Rodolfo H. Torres

Coherent light scattering by blue feather barbs

The structural colours of avian feather barbs are created by the scattering of light from the spongy matrix of keratin and air in the medullary layer of the barbs,. However, the precise physical mechanism for the production of these colours is still controversial,,,. Here we use a two-dimensional (2D) Fourier analysis of the spatial variation in refractive index of the blue feather barbs of the plum-throated cotinga (Cotinga maynana, Cotingidae) to show that the colour is produced by constructive interference between light waves scattered coherently by the nanostructured keratin-air matrix of the barbs.

Structural colouration of avian skin: convergent evolution of coherently scattering dermal collagen arrays

SUMMARY Structural colours of avian skin have long been hypothesized to be produced by incoherent (Rayleigh/Tyndall) scattering. We investigated the colour, anatomy, nanostructure and biophysics of structurally coloured skin, ramphotheca and podotheca from 31 species of birds from 17 families in 10 orders from across Aves. Integumentary structural colours of birds include ultraviolet, dark blue, light blue, green and yellow hues. The discrete peaks in reflectance spectra do not conform to the inverse fourth power relationship predicted by Rayleigh scattering. The dermis of structurally coloured skin consists of a thick (100–500 μm) layer of collagen that is usually underlain by a layer of melanin granules. Transmission electron micrographs (TEMs) of this colour-producing dermal collagen layer revealed quasi-ordered arrays of parallel collagen fibres. Two-dimensional (2-D) Fourier analysis of TEMs of the collagen arrays revealed a ring of peak spatial frequencies in the spatial variation in refractive index that are the appropriate size to make the observed ultraviolet–yellow colours by coherent scattering alone. One species, Philepitta castanea (Eurylaimidae), has exceptionally ordered, hexagonal arrays of collagen fibres that produce a hexagonal pattern of spatial frequency peaks in the power spectra. Ultraviolet, blue, green and yellow structural colours of avian skin are produced by coherent scattering (i.e. constructive interference) by arrays of collagen fibres in the dermis. Some yellow and orange skin colours are produced with a combination of structural and pigmentary mechanisms. These combined colours can have reflectance spectra with discrete peaks that are more saturated than hues produced by carotenoid pigments alone. Bluish facial skin from two species of Neotropical antbirds (Thamnophilidae) are nanostructurally too small to produce visible light by coherent scattering, and the colour production mechanism in these species remains unknown. Based on the phylogenetic distribution of structurally coloured skin in Aves, this mechanism of colour production has evolved convergently more than 50 independent times within extant birds.

Anatomically diverse butterfly scales all produce structural colours by coherent scattering

SUMMARY The structural colours of butterflies and moths (Lepidoptera) have been attributed to a diversity of physical mechanisms, including multilayer interference, diffraction, Bragg scattering, Tyndall scattering and Rayleigh scattering. We used fibre optic spectrophotometry, transmission electron microscopy (TEM) and 2D Fourier analysis to investigate the physical mechanisms of structural colour production in twelve lepidopteran species from four families, representing all of the previously proposed anatomical and optical classes of butterfly nanostructure. The 2D Fourier analyses of TEMs of colour producing butterfly scales document that all species are appropriately nanostructured to produce visible colours by coherent scattering, i.e. differential interference and reinforcement of scattered, visible wavelengths. Previously hypothesized to produce a blue colour by incoherent, Tyndall scattering, the scales of Papilio zalmoxis are not appropriately nanostructured for incoherent scattering. Rather, available data indicate that the blue of P. zalmoxis is a fluorescent pigmentary colour. Despite their nanoscale anatomical diversity, all structurally coloured butterfly scales share a single fundamental physical color production mechanism - coherent scattering. Recognition of this commonality provides a new perspective on how the nanostructure and optical properties of structurally coloured butterfly scales evolved and diversified among and within lepidopteran clades.

