The tempo-spatially modulated polarization interference imaging spectroscopy (TSMPIIS) is a promising technology, which is wider and wider used in many fields such as the environmental hazards assessment, agriculture, mineral exploration, urban study and so on [1–5]. As a novel means for remotely detecting the spectral information of the object, the TSMPIIS can simultaneously detect and identify the spatial, spectral and polarization information of the object on the basis of its physical properties [6–10]. Not only can this technology greatly improve the ability for people to understand the objective world, but it also provides a sharp tool for detecting the abundant information of the object from multiple perspectives, which achieves the unity of the observed object from the specific image cognition to the abstract logical cognition [11].

In order to overcome to the drawbacks of the slit and the dynamic scanning, the polarization interference imaging spectrometer based on a Savart polariscope interference imaging spectrometer (SPIIS) is proposed based on the TSMPIIS [12–15]. Due to the uncontrollable factors existing in the imaging process, including the observation scale, angle, and complex background, the spectrum extracted from the SPIIS image is rarely pure, which is named original spectrum. In order to make the quantitative studies of the spectral information of the object, an accurate spectral calibration of the sensor data must be performed [16–19]. The spectral calibration for the SPIIS mainly includes the spectral range and spectral resolution calibrations together with the lateral displacement and spectral line position precision calibrations, where the spectral resolution and spectral line position are interconnected with the lateral displacement. Therefore, if one of these parameters is calibrated using a high precision monochromator or laser, the other parameters will be known [20–23].

There are two major tasks being carried out in this paper, and the first task is the spectral line position calibration on the basis of the monochromator, integrating sphere and spectroradiometer while the second task is the spectral resolution calibration based on the laser, beam expander and spectroradiometer.

The outline of this paper is as follows: The spectrum calibration system of the SPIIS is firstly given in section 2, and then the spectral line position calibration together with the spectral resolution calibration are performed in sections 3.1 and 3.2, in order to improve the spectral quality of the SPIIS and reflect its detecting ability. To evaluate the spectral performance, the methodology and measurement for the spectrum quality calibration of the SPIIS are studied in detail, which use the linear interpolation, cubic spline interpolation, and piecewise cubic interpolation methods to achieve the spectral line position calibration of the restored spectrum. Taking the quadratic fitting coefficient as the wavelength calibration coefficient, the calibrated equation is obtained.

According to the imaging principle of the SPIIS, a set of suitable spectral calibration system is developed using the high precision monochromator and laser, and the experimental device is depicted in Fig. 1. The light beam from the monochromator or laser firstly passes through the integrating sphere or beam expander, and then full of the pupil of the SPIIS by adjusting the optical axis of the integrating sphere or beam expander coincided with that of the SPIIS. The area source of the integrating sphere, whose uniformity is better than 1% is chosen as the standard measured signal. The spectroradiometer (Field Spec 3 VNIR; ASD Inc., VA, USA), with the national measurement standard as a standard transmission, is placed in the export of the integrating sphere or beam expander, which may monitor different radiant brightness levels for a radiant calibration. Here, a short description of the configuration and property of the SPIIS is given, which is composed of the fore-optics system, followed by the polarizer, Savart polariscope and analyzer together with the imaging system [13].

Figure 1. Spectral quality calibration, where (a) spectral line position calibration based on a monochromator and (b) spectral resolution calibration based on a laser.

3.1. Spectral Line Position Calibration

In this section, the spectral line position calibration is performed. The spectral line position calibration refers that the monochromatic light with a known center wavelength position and narrow wavelength bandwidth is adopted to determine the maximum optical path difference of the interferometer, and the restored spectrum is obtained by the Fourier transform, while the standard spectrum is acquired by the spectroradiometer under the same conditions; Finally, the relationship between the wavelength at the peak of the standard spectrum and the pixel position at the maximum of the restored spectrum is established. According to the spectral calibration principle of the SPIIS, the experimental process of the spectral line position calibration is shown in Fig. 2. The monochromatic light is generated by the monochromator and expanded by the integrating sphere with its diameter of 2.5 inches, and its output light source is the halogen lamp with its power of 125 W. A spectral response function comes from one pixel of the SPIIS, and the wavelength at the peak of the response function is located on the center of the pixel, which means that the relationship between the scanning wavelength and the response value of the detector pixel is established. When all the pixels are dealt with above process, the wavelength positions are mapped to the focal plane of the SPIIS.

Figure 2. Spectral line position calibration experiment.

Meanwhile, the spectroradiometer is aligned with the output port of the integrating sphere, and the spectral image is acquired as the standard data.

The wavelength of the monochromatic light is selected as 480–960 nm, and 10 interferograms are acquired by the SPIIS every 10 nm. By the Fourier transform, the restored spectrum is obtained, while the standard spectrum is acquired by the spectroradiometer under the same conditions. The wavelength at the peak of the standard spectrum and the pixel position at the maximum of the restored spectrum are recorded, respectively, as shown in Table 1. From Table 1, the measured wavelength by the spectroradiometer is smaller than the theoretical wavelength exported by the monochromator, and the error is in a range of [0.00 nm, 8.70 nm], mainly due to the spectroradiometer not docking with the exit slit of the monochromator.

