Design considerations for the absolute testing approach of ...lunazzi/temp/artigo13 OptExp/referenciasOSA13... · Design considerations for the absolute testing approach of aspherics

Design considerations for the absolute testing approachof aspherics using combined diffractive optical elements

Gufran S. Khan,1,2,* Klaus Mantel,1 Irina Harder,1 Norbert Lindlein,1 and Johannes Schwider1

1Institute of Optics, Information and Photonics (Max-Planck Research Group),Friedrich-Alexander-University Erlangen-Nuremberg, Staudtstrasse 7�B2, 91058 Erlangen, Germany

2Central Scientific Instruments Organisation, 160030 Chandigarh, India

*Corresponding author: [email protected]

Received 31 May 2007; revised 8 August 2007; accepted 10 August 2007;posted 13 August 2007 (Doc. ID 83446); published 27 September 2007

Aspheric optical surfaces are often tested using diffractive optics as null elements. For precise measure-ments, the errors caused by the diffractive optical element must be calibrated. Recently, we reported firstexperimental results of a three position quasi-absolute test for rotationally invariant aspherics by usingcombined-diffractive optical elements (combo-DOEs). Here we investigate the effects of the DOE sub-strate errors on the proposed calibration procedure and present a set of criteria for designing an optimizedcombo-DOE. It is demonstrated that this optimized design enhances the overall consistency of theprocedure. Furthermore, the rotationally varying part of the surface deviations is compared with therotationally varying deviations obtained by an N-position averaging procedure and is found to be in goodagreement. © 2007 Optical Society of America

OCIS codes: 120.0120, 120.3180, 120.4800, 220.6650, 050.1970.

1. Introduction

The improved performance capability of asphericswith a reduced number of elements has been dis-cussed by optical engineers for quite a long time.Therefore, aspheric surfaces have become inevita-ble in many optical systems. For example, the useof aspherics instead of spherical surfaces reducesthe number of optical surfaces in microlithographysystems. In photographic optics, telescopic systems,head-up displays and helmet mounted displays, wherevolume and weight constraints are significant, as-pherics are often the only option. Technological ad-vances in machines and tools have made it possible toproduce aspherics of high surface quality. However,an equally precise metrology procedure is still ex-tremely difficult.

The metrology of aspherics is commonly performedby using a refractive or reflective null element in astandard interferometer setup [1]. This elementworks as a compensator for the deviation of the as-phere from a sphere. The production of such a null

element for each specific aspheric is time consumingand expensive. In addition, it is difficult with complexaspheres. Computer-generated holograms (CGHs),which can also be termed as diffractive optical ele-ments (DOEs), provide a reasonable alternative andhave long been in use as null elements for aspherics[2–5]. Over the years, different possibilities for insert-ing the DOE into the interferometer setup have beeninvestigated and their relative advantages and dis-advantages were reported [6]. The architecture withthe DOE in the object arm just before the specimen,compensating the aspheric wavefront, provides ro-bust interferometer architecture. Additionally, theDOE can be designed independently of the setupsince the entire interferometer optics need not beincluded. However, in such an architecture where theDOE is placed in the object arm, the quality of theDOE substrate plays a critical role. Therefore, it isimperative either to know the DOE deviations before-hand so that they can be subtracted from the finalresults or to apply suitable calibration procedures. Toachieve a high accuracy measurement, an absolutecalibration of the interferometric system is required,including the removal of errors introduced by the nullelements.

