Visible, near-infrared dual-polarization lidar based on polarization cameras: system design, evaluation and atmospheric measurements

Zheng Kong; Jiheng Yu; Zhenfeng Gong; Dengxin Hua; Dengxin Hua; Liang Mei

doi:10.1364/OE.463763

1. Introduction

Aerosol, as a critical role in the atmosphere, can scatter and absorb solar radiation affecting the global radiation budget and global climate [1–3]. Moreover, aerosols with an aerodynamic diameter smaller than 2.5 µm can be inhaled in the body, causing a great impact on human health [4,5]. Lidar is an active remote sensing technique that can be used to investigate atmospheric parameters from the ground up to the top of the atmosphere, contributing to our knowledge of Earth’s atmosphere, especially the climatic effects from aerosols [6,7]. Dual/multiple-wavelength polarization lidar has been widely employed for acquiring optical and microphysical properties of atmospheric aerosols such as particle shapes and sizes [8–12]. Since 2006, the space-borne dual-wavelength Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) has provided extensive observations of clouds and aerosols [13–15]. On the other hand, many ground-based polarization lidar systems have also been developed and widely employed for characterizing aerosol optical properties, analyzing long-range aerosol transportation and discriminating cloud/aerosol types, etc.

In 2012, David et al. designed an ultraviolet-visible (UV-VIS) polarization lidar for aerosol monitoring, which can retrieve low particle depolarization ratios and specifically address the light backscattered by spherical and non-spherical tropospheric aerosols [16]. Meanwhile, Groß et al. employed dual-wavelength Raman-depolarization lidar systems to characterize volcanic aerosols [17] and long-range transported Saharan dust across the Atlantic Ocean [18]. In 2017, Haarig et al. performed triple-wavelength polarization lidar measurements for Saharan dust layers at Barbados and concluded that most of the irregularly shaped coarse-mode dust particles have sizes around 1.5–2 µm according to the spectral features of the depolarization ratio [19]. They have also demonstrated the potential of a triple-wavelength polarization Raman lidar for “3 + 3+3 profiling” (three backscatter coefficients, three extinction coefficients and three depolarization ratios) [20]. In 2020, Huang et al. developed a dual-polarization lidar system and showed that the volume depolarization ratios at 355 nm and 532 nm can be markedly different for typical types of aerosols and clouds [21]. In 2021, Bohlmann et al. conducted depolarization ratio measurements on atmospheric pollen by a multiwavelength Raman polarization lidar Polly^XT at 355 and 532 nm and a co-located Halo Doppler lidar at 1565 nm, showing wavelength dependence of the depolarization ratio owing to spruce pollen [22,23].

The pulsed dual-polarization lidar requires four polarization channels to implement co- and cross-polarization detection at two wavelengths apart from a high-performance Nd: YAG laser, leading to increasing complexity and cost of the lidar system. This is also a major reason that the most widely deployed lidar is still the single-wavelength lidar (with polarization channels) such as the micro-pulse lidar (MPL) and the ceilometer, despite the remarkable capability of the dual/multiple-wavelength polarization lidar technique for aerosol classification studies. Moreover, the gain ratio at each wavelength, which refers to the difference in the optoelectronic gain between the co- and cross-polarized receiver channels, requires additional polarization optics for frequent calibration as it may be sensitive to measurement conditions [24,25]. The misalignment (offset) angle between the laser’s polarization plane and the incident plane of the polarization beam splitter, which can affect the measurement accuracy of the depolarization ratio, is also difficult to evaluate [26]. Thus, developing a robust, low-cost and low-maintenance dual-polarization lidar for field-deployed atmospheric aerosol sensing is still of great interest.

Recently, imaging lidar techniques have been developed for atmospheric pollution monitoring [27–30] and aerosol polarization studies, where the 808-nm polarization measurement was achieved by alternately transmitting two orthogonally polarized laser beams into the atmosphere and measuring the backscattering signals based on the time-multiplexing scheme [31,32]. Nevertheless, the influence of the polarization crosstalk and the offset angle on the depolarization ratio is still difficult to evaluate. In 2021, a single-band polarization lidar, employing a linearly polarized continuous-wave 450-nm laser diode and a polarization camera, has been demonstrated for all-day accurate retrieval of the atmospheric depolarization ratio [33,34], which is not able to obtain the spectral characteristics of the depolarization ratio and the extinction coefficient.

In the present work, we proposed and designed a visible, near-infrared (VIS-NIR) dual-polarization imaging lidar for all-day field measurements of atmospheric aerosols by employing two high-power laser diodes (458 nm and 808 nm) as light sources and two polarization cameras as detectors. The polarization camera, featuring of high-integration, perfect alignment and real-time polarization measurement, integrates four-directional wire-grid polarizers (e.g., 0°, 45°, 90° and 135°) formed on the photodiodes of the image sensor chip [35–38]. The backscattering lidar signals of the transmitted linearly polarized laser beam can thus be detected by the polarization camera with a four directional polarization image captured in a single frame. The linear volume depolarization ratio (LVDR) and the offset angle can be precisely calculated from four-directional polarized backscattering signals at each wavelength. A rotating linear polarizer (RLP) method has been proposed and demonstrated for one-time measurement of the extinction ratio of the polarization camera, which can be used for eliminating the crosstalk between different polarization channels. Atmospheric aerosol measurements have been carried out to verify the system performance and investigate the aerosol optical properties, such as the aerosol extinction coefficient, the linear particle depolarization ratio (LPDRs), the color ratios and the ratio of LPDR.

2. VIS-NIR dual-polarization imaging lidar

2.1 Optical setup

The system schematic, picture and specifications of the VIS-NIR dual-polarization imaging lidar are shown in Fig. 1(a), 1(b) and Table 1, respectively. The lidar system consists of an enclosed transmitter and a detachable large focal receiver. The transmitter mainly includes two high-power laser diodes, optical components (polarizer, beam splitter, collimator, etc.) and electronics. All optics and electronics are placed on an aluminum-alloy base plate (ABL) and enclosed by a metal rain-proof cover with a 90-mm diameter optical window. A 458 nm laser diode with 4.75-W output power and an 808 nm laser diode with 5-W output power are used as light sources. Each laser diode is installed on a customized aluminum alloy mount with a high-power thermoelectric cooler and a cooling fan. The emitted laser beam by the 458 nm laser diode is reshaped by a pair of cylindrical lenses to reduce the divergence of the laser beam along the fast axis, matching the receiving angle of the collimating lens. The emitted laser beam by the 808 nm laser diode passes through the collimating lens directly since the divergence of the original laser beam is relatively small. The fast axis of the 458 nm laser diode and the slow axis of the 808 nm laser diode are oriented perpendicularly to the ABL to optimize the range resolution. Meanwhile, the polarization directions of the two laser beams are consistent with the slow axes of the laser diodes. As shown in Fig. 1(a), two linear polarizers at 458 nm and 808 nm are utilized to improve the degree of linear polarization (DoLP) of the transmitted laser beams. The transmission axis of the linear polarizer should be consistent with the polarization state of the emitted laser beam. According to previous studies [39], the extinction ratios of 450 nm and 808 nm laser diodes are about 100:1 and 68:1, respectively. Thus, the DoLP of the transmitted laser beam at 458 nm and 808 nm can both exceed 99.99% after passing through a high extinction ratio linear polarizer (e.g. >2000:1 at 450 nm and >1500:1 at 808 nm). The laser beams are coupled by a dichroic mirror placed with 45° tilted inside a cube, which are then folded by an elliptical mirror and transmitted into the atmosphere after collimated by a 76-mm diameter achromatic lens (f = 400 mm). The optical components as well as the corresponding mounts are fixed on a rotatable mounting plate. The overlapping field of view (FOV) between the receiving telescope and the transmitted laser beam can thus be aligned by manually adjusting the rotatable mounting plate. The rotatable mounting plate can be locked on the ABL after the lidar system is well aligned.

