Quantifying the refractive index dispersion of a pigmented biological tissue using Jamin–Lebedeff interference microscopy

Jamin–Lebedeff polarizing interference microscopy is a classical method for determining the refractive index and thickness of transparent tissues. Here, we extend the application of this method to pigmented, absorbing biological tissues, based on a theoretical derivation using Jones calculus. This novel method is applied to the wings of the American Rubyspot damselfly, Hetaerina americana. The membranes in the red-colored parts of the damselfly’s wings, with a thickness of ∼2.5 µm, contain a pigment with maximal absorption at ∼490 nm and a peak absorbance coefficient of ∼0.7 µm−1. The high pigment density causes a considerable and anomalous dispersion of the refractive index. This result can be quantitatively understood from the pigment absorbance spectrum by applying the Kramers–Kronig dispersion relations. Measurements of the spectral dependence of the refractive index and the absorption are valuable for gaining quantitative insight into how the material properties of animal tissues influence coloration. Scientists have used Jamin–Lebedeff interference microscopy to measure the refractive index of strongly pigmented biological tissue. Doekele Stavenga and co-workers from the University of Groningen in the Netherlands used the technique to analyze the refractive index, and its variation with wavelength, of the red wings of the American Rubyspot Damselfly. Until now, this microscopy scheme has been applied only to transparent, unpigmented tissue. The researchers report that the high density of pigment in the damselfly wing results in a significant anomalous dispersion of its refractive index, with the index increasing for longer wavelengths. This is believed to enhance the reflectivity in the red spectral region and thus improve the colour contrast of the wing decorations. Similar contrast shown effects are thought to be present in the eyes of some crustaceans.


INTRODUCTION
Pigmentary coloration occurs in tissues in which pigments selectively absorb light in a certain wavelength range while light outside this absorption band is scattered, thus coloring the tissue. Pigmentary coloration is common in many animals; for instance, blue-absorbing carotenoids produce the yellow color of the feathers of many songbirds, 1,2 and the ultraviolet-and blue-absorbing pterins in pierid butterfly wing scales produce white (Small White, Pieris rapae) and yellow (Eastern Pale Clouded Yellow, Colias erate) wings, respectively. 3 Due to their usually high concentration, the pigments act as effective highpass spectral filters. However, because absorption is intimately linked to the refractive index, a strongly absorbing pigment can increase the refractive index and can thus strongly influence the coloration.
A quantitative understanding of the coloration of a tissue requires an analysis of both the spatial distribution and the optical properties of the tissue components. For anatomy, well-established methods include light and electron microscopy, and for pigment characterization, several spectrophotometry approaches exist. 4 For microscopic objects, refractive index matching fluids are often used, with Jamin-Lebedeff interference microscopy being one of the most sensitive methods. Thus far, such investigations have been conducted only on transparent, unpigmented biological tissues. [5][6][7] In the visible wavelength range, transparent materials have a fairly constant refractive index, i.e., they exhibit little dispersion; thus, refractive index measurements are commonly restricted to one wavelength. However, Jamin-Lebedeff interference microscopy is also suitable for a broad wavelength range, as we recently demonstrated using bird feathers and butterfly wing scales, where we determined the spectral dependence of the refractive index of the main components, keratin and chitin. 8 Here, we show that the Jamin-Lebedeff interference microscopy method can also be applied to pigmented tissues, where the dispersion is not insignificant. We first present a theoretical derivation for the measurements, and we subsequently demonstrate the method using the densely pigmented, red wing regions of the American Rubyspot, Hetaerina americana, a damselfly found throughout North America.

Animals
Specimens of the American Rubyspot, Hetaerina americana, were obtained from the National Museum of Natural History (Naturalis, Leiden, The Netherlands; curator Dr J van Tol). Small sections were cut from the wings for Jamin-Lebedeff interference microscopy and microspectrophotometry.

