Abstract

Tight focusing properties of a circular partially coherent Gaussian (CPCG) beam with linear polarization have been studied based on vectorial Debye theory. Expressions for the intensity distribution and degree of coherence near the focus are derived. Numerical calculations are performed to show the intensity distribution and degree of coherence of the CPCG beam in the focal region. It is interesting to find that after focusing the CPCG beam through a high numerical-aperture objective we can obtain a super-length optical needle (>12λ) with homogeneous intensity along the propagation axis and wavelength beam size (λ). Moreover, the numerical calculations of coherence illustrate that, in the range of full width at half-maximum of the optical needle, for any two of the parallel electric field components of the optical needle the coherence is close to 1, but for any two of orthometric electric field components the value of coherence is between 0.4 and 0.9. Such a non-diffracting optical needle may have potential applications in atom optical experiments, such as in atom traps and atom switches.

© 2018 Optical Society of America

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References

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2017 (7)

2016 (2)

Z. S. Man, C. J. Min, L. P. Du, Y. Q. Zhang, S. W. Zhu, and X. C. Yuan, “Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes,” Opt. Express 24, 874–882 (2016).
[Crossref]

C. M. Sundaram, K. Prabakaran, P. M. Anbarasan, K. B. Rajesh, and A. M. Musthafa, “Creation of super long transversely polarized optical needle using azimuthally polarized multi Gaussian beam,” Chin. Phys. Lett. 33, 64203–64206 (2016).
[Crossref]

2015 (1)

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9, 231–240 (2015).
[Crossref]

2014 (2)

J. W. Charles, K. Prabakaran, K. B. Rajesh, and H. M. Pandya, “Generation of sub wavelength super long dark channel using azimuthally polarized annular multi-Gaussian beam,” Opt. Quantum Electron. 46, 1079–1086 (2014).
[Crossref]

X. Y. Weng, X. M. Gao, H. M. Guo, and S. L. Zhuang, “Creation of tunable multiple 3D dark spots with cylindrical vector beam,” Appl. Opt. 53, 2470–2476 (2014).
[Crossref]

2013 (3)

B. Gu, J. L. Wu, Y. Pan, and Y. P. Cui, “Achievement of needle-like focus by engineering radial-variant vector fields,” Opt. Express 21, 30444–30452 (2013).
[Crossref]

T. Liu, J. B. Tan, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[Crossref]

L. Z. Rao, H. C. Lin, and Q. Q. Sun, “Spatial correlation properties of tightly focused J0-correlated azimuthally polarized vortex Beams,” Chin. Phys. Lett. 30, 054211 (2013).
[Crossref]

2012 (7)

Z. Y. Chen, L. M. Hua, and J. X. Pu, “Tight focusing of light beams: effect of polarization, phase, and coherence,” Prog. Opt. 57, 219–260 (2012).
[Crossref]

Y. M. Dong, F. Wang, C. L. Zhao, and Y. J. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[Crossref]

J. X. Li, Y. I. Salamin, B. J. Galow, and C. H. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85, 063832 (2012).
[Crossref]

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Z. Y. Chen and D. M. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012).
[Crossref]

S. N. Khonina and I. Golub, “Enlightening darkness to diffraction limit and beyond: comparison and optimization of different polarizations for dark spot generation,” J. Opt. Soc. Am. A 29, 1470–1474 (2012).
[Crossref]

K. L. Hu, Z. Y. Chen, and J. X. Pu, “Generation of super-length optical needle by focusing hybridly polarized vector beams through a dielectric interface,” Opt. Lett. 37, 3303–3305 (2012).
[Crossref]

2011 (2)

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36, 4104–4106 (2011).
[Crossref]

L. N. Guo, Z. L. Tang, C. Q. Liang, and Z. L. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43, 895–898 (2011).
[Crossref]

2010 (1)

2009 (2)

2008 (2)

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

Z. M. Zhang, J. X. Pu, and X. Q. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[Crossref]

2007 (1)

2005 (1)

2004 (1)

2002 (1)

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[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. London Ser. A 253, 358–379 (1959).
[Crossref]

Anbarasan, P. M.

