Mueller matrix spectroscopic ellipsometry study of chiral nanocrystalline cellulose films

Dried bacterial cellulose membranes were milled to pass through a 0.5 mm screen to ensure uniform particle size and to increase the surface area. The milled pulp was hydrolyzed in 8.75 ml of a sulfuric acid solution per gram of pulp at a concentration of 64 wt% at 50 °C for 0.5 h under vigorous stirring. The bacterial cellulose suspension was then diluted with cold water (10 times the volume of the acid solution) to stop the hydrolysis, and allowed to settle overnight. The clear top layer was decanted, and the remaining cloudy layer was centrifuged at 6000 rpm for 10 min (Jouan C3i-CR3i multifunction centrifuge). The supernatant was decanted, and the resulting thick white suspension was washed three times with water to remove water soluble cellulose materials. The thick white suspension was dialyzed against water for one to four days in cellulose membrane tubes (12 000–14 000 molecular weight cut-off). The suspension was diluted to the desired concentration and then dispersed by ultrasound treatment in a sonicator (Sonics Vibra-Cell VC 505 500 W 20 kHz) with a 6 mm diameter probe. Typically, 50 ml of a 3.0 wt% NCC suspension was placed in a 100 ml plastic tube and sonicated at 60% of the maximum power (300 W). Prolonged sonication (to an energy input of over 700 J g⁻¹ of NCC) was performed in an ice bath to prevent desulfation caused by the suspension heating. The suspension (8 ml) was transferred to a polystyrene Petri dish (diameter of 45 mm) and allowed to evaporate under ambient conditions until yielding solid films with chiral nematic organization.

The most general description of the polarization and depolarization capabilities of a sample requires the use of Stokes vectors and Mueller matrices [23]. A Stokes vector has four components (I, Q, U, V) which are expressed in terms of the irradiances of six basic polarization states: I_p, I_s, I_+45°, and I_−45° corresponding to linear polarization parallel (p), perpendicular (s), at +45° and at −45° relative to the plane of incidence, respectively; and the other two corresponding to right- and LHCP light I_R and I_L, respectively. The explicit form of the Stokes vectors is

$$\begin{eqnarray}&&{\bf{S}}=\left[\begin{array}{c}I\\ Q\\ U\\ V\end{array}\right]=\left[\begin{array}{c}{I}_{p}+{I}_{s}\\ {I}_{p}-{I}_{s}\\ {I}_{+{45}^{^\circ }}-{I}_{-{45}^{^\circ }}\\ {I}_{R}-{I}_{L}\end{array}\right].\end{eqnarray} \tag{ 1 }$$

In the Mueller–Stokes description, the light-sample interaction is accounted for by a linear relationship between the Stokes vectors of incident (S_i) and outgoing (S_o) light beams, through a 4 × 4 Mueller matrix (M),

$$\begin{eqnarray}{{\bf{S}}}_{{\rm{o}}}={\bf{M}}{{\bf{S}}}_{{\rm{i}}}=\left[\begin{array}{cccc}{m}_{11} & {m}_{12} & {m}_{13} & {m}_{14}\\ {m}_{21} & {m}_{22} & {m}_{23} & {m}_{24}\\ {m}_{31} & {m}_{32} & {m}_{33} & {m}_{34}\\ {m}_{41} & {m}_{42} & {m}_{43} & {m}_{44}\end{array}\right]{{\bf{S}}}_{{\rm{i}}}.\end{eqnarray} \tag{ 2 }$$

In this work, normalized Mueller matrices (m₁₁ = 1) and Stokes vectors S_i (I_p + I_s = 1) are used.

After interaction with the sample, the light beam (reflected or transmitted) in general will emerge partially polarized with a degree of polarization P given by [23],

$$\begin{eqnarray}&&P=\displaystyle \frac{\sqrt{{Q}^{2}+{U}^{2}+{V}^{2}}}{I}.\end{eqnarray} \tag{ 3 }$$

The polarized component of the emerging beam in general will be elliptically polarized with ellipticity e [17],

$$\begin{eqnarray}&&e=\,\tan \left(\displaystyle \frac{1}{2}{\rm{\arcsin }}\,\displaystyle \frac{V}{\sqrt{{Q}^{2}+{U}^{2}+{V}^{2}}}\right),\end{eqnarray} \tag{ 4 }$$

where −1 ≤ e ≤ +1. The extrema correspond to left- and right-handed circularly polarized (RHCP) light, respectively; for linearly polarized light, e = 0, and other values of e are for elliptical polarization. The major axis of the polarization ellipse is located at an azimuthal angle φ measured from the plane of incidence and is given by [23],

$$\begin{eqnarray}&&\varphi =\displaystyle \frac{1}{2}{\rm{\arctan }}(U/Q).\end{eqnarray} \tag{ 5 }$$

