Two-dimensional Fourier analysis of the spongy
medullary keratin of structurally coloured
feather barbs
Richard O. Prum1*, Rodolfo Torres2, Scott Williamson1 and Jan Dyck3
1

Natural History Museum and Department of Ecology & Evolutionary Biology and 2Department of Mathematics, University of Kansas,
Lawrence, KS 66045-2454, USA (prum@ukans.edu; rtorres@math.ukans.edu)
3Institute of Population Biology, Universitetsparken 15, DK-2100 Copenhagen, Denmark
We conducted two-dimensional (2D) discrete Fourier analyses of the spatial variation in refractive index
of the spongy medullary keratin from four di¡erent colours of structurally coloured feather barbs fromthree species of bird: the rose-faced lovebird, Agapornis roseicollis (Psittacidae), the budgerigar, Melopsittacus
undulatus (Psittacidae), and the Gouldian ¢nch, Poephila guttata (Estrildidae). These results indicate thatthe 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 Fouriercomponents of spatial variation in refractive index in the tissue are of the appropriate size to produce theobserved colours by constructive interference alone. The peaks of the predicted re£ectance spectra calculated from the 2D Fourier power spectra are congruent with the re£ectance spectra measured by using
microspectrophotometry. The alternative physical models for the production of these colours, theRayleigh and Mie theories, hypothesize that medullary keratin is an incoherent array and that scatteredwaves are independent in phase. This assumption is falsi¢ed by the ring-like Fourier power spectra ofthese feathers, and the spacing of the scattering air vacuoles in the medullary keratin. Structural coloursof avian feather barbs are produced by constructive interference of coherently scattered light waves fromthe optically heterogeneous matrix of keratin and air in the spongy medullary layer.
Keywords: constructive interference; feathers; Fourier analysis; structural colour
However, there are colour-producing biologicalstructures that lack lamellar organization and to which
multilayer re£ector models do not apply. Researchers have
tried to use Bragg's Law for some structures in which thescatterers are organized into nearly crystalline arrays (see,for example, Durrer 1962; Durrer & Villiger 1966, 1970;
Prum et al. 1994). However, these applications are only
approximate: the size and distance between scatterers
varies, and each scatterer is not negligibly small indiameter in comparison to visible light (Land 1972).
Current models are even less appropriate for biologicalstructures in which the scatterers are in a quasi-random,less-than-perfect array, as in the structurally coloured
avian feather barbs examined here. In fact, the distribution of the scattering air vacuoles in the medullary
keratin of structurally coloured feather barbs is so
random that these structural colours are generally
hypothesized to be produced by incoherent Rayleigh or
Mie scattering instead of by coherent constructive interference (see, for example, Fox 1976; Finger 1995).
Recently, we conducted a two-dimensional (2D)
Fourier analysis of the medullary keratin of the brilliantly
blue feather barbs of the plum-throated cotinga (Cotinga
maynana) (Prum et al. 1998). Our ¢ndings demonstrated
that, despite its lack of lamellar or crystal-like organization, the spatial variation in refractive index of the
medullary keratin of Cotinga is highly nanostructured, and

1. INTRODUCTION

Structural colours of organisms are produced by opticalinteractions of light with biological structures that are
similar in size to the wavelengths of visible light. Colourproducing biological nanostructures are extremely diversein composition, but most are uni¢ed by a common structural organization and physical mechanism. Most function
as multilayer re£ectors in which two or more media withdi¡erent refractive indices are organized in alternating
parallel layers (Land 1972; Macleod 1986). Multilayerre£ectors produce colours by constructive interference
among light waves scattered coherently from the interfaces
between media of di¡erent refractive indices. Physicalmodels of `ideal' re£ectors predict that the optical thickness(the refractive indexthe actual thickness) of each layer
should be equal to one-quarter of the wavelength of
maximum re£ectance (Land 1972; Macleod 1986). Such`quarter-wave' stacks have been identi¢ed in structurally
coloured tissues from many animal phyla (see, for example,
Land 1972) and a diversity of plants (Lee 1997). Recent
research has also documented that various `non-ideal'
systems, in which optical thickness of layers varies, canproduce structural colours that may be functionally superior
and adaptive for particular organisms (Parker et al. 1998).
*

Author for correspondence.

