Holographic optical trapping

Figure 1: Two beams of light focus to two optical tweezers, and also form an interference pattern at the lens' input pupil. The same traps can be created from a single input beam by placing an equivalent hologram in the input pupil.

Multiple beams of light all passing through the objective's input pupil with their own angle of incidence and degree of collimation create a configuration of optical traps, as shown in Fig. 1. If these beams are mutually coherent, they form an interference pattern in the input pupil, with fields of the form

$$\psi(\vec{\rho}) = u(\vec{\rho}) \, \exp\left( - i \, \varphi(\vec{\rho}) \right),$$ (1)

at point $\vec{\rho}$. Were the same pattern of amplitude modulations, $u(\vec{\rho})$, and phase modulations, $\varphi(\vec{\rho})$, imposed on the wavefronts of a single incident beam as it passed through the input pupil, the modified beam also would create the same trapping pattern. This is the principle behind holographic optical trapping.

Creating multiple optical traps does not require a fully complex hologram. Because optical trapping relies only on gradients in the intensity, and not on the phase, even quite complicated three-dimensional configurations of optical traps can be specified with just $\varphi(\vec{\rho})$, leaving the amplitude profile $u(\vec{\rho}) = u_0(\vec{\rho})$ of the input beam unchanged. The phase-only diffractive optical element (DOE) encoding a particular pattern of traps is an example of a class of holograms known as kinoforms. The trick, then, is to compute the kinoform that projects a particular pattern of traps.

Several algorithms have been proposed for seeking holograms that most accurately and most rapidly approximate desired trapping patterns. The fastest is to compute the phase associated with a linear superposition of the desired beams, and to simply discard the associated amplitude variations (5). Such straightforward superposition is surprisingly effective, particularly if the beams are chosen to have random relative phases. The resulting trapping pattern tends to be marred, however, by large numbers of ``ghost'' traps at symmetry-dictated positions, and also by large variations in the traps' intensities from their design values. For many applications, however, the resulting performance is more than adequate, and the ease of computation facilitates real-time interactive control.

Superposition also provides an outstanding starting point for refinement algorithms. Iterative refinement schemes based on the Gerchberg-Saxton and adaptive additive algorithms (11) improve all aspects of the holograms' performance (12,6), although at substantial computational cost, particularly for three-dimensional trapping patterns. A modified adaptive-additive algorithm that calculates fields only at the traps' locations (7) is far more efficient, but also less effective at suppressing ghost traps. More recently, direct search algorithms have been shown to yield substantially more accurate DOE estimates (8) and also can be far more efficient if started from the randomly-phased superposition (8) rather than from a random phase field (12).

The field due to an array of $M$ discrete point-like traps located at $\{\vec{r}_m\}$ can be approximated by

$$\psi(\vec{r}) = \sum_{m=1}^M \psi_m \, \delta(\vec{r}- \vec{r}_m)$$ (2)

$$\psi_m = \alpha_m \, \exp(-i \phi_m),$$ (3)

where $\alpha_m$ is the relative amplitude of the $m$-th trap, normalized by $\sum_{m=1}^M \left\vert \alpha_m \right\vert ^2 = 1$. The relative phases, $\phi_m$, generally are assigned randomly, but also may be specified for particular applications. In most practical implementations, such as that depicted in Fig. 2, the DOE, $\varphi(\vec{\rho})$, encoding the traps also is discretized into an array of $N$ phase pixels $\varphi_j$ located at $\vec{\rho}_j$. Consequently, the complex field at each trap can be described by a nonlinear transformation of the input phase

$$\psi_m = \sum_{j=1}^N T_{m,j} \, u_j \, \exp(i \varphi_j),$$ (4)

where the transfer matrix $T_{m,j}$ describes the coherent propagation of light from pixel $j$ on the DOE to trap $m$ in the focal plane, given the input beam's amplitude profile, $u_j = u_0(\vec{\rho}_j)$. In our implementation, the amplitude profile is approximated by the Heaviside step function $u_0(\vec{\rho}) = \Theta(\rho - R)$, where $R$ is the radius of the optical train's aperture.

The transfer matrix for a two-dimensional configuration of conventional optical tweezers is given in scalar diffraction theory by (13,8)

$$T^{(0)}_{m,j} = \frac{1}{\lambda f} \, \exp\left(-i \frac{2\pi \vec{r}_m \cdot \vec{\rho}_j}{\lambda f} \right).$$ (5)

More generally, the transfer matrix can take the form

$$T_{m,j} = \prod_{k=0}^{K_m} T^{(k)}_{m,j},$$ (6)

where the additional $K_m$ contributions, $T^{(k)}_{m,j}$, describe wavefront-shaping operations specific to the $m$-th trap. For example, if the DOE displaces the $m$-th trap by a distance $z_m$ along the optical axis, then

$$T^z_{m,j} = \exp\left(i \, \frac{\pi \rho_j^2 z_m}{\lambda f^2}\right)$$ (7)

returns its image to the focal plane for analysis (8,7). More dramatic transformations implemented with Eq. (6) will be described in Sec. 5.

