A surface layer adaptive optical system error analysis method

Abstract

The application discloses a ground-layer adaptive optical system error analysis method, and relates to the technical field of astronomical telescopes adaptive optics, which is characterized in that: a first structure function of residual wavefront generated by un-compensated atmospheric turbulence error is calculated; a second structure function of residual wavefront generated by instrument error of the ground-layer adaptive optical system is calculated; the first structure function and the second structure function are linearly superposed to obtain a total performance structure function of the ground-layer adaptive optical system; a long-exposure optical transfer function is calculated according to the total performance structure function, and Fourier transform is performed on the long-exposure optical transfer function to obtain a point spread function of the ground-layer adaptive optical system; wherein the point spread function is used for analyzing error and performance of the ground-layer adaptive optical system.

Description

Technical Field

[0001] This invention relates to the field of adaptive optics technology for astronomical telescopes, and more specifically, to an error analysis method for adaptive optics systems at the Earth's surface. Background Technology

[0002] With the development of modern ground-based large-aperture and extremely large-aperture astronomical telescopes, adaptive optics systems have become an indispensable core component to overcome wavefront distortion caused by atmospheric turbulence. Among these, surface-level adaptive optics systems utilize multi-wavefront sensor (WFS) tomography to detect and primarily correct for low-level atmospheric turbulence near the Earth's surface using deformable mirrors. Although this system does not aim for extreme diffraction-limited resolution, it significantly improves the energy concentration of images over a very large field of view, i.e., improves the system's natural seeing, making it an important operating mode for modern large survey telescopes to enhance observational efficiency. Accurately predicting the long-exposure point spread function (PSF) under different atmospheric turbulence profiles and hardware parameters is crucial during the system design, error allocation, and performance evaluation stages.

[0003] Currently, performance evaluation of surface adaptive optics systems primarily relies on Monte Carlo-based end-to-end physical optics numerical simulation methods, such as OOMAO. This method requires pre-generating a large number of multi-layered, dynamically evolving atmospheric random phase screens and iteratively calculating the diffraction and propagation of light waves, wavefront sensing, and the fitting process of deformable mirrors. However, the computation time for such end-to-end simulations often increases dramatically with the fourth power of the telescope aperture. For large field-of-view, large-aperture telescopes, obtaining statistically significant long-exposure point spread functions typically requires hours or even days for a single simulation. This massive consumption of computational resources makes it completely unsuitable for the engineering requirements of large-scale parameter space scanning and global optimization needed in the early stages of system design.

[0004] Furthermore, in physical optics numerical simulations, atmospheric residual errors and various instrument underlying errors (such as Hartmann detection noise of wavefront sensors, wavefront reconstruction aliasing errors caused by finite sampling, servo delay errors of control systems, and non-common-path errors between optical paths) are deeply coupled in the final wavefront distortion. This simulation process, characterized by its "black box" nature, makes it difficult for engineers to isolate individual error sources for retrospective analysis of spatial frequency characteristics, thus failing to provide precise quantitative guidance for extremely stringent telescope error allocation schemes. To circumvent computational bottlenecks, the engineering community sometimes resorts to the traditional static error variance superposition method. However, this rough estimation based on a single variance can only obtain the lower limit of the system's Strell ratio, completely ignoring the distribution characteristics of various errors in the spatial frequency domain, and failing to provide the final image containing morphological features such as high-frequency halos and low-frequency core sharpness.

[0005] While existing frequency domain analytical simulation methods offer high computational efficiency, most are developed based on the full-band correction assumptions of single-conjugate adaptive optics systems. The core physical characteristic of surface-level adaptive optics systems is partial correction over multiple atmospheric layers, resulting in extremely complex statistical characteristics of their residual phase. Current frequency domain analytical techniques lack a mechanism that can rigorously mathematically unify and decouple this multi-layered partial correction transfer function from the highly complex underlying instrument system errors. Summary of the Invention

[0006] The purpose of this invention is to provide an error analysis method for surface layer adaptive optics systems, which solves the problems of high computational resource consumption and difficulty in independently tracing the source of errors in existing technologies during simulation.

