Abstract

Statistical correlations of the light emitted by a partially coherent source can produce frequency shifts in the spectrum observed in the far field if the correlation function of the emitted radiation does not satisfy a certain scaling law. A Fourier achromat is used to generate a secondary source in which the degree of spectral coherence is independent of wavelength; i.e., it violates the scaling law. The spectrum detected in the far zone of the secondary source is, in general, found to be displaced in frequency and distorted relative to the spectrum measured at the secondary source. The displacement can be toward the higher frequencies or the lower frequencies depending on the direction of observation.

© 1988 Optical Society of America

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References

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  1. E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
    [Crossref] [PubMed]
  2. G. M. Morris, D. Faklis, Opt. Commun. 62, 5 (1987).
    [Crossref]
  3. E. Wolf, Nature 326, 363 (1987).
    [Crossref]
  4. E. Wolf, Opt. Commun. 62, 12 (1987).
    [Crossref]
  5. E. Wolf, Phys. Rev. Lett. 58, 2646 (1987).
    [Crossref] [PubMed]
  6. M. Bocko, D. H. Douglass, R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
    [Crossref] [PubMed]
  7. See, for example, E. Wolf, J. Opt. Soc. Am. 72, 343 (1982); J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 80; M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 503.
    [Crossref]
  8. W. H. Carter, E. Wolf, J. Opt. Soc. Am. 67, 785 (1977).
    [Crossref]
  9. G. M. Morris, Appl. Opt. 20, 2017 (1981).
    [Crossref] [PubMed]
  10. C. Brophy, Opt. Commun. 47, 364 (1983).
    [Crossref]
  11. G. M. Morris, D. L. Zweig, in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

1987 (5)

G. M. Morris, D. Faklis, Opt. Commun. 62, 5 (1987).
[Crossref]

E. Wolf, Nature 326, 363 (1987).
[Crossref]

E. Wolf, Opt. Commun. 62, 12 (1987).
[Crossref]

E. Wolf, Phys. Rev. Lett. 58, 2646 (1987).
[Crossref] [PubMed]

M. Bocko, D. H. Douglass, R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

1986 (1)

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[Crossref] [PubMed]

1983 (1)

C. Brophy, Opt. Commun. 47, 364 (1983).
[Crossref]

1982 (1)

1981 (1)

1977 (1)

Bocko, M.

M. Bocko, D. H. Douglass, R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

Brophy, C.

C. Brophy, Opt. Commun. 47, 364 (1983).
[Crossref]

Carter, W. H.

Douglass, D. H.

M. Bocko, D. H. Douglass, R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

Faklis, D.

G. M. Morris, D. Faklis, Opt. Commun. 62, 5 (1987).
[Crossref]

Knox, R. S.

M. Bocko, D. H. Douglass, R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

Morris, G. M.

G. M. Morris, D. Faklis, Opt. Commun. 62, 5 (1987).
[Crossref]

G. M. Morris, Appl. Opt. 20, 2017 (1981).
[Crossref] [PubMed]

G. M. Morris, D. L. Zweig, in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

Wolf, E.

Zweig, D. L.

G. M. Morris, D. L. Zweig, in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Nature (1)

E. Wolf, Nature 326, 363 (1987).
[Crossref]

Opt. Commun. (3)

E. Wolf, Opt. Commun. 62, 12 (1987).
[Crossref]

G. M. Morris, D. Faklis, Opt. Commun. 62, 5 (1987).
[Crossref]

C. Brophy, Opt. Commun. 47, 364 (1983).
[Crossref]

Phys. Rev. Lett. (3)

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[Crossref] [PubMed]

E. Wolf, Phys. Rev. Lett. 58, 2646 (1987).
[Crossref] [PubMed]

M. Bocko, D. H. Douglass, R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

Other (1)

G. M. Morris, D. L. Zweig, in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

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Figures (2)

Fig. 1
Fig. 1

Experimental configuration for realization of a secondary source with a controlled degree of spectral coherence. An object located in plane I is illuminated using a broadband, partially coherent source that obeys the scaling law. A secondary source with wavelength-independent spatial coherence is formed in plane II through application of the generalized Van Cittert–Zernike theorem. The spectral intensity is measured at the secondary source (plane II) and in the far field of the secondary source (plane III).

Fig. 2
Fig. 2

Spectral shifts produced by a Gaussian-correlated planar source. The spectral intensity is measured at (a) the secondary source and in the far field of the secondary source (b) on axis and (c) u = 20 mm off axis. The peak spectral intensity measured at the off-axis point exhibits a red shift, whereas that measured on axis exhibits a blue shift. The peak of each curve has been normalized to unity.

Equations (14)

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U II ( x ; ν ) = U I ( ξ ; ν ) h ( x , ξ ; ν ) d 2 ξ .
W ( x 1 , x 2 ; ν ) = U ( x 1 ; ν ) U * ( x 2 ; ν ) ,
S ( x ; ν ) = W ( x , x ; ν ) ,
μ ( x 1 , x 2 ; ν ) = W ( x 1 , x 2 ; ν ) [ S ( x 1 ; ν ) S ( x 2 ; ν ) ] 1 / 2 ,
h AFT ( x , ξ ; ν ) = i λ 0 F 0 exp ( i 2 π λ 0 F 0 x · ξ ) ,
W II ( x 1 , x 2 ; ν ) = g ( ξ ; ν ) g * ( ξ ; ν ) W in ( ξ , ξ ; ν ) × h AFT ( x 1 , ξ ; ν ) h AFT * ( x 2 , ξ ; ν ) d 2 ξ d 2 ξ ,
S II ( x ; ν ) = S ( 0 ) ( ν ) T ( ν ) π F 0 2 ( ν 0 ν ) 2 | P ( ν 0 F 2 ν F 0 x ) | 2 × | f ( ξ M ) | 2 | g ( ξ ) | 2 d 2 ξ ,
μ Π ( Δ x ; ν ) = μ II ( Δ x ; ν 0 ) = | f ( ξ M ) | 2 | g ( ξ ) | 2 exp ( i 2 π λ 0 F 0 Δ x · ξ ) d 2 ξ | f ( ξ M ) | 2 | g ( ξ ) | 2 d 2 ξ ,
h ( u , x ; ν ) = i λz exp ( λz u · u ) exp ( i 2 π λz u · x ) ,
S III ( u ; ν ) = Q ( x 1 ) Q * ( x 2 ) W II ( x 1 , x 2 ; ν ) × h ( u , x 1 ; ν ) h * ( u , x 2 ; ν ) d 2 x 1 d 2 x 2 ,
S III ( u ; ν ) = S ( 0 ) ( ν ) T ( ν ) π z 2 | f ( ν F 0 M ν 0 z u ) | 2 | g ( ν F 0 ν 0 z u ) | 2 × | P ( ν 0 F 2 ν F 0 x ) | 2 | Q ( x ) | 2 d 2 x .
S III ( u ; ν ) = A S ( 0 ) ( ν ) T ( ν ) π z 2 × | f ( ν F 0 M ν 0 z u ) | 2 | g ( ν F 0 ν 0 z u ) | 2 ,
S III ( u ; ν ) = K S II ( ν ) ( ν ν 0 ) 2 | f ( ν F 0 M ν 0 z u ) | 2 | g ( ν F 0 ν 0 z u ) | 2 ,
S III ( u ; ν ) = K S II ( ν ) ( ν ν 0 ) 2 × exp [ ( ν F 0 ν 0 z ) ( 1 σ 1 + 1 M 2 σ 2 2 ) u · u ] .

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