Abstract

The simple formula derived in the paper establishes a direct relationship between moire beat patterns and basic grids for minute displacements. The possibilities of finding a basic grid for a desired moire pattern are pointed out. The analysis is illustrated by several examples of Fresnel moire zone plate patterns and concentric equidistant circular moire patterns obtained by changes of scale and rotation. Possible advantages of the practical use of this element are outlined.

© 1991 Optical Society of America

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References

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  1. P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, London, 1969).
  2. L. O. Vargady, “Moire Fringes as Visual Position Indicators,” Appl. Opt. 3, 631–636 (1964).
    [CrossRef]
  3. G. T. Reid, “A Moire Fringe Alignment Aid,” Opt. Laser Eng. 4, 121–126 (1983).
    [CrossRef]
  4. H. H. M. Chau, “Moire Pattern Resulting from Superposition of Two Zone Plates,” Appl. Opt. 8, 1707–1712 (1969).
    [CrossRef] [PubMed]
  5. H. H. M. Chau, “Properties of Two Overlapping Zone Plates of Different Focal Lengths,” J. Opt. Soc. Am. 60, 255–259 (1970).
    [CrossRef]
  6. P. Szwaykowski, K. Patorski, “Moire Fringes by Evolute Gratings,” Appl. Opt. 28, 4679–4681 (1989).
    [CrossRef] [PubMed]
  7. A. W. Lohmann, D. P. Paris, “Variable Fresnel Zone Pattern,” Appl. Opt. 6, 1567–1570 (1967).
    [CrossRef] [PubMed]
  8. G. I. Rogers, “A Geometrical Approach to Moire Pattern Calculations,” Opt. Acta 24, 1–14 (1977).
    [CrossRef]
  9. G. I. Rogers, L. C. G. Rogers, “The Interrelations Between Moire Patterns, Contour Fringes, Optical Surfaces and Their Sum and Difference Effects,” Opt. Acta 24, 15–22 (1977).
    [CrossRef]
  10. J. M. Burch, D. C. Williams, “Varifocal Moire Zone Plates for Straightness Measurement,” Appl. Opt. 16, 2445–2450 (1977).
    [CrossRef] [PubMed]
  11. W. B. Herrmannsfeldt, M. J. Lee, J. J. Spranza, K. R. Trigger, “Precision Alignment Using a System of Large Rectangular Fresnel Lenses,” Appl. Opt. 7, 995–1005 (1968).
    [CrossRef] [PubMed]
  12. P. W. Harrison, “A Laser-Based Technique for Alignment and Deflection Measurement,” Civ. Eng. Public Works Rev. 68, 224–227 (1973).
  13. B. M. New, “Versatile Electrooptic Alignment System for Field Applications,” Appl. Opt. 13, 937–941 (1974).
    [CrossRef] [PubMed]
  14. P. W. Harrison, “Growth of a Practical Laser-Based Alignment Techniques,” in Proceedings, The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., London, 1976).
  15. Z. Jaroszewicz, V. Moreno, S. Bara, “Interferometric Alignment Using Parabolic and Off-Axis Conical Zone Plates,” Appl. Opt. 29, 4614–4617 (1990).
    [CrossRef] [PubMed]
  16. Z. Jaroszewicz, V. Moreno, S. Bara, “Application of Zone Plates in Interferometric Positioning,” Proc. Soc. Photo-Opt. Instrum. Eng. 1121, 246–250 (1990).
  17. R. F. Stevens, “A Zone Plate Interferometer for Pointing,” NPL Report MOM84 (Mar.1987).

1990 (2)

Z. Jaroszewicz, V. Moreno, S. Bara, “Interferometric Alignment Using Parabolic and Off-Axis Conical Zone Plates,” Appl. Opt. 29, 4614–4617 (1990).
[CrossRef] [PubMed]

Z. Jaroszewicz, V. Moreno, S. Bara, “Application of Zone Plates in Interferometric Positioning,” Proc. Soc. Photo-Opt. Instrum. Eng. 1121, 246–250 (1990).

