Method for controlling dispersion nonlinearity of reflective spectrometer based on cartesian coordinate system

Abstract

The application discloses a reflective spectrometer dispersion nonlinearity control method based on a Cartesian coordinate system, which comprises the following steps: modeling a reflective grating spectrometer light path in combination with a Cartesian coordinate system, obtaining the diffraction angles of different wavelengths of a complex light after the complex light is diffracted by the grating, and obtaining the incident point coordinates and corresponding off-axis angles of different wavelengths on a focusing spherical mirror; obtaining the meridian image point coordinates of different wavelengths after the different wavelengths are focused by the spherical mirror based on a generalized Courant formula; and controlling the meridian image point coordinates to be symmetrical about a central wavelength, calculating key structure parameters, and realizing the dispersion nonlinearity control of the spectrometer. The application utilizes the grating principle and off-axis light path characteristics to control the overall dispersion nonlinearity of the spectrometer, and realizes a dispersion nonlinearity control technical scheme of the reflective grating spectrometer with high spectral resolution and a large numerical aperture.

Description

Technical Field

[0001] This invention belongs to the field of precision optical measurement technology, specifically a nonlinear control method for dispersion of a reflective spectrometer based on the Cartesian coordinate system. Background Technology

[0002] Currently, spectral confocal displacement measurement employs a non-contact, high-precision micro-displacement measurement method. Its system architecture mainly consists of a pre-confocal probe and a post-confocal spectrometer. The peak wavelength signal reflected back by the pre-confocal probe is the characteristic peak captured by the post-confocal spectrometer; analyzing this signal can reveal the height information of the sample surface. However, the wavelength-displacement response curve of the dispersive objective in the pre-confocal probe unit exhibits non-linear characteristics, causing the measurement error at the edge of the measurement range to be significantly higher than in the central region, increasing the maximum linear error across the entire range. Similarly, the post-confocal spectrometer needs to achieve lower spectral dispersion non-linearity error while ensuring high spectral resolution. Therefore, the accurate and efficient extraction of the peak wavelength of the confocal spectrum by the spectrometer is a crucial prerequisite for ensuring the accuracy of spectral confocal displacement measurement.

[0003] Traditional Czerny-Turner (optical structure of a reflective grating spectrometer) grating spectrometers primarily focus on high spectral resolution, large numerical aperture, and miniaturization, neglecting the dispersive nonlinearity of the grating. Currently, only echelle grating spectrometers have been studied in terms of the dispersive uniformity of the combined prism material, but the problem of the dispersion uniformity being destroyed by off-axis reflection after uniform dispersion of the dispersive element has not been considered.

[0004] Therefore, to address the above problems, a nonlinear control method for the dispersion of a reflectance spectrometer based on the Cartesian coordinate system is proposed. Summary of the Invention

[0005] To address the aforementioned problems in the existing technology, this invention provides a method for controlling the dispersive nonlinearity of a reflective spectrometer based on the Cartesian coordinate system. This method utilizes the grating principle and off-axis optical path characteristics to control the overall dispersive nonlinearity of the spectrometer, thereby achieving a high spectral resolution and large numerical aperture dispersive nonlinearity control technology for reflective grating spectrometers.

[0006] The technical solution to achieve the above objectives is: A method for nonlinear control of dispersion in a reflectance spectrometer based on Cartesian coordinates, comprising: Step S1: Model the meridional optical path of the reflective grating spectrometer using a Cartesian coordinate system; Step S2: In the established Cartesian coordinate system, with the grating normal and dispersion direction as references, obtain the diffraction angles of different wavelengths of polychromatic light after diffraction by the grating, and calculate the incident point coordinates and corresponding off-axis angles of different wavelengths on the focusing spherical mirror based on key structural parameters; among which, the key structural parameters include the grating incident angle and the off-axis angle of the center wavelength of the focusing spherical mirror. Step S3: Based on the generalized Cottington formula, using the obtained coordinates of the incident points and off-axis angles for different wavelengths, calculate the coordinates of the meridional image points after off-axis reflection of different wavelengths by the focusing spherical mirror. Step S4: Establish a dispersion nonlinear control model to ensure that the coordinates of the obtained metropolitan image points of each wavelength are symmetrically distributed on the image plane about the center wavelength. Then, calculate the key structural parameters to realize the dispersion nonlinear control of the reflective grating spectrometer.

