This dissertation is comprised of two separate topics within the domain of polarization optical engineering. The first topic is a Mueller matrix roots decomposition, and the second topic is polarization computer generated holography. The first four chapters of the dissertation are on the topic of the Mueller matrix roots decomposition. Recently, an order-independent Mueller matrix decomposition was proposed in an effort to organize the nine depolarization degrees of freedom. Chapter 1 discusses relevant Mueller matrix decomposition prior art and the motivation for this work. In chapter 2, the critical computational issues involved in applying this Mueller matrix roots decomposition are addressed, along with a review of the principal root and common methods for its calculation. The choice of the pth root is optimized at p = 10⁵, and computational techniques are proposed which allow singular Mueller matrices and Mueller matrices with a half-wave of retardance to be evaluated with the matrix roots decomposition. A matrix roots algorithm is provided which incorporates these computational results. In chapter 3, the Mueller matrix roots decomposition is reviewed and a set of Mueller matrix generators are discussed. The parameterization of depolarization into three families, each with three degrees of freedom is explained. Analysis of the matrix roots parameters in terms of degree of polarization maps demonstrates that depolarizers fall into two distinct classes: amplitude depolarization in one class, and phase and diagonal depolarization in another class. It is shown that each depolarization family and degree of freedom can be produced by averaging two non-depolarizing Mueller matrix generators. This is extended to provide further insight on two sample measurements, which are analyzed using the matrix roots decomposition. Chapter 4 discusses additional properties of the Mueller matrix roots generators and parameters, along with a pupil aberration application of the matrix roots decomposition. Appendix C, adapted from a conference proceedings paper, presents an application of the matrix roots depolarization parameters for estimating the orientation of a one-dimensionally textured object. The last two chapters are on the topic of polarization computer generated holography. In chapter 5, an interlaced polarization computer-generated hologram (PCGH) is designed to produce specific irradiance and polarization states in the image plane. The PCGH produces a tangentially polarized annular pattern with correlated speckle, which is achieved by a novel application of a diffuser optimization method. Alternating columns of orthogonal linear polarizations illuminate an interlaced PCGH, producing a ratio of polarization of 88% measured on a fabricated sample. In chapter 6, an etched calcite square-wave retarder is designed, fabricated, and demonstrated as an illuminator for an interlaced polarization computer generated hologram (PCGH). The calcite square-wave retarder enables alternating columns of orthogonal linear polarizations to illuminate the interlaced PCGH. Together, these components produce a speckled, tangentially polarized PCGH diffraction pattern with a measured ratio of polarization of 84% and a degree of linear polarization of 0.81. An experimental alignment tolerance analysis is also reported.

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