Abstract

We explore the behavior of a class of fully correlated optical beams that span the entire surface of the Poincaré sphere. The beams can be constructed from a coaxial superposition of a fundamental Gaussian mode and a spiral-phase Laguerre-Gauss mode having orthogonal polarizations. When the orthogonal polarizations are right and left circular, the coverage extends from one pole of the sphere to the other in such a way that concentric circles on the beam map onto parallels on the Poincaré sphere and radial lines map onto meridians. If the beam waist parameters match, the map is stereographic and the beam propagation corresponds to a rigid rotation about the pole. We present an experimental example of how a symmetrically stressed window can produce these beams and show that the predicted rotation indeed occurs when moving through the beams’ focus.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000), http://wwww.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-2-77.
    [CrossRef] [PubMed]
  2. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009),http://www.opticsinfobase.org/abstract.cfm?URI=aop-1-1-1 and references therein.
    [CrossRef]
  3. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric surface,” Opt. Express 9, 490–497 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-490.
    [CrossRef] [PubMed]
  4. D. G. Hall, “Vector-beam solutions of Maxwells wave equation,” Opt. Lett. 21, 9–11 (1996), http://www.opticsinfobase.org/abstract.cfm?URI=ol-21-1-9.
    [CrossRef] [PubMed]
  5. R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427 (1994), http://www.opticsinfobase.org/abstract.cfm?URI=ol-19-7-427.
    [CrossRef] [PubMed]
  6. P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-10-411.
    [CrossRef] [PubMed]
  7. C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).
  8. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing Light to a Tighter Spot,” Opt. Commun. 179, 1 (2000).
    [CrossRef]
  9. R. Dorn, S. Quabis, and G. Leuchs “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  10. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=ao-43-22-4322.
    [CrossRef] [PubMed]
  11. R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-22-7-1420.
    [CrossRef]
  12. G. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-7-4567.
    [CrossRef] [PubMed]
  13. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-11-1726.
    [CrossRef] [PubMed]
  14. W. Chen and Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-7-4106.
    [CrossRef] [PubMed]
  15. K. J. Moh, X.-C. Yuan, J. Bu, S. W. Zhu, and Bruce Z. Gao, “Radial polarization induced surface plasmon virtual probe for two-photon fluorescence microscopy,” Opt. Lett. 34, 971–973 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-971.
    [CrossRef] [PubMed]
  16. N. Hayazawa, “Focused Excitation of Surface Plasmon Polaritons Based on Gap-Mode in Tip-Enhanced Spectroscopy,” Jpn. J. Appl. Phys. 46, 7995 (2007).
    [CrossRef]
  17. K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603 (2006).
    [CrossRef]
  18. N. Moore and M. A. Alonso, “Closed-form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5926.
    [CrossRef] [PubMed]
  19. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-46-1-61.
    [CrossRef]
  20. A. K. Spilman and T. G. Brown, “Stress-induced Focal Splitting,” Opt. Express 15, 8411–8421 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8411.
    [CrossRef] [PubMed]
  21. A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence,” Proc. SPIE, 6667, 666701 (2007).
  22. N. Moore and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A 29, 2211–2218 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=josaa-26-7-1754.
    [CrossRef]
  23. E. G. Sauter, “Gaussian beams and the Poincare sphere,” Microwave Opt. Technol. Lett. 4, 485–486 (1991).
    [CrossRef]

2009 (3)

2008 (2)

2007 (5)

W. Chen and Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-7-4106.
[CrossRef] [PubMed]

A. K. Spilman and T. G. Brown, “Stress-induced Focal Splitting,” Opt. Express 15, 8411–8421 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8411.
[CrossRef] [PubMed]

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-46-1-61.
[CrossRef]

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence,” Proc. SPIE, 6667, 666701 (2007).

