CN105547179A - Measurement method of aspheric equation - Google Patents

Abstract

The present invention discloses a measurement method of an aspheric equation. The method comprises: 1) measuring a measured aspheric annular surface shape view through adoption of a phase-shift interferometer, and measuring a relative displacement from the measured aspheric surface to the last measurement point through adoption of the phase-shift interferometer; 2) calculating and obtaining a 'zero stripe' radius corresponding to the measurement point according to the measured annular surface shape view; 3) calculating and obtaining a rise derivative value corresponding to the measurement point of the measured aspheric surface according to the 'zero stripe' radius and the relative displacement measured by the phase-shift interferometer; 4) calculating and obtaining the derivative expression of the measured aspheric equation according to the rise derivative value corresponding to the measurement point; and 5) calculating and obtaining the measured aspheric equation according to the derivative expression of the measured aspheric equation. The measurement method of an aspheric equation is simple in measurement method, low in cost, easy to operate and the like.

Description

A kind of measuring method of aspherical equation

Technical field

The present invention relates to field of optical detection, specifically provide a kind of measuring method of aspherical equation.

Background technology

In optical design, single sphere can be used for the degree of freedom of optimization to only have radius-of-curvature; And aspheric surface is except vertex curvature radius, also have quadric surface constant and higher order term coefficient.Because aspheric surface to have the degree of freedom of more designs than sphere, so in a lot of optical system, all generally adopt non-spherical element to reduce the complexity of system, and improve the image quality of system.

Current, for aspheric detection, mostly all concentrate on aspheric shape, namely adopt splicing method (annular sub-aperture splicing or circular sub-aperture splicing) or null compensator testing (compensating glass method or calculation holographic method) to come to detect Aspherical-surface testing, seldom pay close attention to the detection to aspherical equation.

Therefore, research and develop a kind of detection method of aspherical equation, become people's problem demanding prompt solution.

Summary of the invention

Given this, the object of the present invention is to provide a kind of measuring method of aspherical equation, obtain aspherical equation to realize measuring when complete unknown aspherical equation.

Technical scheme provided by the invention is specially, and a kind of measuring method of aspherical equation, is characterized in that, comprising:

1) utilize phase-shifting interferometer to measure tested aspheric ring surface shape figure, utilize distance measuring interferometer to measure the relative shift of tested aspheric surface apart from a upper ring surface shape figure measurement point;

2) according to the ring surface shape figure that described phase-shifting interferometer is measured, calculate and obtain at " zero striped " radius corresponding to measurement point;

3) according to described " zero striped " radius, the relative shift of described distance measuring interferometer measurement and the derivative expressions of aspherical equation, iteration optimization method is adopted to calculate tested aspheric equation.

Preferably, the described ring surface shape figure measured according to described phase-shifting interferometer, calculates and obtains at " zero striped " radius corresponding to measurement point, comprising:

If orbicular interference stripe is not communicated with image center, ring surface shape figure adopts numerical differentiation to calculate meet the coordinate of threshold requirement, then adopt least square method to calculate and obtain at " zero striped " radius corresponding to measurement point;

If orbicular interference stripe is communicated with image center, adopt out of focus and the spherical aberration fitting process of Zernike, calculate and obtain at " zero striped " radius corresponding to measurement point.

Further preferably, the out of focus of described employing Zernike and spherical aberration fitting process, calculating acquisition at " zero striped " radius formula that measurement point is corresponding is:

${\mathrm{\ρ}}_i=\sqrt{\frac{3a_2-a_1}{6a_2}}$

Wherein, a ₁represent out of focus coefficient, a ₂represent coefficient of spherical aberration, and described a ₁and a ₂by carrying out Zernike matching acquisition to ring surface shape figure.

Further preferably, the derivative expressions of described aspherical equation is:

$z^{\′}\left(\mathrm{\ρ}\right)=\frac{c\mathrm{\ρ}}{\sqrt{1-c^2{\mathrm{\ρ}}^2}}+\underset{n=0}{\overset{M}{\Σ}}{b}_n{P}_n(\mathrm{\ρ}/{\mathrm{\ρ}}_{\max })$

Wherein, c represents best-fit ball curvature, ρ represents " zero striped " radius, ρ _maxrepresent aspheric surface radial coordinate maximal value, M represents integer, b _nrepresentative polynomial { P _n(x) } coefficient, and polynomial expression { P _n(x) } meet: ${\∫}_0^1{P}_n\left(x\right)P_m\left(x\right)dx={\mathrm{\δ}}_{mn},m\≥0,n\≥0.$

Further preferably, M=6.

