Abstract

Effect of primary spherical aberration on the tight focusing of linearly and circularly polarized Laguerre–Gaussian (LG) beams is studied by using the vectorial Debye integral. Results are presented for the intensity distribution and square of the polarization components. In the case of the linearly polarized LG beam with unit and double topological charge, the presence of aberration reduces the residual intensity at the focal point and spreads the sidelobes. If the beam is circularly polarized, the aberration results in an increase in the size of the dark core along with a reduction in the intensity at the periphery of the bright ring. The effect of aberration is also discussed in the context of the fluorescent spot size in the focal plane of a stimulated-emission-depletion microscope.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]

2007 (4)

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano Lett. 7, 415-420 (2007).
[CrossRef] [PubMed]

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing,” Opt. Lasers Eng. 45, 773-782 (2007).
[CrossRef]

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32, 2357-2359 (2007).
[CrossRef] [PubMed]

2006 (3)

Y. Roichman, A. Waldron, E. Gardel, and D. G. Grier, “Optical traps with geometrical aberrations,” Appl. Opt. 45, 3425-3429 (2006).
[CrossRef] [PubMed]

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 106/1-8 (2006).
[CrossRef]

L. E. Helseth, “Smallest focal hole,” Opt. Commun. 257, 1-8 (2006).
[CrossRef]

2005 (4)

Y. Iketaki, T. Watanabe, M. Sakai, S.-I. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602-033602/9 (2005).
[CrossRef]

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, “Investigation of polarization effects for high numerical-aperture first order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead,” Opt. Express 13, 10440-10447 (2005).
[CrossRef] [PubMed]

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation of a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4, 023006-023017 (2005).
[CrossRef]

2004 (2)

2002 (1)

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

2001 (2)

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]

C. J. R. Sheppard, “High aperture beams,” J. Opt. Soc. Am. A 18, 1579-1587 (2001).
[CrossRef]

1999 (1)

1996 (1)

N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

1994 (1)

1993 (2)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293-2310 (1993).
[CrossRef]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337-347 (1993).
[CrossRef]

1991 (1)

1987 (1)

1974 (2)

A. Yoshida and T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281-292 (1974).

A. Yoshida and T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322-331 (1974).

1966 (1)

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

1959 (2)

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

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Allen, L.

N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

Asakura, T.

A. Yoshida and T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281-292 (1974).

A. Yoshida and T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322-331 (1974).

Biss, D. P.

Bokor, N.

Bossi, M.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 106/1-8 (2006).
[CrossRef]

Brown, T. G.

Chiu, D. T.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano Lett. 7, 415-420 (2007).
[CrossRef] [PubMed]

Chon, J. W. M.

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

Daigoku, K.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Davidson, N.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Dorn, R.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

Ebihara, T.

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation of a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4, 023006-023017 (2005).
[CrossRef]

Edgar, J. S.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano Lett. 7, 415-420 (2007).
[CrossRef] [PubMed]

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]

Fong, C.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano Lett. 7, 415-420 (2007).
[CrossRef] [PubMed]

Fujii, M.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32, 2357-2359 (2007).
[CrossRef] [PubMed]

Y. Iketaki, T. Watanabe, M. Sakai, S.-I. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602-033602/9 (2005).
[CrossRef]

N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, “Investigation of polarization effects for high numerical-aperture first order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead,” Opt. Express 13, 10440-10447 (2005).
[CrossRef] [PubMed]

Gahagan, K. T.

Gan, X.

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

Gardel, E.

Grier, D. G.

Gu, M.

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

Hell, S. W.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 106/1-8 (2006).
[CrossRef]

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]

S. W. Hell and J. Wichmann, “Breaking the diffraction resolution by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780-782 (1994).
[CrossRef] [PubMed]

Helseth, L. E.

L. E. Helseth, “Smallest focal hole,” Opt. Commun. 257, 1-8 (2006).
[CrossRef]

Iketaki, Y.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32, 2357-2359 (2007).
[CrossRef] [PubMed]

N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, “Investigation of polarization effects for high numerical-aperture first order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead,” Opt. Express 13, 10440-10447 (2005).
[CrossRef] [PubMed]

Y. Iketaki, T. Watanabe, M. Sakai, S.-I. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602-033602/9 (2005).
[CrossRef]

Ishiuchi, S.-I.

