Abstract

An overview of the recent developments in the field of cylindrical vector beams is provided. As one class of spatially variant polarization, cylindrical vector beams are the axially symmetric beam solution to the full vector electromagnetic wave equation. These beams can be generated via different active and passive methods. Techniques for manipulating these beams while maintaining the polarization symmetry have also been developed. Their special polarization symmetry gives rise to unique high-numerical-aperture focusing properties that find important applications in nanoscale optical imaging and manipulation. The prospects for cylindrical vector beams and their applications in other fields are also briefly discussed.

© 2009 Optical Society of America

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2008 (8)

2007 (14)

W. Chen, Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE 6450, 64500D (2007).
[CrossRef]

A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J.-C. Weeber, A. Dereux, G. P. Wiederrecht, L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. 32, 2535–2537 (2007).
[CrossRef] [PubMed]

W. Chen, Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007).
[CrossRef] [PubMed]

W. Chen, Q. Zhan, “Field enhancement analysis of an apertureless near field scanning optical microscope probe with finite element method,” Chin. Opt. Lett. 5, 709–711 (2007).

H. Kawauchi, K. Yonezawa, Y. Kozawa, S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32, 1839–1841 (2007).
[CrossRef] [PubMed]

G. Chang, C. J. Divin, C. H. Liu, S. L. Williamson, A. Galvanauskas, T. B. Norris, “Generation of radially polarized terahertz pulses via velocity-mismatched optical rectification,” Opt. Lett. 32, 433–435 (2007).
[CrossRef] [PubMed]

M. Meier, V. Romano, T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[CrossRef]

H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[CrossRef]

X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef] [PubMed]

A. K. Spilman, T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007).
[CrossRef]

B. Hao, J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15, 3550–3556 (2007).
[CrossRef] [PubMed]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[CrossRef] [PubMed]

K. Yonezawa, Y. Kozawa, S. Sato, “Compact laser with radial polarization using birefringent laser medium,” Jpn. J. Appl. Phys., Part 1 46, 5160–5163 (2007).
[CrossRef]

M. A. Ahmed, A. Voss, M. M. Vogel, T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32, 3272–3274 (2007).
[CrossRef] [PubMed]

2006 (14)

V. G. Niziev, R. S. Chang, A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393–8399 (2006).
[CrossRef] [PubMed]

J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
[CrossRef] [PubMed]

M. R. Beversluis, L. Novotny, S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650–2656 (2006).
[CrossRef] [PubMed]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

K. Yonezawa, Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31, 2151–2153 (2006).
[CrossRef] [PubMed]

T. Hirayama, Y. Kozawa, T. Nakamura, S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express 14, 12839–12845 (2006).
[CrossRef] [PubMed]

W. Chen, Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[CrossRef]

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
[CrossRef]

D. P. Biss, K. S. Youngworth, T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45, 470–479 (2006).
[CrossRef] [PubMed]

A. F. Abouraddy, K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef] [PubMed]

N. Bokor, N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre–Gaussian beam,” Opt. Lett. 31, 149–151 (2006).
[CrossRef] [PubMed]

Q. Zhan, “Properties of circularly polarization vortex beams,” Opt. Lett. 31, 867–869 (2006).
[CrossRef] [PubMed]

J. W. Haus, Z. Mozumder, Q. Zhan, “Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr Medium,” Opt. Express 14, 4757–4764 (2006).
[CrossRef] [PubMed]

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance by radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006).
[CrossRef] [PubMed]

2005 (10)

Y. Q. Zhao, Q. Zhan, Y. L. Zhang, Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005).
[CrossRef] [PubMed]

Q. Cao, J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005).
[CrossRef] [PubMed]

R. Borghi, M. Santarsiero, M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005).
[CrossRef]

T. D. Visser, J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
[CrossRef]

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[CrossRef] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[CrossRef]

B. Jia, X. Gan, M. Gu, “Direct measurement of a radially polarized focused evanescent field facilitated by a single LCD,” Opt. Express 13, 6821–6827 (2005).
[CrossRef] [PubMed]

Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005).
[CrossRef] [PubMed]

2004 (11)

C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. 43, 2397–2399 (2004).
[CrossRef] [PubMed]

G. Volpe, D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[CrossRef]

N. Bokor, N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
[CrossRef] [PubMed]

C. J. R. Sheppard, A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
[CrossRef] [PubMed]

S. F. Pereira, A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[CrossRef]

X. Luo, T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84, 4780–4782 (2004).
[CrossRef]

Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004).
[CrossRef]

P. Török, P. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
[CrossRef]

R. Borghi, M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
[CrossRef]

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

Q. Zhan, “Trapping nanoparticles with cylindrical polarization,” Proc. SPIE 5514, 275–282 (2004).
[CrossRef]

2003 (6)

Q. Zhan, “Optical radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5, 229–232 (2003).
[CrossRef]

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[CrossRef] [PubMed]

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

C. Sun, C. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003).
[CrossRef] [PubMed]

Q. Zhan, “Cylindrical polarization symmetry for nondestructive nano-characterization,” Proc. SPIE 5045, 85–92 (2003).
[CrossRef]

