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 possible combinations of pigments and structural features generates a vast palette of available colors [38]. Figure 2.1 shows the macroscopic structure of a bird feather. It is formed by a main shaft that is branched into barbs which are further segmented into barbules. Structural colors are caused by variations of the refractive index of the structure at a scale smaller than the wavelength of light, and the advent of techniques that enable us to look at the subwavelength architecture of the barbules has facilitated the investigation of the mechanisms responsible for the diversity of the coloration effects in these animals [8]. The structure of the barbules is fundamentally responsible for many dazzling effects. Keratin, melanin and air often combine in different patterns that create photonic structures carefully designed to interact with light in certain ways [6]. While amorphous [6, 17] or fibrous [40] distributions of air in a keratin and melanin matrix are responsible for noniridescent structural colors, the regular arrangement of melanosomes (melanized particles) of various geometries in a keratin matrix forms structures with contrasting indices of refraction that generate both hue and iridescence effects [7]. Figure 2.2 samples the variety of structures that can be achieved by modification of the distribution and shape of the melanosomes. We can identify the relationship between the visual characteristics of a selection of bird feathers (figures 2.2a-2.2d) and the TEM images of cross-sections of the barbules present in them (figures 2.2e-2.2h). The glossiness of the feathers of the common crow (figure 2.2a) is originated from arrangements of melanosome rodlets in a keratin matrix that lack periodicity, as shown in figure 2.2e [41]. In contrast, the feathers of the Qualicalys major (figure 2.2b) owe their iridescence to a more regular and dense distribution of the melanin rodlets beneath a thin keratin cortex (figure 2.2f). An ordered distribution of melanosome rodlets in a periodic fashion (figure 2.2g) can generate both hue and iridescence features (figure 2.2c) [13, 18]. Finally, structures formed by air holes in a keratin and melanin matrix (figure 2.2h) are responsible for colored feathers that lack iridescence (figure 2.2d). Compared to the nanostructures formed by filled melanosomes (figure 2.3a), the nanostructures composed of hollow melanosomes shown in figure 2.3b generously broaden the range of possible structures formed, allowing for brighter and more saturated colors [11,36,37]. Although these composite structures are formed by only keratin, melanin and air, many variables are involved in the determination of the color spectrum generated by them. The indices of refraction and absorption coefficients of the keratin and melanin [18]; the periodicity, form and regularity of the structure [14, 37, 41] and the roles of the different materials in the interaction with light [15] are all determinant factors of the coloration spectra of these structures. In order to fully understand them

3 (c) (d) (a) (b)

(g)

(e) (f) (h)

Figure 2.2: Figures 2.2a, 2.2b, 2.2c, and 2.2d show pictures of diﬀerent bird feathers, while pictures 2.2e, 2.2f, 2.2g and 2.2h show the cross section of their respective barbules, as seen in a transmission electron microscope. Figures 2.2e and 2.2f show the structures responsible, respectively, for the glossiness of the common crow (2.2a) and for the iridescence of Quasicalys major (2.2b). Pictures were taken from [41]. The bright reﬂection of the Bird of paradise Lawess parotia occipital feathers (2.2c) is due to the highly ordered arrangement of melanin rodlets in its barbule (2.2g). Figures 2.2c and 2.2g were taken from [16]. Figure 2.2h shows structure formed by air cylinders in a keratin and melanin matrix, responsible for the noniridescent yellowish-green tail feathers of the black-billed magpie (2.2d). Both pictures were taken from [19]. Scale bars of images 2.2e, 2.2f, 2.2g and 2.2h are, respectively, 2 µm, 2 µm, 2 µm and 1 µm.

(a) (b)

Figure 2.3: Filled melanosomes (2.3a) and hollow melanosomes (2.3b), compared. The structure formed by hollow rodlets allows a larger variety of possible scattering structures to be formed. A lower eﬀective index of refraction can be achieved (lower fraction of the volume is packed with a material with high index of refraction) while keeping the thermodynamically favorable close-packing of the structure [37]. Pictures are taken from [36]. Scale bars are 500 nm

4 biologically and physically, it is necessary to develop an intuitive and comprehensive physical model capable of describing their operation. Comparison of the experimental reflectance spectra obtained from the feathers with the reflectance spectra calculated with physical models can provide insight into the role of the different materials in the structure. Moreover, being able to predict how changes in the parameters aforementioned would change the reflectance spectra obtained could allow us, for instance, to determine if and how these structures have been optically optimized during the evolution of the features, and based on which criteria [38]. This could lead us to engineer biomimetic structures efficiently, inspired by the structures that nature elaborated throughout millions of years [42–44]. Inspection of the structures showed in figures 2.2 and 2.3 indicates that a modified multilayer theory of the interaction between light and matter capable of simplifying the treatment of inhomogeneous media could potentially lead to the better understanding of the optical properties of these structures. In the hunt for a model capable of describing and predicting the behavior of light when interacting with these structures, we examine the very basic elements of optical theory.

