Correlations between polycrystalline fabric and the polarization of transmitted light

Wing S. Chan; Joseph Talghader

doi:10.1364/OE.18.003109

1. Introduction

Polarization optical microscopy has been used for many decades to measure the crystal properties of polycrystalline materials and is one of the basic techniques of mineralogy [1]. However, this technique assumes that the sample in question can be handled and prepared for microscopic inspection. If sample preparation is destructive, or if a sample is extremely large and exists in a region where microscopy is not possible, then alternative techniques must be developed to gain information on the crystal size and orientation, often called fabric.

An example of such conditions occurs in glacial ice in the ice sheets of Greenland and Antarctica. Glacial ice, like quartz and many other crystals, is uniaxial. In an ice sheet, pressure builds with depth and ice tends to deform and flow outward. To minimize strain energy, the crystals in the ice will tend to orient so that flow occurs along the basal plane, that is, along a direction perpendicular to the uniaxial c-axis. The current state-of-the-art for analyzing ice crystal structures uses techniques adapted from mineralogy, where a wafer of ice is sliced from an ice core and analyzed with polarized light under different orientations [2–4]. However, this technique cannot obtain a continuous record of fabric with depth as it is impractical and destructive to slice ice cores thousands of times for thin sections. Obtaining a continuous record of layering in an ice sheet is very important to climatology, and any rapid change in crystal structure with depth must be identified so that regions where flow may have disturbed the record can be cataloged. In this application, it is not necessary to obtain a complete record of fabric but rather only to identify rapid changes with depth.

There are at least two ways to handle the analysis of extremely large samples of a polycrystalline material. First, one can drill a core out of the sample and analyze it using transmissive techniques. Alternately, one could use reflective techniques, such as scattering from defects or inclusions in a polycrystalline material, to analyze the core or the hole inside the sample. Such techniques have been used to determine the crystal structure or orientation of particles suspended in air [5–7] or water [8]. In this paper, we describe an example of the first: a non-destructive transmissive method applied to a polycrystalline material to identify changes in the average fabric.

A conceptual diagram of the method is shown in Fig. 1 . Linearly polarized laser light is incident upon a transparent polycrystalline material. After traversing the material, the degree of uniform linear polarization of the output light spot is measured. It is then compared to simulations using Mueller matrices to assess the strength of the fabric. For example, a fabric with c-axes oriented within 10° of a certain axis will have a smaller measured polarization than one with c-axes oriented within 20° of a central axis. Changes in the direction of orientation will likewise have a polarization effect. (Note that the assumption of uniaxial orientation is not necessary.) Note that this method rapidly identifies regions where the fabric changes from one type to another; it does not, nor is it intended to, uniquely identify the exact fabric of the polycrystalline material.

Fig. 1 Conceptual diagram of a transmissive experiment for measuring polarization changes in a birefringent polycrystal. Totally polarized light propagates through a stack of irregularly shaped single crystal grains and its degree of polarization decreases every time it hits the rough boundaries. The crystals are all immersed in index matching oil to minimize scattering and each has an individual rotation stage.

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2. Theory

A detailed description begins with simulating the polarization properties of light as it travels through many crystals, assuming that it enters and exits in air. Both the laser spot and the polycrystalline grains are assumed to be large enough to permit a geometrical optics approach to the problem (millimeter-sized grains and larger). The incident light illuminates one or more crystals at the surface of a polycrystalline material and then the light propagates through crystals in series and in parallel. This approach means that the simulation method can produce accurate correlations for cases where the laser spot size overlaps many crystals or only a fraction of one. Each of the grains has its own c-axis and irregular grain boundaries. The light from one part of the laser spot will in general undergo a different crystal path and thus a different amount of phase retardation than the light from other parts of the spot. We assume such different parts of light are incoherent with one another, and it is therefore appropriate for us to use Mueller matrices and Stokes parameters in our calculations [9].

The Stokes vector of the input light and output light in one part of the light spot are given by

S_{1} = [\begin{matrix} I \\ Q \\ U \\ V \end{matrix}] a n d {S^{'}}_{1} = [\begin{matrix} I^{'} \\ Q^{'} \\ U^{'} \\ V^{'} \end{matrix}]

respectively. It should be noted that all the Stoke’s parameters except I and I’ can take on positive or negative values [9].

