Gaussian process regression-based echelle grating spectrometer modeling method

Abstract

The invention relates to the technical field of spectrometers, and discloses an echelle grating spectrometer modeling method based on Gaussian process regression, which comprises the following steps: in a 275-800nm spectral range, selecting wavelength points at an interval of 1nm by adopting a layered random sampling method, constructing a two-dimensional spectral modeling data set with statistical significance, and obtaining multiple groups of training sample data; respectively constructing an Xmodel regression model between the wavelength and the prism dispersion direction coordinate (X) and a Ymodel regression model between the wavelength and the grating dispersion direction coordinate (Y) by adopting Gaussian process regression; a continuous wavelength sequence is generated with 0.001 nm as the step length, and theoretical spectrum position distribution in the whole wavelength range is obtained; a complete two-dimensional spectrum model is established by constructing a matrix, the rows and columns of the matrix respectively correspond to a Y coordinate and an X coordinate, and the element values of the matrix correspond to wavelength values. According to the method, the establishment complexity of the two-dimensional spectrum model can be simplified, the time required for establishing the model is shortened, and the precision of the two-dimensional spectrum model is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of spectrometers, and in particular to a modeling method of an echelle grating spectrometer based on Gaussian process regression. Background Art

[0002] The characteristic spectrum of a substance is as unique as its "fingerprint," capable of determining its composition and content. Spectrometers analyze spectra to identify and classify substances, finding widespread application in fields such as quality inspection and medicine. Spectral resolution is a crucial parameter for spectroscopic instruments.

[0003] Echelle grating spectrometers offer extremely high resolution, reaching picometer levels. Their core dispersive element is a grating with a large blaze angle and low line density. This grating enables the spectrometer to operate at high diffraction orders, exhibiting extremely high angular dispersion, thereby achieving extremely high resolution for the spectrometer. However, this also results in severe order overlap, necessitating the use of another dispersive element for secondary dispersion in different directions, forming a cross-dispersive optical path. The key to achieving ultra-high spectral resolution is the use of a planar array detector to receive the two-dimensional dispersion spectrum of the echelle grating spectrometer. While the two-dimensional spectrum improves spectral resolution, it also increases the difficulty of wavelength inversion, which in turn determines the accuracy of the instrument's wavelength analysis. Therefore, establishing an accurate two-dimensional spectral model has been one of the most important issues in the study of echelle grating spectrometers.

[0004] Existing research mainly uses theoretical modeling combined with standard light source calibration to solve this problem. There are usually three methods for establishing a two-dimensional spectral model. The first is to establish the coordinates of each wavelength by tracing the geometric optical equations of the échelle grating spectrometer based on the theory of geometric optics. However, this method takes too long to trace the light and is not suitable for the practical application of the échelle grating spectrometer. The second is to construct the physical relationship between wavelength and coordinate based on the grating equation and the dispersion equation of the prism, and then calculate the coordinates of each wavelength to complete the model establishment; this method has a large error. The third is to fit the calculation equation between wavelength and coordinate based on the optical characteristics of the échelle grating spectrometer to achieve the establishment of the model.

[0005] For example, the invention patent application number CN202510486513.6, titled "Method and System for Constructing a Two-Dimensional Spectral Restoration Model for an Echelle Grating Spectrometer," uses geometric optics principles to calculate the light transmission path and imaging position, establishes the relationship between the wavelength in the Y direction of the echelle grating diffraction, the wavelength in the X direction of the prism dispersion, and the position of the detector pixel, and constructs a two-dimensional spectral restoration model for the echelle grating spectrometer. The model is then calibrated using a standard light source, and the optical structure parameters in the model are optimized to obtain the optimal two-dimensional spectral restoration model for the echelle grating spectrometer. However, this method takes a long time to establish.

