**4. One-dimensional photonic structures**

One-dimensional, planar or curved, photonic structures are frequent in nature. Insects, in particular, have frequently evolved this kind of structure for the purpose of coloration, as part of signaling or camouflage strategies. The reason may be that the process of fabrication of the outer part of a cuticle by epidermal cells, layer by layer, is compatible with the formation of such structures, even if we cannot claim at the moment that these mechanisms have been understood in all details.

Many different cases of one-dimensional photonic crystal have been seen in animalia, maybe because this is the most direct way to produce a metallic and/or iridescent color and, in this way, improve specific intra or interspecific functions. We will essentially examine two cases of physical designs: the single layer film and the Bragg mirrors.

The multilayer is the most common type of iridescent structure found in beetles and is also very common in butterflies (Kinoshita et al., 2008; Noyes et al., 2007; Parker et al., 1998). In many cases, these multilayers are composed of alternating layers of chitin and air partially filled with a chitinous compound. This produces a high/low index bilayer. Constructive interferences between light reflected by different layers produce one or several colors. The dominant reflected wavelength can be determined by the thicknesses of the layers and the average refractive index (see formula 11). The wavelengths that are not reflected are transmitted and the transmission spectrum is the exact complement of reflection if the system is considered non-absorbing. The color arising from a multilayer also varies with the angle of observation. As the reflection angle increases (starting from normal), the color shifts to lower wavelengths (blue shift).

The reflectors can be epicuticular (as in cicindelinae or in some chrysomelidae) (Kurachi et al., 2002), while others are endocuticular (Hinton, 1973).

Iridescence in bird feathers comes sometimes from 1D structure. They are located in the barbules like in satin bowerbirds *Ptilonorhynchus violaceus minor* that shows a violet to black iridescence coming from a single layer of keratin on the top of a layer of melanin (Doucet et al., 2006). In European starlings *Sturnus vulgaris*, multiple layers of keratin and melanin give a green-blue iridescence (Cuthill et al., 1999, Doucet et al., 2006).

#### **4.1 Thin films**

The single self-supported thin film and the optical overlayer covering a substrate have been known since a long time, in the planar and some other geometry. Constructive and

How Nature Produces Blue Color 13

Note that, if the refractive index *n*3 is larger than *n*2 (itself larger than *n*<sup>1</sup> ), the addition of 1 2 at the denominator must be skipped. The dependence of this dominant reflected

which the surface is viewed, is the phenomenon of iridescence, which often signals a

A typical blue color is perceived for a dominant reflected wavelength of 475 nm. Take, specifically, a thin film of thickness 200 nm and refractive index 1.5. The branch 0 *m* reflects infrared radiation, from 1470 nm under normal incidence to 1150 nm under grazing angles. The 1 *m* order extends in the visible, from 490 to 385 nm: providing shortwavelength blue coloration. The 2 *m* and higher orders are deeper in the ultraviolet.

A natural example comes from the study of the iridescent wing of a giant tropical wasp, *Megascolia procer javanensis* (Sarrazin et al., 2008). In this particular case, the wing is shown to be made of rigid structure of melanized chitin, except for an overlayer, on each side of the wing. The overlayer can be shown to act as a transparent interference thin film with a thickness of 286 nm. The refractive index of the material in this layer is not precisely known, so that its analysis requires examining the reflectance spectrum in detail, for various angles of incidence. The substrate supporting this layer is better known, as a solid mix of chitin and melanin. This mix was studied by de Albuquerque (de Albuquerque et al., 2006), including the dispersion related to melanin absorption. This absorption is strong here, as can be seen from the opacity of the wing. An adjustment of the refractive index of the overlayer allows the reflectance spectra to be fitted quite nicely at all incidence angles, and provides a value 1.76 *n* , which is very reasonable. The iridescence is weak: from bluish green near-

The stacking of multiple planar layers (or "multilayers structure") is another type of structure that produces selective reflection. Among these, one important class is the periodic stack, where a group of two layers is repeated a finite number of times. This structure is also known as a Bragg mirror. When the number of periods is large (but, in practice, it does not need to be in excess of, say, 3 or 4 in the kind of structures examined here), the optical response can be approached by assuming an infinite number of periods, which can be called a one-dimensional photonic crystal. In this case, it is not very difficult to

In this limit, the incident frequency is conserved in all scattered waves, but not the wave vector in the stacking direction. A wave with wave number *<sup>z</sup> k* will change this wave number

*z z kkm*

This means that, in a periodic multilayer stack, waves propagate normally, unless their wave vector *<sup>z</sup> k* obeys the above relation. For waves at the conserved incident frequency

2

*a*  (7)

, where *m* is an

,

so that its output value is a choice of any of the quantities *km a <sup>z</sup>* 2

, which means a change of color with the angle under

normal incidence to greenish blue under a grazing incidence.

predict the dominant color that will be reflected.

integer and *a* is the period thickness:

wavelength on the incidence angle

structural color.

