⇦Chinese Physics Letters, 2020, Vol. 37, No. 7, Article code 074201 Direct Spatially Resolved Snapshot Interferometric Phase and Stokes Vector Extraction by Using an Imaging PolarCam Dahi Ibrahim1,2 and Daesuk Kim1* Affiliations 1Division of Mechanical System Engineering, Chonbuk National University, Jeonju 54896, Korea 2Engineering and Surface Metrology Lab, National Institute of Standards, Tersa St., El haram, El Giza, Egypt Received 21 April 2020; accepted 27 May 2020; published online 21 June 2020 Supported by Samsung Research Funding & Incubation Center of Samsung Electronics under Grant No. SRFC-TA1703-11.
*Corresponding author. Email: dashi.kim@jbnu.ac.kr Citation Text: Ibrahim D and Kim D 2020 Chin. Phys. Lett. 37 074201    Abstract We extract the 3D phase $\varDelta$ and the Stokes parameter $S_{3}$ of a transmissive anisotropic object spatially using an interferometric PolarCam. Four parallel interferograms with a phase shift of $\pi$/2 between the images are captured in a single snapshot and then reconstructed by the four-bucket algorithm to extract the 3D phase of the object. The $S_{3}$ is then calculated directly from the obtained 3D phase $\varDelta$. The extracted results of $\varDelta$ and $S_{3}$ were compared with those extracted from the non-interferometric PolarCam and the Thorlabs polarimeter, and the results match quite well. The merit of using the interferometric PolarCam is that no mechanical movement mechanisms are included, and hence the $\varDelta$ and $S_{3}$ of the object can be extracted, with high accuracy and within a part of a second (three times faster than non-interferometric PolarCam and Thorlabs polarimeter methods). Moreover, this method can be applied in the field of the dynamic spectro–interferometric PolarCam and can be implemented using swept-wavelength approaches. DOI:10.1088/0256-307X/37/7/074201 PACS:42.30.Rx, 42.30.Va, 07.60.Ly, 42.30.Kq © 2020 Chinese Physics Society Article Text Optical techniques with fast and precise measurements have become vital technologies in the mass production fields of micro-nano samples, which are used extensively in industry. Polarization measurement systems such as spectroscopic ellipsometry and polarimetry[1–5] can provide highly precise measurement capability. However, they employ mechanical rotating mechanisms to extract the 3D phase $\varDelta$ and the Stokes vector $S$ of an object.[2] Available polarization measurement systems with mechanical rotating mechanisms are considered to be high cost, as well as time consuming to use. For example, the measurement time given by the commercial Muller spectro-polarimeter Poxi-spectra$^{\rm TM}$ (Tokyo instrument Inc., Japan) is greater than 8 s, while the measurement time given by the commercial spectroscopic ellipsometer (J.A. Woollam Co., USA) is around 2 s. Some attempts have been made to decrease the measurement time while maintaining the high sensitivity by combining polarimetry with interferometry.[3] Other attempts have been made to replace scanning approaches with spectral Mach–Zehnder interferometry combined with dual-channel sensing modules.[4,5] Although the system is strongly robust, it has high complexity and requires complicated alignment steps. Recently, Abdelsalam[6] proposed a simple robust polar interferometer based on a one-piece Michelson interferometer to extract the 3D phase ($\varDelta$) of birefringent and nonbirefringent samples with moderate accuracy. In this work, we modified the work reported in Ref. [6], with single shot parallel phase shifting to obtain the 3D phase of the object with high accuracy and in real time. Here the polar interferometer is combined with a PolarCam to extract parallel four interferograms in a single shot with a phase shift of $\pi$/2 between the images. The polar interferometer, which is a one-piece Michelson interferometer, is explained in detail in Ref. [6]. The PolarCam combined with the polar interferometer is a PolarCam$^{\rm TM}$ from 4D Technology Co., with $648\times 488$ pixels and a pixel size of 7.4 µm. The PolarCam includes a micro-polarizer, which is an array containing a pattern of polarizers with four discrete polarizations (0$^{\circ}$, 45$^{\circ}$, 90$^{\circ}$, 135$^{\circ}$), known as a super pixel which is repeated over the entire array.