Abstract

It is shown that, among all partially coherent wave fields having the same informational entropy, the product of the effective widths of the intensity functions in the space and the spatial-frequency domains takes its minimum value for a wave field with a Gaussian-shaped cross-spectral density function. Furthermore, it is shown how this minimum value is related to the informational entropy and how this informational entropy is related to other quantities that can measure the overall degree of coherence of the partially coherent wave field.

© 1986 Optical Society of America

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References

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  1. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  2. M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
    [Crossref]
  3. M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A 1, 711–715 (1984).
    [Crossref]
  4. E. L. O’Neill, T. Asakura, “Optical image formation in terms of entropy transformations,” J. Phys. Soc. Jpn. 16, 301–308 (1961).
    [Crossref]
  5. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 8.
  6. H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–332.
    [Crossref]
  7. M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
    [Crossref]
  8. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [Crossref]
  9. R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, New York, 1953), Vol. 1.
  10. F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).
  11. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

1984 (1)

1983 (1)

1977 (1)

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[Crossref]

1976 (1)

1961 (1)

E. L. O’Neill, T. Asakura, “Optical image formation in terms of entropy transformations,” J. Phys. Soc. Jpn. 16, 301–308 (1961).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Asakura, T.

E. L. O’Neill, T. Asakura, “Optical image formation in terms of entropy transformations,” J. Phys. Soc. Jpn. 16, 301–308 (1961).
[Crossref]

Bastiaans, M. J.

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, New York, 1953), Vol. 1.

Gamo, H.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–332.
[Crossref]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, New York, 1953), Vol. 1.

Mandel, L.

O’Neill, E. L.

E. L. O’Neill, T. Asakura, “Optical image formation in terms of entropy transformations,” J. Phys. Soc. Jpn. 16, 301–308 (1961).
[Crossref]

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 8.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Riesz, F.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Sz.-Nagy, B.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).

Wolf, E.

J. Opt. Soc. Am. (2)

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

J. Phys. Soc. Jpn. (1)

E. L. O’Neill, T. Asakura, “Optical image formation in terms of entropy transformations,” J. Phys. Soc. Jpn. 16, 301–308 (1961).
[Crossref]

Opt. Acta (1)

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[Crossref]

Other (6)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 8.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–332.
[Crossref]

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, New York, 1953), Vol. 1.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

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Figures (1)

Fig. 1
Fig. 1

The entropy-based degree of coherence μent versus the quantity σ that occurs in the uncertainty principle.

Equations (29)

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S ¯ ( u 1 , u 2 , ω ) = S ( x 1 , x 2 , ω ) exp [ i ( u 1 x 1 u 2 x 2 ) ] d x 1 d x 2 .
d x 2 ( ω ) = x 2 S ( x , x , ω ) d x S ( x , x , ω ) d x .
S ( x 1 , x 2 ) = 1 k m = 0 λ m q m ( x 1 k ) q m * ( x 2 k ) .
S ( x 1 , x 2 ) q m ( x 2 k ) d x 2 = λ m q m ( x 1 k ) ( m = 0 , 1 , ) ;
q m ( ξ ) q n * ( ξ ) d ξ = { 1 for m = n 0 for m n ( m , n = 0 , 1 , ) .
λ 0 λ 1 0 .
2 d x d u m = 0 λ m ( 2 m + 1 ) m = 0 λ m ,
exp [ π ξ 2 2 π ( ξ w ) 2 ] = 2 1 / 4 m = 0 ( m ! ) 1 / 2 ( 4 π ) m / 2 w m ψ m ( ξ ) .
ν m = λ m m = 0 λ m ( m = 0 , 1 , ) .
ν 0 ν 1 0 , m = 0 ν m = 1 ,
2 d x d u m = 0 ν m ( 2 m + 1 ) .
m = 0 ν m ( 2 m + 1 ) m = 0 ν m = 1 ,
m = 0 ν m ln ν m .
ν 0 ν 1 0 , m = 0 ν m = 1 , m = 0 ν m ln ν m = constant .
ν m = 2 σ 1 + σ ( 1 σ 1 + σ ) m ( m = 0 , 1 , …; 0 < σ 1 ) ,
exp [ m = 0 ν m ln ν m ] = 2 σ 1 σ 2 ( 1 σ 1 + σ ) 1 / 2 σ .
S ( x 1 , x 2 ) = 1 k m = 0 2 σ 1 + σ ( 1 σ 1 + σ ) m ψ m ( x 1 k ) ψ m * ( x 2 k ) = 1 k 2 σ exp { π 2 k 2 [ σ ( x 1 + x 2 ) 2 + 1 σ ( x 1 x 2 ) 2 ] } .
2 d x d u 1 σ ,
μ ent = exp ( m = 0 ν m ln ν m ) .
μ ent = 2 σ 1 σ 2 ( 1 σ 1 + σ ) 1 / 2 σ ( 0 < σ 1 ) ;
μ p = { m = 0 ν m p } 1 p 1 ( p > 1 ) ,
lim p μ p = ν 0 ,
lim p 1 μ p = exp ( m = 0 ν m ln ν m ) = μ ent ,
d μ p d p 0.
μ p = exp [ f ( p ) g ( p ) ] ,
f ( p ) = ln m = 0 ν m p and g ( p ) = p 1 ;
f ( p ) = d f d p = m = 0 ν m p ln ν m m = 0 ν m p , g ( p ) = d g d p = 1.
lim p 1 f ( p ) g ( p ) = lim p 1 f ( p ) g ( p ) = m = 0 ν m ln ν m .
μ p = { m = 0 ν m p } 1 p 1 = { m = 0 ( ν m q ) p 1 q 1 ( ν m ) q p q 1 } 1 p 1 { ( m = 0 ν m q ) p 1 q 1 ( m = 0 ν m ) q p q 1 } 1 p 1 = { m = 0 ν m q } 1 p 1 = μ q ,

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