1. Introduction

Research on the echelle spectrometer originated more than 70 years ago [1], and early echelle spectrometers were mainly used for astrophysical plasma research [2]. The echelle spectrometer has the advantages of a wide measurement band, high spectral resolution, small volume, non-pollution, non-destructiveness, and high speed in a single exposure [3–6]. Currently, the echelle spectrometer is widely used in environmental detection, semiconductor testing, and material composition detection [7], and has become important in modern spectral analysis.

The echelle spectrometer has a cross-dispersion structure for imaging on an area array CCD. It is a major challenge to design a highly precise instrument when the light is turned many times, many optical elements are involved, the optical path is long, and the imaging surface is large. The accuracy of the echelle spectrometer is mainly affected by the following factors: 1) the deviation of the calculated spot position during spectral calibration; 2) the accuracy of the mathematical model used for ray tracing; 3) error in processing, assembly, and adjustment of optical components; 4) random detector noise; 5) the spectral drift caused by temperature changes; 6) changes in relative positions of structural parts due to stress release and instrument transportation, which lead to a change in the light path, resulting in decreased wavelength accuracy.

At present, research on calibrating the echelle spectrometer consists mainly of the following. Researchers at the Changchun Institute of Optics, Fine Mechanics and Physics have proposed a method for accurately installing and calibrating spectral instruments [8], and discussed an optical design method [9–11]. They proposed a method to improve the accuracy of the mathematical model of ray tracing. To improve the accuracy of the spectrogram model, M.He proposed a method to extract the centroid of the mercury light spot quickly and effectively [12], and R. Zhang proposed using the characteristic light spot of mercury for correction [13] to overcome the influence of processing, assembly, and adjustment of parts. To solve the problem that the equipment is sensitive to temperature changes, Aya Taleb proposed a method for rapidly checking and rectifying calibration using the emission spectrum of standard laser-ablated steel [14].

Most calibration studies on echelle spectrometers have focused on optimizing the theoretical model of ray tracing [15]. In fact, factors 1, 3, 4, 5, and 6 above also have a great influence on the spectral position of the instrument. This paper first calculates the spot position with high accuracy. Then, taking the spot position as the benchmark and examining the minimum comprehensive deviation between the spectrogram model and the spot position, we obtain the real parameters of the instrument to reduce the nonlinear deviation of the spectrogram restoration model, thereby improving the calibration of the echelle spectrometer.

2. Theoretical model of the echelle spectrometer

The echelle spectrometer is a transient-measurement spectrometer that uses an echelle grating as the main dispersion element and combines it with a longitudinal dispersion element to distinguish monochromatic light of different wavelengths and different orders in two-dimensional space. Thanks to the full-wave blaze and high diffraction order of its echelle grating and core beam splitter, the echelle spectrometer simultaneously has the advantages of wide band range and high resolution. However, it is more difficult to obtain directly the wavelength of the echelle spectrometer than that of a conventional grating spectrometer. Therefore, it is necessary to establish a wavelength calibration model, that is, to establish the corresponding relationship between the detector’s two-dimensional imaging coordinates and the wavelength. This process is called spectrogram restoration. The theoretical spectrogram restoration model can be obtained by geometric ray tracing according to the instrumental design parameters. The most widely used optical structure is the research object of our algorithm [16], as shown in Fig. 1.

 

Fig. 1. Optical structure of echelle spectrometer.

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The position coordinates of any wavelength on the two-dimensional image plane are jointly determined by the main dispersion element parameters, the longitudinal dispersion element parameters, and the instrumental system parameters. The relationship can be expressed as:

Therefore, when the design parameters are known, the position of any wavelength on the detector can be obtained according to Eq. (1) and (2), and the theoretical model of the echelle spectrometer is established.

