Determination of Stokes Vector from a Single Image Acquisition

Four Stokes parameters (1852) define the polarization state of light. Measured changes of the Stokes vector of light traversing an inhomogeneous sample are linked to the local anisotropies of absorption and refraction and are harnessed over an increasing range of applications in photonics, material, and space/earth observation. Several independent polarization sensitive measurements are usually required for determination of the all four Stokes parameters, which makes such characterization procedure time‐consuming or requires complex setups. Here, a single‐snapshot approach to Stokes polarimetry in transmission is introduced by use of a four‐polarization camera with the on‐chip integrated polarizers. A quarter‐waveplate is added in front of the sample and is illuminated by a linearly polarized light. This approach is demonstrated by measuring birefringence Δn≈0.012$\Delta n\approx 0.012$ of spider silk of only ≈6μ$\approx 6\nobreakspace \umu$ m‐diameter using microscopy, however, due to its generic nature, it is transferable to other spectral ranges and imaging applications, e.g., imaging from a fast moving satellite or drone or monitoring fast changing events such as phase transitions.


I. INTRODUCTION
Orientation is an important property of materials, underlying the optical 1 , mechanical 2 , and thermal 3 behaviours.Imaging of aligned structures in biological samples [4][5][6] and tissues are linked to various medical conditions and is of the highest interest for the development of new medical instrumentation and bio-scaffolds for tissue regeneration 7 .An optical 3D tomography based on holography is demonstrated for imaging of a 3D micro-object which has combined birefringence and absorption inhomogenieties 8 .Recently, a 3D tomography of an object was extracted from optical images using a tensorial description of light-matter interaction 9 .These microscopic techniques are expected to find a wide range of applications in microscopy.3D tomography can even be made with nonpropagating optical near-fields, using polarisation-controlled input and polarisation analysis at the output for the far-IR and IR wavelengths in a popular attenuated total reflection geometry 10 .Orientation can be revealed from transmission or reflection even under conditions where feature sizes of ∼ 100 nm of an aligned pattern are ∼ 20 times below the diffraction limit, as shown for the IR chemical fingerprinting wavelengths where nanoscale resolution can not be reached 11 .Mechanical stress-induced anisotropies can be imaged by change of characteristic wavenumbers in Raman spectroscopy/scattering as demonstrated for sapphire 12 .Peculiarities of population dynamics of photoexcited carriers inside semiconductors can be revealed by the transient grating method with polarisation sensitive probing or excitation 13 .Apart of science based methods, polarisation analysis is widely adopted in industry for edge and alignment detection 14 , satellite polarisation imaging 15 , an inspection of stress distribution in semiconductors and solar cells 16 .Stokes polarimetry, which determines the variation of the polarisation state across the image of an optical vortex 17 and measurement of the polarisation state at separate RGB colors in the image 18 can be carried out using integrated polarisers and a quarter waveplate.Integration of polarisation elements even to a higher degree of complexity is demonstrated with metamaterials, where the state of polarisation, including a waveplate function can be realized on a flat surface in front of detector 19 .
Here, we introduce a fast single snap-shot polarimetry technique based on the four polarisation (4-pol.)imaging.It was developed for absorption anisotropy (dihroism) mapping 20 and now it is demonstrated for birefringent objects (optical retarders) using a 4-pol.camera with an on-chip integrated wire-grid polarisers.Determination of all four Stokes parameters, which fully describe the polarisation state of propagating light, was carried out in a single image acquisition using a liquid crystal retarder with a λ /4 waveplate retardace inserted before the sample.The amplitude of the retardance and its orientation azimuth were both retrieved from a single measurement.This opens the feasibility for real-time imaging and monitoring of anisotropy changes during phase transitions, e.g., amorphous-to-crystalline changes.

II. SAMPLES AND METHODS
We used a 2 12 -level grey scale CMOS camera (CS505MUP1 Thorlabs) for imaging in the visible spectral range.It has an extinction ratio larger than 100 for the entire visible spectral range with maximum at Ext ≡ T max T min ∼ 400.While the intensity which is defined by the sum of all four images appears as usual, the orientation of linear polarisers is revealed by calculating the azimuth angle θ shi f t = 1 2 arctan 2 for each pixel, where arctan 2 is the 2-argument arc-tangent θ = arctan 2 (y, x) with −π < θ ≤ π, i.e., it returns angle θ in the full [−π, π] range 21 .The images in this study were calculated from original four images without relying of the intensity and azimuth images provided by the software which controls the camera.
Silk from golden orb web spiders Trichonephila plumipes was used in this study; they were collected during the night in Sydney, Australia.Their major ampullate (dragline) silk was collected by forcible spooling 22 .For silk harvesting, spiders were anesthetized using CO 2 , placed ventral side up on a foam platform, and fixed using non-adhesive tape and pins.Then, a single silk thread from the spinnerets was collected under a dissecting microscope and an electronic spool rotating at 1 m/min was used to reel silk threads from spiders 23 .

