Light-Matter Interaction and the Structural Coloration of Birds

Light-Matter Interaction and the Structural Coloration of Birds Carmem Maia Gilardoni Research Paper Top Master in Nanoscience, Zernike Institute for Advanced Materials Supervised by: Prof. Dr. Doekele Stavenga [February-March 2016] Contents 1 Introduction 1 2 Bird Feather Structure 2 3 Physical Models 5 3.1 Maxwell's Equations . .5 3.2 Finite Difference Time Domain (FDTD) Formalism . .6 3.3 Band Structure Formalism . .7 3.3.1 The Eigenvalue Problem in Frequency Domain . .7 3.3.2 The Periodic Dielectric . .8 3.4 Effective Medium Description . 10 3.4.1 Thin Film Theory . 10 3.4.2 The Transfer Matrix . 12 3.4.3 Homogenization of the Refractive Index . 13 4 Application of the Models 16 5 Conclusion 19 Abstract Birds feathers have evolved showing an impressive pallet of colors and optical effects generated by the diverse arrangement of keratin, melanin and air in their barbules. Small changes to only a few parameters in the structure of the barbules are responsible for varied optical features. In order to understand the role of each parameter involved in the description of the photonic structures responsible for the structural coloration of birds, valid physical models for these arrangements must be obtained. We develop a description of the physical models generally applied to the description of the structure of the barbules, specifically the complete solution of Maxwell's equations in the time domain (FDTD) or in the frequency domain (photonic band structure), and the thin-film description of the structure through the effective-medium theory for the refractive index. In the context of the papers analyzed in this report, potential applications of an effective- medium theory for complex photonic structures are identified. The application of these models to different biological structures is examined, and the validity of these applications based on the assumptions inherent to the physical models is investigated. We find that a thorough development of a general effective-medium theory would allow the investigation of these structures under a new perspective, potentially providing quantitative insight into their properties. ii Chapter 1 Introduction Throughout millions of years of evolution, birds have developed the most varied ways of producing bright, often very saturated colors. These mechanisms of coloration have for centuries intrigued the scientific community, and great minds of the optics field such as Newton, Michelson and Lord Rayleigh have all speculated about their origin [1{3]. Particularly, two different mechanisms of color production have evolved: coloration from pigments and structural coloration [4{7]. Although the science of pigments is an equally broad and fascinating area of study, we here limit ourselves to reviewing the mechanisms of structural coloration and the role of nanosctructured media with features at the scale of the wavelength of visible light in producing such varied and intriguing color patterns. Since the pioneering work of Greenwalt [8] in which he elucidates the structural origin of iridescent colors of hummingbird feathers, it has become clear that nanostructured materials with contrasting dielectric constants are responsible for these dazzling color patterns. Furthermore, the development of techniques that allow us to observe the structure of the bird feathers at the nanoscale, such as Atomic Force Microscopy (AFM) and Transmission Electron Microscopy (TEM), has enabled the description of the architecture of these films in detail [9{19]. Thus, the task at hand in order to fully understand the origins and implications of the structural colors of birds is to fully grasp how these structures interact with light. The description of the interactions between light and matter can be performed accurately with the complete solution of Maxwell's equations. In principle, if the structure of the material where light propagates is known, this set of partial differential equations allows us to calculate the behavior of light in the medium without any constraints. Although the equations themselves might not be analytically solvable [20], the development of faster and more robust computational tools has allowed for the implementation of numerical techniques that provide very reliable approximations of the solutions. Indeed, a number of different computational approaches to the description of the propagating electromagnetic fields in matter have been developed [21{30], and they have proved themselves extremely effective tools in elucidating how complex structures can be used to influence the behavior of light. However, solving large sets of partial differential equations is not an easy task, even for computers. Furthermore, as mentioned before, a complete description of the medium must be given in order to attain unique solutions to the problem, including the dielectric constant of the medium as a function of space, and a set of physically valid boundary conditions. It often happens that in the description of biological nanostructures, these parameters might be the very information that is pursued. Maxwell's equations describe how electric and magnetic fields behave in space and in time, and how they are related to each other. Three different computational approaches can be taken when tackling the problem of finding solutions to these equations and describing how light propagates in biological media. The most general solution to the equations yields the value of the fields in every point in space, at every moment in 1 time. Computationally, this is realized in finite-difference time-domain (FDTD) simulations, which model the interaction between light and matter by solving for the electromagnetic field in the time-domain [21], when the dielectric constant of all points in the considered space are known. Alternatively, we can gather information about the propagation of light in matter by taking a plane wave approximation for the electromagnetic fields, and solving Maxwell's equations in the frequency domain [26]. Although this approach does not yield a full description of the fields in space and time, it can provide insight into what frequency components of the electromagnetic fields can propagate in the material, and which ones are fully reflected by it. Finally, the description of thin films dates to long before the Maxwell's equations and is based on simple mathematical constructions such as Snell's and Fresnel's equations [31]. This indicates that there must be a simpler approach to the problem of describing the interaction of light with the medium. This is the role of the mean-field approximation to the interaction of light with matter. In this method, the goal is finding an effective dielectric constant of composite films, which is a task that is often easier said than done [32,33]. However, once an effective refractive index is obtained, the medium can be divided in arbitrarily thin films and we can use simple, fast simulations to describe the reflection and transmission of light through these films when the thickness of the films is much smaller than the wavelength of the light [29, 34]. The description of the effective refractive index of one-dimensional two-phase materials is well established [35]. Inspired by the fundaments of this formalism, the development of models that accurately describe the effective refractive index of more general structures can be a potent tool in modelling how biological photonic structures interact with light. It is the focus of this work to provide an overview of how some mechanisms of structural coloration in bird feathers can be optically modelled using the three different approaches aforementioned. With this in mind, we will provide a description of the physical structure of these bird feathers. An overview of the papers that developed both the formalism and the computational application of the three mathematical models of the interaction between light and matter will be discussed. Special attention will be given to the development of a homogenization theory for the refractive index of periodic structures, since this approach shows the potential to significantly cut computational costs when modelling complex nanostructures. Finally, we will examine why different techniques were applied to the description of bird coloration in the biological literature, and what can be understood about these biological structures due to the physical modelling of their properties. Chapter 2 Bird Feather Structure The coloration of birds is one of the key vehicles of inter- and intra-specific communication in several bird species [36]. Consequently, birds have developed several mechanisms of coloration that enable diversity of the features produced [37]. These mechanisms can be broadly classified according to their origins: diffuse coloration due to light absorption by pigments, and directionally reflected coloration due to nanostructured features of the feathers [7]. The structural mechanism of bird coloration is responsible for the hue of the light reflected from 2 Figure 2.1: Structure of a bird feather, showing the branching of the shaft into barbs, and of the barbs into barbules. Knowledge of the nanostructure of the barbules is fundamental to the description of the interaction of the feathers with light. Picture taken from [39]. the feather, its saturation, brightness and iridescence. These aspects are equivalent, respectively, to the peak wavelength in a reflectance spectrum, its width, intensity and the dependence on the angle of observation. The different mechanisms of coloration are not mutually exclusive, therefore the multiple

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