Abstract

We present an inverse method to engineer uniform-intensity focal fields with arbitrary shape. Amplitude, phase, and polarization states, as adjustable parameters, are used to seek the desired focal fields in the non-iterative computational procedure. Our method can be applied to the cases with low and moderate numerical aperture (NA), in which case the feasibility and validity of our approach have been demonstrated in theory, simulation and experiment, respectively. For the case of higher NA, simulated results based on the Richards-Wolf diffraction integral are shown. We also made some discussions on the experiments with the higher NA. Our method should have wide applications in optical micro machining, optical trapping and so on.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (1)

S. A. Abbas, Q. Sun, and H. Foroosh, “An exact and fast computation of discrete Fourier transform for polar and spherical grid,” IEEE Trans. Signal Process. 65, 2033–2048 (2017).
[Crossref]

2016 (2)

2015 (1)

2014 (2)

2013 (2)

Y. Pan, S. M. Li, L. Mao, L. J. Kong, Y. N. Li, C. H. Tu, P. Wang, and H. T. Wang, “Vector optical fields with polarization distributions similar to electric and magnetic field lines,” Opt. Express 21, 16200–16209 (2013).
[Crossref] [PubMed]

K. Prabakaran, K.B. Rajesh, and T.V.S. Pillai, “Focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam by diffractive optical element,” Optik 124, 5039 (2013).
[Crossref]

2012 (3)

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[Crossref] [PubMed]

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[Crossref] [PubMed]

F. Kenny, D. Lara, O. G. Rodriguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express 20, 14015–14029 (2012).
[Crossref] [PubMed]

2011 (2)

K. Lou, S. X. Qian, X. L. Wang, Y. N. Li, B. Gu, C. H. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express 20, 120–127 (2011).
[Crossref]

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

2010 (2)

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

H. Chen, Z. Zheng, B. F. Zhang, J. P. Ding, and H. T. Wang, “Polarization structuring of focused field through polarization-only modulation of incident beam,” Opt. Lett. 35, 2825–2827 (2010).
[Crossref] [PubMed]

2009 (4)

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

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102, 103902 (2009).
[Crossref]

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[Crossref]

B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009).
[Crossref]

2008 (4)

B. Hao and J. Leger, “Numerical aperture invariant focus shaping using spirally polarized beams,” Opt. Commun. 2811924–1928 (2008).
[Crossref]

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

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16, 7203–7213 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (3)

M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006).
[Crossref]

A. Averbucha, R. R. Coifmanb, D. L. Donohoc, M. Eladd, and M. Israelid, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145–167 (2006).
[Crossref]

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

2005 (1)

Y. Zhao, Q. Zhan, and Y. P. Li, “Design of DOE for beam shaping with highly NA focused cylindrical vector beam,” Proc. SPIE,  5636, 56 (2005).
[Crossref]

2004 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

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. 82, 5251–5254 (2001).
[Crossref]

1998 (1)

A. M. Morales and C. M. Lieber, “A laser ablation method for the synthesis of crystalline semiconductor nanowires,” Science 279, 208–211 (1998).
[Crossref] [PubMed]

1995 (1)

1989 (1)

1985 (1)

1984 (1)

1959 (1)

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

Abbas, S. A.

S. A. Abbas, Q. Sun, and H. Foroosh, “An exact and fast computation of discrete Fourier transform for polar and spherical grid,” IEEE Trans. Signal Process. 65, 2033–2048 (2017).
[Crossref]

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Antosiewicz, T. J.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102, 103902 (2009).
[Crossref]

Averbucha, A.

A. Averbucha, R. R. Coifmanb, D. L. Donohoc, M. Eladd, and M. Israelid, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145–167 (2006).
[Crossref]

Bautista, G.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[Crossref] [PubMed]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 82, 5251–5254 (2001).
[Crossref]

Boruah, B. R.

B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009).
[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. 82, 5251–5254 (2001).
[Crossref]

Chen, H.

