\chapter{On the Absence of Red Structural Color in Photonic Glasses, Bird Feathers and Certain Beetles}
%\label{chap:HighD}
\label{ch:absence_of_red}
\graphicspath{{figures/ch2/}} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

Up to this point, we have not encountered any photonic glasses with saturated yellow, orange, or red color. While systems with angle-independent structural red have been reported~\cite{Takeoka2009, Harun-Ur-Rashid2010, Kumano2011, Gotoh2012, Park2014}, the color saturation for long-wavelength hues is poor compared to that for blue. Interestingly, red angle-independent structural color also appears to be rare in nature. Birds use structural color only for blue and green; red colors in bird feathers come from absorbing pigments~\cite{Saranathan2012}. And while the scales of the longhorn beetle \textit{Anoplophora graafi} have structural colors that span the visible spectrum, there are no saturated red colors -- only a pale purple~\cite{Dong2010}.

\begin{figure}
\centering
\includegraphics{reflection_data_pmma170_pmma240_pmma330}
\caption{\label{fig:reflection_data_pmma170_pmma240_pmma330} Measured reflectivity spectra
  for three similarly-prepared colloidal glasses of poly(methyl
  methacrylate) particles in air.  Insets show photographs of the
  samples and diameters of the particles used to make the samples.
  The purple sample would appear red if not for the high reflectivity
  in the blue, indicated by the arrow.}
\end{figure}

The absence of red photonic glasses does not appear to have been
acknowledged in prior literature, let alone explained. Previous work on photonic
glasses~\cite{Dong2010, Noh2010AdvMat, Liew2011, Dalba2011,
  Dufresne2009, Forster2010} has focused on structures found in nature
and their biomimetic analogues, nearly all of which are blue. Noh and
coworkers~\cite{Noh2010OptExpress, Noh2010PRE} proposed a theoretical
model based on single and double scattering to explain the optical
properties of these blue samples, and Liew and
coworkers~\cite{Liew2011} and Rojas-Ochoa and
coworkers~\cite{Rojas-Ochoa2004} showed that the inter-scatterer
correlations not only give rise to color, but also suppress multiple
scattering. However, if the color were entirely determined by
correlations, one would expect that all colors could be made simply by
linearly rescaling the structure. As we show below, this approach does
not work (Figure~\ref{fig:reflection_data_pmma170_pmma240_pmma330}).

Here we present a model that explains the absence of long-wavelength
structural colors in photonic glasses.  Our model accounts for both
interparticle correlations as well as the scattering behavior of
individual particles. We show that short-wavelength resonances in the
single-particle scattering cross-section, akin to cavity resonances,
introduce a blue peak in the spectrum that changes the hue of a red
structural color to purple. The model, which agrees with measured
spectra from synthetic photonic glasses, provides a framework for
understanding the limitations of current photonic glasses
and enables the design of new systems without those limitations.

\section{Experiment}

To demonstrate that resonant scattering from the interparticle correlations is not sufficient to explain the colors of photonic glasses, we prepare colloidal glasses from poly(methyl methacrylate) (PMMA) spheres of various diameters and study their colors with spectrometry. Figure \ref{fig:reflection_data_pmma170_pmma240_pmma330} shows color photographs and reflection spectra of samples prepared in the same way using three different particle diameters $d=$ 170, 240, and 330 nm.  These samples were prepared by mixing one part of an aqueous suspension containing 1\% w/w carbon black (Cabot) and 2\% w/w Pluronic F108 (BASF) with two parts of a monodisperse suspension of poly(methyl methacrylate) particles at 20\% w/w, centrifuging the mixture at 5200 g for 30 minutes, removing the supernatant, and slip-casting on a gypsum substrate. The final samples are amorphous, dense packings of particles in air.  The small amount of carbon black suppresses multiple scattering, making it easier to see the color~\cite{Forster2010}.

We measure the reflection spectra as a function of wavelength with a
fiber-optic spectrometer (OceanOptics HR2000+) mounted on an optical
microscope (Nikon LV-100).  The samples are illuminated with white
light from a halogen lamp that is collimated by minimizing the
condenser aperture. The divergence angle of the beam is 0.02
radians. The light scattered by the sample is collected by a
50$\times$ objective (Nikon LU Plan Fluor, NA = 0.8) and imaged onto
the detection fiber (OceanOptics QP600-2-UV/VIS). Because the
objective NA is the limiting numerical aperture in our system, our
measurements include light scattered up to a maximum angle
$\theta_{\rm{max}}=\arcsin(0.80)=0.93$ radians. We normalize all
spectra to the reflected intensity from an aluminum mirror, we average
the spectra measured over five different locations on the sample, and
we smooth the spectra using a 50-nm window.

Two peaks are visible in the spectra, one at 515 nm in the $d=$ 240 nm
sample and the other at 710 nm in the $d=$ 330 nm sample.  The ratio
of these peak wavelengths is approximately that of the two particle
sizes $330/240 \approx 710/515$.  The linear scaling is consistent
with the hypothesis that the color is due to constructive interference
of light scattered from correlated regions of particles.  The high
reflectivity of the $d=$ 170 nm sample at short wavelengths suggests a
structural resonance in the ultraviolet (UV), again consistent with
this hypothesis.  Thus the peak wavelength does appear to scale
linearly with the particle size, and hence with the lengthscale of
structural correlations, assuming that the volume fraction and
structure of all of our samples are similar.

