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Helico-conical optical beams: a product of helical and conical phase fronts

Carlo Amadeo Alonzo, Peter John Rodrigo, and Jesper Glückstad

Open Access Open Access

Abstract

Helico-conical optical beams, different from higher-order Bessel beams, are generated with a parallel-aligned nematic liquid crystal spatial light modulator (SLM) by multiplying helical and conical phase functions leading to a nonseparable radial and azimuthal phase dependence. The intensity distributions of the focused beams are explored in two- and three-dimensions. In contrast to the ring shape formed by a focused optical vortex, a helico-conical beam produces a spiral intensity distribution at the focal plane. Simple scaling relationships are found between observed spiral geometry and initial phase distributions. Observations near the focal plane further reveal a cork-screw intensity distribution around the propagation axis. These light distributions, and variations upon them, may find use for optical trapping and manipulation of mesoscopic particles.

©2005 Optical Society of America

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References

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2004 (5)

2003 (4)

2002 (1)

1999 (1)

1998 (1)

J. Courtial, “Self-imaging beams and the Guoy effect,” Opt. Commun. 151, 1–4 (1998).
[Crossref]

1997 (2)

1996 (1)

1995 (1)

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[Crossref]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

1992 (1)

1984 (1)

1974 (1)

Allen, L.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref] [PubMed]

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M.J. Padgett, and M. Babiker , “The Orbital Angular Momentum of Light,” in Progress in Optics39, E. Wolf, ed. (Elsevier, Amsterdam, 1999).
[Crossref]

Babiker, M.

L. Allen, M.J. Padgett, and M. Babiker , “The Orbital Angular Momentum of Light,” in Progress in Optics39, E. Wolf, ed. (Elsevier, Amsterdam, 1999).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259–268 (2004).
[Crossref]

M. V. Berry , “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices…),” in Singular Optics, M.S. Soskin, ed., Proc. SPIE3487, 1–5 (1998).

Biener, G.

Bryngdahl, O.

Campbell, P.A.

P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593.
[Crossref] [PubMed]

Campos, J.

Carcole, E.

Cederquist, J.

Chang, J.-S.

Chattrapiban, N.

Cofield, D.

Cottrell, D. M.

Courtial, J.

J. Courtial, “Self-imaging beams and the Guoy effect,” Opt. Commun. 151, 1–4 (1998).
[Crossref]

Crabtree, K.

Curtis, J. E.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[Crossref] [PubMed]

J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003).
[Crossref] [PubMed]

Cuschieri, A.

P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593.
[Crossref] [PubMed]

Daria, V. R.

V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325 (2004).
[Crossref]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time interactive optical micromanipulation of a mixture of high- and low-index particles,” Opt. Express12, 1417–1425 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1417.
[Crossref] [PubMed]

Davis, J. A.

Dholakia, K.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref] [PubMed]

P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593.
[Crossref] [PubMed]

Frank, T. G.

P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593.
[Crossref] [PubMed]

Gahagan, K. T.

Glückstad, J.

V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325 (2004).
[Crossref]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time interactive optical micromanipulation of a mixture of high- and low-index particles,” Opt. Express12, 1417–1425 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1417.
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, Second Edition (McGraw-Hill, New York, 1996).

Grier, D. G.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[Crossref] [PubMed]

J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003).
[Crossref] [PubMed]

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express12, 1144–1149 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144.
[Crossref] [PubMed]

Hasman, E.

Heckenberg, N. R.

Hershcovitz, O.

Hill III, W. T.

Iemmi, C.

Jeon, J.-H.

Kim, G.-H.

Kleiner, V.

Ko, K.-H

Ladavac, K.

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express12, 1144–1149 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144.
[Crossref] [PubMed]

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.6, 71 (2004), http://www.iop.org/EJ/abstract/1367-2630/6/1/071.
[Crossref]

Lee, J.-H.

Lee, W. M.

Lipson, S. G.

MacDonald, M. P.

P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593.
[Crossref] [PubMed]

Marquez, A.

McDuff, R.

Moed, S.

