# Chapter 4 Web Topics

## Introduction

The wave theory of light gives us precise equations for computing the reflection and refraction of light at the boundary between two materials with different indices of refraction. This unit provides a brief overview of these equations, and links to a number of web resources containing animations and tutorials for these principles.

## Huygens wave principle of reflection and refraction

It is far easier to understand the processes of reflection and refraction by viewing a movie of a light wave as it encounters a boundary. At the website listed below, you can find an animation of Huygens wave principle of reflection and refraction. A few sample points along the boundary of the two media are shown in pink and the wave is shown being reflected and refracted off these points. In reality, there would be an infinite number of points along the way. Christian Huygen (1629–1695), a Dutch scientist, was the first major proponent of the wave theory of light.

http://www.walter-fendt.de/html5/phen/refractionhuygens_en.htm

## Quantitative expressions for the effects of different refractive indices on either side of a boundary

The difference in the refractive indices of the two media at a boundary affects reflection and refraction in several ways. Most of these effects can be described quantitatively. In the equations below, subscript 1 refers to the first medium (from which the wave is coming) and subscript 2 describes the second medium (after the wave crosses the boundary). The refractive indices are indicated by n, and angles relative to the normal are indicated by θ.

Angle of refraction: The angle of refraction depends on the incident angle and the indices of refraction for the two media, according to Snell’s law, where the subscript 1 refers to the first (incident) medium and the subscript 2 refers to the second medium:

n1 sin θ1 = n2 sin θ2

The application at the following website enables the user to vary the angle of incidence and the refractive indices of the two media to compute the angle of refraction:

http://www.walter-fendt.de/html5/phen/refraction_en.htm

Amount of reflected light: More light is reflected at a boundary the greater the contrast in refractive indices. The proportion of light reflected, R, is given by the Fresnel equation:

R = [ (n1 – n2) / (n1 + n2) ]2

Snell’s window: When light from a relatively dense medium (high index of refraction) hits the boundary of a medium with a lower index at a high angle of incidence, it is completely reflected. This point, called total internal reflection, occurs when the angle of refraction is 90° or larger and Snell’s law is undefined. One familiar consequence of this phenomenon, called Snell’s window, is the restriction of the visual field to low angles of incidence when under water looking up at the surface toward objects in the air. As illustrated below, an animal in water that looks up at the surface sees the entire hemisphere above the water condensed into a solid angle of 97° due to refraction. At angles of incidence greater than 48.5° (Brewster’s angle for water), light is reflected back into the water at the surface and this region appears mirror-like.

Index of refraction: The index of refraction can also be expressed as:

Here, N is the density of atoms in the medium and the constant includes the weight and charge of electrons. The density of atoms decreases the speed of propagation, v, (and thus increases n), because the more molecules or atoms there are available for polarization, the larger the counter-field that can be induced in a material by an external electric force. The frequency difference (ω20ω2) reflects the degree to which the driving frequency is similar or not similar to the resonance frequency. When the two are equal, the denominator in the above expression is zero and thus n is infinitely large. This is equivalent to saying that the speed of propagation in the medium at this frequency is zero: everything is absorbed and nothing is transmitted. At driving frequencies close to ωo, transmission will be poor but not zero and there will be complicated phase effects. As the difference in frequencies increases, n drops off quickly. In fact, because the frequency terms are squared, it will not take a very large difference between ω and ωo before n ≈ 1.

Glass and water are transparent substances with indices of refraction greater than 1 (1.5 and 1.33, respectively). Both materials have resonant frequencies close to the visible light range. Glass has a resonant frequency of about 15.0x1014 Hz in the UV range. Shorter wavelength radiation in the visible range (e.g., violet and blue) is closer to this resonant frequency and is therefore refracted at a greater angle when entering and leaving boundaries with air than is longer wavelength radiation (e.g., yellow and red). A beam of white light directed at a glass prism at an angle is refracted at two boundaries and is split into a spectrum of colors, a process called dispersion. The illustration below shows this process and provides the wavelengths and frequencies of different visible light hues. Violet light is clearly closer in frequency to the resonant frequency, and is bent more at both boundaries.

Brewster’s angle: When light from a less dense medium strikes the surface of a denser dielectric medium at an angle, the reflected ray is partially polarized in the plane parallel to the surface. At a critical angle, called Brewster’s angle, it is completely polarized. This phenomenon occurs when the incident and refracted angles add up to 90°. Brewster’s angle can be computed by:

tan θB = n2 / n1

In the illustration below for light incident in air hitting a glass boundary, Brewster’s angle is about 56°. The incident ray contains electric vectors oriented in all possible directions but only two are shown, E1, oriented in the plane of the page, and E2, oriented perpendicular to the page. At the critical angle, only vector E2 will be present in the reflected wave and E1 will predominate in the refracted wave.

Polarized sun glasses that transmit only vertically polarized light are designed to cut the glare of the horizontally plane-polarized light that reflects off of flat substrates without reducing other light substantially. Take your glasses off and rotate them 90°. Now you will see primarily the reflected glare!

Scattered light is also plane polarized when viewed from a 90° angle from the incident beam for similar reasons. The natural world contains complex patterns of polarized light generated by reflection and scattering. Many animals can perceive the plane of polarized light and use the patterns for a variety of functions, including navigation, orientation toward water, prey detection, and even signaling. These issues are taken up in Web Topic 5.2.

Light and color: An excellent resource with good explanations and many useful applets: http://micro.magnet.fsu.edu/optics/lightandcolor/index.html

Reflection and refraction: More animations of Huygens waves: http://scienceprimer.com/reflection-refraction

Radiation and the human body: A good explanation of how different frequencies of radiation interact with biological organisms: http://hyperphysics.phy-astr.gsu.edu/hbase/mod4.html

Other useful physics websites:

• The Physics Classroom has useful tutorials on general physics, motion, forces, work and energy, momentum, electricity and magnetism, light and sound waves, and nuclear physics at the high school and introductory college levels. Find it here: http://www.physicsclassroom.com/.
• The Hyperphysics website contains more advanced tutorials in physics and astronomy at the college level. Find it here: http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html.

