REVIEW OF SCIENTIFIC INSTRUMENTS 88, 113106 (2017)

Aspheric optical surface profiling based on laser scanningand auto-collimation

Hongbo Xie,1 Min Jiang,1 Yao Wang,1 Xiaotian Pang,1 Chao Wang,2 Yongpeng Su,1and Lei Yang1,a)1College of Precision Instrument and Opto-Electronics Engineering, Tianjin University, Key Laboratoryof Optoelectronics Information Technology, Ministry of Education, Tianjin 300072, China2School of Engineering and Digital Arts, University of Kent, Canterbury CT2 7NT, United Kingdom

(Received 13 July 2017; accepted 24 October 2017; published online 14 November 2017)

Nowadays the utilization of aspheric lenses has become more and more popular, enabling highlyincreased degree of freedom for optical design and simultaneously improving the performance ofoptical systems. Fast and accurate surface profiling of these aspheric components is a real demandin characterization and optimization of the optical systems. In this paper, a novel and simple surfaceprofiler instrument is designed and developed to fulfill the ever increasing need of testing the axiallysymmetric aspheric surface. The proposed instrument is implemented based on a unique mappingbetween the position and rotation angle of the reflective mirror in optical path and the coordinateof reflection point on the surface during rapid laser beam scanning. High accuracy of the proposedsurface profiling method is ensured by a high-resolution grating guide rail, indexing plate, and positionsensitive detector based on laser auto-collimation and beam center-fitting. Testing the meridian line ofboth convex and concave surfaces has been experimentally demonstrated using the developed instru-ment. In comparison to tested results from conventional image measuring instruments and coordinatemeasuring machines, coefficient of determination better than 0.999 99 and RMS less than 1.5 µmhave been achieved, which validates the feasibility of this method. Analysis on the systematic erroris beneficial to further improve its measurement accuracy. The presented instrument—essentiallybuilds on the geometrical optics technique—provides a powerful tool to measure the aspheric sur-faces quickly and accurately with stable structure and simple algorithm. Published by AIP Publishing.https://doi.org/10.1063/1.4995685

I. INTRODUCTION

Aspheric lenses have been widely used for decades owingto their unique capability to eliminate optical aberrationsinherent to spherical lens systems and also reduce other opticalaberrations such as astigmatism.1 Compared to the classicalspherical optical design, an aspheric lens makes it possibleto significantly reduce the number of elements and thereforethe size and weight of an optical system.2–4 Characterizingthe desired shape of individual aspheric mirrors or lenses iscrucially important in optical design, processing, and applica-tion.5 In particular, quality control in the fabrication processmakes 3D optical surface profiling of aspheric lenses a realdemand.6

As a typical testing method for aspheric surfaces, a sty-lus profilometer can be used for the rough surface and smoothsurface.7 However, it always takes much time while tracingthe surface and limited by the dimension. Another favorabletesting category is based on interferometry which can measurethe surfaces with very high accuracy.8 Among them, null test-ing methods, such as computer generated hologram (CGH),have been successfully applied to the measurement of dif-ficult surfaces.9,10 The drawback of such metrology methodis that a matching null optics has to be fabricated for everydesign elements, which is highly time and cost consuming.10

a)Electronic mail: yanglei@tju.edu.cn

An alternative way is non-null testing, such as sub-aperturestitching,11 which offers flexible measurement results withacceptable accuracy and allows a general test of asphericlenses in some sense.12,13 Nevertheless, interferometry basedsolutions are usually composed of complex and expensiveoptical structures and always require sophisticated algorithmto recover the whole profiles.14 Consequently, other mea-suring technologies, based on geometric ray-testing, withits unique ability of simple, convenient, and low-cost arestill extremely attractive for fabrication and production ofhigh-quality aspheric lenses.15

Most recently, aiming at profile measurement of opti-cal surfaces, scanning deflectometric systems have been welldeveloped in terms of the geometric relationship between scan-ning rays.15–17 Ishikawa et al. reported scanning deflectometryusing a rotating commercial autocollimator with complemen-tary metal-oxide semiconductor (CMOS) as a light-receivingelement, which successfully measured aspheric lenses withextremely high-accuracy.16 However, rotating the entire auto-collimator limits the measurement range of deflectometry,especially for the optical surfaces with large slope changes.Therefore, an alternative approach to enlarge the rotatingangle and improve the translational and rotational speed ofthe autocollimator is still highly essential to the developeddeflectometric tester.