However, whereas the $d=$ 170 nm sample appears blue and the $d=$ 240
nm sample appears green, the $d=$ 330 nm sample does not appear
red. The reason is that its reflectivity rises toward the blue,
suggesting a second peak at 430 nm or below.  We have observed similar
spectra in samples made of polystyrene and silica particles whenever
the particle size is 250 nm or larger. Dong and
coworkers~\cite{Dong2010} found a similarly-shaped spectrum, with one
peak in the red and another apparent one at short wavelengths, for
purple longhorn beetle scales~\footnote{see the bottom spectrum in
  Figure 1c of \cite{Dong2010}}. Takeoka and
coworkers~\cite{Takeoka2013} also observed a short-wavelength peak in
the spectrum of their amorphous packings of $d=$ 360 nm silica
particles~\footnote{see Figure 1b, pink curve in their paper}.  In
order to understand the absence of red in all of these photonic
glasses, we must understand the origin of this short-wavelength
reflectivity peak.
 
\section{Theory}

\begin{figure}
\centering
\includegraphics{source-detector-angles}
\caption{\label{fig:geometry} Scattering geometry for our model.}
\end{figure}

\begin{figure}
\centering
\includegraphics{structure_factor_phi0p55}
\caption{\label{fig:structurefactor} Structure factor of a colloidal glass with volume fraction $\phi=$ 0.55 calculated from the Ornstein-Zernike equation under the Perkus-Yevick approximation.}
\end{figure}

To explain our observations, we use a single-scattering model where we treat scattering from a particle assembly as the result of two separable processes: scattering from each individual particle, as described by a form factor, and interference between waves scattered from the particles, as described by the structure factor of the glass. In our analysis, we first use the form and structure factor to calculate the differential scattering cross-section of a glass, and then we derive a formula that relates this cross-section to reflectivity, the quantity that we measure. According to our model, structural color in photonic glasses is primarily determined by the peaks of the structure factor. As we shall show, the form factor can either undermine this color or enhance it, depending on the structure of the system and the particle properties. 

The scattering geometry that we model follows the picture presented in the Introduction (\ref{fig:q_diagram}); we show it again here in Figure \ref{fig:geometry}, drawn in terms of a typical experimental setup. Recall that, assuming elastic scattering so that $\abs{\mathbf{k}_{\mathrm{in}}} = \abs{\mathbf{k}_{\mathrm{s}}} = k$, we can relate the magnitude of momentum change of the light field, $q$, to $k$, via Equation \ref{eq:q}:
%
\begin{equation}
q = 2 k \sin(\theta /2) 
\label{eq:q}
\end{equation}
%
where $k = 2\pi n_{\mathrm{eff}} / \lambda$, $n_{\mathrm{eff}}$ is the effective refractive index of the medium and $\lambda$ is the wavelength of light in vacuum. As with the glasses of core-shell particles discussed in the previous chapter, for the effective index of the material we use a weighted average calculated using the Maxwell-Garnett mean-field approximation~\cite{MaxwellGarnett1904}: 
%
\begin{equation}
n_{\mathrm{eff}} = n_{\mathrm{med}}\sqrt{\frac{2n_{\mathrm{med}}^{2} +
    n_{\mathrm{p}}^{2} + 2\phi(n_{\mathrm{p}}^{2} -
    n_{\mathrm{med}}^{2})}{2n_{\mathrm{med}}^{2} + n_{\mathrm{p}}^{2}
    -\phi(n_{\mathrm{p}}^{2} - n_{\mathrm{med}}^{2})}}.  
\label{eq:Maxwell-Garnett}
\end{equation}
%
where $n_{\mathrm{med}}$ is the refractive index of the material surrounding the particles (air, in our case), $n_{\mathrm{p}}$ is the refractive index of the particles, and $\phi$ is the volume fraction occupied by the particles.  %We use the effective index because the particle packings are dense, so that the scattered fields ``see'' an index intermediate between the particle and medium index.  Although the Maxwell-Garnett approximation is typically used when the refractive index variations are much smaller than the wavelength, Vos and coworkers~\cite{Vos1996} showed that it is a good approximation even for photonic crystals, and Forster and coworkers~\cite{Forster2010} have shown the same for photonic glasses.

Assuming perfect monodispersity, we can express the differential
scattering cross-section of a glassy ensemble of particles,
$d\sigma_{\mathrm{glass}} / d\Omega$, as a product of the form factor
$F$ and the structure factor $S$~\cite{Kaplan1994, Rojas-Ochoa2004}:
%
\begin{equation}
d\sigma_{\mathrm{glass}} / d\Omega = (1/ k^2) F S.
\label{eq:diff_scattering_cross-section_glass}
\end{equation}
%
The form factor is related to the differential scattering cross
section of a single particle, $d\sigma / d\Omega$, through $F = (1 /
k^2) d\sigma / d\Omega$ \cite{BohrenHuffman}; we calculate $F$ using
Mie theory~\cite{Mie1908, BohrenHuffman}. The structure factor is the
Fourier transform of the pair correlation function of the
particles~\cite{ChaikinLubensky}; we calculate it using a numerical
solution to the Ornstein-Zernike equation with the Perkus-Yevick
closure approximation~\cite{PercusYevick1958}.  In so doing, we are
assuming that the structure of our particle glasses is close to that
of a hard-sphere liquid. Figure~\ref{fig:structurefactor} shows $S$
calculated for a volume fraction $\phi = 0.55$ as a function of a
dimensionless wavevector $x = q d$, where $d$ is the diameter of the
particles. We use this structure factor in all our calculations.

