CN110095858B - Self-adaptive optical deformable mirror elastic modal aberration characterization method - Google Patents
Self-adaptive optical deformable mirror elastic modal aberration characterization method Download PDFInfo
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Abstract
The invention provides a characterization method for elastic modal aberration of an adaptive optical deformable mirror. The elastic modal aberration characterization method of the invention is derived from the deformation of the resonance mode of the deformable mirror, so that the elastic modal polynomial represents each order aberration of wavefront difference decomposition, and as each modal aberration is the natural mode shape of the deformable mirror, the correction residual error is smaller, and the correction efficiency is higher.
Description
Technical Field
The invention belongs to the technical field of adaptive optics, and particularly relates to a characterization method for elastic modal aberration of an adaptive optics deformable mirror.
Background
The purpose of adaptive optics is to repair the distortion of the optical wavefront due to factors such as atmospheric turbulence. Adaptive optics first detects the wavefront distortion and then corrects the wavefront in real time by means of a small deformable mirror mounted behind the focal plane of the telescope.
To perform the adaptive correction, the wavefront difference u (r, θ) is first deformed into the sum of a series of orthogonal functions, as follows:
in the above formula, unm(r) cos (n θ) is an orthonormal function, i.e., aberration; n represents a symmetry number; m represents the order.
It is now common practice to characterize u by Zernike polynomial aberration characterizationnm(r) cos (n θ), the total optical aberration is the sum of the aberrations of the various orders and can therefore be decomposed by Zernike polynomials, however each aberration decomposed by this method is artificially set. According toAccording to the general principle of adaptive optics, all aberrations which can be produced by applying load to the deformable mirror can be corrected, and on the contrary, aberrations which cannot be produced by applying load to the deformable mirror cannot be corrected, so that residual errors are always difficult to correct in the method.
Disclosure of Invention
In order to solve the technical problem, the invention provides a method for characterizing elastic modal aberration of an adaptive optical deformable mirror. The following presents a simplified summary in order to provide a basic understanding of some aspects of the disclosed embodiments. This summary is not an extensive overview and is intended to neither identify key/critical elements nor delineate the scope of such embodiments. Its sole purpose is to present some concepts in a simplified form as a prelude to the more detailed description that is presented later.
The invention adopts the following technical scheme:
in some optional embodiments, there is provided an adaptive optical deformable mirror elastic modal aberration characterization method, comprising:
constructing a vibration mode differential equation of the deformable mirror, solving the vibration mode differential equation, and deducing an elastic mode polynomial of the deformable mirror;
normalizing the elastic modal polynomial;
the normalized elastic mode polynomial corresponds to each aberration of the wavefront difference decomposition one by one, so that the elastic mode polynomial represents each order of aberration of the wavefront difference decomposition.
In some optional embodiments, the constructing a mode shape differential equation of the deformable mirror, solving the mode shape differential equation, and deriving the elastic mode polynomial of the deformable mirror includes:
establishing a cylindrical coordinate system, wherein the direction of a z coordinate is vertical to the surface of the deformable mirror, r is the radial coordinate of the deformable mirror, and theta is the angular coordinate of the circumferential direction of the deformable mirror;
acquiring each stress component at each position of the deformable mirror;
and integrating the stress components of each position of the deformable mirror along the thickness direction of the deformable mirror to obtain bending moments in each coordinate direction of unit length, further obtaining shearing forces in the r and theta coordinate directions of the unit length of the deformable mirror, further obtaining a vibration mode differential equation of the deformable mirror, and solving the vibration mode differential equation to deduce an elastic mode polynomial of the deformable mirror.
