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Abstract
An iterative local Fourier transform (ILFT)-based high-accuracy wavelength calibration for Fourier transform imaging spectrometer (FTIS) is proposed. The wavelength calibration for FTIS is to determine the relation between the wavelength and the wavenumber position. However, the wavenumber position solved by conventional method is only accurate up to integers restricted by the picket-fence effect of discrete Fourier transform. While the proposed ILFT can increase the accuracy of calculating the wavenumber position by combining the local Fourier transform and a few iterations. In this paper, the method is investigated in theory and then by simulations and experiments. The simulations show that the accuracy of the wavenumber position calculated by the ILFT is increased by 100 times than conventional method with noise, phase error, and non-uniform sampling of optical path difference. And the experimental results indicate that the ILFT decreases the absolute error of wavelength calibration from about 2.03 nm to 0.16 nm. Therefore, the method provides theoretical and technical support for FTIS and promotes the development of superior resolutions therein.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The spectrum of a target can reflect its material composition, making the spectrum an important optical parameter. Fourier transform imaging spectrometer (FTIS) has gradually gained potential for spectral measurement owing to its advantages of high throughput, spatial resolution, and spectral resolution. This technology is widely applied in environmental monitoring, biomedicine, disguise detection, and other fields. Various types of FTISs are developed based on interferometers such as the Michelson interferometer [1–7]; Fabry–Pérot interferometer [8,9]; Sagnac interferometer [10–17]; Mach–Zehnder interferometer [18]; and birefringent interferometer using Savart plate [19–22], Wollaston prism [23–27], or other types [28]. The FTISs first acquire the broadband fringe pattern of the target. Then, the spectral intensity curve of different wavelengths can be calculated by performing Fourier transform (FT) on the broadband fringe pattern. However, the spectral intensity curve calculated via FT reflects a relationship between the spectral intensity and the wavenumber position. The wavenumber position of a certain wavenumber is defined as the abscissa of the first positive sidelobe of the fast Fourier transform (FFT) spectrum of the monochromatic fringe pattern of that wavenumber. As a matter of fact, the wavenumber position cannot be related to the wavenumber or wavelength directly. Wavelength calibration must be performed to obtain the relationship between wavenumber position and wavelength. The curve calculated by FT can then be converted to exhibit a direct relation between the spectral intensity and its wavelength, which is suitable for analysis. The fundamental procedure of wavelength calibration is the calculation of wavenumber position; therefore, the accuracy of calculating wavenumber position directly governs the accuracy of wavelength calibration. Conventional method can only obtain the wavenumber position in the range of integers that is affected by the picket-fence effect in FT. The method adversely affects wavelength calibration and then influence the subsequent radiometric calibration.
To improve the accuracy of wavelength calibration, Brasunas proposed a zero-padding Fourier transform (ZPFT) method for locating the wavenumber position of a He–Ne laser, wherein the accuracy only slightly increased because few zeros were padded to the interferogram [29]. Comisarow also proposed a zero-filling FFT method for the wavelength calibration of FTIS, wherein the accuracy was further increased by padding more zeros than before, albeit resulting in heavy computational burden [30]. To better implement wavelength calibration, Lin proposed a high-precision spectral calibration method. Their method achieved high-accuracy wavelength calibration by approaching the total optical path difference (OPD) iteratively. The method considers the theoretical total OPD as a basis to calculate the wavenumber. The difference between the calculated wavenumber and the actual one is used to correct the total OPD. After iterations, the total OPD can be approximated. The method does not require zero padding and increases the accuracy to a certain degree. However, the iterational calculations are time-consuming [31]. Yang proposed a wavenumber calibration based on method by Lin, wherein the subpixel wavenumber position was acquired by estimation. Yang approximated the spectrum around the peak as a Gaussian distribution. Then, the subpixel wavenumber position can be calculated. This method avoids iterative calculations as well as zero padding, but the estimated position was not very accurate [32].
Zhang proposed a wavelength calibration method to ameliorate the inconsistency of fringe patterns in the row direction based on the Savart interferometer. The method focused on conducting FTIS based on Savart interferometer and achieved good calibration results. However, the wavenumber position calculation was not discussed in detail [33]. Cho also proposed a method to improve the spectral resolution of FTIS based on the Sagnac interferometer via signal padding. The method can balance the tradeoff between spectral resolution and discrete sampling. The spectral resolution as well as the wavelength calibration increased about 10 times than before. However, the accuracy still has potential to increase further by padding more zeros [34]. These methods can raise the accuracy to a certain degree, but cannot simultaneously increase accuracy and lower computational complexity.
