PHYSICAL REVIEW E 83, 051917 (2011)

Direct determination of the refractive index of natural multilayer systems

Shinya Yoshioka* and Shuichi KinoshitaGraduate School of Frontier Biosciences, Osaka University, Osaka 565-0871, Japan

(Received 23 February 2011; published 18 May 2011)

It is well known that the metal-like strong reflection observed in the elytra of some kinds of beetles is producedby multilayer thin-film interference. For the quantitative analyses of the structural colors in these elytra, it isnecessary to know accurate values of the refractive indices of the materials that comprise the multilayer structure.However, index determination is not an easy task: The elytron surface is not flat but curved and usually containsmany irregular bumps, which cause scattering loss. These structural characteristics prevent us from directlyapplying conventional optical techniques for index determination, such as ellipsometry, since these techniquesrequire a perfectly specular surface. In this paper, we report a new experimental procedure that can directlydetermine the refractive indices of individual layers in natural multilayer systems. This procedure involvessemi-frontal thin-sectioning of the sample and subsequent optical examinations using a microspectrophotometer.We demonstrate that the complex refractive index and its wavelength dependence can be successfully determinedfor one kind of beetle.

DOI: 10.1103/PhysRevE.83.051917 PACS number(s): 87.80.−y, 78.20.Ci, 42.25.Hz

I. INTRODUCTION

In the developing field of biomimetics, the structural colorin nature has attracted considerable scientific interest [1–6].Numerous studies have discovered interesting optical systemsin various organisms, for example, in butterfly wing scales[7–10], beetle elytra [11–14], bird feathers [15–19], andiridophores of fish and cuttlefish [20–22]. In addition, somenew experimental techniques have been reported for detailedoptical characterization of the structural colors [23–26].In spite of these advances, there have not been sufficientquantitative analyses even for one of the simplest colorationmechanisms, multilayer optical interference, owing to limitedknowledge of the refractive index of materials that comprisemultilayer systems.

The elytra of some beetles are among the most commonexamples of structural color found in nature. Previous studieshave revealed that electron-dense and electron-lucent layersare stratified beneath the elytron surface and have thicknessescomparable with the wavelength of light [27–31]. Thus, there isnow little doubt that the metallic luster of these elytra is causedby multilayer thin-film interference. Theoretical evaluationof the optical properties of these elytra, such as angle- andpolarization-dependent reflectance, requires accurate valuesof the refractive indices. However, unlike the case of artificialthin-film systems, it is difficult to determine the refractiveindex values for such multilayer systems in beetles. Thisdifficulty is mainly because the elytron surface is not flatbut curved and usually contain numerous small irregularbumps that can cause scattering loss. Moreover, the layerthicknesses may not be uniform over the optically examinedarea. These structural characteristics, which are inherent innatural samples, make it difficult to directly apply standardoptical techniques such as ellipsometry to the elytra of beetles,since such techniques require a large and specular surface ofthe sample.

*syoshi@fbs.osaka-u.ac.jp

Recently, Noyes et al. [30] experimentally measured theangle- and polarization-dependent reflection spectrum of abuprestid beetle and simultaneously analyzed the spectra todetermine the refractive indices of two types of materials thatcomprise the multilayer structure and the thicknesses of a totalof 17 layers. This approach should work in principle. However,a fitting procedure with many adjustable parameters generallyresults in several parameter sets that reproduce similar spectra.In particular, optically absorbent layers or scattering lossresults in the smoothing out of detailed spectral features suchas sharp peaks and dips, making it more difficult to obtain aunique parameter set through a fitting analysis.

To overcome these difficulties, we report an experimentalprocedure to determine the complex refractive index ofnatural multilayer systems and its wavelength dependence.The procedure consists of two experimental steps. First,the multilayer structure is thin-sectioned so that the sectionis nearly, but not perfectly, parallel to the layers (semi-frontal thin-sectioning). Although the individual layers areapproximately 100 nm thick, the multilayer system appearsas a broad stripe pattern in the semi-frontal thin section,the stripe width of which is expanded to more than a fewmicrometers. Second, a microspectrophotometer is employedto quantitatively measure the reflection and transmissionspectra from small regions of the thin section that correspondto individual layers. These spectra are analyzed using thethin-layer interference model to obtain wavelength-dependentcomplex refractive index values. In this paper, we report thatthis method can be successfully applied to the elytron of thejewel beetle Chrysochroa fulgidissima, shown in Fig. 1(a),which is famous for its use as decoration in ancient Buddhistcraft [32].