1989 (1)

1983 (1)

G. T. Reid, “A Moire Fringe Alignment Aid,” Opt. Laser Eng. 4, 121–126 (1983).
[CrossRef]

1977 (3)

G. I. Rogers, “A Geometrical Approach to Moire Pattern Calculations,” Opt. Acta 24, 1–14 (1977).
[CrossRef]

G. I. Rogers, L. C. G. Rogers, “The Interrelations Between Moire Patterns, Contour Fringes, Optical Surfaces and Their Sum and Difference Effects,” Opt. Acta 24, 15–22 (1977).
[CrossRef]

J. M. Burch, D. C. Williams, “Varifocal Moire Zone Plates for Straightness Measurement,” Appl. Opt. 16, 2445–2450 (1977).
[CrossRef] [PubMed]

1974 (1)

1973 (1)

P. W. Harrison, “A Laser-Based Technique for Alignment and Deflection Measurement,” Civ. Eng. Public Works Rev. 68, 224–227 (1973).

1970 (1)

1969 (1)

1968 (1)

1967 (1)

1964 (1)

Bara, S.

Z. Jaroszewicz, V. Moreno, S. Bara, “Interferometric Alignment Using Parabolic and Off-Axis Conical Zone Plates,” Appl. Opt. 29, 4614–4617 (1990).
[CrossRef] [PubMed]

Z. Jaroszewicz, V. Moreno, S. Bara, “Application of Zone Plates in Interferometric Positioning,” Proc. Soc. Photo-Opt. Instrum. Eng. 1121, 246–250 (1990).

Burch, J. M.

Chau, H. H. M.

Harrison, P. W.

P. W. Harrison, “A Laser-Based Technique for Alignment and Deflection Measurement,” Civ. Eng. Public Works Rev. 68, 224–227 (1973).

P. W. Harrison, “Growth of a Practical Laser-Based Alignment Techniques,” in Proceedings, The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., London, 1976).

Herrmannsfeldt, W. B.

Jaroszewicz, Z.

Z. Jaroszewicz, V. Moreno, S. Bara, “Interferometric Alignment Using Parabolic and Off-Axis Conical Zone Plates,” Appl. Opt. 29, 4614–4617 (1990).
[CrossRef] [PubMed]

Z. Jaroszewicz, V. Moreno, S. Bara, “Application of Zone Plates in Interferometric Positioning,” Proc. Soc. Photo-Opt. Instrum. Eng. 1121, 246–250 (1990).

Lee, M. J.

Lohmann, A. W.

Moreno, V.

Z. Jaroszewicz, V. Moreno, S. Bara, “Application of Zone Plates in Interferometric Positioning,” Proc. Soc. Photo-Opt. Instrum. Eng. 1121, 246–250 (1990).

Z. Jaroszewicz, V. Moreno, S. Bara, “Interferometric Alignment Using Parabolic and Off-Axis Conical Zone Plates,” Appl. Opt. 29, 4614–4617 (1990).
[CrossRef] [PubMed]

New, B. M.

Paris, D. P.

Patorski, K.

Reid, G. T.

G. T. Reid, “A Moire Fringe Alignment Aid,” Opt. Laser Eng. 4, 121–126 (1983).
[CrossRef]

Rogers, G. I.

G. I. Rogers, “A Geometrical Approach to Moire Pattern Calculations,” Opt. Acta 24, 1–14 (1977).
[CrossRef]

G. I. Rogers, L. C. G. Rogers, “The Interrelations Between Moire Patterns, Contour Fringes, Optical Surfaces and Their Sum and Difference Effects,” Opt. Acta 24, 15–22 (1977).
[CrossRef]

Rogers, L. C. G.

G. I. Rogers, L. C. G. Rogers, “The Interrelations Between Moire Patterns, Contour Fringes, Optical Surfaces and Their Sum and Difference Effects,” Opt. Acta 24, 15–22 (1977).
[CrossRef]

Spranza, J. J.

Stevens, R. F.

R. F. Stevens, “A Zone Plate Interferometer for Pointing,” NPL Report MOM84 (Mar.1987).

Szwaykowski, P.

Theocaris, P. S.

P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, London, 1969).

Trigger, K. R.

Vargady, L. O.

Williams, D. C.

