CN104297829A - Method for optimum design of planar variable-pitch grating - Google Patents

Abstract

The invention discloses a method for optimum design of a planar variable-pitch grating and relates to the spectrum technical field. By the adoption of the method, the problem that the groove density function of a variable-pitch grating manufactured with existing methods is quite different from an expected groove density function is solved. According to the method, the meridian expected groove density function of the planar variable-pitch grating is obtained by means of a spherical wave exposure system, the deviate focus, meridian coma and spherical aberration of the planar grating are obtained according to the eikonal function of the variable-pitch grating, the variable-pitch grating is made to focus according to a grating equation, the expected groove density function, the value of the deviate focus F20, the value of the meridian coma F30 and the value of the spherical aberration F40, the meridian coma and the spherical aberration are zero, the expected groove density function coefficient optimized spherical wave logging parameters of the planar variable-pitch grating are obtained, the integral function of the square of the difference between expected groove density and designed groove density of each point on the surface of a substrate of the grating serves as an objective function and is optimized, the minimum value is obtained, and then optimum design of the planar variable-pitch grating is achieved. By the adoption of the method, computing time is shortened, and computing efficiency is improved.

Description

The Optimization Design of Varied Line Space Plane Gratings

Technical field

The present invention relates to spectral technique field, the Optimization Design of the grid pitch changing grating be specifically related to.

Background technology

Grid pitch changing grating mainly has following principal feature: self-focusing, aberration correcting capability peace Jiao Chang, can reduce the optical element in optical system, reduce parasitic light, improve diffraction efficiency and resolution.The above-mentioned advantage of Varied Line Space Plane Gratings only has the request for utilization according to spectral instrument, and strict design grating line density function just can be achieved.

The method for making of grid pitch changing grating is divided into two kinds, mechanical scratching and holographic exposure.Compared with ruling grating, holographic grating making is simple, and be convenient to change groove shape, the base type that can make is abundanter, and has without ghost line, the advantages such as parasitic light is low.The making of grid pitch changing grating now mainly contains two kinds of methods, one method uses spherical waves interfere exposure, the point making substrate surface different, with the exposure of different interference angles, forming grid pitch changing grating, can eliminate specific order aberration by reasonably selecting recording parameters.Another kind method uses aspherical wavefront to interfere exposure, and further increase the degree of freedom of recording beam path, theory calculate can be designed and expect the grid pitch changing grating that incisure density function meets completely, eliminates higher order aberratons.But in actual fabrication process, due to it, to make light path comparatively complicated and introduce auxiliary mirror, for meeting design requirement, higher requirement is had to the processing of auxiliary mirror and the debugging of recording parameters, technique not easily realizes, often causes the incisure density function of the grid pitch changing grating made and expect that incisure density function exists larger error.Because spherical wave exposure light path only has four recording parameterses, technique realizes adjustment accuracy relatively easy, and then obtaining the grid pitch changing grating meeting designing requirement, the Varied Line Space Plane Gratings therefore furtherd investigate under spherical wave exposure designs and the optimized algorithm of recording parameters has actual application value and active demand.

Summary of the invention

The present invention is the incisure density function of the grid pitch changing grating solving the making of existing method and expects that incisure density function exists the problem of larger error, provides a kind of Optimization Design of Varied Line Space Plane Gratings.

Step one, utilize spherical wave exposure system, the meridian obtaining Varied Line Space Plane Gratings expects incisure density function, is formulated as:

$n=n_{10}+n_{20}y+\frac{3}{2}{n}_{30}{y}^2+\frac{1}{2}{n}_{40}{y}^3$

In formula, n ₁₀, n ₂₀, n ₃₀and n ₄₀for coefficient expects the coefficient of incisure density function;

Step 2, iconal F according to grid pitch changing grating, obtain the out of focus F of plane grating ₂₀, meridian coma F ₃₀, spherical aberration F ₄₀, F is formulated as:

