1. Introduction

Light carries angular momentum comprised of both a spin component associated with polarization [1, 2], and of an orbital component arising from the spatial profile of light intensity and the phase [3]. Considerable interest in orbital angular momentum arises from its potential use in multiple applications including quantum information processing [4, 5, 6, 7, 8, 9], atomic manipulation [10, 11, 12, 13], micromanipulation [14, 15, 16] and the biosciences [17, 18]. Control of the local and total orbital angular momentum in a beam, as well as the quantum mechanical state of the photons is important in many of these applications [4] as is the spatial distribution and photon density of the beam.

In general, any beam with inclined wavefronts carries orbital angular momentum. Allen and coworkers [3] showed that Laguerre-Gaussian laser beams, which have a helical phase structure described by ϕ = ℓθ, possess a well-defined orbital angular momentum of ℓh̅ per photon. At the center of the helical phase is a singularity where the phase is undefined and the field amplitude vanishes, giving rise to a “dark beam” in the core of the light wave. The orbital angular momentum in a helical beam (also called an optical vortex beam) can be adjusted by either changing the wavefronts’ helicity, ℓ, or by increasing the photon flux. However, since the radius of an optical vortex scales with its helicity [19, 20], applications that require controlled geometry or photon density will be limited to fixed orbital angular momentum (OAM).

This article introduces an OAM-tunable optical vortex which maintains a constant geometry and total intensity during tuning (Fig. 1). The OAM-tunable optical vortex is the resultant beam produced by the overlap of two collinear optical vortex beams of equal helicity but opposite chirality. We show that the local orbital angular momentum density of this complex optical vortex can be smoothly tuned without adjusting the resultant beam’s topological charge, which is shown to be equivalent to the magnitude of the helicity of the two component beams. Surprisingly, despite the rapid variation of the photon flux with azimuthal position, the local orbital angular momentum density for a given interference vortex remains constant. The final result is the decoupling of the orbital angular momentum, the topological charge, the geometry and the power distribution in this class of helical beam.

2. Interference of two oppositely charged optical vortex beams

We experimentally and theoretically address the “interference” optical vortex produced by the superposition of two collinear optical vortices with variable relative amplitudes and equal but opposite helicity, ℓ. The scalar field of a classic optical vortex is u = A(r)e ^(iℓθ) where ℓ is a positive or negative integer and A(r) is real. An interference vortex is described by the sum of two such scalar fields,

with constant, positive, real amplitudes a, b and phase offsets φ_a and φ_b. In agreement with intuitive arguments, the mixing amplitudes a and b govern the relative contribution of each optical vortex to the local and total orbital angular momentum. The quantitative dependence on the mixing amplitudes is derived in Section 4.

Collinear optical vortex beams have been presented in various contexts. The optical cogwheel beam [21, 22] is an interference vortex with a = b (Fig. 1(f)). It carries no orbital angular momentum, and possesses a modulated intensity profile about the azimuthal coordinate and with 2ℓ intensity peaks. Other work shows that the interference of two helical beams with different ℓ-values produces concentric optical vortices of different radii [23, 24]. In a more general approach, Soskin and coworkers address the orbital angular momentum in multiple collinear vortices of equal amplitude and varying topological charges [25]. Maleev and Swartzlander [26] address the intensity distribution and motion of optical phase singularities of noncollinear overlapping optical vortices. Segev [27] presents a theoretical treatment of the orbital angular momentum content in stable necklace-ring beams in a self-focusing Kerr medium. A subclass of necklace-ring beams are equivalent to the interference vortex introduced here. Very recent work has introduced the above formulation to study thermally activated transport on tilted washboard energy landscape potentials [28].

 

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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To investigate the angular momentum content of an interference optical vortex, we create an optical vortex trap using a holographic optical tweezers (HOT) apparatus [15]. Optical traps are particularly useful for measuring the angular momentum content of focused beams [14, 29, 30]. The key component of our HOT apparatus, as previously described [31], is a computer-addressable reflective liquid crystal display called a spatial light modulator (SLM) which imposes the phase profile of the desired interference vortex onto an incoming collimated beam. The phase profile of an interference vortex extracted from Eq. (1) is

where α = (φ_a - φ_b)/(2ℓ). This expression accurately describes the phase for values between -π/(2ℓ) < θ + α < π/(2ℓ). For θ + α = π/(2ℓ) the phase is π/2. This wedge of phase plus a multiple of π is repeated around the axis in 2ℓ slices from 0 < θ < 2π as shown in Fig. 1(a–c). The mixing amplitudes a and b determine the modulation amplitude, c = (a-b)/(a+b). To explicitly calculate the phase at all values of θ , we introduce an extended expression [32].

When creating an optical vortex with HOT, the beam’s amplitude is determined by the input beam, which is typically Gaussian. Theory shows that the discrepancy between the theoretical amplitude and the true amplitude of the beam does not significantly impact the resultant beam’s intensity distribution or the orbital angular momentum [20]. An image of this beam is then transferred to the back focal plane of a high numerical aperture objective lens which tightly focuses the beam into the focal plane of a microscope, where it may be imaged with and without trapped objects.

