# Tower Waveplates Meet Comprehensive Requirements for AOT Laser Polarimeter.

## Bistatic laser polarimeter calibrated to 1% at visible-SWIR wavelengths

## BRIAN G. HOOVER,^{1,*} DAVID A. RUGELY,^{1} CHRISTOPHER M.FRANCIS,^{2} GAL ZEIRA,^{2} AND VICTOR L. GAMIZ^{2}

^{1}Advanced Optical Technologies, Inc., Albuquerque, New Mexico, USA

^{2}USAir Force Research Laboratory, Kirtland Air Force Base, New Mexico, USA

^{∗}hoover@advanced-optical.com

http://www.advanced-optical.com

Abstract: This paper documents the accuracy and precision of the U. S. Air Force Research Laboratory APCL laser polarimeter in arbitrary bistatic geometries at the three laser wavelengths 633nm, 1064nm, and 1550nm. The difference between measured and theoretical-truth Mueller matrices of calibration components is used as the calibration metric and justified relative to block ellipsometer calibration methods. Calibration of thepolarimeter ellipsometry mode is demonstrated first, at quasi-monostatic and large bistatic angles, employing a metallic mirror and a dielectric window as the calibration component, respectively, the latter in order to avoid uncertainty in the retardance of typical metallic mirrors at large incident angles. This uncertainty is demonstrated in measurements of COTS protected-silver mirrorsfrom two vendors, revealing an

approximately λ/8 retardance difference, for reflection through 90◦ , between nominally-identical mirrors from the two vendors. Polarimeter calibration is finally extended beyond ellipsometry by calibrating depolarization measurements using a new technique employing ensembles of polarized states as calibration components.

©2016 Optical Society of America

OCIS codes:(120.2130) Ellipsometry and polarimetry; (120.5410) Polarimetry; (120.3940) Metrology; (230.5440) Polarization-selective devices; (260.5430) Polarization

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_{0}and M_{00}be non-negative

## 1. Introduction

Techniques in active or laser polarimetry, also called Mueller-matrix polarimetry, are gaining popularity for applications in remote sensing, biomedical optics, and plasmonic and nanostructure characterization, among others. Measurement of the Mueller matrix or elements thereof (see Appendix A), which is accomplished by an active or laser polarimeter, is a challenge documented in a large body of metrology literature [1–10]. Part of this challenge stems from under-specification of the optical components (waveplates, polarizers, etc.) needed to make polarization measurements. While many contemporary vendors offer CAD models of lenses, spectral filters, and other common optical components, to our knowledge no vendor currently provides such specifications on polarization optics [11]. As a result, polarimeter measurements and calibrations inherit uncertainties in the properties of their components. This problem could be overcome, and polarimeter calibration, development, and use facilitated, if optics vendors would provide more complete specifications of their polarization optics. The other primary challenge for polarimeter calibration in general geometries is provision of calibration standards or components, ie, components with known theoretical-truth Mueller matrices. While monostatic or quasi-monostatic configurations can be calibrated by reflection from a typical metal mirror, which has little effect on polarization at near-normal incidence, the polarization effects of metal mirrors at large incident angles are generally not well known a priori due to protective coatings and other unknown and possibly proprietary material factors. In Section 4.1.2 dramatic differences are demonstrated between the measured Mueller matrices of standard protected-silver mirrors from two catalog vendors, highlighting the need for better vendor specifications and careful selectionof calibration standards.

The differences between polarimetry and ellipsometry are reviewed in Section 2. Constraints on measurement geometry and sample properties, as well as popularity in the semiconductor industry, have led to highly accurate ellipsometer calibrations [9–10], several of which are compared with our calibration results in Section 5. Without these simplifying constraints, calibration of a general laser polarimeter is more challenging. It has proven especially difficult to calibrate polarimeter measurements of depolarizing samples and materials. While many calibration attempts utilize Spectralon as a depolarizing standard, measurements show that in many geometries Spectralon is not an ideal depolarizer [12]. In Section 4.2 we introduce a depolarization calibration standard constructed as the average of an ensemble of polarized states,analogous to a speckle average, and apply it to demonstrate calibration of the AFRL APCL laser polarimeter to better than 1% overall elemental error.

