Tolerances

Regardless of the fabrication method, any practical phase hologram will deviate from its design and these deviations will degrade its performance. We consider two principal fabrication defects: overall multiplicative error in the phase modulation due to mismatches between wavelength and etch depth, and random noise in the local phase shift due to roughness. To quantify these defects' influence on hologram performance, we define the efficiency, ${\cal E}$, to be the fraction of incident laser power projected into the planned tweezer pattern. For simplicity, we compare the intensity pattern in the focal plane when the actual hologram is illuminated by a uniform plane wave, $\tilde I^f(\vec \rho)$, to the ideal intensity pattern in the focal plane, $I^f_0(\vec \rho) = (2 \pi f/k)^2 T^2(\vec \rho)$. The corresponding efficiency,

$${\cal E} \equiv \frac{\sum_{i=1}^{M^2} T^2(\vec \rho_{i}) \tilde I^f(\vec \rho_{i}) }{\sum_{i=1}^{M^2} T^2(\vec \rho_{i})},$$ (19)

is a less stringent measure of the agreement between the ideal and actual holograms than the error, $\epsilon_n$, since it is possible to have ${\cal E} = 1$ when $\epsilon_n > 0$, but $\epsilon_n = 0$ implies ${\cal E} = 1$.

To give a feel for the results obtained with our methods, we calculate the efficiency of four standard holograms as a function of the severity of the fabrication defects. The four standard holograms are continuous and binary versions of patterns encoding $4\times 4$ and $20 \times 20$ square tweezer arrays, each with the same lattice constant. We calculated all four holograms twenty times, and selected the most efficient hologram from each group to use in the the efficiency studies.

Figure 8: Influence of phase errors on projection efficiency. Symbols indicate numerically calculated efficiencies for continuous and binary holograms encoding $4\times 4$ and $20 \times 20$ square arrays of tweezers.

The phase modulation created by an etched hologram is proportional to the etch depth, Eq. (18). If the etch rate is not precisely controlled, or if the hologram is illuminated with light of the wrong wavelength, the actual phase profile, $\tilde \Phi^{in}(\vec r)$, will differ from the design $\Phi^{in}(\vec r)$ by a scale factor, $\tilde \Phi^{in}(\vec r) = \alpha \Phi^{in}(\vec r)$. As $\alpha$ departs from unity, most of the laser light not contributing to the tweezer array is focused at the central undiffracted spot. Fig. 8 shows the efficiency, ${\cal E}(\alpha)$, of the four standard holograms as a function of $\alpha$. Even the continuous holograms with $\alpha=1$ are not perfectly efficient because the AA algorithm rarely identifies a globally ideal phase modulation. Binary holograms are still less efficient, with ideal efficiencies near 80%. Reassuringly, Fig. 8 suggests that a hologram's efficiency does not depend strongly on precisely matching etch depth to the light's wavelength.

Even if the overall etch depth is carefully controlled, reactive ion-etching creates a rough surface, whose asperities add random fluctuations to the phase profile. We measured the surface topography of our fused silica wafers after etching and found a Gaussian distribution of etch depths, with a standard deviation of 60 nm or $\pi/10$ radians at 532 nm illumination. This roughness is laterally uncorrelated down to length scales of less than 280 nm.

We gauged roughness' influence on the holograms' efficiencies by adding uncorrelated Gaussian noise to the calculated optimal phase profiles,

$$\tilde \Phi^{in}(\vec r) = \Phi^{in}(\vec r) + \eta(\vec r),$$ (20)

where the noise's probability distribution is given by

$$\rho(\eta) = \frac{1}{\sqrt{2 \pi \sigma_\Phi^2}} \exp \left(- \frac{\eta^2}{2 \sigma_\Phi^2} \right).$$ (21)

Fig. 9 shows how the efficiency, ${\cal E}(\sigma_\Phi)$, of the four standard holograms decreases with increasing surface roughness, $\sigma _\Phi$.

Figure 9: Influence of roughness on efficiency. Symbols indicate numerically calculated efficiencies for continuous and binary holograms encoding $4\times 4$ and $20 \times 20$ square arrays of tweezers subject to random Gaussian phase noise of magnitude $\sigma _\Phi$. Solid curves show the corresponding ensemble-averaged predictions from Eq. (24).

Combining Eqs. (4) and (20) yields the electric field profile in the focal plane for a given manifestation of the noise profile in the input plane,

$$\tilde E^f(\vec \rho) = \frac{k}{2 \pi f} \, \int d^2 r \, \exp \... ...c{k}{f} \, \vec r \cdot \vec \rho + i\Phi^{in}(\vec r) + i\eta(\vec r) \right).$$ (22)

Averaging over all possible phase profiles yields

$$\langle \tilde E^f(\vec \rho) \rangle = \exp \left( - \frac{1}{2} \, \sigma_{\Phi}^2 \right) E^f(\vec \rho),$$ (23)

so that

$$\langle {\cal E}(\sigma_{\Phi})\rangle = {\cal E}(0) \exp \left(-\sigma^2_{\Phi}\right).$$ (24)

This result agrees well with numerically calculated efficiencies, as can be seen in Fig. 9. Substituting the measured $\sigma^2_\Phi$ for our etched binary holograms, we estimate that roughness diminishes their efficiencies by a further 10% to roughly 70%.