Polarization, an essential characteristic of light, characterizes the direction in which the electric component of the field oscillates. The state of polarization (SoP) determines the interaction of light with anisotropic, chiral, and magnetized matter and thus forms the basis of diverse optical technologies such as polarization spectroscopy, ellipsometry, sensing, and imaging.1^–5 The most direct method for polarization measurements is to simultaneously detect the amplitude contrast and phase difference between two orthogonal polarization states.6^–9 However, the difficulty in capturing the phase information makes the direct measurement challenging. Conventional methods usually contain a series of intensity measurements using either spatial beam-splitting techniques (division of space) or rotated waveplates (division of time).10 Therefore, complicated optical setups that need precise adjustments are commonly required, making the solutions bulky and time-consuming. In addition, due to the narrowband restrictions of separate elements such as wave plates, the broadband feasibility of traditional polarimeters is drastically hindered. To explore compact, high-speed, and broadband-available polarimetry for polarization detection is urgently demanded in modern optics and informatics.11^,12

Spin depicts the orthogonal circular polarizations of light, i.e., left-handed circular polarization (LCP) and right-handed circular polarization (RCP). With the amplitude contrast and phase difference of opposite spins, one can determine the polarization state of light. In recent years, planar optics composed of plasmonic or dielectric artificial nanostructures have been widely adopted to overcome the bulky size of conventional optical systems.13^–18 Full-Stokes parameter detections can be carried out through the division of focal plane polarimetry with repeatable six polarization-dependent metalenses, which suffer from narrow working bands.19^–21 Polarimetry based on vector holography or multifocal metalenses is implemented for both SoP and spectral detection.6^,22 Matrix Fourier optics is also applied for division of aperture polarimetry, enabling a compact Mueller matrix imaging system.23^,24 Specific polarization or wavelength encoded metasurfaces combined with neural networks enable a high-accuracy full-Stokes polarimetry and high-resolution real-time polarization imaging with a single snapshot.8^,25^,26 Despite impressive progress, the existing methods still suffer from limited resolution in polarization imaging, narrow operating bands, and stick functions after fabrication. Liquid crystal (LC) is considered an ideal candidate for active planar optics thanks to its large optical birefringence over broadband and excellent responsiveness to various external stimuli.16^,27^–30 Thereby, LC-based polarimeters are expected to break the above limitations and release broadband tunable polarimetry with the merits of compact size, high speed, and optical efficiency.

In this Letter, we propose an all-LC polarimetry for polarization detection and imaging. It is cascaded by an LC $q$-plate and an LC polarization grating (LCPG). Via applying proper voltages to the separate elements, matching and mismatching of half-wave conditions are maintained, respectively. The amplitude contrast and phase difference between two orthogonal spins are read directly from the diffraction pattern. The intensity contrast between the ±1st orders and the rotation angle of the dark split in the 0th order reveals the amplitude contrast and phase difference, separately. A broadband tunability from visible to near-IR is demonstrated. An accuracy comparable to a commercial polarimeter is obtained in characterizing the feature points on the Poincaré sphere. Moreover, polarization imaging is verified on both a patterned phase retarder and a birefringent ruler. It supplies a practical solution for broadband polarization detection that overcomes the shortcomings of traditional techniques and may promote advances in polarization spectroscopy, ellipsometry, and many other advanced photonic applications.

Polarized light can be divided into LCP $(|L⟩=\frac{\sqrt{2}}{2}\left[\begin{array}{c}1\\ -i\end{array}\right])$ and RCP $(|R⟩=\frac{\sqrt{2}}{2}\left[\begin{array}{c}1\\ i\end{array}\right])$ components. The polarization state of light is expressed as $$E=a_{\text{L}}{\mathrm{e}}^{i{\phi }_{\text{L}}}|L⟩+a_{\text{R}}{\mathrm{e}}^{i{\phi }_{\text{R}}}|R⟩=a_{\text{R}}{\mathrm{e}}^{i{\phi }_{\text{R}}}(r{\mathrm{e}}^{i\mathrm{\Delta }\phi }|L⟩+|R⟩),$$(1)where $a_{\text{L}}$ and $a_{\text{R}}$ represent the amplitudes of orthogonal spins, $a_{\text{L}}^2+a_{\text{R}}^2=1$, $r=a_{\text{L}}/a_{\text{R}}$, and $r\in [0,\infty )$ are the amplitude contrast; ${\phi }_{\text{L}}$ and ${\phi }_{\text{R}}$ represent the phases of orthogonal spins, $\mathrm{\Delta }\phi ={\phi }_{\text{L}}-{\phi }_{\text{R}}$ and $\mathrm{\Delta }\phi \in [\mathrm{0,2}\pi ]$ depict the phase difference.6 According to Eq. (1), the polarization state is determined only by $r$ and $\mathrm{\Delta }\phi$. Each polarization state corresponds to a certain point on the Poincaré sphere, which can be depicted by a set of three Stokes parameters. The Stokes parameters can be calculated by $$\{\begin{array}{l}{\mathrm{S}}_1=\frac{2r}{1+r^2}\cos \mathrm{\Delta }\phi \\ {\mathrm{S}}_2=\frac{2r}{1+r^2}\sin \mathrm{\Delta }\phi \\ {\mathrm{S}}_3=\frac{1-r^2}{1+r^2}\end{array}.$$(2)

