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Quantitative characterization of iridescent colours in biological studies: a novel method using optical theory

Published:14 December 2018https://doi.org/10.1098/rsfs.2018.0049

    Abstract

    Iridescent colours are colours that change with viewing or illumination geometry. While they are widespread in many living organisms, most evolutionary studies on iridescence do not take into account their full complexity. Few studies try to precisely characterize what makes iridescent colours special: their angular dependency. Yet, it is likely that this angular dependency has biological functions and is therefore submitted to evolutionary pressures. For this reason, evolutionary biologists need a repeatable method to measure iridescent colours as well as variables to precisely quantify the angular dependency. In this study, we use a theoretical approach to propose five variables that allow one to fully describe iridescent colours at every angle combination. Based on the results, we propose a new measurement protocol and statistical method to reliably characterize iridescence while minimizing the required number of time-consuming measurements. We use hummingbird iridescent feathers and butterfly iridescent wings as test cases to demonstrate the strengths of this new method. We show that our method is precise enough to be potentially used at intraspecific level while being also time-efficient enough to encompass large taxonomic scales.

    1. Introduction

    Most interactions between organisms, whether between different species (interspecific) or different individuals of the same species (intraspecific), involve communication. Communication can have different purposes (e.g. warning, camouflage, display) and use different channels (e.g. olfactory, acoustic, visual) [1]. In particular, colour is a specific kind of communication channel that can be produced through two non-mutually exclusive mechanisms: pigmentary colours are generated by the selective absorption of some wavelengths by special molecules called pigments while structural colours are generated by the physical interaction of light with matter, causing dispersion, diffraction or interferences [2].

    Among structural colours, iridescent colours change depending on the illumination or observation angle. They can be produced by interferences of light after reflection by a thin-film or multilayer structure, or diffraction on a grating. Iridescent colours are present in many taxa, and particularly widespread among bony fishes (Actinopterygii), insects, as well as some birds (see detailed review in table 1 for studies on each one of these taxa). Iridescent colours seem to be involved in many important biological processes [123] and their angular dependency is likely under selection to produce complex visual signals [74,87,115,124]. In some cases, however, angular dependency may be selected against [125]. In all those cases, the study of the evolution of iridescent colours requires a precise quantification of the angular dependency. However, the inherent physical complexity of iridescent colours has hampered the development of quantitative methods to fully describe them in the angle space.

    Table 1. Review of the methods used in the literature to study iridescent colours from multilayer or thin-film structures. The criteria we used for studies to be included in the table were the following: (i) at least one quantitative reflectance measurement using a spectrometer, (ii) functioning with white light (no monochromatic illumination), and (iii) the patch measured had to be described as iridescent in the article. A more detailed version of this table, with all angle configurations and colour variables used for each study is available in the electronic supplementary material. The terms ‘constant illumination’, ‘constant collection’, ‘constant angle bisector’ and ‘constant span’ are defined in figure 3d.

    Collapse
    no. measurementsfibre configuration (no. studies)birdsarthropodsothers
    single measurementsingle fixed angle (53)[3–33][34–48]bony fishes [49]; mammals [50]; plants [51–54]
    single measurement relative to the structure orientation (6)—[55–60]—
    multiple measurements along a single lineconstant illumination (5)[61][62–64]bacteria [65]
    constant collection (2)[66][67]—
    constant angle bisector (16)[68–78][79–83]—
    constant span (16)[84–87][88–96]bony fishes [97]; lizards [98,99]
    multiple measurement linesmultiple constant illuminations (4)[100][101,102]bacteria [103]
    multiple constant collections (1)[104]—
    multiple constant spans (1)[105]—
    constant illumination and bisector (3)—[106,107]bacteria [108]
    multiple illumination and bisector (1)—[109]—
    constant illumination and span (3)[110,111][112]—
    constant span and bisector (6)[113–115][116–118]gastropods [119]
    constant illumination, span and bisector (4)[120,121][102,122]—

    We reviewed all studies that performed reflectance measurements of biological samples with iridescent colours produced by a multilayer or a thin-film structure in table 1. We notice two main trends: (i) many studies measure iridescence at a single fixed angle (first row in table 1). In these studies, authors generally remain cautious and warn they are not attempting to measure angle dependency. However, the multilayer or thin film producing iridescent colours may not be parallel to the sample surface [67,80,96,102,109], and the angle between them and the sample surface may vary between species or even between individuals of the same species [105]. Hence, even though the angle of the measuring optical fibres relative to the macroscopic is constant, the angle relative to the structures is not. This jeopardizes any biological interpretation of differences between samples because the effects of many different parameters are intertwined.

    (ii) Other studies take measurements at multiple angles but few attempt to precisely quantify angle dependency (‘Literature review’ folder in electronic supplementary material). Even when angle dependency is quantified, variables never stem from a theoretical approach, which leads to a large diversity of custom variables for each author. This heterogeneity in the methods, variable naming and sign conventions has likely hindered the spread of new concepts and results among researchers working on iridescence in living organisms.

    Osorio & Ham [110] and Meadows et al. [114] started to address this heterogeneity in measurement methods and advocated for the use of a goniometer to reliably measure colour in a controlled angle configuration. However, they did not propose a detailed protocol or statistical tools to study angular dependency. Here, we use the optical laws that govern iridescence to propose a set of parameters to characterize angle dependency of brightness, hue and saturation of iridescent colours. Next, we confirm the validity of these equations for complex biological structures using two highly different groups of organisms well known for their iridescent colours: Trochilidae (hummingbirds) and Lepidoptera (i.e. butterflies and moths), including the iconic Morpho butterflies that harbour large wings with bright iridescent blue colours. The standard framework we propose here makes iridescent colours comparable across taxa and across studies, opening up new perspectives in the study of their biological functions.

