Mueller matrix roots algorithm and computational considerations.

Recently, an order-independent Mueller matrix decomposition was proposed in an effort to elucidate the nine depolarization degrees of freedom [Handbook of Optics, Vol. 1 of Mueller Matrices (2009)]. This paper addresses the critical computational issues involved in applying this Mueller matrix roots decomposition, along with a review of the principal matrix root and common methods for its calculation. The calculation of the pth matrix root is optimized around p = 10(5) for a 53 digit binary double precision calculation. A matrix roots algorithm is provided which incorporates these computational results. It is applied to a statistically significant number of randomly generated physical Mueller matrices in order to gain insight on the typical ranges of the depolarizing Matrix roots parameters. 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.

Square-wave retarder for polarization computer-generated holography.

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.

Polarization synthesis by computer-generated holography using orthogonally polarized and correlated speckle patterns.

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 the 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. The demonstrated technique can be applied to designs for arbitrary irradiance and polarization states in the image plane.

Optimization of Mueller matrix polarimeters in the presence of error sources.

Methods are presented for optimizing the design of Mueller matrix polarimeters and and in particular selecting the retardances and orientation angles of polarization components to ensure accurate reconstruction of a sample's Mueller matrix in the presence of error sources. Metrics related to the condition number and to the singular value decomposition are used to guide the design process for Mueller matrix polarimeters with the goal of specifying polarization elements, comparing polarimeter configurations, estimating polarimeter errors, and compensating for known error sources. The use of these metrics is illustrated with analyses of two example polarimeters: a dual rotating retarder polarimeter, and a dual variable retarder polarimeter.

Mueller matrix retinal imager with optimized polarization conditions.

A new Mueller matrix polarimeter was used to image the retinas of normal subjects. Light from a linearly polarized 780 nm laser was passed through a system of variable retarders and scanned across the retina. Light returned from the eye passed through a second system of retarders and a polarizing beamsplitter to two confocal detection channels. Optimization of the polarimetric data reduction matrix was via a condition number metric. The accuracy and repeatability of polarization parameter measurements were within +/- 5%. The magnitudes and orientations of retardance and diattenuation, plus depolarization, were measured over 15 degrees of retina for 15 normal eyes.

Oxygen Saturation Measurements of Blood in Retinal Vessels during Blood Loss.

We describe a noninvasive technique and instrumentation for measuring the oxygen saturation of blood in retinal arteries and veins. The measurements are made by shining low-power lasers into the eye, and scanning the beams across a retinal blood vessel. The light reflected and scattered back out of the eye is collected and measured. The oxygen saturation of blood within the vessel is determined by analyzing the vessel absorption profiles at two wavelengths. A complete saturation measurement can be made in less than 1 s, allowing real-time measurement during physiologic changes. The sensitivity of this measurement technique to changes in retinal saturation has been demonstrated through a series of pilot studies in anesthetized swine. We present data indicating that retinal venous oxygen saturation decreases during ongoing blood loss, demonstrating a potential application of an eye oximeter to noninvasively monitor blood loss. © 1998 Society of Photo-Optical Instrumentation Engineers.

Buchdahl's glass dispersion coefficients calculated in the near infrared.

Buchdahl's chromatic coordinate provides a rapidly converging polynomial form for describing the dispersion of glass. A method of generating Buchdahl's chromatic coordinate coefficients with a Code V sequence file is presented. The quadratic model coefficients for eighty-two glasses at a base wavelength of 0.830 microm are listed. When evaluating Buchdahl coefficients across different wavelength ranges (i.e., infrared or ultraviolet), the chromatic coordinate constant alpha can be changed to improve convergence of the refractive-index model.

Buchdahl's glass dispersion coefficients calculated from Schott equation constants.