TABLE 1. Spectral line position calibration data.

Theoretical Wavelength (nm)	Wavelength at the Peak of the Standard Spectrum (nm)	Pixel Position at the Maximum of the Restored Spectrum
490	490.0	1
500	498.5	116
510	509.6	111
520	518.9	106
530	528.5	101
540	538.5	97
580	576.6	80
620	617.1	77
660	656.1	54
700	696.1	43
740	736.5	34
780	776.6	25
820	821.3	17
860	858.3	11
870	864.8	9
880	878.2	8
890	885.0	7
900	898.9	5
910	906.0	3
920	913.3	2
930	928.1	1
940	935.7	1
950	943.5	1
960	951.3	1

Because of a lot of wavelength data used, the fitting polynomial can obtain higher index, which makes the fitting results close to the true curve. Considering the practical application requirement and convenience, the fitting index should not be too high. According to Table 1, the polynomial exponents are selected as 1, 2, 3, and 4, respectively, and the fitting results are shown in Fig. 3.

Figure 3. Fitting curve of the wavelength changing with the pixel, where (a) a polynomial fitting, (b) quadratic polynomial fitting, (c) cubic polynomial fitting, and (d) quartic polynomial fitting.

From Fig. 3, for a polynomial fitting, a small amount of original data points is distributed on the right of the fitting curve while most of them are on the left. However, for quadratic, cubic, and quartic polynomial fitting, most of original data points are located on the fitting curve, and only a small amount of them is located on the right of the fitting curve. The polynomial fitting errors are shown in Table 2.

TABLE 2. Polynomial fitting error for the spectral line position calibration data.

Theoretical Wavelength (nm)	Different Fitting Error (nm)
	A Polynomial Fitting	Quadratic Polynomial Fitting	Cubic Polynomial Fitting	Quartic Polynomial Fitting
500	−2.7787	3.8210	61.7818	17.2324
510	34.4938	0.0490	59.2238	7.2213
520	26.3763	3.6020	55.5108	1.1274
530	18.2588	2.4800	50.7178	−2.1338
540	13.7648	3.0964	48.1562	−1.3606
580	−7.8347	−1.6034	28.4862	3.1790
620	−18.5637	−4.2740	13.2268	6.7964
660	−22.0457	−5.0608	2.8040	4.7945
700	−21.9042	−5.8642	−4.9602	−2.6435
740	−14.5157	−1.6788	−4.5620	−5.9338
780	−7.1272	0.3196	−4.1638	−10.2794
820	3.8848	4.7044	0.9882	−7.3661
860	22.1438	16.8590	15.2638	8.4793
900	40.4028	28.0416	30.0362	29.4295
910	43.1558	28.2198	31.7598	34.5568
920	49.5323	33.2640	37.6492	42.4418
930	55.9088	38.2900	43.5578	50.5548

From Table 2, the accuracy of the quadratic polynomial fitting is the highest, and it’s fitting error is better than −5.8642 nm in the wavelength range of [500 nm, 820 nm]. Taking the quadratic fitting coefficient as the wavelength calibration coefficient, the calibrated equation is obtained:

where the pixel n ∈ [1,116].

In order to make the fitting curve of the spectral line position calibration more accurate, the variations of the wavelength λ with the pixel n by the linear interpolation, cubic spline interpolation and piecewise cubic interpolation processing are obtained, respectively, as shown in Fig. 4. For the linear interpolation, cubic spline interpolation and piecewise cubic interpolation methods, most of original data points are on the curve interpolation, in addition to a few data points near zero.

Figure 4. Fitting curve of the wavelength ^λ changing with the pixel n by different interpolations.

The output wavelengths of seven monochromatic light produced by the monochromator are 550 nm, 600 nm, 680 nm, 720 nm, 760 nm, 800 nm, and 840 nm, respectively, and the interferograms are acquired by the SPIIS. By the data processing, the restored spectra are obtained, and the conversion flow chart from the original interferogram to restored spectrum is described in Fig. 5. Where the data denoising means that the sampled interference data must be filtered by differential filtering to recover the real spectrum accurately; The phase correction refers that adopting the convolution to correct some interferogram with high asymmetry, which can obtain the more symmetrical interferogram; the data toe-cutting refers to the suppression of the false signal of toe in the recovery spectrum due to the limited scanning range of the optical path difference between two branches of the spectrometer. The spectral line position calibration by the quadratic polynomial fitting is performed, and the calibration results are shown in Fig. 6. Where the quadratic, cubic, quartic polynomial fitting are respectively adopted for the monochromatic light with its wavelength 840 nm.

Figure 5. Conversion flow chart from the original interferogram to restored spectrum.

Figure 6. Interferogram and restored spectrum after the spectral line position calibration, where (a) for 550 nm, (b) for 600 nm, (c) for 680 nm, (d) for 720 nm, (e) for 760 nm, (f) for 800 nm, (g) interferogram for 840 nm, (h) quadratic polynomial fitting for 840 nm, (i) cubic polynomial fitting for 840 nm, and (j) quartic polynomial fitting for 840 nm.