[0007] Preferably, in step S1, the Cartesian coordinate system modeling includes: Coordinate system correspondence rule: The normal and dispersion direction of the grating in the meridional optical path of the reflective grating spectrometer are respectively aligned with the X-axis and Y-axis of the Cartesian coordinate system; Origin setting: Set the center of the grating surface as the origin of the Cartesian coordinate system, and use it as the reference point for the collimating and focusing optical paths before and after the grating. Optical path model establishment: For the optical path in the Cartesian coordinate system, the incident light of the grating is located in the first quadrant. Coordinate function models and angle function models are established for the optical path before diffraction and for each wavelength optical path after diffraction.

[0008] Preferably, in step S2, in the reflective grating spectrometer structure, to satisfy the astigmatism elimination condition, the off-axis angle of the collimating spherical mirror in the collimating optical path in front of the grating is expressed by the dispersion nonlinearity control parameter: ; In the formula, The off-axis angle of the collimating spherical mirror. To focus the radius of curvature of the spherical mirror, Let be the radius of curvature of the collimating spherical mirror. The center wavelength is the off-axis incident angle of the spherical mirror. The incident angle of the grating, The center wavelength, is the grating constant.

[0009] Preferably, step S2 includes: Obtain the diffraction angles for different wavelengths: For the polychromatic light band, based on the grating equation, the relationship between the grating incident angle and the center wavelength diffraction angle, as well as the relationship between any wavelength and the center wavelength diffraction angle, are determined: ; In the formula, For wavelength, The diffraction angle is the center wavelength. Calculate the coordinates of the incident point on the focusing spherical mirror: The center wavelength is at the coordinates of the incident center of the focusing spherical mirror. satisfy: ; In the formula, This is the distance between the grating and the collimating spherical mirror; Calculate the coordinates of the incident point for any wavelength: Any wavelength at the incident center of the focusing spherical mirror The theoretical model of coordinates satisfies: ; ; In the formula, For any wavelength of the grating, the diffraction angle is... The center wavelength is the off-axis incident angle of the spherical mirror. The X-coordinate is the center position of the center wavelength on the focusing spherical mirror. There are two different structures. This refers to a non-cross-structure reflection grating spectrometer. This refers to a cross-structure reflection grating spectrometer; Calculate the distance between the incident center and the grating face center: The theoretical model for the distance between the incident center of the focusing spherical mirror and the center of the grating surface for any wavelength satisfies: ; Calculate the corresponding off-axis angle: The theoretical model for the off-axis angle cosine function of an incident ray of any wavelength on a focusing spherical mirror satisfies: ; In the formula, For any wavelength, the off-axis angle of the focusing spherical mirror; Through the above steps, after grating diffraction, the coordinates of the incident point of any wavelength on the focusing spherical mirror are obtained. and off-axis angle This enables precise calculation of the entire wavelength optical path within the operating band.

[0010] Preferably, step S3 specifically includes: The reflective grating spectrometer calculates the off-axis collimating and focusing optical paths based on the generalized Cottington formula, obtaining the meridional image distance of any wavelength after off-axis reflection by the focusing spherical mirror. : ; In the formula, Let be the distance between the slit and the collimating spherical mirror. This is the distance between the collimating spherical mirror and the grating; Combined with the incident point coordinates of the focusing spherical mirror Off-axis angle and meridian distance Calculate the coordinates of the meridional image point after off-axis reflection of any wavelength. : ; Through the above steps, the meridional coordinates of each wavelength on the image plane are calculated comprehensively, laying the foundation for subsequent control of image point symmetry about the center wavelength and reduction of dispersion nonlinearity error.

