Three-dimensional holographic ring traps

Yohai Roichman and David G. Grier
Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003

Date: October 25, 2006

Abstract:

We describe a class of diffractive optical elements intended for use in holographic optical trapping systems that project ring-like optical traps. Unlike optical vortex traps, which carry orbital angular momentum, these ring-like traps implement a force-free one-dimensional potential energy well for micrometer scale objects. Also unlike most optical vortices or ring-like Bessel beams, these ring traps have strong enough axial intensity gradients to trap objects in three dimensions.

Optical tweezers (1) have become indispensable tools for research and development in biology, physics, chemistry and engineering (2). Typically formed by focusing a Gaussian laser beam with a high-numerical-aperture lens, they excel at manipulating micrometer-scale objects. This Letter describes a new class of ring-like optical traps created with shape-phase holography (3) and the holographic optical trapping technique (4,6,5) that can move microscopic objects along closed trajectories in three dimensions. Holographic ring traps broadly resemble optical vortices (9,8,7) but feature qualitatively better trapping characteristics and independent control over the trap's shape and force profiles. This flexibility creates new opportunities for fundamental research (10,11), materials processing (12,13) and micro-opto-mechanics (15,14).

An optical vortex is created by focusing a helical mode of light (16,17), whose field,

$$\psi(\vec{\rho}) = u_0(\rho) \, e^{i \ell \theta}$$ (1)

is characterized by the integer-valued winding number, $\ell$ . Here, $\vec{\rho} = (\rho,\theta)$ is the polar coordinate relative to the optical axis, and $u_0(\rho)$ is a real-valued radially symmetric amplitude profile. In many implementations, $u_0(\rho)$ is a Gaussian and the helical phase profile is imposed by a mode converter, such as a phase-only hologram. A helical beam focuses to a ring of radius $R_\ell \propto \ell$ because destructive interference along the beam's central screw dislocation suppresses its axial intensity (18,20,19). Objects in an optical vortex experience a torque (21) because each photon in a helical beam carries orbital angular momentum $\ell \hbar$ (16,17). These properties provide the basis for a wide range of applications. Despite their utility, optical vortices' performance can be qualitatively improved by applying scalar diffraction theory. The result is a new class of highly effective and flexible holographic ring traps.

An optical ring trap in the focal plane of a lens of focal length $f$ is characterized by its radius, $R$ , its azimuthal amplitude profile, $a(\phi)$ , and its azimuthal phase profile, $\eta(\phi)$ . The associated field in the lens' input plane is given by the Fresnel transform (22)

$$\psi(\vec{\rho}) = \int \frac{d^2r}{f\lambda} \, a(\phi) \, \delt... ...hi)} \, \exp\left( i \frac{\pi}{\lambda f} \, \vec{r} \cdot \vec{\rho} \right),$$ (2)

where $\lambda$ is the wavelength of light, and where we have dropped irrelevant phase terms. Integrating over the radial coordinate $r$ yields

$$\psi(\vec{\rho}) = \frac{R}{f\lambda} \, \int_0^{2\pi} d\phi \, a... ...\, \exp\left( i \frac{\pi}{\lambda f} \, R\rho \, \cos(\phi - \theta) \right) .$$ (3)

Substituting $a(\phi) = 1$ and $\eta(\phi) = \ell \phi$ to create a uniform ring carrying orbital angular momentum yields

$$\psi(\vec{\rho}) = \psi_0 \, J_\ell\left(k\rho \right) \, e^{i \ell \theta},$$ (4)

where $k = \pi R / (\lambda f)$ and $J_\ell(k\rho)$ is the $\ell$ -th order Bessel function of first kind.

Figure 1: Shape-phase hologram encoding a ring trap with radius $R = 6~\ensuremath{\unit{\mu m}}\xspace$ and topological charge $\ell = 20$ . (a) Phase profile, $\varphi(\vec {\rho})$ . (b) Shape function, $S(\vec {\rho})$ , selected probabilistically.

Figure 2: (a) Optical vortex with $\ell = 60$ . (b) Holographic ring trap with $R = 3~\ensuremath{\unit{\mu m}}\xspace$ and $\ell = 20$ . Scale bars indicate 5  $\unit{\mu m}$ . Color bars indicate the relative intensity scale. Although both traps have the same radius, the ring trap is more sharply focused in all three dimensions.

A hologram transforming a Gaussian beam into a ring trap would have to modify both the amplitude and phase of the incident light according to Eq. (4). The field's amplitude, however, depends only on $\rho$ , and its phase depends only on $\phi$ . This separation into two linearly independent one-dimensional functions lends itself to implementation as a phase-only hologram by shape-phase holography (3). We previously have applied shape-phase holography in Cartesian coordinates to create holographic line traps (23,3). When implemented in polar coordinates, the shape-phase hologram for a ring trap takes the form

$$\varphi_{SP}(\vec{\rho}) = S(\vec{\rho}) \, \varphi(\vec{\rho}) + \left[1 - S(\vec{\rho})\right] \, q(\vec{\rho}),$$ (5)

where

$$\varphi(\vec{\rho}) = \left\{\ell \theta + \pi \, H\left(-J_\ell(k\rho)\right) \right\} \bmod 2 \pi,$$ (6)

is the phase of $\psi(\vec{\rho})$ from Eq. (4), incorporating the Heaviside step function, $H(x)$ , to ensure that the amplitude profile, $u(\vec{\rho}) = \left\vert J(k\rho) \right\vert$ , is non-negative. The binary shape function, $S(\vec {\rho})$ , approximates the continuous variations in $u(\rho)$ by assigning an appropriate number of pixels to $\varphi_{SP}(\vec{\rho})$ at radial coordinate, $\rho$ . The unassigned pixels are given values from a second hologram, $q(\vec{\rho})$ , that diverts the extraneous light.

