## Abstract

Materials showing birefringence and polarization selective absorption (dichroism) affect the polarization state of incoming light in a peculiar way, quite different from the one exhibited by phase retarders like waveplates. In this paper, we report on the characterization of a Polymer LIquid CRYstal Polymer Slices (POLICRYPS) diffraction grating used as a dichroic phase retarder; the dichroic behaviour of the grating is due to the polarization-dependent diffraction efficiency. Experimental data are validated with a theoretical model based on the Jones matrix formalism, while the grating behavior is modeled by means of the dichroic matrix. In this way, the birefringence of the analyzed structure is easily obtained. For comparison purposes, also two systems different from POLICRYPS have been fabricated and tested.

© 2010 OSA

## 1. Introduction

Polarized light impinging on a birefringent optical system undergoes, in general, a modification of its polarization state. This behaviour is well known since the very first experiments performed on calcite crystals by the Danish scientist Erasmus Bartholinus in 1669 [1]. Since then, birefringent materials have been largely investigated and employed in a multitude of devices of common use in scientific laboratories (e.g. waveplates or polarizers) but also in technological devices, which exploit birefringence in a smart way. In particular, LCD TVs, mobile phones or other Liquid Crystal (LC) based portable devices have become very popular due to the flexibility of many of the physical properties of LCs, which can be used to influence the polarization state of light as well, when utilized as birefringent materials. Since Nematic Liquid Crystals (NLCs) exhibit both an ordinary (*n _{o}*) and an extraordinary (

*n*) refractive index, depending on the film thickness, a relative phase difference can be introduced between the ordinary and the extraordinary components of a light beam propagating in such media. In an ideal system, where NLC molecules are homogeneously aligned in a given direction, this phase difference can be calculated as

_{e}*δ=2πLΔn*where

_{LC}/λ*L*is the sample thickness,

*λ*is the impinging light wavelength and

*Δn*the value of the birefringence. However the real value of this quantity depends, in fact, on the order degree of the sample and it is more appropriate to consider an average value

_{LC}=n_{e}- n_{o}*Δn*. The value of

_{avg}≤Δn_{LC}*Δn*can also be varied by applying an external electric field

_{avg}**perpendicular to the glass slabs of a planar LC cell; in this case, the NLC director reorients along the direction of**

*E***and this influences the birefringence of the NLC film, thus determining the effective phase retardation of an impinging beam.**

*E*In a previous paper [2], we have explained how some drawbacks can arise when exploiting above mentioned effects to fabricate a device for real applications. In fact, the LC film is quite sensitive to temperature variations and this limits the use of the device when driven by high power lasers. In the same ref [2], we have suggested a way to overcome the problem by using POLICRYPS diffraction gratings [3]. These structures show a very sharp and precise morphology, constituted by a periodicity of polymer slices that confine and stabilize thin films of NLC molecules, influencing also their alignment. When employed as phase retarders, POLICRYPS gratings show peculiar properties, due to the circumstance that such structures behave at the same time both as diffraction gratings and as birefringent media. This consideration brought us to investigate the characteristics of POLICRYPS in details. The result, reported in the present paper, is a model of general validity, which can account for the behavior of a light beam that undergoes to both a polarization selective absorption and a modification of its polarization state.

## 2. The POLICRYPS grating as a dichroic absorber

When polarized light impinges on a diffraction grating, the number and intensity of diffracted orders depend on several grating parameters, like fringe spacing *Λ*, thickness *L* and refractive index modulation *Δn _{gr}* along the structure, but also on features related to the impinging light, like wavelength

*λ*and incidence angle

*α*. In order to estimate the number of diffracted orders, the standard grating equation can be used [4], while the intensity of each diffracted order can be obtained by exploiting both Kogelnik’s Coupled wave theory [5], or the Rigorous coupled wave analysis [6], depending on the needed degree of accuracy of results. It has been already observed in the past [7] that, in a POLICRYPS diffraction grating, very well aligned NLC films are confined by polymer slices, the molecular director being oriented, in average, perpendicularly to the polymeric material. This orientation of the birefringent material has an influence on the grating diffraction efficiency

*η*, which strongly depends on the angle

*θ*between the light electric field

**and the average orientation of the NLC director (optical axis of the grating). If we roughly define the refractive index modulation along the grating structure as**