The Mueller matrix measurements were performed with a dual rotating compensator ellipsometer (RC2, J. A. Woollam Co., Inc.) in the wavelength (λ) range 245–1000 nm. Since the films are inhomogeneous and not flat, focusing probes were used to achieve a beam spot with size below 100 μm. Images of the area measured were acquired with a CCD camera attached to the RC2 system. More details about the instrument can be found in [17–22]. Measurements in transmission mode were done at normal incidence whereas in reflection mode, measurements were performed at angles of incidence (θ) between 20° and 75° in steps of 5°. The transmitted irradiance of unpolarized light at normal incidence on areas of 2 mm in diameter was measured in the spectral range of 240–840 nm using a FilmTek 3000 system (SCI, Inc.). The transmittance of circular right- and left-handed polarized light was also measured using commercial plastic-sheets filters (Edmund Optics), which show good performance in the 400–700 nm wavelength range as evaluated with the RC2 system. The thickness of the film was determined from cross-section scanning electron microscopy (SEM) images using a JXA-8530F system (JEOL).

The Mueller-matrix spectroscopic ellipsometry data of an NCC chiral film are shown as function of wavelength and angle of incidence in the contour map in figure 1(a). A careful inspection provides the following relationships for all angles of incidence: m₁₂ = m₂₁, m₁₃ = −m₃₁, m₁₄ = m₄₁, m₂₃ =−m₃₂, m₂₄ = m₄₂, and m₃₄ = −m₄₃. As was earlier reported for beetle cuticles [20], these symmetries in M result from the constraint for cross polarization coefficients in chiral systems r_ps = −r_sp. The band of selective reflection of left-handed polarized light is identified by the negative values of m₄₁ (yellow–red regions). At θ = 20°, this band is centered at 516 nm and shifts to shorter wavelengths at larger angles of incidence. It is known that at oblique incidence the wavelength of selective reflection is given by ${\lambda }_{m}={n}_{{\rm{av}}}{\rm{\Lambda }}\,\cos \,{\theta }_{{\rm{t}}},$ where θ_t is the angle of wave propagation inside the helicoidal structure determined from Snell's law ${n}_{{\rm{a}}}\,\sin \,\theta ={n}_{{\rm{av}}}\,\sin \,{\theta }_{{\rm{t}}}$ [24]. Therefore, taking n_a = 1 and n_av = 1.54 [25] the corresponding pitch is 343 nm. At wavelengths outside the band of selective reflection, M shows a behavior typical for dielectric materials [20]. For comparison, the data of M from the cuticle of the beetle C. mutabilis [21] is shown in figure 1(b). It can be noticed the similarities between the contour maps in figure 1 which shows that the polarization and depolarization properties of NCC chiral films and beetle cuticles are qualitatively similar.

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As was mentioned above, NCC chiral films are characterized by a mosaic-like texture as can be observed in the image on the left in figure 2. This texture identifies the multidomain type and is characterized by both, random helix direction and pitch length distribution. Single-domain structures are characterized by the presence of a single helical structure selectively reflecting a well-defined band of wavelengths. In the image of the NCC chiral film in figure 2, we can see domains of different colors (blue, green, red), which define it as the multidomain type. Therefore, measurements at different places show selective reflection bands with different spectral characteristics. On the other hand, the image on the right in figure 2 corresponds to the cuticle of the beetle C. mutabilis where a more uniform color is seen. From the latter observation, it can be tempting to assume that the cuticle of beetles is of the single-domain type. However, the cuticle of C. mutabilis is comprised of chiral layers of different pitch, being larger near the surface.

