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

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

2009 (2)

2008 (2)

2007 (5)

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]

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).

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]

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 (Stuttg.) 104, 175–177 (1997).

1996 (1)

1994 (1)

1991 (1)

E. G. Sauter, “Gaussian beams and the Poincare sphere,” Microw. 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.

Bruce, Z.

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]

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.

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,” Microw. Opt. Technol. Lett. 4, 485–486 (1991) .
[CrossRef]

Sheppard, C. J. R.

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]

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

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-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]

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 (Stuttg.) 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.

Appl. Opt. (2)

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)

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]

Microw. Opt. Technol. Lett. (1)

E. G. Sauter, “Gaussian beams and the Poincare sphere,” Microw. 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://www.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]

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]

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]

Opt. Lett. (4)

Optik (Stuttg.) (1)

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

Other (1)

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]

Supplementary Material (1)

» Media 1: MOV (1586 KB)     

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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)]},

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