CN102053352B - Design method of synchrotron radiation curved-edge focusing mirror with maximum application range - Google Patents

Abstract

The invention discloses a method for obtaining a synchrotron radiation curved-edge bent focusing mirror with the maximum application range, which comprises the following steps: establishing an ideal elliptical mirror surface equation under focusing conditions; establishing a mirror body bending flexibility differential equation to obtain an inertia moment distribution design function meeting the requirement that the shape of the bent surface is an ideal ellipse, wherein the function contains a bending moment distribution relative slope design value kMd to be designed; when the focusing conditions deviate from the design value, calculating and adjusting the bending moment of the focusing mirror to deviate from the design value so as to be adapted to the variation of the focusing conditions; and calculating the error mean square root of the total remaining surface shape after the bending moment is adjusted, solving the kMd value which allows the absolute value of the mean square root to the derivative with respect to a certain focusing condition is minimum, then substituting the kMd value into the inertia moment distribution design function to obtain the geometric design of the mirror body optimized according to the focusing condition; and determining the focusing mirror parameters and bending the focusing mirror according to the calculation result. When the focusing condition is changed, the focusing mirror disclosed by the invention can greatly eliminate the surface shape error by adjusting the bending moment, thereby maximizing the application range of the curved-edge bent focusing mirror with respect to a certain focusing condition.

Description

Obtain the method for the bent limit of maximum scope of application synchrotron radiation focus lamp

Technical field

The present invention relates to a kind of method that obtains the bent limit press-bending focus lamp of synchrotron radiation of the maximum scope of application, belong to synchrotron radiation light beam line engineering, synchrotron radiation optical technology field.

Background technology

One of advantage of synchrotron radiation is a high brightness.Brightness refers generally to the photon number density in the phase space, and synchrotron radiation photon flux is high, the little characteristics that bring up high brightness of phase space volume.According to Liouville's theorem, not sacrifice under the photon flux prerequisite, brightness can't improve.When the line width of compression light beam, its angular breadth will increase; Otherwise when the compression angle width makes light beam become more collimation, its line width will increase.Yet different experiments is different to the phase space shape need of light beam, and for example the fluorescence micro-zone analysis requires little spot size, and the macromolecule crystal diffraction experiment requires small luminous spot dimension and collimation or the like preferably simultaneously.The demand to light beam small size hot spot, high flux surface density has been satisfied in the generation of microfocus device.

Adopt more microfocus device to be broadly divided into four types at present on the synchrotron radiation bunch: the one, Kirkpatrick-Baez mirror (being called for short the K-B mirror) microfocus device; The 2nd, catheter type microfocus device is divided into the single tube lens again and integrates the kapillary lens; The 3rd, the combination refractor focalizer that declines; The 4th, pure diffraction type focalizer mainly contains zone plate, Laue multilayer film.The K-B mirror is the focusing image-forming system with their naming that P.Kirkpatric and A.V.Baez at first propose; It is with the good characteristic of many catoptrons such as high-transmission efficient (＞70%), no chromatic dispersion and radiation hardness; And the advantage that is easy to many K-B structures such as realizing, aberration is very little on the technology, become the current microfocus device that the most extensively adopts.Shown in Fig. 1 a and Fig. 1 b; Mirror M 1, M2 that level and vertical focusing were placed, were responsible for respectively to the K-B mirror by two independent orthogonal form, and the light beam that light source source sends focuses to picture point focus via the reflection of mirror M 1, M2; Mostly reflecting mirror surface shape is cylinder, and its imaging formula is:

$\frac{1}{p}+\frac{1}{q}=\frac{1}{f}---\left(1\right)$

Wherein, p be light source to the catoptron distance, also be spacing, q is that catoptron arrives and focuses on the picture point distance, also is image distance, f is the focal length of catoptron.Owing to be used for the K-B mirror of microfocus bigger pantograph ratio to be arranged; Incident angle will be limited in the angle of total reflection; In order to guarantee bigger reception, bigger mirror length is arranged again, thereby extensive at present employing reduces aberration near the face shape of desirable elliptic cylinder as far as possible.Directly machine-shaping elliptic cylinder mirror cost is very expensive, and focal length can't be regulated.And utilize bending technology to obtain the elliptic cylinder mirror level crossing, greatly reduce the difficulty of processing of mirror, and can realize focal length adjusting within the specific limits.

