Rapid Discrimination of Crystal Polymorphic Forms Using Nonlinear Optical Stokes Ellipsometric Microscopy
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1 Rapid Discrimination of Crystal Polymorphic Forms Using Nonlinear Optical Stokes Ellipsometric Microscopy Paul D. Schmitt, Emma L. DeWalt, Ximeng Y. Dow, and Garth J. Simpson* *Corresponding uthor: Department of Chemistry, Purdue University, West Lafayette, IN. Supporting Information Example single color Fourier coefficient images (Figure SI-1), images of in-plane crystal orientation from the OrientationJ plug-in (NIH ImageJ) (Figure SI-2), polarized laser transmittance images (Figure SI-3), and an analytical/ graphical explanation of the process of linear discriminant analysis S-1
2 Figure SI-1: Raw 32-bit SHG image (upper left, only 1 of 10 polarizations shown) and 5 Fourier coefficient images (converted to voltages) -E shown on red, green, blue, cyan, and magenta color scales (respectively). Merging of these 5 independent images into one 5 color image enables single-image visualization of the data set. Together with knowledge of the scale of each color s lookup table, this single 5 color image contains the entirety of the polarization-dependent information recovered from the NOSE the measurement. S-2
3 Figure SI-2: Output of orientation analysis via OrientationJ for a single FOV for each form of D- Mannitol. The hue represents the in-plane rotation angle for each pixel according to the look-uptable at right, with pixel intensity retained from the original images. Images are shown on different brightness scales for ease of visualization. The calculated values of phi are averaged across each crystal and then fed into an algorithm for recovery of additional orientation angles (theta, psi), overall SHG intensity, and the unique, non-zero, local-frame tensor elements. Figure SI-3: Example polarized laser transmittance images for orthorhombic (left) and monoclinic (right) forms of D-Mannitol. Collection of the polarized laser transmittance is simultaneous with the acquisition of SHG, and allows recovery of information describing sample birefringence. The simultaneous acquisition of these two imaging modalities (laser transmittance and SHG) also ensures perfect registry between the two images. S-3
4 Description of Linear Discriminant nalysis (LD): LD was used to identify a single projected dimension that best resolved the sample response (polynomial coefficients, local-frame tensor elements, etc.) from each class (monoclinic and orthorhombic, and respectively). Data are treated as vectors in a multi-dimensional space, where the number of vector elements (e.g., 10 for the recovered polynomial coefficients) dictates the dimensionality. In LD, the classes undergo optimal separation through maximizing the value of the Fisher linear discriminant, J, given in equation SI-1. 1,2 ( ) Jw ( ) s s (Equation SI-1) Here, and are the scalar projections of the mean data vectors for each class onto the projected dimension ( w ), and s and s are the variances of each class after projection onto the same axis. The axis w was found through solving the eigenvector-eigenvalue equation shown below: S S w Jw (Equation SI-2) 1 W Where the matrices S and W given by the equations below S represent the within and between class variance (respectively), and are S ( ) ( ) T (Equation SI-3) With an analagous expression for N 1 T S S S ; S (( x ) ) (( x ) ) (Equation SI-4) W W W W i i N 1 i 1 relevant class ( N total vectors in class ), and S W not shown here explicitly. x i indicates the i th vector from the is the average vector from class. graphical representation of LD for two classes of 2-dimensional data is shown below. S-4
5 Figure SI-4: Left: Raw data to undergo analysis by LD. 5 2-dimensional vectors (arranged together into matrices) make up each class of data, and. Middle: 2-dimensional plot of the data, along with the Fisher discriminant found from solving equations SI-2 through SI-4. Right: The same data post-projection onto the Fisher discriminant. The data are plotted against themselves for ease in visualizing the separation. S-5
6 Cited References: (1) Fisher, R.. nn. Hum. Genet. 1936, 7 (2), (2) McLachlan, G. J. Discriminant nalysis and Statistical Pattern Recognition; John Wiley & Sons: Hoboken, New Jersey, S-6
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