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

2002 (8)

2001 (3)

F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
[CrossRef]

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
[CrossRef]

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

2000 (2)

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

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

1999 (2)

E. M. Furst, A. P. Gast, “Micromechanics of dipolar chains using optical tweezers,” Phys. Rev. Lett. 82, 4130–4133 (1999).
[CrossRef]

V. G. Niziev, A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[CrossRef]

1998 (2)

H. Kano, S. Mizuguchi, S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B 15, 1381–1386 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

1997 (1)

Y. B. Du, P. Tong, “Light scattering properties of paramagnetic particles,” J. Chem. Phys. 107, 355–362 (1997).
[CrossRef]

1996 (2)

1995 (1)

C. Ye, “Construction of an optical rotator using quarter-wave plates and an optical retarder,” Opt. Eng. 34, 3031–3035 (1995).
[CrossRef]

1994 (1)

1992 (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

1990 (1)

1989 (1)

R. Yamaguchi, T. Nose, S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. Part 1 28, 1730–1731 (1989).
[CrossRef]

1987 (1)

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

1981 (1)

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

1972 (2)

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

Y. Mushiake, K. Matzumurra, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[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, 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]

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris. 110, 1251–1253 (1890).

A-Ameur, K.

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy, K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef] [PubMed]

Ahmed, M. A.

Alonso, M. A.

Anderson, E.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Baykal, Y.

Beversluis, M. R.

M. R. Beversluis, L. Novotny, S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650–2656 (2006).
[CrossRef] [PubMed]

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (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, 5251–5254 (2001).
[CrossRef] [PubMed]

Biener, G.

Biss, D. P.

Bisson, J. F.

J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
[CrossRef] [PubMed]

Bokor, N.

Bomzon, Z.

Borghi, R.

R. Borghi, M. Santarsiero, M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005).
[CrossRef]

R. Borghi, M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
[CrossRef]

Bouhelier, A.

Brown, D. E.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Brown, T. G.

Bruyant, A.

Bu, J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

Burge, R. E.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

Cai, Y.

Cao, Q.

Q. Cao, J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005).
[CrossRef] [PubMed]

Carrasco, S.

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
[CrossRef]

Chang, G.

Chang, R. S.

Chen, H.

H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[CrossRef]

Chen, W.

W. Chen, Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE 6450, 64500D (2007).
[CrossRef]

W. Chen, Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007).
[CrossRef] [PubMed]

W. Chen, Q. Zhan, “Field enhancement analysis of an apertureless near field scanning optical microscope probe with finite element method,” Chin. Opt. Lett. 5, 709–711 (2007).

W. Chen, Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[CrossRef]

W. Chen, Q. Zhan, “Realization of evanescent Bessel beam via surface plasmon interference excited by radially polarized beam” Opt. Lett. (to be published).

Cheng, J.

J. Cheng, X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
[CrossRef]

J. Cheng, A. Volkmer, X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002).
[CrossRef]

Cheng, W.

W. Cheng, J. W. Haus, Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE7200 (to be published).

Chipman, R. A.

Chong, C. T.

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

Choudhury, A.

Colas des Francs, G.

Courjon, D.

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[CrossRef]

Davidson, N.

de Saint Denis, R.

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[CrossRef]

Dereux, A.

Ding, J.

Divin, C. J.

Dorn, R.

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

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

Du, Y. B.

Y. B. Du, P. Tong, “Light scattering properties of paramagnetic particles,” J. Chem. Phys. 107, 355–362 (1997).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Eberler, M.

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

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Eyyuboglu, H. T.

Fang, N.

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef] [PubMed]

Feurer, T.

M. Meier, V. Romano, T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[CrossRef]

Foley, J. T.

T. D. Visser, J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
[CrossRef]

Ford, D. H.

Fourkas, J. T.

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
[CrossRef]

Furst, E. M.

E. M. Furst, A. P. Gast, “Micromechanics of dipolar chains using optical tweezers,” Phys. Rev. Lett. 82, 4130–4133 (1999).
[CrossRef]

Galvanauskas, A.

Gan, X.

Garmire, E.

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

Gast, A. P.

E. M. Furst, A. P. Gast, “Micromechanics of dipolar chains using optical tweezers,” Phys. Rev. Lett. 82, 4130–4133 (1999).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Glöckl, O.

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

Gori, F.

F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
[CrossRef]

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

Gouy, L. G.

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris. 110, 1251–1253 (1890).

Graf, T.

Grosjean, T.

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[CrossRef]

Gu, M.

Guattari, G.

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

Guo, C. S.

Hall, D. G.

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
[CrossRef] [PubMed]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Hao, B.

Hasman, E.

Haus, J. W.

J. W. Haus, Z. Mozumder, Q. Zhan, “Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr Medium,” Opt. Express 14, 4757–4764 (2006).
[CrossRef] [PubMed]

W. Cheng, J. W. Haus, Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE7200 (to be published).

Hecht, B.

L. Novotny, B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).
[CrossRef]

Hell, S.

Helseth, L. E.

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
[CrossRef]

Hershcovitz, O.