**4.2 Bragg mirrors** 

this implies that

destructive interference of light waves in thin films (soap bubbles or oil films on water) show colorful patterns. The interference occurs between light waves reflecting off the top surface of a film with the waves multiply reflected from the bottom surface. In order to obtain a nice colored pattern, the thickness of the film has to be on the order of the wavelength of the incident light.

Fig. 5. Interference in a planar homogeneous slab. The phase shift between one reflected wave and its successor determine the intensity, for a given incidence angle and wavelength.

A single thin film illuminated from air reflects light at a wavelength as a Fabry-Pérot etalon. For a given thickness *d* of the film and incidence angle , the phase delay between the two first successively emerging rays from the multiply reflected beams is (we assume, for instance, a dense slab 21 23 *nn nn* and and we name the angle of refraction inside the film: 2 1 *n tn* sin sin )

$$\Delta\phi = \left\{ n\_2 \frac{2\pi}{\lambda} \left( \frac{2d}{\cos t} \right) \right\} - \left\{ \left[ \pi \right] + n\_1 \frac{2\pi}{\lambda} (2d \tan t \sin \theta) \right\} \tag{4}$$

The first term is the phase change of the transmitted electric field wave travelling one round trip in the film before its next exit, while the last term is the progress of the reflected beam in air before joining back the other path wave. The "optional" half-wavelength phase delay (which can be written [ ] or[ ] , without consequences) occurs only when an electric wave reflects on a medium with higher refractive index. All other phase delays between successive emerging rays are the same. Maximal reflections occur when all the exiting beams are in phase, which means *m*2 , where *m* is an integer. This condition allows determining, under a specific incidence angle, the reinforced wavelengths, which turn out to be

$$\mathcal{A} = \frac{2n\_2 d}{m + \left[\frac{1}{2}\right]} \text{cost} \tag{5}$$

Or, equivalently,

$$\lambda = \frac{2d\sqrt{n\_2^2 - n\_1^2 \sin^2 \theta}}{m + \left[\frac{1}{2}\right]} \tag{6}$$

destructive interference of light waves in thin films (soap bubbles or oil films on water) show colorful patterns. The interference occurs between light waves reflecting off the top surface of a film with the waves multiply reflected from the bottom surface. In order to obtain a nice colored pattern, the thickness of the film has to be on the order of the

Fig. 5. Interference in a planar homogeneous slab. The phase shift between one reflected wave and its successor determine the intensity, for a given incidence angle and wavelength.

the two first successively emerging rays from the multiply reflected beams is (we assume, for instance, a dense slab 21 23 *nn nn* and and we name the angle of refraction inside the

The first term is the phase change of the transmitted electric field wave travelling one round trip in the film before its next exit, while the last term is the progress of the reflected beam in air before joining back the other path wave. The "optional" half-wavelength phase

electric wave reflects on a medium with higher refractive index. All other phase delays between successive emerging rays are the same. Maximal reflections occur when all the

condition allows determining, under a specific incidence angle, the reinforced wavelengths,

2 1 2

cos *n d <sup>t</sup>*

222 2 1 2 sin 1 2

    2

*m*

 

> *dn n m*

 *m*2

 

<sup>2</sup> <sup>1</sup> <sup>2</sup> 2 tan 2 2

*n dt* sin *<sup>d</sup>*

, the phase delay between

, without consequences) occurs only when an

(5)

, where *m* is an integer. This

(6)

as a Fabry-Pérot

(4)

A single thin film illuminated from air reflects light at a wavelength

cos

*t*

 or[ ] 

etalon. For a given thickness *d* of the film and incidence angle

*n*

exiting beams are in phase, which means

wavelength of the incident light.

film: 2 1 *n tn* sin sin

which turn out to be

Or, equivalently,

)

delay (which can be written [ ]

Note that, if the refractive index *n*3 is larger than *n*2 (itself larger than *n*<sup>1</sup> ), the addition of 1 2 at the denominator must be skipped. The dependence of this dominant reflected wavelength on the incidence angle , which means a change of color with the angle under which the surface is viewed, is the phenomenon of iridescence, which often signals a structural color.