II. EXPERIMENTAL METHODS

Samples were supplied by Prof. Hirowatari at OsakaPrefecture University. Since the elytron was already driedand solid, it was directly embedded in epoxy resin (Quetol-812, Nisshin EM) without dehydrating or fixing procedures.After the sample (the red striped part of the elytron) was

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SHINYA YOSHIOKA AND SHUICHI KINOSHITA PHYSICAL REVIEW E 83, 051917 (2011)

(b)(a)

FIG. 1. (color) (a) Jewel beetle Chrysochroa fulgidissima and (b)an electron micrograph of the multilayer system beneath the elytronsurface [the red striped part in (a)]. Scale bar: 500 nm.

thin-sectioned at an approximate thickness of 70 nm using anultramicrotome (Reichert, UltracutN), it was observed usinga transmission electron microscope (TEM, Hitachi H-7650).The thickness of the section was more accurately determinedlater, as described in Sec. IV. A staining process for TEMobservations was carried out so as to analyze the actualelectronic density of the multilayer structure.

We used a microspectrophotometer that consisted ofa fiber-optic spectrometer (Ocean Optics Inc., USB 2000) anda microscope (Olympus, BX51) equipped with two lamps−axenon lamp for epi-illumination and a tungsten lamp fortransmission illumination. The optical system was similar tothat described previously [33]. A dry objective (Olympus,MPLFLN100x, NA 0.90) was used to measure the reflectionspectrum in air. On the other hand, an oil-immersion objective(Olympus, UPLSAPO100xO, NA 1.40) was used to measurethe transmission spectrum for the sample section immersedin immersion oil (Olympus) so as to minimize reflectionloss and to more accurately analyze optical absorption.Since a 50-μm-diameter fiber was used for the fiber opticspectrometer, the spectrum was obtained supposedly for acircular region with a diameter of 0.5 μm, estimated onthe basis of geometrical optics. Since this diameter wascomparable with the spatial resolution obtained using theRalyleigh criterion, a deconvolution procedure was generallynecessary to precisely obtain the position-dependent spectrum.In this study, however, the examined thin section was uniformover a much wider area than the spatial resolution; hence,the deconvolution procedure was not carried out. As thereflectance standard, we employed a broadband dielectric mul-tilayer reflector covering a wavelength range of 300–2000 nm(Sigma Koki, TFMS-30C05-3/20). The microscope wasequipped with a two-dimensional piezo-positioning stage[Sigma Koki, SFS-120XY(WA)]. Both the stage and spec-trometer were controlled by the same computer so that thespectrum was recorded for each 200-nm step during the lateralscan of the sample over several dozen micrometers. To makethe spectral analysis easier, a 400-μm-diameter pinhole wasplaced on the aperture stop of the microscope to restrict theangular range of illumination. At the largest illumination angle,

11◦ under this condition, we theoretically confirmed that thereflectance spectral shape was almost the same as the case ofnormal incidence for unpolarized light.

In order to estimate the thickness of the semi-frontal thinsection, it is necessary to know the refractive index of the epoxyresin in which the sample elytron was embedded. We cut oneof the resin blocks into a triangular-prism-like shape with aheight of approximately 3 mm, and we carefully polished itssurfaces using lapping films. Then, the conventional minimum-deviation-angle method was employed to determine the indexvalues at several wavelengths of light emitted from a mixedgas laser beam source (Spectra Physics, Stabilite 2018). It isnoted that the bulk resin block appeared translucently yellow,indicating the optical absorption of blue light. We measured thetransmission spectrum of the resin block and confirmed thatthe imaginary part of the refractive index was so small thatthe index could be considered virtually real for the spectralanalysis in this study. The refractive index of the immersionoil was also measured using the minimum-deviation-anglemethod by containing the immersion oil in a triangular glasscell.

