analysis of a scanning pentaprism system for measurements ... · analysis of a scanning pentaprism...

Analysis of a scanning pentaprism system formeasurements of large flat mirrors

Julius Yellowhair1,* and James H. Burge2

1Currently with Sandia National Laboratories, 1515 Eubank SE, Albuquerque, New Mexico 87123, USA2College of Optical Sciences, University of Arizona, 1630 East University Boulevard, Tucson, Arizona 85721, USA

*Corresponding author: [email protected]

Received 22 February 2007; revised 26 July 2007; accepted 29 July 2007;posted 8 August 2007 (Doc. ID 80354); published 3 December 2007

The optical surface of a large optical flat can be measured using an autocollimator and scanningpentaprism system. The autocollimator measures the slope difference between a point on the mirror anda reference point. Such a system was built and previously operated at the University of Arizona. Wediscuss refinements that were made to the hardware, the alignment procedure, and the error analysis.The improved system was demonstrated with a 1.6 m flat mirror, which was measured to be flat to 12 nmrms. The uncertainty in the measurement is only 9 nm rms. © 2007 Optical Society of America

OCIS codes: 120.3940, 220.4840.

1. Introduction

In optical surface metrology, systems with penta-prism(s) are used where conventional interferometrictesting would otherwise be difficult or limited [1–4].Two commonly used methods for testing flat mirrorsinterferometrically are the Fizeau [5] and Ritchey–Common [6] tests. The Fizeau test uses mostly com-mercial Fizeau interferometers, which are generallylimited to 10 cm apertures. Thus commercial Fizeauinterferometers are insufficient for full surface test-ing of large flat mirrors without performing sub-aperture testing and stitching. The accuracy andefficiency of the measurements diminish as the sub-aperture becomes smaller compared to the size of thetest surface. The Ritchey–Common test can measurefull flat surfaces, but the test requires a referencespherical mirror larger in size than the flat mirrorunder test. On a large scale this test is difficult andcumbersome to set up. The scanning pentaprism testovercomes these limitations and can measure loworder aberrations on any large flat mirror.

In this paper we describe and analyze the scanningpentaprism test, a highly accurate optical slope test,to measure flatness in very large mirrors. The main

components of the test system are two pentaprismsand a high resolution electronic autocollimator. Thetest system builds on a previous scanning system,which was designed to deflect the autocollimatorbeam up to a flat mirror suspended above the test site[1,2]. In contrast, our refined system deflects thebeam down to where the large flat mirror is sup-ported on a polishing table and whiffletree-type sup-port. We use a second autocollimator as part of anactive feedback control to maintain angular align-ment of one prism during scanning operation. Themain improvements to the previous system are inthe mechanical hardware to provide more stabilityand to maintain alignments during scanning, analignment procedure to increase the accuracy of thesystem, and error analysis to help establish the sys-tem performance. The system performance is onlylimited in accuracy by second order error influencesdue to coupling between misalignments and motionsin the autocollimator, prisms, and the test surface.Beam errors also couple with lateral motion of thescanning prism to cause additional second order slopeerrors. The refinements to the system help minimizethese errors. We provide an error analysis to quantifythe measurement errors. Using the results from theerror analysis, we show through Monte Carlo simu-lations that over a 2 m flat the accuracy for measur-

0003-6935/07/358466-09$15.00/0© 2007 Optical Society of America

8466 APPLIED OPTICS � Vol. 46, No. 35 � 10 December 2007

ing power is about 9 nm rms with the improvedsystem.

Pentaprisms deflect light beams at a constant an-gle (nominally by 90°) regardless of its orientation inthe line-of-sight direction. Various optical test sys-tems and devices involving pentaprisms make use ofthis unique property. Systems involving pentaprismshave been developed to measure optical surface er-rors by measuring surface slopes [1–4]. Simply inte-grating the slope data gives surface height profiles.Through multiple measurements of the optical sur-face in different directions, a full synthetic surfacemap can be obtained. Other systems with pentap-risms are used to aid in optical alignments or to makelow order error corrections to deformable optical sur-faces. Sensitive optical alignments and error correc-tions are made possible by the capability of multiplepentaprisms to project collimated and nominally par-allel reference beams onto optical surfaces. Due to theusefulness of pentaprism systems, efforts have beenmade to understand the error influences and buildsystems that are less sensitive to alignment and mo-tion errors [7,8].