[7–9] The corresponding four interferograms in a single snapshot are ($I_{0}$(0$^{\circ}$), $I_{45}$(90$^{\circ}$), $I_{90}$(180$^{\circ}$), $I_{135}$(270$^{\circ}$)), respectively. A quarter wave plate (QWP) between the polar interferometer and the PolarCam is used to make uniform the irradiance in the four interferograms in the single snapshot. The four-bucket algorithm was used to reconstruct the 3D phase ($\varDelta$) of the object in real time.[10] The Stokes parameter $S_{3}$ is then calculated directly from the obtained 3D phase ($\varDelta$). Here, we focused on calculation of the $S_{3}$ parameter because it is not provided by the PolarView$^{\rm TM}$ polarimetry software attached with the PolarCam. The extracted results of the $\varDelta$ and $S_{3}$ of the object (QWP) were compared with those extracted by the non-interferometric PolarCam and by the commercial Thorlabs polarimeter, and the results match quite well. The results of $\varDelta$ extracted by the proposed interferometric PolarCam were also compared with those extracted by the angular spectrum method (ASM),[6] and the comparison shows that the results of $\varDelta$ extracted by the proposed interferometric PolarCam are more accurate compared to the results reported in Ref. [6] (see Fig. 7). We attribute the better phase accuracy to the superior performance of the parallel phase shifting technique used in reconstruction. Moreover, the proposed system has no mechanical movement mechanisms and hence can extract the 3D phase ($\varDelta$) and $S_{3}$ of the object with high accuracy in real time. We believe that this technology will open paths to potential applications in the field for the dynamic spectro–interferometric PolarCam, which can be implemented by using swept-wavelength methods. The proposed interferometric PolarCam scheme with no mechanical movement is depicted in Fig. 1. A photograph of the experimental interferometric PolarCam is shown in Fig. 2. The light from a laser diode of wavelength $\lambda = 635$ nm is expanded by a beam expander and then linearly polarized by a polarizer $P_{1}$ set at an angle 45$^{\circ}$. The polarized light passes through a nonpolarizing beam-splitter (NPBS) to illuminate the polar interferometer. The one-piece interferometer is comprised of a polarizing beam-splitter (PBS) and two identical flat mirrors with flatness $\lambda$/10. The one-piece interferometer was explained in detail in Ref. [6].
cpl-37-7-074201-fig1.png
Fig. 1. Schematic diagram of the proposed interferometric PolarCam. $P_{1}$ and $P_{2}$: polarizers fixed at 45$^{\circ}$ with the optical axis; M: mirror; PBS: polarizing beam splitter; NPBS: nonpolarizing beam splitters; QWP: quarter wave plate set at $-45^{\circ}$; P: parallel component of light; S: orthogonal component of light; $\lambda$: wavelength.
cpl-37-7-074201-fig2.png
Fig. 2. Photograph of experimental interferometric PolarCam.
The two mirrors of the interferometer are mounted carefully in the two arms of the interferometer at equidistant lengths from the PBS. Two orthogonal polarized beams, namely p and s, are generated by the PBS and reflected from the two mirrors with reflections $r_{\rm p}$ and $r_{\rm s}$. Here, the polarization states, due to reflection or transmission through the optical components, are assumed negligible. The two orthogonal reflected beams pass through the PBS and are then reflected by the NPBS to a second linear polarizer $P_{2}$ set at an angle 45$^{\circ}$. Note that the interference does not occur between the two orthogonal polarized beams, so a polarizer with 45$^{\circ}$ was utilized to transfer one of the two orthogonal polarized beams to match with the other to obtain the interference fringes. The obtained four interferograms ($I_{0}$, $I_{45}$, $I_{90}$, $I_{135}$) in a single snapshot captured by the PolarCam are shown in Fig. 3. A QWP set at $-45^{\circ}$ with the fast axis was used in front of the PolarCam to make uniform the irradiance in the four interferograms in the single snapshot.[11–16] Here, the object is positioned before the one-piece interferometer in front of the $P_{1} $,[17] as shown in Figs. 1 and 2.
cpl-37-7-074201-fig3.png
Fig. 3. Parallel four interferograms in a single snapshot captured at the azimuth angle of the object (QWP) $\theta =-45^{\circ}$.