In some optional embodiments, the process of acquiring stress components of the deformable mirror comprises:
and obtaining the strain component of each point of the deformable mirror by the deflection of the deformable mirror, as shown in formula 1:
formula 1:
in formula 1, z is the coordinate of the vertical direction of the deformable mirror, r is the radial coordinate of the deformable mirror, theta is the angular coordinate of the circumferential direction of the deformable mirror, and wz(r) deformable mirror deflection;
when the deformable mirror is formed by bonding two layers of different materials, the stress components at each position of the top layer material are as follows:
formula 2:
in the formula 2, EsAnd vsRespectively representing the Young's modulus and Poisson's ratio, epsilon, of the top layer materialr、εθ、γrθStrain components of each point of the deformable mirror;
when the deformable mirror is formed by bonding two layers of different materials, the stress components of the bottom layer material at all positions are as follows:
formula 3:
in formula 3, EpAnd vpRespectively, the Young's modulus and Poisson's ratio of the substrate material, epsilonr、εθ、γrθStrain components of each point of the deformable mirror;
when the deformable mirror is made of the same material, let E in formula 2 and formula 3s＝Ep，vs＝vp。
In some optional embodiments, the integrating the stress components of the deformable mirror along the thickness direction of the deformable mirror obtains bending moments in each coordinate direction of unit length, and further obtains shearing forces in r and θ coordinate directions of the unit length of the deformable mirror, so as to obtain a vibration mode differential equation of the deformable mirror, and the process of solving the vibration mode differential equation to derive the elastic mode polynomial of the deformable mirror includes:
defining the neutral plane of the deformable mirror, and setting the thickness from the neutral plane to the upper top surface of the deformable mirror as hm1The thickness from the neutral plane to the lower bottom surface of the deformable mirror is hm2Then h ism1And hm2Expressed as:
formula 4:
in the formula 4, EsExpressing the Young's modulus of the top layer material, EpExpressing Young's modulus, h, of the underlying materialsThickness of the top layer material, hpIs the thickness of the underlying material;
and integrating the stress components at each position of the deformable mirror along the thickness direction of the deformable mirror to obtain the bending moment in each coordinate direction of unit length, wherein the formula is as follows:
formula 5:
in formula 5, DsAnd DpEquation 6 is satisfied as follows:
formula 6:
in formulas 5 and 6, σr、σθ、τrθThe stress components of the deformable mirror are all parts; h ism1The thickness from the neutral surface to the upper top surface of the deformable mirror; h ism2The thickness from the neutral surface to the lower bottom surface of the deformable mirror; z is a coordinate in the vertical direction of the deformable mirror; r is the radial coordinate of the deformable mirror; theta is the angular coordinate of the deformable mirror in the circumferential direction; w is az(r) deformable mirror deflection; v. ofsRepresenting the poisson's ratio of the top layer material; v. ofpRepresenting the poisson ratio of the bottom layer material; esRepresenting the material of the top layerA modulus; epRepresenting the Young's modulus of the underlying material; h issIs the thickness of the top layer material; h ispIs the thickness of the underlying material;
when the deformable mirror is made of the same material as the whole, let E in formula 4, formula 5 and formula 6s＝Ep，vs＝vp，h＝hs+hp；
The r and theta coordinate direction shearing force of the unit length of the deformable mirror is further obtained by the formula 6, and the formula 7 is as follows:
formula 7:
in formula 7, Mr、Mθ、MθrBending moment in each coordinate direction for the unit length of the deformable mirror;
further obtaining the equivalent shearing force, as formula 8:
formula 8:
in formula 8, Qr、QθShear forces in the r and theta coordinate directions of the unit length of the deformable mirror; mr、Mθ、MθrBending moment in each coordinate direction for the unit length of the deformable mirror;
then the motion control equation of the deformable mirror is as follows:
formula 9:
in formula 9, QrQθShear forces in the r and theta coordinate directions of the unit length of the deformable mirror;
in formula 9, ξ ═ ρshs+ρphpFormula 10 is obtained from formula 5, formula 7 and formula 9 as follows:
formula 10:
in the formula 10, wz(r) is a deformable mirrorDeflection; xi ═ pshs+ρphp；
D＝Ds+Dp，Ds、DpAs shown in formula 6;
since simple vibration, the deflection is expressed as a complex number in formula 11:
formula 11: w is az(r，θ，t)＝Re{Wz(r，θ)exp(iωt)}；
Equation 12 is obtained from equation 10 as follows:
formula 12: d ^2▽2Wz-ω2ξWz＝0；
Let equation 12 solve for equation 13, as follows:
formula 13: wz(r，θ)＝un(r) cos (n θ); n represents a symmetry number;
substitution of formula 13 for formula 12 yields formula 14 as follows:
formula 14:
in the formula (14), the compound represented by the formula,
the solution of equation 14 is written as equation 15, as follows:
formula 15: u. ofn(r)＝A1，nJn(λr)+A2，nYn(λr)+A3，nIn(λr)+A4，nKn(λr)；
In formula 15, λ4＝ω2ξ/D；Jn(λr)、Yn(λr)、In(λr)、Kn(λ r) is an n-order first class, an n-order second class, a modified n-order first class, and a modified n-order second class Bessel function, respectively;
in formula 15, Yn(lambda r) and Kn(λ r) is infinite, let A2，n＝A4，nWhen 0, equation 15 is rewritten to equation 16 as follows:
formula 16: u. ofn(r)＝A1，nJn(λr)+A3，nIn(λr)；
A matrix equation is obtained, as shown in equation 19:
from equation 19, it can be seen that the two unknowns A to be solved for1,n、A3,nAll depend on λ, giving the trivial solution of equation 19, then equation 20:
formula 20:
a series of lambda values are obtained from equation 20, and the values of each lambda are substituted into equation 19 to obtain two unknowns A1,n、A3,nWherein one unknown may be represented by another unknown, A3,nFrom A1,nMultiplied by a correlation coefficient.