2. System Design and Development

A. Test Concept

In this subsection we discuss the test concept anddesign of the scanning pentaprism system. A sche-matic diagram of the test setup is shown in Fig. 1.Two pentaprisms were coaligned to a high resolutionelectronic autocollimator. Both prisms deflected thecollimated beam from the autocollimator nominallyby 90° to the mirror surface. The first prism had acoupling wedge that transmitted about 50% of thebeam to the second prism. The beams reflected offthe test surface and returned to the autocollimator,where small angle deviations revealed slope errors inthe mirror surface.

The test system can be used in either a scanning ora nonscanning mode. In the scanning mode, the first(reference) prism remained fixed, while the second(scanning) prism was translated over the mirror sur-face sampling the mirror every few centimeters.Since the autocollimator can only measure one returnsignal at a time, we used electronically controlledshutters to alternately select the reference path and

the scanning path. By taking a difference betweenthe two prisms, the measurements became insensi-tive to the motions of the autocollimator and testmirror in the measurement direction. An integrationof the slope data provided surface height profilesalong the scanning direction. We used an alternativemethod of fitting low order slope functions derivedfrom the Zernike polynomials to the slope datathrough a least squares calculation [9]. We deter-mined the Zernike coefficients after the fit and recon-structed a surface topology map using slope data fromseveral scans at different orientations over the mir-ror. In the nonscanning or staring mode, both prismsremained fixed, and the flat mirror under test wasrotated while the slope data were acquired continu-ously. Typically, we positioned the reference prism atthe edge of the mirror and the scanning prism at thecenter of the mirror. The nonscanning mode mea-sured � dependent aberrations such as astigmatism(2� dependence) and trefoil (3� dependence). In Sec-tion 3, we provide results from both modes of opera-tion.

The pentaprism system allowed slope determina-tion only in the scan or measurement direction. Wecall this direction the line-of-sight pitch or in-scandirection. The degrees of freedom (pitch, yaw, androll) for the main test components are defined in Fig.2. Coupling between misalignments and motions ofthe components in these degrees of freedom causedsecond order errors to the beam line of sight. Sourcesof line-of-sight errors up to second order are listed inTable 1. The line-of-sight pitch (scan direction) varieslinearly with autocollimator and test surface pitchangles, and quadratically with other angular param-eters. The first order error from the autocollimatorand test surface pitch motions were common toboth prisms, so these errors were eliminated by per-forming difference measurements between the twoprisms. Careful system alignment and active controlof the scanning prism minimized the second ordererrors.

Fig. 1. Schematic of the scanning pentaprism test system.Fig. 2. Coordinate system and definition of the degrees of freedomfor the autocollimator, scanning pentaprism, and test surface.

10 December 2007 � Vol. 46, No. 35 � APPLIED OPTICS 8467

1. Pentaprism Pitch Motion SensitivitySmall pitch motion of the pentaprism was expected.This motion, however, had no effect on the beamdeviation in the line-of-sight pitch direction. This isthe beauty of the pentaprism: the input beam is de-viated at a constant (nominally) 90° angle, indepen-dent of small pitch errors. We took full advantage ofthis unique property.

2. Pentaprism Yaw and Roll Motion SensitivitiesThe in-scan (vertical) and cross-scan (horizontal) an-gles, as seen by the autocollimator, were coupled forpentaprism yaw and roll motions. The dependence islinear for prism yaw motion and quadratic for rollmotion [7,8]. When the prism was aligned, thespot motion was the same for small prism roll andyaw motions. The in-scan measurement was thenperpendicular to the spot motion caused by motionsin prism roll and yaw. The noise over the amountof motion in roll and yaw gave the accuracy of thealignment. For example, if the prism yaw and rollalignments were maintained to 50 �rad, the in-scanmeasurements varied by less than 10 nrad (discussedin Section 4).

