Abstract

A new technique of optical interferometry based on the statistics of the fully developed speckle field is proposed. It is revealed that the complete randomness of the speckle phase can play the role of a standard phase in a statistical sense, and the phase of the object under testing can be derived in a statistical way, in contrast to conventional interferometry. The technique is first described in relation to the phase-shifting interferometry and the compensation problem for the phase-shift error. Next the method is generalized as an independent interferometric technique.

© 1991 Optical Society of America

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References

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  1. J. H. Bruning, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, D. R. Herriott, Appl. Opt. 13, 2693 (1974).
    [CrossRef] [PubMed]
  2. P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
    [CrossRef]
  3. J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.
  4. H. Kadono, T. Asakura, J. Opt. Soc. Am. A 2, 1787 (1985).
    [CrossRef]
  5. H. Kadono, N. Takai, T. Asakura, J. Opt. Soc. Am. A 3, 1080 (1986).
    [CrossRef]
  6. Y. Y. Hung, Opt. Eng. 21, 391 (1982).

1986 (1)

1985 (1)

1982 (2)

Y. Y. Hung, Opt. Eng. 21, 391 (1982).

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

1974 (1)

Asakura, T.

Brangaccio, D. J.

Brown, N.

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

Bruning, J. H.

Gallagher, J. E.

Goodman, J. W.

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

Hariharan, P.

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

Herriott, D. R.

Hung, Y. Y.

Y. Y. Hung, Opt. Eng. 21, 391 (1982).

Kadono, H.

Oreb, B. F.

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

Rosenfeld, D. P.

Takai, N.

White, A. D.

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

Opt. Eng. (1)

Y. Y. Hung, Opt. Eng. 21, 391 (1982).

Other (1)

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

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Figures (3)

Fig. 1
Fig. 1

Evaluated phase ϕ′ from Eq. (3) in terms of the existence of a phase-shift deviation with −ψ = ψ3 = 2.20 for the desired phase shift ψ = π/2.

Fig. 2
Fig. 2

Dependence of the probability-density distribution of the evaluated speckle phase ϕ′ on the phase-shift deviations Δψs and Δψa.

Fig. 3
Fig. 3

Optical system of a statistical interferometer. M’s, mirrors; L’s, lenses; CCD, charge-coupled-device camera.

Equations (18)

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I i ( x ) = I 0 ( x ) { 1 + b ( x ) cos [ ϕ ( x ) + ψ i ] }             for i = 1 , 2 , 3 .
tan ϕ = ( I 1 - I 2 ) ( cos ψ 3 - 1 ) - ( I 3 - I 2 ) ( cos ψ 1 - 1 ) ( I 1 - I 2 ) sin ψ 3 - ( I 3 - I 2 ) sin ψ 1 .
tan ϕ = I 1 - I 3 I 1 + I 3 - 2 I 2 cos ψ - 1 sin ψ .
p ϕ ( ϕ ) = 1 / 2 π             for - π ϕ < π .
ψ 1 = - ψ - Δ ψ s + Δ ψ a , ψ 2 = 0 , ψ 3 = ψ + Δ ψ s + Δ ψ a .
p ϕ ( ϕ ) = p ϕ ( ϕ ) d ϕ / d ϕ = 1 2 π | cos 2 ( ϕ + Δ ψ a + β ) Γ cos β cos 2 ϕ | = 1 2 π | Γ cos β ( 1 / 2 ) ( Γ 2 - 1 ) cos 2 ϕ + Γ sin β sin 2 ϕ + ( Γ 2 + 1 ) / 2 | ,
Γ = sin ( ψ + Δ ψ s ) γ cos ψ - 1 sin ψ ,
γ = [ cos 2 ( ψ + Δ ψ s ) - 2 cos ( ψ + Δ ψ s ) cos Δ ψ a + 1 ] 1 / 2 ,
cos β = [ cos ( ψ + Δ ψ s ) - cos Δ ψ a ] / γ , sin β = sin Δ ψ a / γ .
p ϕ ( ϕ ) = α π [ ( α 2 - 1 ) cos 2 ϕ + ( α 2 + 1 ) ] ,
α = sin ( ψ + Δ ψ s ) cos ( ψ + Δ ψ s ) - 1 cos ψ - 1 sin ψ .
p ϕ ( ϕ ) = Γ π [ 2 Γ tan β sin 2 ϕ + ( Γ 2 + 1 ) ] .
T = 1 2 π - π π cos 2 ϕ p ϕ ( ϕ ) d ϕ = π ( Γ 2 - 1 ) / 2 Γ cos β ,
U = 1 2 π - π π sin 2 ϕ p ϕ ( ϕ ) d ϕ = π tan β ,
V = 1 2 π - π π 1 p ϕ ( ϕ ) d ϕ = π ( Γ 2 + 1 ) / Γ cos β .
Γ 2 = ( V + 2 T ) / ( V - 2 T ) .
cos ( ψ + Δ ψ s ) = ( Γ 2 - 2 ) / ( Γ 2 + 2 ) , sin Δ ψ a = - ( 2 U / π ) 2 / ( Γ 2 + 2 ) ,
= ( cos ψ - 1 ) / sin ψ .

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