Coherent Backscattering of Light by an Anisotropic Biological Network

Coherent backscattering of light by an anisotropic biological network

∗ Gianni Jacucci1, , Olimpia D. Onelli 1, Antonio De Luca 2,3, Jacopo Bertolotti 4, ∗ Riccardo Sapienza 5, , and Silvia Vignolini 1

Abstract

The scattering strength of a random medium relies on the geometry and spatial distribution of its components as well as on their refractive index. Anisotropy can therefore play a major role in the optimisation of the scattering efﬁciency both in biological and synthetic materials. In this work, we show that by exploiting the coherent backscattering phenomenon it is possible to characterise the optical anisotropy in the Cyphochilus beetle scales without the need of changing their orientation or their thickness. For this reason, such a static and easily accessible experimental approach is particularly suitable for the study of biological specimens. Moreover, the estimation of the anisotropy in the Cyphochilus beetle scales might provide inspiration for improving the scattering strength of artiﬁcial white materials.

I. Introduction Nature, however, provides a different approach that can serve as an inspiration for An object is opaque white when the light the manufacturing of polymeric, thin but still incident on it undergoes multiple scattering opaque white materials. [6, 7, 8] In partic- events before exiting the medium, i.e. when ular, the brilliant whiteness shown by the the object is optically thick. [1] The opti- Cyphochilus beetle is known to be generated cal thickness is defined as the ratio between by multiple scattering of light inside the ex- the physical thickness of an object and the tremely thin scales (≃ 7 µm thick) that cover transport mean free path (ℓt), namely the dis- its exoskeleton (Figure 1a-b). [9, 10, 11, 4, 12] tance that light travels before losing informa- The beetle intra-scale structure is composed of tion about its starting propagation direction a nanostructured network of chitin filaments [2, 3]. Commonly, ℓt is of the order of tens with a filling fraction of around 45%.[13] The of microns in low-refractive index white mate- chitin fibers inside the beetle scales are organ- rials. [4, 5] Therefore, opacity is only achieved ised anisotropically, i.e. mainly oriented par- for relatively large thicknesses (of the order allel to the surface of the scales (Figure 1c). of hundreds of microns) to allow a sufficient This structural anisotropy increases the scat- number of scattering events. tering strength in the orthogonal direction to the scale surface, at expense of the in-plane Gianni0 Jacucci, Dr. Olimpia Onelli, Dr. Silvia Vi- gnolini, Department of Chemistry, University of Cam- scattering, which is not as relevant for the to- bridge, Lensfield Road, Cambridge CB2 1EW, UK; ∗e-mail: tal reflectance of the insect. [12] With such [email protected] morphology and geometry, the beetle achieves Antonio De Luca, Department of Physics, University of a high total reflectance (about 70% over the Calabria, Rende (CS), via Pietro Bucci, 87036, Italy whole visible range) with a thin, lightweight, Jacopo Bertolotti, Department of Physics and Astronomy, and anisotropic network made of low refrac- University of Exeter, Stocker Road, Exeter EX4 4QL, UK tive index material. [4, 13, 12] Riccardo Sapienza, The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2BW, In the recent years a number of different United Kingdom; ∗ e-mail: [email protected] techniques have been used to characterise