Hierle, R.

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[CrossRef]

Hiller, J. M.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Hirayama, T.

Hong, M. H.

Hua, J.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Huang, C.

Ignatovich, F.

Ishihara, T.

X. Luo, T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84, 4780–4782 (2004).
[CrossRef]

Jackel, S.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[CrossRef] [PubMed]

Jahns, J.

Q. Cao, J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005).
[CrossRef] [PubMed]

Jia, B.

Kano, H.

Kawata, S.

Kawauchi, H.

Kimball, C. W.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Kimura, W. D.

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Kleiner, V.

Kozawa, Y.

Lai, W. J.

Lee, H.

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef] [PubMed]

Leger, J.

Leger, J. R.

Lerman, G. M.

Leuchs, G.

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

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

Levy, U.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Li, J.

J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
[CrossRef] [PubMed]

Li, Y. P.

H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[CrossRef]

Y. Q. Zhao, Q. Zhan, Y. L. Zhang, Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005).
[CrossRef] [PubMed]

Lim, B. C.

Lin, Q.

Lipson, S. G.

Liu, C.

Liu, C. H.

Liu, Z.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[CrossRef] [PubMed]

Low, D. K. Y.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

Lukyanchuk, B.

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

Lumer, Y.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[CrossRef] [PubMed]

Luo, X.

X. Luo, T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84, 4780–4782 (2004).
[CrossRef]

Machavariani, G.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[CrossRef] [PubMed]

Marhic, M. E.

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

Matzumurra, K.

Y. Mushiake, K. Matzumurra, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
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Appl. Phys. A (1)

M. Meier, V. Romano, T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
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Appl. Opt. (7)

Appl. Phys. Lett. (1)

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J. Opt. A (1)

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J. Cheng, X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
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J. Opt. Soc. Am. A (2)

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

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
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Jpn. J. Appl. Phys., Part 1 (1)

K. Yonezawa, Y. Kozawa, S. Sato, “Compact laser with radial polarization using birefringent laser medium,” Jpn. J. Appl. Phys., Part 1 46, 5160–5163 (2007).
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Nano Lett. (2)

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
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L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
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Nat. Photonics (1)

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
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Nature (1)

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Opt. Commun. (3)

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Opt. Express (2)

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Q. Zhan, “Trapping nanoparticles with cylindrical polarization,” Proc. SPIE 5514, 275–282 (2004).
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Other (10)

W. Chen, Q. Zhan, “Realization of evanescent Bessel beam via surface plasmon interference excited by radially polarized beam” Opt. Lett. (to be published).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and Gratings (Spinger-Verlag, 1988).

W. Cheng, J. W. Haus, Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE7200 (to be published).

M. Gu, ed., Advanced Optical Imaging Theory, Vol. 75 of Springer Series in Optical Sciences (Springer-Verlag, 1999).

For example, the vortex phase plates from RPC Photonics (http://www.rpcphotonics.com/).

For example, the ZPol from Nanophoton (http://www.nanophoton.jp/) and the radial polarizer from ARCoptix (http://www.arcoptix.com/).

For example, J. M. Senior, Optical Fiber Communications (Prentice Hall, 1992).

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

L. Novotny, B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).
[CrossRef]

C.-C. Shih, “Radial polarization laser resonator,” U.S. patent 5,359,622 (Oct. 25, 1994).

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Figures (49)

Fig. 1
Fig. 1

Spatial distribution of instantaneous electric vector field for several conventional modes and CV modes: (a) x-polarized fundamental Gaussian mode; (b) x-polarized HG 10 mode; (c) x-polarized HG 01 mode; (d) y-polarized HG 01 mode; (e) y-polarized HG 01 mode; (f) x-polarized LG 01 mode; (g) radially polarized mode; (h) azimuthally polarized mode; (i) generalized CV beams as a linear superposition of (g) and (h).

Fig. 2
Fig. 2

Formation of radial and azimuthal polarizations using linear superposition of orthogonally polarized HG modes.

Fig. 3
Fig. 3

Diagram of a ruby laser that generates CV beam output. The optical axis of the calcite crystal is parallel to the resonator axis. The combination of the calcite crystal, the telescope, and the aperture and stop provides the mode discrimination to force the laser operation in CV polarization mode. Figure adapted with permission from [1]. © 1972 American Institute of Physics.

Fig. 4
Fig. 4

Nd:YAG laser that generates CV beams by using axial dichroism created by a conical Brewster prism (CBP). The structure of the conical Brewster prism is shown to the right. Figure adapted with permission from [20].

Fig. 5
Fig. 5

Yb:YAG thin-disk laser that generates CV beam outputs using a circular multilayer polarizing grating end mirror. The output beam profile and polarization pattern are shown to the right. Figure adapted with permission from [21].

Fig. 6
Fig. 6

Diagram of a laser cavity design with Sagnac interferometer as an end mirror to create CV beam output. BS, beam splitter. Figure reprinted with permission from [22].

Fig. 7
Fig. 7

Mach–Zehnder interferometer setup to verify the spiral Berry’s phase given in Eq. (3.2). The Berry’s phase is shown in the lower right-hand corner.

