## 2 Fundamental Principles
### Mathematical Formulation
Substituting Equations (2) and (3) for the reference, \(E_{R}\) and signal, \(E_{S}\) components in Equation (5) will yield Equation (7). Here, the light source intensity spectrum is denoted as \(S(k)=|s(k)|^{2}\). This represents a modulating signal term combined with a carrier signal term \"DC term\" to produce the modulated signal shown in Figure 1d. \[ \begin{array}{ll}I_{D}(k)&=\frac{\rho}{4}S\left(k\right)\left[R_{R}+\sum_{n =1}^{N}R_{Sn}\right]\\ &+\frac{\rho}{2}\left[S\left(k\right)\sum_{n=1}^{N}\sqrt{R_{R}R_{Sn}}(cos \left[2 kn\left(z_{R}-z_{Sn}\right)\right])\right]\\ &+\frac{\rho}{4}\left[S\left(k\right)\sum_{n\
eq m=1}^{N}\sqrt{R_{Sn}R_{Sm}}( cos\left[2 kn\left(z_{Sn}-z_{Sm}\right)\right])\right],\end{array}  \tag{7}\]  
where \(\rho\) is the responsivity of the line scan camera, \(R_{R}\) is the reflectance of the reference, \(R_{S}\) is the reflectance of the sample reflector, \(2(z_{R}-z_{Sn})\) is the round trip optical path difference (OPD) between the reference and \(n^{\text{th}}\); and \(m^{\text{th}}\). sample reflectors. It can be seen from Equation (7) that the intensity of the signal is composed from the sum of three terms:  
The 1 st term is a constant \"DC\" signal, independent of changes in the OPD between the reference and sample signals. This term dominates the measured signal with an amplitude proportional to the power reflectivity of the reference and sample, scaled by the light source optical intensity as a function of wavenumber. This dominance can be reduced through changing the beam splitter ratio such that a greater percentage of the signal is directed towards the sample arm of the interferometer. Additionally, as the reference signal optical power is typically far greater than the samples returned signal optical power, this term will limit the signal strength before the spectrometer detector becomes saturated; by reducing the reference signal, the signal strength can be increased without saturation. The 2 nd term represents the reflected signals from the measured sample. The signal magnitude contribution from this term is typically far less than the DC signal contribution, due to the optical intensity being proportional to the square root of the reference and sample reflectivity. The 3 rd term represents the interference between different sample reflectors for a system where the reference and sample paths are separate as in Figure 1a. This term will appear as additional peak artefacts in the Fourier domain of the measured signal; reduction can be achieved through increasing the reference signal power through changing the beam splitter ratio directing light along each path. As shown in Equation (7), the 2 nd term signal amplitude is dependent on the reflectivity of the reference and sample whilst the 3 rd term is dependent on only the sample signal reflectivity. However, for a common-path configuration, this represents the reflected signals from the measured sample. Depth information is obtained by performing a Fourier transform on Equation (7) to form Equation (8), allowing for determination of the modulating frequencies corresponding to sample reflector locations:  
\[i_{D}(z) =\frac{\rho}{8}\gamma(z)[R_{R}+R_{S1}+R_{S2}+...]+\frac{\rho}{4} \gamma(z)*\sum_{n=1}^{N}\sqrt{R_{R}R_{Sn}}(\delta[z\pm 2(z_{R}-z_{Sn})])\] \[+\frac{\rho}{8}\gamma(z)*\sum_{n\
eq m=1}^{N}\sqrt{R_{Sn}R_{Sm}}( \delta[z\pm 2(z_{Sn}-z_{Sm})]),\]  
where \(i_{D}(z)\) is intensity recorded as a function of OPD and \(\gamma\left(z\right)=FT\{S(k)\}\) is the coherence function manifesting as a convolution of the Dirac delta function which represents frequency elements of the Fourier transformed signal at set bins. Hence, this term determines the point spread function (PSF) over which the spread of interference fringes occur for a set sample reflector.