In this work, we proposed and demonstrated a laserscanning and auto-collimation tester (LSACT) and its light

0034-6748/2017/88(11)/113106/7/$30.00 88, 113106-1 Published by AIP Publishing.

113106-2 Xie et al. Rev. Sci. Instrum. 88, 113106 (2017)

recognizing and locating ability is accomplished by a home-made autocollimator, being composed of a reflective mirror(RM) and a position sensitive detector (PSD). Compared tooperating the whole autocollimator, rotating a reflectivemirror provides faster scanning and larger rotatory angle.Instead of using a CMOS detector, a PSD with higher res-olution is employed to capture the position of the scanninglight.

As a feasible measurement technique for characterizingaspheric optical surfaces with axially symmetric structure,LSACT was developed by means of geometrical relationshipsbetween scanning beams and meridian line of the surface undertest. In LSACT, collimated laser beam illuminates one point onthe surface under test. The spatial coordinate of the point canbe uniquely determined from the position and rotation angle(can be defined as pose) of a RM that guarantees perfect nor-mal incidence to the optical surface. The entire meridian line ofthe surface can be reconstructed through laser beam scanning.Accurately measuring the rotation angle of the reflectionmirror during each scanning step is crucial to achieve high-accuracy surface profiling. This is ensured by using a high-resolution PSD which can precisely judge the angle of thereflected scanning beam.

In view of the auto-collimation and beam center-fittingprinciple, when the reflected scanning beam is exactly locatedat the center of the PSD, totally coinciding incident andreflected beams definitely determine the relationship betweenthe position of the scanned point on the surface and the RM.To measure the shape of the surface, a close form calculationmodel has been developed utilizing the movement coordinateand rotary angle of the RM. Therefore, our method completelyeliminates the need for calibration plates or reference surfacesand features compact structure and simple algorithm. Further-more, the presented approach provides a versatile solution forvarious aspheric surfaces (concave or convex) with differentsizes.

The remainder of this article is structured as follows.In Sec. II, we first describe the operation principle of theproposed LSACT system. We apply it in Sec. III to studythe algorithm of performing the profile using experimen-tal parameters. More experimental details are given inSec. IV, where results of measuring two different asphericcomponents are reported. Finally, we summarize our find-ings and discuss implications and future directions for thiswork.

II. PRINCIPLE AND METHOD

Figure 1 illustrates diagrammatically the key principle onwhich the tester is based. It mainly consists of a laser, a PSD,a RM, a scanning device, and the surface fixed at the clamp-ing plane. The instrument is designed basically to measurethe contour profile of the fixed surface utilizing the coordi-nate of discrete points, originated by recording the horizontalmovement and rotation of the reflective mirror in the arrange-ment. Without loss of generality, we assume that the surfaceis axially symmetric. Measuring the aspheric surface can besimplified by depicting the outline of the meridian plane andthen symmetrically rotating it.

FIG. 1. Schematic diagram of the laser scanning and auto-collimation testerfor the aspheric surface. RM, reflective mirror; GGR, grating guide rail; IP,indexing plate; BE, beam expander; PSD, position sensitive detector; F, filter;PBS, polarization beam splitter; λ/4, quarter-wave plate. The principle weshould follow is recording the rotation angle of the RM when guaranteeingthe scanning beam vertically incident on the surface and comes back alongthe original optical path.

In practice, the p-polarized 532 nm light beam emittedfrom the laser (Model: MGL-S-532-B), passing through thepolarization beam splitter (PBS) and λ/4 plate, incidents onthe surface by a RM and comes back to the RM again afterreflected by the surface. Then, after properly adjusted by theλ/4 plate, the reflected s-polarized light passes through thefilter (F) and beam expander (BE) and directly hits on the PSD.On account of the auto-collimation and beam center-fittingprinciple, the scanning beam can return along the original pathin the context of the light beam being aligned with the PSDcenter, when the RM rotates a special angle defined as β (seeFig. 1). In order to reduce the spot size of light injecting on thePSD, a BSL with 4× Kepler telescope structure, whose exit-pupil diameter is less than 9.5 mm, is introduced to the system.A photograph of the constructed LSACT in the lab is shown inFig. 2.

Before measurement, two alignment procedures assuringthe coincidence of the rotation axis for a RM, IP, and motor andguaranteeing the direction of GGR being parallel to the clamp-ing plane should be focused on. In the first procedure, threephotodiodes with small active-area (First Sensor, PC5-6-TO5)are put in different locations in the x-y plane and connected toan oscilloscope (Tektronix TPS2014) for capturing the output

FIG. 2. A photograph of the experimental setup of the LSACT in the lab.