Direct search refinement starts from an estimate $\varphi_j$ for the DOE, and the associated fields $\psi_m$ calculated with Eq. (4). If the DOE exactly encoded the desired trapping pattern, then the calculated amplitudes $\left\vert \psi_m \right\vert$ would agree with the design values, $\alpha_m$. The algorithm seeks to minimize actual discrepancies between $\left\vert \psi_m \right\vert$ and $\alpha_m$. Following Meister and Winfield (14), we adopt the error function

$$E = - \left< \left\vert \psi_m \right\vert ^2 \right> + \gamma \,... ...ight\vert ^2 \alpha_m^2 \right>}{\left< \alpha_m^4 \right>} \right)^2 \right>},$$ (8)

where the weighting factor $\gamma$ sets the relative importance attached to diffraction efficiency ( $\gamma = 0$) and fidelity to design ( $\gamma > 0$). Improvements are sought by selecting pixels at random, changing their phase values, recomputing the fields, and retaining only those proposed changes that improve the performance. Because the relationship between $\left\vert \psi_m \right\vert$ and $\varphi_j$ is inherently nonlinear, the search proceeds sequentially, and the process continues until $E$ is reduced to an acceptable level.

Convergence starting from a randomly-phased superposition typically is achieved with a single pass through the array, for a total of $MN$ operations. This is comparable in computational cost to the initial superposition and so roughly doubles the total cost of the computation. The benefits of can be disproportionately large, as a practical example illustrates.

Figure 2: Schematic implementation of holographic optical traps. An expanded laser beam is reflected by a liquid crystal spatial light modulation, which imprints a computer-generated hologram onto its wavefronts. The $200 \times 200$ pixel region of a CGH shown encodes a pattern of 119 optical tweezers in a quasiperiodic arrangement. The phase hologram is relayed to the input pupil of an objective lens that focuses it into holographic optical traps, shown here trapping 1.5  $\unit{\mu m}$ diameter colloidal spheres in water.

We project holographic optical traps with the system shown schematically in Fig. 2. Light from a frequency-doubled Nd:YVO$_4$ laser (Coherent Verdi) is expanded to fill the face of a reflective liquid crystal spatial light modulator (SLM) (Hamamatsu X8267-16 PPM), which can impose a phase shift between 0 and $2 \pi$ radians at each pixel in a $768 \times 768$ array. The phase-modified beam is relayed to the input pupil of a 100$\times$, NA 1.4, Plan Apo oil immersion objective mounted in a Nikon TE-2000U inverted optical microscope, which focuses the light into optical traps. Because the SLM's face lies in a plane conjugate to the objective's input pupil, the effect is the same as if the DOE were placed in the input pupil, as in Fig. 1. The benefit of this arrangement is that the trapped sample can be imaged onto a CCD camera using the microscope's standard optical train, with the imaging illumination passing through the dichroic mirror used to direct the trap-forming laser.

In practice, not all of the input beam is diffracted by the SLM, and the undiffracted portion ordinarily would form a bright trap right in the middle of the field of view. To counter this, we adjust the beam expander so that the SLM is illuminated with a slightly converging beam. Projecting optical traps into the microscope's focal plane therefore requires the traps to be displaced along the optical axis with the computed DOE. The undiffracted beam therefore focuses into a different plane within the relay optics than the intended traps, and so can be blocked with a spatial filter without disrupting the traps. Displacing the trapping plane has the additional benefit of projecting most residual ghost traps out of the sample volume.

The result can be seen in the typical images in Fig. 2. Here, an eight-bit CGH imprinted on the input beam by the SLM creates the pattern of focal spots in the intermediate focal plane, which is shown trapping colloidal spheres dispersed in water. This particular quasiperiodic arrangement of 119 optical traps is particularly challenging because it lacks reflection symmetry about the optical axis. As a result, a typical DOE computed by superposition alone suffers from more than 50 percent root-mean-squared (RMS) relative deviations from design amplitudes. Imaging photometry and measurements of the traps' potential energy wells by particle tracking (8,15) confirm that the DOE refined by direct search is uniform to within 5 percent, a factor of ten improvement.

Figure 3: Two views of a rotating icosahedron of colloidal spheres created with dynamic holographic optical tweezers.

Our implementation of dynamic holographic optical trapping permits full three-dimensional manipulation over a $100 \times 100 \times 40~\unit{\ensuremath{\unit{\mu m}}\xspace ^3}$ volume. Micrometer-scale colloidal spheres are readily stacked five or more deep along the axial direction, with three-dimensional quasicrystals consisting of hundreds of spheres having recently been demonstrated (16). The arrays' fidelity to design intensities, and the DOE's overall efficiency fall off as the arrays become increasingly complicated. How design complexity affects implementational efficacy has yet to be worked out.

Demonstrations of three-dimensional control (5,12) such as the rotating icosahedron in Fig. 3, reveal that objects can be organized into vertical stacks along the optical axis. Three-dimensional assemblies consisting of hundreds of spheres in asymmetric configurations up to nine layers deep recently have been demonstrated (16). Still larger areas and depths can be accessed, at least in principle, by creating time-shared three-dimensional trapping patterns (17). The resulting structures can be made permanent, for example by gelling the suspending fluid (18,19,16).