Formulation of moiré fringes based on spatial averaging

Optik 122 (2011) 510–513

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Formulation of moiré fringes based on spatial averaging M. Abolhassani ∗ Department of Physics, University of Arak, Code 38156, Arak, Iran

a r t i c l e

i n f o

Article history: Received 25 August 2009 Accepted 26 March 2010

PACS: 42.30.Ms 42.30.SyMoiré fringe Spatial averaging Grid noise

a b s t r a c t The use of moiré technique in measurements often involves locating the position of moiré fringe and in some cases determining its profile. Due to intensity fluctuation in the fringe pattern, these measurements are accompanied by some errors. It is possible to define a smoothed version of the original fringe pattern and then formulate the related subject in accordance with the characteristics of this new pattern. This procedure reduces these types of errors and gives a well-defined profile of the fringes. In this paper a formulation of moiré phenomenon based on spatial averaging is presented which, without ignoring any of frequencies, leads to a smooth profile. © 2010 Elsevier GmbH. All rights reserved.

1. Introduction The study of moiré effect and its applications is an important area in applied optics [1,2]. Moiré pattern is generated by superposition of two periodic or quasi-periodic structures, and has a coarser structure compared to the original ones. A frequency appears on this pattern that is much smaller than the fundamental frequencies of the original structures. According to experimental results [3], moiré pattern is defined as a linear combination of this frequency component and its harmonics. In addition to these frequencies there exist other frequencies in the generated pattern. Due to the presence of these components the pattern profile becomes nonperiodic in the direction normal to the moiré fringes, fluctuates and depends on the location where it is determined. These frequency components are called grid noise. Whether the noise structure is observable or not depends on the characteristics of the detecting system. In another paper [4] it has been experimentally verified that scanning of the resultant moiré patterns by a slit whose width is almost equal to the grating period in the direction perpendicular to the moiré fringes, leads to a profile which is in good agreement with the one which is achieved by formula of the pattern without its grid noise terms. These experiments have been performed by several two perfect binary gratings, which produce straight-line moiré fringes, with various opening-ratio pairs. It is shown mathematically in this paper that the longitudinal averaging of intensity along each curve parallel to the moiré fringe and then devoting this amount to all points of that curve leads to a

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pattern that its function is exactly identical to the function that has been derived in Ref. [4]. In other words, the equation introduced in Ref. [4] is an accurate one provided that the real pattern is altered by a pattern where its intensity in every point is equal to the average intensity of all points belonging to a curve parallel to the moiré fringes and passing through that point in the real pattern. This procedure is equivalent to the one that has been introduced in Ref. [4] with an infinitesimal scanning slit width; meanwhile this procedure is is not restricted to straight-line moiré fringes but it is applicable to curved moiré fringes too. By this procedure we can get a pattern that is smooth, and also can relate its characteristics to the measuring parameters with a more accuracy. This technique has been used in moiré metrology to some extent in the past [5]. In many applications of moiré technique, such as moiré topography, grid noise removal is indeed an essential problem. For this purpose different methods are used. In one of these methods grid noise can be removed by simultaneously translating the two gratings in their own plane which is equivalent to time averaging of moving pattern [6]. Another method equivalent to this method has been introduced in which grid noise can be removed without moving any part [7]. Grid noise removal has been performed also by spatial averaging along the fringes [8], but this operation has been executed by using a computer and has not been formulated particularly in moiré fringe technique. It seems that a formulation of moiré fringes based on spatial averaging can affect on simplicity and accuracy of quantitative analysis of experimental results. Another formulation of moiré fringes based on spatial averaging through cross-correlation function of two grating functions has been introduced in the literature [9], but the present approach is a more natural and obvious one. In the next section, first straightline and then curved moiré fringes are treated by this method. Though the first case can be thought of a special case of the sec-

M. Abolhassani / Optik 122 (2011) 510–513

511

Indicial equation of these fringes is [4] M x cos  − y sin  − x = p; d2 d1

p ∈ Z.

(4)

These fringes make an angle ˛ with x-axis by denoting Eq. (4) is equal to tan ˛ = cot  −

Md2 . d1 sin 

(5)

We turn the coordinate system as much as ˛ −  / 2 so that moiré fringes become parallel to y-axis. This new system is called x y . Coordinates of a point in the two systems are related to each other as follows: x = x sin ˛ + y cos ˛,

y = −x cos ˛ + y sin ˛.

