The Determination of the Absolute Contours of Optical Flats

William Primak

Emerson's procedure (the one in use at the National Bureau of Standards) for determining the absolutecontours of optical flats was refined to increase the precision of the method and the speed of taking readings.The fringes were scanned over a photoelectric detector and the intensity profile presented on an oscillo-scope. Setting accuracy of a fraction of a hundredth of a fringe was achieved. The data were gatheredon a data logger. A complete set of data was obtained in about 5 min. The precision of determining aset of differences of two plates was -d fringe and the precision achieved in determining the absoluteplate contours is estimated as 0.005 fringe. The problem of extending the method to 0.002 fringe isdiscussed.

IntroductionThe problem of determining the true absolute con-

tours of optical flats is discussed by Emerson.' Themethod devised by him is currently the one employedat the National Bureau of Standards for the certifica-tion of optical flats.2 The absolute contours of thestandard flats were determined by the usual method ofintercomparing three sets of flats and solving the threesets of differences in order to obtain the absolute contourof each flat. Now intercomparisons are made betweenthe standard flats and unknown flats whose contour iscalculated from the differences and the known contourof the standard flat. The contour differences aredetermined by measuring the deviation of a Fizeaufringe from straightness when the plates are transportedunder a Fizeau fringe viewer. Three quantities mustbe measured: the position of the flat, the spacing of thefringes, and the deviation of the fringe from straightness.The fringe position is read visually by setting a crosshair on the fringe. This operation is a surprisinglyprecise one when the cross hair is set on the dark fringe'and can be accomplished to -l fringe by the averageperson after relatively little experience.

Emerson showed that even for thick plates the de-flection produced by mounting the plates is severe whenmaking measurements to this precision. It is easy tomake permanent changes in contour of the plates by themounting procedure. Thus, he cites as an example,permanent changes in contour produced by their ownweight when 25-cm diam flats were balanced on three

The author is with Argonne National Laboratory, Argonne, Illi-nois 60439.

Received 26 May 1967.This work was performed under the auspices of the U.S. Atomic

Energy Commission.

0.32-cm paper disks placed at 5% of the radius from thecenter. In most scientific applications, plates are usedin different orientation from that tested at the NationalBureau of Standards and are rarely mounted with suit-able care. They therefore consider that their certifica-tion represents superfluous precision for most scientificapplications. There is little question but that it isdesirable to make tests in the mounting that will beused.* In optical shops, flatwork is usually tested bysetting the unknown on a test flat (or a test flat on theunknown). A skilled workman may miss an error ofT-0 fringe by this technique; although if he is told wherethere is an error, he may be able to detect as little as

a fringe. Thus, the certification given by the NationalBureau of Standards meets even the most precise needsof the optical industry.

In recent years, in connection with the developmentof lasers, there has arisen a demand for pairs of plateswith fractional 0.01 fringe differences. The procedure,at first, was to take larger plates and select areas thatwould meet the requirements. Several manufacturershave offered plates whose flatness has been stated asbeing within 0.01 fringe. However, it is obvious fromthe above that such a specification is meaningless with-out careful consideration of mounting and thermalenvironment. Fractional 0.01-fringe sensitivity hasbeen attained in scanning of Fabry-Perot etalons, butthese methods tend to possess undesirable features, inprinciple, which would relegate them to specific ap-plications. (For two ingenious recent methods seeRef. 4.) First is the deformation of the plates byspacers and mountings which must generally be rigid

* This is summarized from a private discussion with members ofthe Engineering Metrology Section held in September 1963. It isnot known whether these are their current views.