The method that calculating K-B focuses on desirable minute surface face shape is following:

Like Fig. 2, p is a spacing, and q is an image distance, also is that the definition of p, q is identical with formula (1), and minute surface center light grazing angle is θ, maximum grazing angle θ on the minute surface _MaxAt the mirror terminal position, we are initial point with the mirror center, are that x axle, vertical centre minute surface direction are set up coordinate system for the y axle along the minute surface length direction, and obtain the object point coordinate for (pcos θ, psin θ), and picpointed coordinate is (qcos θ, qsin θ).Through certain point on the minute surface (x, light light path expression formula y) is:

$s=\sqrt{{(x+p\cos \mathrm{\θ})}^2+{(y-p\sin \mathrm{\θ})}^2}+\sqrt{{(x-q\cos \mathrm{\θ})}^2+{(y-q\sin \mathrm{\θ})}^2}---\left(2\right)$

According to Fermat principle, light path is the shortest, and s is 0 to the total differential of x, obtains desirable elliptic equation:

$\mathrm{el}(p,q,\mathrm{\θ};x)=\frac{(p+q)((p-q)x\cos \mathrm{\θ}+2(-\mathrm{pq}+\sqrt{\mathrm{pq}(\mathrm{pq}-x^2-\mathrm{px}\cos \mathrm{\θ}+\mathrm{qx}\cos \mathrm{\θ})}))\sin \mathrm{\θ}}{-{(p+q)}^2+{(p-q)}^2{\sin }^2\mathrm{\θ}}---\left(3\right)$

It is theoretical to bend based on mechanics of materials beam, and the press-bending amount of deflection satisfies the following differential equation

$y^{\′\′}\left(x\right)=\frac{M\left(x\right)}{\mathrm{EI}\left(x\right)}---\left(4\right)$

Wherein, x is the position on the beam, and y (x) is the press-bending amount of deflection, and M (x) is a moment of flexure, and I (x) is a moment of inertia, and E is a Young modulus.In order to make focus lamp press-bending face shape for oval, apply moment of flexure at mirror body two ends, two ends apply moment of flexure then on the mirror moment of flexure distribution M (x) be linear function M (x)=M ₀(l+k _MX), M ₀, k _MBe respectively the relative slope that mirror center moment of flexure and moment of flexure distribute.The physical definition formula of the moment of inertia I of mirror x place (x) is

wherein; W (x) is the width of mirror at the x place, and T (x) is the thickness of mirror at the x place.For the equal thickness mirror, T (x) is a constant, and then W (x) is directly proportional with I (x).

Guaranteeing M _d(x)=M _0d(l+k _MdX)＞0 and under the approximate condition of beam, M no matter _0dAnd k _MdGet what design load (all parameters of subscripting d are all represented its design load), the design of mirror body moment of inertia distribution function I (x) only need be satisfied:

$I\left(x\right)=\frac{M_{0d}(1+k_{\mathrm{Md}}x)}{E\×{\mathrm{el}}^{\′\′}(p_d,q_d,{\mathrm{\θ}}_d;x)};$

The face shape that can guarantee to bend (is y (x)=el (p for desirable ellipse _d, q _d, θ _dX)), also promptly, the focused condition design load p that is being designed _d, q _dAnd θ _dThe ideal focusing of the no face shape error of following realization.

After the mirror body that designs machined, I (x) just decided, if focused condition p, q or θ change and off-design value (desirable ellipse is the off-design value also) at this moment, can adjust focus lamp moment of flexure off-design value (adjustment M ₀And k _MThe off-design value) adapt to the variation of focused condition:

M ₀＝E×I(0)×el″(p，q，θ；0)，

$k_M=\frac{I^{\′}\left(0\right)}{I\left(0\right)}+\frac{{\mathrm{el}}^{\′\′\′}(p,q,\mathrm{\θ};0)}{{\mathrm{el}}^{\′\′}(p,q,\mathrm{\θ};0)}---\left(6\right)$

Through adjustment moment of flexure off-design value, face shape three rank of can realizing bending are similar to new desirable oval, but still have the face shape error on >=4 rank, promptly remain face shape error, and this residue face shape error is along with focused condition increases the increase of design load bias.

