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A connection between Mueller and Jones representations for polarization analysis in nonlinear optics for structures buried in turbid media is discussed and applied to second harmonic generation (SHG) imaging of thick collagenous tissue samples. Polarization analysis for buried interfaces and structures is complicated by heterogeneous phase retardance as incident light fields approach the focal plane. Further complexities arise in nonlinear optics, where multiple incident fields interact to form a unique polarization-dependent response. The Stokes-Mueller framework provides sufficient generality to describe fields composed of many superimposed polarizations, i.e., depolarized or partially polarized light. However, the Stokes-Mueller framework is intrinsically incompatible with Jones/Cartesian representations of nonlinear optical phenomena most commonly used to relate recovered tensor elements back to molecular-scale structure and orientation. To address this challenge, a mathematical framework bridging the more general Stokes-Mueller framework to the more utilitarian Jones/Cartesian framework was developed. This framework was previously applied to the limiting case of completely depolarized light, predicting SHG intensity emitted by z-cut quartz and enabling direct recovery of collagen orientation in thin tissue samples. In the presented work, the Stokes-Mueller framework was applied to imaging thick tissue sections, where native turbidity of the tissue induces significant depolarization of the incident fundamental beam. By modulating the incident polarization state rapidly with an electro-optic modulator, ten images of unique incident polarization were simultaneously acquired. The Stokes vectors for the incident fundamental light and second harmonic generation were measured, and the Stokes-Mueller framework was utilized to fit to the underlying Jones tensor elements for collagen, which are directly related to molecular-scale structure and orientation.
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