From Fig. 6, there is a certain gap between the center positions of the theoretical and restored wavelengths, and the error varies from 0.8 nm to 7.2 nm, which is mainly caused by different shear displacements of two beams light. For the monochromatic light with its wavelength of 840 nm, the quadratic, cubic, and quartic polynomial fitting errors are 12.5 nm, 9.4 nm, and 1.1 nm, respectively, and the quartic polynomial fitting error is minimum. In the practical application, the position accuracy of the center wavelength can be improved by reducing the spectral response range of the instrument.

3.2. Spectral Resolution Calibration

The experiment facility of the spectral resolution calibration for the SPIIS is shown in Fig. 7, and the laser with the narrow spectral line width is chosen as a standard source. In experiment, firstly, adjust the optical axes of the laser and beam expander to be consistent with that of the SPIIS. Afterwards, regulate the beam expander to ensure that the entrance pupil of the SPIIS is filled with the laser beam. Finally, the interference images of different monochromatic lights are acquired by the CCD detector, and the restored spectrum is obtained by the dark background removed, filtering, apodization, phase correction, and Fourier transform.

Figure 7. Spectral resolution calibration experiment.

In the data processing of the interference image for the monochromatic light, the wavelength width at half of the peak value of the restored spectral curve (FWHM) is defined as the spectral resolution.

The deviation between the restored and theoretical spectral resolution is calculated:

where δλ′ and δλ represent the restored spectral resolution and theoretical spectral resolution, while λ₁ and λ₂ stand for the wavelengths of the restored spectral curve at the full width at half maximum (FWHM). In the spectral resolution calibration, six monochromatic lights with wavelengths of 525 nm, 543 nm, 571 nm, 760 nm, 810 nm and 850 nm, respectively, are selected, and the restored spectra are shown in Fig. 8. According to the image data recorded by Fig. 8, the spectral resolution of different wavelengths is shown in Table 3. The theoretical spectral resolution is calculated based on the maximum optical path difference (OPD) with its value of 57.08 μm. If the wavelength of the monochromatic light is in a range of [525.00 nm, 633.00 nm], the laser is chosen as the calibration source, while the monochromator is set as the calibration source when the wavelength changing from 720.00 nm to 850.00 nm.

Figure 8. Restored spectra corresponding to different wavelengths of (a) 525 nm, (b) 543 nm, (c) 571 nm, (d) 633 nm, (e) 720 nm, (f) 760 nm, (g) 810 nm, and (h) 850 nm.

TABLE 3. Spectral resolution of different monochromatic lights.

Wavelength of the Monochromatic Light (nm)	Theoretical Spectral Resolution (nm)	Restored Spectral Resolution (nm)	Deviation of the Spectral Resolution (nm)
525.00	2.40	2.50	0.10
543.00	2.60	2.70	0.10
571.00	2.80	2.90	0.10
633.00	3.50	3.90	0.40
720.00	4.54	4.60	0.06
760.00	5.06	6.10	1.04
810.00	5.75	6.60	1.05
850.00	6.33	7.10	1.77

From Table 3, the restored spectral resolution is greater than the theoretical value, and the deviation between them becomes larger with the wavelength increasing, which is mainly caused by the structural design of the SPIIS, the rationality of the spectral restoration algorithm together with the selection of the maximum OPD. In addition, both the laser broadening and the monochromator broadening also affect the restored spectral resolution.

The experimental scheme of the spectral quality calibration for the SPIIS based on the monochromator, laser, integrating sphere, beam expander and spectroradiometer is proposed in this paper. In the spectral line position calibration, the linear interpolation, cubic spline interpolation and piecewise cubic interpolation techniques are adopted, and the calibration precision of the quadratic polynomial fitting is the highest, whose fitting error is better than −5.8642 nm for the wavelength of [500 nm, 820 nm]. Taking the quadratic fitting coefficient as the wavelength calibration coefficient, the exact expression of the wavelength changing with the sampling points is obtained: λ = 0.0135n² − 5.0189n + 896.7154, where n ∈ [1,116].

With the spectral resolution, the laser is chosen as a calibration source for the wavelength of [525.00 nm, 633.00 nm], and the deviation between the restored and theoretical spectral resolution is less than 0.5 nm. However, if the wavelength changing from 720.00 nm to 850.00 nm, the monochromator is used as a calibration source, and the deviation of the restored spectral resolution relative to the theoretical spectral resolution is larger than 1 nm. In a word, the restored value of the spectral resolution is greater than the theoretical value, and the largest deviation is 1.77 nm, which is mainly caused by the structural design of the SPIIS, the rationality of the spectral restoration algorithm and the selection of the maximum OPD.

Natural Science Foundation of Shandong Province (Grant no. ZR2022MF266); Innovation Center for Feng Yun Meteorological Satellite (Grant no. FY-APP-ZX-2022.0206).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.