RU1841096C - Deformable mirror - Google Patents

Abstract

FIELD: physics, optics.

SUBSTANCE: invention relates to optical instrument-making, particularly deformable mirrors. The deformable mirror includes an elastic plate with a reflecting front surface and a variable-thickness profile of the back surface, attached on the periphery to a chassis, as well as deformation means linked to the centre of the plate. The thickness profile of the plate satisfies the expression: $$\frac{dy}{dx}=\frac{32\frac{c^2}{a^2}-n^2(n-2)x^{n-3}{(\alpha +\beta \sin y)}^3}{3n(n-1+\nu )\beta \cos y{(\alpha +\beta \sin y)}^2{x}^{n-1}},$$

where ν is Poisson's ratio; $$a$$

- is the diameter of the deformable mirror; c is the diameter of the deformation means in contact with its back surface; $$x=\frac{r}{a},$$

where r is the current coordinate along the surface of the deformable mirror; n is a given constant 1≤n≤20; $$\beta =\mathrm{0,5}\frac{h_{\min }-h_{\max }}{h_{\max }},$$

$$\alpha =\mathrm{0,5}\frac{h_{\min }+h_{\max }}{h_{\max }},$$

where h_min, h_max denote the minimum and maximum thickness of the profile of the back surface of the plate, determined structurally; $$y=\arcsin \left[\frac{h(x)}{h_{\max }\beta }-\frac{\alpha }{\beta }\right],$$

where h(x) is the current thickness of the profile of the plate.

EFFECT: high accuracy of aberration compensation.

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Description

The invention relates to optical instrumentation, in particular to deformable planoid mirrors designed to compensate for aberrations in mirror optical systems.

Mirrors with a complex aspherical profile of the reflecting surface, obtained from an initially flat mirror, are called planoidal (D. D. Maksutov, Astronomical Optics, Nauka, 1979, p. 319). Mirror optical systems with such mirrors are among the most high-quality ones in the image they form (G. M. Popov, "Modern Astronomical Optics", Nauka, 1988, p. 145).

It is known that a deformable mirror is used as a secondary mirror in the Cassegrain mirror system, containing an elastic plate with a reflective front layer, peripherally connected to the bearing body, means of its deformation, for example, a column of piezoelectric material, connecting the center of the elastic plate and the base of the bearing body (French Patent No. 2.519.151, M.C. ⁴ G02B 23/06).

The quality of compensation of the aberrations of the mirror system is determined by the shape of the reflecting surface of the deformable planoid mirror.

In an analog device, as shown in the description, the shape of the reflective surface after deformation is a profile close to parabolic.

The disadvantages of the known design are:

- the need to use primary mirrors with aspherical surfaces in such a mirror system with such mirrors to ensure a minimum of spherical aberration and higher order aberrations, which leads to an increase in labor input and a decrease in the manufacturability of the structure as a whole;

- low quality compensation of aberrations of the optical system since only one of the lower aberrations is eliminated - defocusing; (see Fig. 3, curve II);

- the lack of strict repeatability and certainty of the profile of the reflecting surface due to the strong influence of not one but a set of design parameters: the method of pinching the elastic plate, the properties and type of its material, as well as the configuration of the connection between the end face of the deforming drive and its rear side.

A known design of a deformable planoid mirror, selected as a prototype, containing an elastic plate with a reflective coating, pinched along the contour in the bearing housing, the means of its deformation associated with its center, while its thickness varies according to the equation:

$$\frac{\text{h}}{\text{R}}=-{\left[\text{12 (1}-\text{ν) Ω}\frac{\text{P}}{\text{E}}\right]}^{\text{1/3}}{{\text{(iρ}}^{\text{2}}+{\text{jlnρ}}^{\text{2}}+\text{k)}}^{\text{1/3}}\text{(one)}$$

where E and ν are Young's modulus and Poisson's ratio, respectively;

R is the radius of the deformable mirror;

, $$\Omega =\frac{r}{\mathrm{four}R}$$

,

where r is the radius of curvature formed by the deformable mirror;

P is the load per unit surface;

ρ is the normalized coordinate along the surface;

i, j, k are constant coefficients, the value of which is 1, 0, -1 depending on the load and support conditions (French Patent No. 2,343.262, M. Cl ⁴ G02B 5/10).