C. M. Sundaram, K. Prabakaran, P. M. Anbarasan, K. B. Rajesh, and A. M. Musthafa, “Creation of super long transversely polarized optical needle using azimuthally polarized multi Gaussian beam,” Chin. Phys. Lett. 33, 64203–64206 (2016).
[Crossref]

Banzer, P.

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9, 231–240 (2015).
[Crossref]

Beversluis, M. R.

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

Brown, T. G.

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

Cai, Y. J.

Y. J. Cai, Y. H. Chen, J. Y. Yu, X. L. Liu, and L. Liu, “Chapter three-generation of partially coherent beams,” Prog. Opt 62, 157–223 (2017).
[Crossref]

C. Ping, C. H. Liang, F. Wang, and Y. J. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25, 32475–32490 (2017).
[Crossref]

Y. M. Dong, F. Wang, C. L. Zhao, and Y. J. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[Crossref]

Charles, J. W.

J. W. Charles, K. Prabakaran, K. B. Rajesh, and H. M. Pandya, “Generation of sub wavelength super long dark channel using azimuthally polarized annular multi-Gaussian beam,” Opt. Quantum Electron. 46, 1079–1086 (2014).
[Crossref]

Chen, W. B.

Chen, Y. H.

Y. J. Cai, Y. H. Chen, J. Y. Yu, X. L. Liu, and L. Liu, “Chapter three-generation of partially coherent beams,” Prog. Opt 62, 157–223 (2017).
[Crossref]

Chen, Z. Y.

Chong, C. T.

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

Cui, Y. P.

de Sande, J. C. G.

Ding, C. L.

Dong, Y. M.

Y. M. Dong, F. Wang, C. L. Zhao, and Y. J. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[Crossref]

Du, L. P.

Foreman, M. R.

Fourmaux, S.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Friberg, A. T.

Galow, B. J.

J. X. Li, Y. I. Salamin, B. J. Galow, and C. H. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85, 063832 (2012).
[Crossref]

Gao, X. M.

Golub, I.

Gong, L. P.

Gori, F.

Gu, B.

Gu, M.

M. Gu, Advanced Optical Imaging Theory (Springer, 1999).

Guo, H. M.

Guo, L. N.

L. N. Guo, Z. L. Tang, C. Q. Liang, and Z. L. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43, 895–898 (2011).
[Crossref]

Hu, K. L.

Hua, L. M.

Z. Y. Chen, L. M. Hua, and J. X. Pu, “Tight focusing of light beams: effect of polarization, phase, and coherence,” Prog. Opt. 57, 219–260 (2012).
[Crossref]

Kaivola, M.

Karthik, V.

K. Prabakaran, P. Sangeetha, V. Karthik, K. B. Rajesh, and A. M. Musthafa, “Tight focusing properties of phase modulated radially polarized Laguerre Bessel Gaussian Beam,” Chin. Phys. Lett. 34, 054203 (2017).
[Crossref]

Keitel, C. H.

J. X. Li, Y. I. Salamin, B. J. Galow, and C. H. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85, 063832 (2012).
[Crossref]

Khonina, S. N.

Kieffer, J. C.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Koivurova, M.

Lajunen, H.

Légaré, F.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Leger, J. R.

Leuchs, G.

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9, 231–240 (2015).
[Crossref]

Li, J. X.

J. X. Li, Y. I. Salamin, B. J. Galow, and C. H. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85, 063832 (2012).
[Crossref]

Liang, C. H.

Liang, C. Q.

L. N. Guo, Z. L. Tang, C. Q. Liang, and Z. L. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43, 895–898 (2011).
[Crossref]

Lin, H. C.