In the structure factor, the peak at $x_{o}$ corresponds to the
wavevector $q_{o} = 2 \pi/ d_{\mathrm{avg}}$, where $d_{\mathrm{avg}}$
is the average center-to-center spacing between nearest neighbors.
The peak wavevector $q_{o}$ gives rise to constructive interference and
color, because it sets the relative phase difference between light
waves scattered from neighboring particles.  Resonant scattering
occurs when this phase difference is an integer multiple of $2\pi$.
The wavelengths at which this happens can be determined from
Equation~\ref{eq:q}:
\begin{equation}
\lambda = (4 \pi n_{\mathrm{eff}} d / x_{\mathrm{o}}) \sin(\theta /2). 
\label{eq:resonant_wavelength}
\end{equation}

To describe the intensity of scattered light that reaches the
detector, we integrate the differential scattering cross-section over
the solid angle corresponding to the numerical aperture (NA) of our
detector, taking into account transmission and refraction at the
air-sample interface:
%
\begin{equation}
\sigma_{\mathrm{detected}} = 1/k^{2} \int_{\phi =
  0}^{2\pi}\int_{\theta_\mathrm{min}}^{\pi} T_{\mathrm{s-a}} F S
\sin(\theta)\, d \theta\, d \phi 
\label{eq:detected_cross-section_glass}
\end{equation}
%
where $T_{\mathrm{s-a}}$ is the Fresnel coefficient for transmission
at the interface, $\theta_{\mathrm{min}} = \pi -
\arcsin(\mathrm{NA}/n_\mathrm{p})$ is the minimum scattering angle
that we detect, and $\phi$ is the polar angle in the plane
perpendicular to the scattering plane. There is no dependence on
$\phi$ because the particles are spherical and the structure
is isotropic.

To compare our calculations to our measurements we now derive a
relation between $\sigma_{\mathrm{detected}}$ and the measured
reflectivity, $R$. To do this we must account for extinction of light
as it propagates through the sample; the intensity of light scattered
from layers close to the surface is higher than the intensity of light
scattered deeper in the sample because of attenuation by
scattering. Under the assumption of single scattering, this
attenuation scales exponentially with depth following Beer's Law,
$I(x) = I_{\mathrm{o}} e^{-\rho \sigma x}$, where $\rho$ is the number
density of particles, $\sigma$ is the scattering cross-section for the
full solid angle ($0\leq\phi\leq2\pi$, $0\leq\theta\leq\pi$), and $x$
is the distance light has propagated in the
medium~\cite{BohrenHuffman}. If the glass consists of slices of
infinitesimal thickness $\delta x$, the total reflected intensity $I$
is the sum of the intensities $\delta I$ reflected from each slice:
$\delta I = I(x) \sigma_{\mathrm{detected}} \rho\, \delta x$, where
$\sigma_{\mathrm{detected}}$ is given by
Equation~\ref{eq:detected_cross-section_glass}. After integrating both
sides and including the Fresnel coefficients for transmission
($T_{\mathrm{a-s}}$) and reflection ($R_{\mathrm{a-s}}$) at the air-sample
interface, we find
%
\begin{equation}
R = T_{\mathrm{a-s}} \frac{\sigma_{\mathrm{detected}}}{\sigma} (1 - e^{-\rho \sigma l}) + R_{\mathrm{a-s}}
\label{eq:diffuse_reflection}
\end{equation}
% 
where $l$ is the optical thickness of the sample, or the maximum
distance that light can propagate in it.

The reflectivity for a glass of poly(methyl methacrylate) spheres at a
volume fraction $\phi=0.55$, as calculated according to
equation~\ref{eq:diffuse_reflection}, is shown in Figure
\ref{fig:rescaled_theory}. We have omitted the Fresnel coefficient for
reflection at the air-sample interface to better illustrate the
features that arise from scattering from the bulk colloidal glass.
The plot is shown as a function of the scaled particle size $kd$,
where $k = 2 \pi n_\mathrm{p} / \lambda$. All terms in Equation
\ref{eq:diffuse_reflection}---except for the Fresnel reflection
coefficient, which adds an offset in amplitude---scale with
$kd$. Thus this master curve describes reflectivity from a glass made
of any particle size (of the same material and volume fraction),
assuming dispersion is small. Depending on the particle sizes and
refractive indices, different features of the curve will fall within
the visible range; here we mark the edges of our detection range for
$d=334$ nm with the blue (425 nm) and red (800 nm) vertical dashed
lines.

