CN110095858A - Self-adapting optical distorting lens Elastic mode aberration characterizing method - Google Patents
Self-adapting optical distorting lens Elastic mode aberration characterizing method Download PDFInfo
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Abstract
The present invention provides a kind of self-adapting optical distorting lens Elastic mode aberration characterizing method.Elastic mode aberration characterization method of the invention is derived from the deformation of distorting lens resonance mode itself, make each rank aberration of Elastic mode polynomial repressentation wavefront Difference Solution, since every mode aberration is all the natural vibration shape of distorting lens itself, correction residual error is smaller, corrects more efficient.
Description
Technical field
The invention belongs to adaptive optical technique field more particularly to a kind of self-adapting optical distorting lens Elastic mode aberrations
Characterizing method.
Background technique
The purpose of adaptive optics is to repair the distortions of the factors to light wave wavefront such as atmospheric turbulance.Adaptive optics first has to
Wavefront distorting event is detected, it is then real-time in face of wavefront by being mounted on one piece of small-sized deformable mirror at telescope focal plane rear
It is corrected.
Adaptive corrective is carried out, resolves into a series of sum of orthogonal functions to wavefront difference u (r, θ) deformation first, it is as follows
Formula:
In above formula, unm(r) cos (n θ) is orthonormal function, i.e. aberration；N indicates symmetry number；M indicates order.
Usually way is to characterize u using zernike polynomial aberration characterization method at presentnm(r) cos (n θ), the total aberration of optics are
The sum of each rank aberration, therefore be that can be decomposed by zernike polynomial, however each aberration that this method is decomposed all is artificial
Setting.According to adaptive optics General Principle, all aberrations that applies load and can manufacture to distorting lens itself can be rectified
Just, it conversely, applying the aberration that load cannot be manufactured to distorting lens itself, then cannot correct, therefore, this method is often deposited
It is difficult to correct in some residual errors.
Summary of the invention
In order to solve the above technical problems, the present invention provides a kind of self-adapting optical distorting lens Elastic mode aberration characterization side
Method.In order to which some aspects of the embodiment to disclosure have a basic understanding, simple summary is shown below.The summary portion
Dividing is not extensive overview, nor to determine key/critical component or describe the protection scope of these embodiments.It is unique
Purpose is that some concepts are presented with simple form, in this, as the preamble of following detailed description.
The present invention adopts the following technical scheme:
In some alternative embodiments, a kind of self-adapting optical distorting lens Elastic mode aberration characterizing method is provided, is wrapped
It includes:
The vibration shape differential equation for constructing distorting lens, solves the vibration shape differential equation, derives the bullet of distorting lens
Property modal polynomials；
The Elastic mode multinomial is normalized；
Each aberration of Elastic mode multinomial and wavefront Difference Solution after normalization corresponds, and keeps Elastic mode multinomial
Each rank aberration of formula expression wavefront Difference Solution.
In some alternative embodiments, the vibration shape differential equation of the building distorting lens, to the vibration shape differential equation
It is solved, derives that the polynomial process of the Elastic mode of distorting lens includes:
Cylindrical coordinate, z coordinate direction vertical deformation mirror surface are established, r is distorting lens radial coordinate, and θ is distorting lens circumference
Direction angular coordinate；
Obtain distorting lens each components of stress everywhere；
To distorting lens, each components of stress integrate to obtain each coordinate direction of unit length along distorting lens thickness direction everywhere
Moment of flexure, further obtains r the and θ coordinate direction shearing of distorting lens unit length, so that the vibration shape differential equation of distorting lens is obtained,
The vibration shape differential equation is carried out to solve the Elastic mode multinomial for deriving distorting lens.
