Modular Optical Diodes for Circular Polarization Based on Glass‐Supported Cellulose Nanocrystal/Polyethylene Glycol Composite Films

Depending on processing methods and conditions, cellulose nanocrystal (CNC) films exhibit linear birefringence or selective Bragg reflection. The latter means reflection of light with left‐handed circular polarization (LCP) due to CNC in a helicoidal microstructure of the same handedness. Herein, glass‐supported CNC/polyethylene glycol (CNC/PEG) composite films with PEG concentrations in the range of 0–30% w/w with selective Bragg reflection at wavelengths from 440 to 550 nm are prepared. A modular device comprised of a dip‐coated birefringent CNC‐glass sample sandwiched between two CNC/PEG‐glass samples shows different responses to light with LCP and to light with right‐handed circular polarization (RCP). The device suppresses selective Bragg reflection from the rear (front) CNC/PEG sample for incident light with LCP (RCP), even when the birefringent film does not meet the condition for a halfwave plate. This behavior resembles the performance of optical diodes for circular polarization. Polarization properties of composite films and optical diodes in terms of degree of polarization and ellipticity are discussed within the Stokes–Mueller formalism. Electromagnetic simulations of Mueller matrices reveal the equivalence of modular and in‐tandem film approaches of optical diodes.


Introduction
Optical diodes are characterized by a unidirectional propagation of electromagnetic waves in a narrow spectral band taking advantage of nonlinear optical effects as in microring resonators [1,2] or one-dimensional photonic crystal structures. [3,4]By exploiting the selectivity of chiral nematic liquid crystals to reflect circularly polarized light of the same handedness as the liquid crystal, another type of optical diode was developed by sandwiching a nematic liquid crystal layer between two cholesteric layers. [5]he latter device was inspired by the cuticle of the scarab beetle Chrysina resplendens (Boucard 1875, formerly Plusiotis resplendens), which is comprised of a unidirectional layer of chitin-uric acid fibrils sandwiched between two layers of fibrils in a left-handed helicoidal structure. [6]iomimetic reflectors have also been fabricated using self-organizing, self-aligning liquid crystal polymers. [7]10] Slow evaporation leads to solid films comprised of CNC in a left-handed helical arrangement. [8,11]That is, CNC are preferentially oriented in pseudoplanes (with ordinary n o and extraordinary n e refractive indices), that preferential orientation twist between adjacent pseudoplanes.The structural pitch Λ (distance to complete a 360°turn) and the in-plane average refractive index n av = (n o þ n e )/2 determine the wavelength λ B = n av Λ for reflection of light with lefthanded circular polarization (LCP).This is referred to as the circular Bragg phenomenon or selective Bragg reflection. [12]Among methods to control λ B , the addition of salts produces a blueshift, [13] whereas a redshift results in application of ultrasonication power. [14]17][18][19][20] CNC-based films have also been used to mimic the reflection properties of the scarab beetle C. resplendens.A hyper-reflective nanocomposite film was fabricated by embedding a uniaxially oriented polyamide-6 layer between two (CNC)/polyethylene glycol diacrylate layers. [21]Impregnation of iridescent CNC films with a nematic liquid crystal reflecting light both with LCP and right-handed circular polarization (RCP) has also been investigated. [22]Highly reflective CNC-organosilica films embedding a uniaxial bacterial cellulose film have been reported. [23]A nematic-like phase, spontaneously intercalated into left-handed helicoids by rapid gelation of CNC suspensions, produced reflection of light with both LCP and RCP, that is, an ambidextrous response. [24][27][28] The referenced works evidence the interest in CNC-based devices for the manipulation of circular polarization of light inspired by the cuticle of the scarab beetle C. resplendens.In those works, the studies are primarily based on irradiance measurements in reflection mode.However, for a complete description of polarization properties of such complex structures it is necessary to use methodology based on the Stokes-Mueller formalism.
In this work, the optical response of a CNC-based modular device that resembles the performance of optical diodes for circular polarization is investigated.Glass-supported CNC/PEG composite films and dip-coated CNC birefringent film on glass constitute the modules of the device.Polarization properties in the transmission mode of modules and optical diodes are discussed within the Stokes-Mueller formalism.Equivalence between modular and in-tandem film design approaches of optical diodes is investigated by electromagnetic simulations of Mueller matrices.

