Open Access
Add to BibSonomy
Abstract
We demonstrate calibration and operation of a Mueller matrix imaging microscope using dual continuously rotating anisotropic mirrors for polarization state generation and analysis. The mirrors contain highly spatially coherent nanostructure slanted columnar titanium thin films deposited onto optically thick titanium layers on quartz substrates. The first mirror acts as polarization state image generator and the second mirror acts as polarization state image detector. The instrument is calibrated using samples consisting of laterally homogeneous properties such as straight-through-air, a clear aperture linear polarizer, and a clear aperture linear retarder waveplate. Mueller matrix images are determined for spatially varying anisotropic samples consisting of a commercially available (Thorlabs) birefringent resolution target and a spatially patterned titanium slanted columnar thin film deposited onto a glass substrate. Calibration and operation are demonstrated at a single wavelength (530 nm) only, while, in principle, the instrument can operate regardless of wavelength. We refer to this imaging ellipsometry configuration as rotating-anisotropic-mirror-sample-rotating-anisotropic-mirror ellipsometry (RAM-S-RAM-E).
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical polarimetry is a measurement technique used to determine how the polarization properties of light have altered upon transmission through, reflection off, or scattering from a sample or a given optical system. The general approach is to probe an unknown sample with light of known polarization and to determine the state of light polarization after interaction with a sample or optical system. Typically, the intensity of the light is recorded and calibrated polarizing optical components are used to modulate the recorded intensity. Appropriate mathematical model descriptions then permit identification of the polarization changes upon comparison between recorded intensity data and model calculated data. For non-depolarizing samples or optical systems, the resulting change in polarization can be represented in terms of complex valued $2\times 2$ matrices known as Jones matrices. [1] However, the most general representation of the polarization properties of a sample or optical system is the Mueller matrix. [2] Mueller matrices are able to represent fully polarizing, partially polarizing, or fully depolarizing samples or optical systems in the form of a real valued $4\times 4$ matrix, and mathematically connect the light polarization representing Stokes vectors of the incident and exiting light beams. [3,4]
Polarimetric imaging has seen use in film measurement and metrology [5–19], studying liquids [20–23], characterizing the properties of optical components and systems [24–34], studying biological systems [23,35–44], and has shown potential as a medical diagnostic tool [45–63]. Polarimetry performed in specular reflection at oblique light incidence to a sample or optical system under investigation is typically termed ellipsometry. While ellipsometry is most often used to spectroscopically resolve the polarization properties of bulk samples, it has also been used to spatially resolve polarization properties. Imaging ellipsometers may measure $\Psi$ and $\Delta$, or more generally the ratio of orthogonal polarization states [7,12,18,64–70] or report all Jones and/or Mueller matrix elements [11,17,71]. Imaging optics and laterally resolved detectors have also been used to produce images of polarization properties of a sample or optical system as a function of the steric angle of incidence, for example in conical refraction [11,72,73]. Instruments that record polarimetric images at normal incidence to the sample or specimen image plane are typically referred to simply as polarimeters or Mueller matrix imaging polarimeters. Many different arrangements have been demonstrated, both in transmission [11,27,59,63,74–88] and reflection [11,27,59,63,74–88].
Numerous apparatuses, their calibration, and use have been reported for measurement of Jones and/or Mueller matrix elements of samples and/or optical systems [89]. A fundamental consideration is the type, amount, and arrangement of optical polarizing components in an ellipsometer or polarimeter instrument, and the resulting information that can be gained and eventually imaged. For example, a simple ellipsometry measurement which can result in the measurement of $\Psi$ and $\Delta$, which represent the change in amplitude and phase of the ratio between the electric field components of $p$ and $s$ polarized light, respectively, may require two linear polarizers only, and both parameters can be determined except for the sign of $\Delta$. The same setup would be able to determine the Jones matrix elements normalized by one of its on-diagonal element, and again without the sign of all three phase parameters [90], or alternatively, the upper 3$\times$3 block elements of the Mueller matrix normalized by element $\mathrm {M}_{11}$ [91]. An overview of accessible Mueller matrix elements using ideal rotating-element polarizer and retarder elements, and/or photoelastic modulator-element configurations is given by Hauge [89,92].
An early example of imaging ellipsometry was then demonstrated by Erman et al. in the form of a rotating analyzer ellipsometer in which the probe beam was focused to a small region of the sample. The sample could then be translated in the x and y directions in order to spatially resolve the ellipsometric parameterss $\Psi$ and $\Delta$ [93]. Erman et al. then used this instrument to produce ellipsometric images of GaInAs metal insulator semiconductor structures between various steps in the production process [18]. Cohn et al. then expanded this concept using a charge coupled device (CCD) camera as the detector in a polarizer-compensator-sample-analyzer (PCSA) imaging ellipsometer [64,65,94,95], dispensing with the need to scan the sample surface to acquire ellipsometric images. Chipman and Pezzaniti then expanded these concepts to measure full Mueller matrix images of samples and optical systems using a fixed polarizer and analyzer with dual rotating compensators [24–27,96,97]. Dayton et al. then demonstrated an imaging polarimeter using a polarizer, analyzer, and four liquid crystal phase retarders to measure full Mueller matrix images in transmission, eliminating the need to physically rotate optical components [98]. Han et al. then demonstrated an imaging ellipsometer capable of measuring $\Psi$ and $\Delta$ using a fixed polarizer and analyzer with a photoelastic modulator [68]. Alali et al. then demonstrated a full Mueller matrix imaging instrument using a fixed polarizer and analyzer along with four photoelastic modulators [99]. Arteaga et al. then improved the dual rotating compensator Mueller matrix imaging technique by continuously rotating both compensators during a measurement [75], a technique that had previously been used in commercially available spectroscopic (non-imaging) ellipsometers.
To our knowledge, all previously described Mueller matrix imaging instruments have relied on transmissive optics for polarization state modulation and analysis. Moreover, in order to recover the full 4$\times$4 Mueller matrix of a sample at a single wavelength without sacrificing lateral resolution, all such instruments require at least four polarization modifying components such as linear, elliptical, and/or circular polarizers, waveplates, photoelastic modulators, etc. [11,17,27,45,47,59,63,71,74,75,77–85,88,98,100–106]. We have recently demonstrated that anisotropic mirrors, when rotated about their surface normal and without additional polarizing components, can provide sufficient modulation of the Stokes parameters to serve as both polarization state generator and polarization state analyzer in a Mueller matrix polarimeter or ellipsometer [107]. Here, this concept has been augmented with imaging optics to create the first all-element Mueller matrix imaging microscope to operate using only two polarization modulating optical components. Aside from the optical and mechanical simplicity of this arrangement, benefits of our setup include the quasi unlimited spectral range of its applicability and the avoidance of incoherent interference within transmissive optical components whose optical thickness is outside the coherence length of the source. Such interference affects proper ellipsometric measurements if not calibrated and/or avoided altogether. Backside reflections introduce multiple probe beams with intensity, polarization, and phase properties that deviate from the first transmitted beam, for example, within retarder or circular polarizer waveplates, or within polymer-backed wire-grid polarizers. In our work here, the anisotropic mirrors use nanoscopic metal columns which produce sufficient anisotropy for modulation of the Stokes parameters of light reflected from their surfaces upon rotation, while dimensions of the nanostructured films are much less than the wavelength used here. Hence, incoherent interference effects within the polarizing optical components are of no concern in our approach.