In some alternative embodiments, the process for obtaining distorting lens each components of stress everywhere includes:
Each components of strain of distorting lens are obtained by distorting lens amount of deflection, such as formula 1:
Formula 1:
In formula 1, z is distorting lens vertical direction coordinate, and r is distorting lens radial coordinate, and θ is distorting lens circumferencial direction angle seat
Mark, wzIt (r) is distorting lens amount of deflection；
When distorting lens is bonded by two layers of different materials, each components of stress are quilting material everywhere, such as formula 2:
Formula 2:
In formula 2, EsAnd vsRespectively indicate quilting material Young's modulus and Poisson's ratio, εr、εθ、εrθIt is answered for each point of distorting lens
Variation amount；
When distorting lens is bonded by two layers of different materials, each components of stress are primer everywhere, such as formula 3:
Formula 3:
In formula 3, EpAnd vpRespectively indicate primer Young's modulus and Poisson's ratio, εr、εθ、εrθIt is answered for each point of distorting lens
Variation amount；
When distorting lens generally same material, E in formula 2 and formula 3 is enableds=Ep, vs=vp。
In some alternative embodiments, it is described to distorting lens everywhere each components of stress along distorting lens thickness direction integrate
The each coordinate direction moment of flexure of unit length is obtained, r the and θ coordinate direction shearing of distorting lens unit length is further obtained, thus
The vibration shape differential equation of distorting lens is obtained, the vibration shape differential equation solve and derives that the Elastic mode of distorting lens is polynomial
Process includes:
The neutral surface of distorting lens is defined, if top thickness is h on neutral surface to distorting lensm1, neutral surface to distorting lens bottom
Face with a thickness of hm2, then hm1And hm2It indicates are as follows:
Formula 4:
In formula 4, EsIndicate quilting material Young's modulus, EpIndicate primer Young's modulus, hsFor the thickness of quilting material
Degree, hpFor the thickness of primer；
To distorting lens, each components of stress integrate to obtain each coordinate direction of unit length along distorting lens thickness direction everywhere
Moment of flexure, such as formula 5:
Formula 5:
In formula 5, DsAnd DpMeet formula 6, as follows:
Formula 6:
In formula 5 and formula 6, εr、σθ、σrθFor distorting lens each components of stress everywhere；hm1It is thick for top surface on neutral surface to distorting lens
Degree；hm2For neutral surface to the thickness of distorting lens bottom surface；Z is distorting lens vertical direction coordinate；R is distorting lens radial coordinate；θ is
Distorting lens circumferencial direction angular coordinate；wzIt (r) is distorting lens amount of deflection；vsIndicate quilting material Poisson's ratio；vpIndicate primer Poisson
Than；EsIndicate quilting material Young's modulus；EpIndicate primer Young's modulus；hsFor the thickness of quilting material；hpFor backsheet
The thickness of material；
When distorting lens generally same material, the E in formula 4, formula 5 and formula 6 is enableds=Ep, vs=vp, h=hs+hp；
R the and θ coordinate direction shearing that distorting lens unit length is further obtained by formula 6, such as formula 7:
Formula 7:
In formula 7, Mr、Mθ、MθrFor each coordinate direction moment of flexure of distorting lens unit length；
Equivalent shear force is further obtained, such as formula 8:
Formula 8:
In formula 8, Qr、QθIt is sheared for r the and θ coordinate direction of distorting lens unit length；Mr、Mθ、MθrFor distorting lens unit length
Each coordinate direction moment of flexure；
The then motion control equation of distorting lens, such as formula 9:
Formula 9:
In formula 9, Qr、QθIt is sheared for r the and θ coordinate direction of distorting lens unit length；
In formula 9, ξ=ρshs+ρphp, formula 10 is obtained according to formula 5, formula 7 and formula 9, as follows:
Formula 10:
In formula 10, wzIt (r) is distorting lens amount of deflection；ξ=ρshs+ρphp；D=Ds+Dp,
Ds、DpSuch as formula 6；
Due to being simple harmonic oscillation, amount of deflection is formula 11 with complex representation:
Formula 11:wz(r, θ, t)=Re { Wz(r, θ) exp (i ω t) }；
Formula 12 is obtained according to formula 10, as follows:
Formula 12:
If the solution of equation 12 is formula 13, as follows:
Formula 13:Wz(r, θ)=un(r)cos(nθ)；N indicates symmetry number；
13 substitution formula 12 of wushu obtains formula 14, as follows:
Formula 14:
In formula 14,
The solution of equation 14 is write as the form such as formula 15, as follows:
Formula 15:un(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) be respectively the n rank first kind, the second class of n rank,
Correct the n rank first kind and amendment n rank bessel function of the second kind；
In formula 15, Yn(λ r) and Kn(λ r) be it is infinite, enable A2,n=A4,n=0, then formula 15 rewrites an accepted way of doing sth 16, as follows:
Formula 16:un(r)=A1, nJn(λr)+A3, nIn(λr)；
Matrix equation is obtained, such as formula 19:
As can be seen that two unknown number A to be asked from equation 191,n、A3,nλ is all relied on, formula 19 has trivial solution, then
There is formula 20:
Formula 20:
The value of λ a series of is obtained according to formula 20, the value of each λ is substituted into formula 19, obtains two unknown number A1,n、A3,nOne
As related solution, one of unknown number can indicate with another unknown number, A3,nBy A1,nIt is indicated multiplied by related coefficient.