## 2. Polarimetry versus ellipsometry

Ellipsometry is almost exclusively a laboratory technique for characterization of material systems for which an invertible electromagnetic-wave eflectance or transmittance model exists. While polarimetry can also be applied to characterization, it is more often applied to remote sensing, often n a field instrument, for discrimination or classification of objects with arbitrary polarized reflectance or transmittance characteristics based on omparison with a known database ofpolarization signatures. Ellipsometry employs model-based algorithms while polarimetry often employs data-based algorithms from the field of machine-learning [13–14]

A general laser polarimeter can be operated in ellipsometer mode by constraining the measure-ment geometry and the properties of the sample under est. In conventional ellipsometry the sample is assumed to be planar, homogeneous (non-scattering), and isotropic (non-birefringent), and the easurement geometry is assumed to conform to the law of reflection, ie θr=θi. These constraints still admit a wide variety of sample classes, including nstressed thin films and layered media, relevant for instance to semiconductor and optics manufacturing.

Under the conventional ellipsometry assumptions polarized reflection is completely described by the Fresnel formulae, which specify the complex igenvalues RsandRp,commonly called the Fresnel reflection coefficients. The corresponding Mueller matrix is

where the subscript denotes ellipsometry. The real and imaginary parts of Rsand Rpimply four unknowns in Eq. (1), but the form in which they appear allows reduction to 2 unknowns for normalized data by defining the ellipsometric angles Ψ and ∆ as

A bit of complex algebra shows that Eq. (2) allows the Mueller matrix of conventional ellipsometry to be expressed as

Ellipsometry can be generalized by relaxing the conventional material assumptions and applying more detailed electromagnetic-wave reflectance or transmittance models. Materials with slight inhomogeneity can be accommodated by applying effective-medium theory to define a virtual homogeneous material to be measured in the direction θr=θi[15], maintaining Eq. (3). Generalized ellipsometry accommodates general birefringent materials by replacing the Fresnel formulae (Eqs. (1)-(3)) with more general formulations of Maxwell’s equations [16–17], and can be expressed in terms of non-depolarizing Mueller matrices [18].

Data-based polarimetry algorithms are needed in the absence of sufficiently generalized models of electromagnetic-wave reflection, transmission, and scattering in particular. Inhomogeneous, scattering, or random media cause depolarization, which is difficult to model rigorously. Ellip- someter calibrations that do not encompass depolarizing media are inadequate for polarimetry applications where depolarization is dominant, including most remote-sensing applications.

## 3. Calibration methods

All of the calibrations demonstrated in this paper are examples of component-wise calibration, wherein each optical component of the polarimeter has a model Mueller matrix [8–9]. Requiring the polarimeter to achieve less than 1% overall error in a component-wise calibration implies

that each optical component performs tol within 1% of its associated theoretical-truth Mueller matrix. Component-wise calibration should be distinguished from block calibration , which has been prescribed for polarimeters for instance by Compain et al. [3, 10] and wherein the Mueller matrices of the individual components need not be known. Block calibrations measure a set of calibration elements and apply mathematical algorithms to compensate for unknown errors in the polarimeter components. While component-wise calibrations demand higher-quality components, they also allow effects and errors in each component to be isolated and eliminated, thereby improving calibration results over time. Components of the APCL polarimeter, particularly waveplates, have been qualified through this approach as noted in Section 4. Justification for component-wise calibration is based on the belief that better polarization components, ie components that perform closer to their theoretical truths, will ultimately produce more accurate polarization measurements. Further thoughts on the two polarimeter calibration approaches are noted in Section 5.