Moon, H.-J.

Moreno, I.

Niv, A.

Padgett, M. J.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref] [PubMed]

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.6, 71 (2004), http://www.iop.org/EJ/abstract/1367-2630/6/1/071.
[Crossref]

Padgett, M.J.

L. Allen, M.J. Padgett, and M. Babiker , “The Orbital Angular Momentum of Light,” in Progress in Optics39, E. Wolf, ed. (Elsevier, Amsterdam, 1999).
[Crossref]

Prentice, P. A.

P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593.
[Crossref] [PubMed]

Rodrigo, P. J.

V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325 (2004).
[Crossref]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time interactive optical micromanipulation of a mixture of high- and low-index particles,” Opt. Express12, 1417–1425 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1417.
[Crossref] [PubMed]

Rogers, E. A.

Rotschild, C.

Roy, R.

Sibbett, W.

P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593.
[Crossref] [PubMed]

Simpson, N. B.

Smith, C. P.

Soskin, M.S.

M.S. Soskin and M.V. Vasnetsov , “Singular Optics,” in Progress in Optics42, E. Wolf, ed. (Elsevier, Amsterdam, 2001).

Spalding, G. C.

P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593.
[Crossref] [PubMed]

Swartzlander, G. A.

Tai, A. M.

Tao, S. H.

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Vasnetsov, M.V.

M.S. Soskin and M.V. Vasnetsov , “Singular Optics,” in Progress in Optics42, E. Wolf, ed. (Elsevier, Amsterdam, 2001).

White, A. G.

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.6, 71 (2004), http://www.iop.org/EJ/abstract/1367-2630/6/1/071.
[Crossref]

Yuan, X.

Yzuel, M. J.

Zommer, S.

Appl. Opt. (7)

Appl. Phys. Lett. (1)

V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325 (2004).
[Crossref]

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

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259–268 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (3)

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

J. Courtial, “Self-imaging beams and the Guoy effect,” Opt. Commun. 151, 1–4 (1998).
[Crossref]

Opt. Lett. (5)

Phys. Rev. Lett. (1)

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[Crossref] [PubMed]

Other (8)

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express12, 1144–1149 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144.
[Crossref] [PubMed]

P. A. Prentice, M. P. MacDonald, T. G. Frank, A. Cuschieri, G. C. Spalding, W. Sibbett, P.A. Campbell, and K. Dholakia, “Manipulation and filtration of low-index particles with holographic Laguerre-Gaussian optical trap arrays,” Opt. Express12, 593–600 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-593.
[Crossref] [PubMed]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time interactive optical micromanipulation of a mixture of high- and low-index particles,” Opt. Express12, 1417–1425 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1417.
[Crossref] [PubMed]

M. V. Berry , “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices…),” in Singular Optics, M.S. Soskin, ed., Proc. SPIE3487, 1–5 (1998).

M.S. Soskin and M.V. Vasnetsov , “Singular Optics,” in Progress in Optics42, E. Wolf, ed. (Elsevier, Amsterdam, 2001).

L. Allen, M.J. Padgett, and M. Babiker , “The Orbital Angular Momentum of Light,” in Progress in Optics39, E. Wolf, ed. (Elsevier, Amsterdam, 1999).
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.6, 71 (2004), http://www.iop.org/EJ/abstract/1367-2630/6/1/071.
[Crossref]

J. W. Goodman, Introduction to Fourier Optics, Second Edition (McGraw-Hill, New York, 1996).

Supplementary Material (2)

Media 1: AVI (986 KB)     
Media 2: GIF (268 KB)     

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Figures (11)
Fig. 1.
Fig. 1. Setup for generation of phase-encoded beams and corresponding Fourier transforms.

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Fig. 2.
Fig. 2. Typical phase masks encoded onto the SLM with (a) K=1, (b) K=0. Lighter gray-level indicates greater phase. In pseudo-color unwrapped representation in (c) K=1 and (d) K=0, red has highest phase while blue has lowest.

