Abstract

The best method for investigating moiré phenomena in the superposition of periodic layers is based on the Fourier approach. However, superposition moiré effects are not limited to periodic layers, and they also occur between repetitive structures that are obtained by geometric transformations of periodic layers. We present in this paper the basic rules based on the Fourier approach that govern the moiré effects between such repetitive structures. We show how these rules can be used in the analysis of the obtained moirés as well as in the synthesis of moirés with any required intensity profile and geometric layout. In particular, we obtain the interesting result that the geometric layout and the periodic profile of the moiré are completely independent of each other; the geometric layout of the moiré is determined by the geometric layouts of the superposed layers, and the periodic profile of the moiré is determined by the periodic profiles of the superposed layers. The moiré in the superposition of two geometrically transformed periodic layers is a geometric transformation of the moiré formed between the original layers, the geometric transformation being a weighted sum of the geometric transformations of the individual layers. We illustrate our results with several examples, and in particular we show how one may obtain a fully periodic moiré even when the original layers are not necessarily periodic.

© 1998 Optical Society of America

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References

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  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, UK, 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. O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. 65, 685–694 (1975).
    [CrossRef]
  10. I. Amidror, R. D. Hersch, “Fourier-based analysis of phase shifts in the superposition of periodic layers and their moiré effects,” J. Opt. Soc. Am. A 13, 974–987 (1996).
    [CrossRef]
  11. See, for example, K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), pp. 16–21.
  12. Note that the term curvature is defined in mathematics in a different way; see, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 86.
  13. The equivalence between a 2D mapping from ℝ2 onto itself and a coordinate change in ℝ2 is discussed and illustrated in R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 133–140.
  14. If g1(x, y) and g2(x, y) are dependent, for instance, g2(x, y)=g1(x, y)2, then the 2D transformation g(x, y) is degenerate, and it maps ℝ2 into a 1D curve in ℝ2; see R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 155.
  15. See, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 154–155. If g1(x, y) and g2(x, y) satisfy also the Cauchy-Riemann conditions (a) ∂g1/∂x=∂g2/∂y,∂g1/∂y=-∂g2/∂x or (b) ∂g1/∂x=-∂g2/∂y,∂g1/∂y=-∂g2/∂x, then the transformation g(x) is conformal [see R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 166–167], and it maps the straight lines x=constant,y=constant into curve families x′=constant and y′=constant, which intersect at right angles. This orthogonality is clearly stronger than the mere independence of g1(x, y) and g2(x, y); and indeed, condition (a) implies J(x, y)>0, and condition (b) implies J(x, y)<0. Such an orthogonality is not required for our needs [see for instance Fig. 2(b)], but it is advantageous; for example, it guarantees that the two curvilinear gratings that together form our curved grid r(x, y) do not generate moirés between themselves within the curved grid itself.
  16. A. W. Lohmann, D. P. Paris, “Variable Fresnel zone pattern,” Appl. Opt. 6, 1567–1570 (1967).
    [CrossRef] [PubMed]
  17. A. Zygmund, Trigonometric Series (Cambridge U. Press, Cambridge, UK, 1968), Vol. 1, p. 36.
  18. Note that this particular case has already been proposed in Ref. 16.

1996 (1)

1979 (1)

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

1975 (1)

1974 (1)

1970 (1)

1967 (1)

1964 (2)

Amidror, I.

Bryngdahl, O.

Courant, R.

Note that the term curvature is defined in mathematics in a different way; see, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 86.

See, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 154–155. If g1(x, y) and g2(x, y) satisfy also the Cauchy-Riemann conditions (a) ∂g1/∂x=∂g2/∂y,∂g1/∂y=-∂g2/∂x or (b) ∂g1/∂x=-∂g2/∂y,∂g1/∂y=-∂g2/∂x, then the transformation g(x) is conformal [see R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 166–167], and it maps the straight lines x=constant,y=constant into curve families x′=constant and y′=constant, which intersect at right angles. This orthogonality is clearly stronger than the mere independence of g1(x, y) and g2(x, y); and indeed, condition (a) implies J(x, y)>0, and condition (b) implies J(x, y)<0. Such an orthogonality is not required for our needs [see for instance Fig. 2(b)], but it is advantageous; for example, it guarantees that the two curvilinear gratings that together form our curved grid r(x, y) do not generate moirés between themselves within the curved grid itself.