The preset distribution of the thickness of the elastic plate of the deformable mirror reduces the influence of other design parameters on its profile and makes it possible to reproduce the specified parabolic or spherical profile of its surface with high accuracy.

The disadvantages of the known design are:

- the need to use primary mirrors with aspherical surfaces in the mirror system to provide compensation for higher-order spherical aberration and aberration;

- low quality compensation of higher aberrations of the optical system.

The aim of the invention is to increase the accuracy of compensation for aberrations in a mirror optical system.

This goal is achieved by the fact that in a deformable mirror containing an elastic plate with a reflecting front surface and a thickness profile of the back surface fixed on the periphery on the supporting body, as well as deformation means connected to the center of the plate, the plate thickness profile is made according to the law with the given expression:

$$\frac{\text{dy}}{\text{dx}}=\frac{\text{32}\frac{{\text{c}}^{\text{2}}}{a^{\text{2}}}-{\text{n}}^{\text{2}}\text{(n}-{\text{2) x}}^{\text{n}-\text{3}}{\text{(α}+\text{βsiny)}}^{\text{3}}}{\text{3n (n}-\text{one}+\text{ν) β cozy (α}+{\text{βsiny)}}^{\text{2}}{\text{x}}^{\text{n-1}}}\text{(2)}$$

where ν is the Poisson's ratio,

a is the diameter of the deformable mirror;

c is the diameter of the deforming means in contact with its rear surface;

$$\text{x}=\frac{\text{r}}{a}$$

where r is the current coordinate along the surface of the deformable mirror;

n is a given constant 1≤n≤20;

, $$\alpha =\frac{h_{\min }+h_{\max }}{h_{\max }}\cdot \mathrm{0,5}$$

$$\beta =0.5\frac{h_{\min }-h_{\max }}{h_{\max }}$$

, $$\alpha =\frac{h_{\min }+h_{\max }}{h_{\max }}\cdot 0.5$$

where h _min , h _max - the minimum and maximum thickness of the plate profile, specified from design considerations,

$$\text{y}=\text{arcsin}\left[\frac{\text{h (r)}}{{\text{h}}_{\text{max}}\text{β}}-\frac{\text{α}}{\text{β}}\right]$$

where h (r) is the profile of the current plate thickness.

The specified profile of variable thickness of the plate (2) can be implemented by traditional methods on metalworking equipment (I. A. Druzhinsky, "Complex surfaces", Engineering, 1985, p. 241). The resulting profile of the thickness of the elastic plate during its deformation by a centrally applied force makes it possible to reproduce a form of the form W _o X ⁿ , where W _o is the maximum deflection of the plate, which makes it possible to compensate for all the calculated aberrations of mirror optical systems. The constant n is determined based on the calculation of the total wave aberration of the mirror system.

When compensating for spherical aberration n = 4 (D. D. Maksutov, Astronomical Optics, Nauka, 1979, p. 319). In addition, relation (2) sets the profile of the thickness of the elastic plate with the possibility of limiting its changes to the values of h _min , h _max specified at the design stage.

Comparative analysis with the prototype shows that the inventive deformable mirror has a profile that varies in thickness, in accordance with a given ratio (2).

Thus, the claimed deformable mirror meets the criterion of "novelty." Comparison of the claimed solution not only with the prototype, but also with other technical solutions in this technical field did not allow us to identify in them the features that distinguish the claimed solution from the prototype, which allows us to conclude that the claimed technical solution meets the criterion of "significant differences".

The ability to compensate for the higher aberrations using the inventive deformable mirror, which improves the quality of mirror optical systems in comparison with the compensation of low aberrations by the prototype mirror, allows us to conclude that the claimed solution meets the criterion of "positive effect".