In this paper, we demonstrate calibration and operation of a Mueller matrix imaging microscope using dual continuously rotating anisotropic mirrors for polarization state generation and analysis. We discuss calibration using straight-through-air, linear polarizer, and linear retarder measurements. We demonstrate spatially resolving cababilities by measuring a commercially available (Thorlabs) birefringent resolution target and a spatially patterned titanium slanted columnar thin film. All data are shown for a single wavelength (530 nm) only. However, the instrument can operate regardless of wavelength. We refer to this imaging ellipsometry configuration as rotating-anisotropic-mirror-sample-rotating-anisotropic-mirror ellipsometry (RAM-S-RAM-E).
2. Instrument description
2.1 Optical path
The optical path of the instrument is shown in Fig. 1. The collimated output from a fiber coupled light emitting diode (LED; CREE XP-E green 530 nm) is used as the source. The collimated beam is reflected from the first rotating anisotropic mirror (RAM1) at an incident angle of 45 degrees. A field stop placed directly after the first rotating anisotropic mirror reduces the beam size to approximately 5 mm in the object plane. The beam then travels through the sample and is collected by an infinity corrected objective lens (Mitutoyo MY5X-802). The light exiting the objective is reflected from a second rotating anisotropic mirror (RAM2) at an incident angle of 45 degrees. This beam is then focused onto the image sensor of a monochrome camera (FLIR BFS-U3-04S2M-CS) by an achromatic doublet lens with a focal length of 75 mm. The pixel size is $6.9\times 6.9$ $\mu$m$^{2}$. This optical arrangement provides a uniform $1.88\times$ magnification.
Fig. 1. Left: Optical path of the Mueller matrix imaging microscope. The microscope is composed of a 530 nm fiber coupled light emitting diode (LED), fiber collimating lens (FCL), rotating anisotropic mirrors (RAM1 and RAM2), field stop (FS), objective lens (OL), tube lens (TL), and camera (CAM). Right: 3D view of FCL, RAM1, and FS.
2.2 Polarization components
The polarizing anisotropic mirrors consist of an optically thick titanium film that is coated with a thin (100 nm) top layer of highly spatially coherent titanium slanted columns [108–112]. The titanium slanted columnar thin film (SCTF) top layer was grown using glancing angle deposition. The SCTFs for both anisotropic mirrors were grown using electron beam evaporation under the same growth conditions. Titanium was chosen for simplicity and using different materials for the underlying thick film and/or columnar film may result in higher signal to noise ratio. A detailed investigation of optical performance parameters using various structural properties of SCTF-based anisotropic mirrors is outside the scope of this paper. For the anisotropic mirrors used in this work, highly mechanically stable flat quartz substrates with a thickness of 6 mm were used. This was done to minimize mechanical distortion of the mirror surfaces when applying minimal required mechanical strain upon mounting the substrates onto their rotation stages. An example of a Ti SCTF is depicted in Fig. 2 where top view and side view scanning electron microscopy (SEM) images are shown. The SCTF in Fig. 2 was deposited onto a silicon substrate for easy cleavage and side view investigation. This film was grown under the same conditions of the SCTFs used for RAM1 and RAM2. We note that exact structural parameters are not relevant because the optical properties of each mirror are accurately determined during the calibration process described in our work here. No further polarizing element is used in our imaging setup.
Fig. 2. SEM images of titanium slanted columnar thin film grown on silicon substrate. Cross section of cleaved side of silicon wafer (left) and top down (right) views.
The mirrors were measured on a J.A. Woollam Co. RC2 Mueller matrix ellipsometer at an incident angle of 45 degrees and at multiple azimuthal angles in steps of one degree. The RC2 is a dual continuously rotating compensator Mueller matrix ellipsometer with a wavelength range from 193 nm to 1690 nm that measures the full $4\times 4$ Mueller matrix. The result of this measurement is shown in Fig. 3. The first column indicates strong polarization modulating capability of the anisotropic mirrors for unpolarized light over a broad spectral range. Mueller matrix measurements of the anisotropic mirrors are not required to calibrate the instrument.
Fig. 3. Mueller matrix elements for one of the anisotropic mirrors measured using a spectroscopic ellipsometer at an incident angle of 45 degrees. All Mueller matrix elements are normalized to $\mathrm {M_{11}}$. The x-axis indicates the azimuthal angle of the SCTF mirror in degrees and the y-axis indicates the wavelength in nanometers.
2.3 Coordinate system
The coordinate system traditionally used in ellipsometry is defined by the plane containing the center lines of the beams of light incident to and reflected from the sample plane. Light with an electric field vector parallel with the plane of incidence ($p$-polarized) corresponds to horizontal polarization in the Cartesian frame of reference as viewed from the detector. Light with an electric field vector perpendicular to the plane of incidence ($s$-polarized) corresponds to vertical polarization in the Cartesian frame of reference of the detector. As shown in Fig. 1, the $x$-axis of the coordinate system is parallel with the incident plane of the two rotating anisotropic mirrors and the $y$-axis is perpendicular to this plane. A counter-clockwise rotation of an object (sample or optical system) in the object plane, from the perspective of the detector, corresponds to a counter-clockwise (positive valued) rotation of the Mueller matrix of the object. The operating principle of the Mueller matrix image measurement presented here is very similar to the Mueller matrix ellipsometer using dual continuously rotating anisotropic mirrors described previously. [107] Here, the first rotating anisotropic mirror (RAM1) in Fig. 1 acts as polarization state image generator (PSIG). The second rotating anisotropic mirror (RAM2) acts as polarization state image detector (PSID). In this configuration, using two rotating anisotropic mirrors, we have shown previously that the full Mueller matrix of a given sample can be measured.
2.4 Measurement cycle
Both RAM1 and RAM2 are continuously rotated by motorized stages during a measurement. The PSIG mirror (RAM1) completes one full revolution and the PSID mirror (RAM2) completes five full revolutions for a single measurement cycle. Incremental encoders with an index signal are attached to the motors driving each anisotropic mirror. This allows the motor positions to be synchronized before starting a measurement. The synchronization is to ensure that when RAM1 is at its previously identified (indexed) zero position, and depending on the rotational speed ratio between the two rotation stages, RAM2 will also be at its zero position accordingly. For example, in a 3:5 rotational speed ratio, each time RAM1 has completed three revolutions, RAM2 would be completing its fifth revolution. Thus on the following image, which would be the first in the next acquisition cycle, both motors would start again at their indexed zero positions. Images are acquired at fixed angular intervals so that the angular positions of the PSIG and PSID mirrors are precisely known for every image recorded. The encoder attached to the second rotating anisotropic mirror generates a clock signal that is then divided such that a fixed number of equally spaced pulses are generated during a measurement cycle. This divided clock signal is then connected to one of the general purpose input output (GPIO) pins on the camera. The camera is configured to trigger a frame exposure at the rising edge of this clock signal, similar to Aspnes et al. [113]. The camera uses a global shutter so all pixels are exposed simultaneously. The presence of RAM2 in the imaging path results in a horizontal mirroring of the image formed on the camera’s sensor. This is corrected in software when each frame is recorded by applying a 180 degree out of plane rotation about the y-axis.