Fig. 1. (a) The architecture and (b) the picture of the VIS-NIR dual-polarization imaging lidar system. (c) The timing diagram of the polarization cameras and the modulation signal of the laser diodes, t₁ is the exposure time of the cameras. (d) Laser beam images recorded at 0° polarized channel at 458 nm (top) and 808 nm (bottom).

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Table 1. System specifications of the VIS-NIR dual-polarization imaging lidar

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The receiver mainly includes a Maksutov-Cassegrain telescope, a dichroic mirror, optical filters and two polarization cameras. The backscattering signals at 458 nm and 808 nm collected by the Maksutov-Cassegrain telescope are separated by a 45-degree tilted dichroic mirror and are then detected by two polarization cameras, which are referred to as the VIS-camera (458 nm) and the NIR-camera (808 nm), respectively. A 458 nm interference filter and an 808 nm interference filter are placed before the corresponding polarization cameras to suppress the sunlight background. Each polarization camera can capture a four-directional (0°, 45°, 90° and 135°) polarization image in a single frame. The receiver is protected by a customized metal cover with a 150-mm diameter optical window to protect the receiver from dust and rain.

A mini-computer has also been placed inside the transmitter for system controlling and real-time measurements. The whole lidar system, with the capability of unattended outdoor measurements, weighs about 25 kg and is mounted on a tripod with a heavy-duty gimbal, allowing flexible measurements at different orientations (horizontal 0∼360°, vertical 0∼90°).

2.2 Lidar signal acquisition, processing and inversion

As shown in Fig. 1(c), the trigger signal from the VIS-camera is fed to the NIR-camera as an external trigger for synchronized image acquisitions during measurements. Meanwhile, the trigger signal from the VIS-camera is also input into a Johnson counter, which generated a frequency-divided signal to modulate the intensities of the two laser diodes. Each polarization camera can alternately capture laser beam images and atmospheric background images in a region of interest (ROI, 2448×800 pixels) for dynamic background subtraction. The laser beam image can be decomposed into four polarization images with a dimension of 1224 (H) × 400 (V). The laser beam images recorded at 0° polarized channel at 458 nm and 808 nm are shown in Fig. 1(d). The original lidar signals at four directional polarized channels can be obtained by vertically binning the pixel intensities for each polarization image. The lidar signal is preprocessed by an adaptive Savitzky–Golay filter and signal interpolation to improve the signal-to-noise ratio (SNR) [40]. The exposure time of the polarization camera automatically varies between 100-500 milliseconds according to the total incident light intensity and the averaging number is changed accordingly to achieve the same response time for each lidar profile (e.g., 1 minute).

In the present lidar system, the cameras are placed parallel to the lens plane. This implies that the two polarization cameras can only obtain a clear image of the transmitted laser beam in a certain region. In generally, the telescope is focused on the far range to optimize the spatial resolution at the far range. Therefore, the laser beam image at a close range may be blurred due to out-of-focus imaging, leading to signal distortion in the near range (100$\sim$220 m). The corresponding distance of a given pixel can be determined according to geometrical optics [33]. The blind detection range is about 220 m.

In the present polarization imaging lidar system, the offset angle refers to the misalignment angle between the polarization plane of the emitted laser beam and the polarization direction of the 0° polarized channel of the polarization camera. The offset angle can be optimized to 0° by equalizing the backscattering signals at 45° and 135° polarized channels during system alignment. Even though there might be a small deviation to 0°, the offset angle ($\theta $) can also be precisely calculated from the four polarized backscattering signals based on the Stokes-Mueller formalism [39] (See Appendix A)

(1)$$\tan 2{\theta ^\lambda } = \frac{{V_2^\lambda ER_{135^\circ }^\lambda ({ER_{45^\circ }^\lambda + 1} )- ER_{45^\circ }^\lambda ({ER_{135^\circ }^\lambda + 1} )}}{{V_2^\lambda ER_{135^\circ }^\lambda ({ER_{45^\circ }^\lambda - 1} )+ ER_{45^\circ }^\lambda ({ER_{135^\circ }^\lambda - 1} )}} \times \frac{{ER_{0^\circ }^\lambda ({ER_{90^\circ }^\lambda - 1} )+ V_1^\lambda ER_{90^\circ }^\lambda ({ER_{0^\circ }^\lambda - 1} )}}{{ER_{0^\circ }^\lambda ({ER_{90^\circ }^\lambda + 1} )- V_1^\lambda ER_{90^\circ }^\lambda ({ER_{0^\circ }^\lambda + 1} )}}. $$

Then, the LVDR (${\delta _v}$) can be given by

(2)$$\delta _v^\lambda = \frac{{ER_{0^\circ }^\lambda (V_1^\lambda ER_{90^\circ }^\lambda - 1) - ER_{90^\circ }^\lambda (ER_{0^\circ }^\lambda - V_1^\lambda ){{\tan }^2}{\theta ^\lambda }}}{{ER_{90^\circ }^\lambda (ER_{0^\circ }^\lambda - V_1^\lambda ) + ER_{0^\circ }^\lambda (1 - V_1^\lambda ER_{90^\circ }^\lambda ){{\tan }^2}{\theta ^\lambda }}}, $$

Here $E{R_{0^\circ }}$, $E{R_{90^\circ }}$, $E{R_{45^\circ }}$ and $E{R_{135^\circ }}$ are referred to as the extinction ratios at 0°, 90°, 45° and 135° polarized channels, respectively. ${V_1}$ and ${V_2}$ are defined by $V_1^\lambda = {{({I_{90^\circ }^\lambda \eta_{0^\circ }^\lambda } )} / {({I_{0^\circ }^\lambda \eta_{90^\circ }^\lambda } )}}$ and $V_2^\lambda = {{({I_{135^\circ }^\lambda \eta_{45^\circ }^\lambda } )} / {({I_{45^\circ }^\lambda \eta_{135^\circ }^\lambda } )}}$, respectively. Here ${I_{0^\circ }}$, ${I_{90^\circ }}$, ${I_{45^\circ }}$ and ${I_{135^\circ }}$ are the measured signal intensities at four polarized channels, respectively. ${\eta _{0^\circ }}$, ${\eta _{90^\circ }}$, ${\eta _{45^\circ }}$ and ${\eta _{135^\circ }}$ are the relative quantum efficiencies (QEs) at four polarized channels, respectively. The subscript $\lambda $ indicates the wavelength of 458 nm or 808 nm.

