CN103744173A - Telescope secondary mirror position correction method based on optical spot definition function - Google Patents

Abstract

A method for correcting the position of the secondary mirror of a telescope based on the spot sharpness function. The steps are as follows: (1) Taking the on-axis field of view spot clarity as the objective function, the translation of the secondary mirror along the optical axis and the translation and rotation in two directions perpendicular to the vertical optical axis as variables, through an iterative algorithm, the objective function appearance The position of the extremum secondary mirror. If all variables are iterated simultaneously, the objective function may converge to a local extremum. Therefore, considering that each iterative process includes two iteration steps in the direction of the optical axis and the direction perpendicular to the optical axis, the objective function can converge to the global extremum; (2) The average value of the sharpness function of multiple off-axis field of view spots is used as The objective function is to make the secondary mirror rotate around the zero coma point of the system in two orthogonal directions, iteratively seek the extremum of the objective function, and realize the correction of the relative position of the secondary mirror. This method does not require wavefront detection and reconstruction, so that the adjustment of the telescope system has an objective standard, and can monitor the imaging quality of the telescope in real time.

Description

A Position Correction Method for Telescope Secondary Mirror Based on Facula Sharpness Function

technical field

The invention relates to a method for correcting the position of a secondary mirror of a telescope based on a light spot definition function. This method is suitable for the adjustment and online adjustment of the telescope system.

Background technique

Ideally, the relative positions of the primary and secondary mirrors of the telescope are fixed. However, due to the influence of temperature, gravity and other factors during operation, the relative positions of the primary and secondary mirrors will change. Taking the central vertex of the primary mirror as the origin of the reference coordinate system, the change of the system structure includes the translation of the secondary mirror along the optical axis, the translation in the vertical direction of the optical axis, and the rotation in the vertical direction of the optical axis. Changes in the relative positions of the primary and secondary mirrors will cause certain aberrations in the telescope. These mainly include defocus and spherical aberration produced by translation along the optical axis, coma and astigmatism produced by translation or rotation perpendicular to the optical axis. For a small field of view telescope system, the influence of astigmatism on imaging can be ignored, and the on-axis field of view is mainly composed of coma. By adjusting the secondary mirror, the aberration of the field of view on the axis can be corrected, that is, the aberration of the telescope system can be considered to be corrected. However, as the field of view of the telescope system increases, astigmatism has become non-negligible. Only correcting the aberration of the field of view on the axis cannot realize the correction of the aberration of the entire field of view. Therefore, in the aberration correction of the large field of view telescope system, it is necessary to consider both the on-axis field of view and the off-axis field of view.

Currently, there are various methods for aberration correction using on-axis and off-axis fields of view. For example, the azimuth and eccentricity of the far-field spot in the off-axis field of view are used to calculate the wavefront aberration coefficient, so as to obtain the rotation angle around the zero coma point (see Collimation of Fast Wide-Field Telescopes, BRIAN A.MCLEOD, 1996) ; Use the wavefront detector to obtain the wavefront aberration coefficient, so as to calculate the rotation angle around the zero coma point (see Final alignment of the VLT, L.Noehte and S.Guisard, 2000); use the sensitivity matrix to establish the offset and The relationship between the aberration coefficients is reversely resolved by measuring the aberration coefficients (see Reverse-optimization Alignment Algorithm using Zernike Sensitivity, Eugene D. Kim, et al, 2005). The above methods all obtain the wavefront aberration coefficient indirectly or directly, so as to guide the correction. The disadvantage of the first method is that the calculation complexity is high, and the correction result is greatly affected by the calculation accuracy. The second method requires a wavefront detector, etc., which increases the complexity of the system. The matrix relationship established by the third method is an approximation when the error is small, and there is a large deviation in the matrix relationship when the error is large, and the acquisition of aberration coefficients increases the complexity of the system. It can be seen from the published literature that the aberration correction method of the telescope system is mainly related to the wavefront aberration coefficient, which makes the system complex and increases the difficulty of engineering implementation.

Contents of the invention

The technical problem of the present invention is: to overcome the deficiencies of the prior art, to provide a method for correcting the position of the secondary mirror of the telescope based on the spot definition function. Simple, easy engineering implementation features.

