Abstract

The difference between the phase shift occurring at the object surface owing to displacement and the phase shift occurring at the observation plane of the imaging system of the interferometer is studied. Analytical expressions for the phase shift for a number of surface displacements are found. From these expressions it is found that the difference between the phase shifts at the object and the observation planes depends on the number of speckle-correlation modes in the observation plane and the product between the relative aperture and the relative defocus of the imaging system. For general displacement the results indicate that the accuracy of a phase-shift measurement with a small-aperture interferometer is limited only by the number of speckle-correlation modes at the observation plane for the case of a focused system. For a large-aperture interferometer the phase shift at the observation plane becomes sensitive to defocusing of the imaging system. Agreement between theory and experiments is observed.

© 1998 Optical Society of America

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References

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  1. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
  2. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).
  3. H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. A 10, 316–323 (1993).
    [CrossRef]
  4. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  5. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. A 4, 1931–1948 (1987).
    [CrossRef]
  6. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (Elsevier, New York, 1988), Chap. 5, pp. 350–393.
  7. M. V. Klein, T. E. Furtag, Optics (Wiley, New York, 1986).

1993 (1)

H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. A 10, 316–323 (1993).
[CrossRef]

1987 (1)

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. A 4, 1931–1948 (1987).
[CrossRef]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (Elsevier, New York, 1988), Chap. 5, pp. 350–393.

Furtag, T. E.

M. V. Klein, T. E. Furtag, Optics (Wiley, New York, 1986).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Grum, T. P.

H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. A 10, 316–323 (1993).
[CrossRef]

Hanson, S. G.

H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. A 10, 316–323 (1993).
[CrossRef]

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. A 4, 1931–1948 (1987).
[CrossRef]

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).

Klein, M. V.

M. V. Klein, T. E. Furtag, Optics (Wiley, New York, 1986).

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).

Yura, H. T.

H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. A 10, 316–323 (1993).
[CrossRef]

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. A 4, 1931–1948 (1987).
[CrossRef]

J. Opt. Soc. A (2)

H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. A 10, 316–323 (1993).
[CrossRef]

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. A 4, 1931–1948 (1987).
[CrossRef]

Other (5)

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (Elsevier, New York, 1988), Chap. 5, pp. 350–393.

M. V. Klein, T. E. Furtag, Optics (Wiley, New York, 1986).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).

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Figures (7)

Fig. 1
Fig. 1

Simple imaging system consisting of a single lens with a positive focal length f. The radius of the Gaussian lens aperture is σ. The lens and the aperture are placed at a distance L 1 from the object surface. The observation plane is at a distance L 2 behind the lens.

Fig. 2
Fig. 2

Clean imaging system consisting of two lenses with positive focal lengths f 1 and f 2. The lenses are separated by the sum of their focal lengths. A Gaussian aperture of radius σ f is placed at the focal point between the lenses. When the clean imaging system is ideally focused, the object is located at a length f 1 in front of the first lens and the observation plane is located at a length f 2 behind the second lens. In this figure the system is defocused by movement of the observation plane a distance Δ s away from focus.

Fig. 3
Fig. 3

Clean imaging system defocused by movement of the object surface by a distance Δ o away from focus.

Fig. 4
Fig. 4

Percent deviation from unity, defined as 100(1 - η), of the phase-correction coefficient η. The plot is shown as a function of the product of the relative aperture μ and the relative defocus ξ. Plots are shown for different numbers of speckle-correlation modes N.

Fig. 5
Fig. 5

Percent relative deviation of the phase shift in the observation plane as a function of the distance from the centrum of the membrane (see text). The plot is made for different values of the product μξ and a large number of speckle-correlation modes, N = 10,000.

Fig. 6
Fig. 6

Percent relative deviation of the phase shift in the observation plane as a function of the distance from the centrum of the membrane (see text). The plot is made for different values of the number of speckle-correlation modes N for a focused imaging system, ξ = 0.

Fig. 7
Fig. 7

Theoretical percent deviation from unity, given by 100(1 - η), of the phase-correction coefficient η, (curve) plotted together with the experimental data (diamonds). The plot is shown as a function of the relative defocus ξ. The theoretical curve is plotted with a relative aperture of μ = 0.52 and an infinite number of speckle-correlation modes N.

