Peculiar behaviour of optical polarization gratings in light-sensitive liquid crystalline elastomers

The angular dependence of the diffraction efficiency of volumetype holographic gratings recorded in a single-domain light-sensitive liquid crystalline elastomer was investigated. Usually this dependence is expected to be very similar for intensity gratings and for polarization gratings. However, our measurements resolved a profound difference between the two types of the gratings: a typical Bragg peak of the diffraction efficiency is observed only for intensity gratings, while polarization gratings exhibit a profound dip at the Bragg angle. The appearance of this dip is explained by strongly anisotropic optical absorption of the actinic light during the recording process. ©2016 Optical Society of America OCIS codes: (050.1950) Diffraction gratings; (050.1930) Dichroism; (160.5335) Photosensitive materials. References and links 1. T. Todorov, L. Nikolova, and N. 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Introduction
Optical polarization gratings are periodic holographic structures that are recorded by optical interference patterns at constant intensity but spatially varying polarization states of the optical field [1,2].They exhibit many interesting properties that can be applied in devices for detection and manipulation of the optical polarization state, such as polarization filters and converters, polarizing beam splitters and polarization-sensitive optical data storage units [3][4][5][6][7].Liquid crystalline (LC) materials are particularly suitable for fabrication of polarization gratings, because they possess a collective molecular response in combination with a strong optical anisotropy [8,9].Polarization-responsive recording in liquid crystalline materials is usually based on optical-field-induced reorientation of the mesogenic molecules [10][11][12].Another common mechanism that leads to polarization-type LC gratings is holographic or some other type of patterning of surface layers that control the LC alignment [13][14][15][16][17][18][19][20].In both cases, photoresponse is typically achieved by incorporation of photoisomerizable chemical compounds, mostly azobenzene derivatives [21].
Liquid crystalline elastomers (LCEs) are soft materials that combine liquid crystallinity with rubber elasticity.Light-sensitive LCEs exhibit a huge opto-mechanical response, which is a direct consequence of the specific coupling mechanisms present in these materials [22][23][24].Exposed to optical irradiation, they can change size and shape, flip between different shapes, oscillate, or even move on the supporting surface [25][26][27][28][29][30][31][32].Another interesting feature of light-sensitive LCEs is their strong opto-optical response that is observed as a large photoinduced modification of the optical birefringence.This can be exploited for fabrication of tunable volume-type optical diffraction structures that can be regulated by mechanical strain or by temperature modifications [33][34][35][36][37][38][39].
Our recent papers have reported investigations on LCE-based diffraction gratings fabricated by intensity-modulated interference patterns [40][41][42][43], while this paper reports on polarization gratings.We show that polarization gratings exhibit a similar magnitude of the diffraction efficiency as intensity gratings.However, in contrast to intensity gratings, they display an unusual dip, i.e. a depression structure, instead of the expected Bragg reflection peak.By extending our theoretical model developed for intensity gratings [40], we show that this dip is a consequence of anisotropic absorption of actinic light during the recording process.
Optical transmission gratings were recorded by placing the LCE film in the interference field of two expanded laser beams with a wavelength of either λ r = 351 nm or 364 nm, which resulted in formation of transmission gratings with grating spacing Λ of 1.6 or 1.7 μm, respectively.Recording beams of equal intensity (8 mW/cm 2 ) entered the film symmetrically with respect to the surface normal.The beams were linearly polarized.Their polarization directions with respect to the nematic director n are depicted in Fig. 1.For recording intensity gratings n was set parallel to the s-polarization direction and the recording beams were superposed in parallel (s-s or in p-p) polarization combination.For recording of polarization gratings n was set at 45° with respect to the s-and p-polarization directions and the recording beams were superposed in perpendicular (s-p) polarization combination.The recording time for all gratings was 10 min.Optical diffraction properties of the gratings were