A typical blue color is perceived for a dominant reflected wavelength of 475 nm. Take, specifically, a thin film of thickness 200 nm and refractive index 1.5. The branch 0 *m* reflects infrared radiation, from 1470 nm under normal incidence to 1150 nm under grazing angles. The 1 *m* order extends in the visible, from 490 to 385 nm: providing shortwavelength blue coloration. The 2 *m* and higher orders are deeper in the ultraviolet.

A natural example comes from the study of the iridescent wing of a giant tropical wasp, *Megascolia procer javanensis* (Sarrazin et al., 2008). In this particular case, the wing is shown to be made of rigid structure of melanized chitin, except for an overlayer, on each side of the wing. The overlayer can be shown to act as a transparent interference thin film with a thickness of 286 nm. The refractive index of the material in this layer is not precisely known, so that its analysis requires examining the reflectance spectrum in detail, for various angles of incidence. The substrate supporting this layer is better known, as a solid mix of chitin and melanin. This mix was studied by de Albuquerque (de Albuquerque et al., 2006), including the dispersion related to melanin absorption. This absorption is strong here, as can be seen from the opacity of the wing. An adjustment of the refractive index of the overlayer allows the reflectance spectra to be fitted quite nicely at all incidence angles, and provides a value 1.76 *n* , which is very reasonable. The iridescence is weak: from bluish green nearnormal incidence to greenish blue under a grazing incidence.

#### **4.2 Bragg mirrors**

The stacking of multiple planar layers (or "multilayers structure") is another type of structure that produces selective reflection. Among these, one important class is the periodic stack, where a group of two layers is repeated a finite number of times. This structure is also known as a Bragg mirror. When the number of periods is large (but, in practice, it does not need to be in excess of, say, 3 or 4 in the kind of structures examined here), the optical response can be approached by assuming an infinite number of periods, which can be called a one-dimensional photonic crystal. In this case, it is not very difficult to predict the dominant color that will be reflected.

In this limit, the incident frequency is conserved in all scattered waves, but not the wave vector in the stacking direction. A wave with wave number *<sup>z</sup> k* will change this wave number so that its output value is a choice of any of the quantities *km a <sup>z</sup>* 2 , where *m* is an integer and *a* is the period thickness:

$$k\_z' = k\_z + m\frac{2\pi}{a} \tag{7}$$

This means that, in a periodic multilayer stack, waves propagate normally, unless their wave vector *<sup>z</sup> k* obeys the above relation. For waves at the conserved incident frequency , this implies that

How Nature Produces Blue Color 15

*Hoplia coerulea*, has evolved a cuticle bearing scales Fig. 6 (left). The inner region of these scales is structured to filter out a spectacular blue-violet iridescence on reflection (Vigneron et al., 2005). The cuticle, as seen in scanning electron microscopy is shown in Fig. 6 (right). The scales are attached by a single peripheral point to the underlying cuticle. These scales

The structure in each scale can be interpreted as a stack of some 20 sheets, roughly parallel to the cuticle. Each sheet is actually composed of a very thin plate of bulk chitin, bearing, on one side, a network of parallel rods with a rectangular section. The lateral corrugation associated with the rods has a period of 170 nm, just too small to produce the diffraction of light in the visible range. This acts as a zero-order grating: for visible wavelengths, the rods array appears to be a homogeneous layer, and the concept of an average refractive index is adequate. The average refractive index of the whole structure was evaluated to *n* 1.4 for unpolarized light near normal incidence. As the vertical period turns out to be 120 nm + 40 nm =160 nm, it fulfills perfectly the conditions described above for the production of weakly iridescent blue. The *Hoplia coerulea* structure gives some iridescence, ranging from blue to violet, and effectively behaves as a flat multilayer structure, in spite of the lateral structuring of the rods layers. This structure was recently shown to have an optical response modifiable in presence of humidity, because water can infiltrate the voids. Strangely, the structure's

Under a crude approximation, the structure carried by the ridges of *Morpho* butterflies (for instance *M. menelaus*) can be viewed as a stack of alternating chitin and air layers, with a period of the order of 180 nm and an average refractive index well under 1.4 (Berthier et al., 2003, Berthier et al., 2006). This can explain the normal-incidence bright blue coloration and the shift of the reflected wavelength to the violet as the angle of incidence increases. This simple model has limits: it is unable to explain the off-specular variation of the scattering and the polarization effects observed in the directional reflectance pattern (Berthier, 2010). Plants can also produce coloring multilayers for displaying a blue coloration. Examples can be found in the genus *Selaginella*, for example *S. willdenowii* and *S. uncinata* (Lee, 1997). These plants live in the understory of Central- and South-American rainforests and, strangely,