## 4.2 Quantifying Light and Color

It is important to distinguish two types of light measurements: radiance and irradiance. Both are measures of radiant flux (energy per unit time). The two differ in the acceptance angle of the sensor (Figure 1).

Figure 1: Light sensors. (A) Cosine irradiance collector and (B) radiance collector.

Irradiance is the total amount of light incident on a surface, and includes scattered (diffuse) light as well as direct light. It is measured by an instrument that collects the light from a 180° solid angle. Radiance, on the other hand, is the flux of energy emitted from a specific radiant area such as the sun or an animal’s body or signal patch. Only the light that travels directly from the source area to the receiver is measured using a tube-like or telescoping instrument that cuts out scattered light. The solid angle over which the instrument is measuring must be specified. The units for these different measures are shown in the table below.

 Name Measurement Unitsa Radiant flux Flux (flow per unit time) Photons sec–1 Irradiance Flux density at a surface Photons sec–1 m–2 Radiant intensity Flux per unit solid angle Photons sec–1 sr–1 Radiance Flux per unit solid angle per unit area Photons sec–1 sr–1 m–2

a The measure “sr” means steradian, the unit of solid angle. There are 4π steradians in the complete solid angle of a sphere.

The general term for light measuring instruments is radiometer. These photoelectric devices record radiant energy or power (in joules sec–1, or watts), whereas the relevant unit for animal vision is quantum flux, the number of photons per unit time. Since a photon’s energy is related to its wavelength, an energy unit of red light represents more photons than the same unit of blue light. It is relatively easy to convert an energy flux measure into a photon flux measure with a wavelength-specific equation. The photon flux for a single wavelength in micro moles of photons m–2 s–1 is given by Q(λ) = 0.00835191E(λ), where E is energy flux in watts m–2 (Endler 1990). This conversion may not be necessary, since most light measurements are given as percentages or proportions relative to a standard; the standard and any other components of a visual signaling system must also be measured in the same units.

## Wavelength-specific measurements

Both radiance and irradiance measures can be wavelength-specific by making separate serial measurements on a narrow range of wavelengths. The instrument that does this job is called a spectrometer. While there are several types of spectrometers available, most animal communication researchers now use the diode-array type because it is fast, lightweight, possesses no moving parts, and has low power requirements, so is easy to run on batteries for field use. A popular model is illustrated in Figure 2. The light signal from the sensor is brought into the instrument with a fiber optic cable, conditioned, and spread into a rainbow of colors with a diffraction grating. The dispersed light falls on an array of photodiodes, each of which responds only to the narrow range of wavelengths impinging on it. The diodes are connected to a charge-coupled device (CCD) that produces a voltage. Voltages from each diode are converted to digital counts and sent to a dedicated computer with spectrum analysis software. The acquired spectrum is immediately displayed on the screen and saved in standard computer files.

Figure 2. Diode-array spectroradiometer (Ocean Optics USB4000). Light enters the spectrometer (1) from a fiber optic cable connected to a probe. The amount of entering light is controlled with an adjustable slit (2) and a filter (3) restricts the wavelength range. The filtered light then encounters a collimating mirror (4) that focuses the beam on a diffraction grating (5). Like a prism, this grating scatters the different wavelengths onto a focusing mirror (6) that directs the light via a collector lens (7) onto the detector (8). The detector converts the optical signal to digital format. (From Ocean Optics 2008, http://www.oceanoptics.com.)

Photodiodes exploit the photovoltaic effect at the interface between a semi-conductor and a metal. Electron transitions generate either a voltage in the cell or a decrease in resistance in an electrical circuit with an external voltage source. Diodes measure accumulated radiant energy (joules) and can be made very small. A silicon photodiode permits linear responses over 10 orders of radiant power between 100 and 1100 nm. A typical diode-array instrument contains an array of up to 1000 photodiodes. The wavelength interval that can be measured is fixed by the width of the diodes and therefore cannot be varied. However, a single scan takes a fraction of a second, and the diodes can then be discharged and prepared for a second scan. Several scans may be made and averaged in a matter of seconds. For further information see Andersson and Prager (2006).

## Sampling setup

To make accurate and reliable measurements of a color spectrum from a live animal color patch or sample, several important decisions and steps must be made.

First, an appropriate light source must be obtained. The light source must provide strong, stable, and spectrally smooth light over the desired range of wavelengths. A tungsten-halogen source works well for a wavelength range of 380–700 nm. If the color patch turns out to have a strong reflectance in the UV range, then a specialized UV source should be used. Currently available UV light sources require quite a bit of power and cannot be run from batteries.

Second, the geometry of angles for the light source and the reflection probe relative to the sample surface must be determined. This geometry is of crucial importance for the measurement of structural colors, which depends on the angle of illumination and the angle of viewing. There are several possible configurations, illustrated in Figure 3. The most reliable and recommended arrangement for pigment-based colors is coincident normal (CN), where both the light source and probe are directly above the sample at normal angle. Coincident oblique (CO) moves the both light and probe together to an angle of 45° relative to the sample surface. In oblique normal (ON) the light source derives from a 45° angle but the probe is directly above at the normal angle. For some structurally-based color mechanisms, one may desire to measure the specular reflectance at different angles, so the light source and probe are set at equivalent angles to either side of the normal, e.g., oblique opposite (OO). Finally, a diffuse light source can be established with either multiple light sources, as in diffuse normal (DN), or with an integrating sphere (IS).

Figure 3: Geometries for angles of illumination and observation. The light blue box represents the sample being measured, the light-measuring probe is shown in dark blue, and the illumination lamp is yellow. Black arrows show axes of incident light and specular reflectance; gray arrows show diffuse reflectance. Coincident normal (CN) has both probe and light at normal incident angle; coincident oblique (CO) has both probe and light at some specified angle. Oblique normal (ON) uses light at an oblique angle and probe at normal angle measuring only diffuse reflectance. Oblique opposite (OO), with light and probe at equal but opposite angles from the normal, is used to specifically measure specular reflectance. Diffuse normal (DN) uses diffuse illumination. Integrating sphere (IS) is a second way to generate diffuse lighting. Here, the black bar is a baffle to prevent reflected light from directly entering the probe. (After Andersson and Prager 2006.)