Chapter 3

Physical Models

3.1 Maxwell’s Equations

When the Maxwell’s equations that describe how an electromagnetic wave propagates through a crystal are written in particular ways, some of the information they carry becomes easily accessible. The reformulation of the equations with the intent to reveal diﬀerent parameters has been done in detail [21, 45]. Here, we describe the Maxwell’s equations as they will be employed to ﬁnd their solutions, using the suitable parameters for the biological media of interest. Maxwell’s equations in cgs units are given by:

∇ · B = 0, (3.1a) ∇ · D = 4πρ, (3.1b) 1 ∂B ∇ × E + = 0, (3.1c) c ∂t 1 ∂D 4π ∇ × H − = J (3.1d) c ∂t c where E and B are the electric and magnetic fields, D and H are the displacement field and the magnetic induction field. Moreover, ρ is the free charge density and J is the free current density. In the biological media of interest in this work the electromagnetic fields are weak, such that it is appropriate to assume that

5 the response of the medium to the radiation is linear and that the materials are isotropic and non-magnetic. Furthermore, we assume the absence of free charges and free currents. Following these considerations, the set of equations in 3.1 are signiﬁcantly simpliﬁed [20]:

∇ · H(r, t) = 0, (3.2a) ∇ · ε(r, t)E(r, t) = 0, (3.2b) 1 ∂H ∇ × E + = 0, (3.2c) c ∂t ε(r, t) ∂E ∇ × H − = 0 (3.2d) c ∂t

In equations 3.2, information about the material is contained in the dielectric constant ε(r, t), while the description of the evolving fields is given by H and E. When boundary conditions are imposed, the solution of these equations becomes non-trivial and often impossible to find analytically. It is the numerical solution of the set of coupled partial differential equations 3.2 that we aim for when we introduce the different techniques that follow.

3.2 Finite Diﬀerence Time Domain (FDTD) Formalism

In 1966, Yee proposed an iterative approach to the solution of equations 3.2 that allowed the direct visualization of the H and E fields at all times [21]. He considered a simulation volume that includes the regions of interest (interacting medium, and region where the fields propagate), with perfectly absorbing boundaries. This volume is then divided into a grid of small volume elements ∆x∆y∆z in Cartesian coordinates, such that the functions of interest vary little within the volume elements and the differentials of equations 3.2 can be approximated by finite differences. A point in space (x, y, z) can be denoted by its grid element, such that (x, y, z; t) = (i∆x, j∆y, k∆z; n∆t) and a function of space and time can be written as F n(i, j, k) ≡ F (i∆x, j∆y, k∆z; n∆t). In Cartesian coordinates, each of the equations 3.2c and 3.2d yields a set of three coupled differential equations, one for each space coordinate. For the z component of the magnetic induction field, for instance, we obtain

1 ∂H ∂E ∂E z = x − y (3.3) c ∂t ∂y ∂x Approximating the derivatives in equation 3.3 by ﬁnite diﬀerences as suggested above gives

n+1/2 1 1 n−1/2 1 1 1 Hz (i + , j + , k) − Hz (i + , j + , k) 2 2 2 2 = c ∆t (3.4) En(i + 1 , j + 1, k) − En(i + 1 , j, k) En(i + 1, j + 1 , k) − En(i, j + 1 , k) x 2 x 2 − y 2 y 2 ∆y ∆x

Similar equations can be constructed for the two remaining components of the H field, and for the components of the E field. It becomes apparent from equation 3.4 that we can calculate the value of the magnetic induction field components in a point of space (x0, y0, z0) at time t+∆t/2 if we know the values of the magnetic induction field in the same point at a previous time t − ∆t/2, and the value of the electric field in the vicinity of point (x0, y0, z0) at time t. In an analogous manner, we can calculate the values of the electric field components

6 at a particular point, at time t + ∆t. To resolve this, we must know the value of the electric field components in that point at a previous time t and the value of the magnetic induction field in its vicinity at time t + ∆t/2. Furthermore, we can take volume elements small enough that the dielectric constant ε(r, t) is constant within the individual elements. In this way, the electric and magnetic fields can be solved iteratively for every point in space, at every moment in time. In this development, no constraints were assumed to the components of the fields. Thus, in principle, we could describe any incoming wave of random polarization and angle of incidence upon the surface. Furthermore, we applied no restriction to the shape of the dielectric constant of the medium, which means that any general structure could be considered as a simulation volume, as long as the dielectric constants of the individual constituents are known. The possibility of complex dielectric constants accounts for absorbing media. Moreover, the constraint of a linear response of the medium could be lifted, and a nonlinear optical response could be modelled in a similar way [46]. This freedom of choice of the parameters present in Maxwell’s equations constitutes one of the main ad- vantages of this method. Interaction of light with arbitrary finite volumes can be calculated, and the values of the time-dependent evolution of fields are obtained in real-space. Thus, information such as penetration depth, reflected and transmitted intensity and the far-field scattering intensities are easily obtained. However, these same factors are responsible for the main disadvantage of this method: the lack of constraints implicates extensive computational costs. Solving the problem of the interaction of a two-dimensional structure with light of a broad range of frequency components and of all polarizations can require huge computer power and the calculations are often performed in parallel in supercomputers [16]. Since Yee’s article describing the method in 1966, refinements of the iterative algorithms in order to find stable solutions which require relatively low computer power have had ample space in the literature [22,23,47]. Additionally a large number of different softwares are nowadays available to solve massive sets of Maxwell’s equations in this way, such as TDME3D, GMES, MEEP and Angora [48–50].

3.3 Band Structure Formalism

3.3.1 The Eigenvalue Problem in Frequency Domain

When analyzing the interaction of light with the two dimensional photonic structures responsible for the structural coloration of birds, we are often not interested in the time-evolution of the fields. Thus, we can diminish the computational costs associated with solving the Maxwell’s equations by adopting the band structure formalism. In this approach, we look at how the interaction of light with the medium of interest depends on the frequency of the incoming light, rather than the specific values of the fields in all points of space and time. The photonic band structure formalism has been extensively developed in the literature in the last 30 years [25, 26, 51, 52]. Here, we delineate the main aspects of the derivation. The derivation relies on the fact that the electric field and the magnetic induction field can be separated into a part that varies in space and a part that carries the time-dependence, as in equation 3.5. In this context, these fields can be described as an infinite sum of orthogonal Fourier components that vary harmonically in time. Fourier analysis guarantees that any solution of the equations can be expanded in this way [45].

