CN112985776A - Method for detecting optical parameters of optical system with any wavelength - Google Patents

Method for detecting optical parameters of optical system with any wavelength Download PDF

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CN112985776A
CN112985776A CN202110186511.7A CN202110186511A CN112985776A CN 112985776 A CN112985776 A CN 112985776A CN 202110186511 A CN202110186511 A CN 202110186511A CN 112985776 A CN112985776 A CN 112985776A
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wavelength
formula
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optical system
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CN112985776B (en
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张齐元
王芳
韩森
王浩宇
朱大勇
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SUZHOU H&L INSTRUMENTS LLC
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    • G—PHYSICS
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    • G01M11/02—Testing optical properties
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
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    • G01M11/0242—Testing optical properties by measuring geometrical properties or aberrations
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Abstract

The invention provides a method for detecting optical parameters of any wavelength of an optical system, which belongs to the technical field of optical detection and is used for detecting optical parameters of multiple wavelengths of the optical system, and is characterized by comprising the following steps: step S1, detecting the optical system by optical instrument to obtain the wavelength λ of the optical system at R kinds of wavelength1～λrReference optical parameter f (λ)1)、f(λ2)、······f(λr) (ii) a Step S2, substituting the optical parameters obtained in step S1 into the formula:
Figure DDA0002941986520000011
where R is 3 or more when D is 0 (i.e., only the first 3 items), R is 4 or more when D is not equal to 0, and n is max { X ≧1,X2Calculate the values of parameters A, B, C and D; step S3, substituting A, B, C and D obtained by calculation into formula
Figure DDA0002941986520000012
In, the calculated wavelength is λmThe optical parameters of the optical system of (1).

Description

Method for detecting optical parameters of optical system with any wavelength
Technical Field
The invention belongs to the technical field of optical detection, and particularly relates to a method for detecting optical parameters of any wavelength of an optical system.
Background
The optical parameters of the transmission optical system at any wavelength in a certain waveband range can be predicted by using optical parameters of several known wavelengths through the functional relationship between the optical parameters and the wavelengths, such as patents 201710615484.4, 201810003808.3 and 201910728724.0, which propose to calculate parameters of the transmission wavefront and the focal length of any wavelength of the optical system.
Taking the transmitted wavefront of an optical system as an example, multi-wavelength transmitted wavefront detection is a detection method for predicting the transmitted wavefront of any wavelength in a certain waveband range by using a few transmitted wavefronts with specific wavelengths through the functional relationship between the wavefront and the wavelength. Since Zernike polynomials have orthogonality within the unit circle and some correlation with seidel aberrations, the transmitted wavefront can be described using a combination of Zernike polynomials.
The transmission system wavefront Zernike coefficients as a function of wavelength can be expressed by apochromatic property formula (ACF formula) which can express a monotonous curve and a curve with one inflection point, and Conrady formula which can express a monotonous curve, a curve with one inflection point, and a curve with at most two inflection points. Wherein most Zernike coefficient-wavelength curves in the monochromatic system are monotonous curves, and in the achromatic system, part of Zernike coefficient-wavelength curves are monotonous curves, and the rest are curves with an inflection point. Therefore, many Zernike coefficient curves in an achromatic system can be predicted by using Conrady formula.
In practical application, the Zernike coefficient-wavelength curves of the monochromatic system and the partial achromatic system can be predicted by Conrady formula. The Conrady formula can be calculated in two ways, the first way is to calculate a curve by solving, and since the Conrady formula has three unknowns a, B, and C, at least three sets of data are required to determine the formula, so as to draw a Zernike coefficient-wavelength curve (in patents 201710615484.4, 201810003808.3, and 201910728724.0, the curve is predicted by solving). The other is to calculate the curve by fitting, and fit the Conrady formula curve by a mathematical tool (e.g., Matlab) using several data (the number of fitting coefficients is greater than or equal to) according to a certain algorithm, such as the most commonly used least square method. Theoretically, the Conrady formula can calculate the Zernike coefficient-wavelength curve with high precision by solving. However, in actual measurement, due to measurement errors, sometimes the curve obtained by directly calculating the curve by using a solution method is greatly changed, but the curve calculated by using a method of fitting the Conrady formula can only obtain a monotonic curve, so that the fitting method is only applicable to most monochromatic systems (see the literature, estimated transmitted waves from in a broad band and wide band on Zernike coeff devices [ J ]. Journal of Optics,2019,21(9):095601), and the Zernike coefficient-wavelength curve of the chromatic aberration system cannot be effectively predicted. Due to the measurement error inevitably existing in the actual measurement, the prediction effect consistent with the theoretical calculation cannot be realized in many cases, and the calculated curve precision is poor.