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2. Research Background

Over the past few years, efforts have been made tocalibrate interferometric systems by the use ofcombined-diffractive optical elements (combo-DOEs).In 1998, Schwider [7] proposed the use of such ele-ments for the calibration of the interferometer andthe assessment of the DOE error as a whole. Thecalibration procedure works similarly to the threeposition test for spherical surfaces [8], which requiressurface measurements at two positions, a basic and a180° rotated position, followed by a measurement inthe cat’s-eye position. The combo-DOE carries theinformation for the aspheric as well as for the best fitsphere with a slight linear offset, where the sphericalwavefront is used for the cat’s-eye measurement. Thebasic idea is that the structural errors due to thelimited accuracy of the lithographic tools are nearlyidentical for the aspherical and spherical structure ofthe DOE, if both waves have similar curvature. Inaddition, it can be assumed that variations in theoptical thickness of the DOE substrate impair bothwavefronts in the same manner. Beyerlein et al. [9]have explored the two possibilities for the design ofsuch combo-DOEs, namely sliced and superposedstructures, and have estimated the influence of dis-turbing diffraction orders in both cases. Implement-ing the idea of combo-DOEs, and further developingthe absolute procedure, Reichelt et al. [10] performedthe calibration of such DOEs by using the fact that thewave aberration due to lithographic structure errorschange sign when the �1st order is used instead of the�1st order.

Another calibration approach is an N-positionrotationally averaging method [11]. Based on thisapproach, Freimann et al. [12] have developed an ab-solute test procedure for noncomatic aspheric surfaceerrors by measuring the aspheric in 12 rotated posi-tions. However, this procedure does not calibrate therotationally invariant part of the surface deviation.

Recently, we reported our first experimental dem-onstration of the three position test of aspherics[7,13]. Both design possibilities discussed by Beyer-lein et al. [9] were investigated and it was observedthat the sliced DOE shows less impact of the disturb-ing diffraction orders on the absolute measurement.However, residual astigmatic aberrations were ob-served from a consistency test. In this paper, wefurther investigate the cause of the observed astig-matism in the consistency test and determine thefactors that are limiting the accuracy. DOEs are flatelements and do not fulfill the sine condition [14]. Ourinvestigations show that the tilt of the DOE and itsorientation relative to the offset direction of the cal-ibrating spherical wavefront are the main reasons forthe residual astigmatism in the consistency test.

The basic assumption of this procedure is that theDOE substrate and the lithographic errors affectthe two wavefronts identically and we call this thematching condition. This can easily be understood ifone considers first a DOE with a very coarse diffrac-tive structure. In this case, the smallest period p shall

be so big that a lithography-type error �p will pro-duce a negligible wavefront error, �W�� p�p.With increasing spatial frequency, � � 1�p, a similarstatement can be made concerning neighboring struc-tures for the spherical and the aspheric wavefront,provided the spatial frequencies are not too different.Then a calibration error of the order �W�� sphere � �asphere��p will limit the achievable calibra-tion accuracy. The remaining problem in the contextof the dual encoding of the wavefronts is that thecalibrating spherical wavefront is different from theaspheric one and additionally has a slight linear off-set, thus giving rise to the quasi-absolute nature ofthe test. In this work, we discuss the design consid-erations of the combo-DOE and present an optimizeddesign that better satisfies the matching conditionand improves the consistency of the procedure. How-ever, this inherent quasi-nature still remains the lim-iting factor on the accuracy of the procedure. Theextent of this limitation will also be discussed.

We first recall the quasi-absolute three positiontest procedure in Section 3. The design considerationsof combo-DOE are discussed in Section 4. Here wediscuss all the critical design parameters and dem-onstrate simulation results that help us design anoptimized combo-DOE. Section 5 contains the exper-imental results of the quasi-absolute test. A compar-ison with an N-position rotational averaging test isalso presented. Limitations, advantages, and possibleimprovements are discussed in Section 6, and finalconclusions are eventually drawn.

3. Measurement Principle and Combo-DOE

Here, we present a brief description of the measure-ment procedure. Further details can be found in ourearlier publications [5,7,9,13]. Figure 1 depicts therequired three positions for the absolute measure-ment of a rotationally invariant asphere. The posi-tions are: (1) a basic position, (2) a 180° rotatedposition of the specimen, and (3) the cat’s-eye positionwhere a mirror is placed at the focus of the sphericalwavefront. The aspheric wavefront is used for themeasurements at the first two positions. By usingthese three measurements, the surface deviations ofthe aspheric can be separated from the systematicerrors of the setup. The combo-DOE is tilted by asmall angle to avoid on-axis reflections from its frontand back surface. The distance between the combo-DOE and the aspheric under testing should be on theone hand as small as possible to have an unequivocalcorrespondence between points on the aspheric and

Fig. 1. Measurement principle: three positions for the quasi-absolute measurement of aspheric surfaces.

