Unification of formulation of moiré fringe spacing inparametric equation and Fourier analysis methods

MohammadAbolhassani* and MahmoodMirzaeiDepartment of Physics, Arak University, Sardasht, Arak, Postal Code 38156, Iran

*Corresponding author: m-abolhassani@araku.ac.ir

Received 8 May 2007; revised 31 July 2007; accepted 3 August 2007;posted 7 August 2007 (Doc. ID 82438); published 9 November 2007

Inspections of moiré fringe characteristics, such as period and orientation, conventionally are done by twoapproaches; namely, parametric equation and Fourier analysis methods. In some cases these methods yielddifferent results. This inconsistency is removed by revising the derivation of the indicial equation for moiréfringes by the parametric equation method. © 2007 Optical Society of America

OCIS codes: 120.4120, 120.2650, 070.4790.

1. Introduction

The term moiré comes from French, where it refers towatered silk. The moiré silk consists of two layers offabric pressed together. As the silk bends and folds, thetwo layers shift with respect to each other, causing theappearance of interfering patterns. Natural moiré phe-nomena can be seen in daily life, for example, in thefolds of a moving nylon curtain or in looking throughparallel wire-mesh fences. Generally, superposition oftwo or more periodic (or quasi-periodic) structuresleads to a coarser structure, named moiré pattern ormoiré fringe.

In this work we restrict ourselves to two straight-line gratings, where their superposition causes astraight-line moiré pattern. Period and orientation ofthese fringes depend on periods of two gratings andtheir mutual orientation. For a well-defined moiréfringe, period of a grating must be (or almost) an inte-ger multiple of that for the other one, and the anglebetween their bands must be small. Moiré techniquehas been applied widely in different fields of scienceand engineering, such as metrology and optical testing[1–8]. Most of the measurements by moiré techniqueinvolve period measurement of the generated moiréfringes. Thus, knowing the correct relation betweenmoiré fringe spacing and the periods of the superim-posed gratings is of prime importance.

Various methods are suggested in the literature formodeling and analyzing the moiré phenomenon. Thetwo main ones are the parametric (or indicial) equa-tion method and the Fourier analysis approach. Al-though the main idea for inspection of the topic by theparametric equation method is pointed out in theliterature, presentations of this method for moiré pat-tern description are limited to the case that the pe-riods of the two gratings are equal (or almost equal)[1]. Hence in this case, these two models yield thesame results; otherwise, their results are different.

In this study, in addition to deriving the Fourieranalysis results with a different approach, the para-metric equation method is modified such that the re-sults of the two methods match in all cases. Thus ourwork can be considered as a modification of the para-metric equation method introduced in Ref. [1]. It isworth mentioning that the modified parametric equa-tion method and conventional parametric equationmethod will not give any information about the profileand visibility of the moiré fringes.

2. Moiré Pattern Analysis

Figure 1 shows two straight-line gratings. Lines ofthe first are parallel to the y axis, and those of thesecond make small angle � with the y axis. Periods ofgratings are d1 and d2, respectively, and satisfy thefollowing condition:

d1 � Md2; M � Z. (1)0003-6935/07/327924-03$15.00/0© 2007 Optical Society of America

7924 APPLIED OPTICS � Vol. 46, No. 32 � 10 November 2007

By superposition of these two gratings a moiré pat-tern is formed. We review the Fourier analysis andparametric equation methods and make some modi-fications in the latter method for correct recognitionof moiré pattern.

A. Fourier Analysis Method

We study the Fourier analysis method and derive therelation for the moiré fringes period with an approachthat is slightly different from the one explained inRef. [9]. The transmittances of the two gratings aredescribed by

T1(x, y) � �m���

�

am exp�im 2�

d1x�, (2)

T2(x, y) � �n���

�

bn exp�in 2�

d2�x cos � � y sin ��, (3)

where am and bn are Fourier expansion coefficients oftwo gratings. The resultant transmission for overlap-ping is given by the product T1 and T2:

TM � T1(x, y)T2(x, y)

� �m���

�

�n���

�

ambn expi2���md1

�n cos �

d2�x

�n sin �

d2y�. (4)

This function is a linear combination of elementaryfunctions of the form exp�i2��fXx � fYy� . For anyparticular frequency pair �fX, fY� the correspondingelementary function has a frequency f given by [10]

f � �fX2 � fY

2�1�2. (5)

Thus the following frequencies exist in TM:

fmn � �m2

d12 �

n2

d22 �

2mn cos �

d1d2�1�2

. (6)

We need to find the smallest nonzero frequency inthis set for finding the moiré period dM. The requiredfrequency depends on d1, d2 and �. If for m0 and n0 thefrequency fm0n0

has minimum nonzero value, then

dM �1

fm0n0

�d1d2

�n02d1

2 � m02d2

2 � 2m0n0d1d2 cos �. (7)

