Abstract

Coherence functions on infinitesimal intervals are measured sequentially by detecting intensities after interfering with neighboring wave fields. They are combined arithmetically so that the coherence function over any desired interval is obtained. This method has special features: (i) the effect of turbulence in front of the detector can be eliminated; and, (ii) the coherence function over any distance can be obtained without physically expanding the span of the arm of the interferometer. The principal algorithm of coherence function derivation, discussions about the effectiveness and limitation of the method, and an application of the method are described.

© 1980 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1970), Chap. 10.
  2. M. V. R. K. Murty, “Interference between Wavefronts Rotated or Reversed with Respect to Each Other and its Relation to Spatial Coherence,” J. Opt. Soc. Am. 54, 1187–1190 (1964).
    [Crossref]
  3. L. Mertz, Transformations in Optics (Wiley, New York, 1965), Chap. 4.
  4. H. W. Wessely and J. O. Bolstad, “Interferometric Technique for Measuring the Spatial-Correlation Function of Optical Radiation Fields,” J. Opt. Soc. Am. 60, 678–682 (1970).
    [Crossref]
  5. M. Bertolotti, M. Carnevale, L. Muzii, and D. Sette, “Reversing-Front Interferometer for Phase-Correlation Measurements in the Turbulent Atmosphere,” Appl. Opt. 9, 510–512 (1970).
    [Crossref]
  6. A. Labeyrie, “Attainment of Diffraction Limited Resolution in Large Telescopes by Fourier Analysing Speckle Patterns in Star Images,” Astron. Astrophys. 6, 85–87 (1970).
  7. J. W. Goodman and et al., “Wavefront-Reconstruction Imaging Through Random Media,” Appl. Phys. Lett. 8, 311–313 (1966).
    [Crossref]
  8. V. I. Tatarski, “The effects of the turbulent atmosphere on wave propagation,” (U. S. Dept. of Commerce, TT-68-50464, Springfield, Va., 1971).
  9. A. Ishimaru, “Phase fluctuation in a turbulent medium,” Appl. Opt. 16, 3190–3192(1977).
    [Crossref] [PubMed]

1977 (1)

1970 (3)

1966 (1)

J. W. Goodman and et al., “Wavefront-Reconstruction Imaging Through Random Media,” Appl. Phys. Lett. 8, 311–313 (1966).
[Crossref]

1964 (1)

Bertolotti, M.

Bolstad, J. O.

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1970), Chap. 10.

Carnevale, M.

Goodman, J. W.

J. W. Goodman and et al., “Wavefront-Reconstruction Imaging Through Random Media,” Appl. Phys. Lett. 8, 311–313 (1966).
[Crossref]

Ishimaru, A.

Labeyrie, A.

A. Labeyrie, “Attainment of Diffraction Limited Resolution in Large Telescopes by Fourier Analysing Speckle Patterns in Star Images,” Astron. Astrophys. 6, 85–87 (1970).

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), Chap. 4.

Murty, M. V. R. K.

Muzii, L.

Sette, D.

Tatarski, V. I.

V. I. Tatarski, “The effects of the turbulent atmosphere on wave propagation,” (U. S. Dept. of Commerce, TT-68-50464, Springfield, Va., 1971).

Wessely, H. W.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1970), Chap. 10.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. W. Goodman and et al., “Wavefront-Reconstruction Imaging Through Random Media,” Appl. Phys. Lett. 8, 311–313 (1966).
[Crossref]

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of Diffraction Limited Resolution in Large Telescopes by Fourier Analysing Speckle Patterns in Star Images,” Astron. Astrophys. 6, 85–87 (1970).

J. Opt. Soc. Am. (2)

Other (3)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1970), Chap. 10.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), Chap. 4.

V. I. Tatarski, “The effects of the turbulent atmosphere on wave propagation,” (U. S. Dept. of Commerce, TT-68-50464, Springfield, Va., 1971).

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Figures (9)

FIG. 1
FIG. 1

Schematic imaging system that includes the turbulence.

FIG. 2
FIG. 2

Comparison of standard deviations of phases that are obtained by the conventional and the new methods.