The third decision involves the white standard. All reflectance measurements are made relative to a standard and reported as proportional or percentage values. The white standard is a reference surface that is measured with the same illumination set-up as the sample. The two spectra from the white reference and the sample are then compared. The reference surface should have a high reflectance (close to 100%) over all wavelengths of interest. To make highly accurate measurements, it is also important to remove the effects of electronic noise in the system from both the sample spectrum and the reference spectrum by subtracting what is called “dark currents.” The reflectance of the sample is then computed as R = (ARD) / ARrD), where AR is the illuminated sample, ARr is the illuminated white reference, and D is dark current noise (Andersson and Prager 2006).

## Quantifying brightness, hue, and saturation

Once one has obtained a reflectance spectrum from an animal patch, the next step is to extract useful measures of brightness, hue, and saturation. Researchers use a variety of methods for extracting these values, so they are not always comparable.

Raw reflectance spectra are often jittery curves, and repeated measurements from the same or different individuals may show some variation. Individual curves may be smoothed using a moving average or cubic spline, and several curves may be averaged. Subsequent measurements may be based on these smoothed or averaged curves.

The commonly used methods and formulae for computing brightness, hue, and saturation are illustrated in Figure 4. For more details, the reader is referred to Montgomerie (2006). Brightness is typically the area under the curve over a specified range of wavelengths, but mean brightness and maximum brightness values may also be used. Hue is the location of the wavelength with the greatest reflectance value. Saturation is the most difficult property to measure consistently. Four popular measures include the slope, the ratio of the sum of the low-reflectance region to the sum of the whole region, the ratio of the maximum reflectance to the lowest reflectance, and the difference between the maximum reflectance and lowest reflectance.

Figure 4: Formulae for extracting color measures from spectrographs. Saturation measures: S1 (reflectance ratio) is the sum of reflectance values integrated over the lowest reflectance region (X) divided by sum of reflectance integrated over the entire wavelength range (X + Y) = B1; S2 (spectral saturation) is the ratio of highest to lowest reflectance values; S4 (spectral purity) is the greatest negative slope on the curve; and S6 is the difference between maximum and minimum reflectance values. Brightness measures: B1 (total reflectance) summed over the appropriate range of wavelengths; B2 is the average of these values or B1 divided by nw, the number of wavelength intervals included in the sum; and B3 is the maximum reflectance value. The best measure of hue for a unimodal curve is the wavelength with the maximum reflectance. For cutoff pigments such as carotenoids, the hue is measured as the wavelength at which reflectance is halfway between its minimum and maximum values within the range of 450–700 nm, often called λ[R50]. Other measures may be more appropriate for a bimodal curve. (After Montgomerie 2006.)

It is often informative to plot the absorbance (or absorptance) curve for a color patch, especially one derived from a pigment mechanism. Absorbance is again a relative measure, and can be generated by subtracting the reflectance curve from the white standard curve.

There are some caveats on the meaning of these measurements for structural versus pigmentary colors (Andersson 1999; Andersson and Prager 2006). Carotenoid and other pigments are subtractive colorants, in that they absorb certain wavelengths within the visible range. Structural colors are produced by additive and spectrally selective modifications of surface reflectance using nanoscale structures. Differences in the quality of these two types of color signals will have different consequences for some of the color property measurements. For example, as the concentration of a color-producing pigment increases, more light will be absorbed, and overall brightness will actually decrease. On the other hand, saturation should increase, and the hue may change as well. But, for very highly saturated pigments, further increases in concentration will not change the saturation value. For structural colors, increasing the thickness of multi-layer stacks should result in an increase in brightness, and increasing the regularity of the stack should increase the saturation of the color. Changes in the size and spacing of scatters will change the hue. Therefore, the types of color characteristics we measure as potential indicators of sender quality are expected to differ for the two color mechanisms.

## Using photography

A very simple alternative to the expensive and computationally intensive spectroradiometer technique is to use digital photography for quantifying color parameters. The only standardizing strategy is to take comparable digital photographs of animals from a specified distance and under a controlled source of illumination (or flash). Photographs can then be analyzed using graphic software applications such as Adobe PhotoshopTM, CanvasTM, SigmaScan ProTM, Corel PaintTM, or ImageJTM to give mean values of hue, saturation, and brightness for user-specified color patches. This method will work only for animal color signals that fall within the human vision color range, where RGB color space is appropriate. A good example of the effective use of this technique can be found in Kilner’s (1997) study of mouth coloration in nestling birds.

## Literature Cited

Andersson, S. 1999. Morphology of UV reflectance in a whistling-thrush: implications for the study of structural colour signalling in birds. Journal of Avian Biology 30: 193–204.

Andersson, S. and M. Prager. 2006. Quantifying colors. In Bird Coloration (Hill, G. E. and K. J. McGraw, eds.), pp. 41–89. Cambridge, MA: Harvard University Press.

Endler, J. A. 1990. On the measurement and classification of color in studies of animal color patterns. Biological Journal of the Linnean Society 41: 315–352.

Kilner, R. 1997. Mouth colour is a reliable signal of need in begging canary nestlings. Proceedings of the Royal Society of London Series B-Biological Sciences 264: 963–968.

Montgomerie, R. 2006. Analyzing colors. In Bird Coloration (Hill, G. E. and K. J. McGraw, eds.), pp. 90–147. Cambridge, MA: Harvard University Press.

OceanOptics, Inc. 2008. USB4000 Fiber Optic Spectrometer Installation and Operation Manual. Dunedin, FL: Halma Group Company.

## Introduction

Coherent scattering of a narrow range of wavelengths requires spatial periodicity in refractive index in one or more dimensions. Figure 1 illustrates idealized structures with periodicity in one, two, and three dimensions.

Figure 1. One, two, and three-dimensional array structures composed of high and low refractive index materials. Arrows show the vector directions with periodic variability.

