Abstract

The intensity and the state of coherence are examined in the focal region of a converging, partially coherent wave field. In particular, Bessel-correlated fields are studied in detail. It is found that it is possible to change the intensity distribution and even to produce a local minimum of intensity at the geometrical focus by altering the coherence length. It is also shown that, even though the original field is partially coherent, in the focal region there are pairs of points at which the field is fully correlated and pairs of points at which the field is completely incoherent. The relevance of this work to applications such as optical trapping and beam shaping is discussed.

© 2008 Optical Society of America

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References

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  1. B. Lü, B. Zhang, and B. Cai, "Focusing of a Gaussian Schell-model beam through a circular lens," J. Mod. Opt. 42, 289-298 (1995).
    [CrossRef]
  2. W. Wang, A. T. Friberg, and E. Wolf, "Focusing of partially coherent light in systems of large Fresnel numbers," J. Opt. Soc. Am. A 14, 491-496 (1997).
    [CrossRef]
  3. A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, "Focal shifts of converging diffracted waves of any state of spatial coherence," Opt. Commun. 196, 1-7 (2001).
    [CrossRef]
  4. T. D. Visser, G. Gbur, and E. Wolf, "Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus," Opt. Commun. 213, 13-19 (2002).
    [CrossRef]
  5. L. Wang and B. Lü, "Propagation and focal shift of J0-correlated Schell-model beams," Optik (Stuttgart) 117, 167-172 (2006).
    [CrossRef]
  6. D. G. Fischer and T. D. Visser, "Spatial correlation properties of focused partially coherent light," J. Opt. Soc. Am. A 21, 2097-2102 (2004).
    [CrossRef]
  7. L. Rao and J. Pu, "Spatial correlation properties of focused partially coherent vortex beams," J. Opt. Soc. Am. A 24, 2242-2247 (2007).
    [CrossRef]
  8. G. Gbur and T. D. Visser, "Can spatial coherence effects produce a local minimum of intensity at focus?" Opt. Lett. 28, 1627-1629 (2003).
    [CrossRef] [PubMed]
  9. M. Born and E. Wolf, Principles of Optics, 7th ed. (expanded) (Cambridge Univ. Press, 1999).
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  11. E. H. Linfoot and E. Wolf, "Phase distribution near focus in an aberration-free diffraction image," Proc. Phys. Soc. London, Sect. B 69, 823-832 (1956).
    [CrossRef]
  12. J. T. Foley, "Effect of an aperture on the spectrum of partially coherent light," J. Opt. Soc. Am. A 8, 1099-1105 (1991).
    [CrossRef]
  13. E. Wolf and D. F. V. James, "Correlation-induced spectral changes," Rep. Prog. Phys. 59, 771-818 (1996).
    [CrossRef]
  14. K. T. Gahagan and G. A. Swartzlander, Jr., "Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap," J. Opt. Soc. Am. B 16, 533-537 (1999).
    [CrossRef]
  15. J. Arlt and M. J. Padgett, "Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam," Opt. Lett. 25, 191-193 (2000).
    [CrossRef]
  16. F. Gori, G. Guattari, and C. Padovani, "Modal expansion for J0-correlated Schell-model sources," Opt. Commun. 4, 311-316 (1987).
    [CrossRef]
  17. E. Wolf, "New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources," J. Opt. Soc. Am. 72, 343-351 (1982).
    [CrossRef]
  18. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

2007

2006

L. Wang and B. Lü, "Propagation and focal shift of J0-correlated Schell-model beams," Optik (Stuttgart) 117, 167-172 (2006).
[CrossRef]

2004

2003

2002

T. D. Visser, G. Gbur, and E. Wolf, "Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus," Opt. Commun. 213, 13-19 (2002).
[CrossRef]

2001

A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, "Focal shifts of converging diffracted waves of any state of spatial coherence," Opt. Commun. 196, 1-7 (2001).
[CrossRef]

2000

1999

1997

1996

E. Wolf and D. F. V. James, "Correlation-induced spectral changes," Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

1995

B. Lü, B. Zhang, and B. Cai, "Focusing of a Gaussian Schell-model beam through a circular lens," J. Mod. Opt. 42, 289-298 (1995).
[CrossRef]

1991

1987

F. Gori, G. Guattari, and C. Padovani, "Modal expansion for J0-correlated Schell-model sources," Opt. Commun. 4, 311-316 (1987).
[CrossRef]