In some alternative embodiments, the process Elastic mode multinomial being normalized includes:
20 particular solution of formula isM indicates the order of mode, and m depends on λ, when taking A1,n=1, general normalized solution is such as
Formula 21:
Formula 21:
In formula 21,
The utility model has the advantages that Elastic mode aberration characterization method of the invention is derived from distorting lens mode of resonance itself brought by of the invention
The deformation of state makes each rank aberration of Elastic mode polynomial repressentation wavefront Difference Solution, since every mode aberration is all distorting lens
The natural vibration shape itself, correction residual error is smaller, corrects more efficient.
For the above and related purposes, one or more embodiments include being particularly described below and in claim
In the feature that particularly points out.Certain illustrative aspects are described in detail in the following description and the annexed drawings, and its instruction is only
Some modes in the utilizable various modes of the principle of each embodiment.Other benefits and novel features will be under
The detailed description in face is considered in conjunction with the accompanying and becomes obvious, the disclosed embodiments be all such aspects to be included and they
Be equal.
Detailed description of the invention
Fig. 1 be distorting lens edge of the present invention be free state when distorting lens configuration；
Fig. 2 is the configuration of distorting lens of the present invention distorting lens when simple support at edge r=a；
Fig. 3 is distorting lens of the present invention under free boundary condition, the distorting lens deformation function u as the order m=1 of modemn
(r) with radius r variation diagram；
Fig. 4 is distorting lens of the present invention under the conditions of simply support boundary, the distorting lens deformation letter as the order m=1 of mode
Number umn(r) with radius r variation diagram；
Fig. 5 is distorting lens of the present invention under free boundary condition, the deformation as the order m=1 of mode and symmetry number n=0
Function umn(r,θ)；
Fig. 6 is distorting lens of the present invention under free boundary condition, the deformation as the order m=2 of mode and symmetry number n=0
Function umn(r,θ)；
Fig. 7 is distorting lens of the present invention under free boundary condition, the deformation as the order m=1 of mode and symmetry number n=1
Function umn(r,θ)；
Fig. 8 is distorting lens of the present invention under free boundary condition, the deformation as the order m=2 of mode and symmetry number n=1
Function umn(r,θ)；
Fig. 9 is distorting lens of the present invention under free boundary condition, the deformation as the order m=3 of mode and symmetry number n=1
Function umn(r,θ)；
Figure 10 is distorting lens of the present invention under free boundary condition, the deformation as the order m=4 of mode and symmetry number n=1
Function umn(r,θ)；
Figure 11 is distorting lens of the present invention under simply support boundary condition, as the order m=1 of mode and symmetry number n=0
Deformation function umn(r,θ)；
Figure 12 is distorting lens of the present invention under simply support boundary condition, as the order m=2 of mode and symmetry number n=0
Deformation function umn(r,θ)；
Figure 13 is distorting lens of the present invention under simply support boundary condition, as the order m=1 of mode and symmetry number n=1
Deformation function umn(r,θ)；
Figure 14 is distorting lens of the present invention under simply support boundary condition, as the order m=2 of mode and symmetry number n=1
Deformation function umn(r,θ)；
Figure 15 is distorting lens of the present invention under simply support boundary condition, as the order m=3 of mode and symmetry number n=1
Deformation function umn(r,θ)；
Figure 16 is distorting lens of the present invention under simply support boundary condition, as the order m=4 of mode and symmetry number n=1
Deformation function umn(r,θ)；
Figure 17 is Elastic mode deformation function pair under zernike polynomial Z4 and two arbitrary boundary conditions of the invention and when n=0
Than figure；
Figure 18 is Elastic mode deformation function pair under zernike polynomial Z7 and two arbitrary boundary conditions of the invention and when n=1
Than figure；
Figure 19 is zernike polynomial Z5, Z9, Z14 and distorting lens of the present invention under two arbitrary boundary conditions of free boundary condition
And Elastic mode deformation function comparison diagram when n=2, n=3 and n=4；
Figure 20 be zernike polynomial Z12, Z18, Z19 and distorting lens of the present invention carry simply support boundary under the conditions of and n
=2, Elastic mode deformation function comparison diagram when n=3 and n=4.