Fig. 8
Fig. 8

Generation of CV beam using radial analyzer and SPE. A circularly polarized input is used. The cascaded two λ 2 plates rotate the polarization to the desired pattern.

Fig. 9
Fig. 9

A polarization mode converter setup with two SLMs. The first SLM creates the desired phase pattern. The combination of the λ 4 plate and the second reflection-type SLM converts the beam into the spatially variant polarization pattern. The second SLM can be programed to convert a linearly polarized beam into CV beam. Figure adapted with permission from [32].

Fig. 10
Fig. 10

Segmented spatially variant λ 2 plates that can convert linear polarization into CV polarization. Figure reprinted from [34] with permission from Elsevier.

Fig. 11
Fig. 11

Experimental setup that uses a segmented λ 2 plates polarization converter and a near-confocal Fabry–Perot interferometer (NCFPI) as a mode selector to generate CV beams. OD, optical diode; HWP, half-wave plate; PH, pinhole; TL1, TL2, telescope lenses; PC, polarization converter; FL, focusing lens; CL, collimating lens; M, four mirrors; MD, monitor diode; AS, aperture stop; MO, microscope objective; PD, photodiode. Figure reprinted with permission from [4] (http://prola.aps.org/abstract/PRL/v91/i23/e233901). © 2003 American Physical Society.

Fig. 12
Fig. 12

(a) Geometry of spatially variant subwavelength grating. A scanning electron microscope image of the device is also shown. Experimentally measured spatial polarization patterns for (b) right-hand circular polarization and (c) left-hand circular polarization are illustrated. Figure reprinted with permission from [37].

Fig. 13
Fig. 13

An experimental setup using an EO radial polarization retarder (EO-RPR) to generate CV beams. Figure adapted with permission from [38].

Fig. 14
Fig. 14

A common-path interferometer setup implemented with a SLM to generate CV beams and other more complicated vector beams. Several CV beams with different polarization patterns are also shown to the right. Figure reprinted with permission from [41].

Fig. 15
Fig. 15

Illustrations of the polarization patterns for LP 01 and LP 11 modes of optic fiber. It can be seen that the TE 01 and TM 01 modes from the second-higher-order mode group have azimuthal and radial polarization symmetry, respectively.

Fig. 16
Fig. 16

Laboratory setup that generates CV beams by using few-mode optic fiber. The intensity patterns of radial polarization generated with this setup after passing through a linear polarizer are shown to the right.

Fig. 17
Fig. 17

Diagram of a nonmechanical polarization rotator using LC or EO retarders sandwiched between two orthogonally oriented ( λ 4 plates. The Jones matrix of such a device is shown in the figure. It can be seen that such an arrangement provides polarization rotation that is independent of the incident polarization state.

Fig. 18
Fig. 18

Illustration of the polarization pattern at the pupil plane for the radiation from a vertical electric dipole collected by a high-NA objective. If the optical path is reversed, a radial polarization input should recover the propagating components of the vertical dipole radiation and produce a very tight focus.

Fig. 19
Fig. 19

Focusing of a CV beam. f is the focal length of the objective lens. Q ( r , ϕ ) is an observation point in the focal plane.

Fig. 20
Fig. 20

Focal field calculation examples for CV polarization focused by a high-NA ( NA = 0.8 ) objective lens. (a)–(c) 2D plots of the longitudinal component, the radial component, and the total field strength in the rz plane for incident radial polarization. (d) Line scan across the focus (dashed lines). There is no azimuthal component for this situation. A strong and narrower longitudinal component is obtained. (e)–(f) Results for azimuthal polarization. Only azimuthal component exists for this case.

Fig. 21
Fig. 21

Projections of a focal field experimentally measured by using a focal-plane knife-edge technique for a highly focused (a) radial and (b) azimuthal polarization. Actual focal intensity profiles for (c) radial and (d) azimuthal polarizations were obtained by using tomographic reconstruction. A focal spot as small as 0.16 λ 2 has been obtained for radial polarization with an annular aperture. Figures reprinted with permission from [4] (http://prola.aps.org/abstract/PRL/v91/i23/e233901). © 2003 American Physical Society.

Fig. 22
Fig. 22

Molecule orientation imaging by using radial polarization. The output of an argon ion laser is converted into radial polarization through a mode converter. The radially polarized beam is focused by a high-NA lens ( NA = 1.4 ) on to a polymethyl methacrylate (PMMA) thin-film sample imbedded with fluorescent dye. The upper right-hand corner shows the predicted fluorescent emission rate patterns corresponding to different molecule orientations. Corresponding experimental results are shown in the lower right-hand corner. Figures reprinted with permission from [60] (http://prola.aps.org/abstract/PRL/v86/i23/p5251_1). © 2001 American Physical Society.

Fig. 23
Fig. 23

Flattop focusing with generalized CV polarization. The double- λ 2 -plate polarization rotator converts the incident CV beam into generalized CV polarization with the desired pattern for flattop profile creation at the focal plane.