    2. Model

    2.1. Choice of colour variables

    Since we want to produce a general method that would not depend on any specific vision system, we use variables directly derived from spectra, without computing vision models. We define brightness B as the average reflectance over a range between the minimal (λmin) and maximal (λmax) wavelengths (B2 in Montgomerie [126]), saturation S as the full width at half maximum reflectance and hue H as the wavelength at which reflectance is maximal (H1 in Montgomerie [126]). These three variables are represented in figure 1 and are the most common measures of brightness, hue and saturation in studies about iridescence (see the literature review in the electronic supplementary material).

    Figure 1.

    Figure 1. Graphical representation of the variables we used for hue H (wavelength at peak reflectance Rmax; called H1 in Montgomerie [126]), brightness B (average of reflectance over the wavelength range of interest; B2 in Montgomerie [126]) and saturation S (full width at half maximum; no equivalent in Montgomerie [126]). (Online version in colour.)

    2.2. Assumptions and equations

    Our method relies on three assumptions that greatly simplify the equations for brightness, hue and saturation in the angle space. See appendix A for mathematical proofs of the equations and the role of each one of these assumptions:

    (1)

    Small angles (less than or equal to 30°). Outside of this range, the signal due to iridescence is often very low and all that remains is the effect of the underlying pigments, which can be measured through traditional methods. For all thin films, and in some multilayers (depending on chemical composition), it is possible to consider angles up to 45°, as illustrated in the electronic supplementary material. This may help in producing more repeatable parameter estimates. For instance, a 45° angle can correspond to a viewer standing next to the viewed iridescent patch illuminated from above. Many previous studies have in this way mimicked the position of the bird relative to the sun in their measurements [66,87,98,105,114,118].

    (2)

    The orientation of the layers within the multilayer structure is affected by Gaussian noise. Many developmental processes are controlled by a large array of independent factors of small effect each, causing subsequent errors to often be Gaussian (due to the central limit theorem). This assumption is also empirically supported by the results of Gur et al. [127], who looked at the orientation of guanine crystals in neon tetra fishes (Paracheirodon innesi) using wide-angle X-ray scattering. Fitting a Cauchy distribution (fatter tail distribution) instead of a Gaussian distribution yields similar values of parameter estimates. For simplicity, we here only present the results with Gaussian noise.

    (3)

    Multilayers are ideal, i.e. the optical thickness (layer thickness times optical index) of each layer is constant: n1e1 = n2e2. This is a common assumption [36,54,67,97,107,119,128–130] which is thought to be valid for most animal reflectors [131] because it produces the brightest and most saturated signals with a minimal number of layers (but see Schultz & Rankin [35] and Parker et al. [132] for beetles, Kinoshita et al. [133] for neon tetra).

    This set of assumptions allows us to formally derive simple analytic expressions of brightness B, hue H and saturation S (figure 1) in the angle space (Φinc, Φcol). All variables used in this study with their notations and their possible values are listed in table 2 and illustrated whenever possible in figure 2.

    Table 2. List of parameters used in this study, with their domains of definition and their meanings.

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    symbolrangemeaning
    θi[−π2;π2]incident light angle relative to the multilayer
    θr[−π2;π2]reflected light angle relative to the multilayer
    θ1[0;π2]angle between the incident ray and the interface between layers 1 and 2
    θ2[0;π2]angle between the transmitted ray and the interface between layers 1 and 2
    angle between the incident ray and the interface between layers 2 and 1
    mNinterference order/rank
    BR+brightness at a given configuration
    H[λmin; λmax]hue at a given angle configuration
    SR+saturation at a given angle configuration
    BmaxR+maximal brightness value (achieved for specular position)
    t[−π2;π2]angle between the multilayer surface and the sample surface (=tilt)
    γBR+disorder of the layer alignment in the multilayer/angular dependency of brightness
    Hmax[λmin; λmax]maximal hue value (achieved at normal incidence geometry)
    γHR+angular dependency of hue
    nCoptical index of the material
    eR+thickness of the layer(s)
    Φinc[−π2;π2]angle between incidence fibre and sample surface (measured counterclockwise)
    Φcol[−π2;π2]angle between collection fibre and sample surface (measured clockwise)
    const.Rused to denote a constant whose value is not important for the calculations
    Figure 2.

    Figure 2. Schematic of a tilted multilayer (angle between the multilayer and the sample surface or tilt t = 40°) and incoming and reflected light rays relative to the multilayer structure (with angles θi and θr, respectively) and relative to the sample surface (with angles Φinc and Φcol, respectively). There is a relationship involving the tilt t between angles relative to the multilayer structure (θi and θr) and angles relative to the sample surface (Φinc and Φcol): θi = Φinc − t and θr = Φcol + t. The positive direction for each angle is figured by an arrowhead. The multilayer is composed of an alternance of two layers characterized by the optical indices n1 and n2 and their thicknesses e1 and e2. A schematic at a different scale, focusing on the goniometer, is available in the electronic supplementary material. (Online version in colour.)

