Abstract

This paper presents a description of fringe formation in two-wavelength contour holography. Approximations are introduced primarily by restricting the recording wavelength difference with the aid of a tunable dye laser and by observing the contour fringes through a controlled viewing system aperture. Several holographic recording and readout arrangements are presented for which explicit and conveniently interpretable contour fringe formulations are derived. Finally, the results from a set of experiments performed with each of the arrangements are offered as experimental evidence supporting the analysis.

© 1976 Optical Society of America

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References

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  1. B. P. Hildebrand, K. A. Haines, J. Opt. Soc. Am. 57, 155 (1967).
    [CrossRef]
  2. L. O. Heflinger, R. F. Wuerker, Appl. Phys. Lett. 15, 28 (1969).
    [CrossRef]
  3. C. Leonard, J. Varner, J. Calkins, R. Zech, J. Opt. Soc. Am. 60, 1568 (1970).
  4. W. Schmidt, A. Vogel, D. Preussler, Appl. Phys. 1, 103 (1973).
    [CrossRef]
  5. A. A. Friesem, U. Levy, Y. Silberberg, The Engineering Uses of Coherent Optics (Cambridge U. P., London, 1976), p. 175.
  6. B. P. Hildebrand, “A General Analysis of Contour Holography,” Ph.D. dissertation, U. of Michigan (1967).
  7. J. R. Varner “Multiple Frequency Holographic Contouring,” Ph.D. dissertation, U. of Michigan (1971).
  8. B. P. Hildebrand, The Engineering Uses of Holography (Cambridge U. P., London, 1970), p. 401.
  9. J. S. Zelenka, J. R. Varner, Appl. Opt. 7, 2107 (1968).
    [CrossRef] [PubMed]
  10. J. R. Varner, Appl. Opt. 10, 212 (1971).
    [CrossRef] [PubMed]
  11. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]
  12. E. N. Leith, J. Upatnieks, K. A. Haines, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]

1973 (1)

W. Schmidt, A. Vogel, D. Preussler, Appl. Phys. 1, 103 (1973).
[CrossRef]

1971 (1)

1970 (1)

C. Leonard, J. Varner, J. Calkins, R. Zech, J. Opt. Soc. Am. 60, 1568 (1970).

1969 (1)

L. O. Heflinger, R. F. Wuerker, Appl. Phys. Lett. 15, 28 (1969).
[CrossRef]

1968 (1)

1967 (1)

1965 (2)

Calkins, J.

C. Leonard, J. Varner, J. Calkins, R. Zech, J. Opt. Soc. Am. 60, 1568 (1970).

Friesem, A. A.

A. A. Friesem, U. Levy, Y. Silberberg, The Engineering Uses of Coherent Optics (Cambridge U. P., London, 1976), p. 175.

Haines, K. A.

Heflinger, L. O.

L. O. Heflinger, R. F. Wuerker, Appl. Phys. Lett. 15, 28 (1969).
[CrossRef]

Hildebrand, B. P.

B. P. Hildebrand, K. A. Haines, J. Opt. Soc. Am. 57, 155 (1967).
[CrossRef]

B. P. Hildebrand, “A General Analysis of Contour Holography,” Ph.D. dissertation, U. of Michigan (1967).

B. P. Hildebrand, The Engineering Uses of Holography (Cambridge U. P., London, 1970), p. 401.

Leith, E. N.

Leonard, C.

C. Leonard, J. Varner, J. Calkins, R. Zech, J. Opt. Soc. Am. 60, 1568 (1970).

Levy, U.

A. A. Friesem, U. Levy, Y. Silberberg, The Engineering Uses of Coherent Optics (Cambridge U. P., London, 1976), p. 175.

Meier, R. W.

Preussler, D.

W. Schmidt, A. Vogel, D. Preussler, Appl. Phys. 1, 103 (1973).
[CrossRef]

Schmidt, W.

W. Schmidt, A. Vogel, D. Preussler, Appl. Phys. 1, 103 (1973).
[CrossRef]

Silberberg, Y.

A. A. Friesem, U. Levy, Y. Silberberg, The Engineering Uses of Coherent Optics (Cambridge U. P., London, 1976), p. 175.

Upatnieks, J.

Varner, J.

C. Leonard, J. Varner, J. Calkins, R. Zech, J. Opt. Soc. Am. 60, 1568 (1970).

Varner, J. R.