III. RESULTS

It is important in this study that the assembled microspec-trophotometer provides quantitative values of reflectance andtransmittance. In order to confirm this feature, we commer-cially prepared a test thin-film sample, which is made ofa 45.5-nm-thick silicon layer sputtered on a quartz glasssubstrate. It is designed such that the three optical quantities,reflectance, transmittance and absorptance, have comparablemagnitudes in the wavelength range of visible light. Thethickness and complex refractive index of the test samplehave been determined using a spectroscopic ellipsometer (J.A.Woollam, VASE) at NSG Techno-Research Co., Ltd, Japan.Figure 2 shows a comparison between the experimental andtheoretical spectra; the experimental spectrum is determinedusing the assembled microspectrophotometer, while the the-oretical one is calculated using the thin-film interferencemodel with the predetermined refractive index and thickness.Moderate agreement between these spectra indicates thatthe microspectrophotometer works quantitatively. However,a slight systematic deviation is noticed in transmittance forthe wavelength range of 650–700 nm, although the reasonis not presently clear. In addition, for wavelengths over 700nm, the cold filter (infrared cut filter) inside the microscopesignificantly reduces the transmitted light intensity, resultingin noisy behavior.

The color-causing multilayer structure of the jewel beetle’selytron is shown in Fig. 1(b). This image is obtained using aTEM for a sample that is thin-sectioned vertically to the layers.Thus, we can estimate the thicknesses of electron-dense andelectron-lucent layers to be approximately 105 and 135 nm,respectively. On the other hand, when the multilayer systemis thin-sectioned in a semi-frontal way, it appears as a broadstripe pattern with stripe widths of more than 1 μm, as shown inFig. 3(a). The stripes are observed to be wavy. This is becausethe layers are not perfectly flat.

Figure 3(b) shows the same section observed using anoptical microscope under transmission illumination. It is

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400 500 600 7000.0

0.2

0.4

0.6

0.8

1.0R

, 1-T

1-T

wavelength (nm)

R

FIG. 2. Reflectance (R) and transmittance (T ) spectra of a testthin-film sample made of a 45.5-nm-thick silicon layer sputtered onquartz glass substrate. Rough curves are experimentally determinedusing the assembled microspectrophotometer employed in this study.Smooth curves are theoretically drawn according to the thin-filminterference model in which the wavelength-dependent complexrefractive index and thickness of the silicon layer are predeterminedby using a spectroscopic ellipsometer. It is noted that R and 1 − T

are plotted so that the vertical distance between the two curvescorresponds to absorptance.

clearly observed that the stripe pattern corresponds to thatin the electron micrograph. It has long been assumed that theelectron-dense layer in beetles’ multilayer systems is opticallyabsorbent owing to the presence of some pigment [27]. Thegood correspondence between the images in Figs. 3(a) and 3(b)is a direct evidence supporting this assumption. Several smallcircular patterns are also noticed in the micrographs. Theycorrespond to small bumps on the elytron surface, which havebeen clearly observed using a scanning electron microscope[34].

Figure 3(c) shows the optical micrograph observed underepi-illumination. Interestingly, the contrast in the stripe patternin this image appears opposite to that in the image in Fig. 3(b):The dark stripes in Fig. 3(b) appear bright in Fig. 3(c), whilethe bright stripes appear dark. This fact implies that theelectron-dense layer has a higher real part of the refractiveindex than the electron-lucent layer, which causes higherreflectance, resulting in the brightness. We will later confirmthat the imaginary part of the index is not a prominent factorfor higher reflectance, unlike in the case of metal, which has avery high imaginary part.

