Abstract

In this work, we describe the design and implementation of a Mueller matrix imaging polarimeter that uses a polarization camera as a detector. This camera simultaneously measures the first three Stokes components, allowing for the top three rows of the Mueller matrix to be determined after only N = 4 measurements using a single rotating compensator, which is sufficient to fully characterize nondepolarizing samples. This setup provides the polarimetric analysis with micrometric resolution in about 3 seconds and can also perform live birefringence imaging at the camera frame rate by fixing the compensator at a static 45° angle. To further improve the conditioning of the setup, we also give the first experimental demonstration of an optimal elliptical retarder design.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Polarization cameras with sensor-integrated polarizing filters are an easy and cost-effective method to acquire polarization information. The micro-polarizer array is comprised of blocks of four different angled linear polarizers (90$^{\circ }$, 45$^{\circ }$, 135$^{\circ }$ and 0$^{\circ }$) repeating across the camera sensor. Every block of four pixels thus makes up a superpixel that has information about the direction and degree of linear polarization. By combining the intensities measured by each one of the 4 pixels contained in every superpixel, the camera can directly measure the first [$S_0=(I_{0^{\circ }}+I_{90^{\circ }})/2=(I_{45^{\circ }}+I_{135^{\circ }})/2=(I_{0^{\circ }}+I_{90^{\circ }}+I_{45^{\circ }}+I_{135^{\circ }})/4$], second [$S_1=(I_{0^{\circ }}-I_{90^{\circ }})/2$] and third [$S_2=(I_{45^{\circ }}-I_{135^{\circ }})/2$] components of the Stokes vector. The fourth component ($S_3$) cannot be obtained because the camera does not include any compensating optical element to quantify the ellipticity of the incoming radiation.

Although the integration of a micro-polarizer layer above the photodiodes of camera sensors was proposed and implemented several years ago [1,2], polarization cameras have only recently become widely commercially available [3]. For certain applications, polarization sensors can be very advantageous as they permit quick polarimetric imaging of moving objects, since all four directions of polarization are acquired simultaneously. Polarimetric imaging methods have applications in material science [46], studying biological systems [710], and also have potential as medical diagnostic tools [1114].

The Mueller matrix (MM) is the most general form to represent the polarization response of a sample or optical system, and it connects an input and an output Stokes vector. The real valued $4\times 4$ MM not only represents nondepolarizing systems, but also depolarizing ones (i.e. those that produce partially polarized outgoing light from a totally polarized incident one). For a non-depolarizing sample, it is always possible to recover the full $4\times 4$ MM given only 12 elements [15].

When MM polarimetric imaging is performed with microscopic spatial resolution the technique is usually referred as MM microscopy which is a widefield microscopy technique, and can be regarded as a generalization of the usual white light polarization microscope which is routinely used for qualitative imaging, where the sample is placed between crossed polarizers. MM microscopes allow for an exhaustive and quantitative analysis of the polarimetric response of the specimen down to the individual pixel level. However, one usual drawback of MM microscopes is that they typically require several tens of seconds (or even minutes) to measure the $N\geq 16$ images required to obtain the 16-elements of the MM with a sufficient signal-to-noise ratio. This limits their application to static or very slowly evolving samples.

In this work, we describe the implementation of an incomplete MM microscope imaging system based on a polarization camera and a single rotating compensator with an optimal retardance of $132^{\circ }$, capable of directly measuring the top three rows of the MM. Other MM polarimetry setups using polarization sensitive cameras have also recently been proposed [16], but these rely on a division of focal plane technique to acquire data. Comparatively, our setup is simpler to implement requiring only a single camera, has a faster acquisition time, and a lower standard deviation between measurements. The main advantage of our system compared with similar microscopes equipped with a conventional camera is that only $N=4$ measurements are required to measure the top three rows of the MM, with each measurement consisting of a single image taken at a different angular position for the compensator. Compared to the $N=12$ measurements required for a traditional polarimeter to measure the top three rows of the MM, this dramatically reduces the measurement time while still ensuring a complete and highly-sensitive characterization of a non-depolarizing sample. This setup also allows for real time birefringence imaging of transparent samples while keeping the compensator static.

2. Theoretical description

The setup consists of an unpolarized narrow-band light source which passes through a static polarizer and a rotating compensator before illuminating the sample. After transmission through the sample, the light is imaged directly onto the camera sensor, without the need for a second rotating compensator. This system is easily described with Stokes-Mueller calculus as

$$\mathbf{S}_{\textrm{out}}=\mathbf{M}_{S}\mathbf{M}_{C}\mathbf{M}_P\mathbf{S}_{\textrm{in}},$$
where $\mathbf {S}_{\textrm {out}}$ and $\mathbf {S}_{\textrm {in}}$ are the Stokes vectors measured by the camera and Stokes vector of the light source, respectively, $\mathbf {M}_{S}$ and $\mathbf {M}_P$ are the MM of the sample and the MM of a polarizer, respectively, and $\mathbf {M}_{C}$ is the MM of a rotating compensator given by
$$\mathbf{M}_{C}(\theta,\delta)=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & C_{2\theta}^{2}+S_{2\theta}^{2}C_{\delta} & C_{2\theta}S_{2\theta}(1-C_{\delta}) & -S_{2\theta}S_{\delta} \\ 0 & C_{2\theta}S_{2\theta}(1-C_{\delta}) & C_{2\theta}^{2}+S_{2\theta}^{2}C_{\delta} & C_{2\theta}S_{\delta} \\ 0 & S_{2\theta}S_{\delta} & -C_{2\theta}S_{\delta} & C_{\delta}\end{bmatrix},\quad$$
where we have employed the short notation
$$S_{X}\equiv \sin(X),$$
$$C_{X} \equiv \cos(X).$$

The stokes vector measured by the camera ($\mathbf {S}_{\textrm {out}}$) is therefore given by the result of this vector-matrix multiplication (assuming an input Stokes vector with unitary intensity) as

$$\mathbf{S}_{\textrm{out}}=\begin{pmatrix} m_{00}+m_{01}(C_{2\theta}^{2}+C_{\delta}S_{2\theta}^{2})+m_{03}S_{2\theta}S_{\delta}-m_{02}C_{2\theta}S_{2\theta}(C_{\delta}-1) \\ m_{10}+m_{11}(C_{2\theta}^{2}+C_{\delta}S_{2\theta}^{2})+m_{13}S_{2\theta}S_{\delta}-m_{12}C_{2\theta}S_{2\theta}(C_{\delta}-1) \\ m_{20}+m_{21}(C_{2\theta}^{2}+C_{\delta}S_{2\theta}^{2})+m_{23}S_{2\theta}S_{\delta}-m_{22}C_{2\theta}S_{2\theta}(C_{\delta}-1) \\ m_{30}+m_{31}(C_{2\theta}^{2}+C_{\delta}S_{2\theta}^{2})+m_{33}S_{2\theta}S_{\delta}-m_{32}C_{2\theta}S_{2\theta}(C_{\delta}-1) \end{pmatrix}.$$

While a conventional camera only measures the first Stokes component which only contains information on the top row of the MM, the polarization camera can measure the first three components of the Stokes vector which contains information about the first three rows of the MM. Since the fourth Stokes component cannot be measured by the camera, the last row of the MM of the sample can not be obtained directly.