1 Coherent backscattering of light by an anisotropic biological network a b c II. Results and Discussion

To characterise light propagation in the Cyphochilus scales, we performed a coherent backscattering (CBS) experiment. We measured the angular resolved light scattered by the sample around the backscattering direction, which shows a characteristic peak pro- Figure 1: Images of the white beetle at different magni- file. [23] The CBS is the Fourier transform fications: a) photograph of the Cyphochilus of the spatial distribution of light exiting the beetle; b) micrograph of the scales organisa- tion; c) SEM image of the cross-section of a sample in the backscattering direction. [3] Cyphochilus scale showing the interconnected This phenomenon is often understood as an network of chitin filaments which is respon- interference effect that originates from the su- sible for the white appearance of the insect. perposition of a large number of two-wave Scale bar: 1 cm for a, 200 µm for b, 1 µm interference patterns from reciprocal waves. for c. [23, 24, 25, 26, 27] These waves have trav- eled the same optical path inside the medium but in opposite directions (Figure 2a) and are therefore phase-related. The resulting CBS in- anisotropic media, for example, spatially re- tensity distribution has a conical shape whose solved reflectance, [14, 15] imaging diffuse width provides a direct measurement of the transmission, [16, 17, 12] spatio-temporal vi- light transport mean free path of a material. sualisation of transmitted light, [18] and co- [3] herent backscattering. [19, 20, 21, 22] As The experimental setup is shown in Fig- of yet, the accuracy in the determination of ure 2b. A collimated laser diode (peak wave- the in-plane and out-of-plane components of length at 635 nm, spot-size of 2.5 mm, and the light transport mean free math in the output power of 1.2 mW) was used as light Cyphochilus beetle scales has been limited by source. The scattered signal was focused, us- the strong thickness dependency of the exper- ing a parabolic mirror, on a 100 µm core fiber imental techniques used. [4, 12] connected to a spectrometer. The angular res- ≃ In this work, we show that the coherent olution of our setup was 0.25 mrad. To ac- backscattering technique is well suited for quire the CBS line shape, the speckle pattern, studying the scattering properties of biological which occurs due to the high spatial coherence samples, allowing to estimate the anisotropy of the light source, needs to be averaged out. of a system without the need of changing its [26] This was done by placing the sample on orientation and its thickness. Moreover, the a motorised rotation mount whose axis was CBS provides a precise evaluation of the in- the same of the propagation direction of the plane transport mean free path without requir- incoming laser beam. This averaging proce- ing samples with different thicknesses, in con- dure precludes the possibility to investigate a trast with other static and easily accesible tech- potential in-plane, xy, anisotropy. However, it niques. Our experimental results contribute to has been recently shown that the Cyphochilus the understanding of scattering optimisation scales are characterised by an isotropic spec- in the Cyphochilus beetle scales, providing a tral density in the xy plane. [13] For this rea- valuable guide for the development of novel son, our work focused only on the study of an sustainable materials by showing how to ob- out-plane anisotropy. tain a strong optical response whilst using a The enhancement factor of the coherent sig- low-refractive index biopolymer as building nal, i.e. the ratio between the intensity at exact block. backscattering angle and the incoherent back-

2 Coherent backscattering of light by an anisotropic biological network ground, strongly depends on the polarisation. uration, as discussed in Ref. [28]. This is maximised when the single-scattered The measured CBS signal is reported in Fig- photons, which do not have a reciprocal coun- ure 3. The experimental data show a maxi- terpart and therefore contribute only to the in- mum lower than the theoretical value for semi- coherent background, are filtered out. This infinite media of 1 and a rounded top. This de- can be done by acquiring the CBS in the he- viation is a consequence of the small thickness licity conserving channel (HC) (Figure 2). The of the Cyphochilus scales and can be described HC signal was then normalised to the one ac- by the isotropic theory for finite media: [29] quired in a linear non conserving (LNC) config-

[ [( ) ( )] − ν (ν − (ν − α) cosh(2z α) (ν − (ν + α) cosh(2z α) γ = e b (bu) e + e + c 3 cos ν − α 2 2 α ν 2 2 ( ( ) + u ) ( + ) + u u u + sin(bu) − sinh(2z α)+ (α + ν)2 + u2 (ν − α)2 + u2 e (1) (ν − α) cosh((ν − α)b − 2z α) − ν cosh((ν − α)b) + e + (ν − α)2 + 2 u ] (ν + α) cosh((ν + α)b − 2z α) − ν cosh((ν + α)b) + e (α + ν)2 + u2

µ θ ν 1 1 where = cos( ), = 2 (1 + µ ), u = dex (ne) of the chitin network, which was es- kℓt(1 − µ), α = kℓt sin(θ), B = b + 2ze. The timated by the Maxwell-Garnett’s theory. [34] parameters k, ℓt, and b are the wave vector, the The expression of ze used in this work, and isotropic transport mean free path, and the op- in the related literature regarding the study of tical thickness, respectively. For semi-infinite the Cyphochilus beetle, is the one derived for media, i.e. b → ∞, section II reduces to: [29] isotropic systems. This approximation is justi- [ ( )] − α fied by the absence of an analytical formula of 3 α + ν 1 − e 2z γ = (2) the extrapolation length in anisotropic media. c,semi αµν [(α + ν)2 + 2] 2 u [35] The reduction of the theoretical maximum in Using a filling fraction of (45 6)%,[13] we finite media is caused by the suppression of calculated that n = (1.22 0.03), R = (0.32 long light paths, which are responsible for the e 0.04), and z = (1.29 0.11). Finally, using formation of the cusp of the CBS profile for e the extrapolation length found from the ex- semi-infinite media. [2] To accurately deter- pression above, we obtained ℓ = (1.40 0.09) mine the transport mean free path, the effect t µm from the fit shown in Figure 3. of the internal reflections at the scale interface on the light path distribution inside the chitin In the literature, the isotropic theory has network was accounted in the extrapolation been used to obtain information about media length ze = (2/3)(1 + R)/(1 − R).[30, 31, 32] where the anisotropy is in the plane perpen- R is the angle- and polarisation-averaged re- dicular (xy) to the incoming beam (z direction). flection coefficient at the slab interface and [19, 36] This type of anisotropy gives rise to a can be obtained by an angular integration of CBS cone whose line shape differs when ac- the Fresnel’s coefficients. [31, 33] These coef- quired along the x and y directions. [36] It ficients depend on the effective refractive in- has been recently shown that the isotropic the-