Y. Iketaki, T. Watanabe, M. Sakai, S.-I. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602-033602/9 (2005).
[CrossRef]

Jeffries, G. D. M.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano Lett. 7, 415-420 (2007).
[CrossRef] [PubMed]

Kant, R.

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337-347 (1993).
[CrossRef]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293-2310 (1993).
[CrossRef]

Keller, J.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 106/1-8 (2006).
[CrossRef]

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]

Kogelnik, H.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

Leuchs, G.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

Levenson, M. D.

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation of a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4, 023006-023017 (2005).
[CrossRef]

Li, T.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

Matthews, H. J.

Munro, P. R. T.

Padgett, M. J.

N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

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. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Roichman, Y.

Sakai, M.

Y. Iketaki, T. Watanabe, M. Sakai, S.-I. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602-033602/9 (2005).
[CrossRef]

Senthilkumaran, P.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing,” Opt. Lasers Eng. 45, 773-782 (2007).
[CrossRef]

Shelby, J. P.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano Lett. 7, 415-420 (2007).
[CrossRef] [PubMed]

Sheppard, C. J. R.

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, 1986).

Simpson, N. B.

N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing,” Opt. Lasers Eng. 45, 773-782 (2007).
[CrossRef]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing,” Opt. Lasers Eng. 45, 773-782 (2007).
[CrossRef]

Swartzlander, G. A.

Torok, P.

Unno, Y.

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation of a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4, 023006-023017 (2005).
[CrossRef]

Visser, T. D.

Waldron, A.

Watanabe, T.

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32, 2357-2359 (2007).
[CrossRef] [PubMed]

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Y. Iketaki, T. Watanabe, M. Sakai, S.-I. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602-033602/9 (2005).
[CrossRef]

N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, “Investigation of polarization effects for high numerical-aperture first order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead,” Opt. Express 13, 10440-10447 (2005).
[CrossRef] [PubMed]

Y. Iketaki, T. Watanabe, M. Sakai, S.-I. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602-033602/9 (2005).
[CrossRef]

Wichmann, J.

Wiersma, S. H.

Willig, K. I.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 106/1-8 (2006).
[CrossRef]

Wolf, E.

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

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Yoshida, A.

A. Yoshida and T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322-331 (1974).

A. Yoshida and T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281-292 (1974).

Zhao, Y.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano Lett. 7, 415-420 (2007).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (1)

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

J. Microlithogr., Microfabr., Microsyst. (1)

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation of a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4, 023006-023017 (2005).
[CrossRef]

J. Mod. Opt. (3)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293-2310 (1993).
[CrossRef]

N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337-347 (1993).
[CrossRef]

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

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano Lett. 7, 415-420 (2007).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Geometric configuration.

Fig. 2
Fig. 2

Total intensity distribution with contour lines of an x-polarized LG beam focused by an aplanatic lens with α = 75 ° , γ = 1 , and m = 1 : (a) A s = 0.0 , (b) A s = 0.5 , (c) A s = 1.0 ; with m = 2 : (d) A s = 0.0 , (e) A s = 0.5 , (f) A s = 1.0 .

Fig. 3
Fig. 3

Distribution and contour lines of the squares of x-, y-, and z-polarization components at the focal plane of an aplanatic lens with α = 75 ° for an x-polarized Gaussian beam with γ = 1 . For A s = 0.0 : (a) E x 2 , (b) E y 2 , (c) E z 2 ; for A s = 0.5 : (d) E x 2 , (e) E y 2 , (f) E z 2 .

Fig. 4
Fig. 4

Distribution and contour lines of the squares of x-, y-, and z-polarization components at the focal plane of an aplanatic lens with α = 75 ° for x polarized LG beam with m = 1 , and γ = 1 . For A s = 0.0 : (a) E x 2 , (b) E y 2 , (c) E z 2 ; A s = 0.5 : (d) E x 2 , (e) E y 2 , (f) E z 2 .