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on the DOE. On the other hand, there has to be acertain distance between the DOE and aspheric sothat the different diffraction orders of a binary DOEcan be separated [15]. So, in practice a compromisehas to be found, which minimizes the overall mea-surement uncertainty. This is in our case a distanceof 10 mm between the DOE and aspheric.

To encode both wavefronts simultaneously, theDOE has been sliced in 50 �m wide stripes that arealternatively assigned to the spherical and asphericwaves. To achieve the correct alignment of the DOEwithin the interferometer, a reflection grating struc-ture is provided outside the test structure of the DOE[6]. The �1st order of the reflected beam of the align-ment grating has to be adjusted to a fluffed-out fringein order to have the correct tilt of the DOE. Sucha combo-DOE will produce the desired asphericand spherical wavefronts. However, because of thestriped nature, it will generate additional diffractionorders that may disturb the measurements. To avoidthe direct overlapping of the different orders of bothwavefronts, the off-axis direction of the sphericalwavefront is selected to be perpendicular to the grat-ing vector of the grating produced by the stripes.Upon reflection, while retracing the path, the disturb-ing diffraction orders are blocked by a stop at the con-focal plane of the telescopic system in the object arm.

4. Design Considerations for the Combo-DOE:Simulation Studies

There can be many sources of errors, such as (i) aDOE substrate error, (ii) a systematic error of theempty interferometer without the DOE, (iii) a DOEmisalignment error, (iv) an insufficient stability ofthe interferometer (air turbulence, vibration, drift,etc.), and (v) lithographic writing errors. With theaim to design an optimized combo-DOE and to havean estimate of the achievable accuracy of the proce-dure, we performed simulation studies in the pres-ence of the systematic errors (i)–(iii) by using oursoftware package RAYTRACE [16]. All the simulationshave been done using a wavelength of 633 nm. To bevery close to the real experimental situation, themeasured systematic errors of the empty interferom-eter and the substrate deviations have been used asinput in the simulations. Figure 2 displays typicalaberrations in a double pass configuration. A noisyenvironment and the lithographic writing errorshave not been considered in the simulations. For theaspheric under testing, the paraxial radius of curva-ture was 94.84 mm [see Fig. 7(a) for the asphericprofile and its best fit sphere]. An additional smallastigmatism has been added to the specimen to rep-resent the surface deviations. The main reason forperforming the simulations is that a DOE is a flatelement and therefore violates the sine condition. Soin situations where the DOE is misaligned or theincoming wavefront is not ideal, aberrations (mainlycoma) will be present.

When there is no systematic error in the system, thesurface deviations are perfectly reconstructed by thesimulated three position absolute test. However, in

the presence of systematic errors of the empty inter-ferometer and the DOE substrate (in particular), someremaining error is observed. It is not possible to accessaccuracy from a single absolute measurement. There-fore, we investigate the accuracy with a consistencytest. A consistency test is simulated in the followingway: (1) two absolute results are simulated, one byhaving the surface measurements at 0° and 180° as thebasic and rotated positions, and the other one with thesurface measurements at 90° and 270°. The cat’s-eyeposition is the same in both cases; (2) rotate the secondabsolute result by �90° to ensure the same orienta-tion of the coordinate system for the absolute devia-tions; and then (3) take the difference between them.We present the three most critical design parametersthat influence the accuracy.

