Abstract

Optical beams exhibiting a long depth of focus and a minimum spot size can be obtained with the tight focusing of a narrow annulus of radially polarized light, leading to a needle of longitudinally polarized light. Such beams are of increasing interest for their applications, for example in optical data storage, particle acceleration, and biomedical imaging. Hence one needs to characterize the needles of longitudinally polarized light obtained with different focusing optics and incident beams. In this paper, we present analytical expressions for the electric field of such a nearly nondiffracting, subwavelength beam obtained with a parabolic mirror or an aplanatic lens. Based on these results, we give expressions of the transverse and longitudinal full widths at half maximum of the focal lines as a function of the width of the incident annular beam and we compare the performances of the two focusing systems. Then, we propose a practical solution to produce a needle of longitudinally polarized light with a tunable axial extent and a transverse width reaching the theoretical limit of 0.36λ.

© 2012 OSA

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  1. C. Varin, M. Piché, M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(Pt 2), 026603 (2005).
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  2. H. Dehez, M. Piché, Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009).
    [CrossRef] [PubMed]
  3. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
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  4. Y. Zhang, J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009).
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    [CrossRef] [PubMed]
  29. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
    [CrossRef]
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  31. K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, “Tight focusing of double ring shaped radially polarized beam with high NA lens axicon,” Opt. Laser Technol. 43(7), 1037–1040 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  34. H. Kawauchi, Y. Kozawa, S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008).
    [CrossRef] [PubMed]
  35. M. Stalder, M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996).
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2011 (4)

A. April, P. Bilodeau, M. Piché, “Focusing a TM01 beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011).
[CrossRef] [PubMed]

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[CrossRef] [PubMed]

C. Hnatovsky, V. Shvedov, W. Krolikowski, A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[CrossRef] [PubMed]

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, “Tight focusing of double ring shaped radially polarized beam with high NA lens axicon,” Opt. Laser Technol. 43(7), 1037–1040 (2011).
[CrossRef]

2010 (2)

K. Kitamura, K. Sakai, S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010).
[CrossRef] [PubMed]

A. April, M. Piché, “4π Focusing of TM01 beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010).
[CrossRef] [PubMed]

2009 (3)

H. Dehez, M. Piché, Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009).
[CrossRef] [PubMed]

Y. Zhang, J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009).
[CrossRef] [PubMed]

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95(13), 133703 (2009).
[CrossRef]

2008 (3)

J. Stadler, C. Stanciu, C. Stupperich, A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[CrossRef] [PubMed]

H. Kawauchi, Y. Kozawa, S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008).
[CrossRef] [PubMed]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

2007 (4)

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32(8), 976–978 (2007).
[CrossRef] [PubMed]

Y. Kozawa, S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007).
[CrossRef] [PubMed]

V. P. Kalosha, I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007).
[CrossRef] [PubMed]

T. Grosjean, D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

2006 (1)

P. Dufour, M. Piché, Y. De Koninck, N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. 45(36), 9246–9252 (2006).
[CrossRef] [PubMed]

2005 (1)

C. Varin, M. Piché, M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(Pt 2), 026603 (2005).
[CrossRef] [PubMed]

2004 (2)

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
[CrossRef] [PubMed]

N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004).
[CrossRef] [PubMed]

2003 (3)

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

C. Debus, M. A. Lieb, A. Drechsler, A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

D. P. Biss, T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28(11), 923–925 (2003).
[CrossRef] [PubMed]

2001 (3)

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

M. A. Lieb, A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001).
[CrossRef] [PubMed]

2000 (3)

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000).
[CrossRef] [PubMed]

1996 (1)

M. Stalder, M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996).
[CrossRef] [PubMed]

1993 (1)

C. J. R. Sheppard, M. Gu, “Imaging by high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993).
[CrossRef]

1978 (1)

P.-A. Bélanger, M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17(7), 1080–1088 (1978).
[CrossRef] [PubMed]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

1872 (1)

L. Rayleigh, “On the diffraction of object-glasses,” Mon. Not. R. Astron. Soc. 33, 59 (1872).

Anbarasan, P. M.