The technical solution of the present invention is: a method for correcting the position of the secondary mirror of the telescope based on the spot definition function. Its characteristic steps are as follows:

(1) Take the on-axis field of view spot clarity as the objective function, the optical axis direction is the z-axis direction, and the two directions perpendicular to the optical axis are the x and y-axis directions to establish a rectangular coordinate system, then the displacement variable of the secondary mirror includes along the optical axis There are three variables in the axial direction, that is, the translation in the z direction and the translation in the x and y directions of the vertical optical axis. Through iterative calculations, the position of the secondary mirror where the objective function has an extreme value is obtained; each iteration process in the iterative process includes two steps , the first step: translation along the optical axis direction; the second step: translation and rotation vertical to the optical axis direction; the above two steps are regarded as a cycle, so that the objective function converges to the global extremum; as shown in the first step cycle process in Figure 1 Show;

(2) Taking the average value of the sharpness function of multiple off-axis field of view spots as the objective function, make the secondary mirror rotate around the zero coma point of the system in two orthogonal directions, that is, around the zero along the vertical optical axis x, y axis direction The coma point rotates, and the distance Z _cfp between the central vertex of the secondary mirror and the zero coma point satisfies the following formula, where L is the distance between the central vertex of the secondary mirror and the focal plane, and _m2 is the magnification of the secondary mirror, that is, the focal length of the system and the primary mirror The ratio of focal lengths satisfies m ₂ =f, /f ₁ ', f' is the focal length of the telescope system, f ₁ ' is the focal length of the primary mirror, and b _s2 is the conic constant of the secondary mirror. Iteratively solve the extremum of the objective function to realize the correction of the secondary mirror position, as shown in the second cycle process in Figure 1.

${\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; cfp</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}$

(1) Taking the on-axis field of view spot clarity as the objective function, the translation of the secondary mirror along the optical axis and the translation and rotation of the vertical optical axis along the two orthogonal directions as variables, through a certain iterative algorithm, the objective function appears The position of the extremum secondary mirror. If all variables are iterated simultaneously, the objective function may converge to a local extremum. Therefore, iterating the direction of the optical axis and the direction perpendicular to the optical axis can make the objective function converge to the global extremum. (2) Taking the average value of the sharpness functions of multiple off-axis field of view spots as the objective function, make the secondary mirror rotate around the zero coma point of the system along the orthogonal x and y axes, and iteratively solve the extreme value of the objective function to realize Correction of the relative position of the secondary mirror.

The advantage of the present invention compared with prior art is:

(1) The present invention uses the sharpness function of the far-field spot of the telescope system as the objective function of aberration correction, and obtains the objective function value through the CCD and the computer, without the need for wavefront measurement and wavefront reconstruction, reducing the system complexity , so that the adjustment of the relative position of the primary and secondary mirrors of the telescope has an objective standard.

(2) Using the on-axis and off-axis field of view far-field spot image definition function as the objective function, decoupling the vertical optical axis translation and rotation of the secondary mirror can make the off-axis field of view caused by the deviation of the primary and secondary mirrors The astigmatism of the method is corrected, and the method is simple, direct and effective.

(3) There is no need to calculate the azimuth and eccentricity of the far-field spot in the off-axis field of view, and there are no strict requirements on the distribution of the far-field spot in the off-axis field of view. It can be corrected according to different spot distributions to meet the requirements during operation. The actual situation, so it can be used not only for adjustment, but also for online adjustment.

Description of drawings

FIG. 1 is a flow chart of the method for correcting the relative positions of the primary and secondary mirrors of the large-field-of-view high-resolution imaging telescope based on the multi-field-of-view spot sharpness function of the present invention.

Figure 2 is a schematic diagram of the coordinates of the pupil and the field of view.

Figure 3 is a schematic diagram of the point where the axes of the primary and secondary mirrors of the telescope intersect at zero coma.

Figure 4 is a schematic diagram of the distribution of off-axis field of view points of the telescope.

Fig. 5 is the objective function of the on-axis field of view and the convergence curve of the Strehl ratio with iterations.

Figure 6 is the convergence curve of the off-axis field of view objective function.

Figure 7 shows the convergence curves of the Strehl ratio (SR) of the off-axis field of view points and on-axis field of view points selected in Table 2 with iterations.