A. Relation between the Tilt Orientation of the DiffractiveOptical Element and the Offset Direction of the SphericalWavefront

As already mentioned the spherical wavefront has asmall tilt angle that was initially chosen as 1°. Therecan be four choices for the offset direction of thespherical wavefront as shown in Fig. 3. The optical

Fig. 3. Four choices of the linear offset for the spherical wavefrontwith respect to the optical axis of the interferometer: (a) �X direc-tion, (b) �Y direction, (c) �X direction, and (d) �Y direction. DOEhas been tilted around the Y axis.

Fig. 2. (a) Systematic error of the empty interferometer in doublepass configuration (b) DOE substrate error in double pass config-uration. Contour line spacing is ��30.


axis is taken along the Z direction and the combo-DOE is tilted relative to the XY plane, where the Yaxis shall be the rotation axis. The offset of the spher-ical wave can be along the �X, �X, �Y, and �Ydirections. In principle, it is possible to use any ofthese offset directions irrespective of the tilt orienta-tion of the combo-DOE, provided there are no system-atic errors in the setup. But in the presence of DOEsubstrate and�or interferometer deviations, which inreality is the case, it becomes necessary to considerthe relationship between the tilt direction of thecombo-DOE and the offset direction. The DOE sur-face deviations include a typical wedge of 10 arcsec inthe direction of the global tilt of the combo-DOE.

Figure 4 shows the results of a consistency test foreach of these four cases using the same substrateerror for all offset situations. These results indicate aremaining systematic error of the absolute testingprocedure that depends on the relationship betweentilt and the offset direction. The simulation showsthat the best situation is where the offset of thespherical wavefront is in the direction of the surfacenormal of the DOE. The simulation also shows thatthe tilt angle of the spherical wave should be thesame as the tilt of the DOE in the same direction.This sets the amount of the linear offset in the focalplane of the spherical wavefront. In this case, thefocus of the spherical wavefront lies on a straight linethrough the center of the DOE having the direction ofthe normal of the DOE. Since this is the condition

where the cat’s-eye measurement follows the symme-try with respect to the DOE plane, we denote it thesymmetry condition for the cat’s-eye position. In allthe other situations, a higher amount of astigmatismis present because of the broken symmetry of thecat’s-eye measurement.

B. Influence of the Tilt Amount of the Diffractive OpticalElement in the Presence of a Systematic Error of theSetup

Even in Fig. 4(a), which obeys the symmetry condi-tion, one can still see residual astigmatism. This islikely due to a systematic error of the setup, in par-ticular a wedge error in the DOE substrate. Since theDOEs are flat elements and do not fulfill the sinecondition, they introduce coma in a single measure-ment in the presence of a wedge error. The amount ofcoma increases linearly with the increase in theglobal tilt of the DOE. The primary coma is removedas a misalignment error once the misalignment elim-ination algorithm is applied [17], but the higher or-ders remain. To better understand the effects of thewedge error, we simulated a case where no error hasbeen introduced into the setup, except a wedge of0.002°. Even the specimen has been chosen to be anideal surface.

Figure 5 shows the results of the consistency testsfor a varying amount of the DOE tilt from 0.25° to 2°,where the symmetry condition discussed in the ear-lier section has been followed. One can notice a linear

Fig. 4. Simulation results of the described consistency test in allthe four possible cases of the linear offset for the spherical wave-front with respect to the optical axis of the interferometer (a) �Xdirection, (b) �Y direction, (c) �X direction, and (d) �Y direction.Contour line spacing is ��300.

Fig. 5. Simulation results of the consistency test for the varyingamount of global tilt of the DOE from 0.25° to 2° with respect to theinterferometer while following the symmetry condition discussedin the last section [see Fig. 3(a)] (a) 0.25° tilt, (b) 0.5° tilt, (c) 1.0°tilt, and (d) 2.0° tilt. Contour line spacing is ��1000.


increase in the error with the DOE tilt. This situationdemands that one should design a DOE with a min-imum tilt angle. But the amount of tilt depends ontwo other parameters. One is that some amount of tiltis required to avoid the back reflections from the DOEsurfaces going into the interferometer. In our setup,the tilt angle must be at least 0.25°. The minimum tiltangle can be determined from the space bandwidthproduct that is dictated by the CCD camera and theoptical system as a whole. The space bandwidth prod-uct should not be restricted too much because thenonly slowly varying aberrations can be corrected. An-other constraint is that the offset of the sphericalwavefront should be large enough to separate it fromthe on-axis aspheric wavefront. This tradeoff limitsour tilt angle to at least 0.5° with respect to theinterferometer axis.