Our new approach for calibrating depolarization measurements is motivated by techniques in polarized-speckle metrology [8]. The Stokes parameters (aka Stokes vector) is an ensemble- average as defined in Appendix A. In a static speckle pattern the corresponding time averages (as measured by an irradiance detector) are generally random processes over scattering angle or position on an intersecting plane. A single realization of these random processes, for instance as measured within a single speckle cell, may be expressed as

without the ensemble average. The linear transformation from deterministic input Stokes parameters Sitosois a matrix of random processes, denotedm:

In a static speckle pattern each realization of mis non-depolarizing in a time-average. Taking the ensemble average of Eq. (5),

so obviously 〈m〉=Mper Eq. (22). As noted in Appendix A, the average〈. . .〉is over any andall random processes that affect the electromagnetic-field components Exand/or Ey.

To quantify depolarization we deal with the normalized matrices M′≡M/M_{00} and m′≡m/m

where ‖. . .‖ denotes a matrix norm. Eq. (7) implies that a known level of depolarization can be introduced by controlling the distribution of the non-depolarizing matrices m.Furthermore, for calibration purposes the random process that determines the distribution of mcan be generated by any means, however artificial. The polarimeter only measures the resulting depolarization; it cannot distinguish between a random process contrived for calibration and a natural random process exhibited by a sample or material. Depolarization measurements can therefore be calibrated by measuring an ensemble of Mueller matrices generated by simple non-depolarizing components, for example retarder waveplates, that sums to a depolarization matrix. Polarimeter calibration with an example ensemble depolarization calibration component (a depolarization ensemble) is demonstrated in Section 4.2.

Fig. 1. Laser polarimeter in the Air Force Research Laboratory Active Polarimetric Charac-terization Lab (APCL).

## 4. APCL laser polarimeter

The Active Polarimetric Characterization Laboratory (APCL) has been developed over the past decade at the Air Force Reseach Laboratory (AFRL), Kirtland Air Force Base, NM. The APCL features the laser polarimeter/scatterometer illustrated in Fig. 1, which can measure either specular or diffuse transmissive or reflective objects/samples up to 1’ in diameter, from5 ◦ (aka quasimonostatic) to 180◦ (transmission) bistatic angle. The polarimeter measures the object/sample Mueller matrix in any of these geometries at any of the three laser wavelengths 633nm, 1064nm, or 1550nm. The laser sources are a Melles Griot 25 LHP helium-neon, a Crystalaser IRCL-200-1064 NdYAG, and a Keopsys CEFL-TERA-M5-LP EDFL, respectively. While data from the APCL laser polarimeter has been published in several articles [8, 13–14], detailed metrology and calibration results have not been previously published.

The polarimeter is a conventional dual-rotating retarder (DRR) type with both the polarization- state generator (PSG) and polarization-state analyzer (PSA) comprising a fixed calcite polarizer (Karl Lambrecht Corp.) and a rotating zero-order quartz quarter-waveplate (QWP) (Tower Optical Corp.) The PSG employs a Glan-Taylor polarizer, the PSA a Glan-Thompson for wider acceptance angle. Separate pairs of waveplates and polarizers, with laser-line antireflection coatings, are used for each polarimeter wavelength. Three motorized hollow-core rotation stages rotate the crystals with resolutions better than 0.01◦ . The PSA polarizer/analyzer is on a Klinger CC1 rotary stage and controller. This stage does not rotate during typical data acquisition but must be precisely oriented such that the analyzer is crossed with the PSG polarizer, which is on a manual rotary stage oriented to transmit a vertical state. The PSG waveplate, which is nominally quarter-wave, is on an NSK ESA motor and controller, and the PSA waveplate is on a Newport RGV100 stage and XPS controller. As detailed in [8], the data-acquisition routine collects irradiance measurements at 16 discrete, optimized waveplate-angle pairs in order to estimate the Mueller matrix. The data reported here was collected with the waveplate angles that maximize the determinant of the transmitter and receiver matrices as specified in [8]. The rotary stages are relatively precise but not fast–each set of 16 measurements takes about 22s. The detector is a Newport 818-series photodiode (818-SL for visible and 818-IG for NIR-SWIR) on a Newport 2835-C meter in DC-continuous mode with analog and digital filters activated. The stages are

controlled and the meter read by a custom Labview routine, which can also scan the receiver arm in azimuth (horizontal) and the sample chuck in x,y,z,and azimuth, although these stages are stationary for the data reported in this paper.