Proc. R. Soc. Lond. B (1999) 266, 13^22
Received 7 August 1998 Accepted 10 September 1998

13

& 1999 The Royal Society

14

R. O. Prum and others

Fourier analysis of structurally coloured feathers

that this nanostructure is approximately the correct size
to produce the observed blue hue by constructive interference alone. Our results also indicated that the physicalconditions required for the alternative incoherent scattering mechanisms, Rayleigh or Mie scattering, are not
met by this tissue (Prum et al. 1998).
In this paper, we present more detailed Fourier
analyses of feather barbs from several species that
produce a variety of di¡erent colours with a variety of
medullary structures. Our goals are to test the current
alternative hypotheses for the production of these colours(constructive interference compared with Rayleigh andMie scattering), and to present a new method for the
analysis of colour production by biological structures thatlack the lamellar or crystalline organization assumed bymost models of coherent light scattering. Speci¢cally, weapply a 2D Fourier analysis, derived from an electromagnetic physical model of light scattering by thetransparent human cornea (Benedek 1971), to document
the periodicity of spatial variation in refractive index in
the colour-producing keratin matrix of the medullary
layer of structurally coloured feather barbs.
(a) Structural colours of feather barbs

There are three di¡erent classes of colour-producing
structures in feathers. The ¢rst is unspecialized, unpigmented feather keratin, which produces white by incoherent scattering of all visible light waves. The secondclass includes structurally coloured feather barbules,
which produce generally iridescent structural colours byconstructive interference from arrays of melanin granules
and/or air vacuoles suspended in the barbule keratin(Dyck 1971b, 1976, 1978, 1987; Lucas & Stettenheim 1972;
Fox 1976; Durrer 1986). Here, `iridescence' refers only tocolours that change hue with the angle of incidence orobservation.
The third class of colour-producing feather structures
includes the specialized, spongy, medullary layer of feather
barbs, which are the sole subject of the analyses presented
here. The spongy medullary layer is composed of amatrix of keratin rods and air vacuoles of varying shapes
and sizes around a basal layer of melanin granules that
often surround a large air-¢lled nuclear vacuole (Auber
1957; Dyck 1971a,b, 1976, 1978; Lucas & Stettenheim 1972;
Fox 1976; Durrer 1986). This spongy, medullary keratin
matrix is absent in non-colour-producing feather barbs(Dyck 1978). The medullary keratin matrices di¡er frommultilayer re£ectors (Land 1972) in that they are notorganized around any particular axis or plane ofsymmetry (Auber 1957; Dyck 1971a,b, 1976, 1978; Lucas &Stettenheim 1972; Fox 1976). Although medullary keratin
varies in size, extent, and distribution on the obverse andreverse surfaces (upper and lower, respectively) of feather
barbs (see, for example, Auber 1957; Andersson 1996,
1998), the matrix itself is not organized into layers or acrystal-like array or oriented in any predictable way tothe surface of the feather or to the incident light.
Structurally coloured feather barbs produce generally
non-iridescent blue, violet, turquoise, and ultraviolet(UV) hues. These colours are broadly distributed among
birds, and must have arisen numerous times independently within the phylogenetic history of Aves (Auber
1957; Dyck 1971a,b, 1976, 1978; Durrer 1986; Finger et al.
Proc. R. Soc. Lond. B (1999)

Figure 1. Transmission electron micrographs of the spongy
medullary keratin matrix from feather barbs of (a) blue
Agapornis roseicollis, (b) green Agapornis roseicollis, (c) light blue
Melopsittacus undulatus and (d) violet Poephila guttata. Scalebars, 200 nm.

1992; Finger 1995; Andersson 1996). In combination withyellow carotenoid pigments in the barb cortex, structural
barb colours are an essential component of most green
bird plumages (Fox 1976; Dyck 1978).
(b) Alternative hypotheses of structural colour
production in feather barbs

Despite more than a century of research (reviewed in
Dyck 1971a; Fox 1976), the physical mechanism of colour
production by feather barbs is still being debated (Dyck
1971a,b, 1978; Fox 1976; Finger 1995; Andersson 1996).
These colours are produced by light-scattering at theinterfaces of the keratin and air, which di¡er in refractive
index (RI 1.54 and 1.00, respectively; Dyck 1971a). The
alternative hypotheses di¡er in whether the colours result
from incoherent di¡erential scattering of visible wavelengths or from the phase interactions among coherently
scattered waves.
Rayleigh scattering has been the primary, accepted
explanation of structural colours of feather barbs fornearly a century (HÌcker & Meyer 1902; see reviews in
Dyck 1971a; Fox 1976), and is cited in all general ornithological texts and reference works (e.g. Campbell & Lack
1985; Brooke & Birkhead 1991; Gill 1995). The term
`Rayleigh scattering' has been used confusingly by physicists to refer to a number of di¡erent light-scattering
phenomena (Young 1982) but is now generally agreed torefer to an approximate model of light scattering by particles that are smaller than the wavelength of visible light,
including small spheres or molecules (van de Hulst 1981;
Young 1982; Bohren & Hu¡man 1983). The Rayleighmodel predicts that the scattering e¤ciency of a particle
of a given size will be inversely related to the fourthFourier analysis of structurally coloured feathers

R. O. Prum and others 15

Figure 2. Re£ectance spectra of feather barbs of (a) blue Agapornis roseicollis, (b) green Agapornis roseicollis with the yellow
carotenoid pigment removed, (c) light blue Melopsittacus undulatus (Finger 1995), and (d ) violet Poephila guttata.

