Review of optical freeform surface representation ...

Review of optical freeform surfacerepresentation technique and itsapplication

Jingfei YeLu ChenXinhua LiQun YuanZhishan Gao

Jingfei Ye, Lu Chen, Xinhua Li, Qun Yuan, Zhishan Gao, “Review of optical freeform surfacerepresentation technique and its application,” Opt. Eng. 56(11), 110901 (2017),doi: 10.1117/1.OE.56.11.110901.

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Review of optical freeform surface representationtechnique and its application

Jingfei Ye,a,b,* Lu Chen,b Xinhua Li,b,c Qun Yuan,b and Zhishan Gaob

aNanjing University of Information Science and Technology, School of Physics and Optoelectronic Engineering, Department of OptoelectronicEngineering, Nanjing, ChinabNanjing University of Science and Technology, School of Electronic and Optical Engineering, Nanjing, ChinacJinling Institute of Technology, School of Materials Engineering, Nanjing, China

Abstract. Modern advanced manufacturing and testing technologies allow the application of freeform opticalelements. Compared with traditional spherical surfaces, an optical freeform surface has more degrees of free-dom in optical design and provides substantially improved imaging performance. In freeform optics, the repre-sentation technique of a freeform surface has been a fundamental and key research topic in recent years.Moreover, it has a close relationship with other aspects of the design, manufacturing, testing, and applicationof optical freeform surfaces. Improvements in freeform surface representation techniques will make a significantcontribution to the further development of freeform optics. We present a detailed review of the different types ofoptical freeform surface representation techniques and their applications and discuss their properties anddifferences. Additionally, we analyze the future trends of optical freeform surface representation techniques.© 2017 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.56.11.110901]

Keywords: freeform optics; freeform optical design; optical freeform surface; representation techniques.

Paper 171071V received Jul. 11, 2017; accepted for publication Oct. 26, 2017; published online Nov. 11, 2017.

1 IntroductionWith the advancement of modern precision optical fabrica-tion and measurement technology, optical components withdifferent surface types can be gradually realized. Under thiscondition, a spherical surface is not the only choice in currentoptical design. Owing to their abundant degrees of freedomand strong ability of aberration correction, optical asphericand freeform surfaces are strong candidates. From the geo-metrical viewpoint,1 an optical freeform surface has nonrota-tionally symmetric features. From the aspect of fabricationand design,2 an optical freeform surface is regarded as anoptical surface that leverages a third independent axis duringthe fabrication process to form the optical surface with as-designed nonsymmetric features. Both viewpoints revealthe basic feature of the nonrotational symmetry of opticalfreeform surfaces, which is a challenge for the optical com-munity. In the last 10 years, the technologies of multiaxisslow-servo single-point diamond turning and computer-con-trolled small-lap polishing have been further enhanced, andthe manufacturing capacity and metrology of nonrotationallysymmetric optical surface have been improved. Therefore,the application fields of nonrotationally symmetric opticalsurfaces have increased considerably.

Generally, freeform optics can be categorized into noni-maging and imaging freeform optics. Applications of opticalfreeform surfaces in the nonimaging category include beamshaping,3 concentrators,4 and illumination,5 where opticalfreeform surfaces can control the light direction to improveenergy uniformity and efficiency. Examples of applicationsin the imaging category are ultrashort projection lenses,6

head-mounted displays,7 ultraviolet lithographic objectives,8

and off-axis reflective infrared imaging systems,9 where

optical freeform surfaces are used to improve imaging per-formance and compactness. Hence, freeform optics isregarded as the next-generation and ground-breaking tech-nology in modern optics.2,10

However, the question of how to represent or characterizean optical freeform surface remains. Because the optical free-form surface representation technique has a close relation-ship with its design, manufacturing, testing, and finalapplication, it encompasses all the processes used in free-form optics. Namely, the freeform surface representationtechnique is a fundamental and key research topic, and itsenhancement can further facilitate the development of free-form optics. Obviously, the optical freeform surface repre-sentation technique employed in different applications isnot unique, and its selection depends on practical consider-ations. So far, XY-type, Zernike, Q-type, Chebyshev, andLegendre polynomials, the radial basis function, the nonuni-form rational basis spline (NURBS) function, and otherhybrid methods and techniques have been used to representthe optical freeform surface in each specific application.Thompson and Rolland2 described the history and therevolutionary character of optical freeform surfaces. Fanget al.11 presented a comprehensive review on the manufac-turing and measurement of freeform optics, which containedsome information on freeform surface representation tech-niques. Gross et al.12 published an overview of surface rep-resentations for freeform surfaces particularly from theaspect of power spectral density in the midspatial frequencyfor manufacturing. Fähnle et al.13 presented a special sectionon freeform optics to discuss the current developments in allaspects of freeform optics. We believe that an elaborateanalysis of optical freeform surface representation techniques

*Address all correspondence to: Jingfei Ye, E-mail: [email protected] 0091-3286/2017/$25.00 © 2017 SPIE

Optical Engineering 110901-1 November 2017 • Vol. 56(11)

Optical Engineering 56(11), 110901 (November 2017) REVIEW


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http://dx.doi.org/10.1117/1.OE.56.11.110901

http://dx.doi.org/10.1117/1.OE.56.11.110901

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is necessary, as it can provide a large map about the develop-ment of these methods.

In this paper, we present a comprehensive review of thedifferent types of freeform surface representation techniquesand their applications. Section 2 describes several typicalapplications using different types of optical freeform surfa-ces. Section 3 discusses various mathematical functions andtechniques that can be applied to characterize optical free-form surfaces and analyzes methods used to obtain a free-form surface from discrete data points. In Sec. 4, thedifferences between freeform surface representation tech-niques are discussed and compared in detail. Section 5 con-cludes with the state of the art in optical freeform surfacerepresentation and its future trends.

2 Applications of Different Types of OpticalFreeform Surface Characterization

An optical freeform surface has the advantages of improvingthe imaging performance, reducing the number and weight,and further increasing the compactness of optical compo-nents, especially in off-axis imaging optical systems. Thisis owing to its multiple degrees of freedom and off-axis aber-ration correction ability. One of the most typical examples ofa freeform surface was used by Maitenaz14 and Kanolt15 inprogressive multifocal ophthalmic lenses, which date back to1954. It was applied for the correction of presbyopia andoffered the wearers comfortable, relatively clear visionwith no separation between long and short distances,16 asshown in Fig. 1. The profile of the progressive lens is obvi-ously of the nonrotationally symmetric type, which can berepresented by the combination of Zernike polynomials orthe B-spline function.

Another example of freeform surfaces used in an off-axiscatadioptric Polaroid SX-70 camera was introduced byPlummer;17 this might be the first commercial productemploying optical freeform surfaces characterized by XYpolynomials to improve its imaging performance, as dis-played in Fig. 2. Owing to the optical path configurationof off-axis refractive and reflective types, it had more com-pactness. However, off-axis aberrations aggravated its imag-ing quality. Thus, two freeform optical components wereused to correct the off-axis aberration. A freeform corrector(D) with a fourth-order XY polynomial profile was located atthe system stop, which could reduce the astigmatism, coma,and spherical aberration. Another freeform eyepiece lens (G)

with a sixth-order XY polynomial profile was positioned atthe viewfinder to control the astigmatism and field curvature.