Specific embodiment
The following description and drawings fully show specific embodiments of the present invention, to enable those skilled in the art to
Practice them.Other embodiments may include structure, logic, it is electrical, process and other change.Embodiment
Only represent possible variation.Unless explicitly requested, otherwise individual components and functionality is optional, and the sequence operated can be with
Variation.The part of some embodiments and feature can be included in or replace part and the feature of other embodiments.This hair
The range of bright embodiment includes equivalent obtained by the entire scope of claims and all of claims
Object.
In some illustrative embodiments, a kind of self-adapting optical distorting lens Elastic mode aberration characterizing method is provided,
It is to be manually set relative to zernike polynomial, the essence of Elastic mode aberration characterizing method of the invention is derived from mirror itself
The deformation of resonance mode.
Adaptive optics is to compensate each rank aberration by changing the shape of distorting lens, since zernike polynomial is people
For setting, mirror to be made to become aberration represented by zernike polynomial i.e. shape, it is relatively difficult, but mirror to be made to become elasticity
The aberration that mode indicates is easier, because of shape when this shape originally resonates from mirror itself.
Elastic mode aberration characterizing method of the invention includes:
S1: constructing the vibration shape differential equation of distorting lens, solves to the vibration shape differential equation, derives distorting lens
Elastic mode multinomial.
The Elastic mode polynomial equation of distorting lens is derived based on dynamic elasticity.
S2: the Elastic mode multinomial is normalized.Elastic mode multinomial after normalization is by inexhaustible number
Multinomial complete set composition, it there are two variable, r and θ, it is continuous orthogonal inside unit circle.
S3: each aberration of Elastic mode multinomial and wavefront Difference Solution after normalization corresponds, and makes Elastic mode
Each rank aberration of polynomial repressentation wavefront Difference Solution.
Wherein, the process of S1 includes:
Cylindrical coordinate, z coordinate direction vertical deformation mirror surface are established, r is distorting lens radial coordinate, and θ is distorting lens circumference
Direction angular coordinate；
Obtain distorting lens each components of stress everywhere；
To distorting lens, each components of stress integrate to obtain each coordinate direction of unit length along distorting lens thickness direction everywhere
Moment of flexure, further obtains r the and θ coordinate direction shearing of distorting lens unit length, so that the vibration shape differential equation of distorting lens is obtained,
The vibration shape differential equation is carried out to solve the Elastic mode multinomial for deriving distorting lens.
As illustrated in fig. 1 and 2, when distorting lens is bonded by two layers of different materials, respectively quilting material and backsheet
Material, such as quilting material are glass material, and primer is piezoceramic material.Set glass material layer with a thickness of hs, pressure
Electroceramics material layer with a thickness of hp, distorting lens diameter is 2a.When distorting lens generally same material, distorting lens integral thickness
For h, h=h at this times+hp。
Cylindrical coordinate is established, z coordinate direction vertical deformation mirror surface, distorting lens thickness direction is consistent, and r is that distorting lens is radial
Coordinate, θ are distorting lens circumferencial direction angular coordinate.
Assuming that (hs+hp)/a < < 1, then each components of strain of distorting lens are obtained by distorting lens amount of deflection are as follows:
Formula 1:
In formula 1, z is distorting lens vertical direction coordinate, and r is distorting lens radial coordinate, and θ is distorting lens circumferencial direction angle seat
Mark, wzIt (r) is distorting lens amount of deflection.