113106-3 Xie et al. Rev. Sci. Instrum. 88, 113106 (2017)

signal. When the RM is turned 10 rounds continuously, care-fully adjusting the relative position of the RM, IP, and motorguarantees the reflected light incomes into three photodiodesin every round, which indicates the coaxial of the RM, IPand motor. In the second procedure, a standard optical flat islaid on the clamping plane to replace the sample. The RMmounted on the GGR, fixed at 45◦ all the time, is movedfrom the start to end position. Maintaining the coincidenceof the incident and reflected light in the movement of the RMproves the direction of the GGR in parallel with the clampingplane.

Here the measurement process is presented in more detailas follows. First, the electrical motor controlled by a high-precision grating guide rail (GGR) drives the RM to movewith uniform steps. Then, at each point, the RM mountedon the indexing plate (IP) is continuously driven by a rotarymotor. When the returned scanning beam exactly hits the cen-ter position of the PSD, the rotary motor is turned off andthe pose of the RM is simultaneously recorded by the dataacquisition system. Finally, after scanning all the points of thetested surface, the profile of the surface could be describedby utilizing the linear displacement x and rotation angleβ of RM.

For clarity, the geometrical relationship between the scan-ning light and RM will be analyzed. The angle of the laserbeam incident on the RM, the rotation angle of the RM, andthe angle of the tangent line passing point B are defined asi, β, and α in the xoy Cartesian coordinate system, respec-tively, when the initial rotation angle of the RM is definitelyset at 45◦ for each measured point. Considering the reflec-tion law and optical features of the mirror,18 the equationsof i = 1

2 (90◦ � α) and β = 12 (90◦ � 2i) can be obtained.

Then, the relationship of essential angles in the setup is foundto be

α = 2β. (1)

Meanwhile, the diameter of measured aspheric lens can bededuced as

D= 2(L −H

tan(90◦ − 2β)), (2)

where H is the coordinate of center of the RM in the y-axisand L denotes the displacement value of the RM.

Accordingly, provided that the GGR selected with properlength and the rotation angle of the RM exceeding 45◦ areboth satisfied, the commercial aspheric lens commonly rang-ing from 10 mm to 150 mm is within the measurement rangeof our LSACT. What is more, the precision of the PSD, lineardisplacement and rotation of the mirror greatly determine themeasuring accuracy of the surface, when the position resolu-tion, precision of the GGR, and minimum step of the IP areset at 1 µm, 1 µm, and 0.16 in., respectively.

III. ALGORITHM

In this section, in order to find the geometrical relationshipbetween the measured points in the surface and experimen-tal parameters, the exact coordinates of the measured pointare solved under the basic assumption that aspheric contour

consists of massive amount of arcs. More specifically, thearc, successively connecting to form the contour, is part ofthe osculating circle of contour whose centre is located at they-axis.

The geometrical model of the LSACT under considerationis detailed depicted in Fig. 3. Starting with point A0, the RMis moved by the electrical motor with ∆x step once a time,which is then precisely located at special points An (n = 0, 1,2, . . . ). At a fixed location An, the RM reflects the incident lightonto the surface with point Bn and also reflects the light to thedetector with special rotating angle βn. We assume OnBn isthe extension line of AnBn, which intersects the x-axis at Mn

and the y-axis at On, respectively. In addition, On is defined asthe center of the osculating circle, which inscribes the surfaceand goes through point Bn.

In general, the contour line, connecting all the points Bn,accurately plots the meridional plane of the measured compo-nent. Hence the purpose of following calculation is to deducethe expressive coordinate of Bn (xn, yn), which is the inter-section of line OnBn and string Bn�1Bn. Moreover, the slopeof line OnBn is �cot αn, in which αn satisfies αn = 2βn [seeEq. (1)].

As a starting point, B0 is also a critical point that is able toreflect the incident light into the detector at first time. Takinginto consideration the slope of line O0B0 (O0A0) (�cot α0)and coordinate of A0(�L, H), the equation of O0B0 can bewritten as

y=−cot α0(x + L) + H. (3)

Therefore, the coordinate of B0

(H

cot α0− L, 0

)is obtained.