Examples of natural one-dimensional nanostructures include single-layer systems, with a thick keratin layer over a melanin layer, and multi-layer quarter-wave stacks. An example of a two-dimensional structure is the collagen fiber arrays in the colored skin of many birds and mammals. Three-dimensional structures include diffraction lattices and other crystalline arrays. The effective dimensionality of some complex natural structures is often difficult to determine. Given the basic principles of interference physics, the size and spacing of the scattering structures must be consistent with the observed color of coherently scattered light. Electron microscopy is an essential tool for describing the dimensionality of nanostructured surfaces. Traditional transmission electron microscopy of very thin tissue slices only provides a two-dimensional view, although slices can be taken at different angles. Other techniques such as scanning electron microscopy and higher-voltage electron microscopy of thicker slices with tomographic reconstruction give a three-dimensional view of the structure. In this Web Topic unit we show how these observational techniques, along with mathematical modeling, have been used to quantify the nanostructure properties of colored surfaces and define their dimensionality.

## Reflection modeling

One-dimensional structures typically produce iridescent colors, defined as colors that change as a function of viewing angle or source lighting angle. Their properties can therefore be described with TEM images and basic optical ray reflection and refraction equations. The angular dependence of the reflectance spectrum is described with an instrument called a goniometer, as shown in Figure 2.

Figure 2: A goniometer. The object (feather, wing, carapace, etc.) is secured to a flat platform. A protractor centered over the object shows the normal angle (0°) at the top, and gradations up to 90° on each side. A point light source (left) is aimed at the object from a specified angle, and a directional light receptor (right) is also aimed at the object on the other side. The light source and detector are often placed at matching angles, and measurements of brightness and hue are made at a series of angles. Alternatively, the light source can be stationary at zero degrees while the platform is rotated.

If the object is a flat multilayer stack, the bright hue is observed only over a very narrow range of angles, and outside of these angles the object should appear black. In other words, there is a single highly saturated brightness peak. The hue of the colored region should match that predicted by the Bragg equation— λ/4 = nl dl = nh dh —where n is the refractive index of the layer, d is its thickness, and the subscripts l and h refer to the low- and high-density alternating layers. The linear dimensions would be measured with a TEM image along the z-axis; once the wavelength is known, refractive indices from this equation can be estimated and compared to other estimates for these materials derived from similar kinds of measurements. The brilliantly colored throat and chest feathers of hummingbirds typically show this type of structure, and are only visible when the bird is directly facing the receiver. The striking flash effect of this signal can be seen in this video clip at https://academy.allaboutbirds.org/iridescence-revealed-in-the-feathers-of-an-annas-hummingbird/.

If the object has a simpler two-layer nanostructure, consisting of one thick and relatively transparent layer of a material such as keratin over another layer of melanin granules, then thin-film optical interference principles apply. Such objects are often basically dark-colored, with one to six brightness peaks of different hues which may shift as the angle of incidence or viewing is changed (Figure 3).

Figure 3: Iridescent neck feathers of domestic pigeons. Reflectance spectra of (A) green and (B) purple iridescent neck features measured at different angles of incidence. The actual colors at each angle are shown at the top right of each graph. (From Yin et al. 2006. Reprinted with permission from Physical Review.)

The peaks for a given curve are harmonically related and specified by this formula:

It is not immediately obvious which interfaces are contributing to the iridescent color. Figure 4 shows the possible alternative reflection models for such two-layer systems. Model A involves only the two sides of the keratin layer, Models B and C involve the melanin layer with either the top or bottom keratin interface, and Model D combines reflection from all three interfaces.

Figure 4: Four alternative models of constructive thin-film interference in a two-layer feather barbule nanostructure. (After Doucet et al. 2006. Adapted with permission from the Journal of Experimental Biology.)

To figure out which model best applies, predicted reflectance spectra are computed for each model based on the thicknesses of each layer, their refractive indices, attenuation within each layer, and occurrence of half-wave phase shifts at Interfaces 1 and 2. The reflected components in each model are combined additively. For the avian species subjected to this type of analysis, Model A, involving only the keratin layer, with reflection from the top and bottom interface, appears to provide the best fit (Brink and van der Berg 2004; Doucet et al. 2006; Yin et al. 2006). The melanin layer is too dense for a significant amount of light to pass through it. Figure 5 shows these results for the satin bowerbird, whose plumage is black with ultraviolet iridescence.

Figure 5: Observed reflectance curve and the four predicted model curves for the iridescent male satin bowerbird feather. The solid line is the observed curve; the dashed lines are modeled curves. Model A fits the observed curve best, suggesting that only the keratin layer is responsible for the iridescence while the melanin layer serves to define the bottom keratin interface. (From Doucet et al. 2006. Reproduced with permission from the Journal of Experimental Biology.)

## Fourier analysis

The periodicity of any alternating or oscillating pattern can be described with Fourier analysis. In Web Topic 2.4, we show how simple Fourier analysis of sound waves can be used to break down a complex wave into the sum of component sine (or cosine) waves, each with a specific frequency, amplitude, and relative phase. The amplitudes of each of the component sine waves in the Fourier transform give the relative contributions (weighting) of those frequencies to the periodicity of the original data. A plot of the square of the amplitudes for each of the Fourier components versus their frequency is called the Fourier power spectrum. The power spectrum helps us identify the loudest or dominant frequencies in a sound. Fourier analysis can also be applied to a graphical image of patterned light and dark areas. If there is some regularity to the pattern, coherent scattering of a limited range of wavelengths should occur, and the Fourier power spectrum should exhibit a peak corresponding to the wavelength of the observed dominant hue.

