Abstract

The choice of the mathematical form of spatial correlation functions for optical fields is restricted by the constraint of nonnegative definiteness. We discuss a sufficient condition for ensuring the satisfaction of such a constraint.

© 2007 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. P. De Santis, F. Gori, G. Guattari, and C. Palma, J. Opt. Soc. Am. A 3, 1258 (1986).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  4. J. W. Goodman, Statistical Optics (Wiley, 1983).
  5. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  6. F. Gori, Opt. Commun. 46, 149 (1983).
    [CrossRef]
  7. F. Gori and E. Wolf, Opt. Commun. 61, 369 (1987).
    [CrossRef]
  8. F. Gori, Opt. Lett. 4, 354 (1979).
    [CrossRef] [PubMed]
  9. The authors are indebted to A. T. Friberg for calling this point to their attention.
  10. N. Aronszajn, Trans. Am. Math. Soc. 68, 337 (1950).
    [CrossRef]
  11. E. Parzen, Ann. Math. Stat. 32, 951 (1961).
    [CrossRef]
  12. G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics 59 (Society for Industrial and Applied Mathematics, 1990).
    [CrossRef]
  13. S. Saitoh, Integral Transforms, Reproducing Kernels and Their Applications (Addison Wesley Longman, 1997).
  14. A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics (Kluwer Academic, 2003).
  15. For any nonnegative definite kernel there exists a unique functional Hilbert space whose functions are reproduced when they are multiplied by the kernel by means of the scalar product holding for such space. This is the origin of the phrase 'Reproducing kernel Hilbert space.'
  16. F. Gori, Opt. Acta 27, 1025 (1980).
    [CrossRef]
  17. F. Gori and C. Palma, Opt. Commun. 27, 185 (1978).
    [CrossRef]
  18. P. Vahimaa and J. Turunen, Opt. Express 14, 1376 (2006).
    [CrossRef] [PubMed]

2006 (1)

1987 (1)

F. Gori and E. Wolf, Opt. Commun. 61, 369 (1987).
[CrossRef]

1986 (1)

1983 (1)

F. Gori, Opt. Commun. 46, 149 (1983).
[CrossRef]

1980 (1)

F. Gori, Opt. Acta 27, 1025 (1980).
[CrossRef]

1979 (1)

1978 (1)

F. Gori and C. Palma, Opt. Commun. 27, 185 (1978).
[CrossRef]

1961 (1)

E. Parzen, Ann. Math. Stat. 32, 951 (1961).
[CrossRef]

1950 (1)

N. Aronszajn, Trans. Am. Math. Soc. 68, 337 (1950).
[CrossRef]

Aronszajn, N.

N. Aronszajn, Trans. Am. Math. Soc. 68, 337 (1950).
[CrossRef]

Berlinet, A.

A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics (Kluwer Academic, 2003).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

De Santis, P.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1983).

Gori, F.

F. Gori and E. Wolf, Opt. Commun. 61, 369 (1987).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, and C. Palma, J. Opt. Soc. Am. A 3, 1258 (1986).
[CrossRef]

F. Gori, Opt. Commun. 46, 149 (1983).
[CrossRef]

F. Gori, Opt. Acta 27, 1025 (1980).
[CrossRef]

F. Gori, Opt. Lett. 4, 354 (1979).
[CrossRef] [PubMed]

F. Gori and C. Palma, Opt. Commun. 27, 185 (1978).
[CrossRef]

Guattari, G.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Palma, C.

Parzen, E.

E. Parzen, Ann. Math. Stat. 32, 951 (1961).
[CrossRef]

Saitoh, S.

S. Saitoh, Integral Transforms, Reproducing Kernels and Their Applications (Addison Wesley Longman, 1997).

Thomas-Agnan, C.

A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics (Kluwer Academic, 2003).

Turunen, J.

Vahimaa, P.

Wahba, G.

G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics 59 (Society for Industrial and Applied Mathematics, 1990).
[CrossRef]

Wolf, E.

