Birefringence of nanocrystalline chitin films studied by Mueller-matrix spectroscopic ellipsometry

Birefringence of nanocrystalline chitin films

studied by Mueller-matrix spectroscopic

ellipsometry

A. Mendoza-Galvan, E. Munoz-Pineda, Kenneth Järrendahl and Hans Arwin

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

A. Mendoza-Galvan, E. Munoz-Pineda, Kenneth Järrendahl and Hans Arwin, Birefringence of

nanocrystalline chitin films studied by Mueller-matrix spectroscopic ellipsometry, 2016,

Optical Materials Express, (6), 2, 671-681.

http://dx.doi.org/10.1364/OME.6.000671

Copyright: Optical Society of America: Open Access Journals / Optical Society of America

http://www.osa.org/

Postprint available at: Linköping University Electronic Press

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-127060

Birefringence of nanocrystalline chitin films studied by Mueller-matrix spectroscopic

ellipsometry

A. Mendoza-Galván,1* _{E. Muñoz-Pineda,}1 _{K. Järrendahl,}2 _{and H. Arwin}2,3

1_{Cinvestav-IPN, Unidad Querétaro, Libramiento Norponiente 2000, 76230 Querétaro, Mexico} 2_{Laboratory of Applied Optics, Department of Physics, Chemistry and Biology, Linköping University,}

SE-581 83 Linköping, Sweden

3_{hansa@ifm.liu.se}

*_{amendoza@cinvestav.mx}

Abstract: Birefringent chitin films were prepared by a dipping technique from aqueous suspensions of chitin nanocrystals in a nematic liquid crystal phase. In the films, chitin nanocrystals are preferentially oriented along the withdrawal direction. Normal incidence transmission Mueller-matrix (M) spectroscopic ellipsometry measurements as a function of sample rotation were used to investigate the optical birefringence in the spectral range 0.73 to 5 eV. Analysis of eigenvalues and depolarization data reveal that the Mueller matrix corresponds to a pure retarder for photon energies below 4 eV and is depolarizing in the range 4 to 5 eV. By modeling the chitin film as a slab with in-plane anisotropy the birefringence was determined. The determination of birefringence was extended to include the range of 4 to 5 eV by a differential decomposition of M.

OCIS codes: (160.1190) Anisotropic optical materials; (260.1440) Birefringence; (160.4760) Optical properties; (260.2130) Ellipsometry and polarimetry.

1. Introduction

Chitin is a polymer synthetized by several organisms and it is found in insects, algae, the shells of marine crustaceans and other creatures. The polymer is formed by -(1→4)-N-acetyl-D-glucosamine units with chains adopting a 21 helix conformation and depending upon the arrangement of adjacent polymer chains,

chitin can be found in three allomorphs (α, β, and γ) [1]. α-Chitin is the most abundant allomorph with adjacent polymer chains in an antiparallel arrangement. However, isolated chains of chitin are not found in nature but as semi-crystalline fibrils comprised of rod-like shaped chitin crystallites (chitin whiskers) and amorphous regions between bundles of crystallites. Two disaccharide motifs comprise the orthorhombic unit cell of -chitin [2,3], the polymer chain being oriented along the c-axis which is the fiber axis. In marine creatures the crystallite width and length ranges are 10-50 nm and 150-2200 nm [4], respectively,

production of waste from the shell-fishery industry represents a renewable source to bring chitin into applications [7,8]. The knowledge of fundamental quantities is of importance to develop optical applications using such environmentally friendly material.

The astonishing colors originating from chitin-containing micro- and nanostructures in many insects represent a source of inspiration for optical biomimetics [9-11]. Although the refractive index of chitin has been determined for natural materials (butterflies scales) [12] and man-made amorphous thin films [13], much optical data remain unknown. According to its crystalline structure, chitin is expected to be an optically biaxial material. However, macroscopic chitin single crystals are unavailable and access to oriented samples is necessary to determine the birefringence. So far, only one study has been found on the birefringence of insect chitin and a value of 0.00084 at the wavelength 551 nm was reported [14]. The alignment of chitin crystallites has been achieved by other authors through unidirectional rubbing [15] and the application of intense magnetic fields [16]. In those reports, the authors used infrared spectroscopy and X-ray diffraction to study the uniaxially aligned films. Also images of polarized optical microscopy revealed birefringence but it was not quantified. The apparent low value of birefringence of chitin makes necessary the use of accurate spectroscopic methods.