3. Accurate spectral calibration of the echelle spectrometer

In this section, the theoretical spectrogram model is matched, analyzed, and corrected using the spot position of the mercury light to complete the calibration of the echelle spectrometer. The specific contents include: accurately calculating the spot position of the mercury light spot; using triangle matching to automatically identify the corresponding characteristic spot in the mercury light image; modifying the seven model parameters $\alpha _{g}^{*}$, $\alpha _{p}^{*}$, $\theta ^{*}$, $\gamma ^{*}$, $f^{*}$, $\omega ^{*}$, and $\beta ^{*}$ to build spectrogram restoration model A; calculating the residual error between the spot position and spectrogram restoration model A; fitting the residual curve on the basis of the calibrated spot position; and then constructing spectrogram restoration model B. The high-precision calibration process is shown in Fig. 2.

 

Fig. 2. High-precision spectral calibration of an echelle spectrometer.

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3.1 Accurate calculation of the spot position

Calibration of the echelle spectrometer usually involves using the spot position of the element spectrum to correct the position of the theoretical model. Mercury light has many discrete characteristic peaks from $200\,\mathrm{nm}$ to $1000\,\mathrm{nm}$, which is suitable for calibrating the instrument. Precisely calculating the position of the mercury light spot is the key to ensuring a highly precise spectrogram restoration model for the echelle spectrometer. The traditional method uses local windowing and the centroid algorithm to calculate the spot position $(G_{x}, G_{y})$ :

3.1.1 Multiple-integral time fusion algorithm

The energy difference of different mercury lamps is large. For example, when overexposure occurs at a high spot energy such as at $253.652\,\mathrm{nm}$, the energy of the saturated pixel is cut off and the calculated spot position is distorted. When the signal-to-noise ratio is too low at a weak spot energy such as at $794.818\,\mathrm{nm}$, the background noise influences the accuracy of the spot calculation. Therefore, we adopt multiple-integral time fusion to calculate the spot position. The detector integral time is set as $2\,\mathrm{ms}$, $5\,\mathrm{ms}$, $10\,\mathrm{ms}$, $20\,\mathrm{ms}$, and $50\,\mathrm{ms}$. The effectiveness of each spot is judged at these different integral times. Finally, the real positions of different spots are comprehensively calculated using the effective spot. The specific implementation is as follows:

1) We set the detector integral time $p$ to $2\,\mathrm{ms}$.

2) To reduce the influence of random noise in the image, 10 ImageRaw ${ }_{p}{m}$ frames are collected continuously. The corresponding pixels of 10 frame images are accumulated and averaged to obtain ImageCur ${ }_{p}$ :

3) At integral time $p$, the position of the $q$-th light spot is calculated, and the validity is verified according to

If maxvalue $_{p}^{q}<65000$ and maxvalue $_{p}^{q}>8000$, the $q$-th light spot at integral time $p$ is effective; otherwise, the light spot is invalid. The maximum gray value maxvalue ${ }_{p}^{q}$ is

4) Next, we set the camera integral time $p$ to $5\,\mathrm{ms}$, $10\,\mathrm{ms}$, $20\,\mathrm{ms}$, and $50\,\mathrm{ms}$, and repeat steps 2 and 3 to obtain the position of the $q$-th light spot $(G x_{p}^{q}, G y_{p}^{q})$.

5) We calculate the positions $(G x^{q}, G y^{q})$ of nine spectral calibration spots using

3.1.2 Calculation of the adaptive threshold $T_{G}$

The key to accurately calculating the light spot position is to select a reasonable threshold $T_{G}$. The spot calculation accuracy is reduced if $T_{G}$ is too low to suppress the background noise or too high to collect all the effective pixels. The traditional method of calculating the adaptive threshold is

 

Fig. 3. Corresponding image display in the process of finding the threshold $T_{\text {Down }}$.