power of the wavelength: Rayleigh's inverse fourth powerlaw. Accordingly, Rayleigh scattering should produce
blue, violet, or UV colours, depending on the size of thescatterers. Rayleigh scattering can only produce colours
from the smaller end of the visible spectrum, because
larger particles would scatter all wavelengths of light withequivalent e¤ciency, creating a white appearance.
Recently, Mie scattering, in combination with cortical¢ltering, has been proposed as an alternative explanation
Proc. R. Soc. Lond. B (1999)

of structural colour production by feather barbs (Finger
1995). Mie theory is an exact electromagnetic model ofscattering by an isolated homogeneous spherical particle(van de Hulst 1981; Bohren & Hu¡man 1983). Mie andRayleigh theory are not distinct models: Mie theory
reduces to Rayleigh's inverse power law for particles
smaller than the wavelength of visible light that are notstrongly absorbing (van de Hulst 1981; Bohren &Hu¡man 1983).

16

R. O. Prum and others

Fourier analysis of structurally coloured feathers

Figure 3. 2D Fourier power spectra of the spatial variation in refractive index in the spongy medullary keratin of (a) blue
Agapornis roseicollis, (b) green Agapornis roseicollis, (c) light blue Melopsittacus undulatus, and (d) violet Poephila guttata. The spatial
frequency and direction of each power-spectrum value are given by the length and direction, respectively, of a vector from theorigin to that point. The magnitude of each power spectrum value is given by the colour (scale bars on right).

Both Mie and Rayleigh theory are models of incoherent light scattering. An array of scatterers is considered `incoherent' if the phase relations among the lightwaves scattered by the array can be ignored in thedescription of its scattering behaviour (van de Hulst 1981;
Bohren & Hu¡man 1983). An array is incoherent if its
scatterers are separated by distances that are large incomparison with the wavelength of light concerned (vande Hulst 1981; Bohren & Hu¡man 1983). Incoherent scattering mechanisms hypothesize that structural colours offeather barbs are a consequence only of the size of thescatterers and the di¡erential refractive index of the two
media, and that the spatial distribution of the scatterers
and the phase relations of scattered waves are irrelevant.
In contrast, the constructive interference model
hypothesizes that the phase interactions among lightwaves scattered by the keratin ^ air matrix produce thestructural colours of avian feather barbs (Raman 1935;
Dyck 1971a,b, 1976). Constructive interference models take
into account the phase relations among scattered waves,and are thus described as `coherent' scattering models.
Proc. R. Soc. Lond. B (1999)

The multilayer re£ector (Land 1972) and Bragg re£ection(Durrer 1962; Durrer & Villiger 1966, 1970; Prum et al.
1994) models are approximations of coherent scatteringby arrays that are organized into lamellae or crystalline
arrays, respectively.
According to the constructive interference model,
scattered waves that are out of phase will destructively
interfere and cancel out, whereas scattered waves that arein phase will constructively reinforce one another and becoherently re£ected. The phase relations among scatteredwaves are determined by the size and spatial distributionof the scatterers (Benedek 1971; Dyck 1971a,b). The phaserelations between two waves of the same wavelength
scattered by di¡erent scatterers is determined by thedi¡erence between the distances travelled by each of the
waves, called the path-length addition. For almost all
wavelengths in a medium, a given path-length addition isa fraction, or non-integer multiple, of the wavelength.
Such wavelengths will be out of phase after scattering,
and will destructively interfere with one another andcancel out. However, those light waves that are near toFourier analysis of structurally coloured feathers
the size of the path-length addition (or an integer
multiple of it) will travel a complete phase cycle over that
distance, will be in phase with one another after scattering, and will reinforce one another to produce a wavelength-speci¢c constructive re£ection. The opportunity
for constructive interference by a coherent medium isdetermined by the distribution of potential path-length
additions, which is related to the periodicity of the spatial
distribution of the scatterers (Benedek 1971). If thescatterers are randomly distributed, then path-length
additions are also random, and scattering will beincoherent. However, periodic spatial relations among
scatterers will produce a predictable distribution of pathlength di¡erences among scattered waves, and reinforcement of a limited set of wavelengths.
Constructive interference was ¢rst proposed as an
explanation of structural colours in feather barbs by
Raman (1935), who observed that individual feather
barbs can be subtly iridescent. The constructive interference hypothesis was subsequently revived as an alternative to Rayleigh scattering in a series of publications by
Dyck (1971a,b, 1976, 1978). Dyck documented that there£ectance spectra of many blue and green structurally
coloured barbs showed discrete peaks in the visible spectrum, whereas the Rayleigh scattering model predicts acontinued increase in scattering into the UV. Having
falsi¢ed this prediction of the Rayleigh model, Dyck
(1971a,b, 1976, 1978) presented the `hollow cylinder' model,
which hypothesizes that constructive interference fromthe ordered matrix of air vacuoles and keratin in the
medullary layer produces the structural colours of feather
barbs. Dyck (1971a,b, 1976, 1978) demonstrated that, forfeather barbs of a variety of colours and species, the hueof a structural colour is strongly correlated with the sizeof the air vacuoles and keratin bars in the spongy medullary matrix. Although this evidence is supportive ofconstructive interference, Dyck could not address thefundamental question of whether the medullary keratin
matrix is su¤ciently ordered, or periodic, to produce
wavelength-speci¢c structural colours.
(c) Discrete Fourier analysis and structural colour
production

