Abstract

Focusing of vortex beams by a lens with circular aperture in the paraxial scalar Debye regime is analyzed. The amplitude in the focal region can be expressed naturally in terms of higher order Lommel functions of two variables. Using recurrence relationships, these can then be expressed in terms of low-order Lommel functions. The phase variation in the focal region is investigated, showing some interesting behavior of the Gouy phase anomaly.

© 2014 Optical Society of America

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  1. E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).
  2. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  3. E. Wolf, “Light distribution near focus in an error-free diffraction image,” Proc. R. Soc. A 204, 533–548 (1951).
    [CrossRef]
  4. E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. Sect. B 69, 823–832 (1956).
    [CrossRef]
  5. E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. Sect. B 66, 145–149 (1953).
    [CrossRef]
  6. Z. L. Horváth and Z. S. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun. 222, 51–68 (2003).
    [CrossRef]
  7. Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  8. J. F. Nye and M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. A 336, 165–190 (1974).
    [CrossRef]
  9. V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).
    [CrossRef]
  10. G. Indebetouw, “Optical vortices and their applications,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  11. I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
    [CrossRef]
  12. N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein-Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
    [CrossRef]
  13. M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
    [CrossRef]
  14. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular-momentum to absorbing particles from a laser-beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
    [CrossRef]
  15. T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
    [CrossRef]
  16. D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut beams by a high numerical-aperture objective in free space,” Opt. Express 11, 2747–2752 (2003).
    [CrossRef]
  17. C. J. R. Sheppard, “Polarization of beams and highly focused waves,” presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, 2003.
  18. L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229, 85–91 (2004).
    [CrossRef]
  19. C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).
  20. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).
    [CrossRef]
  21. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [CrossRef]
  22. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
    [CrossRef]
  23. W. Tang, E. Yew, and C. Sheppard, “Polarization conversion in confocal microscopy with radially polarized illumination,” Opt. Lett. 34, 2147–2149 (2009).
    [CrossRef]
  24. C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  25. A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, 1895).
  26. J. Walker, The Analytical Theory of Light (C. J. Clay and Sons, 1904).
  27. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1980).
  28. I. S. Gradstein and I. M. Ryshik, Tables of Series, Products, and Integrals (Harri Deutsch, Thun, 1981).
  29. J. Boersma, “On the computation of Lommel’s functions of two variables,” Math. Comp. 16, 232–238 (1962).
  30. G. Goubau and F. Schwering, “On the guided propagation of electromagnetic wave beams,” IEEE Trans. Antennas Propag. 9, 248–256 (1961).
    [CrossRef]
  31. D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef]
  32. D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).
  33. G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
    [CrossRef]
  34. G. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
    [CrossRef]
  35. J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
    [CrossRef]
  36. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22, 849–861 (2005).
    [CrossRef]
  37. C. J. R. Sheppard, “Cylindrical lenses: focusing and imaging: a review [Invited],” Appl. Opt. 52, 538–541 (2013).
    [CrossRef]

2013 (1)

2009 (1)

2005 (1)

2004 (1)

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229, 85–91 (2004).
[CrossRef]

2003 (2)

2001 (1)

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

2000 (2)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef]

1999 (1)

1997 (2)

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

1995 (1)

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

1993 (2)

G. Indebetouw, “Optical vortices and their applications,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

1992 (3)

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein-Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef]

1987 (1)

1984 (1)

1974 (1)

J. F. Nye and M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

1969 (1)

J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
[CrossRef]

1962 (1)

J. Boersma, “On the computation of Lommel’s functions of two variables,” Math. Comp. 16, 232–238 (1962).

1961 (1)

G. Goubau and F. Schwering, “On the guided propagation of electromagnetic wave beams,” IEEE Trans. Antennas Propag. 9, 248–256 (1961).
[CrossRef]

1958 (1)

G. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

1957 (1)

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

1956 (1)

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

1953 (1)

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. Sect. B 66, 145–149 (1953).
[CrossRef]

1951 (1)

E. Wolf, “Light distribution near focus in an error-free diffraction image,” Proc. R. Soc. A 204, 533–548 (1951).
[CrossRef]

1885 (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Almazov, A. A.

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Berry, M.

J. F. Nye and M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

Boersma, J.

J. Boersma, “On the computation of Lommel’s functions of two variables,” Math. Comp. 16, 232–238 (1962).

Bor, Z. S.

Z. L. Horváth and Z. S. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun. 222, 51–68 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

Brown, T. G.

Dainty, J. C.

J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
[CrossRef]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Elfstrom, H.

Engel, E.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Farnell, G.

G. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Freund, I.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Fried, D. L.

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 absorbing particles from a laser-beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

Gan, X.

Ganic, D.

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Goubau, G.