Fundamentals of Stokes-Mueller Formalism
The Stokes-Mueller formalism provides a complete description of the polarization and depolarization properties of light beams and their interaction with matter in reflection or transmission. [29]athematically, light beams are represented by Stokes vectors S = (I,Q,U,V ) T where T means transpose, I represents total irradiance, Q and U represent linear polarization, and V represents circular polarization.One of the advantages of this formalism is its capability to quantify depolarization.From the Stokes vector, the degree of polarization P of a light beam is given by where P = 1 represents a totally polarized beam and P < 1 is a partially polarized beam.In general, the polarized part of the beam will be elliptically polarized, where the orientation ψ of the polarization ellipse is given by tan 2ψ ¼ U=Q and the ellipticity e can be shown to be Thus, for linear polarization e = 0, whereas the extreme values e = À1 and e = 1 correspond to LCP and RCP, respectively.Other values of e represent elliptical polarization.The 4 Â 4 Mueller matrix M = {M ij } is introduced to describe the sample in light-matter interactions.In a transmission measurement, the Stokes vector of the transmitted beam S t is then determined by where S i represents the incident beam.In this work, we use normalized Stokes vectors (I = 1) and normalized Mueller matrices with elements m ij = M ij /M 11 whereby m 11 = 1.

Electromagnetic Modeling of Mueller Matrices
Calculations of Mueller matrices at normal incidence were performed with the CompleteEASE software (J. A. Woollam Co., Inc.) in the spectral range 300-800 nm with a resolution of 1 nm.Chiral films are modeled as anisotropic twisted slices with real-valued principal refractive indices (n 1 ,n 2 ,n 3 ) in an xyz Cartesian coordinate system with the z-axis along the helix axis.The twist is parameterized by the variable azimuth angle ϕ (in degrees) corresponding to the orientation of n 1 with respect to the x-axis as given [30] ϕðuÞ where u is the distance from the bottom of the chiral structure, d is the thickness of the structure, N t is the number of 360°turns, and ϕ 0 is the azimuth offset of the n 1 direction.The sign of N t defines the handedness of the chiral structure with N t > 0 for a left-handed and N t < 0 for a right-handed structure.The helicoidal pitch is given as Λ = d/N t .Briefly, to calculate a Mueller matrix, the 2 Â 2 Jones matrix J = {t ij } (i,j = p,s) of the entire multilayer system is calculated first. [29]The subscripts p and s refer to direction parallel and perpendicular to the plane of incidence.The matrix J relates incident and emerging waves as given by where E i(p,s) and E t(p,s) are the electric fields of the incident and transmitted electromagnetic waves, respectively.Then, the Mueller matrix of the nondepolarizing optical system represented by the Jones matrix in Equation ( 6) can be calculated with the standard procedure M ¼ TðJ ⊗ J Ã ÞT À1 , where ⊗ denotes the Kronecker product, the asterisk means complex conjugation, and T is the matrix relating the Stokes and coherency vectors. [29]The components of the coherency vector are the elements of the coherency matrix in the Jones representation.Since chiral CNC films show multidomain texture, nonuniformity in the helicoidal layer thickness d is assumed to represent a pitch distribution.In practice, Mueller matrices are calculated for several thicknesses in the interval d-Δd to d þ Δd, which corresponds to a pitch distribution between Λ-Δd/N t and Λ þ Δd/N t .A Gaussian weighted average of these Mueller matrices represents the multidomain sample.In this work, Δd/d = 4% is assumed.