Fig. 24
Fig. 24

Example of the flattop focusing generated with the setup shown in Fig. 23. Line scans of the field distribution across the focal plane are shown to the left. A 2D distribution in the rz plane in the vicinity of the focus is shown to the right.

Fig. 25
Fig. 25

3D focus shaping using generalized CV polarization and pupil plane binary DOE. The pupil plane DOE consists of several concentric rings that give either 0 or π phase transmission for normal incidence. The transition points are specified with the corresponding NA in the image space.

Fig. 26
Fig. 26

Example of 3D flattop focus generated with CV beam. (a) 3D plot of the E 2 distribution in the vicinity of the focus. (b) Line scan of (a) along the z axis. A line scan for linear polarization is also shown for comparison. (c) Line scan in the focal plane.

Fig. 27
Fig. 27

Optical bubble created by radial polarization with DOE pupil plane modulation.

Fig. 28
Fig. 28

Generation of optical needle using radial polarization with pupil plane DOE modulation. The total field and the constituting radial and longitudinal components are shown in the column to the right. The electric vector distribution is shown in the plot at the left-hand bottom. It can be seen that the field near the optical axis is almost purely polarized in the longitudinal direction. Figure adapted and reprinted by permission from Macmillan Publishers Ltd, Nature Photonics [64], © 2008.

Fig. 29
Fig. 29

Geometry for calculation of effective phase retardation of tilted wave.

Fig. 30
Fig. 30

Setup for the Gouy phase shift calculation with uniform annular illumination. An x-polarized collimated beam uniformly illuminates the entrance pupil of an aplanatic objective lens. A circular blocking mask is placed at the entrance pupil of the lens to generate annular illumination. The illumination angles of the innermost and outermost rays are θ 1 and θ 2 , respectively. Q ( r , ϕ ) is an observation point in the focal plane, and f is the focal length of the objective. The geometry for the focal field length calculation is also shown. Δ L is the optical path difference between the innermost ray and the outermost ray.

Fig. 31
Fig. 31

Calculations of the Gouy phase shift under uniform annular illumination with NA 2 = sin θ 2 = 0.95 .

Fig. 32
Fig. 32

Vectorial diffraction theory simulation of a tightly focused beam with a Gouy shift of 2 π . Solid curve, axial intensity distribution; dotted curve, Gouy phase shift. A uniform annular illumination with NA 1 = sin θ 1 = 0.5 and NA 2 = sin θ 2 = 0.95 is used in the simulation. Only the positive side of the optical axis is shown.

Fig. 33
Fig. 33

Calculated axial intensity distribution I z (black solid curve), electric field component E z (blue dotted curve) and the Gouy phase shift (red dashed curve) for uniform radial polarization illumination with NA = 0.5 using the vectorial diffraction method. The space between the vertical dashed–dotted lines is wavefront spacing [67].

Fig. 34
Fig. 34

Comparison of the differential Gouy phase shift at the focal plane ( z = 0 ) calculated by both vectorial diffraction simulation and the analytical expression. In this case, uniform illumination is used with variable sin θ 2 = NA and θ 1 fixed at 0 [67].

Fig. 35
Fig. 35

Diagram of SPR excitation with highly focused radial polarization. The entire beam is p polarized with respect to the silver/dielectric interface. A pupil plane image of the reflected beam is shown to the right. A full dark ring is observed that is due to the SPR excitation by radial polarization. The nonuniformity in the beam profile is caused by the difference in reflection coefficients for the s- and p-polarization components of a beam splitter that steers the reflected beam to the CCD camera.

Fig. 36
Fig. 36

Experimental setup for the generation and detection of plasmonic focusing with highly focused radial polarization. PMT, photomultiplier tube; PM, photomask. A Veeco Aurora 3 NSOM with a nanoaperture fiber probe is used to detect the plasmonic field. The insets show the calculated plasmon focal fields at the metal surface for radial and linear polarization incidence, respectively. For linear polarization, because of the mismatch of polarization and the destructive interference of counterpropagating plasmon waves, the field at the focal plane splits into two lobes with much lower peak intensity.

Fig. 37
Fig. 37

Calculated surface plasmon intensity distribution at silver/air interface with radially polarized illumination. (a) Total intensity E 2 ; a homogeneous spot with a strongly enhanced field at the center is obtained. (b) Longitudinal component E z 2 , which is much stronger than E r 2 and dominates the total field distribution. (c) Radial component E r 2 , which has a donut shape. (d) E z 2 distribution, which is proportional to the NSOM signal detected by an apertured fiber probe.

Fig. 38
Fig. 38

Experimental confirmation of evanescent Bessel beam creation via surface plasmon excitation. (a) Logarithmic plot of measured NSOM signal in the near field. (b) Comparison of the measured and calculated transverse profiles of the NSOM signal ( E z 2 ) in the near field. (c) Measurements at different planes to reveal the nondiffracting nature of the evanescent Bessel beam. (d) Exponential decay of the evanescent Bessel beam [82].

Fig. 39
Fig. 39

Plasmonic lens structure with multiple concentric rings illuminated by radial polarization. Surface plasmons generated by each of the rings propagate to the center and constructively interfere to produce a tight plasmonic focus (shown to the right).