Next, the reflectance and transmittance are quantitativelydetermined using a microspectrophotometer during a lateralscan of the sample placed on a piezo-positioning stage. Fig-ure 4(a) shows reflectance at a wavelength of 470 nm, obtainedas a line profile. The measurement is carried out approximatelyalong the black line shown in Fig. 4(c). Corresponding tothe bright and dark stripes, the reflectance value oscillatesbetween about 0.17 and 0.23. On the other hand, transmittancein immersion oil, shown in Fig. 4(b), is found to decrease toapproximately 0.9, where the reflectance is higher, while itbecomes close to 1.0, where the reflectance is lower. Theseresults quantitatively confirm that the electron-dense stripe

(a)

(b)

(c)

FIG. 3. Semi-frontal thin section of the elytron of the jewel beetle.(a) TEM image, (b) optical image under transmission illumination,and (c) optical image under epi-illumination. The rectangle in (b) istilted for comparison with (a). Note that the upper right region is theepoxy resin. Scale bar: 10 μm.

is optically absorbent and the electron-lucent one is almosttransparent.

Figure 5(a) shows the reflectance spectra for three positionsof the semi-frontal section—the epoxy resin, the electron-dense stripes, and the electron-lucent stripes—which areindicated by the three broken lines in Fig. 4(a). The line shapesof the three spectra commonly show that reflectance smoothlyincreases as the wavelength decreases, except for the shortestwavelength region. This is roughly related to the constructiveinterference of the thin-film that occurs at shorter wavelengths.Figure 5(b) shows the transmittance under oil immersion,which is found to decrease as the wavelength becomes shorterfor the electron-dense stripes, while it only slightly does so forthe electron-lucent stripes.

IV. ANALYSIS

A. Thickness determination

The individual layers in the multilayer system can beoptically resolved in the semi-frontal thin section. To deter-mine the refractive index, we consider a general analysis thatcan determine the complex refractive index values at eachwavelength of light. Three fundamental physical parametersof a single layer affect thin-film interference: the thicknessand the real and imaginary parts of the refractive index. We

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0.15

0.20

0.25

0 10 20 30

0.90

0.95

1.00

(b)

ref

lect

ance

(a)

length (μm)

tran

smit

tanc

e

(c)

d e f

FIG. 4. Results of a line scan measurement. (a) Reflectance inair and (b) transmittance in immersion oil, measured at a wavelengthof 470 nm. Three broken lines, denoted by d , e, and f , indicatethe positions of the spectra shown in Fig. 5; position d is in theregion of the epoxy resin, while e and f correspond to electron-lucentand electron-dense stripes, respectively. (c) Optical image under epi-illumimation. The black line approximately indicates the scanned linewhere reflectance and transmittance are measured. Scale bar: 10 μm.

have already obtained two physical quantities, reflectance andtransmittance. Thus, if the exact thickness is known for thethin section examined by a microspectrophotometer, we canestimate the index value at each wavelength. We utilize thereflectance spectrum of the epoxy resin region for thicknessdetermination. By reshaping the epoxy resin block into aprism shape, the refractive index of the resin is determinedby using the minimum-deviation-angle method; the refractiveindex values are shown in Fig. 6. It is found that the index valuen is slightly dependent on the wavelength λ of light accordingto Cauchy’s equation, expressed as follows:

n(λ) = A + B

λ2, (1)

where A and B are constant parameters. With this wavelength-dependent index value, the thickness of the section can bereliably determined through a fitting analysis using the thin-film interference model. As shown in Fig. 5(a), it is actuallyfound that the fitting curve well reproduces the reflectancespectrum for a thickness of 65 nm.

B. Index determination

We consider a single layer with a complex refractive indexn = η + iκ and a thickness d in a medium having a realrefractive index of nm. Reflectance R and transmittance T

of this layer are given in the following formulas, taking opticalinterference into account:

R =∣∣∣∣ r(1 − eiφ)

1 − r2eiφ

∣∣∣∣2

, (2)

400 500 600 700

0.10

0.15

0.20

0.25

400 500 600 7000.8

0.9

1.0

refl

ecta

nce

wavelength (nm)

tran

smit

tanc

e

wavelength / nm

d

e

f

e

f

(a)

(b)

FIG. 5. Reflectance and transmittance spectra for small regionsof the thin section, determined using a microspectrophotometer.(a) Reflectance at the three positions indicated by three broken linesmarked with d, e and f in Fig. 4. From top to bottom, the spectraare for the electron-dense stripes (f) and the electron-lucent stripes(e), and the epoxy resin region (d). A theoretical curve is also drawnaccording to the thin-film interference model to fit the spectrum ofthe epoxy resin part. This fitting analysis determines the thickness ofthe thin section to be 65 nm (see text for details). (b) Transmittancespectra for the electron-lucent (e) and electron-dense (f) stripes.