X H(r, t) = H(r, ω) exp iωt ω X (3.5) E(r, t) = E(r, ω) exp iωt ω

7 where the sum can be taken into an integral when the distributions of frequencies is continuous. Because the modes are orthogonal, when the ﬁelds in 3.5 are substituted into 3.2, the integrals and sums drop out and we get the two curl equations that describe E(r) and H(r):

iω ∇ × E(r) + H(r) = 0 (3.6a) c iω ∇ × H(r) − ε(r)E(r) = 0 (3.6b) c

Simple algebra brings us to the wave equation for the magnetic induction ﬁeld,

1 ω 2 ∇ × ∇ × H(r) = H(r) (3.7) ε(r) c This is an eigenvalue problem for the operator

1 ΘH(r) ≡ ∇ × ∇ × H(r) (3.8) ε(r) with eigenfrequencies proportional to ω2. The operator 3.8 can be shown to be Hermitian [45], so that two modes only differ if their eigenfrequencies differ. The wave equation 3.7 is reminiscent of the Schrödinger equation [53], which suggests that we could take inspiration from that area when approaching the solution of the electromagnetic problem. The analogy with quantum mechanics helps us understand the physical meaning of the dielectric constant ε(r). This parameter carries the information of the medium where the wave function of interest exists and can be compared to the potential energy in quantum mechanics. With this interpretation in mind, it becomes clear that the problem of solving equation 3.7 for the magnetic induction field in a medium with periodic distribution of dielectric constants is closely related to solving the Schrödingerequation for the wavefunctions of electrons propagating in a semiconductor crystal [54].

3.3.2 The Periodic Dielectric

In the special case where the index of refraction of the medium is periodic, we can deﬁne a lattice such that

ε(r) = ε(r + R). As is done in solid state, we can determine primitive lattice vectors ai and primitive reciprocal P lattice vectors bi such that ai ·bj = 2πδij. In real space, the medium is invariant upon a translation R = niai with integer ni; in reciprocal space, the solutions of the wave equation remain invariant when the wave vector P is translated by Q = i vibi, with integer vi. Thus, any wave vector in the reciprocal lattice is equivalent to P a vector k = i kibi in the volume defined by the three primitive reciprocal lattice vectors, called Brillouin zone. In this way, we can apply Bloch’s Theorem and find solutions of the wave equation 3.7 in the form of the product of a plane wave (exp ik · r) and a function uk(r) which carries the periodicity of the medium. Here, each value of k corresponds to an eigenstate of the operator 3.8 with eigenvalue ω(k). We obtain as a result a set of dispersion relations that relate the wave vector of the propagating magnetic induction field to its energy. The dispersion relations can be visualized in band structure diagrams. We observe that equations 3.6 have no intrinsic scale. When the periodicity of the system is expanded or contracted, all of the terms in the equations will change accordingly. Thus, the value of interest is simply the ratio of the scales of the problem, that is, the period-to-wavelength ratio. The symmetries of the periodic structures are of primary importance in the photonic crystal approach. Using

8 Figure 3.1: Model of one-dimensional, two-dimensional and three-dimensional photonic structures. These structures are characterized by the periodicity of the dielectric constant. Figure taken from [45]. symmetries and the Bloch expansion of the field, the eigenvalue problem presented in equation 3.7 is significantly simplified [45]. Photonic crystals can be classified based on the periodicity of the dielectric constant and thus on the symmetry elements of the system, as shown in figure 3.1. One-dimensional photonic structures have dielectric constants that are periodic in one direction, but constant on the planes perpendicular to it. Such a structure is approximated by a series of thin films. Two-dimensional photonic structures have a dielectric constant that is periodic in two-dimensions, but extends itself in the third dimension; that is, ε(r) = ε(x, y). Essentially, these structures are made up of fibers packed in square or hexagonal lattices. As mentioned before, this is a good model for many arrangements responsible for the structural coloration of birds [19, 55–57]. A three-dimensional photonic crystal has dielectric constant that depends on all three coordinates of space. An electromagnetic wave propagating along the plane of periodicity of a two-dimensional photonic structure can show two distinct polarizations, as in figure 3.2c. The two different possible polarizations localize the electric field and the magnetic induction field in different environments. For TM modes, the magnetic field oscillates in the plane of periodicity. Analysis of figure 3.2c shows that in this case, the electric field will be localized in regions where the dielectric constant of the medium is somewhat uniform. In contrast, the electric field of TE modes oscillates in the plane of periodicity and is spread out over regions of high and low dielectric constant. It follows that the dispersion relations of beams with different polarizations will differ. Just as it happens in semiconductors, the periodicity of the medium can create gaps in the frequencies that propagate in the structure. Physically, this means certain values of energies have no corresponding wave vector in the medium. This is the case when the periodicity of the crystal is such that incoming and reflected waves with a certain energy interfere, generating standing waves. Modes with these energies are said to be forbidden in the photonic crystal and cannot propagate in the medium. These gaps are not always present, and they can often be present for only one polarization of the incoming beam. This brings us to the relevance of this approach to the modelling of the interaction between the light and the medium of interest: the information about the modes that cannot propagate into the crystal and are fully reflected by the surface is readily available in a band structure diagram. Computationally, the solution of the problem of finding the photonic band structure of a crystal has benefited tremendously from the extensive development of techniques to solve band structures of semiconductors, based on ab initio algorithms constructed upon the variational theorem. The variational theorem can be applied in order to find eigenmodes that minimize the energy functional of the system [45]. Since the 90’s, several algorithms have been developed to calculate photonic band structures [24, 25, 52, 58] based on the variational theorem.

9 Calculations are usually done by transforming the eigenvalue problem 3.7 into a finite matrix eigenproblem, which is then solved by linear algebra techniques. An initial guess of the components of the field in an appropriate vector basis is given as input to the computer, which applies a set of transformations to the vector in order to minimize the energy functional. These transformations yield then a second guess for the components of the field, and the process is repeated iteratively until the energy converges into the ground state energy. The calculation of the band structures requires significant less computational power than the calculations associated with the FDTD method [46]. Different software are available to perform the task, and the freely available MIT photonic band solver MPB is most widely used [59].