All images were acquired with an exposure time of 2 ms. Longer or shorter exposure times may be used but it is important that the sum of the exposure time and frame readout time is less than the time between subsequent frame trigger pulses. It is also critical that the gain and exposure time are adjusted such that none of the images in a measurement contain saturated pixels. The frame trigger signal was configured to produce $N$ = 128 pulses in a single measurement cycle and the motor speeds were set such that a full set of frames is recorded within a time of 2.5 s. A larger number of frames $N$ may be acquired to improve signal to noise ratio at the expense of larger data sets and longer acquisition time. Image stacks consisting of all $N$ frames in a measurement are saved on a host computer for later reduction to Mueller matrix images.
2.5 Numerical image wobble correction
Unless the surface normal of both RAM1 and RAM2 are perfectly parallel to the axis of rotation of their respective motors, the beam will wobble as a function of the mirror azimuthal angles $\phi _{1}$ and $\phi _{2}$. To reduce wobble, the mirrors are fixed to tip-tilt stages mounted to the motor shafts, allowing for the mirror surface normal to be adjusted as close as possible to parallel with the axis of rotation. Because there will always be residual mechanical misalignment leading to finite wobble of the reflected beam upon mirror rotation, we implemented a numerical procedure to compensate for image shifting. To remove effects of wobble, each image, $(\tilde {x}, \tilde {y})[n]$, taken during a measurement cycle must be corrected (offset) to a known position $(x, y)[n]$ within the camera image plane, $(\tilde {x}, \tilde {y})[n] \rightarrow (x, y)[n]$, where $\tilde {x}, x$ and $\tilde {y}, y$ denote Cartesian coordinates within the focal plane of the camera. The beam will also drift through the object plane as a function of the first mirror $\phi _{1}$ angle. It is assumed that the polarization properties of the beam in the object plane are sufficiently homogeneous such that the effects of the drift due to the first mirror wobble can be ignored.
The problem of image wobble is not unique to our instrument and other approaches to digitally correct image offsets in transmission type instruments have used image registration [88] and more complex techniques involving optical flow [114] to re-align associated pixels in each recorded frame. Because the wobble is repeatable for each frame in the measurement and we did not observe a spatially varying pixel displacement, we opted for a simple registration scheme. The offsets $(\delta x = \tilde {x}-x, \delta y = \tilde {y}-y)[n]$ are determined as the first step of the initial calibration process by recording a measurement of an isotropic sample with high contrast edges within the field of view. The offsets, unique to each image in the measurement, are determined using phase correlation [115]. To accomplish this, the cross-power spectrum is computed by element-wise multiplying the 2D Fourier transform of the current image with the complex conjugate of the 2D Fourier transform of the image it is being compared with. The resulting 2D complex image is then element-wise divided by its element-wise magnitude, resulting in the cross-power spectrum. The $(x, y)$ position of the brightest pixel of the inverse 2D Fourier transform of the cross power spectrum relative to the center of the frame gives the Cartesian offset between the current image and the image it is being compared to
Fig. 4. Images of mean of $n$-stack images before (left) and after (right) realignment of every image in the stack using offsets $(\delta x, \delta y)[n]$ as described in the text. The blurring of the mean of images in the left figure is due to finite wobble of the second anisotropic mirror upon rotation.
3. Theory
The operating principle of a Mueller matrix polarimeter with a detector that only records intensities (i.e., the detector is unable to discriminate between polarization states) is to first modulate the Stokes parameters of light probing a sample. The polarization modulated light exiting the sample is then remodulated and collected by the detector. The Mueller calculus representation of a Mueller matrix ellipsometer or polarimeter operating at a single wavelength is
4. Calibration
4.1 Fit parameters
In order to create the Mueller reduction matrix from Eq. (19), the Fourier coefficient images must be determined for $\vec {G}$ and $\vec {D}$. This is done by measuring a set of samples with different, well known Mueller matrices under the same conditions and using regression analysis to optimize the instrument parameters. The calibration samples chosen for this experiment were air, a polymer linear polarizer (Edmund Optics XP42-200), and a linear retarder (Thorlabs WPQ10M-546). The Mueller matrix of air is simply the identity matrix. The Mueller matrix of an ideal linear polarizer rotated by angle $\theta$ about the optical axis is
5. Results
5.1 Calibration
The images presented in this paper were recorded at the native resolution of the camera (720 pixels wide by 540 pixels tall), realigned to eliminate the effects of wobble, then cropped to 512 by 512 pixels. This crop and the $~1.88\times$ magnification results in a field of view of approximately 1.9 mm along the x and y axes. The images were then further reduced using 2 by 2 mean binning, resulting in a final resolution of 256 by 256 pixels. Each pixel in the resulting images therefore contains an area in the object plane of $7.3\times 7.3\mu$m$^2$. The cropped region and lateral scale is the same for all of the data shown in this paper. The calibration was performed at the same resolution as the binned images, resulting in Fourier coefficient images and a DC offset image with a resolution of 256 by 256 pixels. All parameters were allowed to vary during the initial calibration except for the angular positions of the waveplate measurements. These remained fixed at known values until the RMSE had reached a minimum, at which point they were allowed to vary along with all other parameters in order to fit for small angular offsets.
The Fourier coefficient images determined during the calibration for $\vec {G}$ are shown in Fig. 5. Ideally, all images would be flat and no spatial variations would occur. However, imperfections in calibration samples and our instrument, such as inhomogeneous reflectance of RAM1 due to defects in the SCTF surface, will lead to finite spatial variance in the Fourier coefficients. As seen in the figure, there is recognizable spatial variation in the Fourier coefficients, especially in the higher order harmonic elements.
Fig. 5. Fourier coefficients for RAM1 after calibration. All images are normalized by image $a_{01}$, except for $a_{01}$. The span, $c$, of the color bar is unique for each column of harmonic coefficient pairs. Here, $c_0 = 1.0$ corresponding to column $a_0$, $c_1 = 0.305$ corresponding to columns $a_1$ and $b_1$, $c_2 = 0.256$ corresponding to columns $a_2$ and $b_2$, $c_3=0.025$ corresponding to columns $a_3$ and $b_3$, and $c_4 = 0.023$ corresponding to columns $a_4$ and $b_4$.
The Fourier coefficient images determined during the calibration for $\vec {D}$ are shown in Fig. 6. Compared to the Fourier coefficient images for $\vec {G}$, there is greater spatial variation in the Fourier coefficients particularly in the fourth column images, $a_{4,j}$ and $b_{4,j}$. However, the magnitude of these coefficients is also very small, and decreasing with the order of the Fourier coefficients. It is also notable that features in the fourth order images appear conjugate between the cosine ($a_{4,j}$) and sine ($b_{4,j}$) coefficients. We explain these features with defects on the surface of RAM2, caused by dust particles during the glancing angle deposition, and which result in wandering spots of intensity and polarization variations (defects) that circulate upon RAM2 rotation through the field of view. We note that the four Fourier coefficient images for column $a_0$ in both $\vec {G}$ and $\vec {D}$ represent the rotation-independent components of the respective Stokes vectors for PSIG and PSID, and which are polarized along the $y$-axis of the image plane, which is perpendicular to the incident planes of both RAM1 and RAM2.
Fig. 6. Same as Fig. 5 for RAM2. $c_0 = 1.0$ , $c_1 = 0.541$, $c_2 = 0.509$, $c_3=0.227$, and $c_4 = 0.067$.