\begin{figure}[h]
\centering
\includegraphics[width=8cm]{rescaled_reflection_SF}
\caption{\label{fig:rescaled_theory} Calculated
  reflectivity as a function of $kd$ for a photonic glass of spheres
  at a volume fraction $\phi=0.55$, as calculated from
  Equation~\ref{eq:diffuse_reflection} with the Fresnel reflection
  coefficient omitted. The vertical dashed lines denote the
  $kd$-values that correspond to the range of visible wavelengths we
  detect, 425 nm (blue line) and 800 nm (red line), when the particle
  size is $d=$ 334 nm.}
\end{figure}

\section{Results and Discussion}

The calculated reflectivity reproduces the locations of all of the peaks in our data, as shown in Figure~\ref{fig:data_and_theory_SF_S_F}. The only free parameters are the volume fraction $\phi = 0.55$ and the thickness $l=$ 16 $\mu$m. The peaks predicted by the theory coincide with the peaks in the data for particle diameters $d=$ 238 nm (Figure \ref{fig:data_and_theory_SF_S_F}(a)) and $d=$ 334 nm (Figure \ref{fig:data_and_theory_SF_S_F}(b)); these sizes are in good agreement with the sizes of the particles measured with scanning electron microscopy, 240 nm and 330 nm. The calculated reflectivity for the $d=$ 334 nm system also reproduces the peak in the blue that we observe in the data.

%The model underestimates the amount of light scattered off-resonance, especially at short wavelengths. We attribute this discrepancy to multiple scattering. Since the probability of multiple scattering increases with the scattering cross-section of individual particles, its contribution should be more pronounced at short wavelengths, which is consistent with what we see.

With this model and data at hand we can address our original question: why does the glass made of 330 nm spheres have a resonance in the blue, when we expect the interparticle correlations to give rise only to a resonance in the red? Multiple scattering does not fully answer the question, because it does not predict any resonances -- only a monotonic increase in the scattered intensity with decreasing wavelength. To identify the source of the reflection peaks in this sample, we compare our data to the reflectivities predicted from the form factor alone and, in another comparison, the structure factor alone (Figure \ref{fig:data_and_theory_SF_S_F}(c)). From the shapes of these two curves we immediately see that the blue peak comes from the form factor and the red peak from the structure factor, boosted by another peak in the form factor that occurs at a longer wavelength.

We conclude that the structural colors of our photonic glasses are
determined not only by interference between waves scattered from
correlated particles, but also by resonances in the single-particle
scattering cross-section.  This can also be seen by rescaling the
measured reflectivities by $\sigma_{F,{\mathrm{detected}}}$ and
plotting them as a function of the dimensionless wavevector $kd$
(Figure~\ref{fig:all_rescaled_data}).  Here
$\sigma_{F,{\mathrm{detected}}}$ is the single-particle
differential scattering cross-section, integrated over the angle
subtended by the detector: 
%
\begin{equation}
\sigma_{F,{\mathrm{detected}}} = 1/k^{2} \int_{\phi =
  0}^{2\pi}\int_{\theta_\mathrm{min}}^{\pi} T_{\mathrm{s-a}} F
sin(\theta)\, d \theta\, d \phi 
\label{eq:detected_cross-section_onlyF}
\end{equation}
%
We see that after rescaling the long-wavelength reflectivity peak of
the $d=$ 330 nm sample coincides with the reflectivity peak of the
$d=$ 240 nm sample at the same value of $kd$, showing that these peaks
arise from structural resonances, while the short-wavelength (high
$kd$) peak in the $d=$ 330 nm sample disappears, showing that it
arises from single-particle scattering.

\begin{figure}
\centering
\includegraphics[scale=0.95]{data_and_theory_SF_S_F}
\caption{\label{fig:data_and_theory_SF_S_F} (a),(b)
  Measured (smooth lines) and calculated (dashed lines) reflection
  spectra of colloidal photonic glasses made of poly(methyl
  methacrylate) spheres. Theoretical spectra are calculated from
  Equation~\eqref{eq:diffuse_reflection} with $\phi = 0.55$ and $l =$
  16 $\mu$m.  Particle diameters that best fit the measured peaks are
  238 nm (a) and 334 nm (b), in good agreement with the measured
  particle diameters. (c) Calculated reflection spectrum for a
  photonic glass made from $d=$ 334 nm particles, including only the
  structure factor (thick curve, divided by 10) and only the form
  factor (thin curve).}
\end{figure}

\begin{figure}
\centering
\includegraphics{all_rescaled_data}
\caption{\label{fig:all_rescaled_data} (a) Reflection spectra from
  Figure~\ref{fig:reflection_data_pmma170_pmma240_pmma330} plotted against the dimensionless lengthscale
  $kd$. Note the increased scattering at high $kd$-values (short
  wavelengths) for the $d$ = 330 nm sample. (b) Same as (a) but
  normalized to the single-particle form factor integrated over the
  detected scattering angles, $\sigma_{F, \mathrm{detected}}$, as
  defined in Eq. \ref{eq:detected_cross-section_onlyF}. The increased
  scattering at high $kd$-values disappears, indicating that it is due
  to the form factor. Differences in amplitude of the peaks are likely
  due to differences in the sample thickness $l$.}
\end{figure}

We note that the single-particle resonances responsible for the high
reflectivity at small wavelengths are \emph{not} the so-called ``Mie
resonances.'' Mie resonances are observed in the \emph{total}
single-particle scattering cross-section, obtained by integrating the
differential scattering cross-section over all angles.  The closest
Mie resonance for a 334 nm poly(methyl methacrylate) particle in our
effective medium occurs at 310 nm.  Instead, as we shall show, the
resonances that contribute to the high reflectivity at small
wavelengths occur for scattering angles near backscattering.