November 1967 / Vol. 6, No. 11 / APPLIED OPTICS 1917

- 7-- ~~6

4Z

2

Ro R R2 R3 R4 R5 R6 R7 R8

81 82 83 84 85 86 87

READINGS-

Fig. 1. Representation of the fringe systems.

because spacing requirements tend to be much moresevere than in Emerson's procedure. Second, examina-tion is generally performed at different parts of theauxiliary optical system; and at best, a correction forits aberrations introduces another subtraction with acorresponding loss in precision. In contrast, in Emer-son's procedure the plates are always viewed at thesame point of the auxiliary optical system; hence, itsaberrations do not introduce errors. Third, coatingmay introduce error by nonuniformity in thickness oroptical properties or by straining the plate'; and it is anuisance, aside from the delay involved in its applica-tion, if the plates have to be worked further. Emer-son's procedure uses uncoated plates.

There are two major criticisms of Emerson's pro-cedure. First, the contour is determined along onlyone diameter so that mapping a whole plate becomes alaborious procedure. Second, Emerson's procedure isonly about as precise as the most precise optical workdone at present, whereas it would be desirable to have aprocedure several times as precise in order to permitreliable examination of the work toward the end of thefiguring of the flat. A more precise method would alsohave the advantage of decreasing the labor of obtaininga standard flat. In order to determine the absolutecontour, three determinations are required and differ-ences are taken. Thus, the only way to obtain thenecessary precision is to make the many determinationsrequired to obtain statistical accuracy.

The following paper describes our progress in increas-ing the speed and precision of Emerson's method.

PrinciplesIn order to increase the precision of setting on the

fringes, the photoelectric procedure described byTomkins and Fred' was adapted to reading the fringes.The image of the fringe pattern is swept across a slit infront of a photoelectric cell and the measured lightintensity is placed on the vertical plates of an oscillo-scope. In synchronism with the sweeping device, thereis placed on the horizontal plates of the oscilloscopealternately an increasing and a decreasing ramp. Thusthere are seen two representations of the light intensityof the fringe pattern, one plotted from left to right andthe other from right to left. When, as the plates aretransported, a fringe is centered on the slit, the twopatterns are superimposed. In effect, the slit is thefiduciary mark for the fringe position. In principle,

the precision can be controlled over many orders ofmagnitude by altering the amplification at the oscillo-scope. In practice, the mechanical stability of thesweep system and electrical noise in the photoelectricsystem and the light source limit the practical precision.

To increase the speed of the operation, the readingswere recorded by an automatic reading device. It be-came very desirable to decrease the kinds of readingsrequired. In fact, it was possible to obtain all theinformation required with one kind of reading. In thelanguage of the data processor, the observations werereduced from a three-channel operation to a one-channel operation. The plate spacers were adjusted togive a set of Fizeau fringes approximately at rightangles to the direction of translation of the plates. Inpractice, fringes spaced about 0.5 cm proved convenient.An encoder was placed on the translating screw and thecarriage positions which centered the fringes on the slitwere read upon command. It is readily seen from Fig. 1that for a true wedge the fringes should be seen at theregular intervals shown by the dotted lines. Thereadings found for the actual surface are shown by thesolid lines. We wish to find the error at the point ofthe reading. This is shown in Fig. 1 for reading R3 asE3. The reading of the ith fringe is R,. The difference6i from the reading for the true wedge is

Si = - (i/n)Rn,

where Rn is the reading of the last (nth) fringe.the error Ei in units of fringes is

Then,

Ei = (n/Rn)Si = (n/R)Ri - i.

Thus, it is seen that when all the readings are multi-plied by n/Rn (the slope of the straight line), the read-ings are converted to small differences (the errors) froma set of integers (the fringe count). This would be theprocedure for making a hand calculation. However,most people would prefer to see the errors given, notwith respect to a line drawn through the location of thefirst and last fringes, where the plate is apt to be mostirregular, but rather one drawn nearly tangent to thecentral area; this is easily done in a machine computa-tion. The above procedure may be used for anyarbitrary line passing through the data. A simpleprocedure is to fit a least squares line through thecentral part of the data (e.g., the central third). If thisline has the equation F = A + BR in the coordinatesof Fig. 1, it is readily seen that with respect to thisline, the errors are

Ei = (1/B)(R - A) - i.