Information from the pentaprism yaw motion canbe used to align both the autocollimator roll and theprism roll. Geckeler [8] derives the slope of the auto-collimator angle readings (vertical angle versus hor-izontal angle) from the pentaprism yaw scan as

Mscanyaw � �TS � �AC, (1)

which reveals information on the difference of the rollangles of the autocollimator and the test surface. Theautocollimator roll was aligned relative to the testsurface when the slope, M, became zero. In addition,Geckeler [8] derives the minimum of the parabolafrom the pentaprism roll scan (vertical angle versushorizontal angle) as

Hscanroll � �PP � �AC � 0.5Mscan

yaw , (2)

which depends on the pentaprism and autocollimatoryaw and the result of the yaw scan. Both Eqs. (1) and

(2) have dependence on pentaprism yaw; thus prismyaw can be used to align autocollimator roll [fromEq. (1)] and prism roll [from Eq. (2)].

3. Beam Collimation Errors and Alignment to thePrism MotionThe errors in the autocollimator collimated beam cou-pled with lateral pentaprism motion to cause addi-tional second order slope errors. We estimated thatthis effect caused 280 nrad slope change per 1 mm oflateral prism motion. This linear motion of the scan-ning prism was aligned to less than 0.5 mm along thebeam, so the above effect was limited to 140 nradsurface slope variation (discussed in Section 4).

B. System Hardware

In this subsection, we describe the components of thesystem. This includes the mechanical and opticalhardware and measurement units.

1. Optical RailsA major improvement in stability over the previoussystem was in the mechanical stage used for the scan-ning operation. The new rail system consisted of two2.5 m heavy duty steel rails spaced 20 cm apart andbridged by the pentaprism and autocollimator plat-forms as shown in Fig. 3. The straightness of the railsystem was measured with a laser tracker to betterthan 0.05 mm�m, a significant improvement over theprevious rail system. Thus the advantage of the newrails was that they limited the lateral and angularmotion of the scanning prism, resulting in less reli-ance on the active control system. In addition, thenew rails were less susceptible to vibrations andwarping; vibrations and warping were notable draw-backs of the previous rails. The new rails and theactive control system maintained the alignment ofthe pentaprisms to within 50 �rad in roll and yaw.The new rails rested on a three point kinematicbase. The kinematic base allowed the test system tobe removed and stowed when not in use.

Fig. 3. Solid model of the scanning pentaprism rail system show-ing the mounting platforms and the three-point kinematic base.

Table 1. Contributions to Line-of-Sight Error

Contributions to Line-of-SightPitch (In-Scan Direction)

Contributions to Line-of-SightRoll (Cross-Scan Direction)

�AC �AC

�TS �PP

�PP2 �PP

�AC � �PP �TS

�AC � �PP �AC � �PP

�AC � �TS �AC � �TS

�TS � �PP �AC � �AC

�TS � �PP �AC � �PP

�: pitch AC: autocollimator�: yaw PP: pentaprism�: roll TS: test surface


2. AutocollimatorsWe used two electronic autocollimators. The first wasa high resolution autocollimator (Elcomat 2000 madeby Moller–Wedel) used for the surface slope measure-ments. The Elcomat 2000 provided a 40 mm colli-mated beam, which was projected onto the mirrorsurface using the two pentaprisms (as described pre-viously). The reflected beam from the scanning prismwas detected by the Elcomat, and angle deviationscaused by slope variations in the mirror surface weremeasured. The Elcomat also functioned as part of theactive control system that monitored cross-scan beammotion due to prism roll or yaw motion.

The second autocollimator had less angular reso-lution (model 3700 made by United Detector Tech-nologies). The UDT 3700 autocollimator was alignedto a return flat mirror that was attached to the scan-ning pentaprism assembly (see Fig. 1). The UDT au-tocollimator was the main component of the activecontrol used to monitor the yaw motions of the scan-ning pentaprism. The UDT autocollimator was insen-sitive to roll motion of the pentaprism, but theElcomat sensed the line-of-sight roll, which wascaused by a combination of the prism roll and yawmotions. The prism roll was determined as the dif-ference between the line-of-sight roll and the prismyaw motion. The UDT autocollimator decoupled thetwo prism motions for the Elcomat. The active controlwas then programmed to maintain both the roll andyaw alignments of the scanning prism to 50 �rad.