The object can be positioned before or after the one-piece interferometer. Note that to feature tiny objects, an imaging lens should be inserted in front of the PolarCam for good spatial resolution. Figure 3 shows the parallel four interferograms in a single snapshot captured by the PolarCam at the azimuth angle of the object set at $\theta =-45^{\circ}$. The phase of the snapshot is extracted precisely by the four-bucket algorithm [18] which is derived in detail in the following. From the extracted phase $\varDelta$, the $S_{3}$ which is not provided by the PolarView$^{\rm TM}$ polarimetry software attached with the PolarCam is extracted precisely from the obtained $\varDelta$. The electric field illuminates the interferometer when the object is set at an azimuth angle $\theta = -45^{\circ}$ with amplitudes $u(-45^{\circ})$, $\nu(-45^{\circ})$, and phases $\xi(-45^{\circ})$. $\eta(-45^{\circ})$ in the $x$ and $y$ directions can be expressed as $$ E_{\rm in}^{\rm object} (-45^{\circ})=\left[ { \begin{array}{l} u(-45^{\circ})e^{i\left[ {\xi (-45^{\circ})} \right]} \\ \nu (-45^{\circ})e^{i\left[ {\eta (-45^{\circ})} \right]} \\ \end{array}} \right].~~ \tag {1} $$ The electric field illuminates the interferometer when the object is set at an azimuth angle $\theta \ne -45^{\circ}$ with amplitudes $u(\theta)$, $\nu (\theta)$ and phases $\xi (\theta)$. $\eta (\theta)$ in the $x$ and $y$ directions is expressed as $$ E_{\rm in}^{\rm object} (\theta)=\left[ { \begin{array}{l} u(\theta)e^{i\left[ {\delta_{\rm p} (\theta)+\xi (-45^{\circ})} \right]} \\ \nu (\theta)e^{i\left[ {\delta_{\rm s} (\theta)+\eta (-45^{\circ})} \right]} \\ \end{array}} \right].~~ \tag {2} $$ The output components in the $x$ and $y$ directions, $E_{\rm p}(\theta)$ and $E_{\rm s}(\theta)$, can be written as $$\begin{alignat}{1} E_{\rm p} (\theta)={}&q(\theta)p(\theta)\beta ({\rm NPBS})\beta_{\rm p} ({\rm PBS})M_{\rm p} \\ &\times \beta_{\rm p} ({\rm PBS})\beta ({\rm NPBS})O_{\rm obj} (\theta)p(\theta)E_{\rm in} e^{ikz_{\rm p} },~~ \tag {3} \end{alignat} $$ $$\begin{alignat}{1} E_{\rm s} (\theta)={}&q(\theta)p(\theta)\beta ({\rm NPBS})\beta_{\rm s} ({\rm PBS})M_{\rm s} \\ &\times \beta_{\rm s} ({\rm PBS})\beta ({\rm NPBS})O_{\rm obj} (\theta)p(\theta)E_{\rm in} e^{ikz_{\rm s} },~~ \tag {4} \end{alignat} $$ where $z_{\rm p}$ and $z_{\rm s}$ are the optical path lengths traveled in the p- and s-polarization paths, $\beta$(NPBS) and $\beta$(PBS) are the Jones matrices of NPBSs and PBSs, $M_{\rm p}$ and $M_{\rm s}$ are the Jones matrices of the reflection mirrors in the p- and s-polarization paths and $E_{\rm in}$ is the inlet wave incident to the object. We set this here as a function of wavelength, since its amplitude and phase in the $x$ and $y$ directions are assumed to be varied with the wavelengths, and can be expressed as $$ E_{\rm in} (\lambda)= \begin{pmatrix} E_{x} e^{i\xi (\lambda)} & 0 \\ 0 & E_{y} e^{i\eta (\lambda)} \end{pmatrix},~~ \tag {5} $$ where $E_{x}$ and $E_{y}$ represent the $x$ and $y$ amplitudes of the complex wave vector $E_{\rm in}(\lambda)$ and $\xi (\lambda)$, $\eta (\lambda)$ the corresponding phases. Here we used only one wavelength $\lambda = 635$ nm, taking into consideration that this optical interferometric PolarCam can be applied to the spectro–interferometric PolarCam by the swept-wavelength method. The merit of using the one-piece interferometer is that it keeps the optical path difference ($z_{p }-z_{\rm s}$) of the interferometer zero. The Jones matrices of $p(45)$, $M\beta({\rm PBS})$ and $q(\theta)$, $O_{\rm obj}(\theta)$ in Eqs. (3) and (4) can be written as $$\begin{align} &p(\theta)= \begin{pmatrix} {\cos^{2}\theta} & {\cos \theta\sin \theta} \\ {\sin \theta\cos \theta} & {\sin^{2}\theta} \end{pmatrix},\\ &M_{\rm p} \beta_{\rm p} ({\rm PBS})=\begin{pmatrix} {r_{\rm p} } & 0 \\ 0 & 0 \end{pmatrix},\\ &M_{\rm s} \beta_{\rm s} ({\rm PBS})=\begin{pmatrix} 0 & 0 \\ 0 & {r_{\rm s} } \end{pmatrix},~~ \tag {6} \end{align} $$ $$\begin{align} &q(\theta)=\begin{pmatrix} {\cos \frac{\delta}{2}+i\sin \frac{\delta}{2}\cos (2\theta)} & {i\sin \frac{\delta}{2}\sin (2\theta)} \\ {i\sin \frac{\delta}{2}\sin (2\theta)} & {\cos \frac{\delta}{2}-i\sin \frac{\delta}{2}\cos (2\theta)} \end{pmatrix},\\ &O_{\rm obj} (\theta)= \begin{pmatrix} {\left| {t_{\rm p} (\theta)} \right|e^{i\delta_{\rm p} (\theta)}} & 0 \\ 0 & {\left| {t_{\rm s} (\theta)} \right|e^{i\delta_{\rm s} (\theta)}} \end{pmatrix},~~ \tag {7} \end{align} $$ where $|t_{\rm p}(\theta)|$, $|t_{\rm s}(\theta)|$ represent the amplitudes of the transmission coefficient of the transmissive object, and $\delta_{\rm p}$ and $\delta_{\rm s}$ stand for the phase changes of the transmissive object along the $x$ and $y$ directions. Using Eq. (5), Eqs. (3) and (4) can be rewritten as $$\begin{alignat}{1} &E_{\rm p} (\theta)=\frac{1}{16}q(-45)\begin{pmatrix} {r_{\rm p} } & 0 \\ 0 & 0 \end{pmatrix} O_{\rm obj} (\theta)E_{\rm in} e^{ikz_{\rm p} },~~ \tag {8} \end{alignat} $$ $$\begin{alignat}{1} &E_{\rm s} (\theta)=\frac{1}{16}q(-45)\begin{pmatrix} 0 & 0 \\ 0 & {r_{\rm s} }\end{pmatrix} O_{\rm obj} (\theta)E_{\rm in} e^{ikz_{\rm s} }.~~ \tag {9} \end{alignat} $$ The interferogram intensity in general can be expressed as $$ I(\theta)=(E_{\rm p} (\theta)+E_{\rm s} (\theta))(E_{\rm p} (\theta)+E_{\rm s} (\theta))^{\ast },~~ \tag {10} $$ where * stands for conjugation. The interferogram intensity at an azimuth angle ($\theta =-45^{\circ}$) or the reference interferogram can be expressed as $$\begin{alignat}{1} I^{\rm object}&(-45^{\circ})= \left| {E_{\rm p} } \right|^{2}+\left| {E_{\rm s} } \right|^{2}\\ &+2\gamma \left| {E_{\rm p} } \right|\left| {E_{\rm s} } \right|\cos [{\varPhi}^{\rm object}(-45^{\circ})],~~~ \tag {11} \end{alignat} $$ where $\gamma$ is the coherence function and ${\varPhi}^{\rm object}(-45^{\circ})$ is the total phase of the object at $\theta =-45^{\circ}$, which can be expressed as $$\begin{alignat}{1} {\varPhi}^{\rm object}(-45^{\circ})=2k[ {z_{\rm p}-z_{\rm s}}]+[{\xi(-45^{\circ})-\eta (-45^{\circ})}],~~ \tag {12} \end{alignat} $$ where $k$ is a wavenumber defined by $k = 2\pi /\lambda$, and $\lambda$ is the wavelength of the laser source. The interferogram intensity at an azimuth angle ($\theta \ne -45^{\circ}$) can be expressed as $$\begin{alignat}{1} I^{\rm object}&(\theta)= \left| {E_{\rm p} (\theta)} \right|^{2}+\left| {E_{\rm s} (\theta)} \right|^{2}\\ &+2\gamma \left| {E_{\rm p} (\theta)} \right|\left| {E_{\rm s} (\theta)} \right|\cos [{\varPhi}^{\rm object}(\theta)],~~ \tag {13} \end{alignat} $$ where ${\varPhi}^{\rm object} (\theta)$ is the phase of the object at $\theta \ne -45^{\circ}$, which can be expressed as $$\begin{align} {\varPhi}^{\rm object}(\theta)={}&2k\left[ {z_{\rm p} -z_{\rm s} } \right] +\left[ {\xi (-45^{\circ})-\eta (-45^{\circ})} \right]\\ &+[\delta_{\rm p} (\theta)-\delta_{\rm s} (\theta)].~~ \tag {14} \end{align} $$ The difference in phase between the object at ($\theta \ne -45^{\circ}$) and the reference (object at $\theta =-45^{\circ}$), which is a subtraction of Eq. (14) from Eq. (12), can be expressed as $$ ({\varPhi}^{\rm object}(\theta)-{\varPhi}^{\rm object}(-45^{\circ}))=\varDelta (\theta)=[\delta_{\rm p} (\theta)-\delta_{\rm s} (\theta)].