Disclosure of Invention
In order to solve the above problems, the present invention provides a method for detecting optical parameters of any wavelength of an optical system, which adopts the following technical scheme:
the invention provides a method for detecting optical parameters of any wavelength of an optical system, which is used for detecting the optical parameters of multiple wavelengths of the optical system and is characterized by comprising the following steps:
step S1, detecting the optical system by optical instrument to obtain the wavelength λ of the optical system at R kinds of wavelength1～λrReference optical parameter f (λ)1)、f(λ2)、……f(λr) (ii) a Step S2, substituting the optical parameters obtained in step S1 into the formula:
Figure BDA0002941986500000031
wherein R is not less than 3 when D is 0, R is not less than 4 when D is not equal to 0, and n is not less than max { X ≧1，X2Calculate the values of parameters A, B, C and D; step S3, willCalculated A, B, C and D are substituted into the formula:
Figure BDA0002941986500000032
in, the calculated wavelength is λmThe optical parameters of the optical system of (1).
The method for detecting optical parameters of any wavelength of an optical system provided by the invention can also have the following characteristics, wherein the step S3 comprises the following steps: step S3-1, substituting A, B, C and D obtained by calculation into the formula:
Figure BDA0002941986500000033
(2) in, the calculated wavelength is λmIs measured with respect to a reference optical parameter f (λ) of the optical systemm) Step S3-2, according to the reference optical parameter f (λ) in step S3-1m) Fitting wavelength of lambdamThe optical parameters of the optical system of (1).
The invention provides a method for detecting optical parameters of any wavelength of an optical system, which is used for detecting optical parameters of multiple wavelengths of the optical system and detecting the optical parameters of the optical system and is characterized by comprising the following steps: step T1, respectively obtaining the λ at R wavelengths by measuring with optical instrument1～λPReference optical parameter f (λ)1)、f(λ2)、……f(λp) (ii) a In step T2, R kinds in step S1 are respectively λ1～λPIs selected from the wavelengths of (A) and (B) and Q is respectively lambdah～λgReference optical parameter f (λ) of the wavelength ofh)、……f(λg) Fitting according to a polynomial formula to obtain fitting polynomial coefficients; step T3, the fitting polynomial coefficient in the step S2 is brought into a polynomial formula, and sampling is carried out to obtain the wavelength range within lambdah～λgOf S kinds has a wavelength of λa～λbReference optical parameter f (λ)a)、……f(λb) (ii) a Step T4, converting the wavelength λ in step S11～λPMiddle divided by lambdah～λgReference optical parameters of wavelengths other than the wavelength and the wavelength λ in step S3a～λbIs substituted into the formula
Figure BDA0002941986500000041
When D is equal to 0, R is equal to or more than 3, when D is equal to 0, R is equal to or more than 4, and n is equal to or more than max { X1，X2Calculate the values of parameters A, B, C and D; step T5, substituting A, B, C and D into the formula:
Figure BDA0002941986500000042
in, the calculated wavelength is λmThe optical parameters of the optical system of (1).
The method for detecting the optical parameter of the optical system with any wavelength provided by the invention can also have the following characteristics, wherein the step T5 comprises the following steps: step T5-1, substituting A, B, C and D obtained by calculation into the formula:
Figure BDA0002941986500000043
(2) in, the calculated wavelength is λmIs measured with respect to a reference optical parameter f (λ) of the optical systemm) Step T5-2, according to the reference optical parameter f (λ) in step S3-1m) Calculating the wavelength as lambdamThe optical parameters of the optical system of (1).
The method for detecting the optical parameters of the optical system with any wavelength provided by the invention can also have the characteristics that the optical parameters are transmission wavefront, the reference optical parameters are Zernike coefficients, namely f (lambda) is Z1(λ)、Z2(λ)、……Zk(λ), formula (1) is:
Figure BDA0002941986500000051
the formula (2) is:
Figure BDA0002941986500000052
in the formula (3) and the formula (4), i is an integer ranging from 1 to k, and k is not more than 37.
The method for detecting the optical parameters of the optical system with any wavelength provided by the invention can also have the characteristic that Di＝0，X1＝1，X23.5, i.e. equation (1) becomes: zi(λ)·λn＝Ai·λn+Bi·λ(n-1)+Ci·λ(n-3.5)(5) The formula (2) becomes:
Figure BDA0002941986500000053
Figure BDA0002941986500000054
in the formula (5), n is not less than 3.5, in the formula (5) and the formula (6), i is an integer ranging from 1 to k, and k is not more than 37.