[0011] Preferably, in step S4, by constructing constraints that make the meridional image point symmetrical about the center wavelength, a dispersive nonlinear control theoretical model is established, and the optimal key structural parameters are obtained by solving the model, ultimately minimizing the dispersive nonlinear error of the spectrometer. Specifically, this includes: Establish a dispersive nonlinear control theory model: By controlling two wavelengths symmetrical about the center wavelength, i.e. and The coordinates of the meridional image points satisfy a symmetry relationship, from which the dispersive nonlinear control theoretical model is derived: ; ; In the formula, This is the wavelength difference, ranging from 0 to half the width of the operating band. and These are respectively about the center wavelength difference wavelength, and respectively wavelength and The X-coordinate of the center position on the focusing spherical mirror. and respectively wavelength and The Y-coordinate of the center position on the focusing spherical mirror. The meridional image distance of the center wavelength through the focusing spherical mirror. and Wavelength and The meridional image distance of the focusing spherical mirror and respectively wavelength and The diffraction angle, and respectively wavelength and Off-axis angle on the focusing spherical mirror.

[0012] Compared with the prior art, the beneficial effects of the present invention are: This invention models the optical path of a reflective grating spectrometer using a Cartesian coordinate system, comprehensively capturing the position and angle information of each wavelength in the optical path, thereby achieving accurate modeling of the reflective grating spectrometer and significantly improving the spectrometer's dispersion nonlinear control capability. This invention introduces the generalized Cottington formula to calculate the off-axis collimating optical path and off-axis focusing optical path of the reflective grating spectrometer, and obtains the meridional image point coordinate function after off-axis reflection of different wavelengths by the spherical mirror, thereby improving the controllability of the position of each wavelength on the image plane. The spectrometer dispersion nonlinearity control function constructed in this invention effectively improves the ability to adjust the wavelength image point relationship of the spectrometer by controlling the symmetry of the image point coordinates of different wavelengths about the center wavelength image point coordinates, thereby realizing the control of the spectrometer dispersion nonlinearity. Attached Figure Description

[0013] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used in conjunction with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings: Figure 1 This is a flowchart of a nonlinear control method for dispersion of a reflectance spectrometer based on the Cartesian coordinate system according to the present invention; Figure 2 This is a schematic diagram of the optical path structure of the non-cross-reflection grating spectrometer in this invention; Figure 3 This is a schematic diagram of the optical path structure of the cross-reflection grating spectrometer in this invention. Detailed Implementation

[0014] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0015] like Figure 1 As shown, a nonlinear control method for the dispersion of a reflectance spectrometer based on Cartesian coordinates includes: Step S1: Model the meridional optical path of the reflective grating spectrometer using a Cartesian coordinate system.

[0016] In this embodiment, Cartesian coordinate system modeling includes: Coordinate system correspondence rule: The normal and dispersion direction of the grating in the meridional optical path of the reflective grating spectrometer are respectively aligned with the X-axis and Y-axis of the Cartesian coordinate system; Origin setting: Set the center of the grating surface as the origin of the Cartesian coordinate system, and use it as the reference point for the collimating and focusing optical paths before and after the grating. Optical path model establishment: For an optical path in Cartesian coordinates, the incident ray of the grating is located in the first quadrant. A non-cross-reflection grating spectrometer, such as... Figure 2 As shown, the cross-reflection grating spectrometer is as follows: Figure 3 As shown, coordinate function models and angle function models can be established for the optical path before grating diffraction and for each wavelength optical path after grating diffraction.

[0017] In summary, by combining the optical path of a reflective grating spectrometer with a Cartesian coordinate system image, a specific and systematic model for nonlinear dispersion control of the reflective grating spectrometer can be constructed, thereby improving the ability to adjust the wavelength-image point relationship of the spectrometer.

[0018] Step S2: In the established Cartesian coordinate system, with the grating normal and dispersion direction as references, obtain the diffraction angles of different wavelengths of polychromatic light after diffraction by the grating, and calculate the incident point coordinates and corresponding off-axis angles of different wavelengths on the focusing spherical mirror based on key structural parameters; among which, the key structural parameters include the grating incident angle and the off-axis angle of the center wavelength of the focusing spherical mirror.