(6)

Transmittance of the two superimposed gratings in the new coordinate system is



∞ ∞  

T (x , y ) =

am bn exp

i2

m=−∞n=−∞

Fig. 1. Orientation of the two straight-line gratings in xy plane. Lines of the first grating are parallel to and those of the second make a small angle  with y-axis. They have periods d1 and d2 , respectively.

+ i2

m

cos ˛ +

d1

m d1



n sin( + ˛) x d2

sin ˛ +

 

n cos( + ˛) y d2

.

(7)

From Eq. (5) we get ond, but owing to the possibility of precise analysis of the first case, both cases are treated separately. In Section 3 correctness of this formulation is evaluated by computer-generated gratings.

sin ˛ =

d1 cos  − Md2 , D

cos ˛ =

sin( + ˛) =

d1 − Md2 cos  , D

cos( + ˛) =

2. Theoretical approach Consider two linear amplitude gratings T1 and T2 with periods d1 and d2 , respectively. Lines of the first grating are parallel to yaxis and the lines of the latter make a small angle  with y-axis, as shown in Fig. 1. Besides we suppose

where D = gives T (x , y ) =



M ∈ N.

∞ ∞  

× exp

In this case the transmittances of the two gratings can be described by ∞ 

am exp

m=−∞ ∞

T2 (x, y) =



bn exp

n=−∞

 i2m 

d2



× exp i2

T (x , y )y =

x

d1

 i2n

=

(2)



∞ ∞  



i2

 m

d1

 m + Mn D

sin  D



x

 

sin  y

.

(9)

T (x , y ) dy 0

∞ ∞  

× exp

am bn

+

(D/(sin ))

am bn



(x cos  − y sin ) ,

m=−∞n=−∞

× exp

nd12 − Mmd22 m − Mn cos  + D d1 d2 D

m=−∞n=−∞

where ai and bi are Fourier coefficients of two gratings and a−i = a∗i , b−i = b∗i . By superposition of two gratings moiré pattern is appeared which has the following transmission function T (x, y) = T1 T2 =



i2

This function is not periodic in x direction (normal to moiré fringes) but in y direction is. Now its average is calculated in y direction

2.1. Both gratings are perfect

T1 (x, y) =

(8)

am bn



(1)

In this treatment two cases are considered: (a) both gratings are perfect, which results in straight moiré fringes and (b) one of them is imperfect, which results in curved moiré fringes.

Md2 sin  , D

d12 + M 2 d22 − 2Md1 d2 cos . Eq. (8) put into Eq. (7)

m=−∞n=−∞

d1 ∼ = Md2 ;

d1 sin  , D

i2

sin  × D





nd12 − Mmd22 m − Mn cos  + D d1 d2 D



(D/(sin ))

exp

i2

 m + Mn

0

D

x

 

sin  y

dy . (10)

But



n cos  x − d2

n d2

  sin  y

.

(3)

sin  D





(D/(sin ))

exp 0

i2

 m + Mn D

 

sin  y

dy = ım,−Mn ,

(11)

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M. Abolhassani / Optik 122 (2011) 510–513

thus

where l is the averaging length on  r . Combination of Eqs. (17) and (18) by eliminating y coordinate gives

Tave (x ) ≡ T (x , y )y





∞

=

a−Mn bn exp

 i2n

n=−∞

=

∞ 

−2M cos  + D



d12

+ M 2 d22

x

d1 d2 D

n=−∞

dM





(12)

d1 d2 d12 + M 2 d22 − 2Md1 d2 cos 

.

(13)



Tave (x ) =

a∗Mn bn

n=−∞





n  exp i2 x . dM

(14)

am exp

∞ ∞  

× exp



i2

m

d1



2

dl =

1 + (dy/dx) dx =

∂g M = d1 ∂x



x−

∂f ∂x



−

2

(21)

M ∂f sin  ∂g =− + . d1 ∂y d2 ∂y

cos  , d2

d12 + M 2 d22 − 2Md1 d2 cos 

n d2

d1 sin 



cos  x −

n

(22)

d2

.

By choosing the lowest frequency (nonzero) and ignoring the other ones, loci of points having the same value are derived as cos  sin  M (x − f ) − x+ y = r, d1 d2 d2

r ∈ R.

