Abstract

A scatter plate interferometer is analyzed based on the principle of statistical optics for what is to my knowledge the first time. It is shown that the optical complex amplitude distribution of scattered-direct light that is scattered by the first scatter plate and transmitted through the second scatter plate is equivalent to the distribution of direct-scattered light that is transmitted through the first scatter plate and scattered by the second scatter plate if there are no aberrations in the tested optical elements. Then the mechanism producing interference fringes and fringe contrast are discussed by means of a statistical method. It is shown that the fringe pattern depends on the correlation of the transmittance distributions of the two scatter plates and the aberration of the tested lens. The analysis coincides with experimental phenomena. The method used gives a new viewpoint on the principles of scatter plate interferometry and reveals its statistical nature.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Mallick, "Common path interferometry," in Optical Shop Testing, D.Malacara, ed. (Wiley, 1978), pp. 79-109.
  2. J. M. Burch, "Scatter fringes of equal thickness," Nature 171, 889-890 (1953).
    [CrossRef]
  3. J. M. Burch, "Scatter-fringe interferometry," J. Opt. Soc. Am. 52, 600-605 (1962).
  4. J. M. Burch, "Interferometry with scattered light," in Optical Instruments and Techniques, J.H.Dickson, ed. (Oriel, 1969), pp. 213-243.
  5. R. M. Scott, "Scatter plate interferometry," Appl. Opt. 8, 531-537 (1969).
  6. L. Rubin, "Scatterplate interferometry," Opt. Eng. 19, 815-824 (1980).
  7. D. C. Su, T. Honda, and J. Tsujiuchi, "Aperture of scatter plate and its effect on fringe contrast in a scatter plate interferometer," Opt. Commun. 50, 137-140 (1984).
    [CrossRef]
  8. D. C. Su, N. Ohyama, T. Honda, and J. Tsujiuchi, "A null test of aspherical surfaces in scatter plate interferometer," Opt. Commun. 58, 139-143 (1986).
    [CrossRef]
  9. J. Rasanen, K. M. Abedin, and M. Kawazoe, "Computer simulation of the scatter plate interferometer by scalar diffraction theory," Appl. Opt. 36, 5335-5339 (1997).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 347-356.
  11. D. W. Li, J. B. Chen, and F. P. Chiang, "Statistical analysis of one-beam subjective laser-speckle interferometry," J. Opt. Soc. Am. A 2, 657-666 (1985).
    [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005), pp. 97-102.
  13. J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Relative Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
    [CrossRef]
  14. J. B. Chen and F. P. Chiang, "Statistical analysis of whole-field filtering of specklegram and its upper limit of measurement," J. Opt. Soc. Am. A 1, 845-849 (1984).
    [CrossRef]

2005

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005), pp. 97-102.

1997

1986

D. C. Su, N. Ohyama, T. Honda, and J. Tsujiuchi, "A null test of aspherical surfaces in scatter plate interferometer," Opt. Commun. 58, 139-143 (1986).
[CrossRef]

1985

1984

D. C. Su, T. Honda, and J. Tsujiuchi, "Aperture of scatter plate and its effect on fringe contrast in a scatter plate interferometer," Opt. Commun. 50, 137-140 (1984).
[CrossRef]

J. B. Chen and F. P. Chiang, "Statistical analysis of whole-field filtering of specklegram and its upper limit of measurement," J. Opt. Soc. Am. A 1, 845-849 (1984).
[CrossRef]

1980

L. Rubin, "Scatterplate interferometry," Opt. Eng. 19, 815-824 (1980).

1978

S. Mallick, "Common path interferometry," in Optical Shop Testing, D.Malacara, ed. (Wiley, 1978), pp. 79-109.

1975

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Relative Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
[CrossRef]

1969

J. M. Burch, "Interferometry with scattered light," in Optical Instruments and Techniques, J.H.Dickson, ed. (Oriel, 1969), pp. 213-243.

R. M. Scott, "Scatter plate interferometry," Appl. Opt. 8, 531-537 (1969).

1962

J. M. Burch, "Scatter-fringe interferometry," J. Opt. Soc. Am. 52, 600-605 (1962).

1953

J. M. Burch, "Scatter fringes of equal thickness," Nature 171, 889-890 (1953).
[CrossRef]

Abedin, K. M.

