Diffractographic Dimensional Measurement.Part 2: Profile Measurement

T. R. Pryor, 0. L. Hageniers, and W. P. T. North

Part 1 of this paper discussed a "diffractographic" method by which changes in separation between twoobjects could be measured using changes in the diffraction pattern of the slit-type aperture formed be-tween them. A unique line measurement capability was demonstrated. In this part we discuss profilemeasurement whereby the diffraction pattern formed is used to indicate changes in separation betweenadjacent sets of opposite points, rather than changes between the points themselves. Positions of pointson one object are usually known, in order to test the profile of the second. The method offers manyinteresting advantages in those applications where it may be used, particularly due to the speed, accuracy,and easily automated nature of the process.

Introduction

Profile measurement is utilized in many manufactur-ing industries for part inspection and, optically speak-ing, standard imaging devices such as contour pro-jectors or microscopes are almost universally used forthis purpose. To improve the performance of suchdevices, Lansraux' and Birch2 have described a diffrac-tive technique employing a spatial filter to remove thedc term (and in some cases higher order terms) fromthe edge image of a coherently illuminated object pro-file. An improved edge image results, though the tech-nique does not appear to have been industrially used,perhaps because of the limitations noted by Dew.3

A quite different diffractive technique is that usedby Rogers and Armitage to monitor film strip edgestraightness. 4 In their device, a straight reference edgeis placed next to the film edge to form an aperture whichdiffracts light from two sources whose separation can beadjusted such that slit width at a point can be deter-mined. By successively performing this operation at allpositions along the slit, one may obtain a profile. Theirtechnique, especially as regards use of a reference edge,is similar to that employed here and curiously does notappear to have been developed further. Perhaps thespecialized nature, slow speed, and limited accuracy ofthe device they describe was responsible.

The diffractographic technique described herein pro-duces a projected profile diffraction pattern rather than

an image of the object edge. In most instances, thispattern has been generated along a line of the slit-typeaperture formed between straight reference edge (orsurface) and an edge (or line upon the surface) of a testobject. While visual observations of the profile pat-terns has been extensively used to date, the techniqueseems most suited to automated inspection using tele-vision scans or spatial filtering by master diffractionpatterns. Before several applications are discussed, abrief review of the simple theory utilized will be pre-sented. The reader is referred to Part 1 as well.'

TheoryVirtually all intermediate optical texts present the

single-slit Fraunhofer diffraction intensity envelopeequation' valid where slit width w is large compared towavelenth X:

I = Io(sinfl/, 2), (1)

where ,3 = (r/X)w sinO and Io is the intensity on the pat-tern centerline, 0 = 0. This situation is illustrated inFig. 1, where the zeroes or minima of the pattern areobserved to fall where

w sin. = nX (n= 1, 2,3,. . .),

which for the small angles 0 nearly always used in prac-tice (due to the concentration of diffracted energy in thisregion) becomes

wxn/R = nX,

The authors are with the Department of Mechanical Engineer-ing, University of Windsor, Windsor 11, Ontario.

Received 21 June 1971.CLEA paper 11.5 (Part 2).

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where Xn is the distance from the centerline = 0 to aminimum of order n, and R is the slit-to-observation-plane distance and is much greater than w. (If a lens isused to form the pattern, its focal length replaces R inthe above equation.) The spacing of the fringes is then

314 APPLIED OPTICS / Vol. 11, No. 2 / February 1972

Fig. 1. Characteristics of the diffraction pattern produced alongthe length of a slit.

the value of X,, in Eq. (2) when n = 1, or

= R/w. (3)

The difference in slit width between two positions 1 and2 along the slit can be expressed by

W2-W1 = R[(1/s2) - (1/8 1)]. (4)If the positions of points on one edge (or surface) are

known, then variations in slit width can be related to theother edge (or surface) completing the slit. In otherwords, the test object profile can be obtained from itscontribution to the diffraction pattern produced by theslit formed between it and the reference object.

The discussion above is very similar to that pre-sented in Part 1 and the bounce-off technique (using theaperture between an edge and a surface) describedtherein can be used here as well. In this case, slitwidth w in the equations above becomes twice the aper-ture width and for nongrazing incidence a cosine cor-rection factor is required.