It is each everywhere to obtain quilting material when distorting lens is bonded by two layers of different materials according to theory of elastic mechanics
The components of stress:
Formula 2:
In formula 2, EsAnd vsRespectively indicate quilting material Young's modulus and Poisson's ratio；εr、εθ、γγθIt is answered for each point of distorting lens
Variation amount.
When distorting lens is bonded by two layers of different materials, primer each components of stress everywhere are as follows:
Formula 3:
In formula 3, EpAnd vpRespectively indicate primer Young's modulus and Poisson's ratio；εγ、εθ、γγθFor each point of distorting lens
The components of strain.
When distorting lens generally same material, E in formula 2 and formula 3 is enableds=Ep, vs=vp。
The neutral surface of distorting lens is defined, neutral surface is the neutral surface for neither shortening nor extending, i.e., is not pressurized and not
Tension, neutral surface are the interfaces of drawing zone and compressional zone on round distorting lens.
If top thickness is h on neutral surface to distorting lensm1, neutral surface is to distorting lens bottom surface with a thickness of hm2, then hm1With
hm2It is expressed as formula 4:
Formula 4:
In formula 4, EsIndicate quilting material Young's modulus；EpIndicate primer Young's modulus；hsFor the thickness of quilting material
Degree；hpFor the thickness of primer.
To distorting lens, each components of stress integrate to obtain each coordinate direction of unit length along distorting lens thickness direction everywhere
Moment of flexure, such as formula 5:
Formula 5:
In formula 5, DsAnd DpMeet formula 6, as follows:
Formula 6:
In formula 5 and formula 6, σr、σθ、τγθFor distorting lens each components of stress everywhere；hm1It is thick for top surface on neutral surface to distorting lens
Degree；hm2For neutral surface to the thickness of distorting lens bottom surface；Z is distorting lens vertical direction coordinate；R is distorting lens radial coordinate；θ is
Distorting lens circumferencial direction angular coordinate；wzIt (r) is distorting lens amount of deflection；vsIndicate quilting material Poisson's ratio；vpIndicate primer Poisson
Than；EsIndicate quilting material Young's modulus；EpIndicate primer Young's modulus；hsFor the thickness of quilting material；hpFor backsheet
The thickness of material.
When distorting lens generally same material, the E in formula 4, formula 5 and formula 6 is enableds=Ep, vs=vp, h=hs+hp。
R the and θ coordinate direction shearing that distorting lens unit length is further obtained by formula 6, such as formula 7:
Formula 7:
In formula 7, Mr、Mθr、MθrFor each coordinate direction moment of flexure of distorting lens unit length.
Equivalent shear force is further obtained, such as formula 8:
Formula 8:
In formula 8, Qr、QθIt is sheared for r the and θ coordinate direction of distorting lens unit length；Mr、Mθ、MθrFor distorting lens unit length
Each coordinate direction moment of flexure.
The then motion control equation of distorting lens, such as formula 9:
Formula 9:
In formula 9, Qr、QθIt is sheared for r the and θ coordinate direction of distorting lens unit length.
In formula 9, ξ=ρshs+ρphp, formula 10 is obtained according to formula 5, formula 7 and formula 9, as follows:
Formula 10:
In formula 10, wzIt (r) is distorting lens amount of deflection；ξ=ρshs+ρphp；D=Ds+Dp,
Ds、DpSuch as formula 6.
Due to being simple harmonic oscillation, amount of deflection can be formula 11 with complex representation:
Formula 11:wz(r, θ, t)=Re { Wz(r, θ) exp (i ω t) }；
Formula 12 is obtained according to formula 10, as follows:
Formula 12:
If the solution of equation 12 is formula 13, as follows:
Formula 13:Wz(r, θ)=un(r)cos(nθ)；N indicates symmetry number.
13 substitution formula 12 of wushu obtains formula 14, as follows:
Formula 14:
In formula 14,
The solution of equation 14 is write as the form such as formula 15, as follows:
Formula 15:un(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) be respectively the n rank first kind, the second class of n rank,
Correct the n rank first kind and amendment n rank bessel function of the second kind.