Fig. 6. *Hoplia coerulea* (right). The beetle's cuticle is covered by scales. The scales take the shape of a disk, with a diameter of about 50 µm and a thickness of 3.5 µm. These scales render a blue or violet color. The scanning electron microscope images (right) shows the

are easily removed by breaking this binding.

coloring structure inside the scales (with permission).

materials turn out to be hydrophilic (Rassart et al., 2009).

$$k\_z = \pm \frac{\pi}{a}, \pm 2\frac{\pi}{a}, \pm 3\frac{\pi}{a}... \tag{8}$$

This corresponds to wave number values that match the so-called Brillouin-zone boundaries. For low contrasts of refractive indexes, defining an average refractive index *n* for the whole structure is a good starting point, taking the perturbation point of view. In this average material the following dispersion relation holds,

$$
\rho = \sqrt{k\_y^2 + k\_z^2} \frac{c}{\overline{n}} \tag{9}
$$

where *c* is the light velocity in vacuum and *<sup>y</sup> k* the wave vector component parallel to the layers. This quantity is conserved across the interfaces, so that anywhere in the structure,

$$k\_y = \frac{\alpha}{c} \sin \theta\_i \prime \tag{10}$$

where *<sup>i</sup>* refers to the incidence angle in the incidence medium (with refractive index assumed to be 1). At the zone boundaries, the *<sup>z</sup> k* and *<sup>z</sup> k* modes are degenerate, but with the appearance of the refractive index contrasts, the degenerescence is lifted because of the formation of standing waves that adopt different configurations relative to high and low refractive-index regions. Then, a gap (forbidden frequencies) appears when the unperturbed dispersion curve crosses the zone boundaries, and this occurs for incident wavelengths in narrow bands centered on ( *m* integers) (Vigneron et al., 2006)

$$\mathcal{Z} = \frac{2a\sqrt{\overline{n}^2 - \sin^2 \theta\_i}}{m} \tag{11}$$

An incident light wave with a frequency in these ranges, impinging on the surface of this semi-infinite photonic crystal, will be totally reflected. We note that, on a photonic crystal surface, total reflection can occur under normal incidence, and also when air is the incidence medium, contrasting our usual knowledge of total reflection conditions.

In order to produce blue (say 480 nm) under normal incidence and violet under grazing incidence (say 350 nm - many living organisms have UV vision), this equation fixes the refractive index average (1.46) and the period *a* (162 nm). This, however, still leaves ample flexibility in choosing the actual bilayer that defines a period. These values, calculated for the first gap ( 1 *m* ), are easily produced with biopolymers such as chitin or keratin, with refractive indexes close to 1.56.

It is also possible to produce long-wavelength blue (say 480 nm) under normal incidence with the second gap 2 *m* because the fundamental reflection ( 1 *m* ) would not appear in the visible, but in the infrared, near 960 nm. However, under larger incidences, iridescence may bring in red contribution to the spectrum, turning violet into some extraspectral metameric color in the purple range.

A good example of this structure is provided by the "blue beetle" *Hoplia coerulea*. This structure could have been classified as a two-dimensional photonic crystal, to be described later, but the optical response is in fact close to that expected for a Bragg mirror, for reasons that will become clear in a moment.

, 2 , 3 ... *<sup>z</sup> k aaa*

This corresponds to wave number values that match the so-called Brillouin-zone boundaries. For low contrasts of refractive indexes, defining an average refractive index *n* for the whole structure is a good starting point, taking the perturbation point of view. In this

2 2 *y z <sup>c</sup> k k n*

where *c* is the light velocity in vacuum and *<sup>y</sup> k* the wave vector component parallel to the layers. This quantity is conserved across the interfaces, so that anywhere in the structure,

> *<sup>y</sup>* sin *<sup>i</sup> k c*

assumed to be 1). At the zone boundaries, the *<sup>z</sup> k* and *<sup>z</sup> k* modes are degenerate, but with the appearance of the refractive index contrasts, the degenerescence is lifted because of the formation of standing waves that adopt different configurations relative to high and low refractive-index regions. Then, a gap (forbidden frequencies) appears when the unperturbed dispersion curve crosses the zone boundaries, and this occurs for incident wavelengths in

> 2 2 2 sin *<sup>i</sup> a n m*

An incident light wave with a frequency in these ranges, impinging on the surface of this semi-infinite photonic crystal, will be totally reflected. We note that, on a photonic crystal surface, total reflection can occur under normal incidence, and also when air is the incidence

