Abstract

The Wigner distribution of rotationally symmetric partially coherent light is considered, and the constraints for its moments are derived. Although all odd-order moments vanish, these constraints lead to a drastic reduction in the number of parameters that we need to describe all even-order moments: whereas in general we have N+1N+2N+3/6 different moments of order N, this number reduces to 1+N/22 in the case of rotational symmetry. A way to measure the moments as intensity moments in the output planes of (generally anamorphic) fractional Fourier-transform systems is presented.

© 2003 Optical Society of America

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References

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  1. E. Wigner, Phys. Rev. 40, 749 (1932).
    [CrossRef]
  2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
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    [CrossRef]
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    [CrossRef]
  6. J. Serna, F. Encinas-Sanz, and G. Nemeş, J. Opt. Soc. Am. A 18, 1726 (2001).
    [CrossRef]
  7. M. J. Bastiaans and T. Alieva, J. Opt. Soc. Am. A 19, 1763 (2002).
    [CrossRef]

2002 (1)

2001 (1)

J. Serna, F. Encinas-Sanz, and G. Nemeş, J. Opt. Soc. Am. A 18, 1726 (2001).
[CrossRef]

1995 (1)

1993 (1)

1976 (1)

1968 (1)

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

1932 (1)

E. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Alieva, T.

Bastiaans, M. J.

Encinas-Sanz, F.

J. Serna, F. Encinas-Sanz, and G. Nemeş, J. Opt. Soc. Am. A 18, 1726 (2001).
[CrossRef]

Mandel, L.

Martínez, C.

Martínez-Herrero, R.

Mejías, P. M.

Mukunda, N.

Nemes, G.

J. Serna, F. Encinas-Sanz, and G. Nemeş, J. Opt. Soc. Am. A 18, 1726 (2001).
[CrossRef]

Papoulis, A.

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

Serna, J.

J. Serna, F. Encinas-Sanz, and G. Nemeş, J. Opt. Soc. Am. A 18, 1726 (2001).
[CrossRef]

Simon, R.

Wigner, E.

E. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Wolf, E.

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

J. Serna, F. Encinas-Sanz, and G. Nemeş, J. Opt. Soc. Am. A 18, 1726 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Phys. Rev. (1)

E. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Other (1)

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

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Tables (1)

Tables Icon

Table 1 Ipqrs for Second-Order Moments [m,n=2,0, 1,1, 0,2] and Fourth-Order Moments [m,n=4,0, 3,1, 2,2, 1,3, 0,4]

Equations (11)

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Wx,y,u,v=--×Γx+12x,y+12y,x-12x,y-12y×exp-j2πux+vydxdy.
Wρ cos ϕ,ρ sin ϕ,ζ cosϕ+θ,ζ sinϕ+θ=Wρ,ζ,θ.
μpqrsE=----Wx,y,u,v×xpuqyrvsdxdydudv,
μpqrsE=0002πWρ,ζ,θρp+r+1ζq+s+1dρdζdθ×02πcos ϕpcosϕ+θq×sin ϕrsinϕ+θsdϕ.
Ipqrsθ=1π02πcos ϕpcosϕ+θq×sin ϕrsinϕ+θsdϕ=12πk=0q l=0sqksl-1k×1+-1p+q-k+l1+-1r+s+k-l×Bp+q-k+l+12,r+s+k-l+12×cos θq+s-k-lsin θk+l.
μp0r0outα,β=k=0p m=0rpkrmμp-k,k,r-m,m cosp-k α×sink α cosr-m β sinm β.
μ2000outα,β=μ2000 cos2 α+2μ1100 cos α sin α+μ0200 sin2 α,
μ1010outα,β=μ1001 sinβ-α.
μ4000outα,β=μ4000 cos4 α+4μ3100 cos3 α sin α+6μ2200 cos2 α sin2 α+4μ1300 cos α sin3 α+μ0400 sin4 α,
μ3010outα,β=μ3001 cos2 α+3μ2101 cos α sin α+3μ1201 sin2 αsinβ-α,
3μ2020outα,β=μ4000 cos2 α cos2 β+2μ3100 cos α cos β sinα+β+6μ2200 cos α sin α cos β sin β +3μ2002 sin2β -α+2μ1300 sin α sin β sinα+β+μ0400 sin2 α sin2 β .

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