The errors obtained here are each a completely in-dependent set of data rather than repeated readingsat chosen points of the plates as in Emerson's procedure.This cannot of itself be construed as an advantage ordisadvantage for either procedure because the majorproblem in working at these precisions is the stability ofthe fringe system. In the procedure described here, alinear drift in the fringe system does not affect the re-sults because the readings are taken at relatively con-

1918 APPLIED OPTICS / Vol. 6, No. 11 / November 1967

stant intervals. However, a change in the fringe spac-ing does introduce an error.

Since the abscissae of the data are not fixed, to aver-age the data or to take differences, interpolation isnecessary. Since the data are completely arbitraryand irregular, for they correspond to the arbitraryirregularities in the plate, they cannot be interpolatedby any simple function. For hand computation, thedifferences are plotted and interpolated graphically,usually to regular intervals on the graphs. A digitalprocedure for this operation has been developed and isdescribed below.

Equipment

Light SourceA stable light source is required. The lamp may be

dc or it may be high frequency, greater than 10,000times the sweep frequency, to avoid a pattern on theoscilloscope. If the light is too monochromatic, un-wanted fringe systems will be seen; if insufficientlymonochromatic, fringe contrast is lost. We use anH100-A4 mercury arc lamp which has been modified bythe glassblower who attached two tubulations, one nearthe base the other near the top. A stream of air at100-300 cm3 /min is blown through the bulb (and ex-hausted to a hood because of the danger of mercury lossshould the lamp crack). The arc is operated at 0.5 Afrom a low ripple dc power supply set at 175 V with aseries resistance consisting of two 150-W, 110-Vtungsten incandescent lamps in parallel to act asstabilizing resistors. The assembly is shielded, andshielded leads are used, to decrease noise pickup which isseen on the oscilloscope. An interference filter assem-bly is used to isolate Hg 546 m,4.

Sweep System and DetectionThe sweep system is that described by Tomkins and

Fred.6 It consists of an octagonal prism of which alter-nate faces are blackened. Thus, vision is obtainedthrough two directions at right angles. The prism isrotated by an 1800 rpm synchronous motor. Thedisplacement x as a function of angle a to the normalis

= T sinca 1 - [(1 - sin2a)/(N' - sina)]2},

where T is the prism thickness (2.54 cm) and N is itsrefractive index (1.52). At the maximum sweepingangle, about 26°, the deviation from linearity is aboutfive times as great as the sine function and of oppositesign. This is quite small, about 10%, and is notobjectionable for any purpose. The problem of pro-viding a stable synchronous signal for the horizontaloscilloscope plates is a very difficult one. For the pre-cision required here, it must be stable in phase to 0.001cycle. The rotor of the motor oscillates much morethan this in the field, moves in the bearing assembly ofthe motor, and generally drifts in ways which seemunaccountable. After several years of attempting,unsuccessfully, to secure an electrical signal sufficientlystable in phase from the rotor, we proceeded to use the

same prism to sweep the image of a triangular apertureover a slit in a direction at right angles to the optic axisused for fringe viewing. Critical adjustment was re-quired because of nonuniform sensitivity of the photo-multiplier surfaces. Two triangular apertures wererequired, one opening to the right and the other to theleft; in our system, they are masked alternately by amask rotated by a synchronous motor. To eliminatethe effects of the motor rotor moving about in the bear-ing assembly, it would be desirable to have the twobeams parallel to each other in passing through theprism, but this arrangement is not as easy to construct.

It will be realized that because of imperfections in theprism construction and mounting, two nonidenticalimage pairs are obtained, one by passing from the frontto the back face of the prism (arbitrarily designated)and the other from the back to the front face. Thus,on the oscilloscope, four tracings are seen, two of each ofthe mirror images. Setting is simplified by eliminatingone from each of these sets. This is done by providingonly two openings in the rotating mask and by provid-ing a similar rotating mask in the path of the mercurysource. The rotating mask for the mercury sourcehas an added advantage of decreasing the heating of theplates. The system so arranged can be used to set on afringe rapidly to about 1 . With care, settings to o

have been secured, but occasionally a greater error willappear. It is therefore undesirable to depend on asingle set of readings.