Existing k _MdMethod for designing is to get the moment of flexure of rectangle mirror under this focused condition to distribute, that is:

$k_{\mathrm{Md}}=\frac{3(p_d-q_d)\cos {\mathrm{\θ}}_d}{2p_d{q}_d}\≈\frac{3}{2}(\frac{1}{q_d}-\frac{1}{p_d})---\left(7\right)$

Substitution (5) formula obtains the geometry designs of the focus lamp of existing method:

$I\left(x\right)=$

$\frac{M_{0d}}{E(p_d+q_d)\sin {\mathrm{\θ}}_d}\×(2p_d{q}_d+3(p_d-q_d)\cos {\mathrm{\θ}}_{d}x)\×{\left(\frac{p_d{q}_d-x^2+(q_d-p_d)\cos {\mathrm{\θ}}_{d}x}{p_d{q}_d}\right)}^{3/2}$

$\≈\frac{M_{0d}(2p_d{q}_d+3(p_d-q_d)x){\left(\frac{(p_d+x)(q_d-x)}{p_d{q}_d}\right)}^{3/2}}{E(p_d+q_d)\sin {\mathrm{\θ}}_d}---\left(8\right)$

The focus lamp that this method for designing obtains is near rectangle; Generally can save starting material; The ideal for no face shape error under the specific focused condition of design is oval but this method for designing has only realized press-bending face shape; And do not realize scope of application maximization, when focused condition p, q or θ changed, face shape error increased rapidly.

Summary of the invention

The object of the present invention is to provide a kind of method that obtains the bent limit press-bending focus lamp of synchrotron radiation of the maximum scope of application; Serve as down the ideal ellipse of no face shape error with only can realizing of solving that existing method the exists face shape of bending at the specific focused condition of design, and can not realize that bent limit focus lamp can adapt to the problem that focused condition changes in a big way.

In order to realize the object of the invention, the method that obtains the bent limit of maximum scope of application synchrotron radiation focus lamp provided by the invention may further comprise the steps:

Set up the desirable elliptic equation of minute surface:

$\mathrm{el}(p,q,\mathrm{\θ};x)=\frac{(p+q)((p-q)x\cos \mathrm{\θ}+2(-\mathrm{pq}+\sqrt{\mathrm{pq}(\mathrm{pq}-x^2-\mathrm{px}\cos \mathrm{\θ}+\mathrm{qx}\cos \mathrm{\θ})}))\sin \mathrm{\θ}}{-{(p+q)}^2+{(p-q)}^2{\sin }^2\mathrm{\θ}};$

Set up the mirror body press-bending amount of deflection differential equation:

Wherein, moment of flexure distribution function M (x)=M ₀(l+k _MX), I (x) is the moment of inertia distribution function;

The press-bending face of being met shape is the moment of inertia distribution design function that ideal is oval, meet the approximate mirror body of beam:

$I\left(x\right)=\frac{M_{0d}(1+k_{\mathrm{Md}}x)}{E\×{\mathrm{el}}^{\′\′}(p_d,q_d,{\mathrm{\θ}}_d;x)};$

It is further comprising the steps of: when calculating focused condition off-design value, adjustment focus lamp mirror moment of flexure off-design value is to adapt to the variation of focused condition:

M ₀=E * I (0) * el " (p, q, θ; 0), $k_M=\frac{I^{\′}\left(0\right)}{I\left(0\right)}+\frac{{\mathrm{El}}^{\′\′\′}(p,q,\mathrm{\θ};0)}{{\mathrm{El}}^{\′\′}(p,q,\mathrm{\θ};0)},$ And calculating residue face shape error;

Ask and make the root mean square of this residue face shape error respectively to the minimum k of the derivative absolute value of focused condition p, q, θ _MdValue, the mechanics that promptly bends design, the said moment of inertia distribution design of substitution function is to obtain the geometry designs of the mirror body of optimizing to this focused condition then;