3. Pentaprisms and HoldersThe first pentaprism had a coupling wedge that al-lowed about 50% transmission of the beam to thesecond pentaprism as shown schematically in Fig. 1.We estimated measuring through the first prism withthe wedge caused about 50 nrad rms measurementerrors in the scanning mode (see Section 4).

The prism holders provided secure mounts and re-mote adjustment of prism yaw and roll motionsthrough Picomotor (New Focus) linear actuators (seeFig. 4). The Picomotor actuators were controlledthrough the active control system; any misalign-ments of the prisms during scanning were fixed bysending commands to the Picomotors to drive theprisms back into alignment. The prism assemblieswere mounted on rail cars, which moved on the twoparallel steel rails.

4. ShuttersWe used two electronically controlled shutters, whichwere mounted between the pentaprisms and the testmirror as shown in Fig. 4 or schematically in Fig. 1.The shutters allowed alternating measurements be-tween the prisms. Shutter A was open and shutter Bclosed when measuring through the reference prismoccurred. The shutter states were reversed whenmeasuring through the scanning prism occurred. Afew seconds delay between shutter operation andmeasurement allowed vibrations caused by the shut-ters to damp out.

C. System Integration

The system can be thought of as an assembly of sub-systems, which included the autocollimator system,the reference and scanning pentaprism assemblies,and the electronics including the workstation andsoftware for active control.

The rail system was the foundation upon whichthese subsystems were integrated (see Fig. 5). Theautocollimator system and prism assemblies weremounted on carriage platforms that were attached torail cars. The platforms then moved on the railtracks; however, only the scanning prism assemblywas allowed to translate over the rails. The autocol-limator system and the reference prism assemblywere locked into position at one end of the rails. Theelectronics and cabling for the active control werehoused in a breakout box and mounted underneaththe autocollimator system platform (not visible inFig. 5). The fully integrated system is shown in Fig. 5.The entire system weighed about 200 kg and could belifted with a hoist and positioned kinematically overthe large mirror.

Fig. 4. Pentaprism assemblies integrated into the system. Elec-tronically controlled shutters are located at the exit face of eachprism. The autocollimator system (not visible) is mounted to theleft.

Fig. 5. Fully integrated and operational scanning pentaprismtest system.


D. System Alignment

Accurate system alignment is essential to slope mea-surement accuracy [7,8]. The system alignment wasperformed in several steps described below, startingwith the coarse and then the fine alignment.

(1) The large flat mirror rested on a whiffletree-type support and an air bearing polishing table,which was leveled to gravity. The pentaprism railswere also leveled to gravity; thus tilt between the testsurface and the rails was minimized from the begin-ning.

(2) A standard He–Ne laser was mounted in placeof the measuring autocollimator. The laser wasaligned to the rails by placing a cross-hair target onthe scanning pentaprism assembly, then by pointingthe laser at the cross-hair while sliding the prismassembly back and forth over the rails. After the laserwas aligned to the rails to less than 0.25 mm�m, thescanning prism was parked at the furthest point onthe rails opposite the reference prism.

(3) The cross-hair target was removed, and thelaser was reflected off the prism front faces. Bothprisms were adjusted in yaw until the laser returnedthrough a 2 mm pinhole placed in the laser path (seeFig. 6). This initial alignment of the prisms in yawwas better than 120 �rad with respect to the mea-surement direction.

(4) The roll motion of the reference prism wasaligned by opening the shutter, reflecting the laserbeam off the test mirror, and adjusting the roll of theprism until the reflected laser spot returned throughthe pinhole. The procedure was repeated for the scan-ning prism. At this point the initial alignment ofprism roll was also better than 120 �rad with respectto the measurement direction.

(5) Next, a return flat mirror (5 cm diameter) wasplaced in the path of the laser beam just before thescanning prism. The return mirror was secured andadjusted until the reflected laser beam returnedthrough the pinhole. The laser and the pinhole wereremoved, and the measuring autocollimator was putin place. The autocollimator was aligned to the returnmirror in both the horizontal and vertical axes. Atthis point the autocollimator was aligned to the railand the prisms to better than 120 �rad.