5.2 Air measurement
The calibration was first verified by computing the Mueller matrix image for the straight-through-air measurement. As shown in Fig. 7, the determined Mueller matrix images closely match the identity matrix and we note that the off-diagonal elements are scaled by 25$\times$ to emphasize the spatial variations. Fig. 7 also depicts images of the standard deviations and which are flat, and within and mostly much less than 6% of the Mueller matrix range of $-1\dots +1$. Further refinement of both the optical model, calibration method, and anisotropic mirror quality could further reduce the deviations from the unit Mueller matrix shown in Fig. 7. However, we deem the deviations small enough to consider the Mueller matrix imaging instrumentation operational, and hence investigate further targets.
Fig. 7. Determined Mueller matrix image for the straight-through-air measurement (left) and standard error of mean (right) according to Eq. (26). All images are normalized by image $\mathrm {M}_{11}$, except for image $\mathrm {M}_{11}$.
5.3 Birefringent resolution target
A birefringent resolution target (Thorlabs R2L2S1B) was measured to confirm the imaging capabilities of the instrument. Images of selected regions of interest of the target are shown in Fig. 8, where stripes implemented with 5 cycles/mm and 10 cycles/mm are depicted. The birefringent resolution target consists of a cured liquid crystal polymer between two isotropic glass slides. In essence, the target consists of a waveplate with constant birefringence, i.e., the retardance of the liquid crystal polymer is ideally the same over the entire slide, except for the laser-written areas (stripes and numerals) where the extraordinary direction (fast axis) is different. Equation (23) describes the anticipated ideal Mueller matrix of such a liquid crystal standard target with linear retardance and absence of absorption. Accordingly, only the lower 3$\times 3$ block elements should reveal significant contrast in deviation from the unit Mueller matrix, $\mathbf {M}=diag\{1,1,1,1\}$, where $diag\{\}$ is the unit matrix. Comparison between Fig. 8 and Eq. (23) immediately reveals that phase retardance parameter $\delta$ is near zero or 180 degrees since elements $\mathrm {M}_{42,24}$ and $\mathrm {M}_{43,34}$ are close to zero. Images $\mathrm {M}_{42,24}$ and $\mathrm {M}_{43,34}$ then contrast the different fast axis orientations, cf. Eq. (23). Contrast in the remaining images are very small. Images $\mathrm {M}_{41,14}$ could indicate very small circular dichroism within the target sample, and all other images are close to the noise. Note that the patterned areas in the target sample cannot be seen by eye without use of a polarizer and analyzer, hence, scattering is anticipated to be low. This can be verified in Fig. 8 where no features are apparent in images of element $\mathrm {M}_{11}$.
Fig. 8. Thorlabs birefringent resolution target, 5 cycles/mm (left) and 10 cycles/mm (right). All images are normalized by image $\mathrm {M}_{11}$, except for image $\mathrm {M}_{11}$.
Regression analysis was then used to determine the spatial distribution of retardance ($\delta$) and fast axis orientation ($\theta$) of the 5 cycles/mm Mueller matrix images in Fig. 8. A histogram of both parameters is shown in Fig. 9. The histograms reveal the nearly 180 degrees retardance of the cured liquid crystal polymer and a difference in fast axis orientation of the surrounding area and the laser-written areas of approximately 25 degrees.
Fig. 9. Retardance (top left) and fast axis orientation (top right) images determined using regression analysis. Histograms showing the distribution of retardance (bottom left) and fast axis orientation (bottom right).
5.4 Slanted columnar thin film
A second laterally varying sample was measured consisting of a titanium SCTF grown on a glass slide. A thin stainless steel stencil was placed over the substrate during the deposition process in order to create a large uninterrupted region as well as a striped region of titanium SCTF film. The striped region consisted of varying length stripes with a consistent width of 0.5 mm. Both the bulk and striped regions were measured using our Mueller matrix microscope. Two sample orientations were measured, first with the slanted columns oriented along the $y$-axis of the image plane then with a clockwise in plane rotation of approximately 45 degrees. The resulting Mueller matrix images of the bulk region of the titanium SCTF sample, for both orientations, are shown in Fig. 10.
Fig. 10. Patterned titanium slanted columnar thin film. The titanium columns are oriented along the $y$-axis (left) and rotated 45 degrees clockwise (right). All images are normalized by image $\mathrm {M}_{11}$, except for image $\mathrm {M}_{11}$. Note that the red striped regions in $\mathrm {M_{12}}$ and $\mathrm {M_{21}}$ in the left image and the blue striped regions in $\mathrm {M_{13}}$ and $\mathrm {M_{31}}$ correspond to stripes of titanium SCTF material.
The SCTF is acting as a weak wire-grid polarizer. With the columns oriented along the $y$-axis, the resulting Mueller matrix is that of a weak horizontal polarizer, with significant contrast in elements $\mathrm {M}_{12}$ and $\mathrm {M}_{21}$ and little attenuation along the diagonal, and according to Eq. (22) with $\theta$ approximately 0 or 180 degrees. The in plane rotation of the sample by $-45$ degrees shifts the contrast into elements $\mathrm {M}_{13}$ and $\mathrm {M}_{31}$ , cf. Eq. (22) with $\theta$ approximately $-45$ degrees while the diagonal elements remain near that of the identity matrix. The same SCTF sample also has an unmasked region and which was also measured at the two rotation orientations (Fig. 11). Similar Mueller element data are recognized. For further comparison, the same sample was also measured using a commercially available ellipsometer (J.A. Woollam Co., Inc.; RC2). The RC2 averages over an approximate area of few square millimeters. The measurements were performed in transmission mode at approximately the same in plane orientations as shown in Fig. 11. A histogram of each Mueller matrix element from both sample orientations in Fig. 11 is overlaid with the Mueller matrix data obtained from the RC2 measurement and shown in Fig. 12. It is clear from the figure that there is generally good agreement between the RC2 measurement and the Mueller matrix images provided by our RAM-S-RAM-E instrument. It is also apparent that the SCTF sample contains significant lateral variation of film properties.
Fig. 11. Bulk titanium slanted columnar thin film. The titanium columns are oriented towards the bottom of the frame. All images are normalized by image $\mathrm {M}_{11}$, except for image $\mathrm {M}_{11}$.
Fig. 12. Histogram of bulk SCTF sample Mueller matrix images. Dashed lines correspond to the same sample orientations measured on J.A. Woollam Company RC2 ellipsometer.
Surface imperfections such as dust on the substrate surface will have significant impact on the lateral optical properties of SCTF films. Dust and other imperfections on the surface will accumulate material that would otherwise adhere to the substrate, resulting in both a buildup of material at the dust spot as well as a shadowed region behind the spot (relative to the material flux) where there is little to no material growth. This is evident in the images of both the bulk and striped regions of the SCTF on glass sample shown above. These surface imperfections will also impact the performance of anisotropic mirrors based on SCTFs used in this type of instrument, as evidenced by the elliptically periodic features in the Fourier coefficient images for RAM2. It is then clear that proper surface preparation is key to creating SCTF mirrors with ideal properties for use in this type of instrument. More sophisticated calibration methods may also be able to account for such variations in PSIG and PSID optical properties, such as the inclusion of additional higher order harmonic coefficients for the offending optic.