Detailed ways

The present invention will be described in further detail below.

Firstly, the basic principle of a method for correcting the secondary mirror position of a telescope based on the spot sharpness function is introduced. According to the third-order vector wave aberration theory, for a two-mirror telescope system, the coma and astigmatism produced by the vertical optical axis translation of the secondary mirror or the rotation around the vertex are shown in the wavefront aberration formulas (1) and (2) .

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; coma</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 131</span>}}\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 131</span>}\mathrm{<span\; class="notranslate">\; j</span>}}{\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; ast</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; [</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 222</span>}\mathrm{<span\; class="notranslate">\; j</span>}}{\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 222</span>}\mathrm{<span\; class="notranslate">\; j</span>}}{\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 222</span>}\mathrm{<span\; class="notranslate">\; j</span>}}{\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; \&CenterDot;</span>{\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$

为归一化视场向量，方向由方位角a决定，大小由距坐标中心距离决定。

为归一化光瞳向量，方向由θ决定，大小由距相应的坐标中心距离决定，如图2所示，

为第j面的像差中心偏移量。对于轴上视场即零视场点，

公式（1）中仅包含第二项

即常数项彗差。公式（2）中包含最后一项

即常数项像散。一般情况下，由于公式（2）中后一项相对公式（1）中第二项

很小，可忽略不计。故对于两镜望远镜系统，当发生前述情况时，轴上视场主要产生彗差。由于次镜垂直光轴的平移和旋转均分别对产生影响。因此可以通过次镜垂直光轴的平移和（或）绕其顶点的旋转对轴上彗差进行校正，使望远镜系统

然而对于

参考图2坐标系，通常情况下是由于次镜沿x,y轴平移所产生的彗差和绕x,y轴旋转产生的彗差相互抵消的结果。因此不同时满足

轴上视场得到校正，轴外视场可能产生大量像散。故可得出结论，对于望远镜系统像差的校正需要同时考虑轴上视场和轴外视场。W' _coma is the third-order coma value, W' _ast is the third-order astigmatism value, j is the surface number of the optical element in the optical system, W _131j and W _222j are the third-order coma of the j-th surface system wave aberration and the astigmatism coefficient,

It is the normalized field of view vector, the direction is determined by the azimuth a, and the size is determined by the distance from the coordinate center.

is the normalized pupil vector, the direction is determined by θ, and the size is determined by the distance from the corresponding coordinate center, as shown in Figure 2,

is the offset of the aberration center of the jth plane. For the on-axis field of view, that is, the zero field of view point,

Only the second term is included in formula (1)

That is, the constant term coma. Equation (2) contains the last term

That is, the constant term astigmatism. In general, because the latter item in formula (2) is relatively the second item in formula (1)

Very small, negligible. Therefore, for the two-mirror telescope system, when the aforementioned situation occurs, the on-axis field of view mainly produces coma. Since the translation and rotation of the vertical optical axis of the secondary mirror are respectively make an impact. Therefore, the on-axis coma can be corrected by the translation of the vertical optical axis of the secondary mirror and/or the rotation around its apex, so that the telescope system

However for

Referring to the coordinate system in Figure 2, it is usually the result of the coma aberration generated by the translation of the secondary mirror along the x, y axis and the coma aberration generated by the rotation around the x, y axis cancel each other. Therefore, it is not satisfied at the same time

On-axis fields of view are corrected, off-axis fields of view can produce a large amount of astigmatism. Therefore, it can be concluded that the correction of the aberration of the telescope system needs to consider both the on-axis field of view and the off-axis field of view.

的位置，实现最终的校正。由于绕零彗差点旋转，轴上视场远场光斑几乎不产生任何变化，而轴外视场光斑产生影响。因此拟采用轴外视场远场光斑（如图4）的清晰度函数作为校正的判断依据，当轴外视场的目标函数达到极值，则实现望远镜主次镜相对位置的校正。According to theory, after the on-axis coma is corrected, the central axes of the primary and secondary mirrors of the telescope system intersect at the point of zero coma, as shown in Figure 3, the coma produced by rotation around this point, translation of the vertical optical axis and rotation around the apex is always can cancel each other out. Therefore, after the on-axis field of view is corrected, the secondary mirror can be rotated around the point of zero coma to find

position for the final correction. Due to the rotation around the point of zero coma, the on-axis field of view far-field spot hardly changes, while the off-axis field of view spot does. Therefore, it is proposed to use the sharpness function of the far-field spot in the off-axis field of view (as shown in Figure 4) as the judgment basis for correction. When the objective function of the off-axis field of view reaches the extreme value, the relative position correction of the primary and secondary mirrors of the telescope will be realized.

