Polarimetric characterization of liquid-crystal-on-silicon panels

Mueller matrix imaging polarimetry of liquid-crystal-on-silicon (LCoS) panels provides detailed information useful for the diagnosis of LCoS problems and to understand the interaction of LCoS panels with other projector components. Data reduction methods are presented for the analysis of LCoS Mueller matrix images yielding contrast ratio, efficiency, spatial uniformity, and the calculation of optimum trim retarders. The effects of nonideal retardance, retardance orientation, and depolarization on LCoS system performance are described. The white-state and dark-state Mueller matrix images of an example LCoS panel are analyzed in terms of LCoS performance metrics typical for red-green-blue wavelengths of 470, 550, and 640 nm. Variations of retardance, retardance orientation, and depolarization are shown to have different effects on contrast ratio, efficiency, and brightness. Thus Mueller matrix images can diagnose LCoS problems in a way different from radiometric testing. The calculation of optimum trim retarders in the presence of spatial variations is discussed. The relationship of the LCoS retardance in single-pass (from front to back) to the double-pass retardance (from entrance to exit) is established and used to clarify coordinate system issues related to Mueller matrices for reflection devices.     

Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error

The relationship between system condition and signal-to-noise ratio (SNR) in reconstructed Stokes parameter images is investigated for rotating compensator, variable retardance, and rotating analyzer Stokes vector (SV) polarimeters. A variety of optimal configurations are presented for each class of systems. The operation of polarimeters is discussed in terms of a four-dimensional conical vector space; and the concept of nonorthogonal bases, frames, and tight frames is introduced to describe the operation of SV polarimeters. Although SNR is an important consideration, performance of a polarimeter in the presence of errors in the calibration and alignment of the optical components is also important. The relationship between system condition and error performance is investigated, and it is shown that an optimum system from the point of view of SNR is not always an optimum system with respect to error performance. A detailed theory of error performance is presented, and the error of a SV polarimeter is shown to be related to the stability and condition number of the polarization processing matrices. The rms error is found to fall off as the inverse of the number of measurements taken. Finally, the concepts used to optimize SV polarimeters are extended to be useful for full Mueller matrix polarimeters.     

Polarimetric diagnosis of 193-nm lithography equipment using a mask with newly developed polarization optical elements

Two lithographic test masks, Stokes polarimeter mask and Mueller matrix polarimeter mask, are introduced. Both the masks comprise newly developed thin polarizers and wide-view-angle λ/4 plates. Photomasks are only 6.35 mm in thickness, and the illumination involves the oblique incidence of 20° at the most. Calcite plates thinned to less than 0.1 mm can perform as polarizers only at a wavelength of 193 nm. The combination of quartz and sapphire plates can mitigate the retardation change with angle of incidence. Stokes polarimeter and Mueller matrix polarimeter masks are used for the measurement of the illumination and the projecting optics, respectively.     

Axially resolved complete mueller matrix confocal microscopy

We introduce a technique that is capable of obtaining complete polarization-sensitive three-dimensional images that could reveal unknown anatomical conditions of living tissue that possess polarization-dependent signatures. Previously, the 16 Mueller coefficients were measured independently only by use of two-dimensional imaging techniques. We also present the experimental combination of a depth-resolved confocal imaging system with a complete Mueller matrix polarimeter. To calibrate the system, a double-pass method had to be implemented. We also indicate, experimentally, that the confocal sectioning of the system has a degrading effect on axially resolved Mueller matrix measurements.     

Automated high-speed Mueller matrix scatterometer

A new scatterometer-polarimeter is described. It measures the angular distribution of intensity and of the complete Mueller matrix of light scattered by rough surfaces and particle suspensions. The measurement time is 1 s/scattering angle in the present configuration but can be reduced to a few milliseconds with modified electronics. The instrument uses polarization modulation and a Fourier analysis of four detected signals to obtain the 16 Mueller matrix elements. This method is particularly well suited to online, real time, industrial process control involving rough surfaces and large particle suspensions (an arithmetic roughness or particle diameter of >1 mum). Some results are given.     

Real-time measurement of the polarization transfer function

We present a simple method for measuring the Mueller matrix associated with a scattering medium. Without involving moving parts, four input states of polarization are generated sequentially, and for each of them all four Stokes vector parameters are simultaneously measured for the complete determination of the Mueller matrix. Two liquid-crystal variable retarders are used for controlling the input state of polarization, whereas the measurement of the state of polarization involves phase modulation with a single-pass photoelastic modulator, and Fourier analysis in two polarization channels. The setup is controlled by a computer, allowing for real-time measurement of the Mueller matrix. The method is tested on standard elements such as polarizers and quarter-wave plates, as well as on inhomogeneous particulate systems.     

