Fluidic MEMS were some of the earliest and most commercially successful MEMS devices. The inkjet printer has displaced many of the other printing technologies for desktop and photographic color printing and is now penetrating the high-end digital printing market. An emerging market is developing for biological “lab-on-a-chip” and sensor applications. The same technology that enables printing color documents on a desktop may enable implantable medical devices to monitor internal chemical concentrations such as blood sugar levels and precisely and continuously dose drugs such as insulin on an as-needed basis. Before considering these applications we consider fluidics on micrometer length scales, as many of the phenomena we are used to on the macroscopic length scales, where our intuitions are formed, do not apply on the microscopic length scales of microfluidic devices.

In a multi-project wafer process your layout will be included with other users' layouts that are combined together into a wafer level layout. All of the designs will be fabricated together using the same process. When all of the fabrication steps have been completed the wafer is subdivided into individual dies for each of the users using an abrasive diamond saw. Dicing “streets” are put in at the wafer level layout by the supplier to comprehend the “kerf ” (width) of the saw blade and a safety zone for chipping during the sawing operation. The saw blade and wafer are sprayed with water to keep them cool during the dicing operation, and the sawing process generates residual particles that must be removed in a post-sawing cleaning step. At this stage custom steps can be performed for each individual customer at the die level. Some of the custom steps include sub-dicing of the individual die into smaller pieces, post-processing steps such as sacrificial release and critical point drying, and packaging of the parts. Packaging can include attaching the die to a carrier, wire bonding and sealing the package to protect the parts from the environment and to provide a controlled interface (electrical, thermal, mechanical, and/or optical) between the parts and the environment.

MEMS deformable mirrors (DMs) have been developed for applications in adaptive optics, including astronomy [1], [2], [3], vision science [4], microscopy [5], and laser communications [6]. In astronomy, adaptive optics have been used to overcome the image aberrations caused by the Earth's atmosphere. Light from a distant star, which can be considered a point source because it is so far away, travels through the vacuum of space as a plane wave. When the plane wave enters Earth's atmosphere, the wavefront is distorted due to dynamic changes in the index of refraction of the atmosphere caused by winds and temperature fluctuations. These fluctuations in the index of refraction cause changes in the velocity of the wavefront, so that some portions travel faster than others, leading to the distorted wavefront shown in Figure 8.1. These dynamic distortions are what cause stars to appear to twinkle. When the star is imaged in a telescope, it appears as a fuzzy blob rather than a point of light, as shown in Figure 8.1(a). By measuring the wavefront distortions from the star using a wavefront sensor, the conjugate of the wavefront distortion can be applied to a deformable mirror to correct the image, as shown in Figure 8.1(b) and (c). When a star is used as a reference point source for making wavefront corrections, it is called a “guide-star.” If light from a nearby galaxy travels through the same part of the atmosphere, the guide-star can be used to correct the image of the galaxy, as shown in Figure 8.1(b) [7].

Overview of MEMS fabrication

Microelectromechanical systems (MEMS) fabrication developed out of the thin-film processes first used for semiconductor fabrication. To understand the unique features of the MEMS fabrication process it is helpful to consider the semiconductor fabrication process.

The typical thin films that are deposited include semiconductors (e.g., polysilicon), insulators (e.g., silicon nitride), and metals (e.g., aluminum). In addition, some layers are grown (oxide), diffused, or implanted (dopants) rather than deposited using thin-film techniques. A cross section of a complementary metal oxide semiconductor (CMOS) process that includes six levels of metal is shown in Figure 1.2 [1]. A schematic diagram of one of the first MEMS devices, which used semiconductor processing for fabrication, the resonant gate transistor, is shown in Figure 1.3 [2].

This chapter is an introduction to the semiclassical approach for the Helmholtz equation in complex systems originating in the field of quantum chaos. A particular emphasis will be made on the applications of trace formulae in paradigmatic wave cavities known as wave billiards. Its connection with random matrix theory (RMT) and disordered scattering systems will be illustrated through spectral statistics.

Introduction

The study of wave propagation in complicated structures can be achieved in the high-frequency (or small-wavelength) limit by considering the dynamics of rays. The complexity of wave media can be due either to the presence of inhomogeneities (scattering centers) of the wave velocity or to the geometry of boundaries enclosing a homogeneous medium. It is the latter case that was originally addressed by the field of quantum chaos to describe solutions of the Schrödinger equation when the classical limit displays chaos. The Helmholtz equation is the strict formal analog of the Schrödinger equation for electromagnetic or acoustic waves, the geometrical limit of rays being equivalent to the classical limit of particle motion. To qualify this context, the new expression wave chaos has naturally emerged. Accordingly, billiards have become geometrical paradigms of wave cavities.

In this chapter we will particularly discuss how the global knowledge about ray dynamics in a chaotic billiard may be used to explain universal statistical features of the corresponding wave cavity, concerning spatial wave patterns of modes, as well as frequency spectra.

Introduction

For a two-dimensional enclosure, such as a membrane or the cross section of an infinitely long duct, those with the very simplest shapes (circles, rectangles, spheres, boxes, etc.) with simple uniform boundary conditions, the modes and natural frequencies can be determined analytically. For any other shape they may be determined numerically by a range of mature numerical techniques of which finite element and boundary element analyses are the best known and the most widely studied. Knowing how to calculate the modes and natural frequencies for any particular shape, however, is not the same as understanding how those modes and natural frequencies depend on the shape. Suppose, for example, that we wish to improve the design of a component by optimizing some quantity such as weight, while leaving its natural frequencies unchanged. In the course of such an optimization changes will be made to the shape, whereupon the process of calculating the modes and natural frequencies must begin all over again; at best, part of the mesh can be re-used. Such an analysis cannot tell us where effort can be most or least profitably concentrated.

It turns out that the shapes that can be analyzed are (for good reason) quite untypical compared with arbitrary shapes. The situation mirrors the one that used to prevail in the study of dynamical systems, where linear differential equations were most widely studied because of their solubility, and the fact that other systems showed radically different qualitative behavior was, for a time, ignored.

The short periodic orbit (PO) approach was developed in order to understand the structure of stationary states of quantum autonomous Hamiltonian systems corresponding to a classical chaotic Hamiltonian. In this chapter, we will describe the method for the case of a two-dimensional chaotic billiard where the Schrödinger equation reduces to the Helmholtz equation; then, it can directly be applied to evaluate the acoustic eigenfunctions of a two-dimensional cavity. This method consists of the short-wavelength construction of a basis of wavefunctions related to unstable short POs of the billiard, and the evaluation of matrix elements of the Laplacian in order to specify the eigenfunctions.

Introduction

The theoretical study of wave phenomena in systems with irregular motion received a big impetus after the works by Gutzwiller (summarized in Gutzwiller 1990). He derived a semiclassical approach providing the energy spectrum of a classically chaotic Hamiltonian system as a function of its POs. This formalism is very efficient for the evaluation of mean properties of eigenvalues and eigenfunctions (Berry 1985, Bogomolny 1988), but it suffers from a very serious limitation when a description of individual eigenfunctions is required: the number of used POs proliferates exponentially with the complexity of the eigenfunction. In this way, the approach loses two of the common advantages of asymptotic techniques: simplicity in the calculation and, more important, simplicity in the interpretation of the results.

Based on numerical experiments in the Bunimovich stadium billiard (Vergini & Wisniacki 1998), we have derived a short PO approach (Vergini 2000), which was successfully verified in the stadium billiard (Vergini & Carlo 2000): the first 25 eigenfunctions were computed by using five periodic orbits.