CN102778210B - Aspheric surface absolute detection method based on computer generated hologram - Google Patents

Abstract

An absolute detection method of aspheric surface based on computational holography, including interferometer, standard lens, CGH ₁ , CGH ₂ , tested aspheric mirror and reference spherical mirror; first, CGH ₁ is used to detect the surface shape of the tested aspheric mirror, and the detection results include the system Error, error of CGH ₁ and error of the measured aspheric mirror; then use CGH ₂ to detect the surface shape of the tested aspheric mirror, the test results include system error, error of CGH ₂ and error of the measured aspheric mirror; then use CGH ₁ and CGH ₂ acts at the same time to detect the surface shape of the reference spherical mirror. The detection results include system error, CGH ₁ error, CGH ₂ error, and reference spherical mirror error; among them, the system error and reference spherical mirror error can be calibrated in advance, so through this Through three tests, the errors of CGH ₁ and CGH ₂ can be obtained respectively, and then the absolute test result of the tested aspheric surface can be obtained; this method is ingeniously designed and simple in structure, and provides an effective method for the absolute testing of aspheric surfaces. It has great engineering application value.

Description

An Absolute Detection Method of Aspheric Surface Based on Computational Holography

technical field

The invention belongs to the technical field of optical measurement, and in particular relates to an absolute detection method of an aspheric surface.

Background technique

With the continuous development of modern optics, the application of aspheric surface is more and more extensive. In the detection of aspheric surfaces, the method of relative measurement is mostly used, that is, the measurement results include system errors and errors of the reference surface. If you want to obtain more accurate surface shape information of the aspheric surface, you need to use absolute detection. Absolute detection of aspheric surface, that is, to remove the influence of system error and reference surface error on the measurement result by a certain method, so that the measurement result only includes the error information of the aspheric surface.

At present, there are relatively few methods for absolute detection of aspheric surfaces, and the measurement steps and data processing are relatively complicated. Among them, M. Beyerlein et al. (Mathias Beyerlein, Norbert Lindlein, Johannes Schwider. "Dual-wave-front computer-generated holograms for quasi-absolute testing of aspherics", Applied Optics, 2000, 41(13) :2440-2447) extended the three-position absolute measurement method of the spherical surface to the absolute measurement of the aspheric surface, and proposed a dual-CGH quasi-absolute measurement method for the aspheric surface, that is, the system error is calibrated by the three-step method, and the The error of Dual-CGH on the spherical surface is directly applied to its error on the aspheric surface, and then the aspheric surface is absolutely detected. However, since the spatial frequencies of the two wavefronts are different, there are certain errors in this direct application. In addition, S. Reichelt et al. (Stephan Reichelt, Christor Pruss, Hans J. Tixiani. "Absolute interferometeric test of aspheres by use of twin computer generated holograms", Applied Optics, 2003, 42(22): 4468-4479) from the University of Stuttgart in Germany proposed The current Twin-CGH considers the transition from the error of CGH to spherical surface to the error of CGH to aspheric surface. Use the five-step method to first obtain the error of Twin-CGH for the spherical surface, and then pass the calculation to the error of Twin-CGH for the aspheric surface. The systematic error obtained in this way is more accurate. However, this method requires a total of seven steps of measurement, the steps are cumbersome, and the alignment error of each measurement has a great influence on it.

Contents of the invention

The purpose of the present invention is to provide an absolute detection method of the aspheric surface to realize the absolute detection of the aspheric surface.

The technical solution of the present invention is: an aspheric surface absolute detection method based on computer generated hologram (CGH), including interferometer, standard lens, CGH ₁ , CGH ₂ , measured aspheric mirror and reference spherical mirror, characterized in that first Use CGH ₁ to detect the surface shape of the tested aspheric mirror, and the test results include systematic errors, errors of CGH ₁ and errors of the tested aspheric mirror; then use CGH ₂ to detect the surface shape of the tested aspheric mirror, and the test results include systematic errors, CGH ₂ error and the error of the measured aspheric mirror; then use CGH ₁ and CGH ₂ to act simultaneously to detect the surface shape of the reference spherical mirror, and the detection results include system error, CGH ₁ error, CGH ₂ error, and reference spherical mirror error; among them , the system error and the error of the reference spherical mirror can be calibrated in advance, so through these three tests, the errors of CGH ₁ and CGH ₂ can be calculated respectively, and then the absolute test results of the measured aspheric surface can be obtained.