The LPDR is closely related with particle shapes, which can be retrieved by [41]

(3)$$\delta _p^\lambda \textrm{ = }\frac{{(1 + \delta _m^\lambda )\delta _v^\lambda {{\boldsymbol R}^\lambda } - (1 + \delta _v^\lambda )\delta _m^\lambda }}{{(1 + \delta _m^\lambda ){{\boldsymbol R}^\lambda } - (1 + \delta _v^\lambda )}},{{\boldsymbol R}^\lambda } = \frac{{{\beta _m} + {\beta _p}}}{{{\beta _m}}}, $$

where ${\delta _m}$ is the molecular depolarization ratio, which can be theoretically calculated from the bandwidth of the interference filters, e.g., 0.008 at 458 nm with 2.3 nm bandwidth of narrowband filter, 0.004 at 808 nm with 3 nm bandwidth for present system [42]. R is the backscatter ratio, defined as the ratio of the total backscattering coefficient to the molecular component. The molecular backscattering coefficients can be calculated through known temperature and pressure [43].

The dual-polarization lidar has two elastic channels. The backscattering or extinction coefficients can be retrieved by inverting the lidar equation using the Fernald algorithm [44]. The key challenge for the Fernald algorithm is to determine the boundary value and the aerosol lidar ratio. The Douglas-Peucker algorithm has been employed to find a subinterval range at the far range, where the atmosphere can be considered homogeneous. The boundary value of the aerosol extinction coefficient is obtained according to the slope method. The aerosol lidar ratios at 458 nm and 808 nm, defined by the ratio of the extinction coefficient to the backscattering coefficient of the aerosol, are set to 50 and 60, respectively [45,46].

The color ratio is a parameter characterizing particle size and a large value indicates larger particle size [47]. The color ratio between the 808 nm and 458 nm backscattering coefficients is given by

(4)$$\chi (z) = \frac{{\beta _{808}^{||}(z) + \beta _{808}^ \bot (z)}}{{\beta _{458}^{||}(z) + \beta _{458}^ \bot (z)}}, $$

Here $\beta _\lambda ^{\textrm{||}}(z)$ and $\beta _\lambda ^ \bot (z)$ are the perpendicular and parallel polarized backscattering coefficients, respectively.

3. Evaluation of the polarization crosstalk and the offset angle

Understanding and evaluating potential systematic errors of the VIS-NIR dual-polarization imaging lidar are crucial for accurate LVDR measurement. In the polarization imaging lidar technique, the primary measurement error of the LVDR may be originated from several factors, e.g., the insufficient DoLP of the transmitted laser beam, the offset angle, the discrepancy of relative QEs among four polarized channels and the polarization crosstalk due to finite extinction ratio of the polarization camera. The DoLPs of the transmitted laser beams at 458 nm and 808 nm can be both improved up to 99.99% by employing linear polarizers with high extinction ratios and the difference between the relative QEs of the four polarized channels can be corrected by manufacturer parameters [39]. Thus, the polarization crosstalk introduced by finite extinction ratios and the offset angle are the remaining two factors that should be carefully evaluated.

3.1 Evaluation of the polarization crosstalk for the polarization camera

The finite extinction ratio of the polarization camera can lead to polarization crosstalk between different polarized channels and thus introduce measurement uncertainty on the LVDR. Nevertheless, the influence of the polarization crosstalk on the LVDR could be eliminated if the extinction ratio of the polarization camera can be precisely obtained. Although the manufacturer has provided typical values of the extinction ratio, the polarization characteristic may slightly deviate. Thus, accurate measurement on the extinction ratio of the polarization camera is crucial to the measurement precision of the LVDR.

In this work, a rotating linear polarizer (RLP) method has been proposed for the measurement of the extinction ratio of the polarization camera, as shown in Fig. 2. The extinction ratio for each wavelength should be performed separately by turning off the other laser diode. A rotatable half-wave plate (HWP) is placed in the optical path in the transmitter to adjust the polarization state of the linearly polarized laser beam. A rotatable linear polarizer with several order-of-magnitude higher extinction ratio (>10⁷:1) than that of the polarization camera (<500:1) is installed in front of the polarization camera, which only allows the backscattered light with the corresponding polarization state to pass through. The extinction ratios for different polarized channels can be measured by jointly rotating the linear polarizer and the HWP. The mathematical description and the measurement procedures of the RLP method are described below.

Fig. 2. The schematic diagram for the rotating linear polarizer (RLP) method. (a), (b) and (c) represent three cases for measuring the extinction ratios of the 0°, 45° and 135° polarized channels when the fast axis of HWP is consistent with the polarization plane of the transmitted laser beam and the transmission axis of the linear polarizer in the receiver is aligned to the 0°, 45°, 135° polarized channels of the polarization camera, respectively. (d) represents the case for measuring the extinction ratio of the 90° polarized channel when the fast axis of HWP is rotated to 45° and the transmission axis of the linear polarizer in the receiver is aligned to the 90° polarized channel of the polarization camera.

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The Mueller matrix of the experimental setup can be described as

(5)$${{\mathbf I}_\phi }\textrm{(}\gamma \textrm{,}\psi \textrm{) = }{\eta _\phi }{{\mathbf M}_{\textrm{cam}}}(\phi ){{\mathbf {\rm M}}_{\textrm{LP}}}(\gamma ){\mathbf F}(\pi ){{\mathbf M}_{\textrm{HWP}}}(\psi ){{\mathbf I}_\textrm{L}}. $$

Here ${{\mathbf I}_\textrm{L}}$ is the Stokes vector of the emitted laser beam. ${{\mathbf M}_{\textrm{HWP}}}(\psi )$ is the Mueller matrix of the ideal HWP, $\psi $ is the rotation angle between the fast axis of the HWP and the polarization state of the emitted laser beam. ${\mathbf F}(\pi )$ is the atmospheric scattering matrix with a scattering angle of $\pi $. ${{\mathbf M}_{\textrm{LP}}}(\gamma )$ is the Mueller matrix of the rotatable linear polarizer, $\gamma $ is the rotation angle between the transmission axis of the linear polarizer and the polarization state of the emitted laser beam. The Mueller matrix of the polarization camera can be expressed as the product between the Mueller matrix of the micro-polarizer (${{\mathbf M}_{\textrm{cam}}}(\phi )$) and the relative QE of the pixel below the on-chip micro-polarizer (${\eta _\phi }$), where the subscript $\phi $ refers to 0°, 45°, 90° or 135° polarization angles with respect to the polarization state of the transmitted laser beam. ${{\mathbf I}_\phi }(\gamma ,\psi )$ is the Stokes vector of the light detected by the polarization camera.

Since the DoLP of the transmitted laser beam after passing through the linear polarizer is very high (>99.99%), the Stokes vector of the linearly polarized light can be described by ${{\mathbf I}_\textrm{L}}\textrm{ = }{I_0}{\left[ {\begin{array}{cccc} 1&1&0&0 \end{array}} \right]^\textrm{T}}$. The Mueller matrix of an ideal HWP with a rotation angle of $\psi $ is given by

(6)$${{\mathbf M}_{\textrm{HWP}}}(\psi ) = \left( {\begin{array}{cccc} 1&0&0&0\\ 0&{\cos 4\psi }&{\sin 4\psi }&0\\ 0&{\sin 4\psi }&{ - \cos 4\psi }&0\\ 0&0&0&{ - 1} \end{array}} \right). $$

The Mueller matrix of the linear polarizer with a rotation angle of $\gamma $ is given by

(7)$${{\mathbf {M}}_{\textrm{LP}}}(\gamma ) = \frac{{{T_{\textrm{LP}}}}}{2}\left( {\begin{array}{cccc} 1&{\cos 2\gamma }&{\sin 2\gamma }&0\\ {\cos 2\gamma }&{{{\cos }^2}2\gamma }&{\sin 2\gamma \cos 2\gamma }&0\\ {\sin 2\gamma }&{\sin 2\gamma \cos 2\gamma }&{{{\sin }^2}2\gamma }&0\\ 0&0&0&0 \end{array}} \right). $$

Here, ${T_{\textrm{LP}}}$ refers to the sum of the maximum and minimum intensity transmittances. The attenuation of the linear polarizer is ignored since the extinction ratio is very high (>10⁷:1). The Mueller matrix of the micro-polarizer of the polarization camera at a certain polarization angle is shown in Appendix A.