Further preferably, described iteration optimization method is specially:

1) distance measuring interferometer measurement " opal " position and described ring surface shape is utilized to scheme the distance of middle endless belt position, and distance measurement obtained is as the initial value of best-fit ball curvature c in described tested aspherical equation, and the initial value of foundation best-fit ball curvature c, estimate corresponding rise value z (ρ _i);

2) according to best-fit ball curvature c and rise value z (ρ _i), utilize formula (a) and least square method, calculate and obtain polynomial expression { P _n(x) } coefficient b _n,

$\underset{n=0}{\overset{M}{\Σ}}{b}_n{P}_n(\mathrm{\ρ}/{\mathrm{\ρ}}_{max})=\frac{{\mathrm{\ρ}}_i}{s_i-z\left({\mathrm{\ρ}}_i\right)}-\frac{c\mathrm{\ρ}}{\sqrt{1-c^2{\mathrm{\ρ}}^2}}---\left(a\right)$

Wherein, ρ _irepresent at " zero striped " radius corresponding to measurement point, s ⁱrepresent the relative shift that distance measuring interferometer is measured, ρ represents " zero striped " radius;

3) according to coefficient b _nobtain the derivative expression of tested aspherical equation, and calculate acquisition aspherical equation according to described expression;

4) according to described aspherical equation, recalculate and obtain best-fit ball curvature c and rise value z (ρ _i), will the best-fit ball curvature c and the step 2 that obtain be calculated) the best-fit ball curvature c that substitutes into subtracts each other, acquisition difference;

5) described difference being compared with threshold value, when being greater than threshold value, repeating step 2) ~ step 4), until described difference is less than or equal to threshold value.

The measuring method of aspherical equation provided by the invention, can realize measuring when complete unknown aspherical equation obtaining aspherical equation, enrich the measurable parameter of optical element, had measuring method simple, the advantages such as cost is low, convenient operation.

Accompanying drawing explanation

Fig. 1 is the measuring system schematic diagram of aspherical equation;

Fig. 2 is the annular interference figure that emulation generates;

Fig. 3 obtains " zero striped " position view for calculating;

Fig. 4, for when interference fringe is not communicated with picture centre, meets " zero striped " location map of setting threshold value;

Fig. 5 is for utilizing the best-fit-circle that in Fig. 2, " zero striped " position calculation goes out;

Fig. 6, for when interference fringe is communicated with picture centre, emulates the interferogram generated;

Fig. 7 is the face shape figure that Fig. 6 is corresponding;

Fig. 8 is the annular interference figure of a certain measuring position actual measurement;

Fig. 9 is the face shape figure that Fig. 8 is corresponding;

Figure 10 is the relative displacement of length-measuring interferometer actual measurement and the position relationship of " zero striped " that calculate;

Figure 11 is the rise that the final aspherical equation obtained calculates;

Figure 12 is the deviation of the rise that the rise of profile measurement is measured with discussion method in invention.

Embodiment

With specific embodiment, the present invention is further expalined below, but is not limited to protection scope of the present invention.

Aspherical equation is obtained in order to realize measuring when complete unknown aspherical equation, present embodiment provides a kind of measuring method of aspherical equation, wherein, measurement mechanism used is see Fig. 1, comprise: frequency stabilized laser S1, collimating and beam expanding system S2, Amici prism S3, converging lenses group S4, phase shift reference surface S5, distance measuring interferometer S6, aspheric surface S7, clamping device S8, multidimensional adjusting table S9, imaging lens S10, detector S11, image acquisition units S12, wherein, S1, S2, S3, S4, S5, S10, S11 and S12 constitute phase-shifting interferometer.