The invention is illustrated by drawings:

In FIG. 1 shows a General view of the structural design of the device;

In FIG. 2 shows the design of a manual drive for microdeformations of the reflective surface of the device.

In FIG. 3 presents, exaggerated, the profiles of the reflective surfaces of the analogue and the claimed device when compensating for spherical aberration.

In FIG. 4 shows the design of a mirror optical system with the claimed device.

The deformable mirror contains (see Fig. 1) an elastic plate 1 made of metal, for example, copper, with a reflective front layer 2 obtained by fine diamond turning of the surface of the plate 1, the thickness of which varies along the profile 3, connected by a ring 4 in its the periphery with a supporting body 5 made of a material with the same coefficient of thermal expansion as the plate 1, means for deforming the plate 1 in the form of a manual drive of micro displacements 6, one end of which is connected to the center of the plates 1 and the other is rigidly connected to 5. rpusom micromovings actuator (see. Fig. 2) comprises a nut 7 and 8, the pipe 9 with openings 10 and screw 11 with a conical tail.

The deformable mirror works as follows: when the screw with the tapered tail 11 in the nut 8 rotates, it moves, pushing the walls of the tube 9, which is symmetrically deformed due to the slots 10. With such a deformation (see Fig. 2), the drive length decreases and the nuts 7 and 8 come together. Since one nut 8 is clamped in the bearing housing 5 (see Fig. 1), and the other 7 is connected with the center of the elastic plate 1, the plate is deformed. As you know, (S.P. Timoshenko, "Plates and Shells", Nauka, 1966, p. 334), the deformation of a plate with a profile of variable thickness is described by the equation:

$$\text{D}\frac{\text{d}}{\text{dr}}\left(\frac{\text{d}\varphi }{\text{dr}}+\frac{\varphi }{\text{r}}\right)+\frac{\text{dD}}{\text{dr}}\left(\frac{\text{d}\varphi }{\text{dr}}+\text{ν}\frac{\varphi }{\text{r}}\right)=\frac{\text{one}}{\text{r}}{\displaystyle \underset{\text{0}}{\overset{\text{r}}{\int }}\text{q (r) rdr (3)}}$$

where D is the cylindrical stiffness; $$\varphi =-\frac{\text{dw}}{\text{dr}}$$

w is the deflection of the elastic plate;

ν is the Poisson's ratio;

r is the transverse coordinate.

, где E - модуль Юнга; h - распределение толщины упругой пластины. Для отыскания распределения толщины h(r), позволяющего воспроизвести заданный профиль зеркала w(r), необходимо ввести ограничение h_max≤h(r)≤h_min, что обеспечит его конструктивное воссоздание. Такое ограничение может быть учтено за счет подстановки в (3) вместо h(r) вспомогательной функции „y", связанной с h(r) соотношением:In its turn $$D=\frac{E{\text{h}}^{\text{3}}}{12(\mathrm{one}-{\nu }^2)}$$

where E is Young's modulus; h is the distribution of the thickness of the elastic plate. To find the distribution of thickness h (r), which allows reproducing a given mirror profile w (r), it is necessary to introduce the restriction h _max ≤h (r) ≤h _min , which will ensure its constructive reconstruction. This restriction can be taken into account by substituting in (3) instead of h (r) the auxiliary function “y” associated with h (r) by the relation:

$$\frac{\text{h (r)}}{{\text{h}}_{\text{max}}}=\text{0}\text{,5}\frac{{\text{h}}_{\text{min}}+{\text{h}}_{\text{max}}}{{\text{h}}_{\text{max}}}+\text{0}\text{,5}\frac{{\text{h}}_{\text{min}}-{\text{h}}_{\text{max}}}{{\text{h}}_{\text{max}}}\text{siny}$$

