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Moiré fringes
Optics and Lasers in Engineering 3 (1982)15—27
MOIRE FRINGES
SHUNSUKE YOKOZEKI
Department of Applied Physics, Faculty of Engineering, Osaka University, Yamada-oka 2-1, Suita, Osaka, 565, Japan (Received: 2 November 1981)
ABSTRACT
This review paper clarifies the characteristics of the incoherent-and-multiplicative type moire fringe phenomenon, presents examples of the application of these characteristics to measurement and explains the profile prediction method and the sharpening conditions of moire fringes formed by two binary gratings.
INTRODUCTION
By superimposing two periodic structures under the proper conditions one can obtain a new fringe pattern. This pattern is known as a moire fringe pattern. By looking through the folds of a lace curtain or the railings of a bridge, viewing television, seeing a screened picture in a book and so on, one can easily find examples of moire fringes. The moire fringe phenomenon has obviously been generally known for some time. Ever since Lord Rayleigh1 suggested the ruling test by using the moire fringe phenomenon, this phenomenon has been applied to measurement—in meteorology,2 in strain analysis,3’4 in topography5’6 and in interferometry.7’8 Moire fringes have several characteristics and moire techniques have been evolved to utilise these characteristics effectively. In this paper, we will derive the basic moire fringe formula, explain the characteristics of moire fringes by using that formula, give examples of moire techniques and then describe the profile prediction method and the sharpening conditions to improve the visibility of moire fringes between two binary gratings. 15 Optics and Lasers in Engineering Publishers Ltd, England, 1982 Printed in Northern Ireland
0143-8166/82/0003-0015/$02.75—-©
Applied Science
16
SHUNSUKE YOKOZEKI 9
FORMATION AND CHARACTERISTICS OF MOIRE FRINGES
Moire fringes may be divided into two types—the coherent and the incoherent. Coherent type moire fringes are produced by two illuminating beams generating two superimposed gratings2 or by overlapping real time interference fringes on a recorded fringe pattern.’0 In these cases, we can observe the moire fringes only by means of optical filtering. Incoherent type moire fringes are formed by superimposing two grating-like light irradiance distributions by photographic or video techniques,7’8 or by incoherently illuminating two superimposed transparent gratings. The coherent type moire fringes can be regarded as the interference fringes between two diffraction waves. Therefore, one can treat the formation of coherent type moire fringes in the same way as that of holographic interference fringes. Holographic interferometry has been studied in detail, so we will consider only incoherent type moire fringes in this paper. The incoherent type moire fringe pattern can be formed in three ways—by additive, subtractive and multiplicative superposition.11 The additive and subtractive types of moire pattern are usually converted into beat patterns of low spatial frequency by taking advantage of various non-linear processes to observe the moire fringes easily. It is difficult to quantitatively study the non-linearity of the process utilised. Therefore, in this paper the analysis will be restricted to the case of the multiplicative type moire pattern. Multiplicative type moire fringes can be formed by incoherently illuminating two transparent superimposed gratings. The moire procedures with multiple gratings have few advantages.1113 However, for simplicity, we will consider only the moire pattern between two gratings. Generally, we can obtain moire fringes by superimposing various periodic structures,14 but we make use of straight line gratings where images are deformed by the object or the measurement apparatus. Therefore, we will discuss only the incoherent-and-multiplicative type moire fringes between two straight-line gratings. Let us study two examples of moire fringe formation. Figure 1 shows the moire fringe patterns obtained by overlapping two gratings of the same period, d, for various rotation angles, 6, with calculated moire fringe spacing, D, almost equal to the experimental value. When D > d, we can distinguish the moire fringes in spite of the observable lines of the gratings. Figure 2 illustrates the moire fringes between two binary gratings and the fringe profile obtained geometrically. This profile can be expressed by a triangular waveform periodic function. This suggests that the moire fringe formula will contain the higher harmonics. The experimental results shown in Figs. 1 and 2 imply the procedure for defining the incoherent-and-multiplicative type moire fringe formula as follows: (i) analyse the spatial frequency of the resultant transmittance of two transparent gratings superimposed, (ii) find the component whose spatial frequency is lower than that of two gratings and its higher harmonic compo-
17
MOIRE FRINGES
~
____
~fflffllluIfflHllfffflffll
______________
~Wfflllnwm~
__
&LffllillffllffuJJ~mhIwJfflIwu~,ffl~t~fflJffl
__________________
~ 0=5~,
D=lOd
B=3~,
111111••
..m
0=10,
/
D=6d
Fig. I.
=
~
0=60,
-—
Dd
_________________________
1/
I
0
D=2d
D
—
=
3d
0
=
90°, D
=
0.7d
Moire fringe patterns between two straight line gratings of the same period, d, for various
rotation angles, 0, with calculated moire fringe spacing, D.
