Abstract

The spatial correlation properties in the geometrical focal region of a converging, partially coherent vortex wave field are analyzed. Expressions are derived for a pair of points on the axis of symmetry and for a pair of points in the focal plane. It is found that the longitudinal and transverse coherence lengths in the focal region change with the variation of the topological charge and the normalized coherence length of the vortex field. In addition, the degree of coherence is shown to possess phase singularities.

© 2007 Optical Society of America

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References

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  1. M. S. Soskin and M. V. Vasnetsov, "A comprehensive review of optical vortices," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219-276.
    [CrossRef]
  2. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
    [CrossRef]
  3. G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
    [CrossRef] [PubMed]
  4. G. Gbur, T. D. Visser, and E. Wolf, "'Hidden' singularities in partially coherent wavefields," J. Opt. A, Pure Appl. Opt. 6, S239-S242 (2004).
    [CrossRef]
  5. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, "Phase singularities of the coherence functions in Young's interference pattern," Opt. Lett. 28, 968-970 (2003).
    [CrossRef] [PubMed]
  6. G. Gbur and T. D. Visser, "Coherence vortices in partially coherent beams," Opt. Commun. 222, 117-125 (2003).
    [CrossRef]
  7. G. Gbur and T. D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006).
    [CrossRef]
  8. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905 (2004).
    [CrossRef] [PubMed]
  9. S. H. Tao, X.-C. Yuan, J. Lin, R. E. Burge, "Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges," Opt. Express 14, 535-541 (2006).
    [CrossRef]
  10. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, "Tuning the orbital angular momentum in optical vortex beams," Opt. Express 14, 6604-6612 (2006).
    [CrossRef]
  11. W. Wang, A. T. Friberg, and E. Wolf, "Focusing of partially coherent light in systems of large Fresnel number," J. Opt. Soc. Am. A 14, 491-496 (1997).
    [CrossRef]
  12. 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]
  13. L. H. Helseth, "Optical vortices in focal regions," Opt. Commun. 229, 85-91 (2004).
    [CrossRef]
  14. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Transformation of higher-order optical vortices upon focusing by an astigmatic lens," Opt. Commun. 241, 237-247 (2004).
    [CrossRef]
  15. I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation vortices in partially coherent light: theory," J. Opt. Soc. Am. B 21, 1895-1900 (2004).
    [CrossRef]
  16. I. D. Maleev and G. A. Swartzlander, Jr., "Composite optical vortices," J. Opt. Soc. Am. B 20, 1169-1176 (2003).
    [CrossRef]
  17. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

2006 (3)

2004 (6)

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, "'Hidden' singularities in partially coherent wavefields," J. Opt. A, Pure Appl. Opt. 6, S239-S242 (2004).
[CrossRef]

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]

L. H. Helseth, "Optical vortices in focal regions," Opt. Commun. 229, 85-91 (2004).
[CrossRef]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Transformation of higher-order optical vortices upon focusing by an astigmatic lens," Opt. Commun. 241, 237-247 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation vortices in partially coherent light: theory," J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

2003 (3)

2002 (1)

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

1997 (1)

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Bekshaev, A. Y.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Transformation of higher-order optical vortices upon focusing by an astigmatic lens," Opt. Commun. 241, 237-247 (2004).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Burge, R. E.

Curtis, J. E.

Fischer, D. G.

Friberg, A. T.

Gbur, G.

G. Gbur and T. D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, "'Hidden' singularities in partially coherent wavefields," J. Opt. A, Pure Appl. Opt. 6, S239-S242 (2004).
[CrossRef]

G. Gbur and T. D. Visser, "Coherence vortices in partially coherent beams," Opt. Commun. 222, 117-125 (2003).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, "Phase singularities of the coherence functions in Young's interference pattern," Opt. Lett. 28, 968-970 (2003).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Helseth, L. H.

L. H. Helseth, "Optical vortices in focal regions," Opt. Commun. 229, 85-91 (2004).
[CrossRef]

Lin, J.

Maleev, I. D.

Mandel, L.

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

Marathay, A. S.

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation vortices in partially coherent light: theory," J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation vortices in partially coherent light: theory," J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

Schmitz, C. H. J.

Schouten, H. F.

Soskin, M. S.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Transformation of higher-order optical vortices upon focusing by an astigmatic lens," Opt. Commun. 241, 237-247 (2004).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "A comprehensive review of optical vortices," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219-276.
[CrossRef]

Spatz, J. P.

Swartzlander, G. A.