*E**Δn*, (where

_{gr}=n_{LC}– n_{p}*n*is the effective refractive index of the liquid crystal experienced by a light beam of given polarization, and

_{LC}*n*is the polymer refractive index), light polarized along the nematic director will experience a high value of

_{p}*Δn*(since

_{gr}*n*

_{LC}≈n_{e}); therefore, according to Kogelnik’s theory, the grating will exhibit a high value of diffraction efficiency

*η∝Δn*. In the opposite case (light polarized perpendicularly to the NLC director), the value of

_{gr}*Δn*is low (zero if the liquid crystal is perfectly aligned and

_{gr}*n*) and the grating will exhibit a very low diffraction efficiency. It is worth noting that high

_{p}≈n_{o}*η*values correspond to low intensity

*I*of the transmitted (zero order diffracted) beam and vice versa; the grating exhibits, therefore, a dichroic effect in the transmission of light in a way which is very similar to the behavior of a linear polarizer. In Fig. 1 , the experimentally measured behavior of the diffraction efficiency of a POLICRYPS structure, probed with a linearly polarized beam impinging on the sample at the Bragg angle (θ

_{t}_{B}≈15°), is plotted as a function of the angle

*θ*between the field

**of the probe light and the optical axis of the grating.**

*E*The grating used for this experiment has a thickness *L*=3.03μm and a fringe spacing *Λ*=1.22μm. From Fig. 1, it can be noticed that the diffraction efficiency assumes a very high value for a *p*-polarization of the impinging radiation (*η*≈94%), while it goes down to a very low value when the probe beam is *s*-polarized (*η*≈8%).

## 3. Phase retardation behavior of a birefringent dichroic material

The behaviour of a POLICRYPS as a phase retarder has been considered in a previous paper [2], where the grating has been assimilated to a retardation plate and the Jones Matrix formalism [8] has been exploited to calculate the phase retardation *δ* introduced by the structure. In that case, however, the theoretical treatment did not take into account any dichroic behavior of the sample (which is always effective, as discussed above); being aware of this limitation, the POLICRYPS of Ref [2]. has been designed to exhibit a very low diffraction efficiency (*η≈*3%) at normal incidence. Implementation of a general model can be obtained by taking into account also the dichroic behavior; this is done by multiplying the Jones Matrix of the generic phase retarder by the dichroic matrix *L* given by:

*H*and

*V*parameter depend on the considered material and can reflect a broad range of situations. For example,

*H*≠0 and

*V*= 0 or viceversa, indicate that the material under investigation behaves as a linear polarizer. If

*H*and

*V*have, instead, the same value, the impinging light is absorbed in an isotropic way and there is no dichroism. Implementation of our model has been, then, developed by taking into account that the experimental set-up used for characterizing dichroic/birefringent samples is the one reported in Fig. 2 . Experiments are performed by fixing the position of the sample and rotating the analyzer around the axis of propagation of the probe light (

*z*axis in Fig. 2). In comparison with the technique reported in ref [2], the actual choice represents an improvement which exhibits several interesting features. In particular, if the morphology of the structure is not homogeneous and the system is not perfectly in axis, the new setup, by avoiding any rotation of the sample, enables to prevent that measurements are performed on areas with different physical characteristics, which would give only an average information on the properties of the different areas.

We derive, now, the equations for calculating the complex electric field ${\tilde{E}}_{out}^{}\left(\beta \right)$ and hence the intensity ${I}_{out}\left(\beta \right)$ of light transmitted by the analyzer *A* in our experimental geometry. We define *β* as the angle between directions of analyzer axis and incident polarization (therefore *β*=0 when the axis of the analyzer *A* is parallel to that of the polarizer *P*).

The Jones Matrix of an analyzer put at a generic angle *β* can be written as:

*A(β)*=

*R(-β)AR(β)*for the Jones Matrix $A=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$of an analyzer oriented along the y axis of the reference system; $R\left(\beta \right)=\left(\begin{array}{cc}\mathrm{cos}\beta & \mathrm{sin}\beta \\ -\mathrm{sin}\beta & \mathrm{cos}\beta \end{array}\right)$ is the

*β*-angle rotation matrix. As for the sample, the Jones Matrix of a retardation plate can be written as:

*δ*is the unknown phase retardation (introduced by the plate) between the two orthogonal components

*E*and

_{||}*E*(with respect to the optical axis of the birefringent material) which the impinging wave is decomposed into. If the optical axis of the sample plate is oriented parallel to the first polarizer (