L. Z. Rao, H. C. Lin, and Q. Q. Sun, “Spatial correlation properties of tightly focused J0-correlated azimuthally polarized vortex Beams,” Chin. Phys. Lett. 30, 054211 (2013).
[Crossref]

Lin, J.

T. Liu, J. B. Tan, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[Crossref]

Lindfors, K.

Liu, L.

Y. J. Cai, Y. H. Chen, J. Y. Yu, X. L. Liu, and L. Liu, “Chapter three-generation of partially coherent beams,” Prog. Opt 62, 157–223 (2017).
[Crossref]

Liu, T.

T. Liu, J. B. Tan, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[Crossref]

Liu, X. L.

Y. J. Cai, Y. H. Chen, J. Y. Yu, X. L. Liu, and L. Liu, “Chapter three-generation of partially coherent beams,” Prog. Opt 62, 157–223 (2017).
[Crossref]

Lukyanchuk, B.

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

MacLean, J. P.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Maluenda, D.

Man, Z. S.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martínez-Herrero, R.

Min, C. J.

Musthafa, A. M.

K. Prabakaran, P. Sangeetha, V. Karthik, K. B. Rajesh, and A. M. Musthafa, “Tight focusing properties of phase modulated radially polarized Laguerre Bessel Gaussian Beam,” Chin. Phys. Lett. 34, 054203 (2017).
[Crossref]

C. M. Sundaram, K. Prabakaran, P. M. Anbarasan, K. B. Rajesh, and A. M. Musthafa, “Creation of super long transversely polarized optical needle using azimuthally polarized multi Gaussian beam,” Chin. Phys. Lett. 33, 64203–64206 (2016).
[Crossref]

Novotny, L.

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

Pan, L. Z.

Pan, Y.

Pandya, H. M.

J. W. Charles, K. Prabakaran, K. B. Rajesh, and H. M. Pandya, “Generation of sub wavelength super long dark channel using azimuthally polarized annular multi-Gaussian beam,” Opt. Quantum Electron. 46, 1079–1086 (2014).
[Crossref]

Payeur, S.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Piché, M.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Ping, C.

Piquero, G.

Prabakaran, K.

K. Prabakaran, P. Sangeetha, V. Karthik, K. B. Rajesh, and A. M. Musthafa, “Tight focusing properties of phase modulated radially polarized Laguerre Bessel Gaussian Beam,” Chin. Phys. Lett. 34, 054203 (2017).
[Crossref]

C. M. Sundaram, K. Prabakaran, P. M. Anbarasan, K. B. Rajesh, and A. M. Musthafa, “Creation of super long transversely polarized optical needle using azimuthally polarized multi Gaussian beam,” Chin. Phys. Lett. 33, 64203–64206 (2016).
[Crossref]

J. W. Charles, K. Prabakaran, K. B. Rajesh, and H. M. Pandya, “Generation of sub wavelength super long dark channel using azimuthally polarized annular multi-Gaussian beam,” Opt. Quantum Electron. 46, 1079–1086 (2014).
[Crossref]

Pu, J. X.

Rajesh, K. B.

K. Prabakaran, P. Sangeetha, V. Karthik, K. B. Rajesh, and A. M. Musthafa, “Tight focusing properties of phase modulated radially polarized Laguerre Bessel Gaussian Beam,” Chin. Phys. Lett. 34, 054203 (2017).
[Crossref]

C. M. Sundaram, K. Prabakaran, P. M. Anbarasan, K. B. Rajesh, and A. M. Musthafa, “Creation of super long transversely polarized optical needle using azimuthally polarized multi Gaussian beam,” Chin. Phys. Lett. 33, 64203–64206 (2016).
[Crossref]

J. W. Charles, K. Prabakaran, K. B. Rajesh, and H. M. Pandya, “Generation of sub wavelength super long dark channel using azimuthally polarized annular multi-Gaussian beam,” Opt. Quantum Electron. 46, 1079–1086 (2014).
[Crossref]

Rao, L. Z.