[0019] In the embodiment, in the reflective grating spectrometer structure, to satisfy the astigmatism elimination condition, the off-axis angle of the collimating spherical mirror in the collimating optical path in front of the grating is expressed by the dispersion nonlinearity control parameter: ; In the formula, The off-axis angle of the collimating spherical mirror. To focus the radius of curvature of the spherical mirror, Let be the radius of curvature of the collimating spherical mirror. The center wavelength is the off-axis incident angle of the spherical mirror. The incident angle of the grating, The center wavelength, is the grating constant.

[0020] In this embodiment, step S2 specifically includes: Obtain the diffraction angles for different wavelengths: For the polychromatic light band, based on the grating equation, the relationship between the grating incident angle and the center wavelength diffraction angle, as well as the relationship between any wavelength and the center wavelength diffraction angle, are determined: ; In the formula, For wavelength, The diffraction angle is the center wavelength. Calculate the coordinates of the incident point on the focusing spherical mirror: The center wavelength is at the coordinates of the incident center of the focusing spherical mirror. satisfy: ; In the formula, This is the distance between the grating and the collimating spherical mirror; Calculate the coordinates of the incident point for any wavelength: Any wavelength at the incident center of the focusing spherical mirror The theoretical model of coordinates satisfies: ; ; In the formula, For any wavelength of the grating, the diffraction angle is... The center wavelength is the off-axis incident angle of the spherical mirror. The X-coordinate is the center position of the center wavelength on the focusing spherical mirror. There are two different structures. This refers to a non-cross-structure reflection grating spectrometer, such as... Figure 2 As shown, This refers to a cross-structure reflection grating spectrometer, such as... Figure 3 As shown; Calculate the distance between the incident center and the grating face center: The theoretical model for the distance between the incident center of the focusing spherical mirror and the center of the grating surface for any wavelength satisfies: ; Calculate the corresponding off-axis angle: The theoretical model for the off-axis angle cosine function of an incident ray of any wavelength on a focusing spherical mirror satisfies: ; In the formula, For any wavelength, the off-axis angle of the focusing spherical mirror; Through the above steps, after grating diffraction, the coordinates of the incident point of any wavelength on the focusing spherical mirror are obtained. and off-axis angle This enables precise calculation of the entire wavelength optical path within the operating band.

[0021] Step S3: Based on the generalized Cottington formula, the coordinates of the meridional image points after off-axis reflection of different wavelengths by the obtained incident point coordinates and off-axis angles are calculated.

[0022] In this embodiment, step S3 specifically includes: The reflective grating spectrometer calculates the off-axis collimating and focusing optical paths based on the generalized Cottington formula, obtaining the meridional image distance of any wavelength after off-axis reflection by the focusing spherical mirror. : ; In the formula, Let be the distance between the slit and the collimating spherical mirror. This is the distance between the collimating spherical mirror and the grating; Combined with the incident point coordinates of the focusing spherical mirror Off-axis angle and meridian distance Calculate the coordinates of the meridional image point after off-axis reflection of any wavelength. : ; Through the above steps, the meridional coordinates of each wavelength on the image plane are calculated comprehensively, laying the foundation for subsequent control of image point symmetry about the center wavelength and reduction of dispersion nonlinearity error.

[0023] In summary, by introducing the generalized Cottington formula and combining it with the off-axis incident angle function of different wavelengths after grating diffraction, the meridional image point coordinates of each wavelength after off-axis reflection by the focusing spherical mirror can be comprehensively calculated, thereby improving the accuracy and controllability of the image plane coordinate function of each wavelength.

[0024] Step S4: Establish a dispersion nonlinear control model to ensure that the coordinates of the obtained metropolitan image points of each wavelength are symmetrically distributed on the image plane about the center wavelength. Then, calculate the key structural parameters to realize the dispersion nonlinear control of the reflective grating spectrometer.