In the two-mirror reflector telescope system, the calculation according to the formula shows that the distance Z _cfp between the zero coma point and the vertex of the secondary mirror satisfies the formula (3). Where L is the distance between the center vertex of the secondary mirror and the focal plane, m ₂ is the magnification of the secondary mirror, that is, the ratio of the focal length of the system to the focal length of the primary mirror, satisfying m ₂ =f, /f ₁ ', f' is the focal length of the telescope system, f ₁ ' is the focal length of the primary mirror, and b _s2 is the conic constant of the secondary mirror.

${\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; cfp</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

The above analysis shows that the objective function of the far-field spot at different points of view can be used as the basis for judging the correction of the secondary mirror. . (2) When the central axes of the primary and secondary mirrors intersect at the point of zero coma, the objective function of the field point on the axis has a unique extremum. (3) When the secondary mirror rotates around the point of zero coma, the objective function of the off-axis field of view has a unique extremum. Based on the above conditions, the objective functions such as formulas (5) and (6) can be taken. J is the average radius of the objective function of the field of view point on the axis, which is used for the correction judgment of the field of view point on the axis. I(x, y) is the light intensity distribution of the field of view, where x, y are the normalized coordinates of the detection surface, and x’, y’ are the coordinates of the center of mass, satisfying the formula (4). As the wavefront aberration decreases, the average radius decreases accordingly. J' is the basis for judging the correction of the off-axis field of view point, i is the i-th off-axis field of view point, n is the number of off-axis field of view points, that is, J' is the average radius of the far-field spot in the off-axis field of view. Since there is coupling between the axial correction and the vertical axis correction when correcting the on-axis field of view, it is necessary to iteratively correct the two respectively. The correction process is shown in Figure 1, and the correction steps are as follows. Step (1) refers to the optical axis in Figure 1 directional translation and translation and/or rotation in the direction perpendicular to the optical axis; step (2) refers to the cyclic process of rotating around the zero coma point in Figure 1. The following steps are the specific description of the calibration process in Figure 1.

${\mathrm{<span\; class="notranslate">\; x</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; \&Integral;</span><span\; class="notranslate">\; \&Integral;</span>\mathrm{<span\; class="notranslate">\; xI</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dxdy</span>}}{<span\; class="notranslate">\; \&Integral;</span><span\; class="notranslate">\; \&Integral;</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dxdy</span>}}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; \&Integral;</span><span\; class="notranslate">\; \&Integral;</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dxdy</span>}}{<span\; class="notranslate">\; \&Integral;</span><span\; class="notranslate">\; \&Integral;</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dxdy</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; =</span>\frac{\sqrt{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dxdy</span>}}{<span\; class="notranslate">\; \&Integral;</span><span\; class="notranslate">\; \&Integral;</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dxdy</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; J</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

(1) On-axis field of view correction

Considering that the translation along the optical axis of the secondary mirror is coupled with the translation and rotation perpendicular to the optical axis, an iterative convergence analysis is performed on them respectively. Use Stochastic Parallel Gradient Descent algorithm (SPGD, Stochastic Parallel Gradient Descent) to find the extreme value of the field of view objective function J on the axis. The basic idea is that the system performance optimization index J can be considered as a function of the control parameter U, that is, J=J(U), where U=(u ₁ ,u ₂ ,u ₃ …u _n ), indicating that U is a function of n variables function. Since the objective function changes fastest along the gradient direction, the algorithm uses the product of the variation δJ of the performance index in formula (7) and the random disturbance {δU _j } satisfying the Bernoulli distribution to estimate the gradient of the k-th iteration, formula (7 ) in J(u ₁ +δ u ₁ ,…u _n +δu _n ) represents the objective function after adding random disturbance, and J(u ₁ ,…u _n ) represents the value of the objective function without disturbance. After the kth iteration, the k+1th control parameter is shown in formula (8), u _j ^k+1 is the value of the jth variable after the kth iteration, r is a fixed parameter, and when the objective function converges to the minimum value r is negative, otherwise it is positive. After a certain number of iterations J converges to a stable value, the search for the extremum of the objective function is completed.