L. Z. Rao, H. C. Lin, and Q. Q. Sun, “Spatial correlation properties of tightly focused J0-correlated azimuthally polarized vortex Beams,” Chin. Phys. Lett. 30, 054211 (2013).
[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]

Saastamoinen, T.

Salamin, Y. I.

J. X. Li, Y. I. Salamin, B. J. Galow, and C. H. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85, 063832 (2012).
[Crossref]

Sangeetha, P.

K. Prabakaran, P. Sangeetha, V. Karthik, K. B. Rajesh, and A. M. Musthafa, “Tight focusing properties of phase modulated radially polarized Laguerre Bessel Gaussian Beam,” Chin. Phys. Lett. 34, 054203 (2017).
[Crossref]

Santarsiero, M.

Schmidt, B. E.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Setälä, T.

Sheppard, C.

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

Shi, L. P.

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

Sun, Q. Q.

L. Z. Rao, H. C. Lin, and Q. Q. Sun, “Spatial correlation properties of tightly focused J0-correlated azimuthally polarized vortex Beams,” Chin. Phys. Lett. 30, 054211 (2013).
[Crossref]

Sundaram, C. M.

C. M. Sundaram, K. Prabakaran, P. M. Anbarasan, K. B. Rajesh, and A. M. Musthafa, “Creation of super long transversely polarized optical needle using azimuthally polarized multi Gaussian beam,” Chin. Phys. Lett. 33, 64203–64206 (2016).
[Crossref]

Tan, J. B.

T. Liu, J. B. Tan, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[Crossref]

Tan, Z. L.

L. N. Guo, Z. L. Tang, C. Q. Liang, and Z. L. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43, 895–898 (2011).
[Crossref]

Tang, Z. L.

L. N. Guo, Z. L. Tang, C. Q. Liang, and Z. L. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43, 895–898 (2011).
[Crossref]

Tchervenkov, C.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Török, P.

Turunen, J.

Wang, F.

C. Ping, C. H. Liang, F. Wang, and Y. J. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25, 32475–32490 (2017).
[Crossref]

Y. M. Dong, F. Wang, C. L. Zhao, and Y. J. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[Crossref]

Wang, H. F.

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

Wang, J. M.

Wang, X. L.

Wang, X. Q.

Weng, X. Y.

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. London Ser. A 253, 358–379 (1959).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wozniak, P.

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9, 231–240 (2015).
[Crossref]

Wu, J. L.

Youngworth, K. S.

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

Yu, J. Y.

Y. J. Cai, Y. H. Chen, J. Y. Yu, X. L. Liu, and L. Liu, “Chapter three-generation of partially coherent beams,” Prog. Opt 62, 157–223 (2017).
[Crossref]

Yuan, X. C.

Zhan, Q. W.

Zhang, Y. Q.

Zhang, Z. M.

Zhao, C. L.

Y. M. Dong, F. Wang, C. L. Zhao, and Y. J. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[Crossref]

Zhao, D. M.

Zhu, S. W.

Zhu, Z. Q.

Zhuang, S. L.

Adv. Opt. Photonics (1)

Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1, 1–57 (2009).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101, 041105 (2012).
[Crossref]

Chin. Phys. Lett. (3)

C. M. Sundaram, K. Prabakaran, P. M. Anbarasan, K. B. Rajesh, and A. M. Musthafa, “Creation of super long transversely polarized optical needle using azimuthally polarized multi Gaussian beam,” Chin. Phys. Lett. 33, 64203–64206 (2016).
[Crossref]

K. Prabakaran, P. Sangeetha, V. Karthik, K. B. Rajesh, and A. M. Musthafa, “Tight focusing properties of phase modulated radially polarized Laguerre Bessel Gaussian Beam,” Chin. Phys. Lett. 34, 054203 (2017).
[Crossref]