The equivalence between a 2D mapping from ℝ2 onto itself and a coordinate change in ℝ2 is discussed and illustrated in R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 133–140.

If g1(x, y) and g2(x, y) are dependent, for instance, g2(x, y)=g1(x, y)2, then the 2D transformation g(x, y) is degenerate, and it maps ℝ2 into a 1D curve in ℝ2; see R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 155.

Glatt, I.

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

Hersch, R. D.

Kafri, O.

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

Lohmann, A. W.

Nishijima, Y.

Oster, G.

Paris, D. P.

Patorski, K.

See, for example, K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), pp. 16–21.

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

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, UK, 1969).

Wasserman, M.

Zwerling, C.

Zygmund, A.

A. Zygmund, Trigonometric Series (Cambridge U. Press, Cambridge, UK, 1968), Vol. 1, p. 36.

Appl. Opt. (2)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (1)

Precis. Eng. (1)

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

Other (10)

A. Zygmund, Trigonometric Series (Cambridge U. Press, Cambridge, UK, 1968), Vol. 1, p. 36.

Note that this particular case has already been proposed in Ref. 16.

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

See, for example, K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), pp. 16–21.

Note that the term curvature is defined in mathematics in a different way; see, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 86.

The equivalence between a 2D mapping from ℝ2 onto itself and a coordinate change in ℝ2 is discussed and illustrated in R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 133–140.

If g1(x, y) and g2(x, y) are dependent, for instance, g2(x, y)=g1(x, y)2, then the 2D transformation g(x, y) is degenerate, and it maps ℝ2 into a 1D curve in ℝ2; see R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 155.

See, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 154–155. If g1(x, y) and g2(x, y) satisfy also the Cauchy-Riemann conditions (a) ∂g1/∂x=∂g2/∂y,∂g1/∂y=-∂g2/∂x or (b) ∂g1/∂x=-∂g2/∂y,∂g1/∂y=-∂g2/∂x, then the transformation g(x) is conformal [see R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 166–167], and it maps the straight lines x=constant,y=constant into curve families x′=constant and y′=constant, which intersect at right angles. This orthogonality is clearly stronger than the mere independence of g1(x, y) and g2(x, y); and indeed, condition (a) implies J(x, y)>0, and condition (b) implies J(x, y)<0. Such an orthogonality is not required for our needs [see for instance Fig. 2(b)], but it is advantageous; for example, it guarantees that the two curvilinear gratings that together form our curved grid r(x, y) do not generate moirés between themselves within the curved grid itself.

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Figures (7)

Fig. 1
Fig. 1

Various curvilinear gratings r(x, y) that have a periodic profile waveform of cos(2πfx): (a) straight cosinusoidal grating: cos(2πfx), (b) rotated straight cosinusoidal grating: cos{2πf[x cos θ+y sin θ]}, (c) parabolic cosinusoidal grating: cos{2πf[y-0.15x2]}, (d) circular cosinusoidal grating: cos(2πfx2+y2), (e) arg sinh(x)-shaped cosinusoidal grating: cos{2πf[y-arg sinh(x)]}, (f) cosinusoidally shaped grating: cos{2πf[y-cos(2πfx/4)]}.

Fig. 2
Fig. 2

(a) Periodic binary line grid p(x, y) of example 3, (b) curved binary grid r(x, y) obtained by applying on p(x, y) the 2D nonlinear transformation g(x, y)=[-x-arg sinh(y), y-arg sinh(x)]. Note that both line grids can be seen also as dot screens of white dots on a black background.

Fig. 3
Fig. 3

Some examples of curvilinear gratings r(x, y) having a square-wave periodic profile (with opening ratio τ/T=0.6) and a bending transformation g1(x, y). (a) Parabolic grating: g1(x, y)=y-0.15x2, (b) circular grating: g1(x, y)=x2+y2, (c) circular zone grating: g1(x, y)=(x2+y2)/8.

