Abstract

A novel interferometric method named statistical interferometry is proposed and studied. In the method, in contrast to the conventional deterministic interferometry, the complete randomness of the two interfering light fields, i.e., the random interference of the fully developed speckle fields, plays an essential role and is used as a standard of phase in a statistical sense. Preliminary experiments were conducted to verify the validity of the method, followed by a computer simulation. As an experimental result, the accuracy of the measurements of an out-of-plane displacement was confirmed up to λ/800 by comparison with the heterodyne interferometer. The method has the advantage of simplicity of the optical system required, while at the same time providing high accuracy.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Hariharan, Optical Interferometry (Academic, Sydney, 1985).
  2. H. Kadono, S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” Opt. Lett. 16, 883–885 (1991).
    [CrossRef] [PubMed]
  3. H. Kadono, S. Toyooka, Y. Iwasaki, “Speckle-shearing interferometry using a liquid-crystal cell as a phase modulator,” J. Opt. Soc. Am. A 8, 2001–2008 (1991).
    [CrossRef]
  4. H. Kadono, S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” in Second International Conference on Photomechanics and Speckle Metrology: Speckle Techniques, Birefringence Methods, and Applications to Solid Mechanics, F. Chiang, ed., Proc. SPIE1554A, 718–726 (1991).
  5. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.
  6. R. S. Shirohi, Speckle Metrology, R. S. Shirohi, ed. (Marcel Dekker, New York, 1993).
  7. R. S. Shirohi, J. Burke, H. Helmers, K. D. Hinsch, “Spatial phase-shifting for pure in-plane displacement and displacement derivatives measurement in electronic speckle pattern interferometry (ESPI),” Appl. Opt. 36, 5787–5791 (1997).
    [CrossRef]
  8. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shear interferometry,” Opt. Eng. 37, 1790–1795 (1998).
    [CrossRef]
  9. J. W. Goodman, “Random variables,” in Statistical Optics (Wiley, New York, 1985), Chap. 2.
  10. H. Kadono, T. Asakua, “Statistical properties of the speckle phase in the optical imaging system,” J. Opt. Soc. Am. A 2, 1787–1792 (1985).
    [CrossRef]
  11. H. Kadono, N. Takai, T. Asakua, “Statistical properties of the speckle phase in the diffraction region,” J. Opt. Soc. Am. A 3, 1080–1089 (1986).
    [CrossRef]

1998 (1)

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shear interferometry,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

1997 (1)

1991 (2)

1986 (1)

1985 (1)

Asakua, T.

Burke, J.

Franze, B.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shear interferometry,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

J. W. Goodman, “Random variables,” in Statistical Optics (Wiley, New York, 1985), Chap. 2.

Haible, P.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shear interferometry,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

Hariharan, P.

P. Hariharan, Optical Interferometry (Academic, Sydney, 1985).

Helmers, H.

Hinsch, K. D.

Iwasaki, Y.

Joenathan, C.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shear interferometry,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

Kadono, H.

Shirohi, R. S.

Takai, N.

Tiziani, H. J.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shear interferometry,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

Toyooka, S.

H. Kadono, S. Toyooka, Y. Iwasaki, “Speckle-shearing interferometry using a liquid-crystal cell as a phase modulator,” J. Opt. Soc. Am. A 8, 2001–2008 (1991).
[CrossRef]

H. Kadono, S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” Opt. Lett. 16, 883–885 (1991).
[CrossRef] [PubMed]

H. Kadono, S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” in Second International Conference on Photomechanics and Speckle Metrology: Speckle Techniques, Birefringence Methods, and Applications to Solid Mechanics, F. Chiang, ed., Proc. SPIE1554A, 718–726 (1991).

Appl. Opt. (1)

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shear interferometry,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

Opt. Lett. (1)

Other (5)

J. W. Goodman, “Random variables,” in Statistical Optics (Wiley, New York, 1985), Chap. 2.

P. Hariharan, Optical Interferometry (Academic, Sydney, 1985).

H. Kadono, S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” in Second International Conference on Photomechanics and Speckle Metrology: Speckle Techniques, Birefringence Methods, and Applications to Solid Mechanics, F. Chiang, ed., Proc. SPIE1554A, 718–726 (1991).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

R. S. Shirohi, Speckle Metrology, R. S. Shirohi, ed. (Marcel Dekker, New York, 1993).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Optical system of a Michelson interferometer explaining the principle of statistical interferometry.