Figure 1 illustrates the configuration of the LC-cascaded polarimeter. It is composed of an LC $q$-plate, an LCPG, and a $y$-polarizer. The LC azimuthal orientations in $q$-plate ($\alpha$) and LCPG ($\beta$) follow $\alpha =q\theta$ and $\beta =\frac{\pi x}{\mathrm{\Lambda }}$, respectively, where $q$ is the topological charge, here $q=-0.5$, $\theta$ and $x$ are the angle and axis of the coordinate, and $\mathrm{\Lambda }$ is the period of the LCPG.16 Working at half-wave condition, the $q$-plate will convert the spin of incident light and encode a conjugated spiral phase of ${\mathrm{e}}^{i\theta }/{\mathrm{e}}^{-i\theta }$ to RCP and LCP incidences, separately. Accordingly, the incident plane wave is completely converted to an optical vortex (OV). The electric field is depicted by $$E^{\prime }={\mathit{\text{T}}}_{q\text{-plate}}·E=-i(a_{\mathrm{L}}{\mathrm{e}}^{i{\phi }_{\mathrm{L}}}{\mathrm{e}}^{i2\alpha }|R⟩+a_{\mathrm{R}}{\mathrm{e}}^{i{\phi }_{\mathrm{R}}}{\mathrm{e}}^{-i2\alpha }|L⟩),$$(3)where $T_{q\text{-}\text{plate}}$ is the Jones matrix of the $q$-plate (see Note 1 in the Supplementary Material). The LCPG works deviating from the half-wave condition, leaving the energy dispersed among the 0th and $\pm 1\mathrm{st}$ orders. The electric field of the light passing through the LCPG is depicted by $$E^{\prime }=T_{\mathrm{LCPG}}·E^{\prime }=-i\cos \left(\frac{\mathit{\Gamma }}{2}\right)E^{\prime }-\sin \left(\frac{\mathit{\Gamma }}{2}\right)a_{\mathrm{L}}{\mathrm{e}}^{i{\phi }_{\mathrm{L}}}{\mathrm{e}}^{2i(\alpha -\beta )}|L⟩-\sin (\frac{\mathit{\Gamma }}{2})a_{\mathrm{R}}{\mathrm{e}}^{i{\phi }_{\mathrm{R}}}{\mathrm{e}}^{-2i(\alpha -\beta )}|R⟩,$$(4)where $T_{\mathrm{LCPG}}$ and $\mathit{\Gamma }$ are the Jones matrix and phase retardation of the LCPG, separately. The three items on the right side of Eq. (4) exhibit 0th, $+1\mathrm{st}$, and $-1\mathrm{st}$ orders, subsequently. After being filtered by a $y$-polarizer and propagating a certain distance, the three orders are spatially separated. The intensities for $+1\mathrm{st}$ and $-1\mathrm{st}$ orders are $I_y^{+1}=\frac{p{a}_{\mathrm{L}}^2}{2}{\sin }^2(\frac{\mathit{\Gamma }}{2})$ and $I_y^{-1}=\frac{p{a}_{\mathrm{R}}^2}{2}{\sin }^2(\frac{\mathit{\Gamma }}{2})$, respectively, where $p$ is a parameter related to propagation distance. Although the intensities of $\pm 1\mathrm{st}$ orders are both reduced, the relative ratio $r$ remains the same. Therefore, $r$ can be calculated by the intensity ratio between $\pm 1\mathrm{st}$ orders. The intensity distribution for the 0th order is depicted by $$I_y=\frac{p^{\prime }}{2}{\cos }^2\left(\frac{\mathit{\Gamma }}{2}\right)[a_{\text{L}}^2+a_{\text{R}}^2-2a_{\text{L}}{a}_{\text{R}}\cos (\mathrm{\Delta }\phi -2\theta )],$$(5)where $p^{\prime }$ is a parameter related to propagation distance. Obviously, the intensity is $\theta$-dependent and minima occur at $\gamma =\frac{1}{2}\mathrm{\Delta }\phi$ or $\gamma =\frac{1}{2}\mathrm{\Delta }\phi +\pi$. As presented in Fig. 1, by measuring the angle between the $x$ axis and the intensity minima on OV at 0th order $\gamma$, one can obtain $\mathrm{\Delta }\phi$. Because both $r$ and $\mathrm{\Delta }\phi$ are read out from the diffraction pattern, the corresponding SoP of incident light is obtained.