III. THEORY: IMAGE OF RETARDATION AND ITS AZIMUTH A. Best fit based analysis
Imaging with a 4-pol.camera can show the retardance distribution and azimuth of a retarder by numerical processing of 4 images as shown in this section.The experimental set up for measurements of the three Stokes parameters S 0,1,2 in transmission consist of a linear polarizer, sample (an unknown retarder) and 4-pol.camera.However, an additional quarter wave plate is required for the determination of retardance vector (retardance and its azimuth).The fourth Stokes parameter S 3 can be determined after such modification of setup as shown in this study.
The Mueller matrix formalism was used to describe the experiment.The polarizer at an angle of ϕ is given by Muller matrix M pol : The matrix can be set to correspond to the wire grid polarizers on the pixels of the CMOS detector.Angles ϕ are set on each quadrant of the camera at ϕ = 0, π/4, π/2, 3π/4, respectively.
The sample acting as a retarder plate with the phase retardance of δ and retardance azimuth of θ is given by: This description (Eqn.2) is for the pure birefringent material and does not take into account dichroism or optical activity.This is a limitation of this approach, which is most useful for birefringence dominant cases of imaged objects.A quarter-wave plate is added for the determination of the phase retardance δ and retardance azimuth θ .The quarterwave plate aligned along the −π/4 direction is given by (follows from Eqn. 2): The incident light was set to the horizontal (x-direction) so that it can be described by the Stokes vector S IN = (1, 1, 0, 0).Finally, the output light (Stokes vector) detected by the CMOS 4-pol.camera is the solution of the following matrix equation: The Eqn. 4 defines the all four output Stokes parameters S OUT (S 3 = 0 due to linear polarisers on 4-pol.camera segments).Next, we use the first S 0 , which is intensity, since we measure intensity images in experiment.It has this form, which is a result of a simple matrix multiplication: By fitting intensity (S 0 ) at four angles ϕ, the amplitude, retardance δ and its azimuth θ can be determined.This protocol was used to determine the retardance and its azimuth δ , θ of yellow spider silk fibers, which are examples of a birefringent (retarder waveplate) material at visible wavelengths.By virtue of simultaneous acquisition of four intensity images at different polarisations and addition of a λ /4-plate to the setup, the retardance and its orientation can be obtained from the single measurement.The quality of the best fit was assessed by calculating R 2 residuals for experimentally determined intensity I i of each pixel in the image by (index i = 1 − 4 corresponds to angles ϕ = 0, π/4, π/2, 3π/4): where the fit function is f (ϕ) = Amp × [1 − sin δ sin 2(ϕ − θ )] /2 (see, Eqn. 6).The fit is obtained for two pairs of (δ 1 , θ 1 ) for 0 ≤ δ ≤ π/2 and (δ 2 , θ 2 ) for −π/2 ≤ δ ≤ 0 corresponding to the positive and negative birefringence, respectively.Regardless of δ sign, the values are calculated from the same image and δ 2 = −δ 1 , θ 2 = θ 1 − π/2.The direction of fast vs. slow axis, θ 1 or θ 2 , is determined unequivocally from the cross-Nicol imaging with complimentary color shifting 530 nm λ -plate, see, e.g., for the polyhydroxybutyrate (PHB) bounded spherulite organic crystal 24 .

B. Analytical solution based analysis
The Stokes vector after passing the λ /4-plate at the setting of right-hand circular (RHC; the fast axis of liquid crystal λ /4 plate is at −45 • ) polariser and sample (retarder plate; Fig. 1) is: Experimentally detected intensity by 4-pol.camera is (polariser at ϕ angles): For the 4-pol.angles ϕ, the corresponding intensities reads: and and can be calculated using experimentally determined 4-pol.intensity images for S ′ 0,1,2 as described above.