examined by a low-power (< 1 mW) laser beam at λ p = 633 nm, a wavelength to which the LCE is not sensitive.The probe beam was either s or p polarized.The sample was mounted onto a rotation stage and rotated around an axis perpendicular to the plane of incidence.The angular dependence of the intensities I 0 , I +1 and I -1 of the 0th and the ± 1st order diffraction peaks in the vicinity of the Bragg angle θ B = arcsin(λ p /2Λ) was measured by photodiode detectors.Intensities of higher diffraction orders were negligible.The (relative) diffraction efficiency was calculated as where i = 1, 0, or + 1.By this definition absorption and scattering losses of the probe beam can be disregarded.The optical absorbance A of the LCE film at λ r = 351 nm was so high (> 4) that its dichroism could not be accessed directly.We therefore characterized the dichroism of the azomesogens (J7) incorporated in the film by introducing them into the conventional nematic liquid crystalline mixture E7 (Shijiazhuang Chengzhi Yonghua Display Material Co.).
Commercial glass cells with a thickness of μm (Instec Inc.) and surface coatings inducing planar LC alignment were filled either with pure E7 or with a mixture of E7 and 1 wt% of J7 and their absorbance as a function of the polarization direction of the linearly polarized incident beam was measured at 351 nm.The results are shown in Fig. 2. Both cells exhibit considerable linear dichroism with maximal absorbance A for the polarization state parallel to the nematic director n (Fig. 2(a)).The absorbance difference between both samples (ΔA), which is attributed to the absorbance of the azomesogens, shows a profound number eight shape (Fig. 2(b)).This signifies a strong preferential alignment of the J7 molecules along the direction of n.We presume that a similar alignment occurs also when J7 is incorporated into the LCE matrix.The Bragg peak exhibits the FWHMs of ∼15° for the grating recorded in the s-s configuration and of ∼8° for the grating recorded in the p-p configuration, respectively.This observation indicates that the depth of recording, i.e. the effective grating thickness, is about two times larger for the p-p grating than for the s-s grating [40].This is a consequence of the weaker absorption (larger penetration depth) of the p-polarized radiation at λ r = 351 nm as compared to the s-polarized beam.Anyway, the peak diffraction efficiency for the p-p grating is about two times smaller than for the s-s grating, which is a consequence of the fact that the p-p grating was probed by the p-polarized probe beam (at λ p = 633 nm) while the s-s grating was probed by the spolarized beam.As discussed in our previous paper, due to the relation between the LC order parameter and birefringence, light polarized perpendicular to the nematic director in general exhibits a lower diffraction efficiency than light polarized parallel to the director [42].shows angular dependencies of the diffraction efficiency for the ± 1st diffraction orders for a grating recorded in the perpendicular (s-p) polarization combination and probed by an s-polarized probe beam.The diffraction efficiency of this grating has a similar magnitude as for the gratings recorded by parallel polarization combinations.However, there exists a striking difference in the observed angular dependence -instead of a peak there is a profound dip observed exactly at the Bragg angle.The depth of the dip varied from sample to sample and from recording to recording, but the dip could always be noticed.We also monitored the angular dependence during decay of the grating due to spontaneous cis-to-trans back isomerization and found that the magnitude of the diffraction efficiency was decreasing with time, however, its angular dependence including the dip remained the same.This observation confirmed that the dip was not associated with the overmodulation of the gratings.
To verify the universality of the observed behaviour we recorded the grating by using perpendicularly polarized recording beams at another recording wavelength, namely at λ r = 364 nm.In addition to this, we measured angular dependencies of the diffraction efficiency for p as well as for s polarized probed beams.The results are shown in Fig. 4. One can notice that the dip at the Bragg peak is present also for these cases.In addition, we analysed some LCE samples with different chemical compositions from the one described above and the dip was observed for them too.Hence we propose that the observed behaviour is a general property of polarization gratings recorded in the LCEs.In the following we will show that it can be explained by the large linear dichroism of the material at the recording optical wavelength.

