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Abstract
Recently developed alignment techniques for liquid crystals enable the definition of arbitrary alignment patterns. We present a method to determine these two-dimensional spatial alignment distributions as well as the retardation of thin nematic liquid crystal films. The method is based on quantifying the influence of the birefringence of such a film on light with known input polarization by measuring the Stokes parameters of light. We show that we are able to distinguish arbitrary alignment patterns unambiguously. In addition, we demonstrate the ability to evaluate the homogeneity of the alignment as well as the thickness or birefringence of the film.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
In the liquid crystalline state of matter, elongated molecules, called mesogens, self-assemble to form a macroscopic state with anisotropic optical and mechanical characteristics. The anisotropic optical properties of liquid crystals (LCs) are widely exploited in LC displays, spatial light modulators, projectors, and other light modulation devices [1]. Outside the realm of optics, in the meantime, a large body of research in the fields of soft robotics, biomimetics, and artificial muscles also relies on the mechanical anisotropy encountered in crosslinked liquid crystalline networks [2–5].
The defining features of the anisotropy are the specific mesogens and their alignment, i.e., the direction of the mesogens’ long axes. The choice of mesogen determines the macroscopic arrangement of the molecules, which further depends on the temperature. A very common state is the nematic phase in which all the mesogens’ long axes are aligned roughly in parallel. Many novel applications and research directions, however, require the anisotropy to vary locally, a state that can be achieved by defining mesogens in domains with spatially varying alignment [6,7].
The photoalignment technique enables the alignment of mesogens using polarized light, resulting in nematic liquid crystalline patterns with arbitrarily definable alignment [8,9]. This approach is being utilized in the fields of optical beam shaping to create q-plates, geometric phase optics and LC-based micro-optics [10–12]. While these applications rely on the optical anisotropy of LCs, the field of biomimetics specifically profits from the mechanical anisotropy. Defining the orientation of the mesogens with high resolution enables the spatial programming of artificial muscles. Thus the spatially varying orientation of LCs offers considerable potential for various applications in optics and biomimetics.
2. PROBLEM
For all these applications, achieving and verifying the desired alignment pattern is crucial. To fabricate a specific q-plate, complex alignment patterns need to be realized [13]. An example of such a pattern is radial alignment, i.e., an alignment pattern in which all the mesogens align to a central point [illustrated in Fig. 3(a)]. To verify the achieved alignment, many researchers observe their samples between two crossed polarizers, exploiting the material’s birefringence. In polarization microscopy (POM), areas with mesogens that are parallel to one of the two polarizers do not transmit light, whereas areas where the mesogens are at a different angle rotate the polarization of light, showing as brighter areas. The maximum brightness usually results when the mesogens are aligned at 45° to both polarizers. In the example of a radially aligned pattern, an image such as Fig. 3(b) will be obtained, in which the black lines show as the areas in which the mesogens are parallel to one of the polarizers.
While the images obtained through POM are usually enough to evaluate the alignment quality, they are still ambiguous since the contained information is just modulo 90°, i.e., it is not possible to distinguish an alignment direction of 0° from 90°. For the sample shown in Fig. 3(b), this means it is not possible to tell whether the mesogens are aligned radially or azimuthally, in which all the mesogens are aligned in circles around the center [Fig. 3(d)]. However, the difference between these two opposing alignment states is fundamental for any application.
Adding a quarter-wave plate (QWP) between the crossed polarizers yields a contrast in the full range of 0° to 180°. This is enough to distinguish radial and azimuthal alignment for otherwise identical samples. On the other hand, the obtained images do not yield an absolute alignment angle since the transmitted light also depends on the layer thickness and the strength of the birefringence of the LC film.
While POM is thus not sufficient to verify the alignment direction, other methods to measure the alignment of LCs are well established. Wide-angle and small-angle X-ray scattering can be used to measure the alignment direction. However, these are single-point measurement techniques and thus not helpful in measuring complex alignment patterns [14].
