**5.1 Gratings**

A grating is usually a superficial structure, periodic in a single direction (say *x* ), with a period *b* . The characteristic rule which describes light scattering by a grating arises from the observation that an incident wave with wave vector *<sup>x</sup> k* which travels through a medium with periodicity *b* comes out with a series of possible wave vectors *kkm b x x* 2 , where *m* is a negative, null or positive integer.

Fig. 7. Geometry of a grating, for an incident beam falling at right angle with the parallel lines.

This implies the following relationship between the incidence and emergence angles

$$\frac{\partial}{\partial t}\sin\phi = \frac{\partial}{\partial t}\sin\theta + m\left(2\pi/b\right) \tag{12}$$

And, more explicitly,

16 Photonic Crystals – Introduction, Applications and Theory

display blue on freshly grown shadowed leaves. The blue coloration arises from a onedimensional multilayer in the moistened cellulose of outer cell walls. The refractive index of moistened cellulose is, in the average, 1.45 and the multilayer found has a period of 160 nm (two layers of different refractive indexes and equal thicknesses, 80 nm). We again find the exact conditions to provide a blue coloration with a one-dimensional photonic crystal.

In two-dimensional photonic structures, one only observes a total translational invariance in one dimension, the other two dimensions being structured by refractive index inhomogeneities. We will include gratings in this category, besides fibrous photonic crystals,

Actually, fibrous photonic crystals can be viewed as a combination of a grating and a multilayer. From the symmetry point of view, we can view a 2D photonic crystal as totally invariant in the "fibers" directions and periodic in two directions perpendicular to the fibers. Defining the surface parallel to the fibers, these two directions are adequately defined as the surface plane and the direction of the normal. The periodicity parallel to the surface produces diffraction similar to that produced by a grating, while the periodicity along the normal, deep under the surface, produces a color selection with a Bragg mirror. Being a combination of both, a 2D photonic crystal tends to be more flexible than either a grating or a Bragg mirror to produce a color such as blue. As explained below, a short-period grating will cease diffracting as its associated lateral inhomogeneity is smaller than the shortest visible wavelength. For this wavelength and larger, the grating will act as a homogeneous average material that can only generates a specular reflection. However, with a 2D photonic crystal, the "normal" periodicity can still be there to produce color selection. If, on the other extreme, the normal periodicity is constrained by weak refractive index contrasts or a tight film thickness, the Bragg mirror will not be effective, but the lateral grating can take over and still manages to produce blue. Additionally, intermediate – less easily described – mechanisms involving simultaneously cooperating diffraction and interference color

A grating is usually a superficial structure, periodic in a single direction (say *x* ), with a period *b* . The characteristic rule which describes light scattering by a grating arises from the observation that an incident wave with wave vector *<sup>x</sup> k* which travels through a medium

Fig. 7. Geometry of a grating, for an incident beam falling at right angle with the parallel lines.

2

, where

with periodicity *b* comes out with a series of possible wave vectors *kkm b x x*

both of them being encountered in nature and able to select blue reflectance.

**5. Two-dimensional photonic structures** 

selection add new channels for producing blue.

*m* is a negative, null or positive integer.

**5.1 Gratings** 

$$
\sin \phi = \sin \theta + m \frac{\lambda}{b} \tag{13}
$$

The integer *m* is the diffraction order. The order zero is a reflection, with emergence and incidence angles identical, with no dependence on the incidence wavelength . By contrast, for 0 *m* , the emergence angle changes with the wavelength, which means that an incidence white beam is decomposed in a colored spectrum after being scattered by the grating. Several orders may be simultaneously present, but the actual number depends on the grating period. In order to produce an acceptable emergence angle, the condition 1 sin *m b* 1 sin must be fulfilled. A given wavelength starts appearing in the order 1 *m* (it will then emerge for a grazing illumination90 ) when

$$b = \frac{\lambda}{2} \tag{14}$$

For a larger period, the same wavelength will be observed for a range of incidence angles which contain 90 . This means that we actually can build gratings that produce only "blue" colors, i.e. wavelengths smaller than 490 nm (then including blue, purplish blue and violet), if the period is the rather precisely defined: 245 nm *b* . They must be illuminated under incidence angles larger than 33°.