Two-dimensional Fourier analysis of the spongy medullary keratin of structurally coloured feather barbs

We conducted two–dimensional (2D) discrete Fourier analyses of the spatial variation in refractive index of the spongy medullary keratin from four different colours of structurally coloured feather barbs from three species of bird: the rose–faced lovebird, Agapornis roseicollis (Psittacidae), the budgerigar, Melopsittacus undulatus (Psittacidae), and the Gouldian finch, Poephila guttata (Estrildidae). These results indicate that the spongy medullary keratin is a nanostructured tissue that functions as an array of coherent scatterers. The nanostructure of the medullary keratin is nearly uniform in all directions. The largest Fourier components of spatial variation in refractive index in the tissue are of the appropriate size to produce the observed colours by constructive interference alone. The peaks of the predicted reflectance spectra calculated from the 2D Fourier power spectra are congruent with the reflectance spectra measured by using microspectrophotometry. The alternative physical models for the production of these colours, the Rayleigh and Mie theories, hypothesize that medullary keratin is an incoherent array and that scattered waves are independent in phase. This assumption is falsified by the ring–like Fourier power spectra of these feathers, and the spacing of the scattering air vacuoles in the medullary keratin. Structural colours of avian feather barbs are produced by constructive interference of coherently scattered light waves from the optically heterogeneous matrix of keratin and air in the spongy medullary layer.

Structural colouration of mammalian skin: convergent evolution of coherently scattering dermal collagen arrays

SUMMARY For more than a century, the blue structural colours of mammalian skin have been hypothesized to be produced by incoherent, Rayleigh or Tyndall scattering. We investigated the colour, anatomy, nanostructure and biophysics of structurally coloured skin from two species of primates – mandrill (Mandrillus sphinx) and vervet monkey (Cercopithecus aethiops) – and two species of marsupials – mouse opossum (Marmosa mexicana) and wooly opossum (Caluromys derbianus). We used two-dimensional (2-D) Fourier analysis of transmission electron micrographs (TEMs) of the collagen arrays in the primate tissues to test whether these structural colours are produced by incoherent or coherent scattering (i.e. constructive interference). The structural colours in Mandrillus rump and facial skin and Cercopithecus scrotum are produced by coherent scattering by quasi-ordered arrays of parallel dermal collagen fibres. The 2-D Fourier power spectra of the collagen arrays from Mandrillus and Cercopithecus reveal ring-shaped distributions of Fourier power at intermediate spatial frequencies, demonstrating a substantial nanostructure of the appropriate spatial frequency to produce the observed blue hues by coherent scattering alone. The Fourier power spectra and the observed reflectance spectra falsify assumptions and predictions of the incoherent, Rayleigh scattering hypothesis. Samples of blue Marmosa and Caluromys scrotum conform generally to the anatomy seen in Mandrillus and Cercopithecus but were not sufficiently well preserved to conduct numerical analyses. Colour-producing collagen arrays in mammals have evolved multiple times independently within the two clades of mammals known to have trichromatic colour vision. Mammalian colour-producing collagen arrays are anatomically and mechanistically identical to structures that have evolved convergently in the dermis of many lineages of birds, the tapetum of some mammals and the cornea of some fishes. These collagen arrays constitute quasi-ordered 2-D photonic crystals.

Blue integumentary structural colours in dragonflies (Odonata) are not produced by incoherent Tyndall scattering

SUMMARY For nearly 80 years, the non-iridescent, blue, integumentary structural colours of dragonflies and damselflies (Odonata) have been attributed to incoherent Tyndall or Rayleigh scattering. We investigated the production of the integumentary structural colours of a damselfly – the familiar bluet, Enallagma civile (Coenagrionidae) – and a dragonfly– the common green darner, Anax junius (Aeshnidae) – using fibre optic spectrophotometry and transmission electron microscopy (TEM). The reflectance spectra of both species showed discrete reflectance peaks of ∼30% reflectance at 475 and 460 nm, respectively. These structural colours are produced by light scattering from closely packed arrays of spheres in the endoplasmic reticulum of box-shaped epidermal pigment cells underlying the cuticle. The observed reflectance spectra do not conform to the inverse fourth power relationship predicted for Tyndall/Rayleigh scattering. Two-dimensional (2-D) Fourier analysis of the TEM images of the colour-producing arrays reveals ring-shaped distributions of Fourier power at intermediate spatial frequencies, documenting a quasiordered nanostructure. The nanostructured Fourier power spectra falsify the assumption of spatial independence of scatterers that is required for incoherent scattering. Radial averages of the Fourier power spectrum indicate that the spheres are substantially nanostructured at the appropriate spatial scale to produce visible colours by coherent scattering. However, the spatial periodicity of the arrays is apparently too large to produce the observed colour by coherent scattering. The nanospheres could have expanded substantially (∼50%) during preparation for TEM. Alternatively, coherent light scattering could be occurring both from the surfaces and from structures at the centre of the spheres. These arrays of colour-producing spheres within pigment cells have convergently evolved at least 11–14 times independently within the Odonata. Structural colouration from arrays in living cells has also fostered the convergent evolution of temperature-dependent colour change in numerous odonate lineages.

