Abstract

A class of uncertainty relations for partially coherent light is derived; the uncertainty relations in this class express the fact that the product of the effective widths of the space-domain intensity and the spatial-frequency-domain intensity of the light has a lower bound and that this lower bound is inversely proportional to the overall degree of coherence. The different members of this class of uncertainty relations correspond to different choices of the quantity that measures the overall degree of coherence. One particular uncertainty relation has the property of being compatible with the ordinary uncertainty principle; the corresponding overall degree of coherence might therefore be considered the best possible choice of all the quantities that measure the overall degree of coherence of the light.

© 1984 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, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
    [Crossref]
  4. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [Crossref]
  5. M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
    [Crossref]
  6. W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [Crossref]
  7. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [Crossref]
  8. E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
    [Crossref]
  9. W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
    [Crossref]
  10. E. Collett, E. Wolf, “New equivalence theorems for planar sources that generate the same distributions of radiant intensity,” J. Opt. Soc. Am. 69, 942–950 (1979).
    [Crossref]
  11. H. Gamo, “Thermodynamic entropy of partially coherent light beams,” J. Phys. Soc. Jpn. 19, 1955–1961 (1964).
    [Crossref]
  12. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.
  13. F. Riesz, B. Sz.- Nagy, Functional Analysis (Ungar, New York, 1955).
  14. A. J. E. M. Janssen, “Positivity of weighted Wigner distributions,” SIAM J. Math. Anal. 12, 752–758 (1981).
    [Crossref]
  15. The proof in Appendix A is due to M. L. J. Hautus, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).
  16. A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982).
    [Crossref]
  17. M. J. Bastiaans, “Lower bound in the uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 1320–1324 (1983).
    [Crossref]
  18. The nonincreasing behavior of μq can be proved by showing that its derivative is nonpositive; indeed, with f(t) = t log t(t≥ 0) a convex function, we haveq2(∑m=0∞λmq)1-1/qddq(∑m=0∞λmq)1/q=∑m=0∞f(λmq)-f(∑m=0∞λmq),which is nonpositive by means of Ref. 20, Sec. 1.4.7, Eq. (1).
  19. M. Marcus, H. Minc, Introduction to Linear Algebra (Macmillan, New York, 1965), Sec. 3.5, Theorem 5.9.
  20. D. S. Mitrinović, Analytic Inequalities (Springer-Verlag, Berlin, 1970).
  21. The proof in Appendix C is partly due to J. Boersma, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).

1983 (2)

1982 (1)

1981 (2)

A. J. E. M. Janssen, “Positivity of weighted Wigner distributions,” SIAM J. Math. Anal. 12, 752–758 (1981).
[Crossref]

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[Crossref]

1979 (1)

1978 (3)

1977 (2)

W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[Crossref]

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

1976 (1)

1964 (1)

H. Gamo, “Thermodynamic entropy of partially coherent light beams,” J. Phys. Soc. Jpn. 19, 1955–1961 (1964).
[Crossref]

Bastiaans, M. J.

M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
[Crossref]

M. J. Bastiaans, “Lower bound in the uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 1320–1324 (1983).
[Crossref]

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[Crossref]

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

Boersma, J.

The proof in Appendix C is partly due to J. Boersma, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).

Carter, W. H.

W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
[Crossref]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[Crossref]

Collett, E.

Courant, R.

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

Gamo, H.

H. Gamo, “Thermodynamic entropy of partially coherent light beams,” J. Phys. Soc. Jpn. 19, 1955–1961 (1964).
[Crossref]

Hautus, M. L. J.

The proof in Appendix A is due to M. L. J. Hautus, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).

Hilbert, D.

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

Janssen, A. J. E. M.

A. J. E. M. Janssen, “Positivity of weighted Wigner distributions,” SIAM J. Math. Anal. 12, 752–758 (1981).
[Crossref]

Mandel, L.

Marcus, M.

M. Marcus, H. Minc, Introduction to Linear Algebra (Macmillan, New York, 1965), Sec. 3.5, Theorem 5.9.

Minc, H.

M. Marcus, H. Minc, Introduction to Linear Algebra (Macmillan, New York, 1965), Sec. 3.5, Theorem 5.9.

Mitrinovic, D. S.

D. S. Mitrinović, Analytic Inequalities (Springer-Verlag, Berlin, 1970).

Nagy, B. Sz.-

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

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).

Starikov, A.

Wolf, E.

J. Opt. Soc. Am. (8)

J. Phys. Soc. Jpn. (1)

H. Gamo, “Thermodynamic entropy of partially coherent light beams,” J. Phys. Soc. Jpn. 19, 1955–1961 (1964).
[Crossref]

Opt. Acta (2)

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

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[Crossref]

Opt. Commun. (1)

W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
[Crossref]

SIAM J. Math. Anal. (1)

A. J. E. M. Janssen, “Positivity of weighted Wigner distributions,” SIAM J. Math. Anal. 12, 752–758 (1981).
[Crossref]

Other (8)

The proof in Appendix A is due to M. L. J. Hautus, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).