In this paper, an iterative local Fourier transform (ILFT)-based high-accuracy wavelength calibration for FTIS is proposed. The proposed method for calculating wavenumber position was compared with ZPFT-based method to verify its superior performance. The remainder of the paper is organized as follows. First, the principle of the FTIS as well as the procedure of wavelength calibration and wavenumber position calculation is introduced based on Sagnac interferometer. Then, the theory of wavenumber position calculation based on ILFT is provided in detail. Thereafter, the ILFT-based method is simulated and compared with ZPFT-based method for accuracy and computational complexity. Finally, the proposed method is applied for the wavelength calibration of real FTIS to verify its performance.
2. Theory of high-accuracy wavelength calibration
2.1 FTIS and wavelength calibration
The schematic of the Sagnac FTIS is shown in Fig. 1. A real image of the target is produced by the first objective lens (L1) in the stop (S) region. The light from the real image in S is then collimated by the second objective lens (L2), which is split by the beam splitter (BS) into reflected and the transmitted light rays. The reflected ray is reflected by the upper mirror (M1) followed by the right mirror (M2) successively. Then, it is again reflected by the BS toward the third objective lens (L3). The transmitted light is reflected by M2 followed by M1 successively. Then, it is transmitted by the BS toward L3. The reflected and the transmitted rays are now laterally sheared. The two rays are finally focused by L3 onto the same location at the detector, where they interfere.
In Fig. 1, the red light ray is split into two rays by Sagnac interferometer. Before imaging via lens L3, the two separated red light rays have a lateral shearing distance, which will result in interference due to OPD. The interference intensity I can be expressed with the OPD x and the spectral intensity S as follows:
where $\sigma $ is the wavenumber, defined as $\sigma = 1/\lambda $.Equation (1) indicates that the interference intensity $I(x)$ and the spectral intensity $S(\sigma )$ are a Fourier transform pairs. Therefore, $S(\sigma )$ can be expressed as
According to Eq. (2), the spectral information of different wavelengths can be obtained by performing FT on the broadband fringe pattern. In Eq. (2), the wavenumber and the OPD are continuous and infinite. However, they are both discrete and finite in real FTIS system after acquisition. As a result, the spectrum is also discrete, and is obtained by converting Eq. (2) as follows:According to the theory of discrete Fourier transform (DFT), $N = 1/(\Delta \sigma \Delta x)$. Therefore, Eq. (3) can be rewritten as
When $I(n)$ represents the monochromatic fringe pattern, $S(k)$ changes into the spectrum of the monochromatic light. $S(k)$ possesses only the first positive sidelobe and the first negative sidelobe, which are symmetrical in the frequency domain. The abscissa of the first positive sidelobe will be $k$ (wavenumber position) in the frequency domain. Monochromatic fringe patterns with different wavenumbers have different frequencies, which results in sidelobes having different abscissa values (wavenumber position) in the frequency domain after undergoing FFT. Therefore, a discrete relation between the wavelength and the wavenumber position can be acquired by performing FFT on monochromatic fringe patterns of different wavenumbers. Wavelength calibration can then be achieved by performing curve fitting with the results. The whole wavelength calibration procedure is presented in Fig. 2.As analyzed before, the fundamental process of wavelength calibration in FTIS is to obtain the abscissa of the first positive sidelobe of the Fourier-transformed monochromatic fringe pattern in the frequency domain. For convenience, we define ${\bf K} = {[0,1, \cdots ,N - 1]^T}$ and ${\bf X} = {[0,1, \cdots ,N - 1]^T}$. Substituting them into Eq. (4), the matrix form of Eq. (4) becomes
The wavenumber position of $S({\bf K})$ can be calculated by Equations (5) and (6) outline the procedure of calculating wavenumber position by the conventional method. ${k_{\textrm{FT}}}$ is only accurate up to integers restricted by the picket-fence effect of DFT, resulting in low accuracy in the following wavelength calibration. The computational complexity of the conventional method is as follows, based on the theory of FFT: where $O({\cdot} )$ is the Big-Ω notation.2.2 Wavenumber position calculation based on the ILFT
Considering the problem that conventional wavelength calibration can only obtain the wavenumber position to an accuracy of integers, a wavenumber position calculation method based on ILFT is proposed. The method first achieves a spectrum zooming operation by zero padding (small multiple) followed by local Fourier transform (LFT). The wavenumber position is coarsely located by searching for the abscissa of the first positive sidelobe of the zoom spectrum. Then, a small range of spectrum around the wavenumber position (integer accuracy) can be calculated by performing LFT to further zoom into the spectrum. A high-accuracy wavenumber position can be obtained by repeating the steps above. By performing such iterations, the method can realize high-accuracy wavenumber position calculation with a bit more computation. The method can finally increase the accuracy of the wavelength calibration procedure. The ILFT is an evolution of the ZPFT. The theory of the ZPFT is introduced as follows.