Fig. 4
Fig. 4

Schematic illustration of a (1,-1) moiré (a) between two straight periodic gratings, (b) between two curvilinear gratings. The dotted lines in (a), which represent the (2,-1) moiré, have been omitted in (b) for the sake of clarity.

Fig. 5
Fig. 5

Two circular zone gratings with raised cosinusoidal periodic profiles (1/2)cos(2πfr)+1/2, which have been horizontally shifted from the origin to the points x=1 (a), x=-1 (b), and their superposition (c). The (1,-1) moiré is clearly seen in the superposition (c) in the form of periodic vertical bands.

Fig. 6
Fig. 6

Superposition of two curvilinear gratings whose bending functions are g1(x, y)=arg sinh(x)+x/8 and g2(x, y)=arg sinh(x) so that g1(x, y)-g2(x, y)=x/8. The (1,-1) moiré obtained in their superposition consists of periodic vertical bands as shown in (a). However, this moiré is periodic only when the two layers are superposed center on center, and the slightest shift or rotation between the two layers destroys the periodicity of the moiré, as shown in (b) and (c). Note that the moiré bands seem to be darker in the center of each drawing; this happens because, for practical reasons, the curvilinear gratings were drawn here with a constant linewidth [compare with the correct, varying linewidths in Fig. 2(b)]. (Weak parasite moirés are a result of the production process.)

Fig. 7
Fig. 7

Dot-screen superposition illustrating example 1 of Section 5.B. (a) Curved dot screen r1(x, y) consisting of distorted 1’s, (b) curved dot screen r2(x, y) consisting of small pinholes. The two layers have been distorted by the same nonlinear coordinate transformation g(x, y)=(2xy, x2-y2). As shown in (c), the (1, 0,-1, 0) moiré generated when r2(x, y) is laterally shifted on top of r1(x, y) is purely periodic, although both r1(x, y) and r2(x, y) are not periodic; this periodic moiré consists of a screen of magnified 1’s, whose period and orientation depend on the shift. Note that rotations destroy the periodicity of the moiré, as illustrated in (d).

Equations (96)

