1. INTRODUCTION

Parametric upconversion, such as sum-frequency generation (SFG) [1], has been regarded as an appealing solution for overcoming the detection constraints in infrared (IR) imaging. IR light holds significant usefulness in various cutting-edge areas, such as astronomical observation [2,3], semiconductor inspection [4], and environmental monitoring [5]. However, IR applications face issues of lower efficiency and greater noise of IR detectors compared with their visible (VIS) counterparts [6]. In the upconversion imaging architecture, instead of directly detecting IR fundamental frequency (FF) photons, the FF signal (at frequency ${\omega _1}$) is first transferred to the VIS with the help of another pump light (${\omega _2}$) via a second-order nonlinear (${{\boldsymbol {\hat \chi}}^{(2)}}$) material [7,8]. Then, the generated sum-frequency (SF) photons (${\omega _3} = {\omega _1} + {\omega _2}$, following the energy conservation law) are detected by VIS detectors. The first IR parametric upconversion can be traced back to the work by Midwinter and Warner in 1967 [9]. Over the past decades, parametric upconversion detection has received growing interest, and various important improvements have been made, such as finding new materials for higher upconversion efficiency [10], increasing imaging field of view (FOV) [11], manufacturing high-power IR sources for target illumination [12,13], and so on.

However, the current parametric upconversion techniques are still limited to reconstructing the intensity and spectral information of FF light, while the polarization information is completely lost. Polarization is a fundamental characteristic that describes how the electric field vector of light oscillates. While the intensity and spectrum inform us about the geometries and material compositions, polarization can bring richer information of targets. For example, by analyzing the light polarization changes induced by reflection or transmission from objects, people can infer the heterogeneity inside a material volume and the roughness or texture of a surface, and achieve enhanced contrast for objects that are difficult to distinguish otherwise [14–16]. In this context, polarization imaging has become an emerging technique to enhance many fields of remote sensing, underwater vision, semiconductor inspection, biological analysis, etc. [4,17–22].

 

Fig. 1. Schematic of nonlinear polarization imaging system. (a) Operation principle of the system. SFG can be represented by a Mueller matrix $\hat{\textbf{M}}({{{\hat{{\boldsymbol \chi}}}^{(2)}},{\textbf{S}^{{\omega _2}}}})$, via which ${\textbf{S}^{{\omega _1}}}$ of signal light is reversibly calculated from ${\textbf{S}^{{\omega _3}}}$ of SF light. (b) Sketch of the setup. GT, Glan–Taylor polarizer; PSG, polarization state generator; HWP, half-wave plate; QWP, quarter-wave plate; L, lens; PSA, polarization state analyzer. The polarization ellipse is defined by azimuth ($\phi$) and ellipticity ($\xi$) angles. Intensity images of “L,” “N,” and “house” masks are given at the top. The experimental coordinates were chosen to overlap with the LN principal crystallographic axes.

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Enlightened by the respective merits of the two techniques of parametric upconversion and polarization imaging, further combining them would enable IR polarization detection by VIS detectors, which can consequently benefit many fields of IR applications. In this paper, we demonstrate a novel nonlinear polarization imaging technique that reconstructs the polarization information of FF [near-IR (NIR)] signal light from the upconverted SF (VIS) light. We develop a new nonlinear Stokes–Mueller formalism that correlates the input FF polarization states with SF light. In this algorithmic framework, the incoming FF signal and outgoing SF radiation are represented by $4 \times 1$ Stokes vectors, and the nonlinear SFG process by a $4 \times 4$ Mueller matrix. We further experimentally achieve polarization imaging of different targets using our system, which reconstructs the polarization of the FF signal light with high precision. Our technique could find wide applications in remote sensing and industrial inspection.