Another possible way to measure the alignment is based on exploiting the linear dichroism of certain chemical groups that are part of or attached to the mesogen. While feasible, this method is specific to every mesogen for the choice of wavelengths. Furthermore, it usually requires imaging in the UV or the MIR wavelength ranges, which can be inconvenient and costly. To use visible light, dichroic dyes that follow the alignment of the mesogens can be doped into the material [15], which allows for convenient determination of the alignment direction, but changes the properties of the film. For applications relying on transparency in the visible range, this method is thus not feasible.
A promising approach to quantify the alignment based on birefringence has been demonstrated by Lin et al. [16]. While elegant, the obtained alignment angle in their work is just in a range of 0° to 90°, which is due to the linear polarization they used as an input, similar to POM.
We present here a measurement concept inspired by Ref. [16] to quantify the spatial distribution of mesogen alignment for thin, nematic LC films in the full range of 0° to 180°. The suggested method allows determining the influence of the material’s birefringence on a defined input polarization by measuring the output polarization through Stokes polarimetry. Simplified Mueller calculus enables the calculation of the spatial distribution of alignment direction as well as the retardation of the film. Since the retardation for a single wavelength depends solely on the film thickness and the birefringence of the material, this method also allows for the calculation of the film thickness.
With the system employed here, we are able to measure spatially resolved alignment direction and retardation with a spatial resolution of ${\lt}100\;{{\unicode{x00B5}{\rm m}}}$ over an area of $15 \times 15\;{\rm{mm}}^2$. The accuracy of both alignment direction and retardation measurements depends nonlinearly on the retardation of the film. For LC layers with a retardation that is not close to 180°, the angular resolution is better than 15°. For a retardation of 90°, we demonstrate an accuracy of better than 1°.
3. METHOD
A. Aligning Liquid Crystals
The orientation of mesogens can be achieved by various methods. To obtain arbitrary alignment patterns, the most flexible and established technique is photoalignment. In this approach, a thin layer of molecules with strong linear dichroism is aligned with patterned polarized light. The samples we discuss in this paper were aligned with the azo-dye Brilliant Yellow, using blue light. The actual liquid crystalline material is the reactive mesogen RM257, selected for its large birefringence and good alignment properties [17]. RM257 is dissolved in toluene and spin coated onto the dichroic layer, which defines its alignment [18,19]. The detailed process has been published separately [20].
B. Quantifying Birefringence in Liquid Crystals
Using photoalignment, arbitrarily aligned LC films can thus be fabricated. These films show strong birefringence that can be used to measure the alignment of the mesogens. The thin birefringent film can be modeled as a waveplate characterized by its retardation $\phi$ and the major axis angle $\alpha$. The former represents the maximum phase shift between two orthogonal electric waves on transmission through the waveplate, while the latter stands for the angle between the extraordinary and optical axes. In the case of an LC film, $\alpha$ represents the alignment angle between the major axis of the mesogens and the optical axis.
The polarization of light can be represented by the Stokes parameters; a vector with four components ${\textbf{S}} = ({S_0}, {S_1}, {S_2}, {S_3})$. ${S_0}$ stands for the total intensity, and ${S_1}$ and ${S_2}$ are the linearly and ${S_3}$ the circularly polarized components of intensity. The influence of a waveplate on the polarization of light is then given by
Here, the input polarization ${\hat{\textbf S}}$ is converted to the output polarization ${\textbf{S}}$ by the Mueller matrix of a waveplate with retardation $\phi$ and major axis angle $\alpha$ [21].
If ${\hat{\textbf S}}$ and ${\textbf{S}}$ are known, this system of equations can be numerically solved for $\phi$ and $\alpha$. While this is feasible for a single point, solving it for an area with many points requires too much computational effort to be useful.
While previous work has simplified Eq. (1) by defining a linear input polarization, this resulted in a usable range for $\alpha$ of just 0° to 90° [16], which is not sufficient to quantify the alignment unambiguously as discussed above. However, we can choose circularly polarized light ${\hat{\textbf S}} = ({\hat S_0}, 0, 0, {\hat S_3})$ as input, which simplifies Eq. (1) to
Fig. 1. Output angle of one half of the arccosine (${\alpha _1}$) and arcsine (${\alpha _2}$) versus input angle. Combination of the two functions’ results in an output angle $\alpha$ with the full output range of $- 90^\circ$ to 90°.