In order to produce blue (say 480 nm) under normal incidence and violet under grazing incidence (say 350 nm - many living organisms have UV vision), this equation fixes the refractive index average (1.46) and the period *a* (162 nm). This, however, still leaves ample flexibility in choosing the actual bilayer that defines a period. These values, calculated for the first gap ( 1 *m* ), are easily produced with biopolymers such as chitin or keratin, with

It is also possible to produce long-wavelength blue (say 480 nm) under normal incidence with the second gap 2 *m* because the fundamental reflection ( 1 *m* ) would not appear in the visible, but in the infrared, near 960 nm. However, under larger incidences, iridescence may bring in red contribution to the spectrum, turning violet into some extraspectral

A good example of this structure is provided by the "blue beetle" *Hoplia coerulea*. This structure could have been classified as a two-dimensional photonic crystal, to be described later, but the optical response is in fact close to that expected for a Bragg mirror, for reasons

*<sup>i</sup>* refers to the incidence angle in the incidence medium (with refractive index

 

(8)

(9)

(11)

, (10)

average material the following dispersion relation holds,

narrow bands centered on ( *m* integers) (Vigneron et al., 2006)

medium, contrasting our usual knowledge of total reflection conditions.

where

refractive indexes close to 1.56.

metameric color in the purple range.

that will become clear in a moment.

*Hoplia coerulea*, has evolved a cuticle bearing scales Fig. 6 (left). The inner region of these scales is structured to filter out a spectacular blue-violet iridescence on reflection (Vigneron et al., 2005). The cuticle, as seen in scanning electron microscopy is shown in Fig. 6 (right). The scales are attached by a single peripheral point to the underlying cuticle. These scales are easily removed by breaking this binding.

Fig. 6. *Hoplia coerulea* (right). The beetle's cuticle is covered by scales. The scales take the shape of a disk, with a diameter of about 50 µm and a thickness of 3.5 µm. These scales render a blue or violet color. The scanning electron microscope images (right) shows the coloring structure inside the scales (with permission).

The structure in each scale can be interpreted as a stack of some 20 sheets, roughly parallel to the cuticle. Each sheet is actually composed of a very thin plate of bulk chitin, bearing, on one side, a network of parallel rods with a rectangular section. The lateral corrugation associated with the rods has a period of 170 nm, just too small to produce the diffraction of light in the visible range. This acts as a zero-order grating: for visible wavelengths, the rods array appears to be a homogeneous layer, and the concept of an average refractive index is adequate. The average refractive index of the whole structure was evaluated to *n* 1.4 for unpolarized light near normal incidence. As the vertical period turns out to be 120 nm + 40 nm =160 nm, it fulfills perfectly the conditions described above for the production of weakly iridescent blue. The *Hoplia coerulea* structure gives some iridescence, ranging from blue to violet, and effectively behaves as a flat multilayer structure, in spite of the lateral structuring of the rods layers. This structure was recently shown to have an optical response modifiable in presence of humidity, because water can infiltrate the voids. Strangely, the structure's materials turn out to be hydrophilic (Rassart et al., 2009).

Under a crude approximation, the structure carried by the ridges of *Morpho* butterflies (for instance *M. menelaus*) can be viewed as a stack of alternating chitin and air layers, with a period of the order of 180 nm and an average refractive index well under 1.4 (Berthier et al., 2003, Berthier et al., 2006). This can explain the normal-incidence bright blue coloration and the shift of the reflected wavelength to the violet as the angle of incidence increases. This simple model has limits: it is unable to explain the off-specular variation of the scattering and the polarization effects observed in the directional reflectance pattern (Berthier, 2010).

Plants can also produce coloring multilayers for displaying a blue coloration. Examples can be found in the genus *Selaginella*, for example *S. willdenowii* and *S. uncinata* (Lee, 1997). These plants live in the understory of Central- and South-American rainforests and, strangely,

How Nature Produces Blue Color 17

sin sin 2 *m b c c*

sin sin *m*

 

The integer *m* is the diffraction order. The order zero is a reflection, with emergence and

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

> 2 *b*

. This means that we actually can build gratings that produce only

For a larger period, the same wavelength will be observed for a range of incidence angles

"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

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

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-

1 sin must be fulfilled. A given wavelength

incidence angles identical, with no dependence on the incidence wavelength

*b* 

 

(12)

(13)

(14)

90 ) when

starts appearing in the

. By contrast,

This implies the following relationship between the incidence and emergence angles

 

And, more explicitly,

 1 sin *m b* 

which contain 90

large illumination angles.

**5.2 Two-dimensional photonic crystals** 

under incidence angles larger than 33°.

 

order 1 *m* (it will then emerge for a grazing illumination

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.
