## 1 Introduction

Dispersion engineering is defined in the context of this contribution as the control of electromagnetic radiation through material dielectric properties. The dawn of dispersion engineering can be traced to early attempts to focus electromagnetic radiation in the optical frequency range. Indeed, evidence exists suggesting crystalline lenses were fabricated by ancient Egyptians 1. A significant advance in dispersion engineering occurred in 13th century Italy with the development of spectacles, greatly expanding the reach of the printed word 2.

In modern times, interest in dispersion engineering has been renewed around the concept of negative refraction through isotropic dispersion 3. The mathematical treatment of electromagnetic radiation through Maxwell's equations provided evidence that negative refraction could, *e.g*., produce a perfect lens if a suitable material could be fabricated. The turn of the 20th century saw the mathematics describing negative refraction modernized 4, which in turn catalyzed intense efforts to fabricate new materials and measure negative refraction. Most of these studies have been focused on materials structures with a high degree of fabrication complexity. Recognition of laminate material anisotropy provided the breakthrough concept necessary to fabricate materials exhibiting anomalous dispersion 5, 6 and to produce precisely measurable dispersion effects. Recent dispersion engineering literature describing anisotropic materials systems exhibiting anomalous dispersion of incident light has increased dramatically since 2002 7–13.

In this contribution, we address dispersion engineering through fabrication and study of new nanolaminates containing thin amorphous-metal and solution-processed-oxide films. We have recently described the materials properties and film interfacial chemistry of these highly regular laminated structures 14. The smooth and pristine interfaces between the amorphous metal and oxide have also enabled the realization of high-performance metal–insulator–metal diodes 15. These findings have prompted us to examine their behavior as a new type of anisotropic dielectric medium.

To appreciate the advances in dispersion engineering enabled by these amorphous composite materials, we first describe the differences between isotropic and anisotropic dispersion. The following discussion of dispersion is focused on non-magnetic materials (*µ* = 1). Additionally, the materials described are planar in nature and are represented by an abrupt change in index of refraction in the *z* direction as light passes into the plane of the material. In planar structures with small out of plane scattering, momentum is conserved to a single plane represented by two dimensions. The mathematics used to describe the dispersion effects observed in planar structures are, therefore, presented in two dimensions.

The dielectric response, *ε*, of a planar isotropic material that does not exhibit directional dependence is described as

where *k*_{xy} is the light's momentum component in the plane of the material, *k*_{z} the momentum component orthogonal to the material interfaces, *n* the material's index of refraction, *ω* the angular frequency of the light, and *c* is the speed of light in vacuum. The engineered dispersion of incident light is accomplished solely through the modulation of *ε*, which is a complex number varying with the frequency of the incident light. As light encounters a change in *ε*, as it passes from free space (*ε* = 1) to a material with *ε* ≠ 1, the direction of the light's momentum (** K**) changes due to a magnitude change in the

*z*component of the light's momentum (

*k*

_{z}). The Poynting vector (

**) represents the direction of energy flux, and is coincident with**

*S***in an isotropic dielectric material. Figure 1(a) illustrates the response of light as it passes into an isotropic dielectric material from air. The described dispersive effect, through the use of isotropic dielectric materials with**

*K**ε*≠ 1, is the means through which light was manipulated via dispersion engineering by the ancient Egyptians 1.

The dielectric response of layered, anisotropic, dielectric materials has two components; *ε*_{xy} in the plane of the material interfaces and *ε*_{z} orthogonal to the plane of the material interfaces. Anisotropic dispersion is exhibited only with *T*_{M} polarized light. *T*_{M} polarization stipulates that the magnetic field vector is parallel to the material plane as shown in Fig. 1(b) and (c). The electric field in *T*_{M} polarized light is influenced by both components of the material dielectric response. Two dielectric response components lead to a dispersion equation

where *ε*_{z} is the dielectric response orthogonal to the material plane and *ε*_{xy} is the dielectric response in the material plane. A condition of anisotropic dispersion is that *ε*_{z} ≠ *ε*_{xy}. Isotropic and anisotropic materials may possess dielectric responses with real components being either positive or negative polarity. A negative real component of a dielectric response is typically indicative of a metallic material in which electromagnetic waves decay due to the fast rearrangement of free charge.

The mathematics describing an anisotropic material possessing two distinct dielectric responses, *ε*_{z} and *ε*_{xy}, allow for three distinct dispersion effects based on the polarity of Re(*ε*_{z}) and Re(*ε*_{xy}). Figure 1(b) illustrates anisotropic, elliptical dispersion which occurs when both Re(*ε*_{z}) and Re(*ε*_{xy}) are positive. Anisotropic, elliptical dispersion separates ** K** and

**as light propagates in the anisotropic material. Anisotropic, hyperbolic dispersion, presented in Fig. 1(c), occurs when Re(**

*S**ε*

_{z}) and Re(

*ε*

_{xy}) possess opposite signs. Negative refraction occurs when Re(

*ε*

_{z}) < 0 and Re(

*ε*

_{xy}) > 0, whereas hyperbolic dispersion with positive refraction occurs when Re(

*ε*

_{z}) > 0 and Re(

*ε*

_{xy}) < 0. Anisotropic materials fabricated in this research possess measured reflectance characteristic of both anisotropic, elliptical dispersion, and hyperbolic dispersion with positive refraction.