The fabrication of the LC cascaded polarimeter is briefly described below. UV-ozone-cleaned indium-tin-oxide substrates are spin-coated with photoalignment agent SD1 (NCLCP, China). Two pieces of glass substrates are assembled face to face and separated by epoxy glue doped with $5\text{-}\mu \mathrm{m}$ spacers to form the cell. Afterward, a digital-mask photopatterning system and a multistep partly overlapping exposure are employed to transfer the designed orientations into the LC cells.31Figure 2(a) shows the azimuthal orientations of a $q$-plate with $q=-0.5$ and an LCPG with $\mathrm{\Lambda }=100\mu \mathrm{m}$, respectively. After filling with E7 (NCLCP, China), SD1 locally guides the orientation of LC through intermolecular interactions; thus, the objective geometric phases are successfully encoded to the cells. Figure 2(b) shows the micrographs of the fabricated $q$-plate and LCPG captured with a polarized optical microscope (POM, Nikon 50i, Japan). After the $q$-plate, LCPG, and the $y$-polarizer are stacked, the total thickness of the polarimeter is only 0.8 mm (see Note 2.1 in the Supplementary Material).

To precisely control the phase retardation of the different components, voltage- and wavelength-dependent intensity of the 0th order of the LCPG is characterized with a modified $V-T$ testing system (SCVR1, JCOPTIX, China). A supercontinuum laser (SuperK EVO, NKT Photonics, Denmark) is selected by an acousto-optic filter (SuperK SELECT, NKT, Denmark) and then illuminates the polarimeter for the measurements. According to the diagram shown in Fig. 2(c), the voltages marked by the solid line are selected as $V_1$ for different wavelengths (470 to 1100 nm) to match the half-wave condition (the cell gaps of $q$-plate and LCPG are the same), whereas the voltages marked by the dashed line are selected as $V_2$ to deviate from the half-wave condition. With the selected voltages applied, polychromatic diffraction patterns are generated by the LC-cascaded polarimeter. Figure 2(d) shows the diffraction patterns for $x$ polarization incidences of different colors. Cases from blue to infrared are demonstrated. The diffraction patterns are recorded with a charge-coupled device (CCD, AIC-401GC-USB, JCOPTIX, China). Notably, both the diameters of OVs and diffraction angles increase along with the increase of wavelength due to their intrinsic wavelength dependencies.32^,33 In addition, we also verified $y$ polarization at different wavelengths (see Note 2.2 in the Supplementary Material); the test results are consistent with the incident polarization state, which demonstrates the LC-cascaded polarimeter covering the entire visible spectrum.

To verify the accuracy of the proposed LC-cascaded polarimeter, we rotate wave plates to convert the incident linear polarization to a certain SoP. A half-wave plate is rotated from 0 to 90 deg in 15 deg increments to obtain six separate points on the equator, whereas a quarter-wave plate is rotated from $-45$ to 45 deg in 15 deg intervals to generate two opposite circular polarization and four elliptical polarization states (see Note 2.3 in the Supplementary Material). All generated SoP is detected by the LC cascaded polarimeter and a commercial polarimeter (PAX1000VIS, Thorlabs, United States) for reference. The measured results are both marked on the Poincaré sphere (Fig. 3). A close match is observed between green stars and purple points. The calculated root-mean-square errors (RMSE) for $S_1$, $S_2$, and $S_3$ are 0.052, 0.066, and 0.045, respectively. According to the spherical coordinates $(u,v)$ of preset SoP, the theoretical diffraction patterns are calculated with the Rayleigh–Sommerfeld vector diffraction theory. The experimental results show good agreement with the simulations. It vividly demonstrates the accuracy of the proposed LC polarimetry for polarization detection.