In the last decade, with the development of advanced fab-rication equipment, as well as commercial optical designtools and optical testing technology, the advantages of opti-cal freeform surfaces have been extensively utilized in differ-ent applications such as projection optical systems, head-worn displays, and off-axis reflective optical systems. Inthese applications, the used freeform surface is generally acontinuous and smooth surface characterized by differentanalytical functions. Additionally, a microfreeform-prism-based lens array is employed in the artificial compoundeye18–20 and can realize a large field of view using multipleimaging channels with lower distortion. Each imaging chan-nel is a microfreeform-prism lens represented by XY poly-nomials or Zernike polynomials to correct the aberration inthe corresponding channel. This type of freeform surface isstepped and noncontinuous globally. From the perspective ofthe overall continuity of surface, the optical freeform surfacecan be categorized into two types, as described above.Continuous and smooth freeform surfaces are much morecommon in the design and application of freeform opticalsystems. Therefore, in the following, we discuss applicationsthat use different types of optical freeform surfaces by focus-ing on continuous and smooth freeform surfaces.

A traditional projection system always has a relativelylong throw distance, which involves large unused spaces.On the other hand, if the presenter stands between the screenand the projector, his shadow is shown on the screen. As theaccessory of the traditional projection system, freeform opti-cal elements can obtain an ultrashort throw distance, a largescreen, and high performance projection, as shown in Fig. 3.Hitachi21 reported that a projector (HCP-A8) using a free-form reflective mirror could realize a 60-in. projection screenat a throw distance of only 47 cm; however, the detailed func-tion type of its profile was not revealed. The simultaneousmultiple surface (SMS) method22 was applied by Miñanoand Benítez in the design of a freeform accessory optical sys-tem for ultrashort throw distance projection. The discretedata point cloud characterizing the freeform surface wasobtained by the SMS method and could be reconstructed

Fig. 1 Progressive lens with nonrotationally symmetric freeformsurfaces.

Fig. 2 Schematic diagram of the optical path of the Polaroid SX-70camera with freeform corrector D located at the stop and freeform eye-piece lens G positioned at the viewfinder.


Ye et al.: Review of optical freeform surface representation technique and its application


accurately by analytical polynomials or functions. Yanget al.23 used XY polynomials to obtain a miniaturized andshort projector. Zhuang et al.6 applied odd XY polynomialsrepresenting the freeform reflector together with a field cur-vature correction method to design an ultrashort projector.The introduction of optical freeform surfaces to projectionoptical systems could further enhance projection displaytechnology to achieve an ultrashort throw distance, largerdisplay screen, higher imaging projection performance,miniaturization, and low weight.

Head-worn displays, which can interact with the virtualdigital world and connect with the real world, are widelyused in modern education, medical treatment, entertainment,and military training. The use of a freeform prism as the eye-piece lens for head-worn displays, as shown in Fig. 4, wasreported by Okuyama24 and Takahashi.25 Each profile of thefreeform prism can be represented by XY polynomials orZernike polynomials. Cakmakci and Rolland26 designed adual-element off-axis near-eye optical magnifier by combin-ing a freeform reflector represented by XY polynomials anda diffractive lens. Furthermore, Cakmakci et al.27 applied theradial basis function, instead of XY polynomials, to charac-terize the freeform reflector, which could increase the exitpupil diameter from 8 to 12 mm. Cheng et al.7 reported

a freeform prism-based head-worn display characterizedby XY polynomials, which had the advantages of a fastfocal ratio, large field of view, and see-through properties.Zheng et al.28 designed and analyzed an off-axis see-throughhead-worn display with XY polynomials that had a large eyerelief of over 60 mm. Pan et al.29 demonstrated an off-axistwo-freeform reflector described by XY polynomials with an8-mm exit pupil diameter, which was used to enhance thevisual sensing of slightly visually impaired patient. Thedevelopment of optical freeform surfaces advances head-worn displays in the direction of a large exit pupil diameter,long eye relief, large field of view, high-resolution display,more comfort, and miniaturization.

Optical freeform surfaces are frequently used in off-axisreflective optical systems. Xu et al.30 presented a freeformspectrometer using a toroid reflective surface to correct astig-matism. Rodgers31 designed an off-axis four-mirror systemthat used the off-axis section of an even-order asphere.Furthermore, Nakano and Tamagawa32 designed an off-axisthree-mirror system. The original configuration was ana-lyzed by conic surfaces and further optimized by Zernikepolynomials. Fuerschbach et al.9 presented an off-axisthree-mirror long-wavelength infrared imaging system usingthe off-axis part of a fringe Zernike polynomial surface by

Fig. 3 Freeform optical elements as the accessory of traditional projection optical system.

Fig. 4 Freeform prism in a head-worn display. Fig. 5 Freeform off-axis reflective infrared imaging system.




combining the nodal aberration theory and the full-field dis-play method, as shown in Fig. 5. The use of an off-axis opti-cal path configuration and a freeform surface makes theimaging system more compact and more efficient in cor-recting the off-axis aberration.

Optical freeform surfaces also play a key role in pano-ramic imaging systems with a wide field of view. Maet al.33 presented a panoramic annular imaging systemwith nonsymmetric, rotational, and variable focal length,which had a freeform surface characterized by XY polyno-mials, as displayed in Fig. 6. Zhou and Bai34 designed asmall-distortion panoramic annular lens with a Q-type free-form surface. Previous panoramic imaging systems had aninherent shortcoming, namely, a central blindness withoutthe image. By applying a stitched freeform surface betweenthe central and peripheral regions, a large field-of-view pano-ramic lens without central blindness could be realized.35

Additionally, several other applications employing free-form surfaces have been demonstrated. Alvarez36 appliedtwo complementary freeform optical components repre-sented by third-order XY polynomials to realize a vary-ing-focal-length lens system. Smilie et al.37 fabricated agermanium Alvarez lens for infrared imaging under condi-tions of modern precision optical manufacturing. Hicks38

developed a ray-tracing method to obtain a discrete datapoint cloud for a freeform reflector, which was used todesign a 45-deg field of view and a lower-distortiondriver-side mirror to effectively increase the driver’s fieldof view. Zhu et al.39 presented a direct design method toobtain a discrete data point cloud, which was fitted byXY polynomials for further optimization of an F-theta free-form scanning lens. Its field of view was ∼120 deg, and thescanning error was lower than 1 μm.

The current applications of optical freeform surfacesshow that there are mainly two strategies for characterizingthe freeform surface. First, the freeform surface is expressedas explicit mathematical functions in the freeform opticaldesign by directly optimizing the coefficients. Most of thecommonly used freeform functions have been integratedinto commercial optical design software. Second, a freeformsurface is derived from discrete data points by specific designalgorithms. Additionally, in freeform surface testing, theobtained freeform shape data are also included in a discretedata point cloud. They must be further fitted accurately by

analytical or numerical functions for the next optimization orthe final freeform surface estimation. Among the differentfunction types for representing freeform surfaces, there isa particular impression about the XY-type polynomial free-form surface, which is frequently used in different applica-tions. This can be illustrated briefly using the opticalaberration theory. When the symmetry of an optical systemis not considered, its aberration function can be approxi-mated by expanding it as a combination of XY polynomialsin a Taylor series for a specific field of view. In other words,each term of the XY polynomials corresponds to a specificaberration and can correct it. Similarly, the aberration func-tion can be decomposed as a linear combination of Zernikepolynomials, which is why Zernike polynomials are increas-ingly used. They provide more degrees of freedom and aber-ration correction capabilities in the freeform optical design.The advantages of freeform surfaces make them attractive tooptical engineers for practical applications. In the followingsection, we describe the different freeform surface represen-tation techniques in detail.