L. Z. Rao, H. C. Lin, and Q. Q. Sun, “Spatial correlation properties of tightly focused J0-correlated azimuthally polarized vortex Beams,” Chin. Phys. Lett. 30, 054211 (2013).
[Crossref]

J. Mod. Opt. (1)

T. Liu, J. B. Tan, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[Crossref]

J. Opt. Soc. Am. A (4)

Laser Photonics Rev. (1)

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9, 231–240 (2015).
[Crossref]

Nat. Photonics (1)

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

Opt. Express (7)

Opt. Laser Technol. (1)

L. N. Guo, Z. L. Tang, C. Q. Liang, and Z. L. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43, 895–898 (2011).
[Crossref]

Opt. Lett. (7)

Opt. Quantum Electron. (1)

J. W. Charles, K. Prabakaran, K. B. Rajesh, and H. M. Pandya, “Generation of sub wavelength super long dark channel using azimuthally polarized annular multi-Gaussian beam,” Opt. Quantum Electron. 46, 1079–1086 (2014).
[Crossref]

Phys. Rev. A (2)

J. X. Li, Y. I. Salamin, B. J. Galow, and C. H. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85, 063832 (2012).
[Crossref]

Y. M. Dong, F. Wang, C. L. Zhao, and Y. J. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[Crossref]

Phys. Rev. Lett. (1)

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

Proc. R. Soc. London Ser. A (1)

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]

Prog. Opt (1)

Y. J. Cai, Y. H. Chen, J. Y. Yu, X. L. Liu, and L. Liu, “Chapter three-generation of partially coherent beams,” Prog. Opt 62, 157–223 (2017).
[Crossref]

Prog. Opt. (1)

Z. Y. Chen, L. M. Hua, and J. X. Pu, “Tight focusing of light beams: effect of polarization, phase, and coherence,” Prog. Opt. 57, 219–260 (2012).
[Crossref]

Other (2)

M. Gu, Advanced Optical Imaging Theory (Springer, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (6)

Fig. 1.
Fig. 1. Tight focusing system. Inset (a) is the intensity distribution of the incident linearly polarized CPCG beam, and (b) is the initial coherence of the incident CPCG beam. The parameters of the insets are chosen as λ = 633    nm , ω 0 = 5    mm , and δ = 0.3 ω 0 .
Fig. 2.
Fig. 2. Intensity distributions of the CPCG beam in the focal plane. (a) Total intensity, (b)  x -polarization component, (c)  y -polarization component, (d)  z -polarization component. The parameters for calculation are chosen as λ = 633    nm , ω 0 = 5    mm , NA = 0.9 , f = 1    cm , E 0 = 1 , and δ = 0.3 ω 0 .
Fig. 3.
Fig. 3. Total intensity distributions of the CPCG beam in the ρ - z plane near the focus. The parameters for calculation are chosen as λ = 633    nm , ω 0 = 5    mm , NA = 0.9 , f = 1    cm , E 0 = 1 , and δ = 0.3 ω 0 .
Fig. 4.
Fig. 4. Distributions of coherence of the CPCG beam in the focal plane. (a), (b), (c), (d), (e), and (f) are | μ x x | , | μ y y | , | μ z z | , | μ x y | , | μ x z | , and | μ y z | , respectively. The parameters for calculation are chosen as λ = 633    nm , ω 0 = 5    mm , NA = 0.9 , f = 1    cm , E 0 = 1 , and δ = 0.3 ω 0 .
Fig. 5.
Fig. 5. Curves of coherence distribution and normalized intensity of the CPCG beam in the focal plane: (a) distribution of | μ x x | , | μ y y | , | μ z z | , and normalized intensity; (b) distribution of | μ x y | , | μ x z | , | μ y z | , and normalized intensity. The parameters for calculation are chosen as λ = 633    nm , ω 0 = 5    mm , NA = 0.9 , f = 1    cm , E 0 = 1 , and δ = 0.3 ω 0 .
Fig. 6.
Fig. 6. Distributions of coherence of the CPCG beam in the propagation plane near the focus. (a), (b), and (c) are | μ x x | , | μ y y | , and | μ z z | , respectively; (d), (e), and (f) are | μ x y | , | μ x z | , and | μ y z | , respectively. The parameters for calculation are chosen as λ = 633    nm , ω 0 = 5    mm , NA = 0.9 , f = 1    cm , E 0 = 1 , and δ = 0.3 ω 0 .

