Abstract

Irregularities in optical thickness of a given nondiffuse transparency are statistically inferred using optical heterodyne detection techniques. The modulus and phase of its complex amplitude transmittance can be recovered from a beat photocurrent by rotating the transparency in a signal laser beam. The ensemble averaged autocorrelation of the transmittance is obtained by making correlation measurements of the recovered real and imaginary parts of the transmittance. This autocorrelation function has a nonvanishing imaginary part, so that the spatial spectral profile (i.e., its power spectrum) of the transparency becomes asymmetric.

© 1978 Optical Society of America

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References

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  1. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 286–288.
  2. See, for example, J. C. Dainty, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1976), Vol. 14, pp. 1–46.
    [CrossRef]
  3. Y. Ohtsuka, I. Sasaki, Appl. Phys. 3, 15 (1974).
    [CrossRef]
  4. Y. Ohtsuka, I. Sasaki, Appl. Phys. 7, 265 (1975).
    [CrossRef]
  5. Y. Ohtsuka, in Recent Advances in Optical Physics, Proceedings of the ICO-10, Prague-1975, B. Havelka, J. Blabla, Eds. (Placky U., Prague, 1976).
  6. H. Arsenlault, S. Lowenthal, Opt. Commun. 1, 451 (1970).
    [CrossRef]
  7. T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
    [CrossRef]

1975 (1)

Y. Ohtsuka, I. Sasaki, Appl. Phys. 7, 265 (1975).
[CrossRef]

1974 (1)

Y. Ohtsuka, I. Sasaki, Appl. Phys. 3, 15 (1974).
[CrossRef]

1972 (1)

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

1970 (1)

H. Arsenlault, S. Lowenthal, Opt. Commun. 1, 451 (1970).
[CrossRef]

Arsenlault, H.

H. Arsenlault, S. Lowenthal, Opt. Commun. 1, 451 (1970).
[CrossRef]

Asakura, T.

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 286–288.

Dainty, J. C.

See, for example, J. C. Dainty, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1976), Vol. 14, pp. 1–46.
[CrossRef]

Fujii, H.

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

Lowenthal, S.

H. Arsenlault, S. Lowenthal, Opt. Commun. 1, 451 (1970).
[CrossRef]

Murata, K.

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

Ohtsuka, Y.

Y. Ohtsuka, I. Sasaki, Appl. Phys. 7, 265 (1975).
[CrossRef]

Y. Ohtsuka, I. Sasaki, Appl. Phys. 3, 15 (1974).
[CrossRef]

Y. Ohtsuka, in Recent Advances in Optical Physics, Proceedings of the ICO-10, Prague-1975, B. Havelka, J. Blabla, Eds. (Placky U., Prague, 1976).

Sasaki, I.

Y. Ohtsuka, I. Sasaki, Appl. Phys. 7, 265 (1975).
[CrossRef]

Y. Ohtsuka, I. Sasaki, Appl. Phys. 3, 15 (1974).
[CrossRef]

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 286–288.

Appl. Phys. (2)

Y. Ohtsuka, I. Sasaki, Appl. Phys. 3, 15 (1974).
[CrossRef]

Y. Ohtsuka, I. Sasaki, Appl. Phys. 7, 265 (1975).
[CrossRef]

Opt. Acta (1)

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

Opt. Commun. (1)

H. Arsenlault, S. Lowenthal, Opt. Commun. 1, 451 (1970).
[CrossRef]

Other (3)

Y. Ohtsuka, in Recent Advances in Optical Physics, Proceedings of the ICO-10, Prague-1975, B. Havelka, J. Blabla, Eds. (Placky U., Prague, 1976).

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 286–288.

See, for example, J. C. Dainty, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1976), Vol. 14, pp. 1–46.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

A Mach-Zehnder type of heterodyne interferometer.

Fig. 2
Fig. 2

Schematic representation of the experimental setup.

Fig. 3
Fig. 3

Measured autocorrelation and cross-correlation curves for the sheet glass with no lacquer film.

Fig. 4
Fig. 4

The modulus and argument of Re(q) calculated from the two correlation curves in Fig. 3.