We first consider the single-particle scattering cross-section for
pure backscattering.  This is proportional to the differential scattering
cross-section for $\theta = \pi$. As shown in Figure
\ref{fig:resonance_vs_od}, the backscattering cross-section can have
one or more resonances, and the wavelengths at which these occur
increase linearly with the particle ``optical diameter,'' 
$n_{\mathrm{p}}d$, following the relation
%
\begin{equation}
\lambda_{\mathrm{resonant}} = 2 n_{\mathrm{p}} d/ z
\label{eq:backscattering_resonances}
\end{equation}
%
where $z$ is an integer that corresponds to the order of the
resonance. This suggests that these resonances are akin to those of a
Fabry-P\'{e}rot cavity, where constructive interference occurs for
wavelengths that fit an integer number of times within the round-trip
optical pathlength enclosed by the cavity~\cite{Hecht}. Similar
resonances occur in the differential scattering cross-section for all
angles, as shown in Figure \ref{fig:differential_cross-sections}. Our
calculations show that the resonant wavelength shifts toward the blue
as the angle decreases.  This blue-shift is consistent with the
decrease in the round-trip optical pathlength inside the sphere with
decreasing angle (see inset). When the differential scattering
cross-section is integrated over the detected solid angle ($\theta =
150^\circ - 180^\circ$ for our detection numerical aperture of
0.8, after refraction), these resonances, though broadened, persist
(Figure \ref{fig:data_and_theory_SF_S_F}(c)).  We therefore conclude
that constructive interference of light inside the particles
contributes to enhanced reflection at short wavelengths.

\begin{figure}
\centering
\includegraphics{resonances_vs_od}
\caption{\label{fig:resonance_vs_od} Resonant wavelengths of the
  backscattering cross-section, as calculated from Mie theory, as a
  function of optical diameter $n_{\mathrm{p}} d$ for a refractive
  index contrast $m =$ 1.2, which corresponds to that in our
  experimental system. The resonant wavelength follows the linear
  relation $\lambda = 2 n_{\mathrm{p}} d / z$ (solid lines), where $z$
  is the order of the resonance. Lines correspond to different values
  of $z$ (top: $z=1$, middle: $z=2$, bottom: $z=3$).}
\end{figure}

\begin{figure}
\centering
\includegraphics{pmma330_air_differential_cross-sections-with_sphere_inset}
\caption{\label{fig:differential_cross-sections}
  Single-particle differential scattering cross-sections for various
  scattering angles as a function of wavelength for a 330 nm PMMA
  sphere in a colloidal glass of spheres in air at a volume fraction
  $\phi = 0.55$. The blue-shift in the resonance is consistent with
  the decrease in optical pathlength inside the sphere with decreasing
  angle: The longest possible pathlength $a$ is twice the diameter,
  and $a > b$ for any angle that differs from backscattering.}
\end{figure}

The model underestimates the amount of light scattered off-resonance, especially at short wavelengths. We attribute this discrepancy to multiple scattering. To estimate the relevance of multiple scattering in our samples, we calculate the transport length, $l_{\mathrm{t}}$: the distance light traverses within a sample before its direction is randomized. If the sample thickness is larger than the transport length, then there is multiple scattering. Moreover, interference can lead to reduced values for the transport length on resonance that are associated with resonant backscattering~\cite{Rojas-Ochoa2004}. The transport length provides an alternative way to describe the conditions for structural color: For a colloidal glass to have a specific color, the transport length for this color needs to be shorter than the transport length for the other colors -- otherwise all wavelengths will undergo multiple scattering and mix diffusely, resulting in white. Mathematically, the transport length is given by the following expression:
%
\begin{equation}
l_{\mathrm{t}} = 1 / \rho \sigma_{\mathrm{t}} 
\label{eq:transport_length}
\end{equation}
%
where $\rho$ is the number density of particles and 
\begin{equation}
\sigma_{\mathrm{t}}=1/k^{2}\int_{\phi = 0}^{2\pi}\int_{\theta = 0}^{\pi}FS(1-\cos{\theta})\sin{\theta}\, d\theta\,d\phi
\label{eq:transport_crosssection}
\end{equation}
%
is the transport cross section~\cite{Rojas-Ochoa2004}. Figure~\ref{fig:transport_length} shows $l_{\mathrm{t}}$ as a function of wavelength for our green and purple samples, in comparison with the fitted sample thickness of 16 $\mathrm{\mu}$m that we obtained from the reflectivity calculations shown in Figure~\ref{fig:data_and_theory_SF_S_F}. For both samples the transport length generally increases with wavelength but has a dip on the structure factor resonance, as previously observed~\cite{Rojas-Ochoa2004}. Moreover, in both samples the transport length is smaller than the fitted sample thickness at short wavelengths, consistently with our hypothesis that multiple scattering contributes additional blue light that is not captured by our single-scattering model. Finally, only the green sample comes close to satisfying our condition for color: except for the 400--430 nm regime, its transport length on resonance is shorter than it is for all other wavelengths. The purple sample, on the other hand, fails to satisfy this condition entirely: its transport length on resonance is much longer than the transport length for most of the visible wavelengths, meaning that there is diffuse scattering of all colors in addition to resonant backscattering. This is consistent with the fact that the purple color we see is not very saturated, since multiple scattering contributes a white background to the structural color. The importance of multiple scattering in these samples can be studied in the future by investigating the effect of sample thickness on color. One effective way to control the thickness of the samples is by changing the amount of carbon black in them, that sets the absorption length in the sample -- namely the distance light traverses in them before it is absorbed. To minimize multiple scattering and enhance color saturation, the samples should have enough carbon black to ensure that the absorption length in the sample is shorter than the transport length for the off-resonant wavelengths. 