Concrete measuring process is as follows:

Preliminary work: select appropriate spherical reference mirror, tested aspheric surface and phase-shifting interferometer are placed concentric, by distance measuring interferometer zero setting;

1) move axially tested aspheric surface along phase-shifting interferometer, utilize phase-shifting interferometer to measure tested aspheric ring surface shape figure, utilize distance measuring interferometer to measure tested aspheric surface and move axially distance.

2) according to the ring surface shape figure that phase-shifting interferometer is measured, calculate and obtain at " zero striped " radius corresponding to measurement point;

3) according to the derivative expressions moving axially distance and aspherical equation that " zero striped " radius, distance measuring interferometer are measured, the computing method of iteration optimization are adopted to obtain tested aspherical equation.

This enforcement side in, step 2) computing method as follows:

When the radial any position normal of the spherical wave that phase-shifting interferometer sends and tested aspheric surface is vertical, there will be shown annular interference figure as shown in Figure 2, calculate schematic diagram as shown in Figure 3.P is any point on aspheric surface face, and normal and the optical axis of aspheric surface being crossed P point intersect at a F; If P point is in " zero striped " position, then the coordinate of P point and F point can be expressed as (ρ _i, z (ρ _i)) and (0, s _i), and meet:

$\frac{{\mathrm{\ρ}}_i}{z^{\′}\left({\mathrm{\ρ}}_i\right)}+z\left({\mathrm{\ρ}}_i\right)=s_i---\left(1\right)$

Wherein ρ _irepresent the aspheric surface radial position corresponding to the i-th endless belt gathered, z (ρ _i) represent the aspheric rise in this position, z'(ρ _i) represent the derivative of this position rise, s _irepresent the relative displacement that distance measuring interferometer is measured.

For the ease of calculating, formula (1) can be write as:

$z^{\′}\left({\mathrm{\ρ}}_i\right)=\frac{{\mathrm{\ρ}}_i}{s_i-z\left({\mathrm{\ρ}}_i\right)}---\left(2\right)$

And meet:

$z\left(\mathrm{\ρ}\right)=\∫z^{\′}\left(\mathrm{\ρ}\right)d\mathrm{\ρ}=\∫\frac{{\mathrm{\ρ}}_i}{s_i-z\left({\mathrm{\ρ}}_i\right)}d\mathrm{\ρ}---\left(3\right)$

On the shape figure of face, " zero striped " position meets:

$\frac{\∂F(x,y)}{\∂\mathrm{\ρ}}=0---\left(4\right)$

Wherein F (x, y) represents the face shape of measuring.

Concrete calculating ρ _i(" zero striped " radius) can be divided into two kinds of situations:

Situation one: when interference fringe is not communicated with picture centre in ring surface shape figure, for the ease of calculating, formula (4) can be write as:

$\frac{\∂F(x,y)}{\∂\mathrm{\ρ}}=\frac{\∂F}{\∂x}\frac{\∂x}{\∂\mathrm{\ρ}}+\frac{\∂F}{\∂y}\frac{\∂y}{\∂\mathrm{\ρ}}=\frac{\∂F}{\∂x}cos\mathrm{\θ}+\frac{\∂F}{\∂y}sin\mathrm{\θ}=0---\left(5\right)$

Opposite shape figure adopts numerical differentiation, arranges suitable threshold value, can obtain " zero striped " location map, as shown in Figure 4, adopts least square method to carry out matching, can obtain center and the radius of " zero striped ", as shown in Figure 5.

Situation two: when interference fringe is communicated with picture centre in ring surface shape figure (now phase-shifting interferometer focus is close to the aspherical mirror vertex centre of sphere), measured face shape forms primarily of out of focus and spherical aberration item, as shown in Figure 6, Figure 7, that is:

F(x,y)≈a ₁(2ρ ²-1)+a ₂(6ρ ⁴-6ρ+1)(6)

Wherein a ₁and a ₂represent out of focus and coefficient of spherical aberration respectively, it can by carrying out Zernike matching acquisition to ring surface shape figure.