F＝r _C+r _D+yF ₁₀+y ²F ₂₀+y ³F ₃₀+y ⁴F ₄₀

$F_{20}=\frac{1}{2}[\frac{n_{20}}{n_{10}}+n_{10}\mathrm{m\λ}(\frac{{\cos }^2\mathrm{\α}}{r_1}+\frac{{\cos }^2\mathrm{\β}}{r_2})]$

$F_{30}=\frac{1}{2}[\frac{n_{30}}{n_{10}}+n_{10}\mathrm{m\λ}(\frac{\sin \mathrm{\α}{\cos }^2\mathrm{\α}}{{r_1}^2}+\frac{\sin \mathrm{\β}{\cos }^2\mathrm{\β}}{{r^2}_2})]$

$F_{40}=\frac{1}{8}[\frac{n_{40}}{n_{10}}+n_{10}\mathrm{m\λ}(\frac{4{\sin }^2\mathrm{\α}{\cos }^2\mathrm{\α}-{\cos }^4\mathrm{\α}}{{r_1}^3}+\frac{4\mathrm{si}{n}^2\mathrm{\β}{\cos }^2\mathrm{\β}-{\cos }^4\mathrm{\β}}{{r_2}^3})]$

In above formula, m is secondary+1 grade of the order of diffraction, and α is use angle, and β is angle of diffraction, and r1 is for entering arm lengths, and r2 is for going out arm lengths, and λ is wavelength;

Step 3, according to expecting out of focus F in incisure density function and step 2 in grating equation, step one ₂₀, meridian coma F ₃₀with spherical aberration F ₄₀value, make grid pitch changing grating focus on F ₂₀=0, coma F ₃₀=0 and spherical aberration F ₄₀=0, be namely formulated as:

$\left\{\begin{array}{c}\sin \mathrm{\α}+\sin \mathrm{\β}=n_{10}\mathrm{m\λ}\\ F_{20}=\frac{1}{2}(\frac{{\cos }^2\mathrm{\α}}{r_1}+\frac{{\cos }^2\mathrm{\β}}{r_2}+n_{20}\mathrm{m\λ})=0\\ F_{30}=\frac{1}{2}(\frac{\sin \mathrm{\α}{\cos }^2\mathrm{\α}}{{r_1}^2}-\frac{\sin \mathrm{\β}{\cos }^2\mathrm{\β}}{{r_2}^2}+n_{30}\mathrm{m\λ})=0\\ F_{40}=\frac{1}{8}(\frac{4{\sin }^2\mathrm{\α}{\cos }^2\mathrm{\α}-{\cos }^4\mathrm{\α}}{{r_1}^3}-\frac{4{\sin }^2\mathrm{\β}{\cos }^2\mathrm{\β}-{\cos }^4\mathrm{\β}}{{r_2}^3}+n_{40}\mathrm{m\λ})=0\end{array}\right.$

Above formula is solved, obtains the expectation incisure density function coefficients n of Varied Line Space Plane Gratings ₁₀, n ₂₀, n ₃₀, n ₄₀, described n ₁₀, n ₂₀, n ₃₀, n ₄₀value be formulated as:

$n_{10}=\frac{1}{{\mathrm{\λ}}_0}(\sin \mathrm{\δ}-\sin \mathrm{\γ})$

$n_{20}=\frac{1}{{\mathrm{\λ}}_0}(\frac{{\cos }^2\mathrm{\γ}}{r_C}-\frac{{\cos }^2\mathrm{\δ}}{r_D})$

$n_{30}=\frac{1}{{\mathrm{\λ}}_0}(\frac{{\cos }^2\mathrm{\γ}\sin \mathrm{\γ}}{{r_C}^2}-\frac{{\cos }^2\mathrm{\δ}\sin \mathrm{\δ}}{{r_D}^2})$

$n_{40}=\frac{1}{{\mathrm{\λ}}_0}(\frac{4{\cos }^2\mathrm{\γ}{\sin }^2\mathrm{\γ}}{r_C^3}-\frac{4{\cos }^2\mathrm{\δ}{\sin }^2\mathrm{\δ}}{r_D^3}+\frac{{\cos }^2\mathrm{\γ}}{{r_C}^3}+\frac{{\cos }^4\mathrm{\δ}}{{r_D}^3})$