Similarly, when we utilize the coordinate of A1 (H � ∆x,L), the equation of line O1B1 can be given by

y=−cot α1(x + L − ∆x) + H. (4)

Then the corresponding coordinate of O1 has the general formof (0, H � cot α1(L �∆x)).

FIG. 3. Principle of measuring the optical surface of aspheric using laserscanning and auto-collimation method. A0, A1, A2, . . . represent the positionof RM at each movement while B0, B1, B2, . . . stand for the measured pointson the surface. The inset map is the amplified detail of the part marked by redcircle, in which line A1B1 is perpendicular to line B1P1.

113106-4 Xie et al. Rev. Sci. Instrum. 88, 113106 (2017)

Hence the slope of line B0O1 can be represented by

k1 =cot α1(L − ∆x) − H

Hcot α0

− L. (5)

Next, the focusing issue is how to find the value of∠B0O1M1, which is the intersection angle of lines B0O1 andB1O1. Obviously,

∠B0O1M1 = arctan |k1 − k ′11 + k1k ′1

|, (6)

where k ′1 =− cot α1.For clarity, the detailed geometrical relationship between

line B0B1 and the centre of a circle O1 is plotted in the insetof Fig. 3 (amplified structure marked by red circle). It is wellknown that the angle of osculation is equal to half of the centralangle. Therefore, the angle of osculation ∠P1B1B0 is equal to∠B0O1M1

2 and ∠B1B0M1 = α1 + ∠B0O1M12 .

The equation of line B0B1 is given by

y=M1(x + L −H

cot α0), (7)

where M1 = tan ∠B1B0M1.According to Eqs. (2) and (5), the coordinate of B1,

intersection point of lines B0B1 and O1B1, is transformed into

x1 =H + M1( H

cot α0− L) + (∆x − L) cot α1

M1 + cot α1, (8a)

y1 =HM1 + (M1∆x − M1H

cot α0) cot α1

M1 + cot α1. (8b)

Similar to calculations above, the method is extended toarbitrary measured point, when the coordinate (0, H � cot αn

(L � n∆x)) stands for the centre of a circle On. The slope ofline Bn�1On takes the form

kn =yn−1 − H + cot αn(L − n∆x)

xn−1. (9)

And the slope of line BnOn has the general form

k ′n =− cot αn. (10)

On the basis of the intersection angle theorem, we obtain

∠Bn−1OnBn = arctan |kn − k ′n1 + knk ′n

|. (11)

In this case, the slope of line Bn�1Bn is in the form

Mn = tan(αn +12

arctan |cot αn + yn−1−H+cot αn(L−n∆x)

xn−1

cot αn(yn−1−H+cot αn(L−n∆x))xn−1

− 1|), (12)

where ∠BnBn�1O = αn + 12 ∠Bn�1OnBn.

It is easy to show that the equation of line Bn�1Bn is givenby

y=Mn(x − xn−1) + yn−1. (13)

Clearly, the equation of line BnOn can be written as

y=− cot αn(x + L − n∆x) + H. (14)

After simplifying, the coordinate of measured point Bn

can be performed analytically and results in the followingexpression:

xn =H − yn−1 + Mnxn−1 + (n∆x − L) cot 2βn

Mn + cot 2βn, (15a)

yn =HMn + (yn−1 −Mnxn−1 −MnL + Mnn∆x) cot 2βn

Mn + cot 2βn.

(15b)

Equation (15) is a simple close-form expression thatreveals the functional relationship between the coordinate ofthe scanned point and fundamental coefficients of measure-ment in general form. As geometrical optics principle of theLSACT is being recognized, the theory and algorithm pre-sented here are expected to be valuable in measurement ofaspheric lens that are based on the auto-collimating laserscanning.

IV. EXPERIMENT AND RESULTS

The proposed method of the LSACT was investigatedexperimentally when concave and convex aspheric lenses werecharacterized. The photograph and nominal data of samples areshown in Fig. 4, where the surfaces to be tested are 10.0 mm and15.6 mm in diameter (provided by manufacturer), respectively.The significant parameter H, height of the RM, in algorithmrestoring the shape of surfaces S1 and S2 was set at 38.73 mmand 35.40 mm, respectively. Measuring the meridian line ofsamples S1 and S2 costs 97.2 s and 39.6 s, respectively, andthe measuring speed is roughly 0.5 s per point. Final measuredresults and deviations between the measured data and nominalcurve are shown in Fig. 5, including the restored 3D map ofsamples by rotating the meridian line along the symmetric axis.

