# Interdisciplinary Applied Mathematics

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Compressibility effects resulting in complicated frequency response, and

Onset of rarefaction effects in micron- and submicron-scale clearances.

An example demonstrating the geometric complexity of MEMS is the Digital Micromirror Device of Texas Instruments (DMD) shown in Figure 6.1. The DMD is a MEMS device used in Digital Light Processing applications, and it consists of half a million (848 x 600) to over one million (1280 x 1024) micromirror pixels. The top left figure shows an SEM picture of nine mirrors, while the underlying hinge structure hidden below the mirrors is shown in the top right plot. The SEM images of the mirror substructure for several pixels and a close-up view of a single mirror substructure are presented in the bottom left and right figures, respectively. The DMD has 16 pm x 16 pm pixel geometry with 1 pm pixel separation. Each mirror oscillates ±10 degrees with a response time of 15 ps.

The DMD is packaged in a mostly nitrogen environment under ambient conditions. The presence of gas affects the device performance significantly. A typical response of a row of 840 19 pm size mirror array subject to a

FIGURE 6.1. Texas Instruments Digital Micromirror Device. The upper left view shows nine mirrors, while the upper right view shows the central mirror removed to expose    the    underlying    structure.    The lower    right shows    a close-up    view    of the

mirror substructure, and the lower left view shows several pixels with the mirror removed. (Courtesy of Texas Instruments.)

step pulse under various air pressures is shown in Figure 6.2. Operation in atmospheric pressure results in the desired fast optical response, while reductions in pressure correspond to oscillatory response with slower time decay. The decay rate is an indication of the gas damping effects, which are reduced with decreased pressure. Since the DMD dimensions are fixed and the reference pressure is reduced, the lower pressure data correspond to increased Knudsen regimes, indicating a strong rarefaction effect on the dynamic response of the DMD mirror.

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