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, “Tight focusing of double ring shaped radially polarized beam with high NA lens axicon,” Opt. Laser Technol. 43(7), 1037–1040 (2011).
[CrossRef]

April, A.

A. April, P. Bilodeau, M. Piché, “Focusing a TM01 beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011).
[CrossRef] [PubMed]

A. April, M. Piché, “4π Focusing of TM01 beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010).
[CrossRef] [PubMed]

Bai, J.

Y. Zhang, J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009).
[CrossRef] [PubMed]

Bainier, C.

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32(8), 976–978 (2007).
[CrossRef] [PubMed]

Bélanger, P.-A.

P.-A. Bélanger, M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17(7), 1080–1088 (1978).
[CrossRef] [PubMed]

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[CrossRef] [PubMed]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Bilodeau, P.

A. April, P. Bilodeau, M. Piché, “Focusing a TM01 beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011).
[CrossRef] [PubMed]

Biss, D. P.

D. P. Biss, T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28(11), 923–925 (2003).
[CrossRef] [PubMed]

Bokor, N.

N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004).
[CrossRef] [PubMed]

Brown, T. G.

D. P. Biss, T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28(11), 923–925 (2003).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Courjon, D.

T. Grosjean, D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32(8), 976–978 (2007).
[CrossRef] [PubMed]

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[CrossRef] [PubMed]

Davidson, N.

N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004).
[CrossRef] [PubMed]

De Koninck, Y.

H. Dehez, M. Piché, Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009).
[CrossRef] [PubMed]

P. Dufour, M. Piché, Y. De Koninck, N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. 45(36), 9246–9252 (2006).
[CrossRef] [PubMed]

Debus, C.

C. Debus, M. A. Lieb, A. Drechsler, A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

Dehez, H.

H. Dehez, M. Piché, Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009).
[CrossRef] [PubMed]

Dorn, R.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Drechsler, A.

C. Debus, M. A. Lieb, A. Drechsler, A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

Dufour, P.

P. Dufour, M. Piché, Y. De Koninck, N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. 45(36), 9246–9252 (2006).
[CrossRef] [PubMed]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Galbraith, C. G.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[CrossRef] [PubMed]

Galbraith, J. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[CrossRef] [PubMed]

Gao, L.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[CrossRef] [PubMed]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Gokulakrishnan, K.

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, “Tight focusing of double ring shaped radially polarized beam with high NA lens axicon,” Opt. Laser Technol. 43(7), 1037–1040 (2011).
[CrossRef]

Golub, I.

V. P. Kalosha, I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007).
[CrossRef] [PubMed]

Grosjean, T.

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32(8), 976–978 (2007).
[CrossRef] [PubMed]

T. Grosjean, D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

Gu, M.

C. J. R. Sheppard, M. Gu, “Imaging by high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993).
[CrossRef]

Hnatovsky, C.

C. Hnatovsky, V. Shvedov, W. Krolikowski, A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[CrossRef] [PubMed]

Huang, Z.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95(13), 133703 (2009).
[CrossRef]

Kalosha, V. P.

V. P. Kalosha, I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007).
[CrossRef] [PubMed]

Kawauchi, H.

H. Kawauchi, Y. Kozawa, S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008).
[CrossRef] [PubMed]

Kitamura, K.

K. Kitamura, K. Sakai, S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010).
[CrossRef] [PubMed]

Kozawa, Y.

H. Kawauchi, Y. Kozawa, S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008).
[CrossRef] [PubMed]

Y. Kozawa, S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007).
[CrossRef] [PubMed]

Krolikowski, W.

C. Hnatovsky, V. Shvedov, W. Krolikowski, A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[CrossRef] [PubMed]

Leuchs, G.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Lieb, M. A.

C. Debus, M. A. Lieb, A. Drechsler, A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

M. A. Lieb, A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001).
[CrossRef] [PubMed]

Lin, J.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95(13), 133703 (2009).
[CrossRef]

Lu, F.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95(13), 133703 (2009).
[CrossRef]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Mahadevan, G.