Equations (61)

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I = I avg ( 1 + M   Re γ ) ,
γ = U b U a * | U b | 2 1 / 2 | U a | 2 1 / 2 ,
U j p = - ikt 0 2 π B exp - ikL     u j r × exp - ik 2 B Dp 2 - 2 r · p + Ar 2 d 2 r , j = b ,   a ,
u a r = u b f r ,
ϕ r = d r · K ,
K = k o - k i ,
u b r 1 u b * r 2 = 2 π k 2   I s r 1 + r 2 2 δ r 1 - r 2 ,
I s r = I 0 exp - 2 r 2 r s 2 ,
A = 1 - L 2 f - 2 i   L 2 k σ 2 ,     B = L 1 + L 2 - L 1 L 2 f - 2 i   L 1 L 2 k σ 2 , C = - 1 f - 2 i k σ 2 ,     D = 1 - L 1 f - 2 i   L 1 k σ 2 .
L 2 = f L 1 + Δ L 1 + Δ - f .
A = m ,     B = Δ c - i   2 f 1 f 2 k σ f 2 ,   C = 0 ,     D = 1 m ,
u a r = u b r exp ik β θ r x ,
γ t = | γ t | exp i Φ t ,
Φ t = - 4 β   Im B p x θ ρ 0 2 ,
ρ 0 2 = 8 | B | 2 k 2 r s 2 + 4 k Im BA * .
Φ t = - η β k   p x m   θ ,
η = μ 2 ξ 2 + N N - 1 - 1 ,
μ = σ r s ,
ξ = Δ L 1 + Δ .
N = 1 + k σ r s 2 L 1 2 .
μ = σ f r s ,
ξ = Δ c f 2 .
N = 1 + k σ f r s 2 f 1 2 .
| γ t | = exp - θ 2 θ t 2 ,
1 θ t 2 = 1 2 k β r s 2 2   η ν ,
ν = μ 2 ξ 2 + 1 N - 1 .
u a r = u b R r exp i K · l - R r ,
R = cos   θ R - sin   θ R sin   θ R cos   θ R ,
l = 1 0 0 1 ,
γ R = | γ R | exp i Φ R .
Φ R = - 4   Im B sin θ R p x K y - p y K x k ρ 0 2 ,
| γ R | = exp - 1 - cos   θ R ρ 0 2 ( p 2 + 2 | K ˜ | 2 | B | 2 + 4   Re B K ˜ · p ) ,
Φ R = - η   1 m sin θ R p x K y - p y K x .
| γ R | = exp - 1 2   r s 2 η 1 - cos   θ R k 2 2 μ L 1 2 × r 2 + 2 ν L 1 μ 2 | K ˜ | 2 - 4 ξ L 1 K ˜ · r ,
d r = - z 0 r o 2 r 2 - r o 2 z ˆ ,
u a r = u b r exp - ik β   z 0 r o 2 r 2 - r o 2 .
γ M = | γ M | exp i Φ M ,
Φ M = k β z 0 2 | B | kr o ρ 0 2 + ( 4 kr o Im B ) 2 kr o ρ 0 4 + 16 | B | 4 k β z 0 2   p 2 - 1 ,
| γ M | = exp - 64   Im 2 B | B | 2 k β z 0 2 ρ 0 2 kr o ρ 0 4 + 16 | B | 4 k β z 0 2   p 2 1 + 2 | B | kr o ρ 0 4 k β z 0 2 1 / 2 .
Φ M = k β z 0 r o 2 η 2 r 2 1 + 1 2 r s r o 2 ν η 2 k β z 0 2 - 1 - 1 2 r s r o 2 ν η r 0 2 .
Φ M | N ξ - = k β z 0 .
Φ M | N ξ = 0 = k β z 0 r o 2 r 2 - r o 2 .
u a r = u b r - Δ r exp i K · Δ r .
γ ip = | γ ip | exp i Φ ip exp - i K · Δ r ,
Φ ip = c 1 Δ r 2 + c 2 p · Δ r .
c 1 = 4   Re BA * kr s 2 ρ 0 2 ,
c 2 = k   Re 1 B - 4   Im 1 B Re BA * ρ 0 2 .
c 1 = η   k 2 L 1 1 N - 1 - μ 2 ξ   L 1 L 1 + Δ ,
c 2 = k mL 1 1 ν η 1 N - 1 - μ 2 ξ   L 1 L 1 + Δ - μ 2 ξ .
c 1 | Δ = 0 = k 2 L 1 1 N ,
c 2 | Δ = 0 = k mL 1 N - 1 N ,
c 1 = 0 ,     c 2 = k f ,
c 1 = k 2 f 1   η μ 2 ξ ,
c 2 = k mf 1 μ 2 ξ ν η - 1 .
γ op = | γ op | exp i Φ op exp - i K · z ˆ δ z ,
Φ op = k 2 Re D B - Re D ˜ B ˜ p 2 ,
Φ op = k 2 1 L 2 1 m f   μ 2 ξ + 1 N - 1 μ 2 ξ 2 + 1 N - 1 - μ 2 Δ - δ z L 1 + Δ 1 m f + δ z f + L 1 + δ z L 1 2 1 N - 1 μ 2 Δ - δ z L 1 + Δ 2 + L 1 + δ z L 1 2 1 N - 1   p 2 ,
Φ op = - k 2 1 L 2 f μ 2 ξ δ z ν   p 2 .
p j = f 2 j   λ δ z 1 / 2 ,
Φ op = - k 2 1 f 2 1 m   μ 2 ξ ν - m δ z f 2 + ξ μ 2 m δ z f 2 + ξ 2 + 1 N - 1   p 2 .
η = 1 μ 2 ξ 2 + 1 .

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