G. Goubau and F. Schwering, “On the guided propagation of electromagnetic wave beams,” IEEE Trans. Antennas Propag. 9, 248–256 (1961).
[CrossRef]

Gradstein, I. S.

I. S. Gradstein and I. M. Ryshik, Tables of Series, Products, and Integrals (Harri Deutsch, Thun, 1981).

Gray, A.

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, 1895).

Gu, M.

He, H.

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

Heckenberg, N.

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein-Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Heckenberg, N. R.

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

Hell, S. W.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Helseth, L. E.

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229, 85–91 (2004).
[CrossRef]

Horváth, Z. L.

Z. L. Horváth and Z. S. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun. 222, 51–68 (2003).
[CrossRef]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their applications,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Khonina, S. N.

Klar, T. A.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Kotlyar, V. V.

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Li, Y.

Linfoot, E. H.

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

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. Sect. B 66, 145–149 (1953).
[CrossRef]

Lommel, E.

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Mathews, G. B.

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, 1895).

Matthews, H. J.

McDuff, R.

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein-Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Nye, J. F.

J. F. Nye and M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Rubinstein-Dunlop, H.

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein-Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Rubinsztein-Dunlop, H.

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

Ryshik, I. M.

I. S. Gradstein and I. M. Ryshik, Tables of Series, Products, and Integrals (Harri Deutsch, Thun, 1981).

Saghafi, S.

Schwering, F.

G. Goubau and F. Schwering, “On the guided propagation of electromagnetic wave beams,” IEEE Trans. Antennas Propag. 9, 248–256 (1961).
[CrossRef]

Sheppard, C.

Sheppard, C. J. R.

C. J. R. Sheppard, “Cylindrical lenses: focusing and imaging: a review [Invited],” Appl. Opt. 52, 538–541 (2013).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).
[CrossRef]

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

C. J. R. Sheppard, “Polarization of beams and highly focused waves,” presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, 2003.

Shvartsman, N.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Smith, C. P.

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein-Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Soifer, V. A.

Soskin, M. S.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Tang, W.

Tiziani, H. J.

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

Török, P.

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

Totzeck, M.

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

Turunen, J.

Vasnetsov, M. V.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Vaughn, J. L.

Walker, J.

J. Walker, The Analytical Theory of Light (C. J. Clay and Sons, 1904).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1980).

Wolf, E.

Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

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

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. Sect. B 66, 145–149 (1953).
[CrossRef]

E. Wolf, “Light distribution near focus in an error-free diffraction image,” Proc. R. Soc. A 204, 533–548 (1951).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

Yew, E.

Youngworth, K. S.

Abh. Bayer. Akad. (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Appl. Opt. (2)

Can. J. Phys. (2)

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

G. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

G. Goubau and F. Schwering, “On the guided propagation of electromagnetic wave beams,” IEEE Trans. Antennas Propag. 9, 248–256 (1961).
[CrossRef]

J. Mod. Opt. (2)

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

G. Indebetouw, “Optical vortices and their applications,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Math. Comp. (1)

J. Boersma, “On the computation of Lommel’s functions of two variables,” Math. Comp. 16, 232–238 (1962).

Opt. Commun. (6)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
[CrossRef]

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229, 85–91 (2004).
[CrossRef]

Z. L. Horváth and Z. S. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun. 222, 51–68 (2003).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein-Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Optik (1)

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

Phys. Rev. E (1)

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

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

Proc. Phys. Soc. Sect. B (2)

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

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. Sect. B 66, 145–149 (1953).
[CrossRef]

Proc. R. Soc. A (2)

E. Wolf, “Light distribution near focus in an error-free diffraction image,” Proc. R. Soc. A 204, 533–548 (1951).
[CrossRef]

J. F. Nye and M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

Other (7)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

C. J. R. Sheppard, “Polarization of beams and highly focused waves,” presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, 2003.

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, 1895).

J. Walker, The Analytical Theory of Light (C. J. Clay and Sons, 1904).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1980).

I. S. Gradstein and I. M. Ryshik, Tables of Series, Products, and Integrals (Harri Deutsch, Thun, 1981).

D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

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Figures (7)

Fig. 1.
Fig. 1.

Intensity variation (a) in the focal plane, (b) in the axial core region, and (c) along the shadow edge for focused vortices of charge m=0, 1, 2. For (a) and (c), [(m+1)Im]2 is plotted. For (b), [2m(m+1)!Im/vm]2 is plotted.

Fig. 2.
Fig. 2.

Intensity in the focal region shown as a contour plot: (a) m=1 and (b) m=2.

Fig. 3.
Fig. 3.

Phase variation with kz suppressed, (a) in the axial core, and (b) along the shadow edge. In (a), a phase u/2 is also suppressed, and the phase is taken as zero at the focus. In (b), a phase u/2 is also suppressed, and the zero of phase is taken so that the phase variation is symmetrical about the focal point.