Fig. 2. Setup to measure spatial distribution of liquid crystal alignment and retardation. A red LED, polarizer (Pol1), and quarter-wave plate (QWP1) are used to create circularly polarized input light illuminating the sample. A 4f imaging system maps the sample on a CMOS camera sensor. Another quarter-wave plate (QWP2) is mounted on a motorized rotation stage in front of a second static polarizer (Pol2). Rotating QWP2 while recording the intensity on the sensor enables the measurement of Stokes parameters.
These equations can be solved to yield the retardation $\phi$ and two expressions for the alignment angle $\alpha$, namely,
Both the arcsine and the arccosine are periodic and yield values only between 0° to 180° and ${-}{{90}}^\circ$ to 90°, respectively. Thus this calculation can quantify a retardation only up to 180°, after which it repeats. Since the inverse trigonometric functions for the major axis angle are multiplied by 0.5, the output value ranges for ${\alpha _1}$ and ${\alpha _2}$ are 0° to 90° and ${-}{{45}}^\circ$ to 45°, respectively. Figure 1 illustrates how both functions repeat in intervals of 90°. A combination of the two functions leads to a combined angle $\alpha$, which is ${\alpha _1}$ when ${\alpha _2}$ is positive, and ${-}{\alpha _1}$ when ${\alpha _2}$ is negative (see Fig. 1):
By combining the two equations for ${\alpha _1}$ and ${\alpha _2}$ [Eq. (3)] with Eq. (4), the alignment angle of the mesogens $\alpha$ can be computed in the desired full range of ${-}{{90}}^\circ$ to 90°, while the retardation $\phi$ is in a range of 0° to 180°. Thus, by measuring the Stokes parameters after transmission of circularly polarized light through a LC film, we are able to calculate the alignment direction of the mesogens as well as the retardation of the film.
The retardation $\phi$ of a birefringent sample is determined by the equation
where $\Delta n$ represents the difference between ordinary and extraordinary refractive indices, and $d$ stands for the thickness of the layer. The wavelength $\lambda$ is a constant so that from our measurement of $\phi$ we can obtain information about the birefringence and the film thickness. However, the calculated value of $\phi$ is periodic and repeats after a retardation of 180°. To estimate the usable thickness determination range, at a wavelength of $\lambda = 633\;{\rm{nm}}$ and a birefringence $\Delta n = 0.18$ [17], the maximum measurable thickness ${d_{{\max}}}$ would beThe result of the retardation and thickness calculation thus needs to be put in context, either by having an already precise estimate of the layer thickness or by just using it as a relative intra-sample measure. In addition, the homogeneity of the retardation can be used for quality control for comparison of different samples that are fabricated using the same process.
C. Measuring the Stokes Parameters
In order to measure the spatial distribution of alignment angle and retardation, we need to obtain the spatial distribution of Stokes parameters. Hence, we need to combine circularly polarized input light, an imaging system to map the spatial distribution on a sensor and a method to measure the Stokes parameters. The setup designed for this work is illustrated in Fig. 2.
The setup uses a red LED and a narrow-band filter (Thorlabs LED630L and FL632.8-1) as the light source, and a polarizer (Pol1) and QWP (QWP1) to create circularly polarized light. To verify the circular polarization, another polarizer is rotated in front of a powermeter. We thus find the fluctuation of power to be less than 2%. This justifies the assumption of using an input Stokes vector of ${\hat{\textbf S}} = ({\hat S_0}, 0, 0, {\hat S_3})$, since the linear components are negligible.
The sample is imaged by a 4f system consisting of two identical lenses on a CMOS sensor. The output of the CMOS sensor shows a nonlinear intensity response due to its gamma correction. This effect is included in the calculation by scaling the measured intensity with a calibration fit.
