## 3 Aberration Simulation of Each Imaging Lens  
Before developing the spectrometer, aberration characteristics were simulated using ZEMAX to understand the characteristics of the two imaging lenses, an ADL and a TFL. The ADL was originally designed to reduce spherical and chromatic aberrations and is commonly used as an imaging lens for spectrometers. However, in the ADL, the focal surface is formed as a curved shape. To address this problem, the TFL was designed to have a flat focal plane and significantly reduce the distortion of the focal surface. Among the lens aberrations, simulations for field curvature and astigmatism were performed over full spectral bands, and the results can be seen in Figure 2. As the angle increased, the tangential and sagittal focal surface of the ADL became further away from the flat image plane, resulting in a maximum shifting of \(-\)1.2 mm and \(-\)0.5 mm for the tangential and sagittal ray, as shown in Figure 2a. The maximum difference between the two focal positions was 0.7 mm. The TFL showed improved field  
Figure 1: Spectrometer design parameters. The focal length of the fiber-optic collimator is \(f_{1}\), the incident angle to the transmission gratings is \(\theta_{\mathrm{inc}}\), the diffracted divergence angle is \(\Delta\varphi\), the focal length of the imaging lens is \(f_{2}\), and the length of the line-scan camera (LSC) is \(L_{\mathrm{c}}\). curvature and astigmatic aberration characteristics compared to the ADL. The simulation results for the TFL are shown in Figure 2b, where the tangential focal position was shifted up to 0.28 mm, and the sagittal focal position was moved up to \(-0.12\) mm. Their maximum difference was 0.4 mm. When we considered that the FWHM spectral bandwidths correspond to an incident angle of around 4 degrees, the maximum focal shifts occurred in the tangential ray and were about 270 \(\upmu\)m for the ADL and about 140 \(\upmu\)m for the TFL. The minimum shifts were about 130 \(\upmu\)m for the ADL and about 85 \(\upmu\)m for the TFL in the sagittal ray. The TFL presented a nearly flat focal surface similar to the detector plane. The astigmatism characteristics of each imaging lens did not change significantly with wavelength, and as a result, we can confirm that the chromatic aberration was well corrected for both imaging lenses. If the focal position is not placed on the pixel surface, the visibility of the detected spectral interference signal is poor. In the simulation results, the TFL showed better field curvature than the ADL. Therefore, we can expect that the spectrometer using the TFL will measure interference signals with enhanced visibility and an OCT image obtained with this spectrometer will have better image contrast. When the focal length is different depending on the wavelength in the same lens, this axial aberration is called the chromatic focal shift. Simulations for the chromatic aberration of the two imaging lenses were also performed and the results were calculated for the entire spectral bandwidths, as shown in Figure 3. The ADL showed a focal shift of less than 40 \(\upmu\)m and the TFL had a focal shift of less than 60 \(\upmu\)m. With regard to the FWHM spectral bandwidths, the chromatic focal shifts were less than 15 \(\upmu\)m and 20 \(\upmu\)m, respectively. The ADL was found to have a smaller chromatic focal shift than the TFL. However, the two imaging lenses had similar values. Both imaging lenses showed nearly uniform chromatic focal length, indicating that chromatic aberration was well corrected in both lenses. The chromatic aberration characteristics of these two imaging lenses were in good agreement with the results shown in Figure 2, whereby the change in the focal position change according to the wavelengths was very small. Figure 2: The field curvature and astigmatic aberration characteristics of each imaging lens. T and S mean the tangential and sagittal focal position of (**a**) the achromatic doublet lens (ADL) and (**b**) the telecentric f-theta lens (TFL).