The phase retardation effect on light passing through a given length d of birefringent crystal can be characterized by

M = {[\begin{matrix} R \end{matrix}]}^{- 1} [\begin{matrix} B \end{matrix}] [\begin{matrix} R \end{matrix}]

where R is the rotation matrix and is defined as

R = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & c o s 2 α & s i n 2 α & 0 \\ 0 & - s i n 2 α & c o s 2 α & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

with α as the azimuth angle. B is the retardation matrix and is defined as

B = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & c o s Γ & s i n Γ \\ 0 & 0 & - s i n Γ & c o s Γ \end{matrix}]

and Γ = \frac{2 π d (n_{e} - n_{o})}{λ}

This light from one part of the spot will propagate and experience different amount of birefringence in each crystal grain according to the corresponding M matrix. The light output after propagating over n crystal grains can be given as

{S'}_{1} = M_{n} M_{n-1} M_{n-2} ... M_{2} M_{1} S_{1}

We can write the total light output as a sum across the spot,

{S^{'}}_{t o t a l} = \sum_{i = 1}^{m} {S^{'}}_{i} = [\begin{array}{l} \sum_{i = 1}^{m} {I^{'}}_{i} \\ \sum_{i = 1}^{m} {Q^{'}}_{i} \\ \sum_{i = 1}^{m} {U^{'}}_{i} \\ \sum_{i = 1}^{m} {V^{'}}_{i} \end{array}] = [\begin{array}{l} {I^{'}}_{t o t a l} \\ {Q^{'}}_{t o t a l} \\ {U^{'}}_{t o t a l} \\ {V^{'}}_{t o t a l} \end{array}]

where m is the number of parts across the light spot.

The degree of polarization [9] of the summed output light can be calculated as

D O P_{t o t a l} = \frac{\sqrt{{({Q^{'}}_{t o t a l})}^{2} + {({U^{'}}_{t o t a l})}^{22} + {({V^{'}}_{t o t a l})}^{2}}}{{I^{'}}_{t o t a l}}

The birefringence strength, (n_e - n_o) is 0.009 for quartz.

3. Experiment

It is difficult, if not impossible, to control the orientation of every individual grain of a polycrystalline material during growth. Therefore in order to produce random but known samples for testing, we constructed a system to produce “artificial” polycrystals. These were composed of irregular pieces of quartz individually placed in a small optical mount so that the orientation on each crystal could be changed within a rotation plane (the c-axis of the crystals could be oriented out of this plane in any direction). Multiple crystals were arranged in series to simulate a desired fabric. In order to more fully simulate a real polycrystalline material, the entire crystal assembly was placed in index-matching fluid to eliminate inter-grain reflections and scattering with air, which would not be present in a continuous solid.

The individual grains were obtained by breaking a single crystal from Sawyer Technical Materials into irregular pieces with sizes on the order of millimeters. The boule of single crystal was broken so that the pieces had irregular faces instead of well-defined facets. The thicknesses of the pieces were then measured and sorted into categories of 0.5mm - 1.5mm, 1.5mm - 2.5mm and 3.5mm - 4.5mm respectively. Each of these pieces was a single crystal and had irregularly shaped boundaries, which made them a close representation of a crystal grain inside a polycrystal.

When the individual grains are arranged in a material, the fabric of that material can be represented by a Schmidt plot. The Schmidt plot is a diagram that shows the orientation of all the crystal axes in the sample. It can be envisioned as a hemisphere where the directions of all the c-axes in the sample are directed from the origin through the hemisphere. The hemisphere is collapsed downward to form the Schmidt plot plane. For example a vertically directed c-axis would form a point at the center of the diagram while a c-axis oriented horizontally would form a point at the edge of the circle with a direction matching that of the c-axis. The fabrics of the materials developed in this study are shown in Fig. 2 .

Fig. 2 Schmidt plots showing random c-axes with different orientation spreads in one plane. These plots represent the fabrics whose simulated and experimental results are shown in Figs. 3 and 4.

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We were interested in the effect of grain size (quartz crystal size) and thus the birefringence path length on the polarization. A piece of quartz was put into a flat-bottomed glass tube filled with index matching oil with refractive index 1.548, which matched that of the average of quartz to minimize light scattering. The glass tube was tested to verify that it did not have any birefringent effects itself. The polarized light source was a HeNe laser with wavelength 632.8nm that passed through a linear polarizer. The Stokes parameters of the transmitted light were measured with a quarter waveplate and another linear polarizer according to the method illustrated in [10]. Additional pieces of quartz from the same size category were added and the Stokes parameters of the output light were again measured. The degree of polarization (as measured through a uniform linear polarizer) could then be calculated from the parameters. This process was repeated for seven crystals and this experiment was repeated three times to obtain averages. The same experiment was repeated for the three different crystal thickness (size) category.

The orientation spread among the crystal grains was also a factor of interest in the gradual decrease in degree of polarization. 7 crystal pieces from the 1.5mm - 2.5mm group were fixed onto 7 non-birefringent flat-bottomed glass dishes filled with index matching oil. The principle polarization axis of each crystal piece on the plane of the dish was measured with a pair of crossed linear polarizers and marked on the side of the dish. The polarized light was shined into one of the crystal dishes, and the corresponding Stokes parameters were measured. Another dish was then stacked on top of the first one and we aligned its polarization axis with respect to the first one according to a computer-generated orientation angle within a specified orientation spread. Again, this process was repeated for 7 crystals and this experiment was repeated for 3 times to obtain averages. The same experiment was repeated for 3 different orientation spreads.

