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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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)

New J. Phys. (1)

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]

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. Express (3)

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 (4)

M. V. Berry, �??Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices�?� ),�?? in Singular Optics, M.S. Soskin, ed., Proc. SPIE 3487, 1-5 (1998).

M.S. Soskin and M.V. Vasnetsov, �??Singular Optics,�?? in Progress in Optics 42, E. Wolf, ed. (Elsevier, Amsterdam, 2001).

L. Allen, M.J. Padgett and M. Babiker, �??The Orbital Angular Momentum of Light,�?? in Progress in Optics 39, E. Wolf, ed. (Elsevier, Amsterdam, 1999).
[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.

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.

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.

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.

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.

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.

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 -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).

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.

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.

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.

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.

Equations (12)

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ψ = θ .
ψ ( r , θ ) = θ 2 π r r 0 ,
ψ ( r , θ ) = 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 = θ r 0 .
ψ K = 1 = 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 = 2 π r 0 [ θ cos θ ( r 0 r r ) sin θ ] and ζ = 2 π r 0 [ θ sin θ + ( r 0 r r ) cos θ ] .
ξ = 2 π r 0 [ θ cos θ + sin θ ] and ζ = 2 π r 0 [ θ sin θ cos θ ] .

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