For a complete optical characterization of a sample, different methods are applied depending on complexity of the sample [17]. Standard spectroscopic ellipsometry is a well established technique for isotropic and pseudo-isotropic (anisotropic films in highly symmetric orientations) thin films. For anisotropic materials in arbitrary orientations and for systems with no depolarization, generalized ellipsometry is often applied. However, depolarization introduced by the sample is often unknown and Mueller-matrix ellipsometry should be used.

In this work, we apply Mueller-matrix transmission ellipsometry to study the birefringence of free-standing films produced by a dipping technique from aqueous suspensions of nanocrystalline chitin (NCCh). The transmission mode is used because in the reflection mode the inherent depolarization introduced by the incoherent superposition of reflections from the front and back interfaces is larger. The experimental details are given in Sec. 2. In Sec. 3.1 the structure and optical performance of the films are described. The characteristics of the measured Mueller matrix and its decomposition are discussed in Sec. 3.2. Birefringence is quantified in Sec. 3.3 through non-linear regression analysis by modeling the ordinary and extraordinary effective refractive indices of chitin films with a Cauchy dispersion relation. In Sec. 3.4 a differential decomposition is applied to Mueller matrix data. The concluding remarks are presented in the last section.

2. Experimental

Aqueous suspensions of NCCh were prepared as described in [18]. Briefly, 1 g of practical grade crab chitin (Sigma-Aldrich) was mixed in 20 ml of a boiling alkaline solution of KOH at 20% by weight for 6

h. This treatment eliminates most of the unwanted compounds in the raw material and promotes a partial deacetylation of chitin. Then the solution was brought to neutral pH by washing in distillated water in several cycles of decantation and centrifugation. Acid hydrolysis of chitin was performed with a 3N solution of HCl (20 ml per 1 g of chitin) at 100 °C for 18 h. Hydrogen peroxide at 30% (1 ml per 1 g of chitin) was added half an hour before the acid hydrolysis terminated to eliminate any remaining pigment. The reaction was stopped by dilution in deionized water at 4 °C with ten times the volume before the hydrolysis. This process removes the amorphous regions of the raw semicrystalline material. Finally, the HCl in excess was removed by 2-3 washings assisted by centrifugation at 8000 rpm for 15 min obtaining a white paste of NCCh with a pH of 2. Neutralization was carried out by dialysis in deionized water for 2-3 days. The resulting white paste was used to prepare the NCCh aqueous suspensions adjusting the pH to 3. The films were deposited on glass and fused quartz substrates by dip-coating in a homemade apparatus at withdrawal speeds of 5, 15 and 30 cm/min. During deposition, the draining accumulates material at the bottom part of the substrates. After drying the films deposited at higher withdrawal speeds, that bottom part detaches from the substrate and strips of about 3 cm long and 0.5 cm wide could be removed mechanically using tweezers. In this work we analyze a film obtained at 30 cm/min.

The birefringence of the films was studied by analyzing measured 44 Mueller matrices in transmission mode. A Mueller matrix (M) provides a full description of the polarizing and depolarizing properties of a sample [19]. It relates the Stokes vectors of the incident (Si) and transmitted (St) light

beams by, . 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 i i t m m m m m m m m m m m m m m m m S MS S               (1)

In Eq. (1) the Stokes vectors of the incident and transmitted light beams (assuming propagating along the z-axis) are expressed in terms of a set of six irradiances,

, º 45 º 45                    L R y x y x I I I I I I I I S (2)

where Ix, Iy, I+45º, and I-45º correspond to linear polarization along the coordinate axes x and y, and at +45º

and at -45º from the x-axis, whereas IR and IL correspond to right- and left-handed circularly polarized

light, respectively. In this work we use Mueller matrices which are normalized to total transmittance for unpolarized light, i.e. the first element in the first row of M. Thus we have m11=1 and other elements have

values in the range [-1,1]. Of interest are the diattenuation D



m12,m13,m14



Tand polarizance





T 41 31 21,m ,m m 

P vectors where T means transpose.

The Mueller-matrix measurements were performed with a dual rotating compensator ellipsometer (J. A. Woollam Co., Inc.) at normal incidence in the wavelength () range 245-1690 nm. A motorized sample rotator was used to measure at rotation angles between 0 and 360° in steps of 10°. Complementary characterization included X-ray diffraction data (Rigaku/Dmax2100), polarized infrared measurements (Spectrum GX system/Perkin Elmer), cross sectional scanning electron microscopy (SEM) (JXA-8530F JEOL system), and atomic force microscopy images (AFM) (Nanoscope IV Dimension 3100/Bruker) working in tapping mode.