First, $I({\bf X})$ is padded with $(M - 1)N$ zeros to obtain $I(\bar{{\bf X}})$. M represents the multiples for zero padding. $I(\bar{{\bf X}})$ can be expressed as:
Performing FT on $I(\bar{{\bf X}})$, we obtain the frequency spectrum:
The abscissa of the first sidelobe ${k_{peak}}$ in $S(\bar{{\bf K}})$ is
As the length of the new signal is M times that of the former one, ${k_{peak}}$ is also enlarged by M times. Hence, the wavenumber position calculated via the ZPFT is The M should be big enough to ensure the accuracy of calculating the wavenumber position. Theoretically, M is at least 10,000 for accuracy to four decimal places. The computational complexity ${T_{\textrm{ZPFT}}}$ of this method is still high despite applying FFT when M is large. The expression of ${T_{\textrm{ZPFT}}}$ is ZPFT calculates the spectrum considering all the elements in $I(\textrm{X})$ as a requirement for FFT. However, $I({\bar{X}})$ contains $I(\textrm{X})$ and padded zeros that can be neglected while calculating. As a result, Eq. (9) can be simplified by applying the principles of block matrix.Figure 3 illustrates that the ZPFT calculates a matrix multiplication of two matrices with size $MN \times MN$ and $MN \times 1$. However, the LFT simplifies the multiplication: the spectrum range that needs to be calculated is the green block, which is from $({k_{\textrm{FT}}} - 0.5)M$ to $({k_{\textrm{FT}}} + 0.5)M$. To that end, the row of the first matrix is limited to $({k_{\textrm{FT}}} - 0.5)M$ to $({k_{\textrm{FT}}} + 0.5)M$. Furthermore, the effective part of $I(\bar{{\bf X}})$ is the blue block. As a result, the column of the first matrix is limited to $0$ to $N - 1$. Finally, the size of the first matrix is decreased from $MN \times MN$ to $M \times N$ and the second matrix is decreased from $MN \times 1$ to $N \times 1$. After the matrix multiplication, we obtain the green block, which is the spectrum we need. Therefore, the LFT requires fewer multiplication operations than ZPFT. To further increase the calculation speed, the LFT is executed by iterations. The multiple of zero padding for each iteration is set as m, and the number for iterations is q. Then, m satisfies the following condition:
Applying LFT with ${{\bf K}_1} = {[({k_{\textrm{FT}}} - 0.5)m,({k_{\textrm{FT}}} - 0.5)m + 1, \cdots ,({k_{\textrm{FT}}} + 0.5)m]^T}$, the spectrum of one-time LFT is obtained: The abscissa of the first sidelobe $k_{peak}^1$ of $S({{\bf K}_1})$ is The accuracy of $k_{peak}^1$ is increased by m times that of ${k_{\textrm{FT}}}$. The spectrum range is set to ${{\bf K}_2} = {[(k_{peak}^1 - 0.5)m,(k_{peak}^1 - 0.5)m + 1, \cdots ,(k_{peak}^1 + 0.5)m]^T}$ to perform LFT again for further increasing the accuracy. The spectrum $S({{\bf K}_2})$ around $k_{peak}^1$ is calculated; then, the new abscissa of the first sidelobe $k_{peak}^2$ is The iterations are continuously performed q times. The spectrum range for the q-th LFT is set to ${{\bf K}_q} = {[(k_{peak}^{q - 1} - 0.5)m,(k_{peak}^{q - 1} - 0.5)m + 1, \cdots ,(k_{peak}^{q - 1} + 0.5)m]^T}$. Then, the spectrum $S({{\bf K}_q})$ around $k_{peak}^q$ is calculated. The final abscissa of the first sidelobe $k_{peak}^q$ is
Table 1. Theoretical k under different M.
From Table 1, it can be seen that the accuracy of calculating k will be up to three decimal places when M is chosen as 1000. However, the number of multiplications it requires is $1000 \times 2000 = 2 \times {10^6}$. The amount of calculation is thus very high. To deal with situations wherein M is very large, we propose a method to reduce the calculations while maintaining the same accuracy as the former method. We first choose a small multiple m for zero padding to zoom in the spectrum. Then, the wavenumber position is roughly located, and a small range of frequency spectrum around the wavenumber position (roughly located) can be calculated by performing LFT to the new signal padded with zeros. Then, a more accurate wavenumber position can be obtained by locating the maximum value of the zoom-in spectrum. By repeating the steps above, a wavenumber position with the same accuracy as one-time spectrum zoom (the multiple for zero padding is $M$) can be acquired.
Taking $M = 1000$ as an example, the detailed process of the ILFT is as follows. The small multiple m of zero padding is chosen as 10. Then, the $k = 123.5$ is obtained, which is accurate to one decimal place. Next, 123.5 is chosen as the center for the zoom-in spectrum, ranging from 123.4 to 123.6. Then, $k = 123.46$ is obtained, with an accuracy up to two decimal places. Similarly, 123.46 is chosen as the center for the zoom-in spectrum ranging from 123.45 to 123.47. The final result of k is 123.456, which is accurate up to three decimal places. In this example, a 1000-times zoom-in spectrum can be acquired by three iterations. By three coarse locating, a k with accuracy to three decimal places can be calculated. The number of multiplications required for one iteration is $10 \times 2000 = 2 \times {10^4}$ by applying local FT. The total number of multiplications required for three iterations is $6 \times {10^4}$. Thus, the number of multiplications required now has decreased by two orders of magnitude compared with the former $2 \times {10^6}$. It can be inferred that the bigger M is, the lower the relative amount of computation will be. The procedure delineated above is the high-accuracy wavenumber position calculation method for FTIS based on ILFT. The process is illustrated in Fig. 4.
