Abstract

A Fourier-based approach is presented for the investigation of multilayer superpositions of periodic structures and their moiré effects. This approach fully explains the properties of the superposition of periodic layers and of their moiré effects, both in the spectral domain and in the image domain. We concentrate on showing how this approach provides also a full explanation of the various phenomena that occur because of phase shifts in one or more of the superposed layers. We show how such phase shifts influence the superposition as a whole and, in particular, how they affect each moiré in the superposition individually: We show that each moiré in the superposition undergoes a different shift, in its own main direction, whose size depends both on the moiré parameters and on the shifts of the individual layers. However, phase shifts in the individual layers do not necessarily lead to a solid shift of the whole superposition, and they may rather cause modifications in its microstructure. We demonstrate our results by several illustrative figures.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).
  2. O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).
  3. A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng. 1, 61–69 (1979).
    [CrossRef]
  4. H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970).
    [CrossRef] [PubMed]
  5. P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969).
  6. Y. Nishijima, G. Oster, “Moiré patterns: their application to refractive index and refractive index gradient measurements,” J. Opt. Soc. Am. 54, 1–5 (1964).
    [CrossRef]
  7. G. Oster, M. Wasserman, C. Zwerling, “Theoretical interpretation of moiré patterns,” J. Opt. Soc. Am. 54, 169–175 (1964).
    [CrossRef]
  8. O. Bryngdahl, “Moiré: formation and interpretation,” J. Opt. Soc. Am. 64, 1287–1294 (1974).
    [CrossRef]
  9. I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in colour separation,” J. Electron. Imaging 3, 295–317 (1994).
    [CrossRef]
  10. I. Amidror, “A generalized Fourier-based method for the analysis of 2D moiré envelope forms in screen superpositions,” J. Mod. Opt. 41, 1837–1862 (1994).
    [CrossRef]
  11. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).
  12. R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1988), pp. 79–84.
  13. G. Oster, The Science of Moiré Patterns, 2nd ed. (Edmund Scientific, Barrington, N.J., 1969).
  14. See, for example, A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 117.
  15. See, for example, A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, Orlando, Fla., 1982), Vol. 1, p. 75.
  16. G. Birkhoff, S. MacLane, A Survey of Modern Algebra, 4th ed. (Macmillan, New York, 1977), pp. 237–238.
  17. Although there exist different possible choices of fundamental frequency vectors f1, f2for p(x, y), each of these choices automatically determines a corresponding pair of fundamental period vectors P1, P2[by Eq. (A1)], and hence it determines also the fundamental period parallelogram that they define and the corresponding virtual-grating periods T1, T2. Note that all the different possible choices represent the same 2-D lattices Lfand LPof p(x, y).

1994 (2)

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in colour separation,” J. Electron. Imaging 3, 295–317 (1994).
[CrossRef]

I. Amidror, “A generalized Fourier-based method for the analysis of 2D moiré envelope forms in screen superpositions,” J. Mod. Opt. 41, 1837–1862 (1994).
[CrossRef]

1979 (1)

A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng. 1, 61–69 (1979).
[CrossRef]

1974 (1)

1970 (1)

1964 (2)

Amidror, I.

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in colour separation,” J. Electron. Imaging 3, 295–317 (1994).
[CrossRef]

I. Amidror, “A generalized Fourier-based method for the analysis of 2D moiré envelope forms in screen superpositions,” J. Mod. Opt. 41, 1837–1862 (1994).
[CrossRef]

Birkhoff, G.

G. Birkhoff, S. MacLane, A Survey of Modern Algebra, 4th ed. (Macmillan, New York, 1977), pp. 237–238.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

Bryngdahl, O.

Glatt, I.

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

Hersch, R. D.

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in colour separation,” J. Electron. Imaging 3, 295–317 (1994).
[CrossRef]

Kafri, O.

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

Kak, A. C.

See, for example, A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, Orlando, Fla., 1982), Vol. 1, p. 75.