[0007] The above-mentioned technical objective of the present invention is achieved through the following technical solution:

[0008] This invention provides an error analysis method for an adaptive optics system at the Earth's surface, the method comprising:

[0009] Calculate the first structure function of the residual wavefront resulting from uncompensated atmospheric turbulence errors;

[0010] Calculate the second structure function of the residual wavefront caused by instrument errors in the surface adaptive optics system;

[0011] By linearly superimposing the first structure function and the second structure function, the overall performance structure function of the surface layer adaptive optics system is obtained.

[0012] The long-exposure optical transfer function is calculated based on the overall performance structure function, and a Fourier transform is performed on the long-exposure optical transfer function to obtain the point spread function of the surface adaptive optics system; wherein, the point spread function is used to analyze the error and performance of the surface adaptive optics system.

[0013] In one implementation, the first structure function of the residual wavefront resulting from uncompensated atmospheric turbulence errors is calculated, including:

[0014] Based on the atmospheric turbulence model, the atmospheric phase power spectrum in the i-th layer was determined;

[0015] Based on the phase power spectrum of the i-th atmospheric layer and the error transfer function pre-introduced to describe the filtering process of the surface layer adaptive optics system, the spatial power spectrum of the residual wavefront obtained by the surface layer adaptive optics system at the turbulent i-th atmospheric layer is calculated.

[0016] Based on the spatial power spectrum, the first structure function of the residual wavefront generated by the uncompensated atmospheric turbulence error is calculated.

[0017] In one implementation, the expression for the first structure function is: ;in, For image spatial frequency, For Fourier transform, This represents the spatial power spectrum corresponding to the residual wavefront at atmospheric turbulence. Let r be the pupil plane displacement vector. .

[0018] In one implementation, the instrument error includes wavefront sensor measurement noise error, wavefront reconstruction error, wavefront control system servo hysteresis error, and optical path non-common path error.

[0019] The second structure function of the residual wavefront generated by instrument errors in the surface adaptive optics system is calculated as follows:

[0020] Based on the mode coefficient covariance matrix and mode shape function, the error structure functions of wavefront sensor measurement noise error, wavefront reconstruction error, wavefront control system servo hysteresis error and optical path non-common path error are calculated respectively.

[0021] The error structure functions of wavefront sensor measurement noise error, wavefront reconstruction error, wavefront control system servo hysteresis error, and optical path non-common path error are linearly superimposed to obtain the second structure function.

[0022] In one implementation, the expression for the error structure function is: ;in, , The pupil plane displacement vector; m and n are the Zernike orders. This represents the cross-covariance of the m-th and n-th model coefficients in the time domain; Let be the pattern shape function, representing the normalized integral of the product of the m-th and n-th Zernike polynomials on the pupil plane.

[0023] In one implementation, the expression for the mode coefficient covariance matrix of the wavefront sensor measurement noise error is:

[0024] ;in, This represents the slope measurement variance of each channel of the wavefront sensor, where N represents the number of effective channels. This represents the noise gain factor.

[0025] In one implementation, the expression for the mode coefficient covariance matrix of the wavefront reconstruction error is:

[0026] ;in, The total atmospheric turbulence coherence constant is . Zernike coefficient in The Noll covariance matrix at that time It is the identity matrix. Let D represent the gradient matrix, and D represent the aperture of the telescope. Indicates the detection wavelength.

[0027] In one implementation, the expression for the mode coefficient covariance matrix of the servo hysteresis error of the wavefront control system is:

[0028] ;in, For each Zernike model, the coefficients are determined by the loop response time and the speed of atmospheric turbulence. The total atmospheric turbulence coherence constant is . Zernike coefficient in The Noll covariance matrix at time D represents the aperture of the telescope.

[0029] In one implementation, the expression for the error structure function of the optical path non-common path error is:

[0030] ;in, This represents the factor of change in Zernike aberration coefficients compared to an ideal optical path. and D represents the aperture of the telescope, which corresponds to the constant parameters of each order of Zernike aberration.

[0031] In one implementation, the expression for the long-exposure optical transfer function is: ;in, Let be the optical transfer function of an ideal diffraction-limited telescope system. This represents the statistical measure of the residual wavefront accumulated at the telescope pupil after compensation by the adaptive optics system at the Earth's surface. This is the pupil plane displacement vector.