18
SHUNSUKE YOKOZEKI
hI
Fig. 2.
Schematic representation of the moire fringe profile formed by two binary gratings with the same period and an opening ratio of 05.
nents and (iii) define the basic moire fringe formula with the sum of the bias term and the terms representing the spatial frequency components found in (ii). We will derive the basic formula of moire fringes between two transparent gratings, T1 and T2, whose parameters are illustrated in Fig. 3. In the field of measurement, sinusoidal or binary gratings are usually adopted. We can, therefore, suppose, for simplicity, that the line profiles of the two gratings, T1 and T2, are symmetrical with respect to the origin. The transmittances, T1 and T2, of two gratings may be written as follows: 12~rm ‘I T1(x, y)=a0+~ a~cos 1—~-—[x—f1(x, y)—x1]j T2(x, y) = b0 + ~ b,, cos (~i~){[x f2(x, y) —
—
x2] cos 0— y sin 0}
(1) (2)
where a0, am, b0 and b,5 are Fourier coefficients defining the line profile of two gratings, f1(x, y) and f2(x, y) express the local deformation at the point (x, y) in the direction of the x-axis, x1 and x2 are the displacements of the gratings and d1 and d2 the periods of the gratings. When the two gratings are superimposed, the resultant transmission function, T(x, y), may be given by T1(x, y)T2(x, y). The function will be given by the sum of the bias term and the terms expressing a multiplicity of fringe systems. Considering the conditions characteristic of the
19
MOIRE FRINGES
V -)~E-d1
Fig. 3.
Two straight line gratings.
moire fringe formation: (a) the angle, 0, being very small (b) the function f1(x, y) and f2(x, y) changing slowly d1=Md2
(3)
where M is an integer, we can estimate the spatial frequency of every fringe system of T(x, y). According to the definition given by the experimental results of Figs. 1 and 2, we can obtain the basic moire fringe formula, Tm(X, y): Tm(x,
y)= a0b0+ ~ a~b,cos 2iri x ~[x—f1(x,
y)—x1]—~-{(x—f2(x,y)—x2) cos 0—y sin oi}
(4)
This formula expresses all geometrical parameters of moire fringes—the fringe profile, the fringe spacing, the fringe orientation and the local fringe deformation. By writing X for the argument of the term for i 1, we can obtain the =
20
SHUNSUKE YOKOZEKI
formula T~(X)describing only the moire fringe profile as follows: T9 (X) = a0b0 + ~ a~,b,cos (2iriX)
(5)
We can call this equation the moire fringe profile formula. Assuming that the first term argument (i = 1) of eqn. (4) is equal to 2irr, we can obtain the indicial formula: [x
—
ft (x,
y) x,]
— ~-
{[x
—
f2(x, y) x2] cos 0— y sin 0} = —
r
(6)
where r is an integer. This formula expresses the geometrical parameters with the exception of fringe profile. We can explain the characteristics of moire fringes by using eqns. (4), (5) and (6) as follows: Case 1, (x,=x2=0,f1~0,f2=0) In this case, the indicial formula can be written as follows: for
0=0
and
d,cosO—d2M sinO M x—--——y+—f1(x,y)+r=0 d,d2 d2 d1
d,=Md2
(7)
fortheothercase.
(8)
Equation (7) shows that the moire fringes for 0 = 0 and d, = Md2 are the contours of the deformation function f,(x, y). The moire fringe pattern represented by eqn. (8) is of the deformed grating type; the fringe spacing is 2d~—2Md d1d2/(d~+M 1d2cos 0)L’2 and the local deformation at a point (x, y) 2d~—2Md is Md2f1(x, y)/(d~+M 1d2cos0)~2. Therefore, the relative moire fringe local deformation, Dr. is: Dr=
Deformation M =—f1(x,y) Fringe spacing d1 .
.