(6) The fine alignment was performed iterativelyusing the readout of the autocollimator. While thetest surface remained fixed, typically adjustmentswere made to the autocollimator and the prisms. Ta-ble 1 lists the angular motions that affected the beam

line-of-sight pointing (pentaprism yaw and roll andautocollimator roll). These are the motions that re-quired adjustments to achieve optimal system align-ment. First, the scanning prism was scanned in yawby about �200 �rad, and the autocollimator horizon-tal and vertical angle readings were observed. Thebehavior of this motion is linear on the angle read-ings. A finite slope in the angle readings indicated amisalignment between the autocollimator and testsurface in roll. With the test surface fixed, adjust-ment was made to the autocollimator to minimizethis misalignment. After adjusting the roll of the au-tocollimator, the prism was rescanned in yaw. Roll ofthe autocollimator was aligned when the angle read-ings were no longer coupled for the prism yaw scan.The autocollimator roll was now aligned to betterthan 50 �rad.

(7) Next, the reference prism was scanned in rollby about �200 �rad, and the autocollimator horizon-tal and vertical angle readings were observed. Thebehavior of this motion is quadratic on the anglereadings. Any coupled readings over small prism rollmotion indicated a misalignment of the prism in yaw.Yaw of the prism was adjusted, and the prism wasthen rescanned in roll. The prism yaw was alignedwhen no noticeable coupling remained between theangle readings during the roll scans. The roll motionwas now constrained to the vicinity of the quadraticminimum. This procedure was repeated for the scan-ning prism. Yaw and roll for both prisms were nowaligned to better than 50 �rad. The system was nowaligned and ready to use.

3. System Performance

The scanning pentaprism system can be used in twomodes: scanning or staring (nonscanning). Diagonalsurface scans were performed in the scanning mode(see Fig. 7). In the staring mode, circumferentialscans were performed where both prisms remainedfixed and the test flat rotated continuously while datawere acquired (see Fig. 11).

A. Diagonal Line Scans in Scanning Mode

In the scanning mode, the reference pentaprism re-mained fixed and the scanning prism was translatedacross the diameter of the mirror acquiring a mea-surement every few centimeters. Typically, we sam-pled the 1.6 m flat with 25 to 50 points across withhigher measurement density near the edges to mon-itor the edges during fabrication. A diagram showing

Fig. 6. Initial alignment of the pentaprisms in yaw. The laser isreflected off the front faces of the prisms.

Fig. 7. Pentaprism scan arrangement. This example shows themirror being rotated in 120° steps for each scan.


an example of three scan paths is shown in Fig. 7.This was achieved by rotating the test flat to two otherpositions after the first scan. The three line scans mea-sured the low order Zernike aberrations [9].

A single diagonal line scan measured only rotation-ally symmetric aberrations. Figure 8 shows the resultof a single diagonal line scan on the finished 1.6 mflat mirror. Forward and backward scans were per-formed for the same diagonal line, and the data wereaveraged. Only the slope functions from the symmet-ric Zernike polynomials were fit to the slope data. Thelinear component of the polynomial fit to the slopedata then revealed power in the mirror surface. Thismeasurement yielded 12 nm rms in power.

A radially normalized plot showing the scanningpentaprism slope data in comparison to the Fizeauinterferometer data for the finished 1.6 m flat isshown in Fig. 9. The figure shows that the scanningpentaprism data and interferometer data, first differ-entiated to obtain the surface slope, have no signifi-cant differences, except at the very edge where high

slopes were observed. The rms difference is about160 nrad after removing the edge point, which iswithin the accuracy (see Section 4) of the scanningpentaprism test in the scanning mode. The largeFizeau interferometer did not measure power, so theinterferometer data were adjusted for power beforethe comparison. The Fizeau interferometer used a1 m custom reference flat and subaperture testing tomeasure the 1.6 m flat mirror (Figs. 10 and 11). Thesurface map, shown in Fig. 11, is from the finishedmirror after combining the subaperture measure-ments. The surface map, measured to 12 nm rms,shows mostly zonal errors.