2. NONLINEAR STOKES–MUELLER FORMALISM

We start our study with developing a Stokes–Mueller algebra for retrieving FF polarization by that of the SF wave, which forms the mathematical basis for our nonlinear polarization imaging system. We adopt a four-element Stokes vector ${\boldsymbol S} = [{s_0},{s_1},{s_2},{s_3}{]^T}$ ($T$ denotes matrix transpose) to describe the polarization state of light [23]. The polarization-dependent interaction between light and matter is denoted by a $4 \times 4$ Mueller matrix $\hat{\textbf{M}}$, and the incident polarization state ($\textbf{S}$) relates to the output state (${\textbf{S}^\prime}$) via ${\textbf{S}^\prime} = \hat{\textbf{M}}\textbf{S}$. This Stokes–Mueller formalism acts as a basis for traditional linear optical polarimetry to characterize light polarization [24]. In recent decades, several attempts have been made to apply polarimetry to the nonlinear optics regime, for example, the polarimetric analysis of second harmonic waves has been used to probe the molecular organization and crystallographic symmetry of materials [25–31]. Here, we develop a new nonlinear polarimetric imaging technique that aims to reconstruct FF polarization from that of SF light, as illustrated by Fig. 1(a).

According to nonlinear optics, the generated SF waves are closely related to the second-order nonlinear polarization ${\textbf{P}^{(2)}}$ excited inside the materials [32]:

3. SETUP OF NONLINEAR POLARIZATION IMAGING

Equation (2) suggests that ${\textbf{S}^{{\omega _1}}}$ and ${\textbf{S}^{{\omega _3}}}$ are bijective, namely, one FF polarization is mathematically paired with one SF polarization, and vice versa. This implies that the traditional phase matching strategy in bulk crystals cannot work here because in a phase-matched configuration, a certain combination of FF polarization components would interact constructively, overwhelming the SFG from other FF polarization combinations [32]. Thus, FF light with different polarizations would result in nearly identical SF polarizations, and the phase-matched SFG effectively acts as a polarizer [33], filtering out too much FF polarization information and making reconstruction impossible. In this context, we used an $x$-cut lithium niobate (LN) thin film on a silicon dioxide substrate (${{\rm SiO}_2}$, NANOLN) as the SFG material in our experiment. LN shows ultrabroadband transparency and strong second-order nonlinearity covering the VIS to mid-IR spectral range, and has worked as a feasible material platform for various photonics applications [34–40]. The LN film in our experiment is about 200 nm thick, which is smaller than the SFG coherence length (of the order of micrometers) [41]; thus, the phase matching condition is steadily relaxed, and all FF polarization information is maintained and retrievable.

A schematic of the experimental setup is shown in Fig. 1(b). Laser pulses were generated by a femtosecond laser. The pump and signal beams were temporally and spatially overlapped on the LN film (experimental details are given in Supplement 1). The optical axis of LN is denominated as $z(e)$ axis. The polarization of the pump beam was fixed along the ordinary ($o -$) axis of the LN crystal (Supplement 1). The SF at 404 nm was collected in the forward direction by a lens (75 mm focal length). Short-pass filters (BG40) were inserted in the SF path to block the remaining 808 nm light. The SF light then entered a home-built full-Stokes polarization state analyzer (PSA) comprising a rotating quarter-wave plate (QWP) and a Glan–Taylor (GT) polarizer and was then detected by a charge-coupled device (CCD, iXon Ultra 888). Ninety-one frames of the SF signal were recorded while the QWP was rotated from 0° to ${180^ \circ}$ with a step of ${2^ \circ}$. Through Fourier transformation of the SF intensity as a function of the QWP rotation angle, the SF Stokes parameters could be obtained [42].