(9)
This relative deformation is obtained as the first measured value in many cases of moire measurement. Equations (7) and (8) show that the moire techniques transform the deformed grating pattern into the contour pattern or the deformed grating pattern of variable fringe spacing. This phenomenon is one of the useful characteristics of moire fringes. (7), (8) and (9) 5’6 Equations moire interferometry7 andexpound Talbot the basic principle of function moire topography, interferometry.8 The f 1(x, y) is proportional to the height of the object’s surface in the case of moire topography, to the wavefront aberration in moire interferometry, and to the partial derivative of the wavefront aberration
21
MOIRE FRINGES
in Talbot interferometry. Equations (7) and (9) imply that the increase in sensitivity will be proportional to the number M. This phenomenon is the principle of the fringe multiplication method.15 Case 2, (x 1=x2=0,f1~0,f2~O) The indicial formula for this case will have the form: ~~2(x,y)—f1(x,y)]=r
for
0=0
and
d1=Md2
(10)
d1cosO—d2M sin0 FM cosO 1 x—---—-— y+~—f1(x,y)—-————f2(x, y) I+r=0 for the other case. d1d2 d2 Ld1 d2 i (11) When 0=0 and d,=Md2, we can say that: M cos0 M —ft(x, y)——-———f2(x, y)=—[f1(x, y)—f2(x, y)] d1 d2 d1
(12)
It is shown by eqns. (10), (11) and (12) that we can perform the optimal subtraction by moire techniques. characteristics haveresulting been applied to 68 These and eliminating the error from the detecting the change of objects imperfection of the interferometer.’6’17 Case 3, (f 1=f2=0, 0=0, d1=Md2) The basic formula for this case can be expressed as follows: Tm(x2x1)
a0b0+~
cos 2~ri[~(x2_xi)]
(13)
This equation shows that the irradiance just behind the two gratings will vary periodically with the relative displacement of two gratings, and expounds the basic principle of the commercial linear displacement transducer. Case 4, (0”0,d1=Md2,x2—x1=v0t) When we adopt the system in which the functions f1(x, y) and f2(x, y) do not change in spite of the displacement of the gratings, the basic formula can be written as follows: Tm(X,
y)= a0b0+ ~
~
cos 2iri
~[f2_fi]+~vot}
(14)
where v0 denotes a constant speed. The transmitted irradiance at a point (x, y) will be changed cyclically and the measured value [f2(x, y)—f1(x, y)] will be given by the phase of the cyclical change of irradiance. This phenomenon may be called fringe scanning and is very useful for moire fringe analysis using 18 electronic techniques.
22
SHUNSUKE YOKOZEKI 19
PROFILE PREDICTION AND SHARPENING OF MOIRE FRINGES
In many measurement methods utilising moire techniques, the information on the subject under test is obtained as fringe patterns and the positions of the moire fringes must be determined with great accuracy. The accuracy will be improved as the moire fringes are made sharper. Moire fringe sharpening15 is, therefore, one of the principal techniques of moire measurement. It is difficult to derive the general conditions for moire fringe sharpening, but it will become possible if we limit our consideration to the case of moire fringes produced by two binary gratings (composed of periodically and alternately arranged transparent and opaque lines). Assuming that the transparent line widths of two binary gratings are h, and h 2, and the periods d1 and d2, respectively, we can define the opening ratios, 01 and 02, as follows: 01=h,/d1,
02=h2/d2
(15)
By using eqn. (5) and the opening ratios 01 and 02, the profile formula for the moire fringes between two gratings can be written as: T0(X) = N02/M + (1/M){0302 + 20302
~ [sinc (k03) sinc (kO2fl cos (2irkX)
}
(16)
where M01=N+03, 03 is a decimal, N an integer and sinc(x)= (sin lrx)/(Trx). On the other hand, we consider the periodic function, F(X), represented by
f(X)
Fig. 4.
Definition of function f(X).
23
MOIRE FRINGES
the function, f(X) (Fig. 4): F(X)= m~os f(X- m) =
(A/S){S(S
+ P)+ 2S(S + P) n~1sine
(nS) sine [n(S
+ P)] cos
(2ITnX)} (17)
where m and n are integers. The profile of this function is illustrated in Fig. 5. By considering the analogy between eqns. (16) and (17), we can obtain the relationship between the moire fringe profile parameters (Fig. 6) and the binary grating parameters (0~,02, 03, M and N) as illustrated in Table 1. A-TYPE
2S+P1
F(X)
-
22~
(03+021)
Wo
F(X)
B-TYPE
2S+P>1
(03+02>1)
S’=l-(P+S) S”=P+2S-1 B=(P+2S-1)A/S
A
L>~j
/
Fig. 5.
Function F(X).
B
24
SHUNSUKE YOKOZEKI
><
x
I-.
I—
E
C
~~Q.~’LIR~X Parameters of the moire fringe profile.
Fig. 6.
By adopting the moire fringe profile prediction method using Fig. 6 and Table 1, we can derive the conditions for making the moire fringe constrast unity, the value of Tmax as large as possible and the fringe width (WD or WB of Fig. 5) thinner as follows. Conditions for Dark Fringe Sharpening 03+02= 1,
N = 0, and
02
—~
1
for 03 ~ 02, 03 —+ 1
for 03>02 (18)
Conditions for Bright Fringe Sharpening N=0
and
03=02—+0
(19)
where the arrow sign of A B means that the moire fringe width can be made thinner as the value of A approaches B. The arrow signs are used for deducing the moire fringe sharpening conditions. This means that there is a limit in the fringe sharpening. The first reason for the limit is that the production accuracy of the gratings is finite. The second is that the value of Tmax becomes smaller with fringe sharpening. —*
TABLE 1 RELATIONSHIPS BETWEEN PROFILE PARAMETERS
Parameters Tmax
03
02
(0
3-*N02)/M
03>02 (1 +N)02/M
Tmm
N02/M (03+02—1 +No2)IM
N02/M (03+02— 1+N02)/M
Q
O2~O3
O3~O2
R
03 1~O2
02 1—03
O3+O2~1 03+02>1
O3+O2~1 03+02>1
MOIRE FRINGES
25
Fig. 7. Examples of moire fringe sharpening. The parameters of two binary gratings: (top) O1=02=O’9 and M= 1; (centre) 01 =O~9,O~=O1 and M=1; (bottom) O5=03=O~1 and M=1.