B. Circumferential Scans: Staring Mode

The circumferential scans provided information on �dependent aberrations such as astigmatism (2�) andtrefoil (3�). Tilt between the autocollimator and thetest surface had a 1 dependence. The main aberra-tion measured in this mode was astigmatism.

In the circumferential scans, both pentaprismswere fixed and the flat mirror was continuously ro-tated (see Fig. 11). The slope data were continuouslyacquired for several full rotations of the flat mirror.The data were then averaged.

For the scans shown in Fig. 12(a), the referencepentaprism (A) was parked near the edge of the large

Fig. 8. Surface slope measurements with the scanning pentap-rism system and a low order polynomial fit. A linear component ofthe polynomial fit on the slope data gives information on power inthe surface (12 nm rms).

Fig. 9. Comparison of the scanning pentaprism slope data andthe interferometer slope data.

Fig. 10. Interferometer measurement on the 1.6 m flat mirror.

Fig. 11. Circumferential scans, where both prisms are fixed andthe mirror is continuously rotated, measure astigmatism, andother � dependent aberrations in the mirror surface.


flat mirror, and the scanning pentaprism (B) wasparked at the center of the mirror. These measure-ments were performed while the mirror was in pro-duction. The difference between the scans [Fig. 12(b)]reveals the amount of contributions of low order �dependent aberrations (dominated by tilt).

A least squares fit of the difference data was per-formed using a fitting function of the form

f � a0 � a1 sin�� b1 cos�� a2 sin�2� � b2 cos�2�� a3 sin�3� � b3 cos�3�. (3)

Table 2 lists the � dependent aberrations and theircoefficient values after the fit. The last column shows

the equivalent surface error for each � dependentaberration.

4. Error Analysis

A. Errors to Line-of-Sight Beam Motion

There were several error sources that contributed tothe line-of-sight beam pitch (or in-scan) motion. Be-low, we describe the error sources and quantify theireffect on the beam line of sight.

1. Errors from Angular Motions of thePentaprisms and AutocollimatorTaking the second order terms from Table 1, an ex-pression for the change in the in-scan line of sight dueto misalignments and angular motions of the auto-collimator, pentaprism, and test surface can be de-rived as

�LOS � 2�PP � �PP � �AC��PP � �PP � �TS�� AC��PP � �PP � �PP� � �TS��PP � �PP�� TS��PP � �PP�, (4)

where each � term indicates the variation in autocol-limator, prism, and test surface motions for that an-gle. Equation (4) shows that the motions of the threecomponents couple with misalignments to cause sec-ond order slope errors in the in-scan line of sight.Table 3 shows a summary of the error terms thatcoupled into the measurement in the in-scan direc-tion, and Table 4 shows the amount of contributionfrom each error term in Eq. (4).

2. Mapping ErrorThe slope errors in the polished test surface wereexpected to be less than 2 nrad�mm rms. The posi-tion of the scanning prism (or the beam) on the testsurface was known to 2 mm. The error due to theprism position thus contributed less than 4 nrad rmsto the total error.

3. Thermal ErrorsThe pentaprisms were made from BK7 glass, whichis fairly sensitive to temperature gradients �7.1� 10�6�°C�. A linear temperature gradient in theprisms changed the index gradient and prism geom-etry. The thermal analysis showed that a tempera-ture gradient of 0.01 °C�m in the prisms would causean additional beam deflection in the line-of-sight di-rection of 17 nrad. The time scale for the two modesof operation of the test system was relatively shortcompared to the prisms’ thermal time constant. Inthe scanning mode we estimated a single scan took

Fig. 12. (a) Circumferential scans at the center and edge of thelarge flat mirror, and (b) difference in the center and edge scansand curve fit. The error bars in the scans indicate stability of therotary air bearing table.