, где a - диаметр деформируемого зеркала, величину c - диаметр контакта деформирующего привода с упругой пластиной из (3), имеем:Introducing also the dimensionless variable $$\text{x}=\frac{\text{r}}{a}$$

, where a is the diameter of the deformable mirror, c is the diameter of the contact of the deforming drive with the elastic plate from (3), we have:

$$\frac{dy}{dx}=\frac{-\frac{q{c}^{2}a6(\mathrm{one}-{\nu }^2)}{E{h}_{max}^3\cdot x}-\frac{d}{dx}\left(\frac{d\varphi }{dx}+\frac{\varphi }{x}\right){(\alpha +\beta \sin y)}^3}{\left(\frac{d\varphi }{dx}+\nu \frac{\varphi }{x}\right)3\beta \cos y{(\alpha +\beta \sin y)}^2}\text{(four)}$$

; $$\beta =\text{0}\text{,5}\cdot \frac{{\text{h}}_{\text{min}}-{\text{h}}_{\text{max}}}{{\text{h}}_{\text{max}}}$$

$$\alpha =\text{0}\text{,5}\cdot \frac{{\text{h}}_{\text{min}}+{\text{h}}_{\text{max}}}{{\text{h}}_{\text{max}}}$$

; $$\beta =\text{0}\text{,5}\cdot \frac{{\text{h}}_{\text{min}}-{\text{h}}_{\text{max}}}{{\text{h}}_{\text{max}}}$$

For a centered mirror optical system, the total aberration, symmetric with respect to the optical axis, can generally be expressed by a relation of the form:

$$w\text{'}\text{'}(x)=w_{\max }{x}^n\text{(5)}$$

Therefore, the elastic plate must be bent along the curve w (x) = w ′ (x) to ensure its compensation

where w (x) is the profile of the curved plate.

Given this expression (4), we rewrite:

$$\frac{dy}{dx}=\frac{32\frac{c^2}{a^2}-n^2(n-2)x^{n-3}{(\alpha +\beta \sin y)}^3}{3n(n-\mathrm{one}+\nu )\beta \cos y{(\alpha +\beta \sin y)}^2{x}^{n-\mathrm{one}}}\text{(6)}$$

The desired distribution of thickness is found from the ratio:

$$\frac{\text{h (r)}}{{\text{h}}_{\text{max}}}=\text{α}+\text{βsiny (7)}$$

The calculation of the function y according to relation (6) is carried out on a MK-61 microcalculator using a program from (A.N. Tsvetkov, V.A. Enanechnikov, "Application Programs for Microcomputers", Finance and Statistics, 1984, p. 60) 15 minutes.

The tolerance for technological reproducibility of the profile is determined from the following considerations. As is known, the deflection of an elastic plate in general form can be represented:

$$w(x)=\alpha \frac{q{a}^{\mathrm{four}}12(\mathrm{one}-{\nu }^2)}{E{h}^3(x)}\cdot \beta (x)\text{(8)}$$

where α is a coefficient depending on the initial conditions of the problem, and β (x) is a constant function for given initial conditions.

Differentiating relation (8) with respect to h, we have:

$$\frac{dw}{dh}=-\alpha \cdot \beta (x)\frac{12q{a}^{\mathrm{four}}(\mathrm{one}-{\nu }^2)}{E}\cdot \frac{3}{h^{\mathrm{four}}}\text{(9)}$$

Because the greatest error occurs at x = 0.5, since at x = 0, -1; +1 the boundary conditions are imposed on the deflection, we have:

$$\Delta w_{\max }={-\alpha \cdot \beta (x)|}_{x=0.5}\frac{12q{a}^{\mathrm{four}}(\mathrm{one}-{\nu }^2)}{E}\cdot \frac{3\Delta h}{h^{\mathrm{four}}}\text{(10)}$$

Since, in our case, β (x) is determined by relation (5), we have:

$$\Delta w_{\max }=-{(0.5)}^n{w}_{\max }\frac{3\Delta h}{h}\text{(eleven)}$$

; w_max=±kλAssuming h = 0.5 (h _min + h _max ); $$\Delta w_{\max }=\pm \frac{\lambda }{\mathrm{twenty}}$$