26
SHUNSUKE YOKOZEKI
We prepared the transparent-and-binary type grating with the same period (d1 = d2, M = 1, N = 0, 01 = 03) and the opening ratios of 0~9 and 01 by photographic techniques. We tried moire fringe sharpening with these gratings. The results are shown in Fig. 7. The fringe contrast of Fig. 7(top) is much less than unity. This result is predicted from Fig. 6 and Table 1. Dark fringe sharpening is observed in Fig. 7(centre) and bright fringe sharpening in Fig. 7(bottom). The grating parameters for the case of Figs. 7(centre) and (bottom) meet the fringe sharpening conditions (eqns. (18) and (19), respectively). SUMMARY
We have derived the basic moire fringe formula, the moire fringe profile formula and the indicial formula expressing the geometrical parameters with the exception of the fringe profile. By using these formulae, we have explained the characteristics of the moire phenomenon as follows. The moire phenomenon transforms the deformed grating pattern into the contour line pattern or the variable period deformed grating pattern. This phenomenon can perform optical substraction. Fringe scanning is possible by moving two gratings in relation to each other. By clarifying these characteristics, it might be possible to find new applications for the moire phenomenon. We have presented a Table showing the relationship between the profile parameters and the sharpening conditions of moire fringes produced by two binary gratings. The moire fringe profile prediction method, which uses the Table and the moire fringe sharpening conditions, may be very useful for measurements using the static moire fringe pattern.
REFERENCES
1.
LORD RAYLEIGH, On the manufacture and theory of diffraction gratings, Phil. Mag. S.4, 47
(1874) 81—93. J. M. BURCH, The meteorological applications of diffraction gratings. In: Progress in optics Vol. 2, E. Wolf (Ed.), North-Holland, Amsterdam, 1963, 73—108. 3. P. S. THEOCARIS, Moire fringes in strain analysis, Pergamon Press, Oxford, 1969. 4. A. J. DURELLI and V. J. PARKS, Moire analysis of strain, Prentice-Hall, Englewood Cliffs, N.J., 1970. 5. D. M. M~s.tows,W. 0. JOHNSON and J. B. ALLEN, Generation of surface contours by moire patterns, AppI. Opt., 9 (1970) 942—7. 6. H. TAKASAKI, Moire topography, Appl. Opt., 9 (1970) 1467—72. 7. S. Y0K0zEKI and S. MIHARA, Moire interferometry, Appl. Opt., 18 (1979) 1275—80. 2.
MOIRE FRINGES 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
27
S. Y0K0zEKI, Electronic Talbot interferometer, Optics and Lasers in Engineering, 2 (1981) 13—19. S. YOKOZEKI, Y. KUSAKA and K. PATORSKI, Geometric parameters of moire fringes, Appl. Opt., 15 (1976) 2223—7. C. POLHEMUS and C. CHOCOL, Method for substracting phase errors in an interferometer, Appl. Opt., 10 (1971) 441—2. 0. BRYNGDAHL, Characteristics of superimposed patterns in optics, J. Opt. Soc. Am., 66 (1976) 87-94. 0. BRYNGDAHL, Moire: Formation and interpretation, J. Opt. Soc. Am., 64(1974)1287—94. 0. BRYNGDAHL, Moire and higher grating harmonics, J. Opt. Soc. Am., 65 (1975) 685—94. G. OSTER, M. WASSERMAN and C. ZWERLING, Theoretical interpretation of moire patterns, J. Opt. Soc. Am., 54 (1964) 169—75. D. POST, Sharpening and multiplication of moire fringes, Exp. Mech., 7 (1967) 154—9. S. YOKOZEKI and T. SUZUKI, Modified double-beam interferometer using the moire method, Appl. Opt., 11 (1972) 446—8. S. YOKOZEKI and T. SUZUKI, Shearing interferometer using the grating as the beam splitter. Part 2, Appl. Opt., 10 (1971) 1690—3. 0. INDEBETOUW, Profile measurement using projection of running fringes, Appl. Opt., 17 (1978) 2930—3. S. YOKOZEKI and K. PATORSKI, Moire fringe profile prediction method and its application to fringe sharpening, Appl. Opt., 17 (1978) 2541—7.