3 Coherent backscattering of light by an anisotropic biological network a

n n air e b

Laser source Spectrometer Figure 3: Theoretical fit of the experimental data (blue Input dots) using the isotropic theory for semi- polariser infinite (light blue dashed line) and finite me-

Rotation Sample dia (red solid line). Both curves were nor- Beam arm splitter malised to the maximum value of the semi- inﬁnite theory. The experimental points were θ Iris Detection polariser obtained normalising the signal measured in λ /4 HC conﬁguration to that acquired in the LNC

Parabolic setup. Mirror

Beam dump [19] In the case of the Cyphochilus beetle, the Figure 2: a) Illustration of two counter-propagating anisotropy is in the xz- and yz-plane (defining photons, red and light blue arrows. For sim- z as perpendicular to the surface of the scales) plicity, the scattering centers are represented [12, 13] and therefore the resulting CBS pro- as spheres. b) Schematic representation of the CBS setup in a HC configuration: the red file is isotropic (for light incoming along the z dashed lines define the cone of backscattered direction). Due to the particularly small thick- ≃ intensity, while the gray line represents the de- ness of the scales ( 7 µm), the probing direc- tection rotation. The sample was mounted on tion of the incoming beam cannot be changed. a rotation mount, whose axis was perpendic- As the anisotropy cannot be investigated di- ular to the propagation direction of the laser rectly, to gain insight on the light transport beam, to average over different disorder reali- inside the scales we performed anisotropic sations. Monte Carlo simulations for scalar waves. A Monte Carlo technique is well suited for de- scribing light propagation in disordered me- ory cannot be used to obtain quantitatively re- dia, where the photons paths can be math- liable information about the anistropic light ematically mapped into random walks. [37, transport along perpendicular directions. [35] 2, 3] Monte Carlo simulations have been ex- However, the anisotropy can be qualitatively tensively used both to investigate theoretical estimated as the ratio between the widths of aspects of anisotropic diffusion, [38, 14, 39, the CBS line shapes acquired along the x and 40, 41, 35] and to accurately describe experi- y directions, which can be individually de- mental results regarding light propagation in scribed by the isotropic theory. [36] Similarly, anisotropic media. [19, 15, 22] in the case of birefringent media as nematic Here, the photons paths in the white bee- liquid crystals, the isotropic theory can be tle scales were modeled by a series of random used to describe the CBS line shapes originat- steps. [42] The anisotropy of the system was ing from different polarisation configurations. introduced via a direction-dependent step size,