In some optional embodiments, normalizing the elastic modal polynomial comprises:
formula 20 is characterized by
m represents the order of the mode, m depends on λ, when taking A1,nThe general normalized solution is as in equation 21:
formula 21:
in the formula (21), the compound represented by the formula,
the invention has the following beneficial effects: the elastic modal aberration characterization method of the invention is derived from the deformation of the resonance mode of the deformable mirror, so that the elastic modal polynomial represents each order aberration of wavefront difference decomposition, and as each modal aberration is the natural mode shape of the deformable mirror, the correction residual error is smaller, and the correction efficiency is higher.
For the purposes of the foregoing and related ends, the one or more embodiments include the features hereinafter fully described and particularly pointed out in the claims. The following description and the annexed drawings set forth in detail certain illustrative aspects and are indicative of but a few of the various ways in which the principles of the various embodiments may be employed. Other benefits and novel features will become apparent from the following detailed description when considered in conjunction with the drawings and the disclosed embodiments are intended to include all such aspects and their equivalents.
Drawings
FIG. 1 is a configuration of a deformable mirror of the present invention with the deformable mirror edges in a free state;
FIG. 2 is a configuration of the deformable mirror of the present invention when simply supported at edge r ═ a;
FIG. 3 is a deformation function u of the deformable mirror of the present invention under the condition of free boundary when the order m of the mode is 1mn(r) plot of variation with radius r;
FIG. 4 shows the deformation function u of the deformable mirror of the present invention under the condition of simple supporting boundary when the order m of the mode is 1mn(r) plot of variation with radius r;
FIG. 5 shows the deformation function u of the deformable mirror of the present invention under the condition of free boundary, when the order m of the mode is 1 and the symmetry n is 0mn(r,θ)；
FIG. 6 shows the deformation function u of the deformable mirror of the present invention under the condition of free boundary, when the order m of the mode is 2 and the symmetry n is 0mn(r,θ)；
FIG. 7 shows the deformation function u of the deformable mirror of the present invention under the condition of free boundary, when the order m of the mode is 1 and the symmetry n is 1mn(r,θ)；
FIG. 8 shows the deformation function u of the deformable mirror of the present invention under the condition of free boundary, when the order m of the mode is 2 and the symmetry n is 1mn(r,θ)；
FIG. 9 shows the deformation function u of the deformable mirror of the present invention under the condition of free boundary, when the order m of the mode is 3 and the symmetry n is 1mn(r,θ)；
FIG. 10 shows the deformation function u of the deformable mirror of the present invention under the condition of free boundary, when the order m of the mode is 4 and the symmetry n is 1mn(r,θ)；
FIG. 11 shows the deformation function u of the deformable mirror of the present invention in the state of a simple supporting boundary when the order m of the mode is 1 and the symmetry n is 0mn(r,θ)；
FIG. 12 shows the deformation function u of the deformable mirror of the present invention in the simple support boundary state when the order m of the mode is 2 and the symmetry n is 0mn(r,θ)；
FIG. 13 shows the deformation function u of the deformable mirror of the present invention in the state of a simple supporting boundary when the order m of the mode is 1 and the symmetry n is 1mn(r,θ)；
FIG. 14 shows the deformation function u of the deformable mirror of the present invention in a simple support boundary state, when the order m of the mode is 2 and the symmetry n is 1mn(r,θ)；
FIG. 15 shows the deformation function u of the deformable mirror of the present invention in the simple support boundary state, when the order m of the mode is 3 and the symmetry n is 1mn(r,θ)；
FIG. 16 shows the deformation function u of the deformable mirror of the present invention in the simple support boundary state, when the order m of the mode is 4 and the symmetry n is 1mn(r,θ)；
FIG. 17 is a graph comparing Zernike polynomial Z4 with the elastic modal deformation function under two boundary conditions of the present invention and n is 0;
FIG. 18 is a graph comparing Zernike polynomial Z7 with the elastic modal deformation function under two boundary conditions of the present invention and n is 1;
fig. 19 is a graph comparing the elastic modal deformation functions of zernike polynomials Z5, Z9, Z14 with the deformable mirror of the present invention under two boundary conditions, the free boundary condition, and n 2, n 3, and n 4;
fig. 20 is a graph comparing zernike polynomials Z12, Z18, Z19 with the elastic modal deformation function of the deformable mirror of the present invention under simple support boundary conditions with n 2, n 3 and n 4.