Fig. 5
Fig. 5

Total intensity distribution and contour lines for an LC polarized LG beam with m = 1 and γ = 1 focused by an aplanatic lens with α = 75 ° : (a) A s = 0.0 , (b) A s = 0.5 , (c) A s = 1.0 .

Fig. 6
Fig. 6

Distribution and contour lines of the squares of x-, y-, and z-polarization components at the focal plane of an aplanatic lens with α = 75 ° for LC polarized LG beam with m = 1 , and γ = 1 . For A s = 0.0 : (a) E x 2 , (b) E y 2 , (c) E z 2 ; for A s = 0.5 : (d) E x 2 , (e) E y 2 , (f) E z 2 .  

Fig. 7
Fig. 7

Intensity distribution in the ( y - z ) plane of an LC polarized LG beam with m = 1 and γ = 1 , focused by an aplanatic lens with α = 75 ° . For (a) A s = 0.0 , (b) A s = 0.5 , (c) A s = 1.0 .

Fig. 8
Fig. 8

Variation of the focal point intensity with defocusing of a linearly polarized LG beam with m = 1 , γ = 1 focused by an aplanatic lens with NA = 0.966 . For (a) A s = 0.0 , (b) A s = 0.5 , (c) A s = 1.0 ; for NA = 0.866 : (d) A s = 0.0 , (e) A s = 0.5 , (f) A s = 1.0 ; for a beam with m = 1 , γ = 3 , and for NA = 0.86 : curves labeled a , A s = 0.0 ; b , A s = 0.5 ; c , A s = 1.0 .

Fig. 9
Fig. 9

Radial intensity profile of an LC polarized LG beam with m = 1 , γ = 1 focused by an aplanatic lens with NA = 0.966 . For A s = 0.0 : (a) A d = 0.0 ; A s = 0.5 : (b) A d = 0.6 ; A s = 0.5 : (c) A d = 0.0 ; A s = 1.0 : (d) A d = 0.0 .

Fig. 10
Fig. 10

Normalized intensity profiles for fluorescence spot in the focal plane for (a) A s = 0.0 , (b) A s = 1.5 .

Equations (26)