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r(x, y)=r1(x, y)rm(x, y).
r(x, y)=p(x2+y2)=cos(2πfx2+y2).
g: xyxy,
xy=g1(x, y)g2(x, y),
J(x, y)=g1xg1yg2xg2y
xy=gxy=g1(x, y)g2(x, y)=-x-arg sinh(y)y-arg sinh(x).
r(x, y)=p[-x-arg sinh(y), y-arg sinh(x)]=m=-n=- rect-x-arg sinh(y)-mTτ ×recty-arg sinh(x)-nTτ.
p(x)=n=-cn exp(i2πnfx).
r(x, y)=p[g1(x, y)]=n=-cn exp[i2πnfg1(x, y)].
p(x, y)=m=-n=-cm,n ×exp[i2π(mx/Tx+ny/Ty)].
r(x, y)=p[g1(x, y), g2(x, y)]=m=-n=-cm,n exp{i2π[mg1(x, y)/Tx+ng2(x, y)/Ty]}.
r1(x, y)=m=-cm(1) exp[i2πmg1(x, y)],
r2(x, y)=n=-cn(2) exp[i2πng2(x, y)].
r1(x, y)r2(x, y)=m=-cm(1) exp[i2πmg1(x, y)] ×n=-cn(2) exp[i2πng2(x, y)]=m=-n=-cm(1)cn(2) exp{i2π[mg1(x, y)+ng2(x, y)]}.
mk1,k2(x, y)=m=-cmk1(1)cmk2(2) ×exp{i2πm[k1g1(x, y)+k2g2(x, y)]}.
m1,-1(x, y)=m=-cm(1)c-m(2) ×exp{i2πm[g1(x, y)-g2(x, y)]}.
r1(x, y)=m=-n=-cm,n(1) ×exp{i2π[mg1(x, y)+ng2(x, y)]},
r2(x, y)=m=-n=-cm,n(2) ×exp{i2π[mg3(x, y)+ng4(x, y)]}.
r1(x, y)r2(x, y)
=m=-n=-cm,n(1) exp{i2π[mg1(x, y)+ng2(x, y)]} ×m=-n=-cm,n(2) exp{i2π[mg3(x, y)+ng4(x, y)]}=n1=-n2=-n3=-n4=-cn1,n2(1)cn3,n4(2) ×exp{i2π[n1g1(x, y)+n2g2(x, y)+n3g3(x, y)+n4g4(x, y)]}.
m(k1, k2, k3, k4)+n(-k2, k1,-k4, k3)
=(mk1-nk2, mk2+nk1, mk3-nk4, mk4+nk3),
mk1,k2,k3,k4(x, y)
=m=-n=-cmk1-nk2,mk2+nk1(1)cmk3-nk4,mk4+nk3(2) ×exp{i2π[(mk1-nk2)g1(x, y)+(mk2+nk1)g2(x, y)+(mk3-nk4)g3(x, y)+(mk4-nk3)g4(x, y)]}=m=-n=-cmk1-nk2,mk2+nk1(1)cmk3-nk4,mk4+nk3(2) ×exp(i2π{m[k1g1(x, y)+k2g2(x, y)+k3g3(x, y)+k4g4(x, y)]+n[-k2g1(x, y)+k1g2(x, y)-k4g3(x, y)+k3g4(x, y)]}).
m1,0,-1,0(x, y)
=m=-n=-cm,n(1)c-m,-n(2) exp{i2π[mg1(x, y)+ng2(x, y)-mg3(x, y)-ng4(x, y)]}=m=-n=-cm,n(1)c-m,-n(2) exp(i2π{m[g1(x, y)-g3(x, y)]+n[g2(x, y)-g4(x, y)]}).
p1(x)=m=-cm(1) exp(i2πmx/T)
p2(x)=n=-cn(2) exp(i2πnx/T).
h(x)=p1(x) * p2(x)=1TTp1(x-x)p2(x)dx
h(x)=m=-cm(1)cm(2) exp(i2πmx/T).
r1(x, y)=p1[g1(x, y)]=m=-cm(1) exp[i2πmg1(x, y)],
r2(x, y)=p2[g2(x, y)]=n=-cn(2) exp[i2πng2(x, y)],
r1(x, y)r2(x, y)=m=-n=-cm(1)cn(2) ×exp{i2π[mg1(x, y)+ng2(x, y)]},
m1,-1(x, y)=m=-cm(1)c-m(2) ×exp{i2πm[g1(x, y)-g2(x, y)]}.
p1,-1(x)=m=-cm(1)c-m(2) exp(i2πmx).
p1,-1(x)=p1(x) * p2(-x),
p1,-1(x)=p1(x) * p2(-x);
g1,-1(x)=g1(x)-g2(x).
mk1,k2(x, y)=m=-cmk1(1)cmk2(2) ×exp{i2πm[k1g1(x, y)+k2g2(x, y)]}.
pk1, k2(x)=m=-cmk1(1)cmk2(2) exp(i2πmx),
pk1, k2(x)=subk1[p1(x)] * subk2[p2(x)],
subk[p(x)]=m=-dm exp(i2πmx),
dm=ckm,