    B(Φinc,Φcol)=Bmaxexp−((Φinc−Φcol)/2−t)22γB2,A 4 bis
    H(Φinc,Φcol)=Hmaxcos(γHΦinc+Φcol2)A 14 bis
    and
    S(Φinc,Φcol)=Smax.2.1
    Hereafter, we focus on brightness B and hue H because saturation S is constant no matter the angle configuration. The brightness B(Φinc, Φcol) in the angle space is entirely defined by three parameters: Bmax, t and γB. The tilt t is the angle between the multilayer structure and the sample surface (as illustrated in figure 2). Bmax is the maximum reflectance produced by the multilayer or thin-film structure, reached when the fibres are placed in a symmetrical configuration relative to the normal of the multilayer. γB is the parameter quantifying the disorder in the alignment of the multilayer structure. This disorder in the structure results in a reflected signal that is not purely specular but instead contains a diffuse component, meaning it can be seen at multiple angle configurations. For this reason, from a macroscopic point of view, γB is correlated with the angular dependency of brightness. Earlier studies used a binary classification of iridescent colours depending on the angle range at which the colour was visible (‘diffuse/directional’ in Osorio & Ham [110], ‘wide-angle/flashing’ in Huxley [55], ‘limited view’ of Vukusic et al. [134]). This classification is positively correlated with 1/γB.

    The hue H(Φinc, Φcol) in the angle space is defined by two parameters: Hmax which is the hue at coincident geometry (when using a bifurcated probe for example) and γH is the angular dependency of hue.

    The variations of brightness and hue in the angle space, according to equations (A 4) and (A 14), respectively, are represented in figure 3.

    Figure 3.

    Figure 3. Colour variables (a) brightness, (b) hue, (c) and (d) hue and brightness of an iridescent multilayer (with tilt t ≠ 0) in the angle space relative to the sample surface (Φinc, Φcol). The colour lines in (d) indicate alternative bases: the angle space relative to the multilayer structure (θi, θr) in blue and (Φinc + Φcol = 0, Φinc − Φcol = t) in orange and illustrates the terms ‘constant illumination’, ‘constant collection’, ‘constant angle bisector’ and ‘constant span’ used in table 1 and throughout this article.

    2.3. Angle and notation conventions

    In the rest of this study, we measure the incoming light ray angles (θi and Φinc) counter-clockwise and the outgoing light ray angles (θr and Φcol) clockwise. For both incoming and outgoing angles, the origin is the normal to the structures (θi and θr) or the normal to the sample (Φinc and Φcol). These conventions are represented in figure 2 where the direction of the arrows on angles represents the positive direction. The tilt t corresponds to the angle between the multilayer and the surface of the sample and is defined as t = Φinc − θi = θr − Φcol (see appendix A for more details about t). In other words, t is positive when the multilayer is tilted towards the illumination and negative otherwise (i.e. t is measured clockwise).

    3. Methods

    3.1. Study system: hummingbirds and butterflies

    We used hummingbirds and butterflies (more precisely some Morpho and Papilio species) as study systems. Hummingbirds make an ideal example to test our framework for numerous reasons. First, they belong to a speciose family where all species are iridescent [135], which allows us to work on a large number of species that diverged fairly recently [136]. Upon visual examination, they display highly different types of iridescent colours, with either ‘diffuse’ (usually on dorsal patches) or ‘directional’ (usually on facial or ventral patches) iridescence (sensu Osorio & Ham [110]). In addition, many species have highly tilted multilayers, providing a good test case to estimate the tilt t [110,114]. Finally, most species are available in large numbers in museum collections. We obtained the authorization from the Muséum National d’Histoire Naturelle to carefully cut feathers using surgical scissors. We selected one male from 36 species, evenly distributed across the phylogeny, from which we took feathers on two patches, one diffuse and one directional (sensu Osorio & Ham [110]).

    Because the exclusive use of hummingbirds as a test taxon for a new method has been criticized in previous studies [86], we also test our method on a very different group: butterflies. Butterflies are phylogenetically distant from birds and have different structures producing iridescence. For these reasons, the fact our method works in both taxa is a compelling argument for its universality. We used 17 butterfly species known to have multilayer structures [101,137]. The full list of species we used for our measurements is available in the electronic supplementary material, for both hummingbirds and butterflies.

    The method presented is also valid for whole specimens (whole birds instead of plucked feathers, for example). We nonetheless opted for the use of single feathers to maximize repeatability. Indeed, the precision of the goniometer measurements relies on the fact that the sample is precisely located at the centre of rotation of both fibres, which is more difficult to ensure for whole specimens.

    3.2. Reflectance measurements

    We measured reflectance at various angles using a purpose-built goniometer, following the recommendations of Meadows et al. [114]. The light emitted by a xenon lamp (300 W) over the 300–700 nm range of wavelengths to which birds are sensitive [138] was brought to the sample through an illuminating UV–visible optical fibre collimated to get a 1 mm light spot at normal illumination. Light reflected by the sample was then collected by a second identical collimated optical fibre and conducted toward an Oceanoptics USB4000 spectrophotometer. This set-up allows for a precise independent rotation of the illumination and the collection fibres, necessary for the measurements of iridescent colours.

    Our previous mathematical exploration (detailed in appendix A.2) revealed that hue is constant along the Φinc + Φcol = const. line (constant span) and brightness along the Φinc − Φcol = const. line (constant angle bisector), as illustrated in figure 3. We thus only need to take measurements in two orthogonal directions: in the direction Φinc − Φcol = const. to quantify hue variation and in the direction Φinc + Φcol = const. to quantify brightness variations. This will allow us to infer all parameters controlling hue and brightness, and therefore to potentially compute all values of hue and brightness in the entire angle space (Φinc, Φcol).

    The shape and size of the light spot on the sample depend on the position of the illuminating fibre relative to the sample. As the angle of illumination θi increases, the light spot becomes more and more elongated, according to a θi cosine function. This means the amount of light received by the spectrometer decreases when θi increases, independently of sample characteristics. This can also be empirically observed by taking measurements of the white reference (which is a Lambertian surface, i.e. reflectance does not depend on the angle) at different angles. To control for this, we took white reference measurements at several angle configurations (detailed in the protocol below). The white standard for this study was an Avantes reference tile WS-2. Because this is a diffuse (Lambertian) white reference and because some iridescent colours are very directional (i.e. all reflected light is focused in a single direction), it is expected to sometimes get values of brightness that can be over 100%.