[0006] Due to instrument alignment errors, environmental variations, and the approximations of the theoretical model itself, there is a certain amount of error between the theoretical model and the actual instrument measurement results. Typically, the characteristic wavelength of a standard light source is used as a reference, and the deviation between the theoretical and actual coordinates is fitted using the least squares method, or the parameters of the theoretical model are adjusted to correct the error. However, these methods cannot completely resolve the constraints of model accuracy and modeling complexity. Therefore, it is of great significance to develop a simple, high-precision modeling method for échelle grating spectrometers. Summary of the Invention

[0007] In order to overcome the shortcomings of the existing technology, the purpose of the present invention is to provide a medium-step grating spectrometer modeling method based on Gaussian process regression, which can simplify the complexity of establishing a two-dimensional spectral model, shorten the time required for model establishment, and improve the accuracy of the two-dimensional spectral model.

[0008] The present invention is implemented by the following technical solution: a modeling method of an echelle grating spectrometer based on Gaussian process regression, comprising the following steps: Step 1: Within the 275-800nm ​​spectral range, a stratified random sampling method was used to select wavelength points at intervals of 1nm, with a sampling accuracy of 0.001nm. Ten sample points were randomly selected from each interval. The spatial coordinates of each wavelength within the free spectral region were accurately traced using Zemax optical design software to construct a statistically significant two-dimensional spectral modeling dataset. Multiple sets of training sample data were obtained, each containing four key characteristic parameters: wavelength value (λ), corresponding spectral order (M), prism dispersion direction coordinate (X), and grating dispersion direction coordinate (Y). Step 2: Use Gaussian process regression to construct the X_model regression model between wavelength and prism dispersion direction coordinate (X), and the Y_model regression model between wavelength and grating dispersion direction coordinate (Y); Step 3: In the X_model regression model and Y_model regression model, the experiment uses 80% of the training sample data as the training set and 20% of the training sample data as the test set, and uses the 5-fold cross-validation method for evaluation and verification; Step 4: Based on the trained X_model and Y_model regression models, generate a continuous wavelength sequence in the range of 275–800 nm with a step size of 0.001 nm, and predict the corresponding X and Y coordinates respectively to obtain the theoretical spectral position distribution in the entire wavelength range; Step 5: Establish a complete two-dimensional spectral model by constructing a matrix, where the rows and columns of the matrix correspond to the Y coordinate and the X coordinate respectively, and the element values ​​of the matrix correspond to the wavelength values.

[0009] Compared with the prior art, the present invention has the following beneficial effects: 1. The present invention provides a modeling method for an echelle grating spectrometer based on Gaussian process regression. Independent regression models are established for the prism dispersion direction (X) and the grating dispersion direction (Y). A nonlinear mapping model between wavelength and two-dimensional coordinates is established using a limited amount of wavelength tracing data as training samples. This method can address the constraints of the accuracy and modeling complexity of the two-dimensional spectral model, simplify the complexity of establishing the two-dimensional spectral model, and shorten the time required for model establishment. The error within the 95% confidence interval of the test set is within 0.5 pixels, thereby improving the accuracy of the two-dimensional spectral model.

[0010] 2. The present invention provides a modeling method for an echelle grating spectrometer based on Gaussian process regression. In the X_model regression model, the wavelength value (λ) is selected as the input feature and the X coordinate is selected as the output variable. The model is modeled using a combination of the Matern kernel function and the WhiteKernel. In the construction process of the Y_model regression model, the RationalQuadratic kernel function and the WhiteKernel combination are selected for regression analysis. This method can accurately characterize the spectral distribution characteristics in the prism dispersion direction and the grating dispersion direction, verifying the high-precision prediction performance of the model. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 This is a schematic diagram of the spatial distribution of the training set wavelengths in the two-dimensional image plane in the Gaussian process regression-based modeling method for the echelle grating spectrometer of the present invention; the vertical axis (Y direction) represents the grating dispersion direction, and the horizontal axis (X direction) corresponds to the prism dispersion direction; the red scattered points in the figure represent the selected training set wavelength samples; Figure 2This is a diagram showing the mapping relationship between wavelength and prism dispersion direction coordinates and the error analysis results in the present invention; Figure A on the left shows the nonlinear correspondence between wavelength and prism dispersion direction (X coordinate); Figure B on the right shows how the error between the predicted and actual prism dispersion direction coordinates varies with wavelength; Figure 3 The mapping relationship between wavelength and grating dispersion direction coordinates and the error analysis results in the present invention; Figure A on the left shows the periodic oscillation relationship between wavelength and grating dispersion direction (Y coordinate); Figure B on the right shows how the error between the predicted grating dispersion direction coordinates and the actual coordinates varies with wavelength; Figure 4 It is a schematic diagram of the two-dimensional spectrum model constructed by the present invention. DETAILED DESCRIPTION

