Optimal birefringence distributions for imaging polarimetry

Abstract

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...

Birefringent Fourier filtering for single molecule coordinate and height super-resolution imaging with dithering and orientation

“…The basis of the proposed technique is the placement at the pupil plane of an element referred to as a stressed-engineered optic (SEO), which is a BK7 glass window subjected to forces with trigonal symmetry at its edges 18 , 19 , 23 (see “Methods”). The spatially varying birefringence pattern that naturally results in the vicinity of the force equilibrium point has been shown to be essentially optimal for applications in polarimetry, in the sense that it efficiently encodes polarization information in the PSF’s shape while causing the smallest possible increase in PSF size 24 . This birefringence pattern is described by the following Jones matrix (in the linear polarization basis) in the Fourier plane of the detection path (Fig.…”

“…1 b) where ( u , φ ) are polar pupil coordinates normalized so that u = 1 corresponds to the pupil’s edge, and c is a coefficient that depends on the stress within the SEO and the radius of the pupil being used. This parameter can be chosen to optimize the system’s performance: small c keeps the extension of the PSFs more restricted, but reduces the amount of information they carry about orientation and z displacement, while large c has the opposite effect 18 , 24 . After passing through the SEO, the two circular polarization components are separated to form two images by inserting a quarter-wave plate (QWP) followed by a Wollaston prism (Fig.…”

“…An important feature of the SEO’s birefringence pattern is that it makes this set of PSF components nearly orthogonal while keeping their extension almost as small as possible (in analogy to the case of paraxial polarization 24 ). This approximate orthogonality implies the strong decoupling of the information for each parameter, as will be discussed in the next section.…”

Geometry and topology in 3D polarization

“…We recently proposed imaging polarimetry techniques suitable for objects composed of point-like sources that are sparce enough that their point-spread functions (PSFs) do not overlap. In these techniques, in addition to (2D or 3D) localization of these points, one can retrieve the state of polarization [9,10]. The key is to tailor the PSF by inserting a suitable birefringent mask in the pupil plane of the imaging system.…”

Birefringent distributions tailored for imaging and other applications

“…For example, simple geometric interpretations in terms of the Poincaré hypersphere were given to both the widening of the PSF and the reduction of the Strehl ratio of imaging systems, due to spatially-varying birefringence of its elements [6]. This formalism has also been used for the design of birefringence distributions that are optimal for different goals, such as the generation of bottle beams [11] and of polarimetric systems [12]. In both cases, the optimal distributions have been found to be extremely similar to the stable birefringence pattern that results from subjecting a glass window to a pressure distribution with trigonal symmetry at the edges [5].…”