Investigating the Influence of the Diffraction Effect on Fourier Transform Spectroscopy with Bandpass Sampling

Abstract

Fourier transform spectroscopy with bandpass sampling (BPS-FTS) increases the interferogram sampling step and reduces sampling frequency, thereby reducing system implementation difficulty. However, this technology has only been introduced in principle and lacks analysis of the actual system error. In this paper, we present a theoretical analysis of the optical system diffraction effect, derive the bandpass spectrum bandwidth and interferogram sampling step under the influence of the diffraction effect, and conduct modeling and a simulation analysis with and without the diffraction effect. Through simulation, we found that when the diffraction effect was considered, the relative deviation of the recovered spectrum from the input spectrum was 1.30%, and the spectral similarity was 0.9669. However, without considering the diffraction effect, the relative deviation of the recovered spectrum from the input spectrum was 6.23%, and the spectral similarity was only −0.0056. The simulation results demonstrate that the diffraction effect has a significant influence on BPS-FTS interferogram sampling and recovery spectra. The results of this study have reference significance for their technical realization of BPS-FTS.

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. Finally, we conclude our analysis in Section 5.

2. Introduction to Bandpass Sampling Theory and BPS-FTS

2.1. Bandpass Sampling Theory

In practice, the spectra measured by Fourier transform spectroscopy instruments are usually limited to a finite bandwidth [3,5,6,7]. According to bandpass sampling theory, the bandpass signal sampling frequency is determined by both the signal bandwidth and maximum frequency, which can overcome the limitations of the Nyquist sampling theorem and recover the signal without aliasing [5,6,7,8,9]. We assume that the highest band signal frequency is $f_H$, the lowest frequency is $f_L$, and the signal bandwidth is $D=f_H-f_L$. According to bandpass sampling theory, to prevent aliasing in the sampled signal frequency domain, it is necessary to ensure that the spectral curve obtained after $m$ and $m-1$ translations of the component in the negative frequency domain of the signal cannot overlap with the original positive frequency domain component of the signal in this spectral region. Therefore, the following equation must be satisfied:

$$\{\begin{array}{l}-f_L+(m-1)f_s\le f_L\\ -f_H+m{f}_s\ge f_H\end{array}$$

(1)

where $f_s=2D$.

From Equation (1), it follows that:

$$\frac{2f_H}{m}\le f_s\le \frac{2f_L}{m-1}$$

(2)

where $1\le m\le \lfloor f_H/D\rfloor$, $\lfloor •\rfloor$ are rounded down.

For Fourier transform spectroscopy, the frequency of the interference signal corresponds to the wavenumber $\nu =1/\lambda$ of the spectrum [6]. According to bandpass sampling theory, the interference signal frequency range can be determined by the wavenumber boundary of the spectrum. Assuming that the detected spectrum wavenumber range is ${\nu }_{\min }~{\nu }_{\max }$, then the interferogram sampling step $\Delta x$ can be determined by Equation (3):

$$\frac{m-1}{2{\nu }_{\min }}\le \Delta x\le \frac{m}{2{\nu }_{\max }}$$

(3)

where $m$ is any positive integer equal to or less than ${\nu }_{\max }/\left({\nu }_{\max }-{\nu }_{\min }\right)$.

When $m_{\max }=\lfloor {\nu }_{\max }/({\nu }_{\max }-{\nu }_{\min })\rfloor$, $\Delta x$ is the maximum sampling step, which is given by:

$$\frac{m_{\max }-1}{2{\nu }_{\min }}\le \Delta x_{\max }\le \frac{m_{\max }}{2{\nu }_{\max }}$$

(4)

Substituting $m_{\max }=\lfloor {\nu }_{\max }/({\nu }_{\max }-{\nu }_{\min })\rfloor$ into Equation (4), we can obtain:

$$\Delta x_{\max }\le \frac{1}{2{\nu }_{\max }}\cdot \lfloor \frac{{\nu }_{\max }}{{\nu }_{\max }-{\nu }_{\min }}\rfloor =\frac{1}{2{\nu }_{\max }}\cdot \lfloor \frac{{\nu }_{\max }}{D}\rfloor$$

(5)

From Equation (5), it can be observed that the maximum sampling step is inversely proportional to the bandwidth, which means that the narrower the bandwidth, the larger the maximum sampling step $\Delta x_{\max }$. As bandwidth $D$ converges to ${\nu }_{\max }$, bandpass sampling approaches the Nyquist sampling theorem.