Fig. 40
Fig. 40

Conical glass tip fully coated with thin silver film. The entire radial polarization input is p polarized with respect to the glass/silver interface and creates surface plasmons that propagate to the tip apex.

Fig. 41
Fig. 41

Logarithmic plots of (a) 2D and (b) 3D electric energy density distribution at the end of the probe tip under radial polarization illumination.

Fig. 42
Fig. 42

(a) Calculated E 2 distribution at the focal plane for a radially polarized beam focused by a high-NA objective. For comparison, the corresponding result from a linearly polarized beam incidence under the same focusing condition is also shown. (b) 2D plot of the time-averaged Poynting vector S z in the r z plane for highly focused radial polarization. (c) Line scan of (b) at the focal plane.

Fig. 43
Fig. 43

Generation of optical chain. (a) Incident radial polarization. (b) Pupil plane DOE that modulates both phase and amplitude of the incident radial polarization. (c) One example of optical chain created with the pupil plane DOE shown in (b). (d) Time-averaged Poynting vector S z [89].

Fig. 44
Fig. 44

Field distribution shifts as the phase difference Δ φ between zone I and zone III varies from 0 to 2 π [89].

Fig. 45
Fig. 45

Magnetic field and electric field at the focal plane for a highly focused ( NA = 0.95 ) azimuthal polarization.

Fig. 46
Fig. 46

Calculated absorption coefficients for s polarization and p polarization of mild steel [96].

Fig. 47
Fig. 47

Expected laser machining cross section for radial and azimuthal polarizations [96].

Fig. 48
Fig. 48

Calculated hole cross sections and corresponding absorbed laser fluence for mild steel with azimuthal and radial polarizations. The guiding effect is taken into consideration. From the simulations, it can be seen that azimuthal polarization is more energy efficient in this case. Figure reprinted from [96] with kind permission of Springer Science+Business Media.

Fig. 49
Fig. 49

Calculated hole cross sections and corresponding absorbed laser fluence for brass with azimuthal and radial polarizations. The guiding effect is taken into consideration. From the simulations, it can be seen that radial polarization is more energy efficient in this case. Figure reprinted from [96] with kind permission of Springer Science+Business Media.

Equations (88)