In formula 15, Yn(λ r) and Kn(λ r) be it is infinite, enable A2,n=A4,n=0, then formula 15 rewrites an accepted way of doing sth 16, as follows:
Formula 16:un(r)=A1, nJn(λr)+A3, nIn(λr)；
Here consider two arbitrary boundary conditions.
First arbitrary boundary conditions then have formula 17 as shown in Figure 1, distorting lens edge is free state:
Formula 17:Mr(a)=0, Vr(a)=0；
Second arbitrary boundary conditions then have formula 18 as shown in Fig. 2, distorting lens is simply supported in r=a:
Formula 18:un(a)=0, Mr(a)=0；
Matrix equation can be obtained by two arbitrary boundary conditions, such as formula 19:
As can be seen that two unknown number A to be asked from equation 191,n、A3,nλ is all relied on, formula 19 has trivial solution, then
There is formula 20:
Formula 20:
The value of λ a series of is obtained according to formula 20, the value of each λ is substituted into formula 19, obtains two unknown number A1,n、A3,nOne
As related solution, one of unknown number can indicate with another unknown number.So A3,nIt can be by A1,nMultiplied by related coefficient
To indicate.
20 particular solution of formula isM indicates that the order of mode, m depend on λ.When taking A1,n=1, general normalized solution is such as
Formula 21:
Formula 21:
In formula 21,
Therefore, wavefront difference u (r, θ) can be decomposed into, such as formula 22:
Formula 22:
In formula 22, n indicates symmetry number, and m indicates order.
Carry out numerical simulation explanation.
For being 165mm bimorph deformable mirror with a bore.
The parameter of glass material: Es=190GPa；νs=0.30；ρS=2350kg/m3；hs=1mm.
Piezoelectric material selects PZT-5H, material parameter: Ep=55GPa；νp=0.40；hp=2mm.
Provide the smaller two-dimensional deformation function u of symmetry number nmn(r) and three-dimensional deformation function umn(r,θ).It can from Fig. 3 Fig. 4
To find out deformation function umn(r) radius r is depended on.From formula 19 and formula 20 it is known that for different n value umn(r) mutually just
It hands over.For identical n, umnIt (r) is also orthogonal for different rank m.What is presented from the edge Fig. 3 is free vibration mode.From
Shown in Fig. 4, it can be seen that distorting lens edge is fixed.Fig. 5 to Figure 16 is three-dimensional deformation function figure, umn(r, θ) is orthogonal
Function.Fig. 5 to Figure 10 is similar zernike polynomial Z4, Z11, Z7, Z5, Z9 and Z14 respectively.Figure 11 to Figure 16 is analogous respectively to
Zernike polynomial Z4, Z11, Z7, Z12, Z18 and Z25.
As can be seen from Figure 17 Z4 and distorting lens of the present invention Elastic mode deformation under free boundary condition and when n=0
Function coincide substantially, with distorting lens of the present invention simply support boundary under the conditions of and n=0 Elastic mode deformation function have one it is whole
Body rigid displacement.
As can be seen from Figure 18 Z7 and distorting lens of the present invention are under free boundary condition and n=1 Elastic mode deformation letter
Base sheet is coincide, and with distorting lens of the present invention under the conditions of simply support boundary and n=1 Elastic mode deformation function in peak value and
Difference is obvious on marginal value.
As can be seen from Figure 19 Z5, Z9, Z14 curvature of curve are respectively than distorting lens of the present invention at two kinds of free boundary condition
Elastic mode deformation function curvature is slightly big under boundary condition and when n=2, n=3 and n=4, but variation tendency is almost the same.
As can be seen from Figure 20 distorting lens of the present invention is under the conditions of carrying simple support boundary and when n=2, n=3 and n=4
Curve values are both greater than 0, and in addition to being 0 at r=0 and r=1, and Z12, Z1 and Z19 are negative in edge.