IV. RESULTS
A. Retardance and its azimuth δ , θ from fit of 4-pol.images light onto the sample or at the detector.With a 4-pol.camera, orientational dependence of absorbance can be easily determined with a single measurement 20 .For samples which have distributed birefringence, a retarder plate plus two polarisers are required and can be used in crossed or aligned geometries with the sample rotated between them 25 .From the angular dependence of transmittance, the absorption and birefringence contributions are separated due to the double modulation frequencies of absorbance, ϕ and 2ϕ, respectively 26 .The birefringence mapping requires a larger number of measurements due to a faster oscillation of transmittance vs. polarisation angle.As shown in Sec.III, it is possible to extract retardance and its orientations from the generic expression Eqn. 6 when the additional λ /4 plate is introduced before the sample.

Measurement of orientational properties of an absorbing sample requires only one linear polariser for the incoming
For the orientation of retardance, an optical activity based polariscopy method has been developed for anisotropy mapping using true color coding of the orientation 27 .
Figure 2 shows a summary of the polarisation analysis of a spider silk strand only ∼ 6 µm in diameter using the proposed best fit mode with an additional λ /4-plate added in front of the sample (retarder).The fit of 4-pol.camera images for each single pixel was carried out using Eqn.6.The retardance δ and its azimuth θ inside the silk are retrieved with high fidelity showing a high confidence range of the fit based on residuals R 2 ≈ 0.996 .This high fidelity fit also signifies that there was negligible dichroism and optical activity at the wavelength of illumination, as expected for silk at visible wavelengths.By selecting a region of interest (ROI) on the silk fiber, the maximum retardance δ 1 = 47.2 • is obtained for the central cross section of the fiber.The aximuth angle θ 1 = 4.6 • is obtained and corresponds to the slow axis of silk (along n e along the fiber).
Figure S1 shows a cross section measurement of the silk fiber at λ = 550 nm wavelength.Due to its small diameter, the edges of image are affected by diffraction (imaging was taken with NA = 0.65 and the corresponding resolution was r = 0.61λ /NA = 516 nm).The two cross sections at maxmax intensity and at the baseline varied by only ∼ 4%.The retardance ∆nd = (δ • /360 • ) × λ = 72.1 nm, which defined birefringence ∆n ≡ n e − n o = 0.012 (for d = 6.02 µm); for d = 5.76 µm, ∆n = 0.0125.
The retardance and its azimuth was determined using the best fit to Eqn. 6 for a spider silk fiber that was only ∼ 6 µm in diameter.The birefringence of silk ∆n ≈ 0.012 was determined.The procedure was accomplished by a single image snapshot using a 4-pol.camera.

B. All four Stokes parameters from 4-pol. imaging
A polarisation state is fully determined (a point on Poincare sphere) when all four S 0,1,2,3 Stokes parameters are known.By addition of a circular polariser (RHC) before the sample (retarder), we showed how retardance δ and its azimuth θ can be determined from best fit as described in the previous section.The fourth Stokes component S 3 can be calculated as S 3 = cos δ , while the first three are directly determined from the fit of 4-pol.images.
An analytical method for determination of (δ , θ ) (Eqn.8) was also used to calculate maps of two equivalent solutions for pairs of (δ 1 , θ 1 ) and (δ 2 , θ 2 ), which both satisfy the tan 2θ = −S ′ 1 /S ′ 2 (Fig. 1(b)).Figure 3 shows results calculated from a pair of experimental values of S ′ 1 , S ′ 2 .Solution (δ 2 , θ 2 ) (Fig. 3(b)) corresponds to the real case with positive retardance δ > 0 values (a negative value of δ would be expected for the form-birefringent material, which has ∆n < 0, i.e., n o > n e ).The ambiguity in (δ , θ ) is solved by the fact that n e > n o in silk and that the slow axis is along the fiber.Another conventional method to solve this issue of fast vs. slow axis is to use a color-shifting plate (usually a 530 nm wavelplate) in cross-polarised imaging, which determines positive and negative contributions to the refractive index with blue-red perpendicular directions corresponding to the n e (slow)-n o (fast) orientations of silk 29 .
Finally, with δ , θ determined via fit or from analytical expression using single a image from a 4-pol.camera, all four Stokes parameters can be defined: S 0 (intensity) and S 1,2,3 .Such calculations using an analytical model and a resolution of the uncertainty of the fast-slow axis orientation (Fig. 3(b)) is shown in Fig. 4. If there is no knowledge of the slow/fast axis orientation at the wavelength of interest, both maps (δ 1,2 , θ 1,2 ) can be calculated from the same data without the need to retake an image (Fig. 3).It is noteworthy that S 3 (δ ) is the same when calculated for the two pairs of (δ 1,2 , θ 1,2 ) based on experimentally measured (S ′ 1 , S ′ 2 ).The main advantage of the above presented procedures to determine δ and θ is due to inherent capability to calculate them from a single image from a 4-pol.camera.For example, formation of a crystalline phase of organic crystal polyhydroxybutyrate (PHB) from a cooled amorphous melt (150 • C) takes place with a typical front speed of few micrometers per second for a tens-of-µm thick bounded spherulite between two transparent plates 24 .Figure 5 shows an optical image (intensity) and orientation angle θ maps obtained by the best fit.
This example shows a complex spiraling crystalline pattern of PHB with the spiraling period of ∼ 20 µm revealed via the orientation azimuth.Future studies will be focused on temporal evolution of phase transitions, which can also be imaged at different wavelengths, e.g., at the IR fingerprinting spectral range.The order-disorder front and orientation anisotropy of chemical bonding could be monitored in real time using the two proposed methods based on the fit and via analytical expression using experimentally measured intensities from 4pol.camera images.