~~ \tag {15} $$ The obtained four interference signals are given as follows: $$\begin{alignat}{1} &I_{0} =\left| {E_{\rm p} } \right|^{2}+\left| {E_{\rm s} } \right|^{2}+2\gamma \left| {E_{\rm p} } \right|\left| {E_{\rm s} } \right|\cos [\varDelta (\theta)],~~ \tag {16} \end{alignat} $$ $$\begin{alignat}{1} &I_{45} =\left| {E_{\rm p} } \right|^{2}+\left| {E_{\rm s} } \right|^{2}-2\gamma \left| {E_{\rm p} } \right|\left| {E_{\rm s} } \right|\sin [\varDelta (\theta)],~~ \tag {17} \end{alignat} $$ $$\begin{alignat}{1} &I_{90} =\left| {E_{\rm p} } \right|^{2}+\left| {E_{\rm s} } \right|^{2}-2\gamma \left| {E_{\rm p} } \right|\left| {E_{\rm s} } \right|\cos [\varDelta (\theta)],~~ \tag {18} \end{alignat} $$ $$\begin{alignat}{1} &I_{135} =\left| {E_{\rm p} } \right|^{2}+\left| {E_{\rm s} } \right|^{2}+2\gamma \left| {E_{\rm p} } \right|\left| {E_{\rm s} } \right|\sin [\varDelta (\theta)].~~ \tag {19} \end{alignat} $$ After some manipulations, the $\varDelta (\theta)$ is obtained by dividing the subtraction of Eq. (19) from Eq. (17) and Eq. (16) from Eq. (18) as follows: $$ \varDelta (\theta)=\tan^{-1}\Big[ {\frac{I_{135} -I_{45} }{I_{0} -I_{90} }} \Big].~~ \tag {20} $$ Equation (20) is exactly the same as the formula of the four-bucket algorithm,[18] which confirms the correctness of derivation. The obtained phase is then subtracted from the phase extracted at $\theta = -45^{\circ}$, satisfying that $\varDelta (\theta) = ({\varDelta}_{\rm o}(\theta \ne -45^{\circ}) - {\varDelta}_{\rm o}(\theta =-45^{\circ})$). Figure 3 shows the four parallel interferograms in a single snapshot captured by the PolarCam at the azimuth angle of the object set at $\theta =-45^{\circ}$. The phase of the snapshot is extracted precisely by the four-bucket algorithm,[18] which is derived in detail. As seen in Fig. 3, if there is any change in the azimuth angle $\theta$, the intensities of the four interferograms in the single snapshot are changed, and consequently $\varDelta$ is changed. To check the performance of the proposed interferometric PolarCam in testing the dynamic object, we rotated the object (QWP) automatically. We then captured 35 snapshots at different azimuth angles starting from ($-55^{\circ}$, $-65^{\circ}$,...,$-385^{\circ}$,$-395^{\circ}$) with an interval of 10$^{\circ}$. The extracted phase of the object at $\theta =-45^{\circ}$ is considered as a reference phase and expressed as ${\varDelta}_{\rm o}(\theta =-45^{\circ}$). The phases at the azimuth angles $\theta \ne -45^{\circ}$ of the object were extracted as $\varDelta (\theta) = ({\varDelta}_{\rm o}(\theta \ne -45^{\circ}) - {\varDelta}_{\rm o}(\theta =-45^{\circ}))$. Note that the captured 35 snapshot interferograms were first corrected from coherent noise by using the flat fielding method.[19–22] The modified flat fielding formula used to correct the raw interferogram intensity $I_{\rm R}$ can be expressed as $$ I_{\rm C} =\frac{M(I_{\rm R} -I_{\rm D})}{(I_{\rm p} -I_{\rm s})-I_{\rm D} },~~ \tag {21} $$ where $I_{\rm p}$ and $I_{\rm s}$ are the flat fielding frames at $p$ and $s$ components captured mutually by blocking one to the other, $I_{\rm D}$ is the dark frame captured when there is no light illuminating the PolarCam, and $M$ is the average pixel value of the corrected flat field frame. By using Eq. (20), the 3D phases $\varDelta (\theta) = ({\varDelta}_{\rm o}(\theta \ne -45^{\circ}) - {\varDelta}_{\rm o}(\theta =-45^{\circ}))$ of the 35 snapshots were obtained. The phases were unwrapped by the graph cut method[23] to remove the 2$\pi$ ambiguity. Figure 4(a) shows the phase profiles of the 35 snapshots extracted at pixels (8, 38070, and 57105) and the average of the phase image area of $324\times 235$ pixels. As seen in Fig. 4(a), the phase profiles match quite well with the phase profile extracted by the commercial Thorlabs polarimeter.