A method for the rapid evaluation of Buchdahl's dispersion coefficients at an arbitrary base wavelength given the Schott equation constants is presented. Buchdahl's chromatic coordinate transformation is performed on the Schott equation for the refractive index. A Taylor series expansion of the transformed Schott equation is then equated to the square of the Buchdahl dispersion model. Equations for Buchdahl's dispersion coefficients are then obtained for any base wavelength by matching the power of Buchdahl's chromatic coordinate omega out to fourth order.

Angular dependence of polarizing beam-splitter cubes.

The field-of-view dependence of polarizing beam-splitter cubes has been studied to characterize their behavior in imaging systems such as optical computers and optical correlators and in other applications that involve noncollimated light. Significant polarization aberration is present in polarizing beamsplitter cubes for two reasons: (1) the s- and p-component orientations, which define the polarizing axes, at the beam-splitting interface vary with the direction of propagation, and (2) the performance of the coating is a function of the angle of incidence. We describe the polarization aberration of a polarizing beam-splitter cube in terms of its diattenuation (polarizing efficiency). We use an imaging polarimeter to measure six figures of merit for three polarizing beam-splitter cubes demonstrating typical polarization aberrations. Finally, we derive the Mueller matrix for a polarizing beam-splitter cube in terms of the s and p transmittance and reflectance and the phase retardances, the parameters generally calculated with thin-film analysis programs.

Coating-induced wave-front aberrations: on-axis astigmatism and chromatic aberration in all-reflecting systems.

The coatings used on telescope mirrors and other optical interfaces can have a marked effect on an optical system's image quality. We describe the wave-front aberrations, particularly the defocus and on-axis astigmatism, that are induced by the s- and p-phase shifts of coatings. These coating-induced wave-front aberrations are very small, particularly near the design wavelengths of the coatings, but they can under certain circumstances overshadow the geometric wave-front aberrations of the system. The wave-front aberrations that are induced by reflection-enhanced coatings on an À/1.5 Cassegrain telescope are numerically evaluated as an example. A theory of coating-induced on-axis astigmatism and chromatic aberration is presented, and a simple algorithm that uses polarization ray tracing to calculate coating-induced defocus and astigmatism coefficients in radially symmetric systems is provided.

Polarization aberrations. 2. Tilted and decentered optical systems.

In paper 1 [Appl. Opt. 33, this issue (1994)] we examined the polarization aberrations of rotationally symmetric systems. In this paper we extend polarization-aberrration theory to include two types of tilted and decentered systems composed of rotationally symmetric elements. One type is systems with collinear centers of curvatures but with decentered pupils. The symmetry in such systems permits the analysis to proceed along lines similar to those in paper 1. The other type is systems with arbitrary tilts and decenters. In these systems the field dependencies of the aberrations from each surface are not concentric. The extension is made by use of a polarization-aberration matrix with vector, instead of scalar, arguments.

Polarization aberrations. 1. Rotationally symmetric optical systems.

The polarization in isotropic radially symmetric lens and mirror systems in the paraxial approximation is examined. Polarized aberrations are variations in the phase, amplitude, and polarization state of the electromagnetic field across the exit pupil. Some are dependent on the incident polarization state and some are not. Expressions through fourth order for phase, amplitude, linear diattenuation, and linear retardance aberrations are derived in terms of the chief and marginal ray angles of incidence and the Taylor series expansion coefficients of the Fresnel equations for reflection and transmission at uncoated and thin-film-coated interfaces. Applications to polarization ray tracing are discussed.

Distortion-adjusting optical elements.

Distortion can be corrected in an image by placing a fourth-order aspheric optical element near the image plane. Moving the aspheric surface longitudinally changes the amount of distortion that is added by the aspheric surface without changing the paraxial image. Third-order astigmatism limits the performance of distortion correctors and may be eliminated by adding another fourth-order aspheric surface. Example elements were fabricated by diamond turning and were shown to introduce distortion without significantly degrading image quality. Three arrangements of distortion correctors are discussed: a single-element planoaspheric arrangement, an antisymmetric two-element arrangement, and a biaspheric arrangement in which distortion is not adjustable.

Maximum power of refractive lenses: a fundamental limit.