For Fourier transform spectrometers, although the detected spectrum is a band signal, its bandwidth typically remains relatively wide. Considering atmospheric sounding applications as an example, their detection bands typically cover the medium wave band [10]. If the wavenumber is in the spectral range ${2100 \mathrm{cm}}^{-1}~{3050 \mathrm{cm}}^{-1}$, according to Equation (5), we can obtain the maximum interferogram sampling step in this range as $\Delta x_{\max }\le [1/(2{\nu }_{\max })]\cdot \lfloor 3050/(3050-2100)\rfloor =[1/(2{\nu }_{\max })]\cdot 3$. Compared to the Nyquist sampling theorem, its sampling step is only three times larger. However, if we divide the spectrum in the range ${2100 \mathrm{cm}}^{-1}~{3050 \mathrm{cm}}^{-1}$ into 100 narrow bands, at this point, the bandwidth $D$ of each narrow spectral band is 1% of the original full spectral bandwidth, the corresponding maximum sampling step $\Delta x$ will increase by approximately two orders of magnitude, and the sampling frequency will decrease by approximately two orders of magnitude.

This analysis demonstrates that, according to the bandpass sampling theory, if the broadband spectrum detected by the Fourier transform spectrometer is converted into a series of continuous narrowband spectra, the interferogram sampling step can be increased, while the interferogram sampling frequency and difficulty of instrument development can be reduced. Therefore, BPS-FTS has been proposed internationally. Its basic principles involve using spectral splitting techniques to subdivide the broadband spectrum into several consecutive narrowband spectra, utilizing bandpass sampling theory to obtain the interferogram for each narrowband spectrum, and subsequently recovering all of the narrowband spectra using Fourier transformation to finally synthesize the broadband spectrum.

2.2. BPS-FTS

According to the analysis in Section 2, it can be observed that to fully exploit the advantages of bandpass sampling theory and reduce the number of interferogram sampling points, it is first necessary to subdivide the broadband spectrum into narrowband spectra. According to spectral imaging theory, a grating is an effective element capable of subdividing a broadband spectrum into a continuous narrowband spectrum. If a plane reflection grating (or a transmission grating) is set in the outgoing optical path of a Michelson interferometer, then the parallel interfering beam from the outgoing optical path of the interferometer will be diffracted by the grating after entering the plane reflection grating, forming a continuous monochromatic interfering beam arranged by wavelength, after which it will enter the focusing lens and finally interfere at the image plane. If a line array sensor is used, then a continuous narrowband interferogram can be obtained. The basic schematic is shown in Figure 1.

When the moving mirror moves at a constant velocity back and forth, each sensor pixel outputs a complete interferometric signal corresponding to a narrowband spectrum during a one-way constant linear motion. According to the bandpass sampling theory, as the spectral range corresponding to the interferogram obtained by each pixel decreases exponentially, the interferogram sampling step also increases exponentially. If we perform Fourier transformation on the sampled interferometric signal, we can recover a series of narrowband spectra. By arranging all pixel recovery spectra by wavenumber and superimposing their intensities, the complete spectrum can be obtained.