3. Results and discussion

3.1 Film formation, structure and optical performance

Under the processing conditions the acid-hydrolyzed chitin nanocrystals are partially deacetylated. The deacetylation produces some amino groups at the surface of the chitin nanocrystals. Protonation of some amino groups in the slight acid aqueous suspensions creates a positive charge causing a repulsive force between the NCCh crystals. At the optimal concentration that repulsive force and the rod shape of NCCh set up the conditions for the formation of a nematic liquid crystalline phase in the aqueous suspension. The preferential ordering in the film is driven by drain forces during withdrawal of the substrate from the suspension. X-ray diffraction data confirmed that the films retain the crystalline structure of the raw material, i.e. an orthorhombic unit cell of -chitin with lattice parameters a=4.74 Å, b=18.86 Å, c=10.32 Å and space group P212121 [2,3]. The crystallite width evaluated with the Scherrer formula was 8 nm

which is typical of crab chitin [20]. Infrared transmission measurements with polarized light parallel and perpendicular to the withdrawal direction gave similar spectra to those discussed by other authors [15]. The thickness determined from cross-sectional SEM images was 18.5±0.6 µm. Figure 1 shows a picture of a glass substrate dip-coated as described above and placed between an LCD screen and a linear polarizer sheet. In Fig. 1 the light coming from the LCD screen is linearly polarized at about 45° from the long side of glass substrate (withdrawal direction) and the polarizer sheet was set in the extinction configuration (see dark regions around the sample). This effect was observed in all the deposited films. The in-plane anisotropy is evident and its origin resides in the preferential ordering of chitin nanocrystals and in the intrinsic anisotropy of chitin. As can be observed in the AFM image (phase data) in Fig. 1, rod-shaped particles 100-180 nm long and 20-30 nm wide comprise the film. Given that the NCCh width determined from X-ray data is 8 nm, the particles are indeed bundles of crystallites.

Fig. 1. Left: Picture of a nanocrystalline chitin dip-coated glass substrate placed between the linearly polarized light from an LCD monitor and a linear polarizer in the configuration for extinction. Right: A phase measurement AFM image reveals the preferential alignment of chitin nanocrystals in the film.

3.2 Characteristics of the Mueller matrix and eigenvalue analysis

Figure 2 shows the normal incidence Mueller matrix transmission data of a free-standing nanostructured chitin film. In the contour map in Fig. 2 the radial and angular coordinates correspond to photon energy (in eV) and sample rotation angle (), respectively. Note that a non-linear scale is chosen to emphasize the symmetries between the mij elements which range in different scales. First we observe in Fig. 2 that the

elements of D and P in M are close to zero whereas the other elements show a richer structure. Second we find that M not is diagonal which it would be for an isotropic sample or for uniaxially aligned samples with the optic axis coincident with the z-direction. In Fig. 2 the non-zero off-diagonal elements thus evidence in-plane anisotropy. Other characteristics of M are related to the -dependence: m44 is invariant,

m24, m34, m42, and m43, are periodic with period π whereas a π/2 periodicity is observed in m22, m23, m32,

and m33. Further relationships noted are that m23 is close to m32, m24=-m42, m34=-m43, m22()=m33(+π/4),

m24()=m34(+π/4), and m42()=m43(+π/4). 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 m₄₄ m₃₄ m₂₄ m₁₄ 0 1 2 3 4 5 0 45 90 135 180 225 270 315 0 1 2 3 4 5 -1 -0.9 -0.6 -0.2 -0.05 0 0.05 0.2 0.6 0.91 0 1 2 3 4 5 (eV) m ij  (deg) 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 m₄₃ m₃₃ m₂₃ m₁₃ m₄₂ m₃₂ m₂₂ m₁₂ m₄₁ m₃₁ m₂₁

Fig. 2. Polar contour map of Mueller matrices in transmission mode as function of photon energy and rotation angle of a nanostructured chitin free-standing film.

Regarding the spectral dependence of M in Fig. 2, it is clear that at high photon energies some elements vary between the extreme values 1 and -1. This is more clearly seen in Fig. 3 where the Mueller-matrix spectra at four rotation angles are shown. In these spectra D and P are close to zero in most of the spectral range with some deviation at higher photon energies. The remaining elements of M show strong variation with photon energies mostly above 4 eV. It should be noticed that m22, m23, m32, and m33 at =10°

and 50° overlap with those at =100° and 140°, respectively, due to the π/2 periodicity aforementioned. Also, the π-periodicity of m24, m34, m42, and m43, is evident as well as the invariance of m44 with .

