CN101125411A - Parallel grinding and cutting method for non-axial-symmetry and non-ball-surface optical element - Google Patents

Abstract

A parallel grinding machining method of non-axisymmetrical aspherical optical element relates to a non-axisymmetrical aspherical optical element, providing a parallel grinding machining method of non-axisymmetrical aspherical optical element applying two interlinking axles of three axles of arc grinding wheel which is made of common diamond. The procedures are that: the processing mode is chosen, the processing parameter is set and the surface point locus G of workpiece is calculated with the non-axisymmetrical aspherical surface equation; with the parameter of arc grinding wheel, the central point locus O of grinding wheel is calculated when three axles of the grinder are coupled; the average value z0' of coordinate value is figured out from the lines of z coordinate value of O, and O is replaced with O'; nonlinear equations are established, with xg and zg as primary value and the Guass-Newton iterative algorithm, and the surface point coordinate xg' and zg' of workpiece are reversely worked out with x0 and z0'; then the xg' and zg' values are substituted in the non-axisymmetrical aspherical surface equation, thereby working out the surface point locus G' of workpiece; with the parameter of the arc grinding wheel, the central point locus O1 of grinding wheel is calculated when two axles of three axles of grinder are coupled.

Description

The parallel grinding and cutting method of non-axisymmetric aspheric surface optical element

Technical field

The present invention relates to a kind of non-axisymmetric aspheric surface optical element, especially relate to a kind of parallel grinding and cutting method of non-axisymmetric aspheric surface optical element.

Background technology

Be used for the scanning objective on laser printer (Laser Printing) and the laser marking machine (Laser Marking), must satisfy the requirement that imaging is improved on the constant-speed scanning plane, but because the imaging of traditional optical, resemble high H and become non-linear relation (H=F*tg θ) with scan angle theta, can't satisfy the linear imaging relation, therefore must deliberately this object lens face shape be designed to have the free form surface of " barrel distortion " according to the requirement (H=F* θ) of linear imaging.The object lens of this special distortion are called the free-form surface lens of F θ form of lens.

Non-axisymmetric aspheric surface optical element (F-theta mirror) is the important kind of free form surface optical element, be indispensable vitals in the laser scanning system, the Laser pattern generation system and the laser scanning precision equipment that are widely used in guided missile tracking pointing instrumentation, marking machine, engraving machine, laser printer, facsimile machine, printing machine and are used for making semiconductor integrated circuit.

In order to obtain more clear and accurate optical imagery, satisfy in Medical Devices and military application thereof, the form accuracy and the surface roughness of non-axisymmetric aspheric surface proposed high requirement.Therefore, in order to satisfy the requirement of machining accuracy and feasibility, some can process for example diamond cutting of digitalization processing method of smooth surface, spherical wheel grinding, mixing electric machining and ELID etc. and be suggested in succession.Yet these processing methods exist that efficient is low, face shape be cannot say for sure problems such as card, program complexity, also have some problems in addition aspect processing process.(referring to document: 1, Kuriyagawa, Tsunemoto; Tachibana, Toru; Syoji, Katsuo; Mori, Yukio.Nonaxisymmetric aspheric ceramic mirror machining using arcenvelope grinding method.Transactions of the Japan Society of Mechanical Engineers, Part C, v 63, n 611, Jul, 1997,2532-2537; 2, Gotoh T, Takaya Y, Miyoshi T.Studies on generating free-formsurface from high-density measured point data (1st Report) .JSPE, Japanese Edition, 1998,64 (1): 84～88)

Summary of the invention

The objective of the invention is at the flexible deficiencies of 3 processing methods of the shaping of non-axisymmetric aspheric surface and spherical emery wheel processing method the too high shortcoming of requirement, a kind of parallel grinding and cutting method of non-axisymmetric aspheric surface optical elements of 3 two interlocks adopting common diamond arc diamond wheel is provided instrument and machinery.

