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Optimal annulus structures of optical vortices

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Abstract

An idea of an optimal annulus structure phase mask with helical wavefront is suggested; the resulting helical mode can be focused into a very clear optical vortex ring with the best contrast. Dependences of the optimal annulus width and the radius of the optical vortex ring on topological charge are found. The desired multi-optical vortices as the promising dynamic multi-optical tweezers are realized by extending our idea to the multi-annulus structure. Such multi-optical vortex rings allow carrying the same or different angular momentum flux in magnitude and direction. The idea offers flexibility and more dimensions for designing and producing the complicated optical vortices. For the Gaussian beam illumination, the optimal spot size, which ensures the high energy/power efficiency for generating the best contrast principal ring, is also found.

©2004 Optical Society of America

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Figures (8)

Fig. 1.
Fig. 1. (a) Spoke-like structure phase mask with ℓ - 40 and (b) corresponding image of optical vortex produced.
Fig. 2.
Fig. 2. Intensity distribution of ℓ = 40 optical vortices created by the annulus structure phase mask of R 0 = 256 pixels, along the radial direction when R 1 is (a) 0, (b) 125, (c) 200 and (d) 230 pixels, respectively.
Fig. 3.
Fig. 3. Dependence of the peak intensity of the principal ring (a) and the first subsidiary ring (b) on the annulus width at different topological charges. The solid, dash, short dash, dash-dot, and dash-dot-dot lines are ℓ = 10, 20, 30, 40, and 50, respectively.
Fig. 4.
Fig. 4. Variation of the peak intensity of the principal ring with the spot size of Gaussian beam when R 0 =256 pixels, ℓ = 40 and ΔRRopt = 50 pixels. The solid line is the theoretical values, while the circles are the experimental results.
Fig. 5.
Fig. 5. (a) An example of the helical mode annulus phase mask, (b) simulated intensity pattern of optical vortex, (c) simulated phase pattern of optical vortex, and (d) experimentally observed intensity pattern of optical vortex.
Fig. 6.
Fig. 6. (a) and (c) are two examples of bi-annulus phase masks, and (b) and (d) are the respective optical vortices produced. Bi-optical vortices have the opposite angular momentum directions in Fig. 6(b) and have the same radii and directions in Fig. 6(d).
Fig. 7.
Fig. 7. Observed patterns of optical vortices in experiments for the non-perfect phase masks with the phase variation ranging from zero to 1.5π. (a), (b), (c) and (d) correspond to Fig. 1, Fig. 5, Fig. 6(b) and Fig. 6(d), respectively.
Fig. 8.
Fig. 8. Dependence of the depth of the azimuthal modulation on the phase deviation for the optimal annulus phase mask with R 0 = 256 pixels, ℓ = 40 and ΔRopt ((40) = 50 pixels.

Equations (7)

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u f ρ ϕ = 2 π ( i ) exp ( i ϕ ) 0 R 0 rdr J ( 2 π λf ρr )
= 2 π R 0 2 p ( i ) exp ( i ϕ ) ( 2 + ) Γ ( 1 + ) 1 F 2 [ 1 + 2 , ( 2 + 2,1 + ) ; p 2 ]
Δ R opt ( ) = 1.4043 R 0 0.5363
ρ P ( ) = 2.1140 ( 1 + 0.2458 ) λf π R 0
u f ρ ϕ = 2 π ( i ) exp ( i ϕ ) ( Δ r ) 2 m = 1 N m u 0 ( m Δ r , ω ) J ( 2 πm Δ r λf ρ )
u 0 r ω = A 0 ω exp ( r 2 ω 2 )
ω opt = 1.43 R m
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