3 Mathematical Descriptions of Optical FreeformSurface

According to the different applications of freeform surfacesmentioned in Sec. 2, the common analytical functions usedto represent optical freeform surfaces can be generally cat-egorized as orthogonal polynomials and nonorthogonalfunctions. On the other hand, representation techniquesemployed to obtain an optical freeform surface from discretedata points are also important in direct freeform optics designand in freeform optical testing. For a freeform surface withlarge slope variation in local areas or large global gradient,hybrid or combined representation methods should be con-sidered for characterizing the fine local features of a freeformsurface. In this section, these representation techniques forfreeform surfaces are discussed and analyzed.

3.1 Analytical Functions

3.1.1 Orthogonal polynomials

Orthogonal polynomials,40 such as Zernike polynomials,41

are widely used to analyze optical surface deviations andwavefront aberrations owing to their elegant mathematicalperformance. Here, we describe mathematical orthogonalityand different analytical orthogonal polynomials.

Definition and properties of orthogonal polynomial.Polynomial functions PmðρÞ, PnðρÞ, and the weighted func-tion wðρÞ that are defined over a range ρ ∈ ½a; b� satisfy arelationship

EQ-TARGET;temp:intralink-;e001;326;203

Zb

awðρÞPmðρÞPnðρÞdρ ¼ δmncn; (1)

then PðρÞ is a weighted orthogonal polynomial, where δmn isthe Kronecker delta function. The subscripts m and n are thenonnegative integers. When m ¼ n, δmn ¼ 1, and whenm ≠ n, δmn ¼ 0. cn is the corresponding coefficient,which can be calculated by Eq. (2). When cn ¼ 1, the poly-nomial is the orthonormal polynomial

Fig. 6 Freeform panoramic annular imaging system with a large fieldof view.




EQ-TARGET;temp:intralink-;e002;63;752cn ¼Z

b

awðρÞ½PnðρÞ�2dρ: (2)

The Zernike circle polynomial is a typical orthogonal andcomplete polynomial used to describe the aberration func-tion, fit wavefront data, or represent a freeform surface.When the freeform surface is expanded as a linear combina-tion of Zernike circle polynomials, Zernike circle polynomialhas the following properties.

1. Because of the orthogonality and completeness ofZernike polynomials, the coefficient of one term is in-dependent of the number of used terms.

2. In the expansion, the first term is the piston term,which indicates the mean value of the surface or wave-front function.

3. Except the first term, the mean value of the other termsis zero.

4. Except the first term, the variance of the surface orwavefront function is the sum of the squared coeffi-cients of the other terms.

5. Except the first term, the standard deviation of each ofthe other terms is the corresponding coefficient value.

Obviously, other different-type orthogonal polynomialshave similar properties to those of Zernike polynomialsfor representing an optical freeform surface.

Two typical circular orthogonal polynomials.

Zernike circle orthogonal polynomials

The ordering number of the Zernike circle polynomial fol-lows Noll’s ordering.42 A Zernike polynomial is expressed as

EQ-TARGET;temp:intralink-;e003;63;381Zjðρ;ϕÞ ¼ Zmn ðρ;ϕÞ

¼8<:

ffiffiffiffiffiffiffiffiffiffiffinþ 1

pR0nðρÞ; m ¼ 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðnþ 1ÞpRmn ðρÞ cos mϕ; m ≠ 0 ðevenÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðnþ 1ÞpRmn ðρÞ sin mϕ; m ≠ 0 ðoddÞ

;

(3)

where the subscript j is the ordering number, n andm are thenonnegative integers, n −m is even, and n −m ≥ 0. Rm

n ðρÞis the radial function, which is described by

EQ-TARGET;temp:intralink-;e004;63;257Rmn ðρÞ ¼

Xðn−mÞ∕2

k¼0

ð−1Þkðn − kÞ!k!�nþm2

− k�!�n−m2

− k�!ρn−2k: (4)

The higher-order Zernike polynomials can be obtainedfrom the recurrence relations that they satisfy. The first 28Zernike polynomials are listed in Table 1.

As discussed in Sec. 2, the Zernike polynomial is widelyused. Fuerschbach et al.9 designed a compact off-axis three-mirror long-wavelength infrared imaging system usingfringe Zernike polynomials to further improve its imagingperformance. In theory, any surface or wavefront, regardlessof how complex it is, can be accurately represented by apply-ing a sufficient number of Zernike polynomial terms.However, in practice, only a finite number of terms are usu-ally used for freeform surface analysis.

Q-type orthogonal polynomials

To overcome the numerical deficiency of even-order polyno-mials and improve the manufacturability and testability ofaspheres, Forbes proposed Qcon

n ðu2Þ polynomials for repre-senting strong surfaces with large aspheric deviations andQbfs

n ðu2Þ polynomials for characterizing mild surfaceswith constrained slopes based on Jacobi polynomials.43–46

Furthermore, a Qmn ðu2Þ polynomial was proposed for char-

acterizing optical freeform surfaces.47 The Qmn ðu2Þ polyno-

mial has a similar construction to the Zernike polynomial andrepresents the deviation between the freeform surface and the

Table 1 The first 28 Zernike circle polynomials Z j ðρ;ϕÞ

n m j Z j ðρ;ϕÞ0 0 1 1

1 1 2 2ρ cos ϕ

1 1 3 2ρ sin ϕ

2 0 4ffiffiffi3

p ð2ρ2 − 1Þ

2 2 5ffiffiffi6

pρ2 sin 2ϕ

2 2 6ffiffiffi6

pρ2 cos 2ϕ

3 1 7ffiffiffi8

p ð3ρ3 − 2ρÞ sin ϕ

3 1 8ffiffiffi8

p ð3ρ3 − 2ρÞ cos ϕ

3 3 9ffiffiffi8

pρ3 sin 3ϕ

3 3 10ffiffiffi8

pρ3 cos 3ϕ

4 0 11ffiffiffi5

p ð6ρ4 − 6ρ2 þ 1Þ

4 2 12ffiffiffiffiffiffi10

p ð4ρ4 − 3ρ2Þ cos 2ϕ


p ð4ρ4 − 3ρ2Þ sin 2ϕ


pρ4 cos 4ϕ


pρ4 sin 4ϕ


p ð10ρ5 − 12ρ3 þ 3ρÞ cos ϕ


p ð10ρ5 − 12ρ3 þ 3ρÞ sin ϕ






pρ5 cos 5ϕ


pρ5 sin 5ϕ

6 0 22ffiffiffi7

p ð20ρ6 − 30ρ4 þ 12ρ2 − 1Þ


p ð15ρ6 − 20ρ4 þ 6ρ2Þ sin 2ϕ


p ð15ρ6 − 20ρ4 þ 6ρ2Þ cos 2ϕ






pρ6 sin 6ϕ


pρ6 cos 6ϕ




best-fit sphere along the normal direction. The sag of thefreeform surface along the z-axis is z ¼ fðρ;ϕÞEQ-TARGET;temp:intralink-;e005;63;730

z ¼ cbfsρ2

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − c2bfsρ

2q

þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − c2bfsρ

2q �

u2ð1 − u2ÞXNn¼0

a0nQ0nðu2Þ

þXMm¼1

umXNn¼0

ðamn cos mϕþ bmn sin mϕÞQmn ðu2Þ

�: (5)

In Eq. (5), when m ¼ 0, then Q0nðu2Þ is the same as

Qbfsn ðu2Þ. When m > 0 and m is an integer, Qm

n ðu2Þ charac-terizes the deviation along the normal direction. Here, amn isthe coefficient of the symmetric termQm

n ðu2Þ cos mϕ and bmnis the coefficient of the nonsymmetric term Qm

n ðu2Þ sin mϕ.Let t ¼ u2; then Qm

n ðu2Þ ¼ Qmn ðtÞ. The Jacobi polynomial

Jmn ðtÞ associated with Qmn ðtÞ is described by Eq. (6).

