Moire fringes by evolute gratings

Piotr Szwaykowski and Krzysztof Patorski

Analytical expressions describing the moire fringes formed by two periodic structures consisting of theevolutes of a circle are derived. The circular and radial moire patterns can be formed by contact andnoncontact superimposition of the structures. In the latter case the self-imaging phenomenon is used.Experimental verification and proposals for use are given.

1. Introduction

The moire fringe technique has become one of themost important and elegant methods in optical metrol-ogy. It is widely applied in interferometry, topogra-phy, optical alignment (linear and angular sensing),and experimental mechanics. Basically the techniquerequires two periodic or quasiperiodic structuresplaced in contact or projected one onto another. Theprojection might be realized either using the usualimaging optical system or applying the so-called self-imaging phenomenon.' The latter method has an ad-vantage in that it requires no optical elements andleads to very simple and compact setups. However,self-imaging requires coherent illumination.

Structures that are most often used are the linear orgrid (cross-type) diffraction gratings. Sometimes,however, the use of circular or radial structures is moreconvenient.2 Spiral gratings might also be applied inthe moire fringe systems.3

All structures can be imaged by optical projectionsystems. Linear grid and circular gratings belong to aset of self-imaging objects, and they can be projectedusing the self-imaging phenomenon. Unfortunately,the radial and spiral gratings do not belong to thisgroup of objects. However, they are very useful forcertain precise adjustments, circular motion measure-ments, and aberration testing of optical systems.

The purpose of this paper is to present a new kind ofspiral-shaped structure that can be self-imaged andform unique patterns of moire fringes.

The authors are with Warsaw Institute of Technology, Institute ofDesign of Precise & Optical Instruments, 8 Chodkiewicza Street, 02-525 Warsaw, Poland.

Received 21 July 1988.0003-6935/89/214679-03$02.00/0.© 1989 Optical Society of America.

II. Analysis

Moire-based systems can use either uniform field orfinite fringe detection modes. The first occurs whentwo identical structures are superimposed, the secondwhen the structures are slightly different, or one ofthem is shifted or inclined to introduce a set of refer-ence fringes. The linear fringe set for Cartesian coor-dinates and the circular or radial fringe set for polarcoordinates are of primary importance.

Recently, so-called skewed radial gratings 4 were de-signed that create a set of circular moire fringes thatmove because of the gratings' mutual rotation. Theseskewed radial gratings are suitable for rotation velocitymeasurements. Previously it was shown that super-imposed spiral gratings create a set of circular or radialfringes3 that can also be used for the same purpose.The disadvantage of all these gratings (radial, skewedradial, and spiral) is that they must be either in contactor superimposed by an optical projection system.

As shown in a previous paper,5 a grating consisting ofa set of evolutes of a circle (Fig. 1) meets the self-imaging conditions. It can be proved that two evolutegratings create a set of circular or radial moire fringes.They can also work under the uniform field detectionmode.

The grating consisting of a set of evolutes can bedescribed (in Cartesian coordinates) by the followingequations:

x = r cos(t - k) + rt sin(t -k),

y = r sin(t - k) - rt cos(t -k), (1)

where ro denotes the radius of the inner circle, so is aseparation angle between two adjacent grating lines, tis a parameter, and k = 1,2,... ,27r/o denotes the num-ber of evolutes in the grating.

Let us consider two gratings of the type being super-imposed. To obtain the shape of moire fringes weshould predict the intersection points of both gratinglines. First, we analyze the case when both gratings

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y

x

Fig. 1. Computer-generated plot of a diffraction grating consistingof ten evolutes of a circle.

are the same and their lines run in opposite directions.The intersection points can be obtained from the fol-lowing equations:

ro cos(t - k) + rot sin(t - k) = ro cos(-z - no)

+ roz sin(z + nV'),

ro sin(t - ki) - rot cos(t - k) = ro sin(-z - n4)

+ r0z cos(-z - ny), (2)

where z is the parameter for the second grating.From Fig. 2 it follows that for the intersection points

both parameters t and z are equal. Thus Eqs. (2) canbe simplified into the form

(ro cosy + rot sin-y)(1 - cosls) + (ro siny - rot cos-y) sin1V = 0,

(ro siny - rot cosy)(1 + coske') + (ro cosy + rot siny) sin1V = 0,

(3)

where y t - kp,k + n = 1.Both equations give the same condition, valid for

any of the intersection points i,y:

.C 1-cosIV = x tan(lV,/2),sing1'

= 1sin1V = x tan(1o/2).=1 + csMV

(4)

This means that for a constant value 1 these points lieon a straight line, a dashed line in Fig. 2. If the value of(, is large enough depending on a grating pitch, moirefringes are obtained that run along this direction.