MacLane, S.

G. Birkhoff, S. MacLane, A Survey of Modern Algebra, 4th ed. (Macmillan, New York, 1977), pp. 237–238.

Nishijima, Y.

Oster, G.

Ostromoukhov, V.

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in colour separation,” J. Electron. Imaging 3, 295–317 (1994).
[CrossRef]

Papoulis, A.

See, for example, A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 117.

Patorski, K.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).

Rosenfeld, A.

See, for example, A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, Orlando, Fla., 1982), Vol. 1, p. 75.

Shepherd, A. T.

A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng. 1, 61–69 (1979).
[CrossRef]

Takasaki, H.

Theocaris, P. S.

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969).

Ulichney, R.

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1988), pp. 79–84.

Wasserman, M.

Zwerling, C.

Appl. Opt. (1)

J. Electron. Imaging (1)

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in colour separation,” J. Electron. Imaging 3, 295–317 (1994).
[CrossRef]

J. Mod. Opt. (1)

I. Amidror, “A generalized Fourier-based method for the analysis of 2D moiré envelope forms in screen superpositions,” J. Mod. Opt. 41, 1837–1862 (1994).
[CrossRef]

J. Opt. Soc. Am. (3)

Precis. Eng. (1)

A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng. 1, 61–69 (1979).
[CrossRef]

Other (10)

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969).

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1988), pp. 79–84.

G. Oster, The Science of Moiré Patterns, 2nd ed. (Edmund Scientific, Barrington, N.J., 1969).

See, for example, A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 117.

See, for example, A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, Orlando, Fla., 1982), Vol. 1, p. 75.

G. Birkhoff, S. MacLane, A Survey of Modern Algebra, 4th ed. (Macmillan, New York, 1977), pp. 237–238.

Although there exist different possible choices of fundamental frequency vectors f1, f2for p(x, y), each of these choices automatically determines a corresponding pair of fundamental period vectors P1, P2[by Eq. (A1)], and hence it determines also the fundamental period parallelogram that they define and the corresponding virtual-grating periods T1, T2. Note that all the different possible choices represent the same 2-D lattices Lfand LPof p(x, y).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Geometric location and amplitude of impulses in the 2-D spectrum. To each impulse is attached its frequency vector, which points to the geometric location of the impulse in the spectrum plane (u, v).

Fig. 2
Fig. 2

Binary gratings (a) and (b) and their superposition (c) in the image domain; their respective spectra are the infinite combs shown in (d) and (e) and their convolution (f). Only impulse locations are shown in the spectra, and not their amplitudes. The circle in the center of the spectrum (f) represents the visibility circle. It contains the impulse pair whose frequency vectors are f1f2 and f2f1 and whose indices are (1, −1) and (−1, 1); this is the fundamental impulse pair of the (1, −1) moiré seen in (c). The dotted line in (f) shows the infinite comb of impulses that represents this moiré.

Fig. 3
Fig. 3

(a) The superposition of two identical gratings at a small angle difference gives rise to a (1, −1) moiré. The spectral interpretation of (a) is shown in the vector diagram (b); compare with Fig. 2(f), which shows also impulses of higher harmonics. (c) The superposition of three identical gratings with angle differences slightly away from 120° gives a significant (1, 1, 1) moiré; its vector diagram is shown in (d).

Fig. 4
Fig. 4

Schematic plot of a 1-fold periodic function p(x) in the image domain. A shift of a′ from the origin is equivalent to a shift of a or to an effective shift of t.

Fig. 5
Fig. 5

Schematic plot of the unshifted 2-fold periodic function p(x) in the image domain: (a) a vector a, expressed in terms of the period vectors P1, P2; (b) the same vector a, expressed in terms of the step vectors T1, T2.