Forbes reported several Qmn ðtÞ polynomials with n ¼

0;1; 2;3 and m ¼ 0;1; 2;3; 4;5. In this paper, the first four-terms of Q6

nðtÞ polynomials are listed in Table 2,

EQ-TARGET;temp:intralink-;e006;63;494Jmn ðtÞ ¼8<:

1 − t2; m ¼ 1 and n ¼ 1

ð−1Þnð2nÞ!!2ð2n−1Þ!! J

�−32;m−3

2

�n ð2t − 1Þ otherwise

:

(6)

To obtain the higher-order Q-type polynomials, a recur-rence relation can be applied to improve the computationefficiency.

Different orthogonal polynomials with differently shapedapertures. For the circular aperture or pupil of the opticalsystem, Zernike polynomials provide a superior perfor-mance. In modern optical engineering, more optical systemswith noncircular pupils or optical components with noncir-cular boundaries are being developed. For example, rectan-gular or square optical elements are applied in anamorphic orhigh-power laser optical systems.48 Many optical elementswith square apertures are used in the U.S. NationalIgnition Facility and the China Inertial Confinement Fusionlaser driver. In large telescopes, the giant reflector is stitchedby hundreds of hexagonal mirrors. Freeform optical elementswith noncircular apertures are also implemented in practicalapplications.49

For freeform surface analysis in freeform optical elementswith differently shaped apertures, the correspondingorthogonal polynomials must be applied to the similarlyshaped apertures. Owing to the orthogonality and complete-ness of Zernike polynomials, a freeform surface with noncir-cular aperture can also be expressed as a linear combinationof the corresponding orthogonal polynomials over the sameaperture. Using Zernike polynomials as the basis functions,orthogonal polynomials (such as Zernike square polyno-mials, Zernike rectangular polynomials, Zernike hexagonalpolynomials, Zernike elliptical polynomials, and Zernikeannular polynomials) over the corresponding aperturescan be derived by the Gram–Schmidt orthogonalizationmethod.50 Tatian,51 Barakat,52 Mahajan and Dai,50 Swantnerand Chow,53 Hasan and Shaker,54 Díaz and Mahajan,55 andDíaz and Navarro56 reported analytical Zernike orthogonalpolynomials with differently shaped apertures. Ferreiraet al.57 proposed a rigorous and powerful theoretical frame-work to obtain the orthogonal basis with a conicoid firstmode for surface specification. Liu et al.58 applied two-dimensional (2-D) Chebyshev polynomials to characterizethe “W”-shaped freeform optical elements. Mahajan59 ana-lyzed the aberration of an anamorphic optical system using2-D Legendre polynomials. These two orthogonal polyno-mials were used for square or rectangular apertures. Thedifferences between 2-D Chebyshev and Legendre polyno-mials for representing complex freeform surfaces were com-pared comprehensively by Ye et al.,60 who elaborated on themathematical expressions of several typical orthogonal pol-ynomials for representing freeform surfaces.

Gram–Schmidt orthogonalization method deriving non-circular orthogonal polynomials. For freeform surfaceswith noncircular apertures, such as square, rectangular, ellip-tical, annular, sector, and hexagonal apertures, the corre-sponding orthogonal polynomials can be derived by theGram–Schmidt orthogonalization method,50 which employsa set of complete orthogonal polynomials as basis functionsand uses an iterative transformation method.

For simplicity, a Zernike circle polynomial Zjðx; yÞ in theCartesian coordinate system is written as Zj. The subscript jis a positive integer starting from 1 and the ordering number,Fj denotes the corresponding noncircular orthogonal poly-nomial, and Gj is the intermediate transformation polyno-mial. Generally, the first term is

EQ-TARGET;temp:intralink-;e007;326;257F1 ¼ G1 ¼ Z1 ¼ 1: (7)

Furthermore, if G, Z, and F satisfy the recurrence relationGjþ1 ¼ Zjþ1 þ

Pjk¼1 cjþ1;kFk, then the corresponding non-

circular normalized orthogonal polynomial F is

EQ-TARGET;temp:intralink-;e008;326;198Fjþ1 ¼Gjþ1

kGjþ1k¼ Gjþ1�

1A

Raperture G

2jþ1dx dy

�1∕2 . (8)

The transformation coefficient cjþ1;k is

EQ-TARGET;temp:intralink-;e009;326;139cjþ1;k ¼ −1

A

Zaperture

Zjþ1Fkdx dy; (9)

where A is the area of a regular noncircular aperture inscribedin a unit circle. The area of an annular aperture isAAA ¼ πð1 − ε2Þ, where ε is the annular ratio. The area of

Table 2 The first four-terms of Q6nðtÞ polynomials.

n Q6nðtÞ

0 89ffiffi7

p

1 − 8ð−77þ72tÞ9ffiffiffiffiffiffiffiffi1397

p

2 8ð73645−159432tþ85344t2Þ3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi542196655

p

3 − 8ð−1858285þ6903936t−8322912t2þ3275520t3Þ9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4280409445

p




a square aperture is ASA ¼ 2. The area of a rectangular aper-ture is ARA ¼ 4a

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − a2

p, where a is the half-width of the

rectangular aperture along the x-axis andffiffiffiffiffiffiffiffiffiffiffiffiffi1 − a2

pis the half-

width along the y-axis. The area of a hexagonal apertureis AHA ¼ 3

ffiffiffi3

p∕2. The area of an elliptical aperture is

AEA ¼ πb, where b is the half-minor axis along they-axis, and the half-major axis along the x-axis is unity.

Using recurrence iteration based on the Gram–Schmidtorthogonalization method, analytical orthogonal polyno-mials for differently shaped apertures can be derived. Thesepolynomials can be used for freeform surface representation,wavefront estimation, and aberration analysis.

Two typical Jacobi square orthogonal polynomials. Inthe orthogonal polynomial used for the square aperturederived by the Gram–Schmidt orthogonalization method,the x and y variables of each term are mixed together.The 2-D Chebyshev and Legendre polynomials are twosquare orthogonal polynomials with x and y variables,respectively, which can be applied to characterize anamor-phic freeform surfaces.

Two-dimensional Chebyshev polynomials

The 2-D Chebyshev polynomials are defined by the productsof one-dimensional first-kind Chebyshev polynomials in thex- and y-dimensions. The Chebyshev polynomials TnðxÞ ofthe first kind61 in the x-dimension are orthogonal in the range½−1;1� with a weighted function wðxÞ ¼ 1∕

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − x2

p, as

shown in


Z1

−1

TnðxÞTmðxÞffiffiffiffiffiffiffiffiffiffiffiffiffi1 − x2

p dx ¼

12πδnm for n ≠ 0; m ≠ 0

π for n ¼ m ¼ 0; (10)

where δnm is the Kronecker delta. In the y-dimension, the yvariable is used instead of the x variable to obtain TnðyÞ.Therefore, a 2-D Chebyshev polynomial Cjðx; yÞ isEQ-TARGET;temp:intralink-;e011;63;356Cjðx; yÞ ¼ TnðxÞTmðyÞ; (11)

where the subscript j is the ordering number. The orthogon-ality of 2-D Chebyshev polynomials is


Z1

−1

Z1

−1Cjðx; yÞCj 0 ðx; yÞ

dx dyffiffiffiffiffiffiffiffiffiffiffiffiffi1 − x2

p ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − y2

p¼

0 for j ≠ j 0

K for j ¼ j 0 ; (12)

whereK is a constant value used for the normalization of 2-DChebyshev polynomials

EQ-TARGET;temp:intralink-;e013;63;216K ¼(π2 for n ¼ m ¼ 0

π2∕4 for n ¼ m ≠ 0

π2∕2 for else

: (13)

The normalized 2-D Chebyshev polynomial CNj can bederived by Eq. (14). The first 28 2-D Chebyshev polynomialsare shown in Table 3.