As shown in Fig. 3 to Figure 20, it can be seen that zernike polynomial aberration characterization method and distorting lens of the present invention are in free margins
Elastic mode aberration method under the conditions of boundary is coincide substantially, but with simply support boundary under the conditions of Elastic mode aberration difference ratio
It is larger.Therefore, zernike polynomial aberration characterization method is used to handle aberration table when distorting lens is simple support boundary condition
Sign is not optimal, preferably proposed by the invention Elastic mode aberration characterizing method.
It should also be appreciated by one skilled in the art that various illustrative logical boxs, mould in conjunction with the embodiments herein description
Electronic hardware, computer software or combinations thereof may be implemented into block, circuit and algorithm steps.In order to clearly demonstrate hardware and
Interchangeability between software surrounds its function to various illustrative components, frame, module, circuit and step above and carries out
It is generally described.Hardware is implemented as this function and is also implemented as software, depends on specific application and to entire
The design constraint that system is applied.Those skilled in the art can be directed to each specific application, be realized in a manner of flexible
Described function, still, this realization decision should not be construed as a departure from the scope of protection of this disclosure.
Claims (5)
1. self-adapting optical distorting lens Elastic mode aberration characterizing method characterized by comprising
The vibration shape differential equation for constructing distorting lens, solves the vibration shape differential equation, derives the springform of distorting lens
State multinomial；
The Elastic mode multinomial is normalized；
Each aberration of Elastic mode multinomial and wavefront Difference Solution after normalization corresponds, and makes Elastic mode polynomial table
Each rank aberration of Difference Solution before oscillography.
2. self-adapting optical distorting lens Elastic mode aberration characterizing method according to claim 1, which is characterized in that described
The vibration shape differential equation for constructing distorting lens, solves the vibration shape differential equation, derives that the Elastic mode of distorting lens is more
The process of formula includes:
Cylindrical coordinate, z coordinate direction vertical deformation mirror surface are established, r is distorting lens radial coordinate, and θ is distorting lens circumferencial direction
Angular coordinate；
Obtain distorting lens each components of stress everywhere；
To distorting lens, each components of stress integrate to obtain each coordinate direction moment of flexure of unit length along distorting lens thickness direction everywhere,
R the and θ coordinate direction shearing for further obtaining distorting lens unit length, so that the vibration shape differential equation of distorting lens is obtained, to vibration
The type differential equation carries out solving the Elastic mode multinomial for deriving distorting lens.
3. self-adapting optical distorting lens Elastic mode aberration characterizing method according to claim 2, which is characterized in that described
The process for obtaining distorting lens each components of stress everywhere includes:
Each components of strain of distorting lens are obtained by distorting lens amount of deflection, such as formula 1:
Formula 1:
In formula 1, z is distorting lens vertical direction coordinate, and r is distorting lens radial coordinate, and θ is distorting lens circumferencial direction angular coordinate, wz
It (r) is distorting lens amount of deflection；
When distorting lens is bonded by two layers of different materials, each components of stress are quilting material everywhere, such as formula 2:
Formula 2:
In formula 2, EsAnd vsRespectively indicate quilting material Young's modulus and Poisson's ratio, εr、εθ、γrθFor each point Ying Bianfen of distorting lens
Amount；
When distorting lens is bonded by two layers of different materials, each components of stress are primer everywhere, such as formula 3:
Formula 3:
In formula 3, EpAnd vpRespectively indicate primer Young's modulus and Poisson's ratio, εr、εθ、γrθFor each point Ying Bianfen of distorting lens
Amount；
When distorting lens generally same material, E in formula 2 and formula 3 is enableds=Ep, vs=vp。
4. self-adapting optical distorting lens Elastic mode aberration characterizing method according to claim 3, which is characterized in that described