Detailed Description
The following description and the drawings sufficiently illustrate specific embodiments of the invention to enable those skilled in the art to practice them. Other embodiments may incorporate structural, logical, electrical, process, and other changes. The examples merely typify possible variations. Individual components and functions are optional unless explicitly required, and the sequence of operations may vary. Portions and features of some embodiments may be included in or substituted for those of others. The scope of embodiments of the invention encompasses the full ambit of the claims, as well as all available equivalents of the claims.
In some illustrative embodiments, an adaptive optical deformable mirror elastic modal aberration characterization method is provided, which is artificially set with respect to a Zernike polynomial, and the essence of the elastic modal aberration characterization method of the present invention is derived from the deformation of the resonance modes of the mirror itself.
Adaptive optics compensates for aberrations of various orders by changing the shape of a deformable mirror, and since the zernike polynomial is set artificially, it is difficult to change the mirror to the shape which is the aberration expressed by the zernike polynomial, but it is easy to change the mirror to the aberration expressed by the elastic mode, since this shape is originally the shape when the mirror resonates.
The elastic modal aberration characterization method comprises the following steps:
s1: and constructing a vibration mode differential equation of the deformable mirror, solving the vibration mode differential equation and deducing an elastic mode polynomial of the deformable mirror.
And deducing an elastic mode polynomial formula of the deformable mirror based on an elastic dynamics theory.
S2: normalizing the elastic modal polynomial. The normalized elastic mode polynomial consists of a complete set of an infinite number of polynomials with two variables, r and θ, which are continuously orthogonal inside the unit circle.
S3: the normalized elastic mode polynomial corresponds to each aberration of the wavefront difference decomposition one by one, so that the elastic mode polynomial represents each order of aberration of the wavefront difference decomposition.
Wherein the process of S1 includes:
establishing a cylindrical coordinate system, wherein the direction of a z coordinate is vertical to the surface of the deformable mirror, r is the radial coordinate of the deformable mirror, and theta is the angular coordinate of the circumferential direction of the deformable mirror;
acquiring each stress component at each position of the deformable mirror;
and integrating the stress components of each position of the deformable mirror along the thickness direction of the deformable mirror to obtain bending moments in each coordinate direction of unit length, further obtaining shearing forces in the r and theta coordinate directions of the unit length of the deformable mirror, further obtaining a vibration mode differential equation of the deformable mirror, and solving the vibration mode differential equation to deduce an elastic mode polynomial of the deformable mirror.
As shown in fig. 1 and 2, when the deformable mirror is formed by bonding two layers of different materials, the top layer material and the bottom layer material are respectively, for example, the top layer material is a glass material, and the bottom layer material is a piezoelectric ceramic material. Setting the thickness of the glass material layer to hsThe thickness of the piezoelectric ceramic material layer is hpThe anamorphic mirror has a diameter of 2 a. When the deformable mirror is made of the same material, the whole thickness of the deformable mirror is h, and h is hs+hp。
And establishing a cylindrical coordinate system, wherein the direction of a z coordinate is vertical to the surface of the deformable mirror, the thickness directions of the deformable mirror are consistent, r is the radial coordinate of the deformable mirror, and theta is the angular coordinate of the deformable mirror in the circumferential direction.