FIG. 3
FIG. 3

Optical system used for the experiment.

FIG. 4
FIG. 4

Optical system to superpose two neighboring wave fields.

FIG. 5
FIG. 5

Signal processing for the derivation of coherence function.

FIG. 6
FIG. 6

Detected signals of the intensities of the superposed wave fields: (a) without turbulence; (b) with turbulence.

FIG. 7
FIG. 7

Phases of the differential coherence functions: (a) simulation; (b) without turbulence; (c) with turbulence.

FIG. 8
FIG. 8

Accumulated phase profiles: (a) simulation; (b) without turbulence; (c) with turbulence.

FIG. 9
FIG. 9

Reconstructed images of an object (dashed-line) consisting of two slits: (a) shows the comparison between the experimentally obtained image without turbulence (long-dashed line) and the corresponding image by the simulation (solid line), which is the result obtained numerically by the optical system used under ideal conditions, (b) shows the comparison between experimentally obtained images with (long-dashed line) and without (solid line) turbulence.

Equations (31)

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V 0 ( x ) = K O ( u ) exp [ j k x u / z 0 ] d u ,
V ( x , t ) = V 0 ( x ) exp [ j ϕ ( x , t ) ] ,
Γ υ ( x , x 0 ) = { V 0 ( x ) exp [ j ϕ ( x , t ) ] V 0 * ( x 0 ) exp [ j ϕ ( x 0 , t ) ] } av = V 0 ( x ) V 0 * ( x 0 ) A ( x , x 0 ) ,
A ( x , x 0 ) = { exp j [ ϕ ( x , t ) ϕ ( x 0 , t ) ] } av .
V 0 ( x ) = Γ υ ( x , x 0 ) [ V 0 * ( x 0 ) A ( x , x 0 ) ] 1 .
Γ υ ( x , x + Δ x ) = Γ υ 0 ( x , x + Δ x ) A ( x , x + Δ x ) = | V 0 * ( x ) V 0 ( x + Δ x ) | exp { j [ ϕ 0 ( x + Δ x ) ϕ 0 ( x ) ] } × A ( x , x + Δ x ) ,
| V 0 ( i Δ x ) | = | V ( i Δ x ) | = [ I υ ( i Δ x ) ] 1 / 2 , x = i Δ x , i = 0 , 1 , 2 , .
| ϕ 0 [ ( i + 1 ) Δ x ] ϕ 0 ( i Δ x ) | = cos 1 ( R e { Γ υ [ i Δ x , ( i + 1 ) Δ x ] } A [ i Δ x , ( i + 1 ) Δ x ] | V 0 ( i Δ x ) | | V 0 ( ( i + 1 ) Δ x ) | ) , i = 0 , 1 , 2 , .
ϕ 0 ( n Δ x ) ϕ 0 ( 0 ) = i = 0 n 1 { ϕ 0 [ ( i + 1 ) Δ x ] ϕ 0 ( i Δ x ) } .
Γ υ 0 ( 0 , n Δ x ) = | V 0 ( 0 ) | | V 0 ( n Δ x ) | exp { j [ ϕ 0 ( n Δ x ) ϕ 0 ( 0 ) ] } .
I i , i = ( 1 T ) [ | V ( x i , t ) | 2 + p ( x i , t ) ] d t , x i = i Δ x , i = 0 , 1 , 2 , , n