[0012] The following describes the embodiments of the present invention through specific examples. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments. The details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the following embodiments and features in the embodiments can be combined with each other unless they conflict.

[0013] The purpose of the present invention is to provide a modeling method of an echelle grating spectrometer based on Gaussian process regression to address the defects of the prior art.

[0014] Example 1 The modeling method of the echelle grating spectrometer based on Gaussian process regression includes the following steps: Step 1: First, within the 275-800nm ​​spectral range, wavelength points were selected at 1nm intervals (with a sampling accuracy of 0.001nm) using a stratified random sampling method. Ten sample points were randomly selected from each interval. Using Zemax optical design software, the spatial coordinates of each wavelength within the free spectral region were precisely traced, thereby constructing a statistically significant two-dimensional spectral modeling dataset. This ultimately yielded multiple sets of training sample data, a total of 5250 sets in this embodiment. Due to the cross-dispersion structure employed by the échelle grating spectrometer, its spectral image exhibits a typical two-dimensional distribution. Each wavelength corresponds to a two-dimensional spatial coordinate determined by the grating dispersion direction coordinate (Y) and the prism dispersion direction coordinate (X). In this embodiment, each set of training sample data contains four key characteristic parameters: wavelength (λ), corresponding spectral order (M), prism dispersion direction coordinate (X), and grating dispersion direction coordinate (Y).

[0015] Reference Figure 1As shown in FIG, a schematic diagram of the spatial distribution of the training set wavelength in the two-dimensional image plane in this embodiment; the vertical axis (Y direction) represents the grating dispersion direction, and the horizontal axis (X direction) corresponds to the prism dispersion direction; the red scattered points in the figure represent the selected training set wavelength samples.

[0016] Step 2: Gaussian Process Regression (GPR) was used to construct the X_model regression model between wavelength and the prism dispersion direction coordinate (X), and the Y_model regression model between wavelength and the grating dispersion direction coordinate (Y). In the X_model regression model, the wavelength value (λ) was selected as the input feature and the X coordinate as the output variable. Because the relationship between wavelength and X coordinate is smooth and infinitely differentiable, a combination of the Matern kernel function and the WhiteKernel was used for modeling. The experiment used 80% of the training sample data as the training set and 20% of the training sample data as the test set. The evaluation was carried out using a five-fold cross-validation method, which was repeated five times independently and the best training result was selected.

[0017] Gaussian process regression is one of the random processes. It is essentially a family of distributions defined on the input space, and the joint distribution of sample points of any finite dimension is a multivariate Gaussian distribution. Gaussian process regression is jointly characterized by the mean function and the covariance function (i.e., kernel function): the mean function describes the output expectation corresponding to each input point, and the covariance function (i.e., kernel function) describes the correlation of output values ​​between different input points. The covariance function is the core of the Gaussian process, which defines the similarity between input variables, thereby determining the smoothness, complexity, and prior assumptions of the prediction function. In this embodiment, the mean function is simplified to zero to focus on the kernel function's modeling ability for the data structure. The kernel function is used to construct the covariance matrix of the prior data. The data on the diagonal of the covariance matrix are the coordinates corresponding to the prior data, i.e., the wavelength. The remaining values ​​in the matrix correspond to the correlation between the two data. The entire Gaussian process regression model is constructed through the covariance matrix to predict the data.

[0018] Reference Figure 2 Figure 2 shows the mapping relationship between wavelength and prism dispersion direction coordinates in this embodiment and the error analysis results. Figure A on the left shows the nonlinear correspondence between wavelength and prism dispersion direction (X coordinate). Experimental results show that the error within the 95% confidence interval of the test set is within 0.5 pixels (6.5µm), indicating that the model has high prediction accuracy.

