JP2641348B2 - Reflector production method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION The present invention minimizes thermal deformation caused by a difference in the coefficient of thermal expansion between partial mirrors when a reflector used for a reflection telescope or the like is constructed by bonding partial mirrors. Thus, the present invention relates to a method for producing a reflecting mirror in which partial mirror materials are arranged and bonded.

[0002]

2. Description of the Related Art There is no clear method for producing a conventional reflecting mirror, and the arrangement of partial mirror materials is intuitively determined and bonded in accordance with individual cases.

FIG. 6 is a perspective view showing a reflecting mirror formed by laminating partial mirrors. A plurality of regular hexagonal partial mirrors 2 (hereinafter, referred to as a stack) as shown in FIG. It is configured as shown in FIG. The surface of the reflecting mirror 1 is, for example, a parabolic surface or a hyperbolic surface with an accuracy of about 1/100 of the observation wavelength so as to reflect and focus electromagnetic waves such as visible light and infrared light arriving from the celestial body. Polished.

If the surface of the reflecting mirror 1 is set to a perfect paraboloid or the like, the incident electromagnetic wave from the celestial body is geometrically focused at one point (focus), but actually, the diffraction of light is performed. Due to the phenomenon, the diameter of the star image of the celestial body does not become zero, and there is a theoretical limit determined by the aperture D of the reflecting mirror 1 and the wavelength λ of the incident electromagnetic wave.

The theoretical limit FWHM (Full Width at Ha)
lf Maximum) is generally expressed as:

[0006]

(Equation 1)

This FWHM is the width when the intensity in the light intensity distribution becomes half of the maximum intensity as shown in FIG. Therefore, the theoretical limit of the size of the star image is determined by the diameter D of the reflecting mirror 1 and the wavelength λ of the incident electromagnetic wave. The larger the diameter D, the smaller the size and the higher the light-gathering power. Increasing the diameter is important for improving the resolution, improving the detection limit, and shortening the exposure time by reducing the size of the star image.

However, actually, since the thermal expansion coefficient of the stack 2 is not zero, when the temperature changes, the reflecting mirror 1 undergoes thermal deformation. Here, if the thermal expansion coefficients of the respective stacks 2 are equal, the respective stacks 2 are deformed in a similar shape, so that only the focal position of the reflecting mirror 1 moves and the imaging accuracy does not deteriorate. Since the coefficient of thermal expansion differs for each, the reflecting mirror 1 generates uneven thermal deformation. When the diameter of the reflecting mirror 1 is large, the stack 2
As the number increases, the deformation becomes more complicated and even a small inclination increases the deformation amount.

Therefore, when such thermal deformation occurs, the light incident from the celestial body is scattered, and the star image has an intensity distribution as shown in FIG. 8 (b), resulting in a blurred image. Even if the diameter of the reflecting mirror 1 is increased, the advantages described above cannot be used.

The main factors of the non-uniformity of the coefficient of thermal expansion of the stack 2 causing such non-uniform thermal deformation are as follows.
There is a type in which the gradient of the thermal expansion coefficient in the thickness direction of each stack 2 is different (causes bimetallic deformation), and a type in which the average thermal expansion coefficient of each stack 2 varies.
As a method of minimizing the thermal deformation, a stack arrangement as shown in FIG. 9 can be considered (first conventional example).

In FIG. 9, the numbers ( Δα _{1 to} Δα ₃₇ ) assigned to the respective stacks 2 correspond to the numbers in all the stacks.
Average value of thermal expansion coefficient and average thermal expansion coefficient of each stack 2
Deviation (hereinafter referred to as the coefficient of thermal _{expansion) (Δ α 1 ...... Δ α} 37)
Which have a large coefficient of thermal expansion ( Δα ₁ ≧
Δα ₂ ... Δα ₃₇ ) and are divided into three groups (cross hatching, hatching by dots, and no shading).