1982

1956

E. H. Linfoot and E. Wolf, "Phase distribution near focus in an aberration-free diffraction image," Proc. Phys. Soc. London, Sect. B 69, 823-832 (1956).
[CrossRef]

J. Mod. Opt.

B. Lü, B. Zhang, and B. Cai, "Focusing of a Gaussian Schell-model beam through a circular lens," J. Mod. Opt. 42, 289-298 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

F. Gori, G. Guattari, and C. Padovani, "Modal expansion for J0-correlated Schell-model sources," Opt. Commun. 4, 311-316 (1987).
[CrossRef]

A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, "Focal shifts of converging diffracted waves of any state of spatial coherence," Opt. Commun. 196, 1-7 (2001).
[CrossRef]

T. D. Visser, G. Gbur, and E. Wolf, "Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus," Opt. Commun. 213, 13-19 (2002).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

L. Wang and B. Lü, "Propagation and focal shift of J0-correlated Schell-model beams," Optik (Stuttgart) 117, 167-172 (2006).
[CrossRef]

Proc. Phys. Soc. London, Sect. B

E. H. Linfoot and E. Wolf, "Phase distribution near focus in an aberration-free diffraction image," Proc. Phys. Soc. London, Sect. B 69, 823-832 (1956).
[CrossRef]

Rep. Prog. Phys.

E. Wolf and D. F. V. James, "Correlation-induced spectral changes," Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

Other

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

M. Born and E. Wolf, Principles of Optics, 7th ed. (expanded) (Cambridge Univ. Press, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

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Figures (9)

Fig. 1
Fig. 1

Illustration of the notation.

Fig. 2
Fig. 2

(a) Three-dimensional normalized spectral density distribution and (b) its contours for the case β 1 = 0.02 m , a = 0.01 m and hence ( β a ) 1 = 2.00 . In this example λ = 500 nm , and f = 2 m .

Fig. 3
Fig. 3

(a) Three-dimensional normalized spectral density distribution and (b) its contours for the case β 1 = 3.5 × 10 3 m , a = 0.01 m and hence ( β a ) 1 = 0.35 . The other parameters are the same as those of Fig. 2.

Fig. 4
Fig. 4

(a) Three-dimensional normalized spectral density distribution and (b) its contours for the case β 1 = 2.5 × 10 3 m , a = 0.01 m and hence ( β a ) 1 = 0.25 . The other parameters are the same as those of Fig. 2.

Fig. 5
Fig. 5

Spectral degree of coherence of the field in the aperture μ ( 0 ) ( ρ , ω ) for three different values of ( β a ) 1 , as discussed in the text. In this example λ = 500 nm , a = 0.01 m , and f = 2 m .

Fig. 6
Fig. 6

Spectral degree of coherence μ ( 0 , 0 , 0 ; x , 0 , 0 ; ω ) (solid curve) and spectral density S ( x , 0 , 0 , ω ) normalized to its maximum value (dashed curve) for the case (a), ( β a ) 1 = 2 and (b), ( β a ) 1 = 0.35 . In this example λ = 0.6328 μ m , a = 0.01 m , and f = 0.02 m .

Fig. 7
Fig. 7

Spectral degree of coherence μ ( 0 , 0 , 0 ; x , 0 , 0 ; ω ) (solid curve) and the spectral density S ( x , 0 , 0 , ω ) normalized to its maximum value (dashed curve), for the case (a), ( β a ) 1 = 0.25 and (b), ( β a ) 1 = 0.2 . In this example λ = 0.6328 μ m , a = 0.01 m , and f = 0.02 m .

Fig. 8
Fig. 8

Normalized spectral density, Eq. (39), in the focal plane for the case ( β a ) 1 = 0.13 . In this example λ = 500 nm , a = 0.01 m , n = 2 , and f = 2 m .

Fig. 9
Fig. 9

Eigenvalues λ n versus n for a J 0 -correlated field with a uniform spectral density across the plane of the aperture. Only the points corresponding to integer values of n are meaningful; the connecting lines are drawn to aid the eye.