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, “Tight focusing of double ring shaped radially polarized beam with high NA lens axicon,” Opt. Laser Technol. 43(7), 1037–1040 (2011).
[CrossRef]

McCarthy, N.

P. Dufour, M. Piché, Y. De Koninck, N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. 45(36), 9246–9252 (2006).
[CrossRef] [PubMed]

Meixner, A. J.

J. Stadler, C. Stanciu, C. Stupperich, A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[CrossRef] [PubMed]

C. Debus, M. A. Lieb, A. Drechsler, A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

M. A. Lieb, A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001).
[CrossRef] [PubMed]

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[CrossRef] [PubMed]

Noda, S.

K. Kitamura, K. Sakai, S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010).
[CrossRef] [PubMed]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Piché, M.

A. April, P. Bilodeau, M. Piché, “Focusing a TM01 beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011).
[CrossRef] [PubMed]

A. April, M. Piché, “4π Focusing of TM01 beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010).
[CrossRef] [PubMed]

H. Dehez, M. Piché, Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009).
[CrossRef] [PubMed]

P. Dufour, M. Piché, Y. De Koninck, N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. 45(36), 9246–9252 (2006).
[CrossRef] [PubMed]

C. Varin, M. Piché, M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(Pt 2), 026603 (2005).
[CrossRef] [PubMed]

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[CrossRef] [PubMed]

Porras, M. A.

C. Varin, M. Piché, M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(Pt 2), 026603 (2005).
[CrossRef] [PubMed]

Quabis, S.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Rajesh, K. B.

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, “Tight focusing of double ring shaped radially polarized beam with high NA lens axicon,” Opt. Laser Technol. 43(7), 1037–1040 (2011).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On the diffraction of object-glasses,” Mon. Not. R. Astron. Soc. 33, 59 (1872).

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Rioux, M.

P.-A. Bélanger, M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17(7), 1080–1088 (1978).
[CrossRef] [PubMed]

Rode, A.

C. Hnatovsky, V. Shvedov, W. Krolikowski, A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[CrossRef] [PubMed]

Sakai, K.

K. Kitamura, K. Sakai, S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010).
[CrossRef] [PubMed]

Sato, S.

H. Kawauchi, Y. Kozawa, S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008).
[CrossRef] [PubMed]

Y. Kozawa, S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007).
[CrossRef] [PubMed]

Schadt, M.

M. Stalder, M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996).
[CrossRef] [PubMed]

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Sheppard, C. J. R.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95(13), 133703 (2009).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Imaging by high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993).
[CrossRef]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Shvedov, V.

C. Hnatovsky, V. Shvedov, W. Krolikowski, A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[CrossRef] [PubMed]

Stadler, J.

J. Stadler, C. Stanciu, C. Stupperich, A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[CrossRef] [PubMed]

Stalder, M.

M. Stalder, M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996).
[CrossRef] [PubMed]

Stanciu, C.

J. Stadler, C. Stanciu, C. Stupperich, A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[CrossRef] [PubMed]

Stupperich, C.

J. Stadler, C. Stanciu, C. Stupperich, A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[CrossRef] [PubMed]

Suresh, N. V.

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, “Tight focusing of double ring shaped radially polarized beam with high NA lens axicon,” Opt. Laser Technol. 43(7), 1037–1040 (2011).
[CrossRef]

Török, P.

P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000).
[CrossRef] [PubMed]

Varga, P.

P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000).
[CrossRef] [PubMed]

Varin, C.

C. Varin, M. Piché, M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(Pt 2), 026603 (2005).
[CrossRef] [PubMed]

Wang, H.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95(13), 133703 (2009).
[CrossRef]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

Zhan, Q.

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
[CrossRef] [PubMed]

Zhang, Y.