Fig. 2
Fig. 2

Phasors corresponding to the phase change that is due to the displacement of the object.

Fig. 3
Fig. 3

Three phasors corresponding to the object phase, in which the intermediate phasor is set to be on the real axis.

Fig. 4
Fig. 4

Effects of symmetrical phase term for (a) ΔΨs=2π/50 and (b) ΔΨs=2π/100 and of antisymmetrical phase term for (c) ΔΨa=2π/100 on the PDF pϕ(ϕ) of evaluated speckle phase ϕ. pϕ(ϕ) becomes uniform when (d) ΔΨa=ΔΨs=0.

Fig. 5
Fig. 5

Determination error of the object phase as a function of the number of data samples with standard deviations of noise component of σn=0, 0.01, 0.1, and 0.5.

Fig. 6
Fig. 6

Probability density function pϕ(ϕ) of the evaluated speckle phase for a certain speckle phase ϕ=0 under the existence of intensity noises for standard deviations of noise component of σn=0.01, 0.1, and 0.5.

Fig. 7
Fig. 7

Optical systems of the statistical interferometer and the simultaneously implemented heterodyne interferometer for the measurement of the out-of-plane displacements of a rough surface object: M, mirror; HM, half-mirror; PL, polarizer; PBS, polarizing beam splitter; D, detector; G, ground glass; PZT, piezoelectric transducer.

Fig. 8
Fig. 8

Dependence of the experimentally determined PDF pϕ(ϕ) of the evaluated speckle phase under the condition of the modulation factor γth being (a) 0, (b) 0.01, and (c) 0.05.

Fig. 9
Fig. 9

Displacement of the object given by the PZT. The displacements were measured by the heterodyne interferometer.

Fig. 10
Fig. 10

Experimental results of the measurements of the small displacements of the object. The horizontal and vertical axes are the displacements measured by the heterodyne interferometer and those measured by the present method, respectively.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

I1(x)=I0(x){1+γ(x)cos[ϕ(x)+Ψ1]},
I2(x)=I0(x){1+γ(x)cos[ϕ(x)]},
I3(x)=I0(x){1+γ(x)cos[ϕ(x)+Ψ3]}.
-Ψ1=Ψ3=Ψ.
ϕ(x)=tan-1I1(x)-I3(x)I1(x)+I3(x)-2I2(x)cos Ψ-1sin Ψ.
pϕ(ϕ)=12π(-π<ϕπ).
pϕ(ϕ)=pϕ(ϕ)dϕ(ϕ)dϕ.
pϕ(ϕ)=12πΓ cos β(1/2)(Γ2-1)cos 2ϕ+Γ sin β sin 2ϕ+(Γ2+1)/2,
Γ=sin(Ψ+ΔΨs)γcos Ψ-1sin Ψ,
γ=[cos2(Ψ+ΔΨs)-2 cos(Ψ+ΔΨs)×cos ΔΨa+1]1/2,
cos β=[cos(Ψ+ΔΨs)-cos ΔΨa]/γ,
sin β=(sin ΔΨa)/γ.
Ψ1=-Ψ-ΔΨs+ΔΨa,
Ψ3=Ψ+ΔΨs+ΔΨa.
T=12π-ππcos 2ϕpϕ(ϕ)dϕ=π(Γ2-1)/2Γ cos β,
U=12π-ππsin 2ϕpϕ(ϕ)dϕ=π tan β,
V=12π-ππ1pϕ(ϕ)dϕ=π(Γ2+1)/Γ cos β.
sin ΔΨa=-2Uπ2/(AB)1/2,
cos(Ψ+ΔΨs)=(Γ2-2)(A/B)1/2,
Γ2=(V+2T)/(V-2T),
=(cos Ψ-1)/sin Ψ,
A=U2+π2,
B=U2(Γ2-2)2+π2(Γ2+2)2.
I1=I0[1+γ cos(ϕ+Ψ1)]+n1,
I2=I0[1+γ cos ϕ]+n2,
I3=I0[1+γ cos(ϕ+Ψ3)]+n3,
pϕ(ϕ)|σn=pϕ(ϕ)|σn=0pϕ(ϕ)|σn,ϕ=0.
Neff=ACCD/πrs2,
rs=0.61λdw.

Metrics