Equation (4) can therefore be rewritten for our system as the scalar product of two vectors

$$S_{\textrm{out},i}=\mathbf{\begin{bmatrix}1 \\ (C_{2\theta}^{2}+C_{\delta}S_{2\theta}^{2})\\ -C_{2\theta}S_{2\theta}(C_{\delta}-1) \\ S_{2\theta}S_{\delta}\end{bmatrix}}^\mathrm{T} \begin{bmatrix} m_{i0} \\ m_{i1} \\ m_{i2} \\ m_{i3} \end{bmatrix} \quad \textrm{with} \; i = 0, 1, 2.$$

As the measurement process consists of taking a collection of intensity measurement for $N$ different positions of the compensator, Eq. (5) can be rewritten as

$$\mathbf{S}_{\textrm{out},i}=\mathbf{B}^{T}\mathbf{A}_i,$$
where $\mathbf {S}_{\textrm {out},i}$ is a vector of N elements, $\mathbf {A}_i$ is 4-component vector containing the i-th row of the MM and $\mathbf {B}$ is a matrix with dimension $N\times 4$. If 4 intensity measurements for $S_{\textrm {out},i}$ are made each at different angles of the compensator ($N=4$), $\mathbf {B}$ is a square matrix and by inverting it, it is possible to solve for $\mathbf {A}$. However, 4 measurements is only the minimum number possible and, when time is not a limiting factor, in polarimetry it is often useful to increase the number of acquisitions to $N>4$ to overspecify the calculation and reduce noise. In this case the Moore–Penrose pseudo inverse of $\mathbf {B}$ can be used to extract $\mathbf {A}_i$ as
$$\mathbf{A}_i=(\mathbf{B}\mathbf{B}^{T})^{{-}1}\mathbf{B}\mathbf{S}_{\textrm{out},i},$$
where $\mathbf {B}\mathbf {B}^{T}$ is a matrix of dimension $4\times 4$. By applying this equation to the first three components of the Stokes vector ($i=0,1,2$) one obtains 12 elements (the first three rows) of the sample’s MM.

There are two parameters to control how well-conditioned the calculation is in Eq. (7): the angles chosen for rotating the compensator, and its retardance. Ill-conditioned situations where $\mathbf {B}\mathbf {B}^{T}$ is close to a singular matrix must be avoided to obtain good stability and noise rejection. While good conditioning can be achieved with a continuously rotating compensator [17], a motor capable of turning to discrete angles allows for more freedom in optimizing the setup. The question of finding the optimal angles for a given number of measurements and the optimal compensator retardance has been analyzed in detail in several works [1821], with the angles $\theta = [\pm 15.1, \pm 51.7]$ being optimal for $N=4$ measurements, and $\delta =132^{\circ }$ retardances being optimal for any number of measurements. Setups optimized for $N>4$ measurements have also been previously implemented using computational optimization techniques [22].

With the addition of a polarization camera, this setup has two unique imaging modalities: the first uses a rotating compensator with an optimal retardance of $\delta \sim 132^{\circ }$ turning to discrete angles, and the second uses a static $\delta \sim 90^{\circ }$ retarder capable of live imaging a sample where linear birefringence is the only optical effect. In the next subsections we discuss both imaging modes.

2.1 Optimal MM imaging with a rotating $\delta =132^{\circ }$ compensator

While the optimal retardance for a rotating compensator polarimeter is $\delta \sim 132^{\circ }$, film compensators most appropriate for imaging are typically only available at non-optimal retardances of $\delta \sim 90^{\circ }$ or $180^{\circ }$ which reduces the noise tolerance of the setup. To improve this, our setup uses an elliptical retarder with a linear retardance of $132^{\circ }$, formed by placing two retarders with a constant angle offset between them [23]. This modifies Eq. (1) as

$$\mathbf{S}_{\textrm{out}}=\mathbf{M}_{S}\mathbf{M}_{C}\mathbf{R}(-\phi)\mathbf{M}_{C}\mathbf{R}(\phi)\mathbf{M}_P\mathbf{S}_{\textrm{in}},$$
where $\mathbf {R}(\phi )$ is the rotation matrix at an angle $\phi$, which is the constant offset between the two compensators. This equation can be simplified by making the substitution
$$\mathbf{M}_{C}\mathbf{R}(-\phi)\mathbf{M}_{C}\mathbf{R}(\phi) = \mathbf{R}(\rho)\mathbf{M}_{LR}' = \textbf{M}_E,$$
where $\mathbf {R}(\rho )$ represents a circular retarder and $\mathbf {M}_{LR}'$ is a linear retarder, together forming the elliptical retarder $\textbf {M}_E$. Inserting this into Eq. (8) we get
$$\mathbf{S}_{\textrm{out}}=\mathbf{M}_{S}\mathbf{R}(\rho)\mathbf{M}_{LR}'\mathbf{M}_P\mathbf{S}_{\textrm{in}}.$$

By inspection, this is the same system as in Eq. (1) with just a rotation matrix added, so the angle $\phi$ can be adjusted such that the setup achieves optimal performance when the linear retardance of $\textbf {M}_{LR}'$ is $132^{\circ }$. However, the axial position of $\mathbf {M}_{LR}'$ is unknown, so an offset $\Delta$ needs to be added to the commanded angles of the compensator. To acquire data with $N=4$ measurements, the compensator is therefore rotated to the angles $\theta = [\pm 15.1, \pm 51.7]+\Delta$. The process of finding the required values of $\phi$ and $\Delta$ is given in [23]. For $N>4$ measurements, the optimal angles can be found in previous studies (such as [20]), with the offset $\Delta$ needing to be added.

The importance of this analysis and optimizing the retardance is best explained by using the efficiency of measurement given by [24]

$$\frac{\delta I'}{\delta m_i} = \left(N \sum_{k=1}^{N} \textbf{r}_{ik}^2\right)^{-\frac{1}{2}} \equiv E_i,$$
where $N$ is the number of measurements, $\delta I$ is the noise in the intensity of the measurement and $\delta I' = N^{-1/2}\delta I$, $\delta m_i$ is the error, $\textbf {r} = (\textbf {BB}^T)^{-1}\textbf {B}$, and $E_i$ is the efficiency of measurement. To improve the noise tolerance of the setup, we would like to maximize this value. The efficiency of measurement applied to our system in Eq. (5) is given in Fig. 1. Since each row of the measured MM is given by the same basis vector, all rows will have equal efficiency. It is seen that while the overall ideal measurement with acceptable efficiency between all elements is achieved using an elliptical retarder with an equivalent linear retardance of $\delta = 132^{\circ }$, the single quarter-wave plate (QWP) linear retarder will actually perform better in the last element, as discussed in the next section.

 figure: Fig. 1.

Fig. 1. The calculated efficiency for measuring each element across a row of the MM. Higher values correspond to better conditioned measurements with better noise tolerances in the corresponding column of the MM.