The nonincreasing behavior of μq can be proved by showing that its derivative is nonpositive; indeed, with f(t) = t log t(t≥ 0) a convex function, we haveq2(∑m=0∞λmq)1-1/qddq(∑m=0∞λmq)1/q=∑m=0∞f(λmq)-f(∑m=0∞λmq),which is nonpositive by means of Ref. 20, Sec. 1.4.7, Eq. (1).

M. Marcus, H. Minc, Introduction to Linear Algebra (Macmillan, New York, 1965), Sec. 3.5, Theorem 5.9.

D. S. Mitrinović, Analytic Inequalities (Springer-Verlag, Berlin, 1970).

The proof in Appendix C is partly due to J. Boersma, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).

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

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

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

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Equations (57)

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E [ ϕ ˜ ( x 1 , t 1 ) ϕ ˜ * ( x 2 , t 2 ) ] = S ˜ ( x 1 , x 2 , t 1 - t 2 ) .
S ( x 1 , x 2 , ω ) = S ˜ ( x 1 , x 2 , τ ) exp ( i ω τ ) d τ .
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 ,
d u 2 ( ω ) = u 2 S ¯ ( u , u , ω ) d u S ¯ ( u , u , ω ) d u .
2 π k 2 ( ω ) d x 2 ( ω ) + k 2 ( ω ) 2 π d u 2 ( ω ) ,
2 π k 2 ( ω ) d x 2 ( ω ) + k 2 ( ω ) 2 π d u 2 ( ω ) ,
S ( x 1 , x 2 , ω ) = m = 0 λ m ( ω ) k ( ω ) q m ( x 1 k ( ω ) , ω ) q m * ( x 2 k ( ω ) , ω ) .
λ 0 ( ω ) λ 1 ( ω ) λ m ( ω ) 0.
q m ( ξ , ω ) q n * ( ξ , ω ) d ξ = δ m n             ( m , n = 0 , 1 , ) ,
q m ( ξ , ω ) = n = 0 a m n ( ω ) ψ n ( ξ )             ( m = 0 , 1 , ) ,
exp [ π ξ 2 - 2 π ( ξ - w ) 2 ] = 2 - 1 / 4 n = 0 ( n ! ) - 1 / 2 ( 4 π ) n / 2 w n ψ n ( ξ ) .
2 π k 2 ( ω ) d x 2 ( ω ) + k 2 ( ω ) 2 π d u 2 ( ω ) = m = 0 λ m ( ω ) n = 0 a m n ( ω ) 2 ( 2 n + 1 ) m = 0 λ m ( ω )
n = 0 a l n ( ω ) a m n * = l n         ( l , m = 0 , 1 , ) .
m = 0 λ m n = 0 a m n 2 ( 2 n + 1 ) m = 0 λ m ( 2 m + 1 ) ,
m = 0 λ m ( 2 m + 1 ) m = 0 λ m ,
m = 0 λ m n = 0 a m n 2 ( 2 n + 1 ) m = 0 λ m 1 ,
2 π k 2 d x 2 + k 2 2 π d u 2 1.
[ m = 0 λ m ( 2 m + 1 ) ] ( m = 0 λ m q ) p / q 2 ( 1 + 1 p ) - p ( m = 0 λ m ) p + 1 × ( p 1 , 1 p + 1 q = 1 ) ,
m = 0 λ m n = 0 a m n 2 ( 2 n + 1 ) m = 0 λ m [ ( m = 0 λ m q ) 1 / q m = 0 λ m ] p 2 ( 1 + 1 p ) - p             ( p 1 , 1 p + 1 q = 1 ) ,
μ q = ( m = 0 λ m q ) 1 / q m = 0 λ m             ( q > 1 ) .
( 2 π k 2 d x 2 + k 2 2 π d u 2 ) μ q p 2 ( 1 + 1 p ) - p ,             ( p 1 , 1 p + 1 q = 1 ) .
d x d u μ q p ( 1 + 1 p ) - p ,             ( p 1 , 1 p + 1 q = 1 ) .
d x d u μ ½ .
μ = λ 0 m = 0 λ m .
m = 0 λ m = S ( x , x ) d x = 1 2 π S ¯ ( u , u ) d u .