The polarimeter alignment procedure has been refined over many years. Aspects important for error reduction are noted here. The crystal axes are aligned one at a time, first the PSA polarizer, then the PSA QWP, and finally the PSG QWP, by minimizing detector power. Each motor axis is aligned parallel to the beam axis to ensure that the crystals remain normal to the beam as they rotate. When aligning the crystal axes it is important to limit motor backlash by moving to the null position monotonically. In bistatic geometries the calibration window (see Section 4.1) must be thick enough to produce separate beams reflected from the front and back surfaces, and a ghost-blocking iris must be installed between the window and the PSA to block the latter beam. Unfortunately, the APCL temperature is not well-controlled, and temperature changes alter the polarization properties of the crystals due to thermal expansion. While temperature- induced errors are minimized by using zero-order waveplates, the temperature dependence is accounted for by calibrating the polarimeter over the temperature range of the measurements and temperature-stamping each recorded matrix. Calibrations at different temperatures are used to create a lookup table of waveplate retardances versus temperature, which is called by the data-processing routine to define the transmitter and receiver matrices [8] for transformation of each measured irradiance matrix into a Mueller matrix. The disadvantage of this procedure is the need to stop general data collection and recalibrate when the temperature changes significantly (∼5◦F) to a range not previously calibrated.

Refinement of the alignment procedures has achieved calibration results that depend, to within 1% overall elemental error, on the waveplate retardances only. The calibration routine, which is part of the general data-processing routine coded in Matlab, performs an optimization versus the theoretical-truth matrix (see Section 4.1) over a two-dimensional retardance grid, producing two retardance versus temperature look-up tables that are called for reduction of all subsequent measured matrices. In order to achieve this level of accuracy all of the polarization optics must exhibit Mueller matrices within 1% of their respective theoretical Mueller matrices [11]. Over the past decade polarizers and waveplates from various manufacturers and vendors have been tested, yielding very mixed results, and components meeting this requirement have been found through trial-and-error. These measurements have shown that waveplates are the most error-prone components of the polarimeter, with many being tested and discarded. Tower Optical Corp. has provided batches of waveplates for testing, with the best waveplates being selected for regular use in the polarimeter.

The source laser power is monitored via a component reflection in the transmitter; the source power is recorded for each Mueller matrix and used to eliminate the effects of source fluctuations in absolute reflectance (eg, BRDF) measurements. The source monitor does not currently correct for high-frequency fluctuations that could affect normalized Mueller-matrix measurements like those presented in this paper, although it could be upgraded to do so by recording the source power for each power measurement rather than for each matrix. The APCL polarimeter employs a telecentric aperture relay [8] to achieve high angular or spatial resolution for sample measurements. Details of the aperture relay will not be included here as it is not essential to the results of this paper, although the presented calibrations that include the relay (633nm quasimonostatic) encompass additional requirements on the relay lenses, in particular good anti-reflection coatings and low-stress mounts. Set-screw lens mounts, for example, typically fail to meet polarization requirements

Fig. 2. Error matrix of the APCL laser polarimeter in the quasi-monostatic configuration at 633nm wavelength, including aperture relay. Error bars represent standard deviations over 40 measurements.