A large number of techniques may be used to obtainthe s values (or their inverse, spatial fringe frequency)necessary to determine profile variations using Eq. (4).In general, one can visually observe and measure fringemovements such that w2 - w is determined to 1 m,and null-balance electronic detection should improvethis by an order of magnitude or more. The range ofmeasurement in normal circumstances is roughly 1.5mm, with larger values obtainable if a cylinder lens isused to form the pattern.

General ConsiderationsBecause of the large number of possible profiles,

many of which require specialized reference edge or sur-face arrangements, it is difficult to imagine one universaldevice (such as a contour projector). Some factorsare, however, common to all types and a few of theseare now discussed.

Figure 2 illustrates perhaps the simplest situation,whereby a straight reference edge is placed next to awavy object edge whose profile we wish to check. Sev-eral things can be noted. First, over the length L, slitwidth w varies in a manner indicating profile, as wellas any difference in the separation of the two end posi-tions P and Q. For simplicity, we assume that w has

been adjusted equal to wQ such that no linear slit widthvariation remains. Obviously two other such pointscould have been used for this purpose as well.

In the region A we notice that slit width variesrapidly with y in what would be described more as aroughness than a change in gross profile. Its effectsappear in virtually all diffractograms (see, for example,Part 1) and may also be caused by particles on the sur-face or edge. Surface roughness (or particle sizes)could conceivably be extracted from the diffractionpattern, though this more complicated situation willnot be treated here.

Region B illustrates a section of test edge whose slopevaries significantly in the region being illuminated. Awide variety of local geometries are included in thisclass, most producing various unwanted diffraction ef-fects in many different directions when an ordinary lasersource is used. While several means for obtainingusable and easily interpretable patterns are under in-vestigation, this paper discusses only those slit-typeapertures which have slowly changing curvatures in theregion illuminated.

From consideration of Eq. (4) and Figs. 1 and 2, it isapparent that the smaller w becomes, the more changein pattern fringe spacing results for a small absolutechange in w along the length. However, the accuracyof measuring the changes w2 - w is, in general, inde-pendent of w and therefore very small slits are to beavoided due to positioning and diffraction problems.On the other hand, very large slit widths are usuallynot desirable due to the very large distance R necessaryto form the pattern. If a lens is used for the purpose,this problem is eliminated, but fringe resolution be-comes visually difficult and added complexity results,especially when the whole profile is illuminated (re-quiring a long cylinder lens).

A final general characteristic is that of the diffractionpatterns produced and their detection. Besides thelarge magnification of small profile changes, there aretwo other advantages of such projected patterns. Firstvery high speed readout of the profile information en-coded as fringe frequency can take place using televi-sion cameras or laser scanner/detector combinations,allowing a whole profile length to be effectively quanti-fied in real time. Second, the projected profile dif-fraction pattern, representing one or more positions onthe length L, can be spatially filtered, giving instan-taneous error signals. Several applications will now bediscussed.

REFERENCE EDGE

WI EG

OBJECT EDGE

Fig. 2. Placement of the reference edge and the object for profiledetermination.

February 1972 / Vol. 11, No. 2 / APPLIED OPTICS 315

1- ~~~~Lj

DIFFRACTOGRAM

Fig. 3. Testing of an edge by comparing it to a known standard(in this case a flat mirror). Also shown is a typical diffractogram.

Checking Straight Edges with a Flat

Mirror Reference

In the arrangement of Fig. 2, one might logicallyquestion how the reference edge was determined to bestraight in the first place. Figure 3 illustrates an ap-plication of the bounce-off technique described in Part 1to just this problem. The reference in this case isone-eighth wave flat mirror (assumed perfectly flat) anda diffractogram representing a razor blade edge tested isalso shown. Note the obvious curvature and rough-ness magnification present, using the experimentalvalues Wavg = 30 ,um, and q5 = 150.

The use of a cylinder lens has allowed production of adiffractogram representing undulations along the wholeblade edge. In the same apparatus, we may focus theincident beam to a line such that a very small region ofthe slit is illuminated. The region of the focal line istherefore blown up on the screen and effective magni-fication obtainable at the screen using a 10-mW gaslaser source (X = 0.6328 Mm) can be several thousand inboth x and y directions.