δJ＝J(u ₁ +δu ₁ ,...,u _n +δu _n )-J(u ₁ ,...,u _n ) (7)

${\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \}</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; \&delta;J\&delta;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; cfp</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; decenterx</span>}}{\mathrm{<span\; class="notranslate">\; tilty</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; decenter</span>}}{\mathrm{<span\; class="notranslate">\; tiltx</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

The displacement despace along the optical axis of the secondary mirror and the translation and rotation decenterx, decentery, tiltx, and tilty of the vertical optical axis are used as correction variables, and J in formula (5) is used as the objective function, and the above algorithm is used for iterative convergence analysis. The on-axis field of view is corrected. After the adjustment, the central axis of the secondary mirror and the central axis of the primary mirror intersect at the point of zero coma aberration (as shown in Figure 3), and the relationship satisfies the formula (9), where the units of decenterx and decentery are meters, and the units of tiltx and tilty are degrees. Z _cfp is the distance between the center vertex of the secondary mirror and the zero coma point, decenterx, decentery are the offsets in the x and y directions, respectively, and tiltx and tilty are the rotations around the x and y axes, respectively.

(2) Off-axis field of view point correction

After completing the calibration process (1), it can be seen that the four parameters approximately satisfy the linear relationship in pairs as shown in formula (9). According to the rotation of the secondary mirror around the coma-free point along the x and y directions, and the control variables satisfy the linear relationship in formula (9), it can be considered that the control parameters are simplified from four to two. Using the above stochastic parallel gradient descent algorithm (SPGD), the rotation around the zero-coma point in the x and y directions is used as a variable, and the average value J′ of the radius of the far-field spot in the off-axis field of view in the formula (6) is used as the evaluation function to achieve Correction for off-axis field of view.

Correction example:

The simulation uses a two-mirror reflective RC optical system for correction simulation, and the system parameters are shown in Table 1. The parameters are brought into formula (3) to get Z _cfp =-0.0106. The correction process was simulated by using ZEMAX optical design software and MATLAB programming software. The analysis process is carried out according to the steps in the correction process. Considering the existence of deviation, the astigmatism of the field of view on the axis is almost zero. The initial parameters of the correction are despace=80μm, decenterx=600μm, decentery=880μm, tiltx=0.02°, tilty=0.03°. The translation of the optical axis direction and the other four parameters all use the aforementioned stochastic parallel gradient descent algorithm (SPGD), so that the two are iterated separately. The objective function J gradually converges to the minimum value with iterations, as shown by the dotted line in Figure 5. The Strehl ratio of the on-axis field of view spot gradually increases with iterations, and finally converges to 1, as shown by the solid line in Figure 5, and finally realizes the correction of the on-axis field of view, so that the axes of the primary and secondary mirrors approximately intersect at the zero coma Almost. Finally, rotate around the zero coma point in the x and y directions, and use the stochastic parallel gradient descent algorithm (SPGD) to correct the off-axis field of view. During calibration, the off-axis field of view is normalized, and the edge of the field of view is 1, and the selected field of view points are shown in Table 2. The objective function J' of the off-axis field of view gradually converges to the minimum value with iterations, as shown in Figure 6. The Strehl ratio of different field of view points changes with iterations, as shown in Figure 7, and the average state of the far-field spot at the off-axis field of view reaches the best.