Optical Chirality in CNC/PEG Composite Films
[17][18][19][20] Regarding the effect of PEG concentration in the composite CNC/PEG films, two important results are of special interest for this work.First, it has been demonstrated that λ B is gradually redshifted as the PEG concentration increases from 0 to 30 wt%.At higher PEG concentrations, crystallization of PEG takes place which diminishes the effective PEG concentration for intercalation in the chiral structure and therefore λ B is blueshifted. [15,16,20]Since selection of CNC/PEG composite films was made only as a mean to control λ B , PEG contents were limited up to 30 wt%.
Figure 1a shows near-to-normal incidence images (2.5 Â 2.5 cm 2 ) of CNC/PEG films with different concentration ratios considered and prepared from suspensions stored for twelve days.This storage time was enough to achieve the gel-to-chiral nematic transformation of the suspensions as was reported before by our group. [31]By increasing the PEG content, a redshift of the color is observed.It should be noticed that the apparent color might be influenced by the viewing angle because of the iridescent effect in this type of film.Figure 1b shows transmittance (T ) spectra of unpolarized light at a relative humidity of 33% and the redshift of the spectral location of the minimum (λ B ) with PEG concentration is clearly seen.The steep decrease at short wavelengths is due to the glass substrate.Figure 1c shows λ B as a function of PEG content.The shift in λ B accounts for the enlarged pitch of the chiral structure due to the effective incorporation of PEG at increasing content.Transmittance spectra of films prepared from suspensions with different CNC/PEG ratios and after different times of storage are shown in Figure S1, Supporting Information.A blueshift of λ B with storage time is seen and is due to autocatalyzed acidic desulfation of CNC in the suspension. [32,33]The film thicknesses of samples prepared with different CNC/PEG ratios as determined from cross-section scanning electron microscopy images (Figure S2, Supporting Information) are 6.8 AE 0.1, 6.7 AE 0.2, 11.0 AE 0.4, and 12.9 AE 0.1 μm for 100/0, 90/10, 80/20, and 70/30 CNC/PEG ratios, respectively.
Figure 2 shows the transmittance spectra T L and T R of CNC/PEG composite films for LCP and RCP light, respectively.Spectra in Figure 2a,b correspond to films prepared from suspensions stored for 12 days with CNC/PEG ratios 100/0 and 70/30, respectively.As can be seen, the T L spectra show minima revealing the strong interaction at λ B of LCP light with the helicoidal structure of the films, producing selective Bragg reflection.In contrast, the T R spectra are featureless indicating practically no interaction of RCP light with the structure of the films.A double-band device can be assembled by placing one sample in front of the other, as depicted in Figure 2c.The corresponding T L and T R spectra shown in Figure 2d show a double-band spectrum in T L whereas the flat response of T R is observed once again.The differential response of the films to light with LCP and RCP demonstrates optical chirality.Figure 2 shows spectra measured for incident light from the film side, but they are the same for incidence from the glass (back) side.To quantify polarization properties, the Stokes-Mueller formalism is applied in the next section.

Mueller Matrices and Polarization Properties of CNC/PEG Composite Films
Figure 3 shows normalized Mueller matrices of CNC/PEG composite films with 100/0 and 70/30 ratios measured in transmission at normal incidence.The scale of all Mueller-matrix elements reported here is shown in the left-bottom panel of m 41 .As is known, λ B of CNC/PEG composite films depends on relative humidity, [15][16][17][18] and some variation is expected when compared to irradiance measurements.The Mueller-matrix elements have an indirect physical interpretation.However, the structure of a Mueller matrix reveals dominant polarization properties.In the present case, the 2 Â 2 central block and the fact that m 41 = m 14 reveal the presence of both circular 300 400 500 600 700 800  birefringence and circular dichroism as the dominant polarization properties of the composite films. [34]These two properties are of fundamental importance in chiral systems as has been reported by our group for free-standing and glass-supported CNC films. [35,36]o explain polarization states of light corresponding to the T spectra in Figure 1b, Equation ( 4) is applied to the Stokes vector for unpolarized light S i = (1,0,0,0) T and the Mueller matrices in Figure 3.The Stokes vector of the transmitted beam will be S t = (1, m 21 , m 31 , m 41 ) T and the polarization properties P and e can be calculated from Equation ( 1) and (3), respectively.
As can be seen in Figure 4a, the degree of polarization of the transmitted beam attains a maximum value of about 0.6 at wavelength 450 and 540 nm for composite films 100/0 and 70/30, respectively, with a right-handed character because e > 0. For the 70/30 film, light with nearly RCP (e % 1) is transmitted in the 500-700 nm wavelength range.
The polarization properties of light corresponding to T L and T R spectra in Figure 2a,b are shown in Figure 4b,c, respectively.In this case, the respective incident Stokes vectors to be considered in Equation ( 4) are S L = (1,0,0,-1) T and S R = (1,0,0,1) T .Thus, in both cases the transmitted beam is highly polarized (P % 1) with the same handedness as the incident light, but with an elliptical character as e 6 ¼ 0, AE1.Since e corresponds to the minor/major axes ratio of the polarization ellipse, the inserts in Figure 4b,c illustrate deviations of circularity for e = 0.9 and e = 0.8, respectively.