6. Conclusion
It was previously shown that anisotropic mirrors, when rotated about their surface normal, provide sufficient modulation of all Stokes parameters to operate as PSG and PSA in a Mueller matrix ellipsometer. This concept was expanded upon and we have experimentally shown rotating anisotropic mirrors can be used for polarization state generation and analysis in a Mueller matrix imaging microscope. Misalignment of the mirrors resulted in repeatable wobble of the image plane which was digitally corrected. The microscope was calibrated by measuring the instrument’s response to a air, a linear polarizer at different angles, and a linear retarder at different angles. The previously used Fourier expansion approach was extended to include a spatial dependence on each of the coefficients. While the calibration was performed using homogeneous samples, the Mueller matrix imaging capabilities of the instrument were demonstrated using a birefringent resolution target and a spatially varying titanium slanted columnar thin film deposited onto glass. Regression analysis was used to determine the spatially varying retardance and fast axis orientation of the birefringent resolution target. The Mueller matrix images of the slanted columnar thin film were compared with transmission Mueller matrices measured on a commercial ellipsometer and good agreement was observed.
Funding
National Science Foundation (DMR 1808715, OIA-2044049); Air Force Office of Scientific Research (FA9550-18-1-0360, FA9550-19-S-0003, FA9550-21-1-0259); Knut och Alice Wallenbergs Stiftelse.
Acknowledgments
This work was supported in part by the National Science Foundation (NSF) under awards NSF DMR 1808715 and OIA-2044049, by Air Force Office of Scientific Research under awards FA9550-18-1-0360, FA9550-19-S-0003, and FA9550-21-1-0259, and by the Knut and Alice Wallenbergs Foundation supported grant ’Wide-bandgap semiconductors for next generation quantum components’. Mathias Schubert acknowledges the University of Nebraska Foundation and the J. A. Woollam Foundation for financial support.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R. C. Jones, “A new calculus for the treatment of optical systemsi. description and discussion of the calculus,” J. Opt. Soc. Am. 31(7), 488–493 (1941). [CrossRef]
2. H. Mueller, “Memorandum on the polarization optics of the photoelastic shutter,” Report of the OSRD project OEMsr-576 2, Massachusets Institute of Technology (1943).
3. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Transactions Camb. Philos. Soc. 9, 233–258 (1852). [CrossRef]
4. D. H. Goldstein, Polarized Light, Optical Engineering - Marcel Dekker (Taylor & Francis, 2011).
5. A.-H. Liu, P. C. Wayner, and J. L. Plawsky, “Image scanning ellipsometry for measuring nonuniform film thickness profiles,” Appl. Opt. 33(7), 1223 (1994). [CrossRef]
6. G. Jin, R. Jansson, and H. Arwin, “Imaging ellipsometry revisited: Developments for visualization of thin transparent layers on silicon substrates,” Rev. Sci. Instrum. 67(8), 2930–2936 (1996). [CrossRef]
7. A. Albersdörfer, G. Elender, G. Mathe, K. R. Neumaier, P. Paduschek, and E. Sackmann, “High resolution imaging microellipsometry of soft surfaces at 3 Mm lateral and 5 Å normal resolution,” Appl. Phys. Lett. 72(23), 2930–2932 (1998). [CrossRef]
8. D. Tanooka, E. Adachi, and K. Nagayama, “Color-Imaging Ellipsometer: High-Speed Characterization of In-Plane Distribution of Film Thickness at Nano-Scale,” Jpn. J. Appl. Phys. 40(Part 1, No. 2A), 877–880 (2001). [CrossRef]
9. Q. Zhan and J. R. Leger, “Measurement of surface features beyond the diffraction limit with an imaging ellipsometer,” Opt. Lett. 27(10), 821 (2002). [CrossRef]
10. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45(8), 1688 (2006). [CrossRef]
11. N. A. Beaudry, Y. Zhao, and R. Chipman, “Dielectric tensor measurement from a single Mueller matrix image,” J. Opt. Soc. Am. A 24(3), 814 (2007). [CrossRef]
12. C. Chou, H.-K. Teng, C.-J. Yu, and H.-S. Huang, “Polarization modulation imaging ellipsometry for thin film thickness measurement,” Opt. Commun. 273(1), 74–83 (2007). [CrossRef]
13. L. Asinovski, D. Beaglehole, and M. T. Clarkson, “Imaging ellipsometry: Quantitative analysis,” phys. stat. sol. (a) 205, 764–771 (2008). [CrossRef]
14. Y. H. Meng, Y. Y. Chen, C. Qi, L. Liu, and G. Jin, “An automatic imaging spectroscopic ellipsometer for characterization of nano-film pattern on solid substrate,” phys. stat. sol. (c) 5, 1050–1053 (2008). [CrossRef]
15. U. Wurstbauer, C. Röling, U. Wurstbauer, W. Wegscheider, M. Vaupel, P. H. Thiesen, and D. Weiss, “Imaging ellipsometry of graphene,” Appl. Phys. Lett. 97(23), 231901 (2010). [CrossRef]
16. C. Röling, P. Thiesen, A. Meshalkin, E. Achimova, V. Abashkin, A. Prisacar, and G. Triduh, “Imaging ellipsometry mapping of photo-induced refractive index in As2S3 films,” J. Non-Cryst. Solids 365, 93–98 (2013). [CrossRef]
17. X. Chen, W. Du, K. Yuan, J. Chen, H. Jiang, C. Zhang, and S. Liu, “Development of a spectroscopic Mueller matrix imaging ellipsometer for nanostructure metrology,” Rev. Sci. Instrum. 87(5), 053707 (2016). [CrossRef]
18. M. Erman, M. Renaud, and S. Gourrier, “Spatially Resolved Ellipsometry for Semiconductor Process Control: Application to GaInAs MIS Structures,” Jpn. J. Appl. Phys. 26(Part 1, No. 11), 1891–1897 (1987). [CrossRef]
19. A. J. Hurd and C. Jeffrey Brinker, “Ellipsometric Imaging of Drying Sol-Gel Films,” MRS Proc. 121, 731 (1988). [CrossRef]
20. R. Reiter, H. Motschmann, H. Orendi, A. Nemetz, and W. Knoll, “Ellipsometric microscopy. Imaging monomolecular surfactant layers at the air-water interface,” Langmuir 8(7), 1784–1788 (1992). [CrossRef]
21. M. Harke and H. Motschmann, “On the Transition State between the Oil-Water and Air-Water Interface,” Langmuir 14(2), 313–318 (1998). [CrossRef]
22. D. G. Goodall, G. W. Stevens, D. Beaglehole, and M. L. Gee, “Imaging Ellipsometry/Reflectometry for Profiling the Shape of a Deformable Droplet as It Approaches an Interface,” Langmuir 15(13), 4579–4583 (1999). [CrossRef]
23. D. Peev, T. Hofmann, N. Kananizadeh, S. Beeram, E. Rodriguez, S. Wimer, K. B. Rodenhausen, C. M. Herzinger, T. Kasputis, E. Pfaunmiller, A. Nguyen, R. Korlacki, A. Pannier, Y. Li, E. Schubert, D. Hage, and M. Schubert, “Anisotropic contrast optical microscope,” Rev. Sci. Instrum. 87(11), 113701 (2016). [CrossRef]
24. J. L. Pezzaniti and R. A. Chipman, “Phase-only modulation of a twisted nematic liquid-crystal TV by use of the eigenpolarization states,” Opt. Lett. 18(18), 1567–1569 (1993). [CrossRef]
25. J. L. Pezzaniti, S. C. McClain, R. A. Chipman, and S.-Y. Lu, “Depolarization in liquid-crystal televisions,” Opt. Lett. 18(23), 2071 (1993). [CrossRef]
26. J. L. Pezzaniti, “Cascaded polarizing beamsplitter cubes in imaging systems,” Opt. Eng. 33(5), 1543 (1994). [CrossRef]
27. J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Express 34(6), 1558–1568 (1995). [CrossRef]
28. M. H. Smith, E. A. Sornsin, T. J. Tayag, and R. A. Chipman, “Mueller matrix imaging of GaAs/AlGaAs self-imaging beamsplitting waveguides,” in Optical Science, Engineering and Instrumentation ’97, (San Diego, CA, United States, 1997), p. 47.