\begin{figure}
\centering
\includegraphics[width=8cm]{transport_length}
\caption{\label{fig:transport_length} Calculated transport length as a function of wavelength for the green ($d$ = 238 nm) and purple ($d$ = 334 nm) photonic glasses of poly-methylmethacrylate spheres, using a volume fraction $\phi=0.55$, as calculated from
  Equation~\ref{eq:diffuse_reflection} with the Fresnel reflection
  coefficient omitted. The vertical dashed lines denote the
  $kd$-values that correspond to the range of visible wavelengths we
  detect, 425 nm (blue line on the right) and 800 nm (red line on the left), when the particle
  size is $d=$ 334 nm.}
\end{figure}

Our results show that the absence of red structural color in photonic glasses can be attributed to the blue scattering resonances within the component particles. The single-particle scattering resonances occur in addition to the resonances from interparticle correlations, meaning that structural color in photonic glasses arises from the combination of resonant scattering from the structure \textit{and} from the individual particles.  For blue and green structural colors, these single-particle scattering resonances do not affect our perception of color, because they occur in the ultraviolet.  While these observations are based on glasses of poly(methyl methacrylate) spheres, the same principles apply to most typical colloidal materials, such as polystyrene and silica, whose refractive indices are not significantly different from that of that of poly(methyl methacrylate).  The chitin particles in beetle scales have similar optical properties~\cite{Dong2010}. 

Although the single-particle resonances we observe should also occur in photonic crystals, the structure factor in crystals is much more sharply peaked than in glasses, so that structural resonances can dominate the single-particle resonances and photonic crystals can have structural red color. The price to be paid for this red color is strong angular dependence: changing the angle between source and detector changes the resonant wavelength, leading to iridescence.  Photonic glasses have a lower and broader peak in the structure factor as a consequence of their disorder.  The breadth of the peak allows the glass to maintain nearly the same structural color over a range of angles between the source and the detector.  At the same time, the low peak amplitude (relative to a crystal) makes the color of the glass susceptible to contamination from the single-particle resonances.  Thus we see that there is an inherent tradeoff between angular independence and structural red color. 

Can this tradeoff be broken?  To do so, we would need to manipulate the resonances from two independent processes: single-particle scattering and coherent scattering from the particle assembly.  The characteristic lengthscale for the single-particle scattering resonances is the particle optical diameter $n_{\mathrm{p}}d$ (Equation~\ref{eq:backscattering_resonances}), and for the structural resonances it is the effective interparticle spacing $n_{\mathrm{eff}}d_{\mathrm{avg}}$ (Equation~\ref{eq:resonant_wavelength}).  Our control parameters are therefore the particle diameter $d$, its refractive index $n_{\mathrm{p}}$, the interparticle spacing $d_{\mathrm{avg}}$, and the index of the medium $n_\mathrm{med}$, which determines the effective index $n_\mathrm{eff}$. To make a red photonic glass, we can tune these parameters to either eliminate the second resonance of the form factor entirely, or to blue-shift it into the ultraviolet while keeping the peak of the structure factor at long wavelengths.   

To eliminate the form factor resonances we can take advantage of dispersion, namely the inherent wavelength dependence of the refractive index. Each material has its own dispersion curve, often described by a polynomial of the form $n(\lambda) = \sum_{\mathrm{i}} C_{\mathrm{i}} / \lambda^{2i}$, where $C_{\mathrm{i}}$ are called the ``Cauchy coefficients'' and are usually determined experimentally~\cite{Hecht, refractiveindexinfo}. With some consideration, it is possible to choose a combination of materials for the scatterers and the medium such that the refractive indices have very similar values at short wavelengths but deviate from each other at longer wavelengths, thus rendering the scatterers invisible to light only at short wavelengths. In Figure~\ref{fig:dispersion_trick}(a) we show the dispersion curves for such a combination of commercially available materials: poly-methylmethacrylate and refractive index liquid Cargille A 1.484. As desired, the two dispersion curves come very close at short wavelengths, but then deviate from each other at long wavelengths. As we show in Figure~\ref{fig:dispersion_trick}(b) the reflectivity of a colloidal glass of poly-methylmethacrylate particles in this oil is almost zero at short wavelengths and the form factor resonance is substantially suppressed, allowing the structure factor resonance to dominate -- in contrast to the reflectivity of a similar poly-methylmethacrylate particle in air. A similar dispersion trick has been previously employed in diffusely scattering structures to generate a variety of colors~\cite{Kumano2011}. Note that, in designing such a system, it will be important to adjust the sample thickness to account for the much reduced refractive index contrast and scattering cross section. 