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Step 1: Calculation of $T_{Down}$

Figure 4 shows the flow chart for calculating $T_{\text {Down }}$. First, we calculate the histogram peak $D N_{\text {peak }}$ in the window, where the adjustment range of the threshold $t$ is $(D N_{\text {peak }}, 65536)$, and gradually adjust upward from $D N_{\text {peak }}$. During threshold segmentation, we set to 0 the pixels in window $S$ that are lower than threshold $t$, and otherwise keep them unchanged. This generates a treated image image2value, as shown in Fig. 3(b). The image image2value contains a light spot and a large amount of background noise at the beginning. As threshold $t$ increases, the independent background noise in image2value gradually decreases. The center of the light spot is taken as the connecting area $U$. As shown in Fig. 3(c), when the number of non- 0 pixels in window S outside the main connecting area $U$ is less than or equal to 2, the value of $t$ is $T_{\text {Down }}$.

 

Fig. 4. Flow chart for finding $T_{Down}$.

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Step 2: Calculation of $T_{U p}$

The process of searching for $T_{U p}$ is shown in Fig. 5. When the threshold $t$ is adjusted within the background span, the fluctuation in the spot position ($\Delta x=G x_{t+1}-G x_{t}$) caused by the threshold adjustment is small because the background energy is weak relative to the spot. When the threshold $t$ is adjusted within the light spot span, $\Delta x$ tends to be large. The spot position in the $x$ direction, $G x_{t}$, is calculated at the threshold $t, t \in [T_{\text {Down }}, \max )$, using the Eq. (10).

 

Fig. 5. Flow chart for finding $T_{U p}$.

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Figure 6 shows the calculated position fluctuation of the spot in the $x$ direction during the threshold change. The abscissa is the threshold $t$, the vertical coordinate is $\Delta x$, and the units are pixels. The value of $T_{U p}$ is found through the fluctuation in the light spot centroid, and the position indicated by the red circle in Fig. 6 is $T_{U p}$. This means when the calculation of the light spot position generates a large fluctuation for the first time during the upward adjustment of the threshold $t$, the threshold $t$ has just entered the light spot energy level from the background energy level at this time. After $T_{D o w n}$ and $T_{U p}$ are calculated, the threshold $T_{G}$ for the light spot centroid can be calculated from

 

Fig. 6. Calculation of the centroid fluctuation of the light spot while threshold $t$ changes.

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3.2 Automatic identification of the mercury light spot

After the echelle spectrometer is installed and adjusted, there is usually a deviation between the actual position of the mercury light spot and its position in the theoretical model, and the characteristic spots of mercury light are difficult to identify. However, the relative displacement between the mercury light spot and corresponding characteristic spectrum of the theoretical model changes very little. To improve the automation of the calibration process, triangle matching is used to automatically identify the mercury light spot. Figure 7(a) shows the mercury light spot in the echelle spectrometer, Fig. 7(b) connects multiple light spots to form multiple triangles, and Fig. 7(c) shows the spatial position where the theoretical model corresponds to the detector. The four characteristic lines of the mercury light in Fig. 7(c) at $253.652\,\mathrm{nm}$, $296.728\,\mathrm{nm}$, $404.656\,\mathrm{nm}$, and $435.833\,\mathrm{nm}$ are amplified to obtain Fig. 7(d).

 

Fig. 7. The characteristic spectrum of the mercury light spot is automatically identified during the spectral calibration.

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The steps for automatic recognition of mercury light spot are as follows:

1) We calculate the internal angles $\angle B A C$, $\angle A B C$, and $\angle A C B$ of triangle $\triangle A B C$, which constitute three characteristic spectral lines of mercury at $253.652\,\mathrm{nm}$, $296.728\,\mathrm{nm}$, and $404.656\,\mathrm{nm}$ in Fig. 7(d). Let $u_{0}=\angle B A C$, $v_{0}=\angle A B C$, and $w_{0}=\angle A C B$ in the theoretical model.