Chen, J.

G. Rui, J. Chen, X. Wang, B. Gu, Y. Cui, and Q. Zhan, “Synthesis of focused beam with controllable arbitrary homogeneous polarization using engineered vectorial optical fields,” Opt. Express 24, 23667–23676 (2016).
[Crossref] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Chen, W.

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

Chen, Z.

Chong, C. T.

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

Coifmanb, R. R.

A. Averbucha, R. R. Coifmanb, D. L. Donohoc, M. Eladd, and M. Israelid, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145–167 (2006).
[Crossref]

Cottrell, D. M.

Courtial, J.

Cui, Y.

Dainty, C.

Davis, J. A.

Ding, J.

Ding, J. P.

H. Chen, Z. Zheng, B. F. Zhang, J. P. Ding, and H. T. Wang, “Polarization structuring of focused field through polarization-only modulation of incident beam,” Opt. Lett. 35, 2825–2827 (2010).
[Crossref] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Donohoc, D. L.

A. Averbucha, R. R. Coifmanb, D. L. Donohoc, M. Eladd, and M. Israelid, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145–167 (2006).
[Crossref]

Eladd, M.

A. Averbucha, R. R. Coifmanb, D. L. Donohoc, M. Eladd, and M. Israelid, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145–167 (2006).
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Flowers, S. W.

Foroosh, H.

S. A. Abbas, Q. Sun, and H. Foroosh, “An exact and fast computation of discrete Fourier transform for polar and spherical grid,” IEEE Trans. Signal Process. 65, 2033–2048 (2017).
[Crossref]

Gianino, P. D.

Gibson, G.

Grier, D. G.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Gu, B.

Gu, M.

Guo, C. S.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Hao, B.

B. Hao and J. Leger, “Numerical aperture invariant focus shaping using spirally polarized beams,” Opt. Commun. 2811924–1928 (2008).
[Crossref]

B. Hao and J. Leger, “Polarization beam shaping,” Appl. Opt. 46, 8211–8217 (2007).
[Crossref] [PubMed]

Hao, J.

Hashimoto, Y.

Hnatovsky, C.

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

Hong, M.

Horner, J. L.

Huttunen, M. J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[Crossref] [PubMed]

Israelid, M.

A. Averbucha, R. R. Coifmanb, D. L. Donohoc, M. Eladd, and M. Israelid, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145–167 (2006).
[Crossref]

Jabbour, T. G.

Jordan, P.

Kauranen, M.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[Crossref] [PubMed]

Kawata, S.

Kawata, Y.

Kenny, F.

Kong, L. J.

Kontio, J. M.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[Crossref] [PubMed]

Krolikowski, W.

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

Kuebler, S. M.

Laczik, Z.

Lara, D.

Lasser, T.

Leach, J.

Leger, J.

B. Hao and J. Leger, “Numerical aperture invariant focus shaping using spirally polarized beams,” Opt. Commun. 2811924–1928 (2008).
[Crossref]

B. Hao and J. Leger, “Polarization beam shaping,” Appl. Opt. 46, 8211–8217 (2007).
[Crossref] [PubMed]

Leger, J. R.

Leitgeb, R. A.

Leutenegger, M.

Li, S. M.

Li, X.

Li, Y. N.

Y. Pan, S. M. Li, L. Mao, L. J. Kong, Y. N. Li, C. H. Tu, P. Wang, and H. T. Wang, “Vector optical fields with polarization distributions similar to electric and magnetic field lines,” Opt. Express 21, 16200–16209 (2013).
[Crossref] [PubMed]

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[Crossref] [PubMed]

K. Lou, S. X. Qian, X. L. Wang, Y. N. Li, B. Gu, C. H. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express 20, 120–127 (2011).
[Crossref]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Li, Y. P.

Y. Zhao, Q. Zhan, and Y. P. Li, “Design of DOE for beam shaping with highly NA focused cylindrical vector beam,” Proc. SPIE,  5636, 56 (2005).
[Crossref]

Lieber, C. M.