Equations (17)

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E ( ρ , ψ , z ) = i E 0 λ 0 α 0 2 π exp [ i k z cos θ i k ρ sin θ cos ( ψ ϕ ) ] × P ( ϕ , θ ) K ( ϕ , θ ) sin θ cos θ d ϕ d θ ,
K ( ϕ , θ ) = [ cos θ + sin 2 ϕ ( 1 cos θ ) e x cos ϕ sin ϕ ( cos θ 1 ) e y cos ϕ sin θ e z ] .
W i j ( ρ 1 , ρ 2 , z ) = W i j ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) = E i ( ρ 1 , ψ 1 , z ) E j * ( ρ 2 , ψ 2 , z ) = ( E 0 λ ) 2 0 α 0 α 0 2 π 0 2 π P ( ϕ 1 , θ 1 ) P ( ϕ 2 , θ 2 ) K i ( ϕ 1 , θ 1 ) K j T ( ϕ 2 , θ 2 ) g ( ϕ 1 , θ 1 , ϕ 2 , θ 2 ) × exp [ i k z cos θ 1 i k ρ 1 sin θ 1 cos ( ψ 1 ϕ 1 ) ] cos θ 1 sin θ 1 sin θ 2 cos θ 2 × exp [ i k z cos θ 2 + i k ρ 2 sin θ 2 cos ( ψ 2 ϕ 2 ) ] d ϕ 1 d ϕ 2 d θ 1 d θ 2 , ( i , j = x , y , z ) ,
W 0 x x ( r 1 , r 2 ) = exp ( r 1 2 + r 2 2 ω 0 2 ) sin c ( r 2 2 r 1 2 δ 2 ) ,
F ( ϕ 1 , ϕ 2 , θ 1 , θ 2 ) = P ( ϕ 1 , θ 1 ) P ( ϕ 2 , θ 2 ) g ( ϕ 1 , θ 1 , ϕ 2 , θ 2 ) = exp [ f 2 ( sin 2 θ 1 + sin 2 θ 2 ) ω 0 2 ] sin c ( f 2 sin 2 θ 2 f 2 sin 2 θ 1 δ 2 ) ,
W x x ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) = ( E 0 π λ ) 2 0 α 0 α exp [ f 2 ( sin 2 θ 1 + sin 2 θ 2 ) ω 0 2 ] sin θ 1 cos θ 1 × sin θ 2 cos θ 2 [ ( 1 + cos θ 1 ) J 0 ( k ρ 1 sin θ 1 ) + ( 1 cos θ 1 ) J 2 ( k ρ 1 sin θ 1 ) cos 2 ψ 1 ] × [ ( 1 + cos θ 2 ) J 0 ( k ρ 2 sin θ 2 ) + ( 1 cos θ 2 ) J 2 ( k ρ 2 sin θ 2 ) cos 2 ψ 2 ] × sin c ( f 2 sin 2 θ 2 f 2 sin 2 θ 1 δ 2 ) exp ( i k z cos θ 1 + i k z cos θ 2 ) d θ 1 d θ 2 ,
W x y ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) = ( E 0 π λ ) 2 0 α 0 α exp [ f 2 ( sin 2 θ 1 + sin 2 θ 2 ) ω 0 2 ] J 2 ( k ρ 2 sin θ 2 ) sin 2 ψ 2 × sin c ( f 2 sin 2 θ 2 f 2 sin 2 θ 1 δ 2 ) exp ( i k z cos θ 1 + i k z cos θ 2 ) sin θ 1 cos θ 1 ( 1 cos θ 2 ) × [ ( 1 + cos θ 1 ) J 0 ( k ρ 1 sin θ 1 ) + ( 1 cos θ 1 ) J 2 ( k ρ 1 sin θ 1 ) cos 2 ψ 1 ] sin θ 2 cos θ 2 d θ 1 d θ 2 ,
W x z ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) = 2 i ( E 0 π λ ) 2 0 α 0 α exp [ f 2 ( sin 2 θ 1 + sin 2 θ 2 ) ω 0 2 ] sin θ 1 cos θ 1 cos ψ 2 × exp ( i k z cos θ 1 + i k z cos θ 2 ) sin c ( f 2 sin 2 θ 2 f 2 sin 2 θ 1 δ 2 ) J 1 ( k ρ 2 sin θ 2 ) ( sin θ 2 ) 2 × [ ( 1 + cos θ 1 ) J 0 ( k ρ 1 sin θ 1 ) + ( 1 cos θ 1 ) J 2 ( k ρ 1 sin θ 1 ) cos 2 ψ 1 ] cos θ 2 d θ 1 d θ 2 ,