Assuming that the number of pixels along the grating dispersion direction of the line array sensor is $M$ and the corresponding pixel size is $P$, then the sensor length along the grating dispersion direction is $L_D=M\cdot P$. Let the spectrum detected by the system be $spe(\lambda )$ and its effective band range be ${\lambda }_{\min }\le \lambda \le {\lambda }_{\max }$. The corresponding effective wavenumber range is ${\nu }_{\min }\le \nu \le {\nu }_{\max }$, where ${\nu }_{\max }=1/{\lambda }_{\min }$, ${\nu }_{\min }=1/{\lambda }_{\max }$. Ideally, we typically consider the grating dispersion to be linear, and the wavelength of the dispersion spectrum is uniformly distributed on the sensor image plane. As the sensor is rectangularly sampled, it will divide the broadband spectral signal into $M$ segments of discrete narrowband spectral signals at equal wavelength intervals, and for the $i$th pixel, the wavelength of the narrowband spectral signal covered is shown in Equation (6):

$$\{\begin{array}{l}{\lambda }_{\min }^i={\lambda }_{\min }+(i-1)\cdot \Delta \lambda \\ {\lambda }_{\max }^i={\lambda }_{\min }+i\cdot \Delta \lambda \end{array}$$

(6)

where $\Delta \lambda =({\lambda }_{\max }-{\lambda }_{\min })/M$.

However, for a practical optical system, the image formed by a point object is typically a diffraction pattern determined by the system aperture diaphragm [11,12,13], which leads to crosstalk and aliasing of the spectra between adjacent pixels. In this case, the actual narrowband spectral signal wavelength range corresponding to each pixel is larger than the wavelength range determined by Equation (6). If we perform bandpass sampling on the interferogram of the actual narrowband spectrum according to the spectral range determined by Equation (6), the sampling step at this point does not meet the actual narrowband spectrum bandpass sampling condition, and the spectrum recovered from the sampled interferogram will produce spectral aliasing, resulting in the distortion of the recovered spectrum. However, the existing literature does not include a theoretical analysis of the influence of the diffraction effect. Hence, the problem is easily neglected during technical implementation, and this affects the final spectral accuracy; therefore, it is necessary to conduct an in-depth theoretical analysis and simulation verification of the problem.

3. Analysis of the Influence of the Diffraction Effect

As can be observed from Figure 1, BPS-FTS is a fusion of dispersive and Fourier transform spectroscopy. When the moving mirror is stationary and in the zero optical path difference position, the system is equivalent to a low-resolution grating spectrometer, where each sensor pixel outputs an integrated value of the spectrum with its narrowband. When the mirror moves back and forth, each pixel outputs a narrowband interferogram, and a higher-resolution spectrum is obtained by spectral recovery. However, after grating splitting, the spectral range of the narrowband corresponding to each pixel in the image plane is frequently unknown, and the main reason for this stems from the optical system-diffraction effect. Therefore, we first analyze the diffraction effect from the perspective of grating splitting [12,13,14], and the system-equivalent optical path is shown in Figure 2.

Assuming that the focal length of the imaging mirror is $f$, $d$ is the distance between grating groves, and light of wavelength $\lambda$ is incident on the grating at an angle of incidence $\alpha$, the resulting diffraction angle is $\beta$. Then, for the full broadband spectrum, the range of $\beta$ is $[{\beta }_{\min },{\beta }_{\max }]$. We assume that the central wavelength ${\lambda }_c$ has a diffraction angle ${\beta }_c$ and is imaged at the origin of the image plane coordinates, with the dispersion direction coinciding with the y-axis. According to the geometric relations, the position $y$ of any angle $\beta$ in the image plane can be obtained as follows:

$$\frac{y}{f}=\tan (\beta -{\beta }_c)$$

(7)

Then

$$\beta =\arctan (\frac{y}{f})+{\beta }_c$$

(8)

According to the grating equation [12,13,14,15,16], the relationship between the incident angle, the diffraction angle, and the wavelength can be obtained as follows:

$$d(sin\alpha \pm sin\beta )=n\lambda ,n=0,\pm 1,\pm 2,\cdots$$

(9)

Taking $n=+1$ as the order of diffraction, we obtain:

$$\lambda =d(sin\alpha \pm sin\beta )$$

(10)

Combining Equations (7) and (10), we can obtain the distribution of wavelength $\lambda$ within the detected spectral range $y$ is:

$$y=f\tan \{\arcsin [\pm (\frac{\lambda }{d}-\sin \alpha )]-{\beta }_c\}$$

(11)

Obviously, Equation (11) represents the distribution of the spectrum over the sensor without considering optical diffraction.