[0025] In this embodiment, by constructing constraints that make the meridional image points symmetrical about the center wavelength, a dispersive nonlinear control theoretical model is established, and the optimal key structural parameters are obtained by solving the model, ultimately minimizing the dispersive nonlinear error of the spectrometer. Specifically, this includes: like Figure 2 , Figure 3 middle, , , respectively wavelength , , The coordinates of the incident center on the focusing spherical mirror , , respectively wavelength , , The coordinates on the detector receiving surface, , The center wavelengths are respectively and wavelength Center wavelength and wavelength Coordinate distance on the detector receiving surface, wavelength , , Image point position on the detector , , Due to the combined influence of the grating and the focusing spherical mirror, a certain degree of dispersive nonlinearity error exists when the wavelength differences are equal but the image point distances are unequal. The dispersive nonlinearity of the grating is combined with the off-axis reflection characteristics of the focusing spherical mirror to jointly control the dispersive nonlinearity error of the reflective grating spectrometer. The meridional image point coordinates of each wavelength after off-axis reflection by the focusing spherical mirror after grating diffraction are symmetrical about the central wavelength on the image plane. To achieve optimal dispersion nonlinearity error, the theoretical model for dispersion nonlinearity control is calculated, namely: By controlling two wavelengths symmetrical about the center wavelength, i.e. and The coordinates of the meridional image points satisfy a symmetry relationship, from which the dispersive nonlinear control theoretical model is derived: ; ; In the formula, This is the wavelength difference, ranging from 0 to half the width of the operating band. and These are respectively about the center wavelength difference wavelength, and respectively wavelength and The X-coordinate of the center position on the focusing spherical mirror. and respectively wavelength and The Y-coordinate of the center position on the focusing spherical mirror. The meridional image distance of the center wavelength through the focusing spherical mirror. and respectively wavelength and The meridional image distance of the focusing spherical mirror and respectively wavelength and The diffraction angle, and respectively wavelength and Off-axis angle on the focusing spherical mirror.

[0026] In summary, by controlling the symmetry of the meridional image point coordinates with respect to the center wavelength, the dispersive nonlinear control theoretical model was calculated, key structural parameters were obtained, and the spectrometer was optimized, reducing the spectrometer's dispersive nonlinear error and improving the accuracy of spectral confocal measurements.