Bring formula (6) into formula (4), can obtain:

${\mathrm{\ρ}}_i=\sqrt{\frac{3a_2-a_1}{6a_2}}---\left(7\right)$

In the present embodiment, step 3) calculation optimization method be:

In order to improve measuring accuracy, the derivative expressions of tested aspherical equation is described as:

$z^{\′}\left(\mathrm{\ρ}\right)=\frac{c\mathrm{\ρ}}{\sqrt{1-c^2{\mathrm{\ρ}}^2}}+\underset{n=0}{\overset{M}{\Σ}}{b}_n{P}_n(\mathrm{\ρ}/{\mathrm{\ρ}}_{\max })---\left(8\right)$

Wherein, c is best-fit ball curvature, ρ _maxfor aspheric face diameter coordinate maximal value, M can carry out integer value according to the actual needs, and in the present embodiment, M gets 6, b _nrepresentative polynomial { P _n(x) } coefficient, and polynomial expression { P _n(x) } meet:

${\∫}_0^1{P}_n\left(x\right)P_m\left(x\right)dx={\mathrm{\δ}}_{mn},m\≥0,n\≥0---\left(9\right)$

According to formula (8), aspherical equation can be expressed as

$\begin{array}{c}z\left(\mathrm{\ρ}\right)=\∫z^{\′}\left(\mathrm{\ρ}\right)d\mathrm{\ρ}\\ =\frac{{\mathrm{c\ρ}}^2}{1+\sqrt{1-c^2{\mathrm{\ρ}}^2}}+{\mathrm{\ρ}}_{max}\underset{n=0}{\overset{M}{\Σ}}{b}_n{T}_n(\mathrm{\ρ}/{\mathrm{\ρ}}_{max})\end{array}---\left(10\right)$

Wherein T _n(x)=∫ P _n(x) dx, n>=0.

Make formula (3) equal with formula (10), for the ease of iteration optimization, can be written as:

$\underset{n=0}{\overset{M}{\Σ}}{b}_n{P}_n(\mathrm{\ρ}/{\mathrm{\ρ}}_{max})=\frac{{\mathrm{\ρ}}_i}{s_i-z\left({\mathrm{\ρ}}_i\right)}-\frac{c\mathrm{\ρ}}{\sqrt{1-c^2{\mathrm{\ρ}}^2}}---\left(11\right)$

Concrete iterative optimization procedure is as follows:

(I) distance measuring interferometer measurement " opal " position and described ring surface shape is utilized to scheme the distance of middle endless belt position, and distance measurement obtained is as the initial value of best-fit ball curvature c in described tested aspherical equation, and the initial value of foundation best-fit ball curvature c, estimate tested aspheric rise z (ρ _i);

(II) by distance measuring interferometer measure relative displacement, step 2) in obtain " zero striped " radius and best-fit ball curvature c and rise z (ρ _i), bring formula (11) into, adopt least square method to calculate fitting coefficient { b _n;

(III) formula (10) is utilized to calculate aspheric equation;

(IV) curvature c and the rise z (ρ of best-fit ball is recalculated according to aspherical equation _i), the best-fit ball curvature that the best-fit ball curvature and step (II) that calculate acquisition substitute into is subtracted each other, obtains difference;

(V) described difference compared with threshold value, when being greater than threshold value, repeat step (II) ~ step (IV), until described difference is less than or equal to threshold value, now, the aspherical equation of acquisition is tested aspheric surface side.

Measured for a high order even aspheric surface by said method, this aspheric surface parameter is as shown in table 1:

Table 1 aspheric surface parameter

Utilize step 1) ~ step 3) complete measurement to aspherical equation, result is as shown in Fig. 8 ~ 11, Fig. 8 is the annular interference figure of a certain measuring position, Fig. 9 is the phase diagram corresponding with interferogram, Figure 10 is the relative displacement of length-measuring interferometer measurement and the position of " zero striped ", the aspheric surface rise that Figure 11 utilizes the aspherical equation measuring acquisition to calculate.

In order to verify the reliability of the method, the measurement of rise that also adopted contourgraph to carry out of this aspheric surface, as shown in figure 12, as can be seen from the figure, the two PV value deviation measured is about 1.6 μm to the two deviation measured.

