## 3 LCI System Design Framework
### Lateral Resolution
#### 3.2.1 Operating outside the Region of Focus  
To demonstrate degradation in lateral resolution outside of a focused region, profile measurements were taken at two offsets of 235 \(\upmu\)m and 1020 \(\upmu\)m from a British five pence coin using the setup described in Figure 5. The system was configured as a fully fibre-enclosed implementation of an LCI with a common path for both the reference and sample signal leg, taking advantage of reduced sensitivity to vibrations, thermal fluctuations, and humidity, and removing the requirement for dispersion compensation between signals [49]. The system was un-lensed with the bare cleaved fibre having a numerical aperture of 0.13. This provides a micrometre-level footprint and form factor of the sensor. The system in this configuration has a confocal parameter of 83.5 \(\upmu\)m and a \(1/e^{2}\) transverse resolution of 5 \(\upmu\)m, hence both measured regions will be outside of the focused beam region. Figure 5: Experimental setup of a common path low coherence interferometer. The system components consist of a broadband light source provided by a super luminescent diode (EXS210068-01, Beratron, 850 nm) with a 3-dB bandwidth of 58 nm and an emitting power of 5.14 mW at 160 mA. A single-mode fibre coupler with a splitting ratio of 50:50 for beam splitting and coupling was used, with only one branch of signal output being used as the common path for the signal and reference. A spectrometer (MayaPro2000, Ocean Optics) operating at 125 Hz with a 2048 \(\times\) 64 pixel array, starting wavelength of 756 nm and spectral range of 174 nm with a resolution of 0.21 nm was used, giving a theoretical axial operating range of 2.15 mm before aliasing as defined from Equation (13). A region of the coin was focused on which contains the artisan's initials \"J.C\" to highlight the variation in lateral resolution with the distance of the coin from the fibre. Unilateral profile measurements were taken with the coin being translated at a constant speed of 100 \(\upmu\)m/s on the \(x\)-axis over 1100 \(\upmu\)m with a measurement integration time of 8 ms and a single spectrum captured per datapoint. This translated to a resolution of approximately 5 \(\upmu\)m per datapoint. Steps of 5 \(\upmu\)m were taken on the \(y\)-axis over a range of 800 \(\upmu\)m.  
Figure 6 shows the results from the two measurements, the red circles indicate a predicted spot size calculated from the NA of the fibre for an offset of 235 \(\upmu\)m and 1020 \(\upmu\)m. A clear reduction in the lateral resolution is present between the two offsets which manifests in a blurring of the measured feature. This degradation is a result of the increased spot on the surface of the sample reflecting light back from a larger area of focus. This results in an integration over the illuminated area which can shift the focus to other regions as the primary returned signal. From the measurement data, an increase of 1784% in the theoretical spot size area relating to an increase of 88% in the measured area covered by the \"J.C\" was observed; it is worth noting that an uncertainty metric for this measurement has not been completed. However, the values reflect the reduction in lateral resolution seen in Figure 6, where the achievable resolution is not equal to the spot size; this is due to the optical intensity distribution across the spot and that, with large incident angles to the sample surface, it is less likely for the light component to be recoupled into the system upon back reflection. Although the comparison may appear dramatic for a change in offset of approximately 900 \(\upmu\)m, the lateral scale of measurement should be taken into account which is approximately 700 \(\upmu\)m; thus, for many applications, this introduced blur will not impact measurement data integrity. However, for some small features such as shown in Figure 6, or surface roughness, this may be an unacceptable compromise and thus measurement offset must be taken into account or additional steps such as focussing optics to improve results must be used.