C. Matching Condition and Quasi-Nature of theProcedure

The basic idea behind the combo-DOE test relies onthe fact that the wave aberrations caused by litho-graphic errors are only proportional to the ratio of thepositioning error of the lithographic writing machineand the local period of the DOE. Therefore, two DOEstructures show similar aberrations in the close lat-eral neighborhood. Thus, it is possible to use differentwavefronts because the difference of the aberrationswill be small.

The design, fulfilling the symmetry condition, aswell as the least possible tilt angle [see Fig. 5(b)]situation, presents the least error. Nevertheless, theconsistency simulation shows an astigmatism of ap-proximately ��160. The reason is that a sphericalwavefront, for the measurement in the cat’s-eye po-sition, is different from the aspheric wavefront, andthis introduces the quasi-nature in the three positiontest. This quasi-nature has two components: (1) thecalibrating wavefront at cat’s-eye is different fromthe aspheric one and (2) an offset must be present toseparate it from the aspheric wavefront. Since theseare the basic procedural limitations, we describe thewhole procedure as quasi-absolute.

The procedure requires that the two wavefrontsshould match in the best possible way, which we termthe matching condition. In this section, we explore thecriticality of this condition and try to minimize thequasi-absolute extent of the procedure. Minimizingthe tilt angle of the DOE (see Subsection 4.B) guar-antees a minimum offset of the spherical wavefront,thus a better matching of both wavefronts. So theeffects shown in the last section are also related to thequasi-nature. The smaller the offset angle the betterthe matching condition is fulfilled. Furthermore, thespherical wavefront can either be chosen as a base fitor a best fit to the aspheric. In the base fit case, justthe paraxial radius of curvature of the aspheric waveis taken. In the best fit case, the integral of thesquares of the wavefront differences to a sphericalwavefront has to be minimized. Figure 6 comparesthe consistency results for the two conditions in thepresence of general interferometric and substrate er-

rors. The best fit spherical wavefront is a betterchoice, because the aspheric wavefront departure isthe least in this case. But the lithographic writingerrors become more critical in the high spatial fre-quency region, where writing periods become finerand finer. So in the fitting algorithm it would bebetter to have more weight in the high frequencyregion, which is in turn based on the minimization ofthe relative difference of the writing periods ratherthan the departure of the wavefront itself. However,this cannot be done unless the exact behavior of thefabrication errors is known.

5. Experimental Results and Discussion

To demonstrate the principle, a phase-shiftingTwyman–Green interferometer setup has been used.A rotationally invariant convex asphere of a diameterof 50 mm has been chosen as a test specimen. Themaximum aspheric deformation from the best fitsphere is approximately 0.25 mm (i.e., �400 waves)at the edge (see Fig. 7). Before presenting absolutetest results, it is necessary to discuss the repeatabil-ity, reproducibility, and interferometer drift of thesetup and the misalignment removal capabilities ofthe evaluation process. Repeatability, reproducibil-ity, and drift analysis give an idea about the overallstability status of the interferometer.

Fig. 6. Consistency results when the calibrating spherical wave-front is chosen as (a) base fit to the aspheric and (b) best fit to theaspheric. Contour line spacing is ��1000.

Fig. 7. Specimen under test (a) rotationally invariant asphericsurface and best fit sphere of the chosen specimen and (b) depar-ture of the aspheric surface from its best fit sphere.