According to this method, the stacks 2 belonging to the group having the medium thermal expansion coefficient and the stack 2 belonging to the small group are arranged around the stack 2 belonging to the group having the large thermal expansion coefficient. In this case, the large thermal expansion of the stack 2 belonging to the group having a large thermal expansion coefficient is moderated by the small thermal expansion of the surrounding stacks 2, and the deformation becomes local, and the amount of deformation is intuitively distributed. Can be expected to be much smaller than if it were offset.

[0013] FIG. 10 is a sectional view of a reflection mirror having a Actuator <br/> eta you correct the thermal deformation (second prior art), 1 reflector, the rear surface of the reflecting mirror 1 3 4 is a processing unit for calculating a correction force from the temperature measurement value of the reflection mirror 1 obtained by the temperature sensor 3, 5 is an actuator controller, and 6 is a correction force applied to the reflection mirror 1 to perform thermal deformation. This is the actuator to be corrected.

According to this method, when correcting thermal deformation,
Attempting to correct all deformations would also correct small pitch irregularities, requiring a large correction power, which is impractical, so expanding the deformation into a series of finite or infinite terms, which is a function of spatial frequency, Only the term with the large pitch is extracted and corrected. At this time, the small pitch irregularities that remain without being corrected result in residual deformation of the mirror surface, deteriorating the quality of the star image.

FIG. 11 shows an arrangement of partial mirror materials in which it is intuitively predicted that thermal deformation will be concentrated on a large pitch of unevenness when only a large pitch of unevenness is corrected as described above. In the figure, the numbers ( Δα _{1 to} Δα ₃₇ ) assigned to each stack 2 are the same as those in FIG.

[0016]

Since the conventional method for producing a reflecting mirror is constructed as described above, only the intuitive arrangement of each stack can be determined. There were issues such as not becoming.

SUMMARY OF THE INVENTION The present invention has been made to solve the above problems, and minimizes the thermal deformation of a reflecting mirror surface or the residual thermal deformation remaining after the thermal deformation is expanded into a series of spatial frequencies and corrected. It is an object of the present invention to obtain a method for producing a reflecting mirror.

[0018]

According to a first aspect of the present invention, there is provided a method for manufacturing a reflecting mirror, comprising: sum of squares of displacement at a plurality of sample points on a mirror surface of the reflecting mirror; The deviation between the average value of the thermal expansion coefficient of each partial mirror and the average value of each partial mirror is defined as a component, and the thermal expansion coefficient vector whose component corresponds to the arrangement position of each partial mirror is formulated by a matrix equation. For each eigenvector in the symmetric matrix of the matrix equation, calculate the sum of squares of the displacement when the components of the thermal expansion coefficient vector are arranged in ascending order and in ascending order in ascending order of the component of the eigenvector. The partial mirrors are arranged and bonded according to the component of the thermal expansion coefficient vector that minimizes the sum of squares.

According to a second aspect of the present invention, there is provided a method of manufacturing a reflecting mirror, wherein thermal deformation of the reflecting mirror is developed into a series of finite or infinite terms which are functions of a spatial frequency, and a predetermined term is corrected. The difference between the sum of the squares of the components of the residual deformation vector indicating the amount of residual deformation and the average value of the thermal expansion coefficients of all the partial mirrors and the average value of the thermal expansion coefficients of the respective partial mirrors is defined as a component. Formulates a thermal expansion coefficient vector corresponding to the arrangement position of each partial mirror with a matrix equation, and for each eigenvector in the symmetric matrix of the matrix equation, the thermal expansion coefficient Calculate the sum of squares of the components of the residual deformation vector when the components of the coefficient vector are arranged in ascending and descending order, and calculate the thermal expansion coefficient vector that minimizes the sum of the squares of the components of the residual deformation vector. It is formed by bonding by arranging the partial mirror material in accordance with the component.

According to a third aspect of the present invention, in the method of manufacturing a reflecting mirror, a component of a thermal expansion coefficient vector is a gradient of a thermal expansion coefficient in a thickness direction of each partial mirror material.