Reference Edge Applications

Once we know the profile of an edge using the tech-nique of Fig. 3, we can use the edge as mentioned pre-Viously to check object profiles. A first example wouldbe to reverse the experiment above and use the edge tocheck surface profiles. As mentioned in Part 1, grazingincidence is usually required to achieve sufficient re-flectance and the object surface must be quasi-flat in theregion of the edge. Curved objects can be used, butFresnel-type diffraction often results, complicating in-terpretation.

When the curved objects mentioned above have suf-ficient curvature, they may be used as edges, relative toa reference edge or surface. Several industrially im-portant objects such as roller bearings, shafts, and millrollers fall into this category. Thin sheet materialpassing over a tension roller could also be similarly pro-filed.

A particular example is shown in Fig. 4, together withthe diffractogram produced. Here portions of the sur-face of two camshaft lobes are being compared to a tool-

maker's straightedge. The surfaces (of a reject cam-shaft) show significant, and uneven, crowning.

Nonstraight Reference Edge

In the previous examples the reference edge (or sur-face) has been straight (or flat), thereby implying alimit on object profile variations (i.e., slit-width changes)of around 1.5 mm or less, due to diffraction considera-tions at visible wavelengths. While oblique incidencetechniques could improve this (with an accompanyingloss of resolution), such limitations effectively restrictapplication in this basic form to quasi-straight or flattest objects.

A further class of objects which could be gauged inmuch the same way is those curved objects to which anarc-like reference edge could be compared. In thiscase a nearly constant slit width is utilized and thelaser/detector array pair scanned around the slit.Many other arrangements are of course possible withinthe framework of this example.

Perhaps more interesting is the experiment of Fig. 5.Here an object with a jagged edge profile (in this case ahousekey) is being compared to a master edge profile.Were the aperture formed to be illuminated simul-taneously or scanned by a raw laser beam, diffractionwould occur in many directions, since the aperture axisof symmetry follows the jagged profile itself. Forpresent purposes, such a complex pattern is useless,since easy extraction of quantitative data is almost im-possible.

In the case shown, however, a cylinder lens is used tofocus the radiation into a line upon the aperture, forcingan axis of symmetry in the horizontal direction asshown. To further facilitate observation of the pat-tern, the master can be made such that Wvertical is con-stant at all positions if a perfect test object is used.Therefore, as the key-master aperture is scanned pastthe cylinder lens focal line (or vice versa) deviations areimmediately detected as diffraction pattern changesfrom the standard.

Spatially Filtered Patterns and Moires

A particularly useful characteristic of the projected

DIFFRACTOGRAM

REFERENCEEDGE

CAMSHAFT

Fig. 4. Scheme for profile testing camshaft lobes, showing atypical diffractogram.

316 APPLIED OPTICS / Vol. 11, No. 2 / February 1972

SCREEN

LENS

Fig. 5. Schematic arrangement illustrating the technique usedto test jagged profiles.

fringe patterns produced by the diffractographic tech-nique is that the information concerning profile is en-coded as spatial fringe frequency which may be filteredto obtain photoelectric signals signiflying a match withthe pattern produced by a known edge profile. Al-ternatively, moire fringes proportional to a deviationfrom the filter profile may be produced. In the lattercase, the moir6s can be produced live through a filter,by double exposure on film or by simple overlay of twodiffractograms.

Two examples utilizing these filterable character-istics are presented. In the first, an apparatus arrange-ment similar to that of Fig. 3 was used to check double-edge razor blade shelf-to-edge tolerances as well asgeneral edge straightness. Here, the blade shelf restedon blocks of equal height above a flat mirror and the dif-fraction pattern produced using a designated masterblade was recorded on a photographic plate placed atplane A-A. Thirty other blades of various manufac-ture were then tested in the same arrangement, theirpatterns being projected through the negative masterpattern replaced at plane A-A. The moir6 fringes pro-duced by several such blade edge patterns are shown inFig. 6 (the central portion, = 0.00325, has beenblocked to prevent film splatter). The undesirable ef-fects of dust on the mirror are quite noticeable.