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Fig. 3.
Fig. 3. Intensity distribution at the focal plane with K=1. (a) image captured from CCD, increasing intensity from black to white; (b) numerical simulation, increasing intensity from violet to red; (c) a simple arithmetic spiral is superimposed in red, on top of the numerical simulation in pseudo-color, and the captured image in gray-scale.

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Fig. 4.
Fig. 4. Intensity distribution at the focal plane with K=0. (a) image captured from CCD; (b) numerical simulation; (c) a simple arithmetic spiral is superimposed in red, on top of the numerical simulation in pseudo-color, and the captured image in gray-scale.

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Fig. 5.
Fig. 5. Diffraction from a linear blazed grating encoded on the SLM. The experimentally obtained diffraction orders are labeled as -3, -2, -1, 0, and 1, respectively.

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Fig. 6.
Fig. 6. Spot diagram of local spatial frequencies (ξ′,ζ′) for (a) K=1, and (b) K=0. Density of plotted points approximately corresponds to observable intensity on the focal plane.

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Fig. 7.
Fig. 7. Linear scaling of spirals at the focal with (a) K=1 and (b) K=0. Radial distances from the origin were measured at ϕ=π. Phase functions encoded onto SLM used ℓ -values ranging from 5 to 100, and ro -values of 3.0 mm (red circles), 4.5 mm (green squares), and 5.5 mm (blue triangles).

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Fig. 8.
Fig. 8. (AVI, 1.067 MB) Propagation of focused beam with K=1. Images (a) through (i) cover a total distance of ~40 mm, with (a) being closest to the lens, and (e) at the focal plane.

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Fig. 9.
Fig. 9. (animated GIF, 261 kB) Numerical simulation of focused beam propagation with K=1. Distances from focal plane used to calculate images are (a) -20 mm, (b) -16 mm, (c) -12 mm, (d) -6 mm, (e) 0 mm, (f) 6 mm, (g) 12 mm, (h) 16 mm, and (i) 20 mm.

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Fig. 10.
Fig. 10. (AVI, 1.01 MB) Propagation of focused beam with K=0. Images (a) through (i) cover a total distance of ~40 mm, with (a) being closest to the lens, and (e) at the focal plane.

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Fig. 11.
Fig. 11. (animated GIF, 275 kB) Numerical simulation of focused beam propagation with K=0. Distances from focal plane used to calculate images are (a) -20 mm, (b) -16 mm, (c) -12 mm, (d) -6 mm, (e) 0 mm, (f) 6 mm, (g) 12 mm, (h) 16 mm, and (i) 20 mm.

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Equations (12)

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ψ ℓ = ℓ θ .
ψ ( r , θ ) = ℓ θ − 2 π r ⁄ r 0 ,
ψ ( r , θ ) = ℓ θ ( K − r ⁄ r 0 ) ,
u ( ρ , ϕ ) = ∫ 0 2 π ∫ 0 ∞ circ ( r ⁄ r 0 ) exp [ i ψ ( r , θ ) ] exp [ − i 2 π r ρ cos ( θ − ϕ ) ] r d r d θ .
m θ = ℓ θ ⁄ r 0 .
ψ K = 1 = ℓ θ − m θ r .
ψ K = 0 = − m θ r ,
G ( ξ , ζ ) = ∫ − ∞ ∞ ∫ − ∞ ∞ g ( x , y ) exp [ − i 2 π ( ξ x + ζ y ) ] d x d y ,
g ( x , y ) = a ( x , y ) exp [ i ψ ̅ ( x , y ) ] ,
ξ ′ = 1 2 π ∂ ∂ x ψ ̅ ( x , y ) and ζ ′ = 1 2 π ∂ ∂ y ψ ̅ ( x , y ) .
ξ ′ = ℓ 2 π r 0 [ − θ cos θ − ( r 0 − r r ) sin θ ] and ζ ′ = ℓ 2 π r 0 [ − θ sin θ + ( r 0 − r r ) cos θ ] .
ξ ′ = ℓ 2 π r 0 [ − θ cos θ + sin θ ] and ζ ′ = ℓ 2 π r 0 [ − θ sin θ − cos θ ] .

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