Two-modulator generalized ellipsometry: theory

A new ellipsometer is described that uses two photoelastic modulator-polarizer pairs, where the photoelastic modulators are operating at differing resonant frequencies. The time-dependent intensity of the light beam is extremely complicated but can be analyzed so that all elements of the sample Mueller matrix are obtained. For a given configuration, nine of the Mueller matrix elements can be measured at any one time; the other seven elements are accessible when the azimuthal angles of the photoelastic modulators are changed. The single-configuration measurement is often sufficient to characterize a number of real situations completely, such as film growth in a vacuum environment, anisotropic samples, and simple depolarization.     

Light backscattering polarization patterns from turbid media: theory and experiment

We present both experimental measurements and Monte-Carlo-based simulations of the diffusely backscattered intensity patterns that arise from illuminating a turbid medium with a polarized laser beam. It is rigorously shown that, because of axial symmetry of the system, only seven elements of the effective backscattering Mueller matrix are independent. A new numerical method that allows simultaneous calculation of all 16 elements of the two-dimensional Mueller matrix is used. To validate our method we compared calculations to measurements from a turbid medium that consisted of polystyrene spheres of different sizes and concentrations in deionized water. The experimental and numerical results are in excellent agreement.     

Second-harmonic-generation-polarimeter calibration by means of a quartz plate

A dual-rotating-retarder polarimeter was used to determine the six measurable observables of the first hyperpolarizability tensor. Calibration of such an instrument requires a reference sample dedicated to wavelength conversion. We calibrated our experimental setup by using a quartz-plate sample in a two step procedure: at first the first retarder then the second one. The retardance and ellipticity angle of both retarders were estimated by minimizing a chi(2) function. We estimated the standard deviation of each parameter from noise spreading and performed this calibration procedure for two experimental case studies, i.e., two angular positions of the quartz sample.     

Photopolarimetric measurement of single, intact pulp fibers by mueller matrix imaging polarimetry

A method based on Mueller matrix polarimetry is developed and demonstrated for determining the fibril angle and relative phase retardation of single, intact pulp fibers. The method permits quantitative and nondestructive determination of these parameters from measurements at one wavelength without any fiber alignment. The Mueller matrix of a pulp fiber and its relationship with the fibril angle and phase retardation are described. A nonmodulation method for determining the Mueller matrix is then proposed that is based on a set of intensity data registered by a single detector. Measurements were carried out with single pulp fibers as samples to test the theoretical prediction. The test measurements and results are described and presented.     

Noise distribution of Mueller matrices retrieved with active rotating polarimeters

Two methods used to retrieve Mueller matrices from intensity measurements are revisited. It is shown that with symmetry or orthogonality considerations, numerical inversions of polarimetric equations can be avoided. With the obtained analytical formulas, noise propagation can be analyzed. If the intensity noise is a Gaussian white noise, the noise of Mueller matrices features remarkable properties. Mueller components are mutually correlated according to a scheme that involves decomposition into four blocks of 2x2 matrices. Variances are unequally distributed: the middle 2x2 block has the highest variance, the element on the bottom right has the lowest. These characteristics have been validated on experimental Mueller images of the free space.     

Comparison of quarter-wave retarders over finite spectral and angular bandwidths for infrared polarimetric-imaging applications

We compare three technological approaches for quarter-wave retarders within the context of polarimetric-imaging applications in the long-wave infrared (LWIR) spectrum. Performance of a commercial cadmium sulfide (CdS) crystalline waveplate, a multilayer meanderline structure, and a silicon (Si) form-birefringent retarder are evaluated under conditions of 8-12 μm broadband radiation emerging from an F/1 focusing objective. Metrics used for this comparison are the spectrally dependent axial ratio, retardance, and polarization-averaged power transmittance, which are averaged over the angular range of interest. These parameters correspond to the characteristics that would be observed at the focal-plane array (FPA) detector of an LWIR imaging polarimeter.     

Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators

A dual rotating compensator ellipsometer based on the optical PC1SC2A configuration described by Collins [1, chap. 7.3] has been developed. The systematic errors for this configuration if the compensators are quarter-wave plates have been already studied [2, 3, 4]. Smith [5] has demonstrated that the optimum retardance of a dual-rotating-retarder (DRR) instrument must be equal to 127° compared to the quarter-wave (90°) retarders generally used. In this condition random errors are optimized. The aim of this work is to used such retarders and verify if the systematic errors due to uncertainties of the optical elements (i.e. analyzer, polarizer, first and second compensators) are improved too. For each optical element in different configurations like single or 4-zone average measurements, the systematic errors are given and compared according to the compensators. It is demonstrated that using a 127° instead of quarter-wave retarders coupled with 4-zone averaging measurement is the best configuration for this instrument. These results were confirmed by a statistical study.     