Suppose (h)s+hp)/a<<1, obtaining the strain component of each point of the deformable mirror by the deflection of the deformable mirror as follows:
formula 1:
in formula 1, z is the coordinate of the vertical direction of the deformable mirror, r is the radial coordinate of the deformable mirror, theta is the angular coordinate of the circumferential direction of the deformable mirror, and wz(r) is the deformable mirror deflection.
According to the theory of elastic mechanics, when the deformable mirror is formed by bonding two layers of different materials, the stress components at each position of the top layer material are obtained:
formula 2:
in the formula 2, EsAnd vsRespectively representing the Young modulus and the Poisson ratio of the top layer material; epsilonr、εθ、γrθIs the strain component of each point of the deformable mirror.
When the deformable mirror is formed by bonding two layers of different materials, the stress components at all positions of the bottom layer material are as follows:
formula 3:
in formula 3, EpAnd vpRespectively representing the Young modulus and the Poisson ratio of the bottom layer material; epsilonr、εθ、γrθIs the strain component of each point of the deformable mirror.
When the deformable mirror is made of the same material, let E in formula 2 and formula 3s＝Ep，vs＝vp。
And defining a neutral plane of the deformable mirror, wherein the neutral plane is a neutral plane which is neither shortened nor extended, namely is not compressed or pulled, and the neutral plane is an interface of a stretching area and a compression area on the circular deformable mirror.
The thickness from the neutral plane to the upper top surface of the deformable mirror is set to be hm1The thickness from the neutral plane to the lower bottom surface of the deformable mirror is hm2Then h ism1And hm2Expressed as formula 4:
formula 4:
in the formula 4, EsRepresenting the Young's modulus of the top layer material; epRepresenting the Young's modulus of the underlying material; h issIs the thickness of the top layer material; h ispIs the thickness of the underlying material.
And integrating the stress components at each position of the deformable mirror along the thickness direction of the deformable mirror to obtain the bending moment in each coordinate direction of unit length, wherein the formula is as follows:
formula 5:
in formula 5，DsAnd DpEquation 6 is satisfied as follows:
formula 6:
in formulas 5 and 6, σr、σθ、τr6The stress components of the deformable mirror are all parts; h ism1The thickness from the neutral surface to the upper top surface of the deformable mirror; h ism2The thickness from the neutral surface to the lower bottom surface of the deformable mirror; z is a coordinate in the vertical direction of the deformable mirror; r is the radial coordinate of the deformable mirror; theta is the angular coordinate of the deformable mirror in the circumferential direction; w is az(r) deformable mirror deflection; v. ofsRepresenting the poisson's ratio of the top layer material; v. ofpRepresenting the poisson ratio of the bottom layer material; esRepresenting the Young's modulus of the top layer material; epRepresenting the Young's modulus of the underlying material; h issIs the thickness of the top layer material; h ispIs the thickness of the underlying material.
When the deformable mirror is made of the same material as the whole, let E in formula 4, formula 5 and formula 6s＝Ep，vs＝vp，h＝hs+hp。
The r and theta coordinate direction shearing force of the unit length of the deformable mirror is further obtained by the formula 6, and the formula 7 is as follows:
formula 7:
in formula 7, Mr、Mθ、MθrBending moment in each coordinate direction is the unit length of the deformable mirror.
Further obtaining the equivalent shearing force, as formula 8:
formula 8:
in formula 8, Qr、QθShear forces in the r and theta coordinate directions of the unit length of the deformable mirror; mr、Mθ、MθrBending moment in each coordinate direction is the unit length of the deformable mirror.
Then the motion control equation of the deformable mirror is as follows:
formula 9:
in formula 9, Qr、QθShear forces in the r and theta coordinate directions are the unit length of the deformable mirror.
In formula 9, ξ ═ ρshs+ρphpFormula 10 is obtained from formula 5, formula 7 and formula 9 as follows:
formula 10:
in the formula 10, wz(r) deformable mirror deflection; xi ═ pshs+ρphp；
D＝Ds+Dp，Ds、DpAs shown in equation 6.
Due to simple harmonic vibration, the deflection can be expressed in complex number as formula 11:
formula 11: w is az(r，θ，t)＝Re{Wz(r，θ)exp(iωt)}；
Equation 12 is obtained from equation 10 as follows:
formula 12: d ^2▽2Wz-ω2ξWz＝0；
Let equation 12 solve for equation 13, as follows:
formula 13: wz(r，θ)＝un(r) cos (n θ); n represents a symmetry number.