In formula, γ and δ angle is be respectively the light of spherical wave by grating substrate central point and the angle of grating substrate normal, and and r _cand r _dcommon as spherical wave recording parameters (γ, r _c, δ, r _d);

Step 4, optimization spherical wave recording parameters (γ, r _c, δ, r _d), grating substrate surface each point is expected that incisure density is optimized as objective function with the integrated square function of design groove-width error, and tries to achieve minimum value, realize the optimal design of Varied Line Space Plane Gratings;

Described objective function obj is formulated as:

$\mathrm{obj}=k_1^2+\frac{1}{3}{y_0}^2(2k_1{k}_3+k_2^2)+\frac{1}{5}{y}_0^4(k_3^2+2k_2{k}_4)+\frac{1}{7}{y}_0^6{k_4}^2$

$\left\{\begin{array}{c}{k}_1=n_{10}-n_0\\ k_2=n_{20}-n_1\\ k_3=\frac{3}{2}{n}_{30}-n_2\\ k_4=\frac{1}{2}{n}_{40}-n_3\end{array}\right.$

In formula, y ₀for grating half width, k ₁, k ₂, k ₃, k ₄for design incisure density coefficient and the error expecting incisure density coefficient.

Beneficial effect of the present invention: the present invention intends based on grid pitch changing grating Aberration Theory, adopt the new algorithm that a kind of global optimization combines with local optimum, research is applied to the optimization principles of Varied Line Space Plane Gratings design, design the holographic grid pitch changing grating of high light spectral resolution, be intended to the approach that exploration spherical wave exposure system makes high precision plane grid pitch changing grating.

Grid pitch changing grating Optimization Design of the present invention is compared with traditional multivariable nonlinearity solving equations, the single-goal function formula set up is when optimizing, rational restriction can be done to the recording parameters span solved, the irrationality of optimum results (cannot realize in actual light path) can be avoided like this when calculating, shorten computing time, improve operation efficiency.Adopt computer numerical method to solve, the design parameter of record grid pitch changing grating can be drawn.

Accompanying drawing explanation

Fig. 1 is spherical wave exposure system schematic diagram in the Optimization Design of grid pitch changing grating of the present invention;

Fig. 2 is grid pitch changing grating exposure system light path schematic diagram of the present invention.

Embodiment

Embodiment one, composition graphs 1 and Fig. 2 illustrate present embodiment, the Optimization Design of grid pitch changing grating, before making grid pitch changing grating, has to pass through strict Theoretical Design, were it not for rational Theoretical Design, it is inconceivable for making grid pitch changing grating.Tell about Theoretical Design process below in detail.

From iconal, the optical path difference of any point in substrate is calculated according to spherical wave geometric theory, Fermat principle is applied in iconal, and series expansion is carried out to it, finally obtain the expectation incisure density function of grid pitch changing grating, Fig. 1 is the spherical wave exposure system schematic diagram of grid pitch changing grating.The matching degree of main consideration incisure density in grating meridian ellipse, ignores curve tendency for the time being, makes z=0, grid pitch changing grating is expected incisure density function is reduced to

$n=n_{10}+n_{20}y+\frac{3}{2}{n}_{30}{y}^2+\frac{1}{2}{n}_{40}{y}^3---\left(1\right)$

In formula, n ₁₀, n ₂₀, n ₃₀and n ₄₀for coefficient expects the coefficient of incisure density function; And n ₁₀, n ₂₀, n ₃₀and n ₄₀about spherical wave recording parameters (γ, r _c, δ, r _d) function, described n ₁₀, n ₂₀, n ₃₀and n ₄₀be formulated as:

$\left\{\begin{array}{c}{n}_{10}=\frac{1}{{\mathrm{\λ}}_0}(\sin \mathrm{\δ}-\sin \mathrm{\γ})\\ n_{20}=\frac{1}{{\mathrm{\λ}}_0}(\frac{{\cos }^2\mathrm{\γ}}{r_C}-\frac{{\cos }^2\mathrm{\δ}}{r_D})\\ n_{30}=\frac{1}{{\mathrm{\λ}}_0}(\frac{{\cos }^2\mathrm{\γ}\sin \mathrm{\γ}}{{r_C}^2}-\frac{{\cos }^2\mathrm{\δ}\sin \mathrm{\δ}}{{r_D}^2})\\ n_{40}=\frac{1}{{\mathrm{\λ}}_0}(\frac{4{\cos }^2\mathrm{\γ}{\sin }^2\mathrm{\γ}}{r_C^3}-\frac{{4\cos }^2\mathrm{\δ}{\sin }^2\mathrm{\δ}}{r_D^3}+\frac{{\cos }^4\mathrm{\γ}}{{r_C}^3}+\frac{{\cos }^4\mathrm{\δ}}{{r_D}^3})\end{array}\right.---\left(2\right)$

In formula, γ and δ angle is be respectively the light of spherical wave by grating substrate central point and the angle of grating substrate normal, and and r _cand r _dcommon as spherical wave recording parameters (γ, r _c, δ, r _d);

The iconal F of grid pitch changing grating can be expressed as the function about (y, z):

F＝r _C+r _D+yF ₁₀+y ²F ₂₀+y ³F ₃₀+y ⁴F ₄₀ (3)

$F_{20}=\frac{1}{2}[\frac{n_{20}}{n_{10}}+n_{10}\mathrm{m\λ}(\frac{{\cos }^2\mathrm{\α}}{r_1}+\frac{{\cos }^2\mathrm{\β}}{r_2})]$

$F_{30}=\frac{1}{2}[\frac{n_{30}}{n_{10}}+n_{10}\mathrm{m\λ}(\frac{\sin \mathrm{\α}{\cos }^2\mathrm{\α}}{{r_1}^2}+\frac{\sin \mathrm{\β}{\cos }^2\mathrm{\β}}{{r^2}_2})]$

$F_{40}=\frac{1}{8}[\frac{n_{40}}{n_{10}}+n_{10}\mathrm{m\λ}(\frac{4{\sin }^2\mathrm{\α}{\cos }^2\mathrm{\α}-{\cos }^4\mathrm{\α}}{{r_1}^3}+\frac{4\mathrm{si}{n}^2\mathrm{\β}{\cos }^2\mathrm{\β}-{\cos }^4\mathrm{\β}}{{r_2}^3})]$

In above formula, m is secondary+1 grade of the order of diffraction, and α is use angle, and β is angle of diffraction, and r1 is for entering arm lengths, and r2 is for going out arm lengths, and λ is wavelength;

Wherein F _ijcan be expressed as:

F _ij＝C _ij+n ₁₀mλM _ij (4)

I is more than or equal to the positive integer that 1 is less than or equal to 4, and j is 0;

$\left\{\begin{array}{c}{M}_{10}=1,C_{10}=-\sin \mathrm{\α}-\sin \mathrm{\β}\\ M_{20}=\frac{n_{20}}{2n_{10}},C_{20}=\frac{1}{2}(\frac{{\cos }^2\mathrm{\α}}{r_1}+\frac{{\cos }^2\mathrm{\β}}{r_2})\\ M_{30}=\frac{n_{30}}{{2n}_{10}},C_{30}=\frac{1}{2}(\frac{\sin \mathrm{\α}{\cos }^2\mathrm{\α}}{{r_1}^2}+\frac{\sin \mathrm{\β}{\cos }^2\mathrm{\β}}{{r^2}_2})\\ M_{40}=\frac{n_{40}}{{8n}_{10}},C_{40}=\frac{1}{8}(\frac{4{\sin }^2\mathrm{\α}{\cos }^2\mathrm{\α}-{\cos }^4}{{r_1}^3}+\frac{4{\sin }^2\mathrm{\β}{\cos }^2\mathrm{\β}-{\cos }^4\mathrm{\β}}{{r_2}^3})\end{array}---\left(5\right)\right.$

In formula, M _ijby use point A, B (operation parameter r _a, r _b) coordinate determine, C _ijdetermined by the coordinate of measuring point C, D.F _ijbe the aberration of varied line spacing plane holographic grating, wherein F ₂₀for out of focus, F ₃₀for coma, F ₄₀for spherical aberration, other represent higher order aberratons.Can find out that iconal F is not only relevant with the position of operation parameter A, B 2 by formula (5), but also relevant with the position of measuring point light source C, D.