An example of such a grating is provided by the array of flutes found on the ridges of the scales on the butterfly *Lamprolenis nitida* (Ingram, 2008). This butterfly is special because it is equipped with two types of gratings on the same scale. One, with a large period, produces a full decomposition of the visible white light, when illuminated from the front. All colors from red to green are shown, but in this configuration, blue light is scattered with a very low intensity. The grating responsible for this coloration is shown in Fig. 8, at the tip of the arrow C. The lamellae, repeated at 700 nm spacing, are slanted in such a way as to maximize the emission in the 1 *m* order, from red to green and to reduce the scattering in the 0 *m* order. This can be understood as a blazed grating and the lack of blue in this coloration is the result of the precise slant angle. However, slanted in the reverse direction, the so-called "flutes" are separated by about 235 nm, not far from the period *b nm* 245 mentioned above. The result is, as observed, a grating that produces only a purplish blue color, under large illumination angles.

#### **5.2 Two-dimensional photonic crystals**

Two-dimensional photonic crystals are fibers with two-dimensional periodic variations of the refractive index in the cross-section. In much the same way as with one-dimensional multilayers, the colored reflections originate from the formation of directional band gaps in the photonic band structure of these crystals. Producing blue alone from an ideal structure which fulfills these rules is difficult, because each stack of reticular plane in the two-

How Nature Produces Blue Color 19

Many (but not all) birds show structural coloration at the level of the feather's barbules. A feather is rigidified by a rachis, an array of barbs attached to the rachis, and an array of barbules attached to the barb. The barbules have the topology of a sack, with an envelope (a hard cortex) containing a medullar medium. As in the Peacock's feathers (Zi et al., 2003), the coloring structure on the blue feathers of the magpie lies on the barbule's cortex. It is constituted of elongated melanine cylinders, disposed parallel to each other with the symmetry of a two-dimensional triangular lattice. These cylinders are the scatterers that produce the coherent reflection in the blue. Strangely, the distance between these scatterers is 270 nm, much too large to explain the blue coloration. In fact, simulation shows that such a fibrous crystal should produce a fundamental gap in the near infrared, and a blue scattering as a "harmonic" of this gap. Indeed, a second band of forbidden propagation exists at higher frequency. At this frequency, the diffraction is less dispersive and the blue coloration produced is relatively saturated. The coloration hue could also be spoiled by the addition of red, arising from the line width of the fundamental reflection in the infrared, producing an extraspectral purple. This is not the case: the structured cortex is thin enough to avoid producing long-wavelengths resonances and the result is a dark blue color, easily

Three-dimensional photonic crystals are also encountered in nature, especially in butterflies, weevils and longhorns. In view the very high diversity of living organisms and the frequency of structural colors, it can be speculated that other families of insects will soon reveal a similar evolution. Three-dimensional photonic crystals are periodic in all three dimensions of space. Illuminated by white light in a well-defined direction, an ideal structure produces several colored beams, each of them corresponding to a stack of reticular planes with its appropriate spacing. A good example of the visual effect produced by an ideal photonic crystal is provided by the so-called Brazilian "diamond" weevil, which displays a green color when viewed from a distance, but under an optical microscope, shows individual scales with a variety of very saturated (pure) colors. Most other weevils and longhorns show less iridescence, and this is usually explained in term of orientation disorder: natural three-dimensional photonic crystals most often appear under the form of photonic polycrystals, with well-defined domains

Fig. 10. The internal structure of a scale on a blue area from the cuticle of the weevil *Eupholus schoenherri*. The structure, with a face-centered cubic symmetry, can be described as an

visible under a bright sunshine.

"opal" structure.

**6. Three-dimensional photonic crystals** 

bringing short-range order and long-range orientation disorder.