The ease with which the photonic bands can be computed would indicate that this is an approach far superior to the FDTD method. However, in the calculation of the eigenvalue problem several assumptions were made, which limit the scope of this technique. First and foremost, the expansion of the ﬁelds as Bloch functions depends on the crystal of interest being inﬁnite, and this is not an adequate approximation to the description of surfaces. Furthermore, the dielectric constant ε(r) is assumed to be real, which excludes the treatment of absorbing media. With these limitations in mind, application of this technique to the modelling of biological photonic structures must be done with care.

3.4 Eﬀective Medium Description

3.4.1 Thin Film Theory

The simplest example of a photonic structure is a thin film. The description of thin-films dates from long before Maxwell’s equations, and relies on very simple mathematical concepts. Snell’s equation determines the relationship between the angle of refraction of a beam going through the boundary between two homogenous media (defined with respect to the normal of the surface) and the contrast between their refractive indices:

n1 sin(θ2) = sin(θ1) (3.9) n2

In bird feathers, the refractive index depends on the wavelength of the incident light and can be complex in order to account for absorbing media; furthermore, we can relate the refractive index and the dielectric constant of non-magnetic media: n(r, λ) = pε(r, λ).

As was done in section 3.3, we assume the plane-wave approximation for the propagating fields. Two particular polarizations of light can be distinguished. For TE modes, the H(r) oscillates in the plane of incidence, and the E(r) oscillates perpendicularly to it; this is referred to as s-polarized light. In the TM mode, the E(r) oscillates in the plane of incidence; this is referred to as p-polarized light. From geometrical considerations based on figure 3.2a, we can define boundary conditions for the E(r) and H(r) at the interface between media 1 and 2 that guarantee the continuity of the fields at the boundary. These boundary conditions, combined with equation 3.9 allow us to calculate the Fresnel equations that describe the reflection and transmission coefficients

10 of the electric ﬁeld at the boundary [60]:

n1 cos(θ1) − n2 cos(θ2) rs = (3.10a) n1 cos(θ1) + n2 cos(θ2) 2n1 cos(θ1) ts = (3.10b) n1 cos(θ1) + n2 cos(θ2)

n1 cos(θ2) − n2 cos(θ1) rp = (3.10c) n1 cos(θ2) + n2 cos(θ1)

2n1 cos(θ1) tp = (3.10d) n1 cos(θ2) + n2 cos(θ1)

The reflection and transmission coefficients in equation 3.10 are defined as the ratio of the magnitude of the reflected or transmitted fields, respectively, and the incoming field. The intensity of the reflected beam R can be obtained by taking the ratio between the square of the amplitude of the reflected field and the square of the amplitude of the incoming field. For the intensity of the transmitted beam of a medium with absorptance A, conservation of energy gives R + T = 1 − A, such that

2 Rs,p = |rs,p| (3.11a)

n2 cos(θ2) 2 Ts,p = |ts,p| (3.11b) n1 cos(θ1)

Equations 3.11 make it clear that the intensities of the beam reflected from the boundary between media with contrasting indices of reflection depends on the angle of incidence. Furthermore, this dependence varies for s- and p-polarized light. We emphasize, as can also be seen from equations 3.10 and 3.11, that the reflectivity of a boundary increases with the contrast between the refractive indices of the two media. When a second boundary is considered, interference effects are introduced into the problem. We can analyze this problem with the help of figure 3.2a, which depicts a layer of thickness d and refractive index n2 sandwiched between media with refractive indices n1 and n3. The amplitudes and the intensities of the fields reflected and transmitted at each interface is given by equations 3.10 and 3.11. However, the intensities of these fields cannot be simply added, since interference effects will change how the fields reflected from the different interfaces interact. The optical path of a field with wave vector k propagating for a distance ∆r in a medium with refractive index n is given by φ = nk∆r. The optical phase difference between two beams represents then a phase difference gained when the two fields traverse different paths in space. If a fraction of the beam is reflected from the first interface (between media 1 and 2) and another fraction is reflected from the second interface (between media 2 and 3), the phase between them is then given by φ = n2(2π/λ)(2d cos(θ2)). On one hand, they will interfere constructively if φ = 0 and the intensity of the reflected beam will be significantly enhanced; on the other hand, they will interfere destructively if φ = π, and the intensity of the reflected beam will be diminished. Thus, we see that the reflectance spectrum will show various minima at values of wavelength for which the condition n2d cos(θ2) = (2m + 1)λ/4 is satistfied, and maxima if instead n2d cos(θ2) = mλ/2 is satisfied, where m is an integer. The dependence of the optical path difference on the wave vector and on the angle of incidence of the incoming light is responsible for effects such as iridescence. Iridescent structures have a reflectance spectrum that depends on the angle of incidence. In other words, different wavelengths will be strongly reflected for different angles between the surface and the incoming beam.