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2.2 Live birefringence imaging with a static $\delta =90^{\circ }$ compensator

The birefringence of a transparent sample can be obtained from elements $m_{13}$ and $m_{23}$, which are in the last column of the MM. The efficiency of measurement calculations in Fig. 1 shows that a $90^{\circ }$ retarder is optimal to measure these elements. Combining these results with the polarization sensitive camera, our setup has the benefit of being able to measure the birefringence of a transparent sample efficiently, and without any moving parts. To demonstrate this, we consider the case of a single QWP retarder $\delta =90^{\circ }$ ($C_{\delta }=0$ and $S_{\delta }=1$) oriented at a fixed angle $\theta =45^{\circ }$ ($C_{2\theta }=0$ and $S_{2\theta }=1$). Equation (4) then simplifies to

$$\mathbf{S}_{\textrm{out}}=\begin{pmatrix} m_{00}+m_{03} \\ m_{10}+m_{13} \\ m_{20}+m_{23} \\ m_{30}+m_{33} \end{pmatrix}.$$

In a transparent sample measured at normal incidence transmission $m_{10}=m_{20}=m_{30}=m_{03}=0$, so the Stokes vector will simply be $\mathbf {S}_{\textrm {out}}^T=(m_{00},m_{13},m_{23},m_{33})$. If the sample is described by a general linear retarder MM, given in Eq (2), then $m_{13} = S_{2\theta }S_\delta$ and $m_{23} = -C_{2\theta }S_\delta$ allowing for the orientation ($\theta$) and retardance ($\delta$) to be solved from $\textbf {S}_{\textrm {out}}$ as

$$\theta=\frac{1}{2}\mathrm{atan2}({-}S_{\textrm{out},1},S_{\textrm{out},2}),$$
$$\delta=\mathrm{asin}\left(\frac{\sqrt{S_{\textrm{out},1}^2+S_{\textrm{out},2}^2}}{S_{\textrm{out},0}}\right),$$
where $\mathrm {atan2}$ stands for the four-quadrant inverse tangent and $\delta$ can take values between 0 and $\pi /2$ radians.

This shows that a linear retarder can be fully characterized with a single measurement, opening the door for real-time birefringence imaging at the frame rate offered by the polarization camera (up to $\sim$ 75 fps with our camera). The case of a sample represented as a linear retarder MM is a rather specific situation but, experimentally, it is of particular relevance in many samples of interest in research or industry (crystals, polymers, liquid crystals, strained glasses, some biological structures, etc) where linear birefringence is the unique optical effect.

3. Instrument description

A photo of the setup, as well as a schematic detailing the system optics can be seen in Fig. 2. The polarization state generator between the light source and the sample is composed of two elements: a polarizer (high-contrast linear polarizing film, Edmund optics) and an achromatic compensator composed of either one or two wave plates (QWP polymer retarder film, Edmund Optics). To control the angle between the two retarders, the second retarder is fixed to a rotation mount (Thorlabs CRM1) connected by cage rods to the first retarder, allowing it to be easily rotated between measurements and removed when live birefringence measurements should be taken. We use film elements because they offer a large aperture, small thickness, and no measurable change in retardance across the aperture of the film.

 figure: Fig. 2.

Fig. 2. Photo (left) and schematic (right) of the MM microscope. The working principle of the polarization camera is shown with the schematic with the layout of the polarizers in each super pixel.

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The compensator is secured on a rotation mount with a resonant Piezolectric motor (Thorlabs ELL14) capable of discrete angle positioning. The sample is placed right after the compensator and long working distance infinite-corrected objectives (Edmund Optics 10X 0.28NA for low magnification, Zeiss 25X 0.5 NA for high magnification) are used for imaging the sample onto the camera (Flir Blackfly BFS-U3-51S5P-C with polarization sensor Sony IMX250MZR). A Raynox DCR-250 lens is used as tube lens. The sensor of the camera is $2448\times 2048$ but each $2\times 2$ set of pixels forms a “superpixel” containing the polarization information, as seen in Fig. 2. Therefore the effective maximum resolution of the $S_{\textrm {out},i}$ Stokes vector components is $1224\times 1024$.

Thanks to the use of a polarization camera, our system has no additional polarizing optical elements between the objective and the camera. This is an especially favorable configuration as optical elements placed in this region can often deteriorate the image quality. For example, many polarimeters employ a second rotating compensator after the objective, but this leads to both mechanical vibrations and unavoidable beam deflections that depend on the rotation and are difficult to compensate [25].

We operate this instrument with two different light sources depending on the application: a white LED light source (Edmund Optics MicroBrite Spot light) combined with a bandpass interference filter when high speed imaging is needed, or a halogen lamp coupled to a monochromator (Horiba iHR320) when it is necessary to have a better spectral resolution. The whole instrument is mounted vertically, following the usual design of transmission microscopes, and takes a very small footprint.

4. Calibration

In rotating compensator polarimeters, the calibration typically consist of determining the retardation of the compensator ($\delta$) and the offset (if any) of its fast axis with respect to the "zero" position of the motors. Both parameters are usually fairly well known following specifications provided by the manufacturer, but the calibration permits a more careful determination. While calibrating, the sample is either removed or replaced with an isotropic sample (such as a clean microscope slide), which reduces Eq. (4) to

$$\mathbf{S}_{\textrm{out}}=\begin{pmatrix} 1 \\ C_{2\theta}^{2}+C_{\delta}S_{2\theta}^{2} \\ -C_{2\theta}S_{2\theta}(C_{\delta}-1) \\ S_{2\theta}S_{\delta} \end{pmatrix}.$$

To calibrate the system we consider the effect errors in the retardation and orientation of the compensator which can respectively presented by $\theta =\theta '+\Delta _{\theta }$ and $\delta =\delta '+\Delta _{\delta }$, where $\Delta _{\theta }$ and $\Delta _{\delta }$ are the errors (assumed to be small) that we have in our data acquisition and that deviate from the real values $\theta '$ and $\delta '$. To first order, the trigonometric functions in Eq. (14) can be written as $S_i\simeq S_{i'}+\Delta _{i}C_{i'}$ and $C_i\simeq C_{i'}-\Delta _{i}S_{i'}$ and the Stokes vector is expanded as

$$\mathbf{S}_{\textrm{out}}=\begin{pmatrix} 1 \\ C_{2\theta'}^{2}+C_{\delta'}S_{2\theta'}^{2} \\ -C_{2\theta'}S_{2\theta'}(C_{\delta'}-1) \\ S_{2\theta'}S_{\delta'} \end{pmatrix}+\begin{pmatrix} 0 \\ -\Delta_{\delta}S_{\delta'}+\Delta_{\delta}S_{\delta'}C_{2\theta'}^2+2\Delta_{2\theta}C_{2\theta'}S_{2\theta'}(C_{\delta'}-1) \\ -\Delta_{2\theta}(C_{\delta'}+1)+2\Delta_{2\theta}(C_{2\theta'}^{2}+C_{\delta'}S_{2\theta'}^{2})+\Delta_{\delta}C_{\delta}C_{2\theta'}S_{2\theta'} \\ \Delta_{2\theta}S_{\delta'}C_{2\theta'}+ \Delta_{\delta}S_{2\theta'}C_{\delta'} \end{pmatrix}.$$
This expression omits all terms that are second order with errors. A direct comparison of the second term of Eq. (15) with Eq. (4) shows that, in the calibration conditions the measurement of certain nonvanishing MM errors is directly the error, in particular:
  • $m_{10}\simeq -\Delta _{\delta }S_{\delta '}\simeq -\Delta _{\delta }$
  • $m_{12}\simeq -2\Delta _{\theta }$
  • $m_{20}\simeq -\Delta _{2\theta }(C_{\delta '}+1)\simeq -\Delta _{2\theta }$
  • $m_{21}\simeq 2\Delta _{\theta }$

Therefore the calibration can be performed by adjusting $\theta$ and $\delta$ until the MM elements $m_{10}$, $m_{12}$, $m_{20}$, $m_{21}$ are made to vanish. While the above calculations were done for a system with only a single linear compensator, when a second one is added, the angle $\phi$ is also adjusted to make the same four terms vanish. The analysis is the same since errors in $\phi$ change both the azimuth angle $\theta$ and linear retardance $\delta$ of the elliptical retarder.