Substitution of formula 13 for formula 12 yields formula 14 as follows:
formula 14:
in the formula (14), the compound represented by the formula,
the solution of equation 14 is written as equation 15, as follows:
formula 15: u. ofn(r)＝A1，nJn(λr)+A2，nYn(λr)+A3，nIn(λr)+A4，nKn(λr)。
In formula 15, λ4＝ω2ξ/D；Jn(λr)、Yn(λr)、In(λr)、Kn(λ r) are the n-th order first class, the n-th order second class, the modified n-th order first class, and the modified n-th order second class Bessel functions, respectively.
In formula 15, Yn(lambda r) and Kn(λ r) is infinite, let A2，n＝A4，nWhen 0, equation 15 is rewritten to equation 16 as follows:
formula 16: u. ofn(r)＝A1，nJn(λr)+A3，nIn(λr)；
Two boundary conditions are considered here.
The first boundary condition is shown in fig. 1, where the deformable mirror edge is in a free state, and has the formula 17:
formula 17: mr(a)＝0，Vr(a)＝0；
A second boundary condition is shown in fig. 2, where the deformable mirror is simply supported at r ═ a, and then has equation 18:
formula 18: u. ofn(a)＝0，Mr(a)＝0；
From both boundary conditions, a matrix equation can be derived, as shown in equation 19:
from equation 19, it can be seen that the two unknowns A to be solved for1,n、A3,nAll depend on λ, giving the trivial solution of equation 19, then equation 20:
formula 20:
a series of lambda values are obtained from equation 20, and the values of each lambda are substituted into equation 19 to obtain two unknowns A1,n、A3,nOne of the unknowns may be represented by another unknowns. Therefore, A3,nCan be prepared from A1,nMultiplied by a correlation coefficient.
Formula 20 is characterized by
m represents the order of the mode, m depending on λ. When getting A1,nThe general normalized solution is as in equation 21:
formula 21:
in the formula (21), the compound represented by the formula,
thus, the wavefront difference u (r, θ) can be decomposed as, for example, equation 22:
formula 22:
in formula 22, n represents a symmetry number, and m represents an order.
Numerical simulation description is performed.
Take a bimorph deformable mirror with an aperture of 165mm as an example.
Parameters of the glass material: es＝190GPa；νs＝0.30；ρS＝2350kg/m3；hs＝1mm。
PZT-5H is selected as the piezoelectric material, and the material parameters are as follows: ep＝55GPa；νp＝0.40；hp＝2mm。
Giving a two-dimensional deformation function u with a small symmetry number nmn(r) and a three-dimensional deformation function umn(r, θ). From fig. 3 and fig. 4, the deformation function u can be seenmn(r) depends on the radius r. From equations 19 and 20, it can be seen that u is the value for different nmn(r) are orthogonal to each other. For the same n, umn(r) pairsM is also orthogonal at different orders. Presented from the edge of fig. 3 is a free vibration mode. From the illustration in fig. 4, it can be seen that the anamorphic mirror edge is fixed. FIGS. 5 to 16 are graphs of the deformation function in three dimensions, umn(r, θ) is an orthogonal function. Fig. 5-10 are similar zernike polynomials Z4, Z11, Z7, Z5, Z9, and Z14, respectively. Fig. 11-16 are similar to zernike polynomials Z4, Z11, Z7, Z12, Z18, and Z25, respectively.
It can be seen from fig. 17 that Z4 substantially matches the elastic modal deformation function of the deformable mirror of the present invention under the free boundary condition and n is 0, and has an integral rigid displacement with the deformable mirror of the present invention under the simple support boundary condition and n is 0.
It can be seen from fig. 18 that Z7 substantially matches the deformable mirror of the present invention under the free boundary condition and n ═ 1 elastic mode deformation function, whereas the deformable mirror of the present invention under the simple support boundary condition and n ═ 1 elastic mode deformation function is clearly different in peak value and edge value.
It can be seen from fig. 19 that the curvatures of the curves Z5, Z9 and Z14 are slightly larger than the curvature of the elastic mode deformation function of the deformable mirror of the present invention under two boundary conditions of the free boundary condition and n is 2, n is 3 and n is 4, but the change trends are basically consistent.