Characterization of Modular Optical Diodes
Figure 5a shows a schematic of a stacked structure of CNC-based films on glass substrates.The structure is comprised of a dip-coated CNC birefringent sample sandwiched between two glass-supported chiral CNC/PEG films.As was previously reported, dip-coating of nonsonicated CNC suspensions is an efficient method to prepare nondepolarizing birefringent CNC films. [27]The Mueller matrix and retardance (δ) of the film used, are shown in Figure S3, Supporting Information.Four cases can be identified in Figure 5a, depending on the polarization state of incident light (LCP or RCP) and incidence from the front or the back sides.The spectra in Figure 5 correspond to CNC/PEG composite chiral films 100/0 and 70/30 identified in the scheme as blue/cyan and red/orange, respectively.Considering the first incident light with LCP from the front side (case 1), the minimum in spectrum T 1 of Figure 5b accounts for light interaction only with the 100/0 film.In contrast, the incidence of LCP light from the back side (case 2) the minimum in T 2 at wavelength 550 nm shows interaction with the 70/30 film.That is, for light with LCP in the forward (backward) direction the device is transparent at wavelength 550 nm (435 nm), but partially blocking for incident LCP light from the back (front) side.This behavior resembles the operation principle of optical diodes for LCP light at wavelengths 550 and 435 nm.For incident RCP light the situation is similar as seen in Figure 5c in the forward (T 3 ) and backward (T 4 ) directions corresponding to cases 3 and 4, respectively.That is, depending on the side of incidence, front or back, LCP (RCP) light interacts with the CNC/PEG composite film at the front (rear) position.This suppression of selective Bragg reflection in one of the films leads to a system that is transparent at certain wavelengths when switching incident light between LCP and RCP states.As a measure of the response for incident light with LCP or RCP in the forward and backward directions, Figure 5d shows transmittance ratios T 1 /T 2 and T 3 /T 4 on dB scale.We call the assembly modular because the components are interchangeable The contrast ratio for forward/backward incidence switching can be increased by depositing larger volumes of suspensions stored for longer times to promote tactoid annealing. [37]igure 6 shows transmittance spectra for an assembly with films 80/20 and 70/30 CNC/PEG ratios from suspensions stored for 40 and 33 days, respectively, sandwiching the birefringent CNC film.These films show the deepest minima in T spectra for unpolarized light among the samples fabricated as shown in Figure S1  Figure 6a, the device shows the double-band spectrum for unpolarized light due to the interaction of the LCP component with the helical structure in both films.The differences in response to the incidence of light with LCP or RCP in the forward/backward directions are shown in Figure 6b,c, respectively, and deeper minima are noticed as compared to those in Figure 5.The inserts in Figure 6b show images of the optical diode as viewed through LCP (top) and RCP (bottom) filters.As explained above, the minimum in the spectrum of T 1 (T 2 ) corresponds to the blue (green) selective Bragg reflection observed in the insert at the top (bottom).In transmission mode, the observed color corresponds to the complementary one observed in the images.The contrast ratios of transmitted light for incidence in the forward and backward directions are shown in Figure 6d, which are more than three times larger than those in Figure 5d.The results shown in Figure 5 and 6 can be explained by the phase-shift introduced by the birefringent CNC film transforming LCP to RCP and vice versa.However, in the spectral range of interest (400-600 nm) for selective Bragg reflection, the retardation decreases from 185°to 120°(see Figure S3b, Supporting Information), which deviates from the ideal value of 180°for a halfwave plate.However, Mueller-matrix modeling of the cuticle of the scarab beetle C. resplendens showed that an exact halfwave plate condition is not necessary to obtain a wealthy of polarization properties in reflection as previously reported. [38]n the next section, the quantification of phase shifts introduced by the birefringent sample is discussed within the Stokes-Mueller formalism.