We can also evaluate how well the angled polarizing filters on the camera sensor perform by using the following metric

$$\beta=\frac{I_{0^{{\circ}}}+I_{90^{{\circ}}}-(I_{45^{{\circ}}}+I_{135^{{\circ}}})}{I_{0^{{\circ}}}+I_{90^{{\circ}}}+I_{45^{{\circ}}}+I_{135^{{\circ}}}}$$
which is zero ($\beta =0$) when there is perfect orthogonality and equal performance between the $0^{\circ }$ and $90^{\circ }$ polarizing filters as well as between the $45^{\circ }$ and $135^{\circ }$. This metric has the advantage that it should not depend on the polarization or the intensity of the incoming radiation.

When the microscope was first completed, $\beta$ initially had a value up to $\approx 0.03$ for some incoming polarizations. This was likely caused by the $45^{\circ }$ and $135^{\circ }$ polarizers performing slightly worse because of their orientations diagonal to the edges of the pixels, rather than parallel. To compensate for this, we subtracted 1.5% of the maximum value from the diagonal polarization elements when using 610 nm light, and 2% of the maximum when using 550 nm light. This made $\beta <0.01$ for any polarization state of the incident light, and had the effect of making the terms along the diagonal of the measured MM equal during calibration, while before the $\beta$ correction the setup consistently had $m_{22} > m_{11}$.

5. Mueller matrix mode

A measurement of the MM of a thin polycrystalline sample containing spherulitic bundles of twisted D-mannitol lamellae taken at 610 nm can be seen in Fig. 3. The sample was grown from the supercooled melt in the presence of the dye Chicago sky blue. Because of the dye’s light absorbing molecules, the crystalline lamella forming the spherulite not only display the strong birefringent character typical from crystalline lamella, but also anisotropic absorption or dichroism (here evidenced by the non vanishing values of $m_{01}\simeq m_{10}$ and $m_{02}\simeq m_{20}$). Details about the sample preparation and the properties of this particular sperulitic material are given in [26].

 figure: Fig. 3.

Fig. 3. Measurement of a spherulite sample taken with $N=4$ measurements using two retarders with a retardance of 81$^{\circ }$ and relative offset $\phi = 21.9^{\circ }$, turning to $\theta = [-48.4^{\circ }, -18.2^{\circ }, 18.4^{\circ }, -85^{\circ }]$.

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Since 610 nm is above the design wavelength of the QWP retarders used, they have a reduced retardance of approximately $81^{\circ }$, so the elliptical retarder was formed by placing them at a relative angle $\phi = 21.9^{\circ }$ to each other. A total of $N=4$ images were taken with the retarders placed at the angles $\theta = [-15.1^{\circ }, 15.1^{\circ }, 51.7^{\circ }, -51.7^{\circ }] + \Delta = [-48.4^{\circ }, -18.2^{\circ }, 18.4^{\circ }, -85^{\circ }]$ where $\Delta = 33.3^{\circ }$. The total measurement time is only $\sim 3s$, which is significantly faster than normal systems requiring $N=16$ measurements (or $N=12$ in cases of systems not measuring the last row of the MM). Moreover, this acquisition speed is only limited by the turning rate of the motor, and could be improved through physical changes to the setup such as placing pre-angled compensators inside a quick filter wheel mount.

The standard deviation of the measurement is shown in Fig. 4 with 10 complete measurements of the MM taken with both (a) the elliptical retarder, and (b) a single 81$^{\circ }$ retarder. The top row of both results has a lower standard deviation than the other rows by a factor of $\sqrt {2}$ since all four pixels in the superpixel are used for light collection, while the remaining two rows only use two pixels. The elliptical retarder performs better for the first three columns, while a single retarder has a lower standard deviation for the last column. These results match with the expected efficiency of measurement for each row given by Fig. 1.

 figure: Fig. 4.

Fig. 4. Measured standard deviation after 10 measurements of the same spherulite sample. (a) Using a dual QWP elliptical retarder with $\phi = 21.9^{\circ }$. (b) Using a single QWP compensator.

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Another capability of our system is seen in Fig. 5, which is an image of potato starch grains under similar measurement conditions as in the previous example. The sample was prepared by rubbing a slice of potato on a microscope slide and adding a drop of index-matching oil before covering it with a glass coverslip. The small starch grains, once immersed in oil, are approximately non-depolarizing which allows the bottom row on the Muller matrix to be found analytically through post-processing [15,27]. The typical symmetries of linearly birefringent samples ($m_{32} = -m_{23}$ and $m_{31} = -m_{13}$) are shown and the Maltese cross like patterns observed in these elements are consequence of the structure of the starch grains: the grain has a concentric structure of alternating amorphous lamellae made of amylose and crystalline lamealle made of amylopectin both growing radially from the center of grain. This example helps to demonstrate the micrometric spatial resolution achievable by our setup.

 figure: Fig. 5.

Fig. 5. Measured MM of potato starch showing Maltese cross pattern in each grain. Since the sample is approximately non-depolarizing, the last row of the MM is can be calculated with an algebraic method based on the top three rows [15].

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6. Live birefringence imaging

Imaging of transparent birefringent specimens is one of the most popular applications in polarimetry. For microscopic imaging systems, the PolScope implementation is one of the most well-known solutions to measure both the birefringence magnitude and orientation [28,29]. The optical design of the PolScope builds on the traditional polarized light microscope introducing two essential modifications: the specimen is illuminated with nearly circularly polarized light, and the traditional compensator is replaced by an electro-optic universal compensator based on two liquid crystal devices and a linear polarizer.

The live birefringence imaging mode of our microscope offers similar possibilities to the PolScope but in a simpler optical layout since, as described in section 2.2, it only requires a static QWP and the polarization camera. As shown in Eq. (13), the measurement of the retardation $\delta$ and the azimuth angle $\theta$ relies on the determination of MM elements $m_{13}$ and $m_{23}$, which according to Fig. 1 is optimal when a single QWP is used.

To demonstrate this capability of our setup, we imaged the growth of a thin resorcinol polycrystalline sample growing from the melt on a glass slide at 550 nm, where the compensator had a retardance $\delta \simeq 90^{\circ }$. Figure 6 shows several frames at 100ms increments, showing both the magnitude and the direction of the crystals birefringence. It is seen that the crystal orientation is maintained along the length of the growth direction of the crystal, which would be expected. Visualization 1 shows these results as an animation.

 figure: Fig. 6.

Fig. 6. Frames from live imaging of resorcinol crystal growth showing both the magnitude in radians (top) and direction in degrees (bottom) of birefringence (see Visualization 1). The images are shown at 100ms intervals, although a faster frame rates are possible.