Fig. 6
Fig. 6

Two-grating (1, −1) moiré of Figs. 3(a) and 3(b) and its phase shifts that are due to a shift in grating A. The origin of each image is indicated by a cross. (a) Both gratings and the moiré are in their initial phase. (b) Grating A is shifted by 1/4 period to the right; the moiré is consequently shifted 1/4 moiré period downward. (c) Grating A is shifted by 1/2 period to the right; the moiré is consequently shifted 1/2 moiré period downward. (d) Grating A is shifted by 3/4 period to the right; the moiré is consequently shifted 3/4 moiré period downward.

Fig. 7
Fig. 7

(1, −1) and (1, −2) moirés between two gratings and their plase shifts that are due to a shift in grating B. (a) Both gratings and the moirés are in their initial phase. (b) Grating B is shifted by 1/4 period to the right; consequently, the (1, −1) moiré is shifted by 1/4 of its period, while the (1, −2) moiré is shifted by 1/2 of its period (note that the cross and the lines indicating the moiré directions have not been moved).

Fig. 8
Fig. 8

Three-grating (1, 1, 1) moiré of Figs. 3(c) and 3(d) and its phase shifts that are due to a shift in grating A. (a) The three gratings and the moiré are in their initial phase. (b) Grating A is shifted by 1/4 period to the right; consequently, the moiré is shifted 1/4 moiré period in the main direction of the moiré.(c) Grating A is shifted by 1/2 period to the right; consequently, the moiré is shifted 1/2 moiré period in the main direction of the moiré. (d) Grating A is shifted by 3/4 period to the right; consequently, the moiré is shifted 3/4 moiré period in the main direction of the moiré.

Fig. 9
Fig. 9

Perpendicular (1, 0, −1, 0) and (0, 1, 0, −1) moirés between two grids (four gratings) and their phase shifts that are due to a shift in the first grid (gratings A and B). (a) Both grids and the two perpendicular moirés are in their initial phase.(b) The first grid (gratings A and B) is shifted by 1/2 period to the right; consequently, the horizontal moiré is shifted 1/2 moiré period downward. (c) The first grid (gratings A and B) is shifted by 1/2 period upward; consequently, the vertical moiré is shifted 1/2 moiré period to the right. (d) The first grid (gratings A and B) is shifted by 1/2 period to the right and 1/2 period upward; consequently, the horizontal moiré is shifted 1/2 moiré period downward, and the vertical moiré is shifted 1/2 period to the right.

Fig. 10
Fig. 10

Magnification of the superposition of three grids (six gratings) with identical periods and equal angle differences of 30°. This is an example of an almost-periodic superposition. In (a) all the grids are superposed in their initial phase, while in (b) the grid A has been shifted by 1/2 period in its two primary directions; note the substantial change in the form of the microstructure that is due to this phase shift.

Fig. 11
Fig. 11

Schematic plot of (a) the 2-fold periodic (skew-periodic) function p(x, y) in the image domain and (b) its skewed impulse nailbed in the spectral domain. The gray parallelogram A in the image domain represents a one-period element (tile) of p(x, y). P1 and P2 are segments of this parallelogram that coincide with the period vectors P1 and P2. It can be shown that the areas of parallelograms A and B are reciprocal: B ¯ = 1/Ā.

Fig. 12
Fig. 12

Magnified view of the main period parallelogram of Fig. 11(a), showing the period vectors Pi and the step vectors Ti.

Equations (48)

Equations on this page are rendered with MathJax. Learn more.