    The detailed protocol we used for our measurements is similar to Waldron et al. [118] and inspired from Osorio & Ham [110] and Meadows et al. [114]. A detailed walk-through of the measurement protocol is presented in box 1, and a worked example is available in the electronic supplementary material.

    Box 1. Measurement protocol.

    (1)

    Move one of the two fibres of the goniometer to find the position where you get a signal of maximal intensity. This position depends on the tilt t of the multilayer and is therefore different for every sample. Once this is done, this means the angle bisector of the two fibres is close to the normal to the multilayer structure (red line in figure 2).

    (2)

    While keeping the same angle bisector, take measurements at different angular spans (orange line Φinc − Φcol = t in figure 3d). These measurements will be used to estimate hue parameters. To have a sample size large enough for reliable estimation and to stay at small angles, we recommend measurements at (Φinc, Φcol) ∈ {(t + 5°, t + 5°), (t + 10°, t + 10°), (t + 15°, t + 15°), (t + 20°, t + 20°), (t + 25°, t + 25°)}.

    (3)

    Take measurements while keeping the angular span between the two fibres constant (e.g. Φcol − Φinc = 20°) and moving the angle bisector (if you cannot do this, because for example, one of your fibres is not mobile, see appendix B.2). This will be used to estimate parameters related to brightness. We recommend three measurements on each side of the supposed normal to the multilayer structure (seven measurements in total) and a span of 20°: (Φinc, Φcol) ∈ {(t − 5°, t + 25°), (t°, t + 20°), (t + 5°, t + 15°), (t + 10°, t + 10°), (t + 15°, t + 5°), (t + 20°, t + 0°), (t + 25°, t − 5°)}. Depending on how directional your sample is, it may be needed to increase the resolution of the measurement grid and only move the angle bisector of 2.5° or 5° at each step.

    (4)

    Take white reference measurements with the same angular spans as before but using the normal to the goniometer as angle bisector (same measurements as in 2 but with t = 0°). If you have followed our advice for measurements, you should now take white measurements at (Φinc, Φcol) ∈ {(5°, 5°), (10°, 10°), (15°, 15°), (20°, 20°), (25°, 25°), (30°, 30°)}.

    (5)

    Take white reference measurements with a constant span but various angle bisectors (same measurements as in 3 but with t = 0°). If you have followed our advice of three measurements on each side to the supposed normal to the multilayer structure and a span of 20°, you should now take white measurement at (Φinc, Φcol) ∈ {( − 5°, 25°), (0°, 20°), (5°, 15°), (10°, 10°), (15°, 5°), (20°, 0°), (25, − 5°)}.

    We repeated each measurement twice, on different days, by two different experimenters for hummingbirds and butterflies. We performed statistical analyses after the completion of the measurement session to prevent experimenter bias.

    3.3. Statistical analyses

    As explained in the previous section, the angle configuration changes the shape of the light spot and thus the total possible amount of light collected by the collection fibre. To address this issue, we first pre-processed spectra to normalize count data using the appropriate reference white spectrum (script available in the electronic supplementary material). Resulting csv files were then imported in pavo R package [139]. Hue values were discarded (i.e. converted to NA) when brightness was lower than 8.5% because hue is not defined for black colours.

    Iridescence parameters can be estimated using various methods, including least-squares optimization and Bayesian nonlinear regression. We used a least-squares optimization as it is more common in biological sciences. We tested the Bayesian approach as well but it returned similar results and it is therefore not presented here.

    We used two indices to estimate the variability of the parameters resulting from our method: (i) relative standard deviation (RSD, also called coefficient of variation or CV) as the standard deviation divided by the absolute value of the mean. (Absolute) standard deviation (SD) is a common measure of the noise in a dataset. RSD is a way to quantify the signal-to-noise ratio. Because it is normalized by the mean value of the parameter, it is dimensionless and can be compared between parameters. It represents the precision of the experimental and statistical framework and does not depend on the sample population. (ii) Repeatability as the intra-class coefficient (ICC) computed with the rptR package [140]. ICC assesses whether the method allows one to discriminate individual samples among the population by comparing intra- and inter-sample standard deviation. ICC is therefore highly dependent on the sample population and on the biological question.

    RSD and ICC complement each other. A very precise method can still lead to non-repeatable measurements if there is no variability in the population. Conversely, a coarse method can work well enough to discriminate between samples and be repeatable if the variability between samples is high.

    4. Results and discussion

    Spectra from measurement along the ‘constant span’ (Φinc + Φcol = 20°) and ‘constant angle bisector’ (Φinc − Φcol = const.) lines after correction by the appropriate white reference are displayed in figure 4 for the iridescent blue of the breast of the hummingbird Heliomaster furcifer. We also show values of hue H and brightness B along these two measurement lines as well as the result from parameter estimation.

    Figure 4.

    Figure 4. Spectra (a,b) and corresponding values of brightness (c,d) and hue (e,f) at different angle configurations for the breast patch of the hummingbird Heliomaster furcifer along the Φinc − Φcol = const. (a,c,e; data points with round shape) and Φinc + Φcol = const. (b,d,f; data points with square shape) lines. Colours correspond to the conversion of the spectra in human vision using the CIE10 visual system. As expected, brightness is constant when Φinc − Φcol = const. and has a Gaussian shape when Φinc + Φcol = const. Conversely, hue has a cosine shape when Φinc − Φcol = const. and is constant when Φinc + Φcol = const. The red lines correspond to the fit of the functions after parameter estimation, with the values of the parameters. The R script to produce this figure is available in electronic supplementary material.