When considering the optical diffraction effect, we assume that the system aperture diaphragm is circular with a radius of $a$. According to the Fraunhofer circular aperture diffraction theory, at this stage, for any point $y$ on the sensor, the corresponding light of wavelength $\lambda$ forms a diffraction spot on the image plane (typically its central bright spot is called an Airy disk) with the center point at $y$. Then, the intensity of light at a distance $r$ from the center can be expressed as follows [12,13]:

$$S(r)=S_0{\left[\frac{2J_1\pi \frac{2\pi ar}{\lambda f}\pi }{\frac{2\pi ar}{\lambda f}}\right]}^2$$

(12)

where $J_1$ is a first-order Bessel function and $S_0$ is the light intensity at the center of the Airy disk. The intensity distribution of the diffraction pattern is shown in Figure 3, where $r=\sqrt{x^2+y^2}$. It can be observed that the intensity of the diffraction pattern decays with the increase in $r$ in a sharp oscillation, showing a ring-like distribution between light and dark.

Table 1. Intensity distribution of each ring of the diffraction pattern.

	r	Peak	Percentage
1st Maximum	0	1.0	83.9%
1st Zero	$1.22f\lambda /(2a)$	0.0	-
2nd Maximum	$1.635f\lambda /(2a)$	0.0175	7.1%
2nd Zero	$2.233f\lambda /(2a)$	0.0	-
3rd Maximum	$2.679f\lambda /(2a)$	0.0042	2.8%
3rd Zero	$3.238f\lambda /(2a)$	0.0	-
4th Maximum	$3.699f\lambda /(2a)$	0.0018	1.5%
4th Zero	$4.24f\lambda /(2a)$	0.0	-
5th Maximum	$4.72f\lambda /(2a)$	0.00078	1.0%

provides the $r$ values corresponding to different light and dark rings and the ratio of their intensity values to the total incident light intensity [11,12,13,17,18].

For spectrometers, at least 80% of the Airy disk energy is typically required to fall within the size range of the sensor pixel in the optical design process. Taking 84% of the diffraction pattern energy falling within just one pixel as the benchmark, by covering 95–99% of the signal power of the spectrum as the criterion, it can be observed from Table 1 that up to the fourth dark ring of the diffraction pattern, the energy occupied represents 95.3% of the whole diffraction pattern energy. Limited by the size of the pixel, we can approximate that the energy outside the fourth dark ring has little effect on other pixels and can be regarded as negligible noise. Then, the effective coverage of light with any wavelength $\lambda$ on the image plane is:

$$r\le \frac{4.24f\lambda }{2a}$$

(13)

Obviously, the $r$ value is larger than ½ of the pixel size. Without loss of generality, when we analyze with the $i$th pixel of the sensor, then the distance of the boundary of the pixel along the y-axis from the origin is:

$$\{\begin{array}{l}{y}_{\min }^i=(i-\frac{M}{2}+1)P\\ y_{\max }^i=y_{\min }^i+P\end{array}$$

(14)

According to the previous analysis, the spectral energy that the pixel can receive is the sum of all spectral energies in the wavelength range determined by Equation (13). As shown in Figure 4, the red and black curves represent two boundary wavelengths, and the corresponding fourth dark ring of the diffraction pattern is at the boundary position of pixel $i$. At this point, any wavelength spectrum located in the two wavelength ranges will contribute to the energy of pixel $i$.