[0027] Finally, it should be noted that the above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for nonlinear control of dispersion in a reflectance spectrometer based on Cartesian coordinates, characterized in that, include: Step S1: Model the meridional optical path of the reflective grating spectrometer using a Cartesian coordinate system; Step S2: In the established Cartesian coordinate system, with the grating normal and dispersion direction as references, obtain the diffraction angles of different wavelengths of polychromatic light after diffraction by the grating, and calculate the incident point coordinates and corresponding off-axis angles of different wavelengths on the focusing spherical mirror based on key structural parameters; among which, the key structural parameters include the grating incident angle and the off-axis angle of the center wavelength of the focusing spherical mirror. Step S3: Based on the generalized Cottington formula, using the obtained coordinates of the incident points and off-axis angles for different wavelengths, calculate the coordinates of the meridional image points after off-axis reflection of different wavelengths by the focusing spherical mirror. Step S4: Establish a dispersion nonlinear control model to make the coordinates of the obtained meridional image points of each wavelength symmetrically distributed about the center wavelength on the image plane, and then calculate the key structural parameters to realize the dispersion nonlinear control of the reflective grating spectrometer. In step S1, the Cartesian coordinate system modeling includes: Coordinate system correspondence rule: The normal and dispersion direction of the grating in the meridional optical path of the reflective grating spectrometer are respectively aligned with the X-axis and Y-axis of the Cartesian coordinate system; Origin setting: Set the center of the grating surface as the origin of the Cartesian coordinate system, and use it as the reference point for the collimating and focusing optical paths before and after the grating. Optical path model establishment: For the optical path in the Cartesian coordinate system, the incident light of the grating is located in the first quadrant. Coordinate function models and angle function models are established for the optical path before diffraction and for each wavelength optical path after diffraction. In step S2, in the reflective grating spectrometer structure, to satisfy the astigmatism elimination condition, the off-axis angle of the collimating spherical mirror in the collimating optical path in front of the grating is expressed by the dispersion nonlinearity control parameter: ； In the formula, The off-axis angle of the collimating spherical mirror. To focus the radius of curvature of the spherical mirror, Let be the radius of curvature of the collimating spherical mirror. The center wavelength is the off-axis incident angle of the focusing spherical mirror. The incident angle of the grating, The center wavelength, The grating constant; Step S2 includes: Obtain the diffraction angles for different wavelengths: For the polychromatic light band, based on the grating equation, the relationship between the grating incident angle and the center wavelength diffraction angle, as well as the relationship between any wavelength and the center wavelength diffraction angle, are determined: ； In the formula, For wavelength, The diffraction angle is the center wavelength. Calculate the coordinates of the incident point on the focusing spherical mirror: The center wavelength is at the coordinates of the incident center of the focusing spherical mirror. satisfy: ； In the formula, This is the distance between the grating and the collimating spherical mirror; Calculate the coordinates of the incident point for any wavelength: Any wavelength at the incident center of the focusing spherical mirror The theoretical model of coordinates satisfies: ； ； In the formula, For any wavelength of the grating, the diffraction angle is... The center wavelength is the off-axis incident angle of the focusing spherical mirror. The X-coordinate is the center position of the center wavelength on the focusing spherical mirror. There are two different structures. This indicates a non-cross-structure reflection grating spectrometer. This refers to a cross-structure reflection grating spectrometer; Calculate the distance between the incident center and the grating face center: The theoretical model for the distance between the incident center of the focusing spherical mirror and the center of the grating surface for any wavelength satisfies: ； Calculate the corresponding off-axis angle: The theoretical model for the off-axis angle cosine function of an incident ray of any wavelength on a focusing spherical mirror satisfies: ； In the formula, For any wavelength, the off-axis angle of the focusing spherical mirror; Through the above steps, after grating diffraction, the coordinates of the incident point of any wavelength on the focusing spherical mirror are obtained. and off-axis angle This enables precise calculation of the entire wavelength optical path within the working band; Step S3 specifically includes: The reflective grating spectrometer calculates the off-axis collimating and focusing optical paths based on the generalized Cottington formula, obtaining the meridional image distance of any wavelength after off-axis reflection by the focusing spherical mirror. : ； In the formula, Let be the distance between the slit and the collimating spherical mirror. This is the distance between the collimating spherical mirror and the grating; Combined with the incident point coordinates of the focusing spherical mirror Off-axis angle and meridian distance Calculate the coordinates of the meridional image point after off-axis reflection of any wavelength. : ； Through the above steps, the meridional coordinates of each wavelength on the image plane are calculated comprehensively, laying the foundation for subsequent control of image point symmetry about the center wavelength and reduction of dispersion nonlinearity error. In step S4, by constructing constraints that make the meridional image points symmetrical about the center wavelength, a dispersive nonlinear control theoretical model is established, and the optimal key structural parameters are obtained by solving the model. This ultimately minimizes the dispersive nonlinear error of the spectrometer. Specifically, this includes: Establish a dispersive nonlinear control theory model: By controlling the wavelength that is symmetrical about the center wavelength, i.e. and The coordinates of the meridional image points satisfy a symmetry relationship, from which the dispersive nonlinear control theoretical model is derived: ； ； In the formula, This is the wavelength difference, ranging from 0 to half the width of the operating band. and These are respectively about the center wavelength difference wavelength, and respectively wavelength and The X-coordinate of the center position on the focusing spherical mirror. and respectively wavelength and The Y-coordinate of the center position on the focusing spherical mirror. The meridional image distance of the center wavelength through the focusing spherical mirror. and respectively wavelength and The meridional image distance of the focusing spherical mirror and respectively wavelength and The diffraction angle, and respectively wavelength and Off-axis angle on the focusing spherical mirror.

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