    4.1. Relative error and repeatability

    Variability and repeatability results are summarized in table 3. We find low values of RSD for hue-related variables for both hummingbirds and butterflies, indicating that our framework provides precise estimations of parameters. For brightness-related parameters, RSD is higher, as is usually the case, even for non-iridescent colours [141–143]. Despite relatively high RSD, all values for brightness remain repeatable, expected tilt t for butterflies because of a low inter-species variability, as demonstrated by the low value of SD.

    Table 3. Repeatability (ICC with likelihood ratio and permutation p-values) and standard deviations (SD and RSD) of iridescence parameters for hummingbirds and butterflies.

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    taxonvariableparam.meanSDRSD (%)ICCp (likel.)p (perm.)
    hummingbirdsbrightnessBmax36.6047.5414.790.947<0.00010.001
    t14.6118.217.4280.968<0.00010.001
    γB13.677.8511.190.8750.00090.002
    hueHmax556.8065.660.30040.997<0.00010.001
    γH0.640.182.2810.6890.0280.098
    butterfliesbrightnessBmax148.8099.786.910.936<0.00010.001
    t2.944.8332.960.2680.180.098
    γB5.355.124.760.769<0.00010.004
    hueHmax492.6927.870.24840.993<0.00010.001
    γH0.730.142.9930.853<0.00010.001

    4.2. Correlation between parameters

    4.2.1. Correlation between Bmax and γB

    Madsen et al. [105] noticed a negative relationship between brightness angular dependency and maximum brightness. From an evolutionary point of view, this means there is a trade-off between the signal brightness at a given angle and the range of angle at which it is not black (i.e. directionality sensu Osorio & Ham [110]).

    This correlation can also be proved theoretically. Indeed, the total energy of light that is reflected by the sample cannot exceed the received light energy. In other words, if absorption is similar across samples, the total brightness reflected in all directions is constant across samples:

    ∫∫B(Φinc,Φcol) dΦinc dΦcol=const.4.1
    The value of this double integral is known (B(Φinc, Φcol) is a bivariate Gaussian function) and when we compute it, we find
    Bmax2πγB2=const.4.2
    and
    Bmax∝1γB.4.3
    We indeed find a positive correlation between Bmax and 1/γB in the empirical data (F = 147.0742, d.f. = 1, p < 0.0001), illustrated in figure 5. We also notice an effect of the taxon (butterflies or hummingbirds) on the slope of the correlation (F1 = 8.3198, p = 0.0057). Because the link between Bmax and 1/γB was proven when ignoring absorption (equation (4.3)), this may suggest that absorption is higher in hummingbirds than in butterflies.
    Figure 5.

    Figure 5. Correlation between Bmax and directionality 1/γB. The dots are the data points. The lines show the result of the generalized linear model. (Online version in colour.)

    4.2.2. Correlation between angular dependency for hue γH and other parameters

    Osorio & Ham [110] found that γH and γB are negatively correlated among 15 bird species from different families. We do not find support for such correlation for either the hummingbirds or the butterflies (F1 = 3.1994, p = 0.074; figure in electronic supplementary material). Additionally, as discussed later in appendix B.3.2, many studies use variables that are correlated to Hmax to quantify hue angular dependence. On the contrary, we find that the parameters used in our method, Hmax and γH, are not correlated (F1 = 0.5167, p = 0.47; figure in electronic supplementary material).

    5. Conclusion

    Using both a theoretical and an experimental approach we find that hue and brightness can be easily characterized for all angle configurations using a set of five parameters (Hmax and γH for hue; Bmax, t and γB for brightness). Additionally, we show that a relatively small number of measurements is sufficient to reliably estimate these parameter values. This is made possible by the fact that hue is constant when the angular span between the two fibres remains constant (Φinc − Φcol = const.), and that brightness is constant for small angles as long as the angle bisector remains in the same position (Φinc + Φcol = const.) (as illustrated in figures 3 and 4). These properties have been previously noticed empirically for hue H1 by Osorio & Ham [110] on 15 bird species sampled from different families and Meadows et al. [114] on Calypte anna. Without being formalized, it had been illustrated for brightness in Eliason & Shawkey [104] and Stavenga et al. [77] for B3 as well as Stavenga et al. [78] for B1.

    Our contribution unlocks new perspectives for studies on iridescent colours, such as the evolution of complex visual signals leveraging angular dependency properties of iridescent colours.

    The proofs for the equation in this article are based on the multilayer theory. However, it is possible that parts of it may work for iridescence from diffraction gratings. Future studies should aim at integrating iridescence from diffraction into our framework. This would allow for a standard set of variables to describe iridescence, no matter its physical origin. Further investigation is also required to assess whether it is possible to relax some of the assumptions made in the paper under certain conditions.

    Data accessibility

    Data used in this study as well as scripts to apply the described method are available in the electronic supplementary material.

    Authors' contributions

    H.G. conducted the study (model construction, data analysis) and wrote the first version of this manuscript. H.G. performed measurements on hummingbirds and D.G. on butterflies. W.D.d.M. designed and built the goniometer. C.A., D.G., M.E. and S.B. contributed to the design of the goniometer. C.A., D.G. and W.D.d.M. helped with measurement protocol. D.G. and M.E. participated in the discussion for biological significance and pitfalls. All authors contributed to the final version of this article.

    Competing interests

    The authors declare no competing interests.

    Funding

    No funding has been received for this article.

