Light, in its various forms as a ray, an electromagnetic wave, and a quantum field, possesses fundamental properties such as intensity, phase, polarization, and wavelength [1]. These properties play a crucial role in understanding the behavior and interaction of light with matter [2,3]. Among them, polarization elucidates the vectorial nature of the electrical field oscillation, which can provide abundant information about sample structure [4,5], handedness of chiral molecules [6,7], surface and interface properties [8,9], etc. Spectroscopic measurement can reveal material composition [10], particle velocity and morphology [11,12], as well as chemical reaction kinetics [13,14]. Spectropolarimetry, which synergistically combines the strengths of both techniques by simultaneously measuring the polarization and spectrum, represents a more comprehensive approach. Accordingly, it has found wide applications in various fields, including chiral metamaterials [15], biomedicine [16,17], polarization remote sensing [18], astrophysics [19,20], and thin-film characterization [21,22].

As depicted in Figs. 1(a)–1(c), traditional spectropolarimeters can be divided into three categories: the division of space (DoS) method, division of time (DoT) method, and channeled method. In the DoS method, beam splitters and dispersive components are employed to separate and distinguish the light with different polarizations and spectrum [23–28]. In DoT method, the discrimination of polarizations is realized by utilizing rotating compensator or active elements, while the separation of spectral components is achieved using tunable filters [29–32]. The general channeled method makes use of polarization interference to amplitude modulate the polarization information onto either spectral or spatial carrier frequencies, from which the desired polarization information can be extracted [33–37]. Among them, the spectral-channeled polarimetry leverages multiple-order retarders to generate spectral carrier frequencies, and a spectrometer is used to measure the channeled spectrum [38–42]. Each method has its own advantages and disadvantages. For example, the DoT method is comparatively easier to implement, but it is susceptible to instability due to the presence of rotating or active elements. Moreover, it is not well-suited for scenarios involving rapidly changing inputs or high capacity processing. Although both the DoS and channeled methods allow for simultaneous measurements, the former may experience lower signal levels and involve complex assemble processes, while the latter often involves complex data inversion processes and is limited by the poor temperature stability of multi-order retarders [43,44]. Furthermore, these conventional methods are usually bulky due to the inherent drive devices, multiple measurement paths, and complex combinations of mechanisms for polarization and spectrum detection [44].

As an emerging class of planar diffractive optical elements (DOEs), metasurfaces can function as an ultracompact platform for polarization measurement [45–48]. As shown in Fig. 1(d), certain metasurface designs can further achieve spectropolarimetry by steering distinct input polarizations and spectral components into different spatial directions in the far field [49,50]. However, these devices have encountered challenges such as low efficiency, limited bandwidths, and poor accuracy [51,52]. Recently, a tunable liquid crystal metasurface was reported for computational spectropolarimetry [32]. However, it is still a type of DoT scheme, and the computational reconstruction was also time consuming. So far, achieving a fast, accurate, and broadband polarization measurement with a simple setup remains a significant challenge.

The vector beams (VBs) with varying polarization states across the wavefront [53–56] have attracted significant interest in a wide range of applications [57–59], such as but not limited to simple and single-shot polarization measurement [60–70]. Since their inception, VBs have evolved to encompass increasing polarization complexities. However, it is worth noting that such complexity may not be necessary in the field of polarization measurement. Many existing VBs polarimeters exhibit substantial polarization redundancy, as a section or even several points on the cross section of VBs can provide ample information for analyzing all the polarization data. This redundancy results in duplicated information in the spatial domain. To enhance spatial utilization, Suzuki et al. employed anisotropic gratings to generate ring-type diffraction [71], thereby exploiting the radially redundant spatial region for spectral measurements and enabling polarization measurement at multiple wavelengths. Nonetheless, their approach cannot distinguish polarizations of the same ellipticity but opposite handedness. To achieve broadband full polarization measurement, we propose a new approach by coding the spectrum and polarization information into polychromatic vector beams (PVBs). As depicted in Fig. 1(e), a custom-designed DOE and a vortex retarder are utilized to convert the incident polychromatic waves into PVBs: the former diffracts light of different wavelengths into concentric circles of different radii (ring rainbow-like diffractions), while the latter codes their polarization information into azimuthal-variant vector fields, as opposed to the rotating compensator utilized in the DoT method. The generated PVBs provide all the information for deducing the initial unknown broadband polarization, which can be achieved by mapping the PVBs onto an intensity image using an analyzer.