In Figure 4, let the wavelengths corresponding to the black and red curve be ${\lambda }_l^i$ and ${\lambda }_r^i$, respectively, and the positions of the centers of the two curves on the sensor image plane are denoted by $y_l^i$ and $y_r^i$, respectively. Then we can obtain:

$$\{\begin{array}{l}{y}_l^i=y_{\min }^i-\frac{4.24f{\lambda }_l^i}{2a}\\ y_r^i=y_{\max }^i+\frac{4.24f{\lambda }_r^i}{2a}\end{array}$$

(15)

Combining Equations (11), (14) and (15), we can obtain wavelengths ${\lambda }_l^i$ and ${\lambda }_r^i$, corresponding to the black and red curves, and all spectra in this wavelength range contribute to the energy of the $i$th pixel. The wavenumber range corresponding to this wavelength range is $[{\nu }_r^i,{\nu }_l^i]$, where ${\nu }_r^i=1/{\lambda }_r^i$, ${\nu }_l^i=1/{\lambda }_l^i$. Obviously, this spectral range is larger than that expressed in Equation (6), which is mainly due to the optical diffraction effect. Next, we analyze the influence of the diffraction effect on the bandpass sampling interferogram and recovered spectrum.

For light with wavelength λ, the distribution of its diffraction pattern at the image plane is shown in Figure 5. Taking the center of the Airy disk as the center of the circle, given any $r$, the diffraction pattern intensity value at $r$ is $S(r)$. Then, within the upper and lower boundaries determined by the sensor pixels in Figure 5, the corresponding energy is given by the line integral to the diffraction pattern light intensity over the arc segment $l(r)$; that is:

$$E_l(r)={\displaystyle {\int }_{l}S(r)dl}$$

(16)

where $l(r)$ is determined as follows:

$$l(r)=2rarc\sin (\frac{p}{2r})$$

(17)

Given an arbitrary pixel $i$, assuming that the pixel size is $P$, and the distance between the pixel center and the Airy disk center is $\Delta d$, for the diffraction pattern of light wavelength $\lambda$, the energy in this pixel can be approximately calculated as follows:

$$B_{\Delta d}^i(\lambda )={\displaystyle {\int }_{\Delta d-P/2}^{\Delta d+P/2}{E}_l(r)dr}$$

(18)

According to Equation (18), the actual spectrum corresponding to any pixel $i$ can be obtained. Combined with the basic theory of FTS [1,10,19,20,21,22], we can obtain the interference signal intensity corresponding to any pixel $i$ as follows:

$$I^i(x)=\frac{1}{2}{\displaystyle \underset{{\nu }_r^i}{\overset{{\nu }_l^i}{\int }}{B}^i(\nu )[1+cos(2\pi \nu x)]d\nu }$$

(19)

where $x$ is the optical path difference. We use bandpass sampling theory to discretize interferogram sampling. We assume that the maximum optical path difference of the system is $L$, and the optical path difference sampling step that satisfies the bandpass sampling theory is $\Delta x$. Then, according to Equation (4), the maximum sampling step of the interferogram of the $i$th pixel of the sensor after considering the diffraction effect is:

$$\Delta x_{\max }^i=\frac{m_{\max }^i}{2{\nu }_l^i}$$

(20)

where $m_{\max }^i=\lfloor {\nu }_l^i/({\nu }_l^i-{\nu }_r^i)\rfloor$. It can be observed that for any pixel $i$, the maximum sampling step of the corresponding interferogram is directly related to the maximum wavenumber and wavenumber bandwidth of the spectrum. In fact, optical systems always have diffraction effects. Therefore, the actual spectral bandwidth corresponding to a pixel is typically several times the ideal spectral bandwidth, and the difference between the maximum wavenumbers is usually small, which will lead to several times the difference in the interferogram sampling step. Through the simulation analysis in Section 5, it can be observed that the influence of several times the difference on the recovered spectrum is fatal. To ensure that there is no spectral aliasing in the recovered spectrum, it is necessary to comprehensively consider the diffraction effect of the optical system, and accurately determine the spectral wavenumber range corresponding to each pixel to eliminate the influence of the diffraction effect.