=10º =50º =100º =140º m 12 m13 m14 m₂₁ m₂₂ m23 m24 m₃₁ m_{32 m} 33 m 34 1 2 3 4 5 -1 0 1 E (eV) m₄₁ 1 2 3 4 5 E (eV) m₄₂ 1 2 3 4 5 E (eV) m₄₃ 1 2 3 4 5 E (eV) m₄₄

Fig. 3. Normal incidence Mueller matrices of a free-standing nanostructured chitin film measured at four rotation angles. Notice that some curves completely overlap making them impossible to distinguish.

All the observations listed in the previous paragraphs on the structure of M in Figs. 2 and 3 indicate, in first approximation, that M might correspond to a linear retarder plate of thickness d with ordinary (no)

and extraordinary (ne) refractive indices and retardance  2d



n_on_e



. A retarder with retardance 

and azimuth  has a Mueller matrix according to [20],









. cos 2 cos sin 2 sin sin 0 2 cos sin 2 cos cos 2 sin 1 cos 2 cos 2 sin 0 2 sin sin 1 cos 2 cos 2 sin 2 sin cos 2 cos 0 0 0 0 1 2 2 2 2 R                                         M (3)

However, Eq. (3) corresponds to a Mueller-Jones matrix, which means that it is non-depolarizing because it was deduced from the corresponding Jones matrix of a rotated linear retarder. On the other hand, M in

Fig. 2 was experimentally determined and it is necessary to investigate how well Eq. (3) represents the data. This can be done by an objective measure of the degree of polarimetric purity (P) (also called depolarization index) of the system which is given by [21],





_{1 ,} 3 1 2 1 2 11 T                    m tr P M M (4)

where tr stands for trace. The complement of P is referred to as the depolarizance (D=1-P). Thus, P=1 corresponds to a Mueller-Jones matrix and P=0 to an ideal depolarizer. In the present case, P is invariant

with  as shown in Fig. 4(a) for four values of . Furthermore, for photon energies below 4 eV, P is close to unity (the average value is 0.998±0.002), and therefore it is concluded that below 4 eV the NCCh film is a pure system and can be described with a Mueller-Jones matrix. This fact and symmetries observed in Figs. 2 and 3 support that MR given by Eq. (3) is a possible description of M up to 4 eV. For photon

energies higher than 4 eV, P deviates from unity and for totally polarized incident light the transmitted beam emerges partially polarized. Among sources of depolarization, light scattering from the bundles of crystallites during the pass of the beam through the film might contribute. Certainly, this effect takes place at all photon energies but with higher impact on P at shorter wavelengths (higher photon energies).

Additionally, it is possible to investigate the depolarization properties of M through a so-called Cloude decomposition [22]. In this decomposition M is expressed as the sum of up to four non-depolarizing Mueller matrices Mj, weighted with the eigenvalues j of the covariance matrix 









ijmij i j

C ,

where mij are the elements of M, i are Pauli matrices and  stands for the Kronecker product. The

Cloude decomposition is expressed as,

4 4 3 3 2 2 1 1M M M M M    (5)

Figure 4(b) shows the spectral dependence of the eigenvalues of C corresponding to M measured at

=0°. They are representative for all rotation angles because the eigenvalues are -invariant. For photon energies lower than 4 eV, 1=1 (the average value is 0.9988±0.0015) and 2=3=4=0 which confirms the

non-depolarization properties of M in this spectral range. At photon energies higher than 4 eV, 1

decreases with photon energy whereas 2, 3, and 4 increase but only 2 increases significantly. Thus, M

is a physically realizable Mueller matrix because all four eigenvalues of C are positive. The occurrence of more than one non-zero eigenvalue demonstrates the incoherent superposition of polarization from at least two different optical elements representing a depolarizing M. Therefore, the electromagnetic modeling in the next section is restricted to the spectral range below 4 eV where M is a pure Mueller-Jones matrix. The complete spectral range is analyzed by a differential decomposition [23.24] in a subsequent section.

1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 P 

Photon Energy (eV) =0° =60° =120° =180° (a) 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 ₁ 2 ₃ ₄ E ige nva lue

Photon Energy (eV) (b)

Fig. 4. (a) Polarimetric purity of Mueller matrices measured at four rotation angles and (b) representative eigenvalues spectra of the covariance matrix corresponding to M. Note that

2=3=4=0 at photon energies below than 4 eV.