The PVB-based spectropolarimeter is experimentally demonstrated by measuring various broadband polarized beams. The measured Stokes parameters in visible light range (400–700 nm) show minor RMS errors (<1%), which is comparable to the accuracy of the excellent single-wavelength full Poincaré polarimeter [64]. Our spectropolarimeter also achieves a good spectral resolution of less than 0.8 nm, and it has an output rate of approximately 100 Hz. We further study the mutarotation process of glucose in an aqueous solution: the time varying optical rotation dispersion curves of the glucose solution are measured, and the optical rotation dispersion of two anomers, namely, α-D-(+)-glucose and β-D-(+)-glucose, is also measured. The proportional changes of the two anomers during the reaction process are monitored, and the measured results agree well with existing chemical knowledge.

The incident broadband beam with unknown polarizations can be described by a wavelength-dependent Stokes vector S(λ)=[S0(λ),S1(λ),S2(λ),S3(λ)]T. To measure S(λ), the incident light is firstly dispersed into a rainbow donut beam with different wavelength at different radii through a specially designed DOE [see Fig. 1(e)]. Therefore, the wavelength-dependent Stokes vector S(λ) is mapped into a radius-dependent S(r). The relation between wavelengths and radii can be established by the dispersive regularities of DOE. Then a vortex retarder is employed to encode the donut beam into a PVB, whose Stokes vector can be described by S(φ,r), where φ is the azimuth angle. The linear polarizer is added between the vortex retarder and camera to analyze the PVB into a spatially varying intensity distribution. The recorded intensity I(φ,r) is a function of the radius r and the azimuthal angle φ, which can be described by the Mueller–Stokes formula as I(φ,r)=a→(φ,r)·S(λ), where a→(φ,r)=[a0(φ,r),a1(φ,r),a2(φ,r),a3(φ,r)] is the first row of the total Mueller matrix of the PVB-based spectropolarimeter, and we term it modulated vector. For broadband incident polarized light, its polarization and spectrum are encoded in the measured light intensity as In×1(φi,rj)=a→n×4(φi,rj)·S4×1(λj)(i=1,2,…,n),(1)where j(j=1,2,…,m) is the sequence number of the m spectral channels (represented by radius rj or wavelength λj); In×1(φi,rj) is an n×1 vector, corresponding to the intensities measured at n different azimuth angles with a radius of rj; and a→n×4(φi,rj) is an n×4 matrix consisting of n corresponding modulated vectors at radius rj. By pre-calibrating the modulated vector a→(φ,r), the polarization of the incident unknown broadband light can be simultaneously measured from a single intensity image by using Eq. (1). For each spectral channel, in principle, four independent intensities at different azimuth angles are sufficient to solve the least squares problem. However, in practical experiments, the number of measurements (n) is usually much larger. This helps to reduce experimental measurement errors and the impact of noise and to improve the precision of the Stokes parameters by averaging over more data points, thus ensuring more reliable results. However, it is important to note that increasing n also increases computational time, as more measurements need to be processed. Therefore, a balance must be kept between using enough measurements to minimize errors and keeping the computational cost manageable. In our experiments, n=180 was chosen to provide high precision while maintaining reasonable processing time.

The feasibility of achieving dispersion of broadband light into circular rainbow rings can be explored through various optical elements. Traditional solutions, such as rotationally symmetric prisms such as axicons, may appear intuitively viable. However, these dispersive elements usually suffer from bulky configurations and low dispersive capabilities. On the contrary, DOEs possess the ability to intricately shape wavefronts using a single, compact optical component [72–74], making them ideal for achieving excellent axisymmetric dispersion mentioned above.