The present invention includes following steps:

1) selects processing mode, set machined parameters, utilize the non-axisymmetric aspheric surface surface equation to calculate surface of the work locus of points G (x _g, z _g, y _g) step;

2) go out grinding machine x axle, y axle and 3 interlock emery wheels of z axle central point locus O (x in conjunction with the arc diamond wheel calculation of parameter ₀, z ₀, y ₀) step;

3) to grinding machine x axle, y axle and 3 interlock emery wheels of z axle central point locus O (x ₀, z ₀, y ₀) the z coordinate figure that respectively is listed as average, z coordinate figure mean value z ₀', with 3 interlock emery wheel central point locus O (x ₀, z ₀, y ₀) become O ' (x ₀, z ₀', y ₀) step;

4) set up Nonlinear System of Equations, with x _g, z _gBe initial value, adopt the Guass-Newton iterative algorithm, utilize x ₀, z ₀' counter asking obtains surface of the work point x _g' coordinate and z _gThe step of ' coordinate;

5) substitution non-axisymmetric aspheric surface surface equation is calculated surface of the work locus of points G ' (x _g', z _g', y _g') step;

6) go out 3 two interlock emery wheel central point locus O in conjunction with the arc diamond wheel calculation of parameter ₁(x ₁, z ₁, y ₁) step, thereby realize the 3 two interlock high accuracy processing of non-axisymmetric aspheric surface workpiece.

Described processing mode is the partitioning scheme of procedure interpolated point, can be divided into etc. the step-length mode and etc. the arc length mode.Promptly the processing length of workpiece is on average cut apart by equidistance etc. the step-length mode at the x of workpiece axle and z direction of principal axis; Promptly the finished surface arc length of workpiece is on average cut apart by equal arc length etc. the arc length mode at the x of workpiece axle and z direction of principal axis.

Described non-axisymmetric aspheric surface surface equation can be expressed as:

$y(x,z)=-R_x+\sqrt{{R_x}^2+x^2}+\frac{{\mathrm{<figure-callout\; id="2"\; label="C\; s\; z"\; filenames="A20071000930600111.png,A20071000930600121.png"\; state="\{\{state\}\}">C</figure-callout>}}_s{z}^{}{}^2}{1+\sqrt{1-(1+k){C_s}^2{z}^2}}$

In the formula,

The x axle is the non-axisymmetric aspheric surface work spindle, and the z axle is a non-axisymmetric aspheric surface workpiece countershaft;

C _s＝1/R _s

R _s＝-R _z+Ax ²+Bx ⁴+Cx ⁶+Dx ⁸+Ex ¹⁰+Fx ¹²

In the formula, Rx---aspheric surface main shaft base radius, Rz---aspheric surface countershaft base radius, Rs---aspheric surface countershaft radius, A, B, C, D, E, F---aspheric surface countershaft coefficient, k---asphericity coefficient.

Described Nonlinear System of Equations is:

$\left\{\begin{array}{c}{x}_0={x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{II}}\mathrm{<figure-callout\; id="0"\; label="R\; z"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">R</figure-callout>}& \\ {z_{}}^{}{{}_0}^{\&prime;}={z_g}^{\&prime;}+\frac{c}{I}r\end{array}\right.\&DoubleRightArrow;\left\{\begin{array}{c}{x}_0-({x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{II}}R)=0\\ {{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">z</figure-callout>}}_0}^{\&prime;}-({z_g}^{\&prime;}+\frac{c}{I}r)=0\end{array}\right.$

In the formula, with x _gAnd z _gBe x _g' and z _g' initial value, adopt the Guass-Newton algorithm to carry out iterative.When satisfying $|x_0-({x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{II}}R)=0|<{10}^{-6}$ With $|{{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">z</figure-callout>}}_0}^{\&prime;}-({z_g}^{\&prime;}+\frac{c}{I}r)=0|<{10}^{-6}$ The time, i.e. target function value f＜10 at each point place ^-6The time termination of iterations, obtain x _g' and z _g'.

The present invention adopts the parallel grinding and cutting method on the grinding machine that is used for the precision plane grinding, can realize the high-precision 3 two interlock processing to the non-axisymmetric aspheric surface optical element, mainly be to adopt diamond arc diamond wheel arc envelope mode on linear interpolation on the x axle and z axle to carry out.The parallel grinding and cutting principle is to make the cutting direction of emery wheel and the direction of motion identical (being that grinding wheel speed is parallel with the work speed direction) of workpiece carry out grinding.The grinding direction of common horizontal grinding medium plain emery wheel and the direction of feed of workpiece meet at right angles, and the wearing and tearing of grinding medium plain emery wheel concentrate on a bit.