[6] Figure 4(b) shows the $S_{3}(\theta)$ profiles of the 35 snapshots extracted at pixels (8, 38070, and 57105) and the average of the $S_{3}$ image area of $324\times 235$ pixels. Figure 4(c) shows the extracted $\varDelta (\theta$) profiles of (b). Figure 4(d) shows the extracted $S_{3}(\theta)$ profiles of (a). Once $\varDelta (\theta)$ is extracted, the $S_{3}(\theta)$, which is not provided by the PolarView$^{\rm TM}$ polarimetry software attached with PolarCam, can be calculated as follows:[11] $$ S_{3} (\theta)=2\sqrt {I_{0} I_{90} } \sin (\varDelta (\theta)),~~ \tag {22} $$ where $I_{0}$ and $I_{90}$ are the interferogram intensities at phase shifts 0 and 90, respectively. Here we focused on calculation of $S_{3}$ because it is not provided by the PolarViewTM polarimetry software attached with the PolarCam. However, the parameters $S_{0} = (I_{0} + I_{90})$, $S_{1} = (I_{0}-I_{90})$, and $S_{2} = (I_{45}-I_{135})$ are already provided by the PolarViewTM polarimetry software attached with the PolarCam.[9] These elements are often normalized to the value of $S_{0}$ so that they have values between $+$1 and $-1$. The advantage of the proposed interferometric PolarCam in comparison with the Thorlabs polarimeter is that no mechanical movement mechanisms included in the proposed interferometric PolarCam. The mechanism of the Thorlabs polarimeter is explained in Ref. [6]. Note that the phases of the object extracted by the Thorlabs polarimeter were calculated as $\varDelta (\theta) = \tan^{-1}(-S_{3}(\theta)/S_{2}(\theta))$, where $S_{2} = (I_{45}-I_{135})$, and $S_{3} = (I_{\rm LHC}-I_{\rm RHC}) $.[9] Also, the Thorlabs polarimeter uses a single pixel photo diode, while the proposed interferometric PolarCam uses a PolarCam to extract the 3D phase. To show the performance of the proposed interferometric PolarCam in testing the dynamic object, we compared the extracted results of phases $\varDelta (\theta)$ and $S_{3}(\theta)$ extracted by the interferometric PolarCam with those extracted by the non-interferometric PolarCam. The schematic diagram of the non-interferometric PolarCam is shown in Fig. 5. In Fig. 5, light from a laser diode of a wavelength $\lambda = 635$ nm is used after collimation to illuminate the PolarCam. The object (QWP) is positioned between the linear polarizer P(45$^{\circ}$) set at an angle 45$^{\circ}$ and the PolarCam. We captured 35 snapshots in the absence of the rotating QWP at the different azimuth angles start from ($-55^{\circ}$, $-65^{\circ}$,...,$-385^{\circ}$, $-395^{\circ}$) to extract the $S_{2}(\theta)$.
cpl-37-7-074201-fig4.png
Fig. 4. (a) Extracted $\varDelta (\theta)$ in degrees extracted by the proposed interferometric PolarCam. (b) Extracted $S_{3} (\theta)$ by the non-interferometric PolarCam. (c) Extracted $\varDelta (\theta)$ profiles of (b). (d) Extracted $S_{3}(\theta)$ profiles of (a).
cpl-37-7-074201-fig5.png
Fig. 5. Schematic diagram of the non-interferometric PolarCam.
cpl-37-7-074201-fig6.png
Fig. 6. (a) Single snapshot at $\theta = -45^{\circ}$ for $S_{2}$ extraction, where $S_{2}= (I_{45}-I_{135})$. (b) Single snapshot at $\theta = 45^{\circ}$(RHC). (c) Single snapshot at $\theta = -45^{\circ}$(LHC) for $S_{3}$ extraction, where $S_{3 }=I_{\rm LHC} - I_{\rm RHC}$.