MOIRE FRINGES
SHUNSUKE YOKOZEKI
Department of Applied Physics, Faculty of Engineering, Osaka University, Yamada-oka 2-1, Suita, Osaka, 565, Japan (Received: 2 November 1981)
ABSTRACT
This review paper clarifies the characteristics of the incoherent-and-multiplicative type moire fringe phenomenon, presents examples of the application of these characteristics to measurement and explains the profile prediction method and the sharpening conditions of moire fringes formed by two binary gratings.
INTRODUCTION
By superimposing two periodic structures under the proper conditions one can obtain a new fringe pattern. This pattern is known as a moire fringe pattern. By looking through the folds of a lace curtain or the railings of a bridge, viewing television, seeing a screened picture in a book and so on, one can easily find examples of moire fringes. The moire fringe phenomenon has obviously been generally known for some time. Ever since Lord Rayleigh1 suggested the ruling test by using the moire fringe phenomenon, this phenomenon has been applied to measurement—in meteorology,2 in strain analysis,3’4 in topography5’6 and in interferometry.7’8 Moire fringes have several characteristics and moire techniques have been evolved to utilise these characteristics effectively. In this paper, we will derive the basic moire fringe formula, explain the characteristics of moire fringes by using that formula, give examples of moire techniques and then describe the profile prediction method and the sharpening conditions to improve the visibility of moire fringes between two binary gratings. 15 Optics and Lasers in Engineering Publishers Ltd, England, 1982 Printed in Northern Ireland
0143-8166/82/0003-0015/$02.75—-©
Applied Science
16
SHUNSUKE YOKOZEKI 9
FORMATION AND CHARACTERISTICS OF MOIRE FRINGES
Moire fringes may be divided into two types—the coherent and the incoherent. Coherent type moire fringes are produced by two illuminating beams generating two superimposed gratings2 or by overlapping real time interference fringes on a recorded fringe pattern.’0 In these cases, we can observe the moire fringes only by means of optical filtering. Incoherent type moire fringes are formed by superimposing two grating-like light irradiance distributions by photographic or video techniques,7’8 or by incoherently illuminating two superimposed transparent gratings. The coherent type moire fringes can be regarded as the interference fringes between two diffraction waves. Therefore, one can treat the formation of coherent type moire fringes in the same way as that of holographic interference fringes. Holographic interferometry has been studied in detail, so we will consider only incoherent type moire fringes in this paper. The incoherent type moire fringe pattern can be formed in three ways—by additive, subtractive and multiplicative superposition.11 The additive and subtractive types of moire pattern are usually converted into beat patterns of low spatial frequency by taking advantage of various non-linear processes to observe the moire fringes easily. It is difficult to quantitatively study the non-linearity of the process utilised. Therefore, in this paper the analysis will be restricted to the case of the multiplicative type moire pattern. Multiplicative type moire fringes can be formed by incoherently illuminating two transparent superimposed gratings. The moire procedures with multiple gratings have few advantages.1113 However, for simplicity, we will consider only the moire pattern between two gratings. Generally, we can obtain moire fringes by superimposing various periodic structures,14 but we make use of straight line gratings where images are deformed by the object or the measurement apparatus. Therefore, we will discuss only the incoherent-and-multiplicative type moire fringes between two straight-line gratings. Let us study two examples of moire fringe formation. Figure 1 shows the moire fringe patterns obtained by overlapping two gratings of the same period, d, for various rotation angles, 6, with calculated moire fringe spacing, D, almost equal to the experimental value. When D > d, we can distinguish the moire fringes in spite of the observable lines of the gratings. Figure 2 illustrates the moire fringes between two binary gratings and the fringe profile obtained geometrically. This profile can be expressed by a triangular waveform periodic function. This suggests that the moire fringe formula will contain the higher harmonics. The experimental results shown in Figs. 1 and 2 imply the procedure for defining the incoherent-and-multiplicative type moire fringe formula as follows: (i) analyse the spatial frequency of the resultant transmittance of two transparent gratings superimposed, (ii) find the component whose spatial frequency is lower than that of two gratings and its higher harmonic compo-
17
MOIRE FRINGES
~
____
~fflffllluIfflHllfffflffll
______________
~Wfflllnwm~
__
&LffllillffllffuJJ~mhIwJfflIwu~,ffl~t~fflJffl
__________________
~ 0=5~,
D=lOd
B=3~,
111111••
..m
0=10,
/
D=6d
Fig. I.
=
~
0=60,
-—
Dd
_________________________
1/
I
0
D=2d
D
—
=
3d
0
=
90°, D
=
0.7d
Moire fringe patterns between two straight line gratings of the same period, d, for various
rotation angles, 0, with calculated moire fringe spacing, D.