29. J. Wolfe and R. A. Chipman, “Reducing symmetric polarization aberrations in a lens by annealing,” Opt. Express 12(15), 3443 (2004). [CrossRef]
30. Y. F. Chao and K. Y. Lee, “Index Profile of Radial Gradient Index Lens Measured by Imaging Ellipsometric Technique,” Jpn. J. Appl. Phys. 44(2), 1111–1114 (2005). [CrossRef]
31. D. Mawet, E. Serabyn, K. Liewer, C. Hanot, S. McEldowney, D. Shemo, and N. O’Brien, “Optical Vectorial Vortex Coronagraphs using Liquid Crystal Polymers: Theory, manufacturing and laboratory demonstration,” Opt. Express 17(3), 1902 (2009). [CrossRef]
32. L. M. S. Aas, I. S. Nerbø, M. Kildemo, D. Chiappe, C. Martella, and F. Buatier de Mongeot, “Mueller matrix imaging of plasmonic polarizers on nanopatterned surface,” in SPIE Optical Metrology, P. H. Lehmann, W. Osten, and K. Gastinger, eds. (Munich, Germany, 2011), p. 80822W.
33. C.-J. Yu, Y.-T. Tseng, K.-C. Hsu, and C. Chou, “Two-dimensional cell parameters measurement of a twisted nematic liquid crystal device by using imaging ellipsometer,” in SPIE OPTO, B. Witzigmann, M. Osinski, F. Henneberger, and Y. Arakawa, eds. (San Francisco, California, USA, 2012), p. 82551T.
34. G. López-Morales, M. D. M. Sánchez-López, Á. Lizana, I. Moreno, and J. Campos, “Mueller Matrix Polarimetric Imaging Analysis of Optical Components for the Generation of Cylindrical Vector Beams,” Crystals 10(12), 1155 (2020). [CrossRef]
35. C. Ye, “Photopolarimetric measurement of single, intact pulp fibers by Mueller matrix imaging polarimetry,” Appl. Opt. 38(10), 1975 (1999). [CrossRef]
36. H. Arwin, “Ellipsometry on thin organic layers of biological interest: Characterization and applications,” Thin Solid Films 377-378, 48–56 (2000). [CrossRef]
37. C. Ye, J. Räty, I. Nyblom, H.-K. Hyvärinen, and P. Moss, “Estimation of lignin content in single, intact pulp fibers by UV photometry and VIS Mueller matrix polarimetry,” Nordic Pulp & Paper Res. J. 16(2), 143–148 (2001). [CrossRef]
38. Z. H. Wang and G. Jin, “A Label-Free Multisensing Immunosensor Based on Imaging Ellipsometry,” Anal. Chem. 75(22), 6119–6123 (2003). [CrossRef]
39. M. Bivolarska, T. Velinov, and S. Stoitsova, “Guided-wave and ellipsometric imaging of supported cells,” J. Microsc. 224, 242–248 (2006). [CrossRef]
40. I. Chamritski, M. Clarkson, J. Franklin, and S. W. Li, “Real-Time Detection of Antigen–Antibody Reactions by Imaging Ellipsometry,” Aust. J. Chem. 60(9), 667 (2007). [CrossRef]
41. X. Zhu, J. P. Landry, Y.-S. Sun, J. P. Gregg, K. S. Lam, and X. Guo, “Oblique-incidence reflectivity difference microscope for label-free high-throughput detection of biochemical reactions in a microarray format,” Appl. Opt. 46(10), 1890 (2007). [CrossRef]
42. P. G. Ellingsen, M. B. Lilledahl, L. M. S. Aas, C. d. L. Davies, and M. Kildemo, “Quantitative characterization of articular cartilage using Mueller matrix imaging and multiphoton microscopy,” J. Biomed. Opt. 16(11), 116002 (2011). [CrossRef]
43. P. G. Ellingsen, L. M. S. Aas, V. S. Hagen, R. Kumar, M. B. Lilledahl, and M. Kildemo, “Mueller matrix three-dimensional directional imaging of collagen fibers,” J. Biomed. Opt. 19(2), 026002 (2014). [CrossRef]
44. H. Arwin, R. Magnusson, E. Garcia-Caurel, C. Fallet, K. Järrendahl, M. Foldyna, A. De Martino, and R. Ossikovski, “Sum decomposition of Mueller-matrix images and spectra of beetle cuticles,” Opt. Express 23(3), 1951 (2015). [CrossRef]
45. A. W. Dreher, K. Reiter, and R. N. Weinreb, “Spatially resolved birefringence of the retinal nerve fiber layer assessed with a retinal laser ellipsometer,” Appl. Opt. 31(19), 3730 (1992). [CrossRef]
46. A. H. Hielscher, A. A. Eick, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” in BiOS ’97, Part of Photonics West, T. Vo-Dinh, R. A. Lieberman, G. G. Vurek, and A. Katzir, eds. (San Jose, CA, 1997), pp. 298–305.
47. J. M. Bueno and M. C. W. Campbell, “Confocal scanning laser ophthalmoscopy improvement by use of Mueller-matrix polarimetry,” Opt. Lett. 27(10), 830 (2002). [CrossRef]
48. J. Chung, W. Jung, M. J. Hammer-Wilson, P. Wilder-Smith, and Z. Chen, “Use of polar decomposition for the diagnosis of oral precancer,” Appl. Opt. 46(15), 3038 (2007). [CrossRef]
49. K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express 16(26), 21339 (2008). [CrossRef]
50. M. F. G. Wood, N. Ghosh, S.-H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Polarized light based birefringence measurements for monitoring myocardial regeneration,” in Optics in Tissue Engineering and Regenerative Medicine III, vol. 7179 (International Society for Optics and Photonics, 2009), p. 717908.
