Method for compensating optical aberrations with a deformable mirror

Abstract

Method for compensating aberrations of an active optical system by a deformable mirror, comprising: (a) formation of a matrix N, binding the optical system aberrations with the displacements of the surface of a deformable mirror, presented as a superposition of natural frequency modes of a mirror simulator having the same geometric shape and the same elasticity characteristics as the deformable mirror but having zero density except for concentrated masses attached at points corresponding to the places of action of the force actuators onto the deformable mirror; (b) formation of the matrix M, binding the coefficients of the natural frequency modes of the mirror simulator with the forces or displacements of the force actuators that deform the mirror; (c) generation of the matrix of forces or displacements for loading the deformable mirror on the basis of the aberration measurement, using the matrices N and M.

Claims

1. Method for compensating aberrations of an active optical system by a deformable mirror, comprising the operations as follows: (a) formation of a matrix N, binding the optical system aberrations with the displacements of the surface of a deformable mirror, presented as a superposition of natural frequency modes of a mirror simulator having the same geometric shape and the same elasticity characteristics as the deformable mirror but having zero density except for concentrated masses attached at points corresponding to the places of action of the force actuators onto the deformable mirror, that are taken into consideration only in the direction of application of forces or displacements deforming the mirror; (b) formation of the matrix M, binding the coefficients of the natural frequency modes of the mirror simulator with the forces or displacements of the force actuators that deform the mirror; (c) generation of the matrix of forces or displacements for loading the deformable mirror on the basis of the aberration measurement, using the matrices N and M.

2. Method of claim 1, characterized in that the matrix F of forces or displacements for the deformable mirror is generated according to the formula:
F=M.sup.1(N.sup.1E) where E is a matrix characterizing the output wavefront aberrations of the optical system.

3. Method of claim 1, characterized in that the matrix F of forces or displacements for the deformable mirror is generated with the use of results of aberration measurements according to the formula:
F=M.sup.1((N.sup.TN).sup.1N.sup.TE) where E is a matrix characterizing the output wavefront aberrations of the optical system.

4. Method of claim 1, characterized in that the matrix F of forces or displacements for the deformable mirror is generated with the use of results of aberration according to the formula:
F=M.sup.1(N.sup.T(N N.sup.T).sup.1E) where E is a matrix characterizing the output wavefront aberrations of the optical system.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) FIG. 1 shows an active optical system of a telescope with a deformable mirror 1 used to compensate aberrations. The telescope comprises as well a wavefront detector 2, software and hardware 3 comprising a mirror simulator, force actuators 4, for example, piezo-actuators.

(2) FIG. 2 shows a mirror used to substantiate that the superposition of orthogonal natural frequency modes of the mirror simulator vibrations entirely describes the quasi-static load of a real deformable mirror.

(3) FIG. 3 shows the numeration of places for adjoining force actuators.

(4) FIGS. 4 to 6 show contour lines of axial displacements of the deformed mirror surface corresponding to the natural frequency modes of the mirror simulator vibrations, respectively the first, 14th and 18th.

(5) FIG. 7 provides nodes of a finite-element grid on the optical surface of the mirror, used for approximation with natural frequency modes.

DETAILED DESCRIPTION

(6) The present invention implements the fact that the quasi-static loading of a real deformable mirror in an active optical system is entirely described by the superposition of orthogonal natural frequency modes of the mirror simulator, where the number of modes is equal to the number of acting forces or displacements exerted with force actuators. As a result, after executing a set of the commands which correspond to the superposition of these modes, no additional displacements arise on the reflective mirror surface that could distort the preset displacements or reduce the accuracy of compensating the wavefront aberrations.

(7) The method for compensating optical aberrations of the present application is carried out as follows:

(8) 1. Initially, with the use of software and hardware, a mirror simulator is formed that has the same geometric shape and the same elastic properties as a deformable mirror but has zero density except for concentrated unit masses attached at points corresponding to the places of action of the force actuators onto the deformable mirror, that are taken into consideration only in the direction of application of forces or displacements deforming the mirror, and that is characterized:

(9) by a set of natural frequency modes of vibrations;

(10) by the matrix N that is a matrix of the sensitivity of the optical system aberrations to displacements of the reflective surface of the deformed mirror, presented as the superposition of natural frequency modes of the mirror simulator vibrations;

(11) by the matrix M that is a matrix of the sensitivity of coefficients of natural frequency modes of the mirror simulator vibrations to the forces or displacements of actuators deforming the mirror.

(12) With the help of the matrix N the aberrations of the optical system are connected to the displacements of the surface of the controlled mirror.

