Figuring algorithm for high-gradient mirrors with axis-symmetrical removal function

Figuring algorithm for high-gradient mirrorswith axis-symmetrical removal function

Changjun Jiao,* Shengyi Li, Xuhui Xie, Shanyong Chen,Dongliang Wu, and Nianhui Kang

National University of Defense Technology, DeYa, Changsha, Hunan Province, 410073, China

*Corresponding author: [email protected]

Received 9 November 2009; revised 5 January 2010; accepted 5 January 2010;posted 6 January 2010 (Doc. ID 119687); published 22 January 2010

Figuring technologies based on intracone and intercone stitching for high-gradient mirrors are discussed.Based on the conventional computer-controlled optics shaping principle, a process model for a single conewith intracone stitching is constructed. With the circular stitching property of the model, a modifiedBayesian-based Richardson–Lucy (RL) algorithm is deduced to deconvolute dwell time for single cone.Building on this algorithm, with the introduction of intercone stitching, a process model for a complexcone is built. Then another modified Bayesian-based RL algorithm is deduced to deconvolute the dwelltime for a complex cone from the properties of intracone stitching and intercone stitching. With a velocityrealization method for dwell time on a spiral path of the cone and the determination criterion of the pathparameter, figuring technologies for single and complex cones are presented. Simulation and experimentdemonstrate that theories and methods discussed can solve key problems of figuring high-gradientmirrors; the figuring technologies are novel methods for high-gradient mirrors and can be used to figuremirrors finely. © 2010 Optical Society of America

OCIS codes: 220.0220, 220.4610, 220.5450.

1. Introduction

With the development of the application technolo-gies, high-gradient optics are to be used in somefields such as x-ray systems and the conformal domeof missiles. In the x-ray field, the working light is in-cident approximately parallel to the optics to avoidbeing absorbed by the material of the optics. Asshown in Fig. 1(a), the typical shape of the x-ray com-ponents is composed approximately of two cones, andthe norms of the cones are both approximately per-pendicular to the axis of the components [1]. The con-formal optics of the dome has perfect aerodynamicsproperties and can reduce air friction significantly.Missiles with conformal domes can have fasterspeeds, shorter flying times, and longer ranges offire, which can enhance their performance. As shownin Fig. 1(b), the typical shape of the conformal opticsis a prolate ellipsoid [2].

For high-gradient mirrors with axis-symmetricalerrors, the circumferential average method [3,4]for low-gradient mirrors can be modified to describethis process as a 1D one, and a shaping model can bebuilt based on a computer-controlled optics shaping(CCOS) principle. With this model, axis-symmetricalerrors of the high-gradient mirrors can be corrected[5]. Key problems in the figuring process of high-gradient mirrors with axis-symmetrical errors areto build a process model such that dwell time thatcan reveal the local characteristics of the surface er-ror could be deconvoluted from this model, and to re-alize the deconvoluted dwell time on some path tocorrect the local error. For the nondevelopability ofthe target surface, the figuring process is nonlinear.So the conventional shaping principle, algorithms fordwell time, and methods for dwell time realizationare not suitable for the case of high-gradient mirrors.

In this paper, a figuring technology for high-gradient mirrors with a (near-) axis-symmetricalremoval function is presented. On the developed pla-nar sector, the process model for a single cone is built

0003-6935/10/040578-08$15.00/0© 2010 Optical Society of America

578 APPLIED OPTICS / Vol. 49, No. 4 / 1 February 2010

with the intracone stitching operation. With dwelltime realization on a spiral path of the cone, the fig-uring technology for a single cone comes into being.Building on this technology, with the introduction ofthe intercone stitching operation, a process model fora complex cone is built. A Bayesian-based dwell timealgorithm is deduced with this model. In the lastsection of this paper, a simulation and experimentare presented to validate these algorithms.

2. Figuring Technology for Single Cone

In this section, on the plane developed from a cone,the intracone stitching is discussed on the basis ofthe conventional CCOS principle. With the stitching,a process model for figuring a single cone is built up.Based on the model, a Bayesian-based iterative algo-rithm is deduced for the deconvolution of dwell time.Moreover, discussed is the dwell time realizationmethod on a spiral path. With these studies, the fig-uring technology for single cone comes into being.

