Determination of basic grids for subtractive moire patterns Salvador Bara, Zbigniew Jaroszewicz, Andrzej Kolodziejczyk, and Vicente Moreno The simple formula derived in the paper establishes a direct relationship between moire beat patterns and basic grids for minute displacements. The possibilities of finding a basic grid for a desired moire pattern are pointed out. The analysis is illustrated by several examples of Fresnel moire zone plate patterns and concentric equidistant circular moire patterns obtained by changes of scale and rotation. Possible advan- tages of the practical use of this element are outlined. Key words: Moire, alignment. 1. Introduction Moire patterns created by mutual displacements of two suitably fitted identical grids have found numer- ous uses in metrology and alignment techniques. The oldest and maybe best known is an example of linear grids to measure long linear displacements 1 ' 2 or to sense small angular displacements. 23 There are also exhaustively discussed moire patterns created by two zone plates, 45 by two evolutes of a circle, and by radial gratingsl 3 as well as masks which by rotation or trans- lation produce Fresnel zone plate (ZP) patterns. 7 -10 Usually the shape of moire fringes is determined in these works by the known equation of the basic grid curves. The purpose of this paper is somewhat different. Taking into account practical applications of some moire patterns it seems to be important to find a broad class of functions giving a desired moire pattern. In subsequent sections we establish such a direct rela- tionship under some simplifying assumptions that specify subtractive, highly distinct moire fringes- moire beats, as well as infinitesimal displacements (in some solutions this condition need not be maintained). A derived relationship is illustrated by examples of grids, which create Fresnel moire zone plate patterns (FMZPPs) with variable focal lengths achieved by changing the scale and by rotation, as well as a concen- Zbigniew Jaroszewicz is with Central Laboratory of Optics, 18 Kamionkowska, P-03-805 Warsaw, Poland; A. Kolodziejczyk is with Warsaw University of Technology, Institute of Physics, 75 Kos- zykowa, P-00-662 Warsaw, Poland; and the other authors are with University of Santiago, Physics Faculty, Optics Group, E-15706 Santiago de Compostela, Galicia,Spain. Received 5 March 1990. 0003-6935/91/101258-05$05.00/0. e 1991Optical Society of America. tric equidistant circular moire pattern (CECMP), also obtainable by changing the scale and by rotation. Of these two the FMZPP seems to be of greater practical importance. It has found a use in the so- called three-point method, where imaging elements with long and variable focal lengths, hardly obtainable as glass lenses, are needed for sensing minute displace- ments in large engineering structures.' 0 - 4 II. Deriving the Formula We start with the representation of the basic refer- ence grid which is given by (or) = k, (1) where k is an integer indexing the family of curves b(r), from which the grid is composed. The second grid, consisting of the same family of curves but dis- placed by a vector Ar, is given by I(r + Ar) = 1, (2) Ibeing an integer. Curves connecting the intersection points of superimposed grids form the moire pattern, and therefore their index m should fulfill the condi- tionl 45 n 1 k n 2 = m, (3) where n,n 2 are nonzero positive integers and m runs over some subsets of integers. Taking into account only the most distinct subtractive moire pattern com- posed from the curves connecting the nearest neighbor intersection points (so-called moire beats), i.e., placing n= = 1 and using the minus in Eq. (3), we obtain C(r + bar) -o) = m - (4) or, assuming only infinitesimal values of the displace- ment vector Ar (which in practice means that they are small compared with the period of the expected moire fringes), the formula describing the moire beats can be rewritten as 1258 APPLIED OPTICS / Vol. 30, No. 10 / 1 April 1991
Transcript
Determination of basic grids for subtractive moire patterns
Salvador Bara, Zbigniew Jaroszewicz, Andrzej Kolodziejczyk, and Vicente Moreno
The simple formula derived in the paper establishes a direct relationship between moire beat patterns andbasic grids for minute displacements. The possibilities of finding a basic grid for a desired moire pattern arepointed out. The analysis is illustrated by several examples of Fresnel moire zone plate patterns andconcentric equidistant circular moire patterns obtained by changes of scale and rotation. Possible advan-tages of the practical use of this element are outlined. Key words: Moire, alignment.