The method for detecting the optical parameters of the optical system with any wavelength can also be characterized in that the optical system is a single-wavelength system, an achromatic system or an apochromatic system.
The method for detecting the optical parameters of the optical system with any wavelength can also have the characteristics that the optical parameters are a focal length, a back focal length, a wave front discrete point, field curvature, distortion or aberration coefficients.
Action and Effect of the invention
The method for detecting the optical parameters of the optical system with any wavelength is used for detecting the optical parameters of the optical system and comprises the steps of firstly detecting the optical system by an instrument to obtain R types of optical parameters with the wavelengths respectively lambda1～λrWith reference to an optical parameter, and then1～λrSubstituting the reference optical parameter into the converted ACF formula
Figure BDA0002941986500000055
After the coefficient of the formula is obtained, the optical coefficient-wavelength curve can be obtained by taking the original formula (ACF formula), thereby predicting the optical parameters of any wavelength in the fixed wavelength range of the optical system. Compared with the conventional fitting method, the method only can accurately fit the monotonous curve with smaller amplitude, and the method can accurately fit the monotonous curve, the curve with one inflection point and the curve with two inflection points, so that the prediction precision is improved. In particular, in actual detection, there is inevitably a measurement errorIn the poor condition, the more accurate prediction of the optical parameters is carried out, and the method has wider application range compared with other methods, so the method of the invention has important significance for the practicability of the multi-wavelength detection technology.
Drawings
FIG. 1 is a Zernike coefficient versus wavelength plot of Z5 fitted using Conrady's equation for an embodiment of the present invention;
FIG. 2 is a Zernike coefficient versus wavelength plot of Z6 fitted using Conrady's equation for an embodiment of the present invention;
FIG. 3 is a Zernike coefficient versus wavelength plot of Z7 fitted using Conrady's equation for an embodiment of the present invention;
FIG. 4 is a Zernike coefficient versus wavelength plot of Z8 fitted using Conrady's equation for an embodiment of the present invention;
FIG. 5 is a Zernike coefficient versus wavelength plot of Z9 fitted using Conrady's equation for an embodiment of the present invention;
FIG. 6 is a Zernike coefficient versus wavelength plot for Z5 predicted using a conversion formula according to an embodiment of the present invention;
FIG. 7 is a Zernike coefficient versus wavelength plot for Z6 predicted using a conversion formula according to an embodiment of the present invention;
FIG. 8 is a Zernike coefficient versus wavelength plot for Z7 predicted using a conversion formula according to an embodiment of the present invention;
FIG. 9 is a Zernike coefficient versus wavelength plot for Z8 predicted using a conversion formula according to an embodiment of the present invention;
FIG. 10 is a Zernike coefficient versus wavelength plot for Z9 predicted using a conversion formula according to an embodiment of the present invention;
FIG. 11 is a Zernike coefficient versus wavelength plot for predicting Z5 using visible light data for an embodiment of the present invention;
FIG. 12 is a Zernike coefficient versus wavelength plot for predicting Z6 using visible light data for an embodiment of the present invention;
FIG. 13 is a Zernike coefficient versus wavelength graph for predicting Z7 using visible light data for an embodiment of the present invention;
FIG. 14 is a Zernike coefficient versus wavelength plot for predicting Z8 using visible light data for an embodiment of the present invention;
FIG. 15 is a Zernike coefficient versus wavelength plot for predicting Z9 using visible light data for an embodiment of the present invention;
FIG. 16 is a Conrady equation fitting graph according to the second embodiment of the present invention;
FIG. 17 is a graph of a transformational fit of a second embodiment of the present invention;
FIG. 18 is a graph of polynomial interpolation and conversion polynomial fitting according to the second embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made with reference to the accompanying drawings.
< example one >
The present invention is a method for detecting optical parameters of an optical system at an arbitrary wavelength, and the first embodiment takes the detection of a transmitted wavefront at an arbitrary wavelength of an optical system as an example.
The embodiment provides a method for detecting any wavelength transmission wavefront of an optical system, which is used for predicting any wavelength transmission wavefront in a certain waveband range, and comprises the following steps:
step S1, using the laser interferometer to detect the optical system respectively to obtain the λ of the optical system at R kinds of wavelength respectively1～λrZ (i.e. Zernike polynomial coefficients)1(λ1)、Z2(λ1)、······Zk(λ1) -. and Z1(λr)、Z2(λr)、······Zk(λr)。
Wherein λ is1～λrArranged from small to large in sequence. In this embodiment, the formula used in this embodiment is the Conrady formula (i.e. formula (4)) after transformation, and the predicted optical parameter is the transmitted wavefront, then λ is greater than or equal to 400 ≦ λ1，λr1700, in other embodiments, 300 λ is 300 λ if the formula used is the transformed ACF formula (i.e., formula (3)) and the predicted optical parameter is the transmitted wavefront1，λr2500, in other embodiments, the coefficient range of the wavelength term in the ACF equation (or Conrady equation) is changed accordingly if used to predict other optical parametersAnd (refer to 201710615484.4, 201810003808.3, 201910728724.0, etc.).