EQ-TARGET;temp:intralink-;e014;63;134CNj ¼Cj∕

ffiffiffiffiK

pCj∕ffiffiffiffiK

p ¼ Cj∕ffiffiffiffiK

ph14

R1−1

R1−1

�Cj∕

ffiffiffiffiK

p �2dx dy

i1∕2 :

(14)

Two-dimensional Legendre polynomials

The 2-D Legendre polynomials are obtained in a similar wayto 2-D Chebyshev polynomials. The Legendre polynomials62

PnðxÞ in the x-dimension are orthogonal over the interval½−1;1� with a weighted function wðxÞ ¼ 1


Z1

−1PnðxÞPmðxÞdx ¼ 2

2nþ 1δnm: (15)

To obtain PnðyÞ, we substitute the x variable in PnðxÞwith the y variable. Thus, the 2-D Legendre polynomials

Table 3 The first 28 2-D Chebyshev polynomials Cj ðx; yÞ.

Polynomial order Cj ðx; yÞ0 C1 ¼ 1

1 C2 ¼ x

1 C3 ¼ y

2 C4 ¼ 2x2 − 1

2 C5 ¼ xy

2 C6 ¼ 2y2 − 1

3 C7 ¼ 4x3 − 3x

3 C8 ¼ ð2x2 − 1Þy

3 C9 ¼ xð2y2 − 1Þ

3 C10 ¼ 4y3 − 3y

4 C11 ¼ 8x4 − 8x2 þ 1

4 C12 ¼ ð4x3 − 3xÞy

4 C13 ¼ ð2x2 − 1Þð2y2 − 1Þ

4 C14 ¼ xð4y3 − 3yÞ

4 C15 ¼ 8y4 − 8y2 þ 1

5 C16 ¼ 16x5 − 20x3 þ 5x

5 C17 ¼ ð8x4 − 8x2 þ 1Þy

5 C18 ¼ ð4x3 − 3xÞð2y2 − 1Þ

5 C19 ¼ ð2x2 − 1Þð4y3 − 3yÞ

5 C20 ¼ xð8y4 − 8y2 þ 1Þ

5 C21 ¼ 16y5 − 20y3 þ 5y

6 C22 ¼ 32x6 − 48x4 þ 18x2 − 1

6 C23 ¼ ð16x5 − 20x3 þ 5xÞy

6 C24 ¼ ð8x4 − 8x2 þ 1Þð2y2 − 1Þ

6 C25 ¼ ð4x3 − 3xÞð4y3 − 3yÞ

6 C26 ¼ ð2x2 − 1Þð8y4 − 8y2 þ 1Þ

6 C27 ¼ xð16y5 − 20y3 þ 5yÞ

6 C28 ¼ 32y6 − 48y4 þ 18y2 − 1




Ljðx; yÞ can be obtained from the product of PnðxÞ andPmðyÞEQ-TARGET;temp:intralink-;e016;63;730Ljðx; yÞ ¼ PnðxÞPmðyÞ: (16)

The orthogonality of 2-D Legendre polynomials isdescribed by


1

4

Z1

−1

Z1

−1Ljðx; yÞLj 0 ðx; yÞdx dy ¼ δjj 0 : (17)

The first 28 2-D Legendre polynomials are listed inTable 4.

3.1.2 Nonorthogonal functions

In addition to the analytical orthogonal polynomials, thereare several nonorthogonal functions applied to characterizefreeform surfaces, such as XY-type polynomials, spline func-tions, and radial basis functions.

XY-type polynomials. In XY-type polynomials, the free-form surface is represented as a base surface with a linearcombination of multiple xmyn monomial terms. The basesurface is usually a conic or anamorphic surface, asshown inEQ-TARGET;temp:intralink-;e018;326;593

zðx;yÞ¼ cxx2þcyy2

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−ðkxþ1Þc2xx2−ðkyþ1Þc2yy2

q þXJj¼2

ajxmyn;

j¼ðmþnÞ2þmþ3n2

þ1; (18)

where cx and cy are the curvature radii in the two correspond-ing directions, and kx and ky are the conic constants. Asshown in Eq. (18), it combines the advantages of an anamor-phic surface and an XY polynomial surface and has moredegrees of freedom for characterizing a freeform surface.

This type of freeform surface is not available in opticaldesign software. However, it can be defined in such softwareusing the feature “user-defined surface” for further optimi-zation of the corresponding freeform optical system.63

Additionally, the off-axis part of the XY-type polynomialis usually used to represent freeform surfaces.

Spline surface. A spline surface is very flexible in com-puter-aided design and freeform surface construction. TheNURBS surface is increasingly being applied to opticaldesign. Its mathematical expression is

EQ-TARGET;temp:intralink-;e019;326;334zðu; vÞ ¼P

ni¼0

Pmj¼0 wi;jBi;kðuÞBj;lðvÞPi;jP

ni¼0

Pmj¼0 wi;jBi;kðuÞBj;lðvÞ

; (19)

where Pi;j is the control function, wi;j is the weighted factor.Bi;kðuÞ is the k’th-order B-spline function along the u-direc-tion in Eq. (20), and Bj;lðvÞ in Eq. (21) is similar to Bi;kðuÞ; itis the l’th-order B-spline function along the v-direction. Thesubscripts i, j, k, and l are nonnegative integers


8<:k¼ 0; Bi;0ðuÞ ¼

1; if ui ≤ u < uiþ1 and ui < uiþ1

0; otherwise

k ≥ 1; Bi;kðuÞ ¼ u−uiuiþk−ui

Bi;k−1ðuÞþ uiþkþ1−uuiþkþ1−uiþ1

Biþ1;k−1ðuÞ;

(20)


8<: l ¼ 0; Bj;0ðvÞ ¼

1; if vj ≤ v < vjþ1 and vj < vjþ1

0; otherwise

l ≥ 1; Bj;lðvÞ ¼ v−vjvjþl−vj

Bj;l−1ðvÞ þ vjþlþ1−vvjþlþ1−vjþ1

Bjþ1;l−1ðvÞ:

(21)

The local feature of a NURBS surface can be controlledby the control function and weighted factor. The freeform

Table 4 The first 28 2-D Legendre polynomials Lj ðx; yÞ.

