The Poincaré sphere is a graphical representation in a three-dimensional space for the polarization of light. Similarly, an optical element with spatially varying birefringence can be represented by a surface on a four-dimensional "Poincaré hypersphere." A projection of this surface onto the traditional Poincaré sphere provides an intuitive geometric description of the polarization transformation performed by the element, as well as the induced geometric phase. We apply this formalism to quantify the effects of birefringence on the image quality of an optical system.

A mathematical extension of the weak value formalism to the simultaneous measurement of multiple parameters is presented in the context of an optical focused vector beam scatterometry experiment. In this example, preselection and postselection are achieved via spatially-varying polarization control, which can be tailored to optimize the sensitivity to parameter variations. Initial experiments for the two-parameter case demonstrate that this method can be used to measure physical parameters with resolutions at least 1000 times smaller than the wavelength of illumination.The concepts of weak value and weak measurement were introduced by Aharonov, Albert and Vaidman in 1988 [1][2][3] as an alternative to the standard measurement formalism of quantum mechanics. For a quantity associated with an operator B, a standard measurement is related to the expected value Φ|B|Φ / Φ|Φ , where Φ is the state vector for the quantum state being measured. Since the state is normalized, the inner product in the denominator is typically taken as unity. Clearly, for Hermitian operators, this expected value is real and limited to the range of values spanned by the eigenvalues of B. On the other hand, weak measurements are based on weak values defined as Φ post |B|Φ pre / Φ post |Φ pre , where Φ pre and Φ post are preselected and postselected states. It is easy to see that there is no bound to a weak value since the denominator can be made arbitrarily small by appropriate preselection and postselection. In fact, weak values need not even be real-valued. Weak measurements have been employed, for example, to measure very small angular deviations with great precision [4][5][6][7].

An optical bottle field containing a three-dimensional intensity null at the focal point can be generated by placing a spatially inhomogeneous birefringent mask at the pupil of an aplanatic high-NA focusing system. We derive the optimal birefringence distribution for which a uniformly polarized input beam is converted into a bottle field with the sharpest possible null in intensity. We show that a stress engineered optical (SEO) window, which has a radially varying retardance, followed by a half-wave plate, performs nearly as well as the optimal solution. Experimental results corroborate that an SEO element can be used to generate a bottle field.

Star test polarimetry is an imaging polarimetry technique in which an element with spatially-varying birefringence is placed in the pupil plane to encode polarization information into the point-spread function (PSF) of an imaging system. In this work, a variational calculation is performed to find the optimal birefringence distribution that effectively encodes polarization information while producing the smallest possible PSF, thus maximizing the resolution for imaging polarimetry. This optimal solution is found to be nearly equivalent to the birefringence distribution that results from a glass window being subjected to three uniformly spaced stress points at its edges, which has been used in previous star test polarimetry setups. IntroductionPolarimetry is the measurement of the polarization state of light and/or the polarization properties of materials. Such measurements are usually characterized in terms of the Stokes parameters and the Mueller matrix, respectively, which are directly accessible from measurements of the intensity. Imaging polarimetry, in which the polarization is measured as a function of position, is particularly important in applications ranging from microscopy to remote sensing.Conventional techniques for Stokes polarimetry require multiple intensity measurements, either through time-sequencing or by splitting different polarization components into several separate detection channels [1,2]. For example, one common method uses a rotating quarter-wave plate (QWP) followed by a fixed linear polarizer, in which the Stokes parameters are deduced from successive intensity measurements with the QWP oriented at different angles [3]. While these techniques can produce highly accurate measurements, they can be relatively complicated and/or time-consuming, generally involving moving parts or multiple beam paths.When a short acquisition time is desirable, rapid polarization measurements may be taken using single-shot polarimetry, in which the Stokes parameters are estimated from a single intensity measurement. A variety of methods exist for single-shot polarimetry, involving gradient index lenses [4], patterned nanoscale gratings [4], or a split aperture composed of multiple polarizers [5]. One particularly simple method, referred to as star test polarimetry, uses a spatially-varying birefringent mask (BM) followed by a uniform polarization analyzer in the pupil plane of an exit-telecentric imaging system. With an appropriately chosen birefringence distribution, the inserted elements can encode full polarization information into the shape of the PSF in the rear focal plane of the lens. This method has been demonstrated experimentally using a stress-engineered optic (SEO), which is a BK7 glass window subjected to stress with trigonal symmetry at its periphery, followed by a circular analyzer [6,7]. Natural applications for this approach are those in which the object is a sparse set of discrete points, such as in astronomy and confocal microscopy. A recent specific application was the real-time monitorin...

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