It can be seen from fig. 20 that the values of the curves for the deformable mirror according to the invention are all greater than 0 under the boundary conditions of simple support with n-2, n-3 and n-4, except for r-0 and r-1, which are 0, and Z12, Z1 and Z19, which are negative at the edges.
As shown in fig. 3 to 20, it can be seen that the zernike polynomial aberration characterization method substantially matches the elastic modal aberration method of the deformable mirror of the present invention under the free boundary condition, but has a larger difference from the elastic modal aberration under the simple support boundary condition. Therefore, the zernike polynomial aberration characterization method is not optimal for processing aberration characterization when the deformable mirror is a simple supporting boundary condition, and is better than the elastic modal aberration characterization method provided by the invention.
Those of skill would further appreciate that the various illustrative logical blocks, modules, circuits, and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both. To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, modules, circuits, and steps have been described above generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the overall system. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present disclosure.
Claims (4)
1. The characterization method of the elastic modal aberration of the adaptive optical deformable mirror is characterized by comprising the following steps of:
constructing a vibration mode differential equation of the deformable mirror, solving the vibration mode differential equation, and deducing an elastic mode polynomial of the deformable mirror; the process comprises the following steps:
establishing a cylindrical coordinate system, wherein the direction of a z coordinate is vertical to the surface of the deformable mirror, r is the radial coordinate of the deformable mirror, and theta is the angular coordinate of the circumferential direction of the deformable mirror;
acquiring each stress component at each position of the deformable mirror;
integrating each stress component at each position of the deformable mirror along the thickness direction of the deformable mirror to obtain bending moment in each coordinate direction of unit length, further obtaining shearing force in r and theta coordinate directions of the unit length of the deformable mirror, thereby obtaining a vibration mode differential equation of the deformable mirror, and solving the vibration mode differential equation to deduce an elastic mode polynomial of the deformable mirror;
normalizing the elastic modal polynomial;
the normalized elastic mode polynomial corresponds to each aberration of the wavefront difference decomposition one by one, so that the elastic mode polynomial represents each order of aberration of the wavefront difference decomposition.
2. The adaptive optics deformable mirror elastic modal aberration characterization method according to claim 1, wherein the process of obtaining each stress component at each position of the deformable mirror comprises:
and obtaining the strain component of each point of the deformable mirror by the deflection of the deformable mirror, as shown in formula 1:
formula 1: epsilonr＝-zwZ，rr，
In formula 1, z is the coordinate of the vertical direction of the deformable mirror, r is the radial coordinate of the deformable mirror, theta is the angular coordinate of the circumferential direction of the deformable mirror, and wz(r) deformable mirror deflection;
when the deformable mirror is formed by bonding two layers of different materials, the stress components at each position of the top layer material are as follows:
formula 2:
in the formula 2, EsAnd vsRespectively representing the Young's modulus and Poisson's ratio, epsilon, of the top layer materialr、εθ、γrθStrain components of each point of the deformable mirror;
when the deformable mirror is formed by bonding two layers of different materials, the stress components of the bottom layer material at all positions are as follows:
formula 3:
in formula 3, EpAnd vpRespectively, the Young's modulus and Poisson's ratio of the substrate material, epsilonr、εθ、γrθStrain components of each point of the deformable mirror;
when the deformable mirror is made of the same material, let E in formula 2 and formula 3s＝Ep，vs＝vp。
3. The method according to claim 2, wherein the integrating of the stress components at each position of the deformable mirror along the thickness direction of the deformable mirror is performed to obtain bending moments in each coordinate direction of unit length, and further to obtain shearing forces in r and θ coordinate directions of the unit length of the deformable mirror, so as to obtain a vibration mode differential equation of the deformable mirror, and the process of solving the vibration mode differential equation to derive the elastic mode polynomial of the deformable mirror includes:
defining the neutral plane of the deformable mirror, and setting the thickness from the neutral plane to the upper top surface of the deformable mirror as hm1The thickness from the neutral plane to the lower bottom surface of the deformable mirror is hm2Then h ism1And hm2Expressed as:
formula 4:
in the formula 4, EsExpressing the Young's modulus of the top layer material, EpExpressing Young's modulus, h, of the underlying materialsThickness of the top layer material, hpIs the thickness of the underlying material;
and integrating the stress components at each position of the deformable mirror along the thickness direction of the deformable mirror to obtain the bending moment in each coordinate direction of unit length, wherein the formula is as follows:
formula 5:
in formula 5, DsAnd DpEquation 6 is satisfied as follows:
formula 6:
in formulas 5 and 6, σr、σθ、τrθThe stress components of the deformable mirror are all parts; h ism1The thickness from the neutral surface to the upper top surface of the deformable mirror; h ism2The thickness from the neutral surface to the lower bottom surface of the deformable mirror; z is a coordinate in the vertical direction of the deformable mirror; r is the radial coordinate of the deformable mirror; theta is the angular coordinate of the deformable mirror in the circumferential direction; w is az(r) deformable mirror deflection; v. ofsRepresenting the poisson's ratio of the top layer material; v. ofpRepresenting the poisson ratio of the bottom layer material; esRepresenting the Young's modulus of the top layer material; epRepresenting the Young's modulus of the underlying material; h issIs the thickness of the top layer material; h ispIs the thickness of the underlying material;
when becomingWhen the whole lens is made of the same material, let E in formula 4, formula 5 and formula 6s＝Ep，vs＝vp，h＝hs+hp；
The r and theta coordinate direction shearing force of the unit length of the deformable mirror is further obtained by the formula 6, and the formula 7 is as follows:
formula 7:
in formula 7, Mr、Mθ、MθrBending moment in each coordinate direction for the unit length of the deformable mirror;
further obtaining the equivalent shearing force, as formula 8:
formula 8:
in formula 8, Qr、QθShear forces in the r and theta coordinate directions of the unit length of the deformable mirror; mr、Mθ、MθrBending moment in each coordinate direction for the unit length of the deformable mirror;
then the motion control equation of the deformable mirror is as follows:
formula 9:
in formula 9, Qr、QθShear forces in the r and theta coordinate directions of the unit length of the deformable mirror;
in formula 9, ξ ═ ρshs+ρphpFormula 10 is obtained from formula 5, formula 7 and formula 9 as follows:
formula 10:
in the formula 10, wz(r) deformable mirror deflection; xi ═ pshs+ρphp；
D＝Ds+Dp，Ds、DpAs shown in formula 6;
since simple vibration, the deflection is expressed as a complex number in formula 11:
formula 11: w is az(r，θ，t)＝Re{Wz(r，θ)exp(iωt)}；
Equation 12 is obtained from equation 10 as follows:
formula 12:
let equation 12 solve for equation 13, as follows:
formula 13: wz(r，θ)＝un(r) cos (n θ); n represents a symmetry number;
substitution of formula 13 for formula 12 yields formula 14 as follows:
formula 14:
in the formula (14), the compound represented by the formula,
the solution of equation 14 is written as equation 15, as follows:
formula 15: u. ofn(r)＝A1，nJn(λr)+A2，nYn(λr)+A3，nIn(λr)+A4，nKn(λr)；
In formula 15, λ4＝ω2ξ/D；Jn(λr)、Yn(λr)、In(λr)、Kn(λ r) is an n-order first class, an n-order second class, a modified n-order first class, and a modified n-order second class Bessel function, respectively;
in formula 15, Yn(lambda r) and Kn(λ r) is infinite, let A2，n＝A4，nWhen 0, equation 15 is rewritten to equation 16 as follows:
formula 16: u. ofn(r)＝A1，nJn(λr)+A3，nIn(λr)；
A matrix equation is obtained, as shown in equation 19:
from equation 19, it can be seen that the two unknowns A to be solved for1，n、A3，nAll depend on λ, giving the trivial solution of equation 19, then equation 20:
formula 20:
a series of lambda values are obtained from equation 20, and the values of each lambda are substituted into equation 19 to obtain two unknowns A1，n、A3，nWherein one unknown may be represented by another unknown, A3，nFrom A1，nMultiplied by a correlation coefficient.
4. The adaptive optics deformable mirror elastic modal aberration characterization method according to claim 3, wherein the process of normalizing the elastic modal polynomial comprises:
formula 20 is characterized by
m represents the order of the mode, m depends on λ, when taking A1，nThe general normalized solution is as in equation 21:
formula 21:
in the formula (21), the compound represented by the formula,
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