Abstract

Diffractive and focusing properties of vector Laguerre–Gaussian beams with obstacle are investigated under tight focusing conditions. Using vector diffraction theory, intensity and polarization distributions near the focus at different orthogonal planes are calculated and analyzed for vector Laguerre–Gaussian beams. It is observed that the beam is able to compensate the distortion produced by obstacles when the size of the obstacle is small. The structural changes in the polarization distribution are not the same in different orthogonal planes. The polarization characteristics of the beam show a significant change when the size of the obstacle is large. A comparative study of the focusing and diffractive properties of vector Laguerre–Gaussian and vector Bessel–Gaussian beams has also been performed.

© 2011 Optical Society of America

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2011 (3)

2010 (4)

2009 (1)

2008 (4)

Z. Zhang, J. Pu, and X. Wang, “Distribution of phase and orbital angular momentum of tightly focused vortex beams,” Opt. Eng. 47, 068001 (2008).
[CrossRef]

V. Karsasek, T. Cizmar, O. Brzobohaty, and P. Zemanek, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef]

E. McLeod and C. B. Arnold, “Subwavelenth direct-write nanopatterning using optically trapped microspheres,” Nature Nanotechnol. 3, 413–417 (2008).
[CrossRef]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
[CrossRef] [PubMed]

2007 (3)

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

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

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

2006 (3)

2005 (2)

T. Cizmar, V. Garces-Chavez, and K. Dholakia, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

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

2004 (2)

S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21, 1192–1197 (2004).
[CrossRef]

V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003(2004).
[CrossRef]

2002 (2)

Z. Bouch, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164(2002).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[CrossRef] [PubMed]

2000 (1)

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[CrossRef]

1998 (1)

1997 (1)

M. Eerelyi, Z. L. Horvath, G. Szabo, and Zs. Bor, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

1987 (1)

F. Gori and G. Guattari, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

1972 (1)

D. Phol, “Operation of a Ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

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. A 253, 358–379 (1959).
[CrossRef]

Agate, B.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Arnold, C. B.

E. McLeod and C. B. Arnold, “Subwavelenth direct-write nanopatterning using optically trapped microspheres,” Nature Nanotechnol. 3, 413–417 (2008).
[CrossRef]

Bor, Zs.

M. Eerelyi, Z. L. Horvath, G. Szabo, and Zs. Bor, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Bouch, Z.

Z. Bouch, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164(2002).
[CrossRef]

Broky, J.

Brown, C. T. A.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Brzobohaty, O.

V. Karsasek, T. Cizmar, O. Brzobohaty, and P. Zemanek, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef]

Cao, G. W.

Chang, R. S.

Chen, H.

Chiu, D. T.

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

Christodoulides, D. N.

Cizmar, T.

V. Karsasek, T. Cizmar, O. Brzobohaty, and P. Zemanek, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef]

T. Cizmar, V. Garces-Chavez, and K. Dholakia, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Comrie, M.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

De Koninck, Y.

Dholakia, K.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

T. Cizmar, V. Garces-Chavez, and K. Dholakia, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003(2004).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[CrossRef] [PubMed]

Ding, J.

Dogariu, A.

Dorn, R.

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

Dufour, P.

Edgar, J. S.

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

Eerelyi, M.

M. Eerelyi, Z. L. Horvath, G. Szabo, and Zs. Bor, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Farrbach, F. O.

F. O. Farrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785(2010).
[CrossRef]

Garces-Chavez, V.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

T. Cizmar, V. Garces-Chavez, and K. Dholakia, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003(2004).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[CrossRef] [PubMed]

Gori, F.

F. Gori and G. Guattari, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Gu, M.

Guattari, G.

F. Gori and G. Guattari, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Gunn-Moore, F.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Horvath, Z. L.

M. Eerelyi, Z. L. Horvath, G. Szabo, and Zs. Bor, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Huang, K.

Ito, A.

Jeffries, G. D. M.

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

Jia, B.

Kang, H.

Karsasek, V.

V. Karsasek, T. Cizmar, O. Brzobohaty, and P. Zemanek, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef]

Kozawa, Y.

Leuches, G.

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

Li, K.

Li, Y. P.

Marienko, I. G.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[CrossRef]

McCarthy, N.

McGloin, D.

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

V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003(2004).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[CrossRef] [PubMed]

McLeod, E.

E. McLeod and C. B. Arnold, “Subwavelenth direct-write nanopatterning using optically trapped microspheres,” Nature Nanotechnol. 3, 413–417 (2008).
[CrossRef]

Melville, H.

V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003(2004).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[CrossRef] [PubMed]

Nesterov, A. V.

Niziev, V. G.

Phol, D.

D. Phol, “Operation of a Ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

Piche, M.

Pu, J.

Z. Zhang, J. Pu, and X. Wang, “Distribution of phase and orbital angular momentum of tightly focused vortex beams,” Opt. Eng. 47, 068001 (2008).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, and G. Leuches, “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. A 253, 358–379 (1959).
[CrossRef]

Rohrbach, A.

F. O. Farrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785(2010).
[CrossRef]

Roskey, D.

V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003(2004).
[CrossRef]

Sato, S.

Shi, P.

Sibbett, W.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[CrossRef] [PubMed]

Simon, P.

F. O. Farrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785(2010).
[CrossRef]

Siviloglou, G. A.

Soskin, M. S.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[CrossRef]

Stevenson, D. J.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Summers, M. D.

V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003(2004).
[CrossRef]

Szabo, G.

M. Eerelyi, Z. L. Horvath, G. Szabo, and Zs. Bor, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Tao, S. H.

Tovar, A.

