## 2 Fundamental Principles
### Mathematical Formulation  
The progress of light through a Michelson configured LCI is shown in Figure 2; offshoots at each stage in the process show key parameters which will be affected by the system components. By understanding the impacting factors on signal quality at each stage of this progression, optimisation of system design and signal processing can be undertaken to maximise achieved signal quality. The theoretical principles of LCI have been detailed in literature for an object with \(M\) discrete back-scattering reflectors [45; 46]. This section collates that information in an end-to-end theoretical breakdown from light source to spatial measurement. The interferometric signal that is generated by the superposition of reference and sample signals is shown in Figure 1d and Equations (2) and (3), respectively; these are composed of the same original signal from the light source represented by Equation (1) but with a phase delay due to the optical path difference between them. \[\mathbf{E}_{0}=s_{0}\left(k,\omega\right)e^{i\left(kz-\omega t\right)}, \tag{1}\]  
\[\mathbf{E}_{R}=\mathbf{A}_{R}e^{i\left(2 kz_{R}\right)};\mathbf{A}_{R}=r_{R} \frac{1}{\sqrt{2}}s_{0}\left(k,\omega\right), \tag{2}\]  
\[\mathbf{E}_{S}=\mathbf{A}_{S}*e^{i\left(2 kz_{S}\right)};\mathbf{A}_{R}=r_{S} \frac{1}{\sqrt{2}}s_{0}\left(k,\omega\right), \tag{3}\]  
\[\mathbf{E}_{D}=\frac{1}{\sqrt{2}}\left(\mathbf{E}_{S}+\mathbf{E}_{R}\right)= \frac{1}{\sqrt{2}}\left[\mathbf{A}_{R}e^{i\left(2 kz_{R}\right)}+\mathbf{A}_{S }e^{i\left(2 kz_{S}\right)}\right], \tag{4}\]  
where \(s_{0}\) represents the light source spectrum as a function of wavenumber \(k\) also known as spatial frequency and angular frequency \(\omega\). The phase offset \(\left(kz\right)\) can be set to zero for the initial signal in Equation (1). The reference signal and sample signal are shown by Equations (2) and (3), respectively, having their corresponding reflectivity represented by \(r_{R}\) and \(r_{S}(z_{S})\), where \(z_{S}\) denotes the path length variable in the sample arm from the beam splitter, the factor \(1/\sqrt{2}\) representing a 50:50 distribution of the optical power of the source signal from the beam splitter. The interferogram signal is composed of the superposition of the returned reference and signal electric fields as shown by Equation (4), the factor \(1/\sqrt{2}\) represents that only half of the returned signal power will be passed towards the detector as shown in Figure 1a; this signal takes the form shown in Figure 1d. To digitise the interferometric signal, an optical detector is used; this comes in the form of a charge-coupled device (CCD) or complementary metal-oxide-semiconductor (CMOS). This means that the recorded intensity \(I\) is proportional to the time averaged electric field multiplied by its complex conjugate. The signal obtained by the spectrometer is shown in Equation (5), where the photocurrent is generated by a square-law detector expressed as a function of spatial frequency. Using the knowledge  
Figure 2: Flowchart of the signal progress throughout a [PERSON] interferometer lit by a broadband light source. The grey boxes indicate effects resulting from each stage. that the modulus of a complex number has the relationship: \(|\bar{z}|\equiv\sqrt{\bar{z}\bar{z}^{*}}\) the following equation for intensity can be determined:  
\[ \begin{array}{ll}I_{D}\left(k,\Delta z\right)&=|\langle\mathbf{E}_{D}\cdot \mathbf{E}_{D}^{*}\rangle|,\\ &=\langle\mathbf{E}_{S}\cdot\mathbf{E}_{S}^{*}\rangle+\langle\mathbf{E}_{R} \cdot\mathbf{E}_{R}^{*}\rangle+2\mathcal{R}\left\{\langle\mathbf{E}_{S}\cdot \mathbf{E}_{R}^{*}\rangle\right\},\end{array}  \tag{5}\]  
where \(\mathcal{R}\) represents the real component and the angled brackets denote a time-average which can be displayed as:  
\[I_{D}\left(k,\Delta z\right)=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{ T}\mathbf{E}_{D}\cdot\mathbf{E}_{D}^{*}dt, \tag{6}\]  
where \(I_{D}\) is the photocurrent generated by the spectrometer.