Compare with horizontal method for grinding, the parallel grinding and cutting processing that the present invention adopts can obtain better suface processing quality, and this is during because of the employing parallel grinding and cutting, and the grinding points of emery wheel moves, and the wearing and tearing of emery wheel reduce.Because the machining locus line of parallel grinding and cutting method straight line and higher working (machining) efficiency and processing flexibility, the processing of non-axisymmetric aspheric surface eyeglass and mould thereof of large-scale medical treatment that is suitable for having relatively high expectations and color copy machine, simultaneously by reducing the quantity of the axle that system can link in the processing, realize the minimum interlock of grinding machine number, utilize 3 two interlock grinding machines that workpiece is realized processing, the spatial movement interpolation complexity of existing 3 interlock processing can be reduced, the control accuracy and the machining accuracy of grinding machine can be obviously improved.

Description of drawings

The grinding machine structural principle schematic diagram that is used for the precision plane grinding that Fig. 1 adopts for the embodiment of the invention.

Fig. 2 is the parallel grinding and cutting method process principle figure of the embodiment of the invention.

Fig. 3 is the left view of Fig. 2.

Fig. 4 is the procedure flow chart of the embodiment of the invention.

Fig. 5 is the partitioning scheme figure of the finished surface of the embodiment of the invention.

Fig. 6 obtains surface of the work point coordinates algorithm pattern for counter the asking of the embodiment of the invention.

The 3 two interlock emery wheel central point locus figure that Fig. 7 obtains by computer programming for the embodiment of the invention.

Fig. 8 is the vertical view of Fig. 7.

The specific embodiment

Following examples will the present invention is further illustrated in conjunction with the accompanying drawings.

The non-axisymmetric aspheric surface system of processing that the present invention adopts comprises 3 interlock grinding machines, circular arc skive, cup emery wheel trimmer and control microcomputer.For realizing high accuracy processing and simple CNC control, adopt the processing mode of x axis linear interpolation and z axle arc envelope, the parallel grinding and cutting method can form parallel machining locus at surface of the work.In the processing, the processing of non-axisymmetric aspheric surface surface of the work is finished in x, y, 3 two interlocks of z axle.

Fig. 1 provides the grinding machine structural principle schematic diagram that is used for the precision plane grinding that the embodiment of the invention adopts, and Fig. 2 and Fig. 3 provide the process principle figure of the parallel grinding and cutting method of the embodiment of the invention.

As shown in Figure 1, workpiece 2 is fixed on the workbench of grinding machine 3, and skive 1 is positioned at directly over the workpiece to be machined 2, and rotating shaft is the z axle.

The parallel grinding and cutting principle makes the cutting direction of emery wheel 1 and the direction of motion identical (being that grinding wheel speed is parallel with the work speed direction) of workpiece 2 carry out grinding referring to Fig. 2 and 3.The grinding direction of common horizontal grinding medium plain emery wheel and the direction of feed of workpiece meet at right angles, and the wearing and tearing of grinding medium plain emery wheel concentrate on a bit.Compare horizontal grinding, parallel grinding and cutting processing can obtain better suface processing quality, because during parallel grinding and cutting, the grinding points of emery wheel moves, and the wearing and tearing of emery wheel reduce, and machining accuracy improves naturally.In Fig. 2, Rx is an aspheric surface main shaft base radius.In Fig. 3, Rz is an aspheric surface countershaft base radius, and r is a diamond arc diamond wheel arc radius, and R is a diamond arc diamond wheel base radius.

Fig. 4 is a procedure flow chart of the present invention.Main implementation step of the present invention is:

1) selects processing mode, set machined parameters, utilize the non-axisymmetric aspheric surface surface equation to calculate surface of the work locus of points G (x _g, z _g, y _g).

Processing mode is the partitioning scheme of procedure interpolated point, can be divided into etc. the step-length mode and etc. the arc length mode.Promptly the processing length of workpiece is on average cut apart by equidistance etc. the step-length mode at the x of workpiece axle and z direction of principal axis; Promptly the finished surface arc length of workpiece is on average cut apart by equal arc length etc. the arc length mode at the x of workpiece axle and z direction of principal axis.

2) go out grinding machine x axle, y axle and 3 interlock emery wheels of z axle central point locus O (x in conjunction with the arc diamond wheel calculation of parameter ₀, z ₀, y ₀).