2) We calculate the positions of all mercury light spots in Fig. 7(a), and calculate the inner corners $u_{i}$, $v_{i}$, and $w_{i}$ of triangle $\Delta _{i}$ after connecting the lines in Fig. 7(b). If triangle $\Delta _{i}$ satisfies any one of the following six conditions, then triangle $\Delta _{i}$ is similar to triangle $\triangle A B C$, and $\Delta _{i}$ is collected into set $\mathrm {S}_{1}$ :

3) In the theoretical model, the length of line segment $AC$ in triangle $\triangle A B C$ is recorded as $L_{A C}$. Triangle $\Delta _{j}'$ in set $S_{1}$ is collected into set $S_{2}$ if the length of the corresponding side $L_{A' C'}$ meets the condition

3) Considering the fourth characteristic line at $435.833\,\mathrm{nm}$ in the theoretical model, we calculate $\angle B A D$ and side length $L_{A D}$. For each triangle $\Delta _{k}''$ of set $S_{2}$ with $A_{k}''$ as a vertex and edge $B_{k}'' A_{k}''$ as the benchmark, we rotate $\angle BAD$ to draw a straight line $L''$, and determine the point $P''$ according to the length of $A D$. We find the nearest mercury light spot near point $P''$ in Fig. 7(a) whose centroid is $D''$. Then, we calculate the side lengths $L_{A D}$, $L_{B D}$, and $L_{C D}$ in Fig. 7(d) according to

4) In the theoretical model, we calculate $\angle C B E$ and the length of line segment $B E$. In Fig. 7(a), we take the $296.728\,\mathrm{nm}$ light spot as the vertex, and take the straight line composed of the $296.728\,\mathrm{nm}$ and $404.656\,\mathrm{nm}$ spots as the benchmark. We rotate $\angle C B E$ to draw a straight line $L$, determine the point $E''$ on the straight line $L$ according to the length of $B E$, and find the nearest light spot in the neighborhood of $E''$. The light spot is just the point corresponding to the $576.96\,\mathrm{nm}$ characteristic line of mercury in the calibration image. The processes for identify the mercury lines at $696.543\,\mathrm{nm}$, $738.393\,\mathrm{nm}$, $794.818\,\mathrm{nm}$, and $912.297\,\mathrm{nm}$ are the same as that for $576.96\,\mathrm{nm}$.

3.3 Modified spectrogram restoration model

The spectral mapping of the echelle spectrometer is affected by factors such as processing, installation, and adjustment. As shown in Fig. 8, the coordinates of the spectrum $\lambda$ in the theoretical model are $S(x, y)$, and an image point of $\lambda$ appears at $S'(x', y')$. Two common methods for correcting spectral deviation are the wavelength solution function and image point coordinates. The wavelength solution function solves and derives the deviation of each parameter in the function and has the advantage of fast speed, but it is too dependent on the parameters in the theoretical model, sensitive to noise, and not robust enough. The image point coordinate method usually carries out a translation transformation and scale transformation on the theoretical model to bring the spectral model closer to the actual position of the light spot [17]. Single-parameter deviation usually causes nonlinear error in the spectral restoration model, and multi-parameter deviation aliasing further increases the error complexity of this model. To reduce the influence of working-parameter deviation on the spectral calibration accuracy, we select the important parameters $\alpha _{g}$, $\alpha _{p}$, $\theta$, $\gamma$, $f$, $\omega$, and $\beta$ for traversal, and a two-step method is adopted to reduce the computation of parameter traversal. The operations are as follows:

 

Fig. 8. Position deviation between the theoretical ray tracing model and actual spectrogram model of the echelle spectrometer.

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Step 1: We optimize the grating incident angle $\alpha _{g}$ and prism incident angle $\alpha _{p}$ in the spectrogram model.

We take the spot position of mercury light as the benchmark for minimizing the comprehensive deviation between the actual spot position and the position in the spectrogram model. The grating incident angle $\alpha _{g}$ and prism incident angle $\alpha _{p}$ are traversed to complete the correction of the spectrogram model parameters, and the optimal values of $\alpha _{g}$ and $\alpha _{p}$ are obtained.