(13) The elements of the matrix N are determined by optical modeling, for example by a numerical evaluation method of ray tracing with sequential ray tracing algorithms. In this case, a licensed software package ZEMAX is used.

(14) The method of numerical modal analysis of a mirror simulator is used to determine the simulator natural frequency modes and a square matrix M.sup.1 that is the reverse of the matrix of the sensitivity of coefficients of natural frequency modes of the simulator vibrations to forces or displacements provided by the force actuators. In this case, a licensed software package for finite-element analysis ANSYS is used. Each column of the matrix M.sup.1 corresponds to one of the natural frequency modes of the mirror simulator vibrations. If the displacement values are assigned to force actuators as commands, the elements of this column are equal to displacements of the finite-element grid nodes determined in the modal analysis, at which concentrated unit masses are attached, in the direction of application of forces or displacements deforming the mirror. If forces are assigned as commands to actuators, the elements of this column represent products of the above-mentioned displacements by the square of natural circular frequency for the corresponding mode of vibrations.

(15) 2. Output wavefront aberrations of the optical system are measured by a wavefront detector 2 shown in FIG. 1, said aberrations being presented, for example, as a sum of the Zernike polynomials. The number of Zernike polynomials is infinite but for practical purposes, a limited number is taken, for example 32 polynomials. Then it is necessary to know 32 coefficients of these polynomials that have linear dimensions, for example in millimeters. These coefficients are determined by a wavefront detector 2. The coefficients form a matrix E under the form of a column vector.

(16) 3. The software and hardware comprising a mirror simulator 3 determine required displacements of the controlled mirror 1 surface, approximated by the superposition of natural frequency modes of the simulator vibrations. The number of these modes is equal to the number of forces or displacements provided by the force actuators. For assigning a specific distribution of the mirror surface displacements, it is necessary to know the coefficients for these modes. The natural modes are used in a non-dimensional form while the coefficients have the same linear dimension as the coefficients at the Zernike polynomials. Coefficients of natural frequency modes in the decomposition of the simulator vibrations form a matrix C that is a column vector.

(17) If the number of natural modes is the same that the above mentioned number of Zernike polynomials, the resulting matrix N is square, according to the above given 3232. If these numbers differ, the resulting matrix is not square. The elements of the matrix N can be any real numbers

(18) In the case when the number of measured parameters for the wavefront aberrations is equal to the number of natural frequency modes of the simulator vibrations, the matrix N inversion is carried out; in the case when the number of measured aberrations is less or more than the number of natural frequency modes of the simulator vibrations, the matrix N pseudo-inversion takes place. Respectively, the superposition of natural frequency modes of the mirror imitator vibrations on the basis of the results of measurement of the wavefront aberrations is determined as follows:
C=N.sup.1E,

(19) or as follows:
C=(N.sup.TN).sup.1N.sup.TE,

(20) or as follows:
C=N.sup.T(N N.sup.T).sup.1E,

(21) where E is a column matrix of the parameters for the wavefront of aberrations, depending on what functions were used for the approximation of aberrations representing coefficients of the Zernike polynomials or differences of the optical path, or local slopes of the wavefront, or coefficients of the natural frequency modes of the simulator vibrations, or other parameters; C is a matrix of natural frequency modes of the mirror simulator vibrations; the upper index T defines the operation of matrix transposition.

(22) 4. The software and hardware containing a mirror simulator 3 determine the forces or displacements that deform the mirror, as a matrix F in accordance with the earlier obtained matrix C, defining the superposition of natural frequency modes of the deformable mirror simulator vibrations and of M.sup.1, that represents a reverse matrix of the sensitivity for the coefficients of natural frequency modes to forces or displacements of force actuators and that is stored by the software and hardware:
i F=M.sup.1C

(23) 5. The mirror is deformed by the force actuators 4 while using the obtained set of forces or displacements.

(24) As an example, a mirror was considered having the structure in the shape of a vase, the finite-element model of which is shown in FIG. 2. The mirror is assembled: a central boss 1 and a ring with arms 2, made of invar are glued to a disk with the optical surface 3, made of glass ceramic. The central boss is fixedly attached. The mirror is loaded by 24 axial forces provided by force actuators. Holes for attaching mechanisms, arranged in pairs on 12 arms are illustrated by arrows in FIG. 2. Numeration of places of attachment is given in FIG. 3.