The operational wavelength range of the DOE was designed to be 400–700 nm. The optical components used in our system have a clear aperture of 1 in. (25.4 mm) and include an achromatic lens with a focal length of 70 mm. To optimize performance, we aim to maximize the diffraction angle for the longest wavelength (700 nm, corresponding to the outermost ring of the rainbow), while ensuring that the diffraction remains within 90% of the clear aperture (to maintain optical components reliability), which corresponds to 22.86 mm. This results in a diffraction angle of 9.27°, and we select 9° in the final design. The size of the DOE is set to 4mm×4mm to accommodate an incident beam with a maximum diameter of 4 mm.

The feasibility of the PVB-based spectropolarimeter is demonstrated by using the configuration shown in Fig. 3. A broadband light (Hamamatsu: EQ-99X-FC), a polarizer (P: Thorlabs, LPVIS100-A), and a quarter-wave plate (Q: Thorlabs, WPQ10E-633) are utilized to generate different incident spectral polarization states. The incident beam is then dispersed into a rainbow donut by the DOE, and an achromatic plano-convex lens (L: focal length f=70mm) is utilized to collimate the divergent rainbow donut beam. A liquid crystal polymer vortex retarder (V: Lbtek, QVR1-633-SP, with a quarter-wave retardance at 633 nm) is utilized to convert the rainbow donut beam into PVBs. Then the PVBs are mapped into a unique spatially varying intensity pattern by an analyzer (A: Thorlabs, LPVIS100-A). Due to the limited diffraction efficiency of the DOE, a certain amount of light will pass through without been diffracted. To suppress its adverse effects, an opaque disk is placed at the center of the analyzer to physically block the transmitted light. Finally, the intensity image is recorded by a scientific CMOS camera (sCMOS: Tucsen, Dhyana95) for further analysis.

Before the measurement, two pre-calibrated procedures need to be performed, namely, the alignment between the wavelengths and the pixel radii and the measurement of the modulated vector a→(r,φ).

The pixels with same radius in the image record the intensity variation of a specific wavelength, from which the polarization information at the corresponding wavelength can be calculated. To achieve broadband spectroscopic polarization measurement, the relation between the wavelengths and the pixel radii needs to be calibrated. According to the imaging formula of the plano-convex lens, the space radius d of the wavelength λ can be calculated as d=f·tan(θ). Further, the wavelength λ as a function of pixel radius r can be obtained by using the dispersive formula of DOE: λ=D·sin[arctan(k·r/f)], where k (unit: mm/pixel) serves as the scale factor for the conversion between pixel radius and space radius in camera imaging. For calibration purposes, single-band bandpass filters (Thorlabs: FKB-VIS-10, FWHM=10nm) at 450, 500, 550, 600, and 650 nm are placed after the broadband light sequentially, and the intensity images are captured to determine the pixel radius for each wavelength, which is shown in Table 2. The pixel radius of each intensity ring and its corresponding wavelength are then used to fit the curve shown in Fig. 4, from which the relation between the wavelengths and the pixel radii can be calculated.Table 2.

Pixel Radius at Each Wavelength

To measure the modulated vector a→(r,φ), the Q in Fig. 3 is replaced by an achromatic quarter-wave plate (Thorlabs, AQWP 10M-580), and six eigenstate polarization states, namely, horizontal (0°), vertical (90°), 45°, 135°, and right (RCP) and left circular polarization (LCP), are generated. Subsequently, the modulated vector a→(r,φ) is derived through linear operation on the six corresponding intensity images [60]. The obtained modulated vector a→(r,φ) is shown in Fig. 5.