The concrete realization steps of the present invention are as follows:

The first step is to place the CGH ₁ (3) at the front focus of the interferometer (1), and use the CGH ₁ (3) to detect the aspheric mirror (5) under test. The measurement result W ₁ includes the system error W _s , CGH ₁ error and the error W _Aspheric of the measured aspheric mirror,

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; Aspheric</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

The second step is to place CGH ₂ (4) at the post-focus position of the interferometer (1), and use CGH ₂ (4) to detect the aspheric mirror (5) under test. The measurement result W ₂ includes the system error W _s , CGH ₂ error and the error W _Aspheric of the measured aspheric mirror, namely

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; Aspheric</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

The third step is to place CGH ₁ (3) at the front focus position of the interferometer, and at the same time place CGH ₂ (4) at the back focus position of the interferometer, and use CGH ₁ (3) and CGH ₂ (4) to align the reference spherical mirror (6) Carry out testing, and the measurement result W ₃ includes the system error and the error of CGH ₁ CGH ₂ error and the error W _Sphere of the reference spherical mirror,

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; sphere</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

In the fourth step, in the above three formulas, the system error W _s and the error W _Sphere of the reference spherical mirror can be calibrated in advance, and they are known quantities. From formula (1) to formula (2), we can get

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

Transformed by formula (3) to get

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; sphere</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Through formulas (4) and (5), the error of CGH ₁ is calculated respectively and CGH ₂ error

In the fifth step, the calculated Substitute into formula (1) or put Substituting into formula (2), the error W _Aspheric ' of the measured aspheric mirror (5) is obtained, which is the absolute detection result of the aspheric surface.

Compared with the prior art, the present invention has the following advantages:

(1) In the system using CGH to detect aspheric surfaces in the present invention, the system error and CGH error are calibrated separately, and then absolute detection is performed on the measured aspheric mirror, so the CGH error can be accurately calibrated.

(2) The absolute detection method of the present invention has an ingenious conception and a simple structure, provides an effective method for the absolute detection of aspheric surfaces, and has great engineering application value.

Description of drawings

Figure 1 is a schematic diagram of placing the CGH ₁ at the front-focus position of the interferometer to detect the aspheric mirror under test;

Figure 2 is a schematic diagram of placing the CGH ₂ at the post-focus position of the interferometer to detect the aspheric mirror under test;

Fig. 3 is a schematic diagram of testing the reference spherical mirror by placing CGH ₁ at the front-focus position of the interferometer and CGH ₂ at the back-focus position of the interferometer.

In each figure, 1. Interferometer, 2. Standard lens, 3. CGH ₁ , 4. CGH ₂ , 5. Measured aspheric mirror, 6. Reference spherical mirror.

Detailed ways

1-3 are schematic diagrams of the three measurements required in the present invention. In Figure 1 and Figure 2, the light beam from the interferometer 1 passes through the standard lens 2 and is incident on the CGH ₁ 3 or CGH ₂ 4, and the CGH changes the spherical wave front into an aspheric wave front to realize the detection of the aspheric mirror 5 under test measurement; in Figure 3, the light beam from the interferometer 1 passes through the standard lens 2, and is incident on the CGH ₁ 3, and the CGH ₁ changes the spherical wavefront into an aspherical wavefront, and then incident on the CGH ₂ 4, which transforms the aspheric The wavefront becomes a spherical wavefront to realize the measurement of the reference spherical mirror 6 . Combined with examples, the measurement process is as follows:

The steps of the absolute measurement method of an aspheric surface are as follows:

The first step, as shown in Figure 1, put the CGH ₁ 3 at the front-focus position of the interferometer 1, use the CGH ₁ 3 to detect the aspheric mirror 5 under test, first adjust the alignment of the interferometer 1 and the CGH ₁ 3, Then adjust the alignment between CGH ₁ 3 and the measured aspheric surface 5, and finally complete the measurement of the measured aspheric surface 5, the measurement result W ₁ includes the system error W _s and the error of CGH ₁ and the error W _Aspheric of the measured aspheric mirror,

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; Aspheric</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