1. Introduction
Moire patterns created by mutual displacements oftwo suitably fitted identical grids have found numer-ous uses in metrology and alignment techniques. Theoldest and maybe best known is an example of lineargrids to measure long linear displacements1'2 or tosense small angular displacements.2 3 There are alsoexhaustively discussed moire patterns created by twozone plates,4 5by two evolutes of a circle, and by radialgratingsl 3 as well as masks which by rotation or trans-lation produce Fresnel zone plate (ZP) patterns.7-10Usually the shape of moire fringes is determined inthese works by the known equation of the basic gridcurves.
The purpose of this paper is somewhat different.Taking into account practical applications of somemoire patterns it seems to be important to find a broadclass of functions giving a desired moire pattern. Insubsequent sections we establish such a direct rela-tionship under some simplifying assumptions thatspecify subtractive, highly distinct moire fringes-moire beats, as well as infinitesimal displacements (insome solutions this condition need not be maintained).A derived relationship is illustrated by examples ofgrids, which create Fresnel moire zone plate patterns(FMZPPs) with variable focal lengths achieved bychanging the scale and by rotation, as well as a concen-
Zbigniew Jaroszewicz is with Central Laboratory of Optics, 18Kamionkowska, P-03-805 Warsaw, Poland; A. Kolodziejczyk is withWarsaw University of Technology, Institute of Physics, 75 Kos-zykowa, P-00-662 Warsaw, Poland; and the other authors are withUniversity of Santiago, Physics Faculty, Optics Group, E-15706Santiago de Compostela, Galicia, Spain.
Received 5 March 1990.0003-6935/91/101258-05$05.00/0.e 1991 Optical Society of America.
tric equidistant circular moire pattern (CECMP), alsoobtainable by changing the scale and by rotation.
Of these two the FMZPP seems to be of greaterpractical importance. It has found a use in the so-called three-point method, where imaging elementswith long and variable focal lengths, hardly obtainableas glass lenses, are needed for sensing minute displace-ments in large engineering structures.' 0 - 4
II. Deriving the Formula
We start with the representation of the basic refer-ence grid which is given by
(or) = k, (1)
where k is an integer indexing the family of curvesb(r), from which the grid is composed. The secondgrid, consisting of the same family of curves but dis-placed by a vector Ar, is given by
I(r + Ar) = 1, (2)
I being an integer. Curves connecting the intersectionpoints of superimposed grids form the moire pattern,and therefore their index m should fulfill the condi-tionl4 5
n1k n 2 = m, (3)
where n,n 2 are nonzero positive integers and m runsover some subsets of integers. Taking into accountonly the most distinct subtractive moire pattern com-posed from the curves connecting the nearest neighborintersection points (so-called moire beats), i.e., placingn= = 1 and using the minus in Eq. (3), we obtain
C(r + bar) -o) = m - (4)
or, assuming only infinitesimal values of the displace-ment vector Ar (which in practice means that they aresmall compared with the period of the expected moirefringes), the formula describing the moire beats can berewritten as
If the curves forming the moire beat patterns are de-scribed as I(r), we are able to establish a direct con-nection between moire beat patterns and the basicgrid:
V4)(r)Ar = (r). (6)
Relying on this formula it is possible not only to deter-mine the shape of the moire pattern, when the basicgrid is known, but also to solve an inverse problem,namely, to find the shape of the basic grid, startingwith the given moire pattern. To wholly establish theformula it is necessary to determine exactly the dis-placement vector. Displacements can be caused inthree ways: by translations, rotations, or by changingthe scale:
Ar=Art+Ar,+Ar,=i(A+ay+#x)+j(Ay-ax+fly). (7)
Translational displacement vector Art is equal to iA,, +jAy. Rotations can be included in the consideration byusing the transformation formula for rotations takenfor infinitesimal values of angle a yielding the rota-tional displacement vector given by Art = iay - jax.Finally, the displacement vector generated by thechange of the scale (which in practice can be achievedby imaging one basic grid onto another with the magni-fication factor slightly different than one) is given byAr8 = -yr-r, where -y is the scaling factor. Within thescope of our approximation y = 1 + fi (where 11 << 1),we obtain Ar, = ifex + j 3y. Since small values of thedisplacement vector were assumed, their mutual influ-ence, as infinitesimal quantities of the second order,can be ignored.