3) to grinding machine x axle, y axle and 3 interlock emery wheels of z axle central point locus O (x ₀, z ₀, y ₀) the z coordinate figure that respectively is listed as average, z coordinate figure mean value z ₀', with 3 interlock emery wheel central point locus O (x ₀, z ₀, y ₀) become O ' (x ₀, z ₀', y ₀).

4) set up Nonlinear System of Equations, with x _g, z _gBe initial value, adopt the Guass-Newton iterative algorithm, utilize x ₀, z ₀' counter asking obtains surface of the work point x _g' coordinate and z _g' coordinate.

After the non-axisymmetric aspheric surface optical element was finished design, its parameter just defined, and its general non-axisymmetric aspheric surface surface equation can be expressed as:

$y(x,z)=-R_x+\sqrt{{R_x}^2+x^2}+\frac{C_s{z}^2}{1+\sqrt{1-(1+k){C_s}^2{z}^2}}---\left(1\right)$

In the formula (1),

The x axle is the non-axisymmetric aspheric surface work spindle, and the z axle is a non-axisymmetric aspheric surface workpiece countershaft.

C _s＝1/R _s

R _s＝-R _z+Ax ²+Bx ⁴+Cx ⁶+Dx ⁸+Ex ¹⁰+Fx ¹² (2)

In the formula (2), Rx---aspheric surface main shaft base radius, Rz---aspheric surface countershaft base radius, Rs---aspheric surface countershaft radius, A, B, C, D, E, F---aspheric surface countershaft coefficient, k---asphericity coefficient.

If d _x, d _zBe respectively x axle and z axle interpolation step-length or arc length, x _I+1=x _i+ d _x, z _I+1=z _i+ d _z

5) substitution non-axisymmetric aspheric surface surface equation is calculated surface of the work locus of points G ' (x _g', z _g', y _g').

With x _I+1=x _i+ d _x, z _I+1=z _i+ d _zSubstitution formula (1) obtains surface of the work locus of points G _Ij(x _g, z _g, y _g) (i=0 ..., m, j=0 ..., n).

6) go out 3 two interlock emery wheel central point locus O in conjunction with the arc diamond wheel calculation of parameter ₁(x ₁, z ₁, y ₁).

For calculating 3 interlock emery wheel central point locus, the normal vector of each processing stand can be calculated by following formula:

$n(a,b,c)=n(-\frac{\&PartialD;y}{\&PartialD;x},1,\frac{\&PartialD;y}{\&PartialD;z})---\left(3\right)$

In the formula (3),

---non-axisymmetric aspheric surface EQUATION x direction method vector,

---non-axisymmetric aspheric surface equation z direction method vector.

Three-shaft linkage emery wheel centrode O (x ₀, z ₀, y ₀) can calculate by formula (4).

$x_0=x_g+\frac{a}{I}r+\frac{a}{\mathrm{II}}R$

${\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">y</figure-callout>}}_0=y_g+\frac{b}{I}r+\frac{b}{\mathrm{II}}R---\left(4\right)$

${\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">z</figure-callout>}}_0=z_g+\frac{c}{I}r$

In the formula (4),

$I=\sqrt{a^2+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20071000930600111.png,A20071000930600121.png"\; state="\{\{state\}\}">b</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="A20071000930600111.png,A20071000930600121.png"\; state="\{\{state\}\}">c</figure-callout>}}^2},$

$\mathrm{II}=\sqrt{a^2+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20071000930600111.png,A20071000930600121.png"\; state="\{\{state\}\}">b</figure-callout>}}^2},$

R---diamond arc diamond wheel arc radius,

R---diamond arc diamond wheel base radius.

By selecting processing mode and setting machined parameters, utilize formula (3), formula (4) and G (x _g, z _g, y _g), can calculate 3 interlock emery wheel central point locus O (x ₀, z ₀, y ₀).