_{⊥}*θ*=0), we have no retardation (

*δ*=0). When the plate is, instead, oriented at an angle

*θ=π/4*the field components have the same amplitude (

*E*) and the plate introduces the maximum retardation. In the Jones matrix sequence representing our system, this choice can be accounted for by applying to

_{||}=E_{⊥}*M*a rotation matrix

*R*of angle

*θ=π/4.*Unfortunately, substituting

*M*with

*R*(-π/4)

*MR*(π/4) would yield an over complication of calculations; a more straightforward procedure is that of leaving the matrix

*M*as it is in Eq. (3) and rotate, instead, the direction of the incoming polarization by the same

*θ = π/4*angle. This corresponds to express the impinging electric field, in terms of Jones Matrices, as ${\tilde{E}}_{inc}^{}=\frac{\sqrt{2}}{2}\sqrt{{I}_{inc}}\left(\begin{array}{c}-1\\ 1\end{array}\right)$. By taking into account this consideration and using matrices

*A(β)*,

*M*and

*L*, the complex electric field of the radiation coming out from the analyzer in the experimental geometry of Fig. 2 is written as:

*β*between the analyzer

*A*and the first polarizer

*P*. In order to evaluate

*H*,

*V*and

*δ*from Eq. (5), we notice that when the analyzer is parallel (

*β=0*) or perpendicular (

*β=π/2*) to the polarizer, the output intensity holds:

*δ*introduced by the sample as:

*I*when the analyzer is put at particular angles (

_{out}*β*= 0, π/2, π/4), it is possible to obtain the values of all parameters appearing in Eq. (5), which are characteristic of the system under investigation. Finally, it is worth noting that, by combining Eq. (6) and Eq. (7) we obtain a relationship between

*H*and

*V*parameters and the measured intensity values utilized for their evaluation:

## 4. Experiment

Knowledge of parameters *H*, *V* and *δ* of a physical system gives information about the system itself; on the other hand, features of different systems can be compared when these parameters are known for each of them. In the following, we report results of the above described experiment, performed on a POLICRYPS grating. In order to compare this system with other birefringent ones which can, eventually, show a dichroic behavior, we have analyzed the characteristic of both a HPDLC grating and a thin film of NLC with planar alignment. For each sample, parameters *H* and *V* and the phase retardation *δ* are calculated by means of Eqs. (6-8), while experimental curves of the intensity *I _{out}*, transmitted by the analyzer for different values of

*β*in the interval

*0≤β≤2π*, are compared with theoretical predictions obtained by means of Eq. (5). Results show a good agreement in all considered cases and confirm the validity of our assumptions. In the following we describe in detail the features of investigated systems and results obtained from experiments.

#### a) POLICRYPS grating

The POLICRYPS structure used for the experiment whose results are reported in Fig. 1 has been utilized to check our model based on the Jones Matrix formalism. The morphology of a typical POLICRYPS grating, as observed at the optical microscope between crossed polarizers, is shown in Fig. 3
. The sharp morphology reflects the almost complete phase separation that is usually achieved in this kind of systems and evidences the absence of LC droplets, which could determine scattering of the impinging light and depolarization of the transmitted light. In particular, the absence of depolarization has been demonstrated in a previous communication [2] where we showed that, by putting the sample between crossed polarizers, when its optical axis is oriented either along the polarizer or the analyzer, no light is transmitted by the system. By applying the Jones Matrix formalism to the POLICRYPS we obtain: *H*=0.727, *V*=0.406 and *δ*=1.26rad. In Fig. 4
, the experimental value of *I _{out}* as a function of the angle

*β*(crosses) is compared with the theoretical behavior predicted by Eq. (5) (solid line). The agreement is quite good within the experimental error. The different values of

*H*and

*V*show that, even at normal incidence, the diffraction efficiency of the POLICRYPS grating is significant.As for the birefringence of the structure, the obtained value of

*δ*yields

*Δn*=0.042. By considering that the periodicity of the grating is much larger than the probe wavelength we can exclude that this considerably high value is due to some kind of form birefringence and hence to the geometrical features of the grating. We are confident, instead, this value indicates that the stabilizing and confining action exerted by polymer slices on the NLC molecules has a direct influence on their alignment.