If Eq. (1) holds and � is a small angle, we must havem0 � �M and n0 � �1; therefore

dM �d1d2

�d12 � M2d2

2 � 2Md1d2 cos �. (8)

B. Modified Parametric Equation Method

The simplest and oldest method for analyzing themoiré pattern is the parametric (indicial) equationmethod. We modify this method such that it can beused for general cases. The parametric equationmethod is based on the lines equations of the originalgratings. If each of the original layers is regarded asan indexed family of lines, the moiré pattern of thesuperposition forms a new indexed family of curves,whose equations can be inferred from the equationsof the original gratings. Lines of the first grating arespecified by

xd1

� h; h � Z, (9)

and those of the second one by

x cos � � y sin �

d2� k; k � Z. (10)

Intersection points of two lines families form a lat-tice in the xy plane. Each point is the intersection ofa line from one grating and a line from the othergrating. The indices of a point, �h, k�, are character-ized by the indices of those two lines, h and k.

In the conventional parametric equation method,moiré fringes are regarded as a family of the indexedlines with the property of

h � k � p; p � Z. (11)

Then moiré fringe period becomes [1]

d1d2

�d12 � d2

2 � 2d1d2 cos �, (12)

which is inconsistent with Eq. (8) for M 1. Thus Eq.(11) needs a modification.

There is an infinite number of line families in thelattice points such that the lines of each family areparallel and equidistant, while in the conventional

Fig. 1. Orientation of the two straight-line gratings in the xyplane. Lines of first grating are parallel to and those of the secondmake a small angle � with the y axis. They have periods d1 and d2,and are indicated by h and k parameters, respectively.

10 November 2007 � Vol. 46, No. 32 � APPLIED OPTICS 7925

parametric equation method, only one family is con-tributed. Moiré fringes is a family of this infinitenumber of families that has the greatest period.

The �M�, N�� line family is defined as the lines join-ing the intersection points of h and k lines whoseindices satisfy

M�h � N�k � p; p � Z, (13)

where M� and N� are constant integers and p runsover the set of integers. For a fixed value of orderedpair �M�, N�� [or equivalently ��M�, �N��], we have aparallel lines family in the xy plane. By substitutingEqs. (9) and (10) in Eq. (13), we get the indicial equa-tion of the �M�, N�� family as

N�x cos � � y sin �

d2� M�

xd1

� p; p � Z. (14)

The distance between two successive lines of thislines family is

d1d2

�N�2d12 � M�2d2

2 � 2M�N�d1d2 cos �. (15)

Due to smallness of � and the condition given in Ref.[1], this expression is maximum when M� � �M andN� � �1. Then the expression given in Eq. (15) re-duces to the one given in Eq. (8). Thus we obtain therelation that the moiré fringe period in the paramet-ric equation method is the same as the one obtainedin Fourier analysis method. Therefore the mentionedinconsistency between two methods is removed.

Both Figs. 2 (a) and 2(b) show two superimposedgratings for the case that the period of one is almosttwice the period of other. The �1, �1� and�2, �1�families are plotted in Figs. 2(a) and 2(b), re-spectively. It is obvious from these figures that the�2, �1� family is the one that is a moiré fringe be-cause it has the greatest period.

3. Conclusions

The parametric equation and Fourier analysis meth-ods in the literature are consistent only when theperiods of the gratings are (almost) equal. We modi-fied the formalism of the parametric equation methodsuch that these two methods are consistent in allcases, i.e., produce the same results. We verify thepredictions of the modified parametric equation andFourier methods by computer simulation and foundthat the results of these methods are consistent withthose of simulation. It is worth mentioning that theresults of the conventional parametric equationmethod are inconsistent with those of computer sim-ulation.

In many experiments by the moiré technique, wemeasure the periods of moiré fringes and relate theirperiods to the parameter of the gratings. It is crucialthat the relations we use be reliable and withoutambiguity. Thus it is important that the two abovementioned methods give us the same results.

The parametric equation method is more suitablein studying the superposition of two structures whenat least one of them is not a straight-line grating.Therefore it is necessary to have a reliable paramet-ric equation method such as ours in these kinds ofstudies.

The authors would like to thank M. Hedayatipourfor his help in editing this manuscript.

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Fig. 2. Two families of infinite �M�, N�� lines families in the latticepoint: (a) �1, �1� line family and (b) �2, �1� line family. The valueof index p in each line family has been calculated by Eq. (13) forcorresponding values of M� and N�.

7926 APPLIED OPTICS � Vol. 46, No. 32 � 10 November 2007