Theoretical model
Superposition of two coherent optical waves with equal amplitudes but orthogonal polarization states leads to optical interference fields with constant intensity but a periodic spatially varying polarization state [45][46][47].At small intersection angles (2θ) of the beams, as it is the case in our recording setup (Fig. 1), the component of the optical field orthogonal to the intersection plane can be neglected (collinear approximation) [48].Consequently, the superposition of s and p polarized beams results in an interference pattern that is varying between circular and linear polarization as depicted in the top line of Fig. 5.This interference pattern can be decomposed into the superposition of two interference sub-patterns: a subpattern of two beams linearly polarized at −45° with respect to the p polarization (direction parallel to n in Fig. 1(b) corresponding to extraordinary polarization) and a sub-pattern of two beams linearly polarized at + 45° with respect to the p polarization (direction perpendicular to n in Fig. 1(b) corresponding to ordinary polarization) [45,49].The extraordinaryextraordinary (e-e) and ordinary-ordinary (o-o) sub-patterns are shifted for Λ/2 along the grating vector (x axis in Fig. 1) [47].
In the LCE material, due to its large linear dichroism, the e-e sub-pattern decreases much faster with sample depth (z axis in Fig. 1) than the o-o sub-pattern.Consequently, the polarization state of the total optical field changes with sample depth and at large depths practically only the o-o sub-pattern subjected to lower absorption survives.Therefore polarization modulation is transformed to intensity modulation.This effect is schematically shown in Fig. 5. Fig. 5. Optical interference field of s and p polarized optical beams as a function of sample depth (see also Fig. 1).Due to linear dichroism the amplitude of the e-e sub-pattern (polarization at −45° with respect to x axis) decreases with sample depth (z axis) much faster than the amplitude of the o-o sub-pattern (polarization at + 45° with respect to x axis).The direction of nematic director n is designated on the left side at the bottom of the image.
Optical absorption in photo-isomerizable materials takes place in a nonlinear manner associated with the difference between the absorption cross sections of the trans and the cis isomers.Cis isomers usually exhibit lower absorption for UV radiation and consequently after the isomerization reaction the material becomes more transparent for the UV light than before it.As a result of this phenomenon, in the beginning of the irradiation process the optical field is present mainly in the surface region of the material.However, with prolonged irradiation, it penetrates deeper and deeper into the volume.This type of "photo-bleaching" is characteristic also for light-sensitive LCEs and was analyzed in one of our previous studies reporting LCEbased optical gratings [40].However, for intensity gratings considered in Refs [40][41][42][43], the anisotropy of optical absorption was not important.If this anisotropy is taken into account, then the rate equation for the relative concentration of the trans isomers c t = C t /(C t + C c ), where C t and C c denote molar concentrations of trans and cis isomers (in units of moles/m 3 ), in [40] is modified to: where γ denotes the conversion efficiency from the electronic excited state to the cis or trans conformation, σ is the absorption cross section, Ψ is a parameter proportional to the optical intensity, and τ is the thermal relaxation time from cis to trans state.The subscripts t and c denote trans and cis states and the indices e and o denote extraordinary and ordinary polarized optical fields, respectively.In Eq. ( 2) it is assumed that light-induced cis-to-trans back isomerization is independent of the polarization state of the optical field, which is reasonable for the chromophores used in our LCE material.In accordance with Eq. ( 2), the components of the imaginary part of the uniaxial optical dielectric tensor ε″, can be expressed as: where k 0 is the magnitude of the optical wave vector in vacuum.
The above described extensions of the theory presented in [40] were used to calculate depth profiles of the optical intensity for extraordinary and ordinary polarized optical fields for different recording times of the grating.The value of (σ t,e /σ t,o ) = 2, in agreement with the measurements shown in Fig. 2 and Fig. 3(a), was used in the calculation.The sample thickness considered in the calculation was 20 μm, which corresponds to the typical effective depth of the gratings for long recording times [40,41].The values of all other parameters were set to be the same as in [40].The optical field was calculated by using a FDTD methodbased software package (WOLFSIM) designed for solving the wave equation in periodic structures of arbitrary anisotropic media [50,51].The result is shown in Fig. 6.Due to the Λ/2 shift between the e-e and o-o interference patterns, the maxima of the extraordinarily polarized field coincide with the minima of the ordinarily polarized field and vice versa.Besides this, due to lower absorption, the ordinarily polarized field penetrates deeper into the sample than the extraordinarily polarized field.Fig. 6.Calculated intensity profiles of extraordinarily and ordinarily polarized optical fields as a function of sample depth for increasing recording times.The recording time linearly increases from t r = 10 s (upper segment) to t r = 140 s (bottom segment).The profile corresponding to one period Λ of the interference pattern is shown in each segment.The ordinarily polarized field penetrates deeper into the sample than the extraordinarily polarized field.Its maxima and minima are shifted for Λ/2 with respect to the extraordinarily polarized field.
Fig. 6).At large sample depths, the ordinary field prevails and consequently the birefringence is most profoundly reduced in the regions of high intensity of the ordinary field (red-coloured regions on the right side of Fig. 6).Due to the Λ/2 spatial shift of these regions, the modulation of the birefringence changes its sign as a function of the sample depth.The spatial dependence of the anisotropy of the real part of the optical dielectric tensor Δε can be described as [40]: where ' e ε and ' o ε denote extraordinary and ordinary components of the real part of the optical dielectric tensor, ε a is the average anisotropy, Δε 1 (z) is the modulation of the anisotropy that depends on the sample depth z, and K g = 2π/Λ is the magnitude of the grating vector.The calculated dependence of Δε 1 (z) for some selected recording times is shown in Fig. 7.One can notice the change of the sign at a depth of about 30% of the sample thickness.The oscillations at the rear side of the sample are the consequence of boundary conditions and limited mesh size.The change of the sign of Δε 1 (z) with sample depth causes destructive interference between the diffraction field of the probe beam generated in surface region of the sample and the diffraction field generated in inner parts of the sample.This effect is most pronounced when the probe beam enters the sample at the Bragg angle θ B .For a weakly diffracted beam the diffraction efficiency of the ± 1st diffraction orders in the vicinity of θ B can be expressed as [47]: where L is the sample thickness and B θ θ θ Δ = − measures a deviation from the Bragg angle.The calculated angular dependence of the diffraction efficiency η(θ) (limited to −0.2 rad < θ < + 0.2 rad) corresponding to the longest recording time considered in Fig. 7 is shown as a red solid line in Fig. 3(b).A profound dip with minimum at θ = θ B is evident.Also in general, a good agreement between the calculated and the measured dependence of η(θ) can be noticed.The largest variations in the observed angular behavior appear in the value of η(θ = θ B ) (Fig. 4).As follows from Eq. ( 5), this value is determined by Having applications in mind, one might wonder how to avoid a detrimental dip at the Bragg angle.As follows from our study, one simply should use either recording wavelengths for which the material exhibits low linear dichroism or employ parallel polarization states.On the other hand, the dip might also be advantageous, e.g., in multiplexing of the usual intensity gratings with polarization gratings.Here, the former can be optimally read out at the Bragg angle whereas the latter in off-Bragg geometry, therefore any "cross-talk" effects can be reduced.