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( 2 + k 2 ) E = 0 ,
E ( x , y , z , t ) = u ( x , y , z ) exp [ i ( k z ω t ) ] .
2 u z 2 k 2 u , 2 u z 2 k u z ,
u ( x , y , z ) = E 0 H m ( 2 x w ( z ) ) H n ( 2 y w ( z ) ) w 0 w ( z ) exp [ i φ m n ( z ) ] exp [ i k 2 q ( z ) r 2 ] ,
d 2 H m d x 2 2 x d H m d x + 2 m H m = 0
u ( r , z ) = E 0 w 0 w ( z ) exp [ i φ ( z ) ] exp [ i k 2 q ( z ) r 2 ] ,
E ( r , ϕ , z , t ) = u ( r , ϕ , z ) exp [ i ( k z ω t ) ] .
1 r r ( r u r ) + 1 r 2 2 u ϕ 2 + 2 i k u z = 0 .
u ( r , ϕ , z ) = E 0 ( 2 r ω ) l L p l ( 2 r 2 ω 2 ) w 0 w ( z ) exp [ i φ p l ( z ) ] exp [ i k 2 q ( z ) r 2 ] exp ( i l ϕ ) ,
x d 2 L p l d x 2 ( l + 1 x ) d L p l d x + p L p l = 0
u ( r , z ) = E 0 w 0 w ( z ) exp [ i φ ( z ) ] exp [ i k 2 q ( z ) r 2 ] J 0 ( β r 1 + i z z 0 ) exp [ β 2 z ( 2 k ) 1 + i z z 0 ] ,
× × E k 2 E = 0 ,
E ( r , z ) = U ( r , z ) exp [ i ( k z ω t ) ] e ϕ ,
1 r r ( r U r ) U r 2 + 2 i k U z = 0 .
U ( r , z ) = E 0 J 1 ( β r 1 + i z z 0 ) exp [ i β 2 z ( 2 k ) 1 + i z z 0 ] u ( r , z ) ,
H ( r , z ) = H 0 J 1 ( β r 1 + i z z 0 ) exp [ i β 2 z ( 2 k ) 1 + i z z 0 ] u ( r , z ) exp [ i ( k z ω t ) ] h ϕ ,
E ( r , z ) = A r exp ( r 2 w 2 ) e i , i = r , ϕ .
E r = H G 10 e x + H G 01 e y ,
E ϕ = H G 01 e x + H G 10 e y ,
E ( r ) = P ( r ) e i , i = r , ϕ ,
P ( r ) = { 1 , r 1 < r < r 2 0 , 0 < r < r 1 } .
E in = e x + j e y = ( cos ϕ e r sin ϕ e ϕ ) + j ( sin ϕ e r + cos ϕ e ϕ ) = e j ϕ ( e r + j e ϕ ) ,
E out = e j ϕ e r .
T = ( cos ( 2 Δ φ ) sin ( 2 Δ φ ) sin ( 2 Δ φ ) cos ( 2 Δ φ ) ) = R ( 2 Δ φ ) ,
T = R ( π 2 ) ( 1 0 0 j ) R ( π 2 ) R ( π 4 ) ( 1 0 0 e j δ ) R ( π 4 ) ( 1 0 0 j ) = j e j δ 2 R ( δ 2 ) ,
E i ( ρ , φ ) = l 0 P ( ρ ) [ cos φ 0 e ρ + sin φ 0 e φ ] ,
e ρ = cos φ e x + sin φ e y ,
e φ = sin φ e x + cos φ e y .
ρ f = g ( θ ) ,
[ l 0 P ( ρ ) ] 2 2 π ρ d ρ = [ l 0 P ( θ ) ] 2 2 π f 2 sin θ d θ ,
P ( θ ) = P ( ρ ) g ( θ ) g ( θ ) sin θ = P ( f g ( θ ) ) g ( θ ) g ( θ ) sin θ .
ρ f = sin θ .
P ( θ ) = P ( f sin θ ) cos θ .
e r = cos θ ( cos φ e x + sin φ e y ) + sin θ e z ,
e φ = e φ = sin φ e x + cos φ e y .
E ( r , ϕ , z ) = i k 2 π Ω a ( θ , φ ) e i k ( s r ) d Ω = i k 2 π 0 θ max d θ 0 2 π a ( θ , φ ) e i k ( s r ) sin θ d φ ,
a ( θ , φ ) = l 0 f P ( θ ) [ cos φ 0 e r + sin φ 0 e φ ] .
s r = z cos θ + r sin θ cos ( φ ϕ ) .
E ( r , ϕ , z ) = i k 2 π 0 θ max d θ 0 2 π l 0 f P ( θ ) [ cos φ 0 e r + sin φ 0 e φ ] e i k ( z cos θ + r sin θ cos ( φ ϕ ) ) sin θ d φ = i A π 0 θ max d θ 0 2 π P ( θ ) [ cos φ 0 e r + sin φ 0 e φ ] e i k ( z cos θ + r sin θ cos ( φ ϕ ) ) sin θ d φ = i A π 0 θ max d θ 0 2 π P ( θ ) [ cos φ 0 ( cos θ cos φ e x cos θ sin φ e y sin θ e z ) + sin φ 0 ( sin φ e x cos φ e y 0 e z ) ] e i k ( z cos θ + r sin θ cos ( φ ϕ ) ) sin θ d φ ,
A = π f l 0 λ .
e r = cos ϕ e x + sin ϕ e y ,
e ϕ = sin ϕ e x + cos ϕ e y .
E ( r , ϕ , z ) = i A π 0 θ max d θ 0 2 π P ( θ ) [ cos φ 0 ( cos θ cos ( φ ϕ ) e r 0 e ϕ sin θ e z ) + sin φ 0 ( 0 e r cos ( φ ϕ ) e ϕ 0 e z ) ] e i k ( z cos θ + r sin θ cos ( φ ϕ ) ) sin θ d φ .
0 2 π cos ( n φ ) e i k r sin θ cos φ d φ = 2 π i n J n ( k r sin θ ) ,
E ( r , ϕ , z ) = E r e r + E z e z + E ϕ e ϕ ,
E r ( r , ϕ , z ) = 2 A cos φ 0 0 θ max P ( θ ) sin θ cos θ J 1 ( k r sin θ ) e i k z cos θ d θ ,
