Structure of Optical Vortices
Abstract.
Helical modes of light can be focused into toroidal optical traps known as optical vortices, which are capable of localizing and applying torques to small volumes of matter. Measurements of optical vortices created with the dynamic holographic optical tweezer technique reveal an unsuspected dependence of their structure and angular momentum flux on their helicity. These measurements also provide evidence for a novel optical ratchet potential in practical optical vortices.
Beams of light with helical wavefronts focus to rings, rather than points, and also carry orbital angular momentum (1); (2); (3) that they can transfer to illuminated objects (4); (5); (3); (6). When focused strongly enough, such helical modes form toroidal optical traps known as optical vortices (4); (7); (8), whose properties present novel opportunities for scientific research and technological applications. For example, optical vortices should be ideal actuators for microelectromechanical systems (MEMS) (9), and arrays of optical vortices (10) have shown a promising ability to assemble colloidal particles into mesoscopic pumps for microfluidic systems. All such applications will require a comprehensive understanding of the intensity distribution and angular momentum flux within optical vortices. This Letter describes measurements of the structure of optical vortices created with the dynamic holographic optical tweezer technique (10), and of their ability to exert torques on trapped materials. These measurements reveal qualitative discrepancies with predicted behavior, which we explain on the basis of scalar diffraction theory.
A helical mode is distinguished by a phase factor proportional to the polar angle around the beam's axis,
(1) 
Here, is the beam's wavevector, is the field's radial profile at position , and is an integral winding number known as the topological charge. All phases appear along the beam's axis, , and the resulting destructive interference cancels the axial intensity. Similarly, each ray in such a beam has an outofphase counterpart with which it destructively interferes when the beam is brought to a focus. Constructive interference at a radius from the optical axis yields a bright ring whose width is comparable to , the wavelength of light. The semiclassical approximation further suggests that each photon in a helical mode carries orbital angular momentum (2), so that the beam can exert a torque proportional to its intensity.
Conventional beams of light can be converted into helical modes with a variety of mode converters (11). Most implementations yield topological charges in the range (2); (6). By contrast, dynamic holographic optical tweezers (10) can create helical modes up to , and so are ideal for studying how optical vortices' properties vary with .
Our system, depicted in Fig. 1, uses a Hamamatsu X7550 parallelaligned nematic liquid crystal spatial light modulator (SLM) (12) to imprint computergenerated patterns of phase shifts onto the wavefront of a linearly polarized TEM beam at from a frequencydoubled Nd:YVO laser (Coherent Verdi). The modulated wavefront is transferred by a telescope to the back aperture of a NA oil immersion objective lens mounted in a Zeiss Axiovert S100TV inverted optical microscope. The objective lens focuses the light into optical traps, in this case a single optical vortex. The same lens also forms images of trapped particles that are relayed to an attached video camera through a dichroic mirror.
The SLM can shift the light's phase to any of 150 distinct levels in the range at each 40 wide pixel in a square array. Imprinting a discrete approximation to the phase modulation onto the incident beam, yields a helical mode with 70 percent efficiency, independent of laser power over the range studied. Conversion efficiency is reduced for by the SLM's limited spatial resolution. Fig. 1(b) is a digital image of an optical vortex reflected by a mirror placed in the objective's focal plane. The unmodified portion of the TEM beam travels along the optical axis and comes to a focus in the center of the field of view. This conventional beam does not overlap with the optical vortex in the focal plane and so does not affect our observations. Fig. 1(c) shows a timelapse multiple exposure of a single 800 nm diameter colloidal polystyrene sphere trapped on the optical vortex's circumference in an 85 thick layer of water between a coverslip and a microscope slide. Angular momentum transferred from the optical vortex drives the sphere once around the circumference in a little under 2 s at an applied power of 500 mW. The image shows 11 stages in its transit at 1/6 s intervals. We studied the same particle's motions at different topological charges and applied powers to establish how helicity influences optical vortices' intensity distribution and local angular momentum flux.
Observing that a single particle translates around the circumference of a linearly polarized optical vortex distinguishes the angular momentum carried by a helical beam of light from that carried by circularly polarized light. The latter causes an absorbing particle to spin on its own axis. Observing that the particle instead translates around the optical axis demonstrates that the angular momentum density in a helical beam results from a transverse component of the linear momentum density, as predicted (2); (6).
Most observed characteristics of optical vortices have been interpreted in terms of the properties of LaguerreGaussian (LG) eigenmodes of the paraxial Helmholtz equation (5); (6). These have a radial dependence
(2) 
where is a generalized Laguerre polynomial with radial index , and is the beam's radius (1). An LG mode with appears as a ring of light whose radius depends on topological charge as (13). Practical optical vortices, including the example in Fig. 1(b), also appear as rings of light and so might be expected to scale in the same way (6). However, the data in Fig. 2 reveal qualitatively different behavior. We obtain from digitized images such as Fig. 1(b) by averaging over angles and locating the radius of peak intensity. Projecting different values of reveals that scales linearly with the topological charge, and not as .
This substantial discrepancy can be explained by considering how the phasemodulated beam propagates through the optical train. The field in the focal plane of a lens of focal length is related in scalar diffraction theory to the field at the input aperture (and thus at the face of the SLM) through a Fourier transform (14). Transforming the helical beam first over angles yields
(3) 
where is the th order Bessel function of the first kind, and is the radius of the optical train's effective aperture. For our system, . Setting for a uniform illumination yields
(4) 
The radius of the principal maximum in is approximated very well by
(5) 
with and . This also agrees quantitatively with the data in Fig. 2.