3.3 Electromagnetic modeling and regression analysis

Quantification of the birefringence was achieved by modeling the film as a uniaxial anisotropic 18.5 µm thick slab with ordinary (no) and extraordinary (ne) effective refractive indices. In principle the chitin

fibrils are biaxial but in a configuration with normal incidence there is not access to the refractive index in the direction normal to the film. In addition, the fibrils may be randomly oriented along their long axis (c-axis) so the in-plane refractive index no may be an average of the indices in the a- and b-directions. In the

model, the optical axis corresponds to the withdrawal direction and is located at an angle 0 from the

x-axis of the laboratory coordinate frame in the measurements. The dispersion relation of no and ne each can

be represented by the Cauchy expression . 4

2 , ,

,e oe oe  oe 

o A B C

n    , where Ao,e, Bo,e, and Co.e are fitting

parameters. However, in the model the refractive indices appear in the retardance term of Eq. (3) as the difference no-ne and, therefore, the only physically meaningful quantities are the differences A=Ae-Ao=,

B=Be-Bo, and C=Ce-Co, i.e. the birefringence



n e no



. Non-linear regression analysis of the

birefringence and 0 was performed with the CompleteEASETM software (J.A. Woollam Co., Inc.). In

addition to best-fit parameter values, the algorithm also delivers correlation and 90% confidence limits data. The best fit Mueller matrix is shown in Fig. 5, and we observe a very good agreement with the experimental data in Fig. 2. In a difference plot (not shown) between the experimental and model M it is found that the elements of D and P differ by less than 0.01. For the remaining elements the differences are also <0.01 except at some azimuths at photon energies above 3.5 eV. From the fitting it was found 0

=-10.01°± 0.01º and the birefringence of the nanocrystalline chitin film is shown in Fig 6. Thus, the measurement at =10° corresponds to the one of a non-rotated retarder. It should be noted that regression at one rotation angle in principle would be sufficient if the sample character was known a priori. However, similar to the case when using multiple angles of incidence in ellipsometric studies on simple samples to reveal model deficiencies, we here use multiple rotation angles to strengthen the analysis by redundant data.

0 1 2 3 4 5 0 45 90 135 180 225 270 315 0 1 2 3 4 5 -1 -0.9 -0.6 -0.2 -0.05 0 0.05 0.2 0.6 0.9 1 0 1 2 3 4 5 (eV)  (deg) m ij 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 m₁₄ m₂₄ m₃₄ m₄₄ m₁₃ m₂₃ m₃₃ m₄₃ m₁₂ m₂₂ m₃₂ m₄₂ m₂₁ m₃₁ m₄₁

Fig. 5. Contour plot of model calculated Mueller matrices as function of rotation angle and photon energy.

In Fig. 6 it can be observed that below 3.5 eV the birefringence is positive with the largest value 0.0032 attained at 0.73 eV which is the low limit of the spectral range used. For comparison, at 2.25 eV ne-no=0.0024 which is almost three times larger than 0.00084, the value reported at 551 nm for insect

chitin [14]. The refractive indices ne and no accounting for the birefringence are effective quantities

because they represent some average between the dielectric tensor of chitin and voids. According to the orthorhombic crystalline structure of chitin its dielectric tensor is given by diag



a,b,c



in the principal

axes frame. Considering that the crystallographic c-axis corresponds to the polymer chain direction (crystallite length), ne would be mainly related to c and no to both a and b. An ongoing investigation is

conducted to determine the dependence of packing density and alignment of crystallites on processing parameters. In the ideal case of perfect alignment and full packing it is expected that ne2=c and

no2=(a+b)/2. If the film is not dense, also structural birefringence will come into play. In case of more

voids, both ne and no will decrease but screening effects will reduce no more than ne. Based on symmetries

of M, its polarimetric purity and the excellent fit to a slab with properties according to Eq. (3), we conclude that M can be accurately described as linear retarder below 4 eV.

0 1 2 3 4 5 -0.012 -0.009 -0.006 -0.003 0.000 0.003 Fitting Diff. decomp. n e -n o

Photon Energy (eV) (a)

Fig. 6. Birefringence (ne-no) of nanocrystalline chitin films determined from non-linear regression

analysis of the Mueller matrices in Fig. 2 and differential decomposition.

