Abstract

The focusing properties of partially coherent Bessel–Gaussian beams passing through a high-numerical-aperture objective are studied based on vectorial Debye theory. Expressions for the intensity distribution, degree of coherence μ, and degree of polarization P are derived near the focus. Numerical calculations are performed to analyze the influences of varying corresponding parameters on the intensity distribution, μ, and P in the focal region. It is shown that the intensity, μ, and P in the focal region are all influenced by varying the effective coherent length Lc of incident beams and maximal angle α determined by the numerical aperture of the objective. Also, the linearly polarized incident field is found to be depolarized after it is focused by a high-numerical-aperture objective.

© 2008 Optical Society of America

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References

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N. Bokor and N. Davidson, Opt. Commun. 279, 229 (2007).
[CrossRef]

T. Grosjean and D. Courjon, Opt. Commun. 272, 314 (2007).
[CrossRef]

G. M. Lerman and U. Levy, Opt. Lett. 32, 2194 (2007).
[CrossRef] [PubMed]

Z. Bomzon and M. Gu, Opt. Lett. 32, 3017 (2007).
[CrossRef] [PubMed]

Q. Zhan, Opt. Lett. 31, 867 (2007).
[CrossRef]

L. Rao and J. Pu, Chin. Phys. Lett. 24, 1252 (2007).
[CrossRef]

2005 (2)

2004 (1)

2003 (1)

2002 (1)

1997 (1)

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Figures (4)

Fig. 1
Fig. 1

Contour plots of the intensity distribution (a) in the focal plane and (b) near the focus. All figures are calculated for n = 1 , λ = 633 nm , β = 0.25 mm 1 , w 0 = 2 cm , f = 1 cm , α = 75 ° , L c = 1 cm .

Fig. 2
Fig. 2

Dependence of intensity distribution in the focal plane on (a) L c and (b) α. Both figures are calculated for ϕ = π 4 . Other parameters for calculation are the same as in Fig. 1.

Fig. 3
Fig. 3

Distribution of degree of coherence in the focal plane: (a) distribution of μ x y , μ x z , and μ y z ; (b) μ x y as a function of L c ; (c) μ x y as a function of α. The parameters for calculation are the same as in Fig. 2.

Fig. 4
Fig. 4

Dependence of distribution of P in the focal plane on (a) L c and (b) α. The parameters for calculation are the same as in Fig. 2.

Equations (11)

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[ E x ( r , ϕ , z ) E y ( r , ϕ , z ) E z ( r , ϕ , z ) ] = i n [ i A [ I n + 0.5 ( I n + 2 e i 2 ϕ + I n 2 e i 2 ϕ ) ] A [ 0.5 ( I n + 2 e i 2 ϕ I n 2 e i 2 ϕ ) ] A [ 0.5 ( I n + 1 e i ϕ I n 1 e i ϕ ) ] ] e i n ϕ ,
W i j ( r 1 , r 2 , z ) = E i * ( r 1 , ϕ 1 , z ) E j ( r 2 , ϕ 2 , z ) , ( i , j ) = ( x , y , z ) ,
W i i ( r 1 , r 2 , z ) = E i * ( r 1 , ϕ 1 , z ) E i ( r 2 , ϕ 2 , z ) , i = x , y , z ,
I p * ( r 1 , z ) I q ( r 2 , z ) = 0 α 0 α A ( θ 1 , θ 2 ) cos θ 1 cos θ 2 sin θ 1 sin θ 2 g ( θ 1 ) g ( θ 2 ) J p ( k r 1 sin θ 1 ) J q ( k r 2 sin θ 2 ) exp [ i k z ( cos θ 2 cos θ 1 ) ] d θ 1 d θ 2 ,
g ( θ i ) = { ( 1 + cos θ i ) p , q = n sin θ i p , q = n ± 1 ( 1 cos θ i ) p , q = n ± 2 ,
I ( r , ϕ , z ) = I x ( r , ϕ , z ) + I y ( r , ϕ , z ) + I z ( r , ϕ , z ) = W x x ( r , r , z ) + W y y ( r , r , z ) + W z z ( r , r , z ) .
W n ( r 1 , r 2 , 0 ) = I 0 A n ( r 1 , r 2 ) exp [ i n ( ϕ 1 ϕ 2 ) ] ,
A n ( r 1 , r 2 ) = J n ( β r 1 ) J n ( β r 2 ) exp [ ( r 1 2 + r 2 2 ) w 0 2 ] exp [ ( r 1 r 2 ) 2 L c 2 ] .
A n ( θ 1 , θ 2 ) = J n ( β f sin θ 1 ) J n ( β f sin θ 2 ) exp [ f 2 ( sin 2 θ 1 + sin 2 θ 2 ) w 0 2 ] × exp [ f 2 ( sin θ 1 sin θ 2 ) 2 L c 2 ] .
μ i j ( r ) = Φ i j ( r ) Φ i i ( r ) Φ j j ( r ) , ( i , j ) = ( x , y , z ) ,
P 2 ( r , ϕ , z ) = 3 2 { I x ( r , ϕ , z ) 2 + I y ( r , ϕ , z ) 2 + I z ( r , ϕ , z ) 2 [ I x ( r , ϕ , z ) + I y ( r , ϕ , z ) + I z ( r , ϕ , z ) ] 2 1 3 } .

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