In this paper, we address the following questions. (i) Is
the structurally coloured spongy medullary keratin
matrix su¤ciently ordered to produce wavelength-speci¢ccolours by constructive interference ? (ii) Is this matrix
the appropriate size to produce the observed structural
colours? (iii) Is the keratin matrix appropriately organized to produce structural colours by incoherent scattering (i.e. Rayleigh or Mie scattering)?
To examine these questions, we apply an explicit electromagnetic model of light scattering and transparency ofthe human cornea created by Benedek (1971). The vertebrate cornea is a quasi-random array of parallel collagen¢bres (Maurice 1984). The cornea is transparent because
the collagen ¢bres that compose it are very small andclose together, resulting in destructive interference among
all scattered wavelengths of visible light (Benedek 1971;
Maurice 1984; Vaezy & Clark 1991, 1993). Previous
research using Bragg's Law to analyse structural colour
production by collagen arrays in the avian skin (Prum etal. 1994) led us to consider applying Benedek's theory ofProc. R. Soc. Lond. B (1999)

R. O. Prum and others 17

corneal transparency to structural colour production by
the spongy medullary keratin matrix of feather barbs: an
optically similar phenomenon on a di¡erent spatial scale.
Benedek (1971) developed an explicit, electromagnetic
model for light scattering by an array of ¢bres. According
to Benedek's theory, in a quasi-random array of scatterers(i.e. a less than perfect lattice), signi¢cant coherent reinforcement is predicted only for those light waves in themedium that are twice the size of the largest componentsof the Fourier transform of the spatial variation in refractive index (Benedek 1971). The discrete Fourier transform
is a basic mathematical tool used to decompose a signal
or image into di¡erent periodic components (Briggs &Henson 1995). Discrete data are transformed into a sumof component sine waves of di¡erent amplitudes andfrequencies (Briggs & Henson 1995). The relative squared
amplitudes of these component waves, called the Fourier
power spectrum, express the contributions of eachfrequency of variation to the original data, and indicate
which frequencies carry the most energy (Briggs &Henson 1995).
Benedek's theory predicts that the Fourier power spectrum of spatial variation in refractive index of a tissue
will predict which wavelengths will be constructivelyre£ected by that tissue. This is because the distribution of
the Fourier coe¤cients of the spatial variation in refractive index is directly related to the distribution of thepath-length additions experienced by light waves
scattered by the tissue. Both 1D and 2D Fourier analysis
have been used to analyse spatial variation in refractive
index in optically transparent, opaque, and white tissuesof the mammalian eye (Vaezy & Clark 1991, 1993; Gisselberg et al. 1991). Here, we apply 2D Fourier analysis to
examine the relation between the spatial variation in
refractive index in the medullary keratin matrix of thefeather barbs and their structural colours. This application of Fourier analysis is derived independently fromelectromagnetic optical theory (Benedek 1971) and iscompletely unrelated to the traditional physical ¢eld of`Fourier optics' in which the Fourier transform is used todescribe the process of di¡raction by a grating or the
resolution of an image by a lens.
2. METHODS(a) Species sampled and observations

We examined structurally coloured feather barbs of fourdi¡erent colours from three di¡erent species: blue and green
contour feathers of the rose-faced lovebird, Agapornis roseicollis(Psittacidae); light blue contour feathers of a captive-bred blue
mutant of the budgerigar, Melopsittacus undulatus (Psittacidae;
KU 46538); and violet-purple contour feathers of the Gouldian¢nch, Poephila guttata (Estrildidae; KU 38025). Contour feathers
were washed and prepared for microscopy and spectroscopy
according to the methods of Dyck (1971a). Carotenoid pigments
were extracted from the green Agapornis feathers as in Dyck
(1971a). Transmission electron micrographs (TEM) wereprepared for Agapornis as by Dyck (1971a), but TEMs of Melopsittacus and Poephila were taken with a Jeol JEM-1200 (¢gure 1).
Re£ectance spectra of intact single feather barbs of Agapornis
and Poephila were obtained with a Zeiss microspectrophotometer
01 using perpendicular incidence, according to the procedure of
Dyck (1978). Percentage re£ectance was calculated by dividing

18

R. O. Prum and others

Fourier analysis of structurally coloured feathers

the re£ectance of the barb by the re£ectance of a white standard
for 30 di¡erent wavelengths at 10 nm intervals between 400 and
700 nm (¢gure 2). The re£ectance spectrum for the light blue
mutant Melopsittacus is from Finger (1995).