From Fig. 4, the ILFT is realized by the following steps:
- (1) As Fig. 1 shows, a Saganc-based FTIS acquires the monochromatic interferogram of a laser. Then, the interference signal $I({\bf X})$ can be obtained by extracting one row of the monochromatic interferogram.
- (3) Taking ${k_{\textrm{FT}}}$ as the center, a small range ${{\bf K}_1}$ around ${k_{\textrm{FT}}}$ is chosen as the spectrum range for spectrum zoom. And ${{\bf K}_1}$ satisfies ${{\bf K}_1} = {[({k_{\textrm{FT}}} - 0.5)m,({k_{\textrm{FT}}} - 0.5)m + 1, \cdots ,({k_{\textrm{FT}}} + 0.5)m]^T}$. Then, the zoom-in spectrum $S({{\bf K}_1})$ can be calculated by performing LFT to the interference signal $I({\bf X})$ with Eq. (18).
- (4) A more accurate wavenumber position $k_{peak}^1$ can be obtained by locating the maximum of the zoom-in spectrum $S({{\bf K}_1})$ with Eq. (19).
- (5) Taking $k_{peak}^1$ as the center, a small range ${{\bf K}_\textrm{2}}$ around $k_{peak}^1$ is chosen is chosen for spectrum zoom. And ${{\bf K}_\textrm{2}}$ satisfies ${{\bf K}_2} = {[(k_{peak}^1 - 0.5)m,(k_{peak}^1 - 0.5)m + 1, \cdots ,(k_{peak}^1 + 0.5)m]^T}$. Then, the zoom-in spectrum $S({{\bf K}_\textrm{2}})$ can be calculated by performing LFT to the interference signal with $I({\bf X})$ Eq. (20).
- (6) A more accurate wavenumber position $k_{peak}^\textrm{2}$ can be obtained by locating the maximum of the zoom-in spectrum $S({{\bf K}_2})$ with Eq. (21).
- (7) The iterations are continuously performed q times. The spectrum range for the q-th LFT is set to ${{\bf K}_q} = {[(k_{peak}^{q - 1} - 0.5)m,(k_{peak}^{q - 1} - 0.5)m + 1, \cdots ,(k_{peak}^{q - 1} + 0.5)m]^T}$. Then, the zoom-in spectrum $S({{\bf K}_q})$ can be calculated by performing LFT to the interference signal $I({\bf X})$ with Eq. (22).
3. Simulations of the proposed method
3.1 Performance in ideal condition
To verify the feasibility of the ILFT, a simulated monochromatic interferogram is tested. The monochromatic interferogram can be expressed as
where the sampling interval of OPD $\Delta x$ is decided by $\Delta x = {\Delta _{\max }}/N$, and ${\Delta _{\max }}$ represents the maximum OPD.Three discrete monochromatic interferograms are simulated, whose detailed parameters are shown in Table 2. The third simulated interferogram is shown in Fig. 5.
Fig. 5. Simulation of monochromatic interferogram. (a) Intensity versus OPD. (b) Intensity versus sampling number. (c) Detailed interference pattern.
Table 2. Parameters of simulation interferogram.
Figure 5(a) is the simulated monochromatic interferogram whose abscissa is the OPD. The interferogram requires discrete sampling because the interference pattern acquired by the detector is discrete. The result of discrete sampling is shown in Fig. 5(b), whose abscissa is the sampling number. The maximum OPD is chosen based on the real FTIS system. The maximum OPD is excessively large to achieve high spectral resolution. As a result, the interference pattern is very dense. The detailed interference pattern is shown in Fig. 5(c) for convenience of understanding.
To verify the performance of the proposed method, the wavenumber position obtained by the ILFT is compared with that obtained by the conventional method and the ZPFT. When M is chosen as 10000, the frequency spectra obtained from the three methods are shown in Fig. 6, using the third row data of Table 2 as an example.
Fig. 6. The frequency spectra of the simulation interference pattern obtained by three methods: (a) the conventional method, (b) ZPFT, and (c) ILFT.