; w _max = ± kλ

where λ is the wavelength, and k is the number of wavelengths that fit in the maximum amplitude of the compensated aberration, we finally have:

$$\Delta h=\frac{h_{\min }+h_{\max }}{120k{(0.5)}^n}\text{(12)}$$

in the particular case of spherical aberration n = 4, and k = 20-30, with h _min = 5 mm h _max = 20 mm, we have:

Δh = 0.17-0.11 mm

Reproduction of a variable profile of an elastic plate with such a tolerance is achieved by mechanical testing of its end surface perpendicular to the axis of rotation and can be provided:

- on a lathe. In this case, the specified profile is provided by the longitudinal-transverse displacement of the caliper;

- on a milling machine a milling cutter with a special profile. Milling can be carried out in several transitions depending on the profile depth h (r).

- on a CNC machine.

The error in reproducing the surface shape by these methods on the walls of the usual degree of accuracy is ± 0.04 mm, which is several times less than the permissible error (12).

The free tolerance (12) for reproducing a given profile h (x) is an important technological condition for the wide use of the stiffness modulation method for deformable mirrors.

The manufacture of an elastic plate with a variable profile defined by relation (2) allows:

- to provide high quality compensation of higher order aberrations with axial symmetry (see Fig. 3 curve 1), which allows mirror optical systems (see Fig. 4), using spherical as the main mirrors, to achieve high quality of the formed image;

- to provide the possibility of using spherical optics in mirror systems, which can significantly increase the manufacturability of the manufacturing process, as well as reduce the cost and time of production.

So, for a real mirror system (see Fig. 4) with parameters: the radius of curvature of the spherical mirror is 494.3 mm, the distance between the primary and secondary mirror is 122.6 mm, the light diameter of the primary mirror is 206 mm.

The ideal profile of the secondary mirror is expressed by the ratio: w (x) = 1.15 · 10 ^-7 x ^3.527 (in mm) (see Fig. 3 curve 1).

If a prototype mirror is used as a secondary mirror, the surface shape will look like curve 2 in FIG. 3 and a residual error with a maximum value of 112 μm will be present in the radiation beam in the plane of the secondary mirror. The residual error when using the inventive mirror is ± 0.15 μm, which meets the quality criterion (DD Maksutov, "Astronomical Optics", 1979, p. 319) at a wavelength of λ≥1 μm.

Claims (1)

A deformable mirror containing an elastic plate with a reflecting front surface and a thickness profile of the back surface fixed peripherally to the bearing body, as well as deformation means connected to the center of the plate, characterized in that, in order to increase the accuracy of aberration compensation, the plate profile is made according to the law:
$$\frac{dy}{dx}=\frac{32\frac{c^2}{a^2}-n^2(n-2)x^{n-3}{(\alpha +\beta \sin y)}^3}{3n(n-\mathrm{one}+\nu )\beta \text{cozy (}\alpha +\beta {\text{siny)}}^{\text{2}}{x}^{n-\mathrm{one}}}$$

Where $$y=\arcsin \left[\frac{h(x)}{h_{\max }\beta }-\frac{\alpha }{\beta }\right]$$

h (x) is the current value of the thickness of the plate profile,
h _min , h _max - respectively, the minimum and maximum thickness of the profile of the back surface of the plate, set structurally,
$$\beta =0.5\frac{h_{\min }-h_{\max }}{h_{\max }}$$

; $$\alpha =0.5\frac{h_{\min }+h_{\max }}{h_{\max }}$$

c is the diameter of the deforming means in contact with the rear side of the mirror,
a is the diameter of the deformable mirror,
n is a given constant 1≤n≤20
$$x=\frac{r}{a}$$

r is the current coordinate along the surface of the mirror,
ν - Poisson's ratio for a given material.

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