18
SHUNSUKE YOKOZEKI
hI
Fig. 2.
Schematic representation of the moire fringe profile formed by two binary gratings with the same period and an opening ratio of 05.
nents and (iii) define the basic moire fringe formula with the sum of the bias term and the terms representing the spatial frequency components found in (ii). We will derive the basic formula of moire fringes between two transparent gratings, T1 and T2, whose parameters are illustrated in Fig. 3. In the field of measurement, sinusoidal or binary gratings are usually adopted. We can, therefore, suppose, for simplicity, that the line profiles of the two gratings, T1 and T2, are symmetrical with respect to the origin. The transmittances, T1 and T2, of two gratings may be written as follows: 12~rm ‘I T1(x, y)=a0+~ a~cos 1—~-—[x—f1(x, y)—x1]j T2(x, y) = b0 + ~ b,, cos (~i~){[x f2(x, y) —
—
x2] cos 0— y sin 0}
(1) (2)
where a0, am, b0 and b,5 are Fourier coefficients defining the line profile of two gratings, f1(x, y) and f2(x, y) express the local deformation at the point (x, y) in the direction of the x-axis, x1 and x2 are the displacements of the gratings and d1 and d2 the periods of the gratings. When the two gratings are superimposed, the resultant transmission function, T(x, y), may be given by T1(x, y)T2(x, y). The function will be given by the sum of the bias term and the terms expressing a multiplicity of fringe systems. Considering the conditions characteristic of the
19
MOIRE FRINGES
V -)~E-d1
Fig. 3.
Two straight line gratings.
moire fringe formation: (a) the angle, 0, being very small (b) the function f1(x, y) and f2(x, y) changing slowly d1=Md2
(3)
where M is an integer, we can estimate the spatial frequency of every fringe system of T(x, y). According to the definition given by the experimental results of Figs. 1 and 2, we can obtain the basic moire fringe formula, Tm(X, y): Tm(x,
y)= a0b0+ ~ a~b,cos 2iri x ~[x—f1(x,
y)—x1]—~-{(x—f2(x,y)—x2) cos 0—y sin oi}
(4)
This formula expresses all geometrical parameters of moire fringes—the fringe profile, the fringe spacing, the fringe orientation and the local fringe deformation. By writing X for the argument of the term for i 1, we can obtain the =
20
SHUNSUKE YOKOZEKI
formula T~(X)describing only the moire fringe profile as follows: T9 (X) = a0b0 + ~ a~,b,cos (2iriX)
(5)
We can call this equation the moire fringe profile formula. Assuming that the first term argument (i = 1) of eqn. (4) is equal to 2irr, we can obtain the indicial formula: [x
—
ft (x,
y) x,]
— ~-
{[x
—
f2(x, y) x2] cos 0— y sin 0} = —
r
(6)
where r is an integer. This formula expresses the geometrical parameters with the exception of fringe profile. We can explain the characteristics of moire fringes by using eqns. (4), (5) and (6) as follows: Case 1, (x,=x2=0,f1~0,f2=0) In this case, the indicial formula can be written as follows: for
0=0
and
d,cosO—d2M sinO M x—--——y+—f1(x,y)+r=0 d,d2 d2 d1
d,=Md2
(7)
fortheothercase.
(8)
Equation (7) shows that the moire fringes for 0 = 0 and d, = Md2 are the contours of the deformation function f,(x, y). The moire fringe pattern represented by eqn. (8) is of the deformed grating type; the fringe spacing is 2d~—2Md d1d2/(d~+M 1d2cos 0)L’2 and the local deformation at a point (x, y) 2d~—2Md is Md2f1(x, y)/(d~+M 1d2cos0)~2. Therefore, the relative moire fringe local deformation, Dr. is: Dr=
Deformation M =—f1(x,y) Fringe spacing d1 .
.