(b) 2D discrete Fourier analysis

We used TEMs of the feather keratin matrix as an estimate of
the spatial variation in refractive index of the tissue. Grey-scalevariation in the images allowed us to distinguish between feather
keratin and air vacuoles, and to estimate the spatial variation in
refractive index in the spongy medullary keratin matrices.
TEM negatives of the spongy medullary keratin of thefeather barbs of Agapornis (20 000), Melopsittacus (30 000)
and Poephila (30 000) were digitized at 300 dpi by means of a
Hewlett-Packard Scanjet 3c (¢gure 1). The digitized TEM
images were analysed with MATLAB (version 5.0; MATLAB
Reference Guide 1992) on a Silicon Graphics Onyx workstation.
The scale of each digitized image (nm per pixel) was calculated
by measuring the number of pixels in the original TEM scalebar in the micrograph. Because the TEM sections are notin¢nitely thin, the TEMs do not accurately depict the sharp
interfaces between the keratin and the air. Furthermore, thekeratin structures frequently stain irregularly, producing
spurious variation in this optically homogeneous medium. To
control for these sources of error, the digitized TEMs wereimage processed to enhance the distinction between the keratinbars and air-¢lled vacuoles (i.e. the spatial function of refractive
index), and to diminish grey-scale variation within eachmaterial (i.e. noise). The pixel intensity of each image wasrescaled in relation to the original variance in intensity. Pixels
more than one standard deviation away from the mean intensity
value were assigned to 0 (black) or 1 (white) and the remaining
variation was rescaled as a linear function between 0 and 1. To
eliminate variation among di¡erent areas of the image, thisvariance rescaling was performed across the entire image on
30 pixel  30 pixel blocks with ¢ve overlapping pixels between
neighbouring blocks (i.e. block processing; Thompson & Shure1995). Pixel intensity was then median-¢ltered: each pixel
received the mean of the surrounding 4 pixel  4 pixel matrix.
The largest available square portion of the keratin matrix wasthen selected for analysis (between 600 and 625 pixels2). Largescale singularities in the images (e.g. cell boundaries or melanin
granules) were avoided when possible, because analysis oflarger-scale variation in structure of the feathers can disrupt thedescription of the nanoscale variation in refractive index of the
medullary keratin matrix itself.
The mean refractive index of the tissue in each image wasestimated by generating a 2-bin histogram of image densityfrom the processed images (Thompson & Shure 1995). The
frequency distributions of the darker and lighter pixels in theblock-processed images were used to estimate the relative
volume of keratin (RI 1.54) and air (RI  1.00) in the image;this relative volume was then used to calculate a weighted mean
refractive index for the tissue (Dyck 1971a). These weighted
means may underestimate the mean refractive index of theentire feather barb, because we did not include the completely
keratinous barb cortex.
The numerical computation of the Fourier transform was donewith the well-established 2D fast Fourier transform (FFT2)
algorithm (Briggs & Henson 1995) in MATLAB (MATLAB
Reference Guide 1992; Thompson & Shure 1995), which is abroadly accepted, standard tool for this kind of analysis. Because
the tissue has no plane of symmetry and is randomly orientated
Proc. R. Soc. Lond. B (1999)

to the surface of the barbs, the orientation of the image within
the matrix is not important (see ½ 3, ¢gure 3). We then calculated the 2D Fourier power spectrum (the distribution of thesquares of the Fourier coe¤cients). The 2D Fourier power spectrum resolves the spatial variation in refractive index in thetissue into its periodic components in any direction from agiven point (¢gure 3). Each value in the power spectrum
reports the magnitude of the periodicity in the original data at
a speci¢c spatial frequency in a given direction from all points
in the original image. The spatial frequency and direction of
any given component in the power spectrum are given by thelength and direction, respectively, of a vector from the origin tothat point. The 2D Fourier power spectra were expressed inspatial frequency (nm ÿ1) by dividing the initial spatial
frequency values by the length of the matrix (pixels in thematrix  nm per pixel). The four quadrants of each power-spectrum matrix were shifted to place the four original corners ofthe matrix in the centre, so that the frequency origin was in
the centre of the image.
We produced predicted re£ectance spectra (¢gure 4) basedon the 2D Fourier power spectra of each tissue (¢gure 3), andthe size and mean refractive index of the images. First, we
produced an estimate of the percentage total Fourier power ofcomponent waves of di¡erent sizes by taking a radial mean of aseries of 0.5 pixel radial intervals of spatial frequency(diameters) for a single quadrant of the power spectrum. (Thepower spectrum of a 2D Fourier analysis is itself 3D; the three
dimensions depict the size, direction, and magnitude of component spatial frequencies). The values of the radial mean of the
2D power spectrum were then rescaled so that the total volume(and total energy) under the rotated radial mean power function
was equal to unity. Next, the relative percentage total power ofFourier components of di¡erent sizes (over all directions) was
estimated by calculating the volumes of radial segments, orshells, of the normalized, rotated, radial mean of the power
spectrum. The inverse of the spatial frequency values were thenmultiplied by twice the mean refractive index of the medium
and expressed in terms of wavelength (nm). The result is a theoretical prediction of the relative magnitude of coherent scattering, or constructive interference, by the tissue that is based
entirely on the spatial variation in refractive index of the tissue.
3. RESULTS(a) Anatomy and microspectroscopy