4 Coherent backscattering of light by an anisotropic biological network i.e. the components of the step vector were tal data with the Monte Carlo simulations, it is sampled from two different negative exponen- possible to disentangle the contribution of ℓxy tial distributions with mean ℓxy and ℓz for the and ℓz to the CBS line shape. In particular, we in-plane and out-of-plane components, respec- obtained a value of ℓxy = (1.40 0.09) µm and tively. The angular component of each of the an optical thickness OT = (6.89 0.13). The random steps was sampled from a distribution data were fitted by minimising the χ2. The of points uniformly distributed on the surface errors in ℓxy and OT were estimated by per- of a unit sphere. [43, 44] The collection of the forming simulations were the two parameters initial and final positions of the walkers that were gradually changed. This procedure was escaped the material from the same face they then repeated for taking into account the un- entered it, that corresponds to reflected pho- certainty in the determination of reflection co- tons, was then used to reconstruct the CBS line efficient (R). shape. [29, 3] The effect of residual absorption The measured optical thickness is in good on the CBS line shape is not considered, due agreement with the total transmission data re- to the negligible absorption of chitin in the vis- ported in the literature. [4, 12] The total trans- ible. [45] The only parameters required for mission (T) for slab geometry media in the dif- our Monte Carlo simulations are the random fusion approximation is: steps distribution, the scales thickness and the 2z reflection coefficient at the scale interface (R). T = e , (3) + A schematic of the simulations parameters is L 2ze illustrated in Figure 4a. where L is the slab thickness. Equation 2 is Figure 4b-c show how the CBS line shape the limit for negligible absorption of the expression derived in Ref. [46]. Using T = is affected both by changing the in-plane (ℓxy) (0.29 0.02), as reported in Ref. [4, 12], and and out-of-plane (ℓz) components of the transport mean free path. In particular, ℓ de- the extrapolation length previously calculated xy = ( ) termines the width of the CBS profile, while we obtained OTlit 6.3 1.2 , consistent the optical thickness, defined as OT = L/ℓ with what we measured. z = (where L is the thickness of the medium), spec- From the measured OT and assuming L ( ) ifies the enhancement of the coherent signal. 7 1 µm, where L and its error represent σ These results can be qualitatively explained the mean and 1 of the distribution reported ℓ = ( ) by the fact that the CBS depends only on in Ref. [12], we obtained z 1.02 0.15 µm. ℓ the distance between the positions of the first The error in z, which is mainly determined by scattering event (when the photon enters the the uncertainty in the thickness of the sample, medium) and the last (when the photon exits can be affected by systematic errors given by the medium). When a large number of pho- surface roughness and curvature. [3, 47, 48] ℓ ℓ tons is considered, these two positions have on Comparing z with xy, the measured opti- average the same z coordinate (which is of the cal anisotropy (OA) is: ℓ order of z) and therefore their distance can be ℓxy − ℓz considered to be z-independent. However, as OA = = (0.37 0.24). (4) ℓz discussed previously for the isotropic theory, when the optical thickness of the medium is Our experimental result is in agreement with small (i.e. when long light paths are not al- the 3D reconstruction of the chitin network re- lowed by the finite thickness of the medium) ported in Ref. [13], which predicts OA ≃ 0.5. the top of the CBS is rounded. The limited influence of the optical thickness on the width of III. Conclusions the CBS line shape allows to obtain a precise value of ℓxy without requiring samples with In conclusion, we demonstrated that the trans- different thicknesses. By fitting the experimen- port mean free path and optical anisotropy of

5 Coherent backscattering of light by an anisotropic biological network a b of changing its orientation and its thickness, 1.35 1 making the CBS a technique particularly suit- 2 5 able for the study of biological specimens. 10

Authors Contribution.

c G.J., O.D.O., A.D.L., R.S., and S.V. designed 1.35 1 L 2 the experiments. G.J., J.B., R.S., and S.V. 5 10 conceived and planned the simulations. G.J.

Z carried out the experiments and the simula- XY tions. All authors provided critical feedback and helped shape the research, analysis and d manuscript.

Acknowledgements.

G.J. thanks A. Lopresti, S.R. Hinestrosa, L. Barberi, and A. Caputo, and for the fruitful discussions and advices on numerical meth- ods. G.J. and O.D.O thank Dr. V.E. Johansen, R. Middleton, Dr. G. Palermo and Dr. G. Kamita for their assistance on experimental techniques.

Data Accessibility. Figure 4: Monte Carlo simulation of the CBS line shape by an anisotropic medium: a) illustration of The raw data and the Monte Carlo code re- the simulation parameters, b-c) varying the garding the publication can be found on the in-plane components and the out-of-plane one online ﬁgshare repository. of the mean free path. For both simulations the thickness of the slab (L) was ﬁxed at 15 µm. d) Fit of the experimental data with Funding. the anisotropic simulation. All the simulation were performed using 1 million photons. The This work was supported by the Euro- simulated curves were normalised to the max- pean Research Council [ERC-2014-STG H2020 imum value of a simulation with OT=1000. 639088], the BBSRC David Phillips Fellowship [BB/K014617/1], the EPSRC [EP/L016087/1, EP/L015978/1, EP/N016920/1], the Lever- light in the Cyphochilus beetle scales can be hulme Trusts Philip Leverhulme Prize and determined by measuring the CBS and that the Leverhulme Trust (No. RPG-2016-129), the results are in agreement with the one pre- the EPSRC (grant nos. EPSRC EP/M027961), dicted in Ref. [13]. Exploiting the CBS ef- the Leverhulme Trust (No. RPG-2014-238), fect provides a measurement of the optical the Royal Society (No. RG140457), and the anisotropy which describes more accurately Nanolase project (PRIN 2012). the scattering properties of the Cyphochilus beetle compared to the results reported in References the literature. In addition, the experimental technique reported here allows to estimate [1] Diederik S Wiersma. Disordered photon- the anisotropy of a system without the need ics. Nature Photonics, 7(7):188–196, 2013.

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