## 2 Spectrometer Design Process  
We intend to show that simply replacing the imaging lens with improved aberration characteristics can significantly improve the imaging performance of the SD-OCT system. Because the performance of the spectrometer varies greatly according to the characteristics of the optical elements, the same optical devices were applied to each spectrometer except for the imaging lens. The imaging lenses that were utilized in the spectrometers were an achromatic doublet lens (ADL, AC254-060-B-ML, THORLABS), which is commonly used in spectrometer development, and a telecentric f-theta lens (TFL) designed to reduce field curvature aberration. To design a spectrometer that is easy to implement, most of the optical components used were commercial products, and a simple optical configuration was selected. The optical design of the spectrometer was done according to the following procedure [35]. The single-mode fiber connected to the spectrometer has a core size of about 6 \(\upmu\)m and the dimensions of a single-pixel of a line-scan camera (LSC, 2048 pixels, CMOS sensor, 140 kHz, SPL2048-140 km, Basler), that is, 10 \(\upmu\)m \(\times\) 10 \(\upmu\)m. Therefore, the amplification of the spectrometer will be less than 1.67 to avoid intensity loss and cross-talk between pixels. Because the amplification of the spectrometer is given by the ratio of the focal length of the collimator and the imaging lens, the focal length of the imaging lens in front of the LSC can be determined from the amplification value. When we consider that the focal length of the collimator (F810 APC-842, THORLABS) is about 36.2 mm, the focal length of the imaging lens is calculated as about 60 mm. The collimated beam with a diameter of 7.8 mm from the fiber-optic collimator is directed to a transmission grating (1800 lines/mm, a center wavelength of 840 nm, WASACHPHOTONICS) at an incident angle of 49.1\({}^{\circ}\). After passing through the transmission gratings, the spectral components are diverged according to the diffraction grating equation. To satisfy the Nyquist sampling theory, the total spectral bandwidth received by the spectrometer must be at least twice the full width at half maximum (FWHM) spectral bandwidth of the light source [36]. For this reason, each edge wavelength is designed to be 778 nm and 900 nm, respectively. The calculated diffraction angle at each edge wavelength is 40.13\({}^{\circ}\) and 59.77\({}^{\circ}\). At the center wavelength of 840 nm, the diffraction angle is 49.11\({}^{\circ}\). This results in a divergence angle of 19.64\({}^{\circ}\) for the designed full spectral bands. This value is also a critical factor in determining the focal length of the imaging lens. When the light beam is incident on a lens with a focal length \(F\) at an angle of \(\theta\), the height of the image at the focal position is the product of \(F\) and \(\theta\) by the ABCD matrix method. As shown in Figure 1, when this is the case, the relation of \(\Delta\varphi\times f_{2}=L_{\mathrm{c}}\) must be satisfied. If the length of the active area of the LSC is 20.48 mm and the divergence angle after the transmission grating is about 19.7\({}^{\circ}\), then the focal length is calculated as 59.7 mm. We can confirm that this value is in good agreement with the value obtained by considering the amplification of the spectrometer. Most existing telecentric f-theta lenses are designed for laser processing using high power lasers with a wavelength of 1060 nm or designed for vision application with visible light. However, it is not appropriate to use this lens because the light source considered in this experiment is not a laser, but a broadband light source with a wide wavelength band of several tens of nm centered around 840 nm.