Abstract

Based on the vectorial Debye theory, the tight focusing properties of partially coherent and circularly polarized vortex beams are investigated. The focused characteristics of right-circular and left-circular polarized partially coherent vortex beams in the focal region are presented and compared by some numerical calculation results. Furthermore, the influences of the source coherence and the numerical aperture of the focusing objective on the tight focusing properties are studied in great detail. It is shown that the coherence and polarization properties of the focused left-circular polarized beam is less influenced by the source coherence and the numerical aperture of the focusing objective than that of the focused right-circular polarized beam. By selecting certain parameters, the widely used flat top beam can be obtained.

© 2009 Optical Society of America

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References

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  1. N. Hayazawa, Y. Saito, and S. Kawata, “Detection, and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
    [CrossRef]
  2. C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145-158 (1982).
    [CrossRef]
  3. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
    [CrossRef] [PubMed]
  4. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high numerical-aperture objective,” Opt. Lett. 33, 49-51 (2008).
    [CrossRef]
  5. N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (2007).
    [CrossRef]
  6. T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
    [CrossRef]
  7. Q. Zhan and R. J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324-331 (2002).
    [PubMed]
  8. D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
    [CrossRef] [PubMed]
  9. S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express 14, 535-541 (2006).
    [CrossRef] [PubMed]
  10. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31, 867-869 (2007).
    [CrossRef]
  11. D. P. Biss and T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12, 384-393 (2004).
    [CrossRef] [PubMed]
  12. I. Cooper, M. Roy, and C. J. Sheppard, “Focusing of pseudoradial polarized beams,” Opt. Express 13, 1066-1071 (2005).
    [CrossRef] [PubMed]
  13. Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
    [CrossRef] [PubMed]
  14. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
    [CrossRef]
  15. M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 1999).
  16. I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).
  17. E. Wolf, “New spectral representation of random sources, and of the partially coherent fields that they generate,” Opt. Commun. 38, 3-6 (1981), Eq. (25).
    [CrossRef]
  18. 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]
  19. 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]
  20. K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561-568 (2005).
    [CrossRef]
  21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  22. T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  23. K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, and A. T. Friberg, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228-231 (2007).
    [CrossRef]
  24. F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel-Dekker, 2000).
    [CrossRef]

2008 (1)

2007 (5)

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (2007).
[CrossRef]

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
[CrossRef]

Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31, 867-869 (2007).
[CrossRef]

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, and A. T. Friberg, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228-231 (2007).
[CrossRef]

2006 (1)

2005 (2)

2004 (3)

D. P. Biss and T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12, 384-393 (2004).
[CrossRef] [PubMed]

N. Hayazawa, Y. Saito, and S. Kawata, “Detection, and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

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

2002 (2)

Q. Zhan and R. J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324-331 (2002).
[PubMed]

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

1997 (1)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

1982 (2)

C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145-158 (1982).
[CrossRef]

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]

1981 (1)

E. Wolf, “New spectral representation of random sources, and of the partially coherent fields that they generate,” Opt. Commun. 38, 3-6 (1981), Eq. (25).
[CrossRef]

1959 (1)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

Biss, D. P.

Bokor, N.

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (2007).
[CrossRef]

Brown, T. G.

Burge, R. E.

Chiu, D. T.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Cooper, I.

Courjon, D.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
[CrossRef]

Davidson, N.

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (2007).
[CrossRef]

Dickey, F. M.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel-Dekker, 2000).
[CrossRef]

Friberg, A. T.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, and A. T. Friberg, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228-231 (2007).
[CrossRef]

K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561-568 (2005).
[CrossRef]

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

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]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Gradysteyn, I. S.

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

Grosjean, T.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
[CrossRef]

Gu, M.

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 1999).

Hayazawa, N.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection, and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Holswade, S. C.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel-Dekker, 2000).
[CrossRef]

Jeffries, G. D. M.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Kaivola, M.

K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561-568 (2005).
[CrossRef]

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Kawata, S.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection, and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

Leger, R. J.

Lin, J.

Lindfors, K.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, and A. T. Friberg, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228-231 (2007).
[CrossRef]

K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561-568 (2005).
[CrossRef]

Mandel, L.

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

Marathay, A. S.

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

McGloin, D.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Palacios, D. M.

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

Priimagi, A.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, and A. T. Friberg, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228-231 (2007).
[CrossRef]

Pu, J.

Roy, M.

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

Saito, Y.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection, and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

Scott Edgar, J.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Setala, T.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, and A. T. Friberg, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228-231 (2007).
[CrossRef]

K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561-568 (2005).
[CrossRef]

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Sheppard, C. J.

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145-158 (1982).
[CrossRef]

Shevchenko, A.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, and A. T. Friberg, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228-231 (2007).
[CrossRef]

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Swartzlander, G. A.

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

Tao, S. H.

Wang, W.

Wang, X.

Wilson, T.

C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145-158 (1982).
[CrossRef]

Wolf, E.

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]

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]

E. Wolf, “New spectral representation of random sources, and of the partially coherent fields that they generate,” Opt. Commun. 38, 3-6 (1981), Eq. (25).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

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

Yuan, X. C.

Zhan, Q.

Zhang, Z.

Zhao, Y.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

N. Hayazawa, Y. Saito, and S. Kawata, “Detection, and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nat. Photonics (1)

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, and A. T. Friberg, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228-231 (2007).
[CrossRef]

Opt. Commun. (3)

E. Wolf, “New spectral representation of random sources, and of the partially coherent fields that they generate,” Opt. Commun. 38, 3-6 (1981), Eq. (25).
[CrossRef]

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (2007).
[CrossRef]

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. E (1)

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

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

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145-158 (1982).
[CrossRef]

Other (4)

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 1999).