We next determine the nonlinear Mueller matrix $\hat{\textbf{M}}({{{\hat{{\boldsymbol \chi}}}^{(2)}},{\textbf{S}^{{\omega _2}}}})$. This process is referred to as calibration of the system, which can be accomplished by resorting to multiple sets of known input and output polarization pairs. The FF polarization states were prepared by a polarization state generator (PSG) consisting of a GT polarizer, a half-wave plate (HWP), and a QWP. For the calibration end, 72 different signal polarization states ${\textbf{S}^{{\omega _1}}}$ that uniformly sample the Poincaré sphere were chosen as input FF polarizations [red dots in Fig. 2(a)]. The generated ${\textbf{S}^{{\omega _3}}}$ values were measured [blue dots on the SF Poincaré sphere in Fig. 2(b)]. The 16 elements of $\hat{\textbf{M}}({{{\hat{{\boldsymbol \chi}}}^{(2)}},{\textbf{S}^{{\omega _2}}}})$ can be numerically determined using a least squares method (Supplement 1). Afterwards, the FF polarization states are retrieved via Eq. (2). For a clearer comparison between the original and retrieved FF polarizations, the three-dimensional Poincaré spheres are expanded into planar maps, as shown in Figs. 2(c) and 2(d), in the forms of azimuth [$\phi = \frac{1}{2}{\rm arctan} (\frac{{{s_2}}}{{{s_1}}})$] and ellipticity [$\xi = \frac{1}{2}{\rm arctan} (\frac{{{s_3}}}{{\sqrt {s_1^2 + s_2^2}}})$] angles, which correspond to the physical orientation and shape of the polarization ellipse, respectively [depicted in Fig. 1(b)]. The overlap between the retrieved polarization states (small gray dots) and the directly measured data (large color dots) in Fig. 2(c) demonstrates the good precision of our technique in reconstructing FF polarization states from SF polarizations.

 

Fig. 2. Calibration of nonlinear polarization imaging system. (a) Experimentally measured Poincaré sphere of FF light (${\omega _1}$). Seventy-two polarization states (red dots) were chosen as input polarizations. (b) Measured SF Poincaré sphere. (c) The FF Poincaré sphere is expanded into a planar map in coordinates of azimuth-ellipticity angles ($\phi - \xi$). The input FF polarization states are presented by color dots, and the retrieved FF values are given by small gray dots. The overlap between them demonstrates the high precision of our system in reconstructing the FF polarizations. (d) $\phi - \xi$ map of the SF sphere. The SF dots are shown by the same color as the corresponding FF dots in (c).

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4. STOKES POLARIZATION IMAGING

We now apply our nonlinear polarization imaging system in polarization imaging, which aims to map the polarization states across a scene of interest. Binary masks with transparent letters of “L” and “N” were used as the targets, which were fabricated by focused ion beam milling through a 200 nm thick opaque gold film supported by a ${{\rm SiO}_2}$ substrate [intensity images of the masks are given in Fig. 1(b)]. The masks were inserted into the signal beam and projected onto the LN film plane. The polarization states of the signal light illuminating the mask were adjusted by the PSG in front. In this configuration, the polarization states were distributed uniformly across the signal beam. After interaction with the pump beam, the polarization information in the signal beam was transferred into the generated SF light. The SF Stokes parameters were analyzed at each pixel, and the SF polarization images were constructed afterwards. Figure 3 gives the results obtained by our system, which are given by different sets of polarization parameters. The first set is the Stokes parameters (${s_i}$), which present the good capability of our system in distinguishing different signal polarizations (more results are given in Fig. S2 of Supplement 1). The Stokes parameters are distributed uniformly inside the letter area, while the background is noisy because no SF signal was detected, resulting from the opacity outside the letter areas. For each mask, the SF images are given in the first column. The retrieved FF images following Eq. (2) are given in the second column, and the directly measured FF results are given in the third column. Images in forms of $\phi$ and $\xi$ are plotted in the last two rows. All the retrieved FF polarization images reproduce the directly measured FF results, indicating the fidelity and precision of our technique in reconstructing FF polarization information.

 

Fig. 3. Polarization images of masks. (a) Results of the “L” and “N” masks with uniform polarization distribution. Outlines of the masks are sketched by dashed lines. A scale bar of 500 µm is given in the bottom-left corner of the first picture. The first three rows give the images of Stokes parameters (${s_1}$, ${s_2}$, and ${s_3}$), and the last two rows show the $\phi$ and $\xi$ images. The measured SF waves, retrieved FF signals, and directly measured FF results are given in different columns. (b) Polarization images of the “house” mask, which contains an inhomogeneous polarization distribution. All retrieved FF images reproduce the directly measured FF results, confirming the precision of our technique.

