Abstract

The intensity probability distribution as well as the cross-spectral density of partially coherent optical fields generated through high-numerical-aperture illuminations are analyzed, and novel effects, not apparent in paraxial optical fields, are described. It is shown that the intensity probability distribution significantly differs from what can be expected from a small-angle analysis, and the number of degrees of freedom for the distribution is higher. It is further shown that the cross-spectral density of a high-angle optical field is a function of the coordinate difference along the propagation direction of the field.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  9. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2006).

2007 (2)

2006 (1)

C. Rydberg, J. Bengtsson, and T. Sandström, J. Microlithography, Microfabrication and Microsystems 5, 33004-1 (2006).

2005 (3)

2002 (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

J. Microlithography, Microfabrication and Microsystems (1)

C. Rydberg, J. Bengtsson, and T. Sandström, J. Microlithography, Microfabrication and Microsystems 5, 33004-1 (2006).

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

Opt. Commun. (1)

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, Opt. Commun. 248, 333 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Proc. SPIE (1)

J. H. Bruning, Proc. SPIE 6520, 652004 (2007).
[CrossRef]

Other (2)

J. W. Goodman, Statistical Optics (Wiley, 2000).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2006).

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Figures (4)

Fig. 1
Fig. 1

Geometry of the coordinate system. The optical field in the plane z = 0 is created by a number of plane waves arriving from the direction ν in the spherical-cap domain D.

Fig. 2
Fig. 2

Numerical simulation of the number of degrees of freedom parameter for the intensity probability distribution of three different polarization states for a circular cap as a function of θ max . Also highlighted in the illustration is the number of degrees of freedom of the intensity probability distribution for θ max = π 2 .

Fig. 3
Fig. 3

Cross-spectral density for of an optical field generated by a spherical-cap illumination of circular polarization with θ max = π 2 along Δ x (or Δ y due to symmetry). The cross-spectral density is different when considering the electric field along the x axis and y axis ( W x x and W y y are equal due to symmetry) or along the z axis ( W z z ) . As a reference the cross-spectral density of a scalar field is also indicated in the illustration.

Fig. 4
Fig. 4

Cross-spectral density of an optical field generated by a spherical-cap illumination of circular polarization with θ max = π 2 along Δ z . Again the cross-spectral density is different when considering the electric field along the x axis and y axis ( W x x and W y y are equal due to symmetry) or along the z axis ( W z z ) . In the scalar approximation the cross-spectral density along Δ z is coherent, and hence W x x ( Δ z ) , W y y ( Δ z ) , and W z z ( Δ z ) are all constant.

Equations (9)

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W i j ( r 1 , r 2 , ν ) = E i * ( r 1 , ν ) E j ( r 2 , ν ) , i , j = x , y , z ,
M = I 2 σ I 2 = Tr ( W ) 2 Tr ( W 2 ) ,
P 3 2 = 3 Tr ( W 2 ) 2 Tr ( W ) 2 1 2 ,
M = 3 2 P 3 2 + 1 .
E ( r ) = D exp [ i r k ] U ( θ , φ ) d S ,
I = E 2 = i = x , y , z D U i ( θ , φ ) d S 2 .
U lin ( θ , φ ) = [ k e y ] U ( θ , φ ) ,
U unp ( θ , φ ) = U 1 ( θ , φ ) e θ + U 2 ( θ , φ ) e φ ,
U circ ( θ , φ ) = U 1 ( θ , φ ) e θ + i U 1 ( θ , φ ) e φ .

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