Abstract

The Wigner distribution function of a circular aperture is determined. Analytic expressions as well as numerical and graphical results are presented.

© 1996 Optical Society of America

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References

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. A. Walther, “Radiometry and coherence,”J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  3. A. Walther, “Propagation of the generalized radiance through lenses,”J. Opt. Soc. Am. 68, 1606–1610 (1978).
    [CrossRef]
  4. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  5. M. J. Bastiaans, “The Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  6. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  7. E. Wolf, “Coherence and radiometry,”J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  8. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).
  9. N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
    [CrossRef]
  10. M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
    [CrossRef]
  11. M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
    [CrossRef]
  12. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970).

1986 (1)

1980 (1)

N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
[CrossRef]

1979 (3)

M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

1978 (3)

1968 (1)

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
[CrossRef]

M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

Marcuvitz, N.

N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
[CrossRef]

Walther, A.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

J. Opt. Soc. Am. (4)

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

Opt. Acta (2)

M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

Opt. Commun. (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Proc. IEEE (1)

N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
[CrossRef]

Other (2)

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

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

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Figures (4)

Fig. 1
Fig. 1

Eye-shaped interval.

Fig. 2
Fig. 2

Function |K(ψ, R)|.

Fig. 3
Fig. 3

Wigner distribution function F(x, 0, u, υ) of a circular source with radius a, for several values of the position variable x: (a) x/a = 0, (b) x/a = 0.95.

Fig. 4
Fig. 4

Scaled eye-shaped interval for x = 0 (circle) and for x = a (parabolas).

Equations (36)

