Abstract

Vector beams are of major importance to tailor tightly focused fields by creating an additional z-polarization component. Till now, mainly focusing properties of fundamental vector beams have been investigated, whereas the knowledge of focused higher-order singular vector fields is still missing. We fill this gap by numerical analysis of these fields, applying their attractive characteristics as including a spatially adjustable amount of radial and azimuthal components. We demonstrate the realization of three-dimensional polarization structures whose total intensity resembles dark stars and bright flowers. Further, we tailor these focal intensity landscapes by modulating the order of incident vector fields. This in turn allows shaping the focus of a light field for specific applications as e.g. advanced microscopy.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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2017 (2)

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7, 42142 (2017).
[Crossref] [PubMed]

2016 (2)

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117, 013601 (2016).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularitites in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

2015 (4)

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius stripes,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref] [PubMed]

M. Taylor, M. Waleed, A. B. Stilgoe, H. Rubinsztein-Dunlop, and W. P. Bowen, “Enhanced optical trapping via structured scattering,” Nature Photon. 9, 669 (2015).
[Crossref]

C. Alpmann, C. Schöler, and C. Denz, “Elegant Gaussian beams for enhanced optical manipulation,” Appl. Phys. Lett. 106, 241102 (2015).
[Crossref]

2013 (2)

2011 (3)

M. R. Dennis, “Fermionic out-of-plane structure of polarization singularities,” Opt. Lett. 36, 3765–3767 (2011).
[Crossref] [PubMed]

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nature Photon. 5, 335–342 (2011).
[Crossref]

X. Pang, T. Visser, and E. Wolf, “Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems,” Opt. Commun. 284, 5517–5522 (2011).
[Crossref]

2009 (2)

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

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

2008 (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,” Nature Photon. 2, 501–505 (2008).
[Crossref]

2007 (1)

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

2006 (3)

2005 (1)

I. Freund, “Cones, spirals, and Möbius strips, in elliptically polarized light,” Opt. Commun. 249, 7–22 (2005).
[Crossref]

2003 (1)

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

2002 (2)

Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41, 4630–4637 (2002).
[Crossref] [PubMed]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
[Crossref]

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] [PubMed]

2000 (1)

1994 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 336, 165–190 (1974).
[Crossref]

Alpmann, C.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularitites in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

C. Alpmann, C. Schöler, and C. Denz, “Elegant Gaussian beams for enhanced optical manipulation,” Appl. Phys. Lett. 106, 241102 (2015).
[Crossref]

Amoruso, S.

J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7, 42142 (2017).
[Crossref] [PubMed]

Andrews, D. L.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Baker, M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Banzer, P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117, 013601 (2016).
[Crossref]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius stripes,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Bauer, T.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117, 013601 (2016).
[Crossref]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius stripes,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Belmonte, A.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Berry, M.

M. Berry and C. Upstill, Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns vol. 18 of Progress in Optics (Elsevier, 1980), chap. IV, pp. 257–346.

Berry, M. V.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 336, 165–190 (1974).
[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] [PubMed]

Bigelow, N. P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Boguslawski, M.

P. Rose, F. Diebel, M. Boguslawski, and C. Denz, “Airy beam induced optical routing,” Appl. Phys. Lett. 102, 101101 (2013).
[Crossref]

Bokor, N.

Boruah, B.

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

Bowen, W. P.

M. Taylor, M. Waleed, A. B. Stilgoe, H. Rubinsztein-Dunlop, and W. P. Bowen, “Enhanced optical trapping via structured scattering,” Nature Photon. 9, 669 (2015).
[Crossref]

Boyd, R. W.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius stripes,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Brown, T.

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] [PubMed]

Bruzzese, R.

J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7, 42142 (2017).
[Crossref] [PubMed]

Cardano, F.

J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7, 42142 (2017).
[Crossref] [PubMed]

Chen, W.

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

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,” Nature Photon. 2, 501–505 (2008).
[Crossref]

Cižmár, T.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nature Photon. 5, 335–342 (2011).
[Crossref]

Davidson, N.

Dennis, M. R.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

M. R. Dennis, “Fermionic out-of-plane structure of polarization singularities,” Opt. Lett. 36, 3765–3767 (2011).
[Crossref] [PubMed]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Singular Optics: Optical Vortices and Polarization Singularities vol. 53 of Progress in Optics (Elsevier, 2009), chap. 5, pp. 293–363.

Denz, C.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularitites in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

C. Alpmann, C. Schöler, and C. Denz, “Elegant Gaussian beams for enhanced optical manipulation,” Appl. Phys. Lett. 106, 241102 (2015).
[Crossref]

P. Rose, F. Diebel, M. Boguslawski, and C. Denz, “Airy beam induced optical routing,” Appl. Phys. Lett. 102, 101101 (2013).
[Crossref]

Dholakia, K.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nature Photon. 5, 335–342 (2011).
[Crossref]

Diebel, F.

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Figures (3)

Fig. 1
Fig. 1

Examples of higher-order singular vector beams with σ12 = ±6: Polarization distribution with red flow lines (left) and phase Φ12 of complex Stokes field (right) of a (a) |σ12 − 2| = 4-fold vectorial flower and (b) 8-fold spider web.

Fig. 2
Fig. 2

Sketch of non-paraxial beam propagation of tightly focused, polarization structured light: (a) Concept image of numerical method, (b) occurence of longitudinal polarization components by focusing RP light fields, (c) focusing of AP field.

Fig. 3
Fig. 3

Dark star and bright flower: Tight focusing (NA = 0.9) of examplary vectorial flower (a) and spider web (c) with σ12 = ±8 resulting in focal field distributions of intensity (|Ex,y,z|2, |E⃗|2 ∈ [0, 1], peak ratio within images) and phase (φx,y,z ∈ [0, 2π]) as shown in (b) and (d), respectively. The size of focal distributions is indicated by the scale bar in (d). Total intensity |E⃗|2 resembles a dark star (bright flower) if a vectorial flower (web) is focused.

Tables (1)

Tables Icon

Table 1 Focal field distribution of tightly focused vectorial flowers (a) and spider webs (b) depending on index σ12 of singularity embedded in incident light field. Peak and mean ratio of each component |Ex,y,z|2 is shown below respective images. Overall focal intensity distribution |E⃗|2 resembles |σ12 − 2|-fold stars/ flowers.

Equations (3)

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E i = [ cos ( σ 12 2 ϕ ) , sin ( σ 12 2 ϕ ) ] T
E t ( θ , φ ) = cos θ [ l 0 , x ( cos θ cos 2 φ + sin 2 φ ( cos θ 1 ) cos φ sin φ sin θ cos φ ) + l 0 , y ( ( cos θ 1 ) cos φ sin φ cos θ sin 2 φ + cos 2 φ sin θ sin φ ) ] .
E ( x , y , z ) = [ E t ( k x , k y ) e i k z z k z / k 0 ] = [ l 0 , x ( k x , k y ) G X ( k x , k y ) ] + [ l 0 , y ( k x , k y ) G Y ( k x , k y ) ] .

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