Tsampoula, X.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Vasnetsov, M. V.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[CrossRef]

Vyas, S.

Wang, H. T.

Wang, X.

Z. Zhang, J. Pu, and X. Wang, “Distribution of phase and orbital angular momentum of tightly focused vortex beams,” Opt. Eng. 47, 068001 (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. A 253, 358–379 (1959).
[CrossRef]

Wright, E. M.

V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003(2004).
[CrossRef]

Yonezawa, K.

Yuan, X.

Zemanek, P.

V. Karsasek, T. Cizmar, O. Brzobohaty, and P. Zemanek, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef]

Zhan, Q.

Zhang, B. F.

Zhang, X. B.

Zhang, Z.

Z. Zhang, J. Pu, and X. Wang, “Distribution of phase and orbital angular momentum of tightly focused vortex beams,” Opt. Eng. 47, 068001 (2008).
[CrossRef]

Zhao, Y.

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

Zheng, Z.

Adv. Opt. Photon. (1)

Appl. Opt. (3)

Appl. Phys. B (1)

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

Appl. Phys. Lett. (4)

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

D. Phol, “Operation of a Ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

T. Cizmar, V. Garces-Chavez, and K. Dholakia, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003(2004).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. Vac. Sci. Technol. B (1)

M. Eerelyi, Z. L. Horvath, G. Szabo, and Zs. Bor, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

JETP Lett. (1)

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[CrossRef]

Nat. Photon. (1)

F. O. Farrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785(2010).
[CrossRef]

Nature (1)

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of optical geometry.

Fig. 2
Fig. 2

Calculated intensity distribution and polarization components for radially polarized Laguerre–Gaussian beam for m = 1 , p = 0 , β 0 = 1.39 in x y plane. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupil plane and at the focal plane are 5.7 × 5.7 mm 2 and 1 × 1 μm 2 , respectively.

Fig. 3
Fig. 3

Calculated intensity distribution and polarization components for radially polarized Laguerre–Gaussian beam for m = 1 , p = 0 , β 0 = 1.39 in x z plane. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupil plane and at the focal plane are 5.7 × 5.7 mm 2 and 1 × 1 μm 2 , respectively.

Fig. 4
Fig. 4

Calculated intensity distribution and polarization components for radially polarized Laguerre–Gaussian beam for m = 1 , p = 0 , β 0 = 1.39 in y z plane. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupil plane and at the focal plane are 5.7 × 5.7 mm 2 and 1 × 1 μm 2 , respectively.

Fig. 5
Fig. 5

Calculated intensity distribution and polarization components for higher-order mode m = 2 , p = 0 , β 0 = 1.59 in x y plane. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupil plane and at the focal plane are 5.7 × 5.7 mm 2 and 1 × 1 μm 2 , respectively.

Fig. 6
Fig. 6

Calculated intensity distribution and polarization components for higher-order mode m = 2 , p = 0 , β 0 = 1.59 in x z plane. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupil plane and at the focal plane are 5.7 × 5.7 mm 2 and 1 × 1 μm 2 , respectively.

Fig. 7
Fig. 7

Calculated intensity distribution and polarization components for higher-order mode m = 2 , p = 0 , β 0 = 1.59 in y z plane. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupil plane and at the focal plane are 5.7 × 5.7 mm 2 and 1 × 1 μm 2 , respectively.

Fig. 8
Fig. 8

Calculated intensity distribution and polarization components for multiring radially polarized Laguerre–Gaussian beam for m = 1 , p = 3 , β 0 = 5 . Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupil plane and at the focal plane are 5.7 × 5.7 mm 2 and 6 × 6 μm 2 , respectively.

Fig. 9
Fig. 9

Calculated intensity distribution and polarization components for radially polarized Bessel–Gaussian beam with β = 0.0005 k and m = 0 for different sizes of obstacles. Field distribution at the pupil plane (a) total intensity distribution, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane (d) total intensity, (e) radial component, (f) azimuthal component, and (g)  longitudinal component. The dimensions at the pupil plane and at the focal plane are 5.7 × 5.7 mm 2 and 3 × 3 μm 2 , respectively.

Equations (9)

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E ( v ) = U ( r , ϕ , z ) exp ( i ω t )
U ( r , ϕ , z ) = u radial i ^ r + u azimuthal i ^ ϕ ,
U ( r , ϕ , z ) = E 0 ω 0 ω ( z ) [ 2 r ω ( z ) ] n ± 1 × L p n ± 1 [ 2 r 2 ω 2 ( z ) ] exp [ r 2 ω 2 ( z ) ] × exp { i k r 2 2 R ( z ) + i k z i [ 2 p + ( n ± 1 ) + 1 ] η ( z ) } × { cos ( n ϕ ) i ^ ϕ sin ( n ϕ ) i ^ r ± sin ( n ϕ ) i ^ ϕ + cos ( n ϕ ) i ^ r } ,
U r , radial ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin ( θ ) cos ( θ ) cos ( ϕ ϕ 0 ) × u radial exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ
U ϕ , radial ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin ( θ ) cos ( θ ) sin ( ϕ ϕ 0 ) × u radial exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ
U z , radial ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin 2 ( θ ) × u radial exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ
U r , azimuthal ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin ( θ ) sin ( ϕ ϕ 0 ) × u azimuthal exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ
U ϕ , azimuthal ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin ( θ ) cos ( ϕ ϕ 0 ) × u azimuthal exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ
U z , azimuthal ( r 0 , ϕ 0 , z 0 ) = 0 ,

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