Burch, J. M.

J. M. Burch, "Interferometry with scattered light," in Optical Instruments and Techniques, J.H.Dickson, ed. (Oriel, 1969), pp. 213-243.

J. M. Burch, "Scatter-fringe interferometry," J. Opt. Soc. Am. 52, 600-605 (1962).

J. M. Burch, "Scatter fringes of equal thickness," Nature 171, 889-890 (1953).
[CrossRef]

Chen, J. B.

Chiang, F. P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005), pp. 97-102.

J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 347-356.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Relative Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
[CrossRef]

Honda, T.

D. C. Su, N. Ohyama, T. Honda, and J. Tsujiuchi, "A null test of aspherical surfaces in scatter plate interferometer," Opt. Commun. 58, 139-143 (1986).
[CrossRef]

D. C. Su, T. Honda, and J. Tsujiuchi, "Aperture of scatter plate and its effect on fringe contrast in a scatter plate interferometer," Opt. Commun. 50, 137-140 (1984).
[CrossRef]

Kawazoe, M.

Li, D. W.

Mallick, S.

S. Mallick, "Common path interferometry," in Optical Shop Testing, D.Malacara, ed. (Wiley, 1978), pp. 79-109.

Ohyama, N.

D. C. Su, N. Ohyama, T. Honda, and J. Tsujiuchi, "A null test of aspherical surfaces in scatter plate interferometer," Opt. Commun. 58, 139-143 (1986).
[CrossRef]

Rasanen, J.

Rubin, L.

L. Rubin, "Scatterplate interferometry," Opt. Eng. 19, 815-824 (1980).

Scott, R. M.

R. M. Scott, "Scatter plate interferometry," Appl. Opt. 8, 531-537 (1969).

Su, D. C.

D. C. Su, N. Ohyama, T. Honda, and J. Tsujiuchi, "A null test of aspherical surfaces in scatter plate interferometer," Opt. Commun. 58, 139-143 (1986).
[CrossRef]

D. C. Su, T. Honda, and J. Tsujiuchi, "Aperture of scatter plate and its effect on fringe contrast in a scatter plate interferometer," Opt. Commun. 50, 137-140 (1984).
[CrossRef]

Tsujiuchi, J.

D. C. Su, N. Ohyama, T. Honda, and J. Tsujiuchi, "A null test of aspherical surfaces in scatter plate interferometer," Opt. Commun. 58, 139-143 (1986).
[CrossRef]

D. C. Su, T. Honda, and J. Tsujiuchi, "Aperture of scatter plate and its effect on fringe contrast in a scatter plate interferometer," Opt. Commun. 50, 137-140 (1984).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. M. Burch, "Scatter-fringe interferometry," J. Opt. Soc. Am. 52, 600-605 (1962).

J. Opt. Soc. Am. A

Nature

J. M. Burch, "Scatter fringes of equal thickness," Nature 171, 889-890 (1953).
[CrossRef]

Opt. Commun.

D. C. Su, T. Honda, and J. Tsujiuchi, "Aperture of scatter plate and its effect on fringe contrast in a scatter plate interferometer," Opt. Commun. 50, 137-140 (1984).
[CrossRef]

D. C. Su, N. Ohyama, T. Honda, and J. Tsujiuchi, "A null test of aspherical surfaces in scatter plate interferometer," Opt. Commun. 58, 139-143 (1986).
[CrossRef]

Opt. Eng.

L. Rubin, "Scatterplate interferometry," Opt. Eng. 19, 815-824 (1980).

Other

J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 347-356.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005), pp. 97-102.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Relative Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
[CrossRef]

S. Mallick, "Common path interferometry," in Optical Shop Testing, D.Malacara, ed. (Wiley, 1978), pp. 79-109.

J. M. Burch, "Interferometry with scattered light," in Optical Instruments and Techniques, J.H.Dickson, ed. (Oriel, 1969), pp. 213-243.

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 (1)

Fig. 1
Fig. 1

Typical simplified setup of a scatter plate interferometer.