11 3.4.2 The Transfer Matrix

The reasoning treated in the previous paragraph can be extended in order to calculate the reflectivity and transmissivity of structures made up of a large number of thin layers with different refractive indices, in the transfer-matrix formalism. We can write the matrix that describes the propagation of an electromagnetic beam through a stack of layers by a system matrix which is simply the product of the matrices that describe the propagation of the beam through the individual layers. The characteristic matrix of a medium with stratified indices of refraction in the zˆ direction is the matrix that relates the x and y components of the electric (or magnetic) field at z = 0 to the x and y components of the field in a plane with constant z. The change to the components of the field when entering medium j can be described by a matrix Dj; the inverse of this matrix −1 describes the changes to the components of the field when exiting medium j, such that Dj Dj+1 describes the changes to the components of the field when going from medium j into medium j + 1. Furthermore, the propagation of the field in the medium j is given by a matrix Pj which describes the phase factor gained. In this way, we can describe the components of the electric field propagating through N layers with complex indices of refraction nj and thickness dj by a transfer matrix M:

" # N−1 M11 M12 −1h Y −1i M = = D0 DjPjDj DN (3.12a) M21 M22 j=1 " # pj pj Dj = (3.12b) qj −qj " # sj 0 Pj = −1 (3.12c) 0 sj

where, for TE polarized light pj = 1 and qj = nj cos (θj); for TM polarized light, pj = cos (θj) and qj = nj; sj = exp (injk cos (θj)dj) [31]. The reﬂectance of the medium composed by the diﬀerent layers is then given by

M 2 R = 21 (3.13) M11 The intensity of the reflectance peak increases with the number of layers due to reflection from the different interfaces and decreases with the absorption by the layers, like is expected intuitively [46]. Since the matrices involved in the simulation simply relate the two components of the electromagnetic field, they are 2x2 matrices. This fact simplifies the linear algebra needed to solve the problem, and substantially reduces the computational effort. The description of structures with a graded refractive index, for instance, can be done in a intuitive way with the transfer-matrix method. Structures with a graded refractive index are structures that have a refractive index that varies in one direction (ˆz) and is homogeneous in the planes perpendicular to it. We can divide the structure in very thin films perpendicular to ˆz and approximate the refractive index of each film by a constant value. Then, the transfer-matrix method can be applied in order to calculate the reflectance spectra of the structure. The simplicity of the description of the problem of reflection from thin layers makes this an inviting approach to analyze the spectra of light reflected from a surface and to engineer surfaces that reflect and transmit light in particular ways. Different spectra can easily be obtained for different values of polarization, different frequencies of light and different angles of incidence simply by taking the appropriate values of reflectivity and transmissivity, and changing the magnitude and direction of the wave vector. Furthermore, the possibility of absorbing media

12 (b) (a) (c)

Figure 3.2: Model of different optical structures. Figure 3.2a shows a thin film sandwiched between two different media, and the behavior of a TM-polarized light beam propagating through the media. Figure 3.2b shows a structure with periodic refractive index in they ˆ direction, and the relevant parameters in the description of an effective-medium theory of such a structure: the periodicity length Λ and the proportion of the volume filled by the different materials, given by w. Figure 3.2c shows a photonic structure for which the dielectric constant is periodic in two dimensions (in the yz plane). The configurations of the two special polarizations of the incoming light are explicited. can be easily accounted for by allowing the refractive index to take complex values. Overall, the technique provides readily available insight into the behavior of light when interacting with a structure, while avoiding the complications of the rigorous methods.

3.4.3 Homogenization of the Refractive Index

The question remains of whether it is possible to find an expression for the effective refractive index of an arbitrary structure in order to use the thin-film approximation to calculate its reflectance spectrum. In the development of section 3.4.1, we have assumed that the refractive index of the individual thin films is homogeneous, and this is not the case for the composite structures. However, we can take inspiration in a number of mean-field approaches to the description of physical phenomena [61, 62] in order to try to homogenize the refractive index of the medium. A successful effective-medium theory for the refractive index would yield an average value for the refractive index of a composite medium. This averaged refractive index would then generate a reflectance spectrum in agreement with experiment. The first thing to consider is that in order for the light to experience an average refractive index when propagating through a medium, the change in refractive index must happen in a scale smaller than the wavelength of the propagating field. A quantitative calculation of the limit of validity of an effective-medium approach is of fundamental importance to the application of the theory, and discussion of the works that address this problem is done further in the text. For periodic one-dimensional structures with period much smaller than the wavelength of the propagating light, the problem of calculating an effective index of refraction for the material has long been solved [28,29,35]. In order to be able to expand the formalism and calculate the effective dielectric constant of more complex structures it is relevant to examine the quantitative formalism applied. Here, we follow the reasoning explained by Lalanne and Lemercier-Lalanne in [29]. Given a periodic one-dimensional structure with period Λ, as shown in figure 3.2b, we can define a grating vector K such that |K| = 2π/Λ. The dielectric constant can be written as ε(y), as long as it shows the periodicity of the structure. In this case, we can expand the dielectric constant also in its Fourier components, such that

13 X ε(y) = εm exp (iKmy) (3.14) m The aim of the effective-medium approach is to describe the propagating electromagnetic wave as a plane p wave with time dependence exp (−iωt) propagating in a medium with effective refractive index neff = (η), where η is a function of the effective dielectric constant, the periodicity of the structure and the wavelength of incoming light η ≡ η(εeff , Λ, λ). In the limit of long wavelengths, that is α = Λ/λ 1, we can assume the validity of the expansion

−1 −2 η = η0 + η1α + η2α + ... (3.15)

When the dimensions of the grating are comparable to the wavelength of the light (α ≈ 1) resonance effects can significantly influence the interaction between the structure and the light. The periodicity of the crystal is transferred to the propagating beam, so that we allow the amplitude of the electric field to vary in the y direction. For a TE beam propagating normally to the grating surface, the amplitude of the electric field is given by

√ X Ex = A(z, y) = exp (i ηkz) Am exp (iKmy) (3.16) m where we used the Fourier expansion of the amplitude A(y). For a plane wave, the curl equations (3.6) derived from Maxwell’s equations are reduced to the Helmhotz’s equation [35]. Thus, the wave function in equation 3.16 must satisfy

∇2A(z, y) + k2ε(y)A(y, z) = 0 (3.17)

Substituting equations 3.14 and 3.16 into 3.17, and performing the diﬀerentiation, we get an inﬁnite set of linear equations for η, such that