[0021]

According to the method for producing a reflecting mirror according to the first, second and third aspects of the present invention, each of the eigenvectors in the symmetric matrix of the matrix equation is made to correspond to the component of the thermal expansion coefficient vector in ascending order of the component of the eigenvector. Calculate the sum of squares of the components of the displacement or residual deformation vector when arranged in ascending order and the order of small values, and calculate the partial mirror material in accordance with the component of the coefficient of thermal expansion that minimizes the sum of the squares of the displacement or residual deformation vector. Are arranged and bonded, the amount of thermal deformation of the mirror surface of the reflecting mirror or the amount of residual thermal deformation after correction is minimized.

[0022]

[Embodiment 1] An embodiment of the present invention will be described below with reference to the drawings. FIG.
Is a plan view showing the reflecting mirror 1 composed of 37 stacks 2, and in the figure, the numbers ( Δ
α _{1 to} Δα ₃₇ ) indicate positions within the reflecting mirror 1 (not the average thermal expansion coefficient).

Next, the operation of the present invention will be described with reference to the flowchart of FIG.

Here, a deviation between the average value of the thermal expansion coefficients of the 37 stacks 2 to be arranged at the stack positions and the average value of the thermal expansion coefficients of the respective stacks 2 (hereinafter referred to as the thermal expansion coefficient)
Δα _1, Δα ₂ in descending order a) that, ......, Δα
₃₇ (Δα ₁ ≧ Δα ₂ ≧... ≧ Δα ₃₇ ), and the displacement at, for example, 1000 thermal displacement measurement points (sample points) set at approximately equal intervals on the mirror surface of the reflecting mirror 1. ΔZ _K
( _{K = 1} , _{..., 1000} ). Therefore, determining the stack position means associating the deviation Δα _i ( _{i = 1, ..., 37} ) with the stack position in FIG.

First, when a thermal expansion coefficient Δα _j ( _{j = 1, ..., 37} ) is given to each of the stacks 2, when the temperature T changes by ΔT by the finite element method, each thermal displacement measurement point is obtained. Since the displacement ΔZ _K ( _{K = 1} , _{..., 1000} ) can be calculated, the thermal expansion coefficient Δα _{j of the} stack 2 ( _{j = 1, ..., 37} )
Δα _{j (j = 1} using 1000 × 37 matrix S which does not depend _{on,
... 37} ) and ΔZ _K ( _{K = 1} , _{..., 1000} ) can be related.

[0026]

(Equation 2)

As can be seen from this equation, the first row of the matrix Sα has the thermal expansion coefficient vectors (Δα ₁ , Δα ₂ ,..., Δ
The _{α 37) (1,0, ......,} 0), the displacement vector ([Delta] Z ₁ when the _{ΔT = 1 ℃, ΔZ 2,} ......, ΔZ 1000) and
Then , similarly, (Δα ₁ , Δα
₂ ,..., Δα ₃₇ ) = (0, 1, 0,..., 0),
ΔZ _K ((0, 0, 1, 0,..., 0))
By calculating _{K = 1} , _{... 1000} ), each row of the matrix Sα can be calculated (step ST1).

Here, the displacement vector U and the thermal expansion coefficient vector α are defined as follows:

[0029]

(Equation 3)

Therefore, Equation 2 can be written as follows.

[0031]

(Equation 4)

The magnitude of the deformation is usually the rms of the displacement.
(Root mean square) in can be evaluated, in this example as follows.

[0033]

(Equation 5)

Thus, it can be seen that the rms can be minimized by minimizing the sum of squares of each displacement Σ (ΔZ _K ) ² , ( _{K = 1} , _{..., 1000} ).

[0035]

(Equation 6)

From equation (6), to minimize thermal deformation,
It is understood that the value of ^t α · ^t S · S · α should be minimized. Here, if R = ^tSS · S , this symmetric matrix R
Is a 37 × 37 symmetric matrix that does not depend on the thermal expansion coefficient vector α (step ST2). Thus, the problem of the optimal arrangement, so as to minimize the ^{‖U‖ 2 = t α · R ·} α, comprising a deviation Δα ₁ ...... Δα ₃₇ the problem arranged in elements of the thermal expansion coefficient vector alpha.