All of the feather barbs examined have an extensive
layer of spongy, medullary keratin that is characterized
by a network of keratin rods and air vacuoles (¢gure 1).
As in other structurally coloured feather barbs (Auber
1957; Dyck 1971a,b, 1976, 1978; Lucas & Stettenheim 1972;
Fox 1976; Durrer 1986), the keratin matrices have noplanes of symmetry and are not orientated to the surfaceof the feather in any predictable manner.
The size and structure of the keratin matrices vary
among the species examined. Agapornis and Melopsittacus
both feature rod-like or columnar keratin bars with relatively uniform diameter, and less uniform air vacuoles ofvariable shape and diameter (¢gure 1a ^ c). This type of
medullary structure (type f, ¢g. 27 of Dyck 1978) is
known from other psittacids, leafbirds (Irenidae) (Dyck
1971b, 1978; Finger et al. 1992), Coracias rollers (Coraciidae;
R. O. Prum, personal observation), and Myiophonas
thrushes (Muscicapidae; Andersson 1996). In Poephila, theFourier analysis of structurally coloured feathers

R. O. Prum and others 19

Figure 4. Predicted re£ectance spectra based on the 2D Fourier power spectra of the medullary keratin of feather barbs from(a) blue Agapornis roseicollis, (b) green Agapornis roseicollis with the yellow carotenoid pigment removed, (c) light blue Melopsittacus
undulatus, and (d) violet Poephila guttata. The predicted re£ectance is given as percentage total Fourier power. (See ½ 2 for details.)

medullary structure is characterized by air vacuoles of
uniform diameter and spacing with a continuous keratin
medium in between (¢gure 1d ). This type of organization(type e, ¢g. 27 of Dyck 1978) has also been found in
Cotinga (Cotingidae; Dyck 1971a; Prum et al. 1998) andMegalaima barbets (Ramphastidae; Dyck 1978).
The re£ectance spectra of all barbs examined showeddiscrete peaks in the visible spectrum between 440 and
540 nm (¢gure 2). The blue Agapornis barbs show a re£ectance peak at 480 nm, whereas the green Agapornis barbs
with the yellow carotenoid pigment extracted show are£ectance peak of 535 nm. The violet breast-feather
Proc. R. Soc. Lond. B (1999)

barbs of Poephila show a distinct re£ectance peak of
450 nm. The light blue barbs of domestic mutant Melopsittacus show a broad re£ectance peak in the visible spectrum at about 440 nm (Finger 1995). The positions of thepeaks in these re£ectance spectra correspond well to thehue or appearance of the feather barbs. Although there£ectance peak of the light blue Melopsittacus barbs is aslightly smaller wavelength than that of the violet Poephilabarbs, the di¡erences between these re£ectance spectra in
the breadth of the peak, or chroma, also contributes tothe observed di¡erences in colour. The violet Poephilabarbs strongly re£ect a narrower band of lower visible

20

R. O. Prum and others

Fourier analysis of structurally coloured feathers

wavelengths, whereas the light blue Melopsittacus barbs
show a broader re£ectance spectrum.
(b) 2D Fourier analysis