To measure the Stokes parameters, we employ a method that can be automated [22,23]. A second QWP (QWP2) on a motorized rotation stage and a static polarizer (Pol2) are inserted in front of the CMOS sensor. By rotating QWP2 a full rotation in $N$ increments of ${\theta _{\rm{s}}}$ and measuring the intensity $I$ at every increment, the following parameters $A - D$ can be obtained:
$A - D$ can then be combined to yield the Stokes parameters as
The procedure for experimental determination of the Stokes parameters is then as follows:
- 1. Calibration of the setup without a sample to verify that the calculated Stokes parameters represent the circular input polarization;
- 2. Insertion of the sample, rotating QWP2 in $N = 36$ steps of ${\theta _{\rm{s}}}{= 10^ \circ}$ and storing the intensity on the sensor $I({\theta _{\rm{s}}})$ at every step;
- 3. Calculation of the intermediate parameters $A,B,C,D$ [Eq. (7)];
- 4. Derivation of the Stokes parameters ${S_0},{S_1},{S_2},{S_3}$ [Eq. (8)];
- 5. Computation of the major axis angles ${\alpha _1}$ and ${\alpha _2}$, and the retardation $\phi$ [Eq. (3)];
- 6. Combination of ${\alpha _1}$ and ${\alpha _2}$ to finally yield the alignment angle $\alpha$ for the full range of $- 90^\circ$ to 90° [Eq. (4)].
4. EXPERIMENTAL RESULTS
A. Alignment
With the presented method, we are able to quantify the spatial distribution of alignment and retardation of thin nematic LC films. As a proof of concept, samples with alignment patterns that deliberately cannot be evaluated unambiguously by POM have been prepared. The samples were fabricated through a combination of photoalignment and spin coating, using the process described in Refs. [19,20].
To emphasize the difference between radial and azimuthal alignment [schematic in Figs. 3(a), (d)], two samples with these alignments were prepared; these are shown between crossed polarizers in Figs. 3(b), (e). Since this technique is periodic after 90°, no difference can be observed. In the same figure, the results of our method for the same two samples are shown [Figs. 3(c), (f)]. They show an acceptable contrast and also indicate that the alignment is not continuous but defined in discrete steps, due to the setup used for the photoalignment procedure.
Fig. 3. Distinguishing radial and azimuthal alignment: the first column illustrates the schematic alignment of the mesogens for both patterns (a), (d). The middle column shows the measurement of LC films between crossed polarizers, in which the two alignment patterns cannot be distinguished (b), (e). The right column displays the measurement of the same samples with the presented method (center), showing a clear contrast (c), (f).
Another sample shown in Fig. 4 seems to be aligned as a checkerboard with two alternating alignments when observed between crossed polarizers [Fig. 4(b)]. The measurement with the presented method [Fig. 4(c)] reveals that there are actually four alternating alignment directions in steps of 45°, illustrated in a magnified way in the same figure [Fig. 4(a)].
Fig. 4. Four-step checkerboard alignment pattern, (a) schematic (magnified for better illustration), (b) imaged between crossed polarizers and (c) measured with the presented method.
To show the versatility of the measurement, another sample was aligned with an image of the Freiburg Minster. The alignment was set to be done in two steps, at 0° and 90°. Between crossed polarizers [Fig. 5(a)], this leads to a pattern that appears to be aligned all in the same direction, though the domain interfaces can be seen. The true alignment measurement [Fig. 5(b)], on the other hand, shows both alignment directions clearly distinguished.
Fig. 5. Alignment pattern of the Freiburg Minster, aligned in two steps at 0° and 90°, (a) imaged between crossed polarizers and (b) measured with the presented method.
All these examples illustrate the shortcomings of POM. Even though POM is a helpful method to judge alignment quality, it is insufficient to distinguish between alignment directions of 0° and 90°. The presented method reveals the real alignment patterns.
B. Retardation
While the alignment angle is defined by the photoalignment setup, the retardation is determined only by the film thickness and the birefringence of the material. Figure 6 shows the calculated retardation of the radial [Fig. 6(a)], azimuthal [Fig. 6(b)], and four-step checkerboard [Fig. 6(c)] samples from Figs. 3 and 4. To better evaluate the results, the mean value $\bar \phi$ and the standard deviation $\sigma$ of the retardation are added for all samples. As is clearly seen, there is a homogeneous retardation across the samples, independent of the alignment direction, which is to be expected since the three samples were prepared with the same process, such that the layer thickness as well as the birefringence are similar and uniform.