According to sections 3.1 and 3.2, we automatically identify and accurately calculate the position of the mercury light calibration spot $(G x^{q}, G y^{q})$, and $\alpha _{g 0}$ and $\alpha _{p 0}$ are the theoretical values of the grating and prism incidence angles. With coordinates centered on $(\alpha _{g 0}, \alpha _{p})$, the space is traversed within a range of $\pm 10$ in the $\alpha _{g}$ and $\alpha _{p}$ directions, as shown in Fig. 9. We calculate the comprehensive deviation $\sigma _{(i, j)}^{1}$ between the actual position $(G x^{q}, G y^{q})$ of spot $q$ and the position $(f_{x}(\alpha _{g}^{i}$, $\theta _{0}$, $\gamma _{0}$, $f_{0}$, $\omega _{0}$, $\lambda _{q})$, $f_{y}(\alpha _{p}^{j}$, $\beta _{0}$, $f_{0}$, $\omega _{0}$, $\lambda _{q}))$ at $\lambda _{q}$ in the spectrogram model according to

 

Fig. 9. Process of traversing parameters $\alpha _{\mathrm {g}}$ and $\alpha _{p}$.

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The parameters $\alpha _{g}$ and $\alpha _{p}$ are within a $\pm 10$ traversal region. The coordinates in the traversal space $(\alpha _{g}^{i}, \alpha _{p}^{j})$ when $\sigma$ $(i, j)^{1}$ is minimum correspond to the optimal grating incidence angle $\alpha _{g}^{*}$ and optimal prism incidence angle $\alpha _{p}^{*}$ of the spectrogram model.

Step 2: We modify the spectrogram model parameters $\theta$, $\gamma$, $f$, $\omega$, and $\beta$.

The parameters $\theta$, $\gamma$, $f$, $\omega$, and $\beta$ also have a great influence on the spectrogram model. To obtain the optimal solutions of these five parameters, $\theta ^{*}$, $\gamma ^{*}$, $f^{*}$, $\omega ^{*}$, and $\beta ^{*}$, three-layer pyramid traversal is proposed to reduce the calculation time:

The three-layer pyramid-five-parameter traversal process is shown in Fig. 10. The first-layer traversal steps are $\theta _{0}$, $\gamma ^s_0$, $f^s_0$, $\omega ^s_0$, and $\beta ^s_0$, and the traversal ranges are $[\theta _0-3\theta ^s_0$, $\theta _0+3\theta ^s_0]$, $[\gamma _0-3\gamma ^s_0, \gamma _0+3\gamma ^s_0]$, $[f_0-3f^s_0, f_0+3f^s_0]$, $[\omega _0-3\omega ^s_0, \omega _0+3\omega ^s_0]$, and $[\beta _0-3\beta ^s_0, \beta _0+3\beta ^s_0]$, respectively. When the parameters $\theta ^{i}$, $\gamma ^{j}$, $f^{k}$, $\omega ^{u}$, and $\beta ^{v}$ are substituted into Eq. (16), the deviation $\sigma ^{2}$ between the calibrated spot position and the position in the spectrogram model is calculated. When $\sigma ^{2}$ is minimum, the parameters $\theta _{1}$, $\gamma _{1}$, $f_{1}$, $\omega _{1}$, and $\beta _{1}$ are calculated. The second-layer traversal ranges are $[\theta _{1}-3 \theta ^{s}_{1}, \theta _{1}+3 \theta ^s_{1}]$, $[\gamma _{1}-3 \gamma ^{s}, \gamma _{1}+3 \gamma _{1}^{s}]$, $[f_{1}-3 f_{1}, f_{1}+3 f_{1}^s]$, $[\omega _{1}-3 \omega _{1}^{s}, \omega _{1}+3 \omega _{1}^{s}]$, and $[\beta _{1}-3 \beta ^{s}_{1}, \beta _{1}+3 \beta _{1}^s]$. Similarly, we calculate $\theta _{2}$, $\gamma _{2}$, $f_{2}$, $\omega _{2}$, and $[\omega _{2}-3 \omega ^{s}, \omega _{2}+3 \omega _{2}^{s}]$, and $[\beta _{2}-3 \beta _{2}^{s}, \beta _{2}+3 \beta ^{s}]$, and we similarly calculate the optimal solutions $\theta ^{*}$, $\gamma ^{*}$, $f^{*}$, $\omega ^{*}$, and $\beta ^{*}$. Thus far, the optimization of the model parameters has been completed, and spectrogram model $A$ is built according to parameters $\alpha _{g}^{*}, \alpha _{p}^{*}, \theta ^{*}, \gamma ^{*}, f^{*}, \omega ^{*}$, and $\beta ^{*}$.