[0032] Compared with the prior art, the present invention has the following beneficial effects:

[0033] This invention addresses the problems of enormous computational resource consumption and difficulty in independently tracing errors in existing surface layer adaptive optics system simulations. It proposes a system error analytical simulation method based on error structure functions and mode covariance matrices. This method abandons the cumbersome process of generating massive dynamic phase screens and progressive ray tracing in traditional end-to-end physical numerical simulations, cleverly transforming the extremely complex spatiotemporal evolution physical process into rapid analytical computation in the pure spatial frequency domain. This reduces the generation time of the long-exposure point spread function for large-aperture, large-field-of-view systems from hours or even days in traditional methods to seconds, breaking the computational bottleneck and greatly improving the efficiency of global scanning and early optimization of system-level parameters. Furthermore, this invention overcomes the theoretical limitation of traditional static error variance superposition methods, which can only roughly estimate the central energy. It creatively incorporates atmospheric multilayer residual errors, wavefront sensor measurement noise, wavefront reconstruction aliasing errors, servo hysteresis, and non-common-path aberrations—complex underlying instrument errors—into the analytical framework of the structure function, achieving complete "white-box" decoupling of various physical errors. This rigorous modeling approach not only accurately transmits and reproduces the spatial morphological characteristics of complex errors such as high-frequency noise halo and low-frequency core asymmetry, comprehensively improving the physical fidelity of system imaging quality prediction, but also provides a solid and quantitative analytical tool for the extremely stringent adaptive optics error allocation scheme and top-level design of modern large astronomical telescopes. Attached Figure Description

[0034] The accompanying drawings, which are included to provide a further understanding of embodiments of the invention and form part of this application, do not constitute a limitation thereof. In the drawings:

[0035] Figure 1 A flowchart illustrating the error analysis method for an adaptive optics system at the Earth's surface provided in an embodiment of the present invention;

[0036] Figure 2 The FWHM distribution map of the full field-of-view PSF of the surface adaptive optics system provided in this embodiment of the invention. Detailed Implementation

[0037] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the embodiments and accompanying drawings. The illustrative embodiments and descriptions of the present invention are only used to explain the present invention and are not intended to limit the present invention.

[0038] It should be noted that the terms "comprising" or "may include" used in the various embodiments of this application indicate the presence of the claimed function, operation, or element, and do not limit the addition of one or more functions, operations, or elements. Furthermore, as used in the various embodiments of this application, the terms "comprising," "having," and their cognates are intended only to indicate a specific feature, number, step, operation, element, component, or combination of the foregoing, and should not be construed as primarily excluding the presence of one or more other features, numbers, steps, operations, elements, components, or combinations of the foregoing, or adding one or more combinations of the foregoing.

[0039] It should be understood that terms such as "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0040] like Figure 1 As shown, this embodiment of the invention provides an error analysis method for an adaptive optics system at the Earth's surface, the method comprising the following steps:

[0041] S101 is the first structure function for calculating the residual wavefront generated by uncompensated atmospheric turbulence errors.

[0042] In this embodiment, based on the Von-Karman atmospheric turbulence model, the first The atmospheric phase power spectrum is expressed as follows: ;in, , For the external scale of atmospheric turbulence, For the first The coherence constant of atmospheric turbulence.

[0043] According to atmospheric stratification theory, atmospheric turbulence can be simplified to the superposition of multiple layers of atmospheric turbulence, with wavefront accumulation and correction occurring at each layer. Therefore, each turbulence layer is statistically independent. Thus, the GLAO system in the [missing information - likely a specific layer or stage]... The spatial power spectrum of the residual wavefront obtained at the laminar turbulence can be expressed as: ;in, For the first Atmospheric phase power spectrum. The introduced error transfer function is related to the GLAO system parameters, the guide star layout, and the turbulence distribution, and is used to describe the filtering process of the GLAO system.

[0044] If the GLAO system is equipped with K satellites with a central field of view angular distance of... The guiding star of the GLAO system is in the i-th atmospheric layer. It can be represented as:

[0045]

[0046] in, For the scientific objective and the inter-satellite angular distance, h is the height of the i-th turbulent layer. Let K be the Airy function, and K represent the number of guiding stars. This indicates the angular distance of the guide star's field of view. For a low-pass filter related to DM spatial resolution, its specific form is: Where d is the size of the actuator pitch on the pupil surface.