To validate the performance of the PVB-based spectropolarimeter, incident beams with various broadband polarization states are generated using different azimuthal combinations of P and Q, and subsequently measured by the spectropolarimeter. To more clearly and visually demonstrate the wavelength and polarization modulation effects of the PVB-based spectrometer, we use a color camera to capture some of the images shown in Figs. 6(a) and 6(b). Figure 6(a) shows the intensity images of a spectrally uniform linearly polarized beam generated by a 30° P, while Fig. 6(b) displays a spectrally varying polarized beam generated by a 40° P and a 5° Q. For the actual calculations, we employed the Tucsen Dhyana95 sCMOS camera, which provides superior image quality and allows for better computational accuracy. The intensity images are then processed to obtain the intensity variations at different radii through a sampling process on corresponding circular regions [77]. The coordinates of the sampling points on each circle have been pre-calculated and are called during image processing to enhance computation speed. The calculated intensity variations within the VIS range (400–700 nm) are shown in Figs. 6(c) and 6(d), respectively, and the broadband polarization states are then calculated and calibrated [78]. The measurement results and measured errors are shown in Fig. 7. More measurements across the Poincaré sphere are performed by generating spectrally varying polarized beams with P fixed at 47° and the Q rotated to 0°, 25°, 50°, 75°, 90°, 115°, 140°, and 165°. The calculated results for the eight measurements (a total of 3272 points) are presented in Fig. 8. The RMS errors of the measured Stokes parameters in VIS range are calculated, and an error of <1% for each vector component is obtained. The results show that we have achieved broadband polarization measurement with exceptional measurement precision across the Poincaré sphere. The measurement accuracy we obtained is comparable to the excellent full Poincaré polarimeter and is much better than the well-established traditional polarimeter and the modern light–matter interactions based polarimeters [64].

Although our proposed PVB-based spectropolarimeter does not contain any mechanical rotating components and demonstrates good mechanical stability, potential sources of mechanical vibration errors still exist in the polarization measurement experiment. To be specific, in this experiment, we use a rotating polarizer and a quarter-wave plate to generate different states of broadband polarized light. This process inevitably introduces mechanical vibrations, which could affect the precision of the measurements. Furthermore, temperature fluctuations could also be a significant source of error. Despite using a zero-order vortex retarder in the system, temperature changes may still cause variations retardance of the vortex retarder, which in turn could impact the accuracy of the measurements.

The spectral resolution is a crucial parameter for the spectropolarimeter; in our proposed system, the spectral resolution can be characterized by the wavelength variation between two radially adjacent pixels, and it can be obtained by calculating the derivative of the wavelength–pixel radius function, which is less than 0.8 nm around the 400–700 nm range. In our experiment, the intensity image processing time is 9.895 ms (using an i7-10700F CPU at 2.90 GHz and MATLAB R2021a software), yielding an output rate of 100 Hz.

Using our PVB-based spectropolarimeter, we study chiroptical effects and the temporal evolution of glucose mutarotation. Glucose mutarotation was chosen as a case study due to the well-documented interconversion between its α- and β-anomeric forms, which leads to changes in optical rotation, making it an ideal example for demonstrating the application of our polarization measurement technique. Its time-dependent nature also showcases the precision and sensitivity of our method in tracking dynamic processes, while its extensive study in chemistry and biochemistry allows for easy comparison and validation against established experimental data.

Upon dissolution of glucose in water, a mixture rapidly forms, comprising both α and β anomers along with the straight-chain form. The α and β anomers undergo a cyclic process of opening to form the carbonyl group, followed by reclosure into either the α or β anomers. This opening and closing mechanism continuously repeats, resulting in an ongoing interconversion between the anomeric forms. At equilibrium, the mixture comprises 36% α-D-glucose and 64% β-D-glucose [79,80].

The experiment is performed at a temperature of 21°C, and the Q in the configuration is replaced by a 4 cm sample cell. First, the modulated intensity image of the distilled water is captured as a reference. Second, a fresh glucose solution of 0.2 g/mL is prepared by dissolving purely crystalline α-D-glucose in distilled water. The solution is immediately measured, and the first image is captured after 4 min of dissolving. Subsequently, 68 sequential images are captured until 291.4 min, and the last image is taken at 18 h after dissolving. Third, the intensity images of the 70 moments and the reference image are processed to calculate the spectral Stokes parameters. The optical rotations are then obtained by calculating the variations in polarization direction between the distilled water reference and the glucose sample. Finally, the optical rotatory dispersion (ORD) curve at each moment is fitted by using the formula [αi]21°Cλ=Ai/λ2+Bi/λ4 (i=1,2,3,…,70) [81], where [αi]21°Cλ is the specific rotation of the glucose solution at wavelength λ and moment i (see Visualization 1). The obtained constants Ai and Bi are plotted and fitted [82] in Figs. 9(a) and 9(b), and then the specific rotation of the glucose solution at any at time can be obtained as [α(t)]21°Cλ=A(t)/λ2+B(t)/λ4.