Wherein, x is for being initial point with the focus lamp center, and along the position coordinates of focus lamp length direction, p, q, θ are respectively spacing, image distance, minute surface center light grazing angle, M ₀Be mirror center moment of flexure, k _MBe the relative slope that moment of flexure distributes, p _d, q _d, θ _dBe respectively the design load of spacing, image distance, minute surface center light grazing angle, M _0dBe mirror center moment of flexure design load, k _MdBe the distribute design load of relative slope of moment of flexure, E is a Young modulus;

Confirm the focus lamp parameter and the focus lamp that bends according to the aforementioned calculation result.

Optimization of the present invention be total residue face type error; Rather than certain single order residue face type error, optimize more accurately, not only can make bent limit press-bending focus lamp under the focused condition that is designed, not have face shape error; When focused condition changes; Can also significantly eliminate face shape error through the adjustment moment of flexure, make residue face shape error root mean square minimum to the derivative absolute value of focused condition, promptly near design attitude, rise the most slowly with the focused condition change in design place.The present invention can realize the scope of application maximization of bent limit focus lamp to certain focused condition like this.

With regard to the principle of focus lamp was classified, the present invention was not only applicable to the bent limit press-bending K-B of synchrotron radiation focus lamp, also was applicable to the bent limit of other synchrotron radiation press-bending types focus lamp.

Structure with regard to focus lamp is classified, and the present invention not only is applicable to the horizontal focusing mirror that need not to consider gravity factor, and is applicable to the other types focus lamp of considering gravity factor.

Description of drawings

Fig. 1 a and Fig. 1 b are respectively schematic side view and the schematic top plan view that shows K-B mirror focusing principle;

Fig. 2 is the desirable face structure synoptic diagram of K-B focus lamp;

Fig. 3 shows the mirror body design of existing method for designing;

Fig. 4 shows the mirror body design of the embodiment of the invention to the maximum scope of application method for designing of spacing p;

Fig. 5 shows the mirror body design of the embodiment of the invention to the maximum scope of application method for designing of image distance q;

Fig. 6 shows the mirror body design of the embodiment of the invention to the maximum scope of application method for designing of grazing angle θ;

Fig. 7 shows different k in the embodiment of the invention _MdP, q, the slope error RMS when θ changes that value is corresponding;

Fig. 8 a shows the slope error RMS when spacing p changes in the embodiment of the invention;

Fig. 8 b shows the slope error RMS when image distance q changes in the embodiment of the invention;

Fig. 8 c shows the slope error RMS when light grazing angle θ in minute surface center changes in the embodiment of the invention.

Embodiment

Below in conjunction with meeting the present invention is explained further details with embodiment.

Basic thought of the present invention is: the moment of flexure relative slope design load k that distributes _MdThe size of the total residue face shape error in the time of can influencing the focused condition change promptly influences the scope of application of focus lamp to focused condition p, q or θ off-design value.Through getting suitable k _Md, make the adjusted residue face shape error of moment of flexure root mean square minimum, thereby make the residue face shape error minimum near the design focused condition the derivative absolute value of focused condition p, q or θ.Therefore everyly utilized the invention of this thought all should belong to protection scope of the present invention.

As previously mentioned, bent limit press-bending mirror has no face shape error under the focused condition that is designed, when focused condition changes, through the adjustment and moment of flexure can eliminate low order (0～3 rank) face shape error, and the 4th and more the face shape error of high-order can't eliminate.The root mean square that maximum scope of application method for designing of the present invention can make total residue face shape error is at design load p _d, q _d, θ _dThe place is minimum to the derivative absolute value of focused condition p, q, θ, promptly near design attitude, rises the most slowly with the focused condition change.Can make the scope of application maximization of focus lamp like this to certain focused condition.