Equations (50)

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U ( r , ω ) = i λ A U ( 0 ) ( r , ω ) e i k s s d 2 r ,
W ( 0 ) ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) ,
W ( r 1 , r 2 , ω ) = 1 λ 2 A W ( 0 ) ( r 1 , r 2 , ω ) e i k ( s 2 s 1 ) s 1 s 2 d 2 r 1 d 2 r 2 .
s 1 = r 1 r 1 ,
s 2 = r 2 r 2 .
s 1 f q 1 r 1 ,
s 2 f q 2 r 2 ,
W ( r 1 , r 2 , ω ) = 1 ( λ f ) 2 A W ( 0 ) ( r 1 , r 2 , ω ) × e i k ( q 1 r 1 q 2 r 2 ) d 2 r 1 d 2 r 2 .
S ( r , ω ) = W ( r , r , ω ) .
S ( r , ω ) = 1 ( λ f ) 2 A W ( 0 ) ( r 1 , r 2 , ω ) e i k ( q 1 q 2 ) r d 2 r 1 d 2 r 2 .
μ ( r 1 , r 2 , ω ) W ( r 1 , r 2 , ω ) [ S ( r 1 , ω ) S ( r 2 , ω ) ] 1 2 .
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = S ( 0 ) ( ω ) e ( ρ 2 ρ 1 ) 2 2 σ g 2 ,
W ( 0 ) ( r 1 , r 2 , ω ) = S ( 0 ) ( ω ) J 0 ( β r 2 r 1 ) ,
W ( r 1 , r 2 , ω ) = 1 ( λ f ) 2 A S ( 0 ) ( ω ) J 0 ( β r 2 r 1 ) × e i k ( q 1 r 1 q 2 r 2 ) d 2 r 1 d 2 r 2 .
r i = ( a ρ i cos ϕ i , a ρ i sin ϕ i , z i ) ( i = 1 , 2 ) .
S coh = lim β 0 W ( r 1 = r 2 = 0 , ω ) = a 4 π 2 S ( 0 ) ( ω ) λ 2 f 2 .
u = k ( a f ) 2 z ,
v = k ( a f ) ρ = k ( a f ) x 2 + y 2 .
S norm ( u , v , ω ) = S ( u , v , ω ) S coh = 1 π 2 0 2 π 0 1 0 2 π 0 1 J 0 { β a [ ρ 1 2 + ρ 2 2 2 ρ 1 ρ 2 cos ( ϕ 1 ϕ 2 ) ] 1 2 } cos [ v ( ρ 1 cos ϕ 1 ρ 2 cos ϕ 2 ) + u ( ρ 1 2 ρ 2 2 ) 2 ] ρ 1 ρ 2 d ρ 1 d ϕ 1 d ρ 2 d ϕ 2 .
r 1 = ( 0 , 0 , z 1 ) ,
r 2 = ( 0 , 0 , z 2 ) .
q 1 r 1 z 1 ( 1 ρ 1 2 2 f 2 ) ,
q 2 r 2 z 2 ( 1 ρ 2 2 2 f 2 ) .
W ( 0 , 0 , z 1 ; 0 , 0 , z 2 ; ω ) = 1 ( λ f ) 2 0 2 π 0 a 0 2 π 0 a S ( 0 ) ( ω ) J 0 { β [ ρ 1 2 + ρ 2 2 2 ρ 1 ρ 2 cos ( ϕ 1 ϕ 2 ) ] 1 2 } × e i k [ z 1 ( 1 ρ 1 2 2 f 2 ) + z 2 ( 1 ρ 2 2 2 f 2 ) ] ρ 1 ρ 2 × d ϕ 1 d ρ 1 d ϕ 2 d ρ 2 .
J 0 { β [ ρ 1 2 + ρ 2 2 2 ρ 1 ρ 2 cos ( ϕ 1 ϕ 2 ) ] 1 2 } = J 0 ( β ρ 1 ) J 0 ( β ρ 2 ) + n = 1 2 [ J n ( β ρ 1 ) J n ( β ρ 2 ) cos [ n ( ϕ 1 ϕ 2 ) ] ] .
W ( 0 , 0 , z 1 ; 0 , 0 , z 2 ; ω ) = f * ( 0 , 0 , z 1 ; ω ) f ( 0 , 0 , z 2 ; ω ) ,
f ( 0 , 0 , z ; ω ) = k f 0 a J 0 ( β ρ ) e i k z ( 1 ρ 2 2 f 2 ) ρ d ρ .
μ ( 0 , 0 , z 1 ; 0 , 0 , z 2 ; ω ) = 1 .