Next, how to arrange the elements of the thermal expansion coefficient vector α that minimizes {U} ² will be described. First, ‖U
‖ ² is a quadratic form, and by its nature, ‖U‖ ² takes the singular value (minimum value, maximum value or inflection point) when the thermal expansion coefficient vector α is parallel to the eigenvector of the symmetric matrix R.
You. Then, check all α that is parallel to the eigenvector
And choose the one that minimizes ‖U‖ ²
No. However, since the thermal expansion coefficient vector α can only change within the range of the rearrangement of the elements, among the thermal expansion coefficient vectors α obtained by rearranging the elements, the one having the direction closest to the eigenvector of the symmetric matrix R is determined.

Further, when obtaining the thermal expansion coefficient vector α whose direction is closest to the eigenvector of the symmetric matrix R, it can be obtained by using the inner product. here,
When the eigenvector of R is x, and the angle between α and x is θ, θ
May be 0 or α that is closest to π may be obtained. However, now, (α, x) = | α‖x | cos θ where (α, x) indicates an inner product. Which gives the maximum or minimum value of α (α, x) closest to x
Should be obtained. Such an arrangement is obtained as follows.

In order to maximize the “lemma” (maximizing the inner product by rearranging the components) (α, x), Δα _i is arranged in ascending order of the components of x in ascending order.

[Proof] The components of the eigenvector x of R are arranged in ascending order.
In, side-by-side from the smaller the Δd _i in the order, the thermal expansion coefficient vector
Make up the torr. At that time,

[0041]

(Equation 7)

Any two terms of the inner product are represented by α _i x _i + α _j x _j
And However, since _{α j = α i + Δα (} Δα ≧ 0) x j = x i + Δx (Δx ≧ 0) and put (magnitude order of the components of x and magnitude order of the components of the alpha correspond, alpha _j is If it is larger than α _i, x _j must be x _i
In this way because it is larger). When the component of α is replaced, these two terms become α _j x _i + α _i x _j . At this time,

[0043]

(Equation 8)

Therefore, when the components are rearranged from Equation 7, the inner product always decreases. Therefore, Equation 7 is the arrangement method that maximizes the inner product.

In order to minimize “lemma” (minimization of inner product by rearrangement of components) (α, x), Δα _i is arranged in descending order in accordance with the order of components of x.

[Proof] The components of the eigenvector x of R correspond in ascending order.
In, side-by-side from the larger the Δd _i in the order, the thermal expansion coefficient vector
Make up the torr. At that time,

[0047]

(Equation 9)

Any two terms of the inner product are represented by α _i x _i + α _j x _j
And However, since _{α j = α i + Δα (} Δα ≧ 0) x j = x i -Δx (Δx ≧ 0) and put (magnitude order of the components of x and magnitude order of the components of the alpha corresponds to the contrary, In this way x _j is always smaller than x _i if α _j is larger than α _i ). When the component of α is replaced, these two terms become α _j x _i + α _i x _j . At this time,

[0049]

(Equation 10)

Therefore, when the components are rearranged from Equation 9, the inner product always increases. Therefore, Equation 9 is an arrangement method that minimizes the inner product.

Therefore, it can be seen that the following procedure may be taken to obtain the optimum arrangement from the above two lemmas. (1) The components of α are arranged in ascending order according to the order of the components of x. This is referred to as α _1. (2) The components of α are arranged in ascending order according to the order of ascending components of x. This is set to α ₂ . (3) α ₁ and α for all eigenvectors
₂ can be. ‖U‖ ² = ^t αRα the calculated may be selected α to minimize the ‖U‖ ² for each (step ST 4, ST 5).

Finally, the stacks 2 are arranged and bonded according to the component of α that minimizes {U} ² (step ST 6 ).