Although optical Fourier principles were developed in the 19th century, Benedek (1971) derived new physical models of coherent light scattering based on electromagnetic principles to examine transparency of the human cornea. This approach has since been adopted by biologists to study color production in animal tissues (Prum et al. 1998, 1999, 2004, 2006; Prum and Torres 2003a, b, 2004; Shawkey et al. 2009). Because any graphical representation of a spatial pattern has at least two dimensions, the Fourier analysis must be expanded to encompass two dimensions. The first step for making this measurement is to obtain a high-contrast gray-scale TEM image of a nanostructured tissue region such as the collagen arrays of bird and mammal skin shown in Figure 6A. The digital image is essentially a matrix of dark and light pixels. Imagine a transect line drawn anywhere across this image. Along this line, one encounters a series of dark pixels followed by a series of light pixels in a repeating pattern. The dark and light cycles are converted into the peaks and troughs of a wave relative to the mean gray level in the whole image. The cyclical pattern of this waveform is measured by distance in units of cycles per nm (spatial frequency), rather than by time in cycles per sec as in a sound wave.

Next, one measures a series of spatial waves radiating out in all directions from each point in the image. Vectors from one such point are shown in Figure 6A. The Fourier component amplitudes of the vectors oriented in the same direction are averaged over all points, and this calculation is repeated for all vector angles. These results are then plotted in the two-dimensional Fourier power spectrum shown in Figure 6B. The center of this graph corresponds to variation at a spatial frequency of zero. Each point reports the magnitude of the periodicity in the original data of a specified spatial frequency in a given direction from all points in the original image. Darker pixels correspond to higher amplitude Fourier components. In this particular example, it is easy to see the regularity in the size and spacing of the dark fibers in A, which results in a circular ring around the origin in B. The two-dimensional plot can now be summarized by taking the radial average of concentric radial bins, or annuli, over all vectors. The outcome of this calculation is graphed in Figure 6C. The high peak at about 0.0065 nm–1 corresponds to the dense ring in the two-dimensional plot in Figure 6B.

The final step is to convert this combined power spectrum into a predicted reflectance spectrum. According to optical Fourier principles, coherently scattered wavelengths from an ordered array should be equal to twice the wavelength of the predominant components of the Fourier transform. Following this model, the spatial frequency averages for each wavelength are inverted and multiplied by twice the average refractive index of the medium and expressed in terms of wavelength. The result is a theoretical prediction of the relative magnitude of coherently scattered light based on the spatial variation in refractive index of the tissue, as shown in Figure 6D. The power spectrum predicts a peak of coherent reflectance at 410 nm, which is consistent with the dark blue color of the tissue (Prum and Torres 2003a, 2004).

Figure 6: Two-dimensional Fourier analysis of a collagen fiber array in the dark blue rump skin of the male mandrill (Mandrillus sphinx). (A) TEM cross-section of the skin showing darker collagen fibers surrounded by lighter mucopolysaccharide (scale bar = 250 nm). White lines show a few of the vectors for Fourier transform analysis emanating from one point. (B) The two-dimensional Fourier power spectrum of the sample in (A) showing the ring-like appearance expected of a periodic two-dimensional structure; note the nearly symmetrical pattern around the origin in the center of the graph. (C) The radial average of power spectra originating from the origin in (B); the gray zone indicates the spatial frequency range visible to mammals. The low peak at a spatial frequency of 0.014, corresponding to the faint outer ring in (B), is in the UV wavelength range. (D) Predicted reflectance spectrum (black) based on the two-dimensional Fourier spectrum in (C) showing a strong peak in the blue wavelength region; the measured reflectance spectrum is in gray. (From Prum and Torres 2004. Reproduced with permission from the Journal of Experimental Biology.)

The performance of this two-dimensional methodology on a one-dimensional laminar array structure is shown in Figure 7 for comparison. Note the linearity of the two-dimensional Fourier power spectrum in Figure 7B. This example still provides a good fit between the predicted and observed reflectance hue (Prum and Torres 2003a.)

Figure 7: A two-dimensional Fourier analysis on a one-dimensional multilayer array structure. The tissue comes from the barbule of a green back-feather of the Splendid Sunbird (Nectarinia coccinigastra). (A) A TEM through the barbule showing multiple layers of melanin and keratin. (B) A two-dimensional Fourier power spectrum; note the lack of circular ring. (C) The power spectrum predicted by Fourier analysis. (D) Measured reflectance of the green feather. (From Prum and Torres 2003a.)

Two-dimensional Fourier analysis applied to a TEM of the spongy matrix of blue and green barb tissues of bird feathers generates a Fourier power spectra such as the one shown in Figure 8. While there is a hint of a ring structure, it is more like a disk. Even disordered materials can produce circular plots.

Figure 8: A two-dimensional Fourier analysis of the blue feather barbs of the rose-faced lovebird, Agapornis roseicollis. (A) A TEM of the quasi-ordered, spongy medulary keratin and air structure in the interior of the barb. Scale bar = 200 mm. (B) A two-dimensional Fourier power spectrum of the same feather barb, showing a slightly oval disk. The magnitude of the power spectrum is given by the color scale bar on the right. (From Prum et al. 1999.)

This spongy tissue actually has a quasi-ordered three-dimensional structure, which cannot be accurately described with conventional two-dimensional visualization and analysis techniques. Newer tomographic techniques have facilitated a more accurate three-dimensional approach. Using intermediate-voltage electron microscopy, a high-power electron beam penetrates thicker slices of tissue and takes images at a series of angles, which then allows the third dimension to be reconstructed. Three-dimensional Fourier analysis can then be used to quantify the spatial frequencies on three dimensions and convert them to a predicted Fourier power spectrum that more precisely matches the observed reflectance spectrum of the tissue (Figure 9). Movies for visualizing the three-dimensional structure can be found at this URL: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2706473/ - supplementary-material-sec.

Figure 9. A three-dimensional analysis of the blue feather barbs of the eastern bluebird Sialia sialis. (A–C) Two-dimensional planes of the tomographic reconstruction of the spongy medulary tissue. The dark areas are keratin and the light areas are air. (D–F) Averaged projections of the three-dimensional Fourier power spectra from the tomographic reconstructions along different the x-, y-, and z-axes, respectively. The ring shape indicates periodic order over short spatial scales. Relative magnitude is indicated by the color map on the right of each graph. (G) Frequency spectrum predicted by the Fourier power spectrum data. The bar color shows corresponding hue. (H) Measured reflectance spectrum from spongy tissue (blue). Black dots with Gaussian fit are predicted spectrum points from (G). (From Shawkey et al. 2009.)