Polarization Properties of Modular Optical Diodes
The modular device shown in Figure 5a is comprised of optical elements in series and its optical performance can be described by the product of individual Mueller matrices.For the device with optical response as shown in Figure 6, the Mueller matrices are M dip , M 70-30 , and M 80-20 which are shown in Figure S3 and S4, Supporting Information.Figure 7 shows the resulting matrices calculated for incident light in the forward direction M 70-30 M dip M 80-20 as well as for incidence from the backside M 80-20 M dip M 70-30 .As is known, the multiplication of Mueller matrices assumes incoherent interaction of electromagnetic waves traveling through the in-series three-module system.As can be noticed, these Mueller matrices exhibit a complicated structure in contrast to the simpler case of Figure 2c  in Figure S5, Supporting Information).Matrix products for samples comprising the device in Figure 5 are shown in Figure S6b, Supporting Information.Next, polarization properties observed in the T spectra shown in Figure 6 are explained with data in Figure 7. Let us first consider the incidence of unpolarized light S u = (1,0,0,0) T .Taking the matrix product with data in Figure 7, the calculated degree of polarization (P) and ellipticity (e) are shown in Figure 8a1 for forward incidence M 70-30 M dip M 80-20 and Figure 8a2 corresponds to backward incidence M 80-20 M dip M 70-30 .As observed, the transmitted beam has relatively high values of P at the two-band spectrum in Figure 6a.However, e has a different sign in the short wavelength band compared to that in the long wavelength band.That is, we see an ambidextrous optical response, with stronger contrast for incidence in the forward direction M 70-30 M dip M 80-20 (Figure a2). Figure 8b1,b2 correspond to the incidence of LCP light from the front and back sides, respectively.In both cases, high degree of polarization P % 1 and close to an RCP state (e % 1) at wavelengths around 500 nm are obtained.Finally, a left-handed (e < 0) transmitted beam with high P % 1 is obtained for incident light with RCP as is shown in Figure 8c1,c2 for forward and backward incidence, respectively.Corresponding spectra of P and e for the device with elements specified in Figure 5 are shown in Figure S7, Supporting Information.
The change of handedness of light after passing each element in the device can be tracked with e of Stokes vectors, as shown in Figure S8 and S9, Supporting Information.The beam transmitted by the first chiral film M chiral1 S L,R keeps the handedness of the incident beam.The action of the birefringent film M dip M chiral1 S L,R changes the handedness.The effect of the second chiral film M chiral2 M dip M chiral2 S L,R does not alter too much the handedness of the beam in most of the spectral range studied, but produces a small change in the aspect ratio of the polarization ellipse.However, selective Bragg reflection of the 70/30 film employed in the device of Figure 6, is strong enough to change the handedness as shown in Figure 8c1, producing a spectral "hole" in handedness at about 513 nm.However, the transmittance is small at that wavelength, as can be seen in Figure 5b.The azimuth of polarization ellipse according to Equation ( 2) is shown in Figure S10, Supporting Information.

Modular Optical Diodes
The experimental results discussed in the previous Section 3.4 can be further supported by the electromagnetic modeling of Mueller matrices of the chiral and birefringent modules comprising the optical diodes, as depicted in Figure 5a.First, for the dipcoated CNC film, analytical inversion of the Mueller matrix [39] revealed linear birefringence as its dominant polarization property.Therefore, the Mueller matrix of a retarder was thus assumed (Equation S1, Supporting Information) with retardation δ = 2πdΔn/λ, where d is the film thickness and Δn = n o -n e the effective linear birefringence.A Cauchy expression was used to model Δn = A þ B/λ 2 , where A and B are fitting parameters.Figure S3, Supporting Information, shows the Mueller matrix M bir , δ, and Δn.
Regarding the Mueller matrices of chiral films, they were modeled as twisted anisotropic slices according to Section 2.2; more details are given elsewhere. [30]Thus, the Mueller matrix M 300 of a helicoidal structure with pitch Λ = 300 nm was calculated with ϕ 0 = 0, d = 12 μm, and N t = 40 in Equation (5).The Mueller matrix of a longer pitch (Λ = 324 nm) chiral structure M 324 was calculated by simply decreasing the number of turns to N t = 37, keeping the values of ϕ 0 and d fixed.In both cases, refractive indices n 1 and n 2 were described with Cauchy expressions n j = A j þ B j /λ 2 where A 1 = 1.59,A 2 = 1.53, and B 1 = B 2 = 100 nm À2 in accordance with the reported birefringence of cellulose Δn = 0.06. [40]It is worth noting that at normal incidence n 3 is undetermined because its orientation coincides with that of the wave vector.As was mentioned in Section 2.2, a 4% inhomogeneity of film thickness was considered to account for lateral variations of the pitch.M 300 and M 324 are shown in Figure S11, Supporting Information.Once Mueller matrices of individual components for an optical diode have been calculated, the response of modular devices as shown in Figure 9a,b, can be analyzed by matrix products M 300 M bir M 324 and M 324 M bir M 300 configurations, which are shown in Figure 9e.Their features are discussed together with the results of the next section.