\begin{figure}
\centering
\includegraphics{indices_and_reflection_pmma_in_air_and_in_cargilleA12_phi0p55}
\caption{\label{fig:dispersion_trick} We can use dispersion to partially index-match the scatterers and the medium and suppress scattering in the blue. (a) Dispersion curves for poly-methylmethacrylate and for a Cargille refractive index liquid with $n=1.486$ at 589 nm. The indices of the two materials are very similiar at short wavelengths and deviate from each other at larger wavelengths. (b) Detection cross section for a glass of poly-methylmethacrylate particles in air and in the same oil as in (a). Because the poly-methylmethacrylate is almost index-matched to the oil at short wavelengths, scattering in that regime is suppressed. For comparison, we also show the reflectivity for a similar glass of particles are in air, divided by 200. In both calculations we use $\phi = 0.55$ and $l = 16 \mathrm{\mu}$m; the particle sizes are 238 nm for the system in oil and 278 nm for the system in air, chosen to yield a resonance at the same wavelength.}
\end{figure}

Another way to eliminate the effect of the form factor resonance on color is by blueshifting it into the ultraviolet and away from the visible spectrum. This can be achieved with particles that have a small optical diameter, as
shown in Figure~\ref{fig:resonance_vs_od}.  The simplest way to reduce
the optical diameter is to use particles with a refractive index
smaller than that of the medium.  In these \textit{inverse} glasses,
the diameter of the particles is about the same as the spacing between
their centers ($d_{\mathrm{avg}} \sim \sqrt{6} d/3$), but their lower
refractive index makes the optical pathlength inside each particle
shorter than the optical pathlength between two particles. As a
result, the form factor resonances are blue-shifted compared to those
of our poly(methyl methacrylate) colloids of similar size, while the
structure factor resonances remain at about the same wavelength. We present our progress towards the realization of such systems in Chapter~\ref{ch:inverse_glasses}. 

Alternatively, blue-shifting the form factor resonance without shifting the structural resonance can also be achieved by decoupling the particle size from the interparticle spacing and using smaller particles as the scatterers. This can be done by packing core-shell particles with a strongly scattering core and a transparent shell, like the ones described in Chapter~\ref{ch:core-shell}. In Chapter~\ref{ch:photonic_balls} we show that this approach indeed enables the creation of full-spectrum, angle-independent structural pigments~\cite{Park2014}. These pigments still suffer from poor color saturation in the red compared to the blue, but, as we discuss, the short-wavelength reflectivity is substantially reduced relative to colloidal glasses made of conventional, homogeneous particles.

One can combine the core-shell and inverse-structure approaches to design a system with a single visible resonance at long wavelengths. In particular, the shell diameter could be chosen such that the interparticle spacing gives rise to a resonance in the red, and the core diameter chosen such that the first-order peak in the form-factor boosts the peak from the structure factor, while the second-order form-factor peak is fully in the ultraviolet.  Our calculations show that such a structure could be made from core-shell particles with air cores and silica shells, embedded in a silica matrix, as shown in Figure~\ref{fig:reflection_red}.

\begin{figure}
\centering
\includegraphics{reflection_red_150_to_775nm_with_inset}
\caption{\label{fig:reflection_red} Calculated reflection for an
  inverse glass of core-shell spheres engineered to scatter most
  strongly in the red. The continuous red curve includes both the
  structure and the form factor, the dashed black curve only the
  structure factor (divided by 2), and the black continuous curve only
  the form factor.  The vertical dashed line marks 425 nm, the
  low-wavelength limit in our spectral measurements. The optimal
  design has air cores with diameter 260 nm and silica shells with
  diameter 280 nm, and the particles are embedded in a silica
  matrix. The reflectivity peak at 632 nm is primarily due to the
  structure factor and determined by the shell diameter. One
  form-factor resonance occurs in the near-IR, boosting the structural
  resonance in the red, and another occurs deep in the ultra-violet (at about
  150 nm), too far away from the visible regime to affect the
color.}
\end{figure}

Curiously, the photonic structures found in bird feathers resemble
these inverse glasses: they often consist of air pockets in a matrix
of $\beta$-keratin~\cite{Noh2010AdvMat, Saranathan2012} that has a
refractive index close to that of
silica~\cite{Leertouwer2011}. However, birds do not seem to have taken
advantage of their inverse structures for colors other than blue.
They rely instead on pigments to acquire yellow, orange and red
colors~\cite{Saranathan2012, Stoddard2011}. Whether this is due to a
physical effect not accounted for in our model or is the result of
evolution or chance remains to be seen.