J. R. Varner, Appl. Opt. 10, 212 (1971).
[CrossRef] [PubMed]

J. S. Zelenka, J. R. Varner, Appl. Opt. 7, 2107 (1968).
[CrossRef] [PubMed]

J. R. Varner “Multiple Frequency Holographic Contouring,” Ph.D. dissertation, U. of Michigan (1971).

Vogel, A.

W. Schmidt, A. Vogel, D. Preussler, Appl. Phys. 1, 103 (1973).
[CrossRef]

Wuerker, R. F.

L. O. Heflinger, R. F. Wuerker, Appl. Phys. Lett. 15, 28 (1969).
[CrossRef]

Zech, R.

C. Leonard, J. Varner, J. Calkins, R. Zech, J. Opt. Soc. Am. 60, 1568 (1970).

Zelenka, J. S.

Appl. Opt. (2)

Appl. Phys. (1)

W. Schmidt, A. Vogel, D. Preussler, Appl. Phys. 1, 103 (1973).
[CrossRef]

Appl. Phys. Lett. (1)

L. O. Heflinger, R. F. Wuerker, Appl. Phys. Lett. 15, 28 (1969).
[CrossRef]

J. Opt. Soc. Am. (4)

Other (4)

A. A. Friesem, U. Levy, Y. Silberberg, The Engineering Uses of Coherent Optics (Cambridge U. P., London, 1976), p. 175.

B. P. Hildebrand, “A General Analysis of Contour Holography,” Ph.D. dissertation, U. of Michigan (1967).

J. R. Varner “Multiple Frequency Holographic Contouring,” Ph.D. dissertation, U. of Michigan (1971).

B. P. Hildebrand, The Engineering Uses of Holography (Cambridge U. P., London, 1970), p. 401.

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Figures (14)

Fig. 1
Fig. 1

General recording and reconstruction geometries: (a) recording; (b) reconstruction.

Fig. 2
Fig. 2

Basic single lens imaging recording arrangement.

Fig. 3
Fig. 3

Holographic setup with telecentric telescope configuration.

Fig. 4
Fig. 4

Optical arrangement for quasi-Fourier holograms.

Fig. 5
Fig. 5

Geometry for recording Fresnel holograms with plane wave illumination and reference beams.

Fig. 6
Fig. 6

Depth contours for Fresnel hologram: Δλ = 0.1 nm; θo = 0.

Fig. 7
Fig. 7

Inclined contour surfaces for Fresnel holograms: Δλ = 0.1 nm; θo = −θ.

Fig. 8
Fig. 8

Effect of viewing aperture location: (a) za = 0; (b) za = 20 cm.

Fig. 9
Fig. 9

Effect of viewing aperture size: (a) Δxa = 0.1 mm, Δya = 8 mm; (b) Δxa = Δya = 4 mm; (c) Δxa = Δya = 10 mm; (d) Δxa = Δya = 26 mm.

Fig. 10
Fig. 10

Contours with single lens imaging: (a) actual object; (b) reconstruction with no contours; (c) Δλ = 0.03 nm; (d) Δλ = 0.1 nm.

Fig. 11
Fig. 11

Contours with telescopic imaging: (a) actual object; (b) inclined contour surfaces.

Fig. 12
Fig. 12

Depth contours with telescopic imaging: (a) Δλ = 0.03 nm; (b) Δλ = 0.06 nm.

Fig. 13
Fig. 13

Experimental recording and readout setup for Fourier transform configuration.

Fig. 14
Fig. 14

Contours on cone: (a) reconstruction with no contours; (b) Δλ = 0.03 nm; (c) Δλ = 0.06 nm; (d) Δλ = 0.09 nm.