 

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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Figure 1(a–c) shows three phase masks calculated using Eq. (2) with c = 1.0,0.2 and 0. An interference vortex has the phase of a classical optical vortex when c = ±1. When c = 0, the phase profile is binary with 2ℓ alternating phase segments of 0 and π, equivalent to the phase of the optical cogwheel. For 0 < |c| < 1, the local curvature of the helical wavefront is no longer constant nor linear; however the phase singularity remains, qualifying the beam as a kind of optical vortex [25].

Figure 1(d–f) shows the corresponding intensity distributions imaged with the HOT setup. For all values of the modulation parameter, c, a ring of light appears with 2ℓ radial intensity fringes that vary with the azimuthal angle, as previously described for both the classical optical vortex trap produced by HOT [33] and the optical cogwheel [21]. Adjusting the relative constant of phase of the two overlapping vortices, α, rotates the intensity fringes about the center of the resultant beam. The intensity distribution between the bright and dark fringes varies as a function of the modulation parameter c, as the measured data show in Fig. 2. Measurements show that the total intensity in the beam is conserved with varying c, and that it is merely redistributed between the dark and bright fringes.

This variation agrees with the theoretical prediction of the intensity distribution of an interference vortex in the image plane of the microscope,

plotted as the solid lines overlapping the data points. The field in the SLM plane, u_iv(r,q) is related to the field in the microscope’s image plane, ũ_iv(r´, q´), by a Fourier transform, which has been shown to preserve the beam’s helical phase and give rise to a radially-dependent amplitude for a single optical vortex [20]. Hence:

where $\tilde{\phi }$ (θ´) ≡ φ(θ) and ∂$\tilde{\phi }$ /∂θ´ ≡ ∂φ/∂θ.

The radius of the interference vortex trap is independent of c and linearly proportional to ℓ like that of a classic optical vortex trap [20]. However, unlike a conventional optical vortex, variations in the local wavefront helicity ∂φ/∂θ do not result in an azimuthally dependent radius R(θ´) ∝ (R ₀ + ∂φ/∂θ) [34]. While this ansatz has been used to dramatically vary the geometry of conventional optical vortices [35], the interference vortex whose local helicity is modulated with 2ℓ peaks (Fig. 2),

 

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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lacks the 2ℓ intensity flower-like lobes that would appear on a conventional vortex. The difference is that the amplitude of an interference vortex depends on the azimuthal coordinate while the amplitude of the optical vortex does not. This apparent limitation is in fact what makes the interference vortex a potentially interesting tool. As is shown in the following section, the vortex’s radius remains constant despite changes in ∂φ/∂θ while its local orbital angular momentum density is tuned smoothly.

3. Smooth variation of the orbital angular momentum

Measurements of the local orbital angular momentum density can be made by studying the velocity of a trapped particle as it spins around the annulus of a helical beam [20, 30, 36, 37, 38]. In practice, only a small percentage of the orbital angular momentum is transmitted through scattering [37]. Nevertheless, the particle speed scales as expected with increases in power, P, and with the predicted orbital angular momentum in the beam which depends on ℓ. In some regimes, the azimuthal intensity variations interfere with the transport of particles around the vortex [20, 28]. This occurs when the variations are large enough to locally trap particles in an optical potential well deeper than k_BT. To avoid this effect, each c-dependent interference optical vortex was filled completely with 2 μm diameter polystyrene microspheres to ensure smooth rotation. Their resulting collective frequency of rotation was determined using video microscopy and particle tracking.

Our experiments show that the interference optical vortex transfers orbital angular momentum to the particles when c ≠ 0. Figure 3(a) shows the period of rotation T measured for four different values of c as a function of the topological charge, Q = 〈∂φ/∂θ〉_θ, given by the beam’s mean helicity [39]. For c > 0, Q = ℓ and for c < 0, Q = -ℓ. Particle rotation is similar to that of a single particle on a classical optical vortex whose period scales as T ~ ℓ² [20] but with an extra dependence on c,

Here, t is constant for all c and ℓ, while s depends on the modulation parameter c. The period changes systematically with different c-values even with fixed ℓ. This is a rare demonstration of the fact that orbital angular momentum and topological charge are conceptually independent [25]. It is possible to have non-zero orbital angular momentum when Q = 0 [40].

Particles spin fastest on a classical vortex (c = ±1) resulting in the shortest period T_min ∝ ℓ². The normalized rotational frequency Ω = T_min/T is constant for all ℓ at fixed c as shown in Fig. 3(b). Each data point constitutes the average of several experimental Ω values at different Q. These observations imply that the average orbital angular momentum can be smoothly varied by changing the modulation parameter c independently of ℓ and P.Meanwhile, ℓ and P uniquely determine the maximum rotational frequency ω ∝ P/ℓ² and the interference vortex’s radius, R ∝ ℓ. An additional advantage is the possibility to tune the particle velocity from full speed to zero without decreasing the intensity such that thermal forces dominate the radial optical gradient force that confines particles to the optical vortex trap.