A second interesting application of the Fourier method was undertaken to examine the wavelength components of the transparent and opaque skin of the eye, i.e., cornea and sclera, respectively. For the cornea to be transparent, the collagen fibers must be very small and ordered to coherently reflect very small wavelengths well outside of the visual range while transmitting light in the visible range. The Fourier spectrum plots shown in Figure 10 illustrate the difference in collagen fiber periodicity very clearly. See Figure 5.14A in the main text for cross section TEMs of these two tissues.

Figure 10: Fourier components for transparent and opaque eye tissues. A comparison of the Fourier components for two regions of the eye: the transparent cornea and the opaque sclera. The cornea reflects only radiation of very small wavelengths, around 75 nm, while transmitting all wavelengths in the visible range. The sclera, in contrast, reflects all wavelengths in the visible range. (After Vaezy and Clark 1994 and Johnsen 2000.)

## Photonics and bandgap modeling

Photonics is the study of light generation, transmission, modulation, amplification, and detection from the particle perspective. The field began around 1960 with the invention of the laser. It arose from solid-state physics and the study of crystal structure using X-ray diffraction patterns. Analogous to the way in which atomic crystals control the movement and spread of electrons through the structure, photonics views the way in which dielectric materials and nanostructured lattices control the propagation of photons of light. The photonics approach uses Maxwell’s electromagnetic wave equations to model light wave transmission, in the same way that electronics uses Schroedinger’s equations of quantum mechanics to analyze electric currents (Joannopoulos et al. 1995).

Any material with periodic variation in high and low dielectric regions over one, two, or three dimensions can be analyzed from this perspective, but the term photonic crystal was first used in two seminal articles by Yablonovitch (1987) and John (1987) to describe two- and three-dimensional structures. Photonic crystals can guide and trap light traveling through the matrix. Depending on the dielectric materials and their arrangement, certain wavelengths or energy bands can pass through the material, called modes, whereas other wavelengths or energy bands are forbidden to pass, called bandgaps. Differential transmission and reflection of wavelengths in the visible range is what produces color. In fact, it is the forbidden wavelengths that are scattered, and this scattering is coherent. Because these higher-dimension structures can selectively pass light waves in different (disjunct) energy regions and directions, some unusual color effects can be produced. Biologists have recently taken up this way of thinking about highly ordered arrays of different dimensions to predict which wavelengths would be scattered and which transmitted. Natural photonic crystals and structures have now been identified for many structurally colored animal integuments. Figure 11 illustrates a bandgap analysis of the iridescent feathers of the peacock’s tail, a relatively simple two-dimensional square structure of melanin rods and airspaces (see also Figure 4.24D in the main text).

Figure 11: Bandgap model of iridescent peacock feathers. (A) Reflectance spectra of the blue, green, yellow, and brown regions of the eye spots of the green peacock Pavo muticus (see Figure 4.24D in main text for photo of the feather and SEM of the barbule nanostructure with a two-dimensional square array of melanin rods and air spaces. (B) Bandgap analysis. The y-axis is frequency in units c/a, where c is the speed of light and a is the lattice constant (140, 150, and 165 nm for blue, green, and yellow regions, respectively, based on measurements of rod and airspace periodicity). The x-axis indicates different wave vectors (the planes of different directions photons can travel through the structure). The Γ–X plane represents waves traveling in a direction normal to the barbule cortex surface, as shown in the bottom right. The lines in the graph show the calculated band structure for electric polarization (solid lines) parallel to the rod direction and magnetic polarization (dashed lines); electric and magnetic polarizations differed little in this two-dimensional structure. The orange blocks show the single partial bandgap region for this structure (for E polarization). A complete bandgap would have extended across all plane directions. This bandgap region corresponds to a reflectance peak in the middle of the visible light range. (C) Reflectance of blue, green, yellow, and brown feather regions predicted by the bandgap model. The peak positions match reasonably well, although the curve shapes are somewhat different. The bandgap position shifts to a higher frequency with increasing angle of incidence, indicating that the color is iridescent. (After Zi et al. 2003.)

More complex bandgap models and advanced visualization techniques are required to analyze three-dimensional structures. One notable study involving bandgap modeling coupled with a novel three-dimensional visualization technique characterized the brilliant green color of a weevil. The three-dimensional structure was assessed by consecutively milling away thin layers (about 30 nm thick) of the structure followed by coating the surface and taking SEM images. The color is generated by a diamond-structured photonic crystal with three bandgap regions, which yield three overlapping spectral peaks in the green frequency range (Galusha et al. 2008). Other studies have used bandgap modeling to explain structural colors produced by several butterfly species (Argyros et al. 2002; Michielsen and Stavenga 2008; Poladian et al. 2009; Saranathan et al. 2010). The Saranathan paper employed small angle X-ray scattering to obtain the three-dimensional structure, a promising technique that yields more precise descriptions of the structure compared to the electron tomographic reconstruction method described earlier.

The details of bandgap modeling are beyond the scope of this online unit. For a good overview of the photonics approach, see the review by Joannopoulos et al. (1997). Also useful are McPhedran et al. (2003), Vukusic and Sambles (2003), Parker (2004), Welch and Vigneron (2007), and Parker (2009). Links to several good web resources, tutorials, and lectures on the physics underlying photonic crystals are listed below.

http://ab-initio.mit.edu/photons/tutorial/

http://www.wisegeek.com/what-is-a-photonic-crystal.htm

## Literature Cited

Argyros, A., S. Manos, M. C. J. Large, D. R. McKenzie, G. C. Cox, and D. M. Dwarte. 2002. Electron tomography and computer visualisation of a three-dimensional ‘photonic’ crystal in a butterfly wing-scale. Micron 33: 483–487.

Benedek, G. B. 1971. Theory of transparency of the eye. Applied Optics 10: 459–473.

Brink, D. J. and N. G. van der Berg. 2004. Structural colours from the feathers of the bird Bostrychia hagedash. Journal of Physics D-Applied Physics 37: 813–818.