Provide the detailed calculated process below:

By (4), (6) formula, can obtain that position total surplus face shape slope error root mean square is on the minute surface:

$\mathrm{Err}={(\underset{L}{\∫}(\∫\frac{M_0(1+k_{M}x)}{E}\mathrm{dx}-{\mathrm{el}}^{\′}(p,q,\mathrm{\θ};x))\mathrm{dx}/L)}^{1/2}---\left(9\right)$

Wherein L is a limit of integration, also is the length of focus lamp projection on the x direction, With (5) formula substitution (9) formula, ask its derivative respectively, and ask the k that makes this derivative absolute value minimum p, q and θ _MdValue.Usually be difficult to try to achieve this derivative and k _MdThe analytical form of value.The present invention uses the minimum value of the absolute value but the method that is not limited to the numerical value iterative computation is differentiated and satisfies the k under this minimum conditions _Md, k _MdThe choosing of numerical value iteration initial value is preferably 0.

More than described mirror body geometry designs, can under different actual demand situation, select to the maximum scope of application of each single focused condition.For the focused condition of general synchrotron radiation microfocus K-B mirror, focus lamp is relatively poor to the adaptability of image distance q, and the scope of application is less.Therefore, consider the common design of optimizing of the scope of application of a plurality of focused conditions, when balance, can sacrifice the scope of application of some other focused conditions and consider optimal design emphatically to image distance q scope like need.

Below with p _d=20.3m, q _d=0.18m, L=0.2m, θ _d=0.002165rad is an example, the marked difference of the method for designing that provides with the embodiment of the invention that is shown specifically existing method for designing, and wherein L is the projected length of focus lamp on the x axle:

(1) utilizes traditional design method: k _Md=8.259m ^-1,

I(x)E/M _0d[m]＝

(23.59 1+8.259x [m]) ((0.18-x [m]) (20.3+x [m])) ^3/2, as shown in Figure 3.

(2) embodiment of the invention is to the maximum scope of application method for designing of p: k _Md=0.04930m ^-1,

I(x)E/M _0d[m]＝

(23.59 1+0.04930x [m]) ((0.18-x [m] (20.3+x [m])) ^3/2, as shown in Figure 4.

(3) embodiment of the invention is to the maximum scope of application method for designing of q: k _Md=-5.556m ^-1,

I(x)E/M _0d[m]＝

(23.59 1+5.556x [m]) ((0.18-x [m]) (20.3+x [m])) ^3/2, as shown in Figure 5.

(4) embodiment of the invention is to the maximum scope of application method for designing of θ: k _Md=-5.527m ^-1,

I(x)E/M _0d[m]＝

(23.59 1-5.527x [m]) ((0.18-x [m]) (20.3+x [m])) ^3/2, as shown in Figure 6.

As previously mentioned, if need to consider the common design of optimizing of a plurality of focused conditions, managing to make do resits an exam considers the optimal design to the q scope.In the present embodiment, the common design of optimizing can simply be taken as the maximum scope of application design that is directed against q described in (3) to p, q and θ.

Fig. 7 shows different k in the embodiment of the invention _MdFocused condition p, q, slope error (slope error) root mean square when θ changes (root mean square is called for short RMS) that value is corresponding.To describe below in the foregoing description when focused condition changes; To the maximum scope of application design of monofocal condition, the common optimal design of the burnt condition scope of application of poly and the residue face shape slope error root mean square contrast of existing design; Shown in Fig. 8 a, Fig. 8 b and Fig. 8 c; Can obviously learn beneficial effect of the present invention, result among Fig. 8 a-8c is summed up tabulation as follows:

Visible from last table, in the present embodiment, the maximum scope of application method for designing that the present invention is directed to certain focused condition is compared with existing method for designing, and the scope of application significantly increases.On the basis that single focused condition is optimized, common optimal design has been sacrificed the originally very abundant p and the θ scope of application of fraction, and has optimized the scope of application of q greatly; Thereby make that the range of adjustment of p, q and θ is all more abundant.

In sum; The method that obtains the bent limit press-bending focus lamp of maximum scope of application synchrotron radiation provided by the invention can make bent limit focus lamp not have face shape error at focused condition design load place; And when focused condition changes; Can realize amplitude peak elimination face shape error through the appropriate change moment of flexure, significantly increase the scope of application of bent limit focus lamp.