D dx. d1 sin 

m + Mn f d1



D d1 sin 





d1

exp i2 0

(23)



m + Mn x dx d1 (24)

Integrand in expression (24) is a periodic function with period d1 . Thus its appropriate limits are 0 and d1 . This choice implies that l = D / sin  (note to Eq. (23)). Eq. (24) put into Eq. (19) gives ∞ 

a∗Mn bn exp(−i2nr).

(25)

n=−∞

  sin  y

dx =

D ım,−Mn . = sin 

Tave (r) =

(x − f ) +

2

(∂g/∂x) + (∂g/∂y) dx. |∂g/∂y|

But from Eq. (17) we derive

(15)

(16)

g(x, y) ≡

Differential of arc length for the curve is



am bn

m=−∞n=−∞

(20)

The integral in Eq. (19) is approximately equal to

where f(x, y) is in-plane distortion of the first grating and is assumed to be a slowly-varying function. Superimposition of two grating leads to moiré fringes having the transmittance T (x, y) = T1 T2 =

(19)

∂g/∂x dy . =− dx ∂g/∂y

exp −i2

[x − f (x, y)],

d1

m=−∞

m + Mn (x − f ) dl. d1

For a given point in the pattern, the corresponding curve is co-direction with vector ( ∂ g / ∂ y)î −(∂ g / ∂ x)ˆj. Then the fringe inclination at a point is equal to

dl ≈

In this case, transmittance of the first grating is described as T1 (x, y) =



r



 i2m

 exp i2

By ignoring the terms proportional to ∂ f / ∂x and ∂ f / ∂y we have

2.2. One of the gratings is imperfect

∞ 

am bn exp(−i2rn)

m=−∞n=−∞



This period that arises is equal to the moiré fringe period [4,10] as should be so. Eq. (12) is rewritten versus moiré fringe period which is more familiar. ∞ 

∞ ∞  



As it is expected, this average results in a periodic function of x with period d1 d2 = = D

1 l ×

nD  exp i2 x . d1 d2

a∗Mn bn

Tave (r) =

This result is similar to that Eq. (14). 3. Computer simulation and results To verify the correctness of this formulation, it is examined, which one of two linear gratings is imperfect. The two linear gratings are made by computer and superimposed to each other; moiré

(17)

For each known value of r (no essentially integer) Eq. (17) represents the equation of a curve. Now we calculate average value of T(x, y) along the curve  r , with equation g(x, y) = r, and devote this value to all points of that curve. In this way a smooth and free of fluctuation pattern is achieved. 1 Tave (r) = l =

1 l

 T (x, y)dl r ∞ ∞  

 am bn

m=−∞n=−∞



× exp i2

m d1

r

(x − f )+

n d2

 cos  x−

n d2

  sin  y

dl, (18)

Fig. 2. Moiré pattern obtained by superposition of two linear amplitude gratings. One of the gratings is distorted.

M. Abolhassani / Optik 122 (2011) 510–513

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Thanks to simple and inexpensive accessibility to electronically recording devices, like CCD, information of a pattern may be recorded and saved as a digital file and then processed. Therefore it is practically easy to smooth the pattern of the original one by averaging its intensity along the fringes. Now we have a reliable and exact equation for smoothed pattern (except the approximations used in determining Eq. (23) and expression (24)). Thus, we can rely on the results which are obtained by moiré metrology more than ever. In order to complete this work, one can execute averaging process practically as is given in Ref. [8], and then compare with the mathematical results of this work. References

Fig. 3. Smoothed version of the moiré fringe pattern shown in Fig. 2.

fringes appear accompanied by grid noise, as in Fig. 2. Fig. 3 shows a pattern that is generated by using Eq. (25). Fourier coefficients used in Fig. 3 are similar to those of superposed gratings in Fig. 2. Comparison of Figs. 2 and 3 shows that the generated pattern by using Eq. (25) is similar to the original pattern but its grid noise has been removed. This similarity verifies that the presented formalism in this work is mathematically correct. 4. Conclusions It was shown mathematically in this paper that formulation of moiré pattern based on spatial averaging of intensity along moiré fringes leads to an equation that is similar to the one which is obtained by ignoring grid noise in the real pattern.

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