Conclusions
Our results demonstrate that a strong linear dichroism at the actinic optical wavelength, which is characteristic for light-sensitive LCEs and also for many other kinds of light-sensitive liquid crystalline materials, can drastically influence the volume-type polarization holographic recording in photo-responsive media.However, up till now this effect was mostly neglected [46,52].In combination with strongly nonlinear recording kinetics associated with photo-bleaching, dichroism can lead to diffractive structures exhibiting various unusual phenomena, such as a profound minimum of the diffraction efficiency at the Bragg angle, which was the subject of our present investigation.
Other features that are worth being studied in LCE-based polarization gratings are polarization properties of the diffracted light and the effect of mechanical strain and temperature on these properties [53].Another open problem that can be conveniently investigated by analysis of optical polarization gratings, is photo-induced alignment in the LCE materials [36,54].Besides this, optical polarization gratings recorded in the LCE media can also open up various new challenges for construction of polarization-sensitive diffractive optical elements that can be regulated by different external stimuli, in particular by mechanical strain.
The diffraction efficiency of a few percent, typically observed in our experiments, is usually too low for practical applications.We want to emphasize that this work focused on a fundamental explanation of the reported phenomenon, and for that reason the same LCE material was used as in our previous studies.This approach allowed us to use material parameters deduced from these studies for our numerical simulations and enhances their soundness.However, the investigated LCE material is far from being optimised from the point of view of diffraction efficiency.As shown in our recent paper, small variations of the chemical structure of the azomesogen moiety and/or suitable clamping of the LCE film can improve the diffraction efficiency for one order of magnitude [42].In addition, further improvements are easily possible by optimizing the recording optical wavelength (with respect to the absorption spectrum of the material), so that the effective thickness of the grating is increased and consequently diffraction occurs in the two-wave Bragg diffraction regime.Another important aspect for applications is also to reduce the grating spacing to about Λ~500 nm, which is an open problem with LCE gratings that needs to be investigated in the future.

