**2. Tyndall diffusion**

Tyndall scattering by relatively distant particles in the range 40 nm – 900 nm generally produces a bluish diffuse coloration.

### **2.1 Theoretical background**

The elastic scattering of light by isolated particles is an important chapter of electrodynamics. The physical mechanism of light diffusion is simple: the electric field which accompanies an incident light beam penetrates and disturbs the polarizable material in the scatterer, which responds by charge oscillations. The sustained acceleration of these oscillating charges produces a reemission of light, at the same frequency, but its directional distribution is much wider than in the incident light. The distribution of the scattered light essentially depends on the polarizability of the material at the incident frequency, but also on the shape of the particle – in just the same way as the relative location of antennas influences the emission direction of radio waves.

An important parameter, for classifying the scattering mechanisms is the ratio between the scatterer's size *r* to the wavelength , *x r* 2 . For 1 *x* , we encounter a mechanism of diffusion called "Rayleigh scattering". This type of light redirection leads to the following distribution of intensities:

$$I = I\_0 \frac{1 + \cos^2 \theta}{2R^2} \left(\frac{2\pi}{\lambda}\right)^4 \left(\frac{n^2 - 1}{n^2 + 2}\right)^2 \left(\frac{d}{2}\right)^6 \tag{1}$$

in which we can assume a total translational invariance. One-dimensional structures are only inhomogeneous in one dimension, with, perpendicularly, complete invariance for two independent translations. These one-dimensional structures are then described as "layered". Thin films, thin film stacks and Bragg mirrors (with the repetition of identical layers) are examples of one-dimensional structures. Two-dimensional structures are totally invariant under a single direction. A straight optic fiber is a two-dimensional structure. A periodic array of parallel fibers, such as the bunch of cilia in ctenophores or the aligned melanin rods in peacock or other bird's feathers, is also a two-dimensional, as well as gratings engraved on flat surfaces. In three-dimensional structures, no direction shows total invariance under

When inhomogeneous, the refractive index can be periodic, in which case the propagation acquires special features that will be examined later. A one-dimensional periodic structure is the basis for a Bragg mirror that produces well-defined reflection bands around specific frequencies. Obtaining blue colors with such a system is relatively tricky and, as will be discussed in this chapter, requires particularly thin layers in order to avoid producing a metameric purple color. Blue two-dimensional photonic crystal also requires special scatterer's spacing and specific conditions: the blue coloration of the wing feathers in the magpies is a very instructive example. Somewhat more complex, when neglecting cross-ribs, microribs and lamellae slant, the *Morpho rhetenor* ribs structure is another example of a twodimensional photonic structure that produces a vivid blue under most directions. Gratings, as found in butterflies can also produce blue iridescence for specific grating periods. Finally,

blue three-dimensional photonic crystals are observed in weevils and longhorns.

Tyndall scattering by relatively distant particles in the range 40 nm – 900 nm generally

The elastic scattering of light by isolated particles is an important chapter of electrodynamics. The physical mechanism of light diffusion is simple: the electric field which accompanies an incident light beam penetrates and disturbs the polarizable material in the scatterer, which responds by charge oscillations. The sustained acceleration of these oscillating charges produces a reemission of light, at the same frequency, but its directional distribution is much wider than in the incident light. The distribution of the scattered light essentially depends on the polarizability of the material at the incident frequency, but also on the shape of the particle – in just the same way as the relative location of antennas

An important parameter, for classifying the scattering mechanisms is the ratio between the

diffusion called "Rayleigh scattering". This type of light redirection leads to the following

<sup>2</sup> 2 2 4 6 1 cos 2 1

 

<sup>0</sup> 2 2 <sup>2</sup> 2 2 *n d I I R n*

  . For 1 *x* , we encounter a mechanism of

(1)

 , *x r* 2 

**2. Tyndall diffusion** 

produces a bluish diffuse coloration.

influences the emission direction of radio waves.

scatterer's size *r* to the wavelength

distribution of intensities:

**2.1 Theoretical background** 

translations. This is the most general geometry for a photonic structure.

where *I* is the intensity scattered at an angle from the incident direction, *R* is the distance from the particle's center, *n* is the refractive index of the particle and *d* its diameter. Typically, for visible light, the diameter of the scatterer should be smaller than about 50 nm to warrant a good quantitative accuracy of the scattering. This expression was probably first derived by John William Strutt (third Baron Rayleigh), based on dimensional ("similitude") arguments (Hoeppe, 1969). Assuming that the scattering is proportional to the number of atoms in the particle – which is the atom concentration times the volume *V* , and inversely proportional to the distance *R* between the particle center and the detector used for measurement, the ratio between the scattered amplitude *A* and the incident amplitude *<sup>A</sup>*0 can be expressed as

$$\frac{A}{A\_0} \text{ or } \frac{V}{R} \mathcal{A}^x \mathcal{C}^y \text{ .} \tag{2}$$

which should be a dimensionless quantity. The light speed *c* and the wavelength should also enter the formula because this is an optical phenomenon, but we do not yet know the exponents *x* and *y* . The right-hand side of the equation, in terms of time [ ] *T* and length[ ] *L* , has dimensions <sup>2</sup> [] [] *<sup>x</sup> y y L T* . For this to become dimensionless, we must have *y* 0 and *x* 2 . This means (as the volume *V* is proportional to the cube of the particle diameter *d* )

$$\frac{I}{I\_0} \propto \left(\frac{A}{A\_0}\right)^2 \propto \frac{d^6}{R^2 \lambda^4} \,. \tag{3}$$