When the fast axis of the HWP is aligned with the polarization state of the transmitted laser beam (i.e.$\psi \textrm{ = }0^\circ $) and the transmission axis of the rotatable linear polarizer is consistent with the 0° polarized channel of the polarization camera (i.e. $\gamma \textrm{ = }0^\circ $), only the parallel component of the backscattering signal can pass through the rotatable linear polarizer and then detected by the polarization camera. The detected backscattering signal intensity is given by

(8)$${i_\phi }(\gamma = 0^\circ ,\psi = 0^\circ ) = {\eta _\phi }\beta {I_0}({2 - d} ){T_{\textrm{LP}}}\frac{{T_\phi ^{\max }\textrm{ + }T_\phi ^{\min }}}{4}[{1 + {D_\phi }\cos 2\phi } ], $$

where $T_\phi ^{\max }$ and $T_\phi ^{\min }$ are the maximum and minimum intensity transmittances of the micro-polarizer of the polarization camera, respectively. ${D_\phi }$ is diattenuation parameter, which is expressed by ${D_\phi }\textrm{ = }{{({T_\phi^{\max } - T_\phi^{\min }} )} / {({T_\phi^{\max }\textrm{ + }T_\phi^{\min }} )}}$.

Thus, the extinction ratio of the 90° polarized channel can be retrieved

(9)$$E{R_{90^\circ }} = \frac{{T_{0^\circ }^{\max }}}{{T_{90^\circ }^{\max }}} \times \frac{{T_{90^\circ }^{\max }}}{{T_{90^\circ }^{\min }}} = \frac{{{i_{0^\circ }}(\gamma = 0^\circ ,\psi = 0^\circ ){\eta _{90^\circ }}}}{{{i_{90^\circ }}(\gamma = 0^\circ ,\psi = 0^\circ ){\eta _{0^\circ }}}}. $$

Here, assuming the maximum intensity transmittance of the 0° polarized channel is equal to that of the 90° polarized channel.

Similarly, the extinction ratio of 135° and 45° polarized channel can be retrieved by rotating the linear polarizer in the receiver to the corresponding orthogonal angles, which are given by

(10)$$E{R_{135^\circ }}\textrm{ = }\frac{{T_{45^\circ }^{\max }}}{{T_{135^\circ }^{\min }}}\textrm{ = }\frac{{{i_{45^\circ }}(\gamma = 45^\circ ,\psi = 0^\circ ){\eta _{135^\circ }}}}{{{i_{135^\circ }}(\gamma = 45^\circ ,\psi = 0^\circ ){\eta _{45^\circ }}}}, $$

(11)$$E{R_{45^\circ }}\textrm{ = }\frac{{T_{135^\circ }^{\max }}}{{T_{45^\circ }^{\min }}}\textrm{ = }\frac{{{i_{135^\circ }}(\gamma = 135^\circ ,\psi = 0^\circ ){\eta _{45^\circ }}}}{{{i_{45^\circ }}(\gamma = 135^\circ ,\psi = 0^\circ ){\eta _{135^\circ }}}}. $$

In principle, the extinction ratio of the 0° polarized channel ($E{R_{0^\circ }}$) can also be obtained by rotating the linear polarizer in the receiver ($\psi \textrm{ = }0^\circ $ and $\gamma \textrm{ = 9}0^\circ $). However, the parallel component of the backscattering signal would be greatly attenuated by the linear polarizer and only cross-polarized backscattering signal can be detected by the polarization camera, leading to a very low SNR and thus large measurement error on $E{R_{0^\circ }}$. On the other hand, the extinction ratio of the 0° polarized channel can be measured by rotating the polarization state of the transmitted laser beam by 90° ($\psi \textrm{ = 45}^\circ $) and rotating the linear polarizer by 90° ($\gamma \textrm{ = 90}^\circ $), which is given by

(12)$$E{R_{0^\circ }}\textrm{ = }\frac{{T_{90^\circ }^{\max }}}{{T_{0^\circ }^{\min }}} = \frac{{{i_{90^\circ }}(\gamma = 90^\circ ,\psi = 45^\circ ){\eta _{0^\circ }}}}{{{i_{0^\circ }}(\gamma = 90^\circ ,\psi = 45^\circ ){\eta _{90^\circ }}}}. $$

The extinction ratio experiments at the 0°, 90°, 45°and 135° polarized channels at 808 nm were carried out for four times under homogenous atmospheric conditions according to the RLP method. During each measurement, the extinction ratios along the measurement path have been spatially averaged to obtain a single value of the extinction ratio. The extinction ratios at 0°, 90°, 45° and 135° polarized channels at 808 nm were 82 ± 2:1, 81 ± 2:1, 71 ± 3:1 and 117 ± 7:1, respectively, which were slightly larger than typical extinction ratios provided by the manufacturer (74:1 at 0°, 74:1 at 90°, 60:1 at 45°, 107:1 at 135°). The systemic error of the LVDR introduced by polarization crosstalk at 808 nm can be reduced to less than 1% even with a small LVDR of 0.05 considering 2% ∼ 6% measurement uncertainties of the extinction ratios. On the other hand, it is worth mentioned that the influence of the measurement uncertainty of the extinction ratio at 0°, 45° and 135° on the LVDR is negligible (<1‰) compared to that at 90° [39]. This is especially true for 458 nm LVDR measurements, where a high extinction ratio of beyond 400:1 can be achieved. Thus, it is only necessary to accurately measure the extinction ratio of the 0° polarized channel, while the manufacture values can be used at other polarization angles. Experiments on the extinction ratio at 458 nm have also been carried out. However, it has been found out that the polarization crosstalk at the 0° or 90° polarization channel is at least two-order of magnitude lower than the incident light. Even though the practical extinction ratio at 458 nm is about 20% deviated from the value provided by the manufacturer (namely 467:1 at 0° and 469:1 at 90°), the measurement error at 458 nm introduced by the polarization crosstalk can be reduced to less than 1% for an LVDR of 0.05. Thus, the 458-nm extinction ratios from the manufacturer are utilized for the retrieval of the offset angle and the LVDR.