For realizing 3 two interlock processing, at x, when the y axle linked interpolation, emery wheel centrode z axial coordinate should remain a constant, so be averaged worth to each row z coordinate

${{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">z</figure-callout>}}_0}^{\&prime;}=\frac{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}{z}_{0i}}{n+1}---\left(5\right)$

So 3 interlock emery wheel central point locus become O ' (x ₀, z ₀', y ₀).Next, utilize X in the emery wheel centrode ₀, z ₀' anti-surface of the work the locus of points of asking, set up following Nonlinear System of Equations:

$\left\{\begin{array}{c}{x}_0={x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{<figure-callout\; id="0"\; label="II\; R\; z"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">II</figure-callout>}}R& \\ {z_{}}^{}{{}_0}^{\&prime;}={z_g}^{\&prime;}+\frac{c}{I}r\end{array}\right.\&DoubleRightArrow;\left\{\begin{array}{c}{x}_0-({x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{II}}R)=0\\ {{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">z</figure-callout>}}_0}^{\&prime;}-({z_g}^{\&prime;}+\frac{c}{I}r)=0\end{array}\right.---\left(6\right)$

In the formula (6), with x _gAnd z _gBe x _g' and z _g' initial value, adopt the Guass-Newton algorithm to carry out iterative.When satisfying $|x_0-({x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{II}}R)=0|<{10}^{-6}$ With $|{{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">z</figure-callout>}}_0}^{\&prime;}-({z_g}^{\&prime;}+\frac{c}{I}r)=0|<{10}^{-6}$ The time, i.e. target function value f＜10 at each point place ^-6The time termination of iterations, obtain x _g' and z _g'.

Substitution formula (1) can be obtained y _g', thereby obtain 3 two interlock surface of the work locus of points G ' (x _g', z _g', y _g'), again by G ' (x _g', z _g', y _g') substitution formula (4) can obtain 3 two interlock emery wheel central point locus O ₁(x ₁, z ₁, y ₁), thereby 3 two interlock high accuracy that realize the non-axisymmetric aspheric surface workpiece are processed.

Fig. 5 represents the partitioning scheme of the finished surface of workpiece 2.d _x, d _zBe respectively x axle and z axle interpolation step-length or arc length, x _I+1=x _i+ d _x, z _i+ 1=z _i+ d _z, substitution non-axisymmetric aspheric surface surface equation formula (1) can obtain the surface point track G of workpiece 2 _Ij(x _g, z _g, y _g) (i=0 ..., m, j=0 ..., n).

Fig. 6 is for the anti-surface of the work point coordinates algorithm pattern of asking, with x _gAnd z _gBe x _g' and z _g' initial value, adopt the Guass-Newton algorithm to carry out iterative.The stopping criterion for iteration of embodiment of the invention setting is target function value f＜10 at each point place ^-6, because of machined parameters unit is mm, when f＜10 ^-6The time, the processing stand path accuracy reaches the nm rank, far above the machine tool motion precision.

When satisfying $|x_0-({x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{II}}R)=0|<{10}^{-6}$ With $|{{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="A20071000930600121.png"\; state="\{\{state\}\}">z</figure-callout>}}_0}^{\&prime;}-({z_g}^{\&prime;}+\frac{c}{I}r)=0|<{10}^{-6}$ The time, i.e. target function value f＜10 at each point place ^-6The time termination of iterations, otherwise x to calculate _g' and z _g' proceed iterative for initial value, finally be met the x of stopping criterion for iteration _g' and z _g'.

The 3 spool two interlock emery wheel central point locus figures of Fig. 7 for obtaining by computer programming, Fig. 8 is the vertical view of Fig. 7.As shown in table 1 with the corresponding machined parameters of Fig. 7.In Fig. 7 and 8, a is 3 two interlock emery wheel central point locus O ₁(x ₁, z ₁, y ₁), b is 3 two interlock surface of the work locus of points G ' (x _g', z _g', y _g').

As mentioned above, the parallel grinding and cutting processing method of 3 two interlock non-axisymmetric aspheric surface workpiece of the present invention has can satisfy the high-precision grinding of finishing the non-axisymmetric aspheric surface workpiece under the minimum situation of the interlock number of axle.

In addition, the present invention is not limited in above-mentioned form of implementation, and nature can carry out various changes in the scope that does not break away from purport of the present invention.