Polynomial order Lj ðx; yÞ0 L1 ¼ 1

1 L2 ¼ ffiffiffi3

px

1 L3 ¼ ffiffiffi3

py

2 L4 ¼ ffiffiffi5

p ð3x2 − 1Þ∕2

2 L5 ¼ 3xy

2 L6 ¼ ffiffiffi5

p ð3y2 − 1Þ∕2

3 L7 ¼ ffiffiffi7

p ð5x3 − 3xÞ∕2

3 L8 ¼ ffiffiffiffiffiffi15

p ð3x2 − 1Þy∕2


pxð3y2 − 1Þ∕2

3 L10 ¼ ffiffiffi7

p ð5y3 − 3yÞ∕2

4 L11 ¼ 3ð35x4 − 30x2 þ 3Þ∕8


p ð5x3 − 3xÞy∕2

4 L13 ¼ 5ð3x2 − 1Þð3y2 − 1Þ∕4


pxð5y3 − 3yÞ∕2

4 L15 ¼ 3ð35y4 − 30y2 þ 3Þ∕8


p ð63x5 − 70x3 þ 15xÞ∕8

5 L17 ¼ 3ffiffiffi3

p ð35x4 − 30x2 þ 3Þy∕8


p ð5x3 − 3xÞð3y2 − 1Þ∕4


p ð3x2 − 1Þð5y3 − 3yÞ∕4


pxð35y4 − 30y2 þ 3Þ∕8


p ð63y5 − 70y3 þ 15yÞ∕8


p ð231x6 − 315x4 þ 105x2 − 5Þ∕16


p ð63x5 − 70x3 þ 15xÞy∕8


p ð35x4 − 30x2 þ 3Þð3y2 − 1Þ∕16

6 L25 ¼ 7ð5x3 − 3xÞð5y3 − 3yÞ∕4


p ð3x2 − 1Þð35y4 − 30y2 þ 3Þ∕16


pxð63y5 − 70y3 þ 15yÞ∕8


p ð231y6 − 315y4 þ 105y2 − 5Þ∕16




shape can be changed flexibly. Chrisp et al.64,65 designed animaging freeform optical system using NURBS freeformsurfaces. The imaging performance of the system wasimproved compared with that obtained when using Zernikepolynomial surfaces. Radial and toroidal NURBS surfaceshave been integrated into commercial optical design soft-ware, which gives optical designers more options. Thenext step should be improving the ray-tracing speed and effi-ciency when using NURBS functions for characterizing free-form surfaces.

Radial basis function. A linear combination of radialbasis functions with the conic surface forms another freeformsurface type, as described by

EQ-TARGET;temp:intralink-;e022;63;598zðx; yÞ ¼ cðx2 þ y2Þ1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðkþ 1Þc2ðx2 þ y2Þ

p þXNn¼1

wnΦn;

(22)

where Φn is the radial basis function and wn is the corre-sponding coefficient. The radial basis function is expressedas Eq. (23) over a supported region with a fixed center

EQ-TARGET;temp:intralink-;e023;63;502ΦnðpÞ ¼ φðkp − pnk2Þ; (23)

where p is a point ðx; yÞ in the supported region, pn is thesupported center ðxn; ynÞ, kp − pnk2 is the Euclidean norm,and φ is the basic function for constructing the radial basisfunction.

Eq. (23) shows that the supported center pn can bechanged in the supported region. Thus, the radial basis func-tion is multicentric. Cakmakci et al.27 designed a head-worndisplay using a Gaussian radial basis function freeform sur-face, which was integrated into optical design software byapplying the “user-defined surface” feature. Moreover, thelocal feature of the freeform surface can be controlled bythe radial basis function.

3.2 Representation Techniques for Deriving anOptical Freeform Surface from Discrete DataPoints

3.2.1 Numerical orthogonal polynomials

The analytical orthogonal polynomials with differentlyshaped apertures derived by the Gram–Schmidt orthogonal-ization method are, in theory, only orthogonal over their con-tinuous domain of definition. Generally, the obtained dataare the discrete data points, where the orthogonality of theanalytical polynomials may be degraded. Thus, a set oforthogonal polynomials that can adapt discrete data pointsare necessary. On the other hand, for a freeform surfacewith a complex aperture, applying the Gram–Schmidt ortho-gonalization method with iterations is rather tedious.Malacara et al.66 presented discrete orthogonal polynomialsfor wavefront analysis with a circular aperture. Dai andMahajan67 proposed a noniterative and fast algorithm toobtain analytical orthogonal polynomials with arbitrarilyshaped aperture. Furthermore, Ye et al.49 presented numeri-cal orthogonal polynomials based on the matrix transforma-tion method for fitting a discrete data point cloud of afreeform surface.

Initially, a numerical orthogonal polynomial is expressedas a linear combination of Zernike polynomials owing to itsorthogonality and completeness, as shown in

EQ-TARGET;temp:intralink-;e024;326;469Flðxn; ynÞ ¼XJj¼1

MljZjðxn; ynÞ; (24)

where Flðxn; ynÞ is the numerical orthogonal polynomial, thesubscript l is the ordering number, and ðxn; ynÞ is the pointcoordinate in the effective region. The number of data pointsin the effective region is N, where n ¼ 1;2; : : : ; N.Mlj is thetransformation coefficient, and J is the number of terms ofthe Zernike polynomial Zjðxn; ynÞ.

For a data point, Eq. (24) can be expanded as


8>>>>>>>><>>>>>>>>:

F1ðxn; ynÞ ¼ M11Z1ðxn; ynÞ þM12Z2ðxn; ynÞþ · · · þM1jZjðxn; ynÞþ · · · þM1JZJðxn; ynÞF2ðxn; ynÞ ¼ M21Z1ðxn; ynÞ þM22Z2ðxn; ynÞþ · · · þM2jZjðxn; ynÞþ · · · þM2JZJðxn; ynÞ...

Fjðxn; ynÞ ¼ Mj1Z1ðxn; ynÞ þMj2Z2ðxn; ynÞþ · · · þMjjZjðxn; ynÞþ · · · þMjJZJðxn; ynÞ...

FJðxn; ynÞ ¼ MJ1Z1ðxn; ynÞ þMJ2Z2ðxn; ynÞþ · · · þMJjZjðxn; ynÞþ · · · þMJJZJðxn; ynÞ

: (25)

The transform matrix M is


M ¼

26666666666664

M11 M12 · · · M1j · · · M1J

M21 M22 · · · M2j · · · M2J

..

. ... ..

. ... ..

. ...

Mj1 Mj2 · · · Mjj · · · MjJ

..

. ... ..

. ... ..

. ...

MJ1 MJ2 · · · MJj · · · MJJ

37777777777775: (26)




Therefore, Eq. (25) can be described in the matrix formEQ-TARGET;temp:intralink-;e027;63;740½F1ðxn; ynÞ F2ðxn; ynÞ · · · Fjðxn; ynÞ · · · FJðxn; ynÞ �

¼ ½Z1ðxn; ynÞ Z2ðxn; ynÞ · · · Zjðxn; ynÞ · · · ZJðxn; ynÞ �

26666666666664

M11 M21 · · · Mj1 · · · MJ1

M12 M22 · · · Mj2 · · · MJ2

..

. ... ..

. ... ..

. ...

M1j M2j · · · Mjj · · · MJj

..

. ... ..

. ... ..

. ...

M1J M2J · · · MjJ · · · MJJ

37777777777775: (27)

For all the data points in the effective region, it can beEQ-TARGET;temp:intralink-;e028;63;57126666666666664

F1ðx1; y1Þ F2ðx1; y1Þ · · · Fjðx1; y1Þ · · · FJðx1; y1ÞF1ðx2; y2Þ F2ðx2; y2Þ · · · Fjðx2; y2Þ · · · FJðx2; y2Þ

..

. ... ..

. ... ..

. ...

F1ðxn; ynÞ F2ðxn; ynÞ · · · Fjðxn; ynÞ · · · FJðxn; ynÞ... ..

. ... ..

. ... ..

.

F1ðxN; yNÞ F2ðxN; yNÞ · · · FjðxN; yNÞ · · · FJðxN; yNÞ

37777777777775

¼

26666666666664

Z1ðx1; y1Þ Z2ðx1; y1Þ · · · Zjðx1; y1Þ · · · ZJðx1; y1ÞZ1ðx2; y2Þ Z2ðx2; y2Þ · · · Zjðx2; y2Þ · · · ZJðx2; y2Þ

..

. ... ..

. ... ..

. ...

Z1ðxn; ynÞ Z2ðxn; ynÞ · · · Zjðxn; ynÞ · · · ZJðxn; ynÞ... ..

. ... ..

. ... ..

.