Fig. 4.
Fig. 4.

Phase variation in the focal region for m=0. The shadow edge is shown as a white line. In (a) a phase kz is suppressed, in (b) an additional u/2, and in (c) an additional u/2. Contours are plotted every π/4. For example, the eight shading colors represent the ranges π to 3π/4, 3π/4 to π/2, and so on, from dark to light. Wrappings from π to π occur at jumps from light to dark color.

Fig. 5.
Fig. 5.

Phase variation in the focal region for m=1. In (a) a phase kz is suppressed, in (b) an additional u/2, and in (c) an additional u/2. Contours are plotted every π/4 in (a) and (b), and every π/2 in (c). Wrappings occur at jumps from dark to light color.

Fig. 6.
Fig. 6.

Phase variation in the focal region for m=2. In (a) a phase kz is suppressed, in (b) an additional u/2, and in (c) an additional u/2. Contours are plotted every π/4. Wrappings occur at jumps from dark to light color.

Fig. 7.
Fig. 7.

Intensity in the focal region for a circular aperture, computed from Eq. (23), with kmax, the maximum value of k given by (a) 0, (b) 1, (c) 2, (d) 3, and (e) 50. Contour values are 0.1, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001.

Equations (23)

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P(ρ,ϕ)=ρmexp(ρ2ρ02)e±imϕ,
U(u,v,ψ)=iNeikz0102π2P(ρ,ϕ)exp[ivρcos(ϕψ)]×exp(12iuρ2)dϕρdρ,
U(u,v,ψ)=2πNeikze±imψ012im+1ρmJm(vρ)×exp(12iuρ2)ρdρ=2πiNeikze±imψIm(u,v),
u=u2iρ02,
Im(u,v)=2im01Jm(vρ)exp(12iuρ2)ρm+1dρ.
Un(u,v)=s=0(1)s(uv)n+2sJn+2s(v),Vn(u,v)=s=0(1)s(vu)n+2sJn+2s(v),
01Jm(vρ)cos[12u(1ρ2)]ρm+1dρ=1u(vu)mUm+1(u,v),01Jm(vρ)sin[12u(1ρ2)]ρm+1dρ=1u(vu)mUm+2(u,v),
Im(u,v)=2imu(vu)meiu/2[Um+1(u,v)+iUm+2(u,v)].
Un(u,v)+Un+2(u,v)=(uv)nJn(v),
Im(u,v)=2u(vu)meiu/2×{[U1(u,v)+iU2(u,v)]iJ0(v)+ik=0m(iu)kJk(v)vk}.
Vn(u,v)=(1)n[Un+2(u,v)+cos(u2+v22u+nπ2)],
Im(u,v)=2u(vu)meiu/2{[Vm(u,v)+iV1m(u,v)]2iexp[i(v2+u2)2u]},
Im(u,v)=2iu(vu)m×{eiu/2[V0(u,v)+iV1(u,v)+k=1m(iu)kJk(v)vk]exp(iv22u)}.
I0=2eiu/2u[U1(u,v)+iU2(u,v)],I1=2eiu/2u(vu){[U1(u,v)+iU2(u,v)]uJ1(v)v},I2=2eiu/2u(vu)2{[U1(u,v)+iU2(u,v)][uJ1(v)v+iu2J2(v)v2]},
I0=2iu{eiu/2[V0(u,v)+iV1(u,v)]exp(iv22u)},I1=2iu(vu){eiu/2[V0(u,v)+iV1(u,v)+iuJ1(v)v]exp(iv22u)},I2=2iu(vu)2{eiu/2[V0(u,v)+iV1(u,v)+iuJ1(v)vu2J2(v)v2]exp(iv22u)}.
U0(u,v)J0(v);U1(u,v)(uv)J1(v)(uv)3J3(v),
I0=2J1(v)v,I1=i2J2(v)v,I2=2J3(v)v.
I0=2iu(eiu/21),I1=2iu(vu)[eiu/2(1+iu2)1],I2=2iu(vu)2[eiu/2(1+iu2u28)1].
U0(u,u)=V0(u,u)=12[J0(u)+cosu]U1(u,u)=V1(u,u)=12sinuU2(u,u)=V2(u,u)=12[J0(u)cosu],
Im=2i01Jm(uρ)exp(12iuρ2)ρm+1dρ=ieiu/2u{1+eiu[J0(u)2k=0mikJk(u)]}.
CUn(u,v)=Un(u,v)+iUn+1(u,v),CUn(u,v)=Un(u,v)+iUn+1(u,v).
CUn(u,v)=s=0(1)s(uv)sJs(v),CVn(u,v)=s=0(1)s(vu)sJs(v).
CU1(u,v)=U1(u,v)+iU2(u,v)=u2eiu/4{j0(u4)J0(v)+k=0ik[jk(u4)+ijk+1(u4)]J2k+2(v)},

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