4.1. Ellipsometer mode

4.1.1. Calibration

In the quasi-monostatic configuration a typical protected-silver mirror (Janos A1510-372) is employed as the calibration mirror. The incident angle is 2.5−2.6◦, at which polarization effects of the mirror are negligible, at least well below systematic errors due to other factors. The normalized theoretical truth matrix in this configuration is

with the negative signs due to the reflection. Figures 2-4 show the error matrices M−M_{mono,} where Mis the measured Mueller matrix, for the APCL laser polarimeter at the wavelengths 633nm, 1064nm, and 1550nm, respectively. With a perfect polarimeter all data points would be zero (as for the normalized M_{00} element). The points represent the systematic errors in the elements while the error bars indicate the standard deviations of the measured elements over 40 matrices. In the quasi-monostatic configuration the overall elemental error is<1% at all wavelengths and <0.6% at the NIR-SWIR wavelengths. The larger systematic errors for the visible wavelength are attributed to the two lenses that constitute the aperture relay between the PSG and the PSA [8], which were not installed for the NIR-SWIR calibrations. The systematic error caused by the aperture relay is evident by comparing calibrations with and without the relay, both of which achieve overall elemental error <1%. In addition to arbitrary reflectors in near-monostatic geometries, transmissive samples are also measured in the quasi-monostatic configuration, mounted preceding the alibration mirror.
Bistatic configurations imply large incident angles on calibration components. At large angles no known optical component provides simple diagonal theoretical-truth matrix like that of Eq. (8). Pure metals exhibit retardance, while metal mirrors may exhibit additional polarization effects, presumably due to protective coatings and/or other unspecified material components

Fig. 3. Error matrix of the APCL laser polarimeter in the quasi-monostatic configuration at 1064nm wavelength. Error bars represent standard deviations over 40 measurements.

Fig. 4. Error matrix of the APCL laser polarimeter in the quasi-monostatic configuration at 1550nm wavelength. Error bars represent standard deviations over 40 measurements.

Fig. 5. Error matrix of the APCL laser polarimeter in the 90◦ bistatic configuration at 1064nm wavelength. Error bars represent standard deviations over 40 measurements.

Section 4.1.2 demonstrates dramatic differences between the polarization properties of nominally identical protected-silver mirrors from two catalog vendors. Due to these uncertainties, metallic mirrors are poor calibration standards for bistatic polarimeters.

Simple dielectric reflectors can serve as calibration standards for bistatic ellipsometers and polarimeters. For instance, mmercial off-the-shelf (COTS) uncoated glass windows, typically high-purity glass with no unspecified coatings, are expected to reflect as nearly ideal diattenuators with extinction axis in the plane of incidence. The normalized Mueller matrix of an uncoated glass window is assumed as the theoretical-truth matrix for bistatic polarimeter calibration:

where qis a function of the glass refractive index per Eq. (1). If we make the assumption that any polarimeter error will break the symmetry of Eq. (9), then the actual value of qis uncritical for calibration purposes. Adopting this assumption, the following bistatic calibration results use M_{01}=M_{10} and M_{22}=M_{33} for the theoretical-truth matrix. Once the calibration matrix is obtained then the value of qcan be inverted to obtain a reasonable estimate of the glass refractive index.

Figures 5-6 show the error matrices M−M_{bi,} where Mis the measured Mueller matrix, for the APCL laser polarimeter in the 90◦ bistatic configuration at the wavelengths 1064nm and 1550nm, respectively. In this configuration the overall elemental error is<0.8%at both wavelengths.

4.1.2. COTS silver mirrors

The contemporary status quo of vendor polarization specifications necessitates calibration of every optical component of a new polarization device or sensor. Assuming components will match their theoretical behavior will often result in expensive troubleshooting during functional testing. This section presents measurements of very standard optical components–commercial off-the-shelf (COTS) protected-silver mirrors–in the APCL 90◦ bistatic configuration at 1064nm

Fig. 6. Error matrix of the APCL laser polarimeter in the 90◦ bistatic configuration at 1550nm wavelength. Error bars represent standard deviations over 40 measurements.

wavelength that highlight the need for component calibration during device or sensor development. Specifically, nominally identical protected-silver mirrors from two major catalog vendors (Vendor A & Vendor B) were found to impress different phase shifts, resulting, depending on the incident

state, in reflected states of opposite handedness or azimuth angle of the polarization ellipse.