In-Tandem Thin-Film Approach
An alternative thin-film approach to build optical diodes for circular polarization comprises a uniaxial birefringent film sandwiched between two chiral films as depicted in Figure 9c,d.
Since the system is a three-film structure on a substrate, coherent superposition of electromagnetic waves cannot be a priori ruled out.That is, the waves traveling in forward and backward directions across the films' interfaces can interfere and the matrix multiplication procedure of the modular case in Figure 5a already discussed, may in principle not apply.Therefore, the Mueller matrix M ch1-bir-ch2 of the three-film system must be calculated as a single matrix.Regarding to Figure 9c, the chiral film attached to the substrate is defined with a graded azimuth orientation as given by Equation ( 5) with ϕ 0 = 0, d = 12 μm, and N t = 40 to describe a helicoidal structure with pitch Λ = 300 nm.Then, the birefringent film is defined as a uniaxial film as described in Figure S2, Supporting Information.On the top, for the second chiral film with pitch Λ = 324 nm, Equation ( 5) is used once again with ϕ 0 = 0, d = 12 μm, and N t = 37.By interchanging the values of N t between the top and bottom chiral films the situation depicted in Figure 9d is obtained.Then, Mueller matrices were simulated with the CompleteEASE software (J. A. Woollam Co., Inc.) to obtain Mueller matrices M 300-bir-324 and M 324-bir-300 configurations shown in Figure S12, Supporting Information.Figure 9e shows a comparison of Mueller matrices of the modular M 300 M bir M 324 (M 324 M bir M 300 ) and thin-film devices M 300-bir-324 (M 324-bir-300 ) devices.As can be noticed, the two approaches give almost identical results.Furthermore, the Mueller matrices provided by electromagnetic modeling (Figure 9e) share most of the features with those derived from experimental data (Figure 7) for the modular device.Furthermore, the transmittance of the three-film system for unpolarized light, left-and right-handed circularly polarized light (Figure S13, Supporting Information) resembles the performance of experimental data shown in Figure 5 and 6.Even more, the degree of polarization and ellipticity for modular and filmbased devices show good agreement as well if Figure 8 and S14, Supporting Information, are compared.

Concluding Remarks
Glass-supported CNC/PEG composite films with PEG concentrations in the range 0-30% w/w were prepared with selective Bragg reflection tunable with PEG concentration.Modular optical diodes were prepared by sandwiching a dip-coated birefringent CNC film between two CNC/PEG composite films of different pitch.This modular assembly suppresses selective Bragg reflection from the rear (front) CNC/PEG sample for incidence of light with LCP (RCP).Tactoid annealing is accomplished by storing the suspensions for over a month leading to an improved contrast ratio of optical diodes.Polarization properties of composite films and optical diodes are discussed within the Mueller-Stokes formalism.Electromagnetic simulations of Mueller matrices reveal equivalence of modular and thin film approaches of optical diodes.