Tao, S. H.

Uhrig, K.

Vasnetsov, M. V.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Transformation of higher-order optical vortices upon focusing by an astigmatic lens," Opt. Commun. 241, 237-247 (2004).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "A comprehensive review of optical vortices," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219-276.
[CrossRef]

Visser, T. D.

G. Gbur and T. D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, "'Hidden' singularities in partially coherent wavefields," J. Opt. A, Pure Appl. Opt. 6, S239-S242 (2004).
[CrossRef]

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]

G. Gbur and T. D. Visser, "Coherence vortices in partially coherent beams," Opt. Commun. 222, 117-125 (2003).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, "Phase singularities of the coherence functions in Young's interference pattern," Opt. Lett. 28, 968-970 (2003).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Wang, W.

Wolf, E.

G. Gbur, T. D. Visser, and E. Wolf, "'Hidden' singularities in partially coherent wavefields," J. Opt. A, Pure Appl. Opt. 6, S239-S242 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, "Phase singularities of the coherence functions in Young's interference pattern," Opt. Lett. 28, 968-970 (2003).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

W. Wang, A. T. Friberg, and E. Wolf, "Focusing of partially coherent light in systems of large Fresnel number," J. Opt. Soc. Am. A 14, 491-496 (1997).
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Yuan, X.-C.

J. Opt. A, Pure Appl. Opt. (1)

G. Gbur, T. D. Visser, and E. Wolf, "'Hidden' singularities in partially coherent wavefields," J. Opt. A, Pure Appl. Opt. 6, S239-S242 (2004).
[CrossRef]

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

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

Opt. Commun. (4)

L. H. Helseth, "Optical vortices in focal regions," Opt. Commun. 229, 85-91 (2004).
[CrossRef]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Transformation of higher-order optical vortices upon focusing by an astigmatic lens," Opt. Commun. 241, 237-247 (2004).
[CrossRef]

G. Gbur and T. D. Visser, "Coherence vortices in partially coherent beams," Opt. Commun. 222, 117-125 (2003).
[CrossRef]

G. Gbur and T. D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Other (3)

M. S. Soskin and M. V. Vasnetsov, "A comprehensive review of optical vortices," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219-276.
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

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

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Figures (5)

Fig. 1
Fig. 1

Illustration of notation.

Fig. 2
Fig. 2

(a) Real and (b) imaginary parts of the degree of coherence μ ( 0 , 0 , 0 ; 0 , 0 , z ) . Other parameters are chosen as f = 20 mm , a = 10 mm , λ = 0.6328 μ m , δ = 0.5 , and σ = 0.4 .

Fig. 3
Fig. 3

Modulus of the degree of coherence, μ ( 0 , 0 , z 1 ; 0 , 0 , z 2 ) . (a) n = 0 , (b) n = 1 , (c) n = 2 , (d) n = 5 . Other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

(a) Modulus of the degree of coherence μ ( 0 , 0 , z ; 0 , 0 , z 2 ) for several values of the normalized coherence length σ, n = 1 . (b) Modulus of the degree of coherence μ ( 0 , 0 , z ; 0 , 0 , z ) for several values of the topological n, σ = 0.4 . Other parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

The degree of coherence μ ( 0 , 0 , 0 ; x , 0 , 0 ) in the geometrical focal plane for different values of the topological charge n and different values of the normalized coherence length σ: (a) σ = 0.1 , (b) σ = 0.4 , (c) σ = 1 , (d) σ = 5 . Other parameters are the same as in Fig. 2.

Equations (22)