Fig. 1 .
Fig. 1.Schematic drawings of the recording configuration for parallel polarization combination (a) and for perpendicular polarization combination (b), respectively.

Fig. 2 .
Fig. 2. Absorbance (at 351 nm) of planarly aligned samples of pure E7 and of E7 mixed with 1 wt% of J7 azomesogens as a function of polarization direction of the linearly polarized incident optical beam (a).Difference ΔA between the absorbances of both samples (b).Vertical arrows in the centre denote the orientation of the nematic director n.

Figure 3 (
Figure3(a) shows the angular dependencies of the diffraction efficiency of the ± 1st diffraction orders as a function of deviation from the Bragg angle⏐θ-θ B ⏐ for gratings recorded either in the s-s or the p-p polarization state combination.The value of ⏐θ B ⏐ was around 12° (externally measured with respect to normal incidence).The Bragg peak exhibits the FWHMs of ∼15° for the grating recorded in the s-s configuration and of ∼8° for the grating recorded in the p-p configuration, respectively.This observation indicates that the depth of recording, i.e. the effective grating thickness, is about two times larger for the p-p grating than for the s-s grating[40].This is a consequence of the weaker absorption (larger penetration depth) of the p-polarized radiation at λ r = 351 nm as compared to the s-polarized beam.Anyway, the peak diffraction efficiency for the p-p grating is about two times smaller than for the s-s grating, which is a consequence of the fact that the p-p grating was probed by the p-polarized probe beam (at λ p = 633 nm) while the s-s grating was probed by the spolarized beam.As discussed in our previous paper, due to the relation between the LC order parameter and birefringence, light polarized perpendicular to the nematic director in general exhibits a lower diffraction efficiency than light polarized parallel to the director[42].

Fig. 3 .
Fig. 3. Diffraction efficiency of the ± 1st diffraction orders as a function of deviation from the Bragg angle for a grating recorded by parallel polarization states (a) and for a grating recorded by perpendicular polarization states (b).Polarization states of the recording and readout (probe) beams are denoted in the images.The red solid line in (b) is the result of a theoretical simulation described in detail in Section 3.

Figure 3 (
Figure 3(b)  shows angular dependencies of the diffraction efficiency for the ± 1st diffraction orders for a grating recorded in the perpendicular (s-p) polarization combination and probed by an s-polarized probe beam.The diffraction efficiency of this grating has a similar magnitude as for the gratings recorded by parallel polarization combinations.However, there exists a striking difference in the observed angular dependence -instead of a peak there is a profound dip observed exactly at the Bragg angle.The depth of the dip varied from sample to sample and from recording to recording, but the dip could always be noticed.We also monitored the angular dependence during decay of the grating due to spontaneous cis-to-trans back isomerization and found that the magnitude of the diffraction efficiency was decreasing with time, however, its angular dependence including the dip remained the same.This observation confirmed that the dip was not associated with the overmodulation of the gratings.To verify the universality of the observed behaviour we recorded the grating by using perpendicularly polarized recording beams at another recording wavelength, namely at λ r = 364 nm.In addition to this, we measured angular dependencies of the diffraction efficiency for p as well as for s polarized probed beams.The results are shown in Fig.4.One can notice that the dip at the Bragg peak is present also for these cases.In addition, we analysed some LCE samples with different chemical compositions from the one described above and the dip was observed for them too.Hence we propose that the observed behaviour is a general property of polarization gratings recorded in the LCEs.In the following we will show that it can be explained by the large linear dichroism of the material at the recording optical wavelength.

Fig. 4 .
Fig. 4. Diffraction efficiencies of the ± 1st diffraction orders as a function of deviation from the Bragg angle for a grating recorded by perpendicularly polarized recording beams (s + p) and readout by s (squares) and by p polarized (circles) probe beams, respectively.

Fig. 7 .
Fig. 7. Modulation of anisotropy of the real part of the optical dielectric tensor as a function of sample depth for different recording times.The relative recording times are denoted in the inset.The resulting calculated angular dependence of the diffraction efficiency of the ± 1st diffraction orders for t r = 100 s is shown as a thick solid line in Fig. 3(b).
on the details of the evolution of Δε(x,z) during the recording process.Further experiments employing varying recording times are needed to resolve these details.