The measured normalized Mueller matrices of the two mirrors at 45◦ angle-of-incidence (AOI) are

over 40 matrices, where the subscript denotes the vendor. To simplify interpretation these can be rounded to

Figure 7 illustrates the polarization ellipses reflected by these mirrors when illuminated by the same circularly-polarized state. The difference between the reflected states, which could obviously result in drastically different information conveyed or measured by a device or sensor, would be difficult to identify without measuring the complete Mueller matrices of the mirrors. The mirror from Vendor A is consistent with Mueller matrices of other protected-silver mirrors measured in this configuration; the mirror from Vendor B has the atypical Mueller matrix.

Fig. 7. Polarization ellipses reflected by nominally-identical protected-silver mirrors from two catalog vendors (A & B) when illuminated at 45◦ AOI by the same circularly-polarized state

Analysis of the matrices of Eq. (12) shows that the mirrors differ by a retardance of approximately λ/8. The cause of the observed phase difference is yet to be determined, although different refractive indices of the protective coatings is considered a candidate.

4.2. Polarimeter mode

The new polarimeter-mode calibration method is demonstrated with a particular depolarization ensemble comprised of states produced by pre-calibrated quarter-wave and half-wave plates. Other depolarization ensembles may be simpler or otherwise preferable to the one specified and demonstrated here. The scope of this section is to demonstrate depolarization calibration with an ensemble calibration element that is relatively simple to implement.

As introduced in Section 3, the depolarization ensemble is a collection of Mueller matrices that sum to the ideal depolarization element

While depolarization ensembles could be derived using systematic linear algebra, the ad hocensemble chosen for initial demonstration comprises the (reflection) identity matrix, which is simply the quasi-monostatic calibration matrix presented in Section 4.1.1, and the combinations of quarter-wave and half-wave retarder matrices listed in Table 1, all of which include the reflection inherent in the APCL quasi-monostatic configuration.

Using the well-known Mueller matrices of half-wave and quarter-wave retarders [11]

it can be shown that

that is, the elements in Table 1 form a depolarization ensemble.

4.2.1. Calibration

The accuracy of the ensemble calibration method depends not only on the inherent accuracy of the individual polarization components (waveplates), but also on the ability to orient the component

c-axes with high accuracy. The half-wave plate used (Tower Optical Z-17.5-A-.500-B-1064) was mounted approximately normal to the beam in an Oriel manual rotary stage with 0.02◦ resolution, while the quarter-wave plate (Tower Optical A025.4DZ 1/4 1064) was mounted in a Klinger CC1 motorized rotary stage on Klinger stepping-motor controller CC1.2 with 0.01◦ resolution. The manufacturer’s tolerance on the waveplate retardance is ±0.005 waves. For the ellipsometer calibration described in Section 4, variations in waveplate retardance (due to manufacturer tolerance and temperature) can be calibrated out, provided the waveplates perform to well within 1% of the theoretical Mueller matrix of a linear retarder [11]. The polarimeter-mode calibration described in this section, with the depolarization ensemble of Table 1, is less tolerant, as retardance variations can prevent Eq. (14) from being satisfied to within the desired accuracy (1% here). Fortunately, the waveplate retardance can be tuned slightly, without causing significant deviation from the theoretical Mueller matrix of a linear retarder, by tilting the waveplate to increase the optical path length through it. For the calibration reported here the quarter-wave plate was tilted to 5-6◦ AOI in order to obtain measured matrices within 1% of the theoretical matrices quoted in Table 1. With this apparatus the measured matrices corresponding to the theoretical matrices in Table 1 were

and

with standard deviations over 40 matrices. Adding these per Eq. (14) demonstrates the measured depolarization ensemble

ie the polarimeter has been calibrated for depolarization measurements to much better than 1% at 1064nm wavelength. Depolarization calibration at other wavelengths can be achieved similarly with sufficiently accurate waveplate components.