A. M. Morales and C. M. Lieber, “A laser ablation method for the synthesis of crystalline semiconductor nanowires,” Science 279, 208–211 (1998).
[Crossref] [PubMed]

Lilly, R. A.

Lou, K.

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[Crossref] [PubMed]

K. Lou, S. X. Qian, X. L. Wang, Y. N. Li, B. Gu, C. H. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express 20, 120–127 (2011).
[Crossref]

Lukyanchuk, B.

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

Mäkitalo, J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[Crossref] [PubMed]

Mao, L.

Morales, A. M.

A. M. Morales and C. M. Lieber, “A laser ablation method for the synthesis of crystalline semiconductor nanowires,” Science 279, 208–211 (1998).
[Crossref] [PubMed]

Neil, M. A. A.

B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009).
[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. 82, 5251–5254 (2001).
[Crossref]

Padgett, M. J.

Pan, Y.

Pillai, T.V.S.

K. Prabakaran, K.B. Rajesh, and T.V.S. Pillai, “Focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam by diffractive optical element,” Optik 124, 5039 (2013).
[Crossref]

Pniewski, J.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102, 103902 (2009).
[Crossref]

Prabakaran, K.

K. Prabakaran, K.B. Rajesh, and T.V.S. Pillai, “Focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam by diffractive optical element,” Optik 124, 5039 (2013).
[Crossref]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Pu, J.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[Crossref]

Qian, B.

Qian, S. X.

Rajesh, K.B.

K. Prabakaran, K.B. Rajesh, and T.V.S. Pillai, “Focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam by diffractive optical element,” Optik 124, 5039 (2013).
[Crossref]

Rao, L.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[Crossref]

Rao, R.

Ren, H.

Richards, B.

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

Rode, A.

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

Rodriguez-Herrera, O. G.

Roichman, Y.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Rui, G.

Sheppard, C.

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

Shi, L.

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

Shvedov, V.

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

Simonen, J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[Crossref] [PubMed]

Sinclair, G.

Sun, B.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Sun, Q.

S. A. Abbas, Q. Sun, and H. Foroosh, “An exact and fast computation of discrete Fourier transform for polar and spherical grid,” IEEE Trans. Signal Process. 65, 2033–2048 (2017).
[Crossref]

Szoplik, T.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102, 103902 (2009).
[Crossref]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Tian, Y. J.

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[Crossref] [PubMed]

Tu, C. H.

Ueki, H.

Venugopalan, P.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Wang, H.

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

Wang, H. T.

Wang, P.

Wang, X.

Wang, X. L.

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[Crossref] [PubMed]

K. Lou, S. X. Qian, X. L. Wang, Y. N. Li, B. Gu, C. H. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express 20, 120–127 (2011).
[Crossref]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Wolf, E.

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

Wróbel, P.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102, 103902 (2009).
[Crossref]

Yao, E.

Yei, P.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[Crossref]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 82, 5251–5254 (2001).
[Crossref]

Yu, Z.

Zeng, T.

Zhan, Q.

Zhang, B. F.

Zhao, Y.

Y. Zhao, Q. Zhan, and Y. P. Li, “Design of DOE for beam shaping with highly NA focused cylindrical vector beam,” Proc. SPIE,  5636, 56 (2005).
[Crossref]

Zheng, Z.

Adv. Opt. Photon. (1)

Appl. Comput. Harmon. Anal. (1)

A. Averbucha, R. R. Coifmanb, D. L. Donohoc, M. Eladd, and M. Israelid, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145–167 (2006).
[Crossref]

Appl. Opt. (5)

Chin. Opt. Lett. (1)

IEEE Trans. Signal Process. (1)

S. A. Abbas, Q. Sun, and H. Foroosh, “An exact and fast computation of discrete Fourier transform for polar and spherical grid,” IEEE Trans. Signal Process. 65, 2033–2048 (2017).
[Crossref]