In this embodiment, the laser interferometer used is a fizeau interferometer, and the Zernike polynomial is a Fringe Zernike polynomial.
Step S2, substituting the Zernike coefficients obtained in step S1 into the formula:
Figure BDA0002941986500000091
in the formula (3), i is an integer ranging from 1 to k, and k is less than or equal to 37; λ is an independent variable, and in this step is λ in step S11～λr；X1、X2、X3Is a specific parameter; a. thei、Bi、Ci、DiIs the parameter to be solved.
When Di is equal to 0 (namely only the first 3 items), R is more than or equal to 3, when Di is equal to 0, R is more than or equal to 4, and n is more than or equal to max { X ≧1，X2}, calculating the parameter Ai、Bi、CiAnd DiThe value of (c).
In this embodiment, the formula (3) is a formula corresponding to the Conrady formula after conversion, specifically, each term in the Conrady formula is multiplied by λnI.e. in formula (1), Di＝0，X1＝1，X2Equation (5) of this example was obtained as 3.5:
Zi(λ)·λn＝Ai·λn+Bi·λ(n-1)+Ci·λ(n-3.5) (5)
wherein n is more than or equal to 3.5.
In other embodiments, the ACF equation (4)) may be used, i.e., the apochromatic property equation is transformed (equation (3) may be obtained), i.e., each term in the ACF equation is multiplied by λnObtaining a transformation formula and then fitting.
Step S3, calculating the obtained Ai、Bi、CiAnd DiSubstituting into formula
Figure BDA0002941986500000092
In, the calculated wavelength is λmZernike coefficient Z of the optical system of1(λm)、Z2(λm)、……Zk(λm). In this embodiment, because DiWhen 0, the formula (4) becomes
Figure BDA0002941986500000093
Step S4, according to Zernike coefficient Z in step S31(λm)、Z2(λm)、……Zk(λm) Fitting wavelength of lambdamThe transmitted wavefront of the optical system of (1). In this embodiment, the reference optical parameter Zernike coefficient is used to fit the transmitted wavefront of the optical system, and in other embodiments, if the reference optical parameter Zernike coefficient is used to predict other optical parameters, the reference optical parameter is the optical parameter to be predicted, that is, the optical parameter of the optical system can be directly obtained in step S3.
Specifically, in this embodiment, 6 fizeau interferometers are used to measure the transmitted wavefront, the measured optical system is a double cemented lens, and the measured wavefront is measured in an inclined state, the wavelengths of the laser light source are 532nm,561nm,632.8nm,671nm,721nm and 1064nm, respectively, and the measurement results are shown in table 1:
TABLE 1Z 5-Z9 Zernike coefficient measurements
Figure BDA0002941986500000101
The measurement results in Table 1 are the measured average values of Zernike coefficients Z5-Z9 of the doublet at 6 wavelengths (Z1-Z4 represent translation, tilt and defocus, and do not generally participate in the fitting).
The maximum value of the error of the measured values is estimated from the multiple measurements, and assuming that the true value of the measurement is between the maximum value and the minimum value, the maximum measurement error of the Zernike coefficients can be expressed as:
Figure BDA0002941986500000102
the maximum measurement error of each Zernike coefficient at 6 wavelengths is obtained according to equation (7), as shown in table 2:
TABLE 2Z 5-Z9 Zernike coefficients maximum measurement error
Figure BDA0002941986500000111
As shown in table 2, since the values of Z7 and Z8 are small by themselves, the measurement errors of Z7 and Z8 are relatively large.
FIG. 1 is a plot of Zernike coefficients versus wavelength for Z5 using Conrady, FIG. 2 is a plot of Zernike coefficients versus wavelength for Z6 using Conrady, FIG. 3 is a plot of Zernike coefficients versus wavelength for Z7 using Conrady, FIG. 4 is a plot of Zernike coefficients versus wavelength for Z8 using Conrady, and FIG. 5 is a plot of Zernike coefficients versus wavelength for Z9 using Conrady.