Claims (6)

1. a measuring method for aspherical equation, is characterized in that, comprising:

1) utilize phase-shifting interferometer to measure tested aspheric ring surface shape figure, utilize distance measuring interferometer to measure the relative shift of tested aspheric surface apart from a upper ring surface shape figure measurement point;

2) according to the ring surface shape figure that described phase-shifting interferometer is measured, calculate and obtain at " zero striped " radius corresponding to measurement point;

3) according to described " zero striped " radius, the relative shift of described distance measuring interferometer measurement and the derivative expressions of aspherical equation, iteration optimization method is adopted to calculate tested aspheric equation.

2. according to the measuring method of aspherical equation described in claim 1, it is characterized in that, the described ring surface shape figure measured according to described phase-shifting interferometer, calculates and obtains at " zero striped " radius corresponding to measurement point, comprising:

If orbicular interference stripe is not communicated with image center, ring surface shape figure adopts numerical differentiation to calculate meet the coordinate of threshold requirement, then adopt least square method to calculate and obtain at " zero striped " radius corresponding to measurement point;

If orbicular interference stripe is communicated with image center, adopt out of focus and the spherical aberration fitting process of Zernike, calculate and obtain at " zero striped " radius corresponding to measurement point.

3. according to the measuring method of aspherical equation described in claim 2, it is characterized in that, the out of focus of described employing Zernike and spherical aberration fitting process, calculating acquisition at " zero striped " radius formula that measurement point is corresponding is:

${\mathrm{\ρ}}_i=\sqrt{\frac{3a_2-a_1}{6a_2}}$

Wherein, a ₁represent out of focus coefficient, a ₂represent coefficient of spherical aberration, and described a ₁and a ₂by carrying out Zernike matching acquisition to ring surface shape figure.

4. according to the measuring method of aspherical equation described in claim 1, it is characterized in that, the derivative expressions of described aspherical equation is:

$z^{\′}\left(\mathrm{\ρ}\right)=\frac{c\mathrm{\ρ}}{\sqrt{1-c^2{\mathrm{\ρ}}^2}}+\underset{n=0}{\overset{M}{\Σ}}{b}_n{P}_n(\mathrm{\ρ}/{\mathrm{\ρ}}_{max})$

Wherein, c represents best-fit ball curvature, ρ represents " zero striped " radius, ρ _maxrepresent aspheric surface radial coordinate maximal value, M represents integer, b _nrepresentative polynomial { P _n(x) } coefficient, and polynomial expression { P _n(x) } meet: ${\∫}_0^1{P}_n\left(x\right)P_m\left(x\right)dx={\mathrm{\δ}}_{mn},m\≥0,n\≥0.$

5. according to the measuring method of aspherical equation described in claim 4, it is characterized in that: M=6.

6. according to the measuring method of aspherical equation described in claim 4, it is characterized in that, described iteration optimization method is specially:

1) distance measuring interferometer measurement " opal " position and described ring surface shape is utilized to scheme the distance of middle endless belt position, and distance measurement obtained is as the initial value of best-fit ball curvature c in described tested aspherical equation, and the initial value of foundation best-fit ball curvature c, estimate corresponding rise value z (ρ _i);

2) according to best-fit ball curvature c and rise value z (ρ _i), utilize formula (a) and least square method, calculate and obtain polynomial expression { P _n(x) } coefficient b _n,

$\underset{n=0}{\overset{M}{\Σ}}{b}_n{P}_n(\mathrm{\ρ}/{\mathrm{\ρ}}_{max})=\frac{{\mathrm{\ρ}}_i}{s_i-z\left({\mathrm{\ρ}}_i\right)}-\frac{c\mathrm{\ρ}}{\sqrt{1-c^2{\mathrm{\ρ}}^2}}---\left(a\right)$

Wherein, ρ _irepresent at " zero striped " radius corresponding to measurement point, s _irepresent the relative shift that distance measuring interferometer is measured, ρ represents " zero striped " radius;

3) according to coefficient b _nobtain the derivative expression of tested aspherical equation, and calculate acquisition aspherical equation according to described expression;

4) according to described aspherical equation, recalculate and obtain best-fit ball curvature c and rise value z (ρ _i), will the best-fit ball curvature c and the step 2 that obtain be calculated) the best-fit ball curvature c that substitutes into subtracts each other, acquisition difference;

5) described difference being compared with threshold value, when being greater than threshold value, repeating step 2) ~ step 4), until described difference is less than or equal to threshold value.