r 1 = ( 0 , 0 , 0 ) ,
r 2 = ( x , 0 , 0 ) .
W ( 0 , 0 , 0 ; x , 0 , 0 ; ω ) = 1 ( λ f ) 2 0 2 π 0 a 0 2 π 0 a S ( 0 ) ( ω ) J 0 { β [ ρ 1 2 + ρ 2 2 2 ρ 1 ρ 2 cos ( ϕ 1 ϕ 2 ) ] 1 2 } × e i k ( ρ 2 x cos ϕ 2 ) f ρ 1 ρ 2 d ϕ 1 d ρ 1 d ϕ 2 d ρ 2 .
W ( 0 , 0 , 0 ; x , 0 , 0 ; ω ) = 2 π ( λ f ) 2 0 a 0 2 π 0 a S ( 0 ) ( ω ) J 0 ( β ρ 1 ) J 0 ( β ρ 2 ) × cos ( k ρ 2 x f cos ϕ 2 ) ρ 1 ρ 2 d ρ 1 d ϕ 2 d ρ 2 .
S ( 0 , 0 , 0 ; ω ) = ( k f ) 2 0 a 0 a J 0 ( β ρ 1 ) J 0 ( β ρ 2 ) ρ 1 ρ 2 d ρ 1 d ρ 2
S ( x , 0 , 0 ; ω ) = 1 ( λ f ) 2 0 2 π 0 a 0 2 π 0 a S ( 0 ) ( ω ) J 0 { β [ ρ 1 2 + ρ 2 2 2 ρ 1 ρ 2 cos ( ϕ 1 ϕ 2 ) ] 1 2 } cos [ k x ( ρ 1 cos ϕ 1 ρ 2 cos ϕ 2 ) f ] ρ 1 ρ 2 d ϕ 1 d ρ 1 d ϕ 2 d ρ 2 .
W n ( 0 ) ( ρ 1 , ρ 2 , ω ) = S ( 0 ) ( ω ) 2 n 2 Γ ( 1 + n 2 ) J n 2 ( β ρ 2 ρ 1 ) ( β ρ 2 ρ 1 ) n 2 ,
S n ( ρ , 0 , ω ) = 1 ( λ f ) 2 A A W n ( 0 ) ( ρ 1 , ρ 2 , ω ) e i k ( ρ 2 ρ 1 ) ρ f d 2 ρ 1 d 2 ρ 2 .
S n ( ρ , 0 , ω ) = 2 ( k a 2 f ) 2 0 1 C ( b ) W n ( 0 ) ( 2 a β b ) J 0 ( 2 k a ρ b f ) b d b ,
C ( b ) = ( 2 π ) [ arccos ( b ) b ( 1 b 2 ) 1 2 ] .
S n ( ρ , 0 , ω ) S coh = 8 S ( 0 ) ( ω ) 0 1 C ( b ) W n ( 0 ) ( 2 a β b ) J 0 ( 2 k a ρ b f ) b d b .
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = n λ n ( ω ) ψ n * ( ρ 1 , ω ) ψ n ( ρ 2 , ω ) .
D W ( 0 ) ( ρ 1 , ρ 2 , ω ) ψ n ( r 1 , ω ) d 2 ρ 1 = λ n ( ω ) ψ n ( ρ 2 , ω ) ,
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = n λ n ( ω ) W n ( ρ 1 , ρ 2 , ω ) ,
ψ n ( ρ , ϕ ) = C n S ( 0 ) ( ω ) [ a n J n ( β ρ ) e i n ϕ + b n J n ( β ρ ) e i n ϕ ] ,
J 0 { β [ ρ 1 2 + ρ 2 2 2 ρ 1 ρ 2 cos ( ϕ 1 ϕ 2 ) ] 1 2 } = k = J k ( β ρ 1 ) J k ( β ρ 2 ) e i k ( ϕ 1 ϕ 2 ) ,
λ n = π a 2 S ( 0 ) ( ω ) [ J n 2 ( β a ) J n 1 ( β a ) J n + 1 ( β a ) ] , ( n = 0 , 1 , 2 , ) .
C n = 1 λ n ( n = 0 , 1 , 2 , ) .
( a 0 + b 0 ) 2 = 1
a n 2 + b n 2 = 1 ( n = 1 , 2 , 3 , ) .
J 0 { β [ ρ 1 2 + ρ 2 2 2 ρ 1 ρ 2 cos ( ϕ 1 ϕ 2 ) ] 1 2 } = J 0 ( β ρ 1 ) J 0 ( β ρ 2 ) + n = 1 2 { J n ( β ρ 1 ) J n ( β ρ 2 ) cos [ n ( ϕ 1 ϕ 2 ) ] } ,
J n ( x ) = ( 1 ) n J n ( x ) ( n N ) .

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