Experimental Section
Preparation of CNC Suspensions: CNC aqueous suspensions were prepared as previously reported. [31]Briefly, ashless filter paper (8 g, Whatman 40) was ground to increase the surface area and then hydrolyzed with 64 wt% sulfuric acid (70 mL, J. T. Baker) at 60 °C for 60 min under mechanical stirring.To stop the reaction, distilled water (700 mL) at 5 °C was added, and the solution was left to rest at 5 °C for 24 h.The top clear layer was decanted, and the bottom part was centrifuged three times for 10 min each at 9000 rpm.The CNC slurry was dialyzed against distilled water with a cellulose membrane (Sigma Aldrich).The water was changed every 24 h until a neutral pH was reached, which took one week.The CNC concentration in the resultant slurry was 7.92 wt%.Polyethylene glycol with a molecular weight of 200 (Sigma Aldrich) in a 20 wt% solution was used.Different ratios of CNC suspension and PEG solution were mixed to obtain a fixed CNC concentration of 6.5 wt% and PEG/(CNC þ PEG) concentrations in the range 0-30 wt%.Suspensions with a 6.5 wt% CNC concentration are suitable for fabrication of photonic films with uniform color in areas of the order of cm 2 . [31]The mixtures were stored at room temperature until used.
Film Coating: The chiral films were shear-coated on glass slides 25 Â 75 mm 2 (Corning 2947) as reported elsewhere. [31,36]Briefly, a volume of 0.3 mL of CNC /PEG suspension stored for 12 days was deposited at one end of a glass slide and then distributed with another glass slide used as a coater plate at a speed of 5 cm min À1 .The substrate remained fixed during deposition.The other two sets of samples were prepared using a volume of 0.5 mL from suspensions stored for 33 and 40 days.The coated substrates were placed in Petri dishes (90 mm diameter) and deionized water droplets (1 mL) were added around the glass slide to promote slow evaporation inside covered dishes.This procedure for slow drying has been reported as effective to increase homogeneity of films. [31]The evaporation took place at room temperature for 3 days.A birefringent CNC film was dip-coated at withdrawal speed 5 cm min À1 as previously reported. [27]haracterization Techniques: Transmittance spectra at normal incidence in the range 250-840 nm were acquired with a FilmTek 3000 system (SCI, Inc., USA).For incidence of light with LCP and RCP, filter sheets (Edmund Optics) operating in the 400-750 nm range were used.A dual rotating compensator ellipsometer (RC2, J. A. Woollam Co., Inc., Lincoln, NE, USA) was used to measure normalized Mueller matrices in transmission mode at normal incidence in the wavelength range 210-1690 nm.The diameter of the collimated probe beam was about 3 mm.Film thicknesses were determined from scanning electron microscopy (SEM) images using a JEOL 7610F equipment.To avoid overcharging, a thin layer of Au-Pd was deposited on samples using a Denton Vacuum Desk V equipment with argon as carrier gas.

Figure 1 .
Figure 1.a) Pictures of CNC/PEG composite films (2.5 Â 2.5 cm 2 ) with concentrations as denoted by inserts.b) Transmittance spectra of unpolarized light for PEG concentrations between 0 and 30% w/w.c) Selective Bragg reflection wavelength as a function of PEG contents.Mean values and standard deviation for three measurements at different positions are shown.

Figure 2 .Figure 3 .
Figure 2. Transmittance spectra using LCP and RCP incident light for: a) a pure CNC film (100/0 CNC/PEG ratio) and b) a 70/30 CNC/PEG composite film.c) Cascade arrangement of two samples with different CNC/PEG ratios.d) Transmittance spectra for a 100/0 sample in front of a 70/30 sample as depicted in (c).

Figure 4 .
Figure 4. Degree of polarization (P) and ellipticity (e) of light transmitted by CNC/PEG composite films with 100/0 and 70/30 CNC/PEG ratios for incidence of a) unpolarized light, b) light with LCP, and c) light with RCP.The inserts in (b) and (c) show dashed circles for e = 1 and deviations of circularity for e = 0.9 and e = 0.8, respectively.Note the break in the scale of the vertical axis in (b).

4 T 1 / T 2 T 3 / T 4 ForwardFigure 5 .
Figure 5. a) Schematics of selective Bragg optical diodes for circular polarization using glass-supported CNC-based films.b) Transmittance spectra for cases 1 and 2 in (a) for a CNC birefringent sample sandwiched between CNC/PEG samples of ratios 100/0 and 70/30; c) transmittance spectra for cases 3 and 4 in (a); d) transmittance ratios on a dB scale.

Figure 6 . 20 MFigure 7 .
Figure 6.Experimental transmittance spectra for a) unpolarized light, b,c) cases 1-4 in Figure 5 for CNC/PEG composite films with ratios 80/20 and 70/30 from 0.5 mL suspensions stored for 40 and 33 days, respectively.d) Transmittance ratios on a dB scale.The inserts in (b) show images of the diode for LCP (top) and RCP (bottom) light.

Figure 8 .
Figure 8. Degree of polarization and ellipticity e for incidence of: unpolarized light (a1,a2), light with LCP (b1,b2), and with RCP (c1,c2) calculated from data in Figure 7. Graphs in the upper and lower rows correspond to incidence in the forward M 70-30 M dip M 80-20 and backward M 80-20 M dip M 70-30 directions, respectively.

Figure 9 .
Figure9.a,b) Schematics of a modular device and c,d) a three-film system.The latter is comprised by a birefringent film sandwiched between two chiral films on glass of pitch 300 and 324 nm in chiral (300 nm)/birefringent/chiral (324 nm)/glass and chiral (324 nm)/birefringent/chiral (300 nm)/glass configurations.e) Comparison of Mueller matrices from electromagnetic modeling of the systems in (a)-(d).