If the $q$-plate is changed to a $q$-plate array (see Note 3.1 in the Supplementary Material), the polarimeter can be used to identify the space-variant polarization fields; in other words, it is suitable for polarization imaging. Part of the $q$-plate array is shown in Fig. 4(a), which implies the resolution of the polarization imaging is $100\mu \mathrm{m}$. We fabricate a patterned phase retarder [Fig. 4(b)] to generate a specific patterned polarization distribution. The polarization ellipses are characterized by the commercial polarimeter and marked correspondingly. The blue and red ellipses represent the $L$- and $R$-handed polarizations, respectively. According to the diffraction pattern [Fig. 4(c)], we can calculate $r$ and $\mathrm{\Delta }\phi$ point by point in a similar way to uniform polarization detection (see Note 3.2 in the Supplementary Material).34Figure 4(d) exhibits the reconstructed polarization distribution according to the results measured by the LC-cascaded polarimeter, which is consistent with the results shown in Fig. 4(b). To further verify the universality of the LC-cascaded polarimeter, it is adopted to measure the polarization distribution of the light passing through a ruler. Figure 4(e) shows the interference color of the ruler with a pair of crossed polarizers, which originates from birefringence induced by stress.35 We select a small region, as marked in Fig. 4(e). Figure 4(f) exhibits the corresponding birefringent color variations under the POM. According to the diffraction pattern [Fig. 4(g)], the polarization distribution is calculated and presented in Fig. 4(h), which is consistent with the variation of interference color.

We proposed an all-LC polarimeter for polarization detection with a total thickness of approximately 0.8 mm. It is composed of an LC $q$-plate, an LCPG, and a $y$-polarizer. The polarimeter can be electrically tuned and thus suits the Stokes measurements in a broad band from visible light to near-infrared. By increasing the LC thickness, the tunable range of the working band can be further extended. Precise characterization of SoP with a compact polarimeter is crucial for various polarization measurements. The accuracy of the proposed polarimeter is comparable to commercial ones. By introducing an LC $q$-plate array, the feasibility of polarization imaging is validated using a patterned phase retarder. This approach provides a simple snapshot method for directly characterizing the phase difference as well as the amplitude ratio between two orthogonal spins. The incorporation of a lens undermines the compactness of the polarimeter; fortunately, it can be perfectly solved by replacing it with a polarization-independent planar lens. The Rayleigh–Sommerfeld vector diffraction theory is available to simulate the LC polarimetry for polarization imaging as well, and an exceptionally high consistency between the input polarization and the measurements is demonstrated (see Note 3.3 in the Supplementary Material).36 The accuracy of polarization imaging can be further enhanced through precalibration or incorporation with deep convolutional neural networks.12 The resolution of polarization imaging can be improved by reducing the size of a single $q$-plate.37 Spectral analysis can be performed based on the angular dispersion of the grating.

An all-LC Stokes polarimetry is proposed for polarization optics. It works on the simultaneous measurement of the amplitude contrast and phase difference of orthogonal spins with a single shot. The electrical tunability of the LC elements enables a wide tunable range of the working band (over 600 nm). The accuracy of the proposed polarimeter is comparable to commercial ones. With a $q$-plate array, polarization imaging is demonstrated in measurements of a patterned phase retarder and stress-induced birefringence of a plastic rule. The polarimeter exhibits the merits of compactness, high efficiency, high accuracy, and widely tunable working band and is promising in various applications such as polarization spectroscopy, ellipsometry, machine vision, and remote sensing.

Guang-Yao Wang received his BS degree from Qingdao University in 2021. He is currently working toward his PhD at the College of Engineering and Applied Sciences, Nanjing University. His research interests include liquid crystal planar optics, holography, and information security.

Han Cao is a PhD candidate working at the College of Engineering and Applied Sciences, Nanjing University. He received his MS degree from Nanjing University. His research interests include liquid crystal polymer planar optics and its engineering applications.

Zheng-Hao Guo is a PhD candidate at the College of Engineering and Applied Sciences, Nanjing University. He received his MS degree from South China Normal University in 2023. His current research interests are liquid crystals and optical simulation.

Chun-Ting Xu received her PhD in optical engineering from the College of Engineering and Applied Sciences, Nanjing University. She is currently a postdoctoral fellow at Nanjing University. Her research interests include chiral liquid crystals and planar photonics.

Quan-Ming Chen obtained his PhD from Nanjing University, in 2023, and now is a postdoctoral fellow at the Southern University of Science and Technology. His research interests include physics and optics in soft photonic crystals.

Wei Hu is a professor at the College of Engineering and Applied Sciences, Nanjing University. He received his PhD from Jilin University, in 2009. His current research interests include liquid crystal materials and optical devices.