r ( x , y ) = r 1 ( x , y ) r 2 ( x , y ) r m ( x , y ) .
R ( u , v ) = R 1 ( u , v ) * * R 2 ( u , v ) * * R m ( u , v ) .
f k 1 , k 2 , k m = k 1 f 1 + k 2 f 2 + + k m f m ,
a k 1 , k 2 , , k m = a k 1 ( 1 ) a k 2 ( 2 ) a k m ( m ) ,
f u k 1 , k 2 , , k m = k 1 f 1 cos θ 1 + k 2 f 2 cos θ 2 + + k m f m cos θ m , f v k 1 , k 2 , , k m = k 1 f 1 sin θ 1 + k 2 f 2 sin θ 2 + + k m f m sin θ m .
f = f u 2 + f v 2 ,             T M = 1 / f ,             φ M = arctan ( f v / f u ) .
T M = T 1 T 2 ( T 1 2 + T 2 2 - 2 T 1 T 2 cos α ) 1 / 2 , sin φ M = T 1 sin α ( T 1 2 + T 2 2 - 2 T 1 T 2 cos α ) 1 / 2
T M = T 2 sin ( α / 2 ) ,             φ M = 90 ° - α / 2.
p ( x ) = m = - n = - c m , n exp [ i 2 π ( m f 1 + n f 2 ) · x ] ,
c m , n = 1 A ¯ A p ( x ) exp [ - i 2 π ( m f 1 + n f 2 ) · x ] d x ,
P ( u ) = m = - n = - c m , n δ [ u - ( m f 1 + n f 2 ) ] .
p ( x - a ) = m = - n = - c m , n exp [ i 2 π ( m f 1 + n f 2 ) · ( x - a ) ] = m = - n = - exp [ - i 2 π ( m f 1 + n f 2 ) · a ] × c m , n exp [ i 2 π ( m f 1 + n f 2 ) · x ] .
ϕ ( u , v ) = f · a = u a + v b .
ϕ m , n = ( m f 1 + n f 2 ) · a = m f 1 · a + n f 2 · a
p ( x - a ) = m = - n = - c m , n exp [ i 2 π ( m f 1 + n f 2 ) · x - i 2 π ( m f 1 + n f 2 ) · a ]
p ( x - a ) = m = - n = - c m , n exp [ i 2 π ( m f 1 + n f 2 ) · x - i 2 π ϕ m , n ] ,
a = n T + t             0 t < T ,
p ( x - a ) = p ( x - n T - t ) = p ( x - t ) .
p ( x - a ) = n = - c n exp [ i 2 π n f ( x - a ) ] = n = - c n exp ( i 2 π n f x - i 2 π n f a ) = n = - c n exp ( i 2 π n f x - i 2 π n ϕ ) .
a = n T + t ,             t = r T             0 r < 1 ,
p ( x - a ) = p ( x - n T - t ) = p ( x - t ) .
ϕ = f · a
p ( x - a ) = n = - c n exp [ i 2 π n f · ( x - a ) ] = n = - c n exp ( i 2 π n f · x - i 2 π n f · a ) = n = - c n exp ( i 2 π n f · x - i 2 π n ϕ ) .
P i · f j = { 1 i = j 0 i j ,
T i = 1 f i f i f i = 1 f i 2 f i .
p ( x - a ) = p 1 ( x - a 1 ) p 2 ( x - a 2 ) = { m = - c m ( 1 ) exp [ i 2 π m f 1 · ( x - a 1 ) ] } × { n = - c n ( 2 ) exp [ i 2 π n f 2 · ( x - a 2 ) ] } = [ m = - c m ( 1 ) exp ( i 2 π m f 1 · x - i 2 π m f 1 · a 1 ) ] × [ n = - c n ( 2 ) exp ( i 2 π n f 2 · x - i 2 π n f 2 · a 2 ) ] = m = - n = - c m ( 1 ) c n ( 2 ) exp [ i 2 π ( m f 1 + n f 2 ) · x - i 2 π ( m f 1 · a 1 + n f 2 · a 2 ) ] = m = - n = - c m ( 1 ) c n ( 2 ) exp [ i 2 π ( m f 1 + n f 2 ) · x - i 2 π ( m ϕ 1 + n ϕ 2 ) ] ,