N. Hayazawa, “Focused Excitation of Surface Plasmon Polaritons Based on Gap-Mode in Tip-Enhanced Spectroscopy,” Jpn. J. Appl. Phys. 46, 7995 (2007).
[CrossRef]

2006 (2)

2005 (1)

2004 (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]

2001 (1)

2000 (2)

1999 (1)

1997 (1)

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

1996 (1)

1994 (1)

1991 (1)

E. G. Sauter, “Gaussian beams and the Poincare sphere,” Microwave Opt. Technol. Lett. 4, 485–486 (1991).
[CrossRef]

Alonso, M. A.

Beckley, A. M.

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence,” Proc. SPIE, 6667, 666701 (2007).

Biss, D. P.

Borghi, R.

Brown, T. G.

Bu, J.

Chen, W.

Choudhury, A.

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]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing Light to a Tighter Spot,” Opt. Commun. 179, 1 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing Light to a Tighter Spot,” Opt. Commun. 179, 1 (2000).
[CrossRef]

Gao, Bruce Z.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing Light to a Tighter Spot,” Opt. Commun. 179, 1 (2000).
[CrossRef]

Greene, P. L.

Hall, D. G.

Hayazawa, N.

N. Hayazawa, “Focused Excitation of Surface Plasmon Polaritons Based on Gap-Mode in Tip-Enhanced Spectroscopy,” Jpn. J. Appl. Phys. 46, 7995 (2007).
[CrossRef]

Jordan, R. H.

Lerman, G.

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]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing Light to a Tighter Spot,” Opt. Commun. 179, 1 (2000).
[CrossRef]

Levy, U.

Moh, K. J.

Moore, N.

N. Moore and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A 29, 2211–2218 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=josaa-26-7-1754.
[CrossRef]

N. Moore and M. A. Alonso, “Closed-form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5926.
[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]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing Light to a Tighter Spot,” Opt. Commun. 179, 1 (2000).
[CrossRef]

Santarsiero, M.

Sauter, E. G.

E. G. Sauter, “Gaussian beams and the Poincare sphere,” Microwave Opt. Technol. Lett. 4, 485–486 (1991).
[CrossRef]

Sheppard, C. J. R.

Spilman, A. K.

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence,” Proc. SPIE, 6667, 666701 (2007).

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-46-1-61.
[CrossRef]

A. K. Spilman and T. G. Brown, “Stress-induced Focal Splitting,” Opt. Express 15, 8411–8421 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8411.
[CrossRef] [PubMed]

Tan, B.

K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603 (2006).
[CrossRef]

Török, P.

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

Venkatakrishnan, K.

K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603 (2006).
[CrossRef]

Youngworth, K. S.

Yuan, X.-C.

Zhan, Q.

Zhu, S. W.

Adv. Opt. Photon. (1)

Appl. Opt. (2)

C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=ao-43-22-4322.
[CrossRef] [PubMed]

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-46-1-61.
[CrossRef]

J. Micromech. Microeng. (1)

K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603 (2006).
[CrossRef]

J. Opt. Soc. Am. A (2)

R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-22-7-1420.
[CrossRef]

N. Moore and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A 29, 2211–2218 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=josaa-26-7-1754.
[CrossRef]

Jpn. J. Appl. Phys. (1)

N. Hayazawa, “Focused Excitation of Surface Plasmon Polaritons Based on Gap-Mode in Tip-Enhanced Spectroscopy,” Jpn. J. Appl. Phys. 46, 7995 (2007).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

E. G. Sauter, “Gaussian beams and the Poincare sphere,” Microwave Opt. Technol. Lett. 4, 485–486 (1991).
[CrossRef]

Opt. Commun. (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing Light to a Tighter Spot,” Opt. Commun. 179, 1 (2000).
[CrossRef]

Opt. Express (7)

P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-10-411.
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000), http://wwww.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-2-77.
[CrossRef] [PubMed]

D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric surface,” Opt. Express 9, 490–497 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-490.
[CrossRef] [PubMed]

W. Chen and Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-7-4106.
[CrossRef] [PubMed]

A. K. Spilman and T. G. Brown, “Stress-induced Focal Splitting,” Opt. Express 15, 8411–8421 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8411.
[CrossRef] [PubMed]

G. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-7-4567.
[CrossRef] [PubMed]

N. Moore and M. A. Alonso, “Closed-form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5926.
[CrossRef] [PubMed]

Opt. Lett. (4)

Optik (1)

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Proc. SPIE, (1)

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence,” Proc. SPIE, 6667, 666701 (2007).