Doucet, S. M., M. D. Shawkey, G. E. Hill, and R. Montgomerie. 2006. Iridescent plumage in satin bowerbirds: structure, mechanisms and nanostructural predictors of individual variation in colour. Journal of Experimental Biology 209: 380–390.

Galusha, J. W., L. R. Richey, J. S. Gardner, J. N. Cha, and M. H. Bartl. 2008. Discovery of a diamond-based photonic crystal structure in beetle scales. Physical Review E 77: 050904.

Joannopoulos, J. D., R. D. Meade, and J. N. Will. 1995. Photonic Crystals: Molding the Flow of Light. Princeton: Princeton University Press.

Joannopoulos, J. D., P. R. Villeneuve, and S. H. Fan. 1997. Photonic crystals: Putting a new twist on light. Nature 386: 143–149.

John, S. 1987. Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters 58: 2486–2489.

Johnsen, S. 2000. Transparent animals. Scientific American 282: 80–89.

McPhedran, R. C., N. A. Nicorovici, D. R. McKenzie, G. W. Rouse, L. C. Botten, V. Welch, A. R. Parker, M. Wohlgennant, and V. Vardeny. 2003. Structural colours through photonic crystals. Physica B 338: 182–185.

Michielsen, K. and D. G. Stavenga. 2008. Gyroid cuticular structures in butterfly wing scales: biological photonic crystals. Journal of the Royal Society Interface 5: 85–94.

Parker, A. R. 2004. A vision for natural photonics. Philosophical Transactions of the Royal Society A-Mathematical, Physical, and Engineering Sciences 362: 2709–2720.

Parker, A. R. 2009. Natural photonics for industrial inspiration. Philosophical Transactions of the Royal Society A-Mathematical, Physical, and Engineering Sciences 367: 1759–1782.

Poladian, L., S. Wickham, K. Lee, and M. C. J. Large. 2009. Iridescence from photonic crystals and its suppression in butterfly scales. J Journal of the Royal Society Interface 6: S233–S242.

Prum, R. O., J. A. Cole, and R. H. Torres. 2004. Blue integumentary structural colours in dragonflies (Odonata) are not produced by incoherent Tyndall scattering. Journal of Experimental Biology 207: 3999–4009.

Prum, R. O., T. Quinn, and R. H. Torres. 2006. Anatomically diverse butterfly scales all produce structural colours by coherent scattering. Journal of Experimental Biology 209: 748–765.

Prum, R. O., R. Torres, S. Williamson, and J. Dyck. 1999. Two-dimensional Fourier analysis of the spongy medullary keratin of structurally coloured feather barbs. Proceedings of the Royal Society of London Series B-Biological Sciences 266: 13–22.

Prum, R. O. and R. H. Torres. 2003a. A Fourier tool for the analysis of coherent light scattering by bio-optical nanostructures. Integrative and Comparative Biology 43: 591–602.

Prum, R. O. and R. H. Torres. 2003b. Structural colouration of avian skin: convergent evolution of coherently scattering dermal collagen arrays. Journal of Experimental Biology 206: 2409–2429.

Prum, R. O. and R. H. Torres. 2004. Structural colouration of mammalian skin: convergent evolution of coherently scattering dermal collagen arrays. Journal of Experimental Biology 207: 2157–2172.

Prum, R. O., T. H. Torres, S. Williamson, and J. Dyck. 1998. Coherent light scattering by blue feather barbs. Nature 396: 28–29.

Saranathan, V., C. O. Osuji, S. G. J. Mochrie, H. Noh, S. Narayanan, A. Sandy, E. R. Dufresne, and R. O. Prum. 2010. Structure, function, and self-assembly of single network gyroid (I4(1)32) photonic crystals in butterfly wing scales. Proceedings of the National Academy of Science USA 107: 11676–11681.

Shawkey, M. D., V. Saranathan, H. Palsdottir, J. Crum, M. H. Ellisman, M. Auer, and R. O. Prum. 2009. Electron tomography, three-dimensional Fourier analysis and colour prediction of a three-dimensional amorphous biophotonic nanostructure. Journal of the Royal Society Interface 6: S213–S220.

Vaezy, S. and J. I. Clark. 1991. A quantitative analysis of transparency in the human sclera and cornea using Fourier methods. Journal of Microscopy-Oxford. 163: 85–94.

Vaezy, S. and J. I. Clark. 1994. Quantitative analysis of the microstructure of the human cornea and sclera using 2-D Fourier methods. Journal of Microscopy-Oxford 175: 93–99.

Vukusic, P. and J. R. Sambles. 2003. Photonic structures in biology. Nature 424: 852–855.

Welch, V. L. and J. P. Vigneron. 2007. Beyond butterflies - the diversity of biological photonic crystals. Optical and Quantum Electronics 39: 295–303.

Yablonovitch, E. 1987. Inhibited spontaneous emission in 3-dimensionally modulated periodic dielectric structures. Journal De Physique 48: 615–616.

Yin, H. W., L. Shi, J. Sha, Y. Z. Li, Y. H. Qin, B. Q. Dong, S. Meyer, X. H. Liu, L. Zhao, and J. Zi. 2006. Iridescence in the neck feathers of domestic pigeons. Physical Review E 74: 5.

Zi, H., X. D. Yu, Y. Z. Li, X. H. Hu, C. Xu, X. J. Wang, X. H. Liu, and R. T. Fu. 2003. Coloration strategies in peacock feathers. Proceedings of the National Academy of Science USA. 100: 12576–12578.

## Introduction

In Chapter 4, we reviewed the diversity of mechanisms that self-advertising animals have adopted to produce colors that are conspicuous against their normal backgrounds. However, most animals do not stop at color production, but add specialized motions that make their color patterns even more conspicuous. Luminescent species are usually nocturnal or live in deep ocean: many turn their lights on and off with species-specific timings to communicate with conspecifics in the dark. Species with neurally controlled chromatophores can vary their color patterns rapidly during a display. Species with iridescent colors typically use movement to shift the viewing angle relative to a receiver and thus change the apparent color. Other species cover colored regions when at rest but expose them for short periods during a display. In this module, we provide examples of each of these different strategies for combining color with motion.