4. Local orbital angular momentum is constant

Our measurements of colloidal rotation on an interference optical vortex do not definitively measure the local orbital angular momentum density in the beam. Given the modulated azimuthal intensity, the angular momentum may vary with position and yet be effectively averaged out by motion on a particle-filled optical vortex. Thus, we consider theoretically how the local orbital angular momentum density varies.

The angular momentum density in a transverse electromagnetic field is M= r×p where the linear momentum density is p = ϵ₀ E×B. Following the lead of Allen [3], a collinear interference optical vortex trap has a scalar field ũ_iv, which is related to the vector potential in the Lorentz gauge by A= ũ_iv exp(-i(kz+ωt)) x̂ with E=-∂A/∂t and B=∇×A. In the paraxial approximation, the time average of the real part of p for any scalar field u is given by

Here ẑ is the unit vector in the z direction and ∇ is the gradient operator in r and θ. Thus, the azimuthal component of linear momentum density for an arbitrary scalar field with inclined wavefronts described by any azimuthally-dependent phase φ(θ) is

Even when the amplitude has an azimuthal dependence, the momentum depends only on the phase and the total intensity of the beam.

The local orbital angular momentum density about the z-axis of an interference OV in the trapping plane is thus

for u = ũ_iv. Despite the complicated functional dependence of the intensity |ũ_iv|² and the local helicity ∂φ/∂θ shown in Fig. 2, they exactly cancel out to give a constant local orbital angular momentum density at a given radius, r´. Two examples are shown in Fig. 2 for c = 0.6 and c = 0.2. The local helicity is an oscillatory function with 2ℓ peaks coincident with the intensity minima of the optical vortex at θ´ + α = nπ/2ℓ, where n is an odd integer between 0 and 2ℓ. While the magnitude of the minimum photon flux decreases with decreasing c, the corresponding local helicity increases proportionally. The result is that the local orbital angular momentum is constant at all azimuthal positions on the interference vortex at a fixed radius.

 

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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The orbital angular momentum carried by a classical optical vortex is M_z(c = 1) ∝ a ²ℓ and M_z(c=-1)∝-b ²ℓ. Thus, the normalized frequency of rotation for a particle around the vortex is given by

The shape of this curve captures the experimentally measured dependence of the normalized frequency on the modulation parameter c. The agreement between theory and data is shown in Fig. 3(b). The solid line represents the theoretical prediction and is equivalent to s ^-1 as defined in Eq. (6). Thus, we have presented a deterministic approach to tune both the local and total orbital angular momentum density in an optical vortex beam of fixed radius and fixed total power.

5. Tunable optical vortex arrays

As described previously [15, 41, 42], one to dozens of classical optical vortices may be created simultaneously and located independently in the focal volume using HOT. Each of these optical vortices may have a unique topological charge and therefore, size and orbital angular momentum content that is controlled by the input parameter ℓ_i, where 0 ≤ i ≤ N and N is the total number of traps. Here it is shown that by replacing the standard input phase for each desired vortex, ϕ_i = ℓ_iθ, with that of an interference vortex (Eq. (2)), we can create arrays of identically-sized optical vortices with smoothly adjustable rotational frequencies. In the example shown here, the phase mask for a system consisting of three vortices is calculated. The three interference vortices are given the same topological charge (ℓ = 8) but are assigned different amplitude modulation values c_i. The total angular momentum of the first vortex was varied by changing c ₁ from -1 to 1 at different times as captured in Fig. 4, while the angular momenta of the two other vortices were set by fixing c ₂ = 0.17 and c ₃ = -1.

When filling these vortices with microspheres, the measured rotational frequency followed the dependency on c established for a single vortex. The standard deviation of the measured values relative to predicted values was in the range of 2%. Changing c ₁ did not influence the rotation of the other two vortices. When c ₁ = 0, the rotation stops completely. Thus, independent control of the speed and direction of angular frequency of particles on multiple interference vortices with fixed diameters in spatially-controlled patterns has been achieved.

6. Conclusion

The collinear superposition of two classic optical vortices with equal helical wavefronts but opposite handedness provides an OAM-tunable class of optical vortices via an independent parameter, c, called the modulating amplitude. This parameter controls the orbital angular momentum content of individual optical vortices without affecting their geometry, which is separately controlled by the topological charge. A closer look at the intensity distribution around a vortex shows that adjusting c changes not only the total angular momentum content but also the depth of the 2ℓ intensity modulations on optical vortices. Surprisingly, despite these local intensity variations, classic electromagnetic theory predicts that the local angular momentum remains constant with azimuthal position in the beam due to compensation by increased local orbital angular momentum. Furthermore, c is the first knob aside from power capable of tuning the optical landscape trapping potential of a fixed-diameter optical vortex trap. This has interesting implications for statistical studies of thermally activated biased transport and the development of novel Brownian ratchets [20, 28]. The extension of these basic results to collections of optical vortices has been demonstrated. Efforts to use optimally positioned groups of optical vortex traps to construct optically-driven micromachines [23] or in-parallel molecular or cell assays [43] may be facilitated by this new degree of control.