Fig. 8. The dorsal wings of male *Lamprolenis nitida* appear matt brown under incident light, normal to the wing surface, but shows various colorations under large incidences. These visual effects are due to the presence of two interspersed gratings on the scales of the rear wings. When illuminated in a postero-anterior direction and observed in backscatter, blue to violet is observed with increasing angle from the wing surface.

dimensional lattice can be considered as a Bragg mirror and the variety of values of stacking periods easily leads to the production of a wide range of colors. The coloration of fibrous organs in some marine animals, such as the Aphrodite sea mouse (Parker et al., 2001) or the Ctenophore *Beroë cucumis* (Welch et al., 2005; Welch et al., 2006), is accompanied with a broad iridescence, covering a spectral range from red to far in the ultraviolet. The high refractive index of water, compared to air, partly explains the iridescence richness, but in both cases, the structure can be viewed as a bunch of parallel fibers and a simple twodimensional photonic crystal. Nature, however has found unexpected ways to produce blue coloration from these fibrous structures and we will give here an account of the way a bird such as the magpie (*Pica pica*) (Vigneron et al., 2006a; Lee, 2010) produces the blue reflection on some of their wing feathers.

Fig. 9. The coloring structure in the tail (a) and wing (b) feather of the common magpie (*Pica pica*). The structure is formed by cylindrical melanine bars distributed to form a hexagonal two-dimensional lattice in the cortex of the barbules. The scattering centers are thin cylindrical cavities in the melanine granules and the color is related to the distance between these centers. The green color on the tail depends on the distance between the granules centers, 180 nm. Strangely, the blue photonic crystal has a larger lattice parameter (270 nm) : the coloration is controlled by a "second gap", at high frequency.

Fig. 8. The dorsal wings of male *Lamprolenis nitida* appear matt brown under incident light, normal to the wing surface, but shows various colorations under large incidences. These visual effects are due to the presence of two interspersed gratings on the scales of the rear wings. When illuminated in a postero-anterior direction and observed in backscatter, blue to

dimensional lattice can be considered as a Bragg mirror and the variety of values of stacking periods easily leads to the production of a wide range of colors. The coloration of fibrous organs in some marine animals, such as the Aphrodite sea mouse (Parker et al., 2001) or the Ctenophore *Beroë cucumis* (Welch et al., 2005; Welch et al., 2006), is accompanied with a broad iridescence, covering a spectral range from red to far in the ultraviolet. The high refractive index of water, compared to air, partly explains the iridescence richness, but in both cases, the structure can be viewed as a bunch of parallel fibers and a simple twodimensional photonic crystal. Nature, however has found unexpected ways to produce blue coloration from these fibrous structures and we will give here an account of the way a bird such as the magpie (*Pica pica*) (Vigneron et al., 2006a; Lee, 2010) produces the blue reflection

Fig. 9. The coloring structure in the tail (a) and wing (b) feather of the common magpie (*Pica pica*). The structure is formed by cylindrical melanine bars distributed to form a hexagonal two-dimensional lattice in the cortex of the barbules. The scattering centers are thin

cylindrical cavities in the melanine granules and the color is related to the distance between these centers. The green color on the tail depends on the distance between the granules centers, 180 nm. Strangely, the blue photonic crystal has a larger lattice parameter (270 nm) :

the coloration is controlled by a "second gap", at high frequency.

violet is observed with increasing angle from the wing surface.

on some of their wing feathers.

Many (but not all) birds show structural coloration at the level of the feather's barbules. A feather is rigidified by a rachis, an array of barbs attached to the rachis, and an array of barbules attached to the barb. The barbules have the topology of a sack, with an envelope (a hard cortex) containing a medullar medium. As in the Peacock's feathers (Zi et al., 2003), the coloring structure on the blue feathers of the magpie lies on the barbule's cortex. It is constituted of elongated melanine cylinders, disposed parallel to each other with the symmetry of a two-dimensional triangular lattice. These cylinders are the scatterers that produce the coherent reflection in the blue. Strangely, the distance between these scatterers is 270 nm, much too large to explain the blue coloration. In fact, simulation shows that such a fibrous crystal should produce a fundamental gap in the near infrared, and a blue scattering as a "harmonic" of this gap. Indeed, a second band of forbidden propagation exists at higher frequency. At this frequency, the diffraction is less dispersive and the blue coloration produced is relatively saturated. The coloration hue could also be spoiled by the addition of red, arising from the line width of the fundamental reflection in the infrared, producing an extraspectral purple. This is not the case: the structured cortex is thin enough to avoid producing long-wavelengths resonances and the result is a dark blue color, easily visible under a bright sunshine.