The 2D Fourier power spectra of all four spongy medullary tissues show a distinct disc pattern, with the largest
power values near the centre of the disc and in a `ring'surrounding the origin (¢gure 3). The clusters of highpower values near the centre come from the lowest spatial
frequencies and are caused by large-scale variations in thegrey scale of the images. The main ring of larger power
values indicates a predominant distribution of power at
spatial frequencies that are the appropriate size scale toproduce coherent re£ections within the visible spectrum.
The low power values at higher spatial frequencies indicatethat these tissues have little periodic spatial variation at
the smallest spatial scales. The power spectra indicate thatthe spatial variation in refractive index in the medullary
keratin of these species is highly ordered on a nanoscale.
Although some of the power spectra are slightly ovoid, thegenerally ring-like pattern indicates that this nanostructure is uniform in all directions (¢gure 3).
The 2D Fourier power spectra were used to predict there£ectance spectra of the feather barbs of each colour andspecies (¢gure 4; see ½ 2). In all four cases, the predictedre£ectance spectra show distinct peaks in the visible spectrum, indicating that the vast majority of the spatial
variation in refractive index is concentrated at spatial
frequencies that should produce constructive re£ections oflight waves within the visible spectrum.
The peaks of the Fourier-predicted re£ectance spectra
are also generally congruent with the observed re£ectancespectra of the same feather barbs (¢gures 2 and 4). Blue
Agapornis barbs have a predicted re£ectance peak of
500 nm (¢gure 4a) and an actual re£ectance peak of
480 nm (¢gure 2a). Green Agapornis barbs with the yellow
carotenoid pigment removed have a predicted re£ectancepeak of 535 nm (¢gure 4b), which is exactly congruent
with the observed peak (¢gure 2b). Light blue Melopsittacus barbs have a predicted re£ectance peak of 480 nm(¢gure 4c) and an observed peak of 440 nm (¢gure 2c).
Violet Poephila barbs show a predicted re£ectance peak of
425 nm (¢gure 4d) and an observed peak of 450 nm(¢gure 2d ). The predicted spectra have peaks within 0^
40 nm of the observed re£ectance peaks.
The congruence between the observed and predictedre£ectance peaks is striking given that there are no
reasons, other than nanostructuring for colour production, to expect anything other than a £at distribution (i.e.
white noise) or inverse exponential distribution (i.e.
inverse frequency-dependent noise) in the Fourier power
spectrum of the refractive index of the medullary keratin
at this spatial scale.
(c) Rayleigh and Mie scattering

The re£ectance spectra of the feather barbs examined
had distinct peaks in the visible spectrum (¢gure 2) anddid not conform to the predictions of Rayleigh's inverse
fourth power law. As Dyck (1971a, 1978) concluded, thisevidence falsi¢es a major prediction of the Rayleigh scattering model.
Futhermore, the physical conditions necessary for incoherent scattering assumed by Mie and Rayleigh theory
Proc. R. Soc. Lond. B (1999)

are not present in the spongy medullary keratin of feather
barbs. In the structurally coloured feather barbs observed
here (and in all those previously described: Auber 1957;
Dyck 1971a,b, 1976, 1978; Durrer 1986; Finger et al. 1992;
Finger 1995; Andersson 1996), the light-scattering air
vacuoles in the spongy medullary keratin matrix aremuch closer together than even the smallest wavelengthsof visible light (55300 nm). Furthermore, the 2D Fourier
power spectra from our analyses demonstrate that theheterogeneities in refractive index are not spatially independent of one another but highly ordered (¢gures 3 and
4). Mie and Rayleigh scattering are not appropriate
models of structural colour production by avian feather
barbs.
4. DISCUSSION

The 2D discrete Fourier analysis of the spongy medullary keratin of structurally coloured feather barbs demonstrates that these are highly ordered, nanostructured
tissues. The nanostructure of the medullary keratin
matrices is nearly uniform in all directions, and is close to
the correct size to produce the observed colours byconstructive interference. These results indicate thatstructural colours of feather barbs are produced byconstructive interference of light waves scattered coherently by the interfaces between keratin and air in the
spongy medullary layer of the barbs.
Previous results demonstrate that these same conclusions apply to the blue feather barbs of Cotinga (Prum et al.
1998). The results reported here indicate that these
conclusions are generalizable to green, blue, and violet
structural colours of feather barbs belonging to bothmajor classes of medullary organization: those withkeratin bars of relatively uniform diameter (e.g. Agapornis,
Melopsittacus) (type f, ¢g. 27 of Dyck 1978). as well as
those with air vacuoles of relatively uniform diameter(e.g. Poephila, Cotinga) (type e, ¢g. 27 of Dyck 1978). The
same physical mechanism is also likely to apply to structurally UV-coloured feather barbs that have the former
type of medullary organization (e.g. Chalcopsitta: Finger etal. 1992; Myiophonus: Andersson 1996, 1998).
Rayleigh scattering has been the predominant explanation of structural colour production by feather barbs fornearly a century (HÌcker & Meyer 1902; see reviews in
Dyck 1971a; Fox 1976), and is cited in all current generalornithology texts and references (e.g. Campbell & Lack
1985; Brooke & Birkhead 1991; Gill 1995). Our results
demonstrate that a critical physical condition required forboth Rayleigh and Mie scattering ö incoherent scattering ö is not met by any structurally coloured feather
barbs, and is directly falsi¢ed by the ring-like Fourier
power spectra of medullary feather keratin (¢gure 3).
Dyck (1971a,b, 1976, 1978) proposed the constructive
interference (originally `hollow cylinder') model as analternative to Rayleigh scattering, but it received limited
acceptance because of the lack of evidence demonstratingthat the spongy medullary keratin is su¤ciently ordered
to produce coherent phase relations among scatteredwaves. However, advances in understanding light scattering by the transparent cornea demonstrate that tissues
need not have a precise lamellar or lattice-like structure
to produce predictable phase relations among scattered