Nano Lett. (1)

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[Crossref] [PubMed]

Nat. Photon. (1)

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

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Opt. Commun. (3)

B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009).
[Crossref]

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

B. Hao and J. Leger, “Numerical aperture invariant focus shaping using spirally polarized beams,” Opt. Commun. 2811924–1928 (2008).
[Crossref]

Opt. Express (9)

G. Rui, J. Chen, X. Wang, B. Gu, Y. Cui, and Q. Zhan, “Synthesis of focused beam with controllable arbitrary homogeneous polarization using engineered vectorial optical fields,” Opt. Express 24, 23667–23676 (2016).
[Crossref] [PubMed]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23, 17701–17710 (2015).
[Crossref] [PubMed]

Y. Pan, S. M. Li, L. Mao, L. J. Kong, Y. N. Li, C. H. Tu, P. Wang, and H. T. Wang, “Vector optical fields with polarization distributions similar to electric and magnetic field lines,” Opt. Express 21, 16200–16209 (2013).
[Crossref] [PubMed]

F. Kenny, D. Lara, O. G. Rodriguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express 20, 14015–14029 (2012).
[Crossref] [PubMed]

K. Lou, S. X. Qian, X. L. Wang, Y. N. Li, B. Gu, C. H. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express 20, 120–127 (2011).
[Crossref]

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002).
[Crossref] [PubMed]

G. Sinclair, J. Leach, P. Jordan, G. Gibson, E. Yao, Z. Laczik, M. J. Padgett, and J. Courtial, “Interactive application in holographic optical tweezers of a multi-plane Gerchberg-Saxton algorithm for three-dimensional light shaping,” Opt. Express 12, 1665–1670 (2004).
[Crossref] [PubMed]

T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16, 7203–7213 (2008).
[Crossref] [PubMed]

M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006).
[Crossref]

Opt. Laser Technol. (1)

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[Crossref]

Opt. Lett. (3)

Optik (1)

K. Prabakaran, K.B. Rajesh, and T.V.S. Pillai, “Focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam by diffractive optical element,” Optik 124, 5039 (2013).
[Crossref]

Phys. Rev. Lett. (5)

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

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102, 103902 (2009).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 82, 5251–5254 (2001).
[Crossref]

Proc. Roy. Soc. A (1)

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

Proc. SPIE (1)

Y. Zhao, Q. Zhan, and Y. P. Li, “Design of DOE for beam shaping with highly NA focused cylindrical vector beam,” Proc. SPIE,  5636, 56 (2005).
[Crossref]

Sci. Rep. (1)

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[Crossref] [PubMed]

Science (1)

A. M. Morales and C. M. Lieber, “A laser ablation method for the synthesis of crystalline semiconductor nanowires,” Science 279, 208–211 (1998).
[Crossref] [PubMed]

Other (1)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

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Figures (12)