pk1, k2(x)=subk1[p1(x)] * subk2[p2(x)].
gk1, k2(x)=k1g1(x)+k2g2(x).
k1g1(x, y)+k2g2(x, y)=g(x, y).
g1(x, y)-g2(x, y)=g(x, y).
k1g1(x, y)+k2g2(x, y)=ax+by+c.
g1(x, y)-g2(x, y)=ax+by+c.
x1, x2,g[(x+x1), y]-g[(x-x2), y]
=a0x+b0y+c0.
g[(x+x1), y]-g[(x-x2), y]
=[a(x+x1)2+by2]-[a(x-x2)2+by2]=a(x2+2xx1+x12-x2+2xx2-x22)=2a(x1+x2)x+a(x12-x22).
p1(x, y)=m=-n=-cm,n(1) ×exp[i2π(mx/Tx+ny/Ty)],
p2(x, y)=m=-n=-cm,n(2) ×exp[i2π(mx/Tx+ny/Ty)].
h(x, y)=p1(x, y) * * p2(x, y)=1TxTyTxTyp1(x-x, y-y) ×p2(x, y)dxdy
h(x, y)=m=-n=-cm,n(1)cm,n(2) ×exp[i2π(mx/Tx+ny/Ty)].
r1(x, y)=p1[g1(x, y), g2(x, y)]=m=-n=-cm,n(1) ×exp{i2π[mg1(x, y)+ng2(x, y)]},
r2(x, y)=p2[g3(x, y), g4(x, y)]=m=-n=-cm,n(2) ×exp{i2π[mg3(x, y)+ng4(x, y)]},
xy=g1(x, y)g2(x, y),xy=g3(x, y)g4(x, y)
r1(x, y)r2(x, y)
=n1=-n2=-n3=-n4=-cn1,n2(1)cn3,n4(2) ×exp{i2π[n1g1(x, y)+n2g2(x, y)+n3g3(x, y)+n4g4(x, y)]},
m1,0,-1,0(x, y)
=m=-n=-cm,n(1)c-m,-n(2) exp(i2π{m[g1(x, y)-g3(x, y)]+n[g2(x, y)-g4(x, y)]}).
p1,0,-1,0(x, y)=m=-n=-cm, n(1)c-m,-n(2) ×exp[i2π(mx+ny)].
p1,0,-1,0(x, y)=p1(x, y) * * p2(-x,-y),
xy=g1(x, y)-g3(x, y)g2(x, y)-g4(x, y),
p1,0,-1,0(x)=p1(x) * * p2(-x);
g1,0,-1,0(x)=g1(x)-g2(x).
mk1,k2,k3,k4(x, y)
=m=-n=-cmk1-nk2,mk2+nk1(1)cmk3-nk4,mk4+nk3(2) ×exp(i2π{m[k1g1(x, y)+k2g2(x, y)+k3g3(x, y)+k4g4(x, y)]+n[-k2g1(x, y)+k1g2(x, y)-k4g3(x, y)+k3g4(x, y)]}).
pk1,k2,k3,k4(x, y)
=m=-n=-cmk1-nk2,mk2+nk1(1)cmk3-nk4,mk4+nk3(2) ×exp[i2π(mx+ny)],
pk1,k2,k3,k4(x, y)
=subk1,k2[p1(x, y)] * * subk3,k4[p2(x, y)],
subr,s[p(x, y)]=m=-n=-dm,n ×exp[i2π(mx+ny)],
dm,n=cmr-ns,ms+nr,
xy=k1g1(x, y)+k2g2(x, y)+k3g3(x, y)+k4g4(x, y)-k2g1(x, y)+k1g2(x, y)-k4g3(x, y)+k3g4(x, y)
xy=k1k2-k2k1g1(x, y)g2(x, y)+k3k4-k4k3g3(x, y)g4(x, y),
x=K1g1(x)+K2g2(x),
pk1,k2,k3,k4(x)=subk1,k2[p1(x)] * * subk3,k4[p2(x)];
gk1,k2,k3,k4(x)=K1g1(x)+K2g2(x).
g1: xyg1(x, y)g2(x, y),g2: xyg3(x, y)g4(x, y),
xyk1g1(x, y)+k2g2(x, y)+k3g3(x, y)+k4g4(x, y)-k2g1(x, y)+k1g2(x, y)-k4g3(x, y)+k3g4(x, y).
xyg1(x, y)-g3(x, y)g2(x, y)-g4(x, y),
g1(x, y)-g3(x, y)=a1x+b1y+c1,
g2(x, y)-g4(x, y)=a2x+b2y+c2.
xy=2xyx2-y2.
[2(x+x0)y]-[2(x-x0)y]=4x0y,
[(x+x0)2-y2]-[(x-x0)2-y2]=4x0x.
k1g1(x, y)+k2g2(x, y)+k3g3(x, y)+k4g4(x, y)
=g(1)(x, y),
-k2g1(x, y)+k1g2(x, y)-k4g3(x, y)+k3g4(x, y)
=g(2)(x, y).
g1(x, y)-g3(x, y)=g(1)(x, y),
g2(x, y)-g4(x, y)=g(2)(x, y).

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