Some latitude remains in selecting the shape function. For holographic line traps, it can be adjusted to minimize intensity variations due to Gibbs phenomenon (3). For a uniform ring trap, $S(\vec {\rho})$ may be selected randomly with probability $P(S(\vec{\rho}) = 1) = \left\vert J_\ell(k\rho) \right\vert / J_\ell(x_\ell)$ , where $x_\ell$ is the location of the first maximum of $J_\ell(x)$ . The angular distribution of pixels in $S(\vec {\rho})$ also may be selected to fine-tune the intensity profile around the ring.

Typical results appear in Fig. 1. The phase pattern's radial rings result from sign changes in $J_\ell(k\rho)$ and determine the trap's radius independent of $\ell$ . In practice, $\phi(\vec{\rho})$ projects a very effective ring trap. The shape function suppresses higher diffraction orders at larger radii by eliminating contributions from pixels near the optical axis. Guo et al. previously reported (24) that blocking the central region of a helical phase hologram similarly improves an optical vortex's performance, and that reducing the effective aperture provides some control over the optical vortex's radius. The shape-phase hologram goes further than this, decoupling the ring's radius from its topological charge without reducing diffraction efficiency. One novel consequence is that holographic ring traps need not carry orbital angular momentum.

The three-dimensional intensity distribution projected by Eqs. (5) and (6) is plotted in Fig. 2, using methods described in Ref. (23). These data demonstrate another substantial benefit of holographic ring traps. Because an optical vortex's radius reflects its wavefronts' topology, its radius, $R_\ell$ , does not vary substantially as the beam is brought to a focus. Without axial intensity gradients to compensate radiation pressure, optical vortices typically cannot trap objects in three dimensions unless a surface or other external force prevents their escape. Holographic ring traps, by contrast, converge to a diffraction-limited focus for $R > R_\ell$ , and thus are true three-dimensional traps.

Figure 3: Colloidal silica spheres 1.5  $\unit{\mu m}$ in diameter captured and translated in three dimensions by a holographic ring trap. Silica spheres naturally sediment onto the lower glass surface, where they diffuse freely. (a) The ring is focused into the spheres' equilibrium plane. (b) The ring of spheres is translated upward by $\Delta z = 10~\ensuremath{\unit{\mu m}}\xspace$ , leaving the sedimented spheres in the lower plane. The scale bar indicates 5  $\unit{\mu m}$ .

The images in Fig. 3 show a ring trap translating micrometer-scale colloidal spheres in three dimensions. These particles are dispersed in a layer of water 40  $\unit{\mu m}$ thick between a glass coverslip and a microscope slide. The sample is mounted on the stage of an inverted optical microscope (Nikon TE-2000U), with the coverslip downward. The dense silica spheres sediment onto the lower surface, where they diffuse freely. When the trap is focused into the spheres' equilibrium plane, Fig. 3(a), trapped spheres have the same bright appearance as nearby free spheres. Mechanically translating the focal plane upward by $\Delta z = 10~\ensuremath{\unit{\mu m}}\xspace$ translates the trapped spheres, but leaves the free spheres behind. The trapped spheres consequently remain in focus, while the others blur. All the while, the trapped spheres circulate around the ring at a rate determined by $\ell$ , the intensity of the light and the distance from the glass surface.

A holographic ring trap also can be translated in three dimensions by adding

$$\varphi_t(\vec{\rho}) = S(\vec{\rho}) \, \left( \vec{\kappa} \cdot \vec{\rho} + \frac{\pi \rho^2 z}{\lambda f^2} \right)$$ (7)

to $\varphi(\vec {\rho})$ (25). Here $\vec{\kappa}$ is the wavevector describing the in-plane translation, and $z$ is the axial displacement. Phase functions correcting for geometric aberrations also can be added to $\varphi(\vec {\rho})$ to improve performance (26). Superimposing the ring's phase function on a conventional holographic trapping pattern creates an array of identical ring traps (27,25). Integrating it into the hologram computation yields heterogeneous patterns of rings and other traps (5,14,25). Arrays of ring traps can create dynamically reconfigurable microfluidic systems (15,14,25), and constitute model systems for nonequilibrium statistical physics (10,11).

Orbital angular momentum displaces light away from the axis of a ring trap, as can be seen in Fig. 2(b). Setting $\ell = 0$ creates diffractionless Bessel beams above and below the ring that terminate at a dark volume around the focus. This light-free volume acts as an optical bottle (28), for dark-seeking objects. Unlike previously reported bottle beams (28), ring-bottles can be projected in arbitrary patterns and sizes.

Finally, holographic ring traps can be sculpted into shapes other than circles by setting $R = R(\phi)$ , in Eq. (3). Unlike modulated optical vortices (19) whose local intensity varies inversely with radius, modulated holographic ring traps can have independently specified intensity profiles.

This work was supported by the National Science Foundation through Grants Number DMR-0451589 and DBI-0629584.