A. Process Model

As shown in Fig. 2(a), the apices of the quasi-polar co-ordinate GPðρ; θÞ are located at the intersection of thetop surfaceand the rotational axisZ. Thepolar angle θis the angle between axis X and rotational radius r.The polar radius ρ is the minimal distance from apoint on the cone to the apices. A single cone can bedefined by three parameters such as the top radiusr1, base radius r2, and leg l of the section isosceles tra-pezoid. The shape parameter p of the cone can be de-fined as p ¼ l=ðr2 − r1Þ. Obviously, p is in the range½1;∞�. When p ¼ 1, the cone degenerates to a plane,and when p → ∞, the cone degenerates to a cylinder.As shown in Fig. 2(b), the sector is the developed

plane of the cone divided along line C1C2. The origi-nal pointOl of the local Cartesian coordinate LCðOl −xyzÞ for the sector is located at the center of the sec-tor. The polar point of the local polar coordinate forthe sector LPðρl; θlÞ is Ol. The inner radius rI, outerradius ro, and the central angle β of the sector arerI ¼ pr1, ro ¼ pr2, β ¼ 2π=p, respectively.The shaping process for the cone would be de-

scribed in the quasi-polar coordinate GPðρ; θÞ andin the local Cartesian coordinate LCðOl − xyzÞ. As-suming that the coordinates of the same point inthese two coordinate systems are ðρ; θÞ and ðx; yÞ,the translation relations between these coordinatescan be expressed as

ρ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2 þ y2Þ

p− rI þ rini ; ð1Þ

θ ¼ p × arctanðy=xÞ;

x ¼ ðρ − rini þ rIÞ cosðθ=pÞ;

y ¼ ðρ − rini þ rIÞ sinðθ=pÞ;

where rini is the minimal radius of the cone inGPðρ; θÞ. For a single cone, rini equals r1. The threefundamental parameters for the figuring processare the removal function R, surface error E, anddwell time T. The translation function for these threeparameters from GPðρ; θÞ to LCðOl − xyzÞ is gp2lc,and the inverse function is lc2gp. The precedingsuperscript of these parameters can be GP and LC,which denote the described coordinate system.

In the remainder of this section, the relationamong dwell time T, removal function R, and surfaceerror E is to be studied. This relation describes theprocess model for a single cone, similar to that ofthe conventional CCOS principle.

Assume that the removal function is axis symmetri-cal. The radius of the removal function is rt. A removalfunction that is near axis-symmetrical would be dealtwith by the circumferential averaged method [6].

As shown in Fig. 3, in the local Cartesian coordi-nate LCðOl − xyzÞ, the region of the surface error Ais ð−β=2; β=2� × ½rI; ro�. The extension regions of sur-face error B and C are ð−β=2; β=2� × ½rI − rt; rIÞ andð−β=2; β=2� × ðro; ro þ rt�, respectively. The extensionregions of convolution computation D and E areðβ=2; β=2þ δβ� × ½rI − rt; ro þ rt� and ð−β=2 − δβ;−β=2Þ×½rI − rt; ro þ rt�, respectively. The additive regionsof inner-cone stitching D’ and E’ are ð−β=2;−β=2þδβ� × ½rI − rt; ro þ rt� and ðβ=2 − δβ; β=2� × ½rI − rt; ro þrt� respectively, where δβ ¼ arcsinðrt=ðrI − rtÞÞ.

The single cone is divided along C1C2 manually, sothe two lines (through polar point and θl ¼ �β=2 inthe local polar coordinate for the sector LPðρl; θlÞ) co-incide with each other, which means that D and D0

Fig. 1. (Color online) Application fields of the high-gradientmirrors: (a) x-ray component, (b) conformal component.

Fig. 2. Description of single cone: (a) single cone, (b) developedsector.