\section{Conclusion}

We have described a physical mechanism that explains the scarcity of
angle-independent structural red color in photonic glasses. Our model
shows that interparticle correlations alone are not sufficient to
understand the reflectivity of photonic glasses. The scattering
behavior of their constituent particles plays an equally
important---and previously unrecognized---role. In particular,
interference of light \textit{inside} the particles can lead to
enhanced scattering at wavelengths other than those related to the
interparticle correlations. Our model describes our experimental
observations well, and it can be used to guide the design of new
photonic glasses with purely red structural color.

To this end, we have shown that it is possible to control the
locations of both the single-particle scattering resonances and the
structural resonances by tuning the refractive indices, particle
sizes, and interparticle distances. Our model predicts that inverse
glasses made of core-shell particles with a low-index core and a
high-index shell that is index-matched to the medium might produce
angle-independent structural red color. If future experiments show
that such structures show poor color saturation in the red, these
results would suggest that another mechanism, such as multiple
scattering, should be accounted for.  If successful, these structures
would complete the palette of visible colors achievable with photonic
glasses, opening the path to their use in practical applications such
as long-lasting dyes or reflective displays.

In the following two chapters, we describe our progress towards angle-independent structural red color with the systems proposed here: core-shell particles with transparent shells, and inverse glasses. 


This chapter is based on the following manuscript: Sofia Magkiriadou, Jin-Gyu Park, Young-Seok Kim, Vinothan N. Manoharan, ``On the Absence of Red Structural Color in Photonic Glasses, Bird Feathers and Certain Beetles,'' Phys. Rev. E 90, 062302 (2014) \cite{Magkiriadou2014}.


%In the following chapter, we use this model to explore the range of colors that can potentially be achieved with colloidal glasses, and we suggest design rules for the creation of highly saturared colors across the visible spectrum. 

%\todo[inline]{fix top margin throughout}
%\todo[inline]{discuss where the theory breaks down and how it should not be used}

%A colloidal glass is illuminated by incident light with wavevector $\mathbf{k}_{\mathrm{in}}$ and scatters light with wavevector $\mathbf{k}_{\mathrm{s}}$.  The angle between $\mathbf{k}_{\mathrm{in}}$ and $\mathbf{k}_{\mathrm{s}}$ is $\theta$.  The vector difference $\mathbf{q} = \mathbf{k}_{\mathrm{s}} - \mathbf{k}_{\mathrm{in}}$ describes the momentum change between the incident and the scattered wave.  The source can emit light into a range of angles defined by the source numerical aperture NA$_{\mathrm{source}} = \sin(\theta_{\mathrm{source}})$, and the detector can detect light coming from a range of angles defined by the detector numerical aperture NA$_{\mathrm{detector}} = \sin(\theta_{\mathrm{detector}})$.
%We assume elastic scattering, so that $|\mathbf{k}_{\mathrm{in}}| = |\mathbf{k}_{\mathrm{s}}| = k = 2\pi n_{\mathrm{eff}} / \lambda$, and 

%\begin{abstract} 
% Colloidal glasses, bird feathers, and beetle scales can all show structural colors arising from short-ranged spatial correlations between scattering centers.  Unlike the structural colors arising from Bragg diffraction in ordered materials like opals, the colors of these photonic glasses are independent of orientation, owing to their disordered, isotropic microstructures.  However, there are few examples of photonic glasses with angle-independent red colors in nature, and colloidal glasses with particle sizes chosen to yield structural colors in the red show weak color saturation. Using scattering theory, we show that the absence of angle-independent red color can be explained by the tendency of individual particles to backscatter light more strongly in the blue.  We discuss how the backscattering resonances of individual particles arise from cavity-like modes, and how they interact with the structural resonances to prevent red.  Finally, we use the model to develop design rules for colloidal glasses with red, angle-independent structural colors.
%\end{abstract}

%Structural color in materials arises from interference of light scattered from inhomogeneities spaced at scales comparable to optical wavelengths. Opals and most other familiar examples of structurally colored materials are ordered~\cite{Sanders1964, Marlow2009}, and as a result, the color of these photonic crystals~\cite{MoldingTheFlowOfLight} depends on their orientation relative to the incident light: they are iridescent. There is another, less well-studied class of materials with angle-independent structural colors. These are called \textit{photonic glasses}~\cite{Ballato2000, Rojas-Ochoa2004, Garcia2007, Garcia2010}, because the inhomogeneities form a random, glassy arrangement with short-ranged order but no long-range order.  As in crystals, the average spacing between neighboring scatterers in a photonic glass is narrowly distributed and determines the resonantly scattered wavelength~\cite{Forster2010, Noh2010AdvMat}. But unlike crystals, photonic glasses are isotropic, so that the condition for constructive interference is independent of orientation.  This coloration mechanism is common in the feathers of birds~\cite{Prum1998, Noh2010AdvMat, Saranathan2012}, whose colors are visually indistinguishable from those of conventional absorbing dyes.  Photonic glasses with structural colors in the visible have also been produced in a variety of synthetic colloidal systems~\cite{Takeoka2009, Ueno2009, Ueno2010, Forster2010, Harun-Ur-Rashid2010, Lee2010, Kumano2011, Gotoh2012, Magkiriadou2012, Takeoka2013}.