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Another example of live birefringence imaging is available in Visualization 2, where we directly measured stretching elastic poly(dimethylsiloxane) (PDMS) substrates suspended in air. In this rubber-like material the elastic deformation induces birefringence, a phenomenon that is usually known as photoelasticity. In this example, the magnitude of the linear birefringence is seen to increase up to values around 0.8 rad and become aligned as the sample is stretched. When the sample is slightly unstretched, it begins to return to its original state, and the magnitude of the birefringence again decreases.

Although in these examples the images are taken every $\sim 100$ ms, our camera is capable of live imaging up to 75 frames per second, barring computational and illumination limitations.

To demonstrate the sensitivity of the live birefringence imaging modality, we took a single measurement of onion peel epidermal cells (Fig. 7). The magnitude of the linear birefringence is seen to be very small inside the cells ($\approx$4 nm), but the setup is still able to accurately determine a direction for the birefringence. This shows that the live birefringence measuring is appropriate even for thin biological samples with low magnitudes of $m_{13}$ and $m_{23}$.

 figure: Fig. 7.

Fig. 7. Live birefringence measurement of a single layer of onion skin showing the intensity image, magnitude of the linear birefringence in radians, and the fast axis orientation in degrees.

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7. Conclusion

We have demonstrated a MM imaging polarimeter that uses a polarization sensitive camera measuring the first three Stokes components to determine the three first rows of the MM of a sample in only $N=4$ measurements. This has the advantage of being much faster than a traditional setup, which requires at least $N=12$ measurements to acquire equivalent information. We were able to analytically solve for the last row of a non-depolarizing sample, and recorded MM micrographs in about 3 seconds. The precision of the measurements was further optimized through the first implementation of an elliptical retarder with the optimal linear retardance for MM polarimetric systems. By placing the retarder in a fixed position, this setup is also capable of live birefringence measurements of transparent samples, potentially opening applications in industry and other fields that require quick measurements of the magnitude and orientation of linear birefringence.

Funding

Ministerio de Ciencia, Innovación y Universidades (RTI2018-098410-J-I00, RYC2018-024997-I) (MCIU/AEI/FEDER, UE).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Visualization 1 and Visualization 2.

References

1. J. Guo and D. Brady, “Fabrication of thin-film micropolarizer arrays for visible imaging polarimetry,” Appl. Opt. 39(10), 1486–1492 (2000). [CrossRef]  

2. G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, “Micropolarizer array for infrared imaging polarimetry,” J. Opt. Soc. Am. A 16(5), 1168–1174 (1999). [CrossRef]  

3. “Sony polarization image sensor,” https://www.sony-semicon.co.jp/e/products/IS/industry/product/polarization.html (2021).

4. J. H. Freudenthal, E. Hollis, and B. Kahr, “Imaging chiroptical artifacts,” Chirality 21(1E), E20–E27 (2009). [CrossRef]  

5. O. Arteaga, S. M. Nichols, and J. Antó, “Back-focal plane Mueller matrix microscopy: Mueller conoscopy and Mueller diffractrometry,” Appl. Surf. Sci. 421, 702–706 (2017). [CrossRef]  

6. A. Le Gratiet, A. Mohebi, F. Callegari, P. Bianchini, and A. Diaspro, “Review on complete mueller matrix optical scanning microscopy imaging,” Appl. Sci. 11(4), 1632 (2021). [CrossRef]  

7. A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021). [CrossRef]  

8. A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019). [CrossRef]  

9. M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014). [CrossRef]  

10. S. Liu, W. Du, X. Chen, H. Jiang, and C. Zhang, “Mueller matrix imaging ellipsometry for nanostructure metrology,” Opt. Express 23(13), 17316–17329 (2015). [CrossRef]  

11. M.-R. Antonelli, A. Pierangelo, T. Novikova, P. Validire, A. Benali, B. Gayet, and A. De 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]  

12. S. Alali and I. A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. Biomed. Opt. 20(6), 061104 (2015). [CrossRef]  

13. Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015). [CrossRef]  

14. J. Ramella-Roman, I. Saytashev, and M. Piccini, “A review of polarization-based imaging technologies for clinical and pre-clinical applications,” J. Opt. 22(12), 123001 (2020). [CrossRef]  

15. R. Ossikovski and O. Arteaga, “Completing an experimental nondepolarizing Mueller matrix whose column or row is missing,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 37(5), 052905 (2019). [CrossRef]  

16. T. Huang, R. Meng, J. Qi, Y. Liu, X. Wang, Y. Chen, R. Liao, and H. Ma, “Fast Mueller matrix microscope based on dual dofp polarimeters,” Opt. Lett. 46(7), 1676–1679 (2021). [CrossRef]  

17. 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]  

18. R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5(5), 681–689 (1988). [CrossRef]  

19. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express 16(15), 11589–11603 (2008). [CrossRef]  

20. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41(4), 619–630 (2002). [CrossRef]  

21. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25(11), 802–804 (2000). [CrossRef]  

22. S. Bian, C. Cui, and O. Arteaga, “Mueller matrix ellipsometer based on discrete-angle rotating Fresnel rhomb compensators,” Appl. Opt. 60(16), 4964–4971 (2021). [CrossRef]  

23. D. Gottlieb and O. Arteaga, “Optimal elliptical retarder in rotating compensator imaging polarimetry,” Opt. Lett. 46(13), 3139–3142 (2021). [CrossRef]  

24. K. Ichimoto, K. Shinoda, T. Yamamoto, and J. Kiyohara, “Photopolarimetric measurement system of Mueller matrix with dual rotating waveplates,” Publications of the National Astronomical Observatory of Japan 9, 11–19 (2006).

25. S. Nichols, “Coherence in polarimetry,” Ph.D. thesis, New York University (2018).

26. X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016). [CrossRef]  

27. C.-Y. Han, C.-Y. Du, and D.-F. Chen, “Evaluation of structural and molecular variation of starch granules during the gelatinization process by using the rapid Mueller matrix imaging polarimetry system,” Opt. Express 26(12), 15851 (2018). [CrossRef]  

28. M. Shribak and R. Oldenbourg, “Techniques for fast and sensitive measurements of two-dimensional birefringence distributions,” Appl. Opt. 42(16), 3009–3017 (2003). [CrossRef]  

29. R. Oldenbourg, “Polarization microscopy with the LC-PolScope,” in Live cell imaging: A laboratory manual (Cold Spring Harbor Laboratory Press, 2005), pp. 205–237.