Translation and MountingAt the National Bureau of Standards the operations

are conducted in a temperature controlled room and apaper and foil shield is used to prevent heat from theoperator and room convection currents affecting theplates.* This is no longer satisfactory at the precisionsought here. The plates are set on an aluminum platewhich is placed on the micrometer slide table. InEmerson's procedure, high precision is not required forthe translation motion, but here it must be measuredto 0.001 cm or better. The plates and the slide are en-closed by hollow walls through which water (regulatedto several hundredths degree) is circulated. It isnecessary not only to maintain the plates at constanttemperature, but the spacers must be maintained atconstant temperature and humidity. Since most of thelight is absorbed in the aluminum plate on which theglass plates sit, it would be desirable to have circulatingwater pass through it, but we have not yet done this.

Optical SystemThe optical system is the conventional Fizeau fringe

viewing system consisting of a collimator, dividingplate, and objective lens, aperture, and screen (slit).The sweeping prism is placed in front of the slit. Theslit and photomultiplier assembly may be rotated tomake it parallel to the image of the fringes. It is there-

* See footnote (page 1917 of this paper).

November 1967 / Vol. 6, No. 11 / APPLIED OPTICS 1919

Data LoggerAn optical encoder is attached to the micrometer slide

shaft. It reads 2 revolution. The screw is a 1-mmscrew. Thus, the readings are about ten times asprecise as required in this application and the last placeis discarded. Readings are taken both on a typewriterand a paper tape. It is essential to have a dual outputif a tape punch is used because of the high failure rateof this device, and the loss of even a single reading in-validates a whole determination by the proceduresdescribed here.

(a)

H H-f

S

DDi

tCC

BB (;C

0) AA(b)

Fig. 2. (a) Elevation. (b) Plan of the optical arrangement. A,air inlet; B, modified H10OA4 lamp; C, metal cooling coil; D,interference filter; E, condenser lens; F, right angle prism; G,aperture; H, I, synchronous shutter driven by synchronousmotor; J collimator lens; K, dividing mirror; L, window; M,interferometer plates; Ar, hollow wall box for thermostat watercirculation; 0, aluminum transportation plate; P carriage table;Q, precision screw attached to encoder; R, telescope lens; S,aperture; T synchronously driven octagonal sweep prism; U,swing mirror; V, alignment viewer; W, slit; X, photomultipliercell; Y rotating mount; AA, filament lamp; BB, condenserlens; CC, right and left wedge apertures; DD, synchronousshutter; EE, projection lens; F, slit; GG, condenser lens;HH, ground glass diffusing screen; JJ, photomultiplier cell.

fore unnecessary to adjust the spacers exactly, a tediousprocedure. A swing-out mirror and viewer is pro-vided in front of the slit for alignment purposes. Sincemost of the plates tested are wedges, the collimatoraperture image on the telescope aperture will shiftfrom one determination to another. The position ofthis aperture, which is used to interrupt the unwantedreflections from the outside faces of the plates, musttherefore be adjusted for each set of plates. Sincephotoelectric setting is used, it may be argued thatsetting on the bright fringes would be equally as satis-factory as setting on the dark fringes. However, theintensity profile of the dark fringes seems to be affectedless by imperfections or unremoved traces of films, andwe have therefore used them in preference.

The arrangement of our system is sketched in Fig. 2.