Fig. 1
Fig. 1 Schematic of the focusing process.
Fig. 2
Fig. 2 Simulated focal field with a shape of rhombus with NA = 0.9 under the ideal condition. (a) shows the total intensity, and (b), (c) and (d) show the radial, azimuthal and longitudinal components, respectively. Any picture has dimensions of 45λ × 45λ.
Fig. 3
Fig. 3 Simulated input field corresponding to the focal field in Fig. 2. (a) the total intensity, (b) the distribution of polarization states, (c) the distribution of phase.
Fig. 4
Fig. 4 Simulated rhombus-shaped focal field with NA = 0.9 under the existing experimental condition. (a) shows the total intensity, and (b), (c) and (d) show the radial, azimuthal and longitudinal components, respectively. Any picture has dimensions of 45λ × 45λ.
Fig. 5
Fig. 5 Simulated input field corresponding to the focal field in Fig. 4 under the existing experimental condition. (a) the total intensity, (b) the distribution of polarization states, (c) the distribution of phase.
Fig. 6
Fig. 6 Simulated focal fields in shapes of triangle [(a)–(d)] and red-cross [(e)–(h)] with NA = 0.9. (a) and (e), the total intensity; (b) and (f), the radial component; (c) and (g), the azimuthal component; (d) and (h), the longitudinal component. Any picture has dimensions of 45λ × 45λ.
Fig. 7
Fig. 7 Simulated input fields corresponding to the focal fields in Fig. 6. (a)–(c) correspond to the triangle shape focal field in Figs. 6(a)–6(d), and (d)–(f) correspond to the red-cross shape focal field in Figs. 6(e)–6(h). (a) and (d) show the total intensity, (b) and (e) show the distribution of polarization states, (c) and (f) show the distribution of phase.
Fig. 8
Fig. 8 Simulated total intensity of rhombus-shaped focal fields with NA = 0.9. From (a) to (d), the dimensions are 60λ × 60λ, 30λ × 30λ, 15λ × 15λ, and 10λ × 10λ, respectively.
Fig. 9
Fig. 9 Simulated input fields corresponding to the focal fields in Fig. 8. (a1)–(a3) correspond to the focal field in Fig. 8(a), (b1)–(b3) correspond to the focal field in Fig. 8(b), (c1)–(c3) correspond to the focal field in Fig. 8(c), and (d1)–(d3) correspond to the focal field in Fig. 8(d). (a1)–(d1) show total intensity, (a2)–(d2) show the distribution of polarization states, and (a3)–(d3) show the distribution of phase.
Fig. 10
Fig. 10 Simulated total intensity of rhombus-shaped focal fields with different NA. All of them have the same size of zero-padding operations to Figs. 4(a)–4(d) correspond to NA = 0.1, 0.3, 0.5, 0.7, with the dimensions of 405λ × 405λ, 135λ × 135λ, 81λ × 81λ, and 58λ × 58λ, respectively.
Fig. 11
Fig. 11 Schematic of experimental setup. SLM—spatial light modulator, L1 and L2—a pair of lenses, λ/4—1/4 wave plates, SF—spatial filter, G—Ronchi phase grating, O—objective, CCD—charge-coupled device.
Fig. 12
Fig. 12 Measured total intensity of focused fields. Images in (a)–(c) show the focal fields in shape of rhombus. Images in (d)–(f) show the focal fields in shape of triangle. Images in (g)–(i) show the focal fields in shape of red-cross. Images in the first column have dimensions of 1500 × 1500 μm2, with NA = 0.1. Images in the second column have dimensions of 250 × 250 μm2, with NA = 0.4. Images in the third column have dimensions of 150 × 150 μm2, with NA = 0.65.

Equations (22)