3.2 Online evaluation of the offset angle

A unique advantage of the polarization imaging lidar is the capability of online measurement of the offset angle. Continuous field measurements as well as statistical analysis on the offset angle have been performed from 23^rd January to 26^th February, 2022. For each wavelength, the four polarized directional backscattering signals with high SNRs during nighttime were utilized to retrieve an offset angle profile, which was spatially averaged along the laser beam path to obtain a single offset angle. The total measurement error of the offset angle includes systematic error and statistic error. The systematic error of the offset angle is mainly due to the uncertainty of the extinction ratio of the polarization camera [39]. Moreover, the statistic error of the offset angle, mainly originating from the random noise of the lidar signals, can be estimated by calculating the standard deviation of the retrieved offset angle profile. According to error propagation, the total measurement error of the offset angle can be estimated by

(13)$${ {\varDelta {\theta^\lambda }} |_{\textrm{total}}} = \sqrt {{{({{{ {\varDelta {\theta^\lambda }} |}_{\textrm{systematic}}}} )}^2} + {{({{{ {\varDelta {\theta^\lambda }} |}_{\textrm{random}}}} )}^2}}. $$

As can be seen from the temporal evolution of the offset angle for one-month consecutive measurements in Fig. 3, the offset angles at 458 nm and 808 nm are -0.13°±0.07° and 0.33°±0.09°, respectively. The temporal variation and the measurement error of the offset angle are both below 0.1°, leading to a negligible error (<1%) even for a small LVDR of 0.01 [24,39]. Thus, the effect on the LVDR from the offset angle, which has been widely existing in conventional polarization lidar techniques, can be online eliminated in almost real time. Moreover, the small temporal variation of the offset angle during the one-month measurement period also proved the reliability of the present lidar system for long-term field operation. It has been found out that the temporal variation of the offset angles at 458 nm and 808 nm had opposite correlation, which is mainly attributed to orthogonal polarization states of the transmitted laser beams at 458 nm and 808 nm. According to Eq. (2), the offset angle can be approximated by

(14)$$\tan 2\theta \approx \frac{{{V_2} - \textrm{1}}}{{{V_2}\textrm{ + 1}}} \times \frac{{1\textrm{ + }{V_1}}}{{1 - {V_1}}} \propto \frac{{I_{135^\circ }^\lambda }}{{I_{45^\circ }^\lambda }} - \textrm{1}. $$

Fig. 3. Temporal evolution of the offset angle for one-month measurements.

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As shown in Eq. (14), the relative relationship between the signal intensities at 45° and 135° polarization states determines whether the offset angle is positive or negative. If the 135° polarization signal is larger than that of 45° polarization, the offset angle is positive; vice versa. Since the transmitted laser beams of 458 nm and 808 nm have orthogonal polarization states, the 45° and 135° polarization channels for 458 nm and 808 nm are flipped. As a result, the temporal variation of the offset angles at 458 nm and 808 nm have opposite correlation.

4. Atmospheric aerosol measurements

Atmospheric aerosol measurements have been carried out to verify the system performance and investigate the aerosol optical properties in Dalian, northern China. The lidar system is deployed on the rooftop of the Graduate Education Building, Dalian University of Technology (DUT), Dalian, China, which is about 33 m above the ground. The PM10 and PM2.5 concentrations, the ambient temperature and the relative humidity (RH) are reported by an air quality monitor (Fairsense, A108) that is located at about 10 m away from the lidar system.

4.1 Dust observation – a case study

Atmospheric vertical sounding has been carried out in March 4^th 2022, and the temporal evolutions of the lidar profiles, the extinction coefficients, the LPDRs, the color ratios and the ratio of LPDR (${{\delta _p^{808}} / {\delta _p^{458}}}$) are shown in Fig. 4. During the measurement period, a large amount of dust was emitted from the Gobi Desert in northern China. The dust aerosols were transported to the Bohai sea and reached Dalian in about 28 hours after the emission, as can be seen from the HYSPLIT backward trajectories shown in Fig. 5(a). Besides, the large variation of the RH at about 3 km altitude (only 3%) also further confirmed the characteristics of the aerosol being dry (see Fig. 5(b)). The whole dust event was observed by the dual-polarization imaging lidar. From 8:00 on March 4^th, dust aerosols appeared beyond 250-m altitude. The dust aerosols started mixing with the local polluted particles at 14:00 on March 4^th.

As can be seen in Fig. 4, the time-height indications have been classified into different regions during the dust event. Before the dust event, the LPDR (0.11 ± 0.02 at 458 nm, 0.13 ± 0.02 at 808 nm) and color ratio (0.67 ± 0.09) for local polluted aerosols near the ground (region A) were small, indicating the presence of spherical particles in the atmosphere. With the occurence of the dust event (see region B), the 458 nm and 808 nm LPDRs were both becoming much larger (0.36 ± 0.03 at 458 nm, 0.33 ± 0.02 at 808 nm), indicating that non-spherical dust particles dominated at higher altitudes. Meanwhile, the ratio of LPDR (0.93 ± 0.08) in region B was smaller than that in region A (1.21 ± 0.12), while the color ratio (0.85 ± 0.12) was larger. During the dust event, the LPDRs of near ground aerosols (0.19 ± 0.03 at 458 nm, 0.22 ± 0.02 at 808 nm in region C) slighly increased comparing to the values before the dust event, which might be caused by the deposition of few dust particles from the upper atmosphere to the near the ground. However, the color ratio and the ratio of the LPDR were both comparable to those in region A, implying that the dust particles have not yet completely sinked to the ground surface. An interesting phenomenon was found that the ratio of LPDRs (1.27 ± 0.15) and color ratio (1.08 ± 0.19) in region D were larger than those in region B, which was mainly attributed to the well mixing of external dust with local polluted aerosols.

Fig. 4. (a) and (b) Time-space map of the total backscattering signals at 458 nm and 808 nm, respectively. (c) and (d) The corresponding time-space map of the extinction coefficients at 458 nm and 808 nm, respectively. (e) and (f) The time-space map of the LPDR at 458 nm and 808 nm, respectively. (g) and (h) The time-space map of color ratio and the ratio of the LPDRs, respectively.

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Fig. 5. (a) HYSPLIT back trajectory on 4^th March 2022 at 04:00 UTC (12:00 local time). (b) Variation in RH with altitude provided by a nearby radiosonde at around 12:00 (local time) on 4^th March 2022.

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4.2 Observation and evaluation of the depolarization ratio and color ratio

Atmospheric measurements have also been performed on a nearly horizontal path with an elevation angle of 3° from 24^th January to 25^th February 2022. Figure 6(a) and 6(b) show the temporal evolution of the total backscattering signals and the LVDR at 458 nm, respectively. During the measurement period, sunny and clear weather dominated, while the haze event and snowing also occurred. Figure 7 shows the typical backscattering profiles, the corresponding extinction coefficient profiles and the LPDR profiles at 458 nm and 808 nm in different measurement conditions. Under clean conditions, the lidar profiles at 458 nm and 808 nm are both almost a straight line with a smaller slope, indicating very homogeneous atmosphere near the ground, as shown by the red triangle line and purple dot-dash line in Fig. 7. Under haze conditions, the retrieving of the boundary value was restrained within 1 km (blue curve with circle in Fig. 7), since the multiple scattering effects would accumulate with the increase of the measurement range. When it was snowing, the LVDR at 458 nm and 808 nm both varied from 0.3 to 0.5, indicating irregular shapes of the snowflake, as shown by dot-dash boxes in Fig. 6(b). Besides, the extinction coefficient was not retrieved during snowing due to the inhomogeneity of the atmosphere caused by falling snowflakes. A strong backscattering echo was observed at around 1.5 km occasionally, e.g., dark green lidar profiles at around 19:43 on 30^th January 2022 in Fig. 7, which originated from the emission of a thermal power station for heating in winter.