4. Simulation Analysis and Verification

To verify the influence of the optical diffraction effect on the BPS-FTS spectrum, we set the pixel size of the simulation analysis to 18μm; the simulated spectral range is 2040 nm~2060 nm, and the corresponding wavenumber range is 4887 cm⁻¹~4902 cm⁻¹. The spectral curve is shown in Figure 6.

For the simulation process, we assume that the optical system is ideal and that no other error effect exists except for optical diffraction. To meet conventional optical design requirements, we assume that the Airy disk radius of the central wavelength spectrum is exactly half the pixel size; that is, $r_0=\mathrm{P}/2$. According to the data in Table 1, we can obtain the radius corresponding to the fourth dark ring in a straightforward manner as follows:

$$r_4=(4.24/1.22)r_0\partial$$

(21)

Because the spectral range is narrow, in the 20 nm band, the diffraction pattern distributions of different wavelength spectra on the image plane are fundamentally the same. To reduce simulation complexity, the intensity distribution of the diffraction pattern of the central wavelength spectrum is uniformly used to approximate the intensity distribution of all other wavelength spectra, and this approximation will not affect the judgment of the final result.

For BPS-FTS, a grating is typically used to subdivide the spectrum. It is assumed that under the effects of the grating, the 20 nm spectrum is uniformly subdivided into 10 narrowband spectra and sampled by 10 pixels, each of which corresponds to a sampling bandwidth of 2 nm. Using an interferometer, an interferogram corresponding to the spectrum is obtained on the image plane. We assume that the Fourier transform spectrometer can achieve a spectral resolution of 0.1 nm and the maximum optical path difference of the interferometer is 2.5 cm. Therefore, we obtain a set of reasonable optical system simulation parameters, as shown in

Table 2. Optical system parameters.

Parameters	Values
Radius of circular hole	0.5706 cm
F-number	3.5986
Number of grating lines	200 lines/mm
Focal length	4.1064 cm

.

The simulation is divided into three parts and is based on the above set of optical parameters. Firstly, the influence of the diffraction effect on spectral distribution is simulated. The influence of the diffraction effect on the interferogram is simulated and analyzed on this basis. Finally, through the simulation of the recovery spectrum, we compare and analyze the influence of the diffraction effect on the recovery spectrum. In the data comparative analysis, some edge spectra are discarded.

(1) Simulation of the diffraction effect on the spectral bandwidth of pixels

In this section, we simulated the distribution of the original input spectrum on each pixel. The spectral bandwidth of each pixel is shown in

Table 3. Bandwidth comparison for each pixel.

Pixel Number	Bandwidth for Each Pixel (cm−1)		Ratio
	Considering the Diffraction Effect	Without Considering the Diffraction Effect	Considering the Diffraction Effect/ Without Considering the Diffraction Effect
1	13.1354	4.8029	2.7349
2	17.9101	4.7926	3.7370
3	21.4248	4.7823	4.4800
4	21.3788	4.7720	4.4800
5	21.3329	4.7618	4.4800
6	21.2871	4.7516	4.4800
7	21.2415	4.7414	4.4800
8	21.1960	4.7312	4.4800
9	17.6710	4.7211	3.7430
10	12.9322	4.7110	2.7451

; the last column shows the ratio of bandwidth both with and without considering the diffraction effect.