Scanning electron microscopy
The ultrastructure of the wing pieces was investigated with an XL-30 ESEM scanning electron microscope (Philips, Eindhoven, The Netherlands). Prior to imaging, the wing pieces were sputtered with palladium.

Spectrophotometry
Reflectance spectra of the differently colored wing areas were measured in air with a bifurcated fiber-optic probe. The instrument comprises six light guides, delivering light from a halogen-deuterium source (AvaLight-D(H)-S; Avantes, Eerbeek, The Netherlands). These illuminating light guides collectively surround a central fiber that acts as a collector of scattered light (spot diameter: ,1 mm), which is delivered to a fiber optic spectrometer (Avantes, AvaSpec-2048). A white diffusing reflectance standard (Avantes WS-2) was utilized as the reference. Using the oscillations in the reflectance spectra, the average wing membrane thickness was derived via a thin film analysis as presented previously. 9 Absorbance spectra of the membranes in between the wing veins were measured with a microspectrophotometer (MSP), which consisted of a xenon light source, a Leitz Ortholux microscope and a spectrometer. The microscope objective was an Olympus 203, NA 0.46. The measurements were performed on wing pieces immersed in a refractive index matching fluid to reduce scattering artifacts. Due to the optics of the glass, the spectral range of the MSP is limited to wavelengths greater than 380 nm. For the Kramers-Kronig calculations, the absorbance coefficient spectrum was extrapolated by assuming that the spectrum consists of two absorption bands, a and b, which can be described by the modified lognormal function: k~a exp {bx 2 1zcxzdx 2 À Á Â Ã , with d53c 2 /8 and x5log 10 (l/l max ) (see Ref. 10). For the a band in Figure 7a, the parameters were a50.374 mm 21 , b5215, c57.07, and l max 5491 nm, and for the b band, a50.598 mm 21 , b5476, c5211.0 and l max 5300 nm.

Jamin-Lebedeff interference microscopy
The wing pieces were placed on a microscope slide and covered by a cover slip after being immersed in a fluid with a refractive index varying between 1.46 and 1.64 (series A of Cargille Labs, Cedar Grove, NJ, USA). The wavelength dependence of the refractive index of each immersion fluid was derived from the refractive index value given for a wavelength of 586 nm and from its Abbe number by using the Cauchy equation n(l)5A1B/l 2 and by calculating the parameters A and B. The microscope slide was mounted on the stage of a Zeiss Universal Microscope set up for Jamin-Lebedeff polarizing interference microscopy. The microscope objective was a Zeiss Pol-Int I 103/0.22. As a light source, we used a halogen lamp, and specific wavelengths were selected by inserting narrowband interference filters (halfwidth: ,15 nm) below the condenser. At each wavelength, photographs were taken at a series of angular positions of the analyzer with a Coolsnap ES monochrome camera (Photometrics, Tucson, AZ, USA). The images were subsequently evaluated with ImageJ, 11 and the obtained sinusoidal intensity profiles were analyzed with Matlab routines.