[0019] Step 3: In the process of constructing the Y_model regression model, due to the order overlapping effect in the grating dispersion direction, the wavelengths of different orders show a periodic oscillation distribution in the Y coordinate direction in the free spectrum region; this characteristic makes it difficult to directly predict the Y coordinate through a single variable (λ); therefore, in the Y_model regression model, wavelength (λ) and spectral order (M) are selected as input features, and Y coordinate is used as the output variable; in order to adapt to this oscillation characteristic, the kernel function uses the RationalQuadratic kernel function combined with the WhiteKernel for regression analysis; the experiment also uses 80% of the training sample data as the training set and 20% of the training sample data as the test set, and adopts the five-fold cross-validation method for evaluation and verification.

[0020] Reference Figure 3 As shown in the figure, the mapping relationship between the wavelength and the grating dispersion direction coordinates and the error analysis results in this embodiment; Figure B on the right shows how the error between the predicted coordinates and the actual coordinates in the grating dispersion direction changes with wavelength. The experimental results show that the error in the 99% confidence interval of the test set is within 0.5 pixels (6.5µm), proving that the model can accurately characterize the spectral distribution characteristics in the grating dispersion direction and verifying the model's high-precision prediction performance in the grating dispersion direction.

[0021] Step 4: Based on the trained X_model and Y_model regression models, a continuous wavelength sequence is generated in the range of 275–800 nm with a step size of 0.001 nm, and its corresponding X and Y coordinates are predicted respectively, thereby obtaining the theoretical spectral position distribution in the entire wavelength range.

[0022] like Figure 4 As shown in , by constructing a matrix, where the rows and columns of the matrix correspond to the Y coordinate and the X coordinate respectively, and the element values ​​of the matrix correspond to the wavelength values, a complete two-dimensional spectral model is established; Figure 4 As can be seen from the figure, the constructed model includes the influence of prism line bending, which is an effect that traditional dispersion-based calculation methods require extremely high computing power to achieve.

[0023] Example 2 The modeling method of the echelle spectrometer is based on Gaussian process regression, in which the Matern kernel function is a generalized form of the radial basis function (RBF) kernel. v The smoothness of the control function, v The smaller the value, the lower the smoothness of the function; when v When it approaches infinity, the Matern kernel function degenerates into the RBF kernel function. Its expression is: (1) Where d (...) Euclidean distance, K V To modify the Bessel function, Γ(v) is the gamma function.

[0024] The RationalQuadratic kernel function can be viewed as an infinite mixture of RBF kernels with different characteristic lengths, which is determined by a length parameter l >0 and a blend parameter α >0 to parameterize, the formula is expressed as: (2) In the formula l is the length scale of the nucleus, α is the scale mixing parameter, d (...) is the Euclidean distance.

[0025] WhiteKernel is used to model independent and identically distributed noise in the observation data. It introduces non-zero terms only on the main diagonal in the covariance matrix, corresponding to the uncertainty of the input points themselves. The introduction of this term improves the model's adaptability to real observation errors.

[0026] Based on the Gaussian process regression algorithm, the present invention establishes independent regression models for the prism dispersion direction (X) and the grating dispersion direction (Y). The former adopts a combination of the Matern kernel function and the WhiteKernel, while the latter adopts a combination of the RationalQuadratic kernel function and the WhiteKernel. A limited number of wavelength tracing data are used as training samples to establish a nonlinear mapping model between wavelength and two-dimensional coordinates. This can solve the constraints of model accuracy and modeling complexity, simplify the model establishment, shorten the time required for establishing the two-dimensional spectral model, and improve the accuracy of the two-dimensional spectral model.