Figure 6(a) shows the spectrum calculated by the conventional method. Therein, the abscissa of the first positive sidelobe is 95, which means $k = 95$. The conventional k is accurate only up to integer values. This k has considerable error compared to the theoretical one. Figure 6(b) shows the spectrum calculated by ZPFT. The k is 95.3508, which is the same as the simulation parameter (95.35) after rounding. It shows that the calculated k is as accurate as the preset k when choosing appropriate M. However, this method calculates the spectrum of $2.048 \times {10^7}$ sampling points. Therefore, the amount of calculation and the required RAM are both large, resulting in long calculation time. Figure 6(c) is the spectrum calculated by the ILFT. The k obtained is 95.3508 from Fig. 6(c). The accuracy of the method is much higher than the conventional one and is at the same level as the ZPFT. However, the ILFT greatly reduced the computational complexity because it only requires to calculate 20 points of spectrum for four times. The spectrum points that the ZPFT calculates about 256,000 times than the ILFT considering one dimension. The ZPFT calculates the multiplication of two matrices size $(2.048 \times {10^7}) \times (2.048 \times {10^7})$ and $(2.048 \times {10^7}) \times 1$ while the ILFT just calculates the multiplication of two matrices sized $20 \times 2048$ and $2048 \times 1$ for four times, considering two dimensions. Evidently, the ILFT is much faster than the ZPFT. Moreover, the k calculated by the ILFT is as accurate as the preset k when choosing appropriate M and right zoom-in frequency band. The simulation results illustrate that the ILFT offers high accuracy and greatly decreases the computational complexity compared to the ZPFT without compromising accuracy. The simulations are consistent with the theory.
To further demonstrate the method’s performance in wavenumber position calculation, the monochromatic interferogram is simulated with noise, phase error, and non-uniform OPD sampling. The k calculated by the ILFT is compared with that by other two methods.
3.2 Performance with effects devices bring
To make the simulations more realistic, we added noise, phase error, and non-uniform OPD sampling to Eq. (26). Herein, the noise is considered first. The monochromatic interferogram with noise is
where $g(n)$ represents the noise intensity for sampling point n.A common noise source in the system is Gaussian noise, which obeys a normal distribution. It has two parameters, mean and standard deviation (SD). The mean of Gaussian noise is zero. SD is acquired by analyzing light intensity from the real detector for authenticity. We utilized an industrial-grade charge coupled device (CCD) for an example, whose model is pointgrey GS3-U3-23S6M-C. The SD of noise in an 8-bit grayscale image is about 5 when the grayscale amplitude of the monochromatic interferogram is about 100. Therefore, the maximum SD of the Gaussian noise was set to 5% of the amplitude of monochromatic interferogram in the simulation. The noise distribution when the SD is 3% of the amplitude of monochromatic interferogram is shown in Fig. 7(a), and the monochromatic interferogram with noise is shown in Fig. 7(b).
Fig. 7. Simulation monochromatic interferogram with noise. (a) Gaussian noise distribution versus OPD. (b) Interference intensity versus OPD.
Then, the monochromatic interferogram is simulated with phase error. In real FTIS, phase error exists for light rays of different wavelengths due to defects in the interferometer or off-center sampling. Based on Eq. (26), the monochromatic interferogram with phase error is
where $\phi (\sigma )$ represents the phase error when wavenumber is $\sigma$.Phase error does not actually change the shape of the interference pattern. It only changes the initial phase.
Next, a monochromatic interferogram with non-uniform OPD sampling is simulated. According to the theory of FTIS, the acquisition of time-sequential interferogram requires scanning in the direction of increasing OPD. However, the OPD requires to be sampled discretely in reality. The uniformity of sampling depends on the performance of the rotation or linear stages. Absolute uniformity in sampling is practically impossible. Thus, based on Eq. (26), the monochromatic interferogram with non-uniform OPD sampling is:
where $u(n)$ is the function of sampling number n. It represents a non-uniform sampling interval.The non-uniform OPD sampling for simulation is based on the spectrum measurement procedure of real FTIS, wherein the experimental sampling interval of OPD basically obeys a normal distribution. The mean of this normal distribution is very close to the theoretical sampling interval, and its SD is about 10% of the theoretical sampling interval. Simulation is performed for a simulated sampling interval with SD of 3% of the theoretical sampling interval. The simulation interferogram under non-uniform OPD sampling is shown Fig. 8.
Fig. 8. Simulated monochromatic interferogram under non-uniform OPD sampling. (a) Sampling interval versus sampling number. (b) Interference intensity versus sampling number.
Figure 8(a) shows the relationship between sampling interval and sampling number. The theoretical sampling interval is set to 1, whereas the simulated sampling interval fluctuates around 1 due to the instability and randomness of the rotation or linear stages. The sampling interval in Fig. 8(a) fluctuates between 0.85 and 1.15. Figure 8(b) shows the monochromatic interferogram simulated with non-uniform OPD sampling.
Combining analyses of Eqs. (27)–(29), the monochromatic interferogram with noise, phase error, and non-uniform OPD sampling can be expressed as
Fig. 9. Simulation monochromatic interferogram with the three device effects. (a) Intensity versus OPD. (b) Intensity versus sampling number. (c) Detailed interference pattern.