In the second step, as shown in Figure 2, put the CGH ₂ 4 at the post-focus position of the interferometer 1, use the CGH ₂ 4 to detect the aspheric mirror 5 to be tested, first adjust the alignment of the interferometer 1 and the CGH ₂ 4, Then adjust the alignment between the CGH ₂ 4 and the measured aspheric surface 5, and finally complete the measurement of the measured aspheric surface 5, and the measurement result W ₂ includes the system error W _s and the error of the CGH ₂ and the error W _Aspheric of the measured aspheric mirror, namely

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; Aspheric</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

In the third step, as shown in Figure 3, put CGH ₁ 3 at the front focus position of the interferometer, and at the same time put CGH ₂ 4 at the back focus position of the interferometer, and use CGH ₁ 3 and CGH ₂ 4 to carry out the reference spherical mirror 6 For detection, first place CGH ₂ 4 at the post-focus position of the interferometer, and adjust the alignment of interferometer 1 and CGH ₂ 4; then place CGH ₁ 3 at the front-focus position of the interferometer, and adjust interferometer 1 and CGH ₁ 3 alignment; then adjust the alignment of the reference spherical mirror 6 and the previous system, and finally complete the measurement of the reference spherical mirror 6, and the measurement result W ₃ includes the error of the system error W _s and CGH ₁ CGH ₂ error and the error W _Sphere of the reference spherical mirror,

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; sphere</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

In the fourth step, in the above three formulas, the system error W _s and the error W _Sphere of the reference spherical mirror can be calibrated in advance, and they are known quantities. From formula (1) to formula (2), we can get

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

Transformed by formula (3) to get

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; sphere</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Through formulas (4) and (5), the error of CGH ₁ is calculated respectively and CGH ₂ error

In the fifth step, the calculated Substitute into formula (1) or put Substituting into formula (2), the error W _Aspheric of the measured aspheric mirror 5 is obtained, which is the absolute detection result of the aspheric surface.

The parts not described in detail in the description of the present invention belong to the well-known technology in the art.

The above is only a specific implementation mode in the present invention, but the protection scope of the present invention is not limited thereto, any partial modification or replacement within the technical scope disclosed in the present invention by anyone familiar with the technology shall cover within the scope of the present invention.

Claims (1)

1. A kind of aspherical surface absolute detection method based on computational holography, the device that described method adopts comprises: interferometer (1), standard lens (2), CGH ₁ (3), CGH ₂ (4), measured aspheric mirror (5) and reference spherical mirror (6), it is characterized in that: concrete realization steps are as follows: The first step is to place the CGH ₁ (3) at the front focus position of the interferometer (1), and use the CGH ₁ (3) to detect the measured aspheric mirror (5). The measurement result W ₁ includes the systematic error W _s , CGH ₁ error and the error W _Aspheric of the measured aspheric mirror, ${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; Aspheric</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ The second step is to place CGH ₂ (4) at the post-focus position of the interferometer, and use CGH ₂ (4) to detect the measured aspheric mirror (5), the measurement result W ₂ includes the system error W _s and the error of CGH ₂ and the error W _Aspheric of the measured aspheric mirror, namely ${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{{\mathrm{<span\; class="notranslate">\; CGH</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; Aspheric</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ The third step is to place CGH ₁ (3) at the front focus position of the interferometer, and at the same time place CGH ₂ (4) at the back focus position of the interferometer, and use CGH ₁ (3) and CGH ₂ (4) to align the reference spherical mirror (6) detect, the measurement result W ₃ includes the error of the system error, CGH ₁ CGH ₂ error and the error W _Sphere of the reference spherical mirror, ${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; CG</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; CG</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; sphere</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$ In the fourth step, in the above three formulas, the system error W _s and the error W _Sphere of the reference spherical mirror can be calibrated in advance and are known quantities. From formula (1) to formula (2), we can get ${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; CG</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; CG</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ Transformed by formula (3) to get ${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; sphere</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; CG</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; CG</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ Through formulas (4) and (5), the error of CGH ₁ is calculated respectively and CGH ₂ error In the fifth step, the calculated Substituting into formula (1) or will Substitute into formula (2) to obtain the error W _Aspheric of the measured aspheric mirror (5), which is the absolute detection result of the aspheric surface.