Abstract

We present a versatile, non-interferometric method for generating vector fields and vector beams which can produce all the states of polarization represented on a higher-order Poincaré sphere. The versatility and non-interferometric nature of this method is expected to enable exploration of various exotic properties of vector fields and vector beams. To illustrate this, we study the propagation properties of some vector fields and find that, in general, propagation alters both their intensity and polarization distribution, and more interestingly, converts some vector fields into vector beams. In the article, we also suggest a modified Jones vector formalism to represent vector fields and vector beams.

© 2012 OSA

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  1. T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
    [CrossRef] [PubMed]
  2. D. N. Christodoulides, “Optical trapping: riding along an airy beam,” Nat. Photonics 2, 652–653 (2008).
    [CrossRef]
  3. K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
    [CrossRef]
  4. H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2011).
    [CrossRef]
  5. A. Abouraddy and K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
    [CrossRef] [PubMed]
  6. S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express 19, 11963–11968 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  11. S. Tripathi, B. J. Davis, K. C. Toussaint, and P. S. Carney, “Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy,” J. Phys. B: At. Mol. Opt. Phys. 44, 015401 (2011).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
    [CrossRef]
  16. M. Bashkansky, D. Park, and F. K. Fatemi, “Azimuthally and radially polarized light with a nematic SLM,” Opt. Express 18, 212–217 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  21. I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, “Complete polarization control of light from a liquid crystal spatial light modulator,” Opt. Express 20, 364–376 (2011).
    [CrossRef]
  22. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley and Sons Ltd., 2007).
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    [CrossRef]
  26. F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express 19, 25143–25150 (2011).
    [CrossRef]

2011 (9)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef] [PubMed]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[CrossRef]

S. Tripathi, B. J. Davis, K. C. Toussaint, and P. S. Carney, “Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy,” J. Phys. B: At. Mol. Opt. Phys. 44, 015401 (2011).
[CrossRef]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef] [PubMed]

H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2011).
[CrossRef]

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express 19, 11963–11968 (2011).
[CrossRef] [PubMed]

S. Vyas, M. Niwa, Y. Kozawa, and S. Sato, “Diffractive properties of obstructed vector Laguerre-Gaussian beam under tight focusing condition,” J. Opt. Soc. Am. A 28, 1387–1394 (2011).
[CrossRef]

F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express 19, 25143–25150 (2011).
[CrossRef]

I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, “Complete polarization control of light from a liquid crystal spatial light modulator,” Opt. Express 20, 364–376 (2011).
[CrossRef]

2010 (2)

M. Bashkansky, D. Park, and F. K. Fatemi, “Azimuthally and radially polarized light with a nematic SLM,” Opt. Express 18, 212–217 (2010).
[CrossRef] [PubMed]

O. Weiss and J. Scheuer, “Emission of cylindrical and elliptical vector beams from radial Bragg lasers,” Appl. Phys. Lett. 97, 251108 (2010).
[CrossRef]

2009 (3)

2008 (1)

D. N. Christodoulides, “Optical trapping: riding along an airy beam,” Nat. Photonics 2, 652–653 (2008).
[CrossRef]

2007 (2)

2006 (1)

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

2005 (1)

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 (1)

1990 (1)

Abdulhalim, I.

Abouraddy, A.

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

Agresti, J.

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef] [PubMed]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1989).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1989).

Bashkansky, M.

Bernet, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef] [PubMed]

Carney, P. S.

S. Tripathi, B. J. Davis, K. C. Toussaint, and P. S. Carney, “Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy,” J. Phys. B: At. Mol. Opt. Phys. 44, 015401 (2011).
[CrossRef]

Chen, J.

Christodoulides, D. N.

D. N. Christodoulides, “Optical trapping: riding along an airy beam,” Nat. Photonics 2, 652–653 (2008).
[CrossRef]

Cizmar, T.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[CrossRef]

Cottrell, D. M.

D’Ambrosio, E.

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef] [PubMed]

Davis, B. J.

S. Tripathi, B. J. Davis, K. C. Toussaint, and P. S. Carney, “Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy,” J. Phys. B: At. Mol. Opt. Phys. 44, 015401 (2011).
[CrossRef]

Davis, J. A.

DeSalvo, R.

Dholakia, K.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[CrossRef]

Ding, J.

Dorn, R.

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

Erkintalo, M.

M. J. Huttunen, M. Erkintalo, and M. Kauranen, “Absolute nonlinear optical probes of surface chirality,” J. Opt. A: Pure Appl. Opt. 11, 034006 (2009).
[CrossRef]

Fatemi, F. K.

Ford, D. H.

Forest, D.

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley and Sons Ltd., 2007).

Furhapter, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Galbraith, C. G.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef] [PubMed]

Galbraith, J. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef] [PubMed]

Gao, L.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef] [PubMed]

Guo, C.-S.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2007).

Hernandez, T. M.

Hodgkinson, I. J.

I. J. Hodgkinson and Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific Publishing Co. Pte. Ltd., 1997).
[CrossRef]

Huttunen, M. J.

M. J. Huttunen, M. Erkintalo, and M. Kauranen, “Absolute nonlinear optical probes of surface chirality,” J. Opt. A: Pure Appl. Opt. 11, 034006 (2009).
[CrossRef]

Iwahashi, S.

Jesacher, A.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Jureller, J. E.

Kauranen, M.

M. J. Huttunen, M. Erkintalo, and M. Kauranen, “Absolute nonlinear optical probes of surface chirality,” J. Opt. A: Pure Appl. Opt. 11, 034006 (2009).
[CrossRef]

Kimura, W. D.

Kitamura, K.

Kozawa, Y.

Kurosaka, Y.

Lagrange, B.