In the figure, 1a, 2a, 3a, 4a and 5a are Zernike coefficient-wavelength curves obtained by Conrady formula fitting using 3 measurement data points (532nm,632.8nm and 721 nm); 1b, 2b, 3b, 4b, 5b are Zernike coefficient-wavelength curves obtained by Conrady formula fitting using 3 measurement data points (532nm,721nm,1064 nm); 1c, 2c, 3c, 4c, 5c are Zernike coefficient-wavelength curves obtained by Conrady formula fitting using 4 measurement data points (532nm,632.8nm,721nm,1064 nm); 1d, 2d, 3d, 4d, 5d are Zernike coefficient-wavelength curves obtained by Conrady formula fitting using 5 measurement data points (532nm,561nm,632.8nm,671nm,721 nm); the Zernike coefficient-wavelength curves obtained by Conrady formula fitting using 6 measurement data points (532nm,561nm,632.8nm,671nm,721nm, 1064nm) are first directly fitted with the Conrady formula to obtain Zernike coefficient-wavelength curves, wherein the Conrady formula fitting is performed using 3 measurement data points (532nm,721nm,1064nm), 4 measurement data points, 5 measurement data points, and 6 measurement data points in table 1, respectively, and the results are shown in fig. 1 to 5.
As shown in fig. 1 to 2, the Zernike coefficient-wavelength curves of Z5 and Z6, which were fitted using Conrady's formula, were substantially the same, and several other predicted Zernike coefficients were substantially consistent with the measurement results. However, as shown in fig. 3 to 5, since Zernike coefficient-wavelength curves of Z7, Z8, and Z9 fitted by using the Conrady formula are very different, and the measured data points are not completely matched with the fitted curves (are also very different from the curve shape simulated by software), the Conrady formula fitting cannot effectively calculate and fit all the Zernike coefficient-wavelength curves, and a better transmitted wavefront prediction result cannot be obtained.
Fig. 6 is a plot of Zernike coefficients versus wavelength for Z5 predicted using the conversion formula, fig. 7 is a plot of Zernike coefficients versus wavelength for Z6 predicted using the conversion formula, fig. 8 is a plot of Zernike coefficients versus wavelength for Z7 predicted using the conversion formula, fig. 9 is a plot of Zernike coefficients versus wavelength for Z8 predicted using the conversion formula, and fig. 10 is a plot of Zernike coefficients versus wavelength for Z9 predicted using the conversion formula, according to an embodiment of the present invention.
Then, the Zernike coefficient-wavelength curves of Zernike coefficients Z5 to Z9 were calculated and predicted by the method in the present example, that is, the above-described steps S1 to S4, in which n in formula (3) is 4.5, and 6 measurement data points (532nm,561nm,632.8nm,671nm,721nm, and 1064nm) were used, and the Zernike coefficient-wavelength curves of Zernike coefficients Z5 to Z9 were obtained as shown in fig. 6 to fig. 10.
As shown in fig. 6 to 10, the Zernike coefficients Z5, Z6, Z7, Z8, and Z9 all have good fitting effects on Zernike coefficient-wavelength curves, and the measured data points are substantially matched with the curves (and are very similar to the curve shape simulated by software, the curve simulated theoretically can only reflect the variation trend of the curve, and the error of the actual measurement system is unknown, so that the simulation curve cannot be directly compared with the measured curve).
Fig. 11 is a plot of Zernike coefficients versus wavelength for Z5 predicted using visible light (532nm,561nm,632.8nm,671nm, and 721nm) data, fig. 12 is a plot of Zernike coefficients versus wavelength for Z6 predicted using visible light data, fig. 13 is a plot of Zernike coefficients versus wavelength for Z7 predicted using visible light data, fig. 14 is a plot of Zernike coefficients versus wavelength for Z8 predicted using visible light data, and fig. 15 is a plot of Zernike coefficients versus wavelength for Z9 predicted using visible light data, according to an embodiment of the present invention.
Finally, because the distance of 1064nm is longer than that of other wavelengths, the measurement data points (532nm,561nm,632.8nm,671nm, and 721nm) of visible light can be used to perform fitting by the method of the present embodiment, and then the 1064nm wavelength measurement data is used to perform verification, thereby further verifying that the Zernike coefficient-wavelength curve of the system under test conforms to the Conrady formula and the validity of (new calculation method) multi-wavelength wavefront detection technology prediction, and the obtained results are shown in fig. 11 to 15.