The change of birefringence in Fig. 6 from positive to negative indicates a stronger dispersion of no

than ne which could be associated to polarization-dependent absorption strengths of non

nonbonding-antibondig (n-*) electronic transitions. These are related to the C=O group in chromophore N-acetyl-D-glucosamine units of chitin that also show circular dichroism at 190-210 nm [25]. Additionally, C=O groups form inter- and intra-molecular hydrogen bonds nearly parallel to a and c axes, respectively [2]. These interactions could affect the n-* transitions in a different way for ordinary and extraordinary electromagnetic waves. Interband electronic transitions are not discarded but the electronic band structure of chitin is unknown.

3.4 Mueller matrix differential decomposition

The differential decomposition establishes that M and its spatial variation along the direction of wave propagation z are related as dMdzmM where m is the differential matrix [23,24]. For homogeneous media, m is independent of z and direct integration gives L=lnM where L=md being d the sample thickness. The matrix L is split into symmetric and asymmetric parts LLmLu where



L GLTG



2 Lm  and





2 T G GL L

Lu  , respectively, and Gdiag



1,1,1,1



. Thus, Lm is a

non-depolarizing matrix containing six elementary properties [24],

. 0 ' 0 ' ' 0 ' 0                     LB LB CD LB CB LD LB CB LD CD LD LD m L (6)

where LB (LD) and LB’ (LD’) are the linear birefringence (dichroism) along the x-y and ±45º axes, CD is the circular dichroism and CB the circular birefringence. On the other hand, Lu is given by [24],

. ' ' 6 5 3 6 4 2 5 4 1 3 2 1                                CDP A p p p p LDP A p p p p LDP A p p p p A u L (7)

where LPD, LPD’, and CDP describe selective depolarization of linearly horizontal, linearly 45º, and circularly polarized light, respectively. The pj elements are related to inhomogeneity in the sample or

time irreversal events [24].

Figure 7 shows the non-depolarizing matrix Lm corresponding to the Mueller matrix measured at

=10º. As was determined from the regression analysis, for this measurement the principal axes and laboratory frame coincide. With the exemption of LB data the scales of other properties is the same as that for CD. The steep increase of LB at 4.73 eV implies a first-order retardance at higher photon energies [26], that is, to determine the actual value of birefringence from LB 2d



none



, we should subtract 2. By

doing this, the data shown in Fig. 6 with the dashed line was obtained. It should be mentioned that the matrix Lu is practically zero for photon energies lower than 4 eV. The diagonal elements monotonically

deviate from zero at higher photon energies due to the depolarization and reach values LPD=0.05, LPD’=0.31 and CDP =0.31 at 5 eV. The preference to preserve linearly polarized light to a higher extent than circularly polarized light might be indicative of a Rayleigh regime for scattering as observed in some biological tissues [27]. For dense packing of fibrils as in cornea, coherent scattering has been suggested to explain experimental data [28]. A more detailed discussion on the source of scattering in the films is beyond of the scope of this work.

-LD -LD' CD -LD CB LB' -LD' -CB -3 0 3 -LB 1 2 3 4 5 -1 0 1 E (eV) CD 1 2 3 4 5 E (eV) -LB' 1 2 3 4 5 E (eV) LB 1 2 3 4 5 E (eV) 

The electromagnetic modeling in Sec. 3.3 implies that a model is imposed on the sample structure and thus implies filtering to some extent, whereby effects of non-ideal sample features are reduced. The differential approach is, on the other hand, analytic and does not distort information and non-ideal effects would propagate through the analysis. We found that the two approaches give the same results which support the correctness of our analysis.

4. Conclusions

Birefringent chitin films were obtained by dip-coating from an aqueous suspension of chitin nanocrystals in a nematic liquid crystal phase. Mueller-matrix spectroscopic measurements on an 18.5 µm thick film were analyzed. Eigenvalues and polarimetric purity show that the measured Mueller matrices represent non-depolarizing samples for photon energies below 4 eV. Non-linear regression modeling confirms that they can be considered as pure retarders. The ordinary and extraordinary effective refractive indices of the anisotropic chitin film were modeled with the Cauchy dispersion relation to determine the birefringence of chitin films in the spectral range of 0.73 to 4 eV. The birefringence was found positive for photon energies below 3.5 eV. A differential decomposition allowed to determinate the birefringence for the whole spectral range.

Acknowledgments

E. Muñoz-Pineda acknowledges the scholarship from Concayt-Mexico. Roger Magnusson is acknowledged for help with eigenvalue decomposition. Enric Garcia-Caurel is acknowledged for helpful discussions on differential decomposition. The Knut and Alice Wallenberg foundation, the Swedish Research Council, Carl Tryggers foundation and the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFO-Mat-LiU # 2009-00971) are acknowledged for financial support.

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