4. Results and discussions

We examined changes in the measured degree of linear polarization with grain size and found that the input light propagating in smaller grains loses its polarization more quickly than when propagating through the same length of larger grains. The light from one region of the laser spot saw a different grain structure from other parts due to the rough boundaries, which gave rise to small differences in the birefringent path length. We interpreted this by treating the irregular boundaries of the crystal grain as surfaces with bumps and troughs. When the totally polarized light propagated in the bulk of one crystal grain, the polarization across the laser spot maintained its coherence. The phase retardations suffered by different regions across the spot were the same since there was only one extraordinary axis. However, different ellipticity of polarization across the spot was introduced when the light reached the grain boundary. If a region of the spot hit a bump at the boundary, it would get an extra retardation on the slow light component than a region that hit a trough. The light then propagated into another crystal grain, which had a different extraordinary axis and another rough boundary, and thus further ellipticity differences between regions were introduced. As the light propagated through the column of crystal grains, the piecewise elliptical polarizations across the light spot were getting more different from each other. Although the light in every region of the spot remained polarized, when viewed as one whole spot, it looked as if it was being depolarized. Mathematically speaking,

{(\sum_{i = 1}^{m} {Q^{'}}_{i})}^{2} + {(\sum_{i = 1}^{m} {U^{'}}_{i})}^{2} + {(\sum_{i = 1}^{m} {V^{'}}_{i})}^{2} \leq {\sum_{i = 1}^{m} ({Q^{'}}_{i})}^{2} + {\sum_{i = 1}^{m} ({U^{'}}_{i})}^{2} + {\sum_{i = 1}^{m} ({V^{'}}_{i})}^{2} = {({I^{'}}_{t o t a l})}^{2}

and therefore

D O P_{t o t a l} \leq 1

We could then see that such summed piecewise polarization over a fixed length of polycrystalline material depended on the number of boundaries in the path. These results are shown in Fig. 3 .

Fig. 3 Degree of polarization versus propagation length for different sized crystals. The data shows that as the crystal size decreases, the measured linear polarization is lost more quickly because there are more crystal transitions to different c-axis orientations. The solid lines represent the numerical solutions of the Mueller matrices simulations for 500 Monte Carlo trials of randomly selected c-axis distributions. The crosses are averages from 3 independent sets of experimental c-axis distributions.

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The spread of the c-axes of the grains was another factor that helped determine the rate of decrease in the measured polarization. As mentioned above, as the light propagated from one grain to another, a change in the c-axis direction widened the ellipticity differences across the laser spot. The larger the change in the c-axis, the faster the polarization appeared to be lost. This effect could be seen in both the simulations and experiments as illustrated in Fig. 4 .

Fig. 4 Degree of polarization versus propagation length for crystal groups whose c-axes had different spreads in orientation. The solid lines are simulated results from 1000 Monte Carlo trials, with grain size fixed at 2mm but with random c-axes within the specified orientation spreads. The crosses are averages from 3 independent experiments with the crystals set to a random orientation within the desired spread.

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5. Conclusions

In conclusion, there are strong correlations between polycrystalline fabric and the rate of loss of linear polarization for light transmitted through a transparent polycrystalline material. Both grain size and the spread of the c-axes have a quantitatively predictable effect on the polarization. These have been theoretically simulated using Mueller matrices and Monte Carlo methods and experimentally tested with quartz crystals. Methods derived from these findings would be useful in rapidly identifying regions of a sample where the fabric rapidly changes from one type to another, such as in reconstructing the folding record of glacial ice.

Acknowledgement

The authors would like to thank Richard Alley, Buford Price, and Ryan Bay for helpful discussions. This work was funded in part by a University of Minnesota Grant-in-Aid for exploratory research.

References and links

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4. D. P. Hansen and L. A. Wilen, “Performance and applications of an automated c-axis icefabric analyzer,” J. Glaciol. 48(160), 159–170 (2002). [CrossRef]

5. K. Sassen, “Ice Crystal Habit Discrimination with the Optical Backscatter Depolarization Technique,” J. Appl. Meteorol. 16(4), 425–431 (1977). [CrossRef]

6. S. Oshchepkov, H. Isaka, J. F. Gayet, A. Sinyuk, F. Auriol, and S. Havemann, “Microphysical properties of mixed-phase and ice clouds retrieved from in situ airborne 'Polar Nephelometer' measurements,” Geophys. Res. Lett. 27(2), 209 (2000). [CrossRef]

7. B. Barkey and K. N. Liou, “Polar nephelometer for light-scattering measurements of ice crystals,” Opt. Lett. 26(4), 232–234 (2001). [CrossRef]

8. J. P. Decruppe, S. Lerouge, and H. Azzouzi, “Rayleigh scattering and flow birefringence measurement in colloidal solutions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 011503 (2005). [CrossRef] [PubMed]

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Correlations between polycrystalline fabric and the polarization of transmitted light

Abstract

1. Introduction

2. Theory

3. Experiment

4. Results and discussions

5. Conclusions

Acknowledgement

References and links

Cited By

Figures (4)

Equations (10)

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