Fourier analysis of structurally coloured feathers
light waves (Benedek 1971; Gisselberg et al. 1991; Vaezy &Clark 1991, 1993).
The spongy, medullary keratin of feather barbs belongs
in a distinct class of structurally coloured tissues thatfunction by constructive interference but which lack alamellar or crystalline organization. The medullary
keratin matrix functions by a physical mechanism similar
to that of multilayer re£ectors and Bragg arrays. Bothsystems have highly organized, nanoscale spatial variationin refractive index that produces coherent scattering.
However, feather barbs di¡er strikingly from these otherstructures in that they are not organized or orientated ona larger spatial scale into layers or crystal-like lattices.
The Fourier power spectra indicate that the nanostructure of the medullary keratin is nearly uniform in all
directions (¢gure 3; Prum et al. 1998). As a consequence,changes in angle of incidence or observation do notchange the overall distribution of path-length additionsor the observed hue. In contrast to many structurally
coloured tissues, feather barbs can produce brilliant huesthat are nearly omnidirectional (Dyck 1971a, 1978). These
functional attributes may explain the diversity, broad
distribution, and multiple evolutionary origins of structurally coloured feather barbs in birds.
(a) Fourier analysis and constructive interference

Developed initially for the study of transparency(Benedek 1971; Gisselberg et al. 1991; Vaezy & Clark 1991,
1993, 1995; Vaezy et al. 1995), these Fourier techniques areexactly suited to the analysis of coherently scattering by
biological tissues, especially structures that cannot be
accurately modelled as multilayer re£ectors. These
include some iridescent avian feather barbules (Durrer1986), vertebrate dermal collagen arrays (Prum et al.
1994), vertebrate and invertebrate iriphores (see, forexample, Bagnara & Hadley 1973), and the mammaliantapetum lucidum (Land 1972). Furthermore, 1D or 2D
Fourier techniques should also converge on current
methods of analysing multilayer re£ectors (Macleod1986), such as those in the invertebrate exoskeleton (Fox
1976; Neville 1993; Parker et al. 1998) and structurally
coloured plants (Lee 1997).
Previous discrete Fourier analyses by Vaezy, Clark andcolleagues have focused on determining whether various
tissues produce coherent scattering within the visible
spectrum (Gisselberg et al. 1991; Vaezy & Clark 1991,
1993, 1995; Vaezy et al. 1995). We have been able toachieve ¢ner resolution of spatial variations in refractive
index by increasing the size of the images analysed from
256 pixels2 to greater than 600 pixels2.
We have made another analytical advance overprevious studies. Vaezy, Clark and colleagues used aradial mean of the 2D Fourier power spectrum as thesummary of the distribution of energy across various
spatial frequencies (Gisselberg et al. 1991; Vaezy & Clark
1991, 1993, 1995; Vaezy et al. 1995). This method is an
accurate way to estimate the distribution of the distance
between scatterers, but it is not an accurate way to
predict the re£ectance spectrum of a tissue. The `power' of
the power spectrum is equal to its volume (the sum of all
power spectrum values). Because the radial mean of the
2D power spectrum has no volume, the radial meanvalues do not express the di¡erential contribution of eachProc. R. Soc. Lond. B (1999)

R. O. Prum and others

21

spatial frequency to the total magnitude of scattering by
the heterogeneities in refractive index of the tissue. The
magnitude of scattering by component spatial frequencies isrelated to the power or volume of those components in the
power spectrum. We therefore calculated the volumes ofcylinders, or shells, for increments of spatial frequency
from the normalized, rotated radial mean of the 2D
power spectrum to estimate more accurately the relative
percentage of total power contributed by each interval of
spatial frequency. In essence, the power values from theradial mean of the 2D power spectrum must be weighted
by their distances from the origin to calculate thevolumes of the shells they circumscribe. This distribution
yields a more accurate description of the re£ectancespectra of the tissues. This result explains why the highpower values at small spatial frequencies near the originof the power spectrum do not contribute substantially tothe magnitude of scattering by the tissue. These components are high-energy, but low-volume in percentage total
power.
This research programme has been aided by stimulating discussions with A. Brush, M. Bar¢eld, G. Barrowclough, R.
Gomulkiewicz, A. Miller, P. Stettenheim and D. Winkler. We
are especially grateful for comments on scattering physics from
C. F. Bohren. The manuscript was substantially improved by
criticisms from three anonymous reviewers. R. Gomulkiewicz
and J. Vogel kindly helped with an earlier attempt at 1D Fourier
analysis of feather barbs. Feather specimens were supplied by
the University of Kansas Natural History Museum (KU)
Division of Ornithology. We are thankful to Tim Buller and the
University of Kansas Department of Mathematics for computing assistance. This research was supported by grants from the
National Science Foundation to R.O.P. (DEB-9318273) and
R.H.T. (DMS-9696267).

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