MethodAt the National Bureau of Standards, spacers were

made by using a paper punch on multipart governmentforms. It was found that, fortuitously, the differentthicknesses of the respective sheets gave a suitablewedge.* Our multipart forms proved unsatisfactory.The spacers for our 10.2-cm diam plates must differ byabout 2 (0.00007 in.) if we wish to observe abouttwenty-five fringes. The paper which is used for mak-ing pads is remarkably uniform in thickness, but it willvary this much from top to middle of a pad. Wechose our sheets in this manner. Punchings were madewith a conventional -in. (6.35-mm) paper punch.Final adjustment of the thickness of the spacers wasmade by rubbing individual punchings between twosheets of bond paper. A set of spacers so made couldbe used for many determinations.

A mask was made which permitted positioning thespacers on an equilateral triangle with apexes at 0.7 ofthe radius.' The second plate was positioned atop thefirst plate in a large aperture Fizeau fringe viewer madewith a surplus aerial photography lens and a large pieceof high quality plate glass as a dividing plate. The pairof plates (interferometer) was set atop a blackenedaluminum plate on three cardboard punchings 6.35-mmdiam located directly below the spacers. The spacerswere adjusted in thickness to give about 2 fringes/cmnearly perpendicular to the intended direction of trans-lation. It is very important to keep the plates properlyoriented with respect to the direction of motion, i.e.,to examine the same points of the plates in each of thedeterminations for the set of three plates. After adjust-ment, the pair of plates sitting on the aluminum diskwas set on the micrometer slide table.

After standing for several hours, three or four sets ofreadings were taken. Only 3 or 4 min are re-quired to take a set of readings for a 10.2-cm diam setof plates. The high speed of taking the readings ob-viates errors from drifting fringes, usually caused bychanges in the moisture content of the spacers.

Readings were taken of the edges of the plates and ofthe fringe positions. The plates were set in the latemorning or early afternoon. Several sets of readingswere secured that day and several more sets the follow-ing morning. Thus, obtaining the absolute contoursof a set of three plates is a three-day undertaking.

* See footnote (page 1917 of this paper).

1920 APPLIED OPTICS / Vol. 6, No. 11 / November 1967

A

KR

Computation and ResultsComputation of the absolute contours for even a

single set of results is a relatively laborious job andtakes about 2 hs. The computations were there-fore programmed for a digital computer with an auto-matic plotter. Fourier series (25 terms) are fittedto the errors, and the operations are performed withthese series. Since the errors are not equispaced,conventional procedures for trigonometric interpola-tion are unsatisfactory. An ad hoc procedure, de-veloped for another purpose, was adapted. The use ofa sine series is the most convenient. To obtain goodconvergence, it is necessary that the errors be zero at theboundaries of the interval, here taken as from 0 to rfor the width of the plate. First, it is necessary toextrapolate the errors to the edge of the plate. Aftertrying several procedures, it was found best to make alinear extrapolation of the errors for the last two fringesadjacent to the edge and, for convenience in the nextstep, to provide an additional artificial point beyond theedge equidistant to the adjacent fringe. Now, valuesof the equation of a line passing through the artificialpoints on the edge are subtracted from the data.Parabolas are fitted to each succeeding set of threepoints starting at every point (the first being the oneoutside the interval), and the contribution of the portionof the parabola within the interval to the Fourier co-efficients is computed. Since the interval is coveredtwice, the Fourier coefficients are then halved. Thesum of this Fourier series and the equation of the linedrawn through the edge points is an excellent represen-tation of the data and is used to perform the subsequentcomputations. For convenience in plotting, the valueof the Fourier series at the center of the plate is com-puted and subtracted from the line. Since the sum oftwo Fourier series is obtained by adding the respectivecoefficients, sets of errors may be averaged by averagingthe Fourier coefficients and the subtracted lines. Sim-ilarly, all the differencing calculations may be made bytaking differences of the averaged coefficients and thesubtracted lines.

It was noted above that a linear drift in the fringepattern does not affect the precision of the method.The precision of setting on a fringe was determined,therefore, by taking sequences of readings of a single

0.02 <a

( 0.04

U- 0.06

0.08

0.100 1 2 3 4 5 6 7 8 9 10

DISTANCE (cm)

Fig. 3. Three sets of errors for the plate C upon plate A and theaveraged Fourier series fit.