[0027] In addition, it should be understood that although this specification is described in terms of implementation methods, not every implementation method contains only one independent technical solution. This narrative method of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment can also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

Claims

1. A modeling method for an echelle spectrometer based on Gaussian process regression, characterized by: The following steps are involved: Step 1: Within the 275-800nm ​​spectral range, a stratified random sampling method was used to select wavelength points at intervals of 1nm, with a sampling accuracy of 0.001nm. Ten sample points were randomly selected from each interval. The spatial coordinates of each wavelength within the free spectral region were accurately traced using Zemax optical design software to construct a statistically significant two-dimensional spectral modeling dataset. Multiple sets of training sample data were obtained, each containing four key characteristic parameters: wavelength value (λ), corresponding spectral order (M), prism dispersion direction coordinate (X), and grating dispersion direction coordinate (Y). Step 2: Use Gaussian process regression to construct the X_model regression model between wavelength and prism dispersion direction coordinate (X), and the Y_model regression model between wavelength and grating dispersion direction coordinate (Y); Step 3: In the X_model regression model and Y_model regression model, the experiment uses 80% of the training sample data as the training set and 20% of the training sample data as the test set, and uses the 5-fold cross-validation method for evaluation and verification; Step 4: Based on the trained X_model and Y_model regression models, generate a continuous wavelength sequence in the range of 275–800 nm with a step size of 0.001 nm, and predict the corresponding X and Y coordinates respectively to obtain the theoretical spectral position distribution in the entire wavelength range; Step 5: Establish a complete two-dimensional spectral model by constructing a matrix, where the rows and columns of the matrix correspond to the Y coordinate and the X coordinate respectively, and the element values ​​of the matrix correspond to the wavelength values.

2. The method for modeling an echelle spectrometer based on Gaussian process regression according to claim 1, wherein: Gaussian process regression is characterized by a mean function and a kernel function. The mean function describes the output expectation corresponding to each input point, while the kernel function describes the correlation between the output values ​​​​of different input points.

3. The method for modeling an echelle spectrometer based on Gaussian process regression according to claim 2, wherein: The mean function is zero, and the covariance matrix of the prior data is constructed through the kernel function. The data on the diagonal of the covariance matrix are the coordinates corresponding to the prior data, i.e., the wavelength, and the remaining values ​​in the matrix correspond to the correlation between the two data.

4. The method for modeling an echelle spectrometer based on Gaussian process regression according to claim 3, wherein: In the X_model regression model, the wavelength value (λ) is selected as the input feature and the X coordinate is selected as the output variable. Since the relationship between the wavelength and the X coordinate is smooth and infinitely differentiable, the Matern kernel function and the WhiteKernel combination are used for modeling.

5. The method for modeling an echelle spectrometer based on Gaussian process regression according to claim 3, wherein: In the process of constructing the Y_model regression model, due to the order overlap effect in the grating dispersion direction, the wavelengths of different orders show a periodic oscillation distribution in the Y coordinate direction in the free spectrum region. This characteristic makes it difficult to directly predict the Y coordinate using a single variable (λ). Therefore, in the Y_model regression model, wavelength (λ) and spectral order (M) are selected as input features, and the Y coordinate is used as the output variable. To adapt to this oscillation characteristic, the RationalQuadratic kernel function and WhiteKernel are combined for regression analysis.

6. The method for modeling an echelle spectrometer based on Gaussian process regression according to claim 4, wherein: The Matern kernel function is passed through the parameter v The smoothness characteristic of the control function is expressed as: （1） Where d (...) Euclidean distance, K V To modify the Bessel function, Γ(v) is the gamma function.

7. The method for modeling an echelle spectrometer based on Gaussian process regression according to claim 5, wherein: The RationalQuadratic kernel function consists of a length parameter l >0 and a blend parameter α >0 to parameterize, its expression is: （2） In the formula l is the length scale of the nucleus, α is the scale mixing parameter, d (...) is the Euclidean distance.

8. The method for modeling an echelle spectrometer based on Gaussian process regression according to claim 4 or 5, characterized in that: WhiteKernel is used to model independent and identically distributed noise in observation data. It introduces non-zero terms on the main diagonal in the covariance matrix, which corresponds to the uncertainty of the input points themselves and is used to improve the model's adaptability to real observation errors.

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