Improved method for calibrating a Stokes polarimeter

We present a method for calibrating a polarization state analyzer that uses a set of well- characterized reference polarization states and makes no assumptions about the optics contained in the polarimeter other than their linearity. The method requires that a matrix be constructed that contains the data acquired for each of the reference polarization states and that this matrix be pseudoinverted. Since this matrix is usually singular, we improve the method by performing the pseudoinversion by singular value decomposition, keeping only the four largest singular values. We demonstrate the calibration technique using an imaging polarimeter based upon liquid crystal variable retarders and with light emitting diode (LED) illumination centered at 472 nm, 525 nm, and 630 nm. We generate the reference polarization states by using an unpolarized source, a single polarizer, and a Fresnel rhomb. This method is particularly useful when calibrations are performed on field-grade instruments at a centrally maintained facility and when a traceability chain needs to be maintained.     

Differential interference contrast microscopy as a polarimetric instrument

Differential interference contrast (DIC) microscopy is shown to be equivalent to an incomplete Stokes polarimeter capable of probing optical properties of materials on microscopic-length scales. The Mueller matrix for a DIC microscope is calculated for various types of samples, and the polarimetric properties for DIC component parts of a spaceflight microscope are spectrally measured. As a practical application, a measurement of the index mismatch between colloidal particles and a nearly index-matched fluid bath was performed.     

Scattering by layered structures with rough surfaces: comparison of polarimetric optical scatterometer measurements with theory

A laboratory model of a layered structure with a rough upper surface (a glass microscope slide cut with a diamond saw) is used to obtain optical polarimetric data. Scatterometer measurements were made of all the Mueller matrix elements associated with light scattered in arbitrary directions. (A preliminary measurement of scattering from a smooth opaque gold film on a silicon wafer was used to validate the calculation of the Mueller matrix elements.) These measurements are compared with corresponding analytical solutions based on the full-wave approach. Physical interpretations of the analytical solutions that account for scattering upon reflection and transmission across rough interfaces are given in a companion paper. The agreement between calculations and measurements suggests that the full wave, polarimetric solutions can provide a reliable database for electromagnetic detection of rough surfaces in remote-sensing applications.     

Rotatable broadband retarders for far-infrared spectroscopic ellipsometry

Rotatable retarders have been developed for applications in spectroscopic, full Mueller Matrix ellipsometry in the far-IR spectral range. Several materials, such as silicon, KRS-5, and a commercial polymer plastic (TOPAS) have been utilized to achieve a fully adjustable retardation between 0° and 90°. Experimental characteristics of the rotatable retarders that utilize three- and four-bounce designs are compared with calculations. We discuss the effect of light focusing on the performance of these rotatable retarders.     

Phase-modulated mueller ellipsometry characterization of scattering by latex sphere suspensions

Phase-modulated Mueller ellipsometry (PMME) is used to probe scattering by suspensions of polystyrene latex spheres, with particle diameters ranging from 400 nm to 3 mum. PMME allows simultaneous measurement of the 16 coefficients of the Mueller matrix. Furthermore PMME measurements can easily be carried out owing to a calibration procedure implemented in a scattering configuration. The measurements performed on low concentrations show good agreement with Mie theory. Moreover size distribution could be obtained with a least-squares method based on a genetic fit algorithm. Experimental evidence of multiple scattering on PMME measurements is also presented.     

Optimized Stokes polarimeters based on a single twisted nematic liquid-crystal device for the minimization of noise propagation

This work evidences the suitability of applying a single twisted nematic liquid-crystal (TN-LC) device to obtain dynamic polarimeters with high accuracy and repeatability. Different Stokes polarimeter setups based on a TN-LC device are optimized, leading to the minimization of the noise propagated from intensity measurements to the Stokes vector calculations. To this aim, we revise the influence of working out of normal incidence and of performing a double pass of the light beam through the LC device. In addition, because transmissive TN-LC devices act as elliptical retarders, an extra study is performed. It analyzes the influence of projecting the light exiting from the TN-LC device over elliptical states of polarization. Finally, diverse optimized polarimeters are experimentally implemented and validated by measuring different states of partially and fully polarized light. The analysis is conducted both for monochromatic (He-Ne laser) and LED light sources, proving the potential of polarimeters based on a single TN-LC device.