F. Gori and E. Wolf, Opt. Commun. 61, 369 (1987).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Ann. Math. Stat. (1)

E. Parzen, Ann. Math. Stat. 32, 951 (1961).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

F. Gori, Opt. Acta 27, 1025 (1980).
[CrossRef]

Opt. Commun. (3)

F. Gori and C. Palma, Opt. Commun. 27, 185 (1978).
[CrossRef]

F. Gori, Opt. Commun. 46, 149 (1983).
[CrossRef]

F. Gori and E. Wolf, Opt. Commun. 61, 369 (1987).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Trans. Am. Math. Soc. (1)

N. Aronszajn, Trans. Am. Math. Soc. 68, 337 (1950).
[CrossRef]

Other (9)

The authors are indebted to A. T. Friberg for calling this point to their attention.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

J. W. Goodman, Statistical Optics (Wiley, 1983).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics 59 (Society for Industrial and Applied Mathematics, 1990).
[CrossRef]

S. Saitoh, Integral Transforms, Reproducing Kernels and Their Applications (Addison Wesley Longman, 1997).

A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics (Kluwer Academic, 2003).

For any nonnegative definite kernel there exists a unique functional Hilbert space whose functions are reproduced when they are multiplied by the kernel by means of the scalar product holding for such space. This is the origin of the phrase 'Reproducing kernel Hilbert space.'

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Equations (19)

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Q ( f ) = W 0 ( ρ 1 , ρ 2 ) f * ( ρ 1 ) f ( ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
W 0 ( ρ 1 , ρ 2 ) 2 W 0 ( ρ 1 , ρ 1 ) W 0 ( ρ 2 , ρ 2 ) .
W 0 ( ξ 1 , ξ 2 ) = F ( ξ 1 + ξ 2 2 ) G ( ξ 1 ξ 2 ) ,
F ( s ) = S M 1 + β s 2 , G ( s ) = e α s 2 ,
W 0 ( ξ 1 , ξ 2 ) = S M e α ( ξ 1 ξ 2 ) 2 1 + β ( ξ 1 + ξ 2 ) 2 4 .
S 0 ( ξ ) = W 0 ( ξ , ξ ) = S M 1 + β ξ 2 ,
μ 0 ( ξ 1 , ξ 2 ) = W 0 ( ξ 1 , ξ 2 ) S 0 ( ξ 1 ) S 0 ( ξ 2 ) = ( 1 + β ξ 1 2 ) ( 1 + β ξ 2 2 ) 1 + β ( ξ 1 + ξ 2 ) 2 4 e α ( ξ 1 ξ 2 ) 2 .
S z ( x ) = 1 λ z F ( σ ) G ̃ ( x σ λ z ) d σ ,
W z ( ) ( x 1 , x 2 ) = 1 λ z F ̃ ( x 2 x 1 λ z ) G ̃ ( x 1 + x 2 2 λ z ) .
μ z ( ) ( x 1 , x 2 ) = exp [ 2 π β x 2 x 1 λ z + π 2 4 α ( x 2 x 1 λ z ) 2 ] .
W 0 ( ρ 1 , ρ 2 ) = p ( v ) H * ( ρ 1 , v ) H ( ρ 2 , v ) d 2 v ,
Q ( f ) = p ( v ) H ( ρ , v ) f ( ρ ) d 2 ρ 2 d 2 v ,
W 0 ( ρ 1 , ρ 2 ) = w ( v ) V * ( ρ 1 v ) V ( ρ 2 v ) d 2 v ,
W 0 ( ρ 1 , ρ 2 ) = F ( ρ 1 + ρ 2 2 ) G ( ρ 1 ρ 2 ) ,
F ( s ) = p ( v ) exp [ 4 α ( s v ) 2 ] d 2 v
W 0 ( ρ 1 , ρ 2 ) = W i ( v 1 , v 2 ) H * ( ρ 1 , v 1 ) H ( ρ 2 , v 2 ) d 2 v 1 d 2 v 2 .
H ( ρ , v ) = τ ( ρ ) exp [ 2 π i v g ( ρ ) ] ,
W 0 ( ρ 1 , ρ 2 ) = τ * ( ρ 1 ) τ ( ρ 2 ) p ̃ [ g ( ρ 1 ) g ( ρ 2 ) ] .
W 0 ( ρ 1 , ρ 2 ) = τ * ( ρ 1 ) τ ( ρ 2 ) p ̃ [ a ( ρ 1 ρ 2 ) ] .

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