Figure 9(a) is the simulated monochromatic interferogram whose abscissa is the OPD. Figure 9(b) represents the relationship between intensity and sampling number, and Fig. 9(c) shows the details of Fig. 9(b). Figure 9(c) illustrates that the interferogram that contains noise, phase error, and the non-uniform OPD sampling becomes very irregular compared with a standard sinusoidal wave. The k is obtained with this interferogram to test the robustness of the ILFT. The k calculated by the three methods and the spectrum of three methods is presented in Fig. 10.
Fig. 10. The frequency spectra of the simulation interference pattern with the three device effects, obtained by (a) the conventional method, (b) ZPFT, and (c) ILFT.
Figure 10(a) shows the spectrum calculated by conventional method. The abscissa of the first positive sidelobe therein is 95, which means $k = 95$. Figure 10(b) is the spectrum calculated by the ZPFT, whose $k = 95.3501$. The value deviates slightly from the theoretical 95.3508, indicating that the three device effects indeed affect the spectrum of the interferogram. However, the k will be 95.35 after rounding, which is as accurate as the preset one. Figure 10(c) is the spectrum calculated by the ILFT, whose $k = 95.3501$. The accuracy of the method is much higher than the conventional one and as good as that of the ZPFT, but the computational complexity is much lower than that of other methods. The simulated interferogram is very close to that acquired by real FTIS. Therefore, the performance of the ILFT verifies its robustness in this situation.
The simulation above proves the applicability of the ILFT in practical conditions. To further analyze the effect that noise, phase error, and non-uniform OPD sampling have on the ILFT, simulations were executed when the three device effects change severally. The results are shown in Fig. 11.
Fig. 11. ILFT performance under different situations. (a) Wavelength position ($k$) for noise with different SDs. (b) Wavelength position ($k$) for different phase errors. (c) Wavelength position ($k$) for different levels of non-uniform OPD sampling.
When the SD of the Gaussian noise varies from 0% to 5% of the amplitude of the interferogram, the k calculated by the ILFT is shown in Fig. 11(a). The difference between the simulation and theoretical k values increases gradually along with an increase in the SD. However, the rounded value of k is still 95.35 within the full range of SD. Thus, we inferred that the ILFT can fully resist the effect of noise with an industrial-grade CCD. We further deduce that the performance of the ILFT would be better with scientific CCD, whose SNR is higher.
When the phase error varies from 0 to 2π, the k calculated by the ILFT is shown in Fig. 11(b); it fluctuates around the theoretical value. However, the rounded value of k still remains at 95.35 when the phase error varies from 0 to 2π. The ILFT performs well with phase error from the simulations.
When the SD of the sampling interval varies from 0% to 10% of the theoretical sampling interval, the k by the ILFT is shown in Fig. 11(c). The difference between the simulated and theoretical k values increases gradually along with the increasing non-uniformity of the sampling interval. However, the rounded value of k remains at 95.35 with the SD of the sampling interval varying from 0% to 10% of the theoretical sampling interval. Thus, non-uniform OPD sampling will not influence the calculation of k by the ILFT to a certain degree. Furthermore, the ILFT will perform better using rotation or linear stages with higher moving uniformity.
The simulations above indicate that the ILFT will perform well when the interferogram has phase error, noise, and non-uniform OPD sampling. When the M value is large enough, the k calculated by the ILFT remains as accurate as the theoretical one. The simulation indirectly demonstrates the reliability of the ILFT for wavenumber position calculation, and by extension for wavelength calibration, in real FTIS.
3.3 Computational complexity
The computational complexity of the ILFT is given in Eq. (24). The computational complexity is compared in Fig. 12 ($N = 2048$, with varying M values).
In Fig. 12, the dotted line represents the ZPFT, the dashed line represents the LFT and the solid line represents the ILFT. Figure 12 shows that the LFT is superior to the ZPFT in computational complexity and the ILFT is superior to the LFT. The curve of the ZPFT is almost parallel to the y axis while the curve of the ILFT is almost parallel to the x axis, indicating the high efficiency of the ILFT and the low efficiency of the ZPFT.
Considering the computational complexity of the conventional method as 1, the computational complexity of the ZPFT is $2.2 \times {10^4}$, that of the LFT is $911.3$, and the ILFT is $3.6$. Evidently, the proposed ILFT method reduces the computational burden.
The running times of ZPFT and ILFT are listed in Table 3, tested on a laptop (Intel Core i7-6700HQ processor 2.60GHz with four cores and eight threads, 64-bit operating system, 16 Gbytes RAM).
Table 3. Running time (ms) of two methods. (N = 2048)
Table 3 shows that the actual running time is consistent with the theoretical analysis. The ILFT is much faster than the ZPFT, and its superiority becomes more significant as M increases.
