## 1 Introduction  
With the advantages of multiple channels, high throughput, and high spectral resolution [1; 2; 3], Fourier transform spectroscopy has been widely used in many fields, such as atmospheric sounding and material analysis. Especially in the former, because of its high sensitivity and high spectral resolution, Fourier transform spectroscopy is among the most ideal technical means for current gas-concentration sounding. The core of Fourier transform spectroscopy technology is the Michelson interferometer or its modified structures, which can be used to obtain the measured substance interferogram for a wide range of optical path differences and obtain a spectrum by Fourier transform recovery. To achieve accurate spectral recovery, interferogram sampling must satisfy the Nyquist sampling theorem [1; 3], which means that the interferogram sampling frequency is greater than twice the highest frequency of the detected spectrum. This leads to shortcomings such as the small interferogram sampling step, a large number of sampling points, and high sampling frequency [4], which increase the complexity of the requirements for interferometer stability and the sampling accuracy of the circuit clock, which significantly increases implementation difficulty. For Fourier transform spectroscopy, it is almost unachievable to break the Nyquist sampling theory limit to significantly increase the sampling step and reduce the sampling rate and the number of sampling points. However, in practice, the spectrum measured by a Fourier transform spectrometer is typically a bandpass signal. At this point, the interferogram can be sampled using bandpass sampling theory, which is more in line with the signal characteristics, overcoming the limitation of the maximum signal frequency, and changing to the limitation of the signal bandwidth to provide greater flexibility for interferogram sampling. Under bandpass sampling constraints, if an effective method can be identified with which to adjust the spectral bandwidth corresponding to the interferogram under certain conditions, it is possible to maximize the interferogram sampling step and reduce the number of sampling points. From this perspective, Fourier transform spectroscopy with bandpass sampling (BPS-FTS) has emerged internationally [5; 6; 7]. This technology disperses a broadband spectrum into a series of continuous narrowband spectra by adding dispersive elements to the interferometric optical path and determines the interferogram of each narrowband spectrum separately to reduce the number of sampling points and the frequency in the whole optical path difference range, thereby reducing the interferometer system realization difficulty. However, the published literature solely focuses on the introduction of the technical principle under ideal conditions and fails to consider the influence of various errors and defects in the actual system on the final, recovered spectral accuracy. Taking the diffraction effect of an optical system as an example, it typically leads to dispersion of the spectrum distribution on the image plane. When bandpass sampling theory is used in this context, the actual spectral bandwidth under the influence of the diffraction effect is significantly wider than the ideal bandwidth. If the spectrum is sampled with the ideal bandwidth, this will lead to severe under-sampling, which results in a serious distortion of the recovered spectrum. Thus, it is necessary to theoretically analyze and verify the influence. Therefore, in this paper, we address the theoretical analysis, simulation, and verification of the influence of the diffraction effect on a BPS-FTS optical system. The paper is organized as follows: In Section 2, we introduce bandpass sampling theory. Section 3 describes BPS-FTS in detail. A theoretical analysis of BPS-FTS under the influence of optical diffraction is performed in Section 4. Simulations are conducted in Section 5 to verify the interferograms and recovered spectra for cases both with and without diffraction effects.