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In order to be physically meaningful, any experimentally determined Mueller matrix must fulfill several requirements. Among others, the system must not over-polarize incident polarized light. In quantitative terms, this means that the depolarizance (D) given by [26],

$$\begin{eqnarray}&&D=1-\left|{\left[\displaystyle \frac{1}{3}\left(\displaystyle \frac{{\rm{t}}{\rm{r}}({{\bf{M}}}^{{\rm{T}}}{\bf{M}})}{{m}_{11}^{2}}-1\right)\right]}^{1/2}\right|,\end{eqnarray} \tag{ 6 }$$

shall be in the range 0 ≤ D ≤ 1. In equation (6), T and tr stand for transpose and trace, respectively and m₁₁ = 1 in our case. Thus, D gives the average measure of the depolarization produced by a system for all incident pure states. For a pure depolarizer D = 1 and D = 0 for a non-depolarizing pure system. Figure 3 shows D of the NCC chiral film in a contour polar representation where the radial and angular coordinates correspond to the wavelength λ and angle of incidence θ, respectively. As can be noticed, D ≤ 1 and M is thus not over-polarizing. The largest values of D are found in the band of selective reflection, which is similar to the observation in the data measured on the cuticle of beetles [20, 21]. However, NCC chiral films are more depolarizing than beetle cuticle.

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Of particular interest are the polarization properties of light reflected for unpolarized incident light with Stokes vector ${{\bf{S}}}_{{\rm{i}}}={[\mathrm{1,0,0,0}]}^{{\rm{T}}}.$ In this case and according to equation (2), the reflected light is given by ${{\bf{S}}}_{{\rm{r}}}\,={[1,{m}_{21},{m}_{31},{m}_{41}]}^{{\rm{T}}},$ i.e. the first column of M. The elements of S_r can then be used in equations (3)–(5) to calculate the quantities defining the polarization properties of the reflected light as is shown in figure 4. As can be noticed at small angles of incidence, P is high in the spectral range of selective reflection and this polarized part of the reflected beam is left-handed because e < 0. For angles of incidence near 55° and outside the Bragg band, the reflected light is almost totally polarized P ≈ 1, with a linear character e ≈ 0 and s-type φ ≈ 90°, which is typical of a dielectric material as in the case of beetle cuticle [20–22].

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Figure 5 shows the Mueller matrix measured in transmission mode at normal incidence. First, we note that m₄₁ = m₁₄ > 0 indicating that right-handed polarized light is transmitted. This is expected because as was discussed in figure 1, the films reflect left-handed polarized light. This complementarity is illustrated by including the elements m₄₁ and m₁₄ measured in reflection mode at angle of incidence 20°. It should be noticed that diagonal and anti-diagonal elements show appreciable values, whereas the remaining ones have values close to zero with some deviations at short wavelengths in m₂₄, m₃₄, m₄₂, and m₄₃ probably due to multiple scattering.

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The depolarizance of the Mueller matrix measured in transmission mode is shown in figure 6. It can be noticed that 0 ≤ D ≤ 0.2 meaning that the M shown in figure 5 is physically meaningful. In addition, at wavelengths larger than 550 nm D is close to zero and the NCC chiral film could be considered as a non-depolarizing pure system. The increasing values of D in the band of selective reflection might be related to the multidomain texture (helicoidal axis orientation and pitch distribution) across the film. On the other hand, at wavelengths shorter than 450 nm D monotonically increases probably due to scattering.

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It is instructive to investigate the polarization properties of light transmitted by the NCC chiral film for different polarization states of the incident beam. Using the Mueller matrix formalism, this implies straightforward calculations using equation (2). We first consider three cases of incident light: unpolarized ${{\bf{S}}}_{{\rm{i}}{\rm{U}}}={[\mathrm{1,0,0,0}]}^{{\rm{T}}},$ LHCP ${{\bf{S}}}_{{\rm{i}}{\rm{L}}{\rm{H}}}={[\mathrm{1,0,0},-1]}^{{\rm{T}}}$ and RHCP ${{\bf{S}}}_{{\rm{i}}{\rm{R}}{\rm{H}}}={[\mathrm{1,0,0,1}]}^{{\rm{T}}}.$ As was noted above for S_iU, the Stokes vector of the transmitted beam is given by the first column of M, whereas for LHCP incident light it is ${{\bf{S}}}_{t}={[1-{m}_{14},{m}_{21}-{m}_{24},{m}_{31}-{m}_{34},{m}_{41}-{m}_{44}]}^{{\rm{T}}}$ and for RHCP incident light the out coming beam has Stokes vector ${{\bf{S}}}_{t}={[1+{m}_{14},{m}_{21}+{m}_{24},{m}_{31}+{m}_{34},{m}_{41}+{m}_{44}]}^{{\rm{T}}}.$ The degree of polarization and ellipticity of the transmitted beam calculated with equations (3) and (4) for each case of incident polarization state are shown in figure 7. In figure 7(a), it can be noticed that unpolarized incident light at 515 nm emerges with a relatively high degree of polarization (P = 0.82) and this polarized part of the transmitted beam has a near circular right-handed character (e = 0.85). In figure 7(b) for LHCP incident light, the degree of polarization appreciably decreases at wavelengths in the band of selective reflection, whereas for RHCP incident light in figure 7(c) the transmitted beam is highly polarized in most of the spectral range. In both cases, the handedness of the incident beam is retained although for LHCP the ellipticity (e = −0.55) considerably deviates from the incident polarization state. For RHCP incident light, the ellipticity of transmitted beam is relatively high (e = 0.84).