[0047] Further, according to Normalization yields the residual phase difference SF for each layer as follows:

[0048] Where L0 represents the outer scale of atmospheric turbulence.

[0049] Furthermore, the SF in each layer is determined according to the turbulence intensity of each layer. Structure function introduced into the atmospheric residual phase difference of all layers The expression is: ; Turbulence intensity at each layer The relationship with the coherence constant of atmospheric turbulence at each layer is as follows: .

[0050] The residual wavefront accumulated at the telescope pupil for the uncompensated atmospheric turbulence component after GLAO system compensation. The corresponding statistical structure function is: ;in, Represents the pupil plane coordinate vector. Indicates the corresponding position The residual wavefront at that location, Indicates the corresponding position The residual wavefront at that location.

[0051] According to the Wiener-Khinchi theorem, the first structure function It can be represented as: ;in, For image spatial frequency, For Fourier transform, This represents the spatial power spectrum corresponding to the residual wavefront at atmospheric turbulence. Let r be the pupil plane displacement vector. .

[0052] Substituting the calculated spatial power spectrum into the statistical structure function yields the first structure function of the residual wavefront generated by the uncompensated atmospheric turbulence error.

[0053] S102, calculate the second structure function of the residual wavefront generated by the instrument error of the surface layer adaptive optics system.

[0054] In this embodiment, for the residual wavefront caused by GLAO instrument error, this embodiment considers four independent error sources: wavefront sensor measurement noise error, wavefront reconstruction error, wavefront control system servo hysteresis error, and optical path non-common path error; the corresponding error structure function (ESF) also remains independent.

[0055] The structure function for each error is represented by the mode coefficient covariance matrix and the mode shape function: ;in, , The pupil plane displacement vector; m and n are the Zernike orders. This represents the cross-covariance of the m-th and n-th model coefficients in the time domain; Let be the pattern shape function, representing the normalized integral of the product of the m-th and n-th Zernike polynomials on the pupil plane.

[0056] It should be noted that the mode coefficient covariance matrix is ​​the one described in this embodiment. , refers to the cross covariance of the m-th and n-th order model coefficients in the time domain. Model coefficients are numerical values ​​of a specific model (such as the Zernike model) that describe systematic errors or wavefront distortions.

[0057] Therefore, for the structure function of each error, by obtaining its covariance matrix, and combining it with... The structure functions for each error can then be obtained.

[0058] Regarding measurement noise error in wavefront sensors, in the GLAO system using the Shaker-Hartmann WFS, the presence of measurement noise causes radial elongation of the measurement spot, thus the measurement noise error manifests as a shift in the spot's centroid. In this case, the centroid errors in the x and y directions are correlated. The elongation is determined by the relative elongation parameter. Quantification, among which That is, the angular range of the guide star visible from the telescope aperture (diameter D) (distance L from the gate and guide star height H). This is the FWHM size of the diffracted Gaussian spot generated by turbulence in the up and down paths. ,in, The gradient matrix, This represents the mode coefficient vector of the real system.

[0059] Since the detection processes of each sub-aperture in the Shaker-Hartmann WFS are physically independent, it is assumed that the variance of the slope measurement for each channel is equal. The covariance matrix of the pure spatial mode caused by WFS noise can be expressed as: Furthermore, WFS noise is filtered by a loop time controller, and in the case of mode control, each mode has its own time response. For a GLAO system with multiple guide satellites, the noise gain factor... The same applies to all modes, showing the decrease (or increase) of the root mean square noise in the closed loop of the system relative to one control cycle. Therefore, This can be further expressed as: ;in, This represents the slope measurement variance of each channel of the wavefront sensor, where N represents the number of effective channels. This represents the noise gain factor.

[0060] Furthermore, combining the formula The structure function of the noise error measured by the wavefront sensor can be approximated as: ;in, The total variance of all correction modes, The characteristic scale of the deformable mirror actuator.