It can be observed from the above calculation results that, owing to the diffraction effect influence, the actual spectral bandwidth corresponding to each pixel is wider than the ideal spectral bandwidth. Due to the boundary effect, the spectral bandwidth and energy corresponding to the four pixels on the edge are lost. Therefore, in this study, only spectral data from the middle six pixels were used in the data comparison analysis, which did not affect the conclusion. Based on the spectral bandwidth data of the middle six pixels, it can be observed that the real spectral bandwidth caused by the diffraction effect is 4.48 times the ideal spectral bandwidth. This will directly affect the determination of the interferogram sampling step, which is expected to have a significant impact on the number of sampling points and spectral recovery. Figure 7 shows the real spectral distribution of each pixel after considering the diffraction effect. On this basis, we simulated and analyzed the interference data sampled under the two conditions, either considering the diffraction effect or not.

(2) Simulation of the interferogram diffraction effect

Under the same maximum optical path difference of the interferometer, the sampling step of the optical path difference for each pixel with and without considering the diffraction effect is shown in

Table 4. Comparison of OPD sampling step for each pixel.

Pixel Number	Maximum OPD Sampling Step (cm)		Ratio
	Considering the Diffraction Effect	Without Considering the Diffraction Effect	Without Considering the Diffraction Effect/ Considering the Diffraction Effect
1	-	-	-
2	-	-	-
3	0.0233	0.1046	4.4945
4	0.0234	0.1048	4.4792
5	0.0234	0.1050	4.4836
6	0.0234	0.1052	4.4880
7	0.0235	0.1054	4.4923
8	0.0236	0.1057	4.4815
9	-	-	-
10	-	-	-

.

The interference data obtained using our simulation are arranged by pixel, as shown in Figure 8 and Figure 9. Figure 8 represents the sampled interferogram corresponding to the case without considering the diffraction effect, and Figure 9 represents the sampled interferogram corresponding to the case considering the diffraction effect. Figure 10 shows the interferogram of the spectrum corresponding to the fifth pixel. It can be observed that the interferogram sampling step after considering the diffraction effect is smaller, with more sampling points and richer details.

(3) Simulation of the diffraction effect on the spectral recovery process

According to the theoretical analysis based on BPS-FTS, we performed spectral recovery of interferograms sampled at two different optical path difference sampling steps with and without diffraction effects. In this process, as the interferogram sampling points obtained according to the bandpass sampling theory are too discrete, the theoretically recovered spectrogram should also consist of some discrete sampling points on the input spectrum. We displayed the spectral shape of the recovered spectrum smoothly, so that the consistency between the recovered spectrum and the input spectrum can be accurately judged. In this paper, the interferometric data are zero-filled in the interferometric dimension during the recovery process. This process corresponds to the data interpolation of the recovered spectral dimensions without changing spectral information. Accordingly, we performed a spectral recovery simulation on the above two interferograms. The recovered spectra of each pixel with and without considering the diffraction effect are shown in Figure 11 and Figure 12, respectively.

We took the recovered spectra of the middle six pixels in both cases and normalized them with the input spectrogram; the comparison is shown in Figure 13. The black line in Figure 13 represents the real input spectrum after considering the diffraction effect, the red line represents the recovered spectrum without considering the diffraction effect, the blue line represents the recovered spectrum after considering the diffraction effect, and the green line represents the curve of the normalized energy collected by each pixel when the system is stationary with 0 optical path difference, such as for a grating spectrometer. It can be observed that if the diffraction effect is not considered, the recovered spectral curve will undergo serious aliasing; hence, only a lower spectral resolution can be achieved. In the case where the diffraction effect is considered, the recovered spectrum is fundamentally the same as the input spectral curve with more obvious details. This is also consistent with our theoretical analysis in Section 4, and the green marker line trend is also consistent with the spectrum trend.

To quantitatively evaluate the recovered spectral accuracy with and without the diffraction effect, a method is used to calculate the relative deviation of the recovered spectrum from the input spectrum, and the spectral correlation mapper (SCM) method is used to calculate the similarity between the recovered and input spectra. Equation (22) details the relative deviation calculation formula as follows [22]:

$$R_{error}=\frac{1}{N}{\displaystyle \sum _{n=1}^N\frac{\left|S_n^{\prime }-S_n\right|}{S_n}}\times 100\%$$

(22)

where $S_n^{\prime }$ is the recovery spectrum, $S_n$ is the standard spectrum, and $N$ is the number of spectral bands.