(9)
This relative deformation is obtained as the first measured value in many cases of moire measurement. Equations (7) and (8) show that the moire techniques transform the deformed grating pattern into the contour pattern or the deformed grating pattern of variable fringe spacing. This phenomenon is one of the useful characteristics of moire fringes. (7), (8) and (9) 5’6 Equations moire interferometry7 andexpound Talbot the basic principle of function moire topography, interferometry.8 The f 1(x, y) is proportional to the height of the object’s surface in the case of moire topography, to the wavefront aberration in moire interferometry, and to the partial derivative of the wavefront aberration
21
MOIRE FRINGES
in Talbot interferometry. Equations (7) and (9) imply that the increase in sensitivity will be proportional to the number M. This phenomenon is the principle of the fringe multiplication method.15 Case 2, (x 1=x2=0,f1~0,f2~O) The indicial formula for this case will have the form: ~~2(x,y)—f1(x,y)]=r
for
0=0
and
d1=Md2
(10)
d1cosO—d2M sin0 FM cosO 1 x—---—-— y+~—f1(x,y)—-————f2(x, y) I+r=0 for the other case. d1d2 d2 Ld1 d2 i (11) When 0=0 and d,=Md2, we can say that: M cos0 M —ft(x, y)——-———f2(x, y)=—[f1(x, y)—f2(x, y)] d1 d2 d1
(12)
It is shown by eqns. (10), (11) and (12) that we can perform the optimal subtraction by moire techniques. characteristics haveresulting been applied to 68 These and eliminating the error from the detecting the change of objects imperfection of the interferometer.’6’17 Case 3, (f 1=f2=0, 0=0, d1=Md2) The basic formula for this case can be expressed as follows: Tm(x2x1)
a0b0+~
cos 2~ri[~(x2_xi)]
(13)
This equation shows that the irradiance just behind the two gratings will vary periodically with the relative displacement of two gratings, and expounds the basic principle of the commercial linear displacement transducer. Case 4, (0”0,d1=Md2,x2—x1=v0t) When we adopt the system in which the functions f1(x, y) and f2(x, y) do not change in spite of the displacement of the gratings, the basic formula can be written as follows: Tm(X,
y)= a0b0+ ~
~
cos 2iri
~[f2_fi]+~vot}
(14)
where v0 denotes a constant speed. The transmitted irradiance at a point (x, y) will be changed cyclically and the measured value [f2(x, y)—f1(x, y)] will be given by the phase of the cyclical change of irradiance. This phenomenon may be called fringe scanning and is very useful for moire fringe analysis using 18 electronic techniques.
22
SHUNSUKE YOKOZEKI 19
PROFILE PREDICTION AND SHARPENING OF MOIRE FRINGES
In many measurement methods utilising moire techniques, the information on the subject under test is obtained as fringe patterns and the positions of the moire fringes must be determined with great accuracy. The accuracy will be improved as the moire fringes are made sharper. Moire fringe sharpening15 is, therefore, one of the principal techniques of moire measurement. It is difficult to derive the general conditions for moire fringe sharpening, but it will become possible if we limit our consideration to the case of moire fringes produced by two binary gratings (composed of periodically and alternately arranged transparent and opaque lines). Assuming that the transparent line widths of two binary gratings are h, and h 2, and the periods d1 and d2, respectively, we can define the opening ratios, 01 and 02, as follows: 01=h,/d1,
02=h2/d2
(15)
By using eqn. (5) and the opening ratios 01 and 02, the profile formula for the moire fringes between two gratings can be written as: T0(X) = N02/M + (1/M){0302 + 20302
~ [sinc (k03) sinc (kO2fl cos (2irkX)
}
(16)
where M01=N+03, 03 is a decimal, N an integer and sinc(x)= (sin lrx)/(Trx). On the other hand, we consider the periodic function, F(X), represented by
f(X)
Fig. 4.
Definition of function f(X).
23
MOIRE FRINGES
the function, f(X) (Fig. 4): F(X)= m~os f(X- m) =
(A/S){S(S
+ P)+ 2S(S + P) n~1sine
(nS) sine [n(S
+ P)] cos
(2ITnX)} (17)
where m and n are integers. The profile of this function is illustrated in Fig. 5. By considering the analogy between eqns. (16) and (17), we can obtain the relationship between the moire fringe profile parameters (Fig. 6) and the binary grating parameters (0~,02, 03, M and N) as illustrated in Table 1. A-TYPE
2S+P1
F(X)
-
22~
(03+021)
Wo
F(X)
B-TYPE
2S+P>1
(03+02>1)
S’=l-(P+S) S”=P+2S-1 B=(P+2S-1)A/S
A
L>~j
/
Fig. 5.
Function F(X).
B
24
SHUNSUKE YOKOZEKI
><
x
I-.
I—
E
C
~~Q.~’LIR~X Parameters of the moire fringe profile.
Fig. 6.
By adopting the moire fringe profile prediction method using Fig. 6 and Table 1, we can derive the conditions for making the moire fringe constrast unity, the value of Tmax as large as possible and the fringe width (WD or WB of Fig. 5) thinner as follows. Conditions for Dark Fringe Sharpening 03+02= 1,
N = 0, and
02
—~
1
for 03 ~ 02, 03 —+ 1
for 03>02 (18)
Conditions for Bright Fringe Sharpening N=0
and
03=02—+0
(19)
where the arrow sign of A B means that the moire fringe width can be made thinner as the value of A approaches B. The arrow signs are used for deducing the moire fringe sharpening conditions. This means that there is a limit in the fringe sharpening. The first reason for the limit is that the production accuracy of the gratings is finite. The second is that the value of Tmax becomes smaller with fringe sharpening. —*
TABLE 1 RELATIONSHIPS BETWEEN PROFILE PARAMETERS
Parameters Tmax
03
02
(0
3-*N02)/M
03>02 (1 +N)02/M
Tmm
N02/M (03+02—1 +No2)IM
N02/M (03+02— 1+N02)/M
Q
O2~O3
O3~O2
R
03 1~O2
02 1—03
O3+O2~1 03+02>1
O3+O2~1 03+02>1
MOIRE FRINGES
25
Fig. 7. Examples of moire fringe sharpening. The parameters of two binary gratings: (top) O1=02=O’9 and M= 1; (centre) 01 =O~9,O~=O1 and M=1; (bottom) O5=03=O~1 and M=1.