One snapshot at $\theta = -45^{\circ}$ for $S_{2}$ extraction is shown in Fig. 6(a). To extract the $S_{3}$, a QWP positioned after the object was used. The QWP is rotated at 45$^{\circ}$(RHC) and 35 snapshots at the different azimuth angles were captured. The QWP was rotated at $-45^{\circ}$(LHC) and 35 snapshots corresponding to the different azimuth angles were captured. The values of the $S_{3}$ values were then calculated by $S_{3}(\theta)= I_{\rm LHC}(\theta) - I_{\rm RHC}(\theta) $,[9] where $I_{\rm LHC}$ and $I_{\rm RHC}$ are the left- and right-hand circular polarization components, respectively. Figures 6(b) and 6(c) show the snapshots at $\theta = 45^{\circ}$(RHC) and at $\theta = -45^{\circ}$(LHC) for $S_{3}$ extraction. The phases are calculated as $\varDelta (\theta) = \tan^{-1}(-S_{3}(\theta)/S_{2}(\theta))$.[24]
cpl-37-7-074201-fig7.png
Fig. 7. Clustered bar graph of some selected phases listed in Table 1 extracted at rotation angles ($-55^{\circ}$, $-135^{\circ}$, $-225^{\circ}$, $-315^{\circ}$, and $-395^{\circ}$).
In this study, we extracted the phase $\varDelta$ and the Stokes parameter $S_{3}$ of the object directly within a part of a second by using the interferometric PolarCam. Parallel four interferograms with a shift of $\pi$/2 between the images were captured in a single shot by the PolarCam and then reconstructed by the four-bucket algorithm. The advantage of using the Polar interferometer is that it has no moving parts compared with the Thorlabs polarimeter. The merit of using the PolarCam, which is combined with the Polar interferometer, is to reduce the time of measurement as well as the noise. The advantage of the proposed interferometric PolarCam is to extract the phase $\varDelta$ and the Stokes parameter $S_{3}$ of the object spatially and within a part of a second. Here the word spatially means that the quantitative values of $\varDelta$ or $S_{3}$ of the object at any pixel of the image should be the same as those shown in Fig. 4. We claim that this advantage will open a route towards potential applications in the field for the dynamic spectro–interferometric PolarCam, which can be implemented by swept-wavelength methods. To see the performance of the proposed interferometric PolarCam (PSM) in comparison with the work published in Ref. [6] we extracted five phases at selected rotation angles ($-55^{\circ}$, $-135^{\circ}$, $-225^{\circ}$, $-315^{\circ}$, and $-395^{\circ}$) and compared them with those extracted by the Thorlabs polarimeter. Note that the proposed method uses the phase shifting method (PSM), while the work reported in Ref. [6] used the angular spectrum method (ASM) in reconstruction. The extracted five phases are listed in Table 1. The clustered bar graph of the extracted five phases listed in Table 1 is shown in Fig. 7. As seen in Table 1 and Fig. 7, the phases extracted by the proposed interferometric PolarCam are better in accuracy as compared with those extracted in Ref. [6] and approach the phase values extracted by the Thorlabs polarimeter.
Table 1. Phases extracted at rotation angles ($-55^{\circ}$, $-135^{\circ}$, $-225^{\circ}$, $-315^{\circ}$, and $-395^{\circ}$) by the interferometric PolarCam (PSM), interferometric CCD camera (ASM) in Ref. [6], and the Thorlabs polarimeter.