Table 1 Telescope parameters in ZEMAX

R(m) Thickness (m) Semi-diameter (m) conic (m) 1 Infinity 5 0.155 0.000 2 -11.040 -4.906 1.200 -1.002 3 -1.358 6.406 0.141 -1.497 IMA -0.631 - 0.081 0.000

Table 2 Normalized off-axis field of view points

fields 1 2 3 4 Positions (0,0.9) (0,-0.8) (0.7,0) (-1,0)

Claims (4)

1. A method for correcting the secondary mirror position of the telescope based on the spot definition function, its characteristic steps are as follows: (1) Taking the on-axis field of view spot clarity as the objective function, the optical axis direction is the z-axis direction, and the two directions perpendicular to the optical axis are the x and y-axis directions to establish a rectangular coordinate system, then the displacement variable of the secondary mirror includes along the optical axis There are three variables in the axial direction, that is, the translation in the z direction and the translation in the x and y directions of the vertical optical axis. Through iterative calculations, the position of the secondary mirror where the objective function has an extreme value is obtained; in the iterative process, each iterative process includes two steps. , the first step: translation along the optical axis direction; the second step: translation and rotation in the vertical optical axis direction; the above two steps are used as a cycle to make the objective function converge to the global extremum; (2) Taking the average value of the sharpness function of multiple off-axis field of view spots as the objective function, make the secondary mirror rotate around the zero coma point of the system in two orthogonal directions, that is, rotate around the x and y directions of the vertical optical axis Zero coma point rotation, where the distance Z _cfp between the center vertex of the secondary mirror and the zero coma point satisfies the following formula (1), where L is the distance between the center vertex of the secondary mirror and the focal plane, and _m2 is the magnification of the secondary mirror, that is, the system The ratio of the focal length to the focal length of the primary mirror satisfies m ₂ =f, /f ₁ ', f' is the focal length of the telescope system, f ₁ ' is the focal length of the primary mirror, and b _s2 is the conic curve constant of the secondary mirror; iteratively solves the objective function extreme value to realize the correction of the position of the secondary mirror; ${\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; cfp</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$ 2. The method for correcting the position of the secondary mirror of the telescope based on the spot definition function according to claim 1, characterized in that: the correction method is suitable for two-mirror reflective or catadioptric telescope systems, including Cassegrain systems, Gehry Gaoli system, RC system. 3. The method for correcting the position of the secondary mirror of the telescope based on the spot definition function according to claim 1, characterized in that: in the steps (1) and (2), the on-axis field of view and the off-axis field of view are respectively The sharpness function of the field spot is used as the objective function, and the extreme value of the objective function is obtained through an iterative algorithm; the iterative algorithm uses a random parallel optimization algorithm to obtain the extreme value of the objective function: including hill climbing method and stochastic parallel gradient descent algorithm (SPGD- Stochastic Parallel Gradient Descent), simulated annealing, genetics, pattern extraction, etc., to achieve the adjustment of the relative position of the secondary mirror; The stochastic parallel gradient descent algorithm is that the objective function J=J(U), where U=(u ₁ ,u ₂ ,…u _n ), represents a total of n variables; since the objective function changes fastest along the gradient direction, the The algorithm uses the product of the variation δJ of the performance index in the following formula (2) and the random disturbance {δu _j } that satisfies the Bernoulli distribution to estimate the gradient of the k-th iteration; after the k-th iteration, the k+1th control The parameters are as in formula (3), r is a fixed parameter; after a certain number of iterations J converges to a stable value, the search for the extreme value of the objective function is completed; δJ＝J(u ₁ +δu ₁ ,...,u _n +δu _n )-J(u ₁ ,...,u _n ) (2) ${\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \}</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; \&delta;J\&delta;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$ 4. the method for correcting the position of the secondary mirror of the telescope based on the spot definition function according to claim 3 is characterized in that: for the correction of the field of view on the axis, the displacement despace along the optical axis direction of the secondary mirror and the translation of the vertical optical axis And rotate decenterx, decentery, tiltx, tilty as the correction variables, and the sharpness function J of the far-field spot as the objective function, perform iterative convergence analysis, and correct the field of view on the axis; after the correction is completed, the central axis of the secondary mirror and the central axis of the primary mirror Intersect at the zero coma point, and the offset relationship satisfies the formula (4), where the units of decenterx and decentery are meters, and the units of tiltx and tilty are degrees; Z _cfp is the distance between the center vertex of the secondary mirror and the zero coma point, decenterx, decentery They are the offsets in the x and y directions, respectively, and tiltx and tilty are the rotations around the x and y axes respectively; ${\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; cfp</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; decenterx</span>}}{\mathrm{<span\; class="notranslate">\; tilty</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; decenter</span>}}{\mathrm{<span\; class="notranslate">\; tiltx</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

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