With the purpose of further testifying the feasibility ofthe presented instrument, samples of S1 and S2 were also mea-sured and profiled utilizing high-precision image measuringinstrument (IMI) and coordinate measuring machine (CMM),respectively. Moreover, the measuring accuracy of IMI (OGP,SmartScope ZIP 250) and CMM (Hexagon, Leitz) can achieve1 µm and 0.5 µm, respectively. In particularly, measuring thesame meridian line in different testers is ensured by startingfrom the marked point on the sample and accurately findingout the vertex of the aspheric lens. Comparison of measured

FIG. 4. Samples of a concave aspheric surface S1 and a convex aspheric S2.According to the datasheets supplied by the manufacturer, the nominal curve

was expressed as y= x2

R(1+

√1− (1+k)x2

R2 )

+ ax4 + bx6 + cx8, where the nominal

coefficients are R = 13.0632, k = �0.572 510, a = 0, b = �9.4106 × 10�9,c = 0 for S1 and R = �11.148, k = �0.255 925, a = 3.981 41 × 10�5, b =�8.873 25 × 10�8, c = 2.631 98 × 10�9 for S2, respectively. The deviationof two samples relative to their best-fit spheres is approximately 30 µm and70 µm, respectively.

113106-5 Xie et al. Rev. Sci. Instrum. 88, 113106 (2017)

FIG. 5. Measurement results of samples S1 and S2 using LSACT. (a) Meridianline of sample S1. Step of GGR ∆x = 0.1 mm, total measured points N = 206.(b) Meridian line of sample S2. Step of GGR ∆x = 0.5 mm, total measuredpoints N = 84. In each picture, red hollow diamond and blue dotted line,corresponding to the primary axis, represent the measured data and nominalcurve, respectively. The deviation between the measured data and nominalcurve is represented by green hollow circle and shown in the secondary axis.The bottom right (top right) inset of (a) [(b)] is the 3D model of sample S1 (S2)restored by the measured meridian line. According to the standard asphericequation expressed in caption of Fig. 4, the fitting coefficients R = 13.071 511,k = �0.571 931, a = 3.124 813 × 10�6, b = �8.723 367 × 10�9, c = 1.699 320× 10�9 for the measured meridian line of S1 and R = �11.173 742, k = �0.265367, a = 3.951 366 × 10�5, b = �7.384 123 × 10�8, c = 2.591 674 × 10�9 formeasured meridian line of S2 are optimized.

results between LSACT and IMI, LSACT and CMM, is shownin Figs. 6 and 7, respectively, including revealing the deviationof every measured points between different methods. In addi-tion, the coefficient of determination C and root mean square(RMS) were also used to assess the proposed method.

Coefficient of determination C, describing the differencebetween two series of measured data, is expressed as

C =

∑Ni=1 PQ −

∑Ni=1 P

∑Ni=1 Q

N√[∑N

i=1 P2 −(∑N

i=1 P)2

N ][∑N

i=1 Q2 −(∑N

i=1 Q)2

N ]

, (16)

where N is the measured points. P and Q represent the coor-dinate matrix of the measured data from LSACT and othertester, respectively. The closer the coefficient C is close to 1,the more similar the two series of data are.

As a normally used indicator to depict the differencesbetween predicted values and observed values, RMS of ourmeasurement can be written as

RMS = {N∑

i=1

(P − Q)2

N}

12 . (17)

FIG. 6. Compared measured results of sample S1 between LSACT and IMI.(a) Measured meridian line of sample S1. The red hollow diamond representsthe measured data from LSACT, while the green dotted line represents themeasured curve from IMI. Coefficient of determination C = 0.999 99 andRMS = 1.496 µm. (b) The deviation of measured data from LSACT to theIMI. Using the standard aspheric equation (shown in caption of Fig. 4), thefitting coefficients R = 13.443 191, k = �0.610 865, a = 4.124 813 × 10�6,b = �9.123 376 × 10�9, c = 4.879 131 × 10�9 for measured meridian line inIMI are obtained.

The measured data of the meridional line of surface S1

and analysis of deviation for measured points are shown inFigs. 6(a) and 6(b), respectively. The coefficient C is as highas 0.999 99 and RMS is 1.496 µm. Meanwhile, the detaileddata of surface S2 are presented in Fig. 7, where the coefficientC is 0.999 99 and RMS is 1.213 µm. It is obvious that themeasured data from LSACT are in agreement with the datafrom other methods, which indicates that the accuracy of newlydesigned LSACT nearly reaches to the conventional surfaceprofiler. On the other hand, the measured data from LSACTalways bigger than the data from CMM and IMI is an existingproblem needing urgent attention.