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Further insight into the optical response of the NCC chiral films is gained when the actual transmittance for each incident polarization state is considered. Figure 8 shows the transmittance measured in the visible range for unpolarized incident light (T_U), LHCP (T_LH), and RHCP (T_RH) incident light. As can be noticed, for unpolarized and LHCP incident light the band of selective reflection is evident. In contradistinction, RHCP incident light is freely transmitted. Also in figure 8 is included the polarized part of the transmittance for each case of incident light, that is, the product of the polarization degree according to equation (3) and the corresponding transmittance P_jT_j (j = U, RH, LH). Thus, the polarized and unpolarized parts are represented by the light-gray and dark-gray areas, respectively. It should be mentioned that in the ideal case of monodomain chiral nematic ordering, the baseline of T_j is expected at about 0.9 for an average refractive index of 1.54 for cellulose with a slight decrease to shorter wavelengths due to the dispersion of the refractive index. Therefore, smaller values of transmittance are indicative of light scattering. In addition, for the ideal case the minima in T_U and T_LH (at λ_min = 505 nm) are expected to be 0.5 and 0, respectively. The larger values obtained result from the multidomain texture. An LHCP incident light of wavelength λ_min will propagate through domains with pitch larger or smaller than Λ = λ_min/n_av with an attenuation decreasing with the distance to λ_min. Also, the LHCP wave will only be partially attenuated in domains with pitch Λ if the number of turns in the helicoids is so small that the domain thickness is less than attenuation length. These two mechanisms may contribute to the non-zero value of T_LH. As a side result, the partial incoherent superposition of these types of contributions depolarizes the transmitted light modifying the ellipticity as well.

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Another case of interest is when the incidence is linearly polarized. As an example, we consider an incident Stokes vector ${{\bf{S}}}_{{\rm{i}}}={[\mathrm{1,}\mathrm{1,}\mathrm{0,}0]}^{{\rm{T}}}.$ In this particular case, the Stokes vector of the transmitted light is ${{\bf{S}}}_{{\rm{t}}}\,={[1+{m}_{12},{m}_{21}+{m}_{22},{m}_{31}+{m}_{32},{m}_{41}+{m}_{42}]}^{{\rm{T}}}.$ The polarization properties of the transmitted beam calculated with equations (3)–(5) are shown in figure 9. As can be noticed in figure 9(a), the degree of polarization is relatively high with a minimum in the band of selective reflection. The polarized part of the transmitted beam is right-handed elliptically polarized in the band of selective reflection and near linearly polarized light e ≈ 0 at larger wavelengths.

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Another property of interest is the optical rotation dispersion (ORD) which results from the difference in the refractive indices for left- and right-handed circularly polarized light. In this case, ORD can be evaluated from the azimuth of the transmitted beam shown in figure 9(b) which shows the change in azimuth for linearly polarized incident light. The maximum rotation is around +20° and −20° at wavelengths slightly smaller than and larger than the selective Bragg resonance wavelength. With a film thickness of 28 μm as determined by SEM we then get a maximum specific rotation of +20/0.028 ≈ +700° mm⁻¹ which is comparable to that found in the cuticle of the beetle Cetonia aurata [27]. Nevertheless, the transmitted light in the spectral band of selective reflection is right-handed ellipticaly polarized as shown in figure 9(a).

In chiral systems, CD is the differential absorption of LHCP and RHCP and is thus a measure of the difference between the absorption coefficients for LHCP (α_L) and RHCP (α_R) incident light. These are determined by assuming a Lambert–Beer dependence of the transmitted irradiances I_L ∼ exp(−α_L d) and I_R ∼ exp(−α_R d), where d is the sample thickness of homogenous materials with natural optical activity where LHCP and RHCP are eigenmodes of propagation [28]. In that case, for LHCP and RHCP incident light the total irradiance of the transmitted light is proportional to the first component of the Stokes vector, that is, I_L = (1 − m₁₄) and I_R = (1 + m₁₄), respectively, and CD can be quantified as,

$$\begin{eqnarray}&&{\rm{C}}{\rm{D}}=\,\mathrm{ln}\left(\displaystyle \frac{1+{m}_{14}}{1-{m}_{14}}\right)=({\alpha }_{L}-{\alpha }_{R})d.\end{eqnarray} \tag{ 7 }$$