51. X. Li and G. Yao, “Mueller matrix decomposition of diffuse reflectance imaging in skeletal muscle,” Appl. Opt. 48(14), 2625 (2009). [CrossRef]
52. M.-R. Antonelli, A. Pierangelo, T. Novikova, P. Validire, A. Benali, B. Gayet, and A. D. Martino, “Mueller matrix imaging of human colon tissue for cancer diagnostics: How Monte Carlo modeling can help in the interpretation of experimental data,” Opt. Express 18(10), 10200–10208 (2010). [CrossRef]
53. A. Pierangelo, A. Benali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Ex-vivo characterization of human colon cancer by Mueller polarimetric imaging,” Opt. Express 19(2), 1582 (2011). [CrossRef]
54. R. Kottaiyan, G. Yoon, Q. Wang, R. Yadav, J. M. Zavislan, and J. V. Aquavella, “Integrated Multimodal Metrology for Objective and Noninvasive Tear Evaluation,” The Ocular Surface 10(1), 43–50 (2012). [CrossRef]
55. Y. A. Ushenko, V. A. Ushenko, A. V. Dubolazov, V. O. Balanetskaya, and N. I. Zabolotna, “Mueller-matrix diagnostics of optical properties of polycrystalline networks of human blood plasma,” Opt. Spectrosc. 112(6), 884–892 (2012). [CrossRef]
56. T. Novikova, A. Pierangelo, A. De Martino, A. Benali, and P. Validire, “Polarimetric Imaging for Cancer Diagnosis and Staging,” Opt. Photonics News 23(10), 26 (2012). [CrossRef]
57. A. Pierangelo, S. Manhas, A. Benali, C. Fallet, J.-L. Totobenazara, M. R. Antonelli, T. Novikova, B. Gayet, A. D. Martino, and P. Validire, “Multispectral Mueller polarimetric imaging detecting residual cancer and cancer regression after neoadjuvant treatment for colorectal carcinomas,” J. Biomed. Opt. 18(4), 046014 (2013). [CrossRef]
58. J. Qi, M. Ye, M. Singh, N. T. Clancy, and D. S. Elson, “Narrow band 3 × 3 Mueller polarimetric endoscopy,” Biomed. Opt. Express 4(11), 2433 (2013). [CrossRef]
59. Y. Wang, H. He, J. Chang, C. He, S. Liu, M. Li, N. Zeng, J. Wu, and H. Ma, “Mueller matrix microscope: A quantitative tool to facilitate detections and fibrosis scorings of liver cirrhosis and cancer tissues,” J. Biomed. Opt. 21(7), 071112 (2016). [CrossRef]
60. A. V. Dubolazov, N. V. Pashkovskaya, Y. A. Ushenko, Y. F. Marchuk, V. A. Ushenko, and O. Y. Novakovskaya, “Birefringence images of polycrystalline films of human urine in early diagnostics of kidney pathology,” Appl. Opt. 55(12), B85 (2016). [CrossRef]
61. J. Vizet, J. Rehbinder, S. Deby, S. Roussel, A. Nazac, R. Soufan, C. Genestie, C. Haie-Meder, H. Fernandez, F. Moreau, and A. Pierangelo, “In vivo imaging of uterine cervix with a Mueller polarimetric colposcope,” Sci. Rep. 7(1), 2471 (2017). [CrossRef]
62. Y. Dong, H. He, W. Sheng, J. Wu, and H. Ma, “A quantitative and non-contact technique to characterise microstructural variations of skin tissues during photo-damaging process based on Mueller matrix polarimetry,” Sci. Rep. 7(1), 14702 (2017). [CrossRef]
63. T. Liu, T. Sun, H. He, S. Liu, Y. Dong, J. Wu, and H. Ma, “Comparative study of the imaging contrasts of Mueller matrix derived parameters between transmission and backscattering polarimetry,” Biomed. Opt. Express 9(9), 4413–4428 (2018). [CrossRef]
64. R. F. Cohn, J. W. Wagner, and J. Kruger, “Dynamic imaging microellipsometry: Theory, system design, and feasibility demonstration,” Appl. Opt. 27(22), 4664 (1988). [CrossRef]
65. R. F. Cohn, J. W. Wagner, and J. Kruger, “Dynamic Imaging Microellipsometry: Proof of Concept Test Results,” J. Electrochem. Soc. 135(4), 1033–1034 (1988). [CrossRef]
66. D. Beaglehole, “Performance of a microscopic imaging ellipsometer,” Rev. Sci. Instrum. 59(12), 2557–2559 (1988). [CrossRef]
67. W. Chegal, Y. J. Cho, H. J. Kim, H. M. Cho, Y. W. Lee, and S. H. Kim, “A New Spectral Imaging Ellipsometer for Measuring the Thickness of Patterned Thin Films,” Jpn. J. Appl. Phys. 43(9A), 6475–6476 (2004). [CrossRef]
68. C.-Y. Han and Y.-F. Chao, “Photoelastic modulated imaging ellipsometry by stroboscopic illumination technique,” Rev. Sci. Instrum. 77(2), 023107 (2006). [CrossRef]
69. L. Jin, Y. Wakako, K. Takizawa, and E. Kondoh, “In situ imaging ellipsometer using a LiNbO3 electrooptic crystal,” Thin Solid Films 571, 532–537 (2014). [CrossRef]
70. F. Huemer, M. Jamalieh, F. Bammer, and D. Hönig, “Inline imaging-ellipsometer for printed electronics,” tm - Technisches Messen 83(10), 549–556 (2016). [CrossRef]
71. S. Liu, W. Du, X. Chen, H. Jiang, and C. Zhang, “Mueller matrix imaging ellipsometry for nanostructure metrology,” Opt. Express 23(13), 17316 (2015). [CrossRef]
72. N. A. Beaudry, Y. Zhao, and R. Chipman, “Multi-angle generalized ellipsometry of anisotropic optical structures,” in Optics & Photonics 2005, J. A. Shaw and J. S. Tyo, eds. (San Diego, California, USA, 2005), p. 588808.
73. C.-J. Tang, R.-S. Chang, and C.-Y. Han, “Using imaging ellipsometry to determine angular distribution of ellipsometric parameters without scanning mechanism,” Opt. Lasers Eng. 77, 39–43 (2016). [CrossRef]
74. L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Near infra-red Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films 519(9), 2737–2741 (2011). [CrossRef]
75. O. Arteaga, M. Baldrís, J. Antó, A. Canillas, E. Pascual, and E. Bertran, “Mueller matrix microscope with a dual continuous rotating compensator setup and digital demodulation,” Appl. Opt. 53(10), 2236–2245 (2014). [CrossRef]
76. S. Y. Berezhna, I. V. Berezhnyy, and M. Takashi, “Dynamic photometric imaging polarizer-sample-analyzer polarimeter: Instrument for mapping birefringence and optical rotation,” J. Opt. Soc. Am. A 18(3), 666 (2001). [CrossRef]
77. K. Bhattacharyya, D. I. Serrano-García, and Y. Otani, “Accuracy enhancement of dual rotating mueller matrix imaging polarimeter by diattenuation and retardance error calibration approach,” Opt. Commun. 392, 48–53 (2017). [CrossRef]
78. M. Borovkova, L. Trifonyuk, V. Ushenko, O. Dubolazov, O. Vanchulyak, G. Bodnar, Y. Ushenko, O. Olar, O. Ushenko, M. Sakhnovskiy, A. Bykov, and I. Meglinski, “Mueller-matrix-based polarization imaging and quantitative assessment of optically anisotropic polycrystalline networks,” PLoS One 14(5), e0214494 (2019). [CrossRef]
79. F. Carmagnola, J. M. Sanz, and J. M. Saiz, “Development of a Mueller matrix imaging system for detecting objects embedded in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 146, 199–206 (2014). [CrossRef]
80. J. H. Freudenthal, E. Hollis, and B. Kahr, “Imaging chiroptical artifacts,” Chirality 21, E20–E27 (2009). [CrossRef]
81. J. C. Gladish and D. D. Duncan, “Liquid crystal-based Mueller matrix spectral imaging polarimetry for parameterizing mineral structural organization,” Appl. Opt. 56(3), 626–635 (2017). [CrossRef]
82. J. C. Gladish and D. D. Duncan, “Spectral imaging polarimeter based on liquid crystal technology,” in SPIE BiOS, V. V. Tuchin, D. D. Duncan, K. V. Larin, M. J. Leahy, and R. K. Wang, eds. (San Francisco, California, USA, 2012), p. 82220X.