Interpretation of the moir6 patterns is easy andquickly done. Using Eq. (3) we can write an equationsimilarto Eq. (4),

We-wa = RX[(s - t)Smst]J,

where s and Sm are the diffraction pattern fringe spac-ings produced at a given position along the edge usingtest and master blades, respectively. Since moir6 fringespacing is defined as6

S = [SmSt/sr - )] ,then using Eq. (5)

IWt - Wm = R/S.

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moir6s. If desired, it must be obtained from considera-tion of the diffraction patterns themselves, as was donein obtaining the profile plots shown in Fig. 6.

For the purposes of industrial gauging, such moir6scould very quickly indicate whether or not a certaintolerance level had been exceeded. In- addition, a veryadvantageous property of such patterns is that theycorrespond only to changes in fringe frequency from thefilter pattern.

A second example of spatial filtering is shown inFig. 7, where a lens is used to image the test patternpassing through the filter onto a photodetector. Thisoptical-null-balance technique produces a distinct dropin signal when a fringe frequency match is obtained,indicating that the test blade and master are equivalentat the position(s) in question. In a basic test of thetechnique, an adjustable edge was uniformly movedthrough a range of slit widths monitored by an LVDTconnected to the x axis of an x-y recorder. Photo-detector voltage was monitored on the y axis, and atypical plot is shown in Fig. S. Not only does a 50%modulation occur at the match, but the curve is quitesymmetric, indicating that equivalent positive or nega-tive departures from the master pattern will producenearly the same detector signals.

Figure 9 is a similar plot over a much larger range ofslit widths. As can be seen, within a considerable dis-tance on either side of the match there are no regionswhich could produce similar (and therefore erroneous)output signals. It would seem possible to extend thisnull balance system to the more general case of variableprofile patterns. In this case a go/no-go signal could beproduced whenever an average fringe frequency dif-ference occurred along the filter. Different rejectioncriteria could be used as needed, with perhaps multipledetectors looking at different sections of the filtered pat-tern.

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MASTERBLADE

_ 10

tj

0

a

I-: -D

(7)

The difference or error in the dimension of the testblade relative to the master blade at the position inquestion is therefore inversely proportional to the moir6fringe spacing. As is apparent from Eq. (7), however,the sign of the error cannot be determined from the

Fig. 6. Moir6 fringe photographs and the slit-width changescalculated from them for several razor blades compared to a

master blade.

February 1972 / Vol. 11, No. 2 / APPLIED OPTICS 317

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I MIRROR

Fig. 7. Arrangement used to illustrate spatial filtering of adiffraction pattern.

Conclusion

Several examples of profile measurement using the

diffractographic slit-type aperture diffraction techniquehave been presented. With the apparatus configura-tions described, some unique advantages result whichallow accurate simultaneous gauging of certain profiles(or minute sections thereof), as well as very fast toler-ance limit checking in some cases. Perhaps the biggestadvantage will prove to be the ease with which accurateprofile measurement can be automated using the tech-nique.

This research was supported by the National Re-search Council of Canada grant A3360.

References

1. G. Lansraux, Metrologia 1, 31 (1965).

2. K. G. Birch, Optica Acta 15, 113 (1968).

3. G. D. Dew, Optica Acta 17, 237 (1970).

4. R. V. Rogers and J. D. Armitage, Opt. Technol. August, 208(1969).

5. D. H. Towne, Wave Phenomena (Addison-Wesley, London,1967), Chap. 12.

6. L. 0. Vargady, Appl. Opt. 3, 631 (1964).

7. T. R. Pryor, 0. L. Hageniers, and W. P. T. North, Appl. Opt.11, 308 (1972).

2-

0 WO- 0.62 mm

0i- 2.5

0

0

2.0

-50 -40 -30 -20 -1 0 10 20 30 40

A w, SLIT WIDTH CHANGE (j m)

Fig. 8. Detector output voltage as a function of slit-widthchange for the arrangement of Fig. 7.

0

0C'

0.

1.0 2.0

w/W,

Fig. 9. Detector output voltage as a function of slit-widthchange for the arrangement of Fig. 7.

318 APPLIED OPTICS / Vol. 11, No. 2 / February 1972

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