This paper explores the maximum optical power, i.e., the shortest focal length, that can be achieved with homogeneous materials given a cylindrical volume and a minimum-diameter constraint. An equation for the maximum optical power for a given lens diameter and two refractive indices is defined. For example, we determine that a 25.4-mm-diameter lens assembly cannot have an effective focal length of

Staircase lens: a binary and diffractive field curvature corrector.

The optical characteristics of a family of diffractive lenses, called staircase lenses, are discussed. These lenses contribute no optical power and the fourth-order wave aberrations Petzval field curvature, distortion, and chromatic aberrations when collimated illumination is used. A diamond-turned staircase lens comprising 150 steps was manufactured. The test results verify the generation of field curvature and longitudinal chromatic aberration.

Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter.

A technique to measure linear diattenuation and retardance spectra of infrared materials in transmission is described. A sample is rotated between two stationary linear polarizers in the sample compartment of a Fourier transform infrared spectrometer. The intensity modulation that results from the rotation of the sample is Fourier analyzed, and the linear diattenuation and linear retardance of the sample are calculated from the Fourier series coefficients for each wavelength. The advantages of this technique include immunity of the measurement to instrumental polarization, to circular diattenuation, and to circular retardance. The rotating sample polarimeter does not require retarders. Compensation for systematic errors from polarizers with diattenuation less than one is included in the data reduction. This technique is useful for the calibration of retarders and polarizers and hence for the bootstrap calibration of more elaborate polarimeters such as Mueller matrix polarimeters. We present as an example of t technique the caliration spectra 3-14 µm of an infrared quasi-achromatic retarder whose fast axis orientation oscillates with wavelength.

Infrared birefringence spectra for cadmium sulfide and cadmium selenide.

Measurements of the birefringence spectra for cadmium sulfide and cadmium selenide from 2.5 to 16.5µm obtained with a rotating sample spectropolarimeter are presented. Because of the similarity in the birefringence spectra for cadmium sulfide and cadmium selenide, a highly achromatic IR retarder can be constructed from a combination of these materials. The ordinary and extraordinary refractive indices for cadmium sulfide are estimated in the region from 10.6 to 15 µm and for cadmium selenide from 10.6 to 16.5 µm by combining these birefringence data with an extrapolation of previous dispersion relations.

HN22 sheet polarizer, an inexpensive infrared retarder.

The popular sheet polarizer, Polaroid HN22, has been measured to be a nearly half-wave retarder in the 3.6-5.4-microm spectral band with a transmittance of approximately 20%. Tuning of the retardance value between 60 degrees and 260 degrees has been demonstrated by tilting of the HN22 sheet with respect to the incident beam. The material's availability, relatively large aperture, large field of view, and low cost make it an excellent candidate for use as an infrared retarder for systems operating in this wave band. Thus HN22 may be employed as an inexpensive half-wave linear retarder and used for rotating the plane of polarization as well as for conversion between circular polarization states.

Calculation of rotationally symmetric multi-focal lens surface.

A numerical method to calculate surfaces for refractive multi-focal intraocular lenses is presented. The computations, performed in a plane containing the axis of rotation of the surface, provide sag values as well as the curvature of the surface between values. Standard ray tracing techniques may then be applied to the calculated surface to verify the optical performance.

Electro-optic coefficient spectrum of cadmium telluride.

A spectrum for the electro-optic coefficient of cadmium telluride measured from 3 to 14 µm is reported. The spectrum shows that the quantity n(3)r(41) has a nearly constant value of 1.09 × 10(-10) m/V over this spectral band, with a slight (5%) dip at the weak absorption band centered at 6 µm. Measurements were performed with an infrared Mueller matrix spectropolarimeter. Transmission spectra of the Mueller matrix were acquired at a set of applied voltages. Retardance spectra were calculated from Mueller matrix spectra, and then the electro-optic coefficient was calculated at each wavelength by a least-squares fit to the resulting retardance as a function of voltage.