## 5. Conclusions

The polarimeter calibration reported in this paper appears to be unmatched. Previous reported ellipsometer calibrations have approached or broken the barrier of 1% overall elemental error only in single configurations. Arteaga, et al. for example reported excellent component-wise calibration of a Mueller-matrix polarimeter based on photoelastic modulators (PEMs), but only in transmission geometry at a single wavelength [9]. Elegant block calibration methods have been developed to compensate for uncertainties in commercial polarization optics. As noted in Section 3, since the individual polarization components need not be measured or verified, block calibration can allow a new polarimeter to obtain data more quickly, but does not allow the effects of individual components to be isolated so that non-conforming components can be systematically replaced. Justification for component-wise calibration is based on the belief that better polarization components, ie components that perform closer to their theoretical truths, will ultimately produce more accurate measurements. Debate between the two calibration approaches should continue until a quantitative or mathematical comparison can show which is superior. The best reported polarimeter block calibration appears to be that of DeMartino, et al., which approached the 1% barrier employing liquid-crystal modulators, but again only in transmission geometry at a single wavelength [10]. The component-wise calibration reported in this paper covers both quasi-monostatic and bistatic reflection geometries at three wavelengths, and also extends beyond ellipsometer calibrations by calibrating against a newly-defined depolarization ensemble.

While one of our objectives is to publicize this capability, another objective is to elevate the standards and specifications of polarimeters and polarization optics. The field of polarization metrology has seen over 20 years of literature on polarimeter calibration, with few previous demonstrations of Mueller-matrix polarimeters under 1% overall elemental error. We have shown that this standard can be achieved through judicious component selection and methodical laboratory procedures. With adequate dedicated resources it could be achieved in much less time than reported here. Multiple polarimeters meeting such high calibration standards need to arise for the field of polarization metrology to mature enough to enable next-generation polarization devices and sensors. This evolution would be accelerated if manufacturers and vendors of polarization optics would provide comprehensive specifications (ie, Mueller matrices) of their products. Such specifications would certainly be worth a premium for developers trying to develop their device or sensor or build a polarization metrology capability quickly.

## A. Mueller matrices

The Stokes parameters (often referred to as the Stokes vector [20]) express the statistical correlations among vector electromagnetic-field components through the field second moments:

where x and y are the typical Cartesian coordinates and 〈. . .〉 denotes the ensemble-average, which is most generally an average over any and all random processes affecting the field components Exand/or Ey. The components of the incident electromagnetic field are denoted E_{xi} and E_{yi.} When the incident field interacts with a material object, and the reflected, transmitted, and/or scattered electromagnetic field generally propagates to a measurement location, the components of the measured electromagnetic field are denoted E_{xo} and E_{yo,} where “o” implies “out”. The output Stokes parameters are then

The Mueller matrix Mis the linear transformation between the incident and output or measured Stokes parameters:

Importantly, the Mueller matrix is a transformation of moments, hence its elements are determin-istic functions of observation variables such as angle and wavelength.

### Funding

The procedures and data reported in this paper were created and obtained for multiple contracts and orders, and the paper was completed as internal R&D. The plurality of funding was provided by Army SBIR contract W909MY-12-C-0023.

### Acknowledgments

Thanks to Pablo Reyes, David Taliaferro, Virgil Kohlhepp, Roger Holten, and Israel Vaughn at AOT for contributions to the APCL. The presented data was obtained under cooperative research and development agreement CRADA 2011-AFRL/RD-03 between the Air Force Research Laboratory, Directed Energy Directorate and Advanced Optical Technologies, Inc. The results and opinions expressed in this paper are those of the authors exclusively and do not represent endorsement of any product, service, or approach by the US government. This paper is approved for public release, distribution unlimited.