As shown in fig. 11 to 15, the Zernike coefficients Z5, Z6, Z7 and Z9 have a good fitting effect on the wavelength curve of the Zernike coefficients, and the predicted Zernike coefficients at 1064nm are basically consistent with the calculated Zernike coefficients at 1064nm, which proves that the Zernike coefficients at far infrared wavelengths can be predicted well by using the visible light measurement data in the limited-width waveband range. Whereas Z8 fits poorly compared to 6 measured data points, demonstrating that selecting the location of the measured data points is very important to the accuracy of the fit curve in the presence of large measurement errors (selecting the entire data fit curve fits better than using only visible light data). An important condition for a more accurate fit of the curve is therefore to try to improve the accuracy of the measurement at each wavelength.
Examples effects and effects
The invention relates to a method for detecting optical parameters of an optical system at any wavelength, which is used for detecting the optical parameters of the optical system. Firstly, an optical instrument is adopted to detect an optical system to obtain R kinds of wavelengths respectively with lambda1～λrReference light ofLearning the parameters, then dividing1～λrSubstituting the reference optical parameter data into a new formula converted by the ACF formula
Figure BDA0002941986500000141
In the present embodiment, a Conrady formula, which is a specific example of an ACF formula, is used to perform transformation to obtain coefficients of the formula, and then the coefficients are brought into the original ACF formula to obtain an optical parameter-wavelength curve, so as to predict optical parameters of an optical system with any wavelength within a certain band range. Compared with the conventional fitting method, the fitting method only can fit the monotonous curve with smaller amplitude, and the fitting method can simultaneously fit the monotonous curve, the curve with one inflection point and the curve with two inflection points, so that the prediction precision is improved. Especially, in actual detection, under the condition that a certain measurement error exists, the method can predict the optical parameters more accurately, and the method has wider application range, so that the method for predicting the curve has important significance for the practicability of the multi-wavelength optical parameter detection technology.
< example two >
The present invention is a method for detecting optical parameters of an optical system at an arbitrary wavelength, and the second embodiment takes the detection of a transmitted wavefront at an arbitrary wavelength of an optical system as an example for explanation.
The second embodiment provides a method for detecting any wavelength transmission wavefront of an optical system, which is used for predicting any wavelength transmission wavefront in a certain waveband range, and includes the following steps:
step T1, respectively obtaining the λ at R wavelengths by measuring with a laser interferometer1～λrZernike coefficient Z of1(λ1)、Z2(λ1)、······Zk(λ1) -. and Z1(λr)、Z2(λr)、······Zk(λr)。
In this embodiment, the laser interferometer used is a fizeau interferometer, and the Zernike polynomial is a Fringe Zernike polynomial.
Step T2, R kinds in step S1Are each lambda1～λrIs selected from the wavelengths of (A) and (B) and Q is respectively lambdah～λgZernike coefficients Z of the wavelengths of1(λh)、Z2(λh)、······Zk(λh) -. and Z1(λg)、Z2(λg)、······Zk(λg) And fitting according to a polynomial formula to obtain fitting polynomial coefficients. Wherein Q is more than or equal to 3, h is more than or equal to 1, and g is more than or equal to r.
In this embodiment, the polynomial formula is
Zi(λ)＝Ei·λ2+Fi·λ+Gi (8)
Step T3, sampling the fitting Zernike coefficients in the step S2 to obtain the wavelength range of lambdah～λgOf S kinds has a wavelength of λa～λbZernike coefficients of;
step T4, converting the wavelength λ in step S11～λrAnd the wavelength λ in step S3a～λbThe Zernike coefficients of (a) are substituted into the formula:
Figure BDA0002941986500000161
wherein i is an integer ranging from 1 to k, k is less than or equal to 37,
when D is presentiWhen the value is 0 (namely only the first 3 items), R is more than or equal to 3, when Di is not equal to 0, R is more than or equal to 4, and n is more than or equal to max { X ≧1，X2}, calculating the parameter Ai、Bi、CiAnd DiThe value of (c).
Step T5, calculating the obtained Ai、Bi、CiAnd DiBringing in
Figure BDA0002941986500000162
In, the calculated wavelength is λmZernike coefficient Z of the optical system of1(λm)、Z2(λm)、……Zk(λm)。
Step S6, according to Z1(λm)、Z2(λm)、……Zk(λm) Fitting wavelength of lambdamThe transmitted wavefront of the optical system of (1).
