Introduction

The spatial arrangement of nanoscale dielectric scattering centers embedded in an otherwise uniform medium can strongly influence propagation of an incident electromagnetic (EM) wave. Exploiting this fact, one may iteratively solve a parameter optimization problem [1] to find a spatial arrangement of identical, non-overlapping scattering cylinders so that the scattered EMwave closely matches a desired target-response. Of course, the efficiency of adaptive algorithms used to find solutions may become an increasingly critical issue as the number of design parameters such as scattering centers increases. However, even for modest numbers of scattering cylinders this method holds the promise of creating nano-photonic device designs that outperform conventional approaches based on spatially periodic photonic crystal (PC) structures.

The configurations considered here consist of either lossless dielectric rods in air or circular holes in a dielectric similar to the majority of quasi two-dimensional (2D) PCs reported in the literature by, for example, [2, 3]. To confirm the validity of the 2D simulations they are compared to full three-dimensional (3D) simulations as well as measurements.

In Section 4.2 the forward problem and a semi-analytic method to compute the electromagnetic field distribution is introduced. The semi-analytic Fourier-Bessel series solutions and a guided random walk routine can be used for electromagnetic device design and the optimization algorithm is described in Section 4.3. The results of these design problems are presented in Section 4.4. The inefficiencies of the optimization routine in conjunction with the particular implementation of the forward solver soon became apparent. The presented types of aperiodic design demand highly efficient forward solvers as well as optimization routines.

Introduction

In this chapter we explore systems whose description lies at the boundary between classical and quantum theory. There are of course many ways to approach this problem. Here, we choose to study the interaction of classical light with small metal particles of arbitrary shape. Specifically, we consider a physical model that is capable of observing the transition from bulk material properties to nanoscale structures, for which quantum effects dominate. We then explore the landscape of possible physical responses of such systems, using optimal design techniques to train our intuition.

The prevalent classical model describing the interaction of visible and infrared electromagnetic radiation with nanoscale metallic clusters is based on Mie theory [1]. This local continuum field model which uses empirical values of a bulk material's linear optical response has been used to describe plasmon resonances in nanoparticles [2–4]. However, such a semi-empirical continuum description necessarily breaks down beyond a certain level of coarseness introduced by atomic length scales. Thus, it cannot be used to describe the interface between quantum and classical macroscopic regimes. Moreover, extensions of Mie theory to inhomogeneous cluster shapes are commonly restricted to low-order harmonic expansions (e.g. elliptical distortions) and so do not exhaust the full realm of possible geometric configurations. In addition, near-field applications, such as surface enhanced Raman scattering [5], are most naturally described using a real-space theory that includes the non-local electronic response of inhomogeneous structures, again beyond the scope of Mie theory.

In the following section we describe a microscopic approach that demonstrates the breakdown of this concept at atomic scales, whereas for large cluster sizes the classical predictions for the plasmon resonances are reproduced.

Introduction

Optimization has a distinguished history in engineering and industrial design. Most approaches, however, assume that the input parameters are precisely known and that the implementation does not suffer any errors. Information used to model a problem is often noisy, incomplete or even erroneous. In science and engineering, measurement errors are inevitable. In business applications, the cost and selling price as well as the demand for a product are, at best, expert opinions. Moreover, even if uncertainties in the model data can be ignored, solutions cannot be implemented to infinite precision, as assumed in continuous optimization. Therefore, an “optimal” solution can easily be sub-optimal or, even worse, infeasible.

There has been evidence illustrating that if errors (in implementation or estimation of parameters) are not taken into account during the design process, the actual phenomenon can completely disappear. A prime example is optimizing the truss design for suspension bridges. The Tacoma Narrows bridge was the first of its kind that was optimized to divert the wind above and below the roadbed [1]. Only a few months after its opening in 1940, it collapsed due to moderate winds which caused twisting vibrational modes. In another example, Ben-Tal and Nemirovski demonstrated that only 5% errors can entirely destroy the radiation characteristics of an otherwise optimized phased locked and impedance matched array of antennas [2]. Therefore, taking errors into account during the optimization process is a first-order effect.

Traditionally, sensitivity analysis was performed to study the impact of perturbations on specific designs and to find solutions that are least sensitive among a larger set of optima.

Dramatic advances in the control of physical systems at the atomic scale have provided many new ways to manufacture devices. An important question is how best to design these ultra-small complex systems. Access to vast amounts of inexpensive computing power makes it possible to accurately simulate their physical properties. Furthermore, high-performance computers allow us to explore the large number of degrees of freedom with which to construct new device configurations. This book aims to lay the groundwork for a methodology to exploit these emerging capabilities using optimal device design. By combining applied mathematics, smart computation, physical modeling, and twenty-first-century engineering and fabrication tools it is possible to find atomic and nanoscale configurations that result in components with performance characteristics that have not been achieved using other methods.

Imagine you want to design and build a novel nanoscale device. How would you go about it? A conventional starting point is to look at a macroscopic component with similar functionality, and consider ways to make it smaller. This approach has several potential pitfalls. For one, with continued reduction in size, device behavior will become quantum in character where classical concepts and models cease to be applicable. Moreover, it is limited by ad hoc designs, typically rooted in our unwillingness to consider aperiodic configurations, unless absolutely mandated by physical constraints. Most importantly this conventional approach misses the enormous opportunity of exploring the full landscape of possible system responses, offered by breaking all conceivable symmetries.

Computational resources, realistic physical models, and advanced optimization algorithms now make it possible to efficiently explore the properties of many more configurations than could be tested in a typical laboratory.

Introduction

Fundamental to most optimal design is the replacement of the actual physical device by high fidelity mathematical models. Models are also used in place of sophisticated sensors and performance measuring instrumentation to avoid the costly and time-consuming process of building a large number of prototype devices fitted with sensors and monitored by the associated test equipment. These mathematical models also help to establish an unambiguous relationship between the design parameters and the performance of the device. This relationship is commonly referred to as the optimal design problem's performance index. Mathematical optimization theory can then be used to characterize local and global optimal designs. Moreover, by formulating the optimal design problem mathematically, we may employ sophisticated mathematical programming techniques to provide an efficient means to search for these local and global optimal designs. This is especially significant if the design space is high or even infinite dimensional. Thus, mathematics plays a central and essential role in solving optimal design problems.

In this chapter we describe how mathematical systems theory can be used to develop a framework for optimal design problems. In particular, we consider the device which is the basis for the optimal design and the environment it is interacting with as a system. The internal quantities that uniquely characterize the status of the system are referred to as the system states. For example, in the electromagnetic waveguide design problem described previously in Chapter 4, the structure of the waveguide, the dielectric Teflon cylinders, and the surrounding space form a system. The state variables of this system include the intensity and directions of the electric and magnetic fields.