A. Interferometer Stability

The repeatability can be derived from the differencebetween two measurements, one right after theother, without any nominal change in the system.This provides information about the sensitivity of thesetup against vibrations and air turbulence. Sincethe specimen has to be rotated by 180° during the testprocedure, the effects of readjustment should also beevaluated. These effects are captured by the repro-ducibility where the difference between two measure-ments involving a readjustment has been taken. Thereproducibility depends on the precision of the me-chanical components in the specimen holder and thecapabilities of the metrologist himself to readjust it tothe fluffed-out fringe condition. Since it is impossibleto readjust the specimen exactly, coherent noise isintroduced [18] limiting the reproducibility in prac-tice to a value of approximately half the repeatability.In our case, the repeatability and reproducibility ofthe setup for single measurements are ��500 and��200, respectively, where � is 633 nm, the wave-length of a He–Ne laser.

If the series of measurements is going to take sometime, which is the case in our procedure, then it isnecessary to know the stability of the interferometerduring this time span. This can be investigated byperforming a drift analysis, which consists of lookingat the difference between the two measurements af-ter a necessary time gap. This has to be carried outwithout touching or changing anything in the setup.The difference between the drift analysis and therepeatability is that the repeatability is a measure ofshort time disturbances, while the drift analysisshows how these deviations evolve with time. Sincewe are going to present the N-position rotational av-eraging test also (in part D of this section), where wetake 36 measurements, which takes approximately 1hour, we performed the drift analysis for the sameduration. The rms values of the difference of the mea-surements with a time gap varying from 30 s to 1 hare presented in Table 1. These are within the rangeof the reproducibility of a single measurement.

B. Removal of Misalignment Aberrations

Misalignment of the specimen relative to the inter-ferometer’s coordinate system plays an importantrole in interferometric metrology in general, but itbecomes more crucial in aspheric metrology becauseof its reduced degree of symmetry. In the presence ofmisalignment, the interference pattern is always asuperposition of the wave aberrations due to the sur-face errors of the test piece, the systematic errors ofthe interferometer, and the aberrations caused by themovement of the aspheric relative to the interferom-

eter frame. The specimen can never be adjusted to theperfect position, so the extent to which misalignmenterrors can be removed must be considered.

The alignment errors due to tilt, decenter, and de-focus of the test surface can be analyzed empirically.For the misalignment removal we have used thestrategy discussed by Young [17]. The misalignmentfunctionals are described by representing 5 degrees offreedom of the aspheric within the interferometer(e.g., three displacements and two tilts) by five mis-alignment vectors. Rotation around the Z axis is notconsidered as a degree of freedom since the surfaceunder test is rotationally invariant. These errorshave been computed mathematically and were thenremoved from the measuring data by a least squaresfit to find out the actual surface errors plus system-atic errors of the interferometer.

To estimate the accuracy of the misalignment elim-ination, several misalignments have been simulatedwith RAYTRACE [16] and have subsequently been elim-inated. The misalignment parameters were chosen tohave a peak-to-valley (P–V) value of the simulatedaberrations close to 2 wavelengths, a value that wecould achieve in the experiment. The same has beenperformed in the experiment with all 5 degrees offreedom. Tables 2 and 3 show the results of the elim-ination process in simulations and in experiment,respectively. Both studies clearly show that the mis-alignment in the Z direction is the most critical oneand requires a much more careful alignment com-pared with the rest of the degrees of freedom.

C. Quasi-Absolute Three Position Test

Using the three position procedure described in theprevious section, the surface deviations of the testspecimen are determined. Figure 8(a) shows the con-tour plot of the absolute deviations of the surface byusing the optimized combo-DOE. The P–V value is1.031 waves and the rms value is 0.244 waves. An x–ypolynomial fit of order 12 has been applied to elimi-nate the noise in the final absolute result. Figure 8(b)shows the reproducibility of the three position testitself, where the difference of two such absolute mea-surements at the same orientation of the specimen

Table 1. Drift Analysis for a Duration of 1 h�

Duration 30 s 1 min 3 min 5 min 10 min 20 min 40 min 60 min

Difference rms (�) 0.0022 0.0027 0.0043 0.0039 0.0044 0.0056 0.0063 0.0072

�Difference has been taken from the initial measurement.