Table 1 (unit: mm)

The workpiece parameter	Workpiece size 176 * 20 (machining locus driven dimension 186 * 22) aspheric surface main shaft base radius Rx=225 aspheric surface countershaft base radius Rz=65.95 asphericity coefficient k=0.45 aspheric surface countershaft coefficient A=0.00068458867854 B=-3.2775429131 * 10 -8 C＝2.442605923×10 -11 D＝-8.8798949282×10 -15 E＝1.5857725875×10 -18 F＝-1.1046744163×10 -22
		The emery wheel parameter	Diamond arc diamond wheel base radius r=60 diamond arc diamond wheel arc radius R=65
The processing step-length	X axle processing step-length dx=2 z axle processing step-length dz=2

Claims (6)

1. the parallel grinding and cutting method of non-axisymmetric aspheric surface optical element is characterized in that may further comprise the steps:

1) selects processing mode, set machined parameters, utilize the non-axisymmetric aspheric surface surface equation to calculate surface of the work locus of points G (x _g, z _g, y _g) step;

2) go out grinding machine x axle, y axle and 3 interlock emery wheels of z axle central point locus O (x in conjunction with the arc diamond wheel calculation of parameter ₀, z ₀, y ₀) step;

3) to grinding machine x axle, y axle and 3 interlock emery wheels of z axle central point locus O (x ₀, z ₀, y ₀) the z coordinate figure that respectively is listed as average, z coordinate figure mean value z ₀', with 3 interlock emery wheel central point locus O (x ₀, z ₀, y ₀) become O ' (x ₀, z ₀', y ₀) step;

4) set up Nonlinear System of Equations, with x _g, z _gBe initial value, adopt the Guass-Newton iterative algorithm, utilize x ₀, z ₀' counter asking obtains surface of the work point x _g' coordinate and z _gThe step of ' coordinate;

5) substitution non-axisymmetric aspheric surface surface equation is calculated surface of the work locus of points G ' (x _g', z _g', y _g') step;

6) go out 3 two interlock emery wheel central point locus O in conjunction with the arc diamond wheel calculation of parameter ₁(x ₁, z ₁, y ₁) step, thereby realize the 3 two interlock high accuracy processing of non-axisymmetric aspheric surface workpiece.

2. the parallel grinding and cutting method of non-axisymmetric aspheric surface optical element as claimed in claim 1 is characterized in that step-length modes such as processing mode is, at the x of workpiece axle and z direction of principal axis the processing length of workpiece is on average cut apart by equidistance.

3. the parallel grinding and cutting method of non-axisymmetric aspheric surface optical element as claimed in claim 1 is characterized in that arc length modes such as processing mode is, at the x of workpiece axle and z direction of principal axis the finished surface arc length of workpiece is on average cut apart by equal arc length.

4. the parallel grinding and cutting method of non-axisymmetric aspheric surface optical element as claimed in claim 1 is characterized in that described non-axisymmetric aspheric surface surface equation is:

$y(x,z)=-R_x+\sqrt{{R_x}^2+x^2}+\frac{C_s{z}^2}{1+\sqrt{1-(1+k){C_s}^2{z}^2}}$

In the formula,

The x axle is the non-axisymmetric aspheric surface work spindle, and the z axle is a non-axisymmetric aspheric surface workpiece countershaft;

C _s＝1/R _s

R _s＝-R _z+Ax ²+Bx ⁴+Cx ⁶+Dx ⁸+Ex ¹⁰+Fx ¹²

In the formula, Rx---aspheric surface main shaft base radius, Rz---aspheric surface countershaft base radius, Rs---aspheric surface countershaft radius, A, B, C, D, E, F---aspheric surface countershaft coefficient, k---asphericity coefficient.

5. the parallel grinding and cutting method of non-axisymmetric aspheric surface optical element as claimed in claim 1 is characterized in that described Nonlinear System of Equations is:

$\left\{\begin{array}{c}{x}_0={x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{II}}R\\ {z_0}^{\&prime;}+{z_g}^{\&prime;}+\frac{c}{I}r\end{array}\right.\&DoubleRightArrow;\left\{\begin{array}{c}{x}_0-({x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{II}}R)=0\\ {z_0}^{\&prime;}-({z_g}^{\&prime;}+\frac{c}{I}r)=0\end{array}\right.$

In the formula, with x _gAnd z _gBe x _g' and z _g' initial value.

6. the parallel grinding and cutting method of non-axisymmetric aspheric surface optical element as claimed in claim 1 is characterized in that adopting the Guass-Newton algorithm to carry out iterative, when satisfying $|x_0-({x_g}^{\&prime;}+\frac{a}{I}r+\frac{a}{\mathrm{II}}R)=0|<{10}^{-6}$ With $|{z_0}^{\&prime;}-({z_g}^{\&prime;}+\frac{c}{I}r)=0|<{10}^{-6}$ The time, i.e. target function value f＜10 at each point place ^-6The time termination of iterations, obtain x _g' and z _g'.

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