References

  • View by:

  1. J. Guo and D. Brady, “Fabrication of thin-film micropolarizer arrays for visible imaging polarimetry,” Appl. Opt. 39(10), 1486–1492 (2000).
    [Crossref]
  2. G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, “Micropolarizer array for infrared imaging polarimetry,” J. Opt. Soc. Am. A 16(5), 1168–1174 (1999).
    [Crossref]
  3. “Sony polarization image sensor,” https://www.sony-semicon.co.jp/e/products/IS/industry/product/polarization.html (2021).
  4. J. H. Freudenthal, E. Hollis, and B. Kahr, “Imaging chiroptical artifacts,” Chirality 21(1E), E20–E27 (2009).
    [Crossref]
  5. O. Arteaga, S. M. Nichols, and J. Antó, “Back-focal plane Mueller matrix microscopy: Mueller conoscopy and Mueller diffractrometry,” Appl. Surf. Sci. 421, 702–706 (2017).
    [Crossref]
  6. A. Le Gratiet, A. Mohebi, F. Callegari, P. Bianchini, and A. Diaspro, “Review on complete mueller matrix optical scanning microscopy imaging,” Appl. Sci. 11(4), 1632 (2021).
    [Crossref]
  7. A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
    [Crossref]
  8. A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
    [Crossref]
  9. M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
    [Crossref]
  10. S. Liu, W. Du, X. Chen, H. Jiang, and C. Zhang, “Mueller matrix imaging ellipsometry for nanostructure metrology,” Opt. Express 23(13), 17316–17329 (2015).
    [Crossref]
  11. M.-R. Antonelli, A. Pierangelo, T. Novikova, P. Validire, A. Benali, B. Gayet, and A. De 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]
  12. S. Alali and I. A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. Biomed. Opt. 20(6), 061104 (2015).
    [Crossref]
  13. Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
    [Crossref]
  14. J. Ramella-Roman, I. Saytashev, and M. Piccini, “A review of polarization-based imaging technologies for clinical and pre-clinical applications,” J. Opt. 22(12), 123001 (2020).
    [Crossref]
  15. R. Ossikovski and O. Arteaga, “Completing an experimental nondepolarizing Mueller matrix whose column or row is missing,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 37(5), 052905 (2019).
    [Crossref]
  16. T. Huang, R. Meng, J. Qi, Y. Liu, X. Wang, Y. Chen, R. Liao, and H. Ma, “Fast Mueller matrix microscope based on dual dofp polarimeters,” Opt. Lett. 46(7), 1676–1679 (2021).
    [Crossref]
  17. 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]
  18. R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5(5), 681–689 (1988).
    [Crossref]
  19. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express 16(15), 11589–11603 (2008).
    [Crossref]
  20. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41(4), 619–630 (2002).
    [Crossref]
  21. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25(11), 802–804 (2000).
    [Crossref]
  22. S. Bian, C. Cui, and O. Arteaga, “Mueller matrix ellipsometer based on discrete-angle rotating Fresnel rhomb compensators,” Appl. Opt. 60(16), 4964–4971 (2021).
    [Crossref]
  23. D. Gottlieb and O. Arteaga, “Optimal elliptical retarder in rotating compensator imaging polarimetry,” Opt. Lett. 46(13), 3139–3142 (2021).
    [Crossref]
  24. K. Ichimoto, K. Shinoda, T. Yamamoto, and J. Kiyohara, “Photopolarimetric measurement system of Mueller matrix with dual rotating waveplates,” Publications of the National Astronomical Observatory of Japan 9, 11–19 (2006).
  25. S. Nichols, “Coherence in polarimetry,” Ph.D. thesis, New York University (2018).
  26. X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
    [Crossref]
  27. C.-Y. Han, C.-Y. Du, and D.-F. Chen, “Evaluation of structural and molecular variation of starch granules during the gelatinization process by using the rapid Mueller matrix imaging polarimetry system,” Opt. Express 26(12), 15851 (2018).
    [Crossref]
  28. M. Shribak and R. Oldenbourg, “Techniques for fast and sensitive measurements of two-dimensional birefringence distributions,” Appl. Opt. 42(16), 3009–3017 (2003).
    [Crossref]
  29. R. Oldenbourg, “Polarization microscopy with the LC-PolScope,” in Live cell imaging: A laboratory manual (Cold Spring Harbor Laboratory Press, 2005), pp. 205–237.

2021 (5)

A. Le Gratiet, A. Mohebi, F. Callegari, P. Bianchini, and A. Diaspro, “Review on complete mueller matrix optical scanning microscopy imaging,” Appl. Sci. 11(4), 1632 (2021).
[Crossref]

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

T. Huang, R. Meng, J. Qi, Y. Liu, X. Wang, Y. Chen, R. Liao, and H. Ma, “Fast Mueller matrix microscope based on dual dofp polarimeters,” Opt. Lett. 46(7), 1676–1679 (2021).
[Crossref]

S. Bian, C. Cui, and O. Arteaga, “Mueller matrix ellipsometer based on discrete-angle rotating Fresnel rhomb compensators,” Appl. Opt. 60(16), 4964–4971 (2021).
[Crossref]

D. Gottlieb and O. Arteaga, “Optimal elliptical retarder in rotating compensator imaging polarimetry,” Opt. Lett. 46(13), 3139–3142 (2021).
[Crossref]

2020 (1)

J. Ramella-Roman, I. Saytashev, and M. Piccini, “A review of polarization-based imaging technologies for clinical and pre-clinical applications,” J. Opt. 22(12), 123001 (2020).
[Crossref]

2019 (2)

R. Ossikovski and O. Arteaga, “Completing an experimental nondepolarizing Mueller matrix whose column or row is missing,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 37(5), 052905 (2019).
[Crossref]

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

2018 (1)

2017 (1)

O. Arteaga, S. M. Nichols, and J. Antó, “Back-focal plane Mueller matrix microscopy: Mueller conoscopy and Mueller diffractrometry,” Appl. Surf. Sci. 421, 702–706 (2017).
[Crossref]

2016 (1)

X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
[Crossref]

2015 (3)

S. Liu, W. Du, X. Chen, H. Jiang, and C. Zhang, “Mueller matrix imaging ellipsometry for nanostructure metrology,” Opt. Express 23(13), 17316–17329 (2015).
[Crossref]

S. Alali and I. A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. Biomed. Opt. 20(6), 061104 (2015).
[Crossref]

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref]

2014 (2)

2010 (1)

2009 (1)

J. H. Freudenthal, E. Hollis, and B. Kahr, “Imaging chiroptical artifacts,” Chirality 21(1E), E20–E27 (2009).
[Crossref]

2008 (1)

2006 (1)

K. Ichimoto, K. Shinoda, T. Yamamoto, and J. Kiyohara, “Photopolarimetric measurement system of Mueller matrix with dual rotating waveplates,” Publications of the National Astronomical Observatory of Japan 9, 11–19 (2006).

2003 (1)

2002 (1)

2000 (2)

1999 (1)

1988 (1)

Alali, S.

S. Alali and I. A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. Biomed. Opt. 20(6), 061104 (2015).
[Crossref]

Antó, J.

O. Arteaga, S. M. Nichols, and J. Antó, “Back-focal plane Mueller matrix microscopy: Mueller conoscopy and Mueller diffractrometry,” Appl. Surf. Sci. 421, 702–706 (2017).
[Crossref]

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]

Antonelli, M.-R.

Arteaga, O.

D. Gottlieb and O. Arteaga, “Optimal elliptical retarder in rotating compensator imaging polarimetry,” Opt. Lett. 46(13), 3139–3142 (2021).
[Crossref]

S. Bian, C. Cui, and O. Arteaga, “Mueller matrix ellipsometer based on discrete-angle rotating Fresnel rhomb compensators,” Appl. Opt. 60(16), 4964–4971 (2021).
[Crossref]

R. Ossikovski and O. Arteaga, “Completing an experimental nondepolarizing Mueller matrix whose column or row is missing,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 37(5), 052905 (2019).
[Crossref]

O. Arteaga, S. M. Nichols, and J. Antó, “Back-focal plane Mueller matrix microscopy: Mueller conoscopy and Mueller diffractrometry,” Appl. Surf. Sci. 421, 702–706 (2017).
[Crossref]

X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
[Crossref]

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]

Azzam, R. M. A.