T =∣∣∣∣ (1 − r2)eiφ/2

1 − r2eiφ

∣∣∣∣2

, (3)

where r is the amplitude reflectance at the interface betweenthe thin layer and the medium and is expressed by Fresnel’sformula as

r = nm − n

nm + n, (4)

where the incidence angle is assumed to be 0◦. The phase φ isdefined as

φ = 4πnd

λ. (5)

The experimentally determined R, T , and d can be usedto determine the index values for each wavelength of lightby numerically solving the above equations for η and κ . Inpractice, the imaginary part κ is first calculated from Eq. (3)with an appropriately assumed value of η. The calculatedvalue κ is then substituted into Eq. (2) to obtain η. Thevalue η is again substituted into Eq. (3) to modify κ , andthe iteration process is continued until the calculated valuesconsistently satisfy the above equations. It should be notedthat the index value of the medium is different in the abovetwo equations, since reflectance is measured in air (nm =1.0), while transmittance is measured in the immersion oil

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400 500 600 7001.51

1.52

1.53re

frac

tive

inde

x

wavelength (nm)

FIG. 6. Refractive indices of the epoxy resin used for embeddingthe sample elytron and the immersion oil. The filled circles andsquares denote the experimental results for a prism-shaped resinblock and the immersion oil contained in a triangular glass cell,respectively, determined by measuring the minimum deviation anglebetween the incident and refracted light at several wavelengths oflasers. Curves are drawn according to Cauchy’s equation, expressedin Eq. (1), which give the best fit to the experimental results, withparameters A = 1.501 and B = 5.271 × 10−3 (μm2) for the epoxyresin and A = 1.498 and B = 6.520 × 10−3 (μm2) for the immersionoil.

having a refractive index that is slightly dependent on thewavelength; nm is approximately 1.520 at 550 nm and itswavelength dependence is described using Eq. (1) as shownin Fig. 6.

Figure 7 shows examples of the calculated index valuesfor the two types of materials that comprise the jewel beetle’smultilayer system. We have examined several thin sectionsand used many different pairs of reflectance and transmittancespectra for the electron-dense and electron-lucent stripes. Thecalculated values are found to moderately agree with eachother. It is quantitatively clarified that the electron-dense layers

1.5

1.6

1.7

1.8

400 500 600 700

0.00

0.05

0.10

0.15

κl

κh

ηl

ηh

real

par

t

(a)

(b)

wavelength (nm)

imag

inar

y pa

rt

FIG. 7. The real (a) and imaginary (b) parts of the wavelength-dependent complex refractive indices for the two types of the ma-terials that comprise the jewel beetle’s multilayer structure, denotedby nh = ηh + iκh and nl = ηl + iκl for high- and low-index layers,respectively. Six gray curves are obtained from six different pairs ofreflectance and transmittance spectra. The black curves are drawnaccording to Eqs. (1) and (6) that approximate the experimentallydetermined index values (see text for details).

0 30 60 90

500

600

700

800

peak

wav

elen

gth

(nm

)

incidence angle (degree)

red part

green part

FIG. 8. Wavelength of the reflectance peak of the elytron of thejewel beetle. The filled circles represent the experimental resultsfor the red striped part (upper) and green part (lower) of theelytron; these results are obtained by the so-called θ -2θ scanmeasurement, except for the data at 0◦, which are obtained by usinga microspectrophotometer. It has been experimentally confirmed thatthe reflection from the small area of the elytron is almost specular,but not perfectly. Curves are theoretically drawn according to theinterference condition, λp = 2(ηhdh cos θh + ηldl cos θl), where ηh(ηl), dh (dl), and θh (θl) are the real part of the refractive index,thickness, and refraction angle of the high-index (low-index) layer.The index value is obtained using Eq. (1) and the thicknesses of thetwo types of layers, dh and dl, are chosen to be 108 and 139 nmfor the red striped part and 75 and 96 nm for the green part,respectively; these values are consistent with the TEM observations.The refraction angles are calculated according to Snell’s law. Since theindex value and refraction angle are dependent on the wavelength,λp is numerically obtained by self-consistently solving the aboveinterference condition. We confirmed that the inclusion of theimaginary part of the refractive index induces only a small shift of afew nanometers in the peak wavelength, by theoretically calculatingthe reflectance spectrum of the multilayer thin-film model of theelytron. It is noted that when a refractive index difference becomesvery large, the wavelength at the maximum reflectance can deviatefrom λp because the spectral peak comes to have a flat-top shape [6].However, such an effect is negligible in the present case with a smallrefractive index difference.

