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Tuning the orbital angular momentum in optical vortex beams

Christian H. J. Schmitz, Kai Uhrig, Joachim P. Spatz, and Jennifer E. Curtis

Open Access Open Access

Abstract

We introduce a method to tune the local orbital angular momentum density in an optical vortex beam without changing its topological charge or geometric intensity distribution. We show that adjusting the relative amplitudes a and b of two interfering collinear vortex beams of equal but opposite helicity provides the smooth variation of the orbital angular momentum density in the resultant vortex beam. Despite the azimuthal intensity modulations that arise from the interference, the local orbital angular momentum remains constant on the vortex annulus and scales with the modulation parameter, c = (a-b)/(a+b).

©2006 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. (a-c)Phase masks of interference vortices with ℓ = 20 and (d-f) their measured intensity for mixing amplitudes: c = 1,0.2 and 0. As c decreases, more intensity is directed from the minima to the 2ℓ maxima on the ring to the maxima. A mixing amplitude c = 0 gives a binary intensity distribution.

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Fig. 2.
Fig. 2. Azimuthal dependence of the normalized intensity, I/(a 2 + b 2) and the normalized phase derivative and local wavefront helicity, (∂φ/∂θ)/#x2113;. The lines represent calculated values using Eqns. 3 and 5 with ℓ = 20 and c = 0.6 (dark lines) and c = 0.2 (light lines). The data symbols represent the measured local intensities whose errors are in the range of the symbol size.

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Fig. 3.
Fig. 3. (a) The measured period T for the modulation parameter c = 1.0,0.3,0.2,0.1 depends parabolically on the mean helicity or equivalently the topological charge, Q.(b) The relative rotational frequency Ω = T min /T as a function of c. The line represents s(c)-1 in Eq. (6), which is expressed analytically in Eq. (10). The maximum rotational speed (c = ±1) was 13 μm/s.

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Fig. 4.
Fig. 4. An optical vortex array consisting of three optical vortices with ℓ = 8. c 1 of vortex 1 is varied between 1 and -1 while vortices 2 and 3 have fixed c 2 = 0.17 and c 3 = -1. The time series shows the independent control of the rotation of three colloidal rings: (a) Ring 1 is slowly rotated counterclockwise, (b) stopped, and (c) rotated clockwise with increasing speed (c,d). The other rings (2,3) are rotated with constant velocity of different magnitude and sign. The bead in the center is fixed by a regular optical trap.

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Equations (10)

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

u i v = A ( r ) [ a e i ( ℓ θ + φ a ) + b e − i ( ℓ θ + φ b ) ] = B ( r , θ ) e i φ ( θ ) ,
φ ( θ ) = arctan [ c tan ( ℓ ( θ + α ) ) ] + ( φ a + φ b ) 2 ,
I ∝ ∣ u ˜ i v ( r ' , θ ' ) ∣ 2 = A ˜ ( r ' ) 2 [ a 2 + b 2 + 2 ab cos ( 2 ℓ ( θ ' + α ) ) ] ,
u ˜ i v ( r ' , θ ' ) = A ˜ ( r ' ) [ a e i ( ℓ θ ' + φ a ) + b e − i ( ℓ θ ' − φ b ) ] = B ˜ ( r ' , θ ' ) e i φ ˜ ( θ ' ) ,
∂ φ ∂ θ = c ℓ cos 2 [ ℓ ( θ + α ) ] + c 2 sin 2 [ ℓ ( θ + α ) ] ,
T = s ( c ) [ 1 + t ℓ 2 ] .
p = ε 0 2 Re [ E * × B + E × B * ] = i ω ε 0 2 ( u * ∇ u − u ∇ u * ) + ω k ε 0 ∣ u ∣ 2 z ̂ .
p = ε 0 ω [ 1 r ∂ φ ∂ θ ∣ u ∣ 2 ] θ ̂ .
M z = ε 0 ω ∂ φ ∂ θ ∣ u ˜ i v ∣ 2 = ε 0 ω 4 a 2 c ℓ ( 1 + c ) 2 A ˜ ( r ' ) 2 = ε 0 ω 4 b 2 c ℓ ( 1 + c ) 2 A ˜ ( r ' ) 2 ,
Ω ( c ) = M z ( c ) M z ( c = ± 1 ) = ± 4 c ( 1 ± c ) 2 .
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