RESULTS AND DISCUSSION
Theory of Jamin-Lebedeff interference microscopy for absorbing media To determine the refractive index of an object with Jamin-Lebedeff interference microscopy, the object is embedded in a reference fluid with a known refractive index. Figure 1 presents a diagram of the system. 5,6 A light beam delivered via the condenser of the microscope first passes through a linear polarizer, P. A birefringent beam splitter, BS, divides the beam into an ordinary and extraordinary ray. The polarization is then reversed by a half-wave plate, H. One ray proceeds unhindered through the immersion fluid (Figure 1, grey), while the other ray passes through the object O. The two rays are subsequently combined by a beam combiner, BC. The joined rays then pass through a Sénarmont compensator, S, which consists of a quarter-wave plate, and finally pass through an analyzer, A.
We assume that the incident beam has one unit power of light intensity after passing through the polarizer P and that the beam splitter BS splits the incident beam into two rays (a and b; Figure 1) with equal intensity. The propagation of a polarized light beam can be treated with the Jones formalism, where the two components of the beam are given by a vector. When the beam passes through an optical element that changes its polarization, this is equivalent to a matrix operation. At level 1 (Figure 1), the two rays are then described by the Jones vectors: 12 When a light beam is monochromatic (wavelength l), a plate with thickness d and refractive index n o causes a phase shift Q o 5kn o d (k52p/l is the wavenumber). The reference medium, with (real) refractive index n r , that surrounds the object introduces a phase shift Q r 5kn r d. Thus, the phase difference between the two rays at level 2 ( Figure 1) is d5k(n r 2n o )d. Generally, the refractive index of the object is complex, n o 5n oR 1in oI , with real and imaginary parts n oR and n oI . The phase shift created by the object is then also complex, or d5d R 1id I 5k(n r 2n oR )d2ikn oI d, i.e., d R 5k(n r 2n oR )d and d I 52kn oI d.
The Jones matrix of the Sénarmont compensator, operating from level 3 to 4 ( Figure 1), is 12 and the Jones matrix of the analyzer rotated over an angle a is Consequently, the Jones vector of the light beam at level 5 ( Figure 1) is The resulting light intensity of the beam is a sinusoidal function of the analyzer angle a with period p (or, 180 6 ) and with amplitude a5exp (d I ) and an offset b5(12a) 2 /4; Da is the phase difference between the reference and object intensity or the angle needed to rotate the analyzer to fully compensate for the phase difference induced by the object: (when the angle is given in degrees Da5180(n r 2n oR )d/l). The phase difference is a linear function of the reference fluid with slope s5pd/l. From the slope, the thickness of the object can be derived: d5sl/p. The amplitude is directly related to the absorption coefficient, k, of a homogeneous object because k52kn oI 522d I /d, which can be derived from the transmittance, T5exp (2kd), or When the imaginary part of the refractive index of the object is negligible, then d is real; thus, a51 and b50. Then, I(a)5cos 2 (a2Da), following the angle-dependent intensity function used for transparent objects (e.g. Ref. 8).
We note that the beam splitter BS and the beam combiner BC (Figure 1) are designed to be identical, and therefore, when the two beams a and b do not have the same amplitude, the formulae derived for the intensity and phase difference remain the same.

Wing anatomy and pigmentation
We have applied the above-described theory to the case of the American Rubyspot, Hetaerina americana, a damselfly of the family Calopterygidae. Most of the damselfly's wings are colorless and transparent, but they are strikingly red-colored near the thorax (Figure 2a and 2b). The wing is a meshwork of wing veins, forming a frame of cells with thin membranes (Figure 2 and 3a). In the red wing parts, both the veins and the membrane between the veins are red pigmented (Figure 2c). Scanning electron microscopy shows that the membranes approximate a thin film, but the surface is often rough due to numerous protrusions (Figure 3b). The membrane thickness varies from below 1 mm to approximately 3 mm from the wing tip to the base near the thorax.
The thin film properties of the membrane are immediately recognizable in the reflectance spectra ( Figure 4). The reflectance spectrum  of the unpigmented wing part (Figure 4a, clear) exhibits strong oscillations; the periodicity yields an average thickness of 0.87 mm. The reflectance measured in the red wing part (Figure 4a, red) is low in the short-wavelength region because the pigment has a peak absorption at ,490 nm (Figure 4b). In the long-wavelength range, the reflectance spectrum shows oscillations corresponding to a thin film with varying thickness; the average thickness in the specific case of Figure 4b was estimated to be 1.9 mm, but at other locations, the average thickness was different (in the range of 1-3 mm).