0

0 2 3 4 5 6 7 8 9 10D ISTA NCE (cm)

Fig. 4. The three averaged Fourier series for the errors of theplates indicated. They have been shifted vertically to avoidconfusion. As computed, the center points on the plates would

have been set to zero.

fringe in the same manner used to obtain the platecontours and taking a linear regression upon thesequence. The probable error of interest in determin-ing the plate contours is that from the averaged Fourierseries. An analogous statistic for the sequence ofreadings is the square root of the error of estimate. F orthree sequences of sixty-six readings each, this variedfrom 6 o to fringe. The analogous statistic for thecarriage setting error was obtained in a similar way bysetting on a drum graduation, and was five to ten timessmaller. The drift in the fringe pattern during thesedeterminations was several times as great as the fringesetting error. If the fringe spacings were alteringappreciably, this might be evidenced in the reproduc-ibility of the error curves.

A typical set of error curves is shown in Fig. 3. It isseen that the errors do not differ from the averagedFourier series by more than 0.002-0.003 fringes for anyexcept the one or two affected by the rapid change incontour near the edge and further affected by the extrap-olation. The readings of the fringes adjacent to theplate edge are rarely reliable, because light scatteredfrom the edge affects the intensity profile (see below).It is concluded that the statistic obtained above forrepeated settings on a single fringe applies to the wholepattern. Thus, taking 6fringe for the precision ofthe averaged Fourier series, since the three sets oferrors are

a =A + B, = B + C,7 = C + A,

where a, , -y are the respective errors as computedhere, and A, B, C are the true plate contour points ofthe respective plates, the absolute contours are givenby

A =a + -, B = a + ,0- 7y, C = 3 + y- a,

and, hence, it is readily seen that the plate contourshave a precision of A fringe.

The application to a set of plates is illustrated in Figs.4 and 5. It is seen that the C plate is very muchpoorer than the others. We have additional sets of

November 1967 / Vol. 6, No. 11 / APPLIED OPTICS 1921

0.02

0.04

0.06

0.08

u,

C 0.

U_0.1

0.

0.

0.1

0.0 1 2 3 4 5 6

DISTANCE (cm)

Fig. 5. The absolute contours of the threetion meeting the tolerance 40.01 fringe bon

tically displaced to avoid conf

determinations on these plates and omade before the equipment reachedof refinement. The general featuresreadily discernable among these det(though the plates were stacked differset the right to left direction was rev(eluded, therefore, that much of the inthe plate contours is real. The D platthan the C plate, but its contour is irA trace of the D-plate irregularity appB-plate contours when the set determthough this trace falls within the stawould be desirable to have a plate of coto determine the contour of the A andis not available at present. It is anticof the irregularity seen near the edgecontours will vanish when a better th

In examining contours of this kind, ito judge the flatness of the plate fromof the optician because the best part oirepresentation may be tilted with respWe have therefore devised a quasi-roto present the contour in a more meaninthe Fourier series are plotted, point5/cm along the graph paper and the pis moved from one point to another.array of errors is saved. Starting withand taking several points on each side csquares fit is made. This process iadding a few points on each side untilwhich is beyond a specified toleranceline. The process is then continued alone until a point on this side is founfrom the last fitted line by more thtolerance. The last line so obtained isthe data and the data are replotted. contour falling within the tolerance i.

, C I and a rectangle of the height of the tolerance is drawnabout the portion meeting the tolerance. If the bestpart of the plate extends to the center of the plate (andthis is usually the case), a very clear visualization of thebest part of the plate is obtained. An example is shownin Fig. 6.