In Fig. 4, segments of different length put into evidence output intensity values for *β*=0 and *β*=π/2 respectively. By means of Eq. (9), these values enable to calculate how large is the dichroism of the birefringent sample.

#### b) Comparison with other soft matter systems

### i) HPDLC grating

Where the HPDLC is concerned, we have fabricated a sample with the same fringe spacing of the POLICRYPS (*Λ*=1.22μm) but a slightly different thickness (*L*=1.9μm instead of 3.03μm). The main difference between HPDLC and POLICRYPS gratings is in their morphology (Fig. 5a
and Fig. 5b). In a typical HPDLC, the liquid crystal nucleates into droplets where the configuration of the LC director is, in general, the bipolar one. Unless particular actions are made on the sample, (e.g. stretching in a particular direction [9]), orientation of the main axis of different droplets is completely random. The HPDLC exhibits, therefore, a quite low value of birefringence, as shown in Fig. 6
.

Calculations made by using Eqs. (6-8) give *H*=0.630, *V*=0.699 and *δ*=0.198rad, which corresponds, for a probe wavelength λ=632.8nm (He-Ne Laser), to a very low birefringence of the structure: *Δn*=*δλ/2πL=*0.010. This result confirms above considerations on the average orientation of the birefringent material. Also in this case, by comparing the output intensity values measured for *β*=0 and *β*=π/2, we get that the dichroism is very limited.

### ii) Thin film of liquid crystals

A thin film of planarly aligned NLC represented, for a long time, the basic system for realizing a tuneable phase retarder. Its simplicity yields, however, its main drawback: the anchoring forces which induce the director alignment are due to the glass substrates, prepared to give a planar orientation of molecules; thus, the NLC alignment can be easily perturbed by external means (e.g. a high power probe beam). A NLC cell can be referred to, therefore, as a standard reference system, to be compared with structures where the NLC material is stabilized by a polymer (HPDLC and POLICRYPS). The sample has been fabricated by filling in with E7 NLC a standard cell (*L*=2.1 μm, produced by EHC Corporation) whose glass substrates have been treated to give a planar alignment of the NLC director. A check of the sample at the optical microscope, between crossed polarizers, evidences a good alignment of the NLC material. Results for the transmitted intensity, obtained by using our technique, are plotted in Fig. 7
. Calculation of *H* and *V* coefficients provides in this case the values *H*=0.719, *V*=0.693, with a retardation *δ*=1.76rad, which corresponds, for a thickness *L*=2.1μm and a probe wavelength *λ*=632.8nm (He-Ne Laser), to a birefringence $\Delta {n}_{avg}=0.147$.

This value is slightly lower than the ideal one *Δn _{LC}*=0.214 (at

*λ*=632.8nm) expected for a perfectly aligned NLC film, and confirms that our technique can be efficiently used also to obtain information about the molecular organization of the birefringent material under investigation.

## 5. Discussion and conclusions

In this paper we have developed an implementation of the Jones Matrix formalism which enables to describe the behavior of light transmitted by structures containing birefringent materials and exhibiting polarization selective absorption of light (dichroism). The idea came from the need of describing the behavior of a POLICRYPS diffraction grating when used as a phase retarder. We realized soon that the technique can have a more general validity, since it can be used to describe any birefringent material which behaves as a dichroic absorber too. In order to check the model, we have chosen, at first, a POLICRYPS diffraction grating and we have designed an experiment devoted to measure the intensity of light transmitted by an analyzer put after the sample. The same experiment has been repeated with other, different, physical systems (HPDLC diffraction grating, thin film of aligned NLC) in order to confirm and compare obtained results. In all considered cases, results show a good agreement with theoretical predictions. Moreover, the evaluation of the birefringence of the different structures has been used for carrying out a comparison of their properties. In particular, we have observed that the degree of order of the birefringent material is very high in the sample containing only an aligned NLC film, while it is lower in geometries where the liquid crystal material is confined and stabilized by a polymer structure (HPDLC and POLICRYPS gratings). It is worth noting that the birefringence value drastically increases (about five times) when we move from the HPDLC grating, where a stochastic distribution of the nematic droplets is present, to the POLICRYPS structure, which contains layers of nematic material homogeneously aligned. This result puts into evidence, once more, the high quality of POLICRYPS and the possibility to exploit these systems in technological applications as switchable phase retarders.

## Acknowledgments

This research has been supported by PRIN 2006 - Umeton - prot. 2006022132_001.

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