A Fourier Tool for the Analysis of Coherent Light Scattering by Bio-Optical Nanostructures

Abstract The fundamental dichotomy between incoherent (phase independent) and coherent (phase dependent) light scattering provides the best criterion for a classification of biological structural color production mechanisms. Incoherent scattering includes Rayleigh, Tyndall, and Mie scattering. Coherent scattering encompasses interference, reinforcement, thin-film reflection, and diffraction. There are three main classes of coherently scattering nanostructures—laminar, crystal-like, and quasi-ordered. Laminar and crystal-like nanostructures commonly produce iridescence, which is absent or less conspicuous in quasi-ordered nanostructures. Laminar and crystal-like arrays have been analyzed with methods from thin-film optics and Braggs Law, respectively, but no traditional methods were available for the analysis of color production by quasi-ordered arrays. We have developed a tool using two-dimensional (2D) Fourier analysis of transmission electron micrographs (TEMs) that analyzes the spatial variation in refractive index (available from the authors). This Fourier tool can examine whether light scatterers are spatially independent, and test whether light scattering can be characterized as predominantly incoherent or coherent. The tool also provides a coherent scattering prediction of the back scattering reflectance spectrum of a biological nanostructure. Our applications of the Fourier tool have falsified the century old hypothesis that the non-iridescent structural colors of avian feather barbs and skin are produced by incoherent Rayleigh or Tyndall scattering. 2D Fourier analysis of these quasi-ordered arrays in bird feathers and skin demonstrate that these non-iridescent colors are produced by coherent scattering. No other previous examples of biological structural color production by incoherent scattering have been tested critically with either analysis of scatterer spatial independence or spectrophotometry. The Fourier tool is applied here for the first time to coherent scattering by a laminar array from iridescent bird feather barbules (Nectarinia) to demonstrate the efficacy of the technique on thin films. Unlike previous physical methods, the Fourier tool provides a single method for the analysis of coherent scattering by a diversity of nanostructural classes. This advance will facilitate the study of the evolution of nanostructural classes from one another and the evolution of nanostructure itself. The article concludes with comments on the emerging role of photonics in research on biological structural colors, and the future directions in development of the tool.

COHERENT SCATTERING OF ULTRAVIOLET LIGHT BY AVIAN FEATHER BARBS

Abstract Ultraviolet (UV) structural colors of avian feathers are produced by the spongy medullary keratin of feather barbs, but various physical mechanisms have been hypothesized to produce those colors, including Rayleigh scattering, Mie scattering, and coherent scattering (i.e. constructive interference). We used two-dimensional Fourier analysis of transmission electron micrographs of the medullary keratin of UV-colored feather barbs of the Blue Whistling Thrush (Myiophonus caeruleus) (Turdidae) to test the alternative hypotheses for production of those UV structural hues. The two-dimensional Fourier power spectra of the tissue reveal a ring-like distribution of peak periodicity at intermediate spatial frequencies (∼0.078 nm −1), which documents that Myiophonus medullary keratin is substantially nanostructured and equivalently ordered in all directions. This nanoscale spatial order falsifies a basic assumption of both the Rayleigh scattering and Mie scattering. A predicted reflectance spectrum based on the Fourier power spectra matches hue of the measured reflectance spectra of the feathers (345 nm). These results demonstrate that the Myiophonus medullary keratin is ordered at the appropriately nanoscale to produce the observed UV hues by coherent scattering.

Symbolic Calculus and the Transposes of Bilinear Pseudodifferential Operators

Abstract A symbolic calculus for the transposes of a class of bilinear pseudodifferential operators is developed. The calculus is used to obtain boundedness results on products of Lebesgue spaces. A larger class of pseudodifferential operators that does not admit a calculus is also considered. Such a class is the bilinear analog of the so-called exotic class of linear pseudodifferential operators and fail to produce bounded operators on products of Lebesgue spaces. Nevertheless, the operators are shown to be bounded on products of Sobolev spaces with positive smoothness, generalizing the Leibniz rule estimates for products of functions.

End-point estimates for iterated commutators of multilinear singular integrals

Iterated commutators of multilinear Calderon-Zygmund operators and pointwise multiplication with functions in