We now observe that rotations and changes of scalecan be more easily treated in polar coordinates, sincechange of scale causes displacements in the radial di-rection and rotation in the angular direction. Bothkinds of displacement can thereby be separated in thepolar coordinates, and Eq. (6) takes the following form:
0(r') (fir + A cos0 + Ay sinO)ar
+ [(r,0)a + I (-Ax sin + A cos0)l =4(rO)ao I r I=
(8)
Neglecting the translation, the above equation can besimplified as follows:
(9)(rO) fr +a(P(r,0) a = 0r0,
and this is used in the examples presented below.We note here that the derived relationship is similar
to the formula describing interference fringes in ZPinterferometers used generally in alignment tech-niques,15"16 although the purpose of introducing it forthe moire fringes phenomenon is slightly different. Inthe case of a ZP interferometer, from the interferencepattern and the known shape of the interfering wave-fronts the range and direction of the displacement canbe determined, whereas the main gain in describingmoire fringes using this formula lies in the possibility
a)
b)
Fig. 1. Equidistant circular moire pattern obtained by change of
of determining the basic grid curves needed to obtain adesired moire pattern.
Ill. Examples
As examples illustrating our derived formula thebasic grids forming CECMP and FMZPP moire pat-terns with variable power were found. In both casesthe grids giving the desired moire pattern by rotationas well as by change of scale are presented. We did notconsider the translation case, since it has already beendiscussed in detail for FMZPPs in the literature.7-10
A. Concentric Equidistant Circular Moire Patterns
In this case Eq. (9) for the change of scale androtation is given by
O91(r,0) Or = alr + a2,
ar,9) a = alr + a2,
where ajr + a2 = m is the CECMP equation.grid equations are then given by
b)Fig. 2. Equidistant circular moire pattern obtained by rotation. Afunction linearly dependent on the angle was added to the basic gridequation ,(x,y) = alr0 + a30: (a) basic grid and (b) resulting moire
pattern.
where f(O) and f(r) are optional functions depending onangle and radius. To demonstrate the basic grid re-sulting from Eq. (lOa), the function 4P(r,O) = (ar/3 +a30 was chosen [Fig. 1(a)], and the resulting moirepattern is presented in Fig. 1(b).
Recently a grid composed from the evolutes of acircle was proposed for obtaining the CECMP bychanging the scale.6 Curves of this grid are as follows:
(rl-rO 1/22 0r2 10/200, (12)
where 00 = 21I/N (N being the number of evolutes) andro is the radius of the circle from which the evolute isunwound. Assuming that (2f/0o)2 << 1, which can beeasily fulfilled, the evolutes can be approximated withthe shape determined by Eq. (a) and the CECMPthus created is given by (r,O) = r 31/0oro). More-over, this particular grid is able to create self-images-an important ability from a practical point of view.
To illustrate the grids creating a CECMP by rota-tion we chose -1 (r,O) = r(aOl/ca + a3), Fig. 2, and CD2(r,O)= 0(alr + a2)/a, Fig. 3. The basic grid equations are nolonger linearly dependent on angle 0 and thereforethey cannot be drawn by subsequent rotation of one
b)Fig. 3. Equidistant circular moire pattern obtained by rotation. Afunction dependent on the radius was added to the basic grid equa-tion (xy) = alr0 + a 2r: (a) basic grid and (b) resulting moire
pattern.
curve by an angle 0 , as in the previous case. As aconsequence, a plot of the grid cannot be drawn with-out discontinuity between dense fringes and coarsefringe areas.