Fig. 6. Measurement results of the retardation $\phi$ of (a) the radial, (b) azimuthal, and (c) four-step checkerboard samples, whose alignment patterns are shown in Figs. 3 and 4. The mean value $\bar \phi$ and the standard deviation $\sigma$ of the retardation are given for each sample.
To evaluate the performance of the retardation measurement, two more experiments were conducted. The first one targets a variation of the thickness of the film and compares the obtained retardation values to thickness measurements with a white light interferometer. The second experiment is aimed at varying the birefringence of the film.
In the first experiment, samples with different film thicknesses were fabricated by varying the concentration of the spin coated LC solution from 5% to 25%, and their retardation was measured with the presented method. For comparison, the film thicknesses were measured with a white light interferometer and converted to a retardation using Eq. (5) and assuming $\Delta n = 0.18$. The results of both measurements are plotted in Fig. 7.
Fig. 7. Retardation for different concentrations of spin coated LC solution. The blue distributions show the measured retardation obtained by the presented method with the median value highlighted. Since the retardation at 25% is larger than 180°, the measurement result was mirrored to the correct range (gray distribution). The red crosses show the retardation values that have been calculated from the thickness measurement, using $\Delta n = 0.18$.
The retardation is linearly proportional to the film thickness [Eq. (5)] for constant birefringence. Our evaluation method yields only $\phi$ in a range of 0° to 180°, beyond which the measurement will yield values returning from 180° back to 0° at $\phi {= 360^ \circ}$. Thus, in Fig. 7, we have to mirror the retardation for an LC concentration of 25% at 180° to put it in the correct range.
The comparison of our measured (blue) with the calculated (red) values (Fig. 7) shows that the former yields lower retardation than expected for thick LC layers prepared with concentrations of ${\gt}15\%$. This is probably due to the assumption of a constant $\Delta n = 0.18$ for the calculation. Thicker layers do not follow the alignment perfectly and thus show lower average birefringence. For the results shown in Fig. 7, choosing $\Delta n \approx 0.16$ corrects for the deviation for thick layers.
Hence, we are not able to tune the layer thickness independently of the birefringence of the sample. However, the opposite is feasible, tuning the birefringence while keeping the layer thickness constant. While the birefringence in crystalline materials is determined by the material and the crystalline structure, LCs can show a continuous range of birefringence. This is because the order of the mesogens can range from full disorder (isotropic) to all mesogens pointing exactly in the same direction. The degree of order thus determines the birefringence.
To vary the order of the LC films, the exposure dose of the photoalignment process can be tuned. A higher exposure dose leads to a higher order of the photoalignment film up to a saturation value that depends on the material. The degree of alignment of the photoalignment film then determines the order and hence the birefringence of the subsequent LC layer.
Samples with different alignment qualities have been fabricated by varying the exposure time of the photoalignment setup from 1 s to 15 s, the rest of the process being equal. The measured retardation is plotted over the exposure time in Fig. 8.
Fig. 8. Measured retardation as a function of photoalignment exposure time. The plot shows the distribution of the retardation with highlighted median values. With increasing exposure time, the degree of alignment improves, leading to larger birefringence and thus larger retardation. For longer exposure times, the birefringence saturates.
For low exposure times, an increase in retardation can be observed, which saturates for longer exposure times. For exposure times of 10 s, the degree of order of our photoalignment process saturates, in this case to $\bar \phi {= 50^ \circ}$, $\sigma {= 1^ \circ}$. We assume $\Delta {n_{{\max}}} = 0.18$, which seems reasonable for the thin film used in this experiment (compare Fig. 7). With this, we can thus calculate our layer thickness to be ${d_\phi} = 490\;{\rm{nm}} \pm 10\;{\rm{nm}}$. With the white light interferometer, we measure the thickness of the LC films to be ${d_{{\rm{WLI}}}} = 520\;{\rm{nm}} \pm 30\;{\rm{nm}}$, which is close to the value obtained through the retardation measurement.