## Examples of movement and color/light displays

### Cephalopods

Soft-bodied cephalopods produce and modulate their body color patterns using chromatophores (see Figure 4.33, Chapter 4). Although this ability was likely first favored as a camouflage adaptation (see first clip below), species in suitably lighted environments have co-opted the system for social and mating signals.

### Fiddler Crabs

Males advertise themselves and their burrows to nearby females by waving their enlarged, and usually contrastingly colored, claw in a stereotyped display. Since some species overlap, each species has its own waving pattern:

### Insects

• Fireflies (species not indicated): Short clip of firefly males advertising in flight during early evening in Hudson Valley, New York.
• Helicopter damselfly (Megaloprepus caerulatus): This species has special markings on the wing tips that generate a fluttering image to attract mates (Schultz and Fincke 2009).
(http://www.arkive.org/helicopter-damselfly/megaloprepus-caerulatus-/video-00.html)
• Butterflies: Butterflies host spectacular color patterns that are made more conspicuous by their fluttering flight. Some species have ultraviolet and iridescent colors that are only visible at certain viewing angles. Species also differ depending on whether both the top and bottom sides of the wings are brightly colored, and which side is exposed when resting. This clip shows natural videography of numerous butterfly species from Bolivia.
• Morpho butterfly (Morpho spp): As documented in Chapter 4 (see Figures 4.18F and 4.29B), Morpho butterflies have iridescent blue coloration on the upper surface of the wings. This color shifts to white/grey at glancing viewing angles and appears to flash in and out when the insect is in flight. Two views are provided below: the first shows a live Morpho opening and closing its wings with a consequent appearance and disappearance of the blue color; the second shows a simulation of how the color changes with viewing angle:

### Lizards

Most lizards have fixed colors in places that are most clearly visible to conspecifics during territorial and mating interactions. Chameleons are unusual in that they can change color using chromatophores. Some species like anoles hide colored patches when not displaying. Some examples:

### Birds

There are a large number of bird species that combine spectacular colors and motion in their displays. However, many perform their displays in flight (e.g. hummingbirds and manakins) making it hard to video-record displaying individuals at sufficiently close range to see the colors. Also, many species include sounds during their displays (e.g., icterids) making it difficult to know how much the colors and motion are important to receivers. Below, we focus on male courtship displays in one group of birds that routinely display while perched at a fixed site, making close-up videography feasible, and are usually silent once females settle nearby. These examples thus show the various ways that color and motion can be combined into a purely visual signal.

Most species in this group of New Guinea and northern Australian birds are promiscuous and with no paternal care. The resulting sexual selection has favored unusually colorful and structurally complex male plumage. Males show off their coloration with stylized courtship movements. Most species call loudly to attract females from afar, but then switch to silent displays when females are close by. Each male prepares a display site by clearing all leaves from a viewing perch for females near to their cleared display saplings or ground court. In the examples below, note how the coloration patterns and movements work together to make the display highly conspicuous:

• Parotia: Species in this genus all display on cleared ground courts in deep forest and usually provide a cleared overhead branch for watching females. Males have a number of wire-like plumes ending in small knobs on their heads. Their display involves a silent ballerina-like dance with head wagging that is exaggerated by the head wires, at the end of which the male pauses briefly, and then rapidly flashes his iridescent throat patch by moving it from side to side.
• Seleucidis melanoleucus: The 12-wired bird of paradise lives in lowland coastal swamps. Males usually display in a spiral around a cleared vertical branch.
(http://macaulaylibrary.org/video/462682)
• Cicinnurus: The two species of this genus listed below both display on vertical saplings or low horizontal branches above a cleared ground court. They first call loudly to attract females, but are relatively silent during displays when females are close by:
• Lophorina superba: The superb bird of paradise lives halfway up the sides of steep mountains in primary forest. When not displaying, males are largely black with little visible color. When displaying, they erect an iridescent turquoise chest shield and surround it with a large black oval generated by their wings and neck cape. Two light-colored iridescent spots are also present in the center of the combined plumage. The male then dances in front of the female. This is such an amazing display that we provide two different clips showing it:
• Ptiloris magnificus: Male magnificent riflebirds have a hidden iridescent turquoise throat patch that can be expanded into a moderately sized chest shield. As in Lophorina, they then surround this with a black oval, here formed by the wings while they dance before the female. During the dance, the male points his beak upwards and tosses his head from side to side to change the iridescent patch viewing angle and make it flash. Females solicit dancing in males by approaching with their heads also pointed up. Two clips of this display:
• Astrapia: Most members of this genus have green or blue-green iridescent throat and crown patches. This close-up of a male A. mayeri shows how such iridescent plumage can change color depending upon the viewing angle.
(http://macaulaylibrary.org/video/465313)
• Paradisaea: Males of this genus tend to display high in treetops and they may form leks with multiple males displaying very closely together. Most have highly elaborate and plumose tails they show off by bending forwards on the branch while extending the flapping wings.
• P. apoda: This clip of the greater bird of paradise shows multiple males on a lek displaying to a visiting female. The display involves a fluffing of the brightly colored tail plumage and softly beating wings that jiggles the tail plumes. Here, the female starts to favor one male who shifts to a second type of display eventually leading to female solicitation and copulation.
(http://macaulaylibrary.org/video/465338)
• P. minor: This clip shows a male lesser bird of paradise calling and then performing several bowing displays that again show off its elaborate tail plumage.
(http://macaulaylibrary.org/video/455265)
• P. rubra: The red bird of paradise lacks the highly plumose tail of the above species, but instead has two very long wiry feathers extending from the tail. The forward bow has largely been replaced by continuous wing waving and side-to-side movements that flip the tail wires back and forth. This sequence also ends with a copulation.
(http://macaulaylibrary.org/video/464532)
• P. rudolphi: Male blue birds of paradise have taken the bow display to its limit: they hang upside down when displaying. As they rotate forward on a perch, they spread out their plumose tails, expand a dark chest shield, and while hanging by their feet, vibrate their whole bodies.