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F ( x , y , u , υ ) = φ ( x + 1 2 x , y + 1 2 y ) × φ * ( x 1 2 x , y 1 2 y ) × exp [ j ( u x + υ y ) ] d x d y ;
F 0 ( x , y , u , υ ) = 1 4 π 2 F m ( x , y , u u , υ υ ) × F i ( x , y , u , υ ) d u d υ ,
φ ( x , y ) = { 1 for 0 x 2 + y 2 a 0 elsewhere .
a 2 ( x 1 2 x ) 2 1 2 y a 2 ( x 1 2 x ) 2 when x a 1 2 x 0 , a 2 ( x + 1 2 x ) 2 1 2 y a 2 ( x + 1 2 x ) 2 when 0 1 2 x a x ,
F ( x , 0 , u , υ ) = 2 ( a x ) 0 exp ( j u x ) d x × 2 a 2 ( x x / 2 ) 2 2 a 2 ( x x / 2 ) 2 exp ( j υ y ) d y + 0 2 ( a x ) exp ( j u x ) d x × 2 a 2 ( x + x / 2 ) 2 2 a 2 ( x + x / 2 ) 2 exp ( j υ y ) d y .
F ( x , 0 , u , υ ) = 2 ( a x ) 0 2 υ sin [ 2 υ a 2 ( x 1 2 x ) 2 ] × exp ( j u x ) d x + 0 2 ( a x ) 2 υ sin [ 2 υ a 2 ( x + 1 2 x ) 2 ] × exp ( j u x ) d x ,
F ( x , 0 , u , υ ) = 2 a x / a 1 2 υ sin ( 2 a υ 1 ξ 2 ) × exp [ j 2 u ( x α ξ ) ] d ξ + 2 a x / a 1 2 υ sin ( 2 a υ 1 ξ 2 ) × exp [ j 2 u ( x a ξ ) ] d ξ
F ( x , 0 , u , υ ) = 16 a 2 R { exp ( j 2 u x ) × x / a 1 sin ( 2 a υ 1 ξ 2 ) 2 a υ × exp ( j 2 a u ξ ) d ξ } ,
F ( x , 0 , u , 0 ) = 16 a 2 R { exp ( j 2 u x ) × x / a 1 1 ξ 2 exp ( j 2 a u ξ ) d ξ } .
F ( x , 0,0,0 ) = 16 a 2 2 [ π 2 arcsin x a x a 1 ( x a ) 2 ] ,
K ( ψ , R ) = 0 ψ exp ( j R cos β ) cos β d β ,
F ( a sin α , 0 , R 2 a sin θ , R 2 a cos θ ) = 16 a 2 2 R R { j exp ( j R sin α sin θ ) [ K ( α θ , R ) ] K * ( α + θ , R ) j π J 1 ( R ) ] } ,
F ( 0,0 , u , υ ) = 4 π a 2 J 1 ( 2 a u 2 + υ 2 ) a u 2 + υ 2 ,
K ( ψ , R ) π R ( 1 + 3 j 8 R ) [ C ( 2 R ( 1 cos ψ ) π ) j S ( 2 R ( 1 cos ψ ) π ) ] × exp ( j R ) + j R 2 cos ψ 1 + cos ψ 2 sin ψ × exp ( j R cos ψ ) ,
F ( x , 0 , u , υ ) F ( x , 0,0,0 ) F ( 0,0,0,0 ) F ( 0,0 , a x a u , a 2 x 2 a υ ) = 4 π a 2 [ 1 2 π arcsin x a 2 π x a 1 ( x a ) 2 ] × J 1 { 2 [ ( a x ) 2 u 2 + ( a 2 x 2 ) υ 2 ] 1 / 2 } [ ( a x ) 2 u 2 + ( a 2 x 2 ) υ 2 ] 1 / 2 .
η 2 + ξ 2 2 x a + x ( ξ 2 ± ξ ) = 1.
φ ( x , y ) = { 1 for 0 | x | a and 0 | y | b 0 elsewhere .
F ( x , y , u , υ ) = 16 ( a | x | ) ( b | y | ) sin 2 u ( a | x | ) 2 u ( a | x | ) × sin 2 υ ( b | y | ) 2 υ ( b | y | )
F ( x , y , u , υ ) = F ( x , y , 0,0 ) F ( 0,0,0,0 ) F ( 0,0 , a | x | a u , b | y | b υ ) ,
I = x / a 1 sin ( 2 a υ 1 ξ 2 ) exp ( j 2 a u ξ ) d ξ .
I = α π / 2 sin ( R cos θ cos φ ) exp ( j R sin θ sin φ ) cos φ d φ .
2 j I = α π / 2 exp [ j R cos ( φ θ ) ] cos φ d φ α π / 2 exp [ j R cos ( φ + θ ) ] cos φ d φ .
I 0 ( θ ) = α π / 2 exp [ j R cos ( φ θ ) ] cos φ d φ ,
I 0 ( θ ) = α θ π / 2 θ exp ( j R cos β ) × ( cos θ cos β sin θ sin β ) d β = cos θ α θ π / 2 θ exp ( j R cos β ) cos β d β sin θ α θ π / 2 θ exp ( j R cos β ) sin β d β = I 1 ( θ ) I 2 ( θ ) .
I 2 ( θ ) = sin θ α θ π / 2 exp ( j R cos β ) sin β d β = sin θ j R { exp [ j R cos ( α θ ) ] exp [ j R sin θ ] } ,
I 1 ( θ ) = cos θ α θ π / 2 exp ( j R cos β ) cos β d β .
K ( ψ , R ) = 0 ψ exp ( j R cos β ) cos β d β
I = 1 2 j [ K ( α θ , R ) K * ( α + θ , R ) j π J 1 ( R ) ] cos θ + j sin θ sin ( R cos α cos θ ) R exp ( j R sin α sin θ ) .
K [ arccos ( 1 s ) , R ] = 0 s exp [ j R ( 1 t ) ] 1 t 1 ( 1 t ) 2 d t = exp ( j R ) 0 s exp ( j R t ) 1 t t ( 2 t ) d t .
L ( s , R ) = 0 s exp ( j R t ) g 0 ( t ) d t t = L [ s , R ; g 0 ( s ) ] ,
R 0 s exp [ j R ( s t ) ] g 0 ( t ) g 0 ( 0 ) t d t = j g 0 ( t ) g 0 ( 0 ) t exp [ j R ( s t ) ] | 0 s j 0 s exp [ j R ( s t ) ] [ g 0 ( t ) g 0 ( 0 ) t ] d t = j g 0 ( s ) g 0 ( 0 ) s j exp ( j R s ) × 0 s exp ( j R t ) g 1 ( t ) t d t ,
g 1 ( t ) = g 0 ( t ) g 0 ( t ) g 0 ( 0 ) 2 t = d g 0 ( t ) d t 1 2 m = 1 d m g 0 ( t ) d t m | t = 0 t m 1 m !
L [ s , R ; g 0 ( s ) ] = g 0 ( 0 ) 0 s exp ( j R t ) d t t + [ g 0 ( s ) g 0 ( 0 ) ] j R s × exp ( j R s ) j R L [ s , R ; g 1 ( s ) ] .
L ( s , R ) = n = 0 ( j R ) n { g n ( 0 ) 0 s exp ( j R t ) d t t + [ g n ( s ) g n ( 0 ) ] j R s exp ( j R s ) } ,
0 s exp ( j R t ) d t t = 2 π R [ C ( 2 R s π ) j S ( 2 R s π ) ] ,
K ( ψ , R ) π R ( 1 + 3 j 8 R ) { C [ 2 R ( 1 cos ψ ) π ] j S [ 2 R ( 1 cos ψ ) π ] } × exp ( j R ) + j R 2 cos ψ 1 + cos ψ 2 sin ψ × exp ( j R cos ψ ) .

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