Appl. Opt. (8)

Civ. Eng. Public Works Rev. (1)

P. W. Harrison, “A Laser-Based Technique for Alignment and Deflection Measurement,” Civ. Eng. Public Works Rev. 68, 224–227 (1973).

J. Opt. Soc. Am. (1)

Opt. Acta (2)

G. I. Rogers, “A Geometrical Approach to Moire Pattern Calculations,” Opt. Acta 24, 1–14 (1977).
[CrossRef]

G. I. Rogers, L. C. G. Rogers, “The Interrelations Between Moire Patterns, Contour Fringes, Optical Surfaces and Their Sum and Difference Effects,” Opt. Acta 24, 15–22 (1977).
[CrossRef]

Opt. Laser Eng. (1)

G. T. Reid, “A Moire Fringe Alignment Aid,” Opt. Laser Eng. 4, 121–126 (1983).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

Z. Jaroszewicz, V. Moreno, S. Bara, “Application of Zone Plates in Interferometric Positioning,” Proc. Soc. Photo-Opt. Instrum. Eng. 1121, 246–250 (1990).

Other (3)

R. F. Stevens, “A Zone Plate Interferometer for Pointing,” NPL Report MOM84 (Mar.1987).

P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, London, 1969).

P. W. Harrison, “Growth of a Practical Laser-Based Alignment Techniques,” in Proceedings, The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., London, 1976).

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Figures (6)

Fig. 1
Fig. 1

Equidistant circular moire pattern obtained by change of scale Φ(x,y) = a1r + a3θ: (a) basic grid and (b) resulting moire pattern.

Fig. 2
Fig. 2

Equidistant circular moire pattern obtained by rotation. A function linearly dependent on the angle was added to the basic grid equation Φ(x,y) = a1rθ + a3θ: (a) basic grid and (b) resulting moiré pattern.

Fig. 3
Fig. 3

Equidistant circular moire pattern obtained by rotation. A function dependent on the radius was added to the basic grid equation Φ(x,y) = a1rθ + a3θ: (a) basic grid and (b) resulting moire pattern.

Fig. 4
Fig. 4

Moire Fresnel ZP pattern obtained by change of scale Φ(x,y) = a1r2 + a3θ: (a) basic grid and (b) resulting moire pattern.

Fig. 5
Fig. 5

Moire Fresnel ZP pattern obtained by rotation. A function linearly dependent on the angle was added to the basic grid equation Φ(x,y) = a1r2θ + a3θ: (a) basic grid and (b) resulting moire pattern.

Fig. 6
Fig. 6

Moire Fresnel ZP pattern obtained by rotation. A function dependent on the radius was added to the basic grid equation Φ(x,y) = a1r2θ + a3r2: (a) basic grid and (b) resulting moire pattern.

Equations (16)

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Φ ( r ) = k ,
Φ ( r + Δ r ) = l ,
n 1 k ± n 2 l = m ,
Φ ( r + Δ r ) Φ ( r ) = m
Φ ( r ) Δ r = m .
Φ ( r ) Δ r = Ψ ( r ) .
Δ r = Δ r t + Δ r r + Δ r s = i ( Δ x + α y + β x ) + j ( Δ y α x + β y ) .
Φ ( r , θ ) r ( β r + Δ x cos θ + Δ y sin θ ) + Φ ( r , θ ) θ [ α + 1 r ( Δ x sin θ + Δ y cos θ ) ] = Ψ ( r , θ ) .
Φ ( r , θ ) r β r + Φ ( r , θ ) θ α = Ψ ( r , θ ) ,
Φ ( r , θ ) r β r = a 1 r + a 2 ,
Φ ( r , θ ) θ α = a 1 r + a 2 ,
( a 1 r / β ) + f ( θ ) + ( a 2 ln r / β ) = n ,
( a 1 r θ / α ) + f ( r ) + ( a 2 θ / α ) = n ,
Φ ( r , θ ) = ( r 2 r 0 2 θ 0 2 r 0 2 ) 1 / 2 θ / θ 0 ,
( a 1 r 2 / β ) + f ( θ ) + ( a 2 ln r / β ) = n ,
( a 1 r 2 θ / α ) + f ( r ) + ( a 2 θ / α ) = n .

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