2 2 X (ε0 − η − j α )Aj + εj−pAp = 0, ∀j (3.18) p6=j In order to get a non trivial solution to the problem, the determinant of the matrix that describes the set of linear equations must be zero. Substituting the expansion 3.15 into the determinant, the following relations can be shown:

η0 = ε0 (3.19a)

η2n+1 = 0 (3.19b) X ε−pεp η = (3.19c) 2 p2 p which explicit the zeroth and second order terms in the expansion of η; furthermore, it states that any correction of odd order is identically null. The process can be repeated for TM polarized light, where the magnetic ﬁeld oscillates in the x direction and the electric ﬁeld has components in the y and z directions. The mathematical derivation yields an expression for η given by

14 2 1 π 2 2 ε2 − ε1 ε0 −2 η = + f (1 − f) 3 α (3.20) a0 3 ε2ε1 a0 where we deﬁne ai as the Fourier components of 1/ε(y). The components present in expressions 3.19 and 3.20 must be obtained from the Fourier series that describe the dielectric constant ε(y) and its inverse 1/ε(y), respectively. For instance, applying the inverse Fourier transform to the function that describes the dielectric constant of the periodic grating shown in ﬁgure 3.2b gives zero order components to the dielectric constant and its inverse that yield

TE η0 = ε0 = fε1 + (1 − f)ε2 (3.21a)

TM 1 1 1 η0 = = f + (1 − f) (3.21b) a0 ε1 ε2

where we define the fraction of the space occupied by the medium of dielectric constant ε1, f ≡ w/Λ. In fact, this result is widely accepted as the first order approximation to the effective dielectric constant for the TE and TM polarized beams propagating normally to a one-dimensional periodic grating [34]. The effective-medium description of one-dimensional structures has been validated by comparison with rigorous calculations [29, 34]. In particular, the approximation of equation 3.21 is a better match to the refractive indices obtained through rigorous calculations for the TE-polarized modes than for the TM-polarized modes. The description of the problem for a two-dimensional periodic structure is somewhat more laborious. Several approaches to the problem have been undertaken, relying both on intuitive analysis and on rigorous calculations. 0 Over 50 years ago, Coriell and Jackson have determined the upper and lower boundaries (respectively 2n2D,max 0 and 2n2D,min) of the effective dielectric constant of a two-dimensional binary system made up of square or cylindrical fibers in a square lattice in the case of small period-to-wavelength ratios [32]. In the last decade of the twentieth century, the development of techniques that allowed the fabrication of subwavelength periodic structures motivated the search for effective-medium description of more complex periodic structures, such as the two-dimensional ones. Bräuerand Bryngdahl, with the intent of designing antireflection gratings, first tried to find an expression for the effective dielectric constant of a two dimensional structure for beams propagating in the plane of periodicity [34]. They found that an appropriate guess for the effective refractive index of a two-dimensional periodic medium was given by

n¯ + 2n0 + 2n0 n2D = 2D,max 2D,min (3.22) 0 5 2 2 wheren ¯ is the average index of refraction calculated based on a weighted average (¯n = f n1 +(1−f )n2). Based on symmetry considerations, the effective refractive index found is independent of the polarization. Although equation 3.22 is not derived rigorously, the authors show that it is an accurate estimation by comparing the reflectivity of the two-dimensional structure obtained with thin film calculation with the results obtained through rigorous applications of the Maxwell’s equations. Lalanne and Lemercier-Lalanne take a more systematic approach to the calculation of the refractive index [29]. Following the same arguments we followed above for the one-dimensional structure, they develop an expression for the coefficients η0 and η2 for a two-dimensional periodic structure with a center of inversion. As the dielectric constant of the medium must be expanded in its Fourier components both in the x and in the y directions, the algebra becomes extensive and will not be discussed in detail here. They find that the zeroth order approximation of the effective dielectric constant depends on the Fourier components of the dielectric function

15 of the medium ε(y, x) and on the ratio between the periodicity in the y and z directions. The calculation is performed for TE and TM-polarized light, and the effective dielectric constants obtained differ. However, for the case where the material has a center of inversion and the periodicity in the y and z directions coincide, so do the effective dielectric constants found, in agreement with the work by Bräuer and Bryngdahl. By comparing the values of reflectivity and transmissivity obtained with the effective-medium approach and obtained through rigorous calculations the authors determined a range of validity for the approximate approach. For a fill factor f = 0.6, accurate results for the reflectivity and transmissivity were obtained up to Λ/λ = 0.15. they concluded that the large errors associated with the effective refractive indices found indicated that the analogy between a two-dimensional periodic structure and a homogeneous film breaks down at large period-to-wavelength ratios, and the plane wave approximation used in the derivation is not appropriate. More recently, the analytical description of these two-dimensional binary periodic structures has been done to exhaustion [29, 63–67]. Several possible lattices have been considered [64]; also, the optical properties of disordered media with randomly shaped or randomly spaced dielectrics have been described [30,68]. Moreover, the validity of the theory has been confirmed experimentally, with a more forgiving range of period-to-wavelength ratio than was previously reported [67]. Nonetheless, it is important to emphasize that all of the mentioned works treat binary structures. When we examine such extensive amount of effort put into describing two-dimensional periodic structures, the absence of literature touching the homogenization of two-dimensional multi-phase structures is astounding. Multi-phase structures are composed of three of more materials, and a new approach must be undertaken in order to homogenize the dielectric constant of the structure. Although the work of Lalanne, by admitting an arbitrary form for the dielectric function of the medium given the existence of a center of inversion, sets a starting point to the challenge, no further work has been found that tries to tackle this problem. As a starting point, we suggest a development inspired by Bräuerand Bryngdahl’s informed guess technique. Using symmetry considerations to find an upper and lower bound to the effective dielectric constant is a major point of concern in the hunt for an effective-medium theory for a multi-phase two-dimensional structure.