Table 2. Aberrations Measured with Circumferential Scans

Aberration Term Fit Function Term(s) Fit Coefficients (�rad) Equivalent Low-Order Surface Error

Piston a0 0.0294 —Tilt a1 sin�� b1 cos�� 0.6328, �1.2888 359 nm rmsAstigmatism a2 sin�2� � b2 cos�2� 0.0366, �0.0043 15 nm rmsTrefoil a3 sin�3� � b3 cos�3� 0.0993, �0.0283 18 nm rms


one hour. For a full measurement, three scans wereperformed that required about three hours to com-plete. In the staring mode, a full measurement typi-cally lasted 10 to 15 min. Only the variation intemperature gradients during the three hour scan inscanning mode and 15 min scan in staring mode con-tributed to the line-of-sight errors. The top level errorbudget allowed for change in the thermal gradient of�0.04 °C�m for the scanning case and �0.02 °C�mfor the staring case. We then expected a line-of-sighterror of up to 34 nrad rms and 17 nrad rms for scan-ning and staring cases, respectively.

4. Errors from Coupling Lateral Motion of thePentaprismsPhase or amplitude variations in the collimated beamdid not affect the system performance to first order,because these effects were common to both pentap-risms. These variations and other beam errors, how-ever, coupled with lateral motion of the scanningprism (relative to the collimated beam) to cause sec-ond order errors. Coupling of phase errors, diffractioneffects, beam nonuniformity, and prism errors withlateral motion of the prisms for the scanning systemwas analyzed by Mallik et al. [1,2]. This analysis for

our system is not repeated here; our system had bet-ter control of the lateral motion of the prisms. Thelateral motion of the scanning prism in our systemwas aligned and maintained to 0.5 mm. The com-bined effect of these errors was then estimated to beless than 80 nrad rms.

5. Combined Random ErrorsTable 5 shows the random errors for the pentaprismstaring and scanning modes separately. There wasan additional error in acquiring the measurementsfrom the scanning pentaprism through the fixed ref-erence pentaprism. This effect is included directly inTable 5.

6. Analysis of Errors Due to Beam DivergenceThere is an additional error to consider that coupledinto lateral motion of the pentaprism. For typicaluses of electronic autocollimators, a collimated out-put beam is assumed. If the beam is slightly diverg-ing, however, the angle readings are shifted by someamount proportional to the divergence angle. In thescanning pentaprism system, this effect of the beamdivergence can couple into lateral motion of the pen-taprism causing a second order error. To quantifythis effect, we devised a simple test. A 13 mm circularaperture was placed at the output port of the mea-suring autocollimator. The aperture was shifted upand down perpendicular to the line of sight by about18 mm from the top edge of the beam to the bottom

Fig. 13. Setup to test for the effect of beam divergence on prismmotion.

Table 3. Budget for Alignment Errors for thePentaprism�Autocollimator System

Parameter Description Tolerance

�PP Initial misalignment of theprism roll

0.13 mrad

��PP Variation in prism roll 0.05 mrad rms�AC Misalignment of the

autocollimator roll relative todirection of motion

0.10 mrad

��AC Variation in autocollimator roll 0.05 mrad rms�PP Initial misalignment of the

prism yaw0.13 mrad

��PP Variation in prism yaw 0.05 mrad rms�TS Misalignment of the test

surface roll relative to thedirection of motion

0.10 mrad

��TS Variation in test surface roll 0.01 mrad rms

Table 4. Misalignment and Perturbation Influences on the Line of Sight

Contribution (Termsfrom Eq. (4))

Amount of Line-of-SightDeviation (nrad rms)

2�PP � ��PP 13��AC � �PP 7��AC � �PP 7��AC � �TS 5

�AC � ��PP 5�AC � ��PP 5�AC � ��TS 1

��TS � �PP 1��TS � �PP 1

�TS � ��PP 5�TS � ��PP 5

Root sum square 20

Table 5. Independent Measurement Errors Assumed to beUncorrelated

Error DescriptionStaring Mode

(nrad rms)Scanning Mode

(nrad rms)

Autocollimator measurementuncertainty (rangedependent)a

140 160

Prism and beam anglevariation

20 20

Mapping error 4 4Thermal effects 17 34Effect of reference prism error 25 50Coupling of lateral motion — 80

Root sum square 145 190

aManufacture’s specification.


edge as shown in Fig. 13, and the change in thevertical angle reading was recorded for the two aper-ture positions.