    Acknowledgements

    We are grateful to C. Doutrelant and two anonymous reviewers for their valuable comments on the manuscript. We would also like to thank the Muséum National d’Histoire Naturelle and in particular J. Fuchs, P. Boussès and A. Previato for letting us sample feathers from museum hummingbird specimens, as well as V. Debat for lending us Morpho specimens to explore Morpho iridescence. Finally, we thank the Living Light conference organizers for putting together this special issue on structural colours in living organisms.

    Appendix A. Mathematical proof of the equations

    A.1. Brightness B in the angle space (Φinc, Φcol)

    For a perfectly regular multilayer, all the reflected signal is focused in the specular direction, at an angle θr equal to the incident angle θi. The brightness B is proportional to the reflected signal intensity, meaning

    B(θi,θr)={B(θi)if θi=θr0   if θi≠θr,A 1
    where B(θi) is defined by the Fresnel factor in the case of a thin-film structure (equation and R code to compute the Fresnel factor available in the electronic supplementary material). However, because we are dealing with small angles (assumption 1), we can approximate B(θi) to a constant Bmax (as illustrated in the electronic supplementary material):
    B(θi,θr)≈{Bmaxif θi=θr0   if θi≠θr.A 2

    But because biological structures are not entirely flat, and because the different layers of the multilayer structure are not perfectly aligned, there is also some amount of light reflected outside of the specular reflection (often referred as diffuse reflection). We thus assume a Gaussian decay of the brightness B around the specular position θi = θr (assumption 2), controlled by a parameter γB related to the disorder of the multilayer:

    B(θi,θr)≈Bmaxexp−((θi−θr)/2)22γB2.A 3
    In the case of a perfectly regular multilayer with no disorder, we have γB = 0 and we find equation (A 2). Conversely, if γB = +∞, the brightness value is the same for all angle configurations, which means we are dealing with a Lambertian surface.

    Additionally, the multilayer structure is not always parallel to the sample surface. It is the case, for example, for hummingbirds included in this study, as well as for Morpho butterflies in Berthier et al. [67], for the rainbow stag beetle, Phalacrognathus muelleri, structures described by Edo et al. [109], six pierid butterflies in Pirih et al. [102], 10 species of butterflies in Wickham et al. [80], and for six species of Heliconius butterflies in Parnell et al. [96]. So the illuminating angle Φinc and the collection Φcol at the macroscopic scale do not necessarily match θi and θr (as illustrated in figure 2). If we denote t the angle between the multilayer surface and the macroscopic sample surface (called tilt hereafter, as in Madsen et al. [105] and Osorio & Ham [110]), we get

    B(Φinc,Φcol)≈Bmaxexp−((Φinc−Φcol)/2−t)22γB2.A 4
    Using equation (A 4), we only have three parameters (Bmax, t and γB) to estimate to be able to reconstruct all values of brightness B in the angle space defined by (Φinc, Φcol). The resulting brightness in this space in plotted in figure 3.

    A.2. Hue H in the angle space (Φinc, Φcol)

    We defined the hue H as the wavelength for which reflectance is maximal. In the context of interferences, it is therefore the wavelength for which reflected light interferes constructively. For a regular multilayer, this happens when

    mH(θ1,θ2)=2(n1e1 cos θ1+n2e2 cos θ2),A 5
    where m is an integer (interference order), θ1 is the angle between the incident light ray and the multilayer structure at the interface between layer 1 and 2, θ2 is the angle between the transmitted ray after going through the first interface between layers 1 and 2 and the multilayer structure, n1 and n2 are the optical indices of the layers, and e1 and e2 the thicknesses of the layers. The products n1e1 and n2e2 are often called optical thicknesses of the layers 1 and 2 (respectively).

    The relationship between θ1 and θ2 is given by Snell’s Law:

    n1 sin θ1=n2 sin θ2.A 6

    Because θ1 ∈ [0; π/2], hue H increases when angle θ1 decreases according to equation (A 5). This means a maximum value for hue Hmax is achieved when θ1 = 0 (in this case θ2 = 0 as well because of Snell’s Law; equation (A 6)):

    mHmax=2(n1e1+n2e2).A 7

    We can replace n1e1 and n2e2 in equation (A 5) using equation (A 7):

    mH(θ1,θ2)=mHmax( cos θ1+ cos θ2)−2(n1e1 cos θ2+n2e2 cos θ1).A 8

    By adding equation (A 8) and equation (A 5), we obtain

    2mH(θ1,θ2)=mHmax( cos θ1+ cos θ2)+2( cos θ1− cos θ2)(n1e1−n2e2).A 9

    We consider here the case of an ideal multilayer, meaning that n1e1 = n2e2 (assumption 3). This allows us to simplify equation (A 9) into

    H(θ1,θ2)=Hmax cos θ1+ cos θ22.A 10
    Because we are working with small angles (assumption 1), Snell’s Law (equation (A 6)) can be approximated by
    θ2≈n1n2θ1A 11
    and
    H(θ1,θ2)≈Hmax cos θ1+ cos (n1/n2)θ12.A 12

    For small angles (assumption 1), this sum of cosine functions can be approximated by a single cosine function with twice the amplitude (numerical proof in the electronic supplementary material):

    H(θ1,θ2)≈HmaxcosγHθ1,A 13
    where γH≈(1+(n1/n2)2)/2 (after identification of the coefficients of the second-order Taylor series expansions in equations (A 12) and (A 13)).

    This reasoning is valid for ideal thin-film structures and multilayers and tells what happens at the specular position. But as explained in the previous section, biological structures are noisy and there is signal outside the specular position. As previously, if there is signal, this means that there is a multilayer for which the position of the fibres is specular. And in this case, we can apply equation (A 13) as well:

    H(Φinc,Φcol)=Hmaxcos(γHΦinc+Φcol2).A 14

    We only need two parameters (Hmax and γH) to plot all hue values in the angle space (Φinc, Φcol) as in figure 3. In the case of non-iridescent colours, we have γH = 0.