1 February 2010 / Vol. 49, No. 4 / APPLIED OPTICS 579

are superimposed, and E and E0 are superimposed.To weaken the edge effect, edge extension shouldbe dealt with in regionsB andC along polar axis withthe Gaussian method that has been discussed in [6].In the developed sector, the region of surface error isA∪B∪C, while the surface error in other regionsshould be zero. The region of dwell time that can re-move material in region A is also A∪B∪C.Mj ðj ¼ 0;…; 2Þis the subregion maker matrix for

regions A∪B∪C, D, and E; j ¼ 0 denotes region D, j ¼1 denotes region A∪B∪C, and j ¼ 2 denotes region E.They have the same size as the surface error matrixE. Elements corresponding to the subregion are ones;other elements are zeros.According to the conventional CCOS principle, in

local Cartesian coordinates LCðOl − xyzÞ, the relationamong E, R, and T is [7],

LCE ¼ LCR ⊗LC T: ð2Þ

The material removal in zones A∪B∪C, D, and E is

GPEj ¼ lc2gpðLCEjÞ ¼ lc2gpðLCE⊙MjÞ; ð3Þ

where, ⊗ denotes matrix convolution and ⊙ denoteselement-size multiplication of matrix the.When the removal function dwells at region

A∪B∪C according to the dwell time T, Region Dand E also have material to be removed for the widthof the removal function. But these two regions areboth virtual, and there are no real materials to be re-moved. Because the cone is divided alongC1C2 manu-ally, the removed materials in region D and E shouldbe added to subregions D0 and E0 of A∪B∪C, respec-tively, according to the material removal additivity,so removed material of the cone with respect to dwelltime T is

LCE ¼ gp2lc�X2

j¼0

rotðGPEj; 2πðj − 1ÞÞ�

≜coneconvðLCR;LC TÞ;ð4Þ

where rotðf ; γÞ denotes rotating f about the rotationalaxis Z with degree γ in the quasi-polar coordi-nates GPðρ; θÞ.

Equation (4) is the relation among the surface er-ror E, removal function R, and dwell time T, and thisrelation describes the process model for the figuringprocess of a single cone.

B. Dwell Time Calculation Algorithm

The problem defined by Eq. (4) is nonlinear. The tra-ditional pulse iteration model is not suitable for thisproblem [8].With the pair of the removal function andshape error for any point on the grid, the advancedmatrixmodel [9] (Carnal–Egert–Hylton, CEHmodel)can describe this problem, and the singular value de-composition method can be used to deconvolute dwelltime based on this model. But the computation costrises with the number of grid points.

In [6], a Bayesian-based iterative algorithm wasdeduced for figuring the linear process. Similar tothe conventional RL iteration [10,11], this algorithmhas a nonnegativity property, which means that theiterative sequences would be larger than zero provid-ed that the first estimate of the dwell time is largerthan zero at any point. The Bayesian-based algori-thm for dwell time is [6,12]

Tkþ1 ¼ Tk⊙

�Rð−x;−yÞRRRdxdy

⊗E

R ⊗ Tk

�: ð5Þ

In Eq. (5), convolution computation appears twice.First, it appears as the simulated removed surfaceerror R ⊗ Tk at the kth iterative dwell time Tk.Second, the correlation between the removal functionand the corrected surface error emerges as the cor-rection factor of the next iterative result. Modifica-tions should be made in these two places accordingto the circular additive property of the stitching pro-cess. With the same processing method as in Eq. (4),the modified iterative algorithm for the single conefiguring process can be expressed as

LCTkþ1 ¼ LCTk

⊙

�coneconv

�LCRð−x;−yÞRRLCRdxdy

;LCE

coneconvðLCR;LC TkÞ

��:

ð6Þ

The modified iterative format—circular RL (CRL)—can be simplified to the RL format described byEq. (5) when the cone is degraded to a plane. CRLis simply a generalized version of Eq. (5). Obviously,the CRL algorithm also has the property of nonnega-tivity. With the surface error used as the first esti-mate T0, the iterative sequences Tk should have the

Fig. 3. Schematic illustration of intracone stitching for singlecone.


nonnegativity property, satisfying the nonnegativitydemand of the dwell time in the figuring process ofmaterial removal.