For the case when the scanning prism was close tothe reference prism [1], the change in the autocolli-mator vertical angle reading was about 10 �rad. Thescanning prism was then moved to the far edge of theflat mirror [2], and performed the aperture shifts. Forthis case, the change in the vertical angle readingwas about 15 �rad. From these measurements weestimated that this effect caused about 280 nradslope change per 1 mm of lateral motion. The prismlinear motion was aligned to less than 0.5 mm alongthe beam, so the effect was limited to 140 nrad sur-face slope variation. This systematic slope variationof �70 nrad corresponds to surface power of less than8 nm rms.

B. Monte Carlo Simulation of the System Performance

A Monte Carlo simulation on the three diagonal sur-face scans separated by 120° was performed to deter-mine the uncertainty distributions of the low orderZernike aberrations using a noise value of 0.3 �radfor a single differential measurement and assuming42 measurement points per scan. The measurementuncertainty of each of the low order aberrations fromthe simulation is listed in Table 6. The simulationresult shows that a 2 m flat mirror can be measuredto 16 nm rms of low order aberrations with the scan-ning pentaprism system after careful system align-ment. The measurement of power has 8 nm rms dueto the systematic effect listed above, limiting the sur-face power measurement to 9 nm rms.

5. Conclusion and Future Work

We provided an analysis of a refined scanning pen-taprism system that measured flatness in large flatmirrors to an accuracy of 9 nm rms; the system hasthe option to measure other low order aberrationsand only � dependent aberrations. The system align-ment procedure was provided in detail. The measure-

ment accuracy was limited only by second orderinfluences from misalignments and autocollimator,pentaprism, and test surface motions, which wereminimized through careful system alignment and ac-tive pentaprism motion control. A Monte Carlo sim-ulation of the system performance was performedbased on the measurement uncertainty estimatedfrom the error analysis. The simulation resultshowed the uncertainty in the measured low orderZernike aberrations, and measurements to 16 nmrms of low order aberrations are achievable for 2 mflat mirrors. The high accuracy of the test systemmakes it ideal for absolute testing of arbitrarily largeflat mirrors. This test system can be used as a finaltest on the surface figure or to guide polishing duringfabrication. The kinematic base allows the test sys-tem to be moved to the polishing table without mov-ing the test flat to a testing fixture and stowed duringpolishing or when not in use.

Absolute calibration of the system was not per-formed. This can be accomplished using a liquid ref-erence surface over very long test paths (e.g., 4 m),where the liquid surface is only limited by the curva-ture of the earth. The result will be excellent charac-terization of the system performance. This task wasleft open for future work.

References1. P. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-m flat

using a pentaprism scanning system,” Opt. Eng. 46, 023602(2007).

2. S. Qian, P. Takacs, G. Sostero, and D. Cocco, “Portable longtrace profiler: concept and solution,” Rev. Sci. Instrum. 72,3198–3204 (2001).

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4. R. D. Geckeler, “ESAD shearing deflectometry: potential forsynchrotron beamline metrology,” Proc. SPIE 6317, 63171H(2006).

5. M. V. Mantravadi, “Newton, Fizeau, and Haidinger inter-ferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley,1992), pp. 1–49.

6. J. Ojeda-Castaneda, “Foucault, wire, and phase modulationtests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992),pp. 265–320.

7. R. D. Geckeler, “Error minimization in high-accuracy scanningdeflectometry,” Proc. SPIE 6293, 62930O (2006).

8. R. D. Geckeler, “Optimal use of pentaprisms in highly accuratedeflectometric scanning,” Meas. Sci. Technol. 18, 115–125(2006).

9. J. Yellowhair and J. H. Burge, “Measurement of optical flat-ness using electronic levels,” Opt. Eng. (to be published).

10. J. Yellowhair, College of Optical Sciences, University ofArizona, 1630 East University Boulevard, Tucson, AZ 85721,R. Sprowl, P. Su, R. Stone, and J. H. Burge are preparing amanuscript to be called “Development of a 1 meter vibration-insensitive Fizeau interferometer.”

Table 6. Measurement Accuracy for the Low-Order ZernikeAberrations with the Scanning Pentaprism System

Zernike AberrationMeasurement Accuracy

(nm rms)

Power 9Cos astigmatism 8Sin astigmatism 8Cos coma 4Sin coma 4Spherical 2Secondary spherical 2

Root sum square 16


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