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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2011 (4)

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

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, and 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]

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

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

2010 (2)

2009 (3)

2008 (3)

2007 (4)

2006 (1)

2005 (1)

C. Varin, M. Piché, and 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)

2003 (3)

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

C. Debus, M. A. Lieb, A. Drechsler, and 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]

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

2001 (3)

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

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

M. A. Lieb and 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)

1996 (1)

1993 (1)

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

1978 (1)

1959 (1)

B. Richards and 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, and 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.

Bai, J.

Bainier, C.

Bélanger, P.-A.

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and 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, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Bilodeau, P.

Biss, D. P.

Bokor, N.

Brown, T. G.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and 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.

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and 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.

De Koninck, Y.

Debus, C.

C. Debus, M. A. Lieb, A. Drechsler, and 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.

Dorn, R.

R. Dorn, S. Quabis, and 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, and 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, and 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, and 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.

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and 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, and 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, and 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, and 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, and 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, and 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, and 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, and 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.

Grosjean, T.

Gu, M.

C. J. R. Sheppard and 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, and 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, and 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.

Kawauchi, H.

Kitamura, K.

Kozawa, Y.

Krolikowski, W.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and 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, and 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, and 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, and 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, and 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 and 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, and 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, and 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, and 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, and 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.

Meixner, A. J.

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and 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.

Novotny, L.

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

Piché, M.

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and 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, and 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, and 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, and 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, and 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, and 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 and 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.

Rode, A.

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

Sakai, K.

Sato, S.

Schadt, M.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and 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, and 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 and 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, and 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, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[CrossRef] [PubMed]

Stadler, J.

Stalder, M.

Stanciu, C.

Stupperich, C.

Suresh, N. V.

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, and 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.

Varga, P.

Varin, C.

C. Varin, M. Piché, and 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, and 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, and 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 and 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, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

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

Zhan, Q.

Zhang, Y.

Zheng, W.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and 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)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and 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, and 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, and 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 and M. Gu, “Imaging by high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993).
[CrossRef]

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

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, and 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, and 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, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

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

Opt. Express (7)

Opt. Laser Technol. (1)

K. B. Rajesh, N. V. Suresh, P. M. Anbarasan, K. Gokulakrishnan, and 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)

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

C. Varin, M. Piché, and 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)

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

R. Dorn, S. Quabis, and 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, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[CrossRef] [PubMed]

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

B. Richards and 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.

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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)

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

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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 .

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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)

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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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