Leger, J. R.

Leuchs, G.

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

Li, Y.

Mackowsky, J. M.

Maurer, C.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Michel, C.

Milione, G.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef] [PubMed]

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef] [PubMed]

Miller, J.

Montorio, J. L.

Moreno, I.

Morgado, N.

Niwa, M.

Noda, S.

Nolan, D. A.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef] [PubMed]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2007).

Park, D.

Park, S.

Pinard, L.

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef] [PubMed]

Quabis, S.

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

Remilleux, A.

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Safrani, A.

Sakai, K.

Sand, D.

Sato, S.

Scherer, N. F.

Scheuer, J.

O. Weiss and J. Scheuer, “Emission of cylindrical and elliptical vector beams from radial Bragg lasers,” Appl. Phys. Lett. 97, 251108 (2010).
[CrossRef]

Simoni, B.

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef] [PubMed]

Takayama, N.

Tarallo, M. G.

Tidwell, S. C.

Toussaint, K. C.

S. Tripathi, B. J. Davis, K. C. Toussaint, and P. S. Carney, “Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy,” J. Phys. B: At. Mol. Opt. Phys. 44, 015401 (2011).
[CrossRef]

S. Tripathi and K. C. Toussaint, “Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection,” Opt. Express 17, 21396–21407 (2009).
[CrossRef] [PubMed]

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

K. C. Toussaint, S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. 30, 2846–2848 (2005).
[CrossRef] [PubMed]

Tripathi, S.

S. Tripathi, B. J. Davis, K. C. Toussaint, and P. S. Carney, “Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy,” J. Phys. B: At. Mol. Opt. Phys. 44, 015401 (2011).
[CrossRef]

S. Tripathi and K. C. Toussaint, “Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection,” Opt. Express 17, 21396–21407 (2009).
[CrossRef] [PubMed]

Vyas, S.

Wang, H.-T.

Wang, X.-L.

Weiss, O.

O. Weiss and J. Scheuer, “Emission of cylindrical and elliptical vector beams from radial Bragg lasers,” Appl. Phys. Lett. 97, 251108 (2010).
[CrossRef]

Willems, P.

Wu, Q. H.

I. J. Hodgkinson and Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific Publishing Co. Pte. Ltd., 1997).
[CrossRef]

Zhan, Q.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

O. Weiss and J. Scheuer, “Emission of cylindrical and elliptical vector beams from radial Bragg lasers,” Appl. Phys. Lett. 97, 251108 (2010).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

M. J. Huttunen, M. Erkintalo, and M. Kauranen, “Absolute nonlinear optical probes of surface chirality,” J. Opt. A: Pure Appl. Opt. 11, 034006 (2009).
[CrossRef]

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

J. Phys. B: At. Mol. Opt. Phys. (1)

S. Tripathi, B. J. Davis, K. C. Toussaint, and P. S. Carney, “Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy,” J. Phys. B: At. Mol. Opt. Phys. 44, 015401 (2011).
[CrossRef]

Nat. Methods (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef] [PubMed]

Nat. Photonics (2)

D. N. Christodoulides, “Optical trapping: riding along an airy beam,” Nat. Photonics 2, 652–653 (2008).
[CrossRef]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[CrossRef]

New J. Phys. (1)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Opt. Express (7)

Opt. Lett. (2)

Phys. Rev. Lett. (3)

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

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef] [PubMed]

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

Other (4)

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2007).

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley and Sons Ltd., 2007).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1989).

I. J. Hodgkinson and Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific Publishing Co. Pte. Ltd., 1997).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Illustration describing the concept of vector field generation. A right circularly polarized Gaussian beam is incident upon a phase plate. Telescope 1 images the phase plate onto the reflective NLC-SLM. Telescope 2 is then used to image the NLC-SLM surface onto the quarter-wave plate which has its fast axis aligned at 135° with respect to the horizontal axis. Light at the output of the quarter-wave plate is a vector field which can be converted to vector beams through spatial filtering or free space propagation. In our experiments, we use a Ti-Sapphire laser operating at 800 nm as the laser source. The phase plate, Lattice Electro Optics, 180° Phase-Half-UF-1006, consists of a 880-nm thick film of SiO2 covering half of a glass window in order to impart a phase shift of 180° to half of the beam incident upon it. Our SLM is a 512 × 512-pixel reflective NLC-SLM (BNS XY Series Nematic SLM) and the quarter-wave plate is QWPO-800-06-4-R10 acquired from CVI Melles Griot.

Fig. 2
Fig. 2

Experimentally obtained polarization analysis of the intensity distribution for single-mode (a–c) and hybrid (d–f) vector fields. Scale bar refers to 1 mm. The direction of the double-sided (black) arrows represents the direction of the transmission axis of the analyzer used.

Fig. 3
Fig. 3

Evolution of the intensity, (a), and polarization, (b), for the quasi-azimuthal and radial-azimuthal vector fields, where the experimental (i) and simulation (ii and iii) data are shown for distances of 0, 10, 100, and 250 cm.

Fig. 4
Fig. 4

(a) Simulated propagation for 30 m is shown for quasi-azimuthal (i) and radial-azimuthal (ii) fields. (b) The effect of spatial filtering for the quasi-radial field, where (i) and (ii) show the experimental and simulated results, respectively. All scale bars are 1 mm.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

JPP(φ)=eiπsgn(ρφ)sng(ρ+πφ)(1001),
J(φ)=eiπsgn(ρφ)sgn(ρ+πφ)[cos((2φ+2α)mod2π2)sin((2φ+2α)mod2π2)].
J(φ)=eiπsgn(ρφ)sgn(ρ+πφ)[e((2φ+2α)mod2π2)ie((2φ+2π)mod2π2)].

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