DiscussionA discussion of the determination of the contour of a

surface in the thousandth fringe region may seemfrivolous to some, because these are molecular dimen-sions. However, those who have worked with fringes,and especially sharpened fringes, have been found nodifficulty in making measurements to this precision.Thus, it is clear that there is a well defined optical sur-face which can be located to this precision by the elec-

; - 9 10 tromagnetic waves. Such a surface bears a simpler7 9 U0 relationship to a physical surface when it is a homo-

geneous dielectric. Although the fringe system em-plates with the sec- ployed here is not as simple as that obtained in thexued. They are ver- Michelson interferometer because there is here some

multiple reflection contribution, the light used is ofrelatively poor monochromaticity and collimation, andthere is present scattered and reflected light from the

n a set A, B, D, other surfaces, yet this fringe system is usually describedits present state as two-beam (in contrast to multiple reflection) fringes.of the plates are The precision attained here in locating the surfaceerminations even associated with this fringe system is among the mostmntly, and in one precise ever achieved for such fringes and approachesersed. It is con- that attained with Fabry-Perot systems. The accuracyegularity seen in of these determinations and the extent to which theye is a little flatter parallel the physical surface are subtle questions whichiuch less regular. have not been explored and are very meaningful inears in the A and view of the known inhomogeneity to a depth of someined is ABD, al- 10 M caused by polishing. Despite these problems, ittistics given. It seems meaningful to discuss the ultimate precision ofmparable quality determining the optical surface associated with theB plates, but one fringe system used here.ipated that some - The refinement of Emerson's procedure for determin-of the A and B ing the absolute contours of optical flats given here

ird plate is used. meets the requirements and capabilities of the bestit is very difficult optical work available today and is possibly somewhatthe point of view beyond present capability. However, it has been theF the plate in this past experience in optical work that if an error can beect to the graph. seen, a further correction can be made. We may antici-tation procedure pate therefore that a further refinement of measurementgful way. When-s are computeden left down as it

The calculatedthe central pointf it, a linear leasts repeated aftera point is foundfrom the fitted

)n the other sided which deviates.an the specifiedS subtracted from'he portion of thes now horizontal

0

0.02

C/CCDzL

0.04

0.06

0.08

0.100 1 2 3 4 5 6

DISTANCE (cm)7 8 9 10

Fig. 6. Example of quasi-rotation for finding the portion of theB plate meeting the specification i 0.005 fringe.

1922 APPLIED OPTICS / Vol. 6, No. 11 / November 1967

I I I I I I I I I

IC I I I I I I I

may be necessary. The basic procedures describedhere have been in use for some six years. Although ithas seemed possible to set on a fringe to better than0.001, probable errors for repeated readings have been

400-80. The problem of stopping the carriage atthe correct point, within microns for fringes spaced0.5 cm, seems beyond the capability of human reflexes.A fivefold increase in precision might result from hav-ing the carriage driven and having a pattern recognitiondevice for the oscilloscope trigger a transfer of the en-coder reading into the memory of the data logger.

The problem of accuracy of the procedure raises otherquestions. It was pointed out above that one of theprinciple advantages of Emerson's procedure is that theobservations are made at one point in the optical sys-tem, and hence the aberrations of the optical system donot affect the results. This is a correct statement forthe external optical system. However, the aberrationsof the plates themselves may introduce an aberrationby projecting the fringe system into a different part ofthe optical system. It is readily seen that if the platesurfaces deviate too much from planeness, the reflectedlight is projected into a different part of the telescopeaperture or is even partially intercepted by the aper-ture.* In effect, we are then examining fringes formedby light no longer normally incident on the plate, andthe fringe system is moved accordingly. A similareffect occurs when there is too large a difference in thecontours of the two plates. Then the interfering rayscome from relatively different portions of the plates thanfrom neighboring areas, and this also causes a shift inthe fringe system relative to neighboring areas. Theseeffects are not serious errors for us because they becomeimportant only when the plates are poor, and they be-come negligible for plates flat to fractional 0.1 fringe.