 

Fig. 10. Three-layer pyramid traversal process.

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We compare the processing time between the conventional method and the three-layer pyramid method under the same search scope and the same traversal step size. At last, we find that the processing time of the three-layer pyramid traversal method is greatly reduced. As shown in Table 1, seven parameters are decomposed into three-layer pyramid-five-parameter traversal (42 min) plus $(\alpha _{g}, \alpha _{p})$ traversal (2.8 min), which meets the required spectral calibration speed for engineering.

3.4 Position compensation in the spectrogram restoration model

After spectrogram restoration model $\mathrm {A}$ is constructed, there is still a certain position deviation between the calibrated spot position and the corresponding spectrum in spectrogram model $\mathrm {A}$. Next, curve fitting is used to further compensate the position in spectrogram model A, and we use a combination method of piecewise fitting and least-squares curve fitting. In the short-wave stage $(200\,\mathrm{nm}$–$696\,\mathrm{nm})$, the fluctuation in the deviation is relatively flat, so piecewise fitting is adopted to further compensate the model deviation. In the long-wave stage $(696\,\mathrm{nm}$–$1000\,\mathrm{nm})$, the fluctuation in the deviation is relatively intense, so least-squares curve fitting is used to compensate the model deviation in this stage. The deviation compensation yields spectrogram model B, which is the final spectrogram model of the echelle spectrometer.

Table 1. Comparison of processing time between conventional five-parameter traversal and three-layer pyramid-five-parameter traversal

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4. Experiments

Accurate calculation of spot position is the key to ensure the spectrogram restoration model of high-precision echelle spectrometer. The traditional spot position calculation adopts local windowing,and the spot position deviation is 0.32 pixels. In contrast, the accuracy of spot position determination algorithm proposed in section 3.1 is 0.1 pixels, which can lay a foundation for modeling.

To verify the improved spectrogram model in this study, the spectrogram model was constructed to correspond to the echelle spectrometer, and compared with the traditional spectrogram model. The traditional approach uses the centroid method to calculate the position of the mercury light spot, and uses the ray tracing method of the research group to construct the spectrogram model. The improved spectrogram model in this paper calculates the spot position according to the method in sections 3.1 and sections 3.2, similarly on the basis of the ray tracing method of the research group, and the parameter correction methods in section 3.3 are added to obtain spectrogram model A. On the basis of spectrogram model A, spectrogram model B is obtained using the position compensation in section 3.4. The nine-point calibration spectrum used to build the spectrogram model consists of $253.652\,\mathrm{nm}$, $296.728\,\mathrm{nm}$, $404.656\,\mathrm{nm}$, $435.833\,\mathrm{nm}$, $576.96\,\mathrm{nm}$, $696.543\,\mathrm{nm}, 738.393\,\mathrm{nm}$, $794.818\,\mathrm{nm}$, and $912.297\,\mathrm{nm}$, and the five test characteristic lines are $365.015\,\mathrm{nm}$, $546.075\,\mathrm {nm}$, $579.067\,\mathrm{nm}$, $763.511\,\mathrm{nm}$, and $811.531\,\mathrm{nm}$. The instruments used in the experiment are shown in the Fig. 11 .The echelle spectrometer is composed of a mirror, collimator mirror, echelle, prism, imaging mirror, and detector. The design parameters of the instrument are shown in Table 2.