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E t = [ E r t E ϕ t E z t ] = cos θ M p [ E r i E ϕ i E z i ] ,
M p = [ cos θ 0 sin θ 0 1 0 sin θ 0 cos θ ] ,
E f = i f λ 0 Ω E t e i ( k z t z f k x t x f k y t y f ) d Ω = i f λ 0 E t e i ( k 0 cos θ z f k 0 cos ϕ i sin θ x f k 0 sin ϕ i sin θ y f ) sin θ d θ d ϕ i ,
E f = i f λ 0 E t e i [ k 0 cos θ z f k 0 cos ϕ i ( r i / f ) x f k 0 sin ϕ i ( r i / f ) y f ] r i f cos θ d r i d ϕ i = i f λ 0 E t e i [ k 0 cos θ z f k 0 ( r i / f ) ( cos ϕ i r f cos φ f + sin ϕ i r f sin ϕ f ) ] r i f cos θ d r i d ϕ i = i f λ 0 [ E t e i k 0 cos θ z f / ( f cos θ ) ] e i k 0 r i r f cos ( ϕ i ϕ f ) / f r i d r i d ϕ i .
E f ( z f = 0 ) = i λ 0 ( E t / cos θ ) e i k 0 r i r f cos ( ϕ i ϕ f ) r i d r i d ϕ i .
E f ( z f = 0 ) = [ E r f ( z f = 0 ) E ϕ f ( z f = 0 ) E z f ( z f = 0 ) ] = i λ 0 p { M p [ E r i E ϕ i E z i ] / cos θ } ,
E t = i λ 0 cos θ k 0 2 E f ( z f = 0 ) e i k 0 r i r f cos ( ϕ i ϕ f ) r f d r f d ϕ f ,
E i = [ E r i E ϕ i E z i ] = i λ 0 cos θ M p 1 p 1 { E r f ( z f = 0 ) E ϕ f ( z f = 0 ) E z f ( z f = 0 ) } ,
M p 1 = [ cos θ 0 sin θ 0 1 0 sin θ 0 cos θ ] ,
p 1 { E z f ( z f = 0 ) } = tan θ p 1 { E r f ( z f = 0 ) } .
E t = [ E x t E y t E z t ] = cos θ M c [ E x i E y i E z i ] ,
M c = [ cos 2 ϕ i cos θ + sin 2 ϕ i ( cos θ 1 ) sin ϕ i cos ϕ i sin θ cos ϕ i ( cos θ 1 ) sin ϕ i cos ϕ i cos 2 ϕ i + sin 2 ϕ i cos θ sin θ sin ϕ i sin θ cos ϕ i sin θ sin ϕ i cos θ ] .
E f = i f λ 0 Ω E t e i ( k z t z f k x t x f k y t y f ) d Ω = i f λ 0 E t e i ( k 0 cos θ z f k 0 cos ϕ i sin θ x f k 0 sin ϕ i sin θ y f ) sin θ d θ d ϕ i = i f λ 0 E t e i [ k 0 cos θ z f k 0 cos ϕ i ( r i / f ) x f k 0 sin ϕ i ( r i / f ) y f ] r i f cos θ d r i d ϕ i = i f λ 0 E t e i [ k 0 cos θ z f ( k 0 / f ) x i x f ( k 0 / f ) y i y f ] 1 f cos θ d x i d y i = i f λ 0 [ E t e i k 0 cos θ z f / ( f cos θ ) ] e i k 0 ( x i x f + y i y f ) / f d x i d y i .
E f ( z f = 0 ) = i λ 0 ( E t / cos θ ) e i k 0 ( x i x f + y i y f ) d x i d y i .
E f ( z f = 0 ) = [ E x f ( z f = 0 ) E y f ( z f = 0 ) E z f ( z f = 0 ) ] = i λ 0 c { M c [ E x i E y i E z i ] / cos θ } ,
E t = i λ 0 cos θ k 0 2 E f ( z f = 0 ) e i k 0 ( x i x f + y i y f ) d x f d y f ,
E i = [ E x i E y i E z i ] = i λ 0 cos θ M c 1 1 { E x f ( z f = 0 ) E y f ( z f = 0 ) E z f ( z f = 0 ) } ,
M c 1 = [ cos 2 ϕ i cos θ + sin 2 ϕ i ( cos θ 1 ) sin ϕ i cos ϕ i sin θ cos ϕ i ( cos θ 1 ) sin ϕ i cos ϕ i cos 2 ϕ i + sin 2 ϕ i cos θ sin θ sin ϕ i sin θ cos ϕ i sin θ sin ϕ i cos θ ] ,
c 1 { E z f ( z f = 0 ) } = tan θ [ cos ϕ i c 1 ( E x f ( z f = 0 ) ) + sin ϕ i c 1 ( E y f ( z f = 0 ) ) ] .
CC = cov ( X , Y ) σ X σ Y ,
p 1 [ E z f ( z f = 0 ) ] = r i 1 ( r i N A ) 2 p 1 [ E r f ( z f = 0 ) ] ,
c 1 [ E z f ( z f = 0 ) ] = r i 1 ( r i N A ) 2 { cos ϕ i c 1 [ E x f ( z f = 0 ) ] + sin ϕ i c 1 [ E y f ( z f = 0 ) ] } .

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