Fig. 6. (a) Time-space map of the total backscattering signal. (b) The corresponding time-space map of the LVDR at 458 nm.

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Fig. 7. (a), (c), (e) Backscattering profiles, the corresponding extinction coefficient profiles and LPDR profiles at 458 nm. (b), (d), (f) Backscattering profiles, the corresponding extinction coefficient profiles and the LPDR profiles at 808 nm in different measurement hours.

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The relationship between the extinction coefficient, LPDR, color ratio and the ratio of LPDR for typical types of aerosols, such as clean continental, polluted continental and dust aerosols were investigated in detail to explore the aerosol optical properties in the urban area of Dalian. The horizontal lidar measurements from 10^th January to 26^th February, 2022, were classified into two types, namely the clean continental aerosol measurement and the polluted continental aerosol measurement to quantitatively evaluate the optical properties of different types of aerosols. The lidar profiles measured with a PM10 concentration less than 40 µg/cm³ and an RH less than 40% were categorized into clean continental aerosol measurements. If the PM10 concentration was larger than 100 µg/cm³ and the RH was more than 60%, the lidar profiles belonged to the polluted continental aerosol measurements. The retrieved extinction coefficient, LPDR and color ratio profiles were spatially averaged along the laser beam path and then temporal averaged in one hour. The lidar profiles recorded on March 4^th, 2022 during the dust event mentioned in Section 4.1 were utilized for statistical studies on the optical properties of dust aerosols.

Figure 8(a) and 8(b) show the relationships between the extinction coefficient and the LPDR for different types of aerosols at 458 nm and 808 nm, respectively. As can be seen, the three aerosol types can also be spatially distinguished in the extinction coefficient-LPDR plot. For clean continental aerosols, the extinction coefficients were 0.07 ± 0.03 km^-1 at 458 nm and 0.04 ± 0.02 km^-1 at 808 nm, and the LPDRs were 0.10 ± 0.03 at 458 nm and 0.12 ± 0.04 at 808 nm. For polluted continental aerosols, the extinction coefficient at 458 nm (0.50 km^-1 ∼ 1.21 km^-1) was much larger than that of 808 nm (0.18 km^-1 ∼ 0.77 km^-1), and the LPDRs were only about 0.02 ± 0.003 at 458 nm and 0.02 ± 0.008 at 808 nm, indicating that the particles were close to spherical during the haze event. For dust aerosols, the LPDRs were 0.34 ± 0.02 at 458 nm and 0.33 ± 0.01 at 808 nm, indicating the atmosphere is dominated by non-spherical dust particles. As shown in Fig. 8(c), the the LPDR plots of 458 nm and 808 nm for clean continental, polluted continental and dust aerosols all distributed around the diagonal of the figure (black dashed line). As can be seen in Fig. 8(d), the color ratios of clean continental aerosols and polluted continental aerosols were 0.50 ± 0.08 and 0.41 ± 0.08, respectively, while the color ratio of dust aerosols was much larger (0.94 ± 0.09), indicating that the extinction coefficient of dust aerosols at 808 nm is comparable to that of 458 nm, and some times even larger. Previous studies have also found out that the extinction coefficient of the dust at 355 nm can be lower than that at 532 nm, which might be resulted from the spectral dependence of the imaginary part of the dust refractive index [48]. These results indicated that the polluted continental aerosols mainly consisted of fine mode particles, while dust aerosols were mainly composed of coarse mode particles [49,50]. The ratio between LPDRs at 808 nm and 458 nm (${{\delta _p^{808}} / {\delta _p^{458}}}$) for the clean continental aerosols, polluted continental aerosols and the dust aerosols were 1.26 ± 0.16, 0.99 ± 0.26, 0.98 ± 0.05, respectively. These promising results indicated that the combination of the extinction coefficient, LPDR and color ratio can efficiently identify different aerosol types.

Fig. 8. (a), (b) The relationships between the extinction coefficient and LPDR at wavelengths of 458 nm and 808 nm, respectively. (c) The relationship between the LPDRs at 458 nm and 808 nm. (d) The relationship between the color ratio and the ratio of the LPDR.

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5. Conclusion

In this work, a VIS-NIR dual-polarization lidar has been proposed and implemented for all-day field measurements of atmospheric aerosols. A rain-proof and compact imaging lidar system with unattended capability was designed by employing two continuous-wave laser diodes (458 nm and 808 nm) as light sources and two polarization cameras with micro-polarizer arrays as detectors, which features low cost and low maintenance. The LVDR and the offset angles can be accurately retrieved from four-directional polarized backscattering signals at wavelengths of 458 nm and 808 nm without employing additional optical components and sophisticated system adjustments.

Evaluations of the polarization crosstalk of the polarization camera and the offset angle have been performed in detail. An RLP method based on the Stokes-Mueller formalism has been proposed and demonstrated for measuring extinction ratios of the polarization camera, which can be used for eliminating the polarization crosstalk of the polarization camera. The measured extinction ratio at 0°, 90°, 45° and 135° polarized channels at 808 nm were 82 ± 2:1, 81 ± 2:1, 71 ± 3:1 and 117 ± 7:1, respectively. The systemic error of the LVDR introduced by polarization crosstalk at 808 nm can be reduced to less than 1% even with an LVDR of 0.05. The extinction ratio at 458 nm can be directly obtained from the manufacturer (namely, 467:1 at 0° and 469:1 at 90°) to reduce the polarization crosstalk effect. The offset angles measured over one month without further adjustment were -0.13°±0.07° at 458 nm and 0.33°±0.09° at 808 nm, which demonstrated good system stability for long-term measurement. The temporal variation and the measurement error of the offset angle, which were both less than 0.1°, caused a negligible error even for a small LVDR of 0.01. Thus, the influence of the offset angle on the LVDR, which has been widely existing in conventional polarization lidar techniques, can be online eliminated in almost real time.

Atmospheric measurements have been carried out to verify the system performance and investigate the aerosol optical properties. During vertical sounding measurements, a dust event transported from the Gobi Desert in northern China has been observed by the dual-polarization lidar system. The spectral characteristic of the aerosol extinction coefficient, the color ratio, LPDR and the ratio of LPDR were studied from atmospheric measurements for one month, from which the clean continental aerosol, the polluted continental aerosol and the dust aerosol can be classified. The measurement results show great potential of employing the present dual-polarization lidar for characterizing aerosol optical properties, discriminating aerosol types and analyzing long-range aerosol transportation.

Appendix A: Derivation of the linear volume depolarization ratio and the offset angle

Since the degree of the linear polarization of the transmitted laser beam after passing through the linear polarizer is very high (>99.99%), the transmitted laser beam can be considered ideal linearly polarized light. Thus, the Stokes vector of the linearly polarized light can be written by

(15)$${{\mathbf I}_\textrm{L}}\textrm{ = }{I_0}{\left[ {\begin{array}{cccc} 1&1&0&0 \end{array}} \right]^\textrm{T}}. $$

Here, ${I_0}$ is the total radiation intensity of the laser diode.

The rotation Mueller matrix of the offset angle ($\theta $) can be described by

(16)$${\mathbf R}(\theta ) = \left( {\begin{array}{cccc} 1&0&0&0\\ 0&{\cos 2\theta }&{ - \sin 2\theta }&0\\ 0&{\sin 2\theta }&{\cos 2\theta }&0\\ 0&0&0&1 \end{array}} \right). $$

The Mueller matrices of the transmitter and the receiver are given by Eq. (17) and Eq. (18), respectively.