26
SHUNSUKE YOKOZEKI
We prepared the transparent-and-binary type grating with the same period (d1 = d2, M = 1, N = 0, 01 = 03) and the opening ratios of 0~9 and 01 by photographic techniques. We tried moire fringe sharpening with these gratings. The results are shown in Fig. 7. The fringe contrast of Fig. 7(top) is much less than unity. This result is predicted from Fig. 6 and Table 1. Dark fringe sharpening is observed in Fig. 7(centre) and bright fringe sharpening in Fig. 7(bottom). The grating parameters for the case of Figs. 7(centre) and (bottom) meet the fringe sharpening conditions (eqns. (18) and (19), respectively). SUMMARY
We have derived the basic moire fringe formula, the moire fringe profile formula and the indicial formula expressing the geometrical parameters with the exception of the fringe profile. By using these formulae, we have explained the characteristics of the moire phenomenon as follows. The moire phenomenon transforms the deformed grating pattern into the contour line pattern or the variable period deformed grating pattern. This phenomenon can perform optical substraction. Fringe scanning is possible by moving two gratings in relation to each other. By clarifying these characteristics, it might be possible to find new applications for the moire phenomenon. We have presented a Table showing the relationship between the profile parameters and the sharpening conditions of moire fringes produced by two binary gratings. The moire fringe profile prediction method, which uses the Table and the moire fringe sharpening conditions, may be very useful for measurements using the static moire fringe pattern.
REFERENCES
1.
LORD RAYLEIGH, On the manufacture and theory of diffraction gratings, Phil. Mag. S.4, 47
(1874) 81—93. J. M. BURCH, The meteorological applications of diffraction gratings. In: Progress in optics Vol. 2, E. Wolf (Ed.), North-Holland, Amsterdam, 1963, 73—108. 3. P. S. THEOCARIS, Moire fringes in strain analysis, Pergamon Press, Oxford, 1969. 4. A. J. DURELLI and V. J. PARKS, Moire analysis of strain, Prentice-Hall, Englewood Cliffs, N.J., 1970. 5. D. M. M~s.tows,W. 0. JOHNSON and J. B. ALLEN, Generation of surface contours by moire patterns, AppI. Opt., 9 (1970) 942—7. 6. H. TAKASAKI, Moire topography, Appl. Opt., 9 (1970) 1467—72. 7. S. Y0K0zEKI and S. MIHARA, Moire interferometry, Appl. Opt., 18 (1979) 1275—80. 2.
MOIRE FRINGES 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
27
S. Y0K0zEKI, Electronic Talbot interferometer, Optics and Lasers in Engineering, 2 (1981) 13—19. S. YOKOZEKI, Y. KUSAKA and K. PATORSKI, Geometric parameters of moire fringes, Appl. Opt., 15 (1976) 2223—7. C. POLHEMUS and C. CHOCOL, Method for substracting phase errors in an interferometer, Appl. Opt., 10 (1971) 441—2. 0. BRYNGDAHL, Characteristics of superimposed patterns in optics, J. Opt. Soc. Am., 66 (1976) 87-94. 0. BRYNGDAHL, Moire: Formation and interpretation, J. Opt. Soc. Am., 64(1974)1287—94. 0. BRYNGDAHL, Moire and higher grating harmonics, J. Opt. Soc. Am., 65 (1975) 685—94. G. OSTER, M. WASSERMAN and C. ZWERLING, Theoretical interpretation of moire patterns, J. Opt. Soc. Am., 54 (1964) 169—75. D. POST, Sharpening and multiplication of moire fringes, Exp. Mech., 7 (1967) 154—9. S. YOKOZEKI and T. SUZUKI, Modified double-beam interferometer using the moire method, Appl. Opt., 11 (1972) 446—8. S. YOKOZEKI and T. SUZUKI, Shearing interferometer using the grating as the beam splitter. Part 2, Appl. Opt., 10 (1971) 1690—3. 0. INDEBETOUW, Profile measurement using projection of running fringes, Appl. Opt., 17 (1978) 2930—3. S. YOKOZEKI and K. PATORSKI, Moire fringe profile prediction method and its application to fringe sharpening, Appl. Opt., 17 (1978) 2541—7.