Rotation angle $-55^{\circ}$ $-135^{\circ}$ $-225^{\circ}$ $-315^{\circ}$ $-395^{\circ}$
Interferometric PolarCam (PSM) $-175^{\circ}$ 1$^{\circ}$ $-181^{\circ}$ 2$^{\circ}$ $-169^{\circ}$
Interferometric CCD camera (ASM) $-183^{\circ}$ 3$^{\circ}$ $-183^{\circ}$ 4$^{\circ}$ $-161^{\circ}$
Thorlabs polarimeter $-172^{\circ}$ 0$^{\circ}$ $-180^{\circ}$ 0$^{\circ}$ $-173^{\circ}$
We argue that the high accuracy of the proposed interferometric method is due to the performance of the parallel phase shifting method in reconstruction. In addition, the proposed interferometric PolarCam system has no mechanical movement mechanisms and hence can extract $\varDelta$ and $S_{3}$ of an object directly within a part of a second (three times faster than the non-interferometric PolarCam). We summarize the study as follows: an interferometric PolarCam system with no mechanical movement to extract the phase $\varDelta$ and the Stokes parameter $S_{3}$ of an object spatially within a part of a second is presented. The obtained results match quite well with those extracted by a Thorlabs polarimeter. Future work involves using this technology to achieve a dynamic spectro–interferometric PolarCam using swept-wavelength approaches. References Dynamic spectro-polarimeter based on a modified Michelson interferometric schemeThe calculation of thin film parameters from spectroscopic ellipsometry dataInterferometric ellipsometerStokes vector measurement based on snapshot polarization-sensitive spectral interferometryRobust snapshot interferometric spectropolarimetryA series of microscope objective lenses combined with an interferometer for individual nanoparticles detectionSPIE ProceedingsSPIE ProceedingsReview of passive imaging polarimetry for remote sensing applicationsTwo-wavelength in-line phase-shifting interferometry based on polarizing separation for accurate surface profilingAnalysis of a micropolarizer array-based simultaneous phase-shifting interferometer4D measurements of biological and synthetic structures using a dynamic interferometerPixelated mask spatial carrier phase shifting interferometry algorithms and associated errorsSingle-shot parallel four-step phase shifting using on-axis Fizeau interferometrySPIE ProceedingsInfrared birefringence spectra for cadmium sulfide and cadmium selenideMulti-object investigation using two-wavelength phase-shift interferometry guided by an optical frequency combHigh-precision 3D surface topography measurement using high-stable multi-wavelength digital holography referenced by an optical frequency combCoherent noise suppression in digital holography based on flat fielding with apodized aperturesSurface microtopography measurement of a standard flat surface by multiple-beam interference fringes at reflectionSingle-shot, dual-wavelength digital holography based on polarizing separationPhase Unwrapping via Graph CutsTwo-frame phase-shifting interferometry for testing optical surfaces
[1] Dembele V, Jin M, Baek B J and Kim D 2016 Opt. Express 24 14419
[2] Jellison G E 1996 Thin Solid Films 290–291 40
[3] Watkins L R 2008 Appl. Opt. 47 2998
[4] Kim D, Seo Y, Jin M, Yoon Y, Chegal W, Cho Y J, Cho H M, Abdelsalam D G and Magnusson R 2014 Opt. Express 22 17430
[5] Kim D, Seo Y, Yoon Y, Dembele V, Yoon J, Lee K and Magnusson R 2016 Opt. Lett. 41 2318
[6] Ibrahim D G A 2019 Curr. Appl. Phys. 19 822
[7] Millerd J E, Brock N J, Hayes J B, North-Morris M B, Novak M and Wyant J C 2004 Proc. SPIE 5531 304
[8] Millerd J, Brock N, Hayes J, Kimbrough B, Novak M, North-Morris M and Wyant J C 2005 Proc. SPIE 5856 14
[9] Tyo J S, Goldstein D L, Chenault D B and Shaw J A 2006 Appl. Opt. 45 5453
[10] Abdelsalam D G and Kim D 2011 Appl. Opt. 50 6153
[11]Azzam R M A and Bashara N M 1976 Ellipsometry and Polarized Light (Amsterdam: North-Holland)
[12] Novak N, Millerd J, Brock N, North-Morris M, Hayes J and Wyant J 2005 Appl. Opt. 44 6861
[13] Toto-Arellano N I 2017 J. Mod. Opt. 64 S20
[14] Kimbrough B T 2006 Appl. Opt. 45 4554
[15] Abdelsalam D G, Yao B, Gao P, Min J and Guo R 2012 Appl. Opt. 51 4891
[16] Brock N, Hayes J, Kimbrough B, Millerd J, North-Morris M, Novak M and Wyant J 2005 Proc. SPIE 5875 58750F
[17] Chenault D B and Chipman R A 1993 Appl. Opt. 32 4223
[18] Abdelsalam D G and Yasui T 2018 Appl. Phys. Lett. 112 171101
[19] Abdelsalam D G and Yasui T 2018 Opt. Lett. 43 1758
[20] Abdelsalam D G and Kim D 2011 Opt. Express 19 17951
[21] Abdelsalam D G, Shaalan M S and Eloker M M 2010 Opt. Lasers Eng. 48 543
[22] Abdelsalam D G, Magnusson R and Kim D 2011 Appl. Opt. 50 3360
[23] Bioucas-Dias J M and Valadão G 2007 IEEE Trans. Image Process. 16 698
[24] Tian C and Liu S 2016 Opt. Express 24 18695