[0061] For wavefront reconstruction error, the centroid offset vector X of each sub-aperture spot in the wavefront sensor is determined by the gradient matrix G and the mode coefficient vector Z, that is: Furthermore, the modal coefficients are obtained by applying the inverse reconstructor matrix, i.e.: The reconstructor almost never perfectly represents the wavefront. Given the statistics of the reconstructor and the atmospheric Zernike coefficients, the covariance matrix caused by wavefront reconstruction errors can be evaluated by the following formula: ;in, The total atmospheric turbulence coherence constant is . Zernike coefficient in The Noll covariance matrix at that time It is the identity matrix. Let D represent the gradient matrix, and D represent the aperture of the telescope. Indicates the detection wavelength.

[0062] For the servo hysteresis error of the wavefront control system, the servo hysteresis error is determined by the loop response time and the speed of atmospheric turbulence. For each model, using a single turbulent layer model moving at wind speed V, the time spectrum function is calculated, multiplied by the loop error suppression function, and integrated to find the coefficients. Without calculating the relative time spectrum function of the Zernike mode, assuming the cross-covariance matrix of the servo hysteresis error is proportional to the Noll matrix, the expression for the covariance matrix of the servo hysteresis error of the wavefront control system is: ;in, For each Zernike model, the coefficients are determined by the loop response time and the speed of atmospheric turbulence. The total atmospheric turbulence coherence constant is . Zernike coefficient in The Noll covariance matrix at time D represents the aperture of the telescope.

[0063] For non-common-path errors, since the probe and imaging optical paths are typically different, non-common-path aberrations will inevitably occur. Typically, non-common-path aberrations are compensated for through careful optical alignment and appropriate offsets in the wavefront sensor. However, this compensation can only be achieved with limited accuracy, so small residual non-common-path aberrations should be considered.

[0064] Aberrations of all orders affect the optical transfer function of a telescope system in complex ways under ideal diffraction-limited conditions. However, small aberrations only cause a slight broadening of the system's point spread function, which can also be described by the error structure function. For small changes in the Zernike aberration coefficients (if the change is a factor of...),... The error structure function for non-common aberrations will become the original... Therefore, the structure function of the optical path non-common path error is as follows: ;in, This represents the factor of change in Zernike aberration coefficients compared to an ideal optical path. and D represents the aperture of the telescope, which corresponds to the constant parameters of each order of Zernike aberration.

[0065] Extensive calculations have shown that, for different Zernike aberrations, the constant parameters... and Maintaining specific patterns. The table below shows some parameters corresponding to Zernike aberrations. and As shown in Table 1 below.

[0066] Table 1 Correspondence table of constants and Zernike aberrations

[0067]

[0068] By linearly superimposing the error structure functions of the wavefront sensor measurement noise error, wavefront reconstruction error, wavefront control system servo hysteresis error, and optical path non-common path error described above, a second structure function is obtained, which can be expressed as: ;in, , , , These are the structure functions of the residual wavefront caused by wavefront sensor measurement noise, wavefront reconstruction error, wavefront control system servo hysteresis error, and optical path non-common path error, respectively.

[0069] S103, the first structure function and the second structure function are linearly superimposed to obtain the overall performance structure function of the surface layer adaptive optics system.

[0070] In this embodiment, the residual wavefront at the telescope pupil plane after compensation by the GLAO system is divided into two parts: the residual wavefront generated by the uncompensated atmospheric turbulence and the residual wavefront generated by the GLAO instrument error. Due to the independence between the original atmospheric turbulence and the system instrument, the statistical structure functions (SF) of the residual wavefront components generated by the two are also independent of each other.

[0071] Therefore, the statistical SF of the residual wavefront after compensation in the GLAO system is expressed as: ;in, The pupil plane displacement vector, is the coordinate vector of the telescope's pupil plane. Corresponding to the residual wavefront accumulated at the telescope pupil after compensation by the GLAO system. , Corresponding to the uncompensated atmospheric turbulence portion, Corresponding to GLAO instrument error.

[0072] Assuming the residual wavefront phase statistics are a stationary random process, then its statistical structure function depends only on , represented as The relationships between the various SFs are as follows: ;in, This represents the first structure function. This represents the second structure function.

[0073] Combining the expression for the second structure function described above, the overall performance structure function of the surface adaptive optics system can be expressed as follows: = .

[0074] S104, calculate the long-exposure optical transfer function based on the overall performance structure function, and perform a Fourier transform on the long-exposure optical transfer function to obtain the point spread function of the surface layer adaptive optics system; wherein, the point spread function is used to analyze the error and performance of the surface layer adaptive optics system.