d x d u μ 2 2 > / .
μ 2 = ( m = 0 λ m 2 ) 1 / 2 m = 0 λ m .
m = 0 λ m 2 = S ( x 1 , x 2 ) 2 d x 1 d x 2 = 1 4 π 2 S ¯ ( u 1 , u 2 ) 2 d u 1 d u 2 .
2 d x d u 1 μ 1.
b m = n = 0 a m n 2 γ n             ( m = 0 , 1 , ) ,
γ 0 γ 1 γ n
n = 0 a l n a m n * = δ l m             ( l , m = 0 , 1 , ) .
h i j = n = 0 a i n a j n * γ n             ( i , j = 0 , 1 , , M , M 0 ) .
ν 0 ν 1 ν m ν M .
ν m γ m             ( m = 0 , 1 , , M ) ,
m = 0 M b m = m = 0 M h m m = m = 0 M ν m m = 0 M γ m .
λ 0 λ 1 λ m ,
m = 0 M λ m b m = λ 0 b 0 + m = 1 M λ m b m = λ 0 b 0 + m = 1 M λ m ( n = 0 m b n - n = 0 m - 1 b n ) = λ 0 b 0 + m = 1 M λ m n = 0 m b n - m = 0 M - 1 λ m + 1 n = 0 m b n = m = 0 M - 1 λ m n = 0 m b n + λ M n = 0 M b n - m = 0 M - 1 λ m + 1 n = 0 m b n = λ M n = 0 M b n + m = 0 M - 1 ( λ m - λ m + 1 ) n = 0 m b n λ M n = 0 M γ n + m = 0 M - 1 ( λ m - λ m + 1 ) n = 0 m γ n = m = 0 M λ m γ m .
m = 0 λ m n = 0 a m n 2 ( 2 n + 1 ) m = 0 λ m ( 2 m + 1 ) .
m = 0 λ m = 1 2 M + 1 [ 2 m = 0 ( M - m ) λ m + m = 0 λ m ( 2 m + 1 ) ] .
m = 0 ( M - m ) λ m 0 m M ( M - m ) λ m ,
0 m M ( M - m ) λ m [ 0 m M ( M - m ) p ] 1 / p × ( 0 m M λ m q ) 1 / q             ( p 1 , 1 p + 1 q = 1 ) ,
0 m M λ m q m = 0 λ m q ,
0 m M ( M - m ) p 1 p + 1 ( M + ½ ) p + 1             ( p 1 ) .
m = 0 λ m ( 2 M + 1 2 p + 2 ) 1 / p ( m = 0 λ m q ) 1 / q + 1 2 M + 1 × [ m = 0 λ m ( 2 m + 1 ) ]             ( p 1 , 1 p + 1 q = 1 ) .
( 2 M + 1 ) p + 1 = 2 ( p + 1 ) [ p m = 0 λ m ( 2 m + 1 ) [ m = 0 λ m q ] 1 / q ] p ,
( m = 0 λ m ) p + 1 1 2 ( 1 + 1 p ) p ( m = 0 λ m q ) p / q × [ m = 0 λ m ( 2 m + 1 ) ]             ( p 1 , 1 p + 1 q = 1 ) .
0 m M ( M - m ) p 1 p + 1 ( M + ½ ) p + 1
f p ( t ) = 1 p + 1 [ ( t + ½ ) p + 1 - ( t - ½ ) p + 1 ] - t p 0             ( t ½ ) ,
g p ( t ) = 1 p + 1 ( t + ½ ) p + 1 - t p 0             ( 0 t ½ ) ,
f p ( t ) = 2 t p + 1 p + 1 n = 1 ( p + 1 2 n + 1 ) ( 1 2 t ) 2 n + 1 .
d d t f p ( t ) = p f p - 1 ( t ) ,
g p ( t ) = t p - 1 ( t 2 p + 1 - t 2 + p 8 ) + p ( p - 1 ) 3 ! t p + 1 ( 1 2 τ ) 3 ( 0 < 1 2 τ < 1 2 t ) .
0 m M ( M - m ) p = m = 0 M ( M - m ) p 1 p + 1 m = 0 M [ ( M - m + ½ ) p + 1 - ( M - m - ½ ) p + 1 ] = 1 p + 1 [ ( M + ½ ) p + 1 - ( M - M - ½ ) p + 1 ] 1 p + 1 ( M + ½ ) p + 1 .
0 m M ( M - m ) p = m = 0 M ( M - m ) p 1 p + 1 m = 0 M - 1 [ ( M - m + ½ ) p + 1 - ( M - m - ½ ) p + 1 ] + ( M - M ) p = 1 p + 1 ( M + ½ ) p + 1 - [ 1 p + 1 ( M - M - ½ ) p + 1 - ( M - M ) p ] 1 p + 1 ( M + ½ ) p + 1 .
q2(m=0λmq)1-1/qddq(m=0λmq)1/q=m=0f(λmq)-f(m=0λmq),

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