Table 2. Residual Aberrations after Misalignment Elimination inSimulations

SimulatedMisalignment

P–V SimulatedField (�)

Residual Aberration

rms (�) P–V (�)

X shift 0.9982 0.0004 0.0026Z shift 0.9996 0.0013 0.0064Y rotation 1.0670 0.0003 0.0024


was taken. The rms value is ��200, which is in therange of the reproducibility of a single measurement.The interferometric errors have been extracted bysubtracting the absolute measurement from a singlemeasurement and are shown in Fig. 8(c).

As discussed in the simulation section, we lookedfor the consistency test as an indication as to theaccuracy of the procedure. Figure 9 compares theresults of the consistency test when: (a) the combo-DOE does not fulfill any of the design conditions andhas a tilt of 1°, (b) the combo-DOE satisfies only thesymmetry condition and has 1° tilt and (c) an opti-mized combo-DOE with 0.5° tilt is taken. These re-sults conform to the predictions of the simulationstudies. The consistency improves by a factor of 10 ifthe symmetry condition is followed and further im-proves by a factor of 2 with an optimized design,where a 0.5° tilt angle and the best fit spherical wavehave been selected.

The occurrence of residual astigmatism of P–Vvalue 0.028 waves in the consistency test even in thecase of the optimized DOE is, we believe, because of

the fact that the procedure is still limited by its quasi-nature and the flat DOEs do not fulfill the sine con-dition. Further simulations yield that the consistencyerror depends on the errors of the empty interferom-eter and on those of the DOE substrate. Therefore, itis expected that the lesser these errors are, the betterthe consistency will be. In particular, the wedge errorin the substrate is very critical as discussed in Sub-section 4.B. Currently, we used commercially avail-able mask plates as substrates, which have a wedgeerror of 10 arcsec. This value can definitely be im-proved by using a higher quality substrate. It is,therefore, left for further investigations to use abetter quality custom made substrate and to demon-strate the effects of substrate quality on the proce-dure.

D. N-Position Test

While the three position test method delivers theabsolute errors of the surface, the rotationally vary-ing deviations can be measured with another method.The surface shape deviations can have rotationallyinvariant as well as rotationally varying errors. Therotationally varying errors can be extracted frommeasurements with the surface being rotated in astepwise manner, a method known as an N-positionrotationally averaging test [11]. The average of Nsuch measurements at the interval of 360�N degreesremoves all rotationally varying errors of the surface:

Wavg �1N �

i�1

N

Wi. (1)

Table 3. Residual Aberrations after Misalignment Elimination in Experiments

Misalignment in �(Approximate)

Residual Aberrations after Misalignment of Approximately

1� 2� 3� 4�

rms (�) P–V (�) rms (�) P–V (�) rms (�) P–V (�) rms (�) P–V (�)

X shift 0.0015 0.0084 0.0034 0.0203 0.0055 0.0374 0.0059 0.0406Y shift 0.0019 0.0093 0.0034 0.0244 0.0043 0.0313 0.0054 0.0401Z shift 0.0051 0.0275 0.0063 0.0314 0.0099 0.0466 0.0233 0.0958X rotation 0.0016 0.0081 0.0014 0.0117 0.0021 0.0149 0.0024 0.0176Y rotation 0.0011 0.0074 0.0012 0.0087 0.0019 0.0140 0.0024 0.0163

Fig. 8. Quasi-absolute three position test procedure. (a) Absolutedeviations of the aspheric surface. Contour line spacing is ��5. (b)Reproducibility of the three position test as a whole. (c) Interfero-metric errors extracted by subtracting the absolute measurementfrom a single measurement. Contour line spacing is ��50.

Fig. 9. Results of the consistency test when the combo-DOE (a)does not fulfill any of the design condition (b) follows only thesymmetry condition (c) is optimized by having the offset angle ofthe spherical wavefront as 0.5°.


