## 2 Diffraction-Based Optical Switching  
To reduce the MEMS mirror weight, the size needs to decrease. However, with tens of microns or less in diameter, using the mirrors as reflectors becomes inefficient. First, the power density is too large, and second, the scattering from the edges of the mirror induces loss. One solution is to distribute the incident beam over several mirrors, and take advantage of the diffraction by using a computer generated hologram. In this embodiment, the MEMS device does not need to have mirrors that change orientation as is the case for 3D MEMS, but pixels that modulate either the amplitude or the phase of the incident beam. The diffraction patterns are computed using a Fraunhofer diffraction integral Equation (1) and a Gerchberg-Saxton type iterative algorithm [14]  
\[E(x_{z},y_{z})=\frac{exp(ikz)}{i\lambda z}exp\left[\frac{ik}{2z}(x_{z}^{2}+y_{ z}^{2})\right]\times\iint_{apeture}E(x_{0},y_{0})exp\left[\frac{-ik}{z}(x_{z}x_{0}+ y_{z}y_{0})\right]dxdy \tag{1}\]  
Example of diffraction patterns are presented in Figure 3.  
If the diffraction pattern is a grating with frequency \(\Lambda\), the beam is diffracted at an angle \(\theta_{d}\) given by the grating equation:  
\[sin(\theta_{d})+sin(\theta_{i})=m\frac{\lambda}{\Lambda}, \tag{2}\]  
where \(\theta_{i}\) is the angle of incidence of the beam, \(m\) the order of diffraction, \(\lambda\) is the wavelength of the light. In this approach, changing the output port of the switch can be done by changing the spatial frequency of the grating displayed by the Spatial Light Modulator (SLM). A decomposition of all possible gratings that could be displayed by the SLM into its Fourier components demonstrates that there could be as many output ports as the number of pixels on the SLM. Figure 2: Schematic representation of a cross section of a three-dimensional (3D) MEMS in butterfly configuration. If more than one input port is needed for the switch, the clear aperture of the SLM can be divided into several sub-apertures, each diffracting the beam coming from a different input port. It has to be noted that in the case of a volume hologram, angular multiplexing can be used to diffract multiple inputs to multiple outputs without crosstalk. However, this type of thick volume hologram cannot yet be realized with the SLM. By using diffraction instead of reflection, other optical functions can be added to the SLM. Examples of such functions are focalization by a Fresnel or Gabor zone plate, aberration correction for coma or astigmatism due to the off-axis diffraction of the beam, variation of the focal length to accommodate a curved focal plane. Another advantage of a diffraction-based optical switch is the possibility for multicasting and signal aggregation. Multicasting happens when the signal from a single input port is distributed over multiple output ports, and aggregation happens when the signal from multiple input ports is combined to a single output port. To demonstrate the potential of an holographic-based optical switch, we used a Texas Instruments DLP (Digital Light Processing) in our initial prototype [15; 16]. The DLP is a MEMS where the mirrors can be tilted in two different directions. In the most popular versions (0.7\({}^{\prime\prime}\) XGA (Extended Graphics Array), and 0.95\({}^{\prime\prime}\) 1080p) the direction of the tilt is \(\pm\)12\({}^{\circ}\). With such a binary mirror operation, the DLP cannot modulate the phase of the light, but it is capable of displaying binary amplitude diffraction patterns that are capable of redirecting the beam. For binary amplitude modulation, scalar theory of diffraction shows that the maximum diffraction efficiency that can be achieved is 10% [14]. However, the advantage of using the DLP is that it updates the diffraction pattern at rates of up to 32,552 Hz. We measured the loss-of-light time for the DLP Discovery 4100 Chipset.