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

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

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel-Dekker, 2000).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Scheme of tight focusing system.

Fig. 2
Fig. 2

Intensity distribution of a partially coherent and circularly polarized vortex beam for different source coherent length L C and different maximal angle α. (a) and (b) Right-circular polarized beam; (c) and (d) left-circular polarized beam. The other parameters are chosen as λ = 633 nm , m = 1 , w 0 = 1 cm , f = 1 cm .

Fig. 3
Fig. 3

μ x y , μ x z , and μ y z distribution of partially coherent and circularly polarized vortex beam on the focal plane. (a) Right-circular polarized beam; (b) left-circular polarized beam. L C = 0.5 cm , α = arcsin 0.9 . The other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Contour distributions of μ x y , μ x z , and μ y z in the propagation plane. (a)–(c) Right-circular polarized beam; (d)–(f) left-circular polarized beam. (a) and (d) μ x y distribution; (b) and (e) μ x z distribution; (c) and (f) μ y z distribution. The other parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

Influence of L C and α on μ x y distribution of the circularly polarized partially coherent vortex beam in the focal plane. (a) and (b) Right-circular polarized beam; (c) and (d) left-circular polarized beam. The other parameters except the varying ones are the same as in Fig. 2.

Fig. 6
Fig. 6

Influence of L C on μ x y distribution of the circularly polarized partially coherent vortex beam near focus. (a)–(c) Right-circular polarized beam; (d)–(f) left-circular polarized beam. (a) and (d) L C = 0.25 cm ; (b) and (e) L C = 0.5 cm ; (c) and (f) L C = 1 cm , α = arcsin 0.9 . The other parameters are the same as in Fig. 2.

Fig. 7
Fig. 7

Influence of α on μ x y distribution of the circularly polarized partially coherent vortex beam near focus. (a)–(c) Right-circular polarized beam; (d)–(f) left-circular polarized beam. (a) and (d) α = 50 ° ; (b) and (e) α = 65 ° ; (c) and (f) α = 80 ° , L C = 0.5 cm . The other parameters are the same as in Fig. 2.

Fig. 8
Fig. 8

Polarization distribution of partially coherent and circularly polarized vortex beam near focus. (a) and (b) Right-circular polarized beam; (c) and (d) left-circular polarized beam. L C = 0.5 cm , α = arcsin 0.9 . The other parameters are the same as in Fig. 2.

Fig. 9
Fig. 9

Influence of L C and α on polarization distribution of partially coherent and circularly polarized vortex beam in the focal plane. (a) and (b) Right-circular polarized beam; (c) and (d) left-circular polarized beam. The other parameters are the same as in Fig. 2.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

E ( r , φ , z ) = [ E x E y E z ] = i k f 2 π 0 α 0 2 π A ( θ ) exp ( im ϕ ) sin θ cos θ × exp [ i k ( z cos θ + r sin θ cos ( ϕ φ ) ) ] × [ ( cos 2 ϕ cos θ + sin 2 ϕ ) ± i cos ϕ sin ϕ ( cos θ 1 ) cos ϕ sin ϕ ( cos θ 1 ) ± i ( cos 2 ϕ + sin 2 ϕ cos θ ) sin θ exp ( ± i ϕ ) ] d ϕ d θ ,
exp [ i k r sin θ cos ( ϕ φ ) ] = l = i l J l ( k r sin θ ) exp [ i l ( ϕ φ ) ] ,
0 2 π exp ( im θ ) d θ = { 2 π if m = 0 0 if m 0 } ,
E ± , x ( r , φ , z ) = i k f 2 0 α A ( θ ) sin θ cos θ exp ( i k z cos θ ) × [ ( 1 + cos θ ) i m J m ( k r sin θ ) exp ( im φ ) + ( cos θ 1 ) i m ± 2 J m ± 2 ( k r sin θ ) exp [ i ( m ± 2 ) φ ] ] d θ ,
E ± , y ( r , φ , z ) = i k f 2 0 α A ( θ ) sin θ cos θ exp ( i k z cos θ ) × [ ± i × ( 1 + cos θ ) i m J m ( k r sin θ ) exp ( im φ ) i ( cos θ 1 ) i m ± 2 J m ± 2 ( k r sin θ ) exp [ i ( m ± 2 ) φ ] ] d θ ,
E ± , z ( r , φ , z ) = i k f 0 α A ( θ ) sin 2 θ cos θ exp ( i k z cos θ ) × i m ± 1 J m ± 1 ( k r sin θ ) exp [ i ( m ± 1 ) φ ] d θ .
E m ( r ) = ( 2 r w 0 ) m exp ( r 2 w 0 2 ) ,
E m ( θ ) = ( 2 f sin θ w 0 ) m exp ( f 2 sin 2 θ w 0 2 ) .
A ( θ 1 , θ 2 ) = ( 2 f 2 sin θ 1 sin θ 2 w 0 2 ) m 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 ] ,
W ( r 1 , r 2 , z ) = [ W x x ( r 1 , r 2 , z ) W x y ( r 1 , r 2 , z ) W x z ( r 1 , r 2 , z ) W y x ( r 1 , r 2 , z ) W y y ( r 1 , r 2 , z ) W y z ( r 1 , r 2 , z ) W z x ( r 1 , r 2 , z ) W z y ( r 1 , r 2 , z ) W z z ( r 1 , r 2 , z ) ] ,
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 ) .
I t ( r , φ , z ) = W ( r , r , z ) = W x x ( r , r , z ) + W y y ( r , r , z ) + W z z ( r , r , z ) = I x ( r , φ , z ) + I y ( r , φ , z ) + I z ( r , φ , z ) .
μ i j ( r ) = W i j ( r ) W i i ( r ) W j j ( r ) ( i , j = x , y , z ) .
P ( 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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