Y. Zhang, J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009).
[CrossRef] [PubMed]

Zheng, W.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95(13), 133703 (2009).
[CrossRef]

Appl. Opt. (2)

P. Dufour, M. Piché, Y. De Koninck, N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. 45(36), 9246–9252 (2006).
[CrossRef] [PubMed]

P.-A. Bélanger, M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17(7), 1080–1088 (1978).
[CrossRef] [PubMed]

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

Appl. Phys. Lett. (1)

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95(13), 133703 (2009).
[CrossRef]

J. Microsc. (1)

C. Debus, M. A. Lieb, A. Drechsler, A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

C. J. R. Sheppard, M. Gu, “Imaging by high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993).
[CrossRef]

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

Y. Kozawa, S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007).
[CrossRef] [PubMed]

P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000).
[CrossRef] [PubMed]

Mon. Not. R. Astron. Soc. (1)

L. Rayleigh, “On the diffraction of object-glasses,” Mon. Not. R. Astron. Soc. 33, 59 (1872).

Nat. Methods (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[CrossRef] [PubMed]

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Opt. Commun. (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

T. Grosjean, D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

Opt. Express (7)

K. Kitamura, K. Sakai, S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010).
[CrossRef] [PubMed]

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
[CrossRef] [PubMed]

Y. Zhang, J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009).
[CrossRef] [PubMed]

M. A. Lieb, A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001).
[CrossRef] [PubMed]

A. April, P. Bilodeau, M. Piché, “Focusing a TM01 beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011).
[CrossRef] [PubMed]

A. April, M. Piché, “4π Focusing of TM01 beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010).
[CrossRef] [PubMed]

Opt. Laser Technol. (1)

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, “Tight focusing of double ring shaped radially polarized beam with high NA lens axicon,” Opt. Laser Technol. 43(7), 1037–1040 (2011).
[CrossRef]

Opt. Lett. (8)

H. Kawauchi, Y. Kozawa, S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008).
[CrossRef] [PubMed]

M. Stalder, M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996).
[CrossRef] [PubMed]

J. Stadler, C. Stanciu, C. Stupperich, A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[CrossRef] [PubMed]

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32(8), 976–978 (2007).
[CrossRef] [PubMed]

D. P. Biss, T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28(11), 923–925 (2003).
[CrossRef] [PubMed]

N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004).
[CrossRef] [PubMed]

V. P. Kalosha, I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007).
[CrossRef] [PubMed]

H. Dehez, M. Piché, Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

C. Varin, M. Piché, M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(Pt 2), 026603 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

C. Hnatovsky, V. Shvedov, W. Krolikowski, A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Other (2)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 3.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 1980).

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Figures (6)

Fig. 1
Fig. 1

The geometry of a) the parabolic mirror and b) the aplanatic lens.In b), the incident rays are refracted by a reference sphere (dashed line) of radius equal to the focal length of the aplanatic lens.

Fig. 2
Fig. 2

Intensity distribution of a needle of longitudinally polarized light, normalized to its maximum value, for α0 = 75° and Δα = 0.01 rad, as computed with Eq. (12).

Fig. 3
Fig. 3

Transverse FWHM of a needle of longitudinally polarized light as a function of the angular thickness of a radially polarized annulus of light focused by (a) a parabolic mirror and (b) an aplanatic lens. Several focusing angles α0 between 45° and 90° are presented (note that α0 = 90° is not achievable when an aplanatic lens is used).

Fig. 4
Fig. 4

The transverse FWHM, as a function of the focusing angle, of a needle of longitudinally polarized light produced by an arbitrary focusing system.

Fig. 5
Fig. 5

Longitudinal FWHM of the focal spot as a function of the angular thickness of a radially polarized annulus of light focused by (a) a parabolic mirror and (b) an aplanatic lens. Several focusing angles α0 comprised between 45° and 90° are presented. The insets give a zoom around practical values of Δα.

Fig. 6
Fig. 6

System to generate radially polarized annulus of light using a lens of focal length f0 and an axicon.

Tables (4)

Tables Icon

Table 1 Expressions of Eqs. (9a)(9d) for a parabolic mirror and an aplanatic lens.

Tables Icon

Table 2 Evolution of the amplitude of the transverse component of the electric field compared to the amplitude of its longitudinal component with the focusing angle, for a fixed angular thickness of Δ α = 0.1  rad .