    A.3. Saturation S in the angle space (Φinc, Φcol)

    A.3.1. Along the ‘constant span’ direction (Φinc + Φcol = const.)

    We know that along the Φinc + Φcol = const. direction (constant span), hue is constant (as shown in equation (A 14) and figure 3b). Using a similar reasoning as in appendix A.1, we find that the reflectance R for a wavelength λ at a given angle configuration (Φinc, Φcol) is given by

    R(Φinc,Φcol,λ)=Rbisector(λ)exp−((Φinc−Φcol)/2−t)22γB2.A 15
    This means that reflectance spectra at all angle configurations along the ‘constant span’ axis (Φinc + Φcol = const.) can be derived by scaling of the spectrum at another angle configuration.

    The saturation S(Φinc, Φcol) is defined as the full width at half maximum of the reflectance spectrum R(Φinc, Φcol, λ). Let us call R the reflectance spectrum at a given angle configuration (Φincpos1,Φcolpos1). Then the saturation S at this configuration is

    S=λ1−λ2,R(λ1)=R(λ2)=Rmax2andλ1>λ2.}A 16

    If the reflectance spectrum R′ at (Φincpos2,Φcolpos2) is equal to R scaled by a factor s, then the saturation S′ is

    S′=λ1′−λ2′,R′(λ1′)=R′(λ2′)=Rmax′2andλ1′>λ2′,}A 17
    where
    R′(λ1′)=R(λ1′)s,R′(λ2′)=R(λ2′)sandRmax′=Rmaxs.}A 18

    From this, we find that

    R(λ2′)s=R(λ1′)s=Rmax2sA 19
    and
    R(λ2′)=R(λ1′)=Rmax2.A 20
    This means that λ′1 = λ1 and λ′2 = λ2. In other words, the full width at half maximum is stable by scaling, which results in the saturation S remaining constant along the Φinc + Φcol = const. axis (constant span).

    A.3.2. Along the ’constant angle bisector’ direction (Φinc − Φcol = const.)

    Additionally, along the Φinc − Φcol = const. axis (constant angle bisector), brightness is constant and only hue changes. This means spectra are translations of one another. The full width at half maximum is also stable by translation so the saturation S remains constant along Φinc − Φcol = const. axis (constant angle bisector).

    A.3.3. In the general case

    All points in the (Φinc, Φcol) space can be reached by a combination of moves along the orthogonal ‘constant span’ (Φinc + Φcol = const.) and ‘constant angle bisector’ (Φinc − Φcol = const.) axes. We just showed the saturation S is constant along these two axes so it is actually constant in the whole (Φinc, Φcol) space.

    Appendix B. Comparison with other methods

    B.1. Measurements at fixed angle configuration

    The angle t between the multilayer structure and the normal to the surface of the feather (tilt) is highly variable between species of the same family (SD = 19.36° in hummingbirds, as reported in table 3). This is in agreement with Osorio & Ham [110] who found tilt values t ranging from − 20° to 40°. Even if the angle configuration (Φinc, Φcol) is constant at the macroscopic scale, the configuration relative to the multilayer structure (θi, θr) may not be constant because of the variation in the tilt t between samples. This means measurements at fixed geometry cannot be compared between samples. For this reason, we warn against measurements of iridescent colours at a fixed angle, even when angular dependency is not studied.

    B.2. Parameter estimation using constant illumination

    Some goniometers only allow for the rotation of the collection fibre while the illumination fibre stays at a fixed position. Measurements realized with a such protocol can still be used with our method but this leads to a loss of statistical power.

    If illumination is provided at a fixed angle Φinc = α:

    B(Φcol)=Bmaxexp−((α−Φcol)/2−t)22γB2=Bmaxexp−(Φcol+2t−α)28γB2.B 1

    So, B(Φcol) is still a normal function of Φcol with the same maximum value Bmax but with parameters t* = 2t − α and γB* = 2γB for mean and standard deviation, respectively.

    Because the estimation of the parameters of a normal function through a regression is more reliable when the standard deviation is low, using anything else than a fixed normal as measurement line, such as a fixed illumination, to study brightness parameters will result in less accurate values.

    Additionally, depending on the exact value of α, it may not be possible to have a fibre configuration where (α + Φcol)/2 = t but the span between the fibres is still less than 90° (small angles assumption). In this case, data points never reach the maximum Bmax, which makes parameter estimation very unreliable.

    Finally, the new value of the mean t* does not have a direct biological and physical interpretation, as opposed to t which is the tilt of the multilayer of thin-film structure.

    For hue, if illumination is at fixed angle α

    H(Φcol)=Hmaxcos(γHα2+γH2Φcol).B 2

    The equation for hue at fixed illumination has a shape different from its general form depending on the span between the fibres, (Φinc + Φcol)/2. There is a constant term in the cosine function and the new term for hue angular dependency is γH* = γH/2. As we explain in the next section, the estimation of the parameters is more reliable for high values of γH. For this reason, the parameter estimation at fixed illumination may not be as precise as along the Φinc + Φcol = const. line.

    B.3. Link with other variables of angular dependency for hue

    B.3.1. Linear regression

    Linear regression instead of cosine regression to estimate Hmax and γH is common [63,75,110,121]. Because the curvature of the cosine function in equation (A 14), defining hue depending on the angular span, is often small, we obtain congruent results using either cosine or linear regression. However, this creates a systematic bias where Hmax is more overestimated for samples with larger angle dependency γH. Indeed, a linear regression overestimates more the intercept value as the curvature of the function increases.