Much of the physics of the Rayleigh scattering is already present in this result, based on this simple reasoning. In particular, the essential point is the so-called "inverse fourth power law", stating that the scattered intensity is inversely proportional to the fourth power of the wavelength. This means that the short wavelengths in the visible white light (violet-blue) are scattered much more efficiently than the long wavelengths (orange-red). The sunlight scattered by small particles appears essentially blue because the solar spectrum contains less violet than blue and because we are less sensitive to violet than to blue.

The Irish physicist John Tyndall contributed to this question as early as in 1860. He noticed the appearance of blue scattering by a vapor of hydrochloric acid, as the particles condensed into larger size droplets and its desaturation, reaching white color, when the particles became too large. Indeed the blue Rayleigh scattering is reinforced as the volume of the scattering center is increased, and continues to do so until the particle becomes larger than the illuminating wavelength. Then, standing waves and resonances start affecting the wavelength dependence of the scattering, giving rise to a much more complex scatter color. Typically, the range of particle sizes that produce a strong blue scattering is between 50 nm to 900 nm and, for this range, where the characteristics of Rayleigh scattering are still qualitatively useful, the scattering is usually called "Tyndall scattering". For spherical particles of small, medium or large sizes, a general treatment exists: Mie scattering (Mie, 1908).

This is not quite the end, as Rayleigh, Tyndall and Mie scattering only describe the scattered intensity by a single isolated particle. When considering aggregated particles,

How Nature Produces Blue Color 7

It is interesting to mention that, in the twenties, Mason attributed all the non-iridescent blue colorations seen in bird feathers to Tyndall scattering but was aware that, in some insects, such coloration could arise from other phenomena (Mason 1923, Mason 1927). As new experimental and imaging techniques developed, new insights showed that blue in bird feathers could also be produced by constructive interference of light waves. Interfaces between keratin and air in the spongy medullar layer of the barbs act as coherent scatterers in that case (Prum et al., 1998; Prum et al., 1999). Blue Tyndall skins also appear in birds. For example, the extinct dodo head skin was found to be showing a diffuse blue color. This skin reveals randomly arranged, fine particles, about the size of the blue light wavelength

Fig. 1. The male dragonfly *Orthetrum caledonicum* (Libellulidae). The blue coloration of the body comes from Tyndall scattering in a waxy layer over the black cuticle (Parker, 2000).

Scattered blues have early been assigned to insects. The scattering occurs in the epidermal cells beneath a transparent cuticle. In the odonate order such as aeschnids, agrionids and libelluloids (*Libellula Pulchella*, *Mesothemis Simplicicollis*, *Enallagma Cyathigerum*, *Aeshnea cyanea*, *Anax walsinghami*) the bright blue diffuse coloration on their body or wings (Mason, 1926; Parker, 2000; Parker, 2005; Veron, 1973) originates from scattering centers under the cuticle. Dragonflies (Mason, 1926) and some other adult insects can also develop a waxy bloom on the surface of their cuticle. The Tyndall effect is then produced by this waxy material and coloration can be destroyed by washing it with a wax solvent (Parker 2000): see

Some butterflies have also been thought to be colored by this mechanism, such as *Papillio zalmoxis* or lycaenids (Huxley, 1976; Berthier, 2006). However, recent research shows that *coherent* interferences could also explain the various observed colors in these butterflies (Wilts et al., 2008; Prum et al., 2006). Tyndall blue has also been recorded in the cuticle of the

(reproduced from GNU free documentation)

(Parker, 2005).

Fig. 1.

things also become more complicated. When the distance between the particles is much larger than the coherence length of the illuminating light, the collective scattering is incoherent, which means that the diffused intensity is merely the sum of the intensities diffused by each scattering center. The incoherent scattering by "Rayleigh particles (in the range of diameters smaller than 50 nm)" and by "Tyndall particles" (in the range 50-900 nm) can still be considered as mechanisms of "Rayleigh" or "Tyndall" scattering, respectively. If the particles are closer, we encounter a case of coherent scattering and we must add vector amplitudes with respective phases rather than intensities. The multiple scattering on nearby particles provides further opportunities for standing waves and resonances and we again lose the inverse fourth power law. The intensity and scatter directions then depend on the spatial distribution of the particles and in particular, the average distance between them.