[0075] In this embodiment, for the GLAO system, the long exposure point spread function (PSF) is typically used to describe the system's correction effect, and its long exposure optical transfer function (OTF) can be approximated as follows: ;in, Let be the optical transfer function of an ideal diffraction-limited telescope system. This represents the statistical measure of the residual wavefront accumulated at the telescope pupil after compensation by the adaptive optics system at the Earth's surface. This is the pupil plane displacement vector.

[0076] In summary, this invention addresses the problems of enormous computational resource consumption and difficulty in independently tracing errors in existing surface adaptive optics system simulations by proposing a system error analytical simulation method based on error structure functions and mode covariance matrices. This method eliminates the cumbersome process of generating massive dynamic phase screens and progressive ray tracing in traditional end-to-end physical numerical simulations, cleverly transforming the extremely complex spatiotemporal evolution physical process into rapid analytical computation in the pure spatial frequency domain. This reduces the generation time of the long-exposure spread function for large-aperture, large-field-of-view systems from hours or even days in traditional methods to seconds, breaking the computational bottleneck and greatly improving the efficiency of global scanning and early-stage optimization of system-level parameters. Furthermore, this invention overcomes the theoretical limitation of traditional static error variance superposition methods, which can only roughly estimate the central energy. It creatively incorporates atmospheric multilayer residual errors, wavefront sensor measurement noise, wavefront reconstruction aliasing errors, servo hysteresis, and non-common-path aberrations—complex underlying instrument errors—into the analytical framework of the structure function, achieving complete "white-box" decoupling of various physical errors. This rigorous modeling approach not only accurately transmits and reproduces the spatial morphological characteristics of complex errors such as high-frequency noise halo and low-frequency core asymmetry, comprehensively improving the physical fidelity of system imaging quality prediction, but also provides a solid and quantitative analytical tool for the extremely stringent adaptive optics error allocation scheme and top-level design of modern large astronomical telescopes.

[0077] In a specific embodiment, the telescope aperture is set to 0.98m; the imaging field of view is 60×60arcsec; the sampling wavelength is 0.55um; the imaging wavelength is 0.705um; the number of guiding stars is 9; the deformable mirror actuators are arranged in a 13×13 configuration; and the atmospheric turbulence model is a 7-layer turbulence model fitted with measured data from the solar observatory site, with parameters for each layer shown in Table 2.

[0078] Table 2 Turbulence Model for Solar Observation Sites

[0079]

[0080] Within a 60×60 arcsec imaging field of view, the PSF at each position was obtained, and the corresponding FWHM (Full Half Width and Full Height) values ​​were calculated to describe the system performance. The FWHM distribution within the 60×60 arcsec field of view was plotted on [the map / plotting]. Figure 2 It should be noted that, Figure 2 In this context, arcsec means arcsecond. Figure 2 (a) represents the system performance without GLAO instrument errors. Figure 2 Part (b) describes the system performance when GLAO instrument errors are included. Figure 2 Part (c) represents the difference between the two. In this example, the error analysis calculation for this surface adaptive optics system takes only about 5 seconds, while traditional algorithms require at least 7 hours. This order-of-magnitude improvement makes it possible to perform tens of thousands of parameter optimization loops (such as finding the optimal guide star altitude or control bandwidth) during the patent design phase. Furthermore, this invention can not only quickly generate PSF images containing GLAO error performance, but also deeply reveal the impact mechanism of each underlying error on imaging quality through quantitative decomposition of the error structure function. The underlying physics-driven analysis capability of this invention is unavailable in traditional black-box simulations and has extremely high engineering guidance value.

[0081] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for error analysis in an adaptive optics system for the Earth's surface, characterized in that the method... include: Calculate the first structure function of the residual wavefront resulting from uncompensated atmospheric turbulence errors; Calculate the second structure function of the residual wavefront caused by instrument errors in the surface adaptive optics system; By linearly superimposing the first structure function and the second structure function, the overall performance structure function of the surface layer adaptive optics system is obtained. The long-exposure optical transfer function is calculated based on the overall performance structure function, and a Fourier transform is performed on the long-exposure optical transfer function to obtain the point spread function of the surface adaptive optics system; wherein, the point spread function is used to analyze the error and performance of the surface adaptive optics system.