(17)$${{\mathbf M}_\textrm{T}}\textrm{ = }{T_\textrm{T}}\left( {\begin{array}{cccc} 1&{{D_\textrm{T}}}&0&0\\ {{D_\textrm{T}}}&1&0&0\\ 0&0&{\sqrt {1 - D_\textrm{T}^2} \cos {\Delta _\textrm{T}}}&{\sqrt {1 - D_\textrm{T}^2} \sin {\Delta _\textrm{T}}}\\ 0&0&{ - \sqrt {1 - D_\textrm{T}^2} \sin {\Delta _\textrm{T}}}&{\sqrt {1 - D_\textrm{T}^2} \cos {\Delta _\textrm{T}}} \end{array}} \right), $$

(18)$${{\mathbf M}_\textrm{R}}\textrm{ = }{T_\textrm{R}}\left( {\begin{array}{cccc} 1&{{D_\textrm{R}}}&0&0\\ {{D_\textrm{R}}}&1&0&0\\ 0&0&{\sqrt {1 - D_\textrm{R}^2} \cos {\Delta _\textrm{R}}}&{\sqrt {1 - D_\textrm{R}^2} \sin {\Delta _\textrm{R}}}\\ 0&0&{ - \sqrt {1 - D_\textrm{R}^2} \sin {\Delta _\textrm{R}}}&{\sqrt {1 - D_\textrm{R}^2} \cos {\Delta _\textrm{R}}} \end{array}} \right). $$

Here T_T, D_T, ${\Delta _\textrm{T}}$ and T_R, D_R, ${\Delta _\textrm{R}}$ are the effective transmittance, diattenuation parameter, and retardance of transmitter and receiver, respectively. In the present work, the transmitter and the receiver are considered ideal. Thus, Eq. (17) and Eq. (18) can be simplified as unit matrices.

The atmospheric scattering volume is assumed as randomly oriented and axially symmetric scatterers, and the scattering matrix can be written by

(19)$${\mathbf F}\textrm{(}\pi \textrm{) = }\beta \left( {\begin{array}{cccc} 1&0&0&0\\ 0&{1 - d}&0&0\\ 0&0&{d - 1}&0\\ 0&0&0&{2d - 1} \end{array}} \right), $$

(20)$$d = \frac{{2{\delta _v}}}{{1 + {\delta _v}}}. $$

Here $\beta $ is the backscattering coefficient and d is the depolarization parameter. ${\delta _v}$ is the linear volume depolarization ratio (LVDR).

The Mueller matrix of the micro-polarizer of the polarization camera at a certain polarization angle (0°, 90°, 45° or 135°) can be deduced

(21)$$\begin{array}{l} {{\mathbf M}_{\textrm{cam}}}(\phi ) = \frac{{T_\phi ^{\max }\textrm{ + }T_\phi ^{\min }}}{\textrm{2}}\\ \left( {\begin{array}{cccc} 1&{{D_\phi }\cos 2\phi }&{{D_\phi }\sin 2\phi }&0\\ {{D_\phi }\cos 2\phi }&{{{\cos }^2}2\phi + Z{{\sin }^2}2\phi \cos \Delta }&{(1 - Z\cos \Delta )\sin 2\phi \cos 2\phi }&{ - Z\sin 2\phi \sin \Delta }\\ {{D_\phi }\sin 2\phi }&{(1 - Z\cos \Delta )\sin 2\phi \cos 2\phi }&{{{\sin }^2}2\phi + Z{{\cos }^2}2\phi \cos \Delta }&{Z\cos 2\phi \sin \Delta }\\ 0&{Z\sin 2\phi \sin \Delta }&{ - Z\cos 2\phi \sin \Delta }&{Z\cos \Delta } \end{array}} \right) \end{array}, $$

where $T_\phi ^{\max }$ and $T_\phi ^{\min }$ are the maximum and minimum intensity transmittances of the micro-polarizer of the polarization camera, respectively. ${D_\phi }$ is diattenuation parameter, which is expressed by ${D_\phi }\textrm{ = }{{({T_\phi^{\max }\textrm{ - }T_\phi^{\min }} )} / {({T_\phi^{\max }\textrm{ + }T_\phi^{\min }} )}}$. ${Z_\phi }$ is defined by ${Z_\phi } = \sqrt {1 - D_\phi ^2} $. $\Delta $ is the retardance of the micro-polarizer.

According above description, the output Stokes vector (${{\mathbf I}_\phi }$) for the present lidar system is given in Eq. (22)

(22)$${{\mathbf I}_\phi }\textrm{ = }{\eta _\phi }{{\mathbf M}_{\textrm{cam}}}(\phi ){{\mathbf {M}}_\textrm{R}}{\mathbf F}(\pi ){{\mathbf M}_\textrm{T}}{\mathbf R}(\theta ){{\mathbf I}_\textrm{L}}. $$

Here, ${\eta _\phi }$ refers to the relative QE of the polarization camera, where the subscript $\phi $ refers to 0°, 45°, 90° or 135° polarization angles with respect to the polarization state of the transmitted laser beam.

The detected backscattering signal at 0°, 90°, 45° or 135° polarization channel, which refers to the first element of the Stokes vector (${{\mathbf I}_\phi }$), can be expressed by Eq. (23) to Eq. (26), respectively:

(23)$${i_{0^\circ }} = {\eta _{0^\circ }}\beta {I_0}\frac{{T_{0^\circ }^{\max } + T_{0^\circ }^{\min }}}{2}[{1 + {D_{0^\circ }}(1 - d)\cos 2\theta } ], $$

(24)$${i_{90^\circ }} = {\eta _{90^\circ }}\beta {I_0}\frac{{T_{90^\circ }^{\max } + T_{90^\circ }^{\min }}}{2}[{1 - {D_{90^\circ }}(1 - d)\cos 2\theta } ], $$

(25)$${i_{45^\circ }} = {\eta _{45^\circ }}\beta {I_0}\frac{{T_{45^\circ }^{\max } + T_{45^\circ }^{\min }}}{2}[{1 + {D_{45^\circ }}(d - 1)\sin 2\theta } ], $$

(26)$${i_{135^\circ }} = {\eta _{135^\circ }}\beta {I_0}\frac{{T_{135^\circ }^{\max } + T_{135^\circ }^{\min }}}{2}[{1 - {D_{135^\circ }}(d - 1)\sin 2\theta } ]. $$

Here the extinction ratio of each polarized channel of the polarization camera can be expressed by $E{R_\phi } = {{\textrm{T}_\phi ^{\max }} / {\textrm{T}_\phi ^{\min }}}$. The diattenuation of the polarization camera (${D_\phi }$) can also be deduced

(27)$${D_\phi } = \frac{{\textrm{T}_\phi ^{\max } - \textrm{T}_\phi ^{\min }}}{{\textrm{T}_\phi ^{\max }\textrm{ + T}_\phi ^{\min }}}\textrm{ = }\frac{{E{R_\phi } - 1}}{{E{R_\phi } + 1}}. $$