(25) As illustrations, the contour lines of axial displacements of the deformable mirror optical surface for first, 14th and 18th natural frequency modes of the mirror simulator vibrations are given in FIGS. 4-6, respectively. The natural frequency for the mode 1 is 30.3852 Hz, for the mode 14 is 485.583 Hz, for the mode 18 is 757.419 Hz. The Figures show that the natural modes represent the particularities of the geometric shape of the mirror: the presence of a central boss and the location of arms.

(26) Table 1 illustrates the elements of the first column of the matrix M.sup.1 for cases of assigning displacements and forces to force actuators.

(27) TABLE-US-00001 TABLE 1 First column of the matrix M.sup.1 Node Control of Point number displacements Control of forces 1 13429 0.26491 9655.62827 13 13469 0.18989 6921.27366 2 27123 0.32932 12003.1693 14 40521 0.23606 8604.02006 3 52634 0.30548 11134.4708 15 66032 0.21897 7981.32624 4 78145 0.19979 7282.29980 16 91543 0.14322 5220.04249 5 103656 0.40573E01 1478.84249 17 117054 0.29083E01 1060.05257 6 129167 0.12952 4720.86947 18 142565 0.92842E01 3383.97757 7 154678 0.26491 9655.62827 19 168076 0.18989 6921.27366 8 180189 0.32932 12003.1693 20 193587 0.23606 8604.02006 9 205700 0.30548 11134.4708 21 219098 0.21897 7981.32624 10 231211 0.19980 7282.29980 22 244609 0.14322 5220.04249 11 256722 0.40573E01 1478.84249 23 270120 0.29083E01 1060.05257 12 282152 0.12952 4720.86947 24 294843 0.92842E01 3383.97757

(28) In the considered example, the needed 24 natural frequency modes of the mirror simulator vibrations for axial displacements of the optical surface multiplied by respective coefficients are represented by their root-mean-square values in the second column of Table 2.

(29) TABLE-US-00002 TABLE 2 Root mean square of natural modes multiplied by coefficients, mm Mode Obtained at application of number Prescribed forces 1 1.30139959E03 1.30139959E03 2 1.81674515E04 1.81674515E04 3 1.77422624E04 1.77422624E04 4 2.41968253E04 2.41968253E04 5 2.37832028E04 2.37832028E04 6 2.93388648E06 2.93388648E06 7 4.41407790E05 4.41407790E05 8 2.47476346E05 2.47476346E05 9 2.40836713E05 2.40836713E05 10 1.21941986E06 1.21941986E06 11 1.04427891E06 1.04427891E06 12 3.93603627E10 3.93603628E10 13 1.81039839E04 1.81039839E04 14 1.32005936E03 1.32005936E03 15 1.74456591E04 1.74456591E04 16 1.46707436E04 1.46707436E04 17 7.01018222E05 7.01018222E05 18 6.82214388E06 6.82214388E06 19 3.40293990E06 3.40293990E06 20 4.03222156E07 4.03222156E07 21 1.89622175E06 1.89622175E06 22 4.18924199E07 4.18924199E07 23 1.50015908E06 1.50015908E06 24 3.30165537E10 3.30165539E10

(30) Table 3 gives the forces found in accordance with the relationship (3).

(31) TABLE-US-00003 TABLE 3 Force number Force, N 1 21.58246324 2 22.62798289 3 7.873509662 4 11.80896372 5 2.46125139 6 7.965528332 7 2.602475619 8 6.978432007 9 15.9334173 10 6.767859234 11 5.888897748 12 5.231199841 13 17.19006065 14 26.02863542 15 24.04734848 16 7.657331756 17 1.160097989 18 4.115367578 19 8.759042553 20 12.21508684 21 13.43810732 22 17.22602362 23 11.97534719 24 8.096216319

(32) Then, to check the results obtained, the finite-element method was used to calculate the displacements of the real mirror optical surface nodes under the effect of forces found, and to approximate them by natural frequency modes of the imitator. FIG. 7 shows nodes of a finite-element grid on the mirror optical surface used for approximation with natural modes. The error of approximation is within machine precision. It means that no additional displacements appeared on the reflective surface that could distort the displacements assigned. The results of the finite-element method calculations are given in the third column of Table 2. As we see, the prescribed superposition of the natural frequency modes of the mirror simulator vibrations on a controlled deformable mirror with an error that is within machine precision.

(33) The given example confirms that the superposition of orthogonal natural frequency modes of the mirror simulator vibrations entirely describes the quasi-static loading of a real deformable mirror under the effect of force actuators. So, the method of the present application enables to eliminate the part of the error of compensation of the active optical system aberrations that is related to the formation of the superposition of modes on the reflective surface of the deformable mirror, needed to compensate wavefront aberrations.