I i , i + 1 = ( 1 T ) [ | V ( x i , t ) + V ( x i + 1 , t ) | 2 + p ( x i , x i + 1 , t ) ] d t , i = 0 , 1 , 2 , , n 1.
V ( x i + 1 , t ) = V 0 ( x i + 1 ) exp [ j ϕ ( x i + 1 , t ) + j α 0 ] ,
Δ I i , i + 1 I i , i + 1 I i , i I i + 1 , i + 1 .
Δ I i , i + 1 = 2 R e { V 0 * ( x i ) V 0 ( x i + 1 ) A ( x i + 1 , x i ) exp [ j α 0 ] } + ( 1 T ) [ p ( x i , x i + 1 , t ) p ( x i , t ) p ( x i + 1 , t ) ] d t .
| V ( x , t ) | | ( 1 T ) p ( x , t ) d t |
Δ I i , i + 1 / 2 ( I i , i I i + 1 , i + 1 ) 1 / 2 A ( x i + 1 , x i ) cos [ ϕ 0 ( x i + 1 ) ϕ 0 ( x i ) + α 0 ] + p i , i + 1 / 2 | V 0 ( x i + 1 ) V 0 ( x i ) |
p i , i + 1 = ( 1 T ) [ p ( x i , x i + 1 , t ) p ( x i , t ) p ( x i + 1 , t ) ] d t
cos 1 ( Δ i , i + 1 2 A ( x i + 1 , x i ) ( I i , i I i + 1 , i + 1 ) 1 / 2 ) cos 1 ( cos [ ϕ 0 ( x i + 1 ) ϕ 0 ( x i ) + α 0 ] + p i , i + 1 2 A ( x i + 1 , x i ) | V 0 ( x i + 1 ) V 0 ( x i ) | ) = ϕ 0 ( x i + 1 ) ϕ 0 ( x i ) + α 0 1 sin [ ϕ 0 ( x i + 1 ) ϕ 0 ( x i ) + α 0 ] × p i , i + 1 2 A ( x i + 1 , x i ) | V 0 ( x i + 1 ) V 0 ( x i ) | cos [ ϕ 0 ( x i + 1 ) ϕ 0 ( x i ) + α 0 ] 2 sin 3 [ ϕ 0 ( x i + 1 ) ϕ 0 ( x i ) + α 0 ] × ( p i , i + 1 2 A ( x i + 1 , x i ) | V 0 ( x i + 1 ) V 0 ( x i ) | ) 2 + .
θ ( n Δ x ) = i = 0 n 1 cos 1 ( Δ I i , i + 1 2 A ( x i + 1 , x i ) ( I i , i I i + 1 , i + 1 ) 1 / 2 ) n α 0 ϕ 0 ( x n ) ϕ 0 ( x 0 ) i = 0 n 1 1 sin [ ϕ 0 ( x i + 1 ) ϕ 0 ( x i ) + α 0 ] × p i , i + 1 2 A ( x i + 1 , x i ) | V 0 ( x i + 1 ) V 0 ( x i ) | .
A ( x i + 1 , x i ) = exp [ γ ( Δ x / l 0 ) 2 ] ,
E { θ ( n Δ x ) } = ϕ 0 ( x n ) ϕ 0 ( 0 ) ,
Var { θ ( n Δ x ) } i = 0 n 1 1 sin 2 [ ϕ 0 ( x i + 1 ) ϕ 0 ( x i ) + α 0 ] × σ p 2 exp [ 2 γ ( Δ x / l 0 ) 2 ] 4 | V 0 ( x i + 1 ) V 0 ( x i ) | 2 ,
Var { θ ( n Δ x ) } C 0 2 n exp [ 2 γ ( Δ x / l 0 ) 2 ] ,
C 0 2 σ p 2 / 4 sin 2 [ ϕ 0 ( x i + 1 ) ϕ 0 ( x i ) + α 0 ] × | V 0 ( x i + 1 ) V 0 ( x i ) | 2 , i = 0 , 1 , 2 , .
Δ x opt = l 0 / 2 γ 1 / 2
Min { Var [ θ ( L ) ] } 2 ( e γ ) 1 / 2 ( L / l 0 ) C 0 2 .
L max l 0 | V 0 | 4 sin 2 α 0 / σ p 2 .
Var [ θ c ( L ) ] 1 sin 2 [ ϕ 0 ( L ) ϕ 0 ( 0 ) + α 0 ] × σ p 2 exp [ 2 γ ( L / l 0 ) 2 ] 4 | V 0 ( L ) V 0 ( 0 ) | 2 ,
Var [ θ c ( L ) ] C 0 2 exp [ 2 γ ( L / l 0 ) 2 ] .
R υ Δ ¯ ( Var { θ c ( L ) } / Var { θ ( n Δ x ) } ) 1 / 2 , L = n Δ x