Z1ðxN; yNÞ Z2ðxN; yNÞ · · · ZjðxN; yNÞ · · · ZJðxN; yNÞ

37777777777775

26666666666664

M11 M21 · · · Mj1 · · · MJ1

M12 M22 · · · Mj2 · · · MJ2

..

. ... ..

. ... ..

. ...

M1j M2j · · · Mjj · · · MJj

..

. ... ..

. ... ..

. ...

M1 J M2 J · · · MjJ · · · MJJ

37777777777775: (28)

Equation (28) can also be written in the matrix form

EQ-TARGET;temp:intralink-;e029;63;303F ¼ ZMT; (29)

where F and Z are two N × J matrices and MT is the transposeof matrix M. Furthermore, the derivation of the numericalmatrix F is described in Ref. 49.

The derivation process of a numerical matrix F formed bythe numerical orthogonal polynomial Flðxn; ynÞ is in gen-eral. According to its practical use, a numerical orthogonalpolynomial can also be decomposed as a linear combinationof other orthogonal polynomials with completeness. Thenumerical orthogonal polynomial can be applied to representthe discrete data point cloud of a freeform surface as well as afreeform surface with a general shaped aperture.

3.2.2 Other representation techniques for handlingdiscrete data points

In the freeform optical design, the lack of an original refer-ence freeform optical configuration is relatively common.The optimization and design methods have been graduallydeveloped. The direct design method for freeform surface

is remarkable. It can be categorized as the aplanatic designmethod,68 partial differential equation method,69 SMSmethod,70 and iterative construction method.71 The directdesign method obtains discrete data points for the freeformsurface, and the discrete data point cloud must be fitted accu-rately by analytical or numerical functions for further opti-mization. In freeform optical testing, the obtained freeformshape data also form a discrete data point cloud. In theory,when the sampling dataset is sufficiently large, analyticalorthogonal polynomials with the corresponding shape ofaperture can be used approximately. Additionally, for a free-form surface with a complex aperture shape, the correspond-ing analytical orthogonal polynomial is usually difficult toobtain. Malacara et al.,66 Dai and Mahajan,67 Ye et al.,49

and Hilbig et al.72 have presented different methods to over-come this issue.

On the other hand, freeform surface or wavefront estima-tion from a slope measurement is an alternative approachused in modern optical testing, such as Shack–Hartmannsensing and phase measuring deflectometry. The measureddata are the gradient-related discrete data point cloud. Themethods used to derive a freeform surface or wavefrontfrom its slope can be classified as zonal73 and modal74




methods. Using the zonal method, Southwell73 analyzed thedifferences of different data sampling types. Li et al.75

applied higher-order truncation errors based on the Taylorseries expansion to increase the estimation accuracy in theSouthwell data sampling type. Huang and Asundi76 usedan iterative compensation method. For freeform surfaceswith complex aperture shapes, the sag data of the freeformsurface derived from the slope could be treated with the dis-crete Fourier transform77 or discrete cosine transform-basedmethods.78 Zou and Rolland79 presented an iterative zonalestimation method with the Gerchberg iteration80 to obtaina freeform surface for general shaped pupils. However,the zonal method is limited by the shape of the aperture,and the iterative method is not very efficient for dynamicallyvarying aperture shapes.

In the modal method, the freeform surface or wavefront isdecomposed as a linear combination of basis functions. Thefinal coefficients characterizing the freeform surface areobtained from the measured slope data. Cubalchini74 pro-posed using the gradient function of Zernike polynomialsas the basis function for fitting the measured gradientdata. The drawback of this approach was that the gradientfunction of a Zernike polynomial is not an orthogonal func-tion, which caused instability in the coefficients. Thus, a set

of polynomials whose gradient functions are orthogonal overthe measured region must be selected. Zhao and Burge81 pro-posed orthonormal vector polynomials in a unit circle.Obviously, for noncircular apertures, Zhao’s method couldbe limited. Mochi and Goldberg82 reported an iterativemethod for obtaining an orthogonal basis function to recon-struct a wavefront from its gradient. In this method, a finiteset of 2-D polynomials that are expected to describe the mea-sured wavefront over the testing domain must be initiallyselected. However, obtaining an orthogonal function overthe aperture is not very easy, especially for complex-shapedapertures. Ye et al.83 presented a numerical orthogonal trans-formation method to obtain numerical orthogonal gradientpolynomials for directly representing the measured gradientdata of a freeform surface with a general shaped aperture.This method was noniterative and efficient and could alsobe applied to the slope-based freeform surface or wavefrontsensing with the dynamically varying aperture shapes.

3.3 Representation Methods for Freeform Surfaceswith Strong Slope Variation

Regarding freeform surfaces with strong slope variations,characterizing fine local features is a key issue for freeform

Fig. 7 A freeform optical element testing experiment. (a) Freeform optical element with steep variation atthe edge of an irregularly shaped aperture, (b) measured data after removing the tilt, (c) fitting error by thefirst 37 Zernike polynomials approximately, and (d) fine reconstruction performance of the proposed com-bination method.




surface estimation. When using only finite polynomial terms,the representation accuracy could be limited. Kaya et al.84

performed a comparative assessment between Zernike poly-nomials and Q-type polynomials for precisely characterizinga freeform surface. To fit an asymmetric local feature withsub-nanometer accuracy, the number of polynomial termswould reach thousands. The calculation of higher-orderZernike or Q-type polynomials is relatively tedious. Trevinoet al.85 selected the first-class Bessel circular functions forcharacterizing a complex corneal surface. Svechnikov et al.86

discussed the lateral resolving capacity of circular Zernike pol-ynomials and the accuracy of representing complex freeformsurfaces, such as Gaussian and Gaussian-like surfaces. Whenmore polynomial terms are applied, the efficiency of the free-form surface estimation is noticeably reduced.

Therefore, the hybrid or combination reconstructionmethod could be an alternative approach to freeform surfacefitting with finite polynomials and high accuracy. The con-cept of the hybrid representation method is similar to that ofsubaperture stitching testing for large-aperture optical surfa-ces. The entire aperture of a freeform surface with strongslope variation is decomposed into multiple overlapping sub-apertures. Every local surface corresponding to the subaper-ture can be reconstructed by Zernike polynomials or otherorthogonal polynomials. Then, the entire freeform surfaceis synthetized by all the local surfaces using specific algo-rithms. Espinosa et al.87 investigated a combination of zonaland modal fitting methods for evaluating an irregular cornealsurface. Its zonal fitting by Zernike circle polynomialwas implemented over a square subaperture rather than over

Table 5 Comparison of typical freeform surface representation techniques.

Freeform surface Representation techniques Advantages Remarks

Mild freeformsurface

Orthogonalpolynomials

Zernike circular polynomials (1) Each term is related toa specific aberration

(1) The definition domain ofthe orthogonal polynomial iscontinuous, not for discretedata points

Q-type polynomials (2) Orthogonality andcompleteness

(2) The derivation ofhigher-order polynomials isnot easy

Zernike polynomials overa regularly noncircular apertureby Gram–Schmidt orthogonalization

(3) One-to-one correspondencewith the aperture shape

2-D Chebyshev polynomials

2-D Legendre polynomials

Nonorthogonalfunctions

XY-type polynomials (1) Related to the aberration Commonly used in freeformoptical design

(2) Global definition of freeformsurface

Spline function (1) Local definition of freeformsurface

Ray-tracing with the designtool is challenging

(2) Flexible in controllingsurface shapes

Radial basis function (1) Local definition of freeformsurface

Must be integrated into thedesign tool using the“user-defined surface” feature

(2) Multicentric feature

Numerical orthogonal polynomials (1) Adapt to general apertureshapes

Numerical orthogonaltransformation method

(2) Adapt to discrete data points

Freeform surfacewith strong slopevariation

Linear combination of hundreds or thousands ofbasis polynomials

Lateral resolving capacity ofbasis polynomials

Computational efficiencyreduced with manypolynomial terms

Hybrid or combination method (1) Local feature characterizedfinely

Local fitting coefficients aremeaningless

(2) Only with finite number ofpolynomial terms




a circular subaperture. Kaya and Rolland88 proposed a hybridmethod combining local fitting in each subaperture and usinga radial basis function as a global approximant for obtainingthe entire surface. Ye et al.89 presented a combination methodfor representing complex freeform surfaces. Every local sur-face was reconstructed by the linear combination of numeri-cal orthogonal polynomials, regardless of the local surfaceaperture shape. The entire freeform surface was derived bythe overlapping averaging approach. As shown in Fig. 7,a freeform optical element over an irregularly shaped aper-ture has steep variations at the edge. When approximatelyfitting it directly by Zernike polynomials over the entire aper-ture, the local fitting error is relatively large. However, itslocal deformations at the edge can be characterized finelyby the proposed combination method.