B. Fresnel Moire Zone Plate Patterns
Fresnel moire zone plate patterns have already beensuccessfully used in a three-point alignment techniquefor accurate measurement of deviations of large engi-neering structures over long distances.10-14 Imagingelements of variable focal length and weak opticalpower are necessary to accomplish the task of focusingthe light at a distance of 10 m and more. Practicallythey cannot be made as a refractive element, and untilnow the FMZPP was the only possible zoom element.In all the solutions encountered in the literature onlythe possibility of creating a FMZPP by translation wasconsidered.
Application of Eq. (9) to the ZP zone distribution*(rO) = a1r2 + a2 gives rise to the following solutions,for change of scale by ,B and for rotation by an angle a,namely:
As an example demonstrating the basic grid given by
Eq. (13a) the function 1(r,O) = (air 2/fl) + a30 was
chosen [Fig. 4(a)] and the resulting moire pattern ispresented in Fig. 4(b). To illustrate the grids neces-sary to create a FMZPP by rotation, the functionPl(r,O) = r2(aO/Oa + a3) was chosen-Fig. 5-and42(r,0) 0(alr2 + a2 )Ia-Fig. 6.
IV. Practical Remarks
Constant a2 assumed in all the solutions affects onlythe phase of the created moire fringes and is thereforenondisturbing. Let us now see that the assumptionabout the infinitesimal value of the displacement is notnecessary for the examples presented, except for thosein which moire patterns caused by a scale change areconsidered (and where a2 is not zero).
All the moire patterns were obtained by successivelydrawing two subsequent plots on one sheet of paper bya plotter. In both cases the second plot was drawnwithin the same borders as the first one. In this way ina case of rotation a region of not very distinct moirefringes, which normally occurs in the area where thecoarse part of the first grid overlaps the dense part ofthe second grid, was replaced by fringes of better con-
trast. This approach is advantageous from an illustra-
tive point of view; however, if we want to take practicaladvantage of this method, a variable angle of rotationmust be introduced and a more careful choice of addi-tional functions f(r) should be made, to achieve similardensity of fringes on both sides of the border, as well astheir similar orientation.
In both cases of a FMZPP formed by the change ofscale and by rotation the aperture of the resultingpattern is constant, contrary to the translation case. AFMZPP formed by the change of scale has found anapplication in a simple moire pattern pointing de-vice.17 Its use as an imaging optical element, however,is of great practical difficulty although, because of thelinear dependence from the angle, no problem occursin areas with decreased contrast moire fringes. Itseems that the use of grids producing a FMZPP byrotation is more practical, although there are greaterdifficulties in the design of such a grid, since there isalways a discontinuity border between areas of denseand coarse fringes.
V. Conclusions
A method for directly determining the basic grid,which when superimposed with its displaced counter-part will create an assumed moire pattern, has been
b)Fig. 6. Moire Fresnel ZP pattern obtained by rotation. A functiondependent on the radius was added to the basic grid equation k(xy)
= ar 2 0 + a2r2: (a) basic grid and (b) resulting moire pattern.
presented in this paper. The method considers themost distinct subtractive moire fringes (moire beats)and assumes displacements that are small comparedwith the moire fringe period, although solutions ob-tained in this way are in some cases applicable also forlarge displacements.
The analysis results were confirmed by experiment,and it may be that the more general shape of the basicgrid creating the Fresnel moire zone plate pattern withconstant aperture is of the greatest practical interest,since it has already found use as an imaging element of
small optical power and variable focal length in align-ment of large engineering structures.
This work was supported by the University of Santi-ago de Compostela, Spain, in accordance with theagreement between this University and the CentralLaboratory of Optics, Warsaw, Poland.
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