To distorting lens, each components of stress integrate to obtain each coordinate direction moment of flexure of unit length along distorting lens thickness direction everywhere, into one
Step obtains r the and θ coordinate direction shearing of distorting lens unit length, so that the vibration shape differential equation of distorting lens is obtained, it is micro- to the vibration shape
Divide equation to carry out solution and derives that the polynomial process of the Elastic mode of distorting lens includes:
The neutral surface of distorting lens is defined, if top thickness is h on neutral surface to distorting lensm1, neutral surface to distorting lens bottom surface
With a thickness of hm2, then hm1And hm2It indicates are as follows:
Formula 4:
In formula 4, EsIndicate quilting material Young's modulus, EpIndicate primer Young's modulus, hsFor the thickness of quilting material, hpFor
The thickness of primer；
To distorting lens, each components of stress integrate to obtain each coordinate direction moment of flexure of unit length along distorting lens thickness direction everywhere,
Such as formula 5:
Formula 5:
In formula 5, DsAnd DpMeet formula 6, as follows:
Formula 6:
In formula 5 and formula 6, σr、σθ、τrθFor distorting lens each components of stress everywhere；hm1For top thickness on neutral surface to distorting lens；hm2
For neutral surface to the thickness of distorting lens bottom surface；Z is distorting lens vertical direction coordinate；R is distorting lens radial coordinate；θ is deformation
Mirror circumferencial direction angular coordinate；wzIt (r) is distorting lens amount of deflection；vsIndicate quilting material Poisson's ratio；vpIndicate primer Poisson's ratio；
EsIndicate quilting material Young's modulus；EpIndicate primer Young's modulus；hsFor the thickness of quilting material；hpFor primer
Thickness；
When distorting lens generally same material, the E in formula 4, formula 5 and formula 6 is enableds=Ep, vs=vp, h=hs+hp；
R the and θ coordinate direction shearing that distorting lens unit length is further obtained by formula 6, such as formula 7:
Formula 7:
In formula 7, Mr、Mθ、MθrFor each coordinate direction moment of flexure of distorting lens unit length；
Equivalent shear force is further obtained, such as formula 8:
Formula 8:
In formula 8, Qr、QθIt is sheared for r the and θ coordinate direction of distorting lens unit length；Mr、Mθ、MθrIt is each for distorting lens unit length
Coordinate direction moment of flexure；
The then motion control equation of distorting lens, such as formula 9:
Formula 9:
In formula 9, Qr、QθIt is sheared for r the and θ coordinate direction of distorting lens unit length；
In formula 9, ξ=ρshs+ρphp, formula 10 is obtained according to formula 5, formula 7 and formula 9, as follows:
Formula 10:
In formula 10, wzIt (r) is distorting lens amount of deflection；ξ=ρshs+ρphp；D=Ds+Dp, Ds、Dp
Such as formula 6；
Due to being simple harmonic oscillation, amount of deflection is formula 11 with complex representation:
Formula 11:wz(r, θ, t)=Re { Wz(r, θ) exp (i ω t) }；
Formula 12 is obtained according to formula 10, as follows:
Formula 12:
If the solution of equation 12 is formula 13, as follows:
Formula 13:Wz(r, θ)=un(r)cos(nθ)；N indicates symmetry number；
13 substitution formula 12 of wushu obtains formula 14, as follows:
Formula 14:
In formula 14,
The solution of equation 14 is write as the form such as formula 15, as follows:
Formula 15:un(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 the n rank first kind, the second class of n rank, amendment respectively
The n rank first kind and amendment n rank bessel function of the second kind；
In formula 15, Yn(λ r) and Kn(λ r) be it is infinite, enable A2,n=A4,n=0, then formula 15 rewrites an accepted way of doing sth 16, as follows:
Formula 16:un(r)=A1, nJn(λr)+A3, nIn(λr)；
Matrix equation is obtained, such as formula 19:
As can be seen that two unknown number A to be asked from equation 191,n、A3,nλ is all relied on, formula 19 has trivial solution, then has formula
20:
Formula 20:
The value of λ a series of is obtained according to formula 20, the value of each λ is substituted into formula 19, obtains two unknown number A1,n、A3,nGeneral phase
Guan Xie, one of unknown number can indicate with another unknown number, A3,nBy A1,nIt is indicated multiplied by related coefficient.
5. self-adapting optical distorting lens Elastic mode aberration characterizing method according to claim 4, which is characterized in that institute
Stating the process that Elastic mode multinomial is normalized includes:
20 particular solution of formula isM indicates the order of mode, and m depends on λ, when taking A1,n=1, general normalized solution such as formula
21:
Formula 21:
In formula 21,
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