The difference between the second embodiment and the first embodiment is that when the R wavelengths in the first embodiment are λ respectively1～λrThe Zernike polynomials are substituted into the formula (3) for fitting, but when no more ideal curve is obtained by fitting (because the multi-wavelength detection technology needs to use as little measurement data as possible, in rare cases, because of less data points, although the new method is used for fitting, part of the curves may not obtain better results), the R wavelengths are respectively lambda1～λrThe Zernike polynomials of (A) are each lambdah～λgFitting the Zernike polynomial of the wavelength by using a polynomial formula, and sampling the formula after fitting the polynomial formula to obtain S types of lambda-wavelengtha～λbThe Zernike coefficients (the parameters and the wavelength curve of the optical system in a larger waveband can be expressed by using an ACF formula or a Conrady formula, but in a smaller waveband range, the polynomial formula has universality, see the literature, "research on the functional relationship between the Zernike coefficients and the wavelengths in the transmitted wavefront", optics report 2018,38(2):170-1～λrThe Zernike coefficients of (a) are jointly substituted into equation (3) for fitting.
Specifically, in the second embodiment, the wavelength data with the wavelengths of 532nm,561nm,632.8nm,671nm and 721nm, i.e. r is 5, λ in step S1, is selected first1＝532nm，λ2＝561nm，λ3＝632.8nm，λ4＝671nm，λ5The 721nm wavelength data was fitted using the method described above, the method of example one, and the Conrady equation to obtain data on the Zernike coefficient Z7 in the Zernike polynomial.
Fig. 16 is a Conrady equation fitting graph of the second embodiment of the present invention, and fig. 17 is a transformation equation fitting graph of the second embodiment of the present invention.
Fig. 16 is a graph of Zernike coefficients Z7 obtained by fitting only the Conrady formula, and fig. 17 is a graph of Zernike coefficients Z7 obtained by the method of example one, i.e., fitting only the conversion polynomial.
As shown in fig. 16 and 17, a curve with an inflection point cannot be obtained by fitting a curve of Zernike coefficients Z7 obtained by using only the Conrady equation and by using the transformation equation.
Fig. 18 is a graph of interpolation and fitting of a conversion polynomial in a polynomial equation according to the second embodiment of the present invention.
Next, in the second embodiment, λ is used first1＝532nm，λ2＝561nm，λ3＝632.8nm，λ4＝671nm，λ5Fitting a polynomial formula under 721nm to obtain a curve with a waveband of 532 nm-721 nm, collecting enough points from the curve, and fitting in formula (4) to obtain Ai、Bi、CiThen, the curve shown in FIG. 18 was obtained by taking in the Conrady equation. As shown in fig. 18, the curve obtained by the method of the second embodiment can be fitted to obtain a curve with an inflection point.
Examples effects and effects
The second embodiment provides a method for detecting any wavelength transmitted wavefront of the optical system, which is different from the first embodiment in that when the R wavelengths in the first step are λ respectively1～λrThe Zernike coefficients are substituted into the formula (1) for fitting, and when an ideal curve cannot be fitted, R wavelengths are respectively lambda first1～λrRespectively of Q kinds in the Zernike coefficients of (a)h～λgFitting the Zernike coefficients of the wavelengths by using a polynomial formula, and sampling the formula after fitting the polynomial formula to obtain S types of lambda-wavelengtha～λbThe Zernike coefficients of (A) and the R wavelengths λ in the step S11～λrThe Zernike coefficients of (a) are jointly substituted into equation (1) for fitting. The second embodiment of the method is applied to the method when only the transformation is usedWhen fitting is carried out, when an ideal curve cannot be fitted or the fitting effect is poor, fitting can be carried out by using a conventional polynomial formula (a good effect can be achieved by using a fitting polynomial with less data in a shorter wave band range, but the effect is poor by using a fitting polynomial with less data in a larger wave band range), more data are obtained and then are brought into a transformation formula for fitting, so that the fitting effect is further improved, and the application of the method is wider.
The above-described embodiments are merely illustrative of specific embodiments of the present invention, and the present invention is not limited to the description of the above-described embodiments.
In the above embodiments, only the transmitted wavefront of the optical system is predicted by the Zernike coefficients by using the method of the present invention, in other embodiments, the method of the present invention may also be used to predict other optical parameters, such as focal length, back focal length, wavefront discrete point, field curvature, distortion or aberration coefficients, etc., and when fitting optical parameters of any wavelength, the desired optical parameters may be directly obtained without obtaining reference optical parameters, such as Zernike coefficients, first, that is, the formula is
Figure BDA0002941986500000191
And
Figure BDA0002941986500000192
the method can be used for predicting the optical parameters f (lambda) of other arbitrary wavelengths in the optical system by only using the detected reference optical parameters f (lambda) of several wavelengths instead of the formula (3) and the formula (4) respectivelym)。
In the above embodiments, the laser interferometer is used to detect the optical parameters, and in other embodiments, other optical instruments may be used to detect different optical parameters.