Therefore, we analyzed the systematic error of the LSACTto demonstrate the factors that cause the measured deviationand impact on the accuracy of the instrument. On account ofthe principle of our instrument, the rotary error of RM EA,position error of RM EP, and measured error of H EH arethe major components of the systematic error. Specifically, therotary error EA is found to be

EA =

√E2

i + E2c + E2

s + (Ed

lt)2, (18)

where Ei is the error of IP (0.16 in.) while Ec is the coaxial errorof RM, IP, and rotary motor (1′). In addition, Es represents

113106-6 Xie et al. Rev. Sci. Instrum. 88, 113106 (2017)

FIG. 7. Compared measured results of sample S2 between LSACT and CMM.(a) Measured meridian line of sample S2. The red hollow diamond representsthe measured data from LSACT, while the orange dotted line represents themeasured curve from CMM. Coefficient of determination C = 0.999 99 andRMS = 1.213 µm. (b) The deviation of measured data from LSACT to theCMM. Using the standard aspheric equation (shown in caption of Fig. 4), thefitting coefficients R = �11.184 673, k = �0.249 573, a = 4.019 932 × 10�5,b = �8.784 430 × 10�8, c = 2.731 465 × 10�9 for measured meridian line inCMM are obtained.

the alignment error concerned with parallelism of GGR andclamping plane (1′). Ed stands for positioning error of PSD(1 µm) and lt denotes the optical path from RM to PSD(∼60 mm).

Meanwhile, the systematic error of the surface profile canbe given by19

σ =√

N√

E2P + (EAH)2 + E2

H , (19)

where N is the total number of measured points. EP and EH

are 1 µm and 2 µm, respectively.Although the position and the rotary error of the RM is

minimized by utilizing high-performance devices and care-ful alignment, the large measured error of height H proba-bly causes the 1-2 µm deviation from our LSACT and otherreliable tester. In order to reduce the cost of designed instru-ment, the height of H, fixed value in whole measurementprocedure, was measured by manual a caliper rather than ahigh-precision rail, which caused the obvious systematic error.The high-precision rail will be used in the y-direction to mea-sure the value of H for the reduction of the systematic errorand improvement accuracy of the instrument in future work.On the other respect, the performance of the LSACT is alsoaffected by several factors such as divergence of the scanning

beam reflected by the surface and stray light incoming into thePSD.

V. CONCLUSION AND DISCUSSION

In conclusion, this article presents a novel and simple tech-nique for the testing of aspheric surfaces using geometricaloptics metrology and analyzes appropriate algorithm used intester for accurately restoring contour of the surfaces. Theabove experimental results, with coefficient of determina-tion and RMS achieving 0.999 99 and 1.5 µm, respectively,prove the capability of proposed LSACT to depict and recon-struct the surface with aspheric shape in rapid rate. More-over, although steep aspheric lens becomes hard to measure,our approach with long-range scanning makes it possible tomeasure samples with nearly 90◦ slope in theory [see Eqs.(1) and (2)]. The reported LSACT, with 350 mm-long GGRand 75 mm maximum height of RM, will allow characteri-zation of 150 mm-diameter steep surface with 75◦ slope inborder.

What is more, as a significant factor to access the perfor-mance of a aspheric surface tester, measuring time is very oftencompromised at the precision. In LSACT, the total measuringtime is closely tied with the diameter of the sample and sam-pling interval. With regard to single sampling point, measuringtime mainly depends on the speed of the step motor and rota-tory motor, response time of PSD and data processing, whichis fixed about 0.5 s. Optimizing the elements that consist ofLSACT can potentially enable 0.3 s measuring time for singlesampling point.

Distinct advantages for this instrument include simple andstable structure as well as high precision. Its highly flexible andlow cost nature makes it a promising metrology technique forfabrication and manufacture of aspheric components, no mat-ter steep or flat. When the sample is put on a rotary table, thenew-design instrument can further be applied to shape spec-ification for other special optical surfaces such as free-formsurface, which facilitates the metrology of optical components.It is worth mentioning that the non-diffracting beam20 can bean alternative solution to combat severe divergence of light inscanning, which will further improve the accuracy of LSACTin future work.21,22

ACKNOWLEDGMENTS

Supported by the Seed Foundation of Tianjin University(No. 2017XZS-0026).

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