The SCM method [23,24] is a derivative of the Pearson’s correlation coefficient and is defined as follows:

$$R_{S,S^{\prime }}=\frac{{\displaystyle \sum _{n=1}^N(S_n^{\prime }-{\mu }_{S_n^{\prime }})(S_n-{\mu }_{S_n})}}{\sqrt{{\displaystyle \sum _{n=1}^N{(S_n^{\prime }-{\mu }_{S_n^{\prime }})}^2{\displaystyle \sum _{n=1}^N{(S_n-{\mu }_{S_n})}^2}}}}$$

(23)

where $S_n^{\prime }$ is the recovered spectrum, ${\mu }_{S_n^{\prime }}$ is the mean value of the recovered spectrum, $S_n$ is the input spectrum, and ${\mu }_{S_n}$ is the mean value of the input spectrum.

The results of the calculations are listed in

Table 5. Calculation results of spectral accuracy evaluation criteria.

	Considering the Diffraction Effect	Without Considering the Diffraction Effect
R error	1.30%	6.23%
SCM	0.9669	−0.0056

.

From these calculation results, we can observe that the relative deviation of the recovered spectrum considering the diffraction effect is smaller than the relative deviation of the recovered spectrum without considering the diffraction effect. Moreover, the spectral similarity between the recovered spectrum considering the diffraction effect and the input spectrum is significantly greater than that without considering the diffraction effect, and its similarity to the input spectrum is close to 1. This demonstrates that in BPS-FTS, the influence of the diffraction effect must be considered.

5. Conclusions

Compared with the traditional Fourier transform spectroscopy technique, BPS-FTS can significantly increase the sampling step and thus reduce the number of sampling points, which leads to a reduction in difficulty for FTS implementation. However, to date, although many studies exist, all of them solely focus on the introduction of technical principles, and there is a distinct lack of analysis of the various influencing factors in the technology implementation process, thus increasing the difficulty with which the technology can be implemented quickly. To address this problem, we focused on theoretical research and simulation analysis of the optical system’s diffraction effect.

Through theoretical analysis and simulation, it can be observed that owing to the influence of the diffraction effect, the narrowband spectral bandwidth corresponding to each sampling pixel is 4.48 times that of the spectral bandwidth without considering the diffraction effect. As a result, the interferogram sampling step, when each pixel does not consider the diffraction effect, is approximately 4.5 times that of the interferogram sampling step when the diffraction effect is considered. Therefore, for each pixel, if the actual interferogram is sampled according to the ideal interferogram sampling step, this leads to interferogram under-sampling, which, in turn, leads to spectral aliasing of the recovered spectrum, resulting in a serious distortion of the spectrum. The above conclusion is also verified by the recovered spectrum simulation analysis. The sampling interferogram-recovered spectrum considering the influence of the diffraction effect is fundamentally consistent with the input spectrum, the relative deviation between them is 1.30%, and the spectral similarity is 0.9669. However, without considering the diffraction effect, the sampling interferogram-recovered spectrum is far from the input spectrum: the relative deviation of the two is 6.23%, and the spectral similarity is only −0.0056. Therefore, in the technical realization process of BPS-FTS, the influence of the diffraction effect of the optical system must be fully considered. In the future, we will conduct further theoretical analysis and research on other errors, such as non-uniform sampling and noise, hoping to promote the implementation and popularization of this technology.

In conclusion, aiming at the phenomenon by which the real narrowband spectral boundary is difficult to define due to the diffraction effect in BPS-FTS, we deduced the real narrowband spectral boundary and solved the problems of interferogram under-sampling and restoring spectral distortion caused by the inaccurate spectral boundary. These results provide effective theoretical and technical support for the implementation of this technology.