ are the input and output optical fields, respectively, represented as a column vector of complex values representing the amplitude and phase of the electric field at specific sampled locations in the plane.

The drawback with the use of multi-mode fibres is that light transmission can be affected by changes in the fibre configuration, such as shape or temperature, leading to a loss in imaging quality. The change in light transmission occurs due to changes in the refractive index profile of the fibre due to bending or temperature.

This project focuses on correcting for changes in light transmission through a multi-mode fibre generated by bending. One of the sources of bending in practical neuroscience imaging experiments is the implantation procedure. As the fibre is flexible, implanting the fibre into an animal brain can lead to changes in shape as it travels toward the target location, being affected by the physical properties of the surrounding tissue. In the case of an anaesthetised animal, this application does not require for the correction to be time limited, as the implantation occurs only once at the beginning of the experiment and the fibre shape will not change during the experiment. Thus, the correction can be applied once before the imaging procedure occurs and the time required to correct will not affect the imaging procedure.

A second application is the deep-brain imaging of active animals. This application requires a fast correction procedure, in order to account for bending introduced by the movement of the animal.

Whether the imaging is achieved in a anaesthetised or active animal, being able to observe the neuronal activity and plasticity at a cellular level will allow for a better understanding of the mammalian brain, which could lead to further developments in the combat against neurological diseases. This can be done through the use of multimode fibres, while taking into account and correcting for any light transmission changes occurring due to bending.

Imaging though a multi-mode fibre (MMF) has already been implemented through the use of a point-scanning approach \cite{mmf_neuron}. In order to obtain an image, a focus is scanned at the distal facet of the fibre, which acts as an excitation beam for use in fluorescence imaging. If tissue containing fluorophores is present at the focus location, they are excited and light at a different wavelength is being emitted through the effect of fluorescence. The emitted light is captured by the MMF and transmitted back to the proximal facet, where an imaging apparatus is present. The amount of emitted light is measured and the value obtained represents the intensity of a pixel in the final image.

In order to achieve focusing capabilities at the distal facet of the MMF, a calibration procedure is performed prior to imaging, where the transmission of light through the MMF is being measured and characterised through the use of a transmission matrix (TM). This information can then be used to calculate the required input optical field which would generate a focus at the output of the fibre. The input optical field is shaped through the use of the spatial light modulator. However, bending introduces changes in the transmission of the light through the MMF, resulting in a loss of focusing capabilities, which leads to a decrease in imaging quality. 

Being able to re-measure the TM would represent the ideal case of a correction procedure, as then the light transmission would be fully characterised again, however, the same calibration approach cannot be used, as it requires access to the distal facet for interferometric measurement. During imaging, the MMF is implanted in the animal, thus, only proximal access would be available.

The first main contribution of the thesis is a correction approach intended to be used with a reflector placed at the distal facet to correct for a focus at the location of the reflector. This was achieved through a sensorless adaptive optics approach, making use of Walsh-Hadamard modes. The amount of light reflected was used as an optimisation parameter and the coefficients of the Walsh-Hadamard modes were individually changed to maximise the amount of reflected light through an iterative process. In order for the reflected light to be used as an optimisation parameter in achieving a focus at the desired location, the reflector must only reflect the light which contributes to the focus. Thus, a reflector must either be very small in size, comparable to the focus size, or some other approach, which has the same outcome, must be used, such as a fluorescent beacon. However, there were two drawbacks observed: the time required to obtain such a correction and the fact that focusing is only achieved at one specific location (at the reflector).

The time constraint, while important for imaging active animals, does not pose a problem for imaging applications at longer time frames, such as neuronal plasticity. However, recovering the focus at multiple reflectors will be required in order to achieve correction over the entire field of view, a requirement for a reasonable quality image to be acquired.

Expanding the locality of the correction can be achieved through the use of Memory Effect (ME) \cite{mem_eff}. This approach is based on the assumption that a small bending does not drastically change the diagonal property of a straight fibre TM, a requirement arising due to assuming that the bent fibre TM is also diagonal, which allows for the reconstruction of the assumed TM from a single focus. However, this approach is limited, as the area in which adequate correction is achieved is dependent on the amount of bending and does not cover the entire distal facet. Thus, multiple reflectors are required to be used, each responsible of a part of the distal facet.

If a sufficient number of reflectors provides an acceptable level of correction across the distal facet, free-space propagation can then be used to generate the required field at the output of the fibre such that a focus at a desired location away from the facet can be generated.

The overall correction procedure is described by the following:
\begin{enumerate}
    \item Regain the ability to focus at the reflectors
    \item Use Memory Effect to regain the ability to control the field of the entire facet
    \item Use free-space propagation to generate excitation foci at desired axial depths
\end{enumerate}

The second main contribution of the thesis is a novel approach to simulating Bragg gratings in multimode optical fibres. The new approach manages to compute the propagation of light through an entire Bragg grating irrespective of its length, by directly solving the system of differential equations. However, some restrictions were placed on the type of grating:
\begin{enumerate}
    \item The grating period must be constant, thus, chirped gratings are not able to be simulated
    \item The cross-section region of the grating within the fibre transversal profile must remain constant, however, there is no restriction in terms of shape or location.
\end{enumerate}

The reason for investigating Bragg gratings is their ability to be used as reflectors within the optical fibre, which could be used in the implementation of the correction procedure described in the first contribution.


%%%
\subsection{Thesis outline}

This thesis is composed of seven chapters, starting with an introduction to the thesis in the first. The second chapter describes the history, introduces the theory and some of the existing literature on which this thesis is based upon, drawing from research areas such as: microscopy, adaptive optics and fibre optics.

Chapter 3 describes the sensorless adaptive optics approach used in the correction procedure. It outlines the theoretical basis and implementation of the algorithm, making use of Walsh modes as an orthogonal basis. Finally, it provides some theoretical and experimental results of the approach. 
Chapter 4 describes the memory effect approach, outlining its use of the diagonal transmission matrix assumption and how it can be used to extend the focus correction area from one position.
Chapter 5 describes how free-space propagation can be used to change the axial position of the focusing plane and combines the methods described in the previous two chapters to coalesce the correction procedure. The chapter provides a numerical analysis of the correction's effectiveness through simulations.

Chapter 6 describes the novel Bragg grating simulation approach in multimode optical fibres. It starts with an existing theoretical description of the light propagation through a Bragg grating, then describes a new approach to solving the entire set of differential equations. The chapter finishes with an outline of the implementation and simulation results of the novel approach.

Finally, chapter 7 represents a conclusion to the thesis which reiterates the achievements, discusses on their potential impact in the research scene and describes a number of directions the work can be improved upon and used in the future.  


\end{document}