Even if the mode converter created a pure LG mode, the optical trapping system's limited aperture, , still would yield a superposition of radial eigenmodes at the focal plane (2), and a comparable linear dependence of on . The superposition of higher modes in our system is evident in the hierarchy of diffraction fringes surrounding the principal maximum in Fig. 1(b).
This linear dependence on leads to scaling predictions for optical vortices' optomechanical properties with which we can probe the nature of the angular momentum carried by helical modes. In particular, a wavelengthscale particle trapped on the circumference of an optical vortex is illuminated with an intensity , where is the power of the input beam, assumingthat the photon flux is spread uniformly around the vortex's circumference in a band roughly thick (see Fig. 2). If we assume that each scattered photon transfers an angular momentum proportional to , then the particle's tangential speed should be proportional to , and the time required to make one circuit of the optical vortex should scale as
(6) 
If the radius had scaled as , then the particle's speed would have been independent of , and the period would have scaled as . Instead, for , we expect .
The data in Fig. 3(a) show that does indeed scale according to Eqs. (5) and (6) for larger values of . For , however, the period is systematically larger than predicted. Similarly, scales with as predicted for lower powers, but increases as increases. In other words, the particle moves slower the harder it is pushed. This unexpected effect can be ascribed to the detailed structure of optical vortices created with pixellated diffractive optical elements; the mechanism presents new opportunities for studying Brownian transport in modulated potentials.
When projected onto our objective lens' input pupil, each of our SLM's effective phase pixels spans roughly . Numerically transforming such an apodized beam reveals a pattern of intensity corrugations, as shown in Fig. 3(a). These establish a nearly sinusoidal potential through which the particle is driven by the local angular momentum flux. We model the intensity's dependence on arclength around the ring as
(7) 
where is the depth of the modulation, and is its wavenumber. For , is approximately independent of .
This modulated intensity exerts two tangential forces on the trapped sphere. One is due to the transferred angular momentum,
(8) 
where we assume a local angular momentum flux of per photon. The prefactor includes such geometric factors as the particle's scattering crosssection. The other is an optical gradient force due to the polarizable particle's response to local intensity gradients:
(9) 
where sets the relative strength of the gradient force. Combining Eqs. (8) and (9) yields the tangential force
(10) 
where we have omitted an irrelevant phase angle, and where . Even if is much smaller than unity, both and can be much larger. In that case, reducing at fixed power increases the depth of the modulation relative to the thermal energy scale , and the particle can become hung up in the local potential minima. The modulated potential thus increases the effective drag.
More formally, a particle's motion along an inclined sinusoidal potential with strong viscous damping is described by the Langevin equation
(11) 
where is the viscous drag coefficient and is a zeromean random thermal force. The associated mobility may be expressed as (15)
(12) 
where is the topological charge at which the modulation reaches at power . Given this result, the transit time for one cycle should be
(13) 
where is the expected period for at in the absence of modulation. The solid curve in Fig. 3(a) is a fit to Eqs. (12) and (13) for , and . The results, , , and , are consistent with the strongly modulated potential shown in the inset to Fig. 3(b). Rather than smoothly processing around the optical vortex, the particle instead makes thermally activated hops between potential wells in a direction biased by the optical vortex's torque.
Replacing with in Eq. (12) yields an analogous result for the period's dependence on applied power for fixed , as shown in Fig. 3(b). Here, is the power at which the modulation reaches . Using and obtained from Fig. 3(a), we find that the sphere's motions above are slower even than our model predicts. The period's divergence at high power is due to a localized “hot spot” on the optical vortex resulting from aberrations in our optical train. Such hot spots have confounded previous attempts to study singleparticle dynamics in helical beams (3). Because hot spots also deepen with increasing power, they retain particles with exponentially increasing residence times. The total transit time becomes (15)
(14) 
The data in Fig. 3(b) are consistent with and . Localization in hot spots becomes comparable to corrugationinduced drag only for powers above and so does not affect the data in Fig. 3(a). Consequently, these data offer insights into the nature of the helical beam's angular momentum density.
In particular, the simple scaling relation, Eq. (6) is remarkably successful at describing a single particle's motions around an optical vortex over a wide range of topological charges. This success strongly supports the contention that each photon contributes to the local angular momentum flux of a helical beam of light, and not only to the beam's overall angular momentum density. It hinges on our observation that the radius of a practical optical vortex scales linearly with its topological charge. The corrugations in apodized optical vortices broadens this system's interest. Not only do they provide a realization of the impossible staircase in M. C. Escher's lithograph “Ascending and Descending”, but they also offer a unique opportunity to study overdamped transport on tilted sinusoidal potentials. As a practical Brownian ratchet, this system promises insights germane to such related phenomena as transport by molecular motors, voltage noise in Josephson junction arrays, and flux flow in typeII superconductors. Preliminary observations of multiple particles on an optical vortex also suggest opportunities to study transitions from jamming to cooperativity with increasing occupation.
This work was supported by a grant from Arryx, Inc., and in part by the MRSEC program of the National Science Foundation through Grant DMR9880595 Equipment was purchased with funds from the W. M. Keck Foundation, and the spatial light modulator was made available by Hamamatsu Corp.
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