Claims (8)

1. A method for detecting optical parameters of any wavelength of an optical system, which is used for detecting optical parameters of multiple wavelengths of the optical system, is characterized by comprising the following steps:
step S1, respectively aligning the optics with optical instrumentThe system detects to obtain that the wavelengths of the optical system at R are respectively lambda1～λrReference optical parameter f (λ)1)、f(λ2)、······f(λr)；
Step S2, substituting the optical parameters obtained in step S1 into the formula:
Figure FDA0002941986490000011
wherein, when D is 0, R is not less than 3,
when D is not equal to 0, R is not less than 4, n is not less than max { X ≧1,X2}，
Calculating the values of parameters A, B, C and D;
step S3, substituting A, B, C and D into the formula:
Figure FDA0002941986490000012
in, the calculated wavelength is λmOf the optical system of (a).
2. The method of detecting an optical parameter of an optical system at an arbitrary wavelength according to claim 1, wherein:
wherein, step S3 includes the following steps:
step S3-1, substituting A, B, C and D obtained by calculation into the formula:
Figure FDA0002941986490000013
in, the calculated wavelength is λmOf the optical system of (a) is determined by the reference optical parameter f (λ)m)，
Step S3-2, according to the reference optical parameter f (λ) in step S3-1m) Fitting wavelength of lambdamOf the optical system of (a).
3. A method for detecting an optical parameter of an optical system at an arbitrary wavelength, the method being used for detecting the optical parameter of the optical system, the method comprising the steps of:
step T1, respectively obtaining the λ at R wavelengths by measuring with optical instrument1～λPReference optical parameter f (λ)1)、f(λ2)、······f(λp)；
In step T2, R kinds in step S1 are respectively λ1～λPIs selected from the wavelengths of (A) and (B) and Q is respectively lambdah～λgOf the wavelength of (a) of (b)h)、······f(λg) Fitting according to a polynomial formula to obtain fitting polynomial coefficients;
step T3, the fitting polynomial coefficient in the step S2 is brought into the polynomial formula, and sampling is carried out to obtain the wavelength range in lambdah～λgOf S kinds has a wavelength of λa～λbOf said reference optical parameter f (λ)a)、······f(λb)；
Step T4, converting the wavelength λ in step S11～λPMiddle divided by lambdah～λgThe reference optical parameter of a wavelength other than the wavelength and the wavelength λ in step S3a～λbIs substituted into the formula:
Figure FDA0002941986490000021
when D is 0, R is more than or equal to 3,
when D is not equal to 0, R is not less than 4, n is not less than max { X ≧1,X2}，
Calculating the values of parameters A, B, C and D;
step T5, substituting A, B, C and D into the formula:
Figure FDA0002941986490000022
in, the calculated wavelength is λmOf the optical system of (a).
4. The method of detecting an optical parameter of an optical system at an arbitrary wavelength according to claim 3, wherein:
wherein the step T5 includes the following steps:
step T5-1, substituting A, B, C and D obtained by calculation into the formula:
Figure FDA0002941986490000031
in, the calculated wavelength is λmOf the optical system of (a) is determined by the reference optical parameter f (λ)m)，
Step T5-2, according to the reference optical parameter f (λ) in step T5-1m) Calculating the wavelength as lambdamOf the optical system of (a).
5. The method for detecting any wavelength optical parameter of an optical system according to claim 2 or claim 4, wherein:
wherein the optical parameter is a transmitted wavefront and the reference optical parameter is a Zernike coefficient, i.e. f (λ) is Z1(λ)、Z2(λ)、······Zk(λ)，
The formula (1) is:
Figure FDA0002941986490000032
the formula (2) is:
Figure FDA0002941986490000033
in the formula (3) and the formula (4), i is an integer ranging from 1 to k, and k is not more than 37.
6. The method of detecting an optical parameter of an optical system at an arbitrary wavelength according to claim 5, wherein:
wherein D is 0, X1＝1，X2＝3.5，
Namely, the formula (1) is:
Zi(λ)·λn＝Ai·λn+Bi·λ(n-1)+Ci·λ(n-3.5) (5)，
the formula (2) is:
Figure FDA0002941986490000041
in the formula (5), n is not less than 3.5, in the formula (5) and the formula (6), i is an integer ranging from 1 to k, and k is not more than 37.
7. The method for detecting any wavelength optical parameter of an optical system according to claim 1 or claim 3, wherein:
wherein the optical system is a single wavelength system, an achromatic system, or an apochromatic system.
8. The method for detecting any wavelength optical parameter of an optical system according to claim 1 or claim 3, wherein:
wherein the optical parameter is a focal length, a back focal length, a wavefront discrete point, a field curvature, a distortion, or an aberration coefficient.
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