have higher index values than electron-lucent layers in boththe real and imaginary parts. Thus, we hereafter refer tothe electron-dense and electron-lucent layers as high- andlow-index layers, respectively. It is found that the real partof the high-index layer, ηh, largely depends on wavelength: itincreases from 1.65 to 1.80 with the decrease in wavelength inthe examined spectral range. The imaginary part κh is alsodependent on wavelength, increasing to about 0.1 for theshortest wavelength. On the other hand, for the low-indexlayer, the real part ηl shows a slight increase from 1.55 to 1.60while the imaginary part κl is found to be very small.

In order to verify the consistency of the determined indexvalues with regard to the optical properties in the case ofthe jewel beetle, we carefully measured the angle-dependentreflectance spectrum. Although the experimental details willbe published elsewhere, the measurement belongs to a typethat is generally referred to as the θ -2θ scan. Figure 8 showsthe wavelength of the reflectance peak plotted against theincidence angle. The dependence on the incidence angle is

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found to be in good agreement with the theoretical curves thatare drawn according to the interference condition.

It may be useful to express the wavelength-dependentindex values using mathematical functions, as such expressioncan probably be used to approximate the index value forother structurally colored beetles. For the real part, it isfound that Cauchy’s equation, expressed by Eq. (1), providesa good approximation with parameters A = 1.56 and B =3.60 × 10−2 (μm2) for the high-index layer and with A = 1.51and B = 1.53 × 10−2 (μm2) for the low-index layer. Withthese parameters, the two curves are drawn in Fig. 7(a)for comparison with experimental results. For the imaginarypart, we empirically found that a simple exponential functionapproximately expresses the wavelength dependence as

κ(λ) = C exp

(− λ

λi

), (6)

with parameters C = 1.62 and λi = 142 (nm) for the high-index layer. The determined imaginary part for the low-indexlayer is much smaller than that for the high-index layer. Thus,κl can be approximated to be zero. It is emphasized that theabove mathematical functions are empirically obtained suchthat they approximate the index values for the wavelengthrange of 400–700 nm examined in this study.

V. DISCUSSION

It is assumed in the above analysis that the spatial regionexamined by the microspectrophotometer purely consists ofone type of material that comprises a high- or low-index layer.This is not always true, however, since the section has a finitethickness. The situation is schematically depicted in Fig. 9;the multilayered structure that consists of two types of layershaving thicknesses Dh and Dl is thin-sectioned nearly parallelto the layers with thickness d. It is shown that within oneperiod of the striped pattern in the thin section, denoted byl, two kinds of materials overlap in some regions, but not inothers. For accurate index determination, it is clearly necessaryto optically examine the regions that do not contain the overlap

l

ψ

Λh

Dl

Dh

d

Λl

FIG. 9. Schematic illustration of semi-frontal thin-sectioning ofthe multilayer structure. d: thickness of the section; Dh and Dl:thicknesses of the high- and low-index layers, respectively; �h and�l: lengths of the regions where the section comprises only high-or low-index layers, respectively; ψ : the tilt angle between thesection and the multilayer; and l: period of the stripe pattern in thesemi-frontal thin section.

using a microspectrophotometer. When the sectioning angle,defined in Fig. 9 as the angle between the planes of the layersand the thin section, is assumed to be ψ , the length of the regionthat consists of only the high- or low-index layer, denoted by�h or �l, respectively, is expressed by the following formula:

�j = Dj

sin ψ− d

tan ψ, (7)

≈ (Dj − d)/ψ, (8)

where j is h or l and ψ is assumed to be small in the secondequation. In the examined semi-frontal thin section, periodl is approximately 10 μm, as shown in Fig. 4(c), and Dh

and Dl are estimated to be approximately 105 and 135 nm,respectively, from the TEM observation. Consequently, theangle ψ is calculated to be 1.4◦ by using the relationsin ψ = (Dl + Dh)/l. Using these values, �h and �l are foundto be 1.7 and 2.9 μm, respectively, which are sufficientlylarger than the smallest area that can be examined by themicrospectrophotometer.