Embodiment 2 FIG. Next, the operation of the invention described in claim 2 will be described with reference to the flow chart of FIG.
This will be described using a chart .

In the present invention, when the thermal deformation is developed in a mode and a predetermined term is corrected, the stacks 2 are arranged so that the deformation remaining after the correction is minimized.

First, the relationship between the displacement vector U and the coefficient of thermal expansion α is, as described above, U = S · α · ΔT, and the residual deformation after correction is obtained by subtracting the correction amount from the displacement vector U. Therefore, to determine the relationship between the residual deformation and the thermal expansion coefficient vector α, the relationship between the correction amount and the thermal expansion coefficient vector α may be determined.

The case where the first to 32nd natural vibration modes are corrected will be specifically described below.

[0057] displacement vector U is, Ru can be expressed as a superposition of the natural oscillation mode of the infinite terms. Natural vibration
The mode can be calculated using the finite element method. Solid
FIG. 3 shows an example of a deformation pattern in the vibrating mode. The deformation pattern q _m of the m-th natural vibration mode is represented by a displacement vector U
When expressed by the displacement q _mi of the same coordinate point as

[0058]

[Equation 11]

[0059] Further, when the expansion coefficients (corresponding to the amplitude of the vibration mode) and A _m, as the displacement vector U is superposition of natural vibration modes, expressed as follows.

[0060]

(Equation 12)

[0061] This No Chi, components up to 32 primary is the correction amount.

[0062]

(Equation 13)

When this is set as follows,

[0064]

[Equation 14]

The amount of correction = Q · A can be written. So,
Instead, the matrix Q is calculated using the finite element method (ST1).
2). Incidentally, the thermal expansion coefficient vector α and the displacement vector U is in linear relation as shown in Equation 4, since in the deployed coefficients A also linear relationship between the displacement vector U, expansion coefficient and the thermal expansion coefficient vector α The number A also has a linear relationship. Therefore, using a certain matrix P, they can be linked as follows.

[0066]

(Equation 15)

Here, P can be calculated as follows. The first line of P represents the thermal expansion coefficient vector α as (1, 0, 0,...).
0) thermal deformation (same as the above-mentioned first line of S )
The a development coefficient obtained by developing in the natural oscillation mode, the thermal deformation is calculated by the finite element method, mode expansion is calculated by a method such as fitting by least squares. The second line is α =
(0, 1, 0,... 0) can be similarly calculated ( ST
13 ).

As a result, the correction amount QA can be linked to the thermal expansion coefficient vector α. Q · A = Q · P · α · ΔT Therefore, the residual deformation vector Uz can be expressed as follows by subtracting the correction amount from the displacement vector U (ST1).
4).

[0069]

(Equation 16)

Therefore, instead of the displacement vector U of the first embodiment,
With use of the residual deformation vectors Uz to, instead of the S
By using Sαz , an optimal arrangement can be obtained as in the first embodiment. Hereinafter, the description is omitted because it is the same as the first embodiment.

The first embodiment according to the first and second aspects of the present invention.
And 2, each stack 2 has a thermal expansion coefficient vector α.
The method of producing a reflecting mirror that minimizes the amount of thermal deformation of the reflecting mirror 1 or the amount of residual thermal deformation after correction using the deviation of the average thermal expansion coefficient of A similar effect can be obtained by using the gradient of the thermal expansion coefficient in the thickness direction of 2.

When the gradient of the thermal expansion coefficient in the thickness direction of each of the stacks 2 is used as the thermal expansion coefficient vector α, after correcting the first to 32nd natural vibration modes of the thermal deformation. FIG. 5 shows an example of bonding with a stack arrangement that minimizes the amount of residual deformation of.

In FIG. 5, the symbols Δα _{1 to} Δα ₃₇ , (Δα ₁ ≧ Δα ₂ ≧... ≧
Δα ₃₇ ) indicates the magnitude of the gradient of the thermal expansion coefficient in the thickness direction of each stack.