83. M. W. Kudenov, M. J. Escuti, N. Hagen, E. L. Dereniak, and K. Oka, “Snapshot imaging Mueller matrix polarimeter using polarization gratings,” Opt. Lett. 37(8), 1367 (2012). [CrossRef]
84. B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43(14), 2824 (2004). [CrossRef]
85. J. M. López-Téllez, R. A. Chipman, L. W. Li, S. C. McEldowney, and M. H. Smith, “Broadband extended source imaging Mueller-matrix polarimeter,” Opt. Lett. 44(7), 1544 (2019). [CrossRef]
86. W. Mickols and M. F. Maestre, “Scanning differential polarization microscope: Its use to image linear and circular differential scattering,” Rev. Sci. Instrum. 59(6), 867–872 (1988). [CrossRef]
87. Y. Otani, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33(5), 1604 (1994). [CrossRef]
88. J. Zhou, H. He, Z. Chen, Y. Wang, and H. Ma, “Modulus design multiwavelength polarization microscope for transmission Mueller matrix imaging,” J. Biomed. Opt. 23(1), 1–8 (2018). [CrossRef]
89. H. Fujiwara, Spectroscopic Ellipsometry (John Wiley & Sons, New York, 2007).
90. M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial tio2,” J. Opt. Soc. Am. A 13(4), 875–883 (1996). [CrossRef]
91. T. Hofmann, U. Schade, W. Eberhardt, C. M. Herzinger, P. Esquinazi, and M. Schubert, “Terahertz magnetooptic generalized ellipsometry using synchrotron and black-body radiation,” Rev. Sci. Instrum. 77(6), 063902 (2006). [CrossRef]
92. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 (1980). [CrossRef]
93. M. Erman and J. B. Theeten, “Spatially resolved ellipsometry,” J. Appl. Phys. 60(3), 859–873 (1986). [CrossRef]
94. R. F. Cohn and J. W. Wagner, “Absolute and random error analysis of the dynamic imaging microellipsometry technique,” Appl. Opt. 28(15), 3187–3198 (1989). [CrossRef]
95. R. F. Cohn, “Evaluation of alternative algorithms for dynamic imaging microellipsometry,” Appl. Opt. 29(2), 304 (1990). [CrossRef]
96. R. A. Chipman and J. L. Pezzaniti, “Analysis of Polarizing Optical Systems for Digital Optical Computing with Symmetric Self Electrooptic Devices,” Tech. rep., University of Alabama in Huntsville (1991).
97. J. L. Pezzaniti and R. A. Chipman, “Determining the operating curves of spatial light modulators with imaging polarimetry,” in SPIE’s 1994 International Symposium on Optics, Imaging, and Instrumentation, J. L. Horner, B. Javidi, and S. T. Kowel, eds. (San Diego, CA, 1994), pp. 271–277.
98. D. C. Dayton, B. G. Hoover, and J. D. Gonglewski, “Full-order Mueller matrix polarimeter using liquid-crystal phase retarders and active illumination,” in International Symposium on Remote Sensing, A. Kohnle and J. D. Gonglewski, eds. (Crete, Greece, 2003), p. 40.
99. S. Alali and I. A. Vitkin, “Optimization of rapid Mueller matrix imaging of turbid media using four photoelastic modulators without mechanically moving parts,” Opt. Express 52(10), 103114 (2013). [CrossRef]
100. I. Berezhnyy and A. Dogariu, “Time-resolved Mueller matrix imaging polarimetry,” Opt. Express 12(19), 4635 (2004). [CrossRef]
101. J. M. Bueno, C. J. Cookson, M. L. Kisilak, and M. C. W. Campbell, “Enhancement of confocal microscopy images using Mueller-matrix polarimetry,” J. Microsc. 235, 84–93 (2009). [CrossRef]
102. A. Carnicer, S. Bosch, and B. Javidi, “Mueller matrix polarimetry with 3D integral imaging,” Opt. Express 27(8), 11525–11536 (2019). [CrossRef]
103. M. Foldyna, M. Moreno, P. R. i Cabarrocas, and A. De Martino, “Scattered light measurements on textured crystalline silicon substrates using an angle-resolved Mueller matrix polarimeter,” Appl. Opt. 49(3), 505 (2010). [CrossRef]
104. B. Le Jeune, J.-P. Marie, P.-Y. Gerligand, J. Cariou, and J. Lotrian, “Mueller matrix formalism in imagery: An experimental arrangement for noise reduction,” in SPIE’s 1994 International Symposium on Optics, Imaging, and Instrumentation, D. H. Goldstein and D. B. Chenault, eds. (San Diego, CA, 1994), p. 443.
105. S. Manhas, J. Vizet, S. Deby, J.-C. Vanel, P. Boito, M. Verdier, A. De Martino, and D. Pagnoux, “Demonstration of full 4×4 Mueller polarimetry through an optical fiber for endoscopic applications,” Opt. Express 23(3), 3047 (2015). [CrossRef]
106. M. H. Smith, P. D. Burke, A. Lompado, E. A. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” in BiOS 2000 The International Symposium on Biomedical Optics, T. Vo-Dinh, W. S. Grundfest, and D. A. Benaron, eds. (San Jose, CA, 2000), pp. 210–216.
107. A. Ruder, B. Wright, D. Peev, R. Feder, U. Kilic, M. Hilfiker, E. Schubert, C. M. Herzinger, and M. Schubert, “Mueller matrix ellipsometer using dual continuously rotating anisotropic mirrors,” Opt. Lett. 45(13), 3541–3544 (2020). [CrossRef]
108. D. Schmidt, A. C. Kjerstad, T. Hofmann, R. Skomski, E. Schubert, and M. Schubert, “Optical, structural, and magnetic properties of cobalt nanostructure thin films,” J. Appl. Phys. 105(11), 113508 (2009). [CrossRef]
109. T. Hofmann, D. Schmidt, A. Boosalis, P. Kühne, C. Herzinger, J. Woollam, E. Schubert, and M. Schubert, “Metal slanted columnar thin film THz optical sensors,” MRS Proc. 1409, mrsf11-1409-cc13-31 (2012). [CrossRef]
110. D. Schmidt, E. Schubert, and M. Schubert, “Optical properties of cobalt slanted columnar thin films passivated by atomic layer deposition,” Appl. Phys. Lett. 100(1), 011912 (2012). [CrossRef]
111. T. Kasputis, M. Koenig, D. Schmidt, D. Sekora, K. B. Rodenhausen, K.-J. Eichhorn, P. Uhlmann, E. Schubert, A. K. Pannier, M. Schubert, and M. Stamm, “Slanted Columnar Thin Films Prepared by Glancing Angle Deposition Functionalized with Polyacrylic Acid Polymer Brushes,” J. Phys. Chem. C 117(27), 13971–13980 (2013). [CrossRef]
112. D. Schmidt and M. Schubert, “Anisotropic Bruggeman effective medium approaches for slanted columnar thin films,” J. Appl. Phys. 114(8), 083510 (2013). [CrossRef]
113. D. E. Aspnes and A. A. Studna, “High Precision Scanning Ellipsometer,” Appl. Opt. 14(1), 220 (1975). [CrossRef]
114. S. Guyot, M. Anastasiadou, E. Deléchelle, and A. De Martino, “Registration scheme suitable to Mueller matrix imaging for biomedical applications,” Opt. Express 15(12), 7393 (2007). [CrossRef]
115. E. De Castro and C. Morandi, “Registration of translated and rotated images using finite fourier transforms,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9(5), 700–703 (1987). [CrossRef]