Equations (41)

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x h i = x a - x a - x v i z a - z v i · z a , y h i = y a - y a - y v i z a - z v i · z a ,
Δ ϕ a ( x o , y o , z o ) = ϕ a 1 - ϕ a 2 = 2 n π ,
ϕ a i = ϕ c i + ϕ h i + ϕ v i ,
Δ ϕ a = r o Δ k + ( k 1 r h 1 - k 2 r h 2 ) - ( k 1 r r 1 - k 2 r r 2 ) + k c Δ r c + k c Δ r a = 2 n π ,
Δ ϕ a = ( r o + r h - r r ) Δ k = 2 n π .
x v = x o + z o tan θ ( λ av λ c - 1 ) , y v = y o , z v = λ av λ c z o ,
α = tan θ ( λ av λ c - 1 ) , β = z a z a - z ¯ o , δ = 1 - β ,
r o = z o cos θ o + x o sin θ o , r h = { [ ( x o - x a ) δ + z o α β ] 2 + ( y o - y a ) 2 δ 2 + z o 2 } 1 / 2 , r r = ( x a δ + x o β + z o α β ) sin θ .
r o = z o cos θ o + x o sin θ o , r h = z o ,
r r = x o β sin θ .
Δ ϕ a = [ z o ( 1 + cos θ o ) + x o ( sin θ o - β sin θ ) ] Δ k = 2 n π .
tan θ inc = - sin θ o = β sin θ 1 + cos θ o
sin θ o = β sin θ = 0.
Δ z o = λ 1 λ 2 ( λ 2 - λ 1 ) ( 1 + cos θ o ) .
Δ ϕ = [ r o + r l - 1 2 f ( x 1 2 + y 1 2 ) + z h - r r ] Δ k = 2 n π ,
Δ ϕ = { 2 z o + ( M sin θ - x s z o - z s ) x o + [ 1 2 ( z o - f ) + 1 2 ( z o - z s ) ] ( x o 2 + y o 2 ) } · Δ k = 2 n π ,
M sin θ - x s z ¯ o - z s = 0 ,
Δ ϕ = { 2 z o + [ 1 2 ( z o - f ) + 1 2 ( z o - z s ) ] ( x o 2 + y o 2 ) } Δ k = 2 n π .
Δ z o = λ 1 λ 2 2 ( λ 2 - λ 1 ) .
x 1 = - M x o ;             y 1 = - M y o ,
Δ ϕ = ( r o + z o - r r ) Δ k = 2 n π ,
r o z o cos θ o + x o sin θ o ,
r r x 1 sin θ r = - M x o sin θ r ,
Δ ϕ = [ z o ( 1 + cos θ o ) + x o ( sin θ o + M sin θ r ) ] Δ k = 2 n π .
tan α = - sin θ o + M sin θ r 1 + cos θ o .
sin θ o + M sin θ r = 0.
Δ ϕ = z o ( 1 + cos θ o ) Δ k = 2 n π .
Δ ϕ a = [ r o + r 1 - 1 2 f ( x 1 2 + y 1 2 ) + z 1 - r h - r r ] Δ k = 2 n π .
Δ ϕ a = r o + z o + z h - f 2 [ z o ( z h - f ) - z h f ] ( x o 2 + y o 2 ) + z o - f 2 [ z o ( z h - f ) - z h f ] ( x h 2 + y h 2 ) + f z o ( z h - f ) - z h f ( x h x o + y h y o ) - r r } Δ k = 2 n π ,
Δ ϕ a = [ r o + z o - z o - f 2 f 2 ( x h 2 + y h 2 ) - 1 f ( x h x o + y h y o ) - r r ] Δ k = 2 n π .
Δ ϕ a = ( r o + z o ) Δ k = 2 n π .
Δ ϕ a = [ z o ( 1 + cos θ o ) + x o sin θ o ] Δ k = 2 n π ,
| Δ ϕ a x a · Δ x a | π and | Δ ϕ a y a · Δ y a | π ,
Δ x a π Δ k | δ sin θ + x a - x o r h δ 2 | - 1 ,
Δ y a π Δ k | y a - y o r h δ 2 | - 1 .
Δ ϕ max - Δ ϕ min π .
Δ ϕ ( x h ) = Δ ϕ o + ( x h 2 2 z - x h sin θ r ) Δ k ,
z o Δ z o ¯ 4 F 1 2 ( 1 2 + 2 F 2 sin θ r + 2 F 2 2 sin 2 θ r ) - 1 for sin θ r D 2 f 2 ,
z o Δ z o ¯ F 1 2 F 2 1 ( sin θ r ) for sin θ r > D 2 f 2 ,
z o Δ z o ¯ 8 F 1 2 .
z ¯ o - z s = 49 cm ;             x s = 9.5 cm ;             sin θ = 0.3 ; z ¯ o = 78.5 cm ;             z h = 24.5 cm ;             f = 182 mm .

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