Let us investigate now the case of two gratings withlines running in the same direction and with slightlydifferent radii r and R of inner circles. The conditionsfor intersecting points are

R cos(t - k) + Rt sin(t - kg) = r cos(z - np) + rz sin(z -np),

R sin(t - kq) - Rt cos(t - kg) = r sin(z - n') - rz cos(z - np).(5)

In this case the parameters t and i in the intersectionpoints are related by the formula (see Fig. 3)

(6)

Fig. 2. Geometry of the system of two evolutes of a circle running in

opposite directions. Both evolutes are separated by the angle (n -k)a.

x

Fig. 3. Geometry of the system of two evolutes (running in the same

direction) of two slightly different radii of inner circles. Both evo-lutes are separated by the angle (ka - np).

z = t + a. (7)

Thus Eq. (5) can be simplified and rewritten aftersome calculations as

(cosy + t siny)(R - r cosfl) = -(r siny - rt cosy) sin:

+ ra sin(ky + fl),

(siny - t cosy)(R - r cosg) = (r cosy + rt sin-y) sin:- ra cos(-y + .), (8)

where y = t-kip, # = (n-k)p + a. From the aboveequations the following conditions fulfilled by any ofthe intersection points !,y are obtained:

i(R - r cosfl) + 5r sink = Rra sin(,y + A),(R - r cosp) -:ir sink = -Rra cos(y + ). (9)

We can introduce here an additional parameter a It follows from Eq. (9) that xc and 5 fulfill a relation

4680 APPLIED OPTICS / Vol. 28, No. 21 / 1 November 1989

R'(1 + 0 = r 2(1 + i2).

fringes obtained with two evolute gratingsopposite directions (separated by the self-

image distance).

Fig. 5. Circular moire fringes obtained with two evolute gratingswith lines running in the same direction illuminated by a spherical

wavefront beam.

x2 + y2 = (rRa)2(10)

(R - r)2 + 2rR sin2fl

If the value of the difference (R - r) is small enough,we obtain moire fringes coinciding with these circles.

If a mutual rotation of both gratings is introduced,we add a new parameter to the value 1o in Eq. (4) andchange the value of a in Eq. (10) to obtain the correctposition of both gratings in the coordinate system.Equations (4) and (10) indicate that the moire fringeswill in both cases be shifted in the direction perpendic-ular to the fringe lines.

Radial-type moire fringes are easy to obtain by plac-ing two identical evolute gratings separated by self-imaging distance with lines running in the oppositedirections into the collimated coherent light beam. Asystem for producing a circular moire fringe set withtwo evolute gratings of slightly different radius romight be achieved by either applying two differentgratings and plane wave illumination or introducingspherical wave illumination into a system consisting oftwo identical gratings.

111. Experimental Work

Gratings consisting of ten evolutes of a circle5 wereused to produce the above described patterns of moirefringes. These gratings were separated by the self-image distance and illuminated by a spatially coherentlight beam. Moire fringes were detected on theground glass plate in the plane of a second structure.The system of radial fringes (Fig. 4) was obtained withboth gratings placed with their lines running in oppo-site directions illuminated by the plane wavefront.The circular fringes (Fig. 5) were observed when bothgratings were illuminated by the spherical wavefrontbeam with their lines running in the same direction.

IV. Conclusions

The evolute gratings were proposed to produce thesystem of radial or circular moire fringes. The self-imaging phenomenon can be applied to superimposethe two gratings making the setup extremely simpleand compact. Another advantage of this method isthe fact that two identical gratings can be used. Themoire fringes produced are of high contrast and can beeasily detected. Both systems of fringes move in thedirection perpendicular to their lines due to the rela-tive rotation of both gratings. Thus the gratings canbe used in measurements of circular motion and het-erodyne detection of interference fringe patterns.

References1. See, for example, W. D. Montgomery, "Self-Imaging Objects of

Infinite Aperture," J. Opt. Soc. Am. 57, 772-778 (1967).2. See, for example, P. Theocaris, Moire Fringes in Strain Analysis

(Pergamon, Oxford, 1969).3. 0. Bryngdahl and W. H. Lee, "Shearing Interferometry in Polar

Coordinates," J. Opt. Soc. Am. 64, 1606-1615 (1974).4. B. Sandler, E. Keren, A. Livnat, and 0. Kafri, "Moire Patterns of

Skewed Radial Gratings," Appl. Opt. 26, 772-773 (1987).5. P. Szwaykowski, "Self-Imaging in Polar Coordinates," J. Opt.

Soc. Am. A 5, 185-191 (1988).

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Fig. 4. Radial moirewith lines running in