Tables Icon

Table 3 Domain of validity of Eqs. (7a) and (7b) for several focusing angles between 45° and 90°. In this domain, the difference between numerical FWHMs and analytical FWHMs is less than 1%. The smaller domain considering transverse and longitudinal FWHMs is given here.

Tables Icon

Table 4 Comparison of the spot size in the focal region of a parabolic mirror and an aplanatic lens for several focusing angles between 45° and 90° and a fixed radial thickness of the incident annulus of light: Δ R / R = 5 % .

Equations (38)

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

E ( r , ϕ , z ) = E o 2 π Ω q ( α ) A ( α , β ) exp ( j k r ) d Ω ,
a ^ ( α , β ) = a ^ x cos α cos β + a ^ y cos α sin β + a ^ z sin α ,
E ( r , ϕ , z ) = E o 2 π 0 2 π α min α max q ( α ) 0 ( α ) a ^ ( α , β ) × exp [ j k ( z cos α r sin α cos ( ϕ β ) ) ] sin α d α d β ,
E r ( r , z ) = j E o α min α max q ( α ) 0 ( α ) sin α cos α exp ( j k z cos α ) J 1 ( k r sin α ) d α ,
E z ( r , z ) = E o α min α max q ( α ) 0 ( α ) sin 2 α exp ( j k z cos α ) J 0 ( k r sin α ) d α ,
E r ( r , z ) = j E o q ( α 0 ) cos α 0 sin α 0 exp ( j k z cos α 0 ) J 1 ( k r sin α 0 ) ,
E z ( r , z ) = E o q ( α 0 ) sin 2 α 0 exp ( j k z cos α 0 ) J 0 ( k r sin α 0 ) .
E r = j E o π Δ α α min α max q ( α ) sin α cos α exp [ ( α α 0 Δ α ) 2 j k z cos α ] J 1 ( k r sin α ) d α ,
E z = E o π Δ α α min α max q ( α ) sin 2 α exp [ ( α α 0 Δ α ) 2 j k z cos α ] J 0 ( k r sin α ) d α .
E r ( r , z ) j E o q ( α 0 ) sin α 0 cos α 0 exp ( z 2 / z 0 2 j k z cos α 0 ) × { [ 1 + j Δ α U r ( α 0 ) ( z / z 0 ) + 1 2 Δ α 2 V r ( α 0 ) ] J 1 ( v ) + 1 4 Δ α 2 v J 2 ( v ) } ,
E z ( r , z ) E o q ( α 0 ) sin 2 α 0 exp ( z 2 / z 0 2 j k z cos α 0 ) × { [ 1 + j Δ α U z ( α 0 ) ( z / z 0 ) + 1 2 Δ α 2 V z ( α 0 ) ] J 0 ( v ) + 1 4 Δ α 2 v J 1 ( v ) } ,
z 0 2 k sin α 0 Δ α = λ π sin α 0 Δ α .
U r ( α 0 ) = q ( α 0 ) q ( α 0 ) + 2 cot ( 2 α 0 ) ,
V r ( α 0 ) = 2 q ( α 0 ) q ( α 0 ) cot ( 2 α 0 ) + q ( α 0 ) 2 q ( α 0 ) ,
U z ( α 0 ) = q ( α 0 ) q ( α 0 ) + 2 cot α 0 ,
V z ( α 0 ) = 2 q ( α 0 ) q ( α 0 ) cot α 0 + q ( α 0 ) 2 q ( α 0 ) ,
E r ( r , z ) j E o q ( α 0 ) sin α 0 cos α 0 exp ( z 2 / z 0 2 j k z cos α 0 ) J 1 ( k r sin α 0 ) ,