Known by analyzing above, aberration F _ijjointly determined by operation parameter and recording parameters.For specific use wavelength, and by reasonably selecting recording parameters can make specific F when operation parameter is determined _ij=0, grid pitch changing grating can be made in the imaging of xy flat focus.But, can not realize in practice eliminating aberration completely at the full service band of spectral instrument, need to eliminate all kinds of aberration targetedly.

According to grating operation parameter, grating equation and formula (4), (5), make grid pitch changing grating out of focus F ₂₀=0, coma F ₃₀=0 and spherical aberration F ₄₀=0, namely

$\left\{\begin{array}{c}\sin \mathrm{\α}+\sin \mathrm{\β}=n_{10}\mathrm{m\λ}\\ F_{20}=\frac{1}{2}(\frac{{\cos }^2\mathrm{\α}}{r_1}+\frac{{\cos }^2\mathrm{\β}}{r_2}+n_{20}\mathrm{m\λ})=0\\ F_{30}=\frac{1}{2}(\frac{\sin \mathrm{\α}{\cos }^2\mathrm{\α}}{{r_1}^2}-\frac{\sin \mathrm{\β}{\cos }^2\mathrm{\β}}{{r_2}^2}+n_{30}\mathrm{m\λ})=0\\ F_{40}=\frac{1}{8}(\frac{4{\sin }^2\mathrm{\α}{\cos }^2\mathrm{\α}-{\cos }^4\mathrm{\α}}{{r_1}^3}-\frac{4{\sin }^2\mathrm{\β}{\cos }^2\mathrm{\β}-{\cos }^4\mathrm{\β}}{{r_2}^3}+n_{40}\mathrm{m\λ})=0\end{array}\right.---\left(6\right)$

The expectation incisure density function coefficients n of grid pitch changing grating can be obtained to this solving equations ₁₀, n ₂₀, n ₃₀, n ₄₀.

Therefore incisure density function is

n(y)＝n ₁₀-n ₂₀y+n ₃₀y ²-n ₄₀y ³ (7)

According to grid pitch changing grating incisure density function (7) formula of being tried to achieve by spectral instrument request for utilization, be optimized design to grid pitch changing grating recording parameters, its method selects suitable recording parameters to make the incisure density function of the grid pitch changing grating made in grating effective coverage, approach (7) formula.But, due to the multi-variable function of grid pitch changing grating incisure density function to be one with recording parameters be variable, need to set up the grid pitch changing grating incisure density objective function using (7) formula as expectation function, if

n _e＝n ₀+n ₁y+n ₂y ²+n ₃y ³ (8)

For spherical wave exposure light path, set up Solving Nonlinear Systems of Equations coefficient n according to formula (1), (2) and (8) ₀, n ₁, n ₂and n ₃, they are about recording parameters (γ, r _c, δ, r _d) function, λ ₀for recording wavelength, namely

$\left\{\begin{array}{c}\frac{1}{{\mathrm{\λ}}_0}(\sin \mathrm{\δ}-\sin \mathrm{\γ})=n_0\\ \frac{1}{{\mathrm{\λ}}_0}(\frac{{\cos }^2\mathrm{\γ}}{r_C}-\frac{{\cos }^2\mathrm{\δ}}{r_D})=n_1\\ \frac{1}{{\mathrm{\λ}}_0}(\frac{{\cos }^2\mathrm{\γ}\sin \mathrm{\γ}}{{r_C}^2}-\frac{{\cos }^2\mathrm{\δ}\sin \mathrm{\δ}}{{r_D}^2})=n_2\\ \frac{1}{{\mathrm{\λ}}_0}(\frac{4{\cos }^2\mathrm{\γ}{\sin }^2\mathrm{\γ}}{r_C^3}-\frac{4{\cos }^2\mathrm{\δ}{\sin }^2\mathrm{\δ}}{r_D^3}+\frac{{\cos }^4}{{r_C}^3}+\frac{{\cos }^4\mathrm{\δ}}{{r_D}^3})=n_3\end{array}\right.---\left(9\right)$