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E 0 ( ρ , ϕ ) = A 0 ( 2 γ ρ ) m L p m ( 2 γ 2 ρ 2 ) exp ( γ 2 ρ 2 ) exp ( i m ϕ ) ,
E 0 ( ρ , ϕ ) = A 0 ( 2 γ ρ ) m exp ( γ 2 ρ 2 ) exp ( i m ϕ ) .
E 0 ( θ , ϕ ) = A 0 ( 2 γ sin θ sin α ) m exp [ γ 2 sin 2 θ sin 2 α ] exp ( i m ϕ ) ,
E 0 ( θ , ϕ ) = A 1 ( θ ) exp ( i m ϕ ) ,
A 1 ( θ ) = A 0 ( 2 γ sin θ sin α ) m exp [ γ 2 sin 2 θ sin 2 α ] .
E ( u , v ) = ( i A λ ) 0 α 0 2 π A 1 ( θ ) P ( θ , ϕ ) A 2 ( θ ) exp ( i m ϕ ) A 3 ( θ , ϕ ) exp [ i v sin α sin θ cos ( ϕ ϕ P ) ] exp ( i u sin 2 α cos θ ) sin θ d θ d ϕ ,
P ( θ , ϕ ) = a ( cos θ cos 2 ϕ + sin 2 ϕ ) + b ( cos θ sin ϕ cos ϕ sin ϕ cos ϕ ) a ( cos θ cos ϕ sin ϕ sin ϕ cos ϕ ) + b ( cos θ sin 2 ϕ + cos 2 ϕ ) a sin θ cos ϕ b sin θ sin ϕ ,
v = k ( x P 2 + y P 2 ) 1 2 sin α , u = k z sin 2 α .
A 3 ( θ ) = exp { i 2 π λ [ A s ( sin θ sin α ) 4 + A d ( sin θ sin α ) 2 ] } ,
E ( u , v ) = ( i A λ ) 0 α 0 2 π A 1 ( θ ) A 2 ( θ ) P ( θ , ϕ ) exp ( i m ϕ ) exp [ i 2 π λ A s ( sin θ sin α ) 4 ] exp { i [ v sin α sin θ cos ( ϕ ϕ P ) + u sin 2 α cos θ ] } sin θ d θ d ϕ .
[ E x ( v x , v y ) E y ( v x , v y ) E z ( v x , v y ) ] = ( i A λ ) 0 α 0 2 π A 1 ( θ ) A 2 ( θ ) exp ( i m ϕ ) exp [ i 2 π λ A s ( sin θ sin α ) 4 ] [ cos θ + sin 2 ϕ ( 1 cos θ ) cos ϕ sin ϕ ( cos θ 1 ) cos ϕ sin θ ] exp [ i v sin α sin θ cos ( ϕ ϕ P ) ] sin θ d θ d ϕ .
I ( v x , v y ) = E x 2 + E y 2 + E z 2 ,
0 2 π e i m ϕ exp [ i v sin α sin θ cos ( ϕ ϕ P ) ] d ϕ = 2 π i m J m ( v sin α sin θ ) exp ( i m ϕ P ) ,
E x = 2 π i m exp ( i m ϕ P ) I m π i m + 2 exp [ i ( m + 2 ) ϕ P ] I m + 2 π i m 2 exp [ i ( m 2 ) ϕ P ] I m 2 ,
E y = i π i m 2 exp [ i ( m 2 ) ϕ P ] I m 2 + i π i m + 2 exp [ i ( m + 2 ) ϕ P ] I m + 2 ,
E z = 2 π i m + 1 exp [ i ( m + 1 ) ϕ P ] I m + 1 2 π i m 1 exp [ i ( m 1 ) ϕ P ] I m 1 ,
I m ( v , u ) = ( i A 2 λ ) 0 α A 1 ( θ ) A 2 ( θ ) A 3 ( θ ) ( 1 + cos θ ) J m [ v sin α sin θ ] exp [ i u sin 2 α cos θ ] sin θ d θ ,
I m ± 1 ( v , u ) = ( i A 2 λ ) 0 α A 1 ( θ ) A 2 ( θ ) A 3 ( θ ) J m ± 1 ( v sin α sin θ ) exp ( i u sin 2 α cos θ ) sin 2 θ d θ ,
I m ± 2 ( ν , u ) = ( i A 2 λ ) 0 α A 1 ( θ ) A 2 ( θ ) A 3 ( θ ) ( 1 cos θ ) J m ± 2 [ v sin α sin θ ] exp [ i u sin 2 α cos θ ] sin θ d θ .
E x = i π i m 2 exp [ i ( m 2 ) ϕ P ] I m 2 + i π i m + 2 exp [ i ( m + 2 ) ϕ P ] I m + 2 ,
E y = 2 π i m exp [ i m ϕ P ] I m + π i m 2 exp [ i ( m 2 ) ϕ P ] I m 2 + π i m + 2 exp [ i ( m + 2 ) ϕ P ] I m + 2 ,
E z = i 2 π i m 1 exp [ i ( m 1 ) ϕ P ] I m 1 + i 2 π i m + 1 exp [ i ( m + 1 ) ϕ P ] I m + 1 .
E ( θ , ϕ ) = A 1 ( θ ) e i m ϕ ( E 1 + e i φ E 2 ) ,
( E x ( v x , v y ) E y ( v x , v y ) E z ( v x , v y ) ) = ( i A λ ) 0 α 0 2 π A 1 ( θ ) cos 1 2 θ exp ( i m ϕ ) A 3 ( θ ) [ [ { cos θ + sin 2 ϕ ( 1 cos θ ) } ± i sin ϕ cos ϕ ( cos θ 1 ) ] [ sin ϕ cos ϕ ( cos θ 1 ) ± i { 1 + sin 2 ϕ ( cos θ 1 ) } ] sin θ ( cos ϕ ± i sin ϕ ) ] exp [ i v sin α sin θ cos ( ϕ ϕ P ) ] sin θ d θ d ϕ .
D v e c t = 1 [ 1 + C ( n E e ) 2 ] ,
I f ( x P , y P ) D ( x P , y P ) I P ( x P , y P ) ,

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