C. Realization Method for Dwell Time

In the spiral scan mode, the polishing tool is trans-lated along the polar axis perpendicular to the work-piece with a constant distance while the workpiecerotates about the Z axis. The case for low-gradientmirrors was discussed in [6,12,13]. As shown inFig. 4(a), the dwell timedensity functionTðρ; θÞ is rea-lized on one spiral path (the spiral parametermay notbeuniform). In thequasi-polar coordinateGPðρ; θÞ, letthe spiral path on the cone be expressed as

ρ ¼ kθ=360; ð7Þ

where k is the pitch of the spiral path, defined as halfof the polar radius between the preceding and subse-quent zone. For the selection criterion, refer to [6,13].In the local Cartesian coordinate LCðOl − xyzÞ ofthe developed sector, the scanning path could be de-scribed as

ρl ¼ ModðθÞkþ kβ α; ð8Þ

where α ¼ θ − 360ModðθÞ=p. The function ModðθÞequals 1 plus the modulus after θ divided by 360.Equation (8) indicates that the scanning path inthe developed sector is still a spiral path of the sametype.To correct local surface error, the dwell time den-

sity function should be realized with varied velocityalong spiral path. As shown in Fig. 4(b), the totaldwell time on the sector with polar radius ρl and po-lar angle dα is

tðρ; θÞ ¼ GPTðρ; θÞρlkdα: ð9Þ

Omitting the transient of the translation system, theresultant velocity between conterminous points onthe spiral path can be computed according to Eq. (9)as

vðρ; θÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpdαÞ2 þ ðkpdα=360Þ2

ptðρ; θÞ : ð10Þ

Because k=360 ≪ 1, Eq. (10) can be approximated as

vðρ; θÞ ≈ pGPTðρ; θÞρlk

¼ 1GPTðρ; θÞrk ; ð11Þ

where r is the rotational radius of the cone at a dweltpoint. Equation (11) indicates that the rotation ofworkpiece approximately equals the resultant velo-city and is the major motion in the spiral scan mode.The resultant velocity is inversely proportional todwell time density, rotational radius, and pitch,and it does not depend on the angular interval dα.Equation (11) is a general version of Eq. (25) in [6]for high-gradient mirrors.

The rotation velocity of the workpiece would be solarge as to exceed the maximum velocity of the sys-tem when the dwell time density or the polar radiusis too small. In this case, the resultant velocityshould be truncated. The total dwell time would belarger than desired with this truncated velocity.However, Eq. (11) also indicates that the resultantvelocity can be adjusted by the pitch of the spiralpath. The selection criteria for the pitch was de-scribed in [6,12,13], which is that the total dwell timeon the zone with rotational radius r and pitch kshould not be smaller than the time consumed inscanning this zone with the truncated velocity. Thiscriteria can deal with large velocity deduced fromsmall dwell time density and small rotational radiusand does not affect the total dwell time.

3. Figuring Technology for Complex Cone

In this section, a complex cone composed of two conesis set as an example to study the intercone stitching.With this operation, the process model for the figur-ing process of a complex cone is built up. Based on themodel, a Bayesian-based iterative algorithm is de-duced for the deconvolution of dwell time. With thesestudies, the figuring technology for a complex conecomes into being.

A. Process Model

As shown in Fig. 5(a), the parameters for one cone ofthe complex cone are r1, r2, l1, and those for anothercone are r2, r3, l2. Assuming that the angle γ betweenthe two generatrices of the cones is so small thatcos γ ≈ 1, tan γ ≈ 0, sin γ ≈ 0.

The definition of the quasi-polar coordinateGPðρ; θÞ for a complex cone is similar to that for a sin-gle cone. As shown in Fig. 5, divide the complex conealong r2 into two single cones designated as cones Iand II. With a method similar to that for a singlecone, local Cartesian coordinates LCðOI − xyzÞ andLCðOII − xyzÞ are defined. So are regions A, B, C,D, E, D0, and E0 for cones I and II. Except for regionsCI and BII, the functions for other regions are thesame as those in a single cone. CI and BII are exten-sion regions of convolution computation, which are

Fig. 4. (Color online) Schematic illustration of dwell time reali-zation: (a) spiral path on a cone, (b) spiral path on developed sectorof cone.


similar to regions D and E. From the intraconestitching discussed in Subsection , the relation be-tween dwell time Tn, removal functionR, and surfaceerror E0

n for any cone of the complex cone is

LCE0n ¼ gp2lc

X2j¼0

rotðGPEjn; 2πðj − 1ÞÞ!