E z ( r , z ) E o q ( α 0 ) sin 2 α 0 exp ( z 2 / z 0 2 j k z cos α 0 ) J 0 ( k r sin α 0 ) .
I ( r , z ) | E ( r , z ) | 2 = | E r ( r , z ) | 2 + | E z ( r , z ) | 2 .
I ( r , z ) I o exp ( 2 z 2 / z 0 2 ) [ J 0 2 ( k r sin α 0 ) + cot 2 α 0 J 1 2 ( k r sin α 0 ) ] ,
| E r | max 2 | E z | max 2 0 , 34 cot 2 α 0 .
I ( r , z ) / I ( 0 , z ) = J 0 2 ( k r sin α 0 ) + cot 2 α 0 J 1 2 ( k r sin α 0 ) = 1 2 .
Longitudinal FWHM z 0 ( 2 ln 2 ) 1 / 2 = λ ( 2 ln 2 ) 1 / 2 π sin α 0 Δ α ,
Δ R R = 2 ( ln 2 ) 1 / 2 [ h ( α 0 + Δ α ) h ( α 0 Δ α ) h ( α 0 ) ] 4 ( ln 2 ) 1 / 2 h ( α 0 ) h ( α 0 ) Δ α ,
R = f 0 γ
Δ R = λ f 0 π w 0
sin α cos α sin α 0 cos α 0 [ 1 + 2 θ cot ( 2 α 0 ) ] ,
sin 2 α sin 2 α 0 ( 1 + 2 θ cot α 0 ) .
E r ( r , z ) = j E o sin α 0 cos α 0 π Δ α exp ( j k z cos α 0 ) q ( α ) [ 1 + 2 θ cot ( 2 α 0 ) ] × exp ( θ 2 Δ α 2 + j k z θ sin α 0 ) J 1 ( k r sin α 0 cos θ ) d θ ,
E z ( r , z ) = E o sin 2 α 0 π Δ α exp ( j k z cos α 0 ) q ( α ) [ 1 + 2 θ cot α 0 ] × exp ( θ 2 Δ α 2 + j k z θ sin α 0 ) J 0 ( k r cos θ sin α 0 ) d θ .
J ν ( x y ) = y ν n = 0 x n ( 1 y 2 ) n ( 2 n ) ! ! J ν + n ( x ) .
q ( α ) = s = 0 q ( s ) ( α 0 ) s ! θ s ,
E r = j E o sin α 0 cos α 0 π Δ α exp ( j k z cos α 0 ) n = 0 s = 0 q ( s ) ( α 0 ) s ! ( k r sin α 0 ) n J n + 1 ( k r sin α 0 ) ( 2 n ) ! ! × θ 2 n + s [ 1 + 2 θ cot ( 2 α 0 ) ] exp ( θ 2 Δ α 2 + j k z θ sin α 0 ) d θ ,
E z = E o sin 2 α 0 π Δ α exp ( j k z cos α 0 ) n = 0 s = 0 q ( s ) ( α 0 ) s ! ( k r sin α 0 ) n J n ( k r sin α 0 ) ( 2 n ) ! ! × θ 2 n + s [ 1 + 2 θ cot α 0 ] exp ( θ 2 Δ α 2 + j k z θ sin α 0 ) d θ .
θ p exp ( θ 2 Δ α 2 + j k z sin α 0 θ ) d θ = π Δ α ( j 1 2 Δ α ) p exp ( z 2 z 0 2 ) H p ( z z 0 ) ,
E r ( r , z ) = j E o q ( α 0 ) sin α 0 cos α 0 exp ( z 2 z 0 2 j k z cos α 0 ) n = 0 s = 0 C n , s × [ H 2 n + s ( z z 0 ) + j Δ α cot ( 2 α 0 ) H 2 n + s + 1 ( z z 0 ) ] J n + 1 ( k r sin α 0 ) ,
E z ( r , z ) = E o q ( α 0 ) sin 2 α 0 exp ( z 2 z 0 2 j k z cos α 0 ) n = 0 s = 0 C n , s × [ H 2 n + s ( z z 0 ) + j Δ α cot α 0 H 2 n + s + 1 ( z z 0 ) ] J n ( k r sin α 0 ) ,
C n , s q ( s ) ( α 0 ) q ( α 0 ) ( j 1 2 Δ α ) 2 n + s ( 1 2 k r sin α 0 ) n s ! n ! .

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