    B.3.2. Difference between two angle configurations with the same angle bisector

    The difference in hue between two angle configurations is sometimes used as a proxy for iridescence [71]. However, it is problematic because it leads to a very high correlation between hue and iridescence, as reported in Dakin & Montgomerie [66] (R2 > 0.95).

    We can prove mathematically this linear correlation. Let us focus on the difference between hue Hpos1 at a given angle configuration (Φinc1,Φcol1) and hue Hmax at coincident geometry (i.e. Φinc + Φcol = θi + θr = 0). It follows from equation (A 14) that defines the hue at any angle configuration that:

    Hpos1−Hmax=Hmax[cos(γHΦinc1+Φcol12)−1].B 3

    From this equation, we see that if γH is constant or displays low variability between samples, Hpos1 − Hmax is proportional to Hmax:

    Hpos1−Hmax∝Hmax.B 4

    We can apply the same reasoning and prove the difference Hpos2 − Hmax between hue Hpos2 at (Φinc2,Φcol2) and Hmax is proportional to Hmax:

    Hpos2−Hmax∝Hmax.B 5

    Thus (doing equations (B 4) and (B 5)), the difference in hue between any two angle configurations (Φinc1,Φcol1) and (Φinc2,Φcol2) is proportional to Hmax:

    Hpos1−Hpos2∝Hmax.B 6

    This correlation between the two variables characterizing hue in the angle space can lead to errors in subsequent statistical inferences. On the opposite and as reported in §4.2.2, the parameters proposed in this study (Hmax and γH) do not have the same issue.

    B.4. Link with other variables of angular dependency for brightness

    We are providing the following comparison with variables that have been previously used in the literature to describe brightness angular dependency. This means that values from previous studies using these variables can still be used in a meta-analysis or a discussion using our new variables Bmax, t and γB. We however explain why they are less precise, less versatile and/or more time consuming than those measured under our unified framework.

    B.4.1. Full width at half maximum and angular breadth

    We have shown brightness is a Gaussian function of standard deviation γB along the line of ‘constant span’ (Φinc + Φcol = const. direction). Many studies previously characterized angular dependency in this direction using the full width at half maximum (hereafter FWHM) [80,102,107,110,113]. For a Gaussian function, there is an easy link between standard deviation and FWHM:

    FWHM=2γB∗2ln⁡2=4γB2ln⁡2≈4.71γB.B 7

    Similarly, some studies use what they call angular breadth [85,86,88–92,112], which they define as the range of angle where brightness is higher than 3% of its maximum (threshold at 10% for [112]):

    ang. breadth=2γB∗4ln⁡10−2ln⁡3=4γB4ln⁡10−2ln⁡3≈10.59γB.B 8

    We see that these variables are proportional to γB in theory. However because they are computed from raw data, without any pre-processing or curve fitting, they are more sensitive to noise.

    B.4.2. Hunter’s specular gloss and integrating sphere

    Multiple studies [75,144,145] use Hunter’s gloss [146], defined by the ratio of specular to diffuse reflectance. This method is convenient because it can easily be achieved using an integrating sphere to capture the needed spectra in two measurements only (one at specular position without the sphere and one with the sphere to capture diffuse and specular reflectance).

    This is equivalent to keeping the illumination at a fixed angle and measuring reflectance at all collection angles. We already know the brightness at the specular position is Bmax. The diffuse reflection is the integral on all angle configurations of the brightness. Hence Hunter’s specular gloss G using the notation defined in this study is

    G=Bmax∫∫B(Φinc,Φcol) dΦinc dΦcol.B 9

    The integral of brightness for every angle configurations is BmaxγB∗2π (integral of the normal with maximum Bmax and standard deviation γB∗), which gives

    G=1γB∗2π=12γB2π.B 10

    However, this is assuming the measurement of Bmax was actually done at the normal to the multilayer (Φinc + Φcol)/2 = t. But there is no way to know whether it is the case without doing several goniometer measurements with different normal positions. Once this is done, γB can be estimated without doing additional integrating sphere measurements.

    B.4.3. Difference/quotient between maximum and another position with the same span

    Some studies [84,86,111] use the difference or the quotient between the brightness at the fibre position where it is maximum and another position. With this approach, they find t and Bmax.

    The difference or the quotient between these two positions can easily be linked to γB because we know that B(Φinc, Φcol) is a normal function of parameters t and γB.

    However, this is very sensitive to noise and measurement error because Bmax and t are estimated with only one data point and γB (or its equivalent variable) with only two data points.

    Appendix C. Structural colours with pigmentary component

    The framework we presented here focuses on purely structural iridescent colours. However many colours integrate both pigmentary and structural components [147,148]. If there is a pigmentary component, it adds constant term Bpigment to brightness B:

    B(Φinc,Φcol)=Birid+BpigmentC 1
    and
    B(Φinc,Φcol)=Bmaxexp−((Φinc−Φcol)/2−t)22γB2+Bpigment.C 2

    This can easily be investigated using our protocol and statistical framework. The only difference is that four parameters (Bmax, t, γB and Bpigment) instead of three need to be estimated by running a nonlinear regression on equation (C 2) instead of equation (A 4).

    There are cases where the structural and pigmentary components of colour act on very different regions of the light spectrum. This happens, for example, in Colias eurytheme [62], where iridescence is restricted to the UV region while the visible region colour is caused by pigments. In this case, our method can be applied directly by restricting the studied wavelength range to the region of interest (this option is available in the code provided in the electronic supplementary material).

    Footnotes

    One contribution of 11 to a theme issue ‘Living light: optics, ecology and design principles of natural photonic structures’.

    Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4275911.

    © 2018 The Author(s)

    Published by the Royal Society. All rights reserved.

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