Refractive index and absorption coefficient
The effect of the absorbing pigment on the refractive index of the wing membranes can be studied with Jamin-Lebedeff interference microscopy.
We therefore cut small sections out of the wing, immersed them in a fluid with a known refractive index, and investigated the sections via interference microscopy with monochromatic light. The image intensities of the wing section and the outer reference area vary sinusoidally with the angular position of the analyzer ( Figure 5). Both the amplitude of the sinusoidal intensity of the wing section relative to the amplitude of the reference and the phase difference are wavelength-dependent (Figure 5d). A series of measurements similar to those shown in Figure 5 were performed on three wing sections from the red wing parts, achieved by immersing the sections in immersion fluids with refractive indices of 1.51, 1.55 and 1.60 (at 586 nm; Figure 6). The image intensity of an area in the wing section and that of a reference area outside the section were measured as a function of the analyzer position for a range of visible wavelengths. Figure 6a (symbols) shows the ratio of the amplitude of the resulting sinusoids calculated for each wavelength. According to Equation (7), the amplitude equals the square root of  Refractive index dispersion of biological pigment DG Stavenga et al 4 the transmittance. The absorbance is defined as D(l)52log 10 [T(l)], and the transmittance spectrum was thus calculated from the absorbance spectrum of Figure 4b with T(l)510 2D(l) (Figure 6a, red line). Figure 6b shows the phase difference between the sinusoids of the wing sections and the reference as a function of wavelength for the three cases of immersion fluids. Linear functions fitted to the phase difference data points for each wavelength, as shown in Figure 6b, yielded a thickness of d52.560.1 mm. The fitted linear functions produced a zero value when the refractive index of the wing membrane equaled the refractive index of the reference: n oR 5n r . This fitting yielded the refractive index spectrum of the wing membrane (Figure 6c, red symbols and curve). The transparent wing area, which was also investigated at a few wavelengths, appeared to have a much lower refractive index (Figure 6c, blue symbols), coinciding well with the dispersion curve obtained for unpigmented chitin. 8 This result suggests that the unpigmented wing regions consist of pure chitin and that the increase in the refractive index of the red-pigmented wing regions is caused by the presence of short-wavelength-absorbing pigment. Figure 6c reveals that the refractive index of the red-pigmented wing membrane has a marked anomalous dispersion. A material generally exhibits anomalous dispersion in the wavelength range of a distinct absorption band. Therefore, it is likely that the measured dispersion curve (Figure 6c) is caused by the absorbing pigment represented in Figure 4b. This assumption can be quantitatively assessed with the Kramers-Kronig dispersion relation (e.g., Refs. 13 and 14): which yields the contribution to the real part of the refractive index, Dn Rj , from band j of the absorption spectrum of the pigment with absorption coefficient k j (l). The absorption coefficient of the wing membrane medium can be derived from the transmittance spectrum, T(l), and from the absorbance spectrum, D(l), with k(l)52ln [T(l)]/d52.3D(l)/d. Using the estimated thickness d52.5 mm, the absorption coefficient spectrum of Figure 7a (red curve) was obtained. To allow for proper Kramers-Kronig calculations, the spectrum was extrapolated to the shorter wavelength range and was assumed to consist of an a and b band, which were fitted with modified lognormal functions (see the section on 'Materials and methods'; Figure 7b, blue curve). Figure 7b shows the contributions to the refractive index calculated from Equation (8). Both absorption bands create a distinct anomalous dispersion curve; for the a band, a peak at ,550 nm and a valley at ,430 nm are obtained. The b band increases the refractive index in the visible wavelength range, where it exhibits normal dispersion, reducing the modulation in the anomalous dispersion curve of the a band. Figure 7b furthermore shows the difference between the refractive indices of the damselfly's red and clear wing membranes (red curve), as calculated from Figure 6c. The latter spectrum is shifted upwards from the calculated dispersion curve by about 0.2-0.3. An obvious interpretation of this upward shift is that it is caused by absorbing bands located at wavelengths of less than 300 nm. These bands could, for instance, arise from specific proteins to which the pigment molecules are bound and that have strong absorption bands in the very short-wavelength range. The nature of the pigment of the damselfly wings is uncertain, but the absorption spectrum resembles that of xanthommatin, 15 a member of the ommochromes, a family of pigments widely encountered among insects. 