3.4 Wavelength calibration simulation
Monochromatic interferograms of different wavelengths were simulated based on Sagnac interferometer for wavelength calibration. The maximum OPD follows the Eq. (31) below.
where ${n_{\max }}$ is the maximum number of detector arrays, p is the length of the pixel size of the detector, d is the lateral shearing distance of the Sagnac interferometer, and f is the focal length of the final imaging lens before the detector.The simulation parameters are chosen based on real Sagnac FTIS. The parameters are set as ${n_{\max }} = 1920$, $p = 5.86$ µm, $d = 0.63$ mm, and $f = 75$ mm. The wavelengths involved in the calibration are 405 nm, 450 nm, 532 nm, 650 nm, 780 nm, 850 nm, 910 nm, and 1050 nm. The interferogram of 630 nm is simulated for analyzing the accuracy of the wavelength calibration result. For authenticity, phase error, Gaussian noise, and non-uniform OPD sampling are all involved in these simulations. The interferograms of 450 nm and 910 nm are shown in Fig. 13.
The wavenumber position k of these interferograms was calculated via the conventional method and the ILFT. The results are listed in Table 4.
Table 4. The wavenumber position k of simulated interferograms derived by two methods.
Table 4 indicates that ${k_2}$ is more accurate than ${k_1}$ for any wavelength. Two curves of wavelength calibration $\lambda (k)$ are fitted respectively with ${k_1}$ and ${k_2}$ (630 nm is not involved). The two $\lambda (k)$ are shown in Fig. 14. The wavelength of laser 630nm is then calibrated by taking the ${k_1}$ (150) and ${k_2}$ (150.015) of 630nm in Table 4 into the two functions $\lambda (k)$.
Fig. 14. Simulated wavelength calibration curves of conventional method and the ILFT. (a) Wavelength versus wavenumber location k. (b) Detailed wavelength calibration curves.
Figure 14(a) shows the two wavelength calibration curves. They seem to be close to each other. Figure 14(b) shows a more detailed plot around 630 nm. Even in Fig. 14(b), the two curves do not diverge much. From Fig. 14(b), the obtained wavelengths were 628.76 nm by the conventional method and 630.22 nm by the ILFT. Thus, the absolute error was decreased from 1.24 nm to 0.22 nm by ILFT. The simulation results demonstrate that ILFT shows higher accuracy for wavelength calibration.
All the simulations indicate that the proposed wavelength calibration method based on ILFT is highly accurate, stable, and efficient in FTIS.
4. Verification experiments
To further verify the validity of the ILFT, we choose an FTIS based on the Sagnac interferometer as the testing system. The focal length of the three objective lenses (L1, L2 and L3) are all 75 mm. The diameter of the BS is 50.8 mm, the its split ratio is 50:50. The diameters of M1 and M2 are all 50.8 mm. The lateral shearing distance of the Sagnac interferometer is set as 0.69 mm. The detector is Point Grey GS3-U3-23S6M-C. The maximum number of detector arrays is 1920. The pixel size of the detector is 5.86µm. Wavelength calibration experiments were conducted on the Sagnac FTIS using lasers with the following wavelengths are 403.6 nm, 452.6 nm, 532.7 nm, 656.8 nm, 786.5 nm, and 858.1 nm.
Figure 15 shows the monochromatic interferograms of two different wavelengths.
Figures 15(a) and 15(b) show the detailed interferograms of 452.6 nm and 656.8 nm, respectively. The k of these interferograms are calculated by conventional method and the ILFT. The results are listed in Table 5.
Table 5. The wavenumber position k solved by two methods.
Table 5 illustrates that ${k_2}$ is more accurate than ${k_1}$ for any wavelength. To verify the validity of the ILFT in wavelength calibration, two curves of wavelength calibration $\lambda (k)$ are respectively fitted with ${k_1}$ and ${k_2}$ (656.8 nm is not involved). Then, the wavelength calibration function $\lambda (k)$ of the two methods can be obtained. We used the wavelength calibration function to calibrate the 656.8-nm laser by taking the k of 656.8nm into $\lambda (k)$ to calculate the wavelength. Comparing the wavelength results from the two methods with the standard value, we analyzed the accuracy of the methods. The wavelength calibration function $\lambda (k)$ of two methods are shown in Fig. 16.
Fig. 16. Wavelength calibration curves of conventional method and the ILFT. (a) Wavelength versus wavenumber location k. (b) Detailed wavelength calibration curves.
Figure 16(a) shows the two wavelength calibration curves. Figure 16(b) shows the detailed version. In Fig. 16, the solid line represents the result from the conventional method while the dashed line represents the result from the ILFT. The two curves seem very similar; however, the calculation results show significantly different results for the two methods in wavelength calibration. Substituting ${k_1}$ and ${k_2}$ of 656.8 nm into the two wavelength calibration curves, the calibrated wavelength of a 656.8-nm laser should be acquired. From Fig. 16(b), the wavelength obtained is 654.77 nm by the conventional method and 656.96 nm by the ILFT. Compared with the standard value, the conventional method has an absolute error of 2.03 nm and a relative error of 0.31%, while the ILFT has an absolute error of 0.16 nm and a relative error of 0.02%. The ILFT decreases the absolute error from 2.03 nm to less than 0.16 nm. These results are quite consistent with the simulations in Section 3.4. Evidently, the ILFT is more accurate. The results comprehensively illustrate the feasibility and accuracy of the ILFT in wavelength calibration experiments for FTIS.