But effective constraint cannot be set up, n when solving formula (9) ₀, n ₁, n ₂and n ₃be difficult to meet the expectation value, or the value that simultaneously meets the expectation but cannot obtain rational recording parameters (γ, r simultaneously _c, δ, r _d).For this reason, multi objective function optimization is reduced to simple target function optimization, expects that incisure density is optimized as objective function with the integrated square function of design groove-width error by grating surface each point, ask its minimum value, obtain desirable recording parameters thus, namely objective function is

Described objective function obj is formulated as:

$\mathrm{obj}=k_1^2+\frac{1}{3}{y_0}^2(2k_1{k}_3+k_2^2)+\frac{1}{5}{y}_0^4(k_3^2+2k_2{k}_4)+\frac{1}{7}{y}_0^6{k_4}^2---\left(10\right)$

$\left\{\begin{array}{c}{k}_1=n_{10}-n_0\\ k_2=n_{20}-n_1\\ k_3=\frac{3}{2}{n}_{30}-n_2\\ k_4=\frac{1}{2}{n}_{40}-n_3\end{array}\right.---\left(11\right)$

In formula, y ₀for grating half width, k ₁, k ₂, k ₃, k ₄for design incisure density coefficient and the error expecting incisure density coefficient.

Composition graphs 2 illustrates present embodiment, configures light path by shown in Fig. 2.Comprise Kr ⁺laser instrument 1, first plane mirror 2, half-reflecting half mirror 3, second plane mirror 4, first pinhole filter the 5, three plane mirror 6, second pinhole filter 7, interference field region 8, grating substrate 9, γ and δ angle are respectively the light of two bundle spherical waves by grating substrate central point and the angle of grating substrate normal.Light beam is sent by laser beam 1, through the first plane mirror 2, two-beam is divided into after half-reflecting half mirror 3, this two-beam is respectively through the second plane mirror 4 and the first pinhole filter 5, two bundle spherical waves are formed after 3rd plane mirror 6 and the second pinhole filter 7, this two bundles spherical wave intersects at a certain angle, and form interference field region 8, this interference field region is exactly the exposure region that we will use.The grating substrate 9 scribbling photoresist is placed in exposure region with certain angle, chooses the suitable time shutter according to light intensity magnitude, after development, just can obtain grating cutting on grating substrate.This experiment selects laser wavelength of incidence to be the Kr of 431.1nm ⁺laser instrument, obtains the grid pitch changing grating of different grating constant by changing γ and δ angle, the grating of what in experiment, we made is 600 lines per millimeters.

In present embodiment, when the monochromatic spherical wave of two bundles meets with certain angle, the region of intersecting at them will form interference field, and produce light and dark interference fringe, this interference field is exactly the exposure region that we will use.If the substrate scribbling photoresist is placed in interference field, so produce cutting at substrate surface by because of light and dark interference fringe.By calculating the position (comprising distance and the angle of wave source point and grating substrate central point) that can draw record spherical wave wave source, theoretical analysis and result of calculation show, the incisure density of grating can be changed by the position changing spherical wave wave source, and then make grating line density reach design load.

According to the method design Varied Line Space Plane Gratings of present embodiment design, the operation parameter of grating is:

Grating line density 600gr/mm, wavelength coverage 50-150nm, enters arm lengths 19000mm, incident angle 87.5 °, detector position 1500mm, resolution 12000.