Baldrís, M.

Benali, A.

Bendandi, A.

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

Bertran, E.

Bian, S.

Bianchini, P.

A. Le Gratiet, A. Mohebi, F. Callegari, P. Bianchini, and A. Diaspro, “Review on complete mueller matrix optical scanning microscopy imaging,” Appl. Sci. 11(4), 1632 (2021).
[Crossref]

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

Brady, D.

Callegari, F.

A. Le Gratiet, A. Mohebi, F. Callegari, P. Bianchini, and A. Diaspro, “Review on complete mueller matrix optical scanning microscopy imaging,” Appl. Sci. 11(4), 1632 (2021).
[Crossref]

Campos, J.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

Canillas, A.

Chang, J.

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref]

Chen, D.-F.

Chen, X.

Chen, Y.

Chipman, R. A.

Cui, C.

Cui, X.

X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
[Crossref]

d’Amora, M.

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

De Martino, A.

Deguzman, P. C.

Dereniak, E. L.

Descour, M. R.

Diaspro, A.

A. Le Gratiet, A. Mohebi, F. Callegari, P. Bianchini, and A. Diaspro, “Review on complete mueller matrix optical scanning microscopy imaging,” Appl. Sci. 11(4), 1632 (2021).
[Crossref]

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

Du, C.-Y.

Du, E.

Du, W.

Duocastella, M.

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

Durfort, M.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

Elminyawi, I. M.

El-Saba, A. M.

Escalera, J. C.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

Freudenthal, J.

X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
[Crossref]

Freudenthal, J. H.

J. H. Freudenthal, E. Hollis, and B. Kahr, “Imaging chiroptical artifacts,” Chirality 21(1E), E20–E27 (2009).
[Crossref]

Garcia-Caurel, E.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

Garnatje, T.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

Gayet, B.

Gil, J. J.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

Giordani, S.

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

Gottlieb, D.

Guo, J.

Guo, Y.

Han, C.-Y.

He, H.

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref]

M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
[Crossref]

He, Y.

Hollis, E.

J. H. Freudenthal, E. Hollis, and B. Kahr, “Imaging chiroptical artifacts,” Chirality 21(1E), E20–E27 (2009).
[Crossref]

Huang, T.

Ichimoto, K.

K. Ichimoto, K. Shinoda, T. Yamamoto, and J. Kiyohara, “Photopolarimetric measurement system of Mueller matrix with dual rotating waveplates,” Publications of the National Astronomical Observatory of Japan 9, 11–19 (2006).

Jiang, H.

Jones, M. W.

Kahr, B.

X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
[Crossref]

J. H. Freudenthal, E. Hollis, and B. Kahr, “Imaging chiroptical artifacts,” Chirality 21(1E), E20–E27 (2009).
[Crossref]

Kemme, S. A.

Kiyohara, J.

K. Ichimoto, K. Shinoda, T. Yamamoto, and J. Kiyohara, “Photopolarimetric measurement system of Mueller matrix with dual rotating waveplates,” Publications of the National Astronomical Observatory of Japan 9, 11–19 (2006).

Le Gratiet, A.

A. Le Gratiet, A. Mohebi, F. Callegari, P. Bianchini, and A. Diaspro, “Review on complete mueller matrix optical scanning microscopy imaging,” Appl. Sci. 11(4), 1632 (2021).
[Crossref]

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

Li, M.

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref]

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Liu, S.

Liu, Y.

Lizana, A.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

Ma, H.

Marongiu, R.

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

Meier, J. T.

Meng, R.

Mohebi, A.

A. Le Gratiet, A. Mohebi, F. Callegari, P. Bianchini, and A. Diaspro, “Review on complete mueller matrix optical scanning microscopy imaging,” Appl. Sci. 11(4), 1632 (2021).
[Crossref]

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S. Nichols, “Coherence in polarimetry,” Ph.D. thesis, New York University (2018).

Nichols, S. M.

O. Arteaga, S. M. Nichols, and J. Antó, “Back-focal plane Mueller matrix microscopy: Mueller conoscopy and Mueller diffractrometry,” Appl. Surf. Sci. 421, 702–706 (2017).
[Crossref]

X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
[Crossref]

Nordin, G. P.

Novikova, T.

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M. Shribak and R. Oldenbourg, “Techniques for fast and sensitive measurements of two-dimensional birefringence distributions,” Appl. Opt. 42(16), 3009–3017 (2003).
[Crossref]

R. Oldenbourg, “Polarization microscopy with the LC-PolScope,” in Live cell imaging: A laboratory manual (Cold Spring Harbor Laboratory Press, 2005), pp. 205–237.

Ossikovski, R.

R. Ossikovski and O. Arteaga, “Completing an experimental nondepolarizing Mueller matrix whose column or row is missing,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 37(5), 052905 (2019).
[Crossref]

Pascual, E.

Paula, F.

X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
[Crossref]

Phipps, G. S.

Piccini, M.

J. Ramella-Roman, I. Saytashev, and M. Piccini, “A review of polarization-based imaging technologies for clinical and pre-clinical applications,” J. Opt. 22(12), 123001 (2020).
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Sabatke, D. S.

Saytashev, I.

J. Ramella-Roman, I. Saytashev, and M. Piccini, “A review of polarization-based imaging technologies for clinical and pre-clinical applications,” J. Opt. 22(12), 123001 (2020).
[Crossref]

Shinoda, K.

K. Ichimoto, K. Shinoda, T. Yamamoto, and J. Kiyohara, “Photopolarimetric measurement system of Mueller matrix with dual rotating waveplates,” Publications of the National Astronomical Observatory of Japan 9, 11–19 (2006).

Shribak, M.

Shtukenberg, A. G.

X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
[Crossref]

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Sweatt, W. C.

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Van Eeckhout, A.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

Vidal, J.

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
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Vitkin, I. A.

S. Alali and I. A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. Biomed. Opt. 20(6), 061104 (2015).
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Wang, X.

Wang, Y.

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref]

Wu, J.

Yamamoto, T.

K. Ichimoto, K. Shinoda, T. Yamamoto, and J. Kiyohara, “Photopolarimetric measurement system of Mueller matrix with dual rotating waveplates,” Publications of the National Astronomical Observatory of Japan 9, 11–19 (2006).

Zeng, N.