¼ coneconvðLCR;LC TnÞ: ð12Þ

The conventional convolution computations are pro-cessed in the local Cartesian coordinates. The intra-cone stitching above and the intercone stitchingbelow are processed in the quasi-polar coordinateGPðρ; θÞ for a complex cone. Similar to the case ofa single cone, the translation functions amongGPðρ; θÞ to LCðO − xyzÞ are signified gp2lc and lc2gp,where, ðrinÞI ¼ r1, ðrinÞII ¼ r1 þ l1.The complex cone is divided along r2 into two single

conesmanually, so regionsCI andC0II [ð−βII=2; βII=2�×

½ðroÞII; ðroÞII þ rtÞ] are superposition regions, and BIIand B0

I [ð−βI=2; βI=2� × ððroÞI − rt; ðroÞI�] are approxi-mately superpositionsprovided that γ ≈ 0. The resultsof conventional convolution of Eq. (12) should betranslated into quasi-polar coordinates GPðρ; θÞ. InGPðρ; θÞ, the result of region CI should be added tothat of C0

II, and the result of BII should be added tothat ofB0

I; then the values of regionsCI andBII shouldbe set to be zero. This operation is intercone stitching,and can be expressed as

LCE ¼ gp2lc

XNn¼1

lc2gpðLCE0nÞ!: ð13Þ

Substituting Eq. (12) into Eq. (13), the integratedcomputation of intracone stitching and interconestitching can be expressed as

LCE ¼ gp2lc�XN

n¼1

�X2j¼0

rotðGPEjn; 2πðj − 1ÞÞ��

≜mconeconvðLCR;LC TÞ; ð14Þ

whereN is the number of single cones in the complexcone. Equation (14) describes the relation among sur-face error E, removal function R, and dwell time T,which builds the process model of figuring complexcone.

B. Dwell Time Calculation Algorithm

Similar to Eq. (6), modifications should be made toEq. (5)in the two convolution computation places.According to intracone and intercone stitching prop-erties of Eq. (14), the modified iterative algorithm fora complex cone is

LCTkþ1 ¼ LCTk⊙

�mconeconv

�LCRð−x;−yÞRRLCRdxdy

;

×LCE

mconeconvðLCR; LCTkÞ

��: ð15Þ

The conventional convolution of Eq. (14) is computedin the local Cartesian coordinates of the cones, andthe intracone and intercone stitching are computedin the quasi-polar coordinates of the complex cone.Obviously, the modified iterative format complexCRL (CCRL) also has the property of nonnegativity.

In the figuring process of low-gradient mirrors, thetarget mirror is approximated by a plane [6], while inthe figuring of high-gradient mirrors, the prolate el-lipsoid of the conformal optics can be approximatedby a complex cone composed of a plane and severalcones. With this approximated complex cone, thealgorithm discussed above can be applied to deconvo-lute dwell time T.

Fig. 5. Schematic illustration of figuring process for complex cone: (a) complex cones composed of two cones, (b) intracone stitching,(c) intercone stitching.


For the deconvoluted dwell time from Eq. (14) for acomplex cone, the realization method on the spiralpath discussed in Subsection 2.C can also be appliedin the case of a complex cone. With these algorithms,the figuring technology for complex cone comesinto being.

4. Simulation and Experiment

The CCRL algorithm, Eq. (15), is a general version ofthe CRL algorithm [Eq. (6)]. It is implied that the va-lidation of CCRL is a sufficient condition for that ofCRL. So, simulation and experiment are made to va-lidate CCRL in this section. For the CCRL algorithm,there are no intrinsically differences between low-gradient mirrors and high-gradient mirrors exceptfor the angle γ. However, it is so small that cos γ ≈1, tan γ ≈ 0, sin γ ≈ 0 for any cases. Thus an experi-ment with low-gradient mirrors can be performedto validate the CRL and CCRL algorithms.In this section, first, simulation of a high-gradient

cone is performed to validate dwell time algorithms.To validate the realization method for dwell time, theCEHmodel is applied to simulate the realization pro-cess of dwell time. Then, an experiment with a low-gradient mirror is performed to validate the figuringtechnology of a complex cone. In this section, allfigures are to be plotted with the cone expanded asa plane.

A. Simulation of High-Gradient Cone

A high-gradient complex cone is chosen as an exam-ple. The cone is assumed to be composed of two simplecones [similar to Fig. 5(a)]. The parameters for thecomplex cone are r1 ¼ 20mm, r2 ¼ 32:68mm, r3 ¼40:5mm, l1 ¼ 30mm, and l2 ¼ 45mm. In the simula-tion process, the removal function is of Gaussiandistribution, and its radius is 10mm.

