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We proposed to speed up the process of measuring optical rotation tensors of crystals by taking advantage of new techniques in polarimetry, especially imaging technologies. We constructed a working Mueller matrix imaging polarimeter based on mechanical light modulation and a CCD camera. The instrument has worked effectively for analyzing linear anisotropies of most organized samples. Likewise, it is effective for samples that have large chiroptical anisotropies and small linear anisotropies such as cholesteric liquid crystals (CLCs). CLCs have huge differential circular reflection bands and circular birefringence (CB) resulting from their twisted mesostructures on the order of the wavelength of visible light. We prepared a mixture of cholesteryl esters (65 wt.% cholesteryloleyl carbonate, 25 wt.% cholesteryl pelargonate, and 10 wt.% cholesteryl benzoate) that is stable in the cholesteric state at room temperature. The crystals reflect different colors in different parts of the sample due to slight differences in the pitch. Mueller matrix micrographs made with 590 nm radiation show, after exponential decomposition of the raw data, circular extinction due to circular differential reflection, and CB. Here, we made vivid the classic De Vries theory (de Vries, H. Rotatory power and other optical properties of certain liquid crystals. Acta Cryst 1951, 4, 219-226) of cholesteric reflection bands, where the sign of the CB changes on either side of the reflection band.

When we began our research in chiroptical imaging a decade ago, we made a commitment to mechanical light modulation in conjunction with cameras. Here, polarization modulation is slow but imaging is fast. Mechanical modulation gave us confidence in the polarization state of the light we were producing. In the interim, the stability of photoelastic modulators (PEMs) has improved considerably. Last year, Arteaga in Barcelona used a dual PEM transmission ellipsometer designed by Jellison and Modine (Jellison, Jr., G. E.; Modine, F. A. Two-modulator generalized ellipsometry: Theory, Appl. Opt. 1997, 36, 8184-8189; A. Two-modulator generalized ellipsometry: Experiment and calibration. Appl. Opt. 1997, 36, 8190-8198. ) to re-measure the CB tensor of alpha-quartz, the first and thus far only use of MM to determine the CB tensor of a crystal (Arteaga, O.; Canillas, A.; Jellison, Jr. G. E. Determination of the components of the gyration tensor of quartz by oblique incidence transmission two-modulator generalized ellipsometry, Appl. Opt. 2009, 48, 5307-5317.). Arteaga further used piezoelectric scanning to image left and right handed domains of benzil. Here, modulation is fast but imaging is slow. In Arteaga’s device, two quartz rotators complement a pair of fixed PEMs to effectively create four PEM positions necessary to fully sample Stokes vector space (the Poincaré sphere). Alternative configurations would involve four fixed PEMs turned on and off in pairs, or four PEMs of different frequency, timed to work in sync.

In order to compare the performance of the New York and Barcelona Mueller matrix imaging polarimeters, we purchased a (001) slab of a commercially grown AgGaS₂ crystal (United Crystal Company). We each collected data on this crystal and established that the standard deviation in the Mueller matrices with our rotating wave plate instrument was 0.001, where the duel PEM was 0.0001 or better. The additional order of magnitude is necessary for pinpointing small chiroptical effects of typical molecular crystals. The duel PEM instrument is also much faster. With our instrument, we can collect the data necessary for one tensor at a single wavelength in two days. With the PEM based instrument, we can collect the full spectrum of Mueller matrices in the same amount of time. Moreover, Arteaga and Canillas have developed an analytical decomposition of experimental Mueller matrices into the equivalent of the differential matrix m (Arteaga, O.; Canillas, A. Analytic inversion of the Mueller-Jones matrices for  homogeneous media, Opt. Lett. 2010, 35, 559-561.). We have implemented their algorithm and it is computationally faster (2 s. vs. 1 hr. for a 1.3 mega-pixel image).

What we have learned has set us toward the design of a new instrument that takes advantage of fast modulation but supercedes the capabilities and speeds of the Barcelona instrument but running 4 PEMs simultaneously. No moving parts are involved. The construction of this device is in process.

With a slight modification of the optics of our imaging polarimeter so as to permit conoscopic illumination, and with the Berreman analysis for non-normal incidence, we can determine the CB or circular dichroism tensor of a crystal with one Mueller matrix image. By supplying a cone of light through the sample, every point in the image plane from the center (normal path) represents a wave vector sampling the crystal at progressively increasing angles. Chipman and coworkers proposed such a transmission ellipsometer for the determination of material dielectric tensors (Beaudry, N. A.; Zhao, Y.; Chipman, R. Dielectric tensor measurement from a single Mueller matrix image. J. Opt. Soc. Am. A 2007, 24, 814-819.) While the authors make no mention of chiroptical properties, the extension of the experiment to CB and circular dichroism is straightforward. 

We have constructed conoscopic Mueller matrix of images of quartz and compared them simulated it first and then made the measurement. The agreement is remarkable for a first try. This technique has many promising applications.