E z ( r , ϕ , z ) = i 2 A cos φ 0 0 θ max P ( θ ) sin 2 θ J 0 ( k r sin θ ) e i k z cos θ d θ ,
E ϕ ( r , ϕ , z ) = 2 A sin φ 0 0 θ max P ( θ ) sin θ J 1 ( k r sin θ ) e i k z cos θ d θ .
g ( θ ) = 2 sin ( θ 2 ) ,
P ( θ ) = P ( f g ( θ ) ) .
g ( θ ) = θ ,
P ( θ ) = P ( f g ( θ ) ) θ sin θ .
g ( θ ) = tan θ ,
P ( θ ) = P ( f g ( θ ) ) ( 1 cos θ ) 3 .
E ( r , ϕ , z ) = i A { [ I 0 + cos ( 2 ϕ ) I 2 ] e x + sin ( 2 ϕ ) I 2 e y 2 i cos ϕ I 1 e z } ,
I 0 = 0 θ max P ( θ ) sin θ ( 1 + cos θ ) J 0 ( k r sin θ ) e i k z cos θ d θ ,
I 1 = 0 θ max P ( θ ) sin 2 θ J 1 ( k r sin θ ) e i k z cos θ d θ ,
I 2 = 0 θ max P ( θ ) sin θ ( 1 cos θ ) J 2 ( k r sin θ ) e i k z cos θ d θ .
T ( θ ) = { 1 for 0 θ < θ 1 , θ 2 θ < θ 3 , θ 4 θ < sin 1 ( NA ) 1 for θ 1 θ < θ 2 , θ 3 θ < θ 4 } ,
E r ( r , ϕ , z ) = 2 A cos φ 0 0 θ max P ( θ ) T ( θ ) sin θ cos θ J 1 ( k r sin θ ) e i k z cos θ d θ ,
E z ( r , ϕ , z ) = i 2 A cos φ 0 0 θ max P ( θ ) T ( θ ) sin 2 θ J 0 ( k r sin θ ) e i k z cos θ d θ .
d φ ( z ) = ( k eff k 0 ) d z = ( k 0 2 k 2 k 0 ) d z ,
d φ G ( z ) d z = + + F ( k x , k y , z ) 2 ( k 0 2 k 2 k 0 ) d k x d k y .
d φ G ( z ) d z = 0 k max 0 2 π F ( k , z ) 2 ( k 0 2 k 2 k 0 ) k d k d ϕ .
φ G = + d φ G ( z ) d z d z = + 0 k max 0 2 π F ( k , z ) 2 ( k 0 2 k 2 k 0 ) k d k d ϕ d z .
k 0 2 k 2 k 0 k 2 2 k 0 .
k 0 2 k 2 k 0 k 0 { 1 1 2 [ ( k k 0 ) 2 ] 1 8 [ ( k k 0 ) 2 ] 2 } k 0 = k 2 2 k 0 k 4 8 k 0 3 .
E ( r , ϕ , z ) = i A π θ 1 θ 2 d θ 0 2 π { [ cos θ + sin 2 ϕ ( 1 cos θ ) cos θ e x sin ϕ cos ϕ ( 1 cos θ ) cos θ e y sin θ cos ϕ cos θ e z ] e i k 0 cos θ z e i ( k x x + k y y ) sin θ cos θ d ϕ } ,
F ( k , z = 0 ) = [ cos θ + sin 2 ϕ ( 1 cos θ ) cos θ ] N ,
N = θ 1 θ 2 d θ 0 2 π cos θ + sin 2 ϕ ( 1 cos θ ) cos θ 2 k 0 2 sin θ cos θ d ϕ = π k 0 2 [ cos 2 θ 1 cos 2 θ 2 ] + π k 0 2 4 [ ( 1 cos θ 2 ) 3 ( 1 cos θ 1 ) 3 ] .
d φ G ( 0 ) d z = k 2 2 k 0 k 4 8 k 0 3 = k x 2 + k y 2 2 k 0 k x 4 + 2 k x 2 k y 2 + k y 4 8 k 0 3 .
φ G = d φ G ( 0 ) d z Δ z = [ k x 2 + k y 2 2 k 0 + k x 4 + 2 k x 2 k y 2 + k y 4 8 k 0 3 ] Δ z ,
Δ z = 2 λ cos θ 1 cos θ 2 λ 2 .
E z ( r , Φ , z ) = i A π θ 1 θ 2 d θ 0 2 π sin θ cos θ e i k 0 z cos θ e i ( k x x + k y y ) sin θ cos θ d ϕ .
F ( k , z = 0 ) = [ sin θ cos θ ] N ,
N = A θ 1 θ 2 d θ 0 2 π sin θ cos θ 2 k 0 2 sin θ cos θ d ϕ = 2 π k 0 2 { 1 3 ( cos θ 1 3 cos θ 2 3 ) + ( cos θ 1 cos θ 2 ) } .
d φ G ( 0 ) d z = 0 k max 0 2 π F ( k x , k y , 0 ) 2 ( k 0 2 k 2 k 0 ) k d k d ϕ = 2 π k 0 3 ( sin 4 θ 2 sin 4 θ 1 ) ( 4 N ) k 0 .
k eff Δ z k 0 Δ z = d φ G ( z ) d z Δ z ,
λ eff = λ 1 + d φ G ( z ) d z λ 2 π .
E r ( r , ϕ , z ) = 2 A θ min θ max P ( θ ) t p ( θ ) sin θ cos θ J 1 ( k 1 r sin θ ) exp [ i z ( k 2 2 k 1 2 sin 2 θ ) 1 2 ] d θ ,
E z ( r , ϕ , z ) = i 2 A θ min θ max P ( θ ) t p ( θ ) sin 2 θ J 0 ( k 1 r sin θ ) exp [ i z ( k 2 2 k 1 2 sin 2 θ ) 1 2 ] d θ ,
F grad = Re ( α ) ɛ 0 E 2 4 ,
F scat , z = n 1 S z C scat c ,
F abs , z = n 1 S z C abs c ,
P ( θ ) = { P 0 if sin 1 ( NA 1 ) θ sin 1 ( NA n 1 ) 0 otherwise } ,
E ( z ) 2 = [ E I ( z ) + E III ( z ) ] [ E I * ( z ) + E III * ( z ) ] = E I ( z ) 2 + E III ( z ) 2 + 2 E I ( z ) 2 E III ( z ) 2 cos δ = E I ( z ) 2 + E III ( z ) 2 + 2 E I ( z ) 2 E III ( z ) 2 cos ( Δ k z z + Δ ϕ ) ,
p m c , E c B , c B E ,
F = ( α m 2 ) B 2 + ω α m ( E × B ) c 2 ,

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