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Next: Adaptive-Additive Algorithm Up: Holographic Tweezer Arrays Previous: Fourier Optics

Phase-Only Holograms

Obtaining a desired wavefront in the focal plane requires introducing the appropriate wavefront in the input plane. Most lasers, however, provide only a fixed wavefront,

$\displaystyle E_0(\vec r) = A_0(\vec r) \, \exp [i \Phi_0(\vec r)].$ (7)

Shaping $ E_0(\vec r)$ into $ E^{in}(\vec r)$ involves modifying both the amplitude and phase at the input plane. Changing the amplitude with a passive optical element necessarily diverts power from the beam and diminishes trapping efficiency. Fortunately, optical trapping relies on the beam's intensity and not on its phase. We can exploit this redundancy by setting $ A^{in}(\vec r) = A_0(\vec r)$ and modulating only the phase of the input beam to obtain the desired trapping configuration.

Several techniques are available for achieving the necessary phase modulation, and some of the associated practical considerations are discussed in Section 6. For the purposes of the present discussion, we will refer to the phase modulating element as a hologram or a diffractive optical element and treat it as if it acts in transmission, as shown in Fig. 1.

After passing through a phase modulating hologram, the electric field in the input plane has a modified wavefront

$\displaystyle E^{in}(\vec r) = E_0(\vec r) \, \exp [i \Phi^{in}(\vec r) ],$ (8)

where $ \Phi^{in}(\vec r)$ is the imposed phase profile. Calculating the phase hologram, $ \Phi^{in}(\vec r)$, needed to project a desired pattern of traps is not particularly straightforward, as a simple example demonstrates.

In a typical application of holographic optical tweezer arrays, the undiffracted beam, $ E_0(\vec r)$, projects a single optical tweezer into the center of the focal plane with output wavefront $ E^f_0(\vec \rho)$, and the goal is to create displaced copies of this tweezer in the focal plane. One possible wavefront describing an array of $ N$ optical tweezers at positions $ \vec \rho_i$ in the focal plane is a superposition of single (non-overlapping) tweezers

$\displaystyle E^f(\vec \rho) = \sum_{i = 1}^N \alpha_i \, E^f_0(\vec \rho - \vec \rho_i),$ (9)

where the normalization $ \sum_{i=1}^N \vert\alpha_i\vert^2 = 1$ conserves energy. $ E^f(\vec \rho)$ may be written as a convolution

$\displaystyle E^f(\vec \rho)$ $\displaystyle = \int d^2 \rho' \, E^f_0( \vec \rho' ) \, T(\vec \rho - \vec \rho')$ (10)
  $\displaystyle \equiv E^f_0 \circ T(\vec \rho)$ (11)

of $ E^f_0(\vec \rho)$ with a lattice function

$\displaystyle T(\vec \rho) = \sum_{i=1}^N \alpha_i \, \delta^{(2)}(\vec \rho - \vec \rho_i).$ (12)

Equations (6) and (8) relate $ E^f(\vec \rho)$ to the associated input wavefront:

$\displaystyle E^{in}_0(\vec r) \, \exp [i \Phi^{in}(\vec r) ]$ $\displaystyle = {\cal F}^{-1} \left\{ E^f_0 \circ T(\vec \rho) \right\}$ (13)
  $\displaystyle = \frac{2\pi f}{k} \, {\cal F}^{-1} \left\{ E^f_0(\vec \rho) \right\} \, {\cal F}^{-1} \left\{ T(\vec \rho) \right\},$ (14)

by the Fourier convolution theorem. The phase modulation needed to achieve the array of optical tweezers then follows from Eq. (8):

$\displaystyle \exp [i \Phi^{in}(\vec r) ] = \frac{2\pi f}{k} \, {\cal F}^{-1} \left\{ T(\vec \rho) \right\},$ (15)

independent of the form of the single tweezer.

The phases of the complex weights, $ \alpha_i$, must be selected so that $ \Phi^{in}(\vec r)$ is a real-valued function. Unfortunately, the resulting system of equations has no analytic solution. Still greater difficulties are encountered in designing more general systems of optical traps, including tweezers which trap out of the focal plane or mixed arrays of conventional and vortex tweezers. Rather than deriving solutions for particular tweezer configurations, we have developed more general numerical methods which we apply in the following Sections to creating planar arrays optical tweezers.


next up previous
Next: Adaptive-Additive Algorithm Up: Holographic Tweezer Arrays Previous: Fourier Optics
David G. Grier 2000-10-27