Subtracting this average from a single measure-ment will give the rotationally varying errors of thesurface:

WNR � Wi � Wavg. (2)

To validate the results obtained by the quasi-absolute three position test, we have performed thisindependent procedure of N-position testing. For thispurpose, 36 measurements have been recorded whilethe specimen was rotated in 10° steps. By subtractinga single measurement from the average of these mul-tiple measurements, the rotationally invariant errorsof the specimen and the systematic errors of the in-terferometer get cancelled out leaving behind the ro-tationally varying errors of the surface. Figure 10compares the rotationally varying errors obtained bythe N-position rotationally averaging test with that ofthe three position test. There is good agreement. Thedifference is shown in Fig. 10(c). The difference has0.029 waves P–V and 0.0038 waves rms, which is ingood agreement keeping in mind that the N-positiontest procedure also has several additional errorsources, as discussed in [12]. The trifoil shape in thedifference [see Fig. 10(c)] is an indication that theinterferometric error, which is also trifoil in our case,has not been eliminated identically in the two meth-ods. Needless to say, neither method is perfect, sinceboth have limitations and sources of errors.

6. Conclusions

In our previous publication [13], we presented theexperimental demonstration of the proposed quasi-absolute three position test for aspherics by using acombo-DOE as a null element. In this paper, the ac-curacy of the procedure is enhanced by optimizing thedesign parameters of the combo-DOE, which is dem-onstrated by the consistency test. It is observed thatthe tilt of the DOE and its orientation, as well as theoffset direction of the spherical wavefront play a crit-ical role in the presence of substrate deviations andsystematic errors of the interferometer.

Further, the rotationally varying deviations ex-tracted using the three position absolute procedureare compared with that of an N-position rotationally

averaging procedure and are found to be in goodagreement. The capability of extracting the completedeviations of the specimen signifies that our proposedtechnique is a promising tool for the interferometriccalibration of aspheric metrology. Another major ad-vantage of our procedure is that we need not knowthe DOE fabrication errors as long as they have aglobal character and are identical in both wavefronts.In principle, this assumption determines the degreeof accuracy of the test. At present, we believe theachieved accuracy is limited by the stability of theinterferometer and the substrate quality. Simulationstudies show that the consistency becomes better asinterferometer errors and substrate deviations godown. Therefore, it is expected that the accuracy willenhance further, once high quality optical compo-nents and DOE substrates are used.

Nevertheless, this procedure is still limited by itsquasi-absolute nature, as the calibrating sphericalwavefront, which has an additional angular offset,deviates from the aspheric one. It can be further en-hanced by having both wavefronts on-axis, thus elim-inating the quasi-contribution of the offset of thespherical wavefront. But in that case, one has to de-vise a scheme to block one wavefront while using theother. Future study will concentrate on a means toincorporate such a wavefront selective mask associ-ated with the combo-DOE. But even after that, thespherical wavefront will remain different from theaspheric one. It is left for further research to examineto what extent this difference limits the overall per-formance. For that purpose one needs to know thelocalized fabrication errors of the DOE in both thewavefronts. Furthermore, it has been observed thatthe misalignment in the z-direction is very critical. Itcan be further improved by having better mechanicalstability and alignment of the specimen, or by im-proving the misalignment elimination algorithm.

Gufran S. Khan thanks the German Academic Ex-change Service (DAAD) and the International MaxPlanck Research School (IMPRS) for providing finan-cial assistance through a research fellowship duringhis stay at the Institute of Optics, Information, andPhotonics (Max-Planck Research Group) Friedrich-Alexander-University of Erlangen-Nuremberg, Ger-many.

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Fig. 10. Comparison of the rotationally varying deviations ofthe aspheric surface extracted by using (a) quasi-absolute threeposition test procedure, (b) N-position averaging procedure, (c)difference of rotationally varying deviations obtained from bothprocedures (a) and (b).


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