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
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M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
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Appl. Opt. (5)

Appl. Sci. (1)

A. Le Gratiet, A. Mohebi, F. Callegari, P. Bianchini, and A. Diaspro, “Review on complete mueller matrix optical scanning microscopy imaging,” Appl. Sci. 11(4), 1632 (2021).
[Crossref]

Appl. Surf. Sci. (1)

O. Arteaga, S. M. Nichols, and J. Antó, “Back-focal plane Mueller matrix microscopy: Mueller conoscopy and Mueller diffractrometry,” Appl. Surf. Sci. 421, 702–706 (2017).
[Crossref]

Biomed. Opt. Express (1)

Chirality (1)

J. H. Freudenthal, E. Hollis, and B. Kahr, “Imaging chiroptical artifacts,” Chirality 21(1E), E20–E27 (2009).
[Crossref]

J. Am. Chem. Soc. (1)

X. Cui, S. M. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shtukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138(37), 12211–12218 (2016).
[Crossref]

J. Biomed. Opt. (1)

S. Alali and I. A. Vitkin, “Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment,” J. Biomed. Opt. 20(6), 061104 (2015).
[Crossref]

J. Opt. (1)

J. Ramella-Roman, I. Saytashev, and M. Piccini, “A review of polarization-based imaging technologies for clinical and pre-clinical applications,” J. Opt. 22(12), 123001 (2020).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. (1)

R. Ossikovski and O. Arteaga, “Completing an experimental nondepolarizing Mueller matrix whose column or row is missing,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 37(5), 052905 (2019).
[Crossref]

Micron (1)

Y. Wang, H. He, J. Chang, N. Zeng, S. Liu, M. Li, and H. Ma, “Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope,” Micron 79, 8–15 (2015).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Publications of the National Astronomical Observatory of Japan (1)

K. Ichimoto, K. Shinoda, T. Yamamoto, and J. Kiyohara, “Photopolarimetric measurement system of Mueller matrix with dual rotating waveplates,” Publications of the National Astronomical Observatory of Japan 9, 11–19 (2006).

Sci. Rep. (2)

A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, J. C. Escalera, M. Durfort, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Polarimetric imaging microscopy for advanced inspection of vegetal tissues,” Sci. Rep. 11(1), 3913 (2021).
[Crossref]

A. Le Gratiet, M. d’Amora, M. Duocastella, R. Marongiu, A. Bendandi, S. Giordani, P. Bianchini, and A. Diaspro, “Zebrafish structural development in Mueller-matrix scanning microscopy,” Sci. Rep. 9(1), 19974 (2019).
[Crossref]

Other (3)

“Sony polarization image sensor,” https://www.sony-semicon.co.jp/e/products/IS/industry/product/polarization.html (2021).

S. Nichols, “Coherence in polarimetry,” Ph.D. thesis, New York University (2018).

R. Oldenbourg, “Polarization microscopy with the LC-PolScope,” in Live cell imaging: A laboratory manual (Cold Spring Harbor Laboratory Press, 2005), pp. 205–237.

Supplementary Material (2)

NameDescription
Visualization 1       Live birefringence measurement of a thin resorcinol sample growing from the melt. The setup is capable of measuring both the magnitude and direction of linear birefringence of the sample without any moving parts.
Visualization 2       Measurement of both the change in magnitude and orientation of the birefringence in real-time. As the sample is stretched, the birefringence becomes uniformly aligned in the same direction across the entire sample.

Data availability

Data underlying the results presented in this paper are available in Visualization 1 and Visualization 2.

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Figures (7)

Fig. 1.
Fig. 1. The calculated efficiency for measuring each element across a row of the MM. Higher values correspond to better conditioned measurements with better noise tolerances in the corresponding column of the MM.
Fig. 2.
Fig. 2. Photo (left) and schematic (right) of the MM microscope. The working principle of the polarization camera is shown with the schematic with the layout of the polarizers in each super pixel.
Fig. 3.
Fig. 3. Measurement of a spherulite sample taken with $N=4$ measurements using two retarders with a retardance of 81$^{\circ }$ and relative offset $\phi = 21.9^{\circ }$, turning to $\theta = [-48.4^{\circ }, -18.2^{\circ }, 18.4^{\circ }, -85^{\circ }]$.
Fig. 4.
Fig. 4. Measured standard deviation after 10 measurements of the same spherulite sample. (a) Using a dual QWP elliptical retarder with $\phi = 21.9^{\circ }$. (b) Using a single QWP compensator.
Fig. 5.
Fig. 5. Measured MM of potato starch showing Maltese cross pattern in each grain. Since the sample is approximately non-depolarizing, the last row of the MM is can be calculated with an algebraic method based on the top three rows [15].
Fig. 6.
Fig. 6. Frames from live imaging of resorcinol crystal growth showing both the magnitude in radians (top) and direction in degrees (bottom) of birefringence (see Visualization 1). The images are shown at 100ms intervals, although a faster frame rates are possible.
Fig. 7.
Fig. 7. Live birefringence measurement of a single layer of onion skin showing the intensity image, magnitude of the linear birefringence in radians, and the fast axis orientation in degrees.

Equations (18)

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S out = M S M C M P S in ,
M C ( θ , δ ) = [ 1 0 0 0 0 C 2 θ 2 + S 2 θ 2 C δ C 2 θ S 2 θ ( 1 C δ ) S 2 θ S δ 0 C 2 θ S 2 θ ( 1 C δ ) C 2 θ 2 + S 2 θ 2 C δ C 2 θ S δ 0 S 2 θ S δ C 2 θ S δ C δ ] ,
S X sin ( X ) ,
C X cos ( X ) .
S out = ( m 00 + m 01 ( C 2 θ 2 + C δ S 2 θ 2 ) + m 03 S 2 θ S δ m 02 C 2 θ S 2 θ ( C δ 1 ) m 10 + m 11 ( C 2 θ 2 + C δ S 2 θ 2 ) + m 13 S 2 θ S δ m 12 C 2 θ S 2 θ ( C δ 1 ) m 20 + m 21 ( C 2 θ 2 + C δ S 2 θ 2 ) + m 23 S 2 θ S δ m 22 C 2 θ S 2 θ ( C δ 1 ) m 30 + m 31 ( C 2 θ 2 + C δ S 2 θ 2 ) + m 33 S 2 θ S δ m 32 C 2 θ S 2 θ ( C δ 1 ) ) .
S out , i = [ 1 ( C 2 θ 2 + C δ S 2 θ 2 ) C 2 θ S 2 θ ( C δ 1 ) S 2 θ S δ ] T [ m i 0 m i 1 m i 2 m i 3 ] with i = 0 , 1 , 2.
S out , i = B T A i ,
A i = ( B B T ) 1 B S out , i ,
S out = M S M C R ( ϕ ) M C R ( ϕ ) M P S in ,
M C R ( ϕ ) M C R ( ϕ ) = R ( ρ ) M L R = M E ,
S out = M S R ( ρ ) M L R M P S in .
δ I δ m i = ( N k = 1 N r i k 2 ) 1 2 E i ,
S out = ( m 00 + m 03 m 10 + m 13 m 20 + m 23 m 30 + m 33 ) .
θ = 1 2 a t a n 2 ( S out , 1 , S out , 2 ) ,
δ = a s i n ( S out , 1 2 + S out , 2 2 S out , 0 ) ,
S out = ( 1 C 2 θ 2 + C δ S 2 θ 2 C 2 θ S 2 θ ( C δ 1 ) S 2 θ S δ ) .
S out = ( 1 C 2 θ 2 + C δ S 2 θ 2 C 2 θ S 2 θ ( C δ 1 ) S 2 θ S δ ) + ( 0 Δ δ S δ + Δ δ S δ C 2 θ 2 + 2 Δ 2 θ C 2 θ S 2 θ ( C δ 1 ) Δ 2 θ ( C δ + 1 ) + 2 Δ 2 θ ( C 2 θ 2 + C δ S 2 θ 2 ) + Δ δ C δ C 2 θ S 2 θ Δ 2 θ S δ C 2 θ + Δ δ S 2 θ C δ ) .
β = I 0 + I 90 ( I 45 + I 135 ) I 0 + I 90 + I 45 + I 135

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