Fig. 5
Fig. 5

The spatial frequency spectrum.

Fig. 6
Fig. 6

Measured autocorrelation and cross-correlation curves for a sheet glass sample with lacquer film on one side.

Fig. 7
Fig. 7

Modulus and argument of Re(q) calculated from the two correlation curves in Fig. 6.

Fig. 8
Fig. 8

Spatial frequency spectrum broadened by the presence of the lacquer film.

Fig. 9
Fig. 9

Phase variations for the sheet glass with no lacquer film.

Fig. 10
Fig. 10

Probability density functions as a function of phase difference between two average points.

Equations (25)

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R ( q ) = lim L ( 1 2 L ) L + L m * ( r 1 ) m ( r 1 + q ) d r 1 ,
R e ( q ) = R ( q )
G ( f ) = R e ( q ) exp ( 2 π i q f ) d q .
E s ( t ) = m ( r υ t ) exp ( 2 π i ν t ) ,
E L O ( t ) = exp i [ 2 π ( ν + ν L O ) t + ζ ] ,
I b ( t ) = A | m ( r υ t ) | cos [ 2 π ν L O t + ζ ϕ ( r υ t ) ] ,
ϕ ( r υ t ) = arg [ m ( r υ t ) ] ,
x ( t ) = A | m ( r υ t ) | cos [ ζ ϕ ( r υ t ) ] ,
y ( t ) = A | m ( r υ t ) | sin [ ζ ϕ ( r υ t ) ] ,
x ( t ) x ( t + τ ) ¯ = A 2 | m ( r υ t ) m [ r υ ( t + τ ) ] | cos { ϕ [ r υ ( t + τ ) ] ϕ ( r υ t ) } ¯ ,
x ( t ) y ( t + τ ) ¯ = A 2 | m ( r υ t ) m [ r υ ( t + τ ) ] | sin { ϕ [ r υ ( t + τ ) ] ϕ ( r υ t ) } ¯ ,
[ x ( t ) x ( t + τ ) ¯ + i x ( t ) y ( t + τ ) ¯ ] / A 2 = m ( r υ t ) m * [ r υ ( t + τ ) ] ¯ = lim T ( 1 2 T ) T T m ( r υ t ) m * [ r υ ( t + τ ) ] d t .
R e ( q ) = lim L ( 1 2 L ) L L m * ( r 1 ) m ( r 1 + q ) d r 1 ,
C ( q ) = | R e ( q ) | = [ C 11 2 ( q ) + C 12 2 ( q ) ] 1 / 2 ,
δ ( q ) = arg [ R e ( q ) ] = tan 1 [ C 12 ( q ) / C 11 ( q ) ] .
R e ( q ) = C ( q ) exp [ i δ ( q ) ] .
R e ( τ ) = x ( t ) x ( t + τ ) ¯ + i x ( t ) y ( t + τ ) ¯ ,
S ( ν ) = R e ( τ ) exp ( 2 π i ν τ ) d τ .
G ( f ) = R e ( q ) exp ( 2 π i q f ) d q ,
G ( f ) = S ( ν / υ ) .
G ( f ) = C ( q ) { cos [ δ ( q ) ] cos ( 2 π f q ) + sin [ δ ( q ) ] sin ( 2 π f q ) } d q ,
R e ( q ) = lim L ( 1 2 L ) L L a ( r 1 ) a ( r 1 + q ) × exp i [ ϕ ( r 1 ) ϕ ( r 1 + q ) ] d r 1 .
exp ( i Δ ϕ ) = P ( Δ ϕ ) cos ( Δ ϕ ) d ( Δ ϕ ) + i P ( Δ ϕ ) sin ( Δ ϕ ) d ( Δ ϕ ) .
C 11 ( q ) = lim L ( 1 2 L ) L L d r 1 × a ( r 1 ) a ( r 1 + q ) P ( Δ ϕ ) cos ( Δ ϕ ) d ( Δ ϕ ) ,
C 12 ( q ) = lim L ( 1 2 L ) L L d r 1 × a ( r 1 ) a ( r 1 + q ) P ( Δ ϕ ) sin ( Δ ϕ ) d ( Δ ϕ ) .

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