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U ( 0 ) ( r , ω ) = E 0 exp ( r 2 w 0 2 ) exp ( i n ϕ ) exp ( i ω t ) exp ( i β ) ,
n = 0 , 1 , 2 , 3 , ,
U ( r , ω ) = i λ S U ( 0 ) ( r , ω ) exp ( i k s ) s d 2 r ,
W ( 0 ) ( r 1 , r 2 , ω ) = U ( 0 ) * ( r 1 , ω ) U ( 0 ) ( r 2 , ω ) .
W ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) ,
W ( r 1 , r 2 , ω ) = ( k 2 π ) 2 S S W ( 0 ) ( r 1 , r 2 , ω ) exp [ i k ( s 2 s 1 ) ] s 1 s 2 d 2 r 1 d 2 r 2 ,
W ( 0 ) ( r 1 , r 2 ) = E 0 2 exp [ ( r 1 2 + r 2 2 ) w 0 2 ] exp [ ( r 1 r 2 ) 2 2 σ g 2 ] exp [ i n ( ϕ 1 ϕ 2 ) ] ,
s 1 f q 1 r 1 ,
s 2 f q 2 r 2 ,
W ( r 1 , r 2 ) = ( E 0 k 2 π f ) 2 S S exp [ ( r 1 2 + r 2 2 ) w 0 2 ] exp [ ( r 1 r 2 ) 2 2 σ g 2 ] exp [ i n ( ϕ 1 ϕ 2 ) ] exp [ i k ( q 1 r 1 q 2 r 2 ) ] d 2 r 1 d 2 r 2 .
μ ( r 1 , r 2 ) = W ( r 1 , r 2 ) [ I ( r 1 ) I ( r 2 ) ] 1 2 ,
I ( r i ) W ( r i , r i ) ( i = 1 , 2 )
q 1 r 1 z 1 ( 1 r 1 2 2 f 2 ) ,
q 2 r 2 z 2 ( 1 r 2 2 2 f 2 ) ,
0 2 π exp [ i n ϕ 1 + r 1 r 2 σ g 2 cos ( ϕ 1 ϕ 2 ) ] d ϕ 1 = 2 π exp ( i n ϕ 2 ) I n ( r 1 r 2 σ g 2 ) ,
W ( 0 , 0 , z 1 ; 0 , 0 , z 2 ) = ( E 0 k a 2 f ) 2 0 1 0 1 exp [ δ ( r 1 2 + r 2 2 ) ] exp [ ( r 1 2 + r 2 2 ) 2 σ 2 ] I n ( r 1 r 2 σ 2 ) exp { i k [ z 1 ( 1 a 2 r 1 2 2 f 2 ) + z 2 ( 1 a 2 r 2 2 2 f 2 ) ] } r 1 r 2 d r 1 d r 2 ,
I ( 0 , 0 , z ) = W ( 0 , 0 , z ; 0 , 0 , z ) = ( E 0 k a 2 f ) 2 0 1 0 1 exp [ δ ( r 1 2 + r 2 2 ) ] exp [ ( r 1 2 + r 2 2 ) 2 σ 2 ] I n ( r 1 r 2 σ 2 ) exp { i k [ z a 2 ( r 1 2 r 2 2 ) 2 f 2 ] } r 1 r 2 d r 1 d r 2 .
q 1 r 1 = 0 .
q 2 = [ r 2 cos ϕ 2 f , r 2 sin ϕ 2 f , ( 1 r 2 2 f 2 ) 1 2 ] ,
q 2 r 2 = r 2 x cos ϕ 2 f .
W ( 0 , 0 , 0 ; x , 0 , 0 ) = ( E 0 2 k 2 2 π f 2 ) 0 a 0 a 0 2 π exp [ ( r 1 2 + r 2 2 ) w 0 2 ] exp [ ( r 1 2 + r 2 2 ) 2 σ g 2 ] I n ( r 1 r 2 σ g 2 ) exp [ i k ( r 2 x cos ϕ 2 ) f ] r 1 r 2 d r 1 d r 2 d ϕ 2 = ( E 0 k a 2 f ) 2 0 1 0 1 exp [ δ ( r 1 2 + r 2 2 ) ] exp [ ( r 1 2 + r 2 2 ) 2 σ 2 ] I n ( r 1 r 2 σ 2 ) × J 0 ( k a r 2 x f ) r 1 r 2 d r 1 d r 2 .
I ( x , 0 , 0 ) = W ( x , 0 , 0 ; x , 0 , 0 ) = ( E 0 k 2 π f ) 2 0 a 0 a 0 2 π 0 2 π exp [ ( r 1 2 + r 2 2 ) w 0 2 ] exp [ ( r 1 2 + r 2 2 ) 2 σ g 2 ] exp [ r 1 r 2 cos ( ϕ 1 ϕ 2 ) σ g 2 ] exp [ i n ( ϕ 1 ϕ 2 ) ] exp [ i k x ( r 1 cos ϕ 1 r 2 cos ϕ 2 ) f ] r 1 r 2 d r 1 d r 2 d ϕ 1 d ϕ 2 = ( E 0 k a 2 f ) 2 l = 0 1 0 1 exp [ ( r 1 2 + r 2 2 ) ( δ + 1 2 σ 2 ) ] J l ( k a x r 1 f ) J l ( k a x r 2 f ) I l + n ( r 1 r 2 σ 2 ) r 1 r 2 d r 1 d r 2 ,

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