Either of Eqs. (7) and (8) is useful to estimate the requiredthickness of the section and the required level of parallelsectioning, depending on the multilayer system in question.It is immediately understood that the section should be thinnerthan the layers for �j to be positive. The reason for sectioningthe red part of the elytron in this study is that the layers areexpected to be thicker than the green part, thus making opticalmeasurements easier.

Earlier studies on insects’ structural colors have assumeddifferent pairs of index values for the optical analyses ofmultilayer interference, for example, 1.40 and 1.73 [35], 1.5and 1.6 [36], and 1.5 and 2.0 [27]. However, as is wellsummarized in a previous paper [30], these values are not basedon completely rigorous measurement techniques and only thereal part of the index is considered. Recently, Noyes et al.carefully measured the angle- and polarization-dependenceof a reflectance spectrum and obtained for a different beetle,Chrysochroa raja, two complex index values, 1.55 + 0.14i

and 1.68 + 0.03i, for the electron-dense and electron-lucentlayers, respectively. The index values obtained in the presentstudy are in good agreement with the above values whenthe real part is compared (ηh � 1.68 and ηl � 1.56) at awavelength of 550 nm; at this wavelength, reflectance ismaximum for C. raja for small incidence angles. However,the results of the two studies are different with regard tothe magnitudes of the real part of the index for the twokinds of layers: the previous study reported a higher real partfor the electron-lucent layer than that for the electron-denselayer, while the present study determines both the real andimaginary parts to be higher for the electron-dense layerthan those for the electron-lucent layer. This difference isa direct consequence of the fact that the electron-densestripe looks brighter under epi-illumination and darker undertransmission illumination. A very recent optical study of thejewel beetle C. fulgidissima assumes index values of the twotypes of materials that have the same magnitude relation asours [37].

In this paper, we propose an experimental procedure ofindex determination for multilayer thin-film systems. Sinceit does not require a large and specular surface, the method

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DIRECT DETERMINATION OF THE REFRACTIVE INDEX . . . PHYSICAL REVIEW E 83, 051917 (2011)

is more suitable when a thin-film system has a restrictedarea and/or contains inhomogeneous structural characteristics.Such systems are prominently found in natural structuralcolors, for example, in many species of beetles that havevarious colors on their sculpted elytron surface [11,14]. Themicrostructure in bird feathers is another such system. In manybird species, the periodic arrangement of melanin granules isknown to cause optical interference. However, the index valuesof these granules have not been experimentally examinedyet. Hummingbirds may be good candidates for such anexamination since the melanin granules in hummingbirdshave been reported to have a platelet shape with a lengthof a few micrometers [38]. A variation of our proposedprocedure can be applied to the helicoidal structure thatcauses circularly polarized reflection, which is found in somespecies of scarabaeid beetles [39–41]. In the case of suchstructural colors, anisotropic material plays an essential rolein the reflection mechanism. If a microspectrophotometer is

polarization-sensitive, the procedure can be applied to themeasurement of anisotropy in the refractive index.

On the other hand, the applicability of our procedure hasa limitation with regard to the process of thin-sectioning.As discussed above, the section should be thinner than anindividual layer in the thin-film system. Using a standardsectioning technique for TEM, the section thickness may beless than 50 nm but probably not less than 10 nm. Moreover, avery thin layer results in weak reflectance. Thus, the sensitivityof the microspectrophotometer becomes another limitation forthe quantitative determination.

ACKNOWLEDGMENTS

This study is supported by the Grant-in-Aid for ScientificResearch Nos. 18740216, 20740242, and 22340121 fromthe Ministry of Education, Culture, Sports, Science andTechnology.

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