The aberrations introduced by the glass of the upperplate and its upper surface are another matter. Theplates given as examples here were made before theimportance of the back face was appreciated and towardthe edges show slopes 1.5 fringes/cm. This would dis-place our fringe images by about (20-cm focal length,40-cm working distance) 0.001 cm, and since our fringesare spaced about 0.5 cm, the error introduced is 0.002fringes. Thus, it is necessary to figure the back faceof a plate examined in this way to a fraction of a fringeif testing is to be extended to 0.001 fringe. It isequally obvious that the error introduced by themotion of the slide is less only by the factor of therefractive index of glass. Thus, for refinement of themethod to 0.001 fringes, the motion of the slide must beplane to about 0.25 At. This is feasible for carefullyhand-scraped work, but it is at about the limit ofmechanical tolerances. The thermal gradients in theslide must be reduced to a fraction of a degree. Whilethese tolerances may seem severe, it should be re-membered that the flatness of the plates is being deter-mined to a precision 1000 times greater than the plane-ness required for the slide motion.

* An additional error may be caused by an unsymmetricalchange in fringe intensity. This is discussed below.

There is another set of errors which is not easilycalculated but is very apparent to the operator. Thefringe setting is obtained by superposition of the fringeprofile on its mirror image. Thus, anything whichserves to distort the intensity of one side of the fringeimage, as compared with other fringe images- leads theoperator to set at a different point on the fringe. Sucha distortion can result from optical imperfections in theupper plate, from a speck of dirt between the plates, orfrom nonuniform films on the plates. This problemafflicts all methods which utilize the intensity of fringes,and the only distinction which arises between thesemethods and the visual procedure employed by Emer-son and used at the National Bureau of Standards isthat the areas observed and the effective integration oflight intensity performed in the several proceduresdiffer. Thus, different kinds of imperfections or filmswill affect the results obtained by the several proceduresa little differently quantitatively; that is, there aredifferent weighting factors for the different errors asthey affect the several methods. It will be appreciatedthat these errors are relatively unimportant when platesare tested to -1- fringe. But if an attempt is made totest them to 0.001 fringe, they become very important.

It is clear from the above discussion that plates whichare intended for test standards whose absolute contoursare to be determined should be free from imperfectionssuch as bubbles, veils, and local absorption. Theyshould be uniform in refractive index, and their backfaces should be figured. The glass should be subjectedto a schlieren examination and carefully examined ina polarimeter using the method recommended byGoranson and Adams.8 At the time this subject waslast examined in our Laboratory (about six years ago),9the only glasses which seemed satisfactory were a highgrade borosilicate crown and Homosil. Althoughinterferometer plates of crown glass are available com-mercially, our own capabilities would preclude ourattempting to figure it to high interferometric specifica-tions because of its high thermal expansion. It isdoubtful whether anyone today is prepared to attemptfiguring anything but a vitreous silica flat in the frac-tional 0.01 fringe region.

References1. W. B. Emerson, J. Res. Natl. Bur. Std. 49, 241 (1952).2. D. B. Spangenberg, Engineering Metrology Section, Metrol-

ogy Division, Institute for Basic Standards, National Bureauof Standards, private communication, June 24, 1965.

3. W. Primak, J. Opt. Soc. Am. 49, 375 (1958).4. F. L. Roesler and W. Traub, Appl. Opt. 5, 463 (1966); T.

Duong, S. Gerstenkorn, and J. M. Hilbert, J. Physique (inpress).

5. A. E. Ennos, Appl. Opt. 5, 51 (1966).6. F. S. Tomkins and M. Fred, J. Opt. Soc. Am. 41, 641 (1951).7. W. Primak, H. Szymanski, and D. Keiffer, J. Appl. Phys. 32,

664 (1961).8. R. W. Goranson and L. H. Adams, J. Franklin Inst. 216, 475

(1933).9. E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellip-

sometry in the Measurement of Surfaces and Thin Films,NBS Misc. Pub. 256 (1964), p. 154.

November 1967 / Vol. 6, No. 11 / APPLIED OPTICS 1923