Equations (40)

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

U S 1 T = C 1 U S 1 ,
U S 1 S = T 1 U S 1 ,
T 1 2 + C 1 2 = 1 ,
U S 1 = A 0 exp ( j k l 0 ) exp [ j k 2 l 0 ( x S 1 2 + y S 1 2 ) ] ,
U 0 T = exp ( j k l 0 ) j λ l 0 exp [ j k 2 l 0 ( x 0 2 + y 0 2 ) ] C 1 U S 1 exp [ j k 2 l 0 ( x S 1 2 + y S 1 2 ) ] exp [ j 2 π λ l 0 [ x S 1 x 0 + y S 1 y 0 ] ] d x S 1 d y S 1 ,
U 0 S = exp ( j k l 0 ) j λ l 0 exp [ j k 2 l 0 ( x 0 2 + y 0 2 ) ] T 1 U S 1 exp [ j k 2 l 0 ( x S 1 2 + y S 1 2 ) ] exp [ j 2 π λ l 0 [ x S 1 x 0 + y S 1 y 0 ] ] d x S 1 d y S 1 ,
U 0 = U 0 T + U 0 S .
U 0 T ( x 0 , y 0 ) = j C 1 A 0 λ l 0 δ ( x 0 , y 0 ) ,
U 0 S ( x 0 , y 0 ) = j A 0 λ l 0 exp [ j k 2 l 0 ( x 0 2 + y 0 2 ) ] T 1 ( x S 1 , y S 1 ) exp [ j 2 π ( x S 1 λ l 0 x 0 + y S 1 λ l 0 y 0 ) ] d x S 1 λ l 0 d y S 1 λ l 0 ,
U 0 T = U 0 T exp ( j ϕ 0 ) ,
U 0 S = U 0 S exp ( j ϕ 0 ) exp [ j k 2 f 0 ( x 0 2 + y 0 2 ) ] exp [ j k ω ( x 0 , y 0 ) ] ,
U S 2 T = C 1 A 0 exp ( j k l 0 ) exp [ j k 2 l 0 ( x S 2 2 + y S 2 2 ) ] ,
U S 2 S = A 0 exp ( j k l 0 ) exp [ j k 2 l 0 ( x S 2 2 + y S 2 2 ) ] exp [ j k ω ( x 0 , y 0 ) ] T 1 ( x S 1 , y S 1 ) exp { j 2 π [ x 0 ( x S 1 λ l 0 + x S 2 λ l 0 ) + y 0 ( y S 1 λ l 0 + y S 2 λ l 0 ) ] } d x 0 d y 0 d x S 1 λ l 0 d y S 1 λ l 0 .
U S 2 = U S 2 T T + U S 2 T S + U S 2 S T + U S 2 S S ,
U S 2 T T = C 1 C 2 A 0 exp ( j k l 0 ) exp [ j k 2 l 0 ( x S 2 2 + y S 2 2 ) ] ,
U S 2 T S = C 1 T 2 ( x S 2 , y S 2 ) A 0 exp ( j k l 0 ) exp [ j k 2 l 0 ( x S 2 2 + y S 2 2 ) ] ,
U S 2 S T = C 2 A 0 exp ( j k l 0 ) exp [ j k 2 l 0 ( x S 2 2 + y S 2 2 ) ] exp [ j k ω ( x 0 , y 0 ) ] T 1 ( x S 1 , y S 1 ) exp { j 2 π [ x 0 ( x S 1 λ l 0 + x S 2 λ l 0 ) + y 0 ( y S 1 λ l 0 + y S 2 λ l 0 ) ] } d x 0 d y 0 d x S 1 λ l 0 d y S 1 λ l 0 ,
U S 2 S S = T 2 ( x S 2 , y S 2 ) A 0 exp ( j k l 0 ) exp [ j k 2 l 0 ( x S 2 2 + y S 2 2 ) ] exp [ j k ω ( x 0 , y 0 ) ] T 1 ( x S 1 , y S 1 ) exp { j 2 π [ x 0 ( x S 1 λ l 0 + x S 2 λ l 0 ) + y 0 ( y S 1 λ l 0 + y S 2 λ l 0 ) ] } d x 0 d y 0 d x S 1 λ l 0 d y S 1 λ l 0 .
U S 2 S T = C 2 A 0 exp ( j k l 0 ) exp [ j k 2 l 0 ( x S 2 2 + y S 2 2 ) ] T 1 ( x S 2 , y S 2 ) .