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We further implement our technique for a more complicated target that contains an inhomogeneous birefringent structure, which introduces multiple polarization components into the signal light. A “house”-shaped mask was used, and a liquid crystal plate (DPP25-B, Thorlabs, not shown in Fig. 1) was inserted in front of the mask. This liquid crystal plate consists of a thin film of a birefringent liquid crystal polymer sandwiched between two glass sheets. The fast-axis orientation and phase retardation of the liquid crystal were manufactured to vary with a periodical pattern inside the plate. Thus, a spatially varying polarization distribution is produced over the cross section of the transmitted signal light. The pseudocolor plots in Fig. 3(b) essentially trace the contour of the spatial birefringence variation of the liquid crystal, and each color band in the images represents a region with the same polarization state. Similar to Fig. 3(a), the retrieved FF polarization images in the second column of Fig. 3(b) show good consistency with the directly measured FF values in the third column.

We then apply our technique to a practical scenario for visualizing the residual stresses in materials [43]. We used a plastic ruler as the target, as shown in Fig. 4(a). The stresses in the plastic impart localized variations and gradients in the material properties, making the material birefringent and nonhomogeneous. As a result, the polarization states of the transmitted light will become nonuniformly distributed. As shown in Fig. 4(b), complex pseudocolor fringe patterns are visible in the polarization images, which correspond to distribution of the internal stresses in the plastic ruler. In contrast, such stress distribution cannot be seen in the traditional intensity photo image in Fig. 4(a). The FOV of the SF images is shown by black circles in Fig. 4(b). The same FOV circles are further drawn in the FF images. The retrieved FF polarization images agree with the directly measured FF results within the FOV circles, confirming the precision of our technique. Notably, polarization patterns can be observed outside the FOV circles in the FF images, suggesting that the FF has a larger FOV than the SF. This can be explained by the fact that the laser beams have a Gaussian intensity profile, and the SFG occurs mainly in the central region of the FF beams.

 

Fig. 4. Polarization imaging of a plastic ruler. (a) Photographs of the ruler. A zoom-in image of the area for polarization imaging is given on the right. The scale bar is 1 mm. (b) The polarization images are given in forms of ${s_i}$, $\phi$, and $\xi$. The scale bar is 750 µm. The measured SF images are given in the first column. The retrieved FF polarization images are given in the second column, which reproduce the measured FF values (third column). Field of view (FOV) is marked by black circles.

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5. CONCLUSION

In conclusion, we have developed a nonlinear Stokes–Mueller formalism and built a nonlinear polarization imaging system based on parametric upconversion, by which we managed to retrieve the polarization information of the incident FF signal light from that of the upconverted SF waves. High-precision FF polarization imaging of various targets is demonstrated using our system. At present, we focus on the degenerate SFG process in which both the pump and probe lights have the same wavelength at the NIR. It is expected that our method is also valid for the non-degenerate SFG process, where the signal photon can be in mid- or long-wavelength IR range and upconverted by visible pump light [44,45], and therefore can be applied to night vision detection, vibrational spectroscopy, and environmental gas spectroscopy. Furthermore, by replacing the coherent signal beam with incoherent light that is naturally emitted/scattered by objects, our technique can be applied to, for example, astronomical observation [2,46–48] and environmental monitoring [49]. However, weaker probe light will result in smaller SFG, which increases the difficulty of detection. To solve this problem, stronger pump light can be used. Furthermore, the SFG efficiency of LN can be enhanced by mechanisms of Fano resonance, guided-mode resonance, or bound states in the continuum at nanoscale [38,39,50,51]. In addition, using a camera with higher sensitivity can be helpful. We can foresee a direct impact of our results on a variety of IR polarimetric applications, such as optical crystallography, chemical sensing, and disease diagnosis.