[0074]

As described above, according to the first, second and third aspects of the present invention, for each eigenvector in the symmetric matrix of the matrix equation, the thermal expansion coefficient vector is made to correspond to the component of the eigenvector in ascending order. Calculate the sum of squares of the components of the displacement or residual deformation vector when the components are arranged in descending order and the order of the components of the thermal expansion coefficient according to the component of the thermal expansion coefficient vector in which the sum of the squares of the components of the displacement or residual deformation vector is smallest Since the partial mirror materials are arranged and bonded, the optimal arrangement of each stack is determined in a broken manner, and the effect of reliably obtaining a reflecting mirror that minimizes the amount of thermal deformation or residual deformation after correction is obtained. is there.

[Brief description of the drawings]

FIG. 1 is a plan view showing an arrangement position of a stack in a reflecting mirror.

FIG. 2 is a flowchart illustrating steps of a method of manufacturing a reflecting mirror according to an embodiment of the present invention.

FIG. 3 is a plan view showing an example of a contour diagram in each of natural vibration modes of the reflecting mirror.

FIG. 4 is a flowchart illustrating steps of a method for producing a reflecting mirror according to an embodiment of the present invention.

FIG. 5 is a plan view showing the result of an embodiment of a method of manufacturing a reflecting mirror according to the present invention;

FIG. 6 is a perspective view showing a configuration of a reflecting mirror.

FIG. 7 is a characteristic diagram showing an intensity distribution of a star image when there is no thermal deformation on a reflecting mirror surface.

FIG. 8 is a cross-sectional view of a reflecting mirror when thermally deformed and a distribution diagram showing an intensity distribution of a star image thereof.

FIG. 9 is a perspective view showing a stack arrangement according to a method for producing a partial mirror material of a reflecting mirror of the first conventional example.

FIG. 10 is a cross-sectional view showing a cross section of a reflecting mirror provided with an actuator for correcting thermal deformation according to a second conventional example.

FIG. 11 is a plan view showing an example of a stack arrangement according to the second conventional example of FIG.

[Explanation of symbols]

 1 reflector 2 stack (partial mirror material)

Claims (3)

(57) [Claims] When a partial mirror material is bonded to form a reflecting mirror, a sum of squares of displacements at a plurality of sample points on a mirror surface of the reflecting mirror and an average value of thermal expansion coefficients of all the partial mirror materials are obtained. The deviation from the average value of the thermal expansion coefficient of each partial mirror is defined as a component, and the thermal expansion coefficient vector whose component corresponds to the arrangement position of each partial mirror is formulated by a matrix equation. For each eigenvector in the matrix, calculate the sum of squares of displacement when the components of the coefficient of thermal expansion are arranged in ascending order and in ascending order in ascending order of the component of the eigenvector. A method for producing a reflecting mirror in which partial mirror materials are arranged and bonded in accordance with the component of the smallest thermal expansion coefficient vector. 2. When a partial mirror material is bonded to form a reflecting mirror, thermal deformation of the reflecting mirror is developed into a series of finite or infinite terms which are functions of spatial frequency, and a predetermined term is corrected. The difference between the sum of the squares of the components of the residual deformation vector indicating the amount of residual deformation and the average value of the thermal expansion coefficients of all the partial mirrors and the average value of the thermal expansion coefficients of the respective partial mirrors is defined as a component. Formulates a thermal expansion coefficient vector corresponding to the arrangement position of each partial mirror with a matrix equation, and for each eigenvector in a symmetric matrix of the matrix equation,
Calculate the sum of squares of the residual deformation vector components when the components of the thermal expansion coefficient vector are arranged in ascending order and in ascending order in association with the components of the eigenvector in ascending order. A method for producing a reflecting mirror in which the partial mirror material is arranged and bonded in accordance with the component of the thermal expansion coefficient vector in which the minimum is obtained. 3. The method according to claim 1, wherein the component of the thermal expansion coefficient vector is a gradient of a thermal expansion coefficient in a thickness direction of each partial mirror material.