U S 2 T S = C 1 C 2 U S 2 S T .
U I = U I T T + U I T S + U I S T + U I S S .
U I T T = C 1 C 2 A 0 λ l 0 M δ ( x i M , y i M ) ,
U I S T = C 2 A 0 λ l 0 M exp [ j k 2 M 2 ( x i 2 + y i 2 ) ] exp ( j k ω ) × T 1 ( x S 1 , y S 1 ) exp [ j 2 π ( x i M x S 1 λ l 0 + y i M y S 1 λ l 0 ) ] d x S 1 λ l 0 d y S 1 λ l 0 ,
U I T S = C 1 A 0 λ l 0 M exp [ j k 2 M 2 ( x i 2 + y i 2 ) ] T 2 ( x S 1 , y S 1 ) exp [ j 2 π ( x i M x S 1 λ l 0 + y i M y S 1 λ l 0 ) ] d x S 1 λ l 0 d y S 1 λ l 0 .
U ̃ I S S = A 0 λ l 0 M exp [ j k 2 M 2 ( x i 2 + y i 2 ) ] × T 1 ( x S 1 , y S 1 ) T 2 ( x S 1 , y S 1 ) exp [ j 2 π ( x i M x S 1 λ l 0 + y i M y S 1 λ l 0 ) ] d x S 1 λ l 0 d y S 1 λ l 0 .
σ T 1 2 = T 1 ( x S 1 , y S 1 ) d x S 1 λ l 0 d y S 1 λ l 0 = 1 λ 2 l 0 2 T 1 2 S Σ ,
σ T 2 2 = T 2 ( x S 1 , y S 1 ) d x S 1 λ l 0 d y S 1 λ l 0 = 1 λ 2 l 0 2 T 2 2 S Σ ,
σ T 1 T 2 2 = T 1 ( x S 1 , y S 1 ) T 2 ( x S 1 , y S 1 ) d x S 1 λ l 0 d y S 1 λ l 0 = 1 λ 2 l 0 2 T 1 2 T 2 2 S Σ ,
I I S T = U I S T 2 = C 2 2 A 0 2 M 2 T 1 2 S Σ ,
I I S T = U I T S 2 = C 1 2 A 0 2 M 2 T 2 2 S Σ ,
I I S S = U I S S 2 = U ̃ I S S 2 = A 0 2 M 2 T 1 2 T 2 2 S Σ .
U I = U I T S + U I S T + U I S S .
I ( x i , y i ) = U I U I * = U I T S 2 + U I S T 2 + U I S S 2 + U I T S U I S T * + U I T S * U I S T + U I T S U I S S * + U I T S * U I S S + U I S T U I S S * + U I S T * U I S S .
U I T S U I S S * = U I T S * U I S S = U I S T U I S S * = U I S T * U I S S = 0 .
U I T S U I S T * = A 0 2 C 1 C 2 T 1 T 2 λ 2 l 2 M 2 S Σ exp ( j k ω ) ,
S Σ = C T 1 , T 2 ( x S 1 , y S 1 ; x S 1 , y S 1 ) exp { j 2 π [ x i M ( x S 1 λ l 0 x S 1 λ l 0 ) + y i M ( y S 1 λ l 0 y S 1 λ l 0 ) ] } d x S 1 λ l 0 d y S 1 λ l 0 d x S 1 λ l 0 d y S 1 λ l 0 ,
C T 1 , T 2 ( x S 1 , y S 1 ; x S 1 , y S 1 ) = T 1 ( x S 1 , y S 1 ) T 2 ( x S 1 , y S 1 ) T 1 T 2 .
I ( x i , y i ) = A 0 2 S Σ M 2 { C 1 2 T 2 2 + C 2 2 T 1 2 + T 1 2 T 2 2 + 2 C 1 C 2 T 1 T 2 S Σ λ 2 l 2 S Σ cos [ k ω ( x i M , y i M ) ] } .
ξ = 2 C 1 C 2 T 1 T 2 C 1 2 T 2 2 + C 2 2 T 1 2 + T 1 2 T 2 2 λ 2 l 2 S Σ S Σ .
ξ max = 2 C 1 C 2 T 1 T 2 C 1 2 T 2 2 + C 2 2 T 1 2 + T 1 2 T 2 2 .

Metrics