 

Fig. 11. Physical picture of instrument.

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Table 2. Design parameters of the echelle spectrometer

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Figure 12 is the contrast curve of the spectral position deviation in the X direction between the actual position of 14 mercury vapor lamp characteristic spots and the traditional spectral model and the updated spectral model A in this paper, Fig. 13 is the contrast curve of the spectral position deviation in the $\mathrm {Y}$ direction between the actual position of 14 mercury lamp characteristic spots and the traditional spectral model and the updated spectral model A in this paper. By comparison, it is found that the deviation between the calculated spot position and the actual spot position of the updated model A is significantly reduced. On the one hand, this paper optimizes the spot extraction algorithm, the threshold is more reasonable, the anti-noise ability is improved, and the target extraction accuracy is higher. On the other hand, the 7 parameters of the spectrogram reduction model A are traced back, and the parameters of the spectrogram reduction model A are closer to the actual value of the instrument. Furthermore, the space-spectral correspondence of the spectral reduction model A is more accurate. We also found that compared with the traditional spectral reduction model, the jitter amplitude of the deviation curve in the X and Y directions of the updated model A becomes gentle with the increase of wavelength, and the gentle curve makes the spectral reduction model A more suitable for curve fitting.

 

Fig. 12. Comparison of deviation in X direction between traditional model and updated model.

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Fig. 13. Comparison of deviation in Y direction between traditional model and updated model.

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The traditional spectrogram model, spectrogram model A, and spectrogram model B were constructed with the nine characteristic spots of mercury light. Table 3 compares the model accuracies using the five test lines. The deviation between the traditional spectrogram model and the spot position is $[-1.717, 1.265]$, the RMSE of X direction is 0.836 and the RMSE of Y direction is 1.199; the deviation between spectrogram model A and the spot position is $[-0.824, 0.448]$, the RMSE of X direction is 0.4 and the RMSE of Y direction is 0.689; the deviation between spectrogram model B and the spot position is $[-0.529, 0.638]$, the RMSE of X direction is 0.299 and the RMSE of Y direction is 0.358. Fig. 14 shows the comparative deviation of RMSE values of different spectrogram restoration models. Obviously the accuracy of spectrogram model A is greatly improved after parameter correction. The jitter of the deviation curve in Fig. 12 and Fig. 13 is gentle in the short-wave stage, which shows that the accuracy of spectrogram model B is greatly improved after curve fitting in the short-wave stage. This test verifies that the deviation of the spectrogram model at 365.015 nm, 546.075 nm, and 579.067 nm is very small, and the model deviation in the long-wave stage (763.511 nm and 811.531 nm) is also improved to a certain extent. The test results show that the accuracy of the spectral restoration model is controlled within 0.3 pixels in a short-wave stage and 0.7 pixels in a long-wave stage.

 

Fig. 14. Comparison of RMSE values between traditional model and updated model.

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Table 3. Deviation comparison between the mercury test spectrum and corresponding positions of three spectrogram models.

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5. Conclusion

This study has enhanced the algorithmic optimization of the echelle spectrometer from two aspects. First, automatic recognition of the mercury light spot was strengthened, and the anti-interference capability was enhanced by a multi-frame accumulation algorithm and adaptive threshold algorithm, which improves the spot calculation accuracy. Second, after the model parameters were optimized, the spectrogram model was reconstructed by fitting the residual curve. By the traditional spectrogram model, the accuracy of the spectral restoration model is controlled within 1.1 pixels in a short-wave stage and 1.8 pixels in a long-wave stage. By contrast, the accuracy of the spectral restoration model is controlled within 0.3 pixels in a short-wave stage and 0.7 pixels in a long-wave stage by the spectrogram model B. An experiment comparing mercury test spectra found that spectrogram model B was significantly more accurate than the traditional spectrogram model, which verified the spectral calibration algorithm. The spectral calibration algorithm meets the engineering requirements for processing speed.