However, in anisotropic depolarizing media, equation (7) is not suitable to use for CD computation because LHCP and RHCP are not propagation eigenmodes. Recently, a differential decomposition was proposed to determine the elementary properties for homogeneous media [29, 30]. That formalism establishes that M and its spatial variation along the direction of wave propagation z are related as dM/dz = mM where m is the z-independent differential matrix. Direct integration gives L = lnM where L = md being d the sample thickness. The matrix L is split into symmetric and asymmetric parts L = L_m + L_u where L_m = (L − GL^TG)/2 and L_u =(L + GL^TG)/2, respectively, and G = diag[1, −1, −1, −1]. Thus, L_m is a non-depolarizing matrix containing six elementary properties [30],

$$\begin{eqnarray}{{\bf{L}}}_{m}=\left[\begin{array}{llll}0 & -{\rm{L}}{\rm{D}} & -{\rm{L}}{\rm{D}}^{\prime} & {\rm{C}}{\rm{D}}\\ -{\rm{L}}{\rm{D}} & 0 & {\rm{C}}{\rm{B}} & {\rm{L}}{\rm{B}}^{\prime} \\ -{\rm{L}}{\rm{D}}^{\prime} & -{\rm{C}}{\rm{B}} & 0 & -{\rm{L}}{\rm{B}}\\ {\rm{C}}{\rm{D}} & -{\rm{L}}{\rm{B}}^{\prime} & {\rm{L}}{\rm{B}} & 0\end{array}\right],\end{eqnarray} \tag{ 8 }$$

where LB (LD) and LB' (LD') are the linear birefringence (dichroism) along the x–y and ±45° axes, CD is the circular dichroism and CB the circular birefringence. On the other hand, L_u is given by [30],

$$\begin{eqnarray}{{\bf{L}}}_{u}=\left[\begin{array}{llll}A & {\rm{\Delta }}{p}_{1} & {\rm{\Delta }}{p}_{2} & {\rm{\Delta }}{p}_{3}\\ -{\rm{\Delta }}{p}_{1} & A-{\rm{L}}{\rm{D}}{\rm{P}} & {\rm{\Delta }}{p}_{4} & {\rm{\Delta }}{p}_{5}\\ -{\rm{\Delta }}{p}_{2}^{} & {\rm{\Delta }}{p}_{4} & A-{\rm{L}}{\rm{D}}{\rm{P}}^{\prime} & {\rm{\Delta }}{p}_{6}\\ -{\rm{\Delta }}{p}_{3} & {\rm{\Delta }}{p}_{5} & {\rm{\Delta }}{p}_{6} & A-{\rm{C}}{\rm{D}}{\rm{P}}\end{array}\right],\end{eqnarray} \tag{ 9 }$$

where LDP, LDP', and CDP describe selective depolarization of linearly horizontal, linearly 45°, and circularly polarized light, respectively. The Δp_j elements are related to inhomogeneity in the sample or time non-reversal events [30]. Although the NCC chiral film shows a multidomain texture i.e. a non-homogeneous media, it is interesting to explore the differential decomposition of its Mueller matrix to evaluate not the intrinsic properties given in equation (8) but their equivalent effective structural parameters.

Figure 10 shows the non-depolarizing matrix L_m corresponding to the transmission Mueller matrix measured at normal incidence shown in figure 5. It is clear that CB and CD are the dominant elementary properties of the NCC chiral film. It should be mentioned that the off-diagonal elements in L_u (not shown) are Δp_j ≈ 0. It is worth mentioning that Δp_j = 0 is a necessary condition for homogeneity in depolarizing media with time-reversal symmetry [30]. Introducing Δα = CD/d, the differential absorption per unit length attains a maximum of 45.22 mm⁻¹ at wavelength 516 nm. On the other hand, diagonal elements of L_u have a similar wavelength dependence as the depolarizance shown in figure 6.

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Current equipment routinely used to quantify CD (known as spectropolarimeters) measure only a few elements of the Mueller matrix (sometimes only one), and often anisotropic artifacts in the system affect the measurements making their interpretation difficult. Also, those equipments are limited to measure samples with low values of CD because in molecular systems the difference between the absorption coefficients α_L and α_R is very small. These limitations are overcome when the Mueller matrix approach is used in chiral films to determine the strong CD signal.