All the simulation-based and experimental results have indicated that the ILFT is capable of highly-accurate wavelength calibration of FTIS. Furthermore, the ILFT offers some advantages.
First, the accuracy of the ILFT-based method in calculating the wavenumber position is extremely high. The conventional method can obtain the wavenumber position accurate only up to integer values, restricted by the picket-fence effect of FFT. The ILFT achieves fast and accurate spectrum zoom by simplifying the matrices participating in Fourier transform and a few iterations. The picket-fence effect can be suppressed with the zoom spectrum by ILFT. The wavenumber position by ILFT can be accurate up to two decimal places when M is chosen as 10000. The improvement of the ILFT is that it can increase the accuracy of calculating the wavenumber position by 100 times than that of the conventional method with noise, phase error and non-uniform sampling of OPD. Consequently, high-accuracy wavelength calibration is enabled by ILFT. The wavelength of a laser is measured via the wavelength calibration curve. The absolute error has decreased to 0.16 nm, which is very accurate for wavelength measurement. Thus, the method is very suitable for wavelength calibration of FTIS for its high accuracy.
Second, the ILFT greatly accelerate the calculation speed. The method first performs the spectrum zoom operation by zero padding (small multiples) and LFT. The wavenumber position was coarsely located by searching for the maximum value of the zoom-in spectrum. Then, a small range of spectrum around this wavenumber position (which has integer-order accuracy) was obtained by performing LFT to further zoom into the spectrum. High-accuracy wavenumber position was then obtained by repeating the steps above. By iteration, the method can realize high-accuracy wavenumber position calculation with a few rounds of computation. For $M = 10000,m = 10$, and the number of iterations required was only 4, which consumes little time. Hence, the ILFT greatly reduces the computational complexity as well as provides the same accuracy as the ZPFT. The method based on ZPFT requires large amounts of zero padding for achieving high accuracy, which increases computational burden despite that the spectrum of interest may only be a fraction of the total spectrum. The simulation indicates that the computational complexity of the ILFT is much lower than that of ZPFT. The experiments illustrate that the ILFT only needs 2.8 ms for zooming into the spectrum to 100,000 times. The calculation speed for ILFT is about 3000 times than the ZPFT. Such short calculation time enables the method to be very suitable for the online calibration for FTIS.
The above descriptions are the advantages of the ILFT, but the method do have its suitable range and applied conditions. The suitable range of ILFT is for calculating high-accuracy wavenumber position in wavelength calibration of FTISs. The FTISs based on Saganc interferometer and one birefringent interferometer have been verified in experiments. The other types of FTISs will be tested with ILFT in the future. Consulting the analysis in section 3.2, the method can be applied in the situation that the SD of noise is no more than 5% of the amplitude of monochromatic interferogram and the SD of the sampling interval is no more than 10% of the theoretical sampling interval. Besides, the value of the phase error contributes little to the deviation of the ILFT. So the ILFT can solve the wavenumber position of interferograms with random phase error.
5. Conclusion
In summary, we have presented an ILFT-based high-accuracy wavelength calibration for FTIS. The ILFT achieves fast and accurate spectrum zoom by simplifying the matrices participating in Fourier transform and a few iterations. The wavenumber position can be obtained with the zoom spectrum by ILFT. Then, the high-accuracy wavelength calibration can be achieved by fitting a curve with the discrete relation of wavenumber position and wavelength. Simulations show that the ILFT can calculate the wavenumber position accurate to two decimals with noise, phase error and non-uniform sampling of OPD to a certain extent. Besides, the ILFT consumes only 2.8 ms for zooming into the spectrum to 100,000 times with a common computer. The calculating speed is much faster than the ZPFT which takes 8829.7 ms in the same condition. Related experiments are carried out with an FTIS based on Sagnac interferometer. The results indicate that the ILFT can decrease the wavelength calibration error from 2.1 nm to 0.16 nm compared with the conventional method. Compared with the ZPFT and the conventional method, the presented ILFT offers improved accuracy in wavenumber position calculation as well as decreased computational complexity. Therefore, the proposed ILFT provides theoretical and technical support for high-accuracy hyperspectral imaging applications. In the future, we will use the ILFT to calibrate other types of FTISs like Savart plate or Wollaston prism-based FTIS to further verify our method.
Funding
National Natural Science Foundation of China (61975079, 61475072, 61405134, 61340007); Priority Academic Program Development of Jiangsu Higher Education Institutions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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