2. The method for error analysis of an adaptive optics system for the Earth's surface layer according to claim 1, characterized in that, The first structure function for calculating the residual wavefront resulting from uncompensated atmospheric turbulence errors includes: Based on the atmospheric turbulence model, the atmospheric phase power spectrum in the i-th layer was determined; Based on the phase power spectrum of the i-th atmospheric layer and the error transfer function pre-introduced to describe the filtering process of the surface layer adaptive optics system, the spatial power spectrum of the residual wavefront obtained by the surface layer adaptive optics system at the turbulent i-th atmospheric layer is calculated. Based on the spatial power spectrum, the first structure function of the residual wavefront generated by the uncompensated atmospheric turbulence error is calculated.

3. The method for error analysis of an adaptive optics system for the Earth's surface layer according to claim 2, characterized in that, The expression for the first structure function is: ;in, For image spatial frequency, For Fourier transform, This represents the spatial power spectrum corresponding to the residual wavefront at atmospheric turbulence. Let r be the pupil plane displacement vector. .

4. The method for error analysis of an adaptive optics system for the Earth's surface layer according to claim 1, characterized in that, The instrument errors include wavefront sensor measurement noise error, wavefront reconstruction error, wavefront control system servo hysteresis error, and optical path non-common path error. The second structure function of the residual wavefront generated by instrument errors in the surface adaptive optics system is calculated as follows: Based on the mode coefficient covariance matrix and mode shape function, the error structure functions of wavefront sensor measurement noise error, wavefront reconstruction error, wavefront control system servo hysteresis error and optical path non-common path error are calculated respectively. The error structure functions of wavefront sensor measurement noise error, wavefront reconstruction error, wavefront control system servo hysteresis error, and optical path non-common path error are linearly superimposed to obtain the second structure function.

5. The method for error analysis of an adaptive optics system for the Earth's surface layer according to claim 4, characterized in that, The expression for the error structure function is: ;in, , The pupil plane displacement vector; m and n are the Zernike orders. This represents the cross-covariance of the m-th and n-th model coefficients in the time domain; Let be the pattern shape function, representing the normalized integral of the product of the m-th and n-th Zernike polynomials on the pupil plane.

6. The method for error analysis of an adaptive optics system for the Earth's surface layer according to claim 5, characterized in that, The expression for the mode coefficient covariance matrix of the noise error measured by the wavefront sensor is as follows: ;in, This represents the slope measurement variance of each channel of the wavefront sensor, where N represents the number of effective channels. This represents the noise gain factor.

7. The method for error analysis of an adaptive optics system for the Earth's surface layer according to claim 5, characterized in that, The expression for the mode coefficient covariance matrix of the wavefront reconstruction error is as follows: ;in, The total atmospheric turbulence coherence constant is . Zernike coefficient in The Noll covariance matrix at that time It is the identity matrix. Let D represent the gradient matrix, and D represent the aperture of the telescope. Indicates the detection wavelength.

8. The method for error analysis of an adaptive optics system for the Earth's surface layer according to claim 5, characterized in that, The expression for the mode coefficient covariance matrix of the servo hysteresis error of the wavefront control system is as follows: ;in, For each Zernike model, the coefficients are determined by the loop response time and the speed of atmospheric turbulence. The total atmospheric turbulence coherence constant is . Zernike coefficient in The Noll covariance matrix at time D represents the aperture of the telescope.

9. The method for error analysis of an adaptive optics system for the Earth's surface layer according to claim 5, characterized in that, The expression for the error structure function of the optical path non-common path error is as follows: ;in, This represents the factor of change in Zernike aberration coefficients compared to an ideal optical path. and D represents the aperture of the telescope, which corresponds to the constant parameters of each order of Zernike aberration.

10. The method for error analysis of an adaptive optics system for the Earth's surface layer according to claim 1, characterized in that, The expression for the long exposure optical transfer function is: ;in, Let be the optical transfer function of an ideal diffraction-limited telescope system. This represents the statistical measure of the residual wavefront accumulated at the telescope pupil after compensation by the adaptive optics system at the Earth's surface. This is the pupil plane displacement vector.

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