Figure 6(a) is the assumed initial surface error dis-tribution with edge extension. The initial RMS (rootmean square) is 0:025 λ (λ ¼ 632:8nm). Figure 6(b)plots the dwell time deconvoluted with CCRL algo-rithm of Eq. (15). This figure indicates that the dis-tribution of dwell time is similar to that of the initialerror. Figure 6(c) is the predicted residual; the RMSis 0:007 λ. There are no special character errors onthe edges of the residual error, which indicates thatthe edge effect deduced is weakened with the Gaus-sian extension on the edges of the target mirror, andthe intracone and intercone stitching methods canmodel the figuring process of complex cone.

The dwell time realization method on a spiral pathand the determination criterion of the path para-meter were validated in [6,12,13]. However, the highgradient of the target mirror differentiates the figur-ing process in this paper from those. In this section,the CEH matrix method is used to validate the dwell

Fig. 6. Simulation result for high-gradient cone: (a) initial surface error, (b) deconvoluted dwell time, (c) predicted residual error, (d)predicted residual error with CEH model.


time realization method. With the CEH method, theshaping model, Eq. (4), is discretized to the matrixmultiplication type,

Em ¼ RtTs: ð16Þ

The elements of dwell time matrix Ts are computedfrom dwell time function T deconvoluted from Eq. (4)

with Eq. (6). The removal matrix Rt is computed bythemethod in [9]. In the simulation process, the pitchof the spiral path is 0:5mm, and the discrete intervalof the polar angle is 0:5°. As shown in Fig. 6(d), thepredicted residual error map with CEH model is si-milar to that in Fig. 6(c). This similarity validates thedwell time realization method in the figuring processof a high-gradient mirror.

B. Experiment with Low-Gradient Mirror

The size of the target Zerodur parabolic sample is160mm in diameter, and the curvature radius onthe vertex is 640mm. Its error map is obtained withdouble reflection, and there is a zone about ⌀52mmin the center that can not be measured. The radius ofthe width of the near-Gaussian removal function isabout 12mm in the ion-beam figuring process. Thisfiguring case is similar to that in [6,12,13]. In the fol-lowing discussion, the parabolic is approximated by acomplex cone composed of three single cones. The twoangles γ1 γ2 between these cones are equal.

Fig. 7. Motion illustration in the experiment of figuring low-gradient mirror.

Fig. 8. Experiment result for low-gradient mirror: (a) initial surface error, (b) deconvoluted dwell time, (c) predicted residual error, (d)surface error after figuring.


In the figuring process, although the target work-piece is low gradient, the relativemotion between theworkpiece and the ion source is the same as that inthe figuring of the high-gradient mirror. That is tosay, the rotation C of the workpiece, the rotation Bof the ion source, and the x and z translations ofthe ion source are interlocked to make the ion sourceperpendicular to the workpiece with a constant dis-tance. The motion illustration is shown in Fig. 7.Figure 8(a) is the initial surface error distribution.

The initial RMS is 0:026 λ. The dwell time is decon-voluted with the CCRL algorithm of Eq. (6). The dis-tribution of the deconvoluted dwell time is shown inFig. 8(b), and the predicted residual error is shown inFig. 8(c). After the figuring run of 58 min, the surfaceerror is reduced to 0:009 λ. As shown in Fig. 8(d),there are no special character errors on the edgesof the residual error. The distribution of the realresidual error is similar to that of the predictedresidual error.The simulation and experiment results above vali-

date that the CCRL (CRL) deconvolution algorithmsand the dwell time realization method can be inte-grated into being a figuring technology for high-gradient mirror.

5. Conclusions

Figuring technologies for high-gradient mirrors withcone stitching methods have been proposed, and thekey problems including a shaping principle based onintracone stitching and intercone stitching, algo-rithms for dwell time calculation, and velocity reali-zation method have been analyzed in detail. Withthese studies, the figuring technologies come intobeing. Figuring simulation and experiment indicate

that these two technologies can be used to figureoptical components of high-gradient finely.

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Figuring algorithm for high-gradient mirrors with axis-symmetrical removal function

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