Can know that by technological general knowledge the present invention can realize through other the embodiment that does not break away from its spirit or essential feature.Therefore, above-mentioned disclosed embodiment with regard to each side, all just illustrates, and is not only.All within the scope of the present invention or the change in being equal to scope of the present invention all comprised by the present invention.

Claims (5)

1. method that obtains the bent limit of maximum scope of application synchrotron radiation focus lamp may further comprise the steps:

Set up the desirable elliptic equation of minute surface:

$\mathrm{el}(p,q,\mathrm{\θ};x)=\frac{(p+q)((p-q)x\cos \mathrm{\θ}+2(-\mathrm{pq}+\sqrt{\mathrm{pq}(\mathrm{pq}-x^2-\mathrm{px}\cos \mathrm{\θ}+\mathrm{qx}\cos \mathrm{\θ})}))\sin \mathrm{\θ}}{-{(p+q)}^2+{(p-q)}^2{\sin }^2\mathrm{\θ}};$

Set up the mirror body press-bending amount of deflection differential equation:

Wherein, moment of flexure distribution function M (x)=M ₀(l+k _MX), I (x) is the moment of inertia distribution function;

The press-bending face of being met shape is the moment of inertia distribution design function that ideal is oval, meet the approximate mirror body of beam:

$I\left(x\right)=\frac{M_{0d}(1+k_{\mathrm{Md}}x)}{E\×{\mathrm{el}}^{\′\′}(p_d,q_d,{\mathrm{\θ}}_d;x)};$

It is characterized in that further comprising the steps of: during focused condition off-design value, adjustment focus lamp mirror moment of flexure off-design value is to adapt to the variation of focused condition:

M ₀=E * I (0) * el " (p, q, θ; 0), $k_M=\frac{I^{\′}\left(0\right)}{I\left(0\right)}+\frac{{\mathrm{El}}^{\′\′\′}(p,q,\mathrm{\θ};0)}{{\mathrm{El}}^{\′\′}(p,q,\mathrm{\θ};0)},$ And calculating residue face shape error;

Ask and make the root mean square of this residue face shape error respectively to the minimum k of the derivative absolute value of focused condition p, q, θ _MdValue, the said moment of inertia distribution design of substitution function is to obtain the geometry designs of the mirror body of optimizing to this focused condition then;

Confirm the focus lamp parameter and the focus lamp that bends according to the aforementioned calculation result;

Wherein, x is for being initial point with the focus lamp center, and along the position coordinates of focus lamp length direction, p, q, θ are respectively spacing, image distance, minute surface center light grazing angle, M ₀Be mirror center moment of flexure, k _MBe the relative slope that moment of flexure distributes, p _d, q _d, θ _dBe respectively the design load of spacing, image distance, minute surface center light grazing angle, M _0dBe mirror center moment of flexure design load, k _MdBe the distribute design load of relative slope of moment of flexure, E is a Young modulus.

2. the method that obtains the bent limit of maximum scope of application synchrotron radiation focus lamp according to claim 1 is characterized in that the root-mean-square value of said residue face shape error is:

$\mathrm{Err}={(\underset{L}{\∫}(\∫\frac{M_0(1+k_{M}x)}{E}\mathrm{dx}-{\mathrm{el}}^{\′}(p,q,\mathrm{\θ};x))\mathrm{dx}/L)}^{1/2};$

Wherein L is the length of focus lamp projection on the x direction,

3. the method that obtains the bent limit of maximum scope of application synchrotron radiation focus lamp according to claim 2; It is characterized in that; After obtaining said residue face shape error root mean square; With the root mean square of the said residue face shape error of said moment of inertia distribution design function substitution, ask respectively to make it to the minimum relative slope design load k of the derivative absolute value of focused condition p, q and θ _Md

4. the method that obtains the bent limit of maximum scope of application synchrotron radiation focus lamp according to claim 3 is characterized in that, uses the method for numerical value iterative computation, asks the minimum value of said derivative absolute value and satisfies the k under this minimum conditions _MdValue, k _MdThe iteration initial value elects 0 as.

5. the method that obtains the bent limit of maximum scope of application synchrotron radiation focus lamp according to claim 3 is characterized in that, if need to three common optimal design of focused condition, then is taken as the maximum scope of application design to q.

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