p ( x - a ) = m = - n = - c m , n exp [ i 2 π ( m f 1 + n f 2 ) · x - i 2 π ( m ϕ 1 + n ϕ 2 ) ] .
p ( x ) = p 1 ( x ) p m ( x ) = [ k 1 = - c k 1 ( 1 ) exp ( i 2 π k 1 f 1 · x ) ] × × [ k m = - c k m ( m ) exp ( i 2 π k m f m · x ) ] = k 1 = - k m = - c k 1 ( 1 ) c k m ( m ) exp [ i 2 π ( k 1 f 1 + + k m f m ) · x ] .
m k 1 , , k m ( x ) = n = - c n k 1 ( 1 ) c n k m ( m ) exp [ i 2 π ( k 1 f 1 + + k m f m ) · x ] .
p 1 ( x - a 1 ) p m ( x - a m ) = { k 1 = - c k 1 ( 1 ) exp [ i 2 π k 1 f 1 · ( x - a 1 ) ] } × × { k m = - c k m ( m ) exp [ i 2 π k m f m · ( x - a m ) ] } = { k 1 = - c k 1 ( 1 ) exp [ i 2 π k 1 f 1 · x - i 2 π k 1 f 1 · a 1 ] } × × [ k m = - c k m ( m ) exp ( i 2 π k m f m · x - i 2 π k m f m · a m ) ] = k 1 = - k m = - c k 1 ( 1 ) c k m ( m ) exp [ i 2 π ( k 1 f 1 + + k m f m ) · x - i 2 π ( k 1 f 1 · a 1 + + k m f m · a m ) ] = k 1 = - k m = - c k 1 ( 1 ) c k m ( m ) exp [ i 2 π ( k 1 f 1 + + k m f m ) · x - i 2 π ( k 1 ϕ 1 + + k m ϕ m ) ] .
n = - c n k 1 ( 1 ) c n k m ( m ) exp [ i 2 π n ( k 1 f 1 + + k m f m ) · x - i 2 π n ( k 1 ϕ 1 + + k m ϕ m ) ] .
m k 1 , , k m ( x - b k 1 , , k m ) = n = - c n k 1 ( 1 ) c n k m ( m ) exp [ i 2 π n ( k 1 f 1 + + k m f m ) · x - i 2 π n ( k 1 ϕ 1 + + k m ϕ m ) ] .
m k 1 , , k m ( x - b k 1 , , k m ) = n = - c n k 1 ( 1 ) c n k m ( m ) exp [ i 2 π n ( k 1 f 1 + + k m f m ) · ( x - b k 1 , , k m ) ] = n = - c n k 1 ( 1 ) c n k m ( m ) exp [ i 2 π n ( k 1 f 1 + + k m f m ) · x - i 2 π n ( k 1 f 1 + + k m f m ) · b k 1 , , k m ] .
( k 1 f 1 + + k m f m ) · b k 1 , , k m = k 1 ϕ 1 + + k m ϕ m .
f k 1 , , k m · b k 1 , , k m = k 1 ϕ 1 + + k m ϕ m .
b k 1 , , k m = ( f k 1 , , k m ) - 1 ( k 1 ϕ 1 + + k m ϕ m ) ,
b k 1 , , k m = T k 1 , , k m ( k 1 ϕ 1 + + k m ϕ m ) ,
ϕ k 1 , , k m = k 1 ϕ 1 + + k m ϕ m .
b 1 , - 1 = T 1 , - 1 ( ϕ 1 - ϕ 2 ) ,
ϕ 1 , - 1 = ϕ 1 - ϕ 2 .
p ( x + x 1 , y + y 1 ) = p ( x , y )             p ( x + x 2 , y + y 2 ) = p ( x , y ) .
P i · f j = { 1 i = j 0 i j .
( P 1 P n ) ( f 1 , , f n ) = [ P 1 · f 1 P 1 · f n P n · f 1 P n · f n ] = [ 1 0 0 1 ]
( P 1 P n ) = ( f 1 , , f n ) - 1 .
T 1 = 1 f 1 2 f 1 T n = 1 f n 2 f n .
v - 1 = 1 v 2 v .
v - 1 · v = v · v - 1 = 1 v 2 v · v = 1 ,
T i = 1 f i 2 f i = f i - 1 .

Metrics