Since the derivatives of both the arcsine and arccosine are nonlinear functions, the accuracy of the technique depends on the retardation of the film. If the retardation approaches 180°, small inaccuracies of the measured Stokes parameters result in large deviations of the measured values of both $\phi$ and $\alpha$ from the real values, as can be seen from Eq. (3). If $|{S_3}| \approx {\hat S_3}$, the value calculated for $\phi$ tends towards infinity. Similarly, the values of ${\alpha _1}$ and ${\alpha _2}$ tend to an undefined value in this case, with ${S_1}$ and ${S_2}$ approaching zero. Thus this method does not yield usable information, due to a large error amplification, when the retardation of the film approaches 180°, i.e., when it acts like a half-wave plate (HWP). Accuracy is best for LC films with low retardation.
To verify these concerns, a QWP and an HWP (for 633 nm) were characterized by our system, at angles varying from 0° to 180°. The retardation of the QWP was measured very accurately over all alignment directions as ${89.8^ \circ} \pm {0.3^ \circ}$. However, the retardation of the HWP was characterized as ${164.3^ \circ} \pm {4.0^ \circ}$, revealing a strong offset, which is almost independent of the alignment direction. Furthermore, the alignment angle of the QWP was characterized with an accuracy of 15°, whereas the alignment angles of the HWP showed very uncorrelated results. Therefore, as presumed, our method has a blind spot for measuring the alignment angles when the retardation is close to a multiple of 180°.
Despite the dependence of accuracy of the retardation on its absolute value, our method is still a valuable tool to evaluate the variation within one sample, as well as between different samples that were fabricated with the same process. This feature is useful to control the quality and homogeneity of LC alignment. Another potential application is the analysis of stress distribution in polymers; the technique can be used to spatially measure the direction and strength of the birefringence with high spatial resolution.
The accuracy of the retardation measurement seems quite low compared to commercial polarimeter systems, which offer ${\lt}1\%$ deviation from the absolute state of polarization (e.g., Meadowlark Polarimeter). However, the fundamental difference is that these polarimeters generate only single-point measurements. The system presented here is an imaging system that reveals the spatial distribution of the polarization of light. As is the case for hyperspectral imaging, gaining spatial information results in a loss of accuracy.
Other systems designed to image the state of polarization with high resolution have been proposed [24] and are commercially available. They are utilized to analyze the stress distribution in polymers. However, they do not quantify the fourth Stokes parameter ${S_3}$, which represents the circular polarization. Therefore, these sensors cannot be used to fully analyze the state of polarization. The outstanding feature of our technique is thus the complete determination of the spatial distribution of the state of polarization, thereby allowing calculation of the spatial distribution of absolute alignment angle and retardation of birefringent materials. Hence, the aim of our method is to enable the analysis of inhomogeneously birefringent samples.
5. CONCLUSION
Many publications in the field of LCs and LC elastomers use POM to evaluate the alignment of their samples, because it is relatively simple to set up, flexible, and affordable, while also being sufficient to show the success and quality of alignment. However, POM is insufficient to determine the absolute alignment direction of LCs unambiguously.
To this end, we have shown here a detailed method to quantify the absolute alignment direction of nematic LC films. The technique is based on quantifying the influence of the birefringence with a known input. Simplified Mueller calculus enables the calculation of the alignment angle of the LC domains as well as the retardation for arbitrary spatial alignment patterns.
Our system allows measurement of the spatial distribution of LC alignment over an area of $15 \times 15\;{\rm{mm}}^2$. The spatial resolution of the system is better than $100\;\unicode{x00B5}{\rm m}$, and the angular resolution is better than 15°. We have shown that the system is capable of distinguishing alignment patterns that cannot be analyzed unambiguously with POM. In addition, the method yields the retardation of the LC film, which gives a good measure of the film’s homogeneity and can be used as inexpensive quality control without further instruments.
Funding
Deutsche Forschungsgemeinschaft (316245751).
Acknowledgment
Portions of this work were presented at the SPIE Photonics Europe in 2020, Measuring the Spatial Distribution of Liquid Crystal Alignment and Retardation Using Stokes Polarimetry.
Disclosures
The authors declare no conflicts of interest.
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