V. CONCLUSIONS AND OUTLOOK
Orientation azimuth of retardance and its value can be extracted without any prior knowledge of orientation from a 4pol.image using polarised illumination of the sample, addition of a λ /4 waveplate, and numerical analysis presented in this study.Importantly, this is can be achieved in a single measurement snap-shot with a 4-pol.camera.With knowledge of the retardance δ , all four Stokes parameters can be calculated, i.e., full characterisation of the light state on a detector is obtained and can be linked to the local retardance and its orientation in the sample.This opens new opportunities for imaging of fast-changing events, e.g., phase changes in crystals under microscopy observation at IR or visible wavelengths as well as polarisation analysis of satellite images where natural illumination has prevalent polarisation in the plane of scattering, while a 4-pol.camera is used for imaging.The single image technique presented here for Stokes polarimetry is expected to find a number of potential applications, also, at different spectral ranges of electromagnetic radiation, e.g., as fibular structure of bamboo shows linear anisotropy at THz spectral range 30 .Since Stokes vector is a 2D projection, the variations inside the sample affect the possibility to decouple a retarded phase due to birefringence from that due to inhomogeneities of thickness or composition.However, for many practical applications where micro-tomed thin slices are imaged, the proposed method should prove useful.If required, true 3D techniques could be applied as recently demonstrated 8,9 .

2θ 1 2θ 2 (S 2 ,FIG. 1 .FIG. 2 .
FIG. 1.(a) Polarisation analysis with four-polarisation (4-pol.)camera (CS505MUP1 Thorlabs).Polarisers are on-chip integrated and the pixel size is 3.45 µm.The anti-clockwise direction when looking into the beam corresponds to positive ϕ angles.The liquid crystal retarder at −45 • defines its fast axis.(b) Conventions for the Stokes parameters S 1,2 and azimuth angles θ 1,2 used in analysis.The ambiguity of slow vs. fast axis definition is due to tan 2θ = −S 1 /S 2 which has solutions 2θ = 2θ − π (or θ = θ − π/2, i.e., the slow or fast axis marked by the dashed line diagonal is not determined unequivocally).A simple cross-Nicol image with a 530-nm color shifter λ -plate can determine slow vs. fast orientation 24 .

FIG. 4 .FIG. 5 .
FIG.4.Maps of Stokes' parameters S 0 (intensity) and normalised S 1,2,3 calculated from analytical model Eqn. 8 (same sample and conditions as in Fig.3).Objective lens NA = 0.65; image taken at λ = 550 nm using bandpass filter.The S 3 = cos δ was calculated from the determined δ ; noteworthy, the S 3 = 0 when calculated for the entire experimental setup with 4-pol.camera detection, i.e., linear polarisers makes S 3 ≡ 0. The background has S 3 = +1 for the RHC illumination set by the LC λ /4-waveplate (Fig.1(a)) and would be S 3 = −1 for the LHC, since definition of S 3 = I RHC − I LHC is by the intensities at two cross circular polarisations I RHC,LHC .
FIG. S1.Image (a) and cross section (b) of the yellow spider silk taken with a 550 ± 10 nm bandpass filter.The maximum intensity cross section is 4.3% larger than the baseline based image cross section.Same sample and imaging conditions as in Fig. 2.