4 DiscussionThe benefits of optical freeform surfaces in optical engineer-ing result from the collaborative development of the manu-facturing, testing, design tools, and application requirementsof freeform surfaces in the past 10 years. Advances in therepresentation techniques of optical freeform surfaces canfurther accelerate the improvement of freeform optics inpractical applications. From the previous analysis of differenttypes of optical freeform surface representation techniques,we have a large map to help understand the general repre-sentation techniques.

Nevertheless, we also need to know how to choose therepresentation technique of a freeform surface in a specificapplication. In general cases, when the freeform surface usedin the freeform optical design or freeform optical elements isunder manufacturing and testing, it can be qualitativelydescribed as a mild or strong freeform surface accordingto the slope variation degree. For example, Gaussian-likelocal areas in a freeform surface can be considered a strongfreeform surface. The freeform surface shown in Fig. 5 couldbe considered a mild freeform surface. For a mild freeformsurface, diverse representation techniques can be employedaccording to its practical use. When considering a freeformsurface with constrained slopes in its design, Q-type polyno-mials could be a good choice. For the estimation of a free-form surface with noncircular regularly shaped aperture,orthogonal polynomials with the corresponding apertureshape or numerical orthogonal polynomials are employed.When local surfaces in a few specific fields must be con-trolled, splines, or radial basis functions may be used. Fora freeform surface with large slope variation, hybrid or com-bination methods could be applied to accurately characterizeits local features. On the other hand, in the process of form-ing the freeform surface (from design to manufacturing totesting), the current technologies of freeform optics alsolimit the choice of freeform surface representation tech-niques in practical applications.

Table 5 presents a comparison of several typical freeformsurface representation techniques and their advantages.Orthogonal polynomials, because of their orthogonality,completeness, and their aberration-related performance,are frequently applied in freeform surface representationand estimation. We must also note that the orthogonalityapplies over a continuously defined domain; namely, thesampled data points in the effective aperture must be suffi-cient. On the other hand, the higher-order polynomial terms

can be derived by iterative recurrence. Regarding nonorthog-onal polynomials, the XY-type polynomial is always used infreeform surface design, as discussed in Sec. 2, owing to itssimplicity compared with orthogonal polynomials. Althoughthe spline function and radial basis function control local fea-tures in the freeform surface representation, their ray-tracingefficiency must be improved. These two functions can beintegrated into design tools by the “user-defined surface”feature. In the method using a linear combination of basepolynomials fitting a strong freeform surface with steeplocal variation with nanometer accuracy, the use of hundredsor thousands of terms is required. Base polynomials haveevident lateral resolving capacity; however, many polyno-mial terms reduce computational efficiency. In this situation,the hybrid or combination method, discussed in Sec. 3.3,could be an alternative approach to freeform surface repre-sentation with finite polynomial terms and high accuracy.Although local fitting coefficients are meaningless for theentire surface, the hybrid or combination method can beapplied to characterize local surface features finely for free-form surface estimation.

5 ConclusionIn the past 10 years, the optical freeform surface represen-tation technique is a hot research topic. The representationtechniques of optical freeform surfaces are diverse and notunique in specific applications. The development of repre-sentation techniques, the requirements of design and appli-cation, and the advancement of manufacturability andtestability of optical freeform surfaces will accelerate forthe improvement of freeform optics. By focusing on opticalfreeform surface representation techniques, we review com-prehensively the current state of the art of representationtechniques and their typical applications. Different typesof representation techniques are discussed and compared,which will provide solutions for how to represent an opticalfreeform surface and which representation technique couldbe used in a specific application.

Generally, a freeform surface is represented as an explicitmathematical function; it is decomposed as a linear combi-nation of basis function added to the base surface. The basesurface is usually a conic surface or sphere. Basis functionscan be XY-type polynomials, Zernike polynomials, Q-typepolynomials, radial basis functions, or NURBS functions.The differences and advantages of these methods are pre-sented in Table 5. For a strong freeform surface with steepslope variation in local areas, a hybrid or combinationmethod could be applied to accurately characterize local fea-tures. In addition, by effectively combining different types offunctions, new functions could be constructed in the futurefor representing optical freeform surfaces. Additionally, thegradient variation of a freeform surface must be consideredwithin the current manufacturing and testing capacities.

Regarding the discrete data points obtained by the directdesign method for an optical freeform surface, the obtaineddiscrete data point cloud characterizing the freeform surfacemust be fitted accurately by an appropriately chosen analyti-cal function to obtain a suitable design starting point forfurther efficient optimization. A similar process for freeformsurface analysis from discrete measured data exists infreeform surface testing. Furthermore, in the design of a free-form optical system with a large field of view, the use of




a sectional or stitched freeform surface is a promising tech-nique. The representation technique, as well as the fabrica-tion and measurement, of this type of freeform surface couldbe challenging.

On the other hand, the functions of most surface typeshave been integrated into commercial optical design soft-ware. However, special functions, such as the radial basisfunction and the stitched surface, are not included. The“user-defined surface” feature of optical design softwareis very powerful and flexible and provides more degreesof freedom for freeform optics design. Nevertheless, theproblem of the ray-tracing speed and optimization efficiencymust be overcome in the future.

AcknowledgmentsThe authors would like to thank the Talent Launch Fund ofNanjing University of Information Science and Technologyand the National Natural Science Foundation of China(Nos. 61377015, 61505080, and 51705271) for financialsupport. The authors also want to thank the editor andreviewers for their valuable comments and suggestions.

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Jingfei Ye received his BS degree and PhD in optics from NanjingUniversity of Science and Technology in 2010 and 2016, respectively.Currently, he is a lecturer in the Department of Optoelectronic Engin-eering at Nanjing University of Information Science and Technology.His current research interests include optical design and testing espe-cially for freeform optics and optical remote sensing especially forlidar.

Lu Chen is currently a PhD candidate at the School of Electronic andOptical Engineering of Nanjing University of Science and Technology.She is working on high-precision metrology and freeform optics.

Xinhua Li is a PhD candidate at the School of Electronic and OpticalEngineering of Nanjing University of Science and Technology. She isalso an associate professor at Jinling Institute of Technology. Hermain research is in the field of visual optics.

Qun Yuan received his PhD in 2014. He is an associate professor atthe School of Electronic and Optical Engineering of Nanjing Universityof Science and Technology. His main interests include optical designand high-precision interferometric metrology.

Zhishan Gao is a professor at the School of Electronic and OpticalEngineering of Nanjing University of Science and Technology. Hismain research interests include optical design, optical testing tech-niques, and visual optics.




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