Supplementary Material (1)

» Media 1: MOV (1586 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

(Media 1) Evolution of the distribution of polarization ellipses for FP beams where ê1 and ê2 are, respectively, (a) RHC and LHC, (b) LHC and RHC, and (c) vertical and horizontal. In all movies, the radial distance is scaled by w(z) and the angle between the x axis and the purple line represents ϕξ(z).

Fig. 2.
Fig. 2.

Experimental arrangement, in which the SEO element is placed in front of a 400 mm focal length lens (L). The illumination source polarization is controlled through a combination linear polarizer/quarter wave plate (P), and passed through an analyzer (A) placed in front of the camera.

Fig. 3.
Fig. 3.

Cross section of the irradiance profile near the beam waist. The total irradiance is shown by the (black) solid line. Dots (red) denote the left hand circular component; Crosses (blue) denote the right hand circular component. The dashed lines represent the best fit Gaussian and Laguerre-Gauss functions, as well as their sum. The beam size of the Gaussian was measured to 0.181±0.002 mm.

Fig. 4.
Fig. 4.

Normalized Stokes maps of the beam sampled at four longitudinal positions within the Rayleigh range. Left: Simulation, based on an ideal superposition with the beam sampled at (a) z = 0; (b) z = 0.3zR; (c) z = 0.67zR; (d) z = zR; Right: Experimental Stokes maps for the beam of Fig. 3 (a) z = 0; (b) z = 60 mm (0.36zR); (c) z = 110 mm (0.67zR); (d) z = 170mm (1.03zR).

Fig. 5.
Fig. 5.

The measured value of tan(ϕξ) deduced from the zeros of s1 plotted as a function of propagation distance z. The linear range is the region over which the experimental beam matches the evolution predicted by LG beam propagation. The dashed line is a linear fit to the central region.

Equations (13)

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

U00(r)=u0ξ(z) exp [ikzρ2w02ξ(z)] ,
ξ(z)=1+i2zkw02=1+izzR,
U01(r)=w02 (x+iy)U00(r)=2 x+iyw0ξ(z) U00 (r).
EP(r;γ)=cosγê1U00(r)+sinγê2 U01 (r) ,
|EP(r;γ)|2=cos2γ|U00(r)|2+sin2 γ |U01(r)|2
=[cos2γ2ρ2w2(z)sin2γ]u02|ξ(z)2exp[2ρ2w2(z)],
EP(r;γ)=(ê1+exp{i[ϕϕξ(z)]}2ρwtanγê2) cos γ u0w0w exp (ρ2w2) ,
S1S0=2ρ̄cos(ϕϕξ)1+ρ̄2,S2S0=2ρ̄sin(ϕϕξ)1+ρ̄2,S3S0=1ρ̄21+ρ̄2 .
J=(ρ)=cos(12cρm2)+isin(12cρm2)(2βϕ),β=2m2,
EP(ρ)=g(ρ)cos(12) ê1 +g(ρ)esin(12) ê2 .
EE(r;p)=(p+p·k2)sin(kr·r)kr·r,EM(r;p)=ip×ksin(kr·r)kr·r,
EP(r;γ)=ikqu0sinh(kq) { cosγ2 [EE(riqẑ;x̂+iŷ2)iEM(riqẑ;x̂+iŷ2)]
+kq2sinγ[EM(riqẑ;ẑ)+iEE(riqẑ;ẑ)]},

Metrics