Chapter 4

Application of the Models

The models mentioned in chapter 3 have been applied extensively to the description of the structure of the barbules of bird’s feathers. Thin-film theory has been applied successfully to the description of the iridescence of black feathers (figure 2.2b) [15,69]. In these works, the authors used a transfer-matrix technique to calculate the reflectance spectra of different models in which different interfaces (air-keratin, keratin-melanin, melanin- keratin) played a role in producing optical features. In [69], the melanin layer is thick and dense, and Doucet found that only the outermost keratin cortex seems to play a role in coherently scattering light. Due to the high extinction coefficient of melanin, it acts as a boundary to form a single thin film of keratin that behaves as

16 described in section 3.4.1. In contrast, in [15], the organization of the melanosomes is somewhat more regular, and the melanin layer beneath the keratin cortex is between 1 and 5 melanosomes thick. Here, Maia et al. varied the thin-film model in order to account for different thicknesses of the melanin layer based on the diameter of the melanosome and the number of melanosomes stacked. The authors found that for a two-melanosome thick layer, the model that considers the reflection from the melanin-keratin boundary converges into the model that only considers reflection from the superior keratin-melanin interface. This indicates that light can penetrate a layer of single melanosomes and, in this case, the interplay between light and the melanin-keratin boundary becomes relevant. In both of these works, the transfer-matrix formalism was applied considering smooth and parallel boundaries between keratin and melanin, and the refractive indices of the layers were taken to be the refractive indices of keratin and melanin without any consideration of the real volumes occupied by these materials. We suggest that a more thorough application of the transfer-matrix formalism, including the homogenization of the refractive index of different layers, could provide further insight into the roles of different parameters, such as the size of the melanosomes, in the coloration mechanisms of these feathers. As was emphasized in section 3.2, the FDTD technique is by far the most flexible, allowing for the calculation of reflectance spectra of any arbitrary structure, but requiring extensive computational efforts. The technique has been applied to the description of the complex structures of the bird of paradise Lawes’s parotia occipital and breast feathers [16]. The occipital feathers have already been presented in figures 2.2c and 2.2g. In these barbules, we see an alternation of single-melanosome and keratin layers. As mentioned above, single layers of melanin are thin enough that light is not extinguished when propagating through them, therefore these structures are likely to originate complex spectra due to multilayer reflections. However, the complexity of the structures requires costly computational methods to be applied. The images obtained with TEM are pixelated, and a refractive index of each pixel is assigned according to its contrast (which is determined by the electron density in the structure). Then, this distribution of indices of refraction is used as the simulation volume in an FDTD calculation, and the respective reflectance spectra are obtained. The calculations must be done in a supercomputer, and approximately 150 Gb of memory are necessary to perform a calculation for an individual wavelength, polarization and angle of incidence. In [18], Stavenga et al. try to model similar structures using the more practical effective-medium theory, in order to validate the value of the refractive index of melanin obtained with Jamin-Lebedeff intererence. Their approach is to divide the barbule structure in lanes approximately 0.5 µm wide and 3.5 µm long. The slices were further divided in layers of height 10 nm, and the contribution of melanin to the refractive index of each layer was determined based on the electron density of the layer. The refractive index obtained through Jamin- Lebedeff interference was fed into a model such as the one described in 3.4.3. The reflectance spectrum for each slice was calculated, and their average yielded a reflectance spectrum for the structure that matched the spectra obtained experimentally. A similar approach applied to the transmittance spectrum yielded the value of the imaginary part of the index of refraction, which accounts for the absorption coefficient of melanin. They found low values for the real part of the refractive index of melanin in these structures (n between 1.7 and 1.8), indicating that the value usually assumed in the literature (n = 2) is too high. The application of the effective-medium theory such as it is performed in [18] is limited since, as described in section 3.4.3, incorporation of more than two indices of refraction in the model is problematic. For this reason, the width of the layers taken (in the work cited, 0.5 µm) must be relatively small. This is troublesome, for instance, if we are to describe the structures formed by hollow melanosomes studied in [37], which are presented in the insets of figures 4.1a and 4.1b. In these structures, some of the 500 nm wide layers would inevitably contain air, keratin and melanin. In order to avoid this complication we could take thinner layers that would then be composite of only two materials. However, these layers would be smaller than the wavelength of the

17 (a) (b)

Figure 4.1: Reflectance spectrum of the feathers of the violet-backed starling (4.1a) and of the wild turkey (4.1b). Dashed lines indicate the reflectance predicted based on a photonic crystal model, while continuous lines are the experimentally obtained spectra. For the wild turkey, a reflectance spectrum upon oblique illumination is also considered (red data). The insets show TEM images of the barbules of the two species. Images were obtained from [37]. Scale bars in the insets are equivalent to 500nm. light, and the homogeneous medium analogy would not be granted [29]. Instead, Eliason et al. prefer to apply a photonic band structure model to describe these arrangements [37]. Exploiting the regularity of the hexagonal close-packed structure of melanosomes in the feathers of the wild turkey and of the violet-backed starling, they calculate the photonic band structure of the crystals formed by the hollow melanosome rodlets in these babrbules. Their intent is to explain the reflectance spectrum of these feathers, which for the wild turkey includes a sharp change in hue with angle of observation, on a structural basis. They conclude that, because hollow melanosomes allow for a greater contrast of indices of refraction and for a larger ratio of materials with low index of refraction even in thermodynamically stable close-packed lattices, these structures provide ample possibility of evolutionary innovation [36]. In this work, albeit the hue of the reflectance peak obtained with the photonic band structure simulation agrees reasonably with the peaks observed experimentally, there are some discrepancies between calculated and experimental spectra, as can be seen from figure 4.1. The authors attribute these discrepancies partly to the absence of empirical values for the indices of refraction of the materials in the structure. Indeed, they assumed the index of refraction of melanin to be n = 2, which, as mentioned before, is likely too high. We emphasize, however, that some discrepancies may come from the application of the photonic crystal model itself. Defects play a huge role in the band structure of photonic crystals [45]. Although simple inspection of the TEM images provided in the work clearly show that these arrangements of melanosomes are periodic, they also explicit that the structure is not perfectly regular and shows a large number of defects, as is expected from most biological systems. There is a clear spread in shape and size of the melanosomes, and in their dislocations from the lattice points. Additionally, the mathematical description of photonic crystals relies on the Bloch expansion of the propagating waves (see section 3.3), which in turn are only valid for infinitely periodic structures. The number of calculated layers of melanosomes in these structures (4.7 for turkeys and 3.5 for starlings) is too low for the unconstrained application of the photonic crystal model. Concerns that the photonic crystal model of these