16 The pigment density is high, with a peak absorption coefficient of ,0.5 mm 21 , resulting in a substantial anomalous dispersion of the refractive index, with a peak value of ,1.595 at 550 nm and a low of ,1.578 at 450 nm (Figure 6c). The local density of the pigment shows some variation and thus results in a variable transmittance. This feature presumably causes the slightly variable wavelength dependence of the sinusoid amplitudes (Figure 5c). The wing membranes are not ideal thin films due to numerous protrusions, which create a rough surface (for similar rough membrane surfaces on dragonfly wings, see Ref. 17). The protrusions are expected to have a negligible effect on the measurements because of the small difference between the refractive indices of the wing membranes and the reference fluid.
Generally, in pigmentary coloration, a short-wavelength-absorbing pigment reduces the reflection/scattering of incident light in the shortwavelength range, leaving the longer wavelength light relatively unobstructed, thus resulting in a yellow, orange or red color, depending on the extent of the pigment's absorption spectrum. Interestingly, a high pigment density causes an increase in the real part of the refractive index in the longer wavelength range, which will enhance the reflectivity of the tissue at the longer wavelengths. Preliminary measurements in several butterfly species suggest that this optical effect of enhancing the color contrast by increasing the refractive index is  (8)).
Refractive index dispersion of biological pigment DG Stavenga et al 5 widespread. The dual effect of pigment absorption, namely, the absorption of short-wavelength light and the enhanced reflection of long-wavelength light, can also be recognized in reflecting pigment cells and tapeta in the eyes of crustaceans [18][19][20] and presumably also in the coloration caused by pigments in cephalopods. 21 We previously studied the pigmented wings of another damselfly, Calopteryx japonica. 22 The pigment in that case is melanin, causing brown-colored wings in the immature female, the mature female and the immature male. However, the mature males have distinctly bluecolored wings. The blue color is present in the wing veins, where melanin is concentrated in layers that alternate with relatively unpigmented layers, thus creating an optical multilayer that strongly reflects in the blue wavelength range. The melanin pigment in this case causes structural coloration. The refractive index dispersion of the melanin layers was found to be normal in the visible wavelength range, that is, the refractive index decreases with increasing wavelength, due to the very broad melanin absorption spectrum, which decreases monotonically with increasing wavelength from the ultraviolet to the infrared region. The strong absorption of the melanin in the very short-wavelength range causes the melanin to endow the tissue with a very high real part of the refractive index, resulting in local refractive index values above 1.8. 22 Consequently, when the melanin is periodically arranged, strong photonic effects can emerge. This possibility is exploited by many animals, e.g., in bird feathers [23][24][25] and beetle cuticles 26,27 (see also Refs. 28 and 29). We note that there are even several examples in which periodically arranged, structurally colored tissue contains additional pigment that selectively filters the reflected light, thus tuning and sharpening the reflectance spectrum. 21,30,31 Pigmentary and structural coloration thus are not strictly separable.

CONCLUSION
Using a newly developed method based on Jamin-Lebedeff interference microscopy, we have investigated the red-pigmented wings of the damselfly H. americana and quantitatively assessed the contribution of the blue-absorbing pigment to the overall refractive index. In all cases of animal coloration, the refractive index of the colored tissue plays a central role, specifically in the very thin tissues of insect wings, butterfly wing scales and bird feather barbules, which often display a strongly saturated pigmentary color. The latter example implies a high pigment density with high refractive indexes as an immediate consequence. As the refractive index depends on wavelength, i.e., featuring dispersion, a quantitative knowledge of the refractive index is key to understanding the resulting coloration. The novel approach based on Jamin-Lebedeff interference microscopy for absorbing media, as outlined in the present paper, allows for straightforward measurements of the refractive index of microscopic objects, which will be valuable for gaining quantitative insight into how the material properties of animal tissues influence coloration.