Through calculating, the recording parameters obtaining Varied Line Space Plane Gratings is:

Measuring point 1 position 1463.88mm, measuring point 1 angle 88.46 °, measuring point 1 position 1322.6mm, measuring point 1 angle 47.27 °.

Varied Line Space Plane Gratings incisure density coefficient optimum results is:

n(y)＝600-0.7869y+6.5573×10 ^-4y ²-3.8229×10 ^-7y ³。

Described in present embodiment is the optimization of grid pitch changing grating, and traditional diffraction grating such as is all at the pitch grating, and they are core parts of various spectral instrument.But along with the progress of science and technology, the particular advantages such as the aberration correction that grid pitch changing grating has, high resolving power and flat Jiao Chang become increasingly conspicuous, make it in field widespread uses such as spatial light spectrometer, plasma diagnostics, synchrotron radiation monochromator, optical fiber communications.Have the grid pitch changing grating of aberration correction function, can obtain high resolving power under glancing incidence condition, this feature will obtain further development and utilization in monochromator and spectrometer, this to grenz ray and extreme ultraviolet radiation significant.

Claims (2)

1. the Optimization Design of Varied Line Space Plane Gratings, is characterized in that, the method is realized by following steps:

Step one, utilize spherical wave exposure system, the meridian obtaining Varied Line Space Plane Gratings expects incisure density function, is formulated as:

$n=n_{10}+n_{20}y+\frac{3}{2}{n}_{30}{y}^2+\frac{1}{2}{n}_{40}{y}^3$

In formula, n ₁₀, n ₂₀, n ₃₀and n ₄₀for coefficient expects the coefficient of incisure density function;

Step 2, iconal F according to grid pitch changing grating, obtain the out of focus F of plane grating ₂₀, meridian coma F ₃₀, spherical aberration F ₄₀, F is formulated as:

F＝r _C+r _D+yF ₁₀+y ²F ₂₀+y ³F ₃₀+y ⁴F ₄₀

$F_{20}=\frac{1}{2}\left[\frac{n_{20}}{n_{10}}{n}_{10}\mathrm{m\λ}(\frac{{\cos }^2\mathrm{\α}}{r_1}+\frac{{\cos }^2\mathrm{\β}}{r_2})\right]$

$F_{30}=\frac{1}{2}[\frac{n_{30}}{n_{10}}+n_{10}\mathrm{m\λ}(\frac{\sin \mathrm{\α}{\cos }^2\mathrm{\α}}{{r_1}^2}+\frac{\sin \mathrm{\β}{\cos }^2\mathrm{\β}}{{r^2}_2})]$

$F_{40}=\frac{1}{8}[\frac{n_{40}}{n_{10}}+n_{10}\mathrm{m\λ}(\frac{4{\sin }^2\mathrm{\α}{\cos }^2\mathrm{\α}-{\cos }^4\mathrm{\α}}{{r_1}^3}+\frac{4{\sin }^2\mathrm{\β}{\cos }^2\mathrm{\β}-{\cos }^4\mathrm{\β}}{{r_2}^3})]$

In above formula, m is secondary+1 grade of the order of diffraction, and α is use angle, and β is angle of diffraction, and r1 is for entering arm lengths, and r2 is for going out arm lengths, and λ is wavelength;

Step 3, according to expecting out of focus F in incisure density function and step 2 in grating equation, step one ₂₀, meridian coma F ₃₀with spherical aberration F ₄₀value, make grid pitch changing grating focus on F ₂₀=0, coma F ₃₀=0 and spherical aberration F ₄₀=0, be namely formulated as:

$\left\{\begin{array}{c}\sin \mathrm{\α}+\sin \mathrm{\β}=n_{10}\mathrm{m\λ}\\ F_{20}=\frac{1}{2}(\frac{{\cos }^2\mathrm{\α}}{r_1}+\frac{{\cos }^2\mathrm{\β}}{r_2}+n_{20}\mathrm{m\λ})=0\\ F_{30}=\frac{1}{2}(\frac{\sin \mathrm{\α}{\cos }^2\mathrm{\α}}{{r_1}^2}-\frac{\sin \mathrm{\β}{\cos }^2\mathrm{\β}}{{r_2}^2}+n_{30}\mathrm{m\λ})=0\\ F_{40}=\frac{1}{8}(\frac{4{\sin }^2\mathrm{\α}{\cos }^2\mathrm{\α}-{\cos }^4\mathrm{\α}}{{r_1}^3}-\frac{4{\sin }^2\mathrm{\β}{\cos }^2\mathrm{\β}{\cos }^4\mathrm{\β}}{{r_2}^3}+n_{40}\mathrm{m\λ})=0\end{array}\right.$

Above formula is solved, obtains the expectation incisure density function coefficients n of Varied Line Space Plane Gratings ₁₀, n ₂₀, n ₃₀, n ₄₀, described n ₁₀, n ₂₀, n ₃₀, n ₄₀value be formulated as:

$n_{10}=\frac{1}{{\mathrm{\λ}}_0}(\sin \mathrm{\δ}-\sin \mathrm{\γ})$

$n_{20}=\frac{1}{{\mathrm{\λ}}_0}(\frac{{\cos }^2\mathrm{\γ}}{r_C}-\frac{{\cos }^2\mathrm{\δ}}{r_D})$

$n_{30}=\frac{1}{{\mathrm{\λ}}_0}(\frac{{\cos }^2\mathrm{\γ}\sin \mathrm{\γ}}{{r_C}^2}-\frac{{\cos }^2\mathrm{\δ}\sin \mathrm{\δ}}{{r_D}^2})$

$n_{40}=\frac{1}{{\mathrm{\λ}}_0}(\frac{4{\cos }^2\mathrm{\γ}{\sin }^2\mathrm{\γ}}{r_C^3}-\frac{4{\cos }^2\mathrm{\δ}{\sin }^2\mathrm{\δ}}{r_D^3}+\frac{{\cos }^4\mathrm{\γ}}{{r_C}^3}+\frac{{\cos }^4\mathrm{\δ}}{{r_D}^3})$

In formula, γ and δ angle is be respectively the light of spherical wave by grating substrate central point and the angle of grating substrate normal, and and r _cand r _dcommon as spherical wave recording parameters (γ, r _c, δ, r _d);

Step 4, optimization spherical wave recording parameters (γ, r _c, δ, r _d), grating substrate surface each point is expected that incisure density is optimized as objective function with the integrated square function of design groove-width error, and tries to achieve minimum value, realize the optimal design of Varied Line Space Plane Gratings;

Described objective function obj is formulated as:

$\mathrm{obj}=k_1^2+\frac{1}{3}{y_0}^2(2k_1{k}_3+k_2^2)+\frac{1}{5}{y}_0^4(k_3^2+2k_2{k}_4)+\frac{1}{7}{y}_0^6{k_4}^2$

$\left\{\begin{array}{c}{k}_1=n_{10}-n_0\\ k_2=n_{20}-n_1\\ k_3=\frac{3}{2}{n}_{30}-n_2\\ k_4=\frac{1}{2}{n}_{40}-n_3\end{array}\right.$

In formula, y ₀for grating half width, k ₁, k ₂, k ₃, k ₄for design incisure density coefficient and the error expecting incisure density coefficient.

2. the Optimization Design of Varied Line Space Plane Gratings according to claim 1, is characterized in that, the spherical wave exposure system described in step one comprises Kr ⁺laser instrument (1), the first plane mirror (2), half-reflecting half mirror (3), the second plane mirror (4), the first pinhole filter (5), the 3rd plane mirror (6), the second pinhole filter (7), interference field region (8), by Kr ⁺laser instrument (1) sends light beam, through the first plane mirror (2), half-reflecting half mirror is divided into two-beam after (3), this two-beam is respectively through the second plane mirror (4) and the first pinhole filter (5), two bundle spherical waves can be formed after 3rd plane mirror (6) and the second pinhole filter (7), this two bundles spherical wave intersects, and forms interference field region (8).