18 structures is unlikely to provide reliable quantitative insight into the system are not absent in the literature [70]. It is important to note, however, that the photonic band structure has been applied to the description of the origins of the coloration of peacock feathers [14], black-billed magpie feathers [19] and duck wing patches [12], with remarkable match between observed and calculated spectra. The structures studied in [14, 19] showed air rodlets immersed in a keratin-melanin matrix (see figure 2.2h), while the structure studied in [12] showed filled rodlets of melanin hexagonally packed in a keratin matrix. Based on calculations of the photonic band diagram of crystals designed from the TEM images of the feathers, the three works associated the color of the structure with its periodicity; furthermore, Eliason et al found a correlation between the size of the melanosomes and thickness of the keratin cortex, which acts as a thin-film, and the color of the duck feathers. It is interesting to consider if new information about the structure could be obtained by applying different physical models to their descriptions. For instance, in [19] the authors emphasize that the specific composition of the keratin-melanin matrix in the structure studied is not yet known, and an effective refractive index the the structure is taken as n = 2. A consolidated description of an effective medium-theory for these structures could provide insight into the proportion of melanin to keratin in these matrices, for instance, by comparison of the spectra calculated for varying filling factors with the experimental reflectance spectrum of the structure. Additionally, in [12], the authors argue that a multilayer description of the structure observed in the barbules of the wing patches of ducks is not adequate since the photonic crystal model shows a better prediction of the polarization features observed in the reflectance spectrum of these feathers. However, application of the effective-medium theory associated with the transfer matrix formalism models the arrangement well, assuming a complete description of the homogenization of the two-dimensional periodic structure. This approach would avoid the assumption of periodic boundary conditions, indispensable in the photonic crystal model.

Chapter 5

Conclusion

Understanding how biological features are responsible for various dazzling optical effects in bird feathers is on itself a complex task. Besides bringing insight into the evolutionary role of several physical parameters, the description of these structures on a physical level can serve as inspiration towards the development of numerous biomimetic structures, such as anti-reflection coatings and photonic crystals used to enhance light extraction in LEDs, for instance [43, 44]. The task of describing the interaction of these media with light can be done either based on the solution of Maxwell’s equations or based on a thin-film approximation of the structure. As described in this work, Maxwell’s equations can be solved in the time-domain or on the frequency domain. On one hand, the time-domain solutions of the problem yield a complete description of the scattering of light from any arbitrary structure, and few assumptions must me made. However, these calculations require extensive computational resources, which hinders its use in investigating the spectral effects of varying parameters, for which several spectra must be generated. On the other hand, the frequency-domain solutions of Maxwell’s

19 equations, obtained through photonic band calculations, require significantly less computational efforts and give clear insight into the frequencies that cannot propagate in a periodic structure and are thus reflected; nevertheless, these solutions rely on the assumption of infinite periodicity of the photonic crystals, and the validity of this assumption in the structure of the barbules could be disputed [70]. Lastly, we reiterate the scope of the effective medium approach. In general, the photonic structures present in the barbules can be sliced into thin layers, for which the multilayer transmittance and reflectance spectra are obtained through the calculation of a transfer-matrix for the system. Implementation of this procedure relies on the existence of an effective refractive index for the thin-films considered. Calculation of this effective refractive index is at the core of the effective-medium theory, and its analytical description for a general structure could enable us to extract significant quantitative information about biological structures from the thin-film model that describes them. The validity of an effective refractive index depends also on the wavelength of light being larger than the features of the structures being described, but this is usually a valid assumption in the cases treated in this work. An effective-medium theory for one-dimensional [29] and for binary two-dimensional [29,34] structures has been developed thoroughly and applied to the description of the arrangements of melanosomes in bird feathers [18]. It is our belief that an effective-medium theory for multi-phase two-dimensional structures could provide a more general description of the photonic effects of barbules composed of three materials than the photonic band model currently applied [37]. Structures formed by hollow melanosomes, for instance, significantly broaden the possible optical effects generated by the structural features of the barbules, but their physical description has only been approached recently in the literature [11, 36, 37], perhaps due to the complexity of the task. Understanding the physical process associated with their great optical diversity, however, could bring relevant insight into how these features evolved. With these considerations, it should be clear that the choice of model used to describe the barbule structure in various species of birds is not unique. The limitations of each mathematical description must be considered when extracting information from the models, and a combination of different approaches can shine light on distinct properties of these structures.

Acknowledgements

I deeply thank Prof. Dr. Doekele Stavenga for all the time dedicated to the improvement of this work. His advice was not only crucial to the development of the paper, but also enlightening. Additionally, I thank Dr. Maxim Pchenitchnikov and Prof. Dr. Ryan Chiechi for the very enriching instruction on the handling of scientiﬁc material. Finally, I would like to thank my brother, Dr. Rafael Maia, for inspiring me to dive into the fascinating world of bird coloration.

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