W y y ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) = ( E 0 π λ ) 2 0 α 0 α exp [ f 2 ( sin 2 θ 1 + sin 2 θ 2 ) ω 0 2 ] sin θ 1 cos θ 1 × sin c ( f 2 sin 2 θ 2 f 2 sin 2 θ 1 δ 2 ) J 2 ( k ρ 1 sin θ 1 ) sin 2 ψ 1 ( 1 cos θ 1 ) sin θ 2 cos θ 2 × exp ( i k z cos θ 1 + i k z cos θ 2 ) J 2 ( k ρ 2 sin θ 2 ) sin 2 ψ 2 ( 1 cos θ 2 ) d θ 1 d θ 2 ,
W y z ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) = 2 i ( E 0 π λ ) 2 0 α 0 α exp [ f 2 ( sin 2 θ 1 + sin 2 θ 2 ) ω 0 2 ] sin 2 ψ 1 ( 1 cos θ 1 ) × sin c ( f 2 sin 2 θ 2 f 2 sin 2 θ 1 δ 2 ) J 1 ( k ρ 2 sin θ 2 ) J 2 ( k ρ 1 sin θ 1 ) cos ψ 2 ( sin θ 2 ) 2 × exp ( i k z cos θ 1 + i k z cos θ 2 ) sin θ 1 cos θ 1 cos θ 2 d θ 1 d θ 2 ,
W z z ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) = 4 ( E 0 π λ ) 2 0 α 0 α exp [ f 2 ( sin 2 θ 1 + sin 2 θ 2 ) ω 0 2 ] J 1 ( k ρ 1 sin θ 1 ) × sin c ( f 2 sin 2 θ 2 f 2 sin 2 θ 1 δ 2 ) J 1 ( k ρ 2 sin θ 2 ) cos ψ 2 ( sin θ 2 ) 2 cos ψ 1 ( sin θ 1 ) 2 × exp ( i k z cos θ 1 + i k z cos θ 2 ) cos θ 1 cos θ 2 d θ 1 d θ 2 .
I x ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) = W x x ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) ,
I y ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) = W y y ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) ,
I z ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) = W z z ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) ,
I total ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) = I x ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) + I y ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) + I z ( ρ 1 , ρ 1 , ψ 1 , ψ 1 , z ) ,
W ( ρ 1 , ρ 2 , z ) = [ W x x ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) W x y ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) W x z ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) W y x ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) W y y ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) W y z ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) W z x ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) W z y ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) W z z ( ρ 1 , ρ 2 , ψ 1 , ψ 2 , z ) ] ,
μ i j ( ρ 1 , ρ 2 , z ) = W i j ( ρ 1 , ρ 2 , z ) / W i i ( ρ 1 , ρ 2 , z ) W j j ( ρ 1 , ρ 2 , z ) .

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