Consider that the solution contains only α-D-glucose at initial state, and very little free aldehyde is present at any time of the reaction [83]. 36% α-D-glucose and 64% β-D-glucose are obtained at equilibrium. Therefore, the mutarotation reaction satisfies the following relation: {Sα(t)·[αα]21°Cλ+Sβ(t)·[αβ]21°Cλ=[α(t)]21°CλSα(t=0)=1,Sα(t→+∞)=0.36Sα(t)+Sβ(t)=1,(2)where Sα(t) and Sβ(t) are the proportions of α-D-glucose and β-D-glucose, and [αα]21°Cλ and [αβ]21°Cλ are the specific rotations of α-D-glucose and β-D-glucose at wavelength λ.

As depicted in Fig. 10(a), the ORD of α-D-glucose, β-D-glucose, and the glucose solution at equilibrium can be calculated using Eq. (2). Additionally, the proportions of the two anomers during the reaction are also calculated and plotted, which are shown in Fig. 10(b). Some of the measured results, [αα]21°CD=110.4°, [αβ]21°CD=19.4° and [α]21°CD=52.2° (where D in [·]21°CD refers to the wavelength of the sodium D-line, i.e., λ = 589.3 nm), are coincident well with the reported results of 112°, 18.7°, and 52.7° [79,83], and the measured ORD of glucose solution at equilibrium also agrees well with the results in Ref. [84]. Furthermore, the temporal evolution of glucose solution mutarotation can be derived from Fig. 10(b). The concentrations of both α and β anomers exhibit exponential changes as the reaction progresses. At 116 min, the proportions of α and β anomers become equal, after which the concentration of the β anomer surpasses that of the α anomer. After approximately 300 min, the reaction gradually reaches an equilibrium state, with the α anomer representing approximately 36% and the β anomer accounting for 64%. Moreover, by calculating the derivative of the anomer concentrations with respect to time, the chemical reaction rate can also be determined.

These results highlight the promising potential of our approach in enabling detailed investigations for temporal evolution of chiral chemical processes and reactions, such as the time-dependent enantiodivergent synthesis proposed by Tu et al. [85]. Other potential applications, such as chiral pharmaceutical purity testing, blood glucose monitoring, and thin-film measurement, could be further explored in the future to enhance the technology’s versatility and impact.

In conclusion, we have demonstrated a single-shot, accurate, and simple approach for broadband polarization measurement. The concept relies on the mapping of any input broadband polarization into polychromatic structured vector beams by using a specifically designed DOE with axisymmetric dispersion and a vortex retarder. After being filtered by a polarizer, the polarization information can be obtained from the azimuthally variant intensity, while the spectral information can be extracted from the radial distributions. Various broadband polarization states are experimentally measured to demonstrate the feasibility of the PVB-based spectropolarimeter. The measured Stokes parameters in visible light range (400–700 nm) show excellent RMS errors of <1%. The PVB-based spectropolarimeter also shows good spectral resolution of <0.8nm, and its output rate can reach about 100 Hz, which is competent for monitoring a rapidly varying situation. We have further studied the mutarotation process of glucose; the measured results are coincident well with the reported results and provide helpful instruction for researching the time evolution of the reaction. This underscores the promise of our approach in facilitating detailed temporal investigations into chiral chemical processes and reactions.

However, the system also faces some limitations, such as the lack of imaging capability, as it currently only supports point-based spectrum and polarization measurements. In addition, the condition number of the system has not yet been optimized, and it can be further enhanced to improve accuracy. There are several potential extensions that can be further explored in the future. (i) The retardation of the vortex retarder can be chosen properly to obtain a minimum condition number (CN) for the system so that the influence of various errors can be minimized, and the measurement accuracy and stability can be further improved. (ii) The measurement speed and accuracy may be further improved by processing the intensity images through deep learning-based algorithms. (iii) The measurement and reconstruction of more complex polarization spectra could be conducted to further highlight the method’s high spectral resolution capabilities. Our research has forged a distinct pathway toward the single-shot, accurate, and simple detection of the broadband polarization, which promises to enhance the capabilities of any application reliant on spectral polarization sensing.