Thus, the follow formulae can be expressed from Eq. (23) to Eq. (26),

(28)$$\frac{{{i_{0^\circ }}}}{{{\eta _{0^\circ }}}} = {I_0}\beta \frac{{T_{0^\circ }^{\max }}}{2}\left( {1 + \frac{1}{{E{R_{0^\circ }}}}} \right)\left[ {1 + \frac{{E{R_{0^\circ }} - 1}}{{E{R_{0^\circ }} + 1}}(1 - d)\cos 2\theta } \right], $$

(29)$$\frac{{{i_{90^\circ }}}}{{{\eta _{90^\circ }}}} = {I_0}\beta \frac{{T_{90^\circ }^{\max }}}{2}\left( {1 + \frac{1}{{E{R_{90^\circ }}}}} \right)\left[ {1 - \frac{{E{R_{90^\circ }} - 1}}{{E{R_{90^\circ }} + 1}}(1 - d)\cos 2\theta } \right], $$

(30)$$\frac{{{i_{45^\circ }}}}{{{\eta _{45^\circ }}}} = {I_0}\beta \frac{{T_{45^\circ }^{\max }}}{2}\left( {1 + \frac{1}{{E{R_{45^\circ }}}}} \right)\left[ {1 - \frac{{E{R_{45^\circ }} - 1}}{{E{R_{45^\circ }} + 1}}(1 - d)\sin 2\theta } \right], $$

(31)$$\frac{{{i_{135^\circ }}}}{{{\eta _{135^\circ }}}} = {I_0}\beta \frac{{T_{135^\circ }^{\max }}}{2}\left( {1 + \frac{1}{{E{R_{135^\circ }}}}} \right)\left[ {1 + \frac{{E{R_{135^\circ }} - 1}}{{E{R_{135^\circ }} + 1}}(1 - d)\sin 2\theta } \right]. $$

Hereby it is assumed that the maximum intensity transmittance of the 0° polarized channel is equal to that of the 90° polarized channel, and the maximum intensity transmittance of the 45° polarized channel is equal to that of the 135° polarized channel. If V₁ and V₂ are defined as

(32)$${V_1} = {{\left( {\frac{{{i_{90^\circ }}}}{{{\eta_{90^\circ }}}}} \right)} / {\left( {\frac{{{i_{0^\circ }}}}{{{\eta_{0^\circ }}}}} \right)}}, $$

(33)$${V_2} = {{\left( {\frac{{{i_{135^\circ }}}}{{{\eta_{135^\circ }}}}} \right)} / {\left( {\frac{{{i_{45^\circ }}}}{{{\eta_{45^\circ }}}}} \right)}}. $$

Then, the V₁ and V₂ can be derived as follows based on Eqs. (28) - (31)

(34)$${V_1} = \frac{{E{R_{0^\circ }}[{(E{R_{90^\circ }} + 1) - (E{R_{90^\circ }} - 1)(1 - d)\cos 2\theta } ]}}{{E{R_{90^\circ }}[{(E{R_{0^\circ }} + 1) + (E{R_{0^\circ }} - 1)(1 - d)\cos 2\theta } ]}}, $$

(35)$${V_2} = \frac{{E{R_{45^\circ }}[{(E{R_{135^\circ }} + 1) + (E{R_{135^\circ }} - 1)(1 - d)\sin 2\theta } ]}}{{E{R_{135^\circ }}[{(E{R_{45^\circ }} + 1) - (E{R_{45^\circ }} - 1)(1 - d)\sin 2\theta } ]}}. $$

The offset angle can be deduced from Eq. (34) and Eq. (35) as follows,

(36)$$\cos 2\theta = \frac{{E{R_{{0^\circ }}}({E{R_{{{90}^\circ }}} + 1} )- {V_1}E{R_{{{90}^\circ }}}({E{R_{{0^\circ }}} + 1} )}}{{({1 - d} )[{E{R_{{0^\circ }}}({E{R_{{{90}^\circ }}} - 1} )+ {V_1}E{R_{{{90}^\circ }}}({E{R_{{0^\circ }}} - 1} )} ]}}, $$

(37)$$\sin 2\theta = \frac{{{V_2}E{R_{{{135}^\circ }}}({E{R_{{{45}^\circ }}} + 1} )- E{R_{{{45}^\circ }}}({E{R_{{{135}^\circ }}} + 1} )}}{{({1 - d} )[{E{R_{{{45}^\circ }}}({E{R_{{{135}^\circ }}} - 1} )+ {V_2}E{R_{{{135}^\circ }}}({E{R_{{{45}^\circ }}} - 1} )} ]}}. $$

Thus, the offset angle can be given without the depolarization parameter d.

(38)$$\tan 2\theta = \frac{{{V_2}E{R_{135^\circ }}({E{R_{45^\circ }} + 1} )- E{R_{45^\circ }}({E{R_{135^\circ }} + 1} )}}{{{V_2}E{R_{135^\circ }}({E{R_{45^\circ }} - 1} )+ E{R_{45^\circ }}({E{R_{135^\circ }} - 1} )}} \times \frac{{E{R_{0^\circ }}({E{R_{90^\circ }} - 1} )+ {V_1}E{R_{90^\circ }}({E{R_{0^\circ }} - 1} )}}{{E{R_{0^\circ }}({E{R_{90^\circ }} + 1} )- {V_1}E{R_{90^\circ }}({E{R_{0^\circ }} + 1} )}}. $$

The V₁ can be expressed by combining Eq. (20) and Eq. (34),

(39)$${V_1} = \frac{{E{R_{0^\circ }}\left[ {(E{R_{90^\circ }} + 1) - (E{R_{90^\circ }} - 1)\frac{{1 - {\delta_v}}}{{1 + {\delta_v}}}\cos 2\theta } \right]}}{{E{R_{90^\circ }}\left[ {(E{R_{0^\circ }} + 1) + (E{R_{0^\circ }} - 1)\frac{{1 - {\delta_v}}}{{1 + {\delta_v}}}\cos 2\theta } \right]}}. $$

Finally, the LVDR can be deduced,

(40)$${\delta _v} = \frac{{E{R_{0^\circ }}({V_1}E{R_{90^\circ }} - 1) - E{R_{90^\circ }}(E{R_{0^\circ }} - {V_1}){{\tan }^2}\theta }}{{E{R_{90^\circ }}(E{R_{0^\circ }} - {V_1}) + E{R_{0^\circ }}(1 - {V_1}E{R_{90^\circ }}){{\tan }^2}\theta }}. $$

Funding

National Natural Science Foundation of China (62075025); Dalian High-Level Talent Innovation Program (2020RQ018); Fundamental Research Funds for the Central Universities (DUT22JC17).

Acknowledgments

The authors greatly acknowledge the valuable help of Teng Ma on the system development.

Disclosures

The authors declare no conflicts of interest.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Component	Model	Specification
Laser sources	Laser Diode	λ₁= 458 nm; Output power: 4.75W;
	Nichia, NUBM08	Divergence: $14^{\circ} ∥ \times 46^{\circ} ⊥$
	Shandong Huaguang	λ₂= 808 nm; Output power: 5W;
	Optoelectronics	Divergence: $6^{\circ} ∥ \times 8^{\circ} ⊥$
Collimation lens	Edmund Optics	Focal length: 400 mm; Diameter: 76 mm
Collimation lens	Achromatic Lens	Focal length: 400 mm; Diameter: 76 mm
Linear polarizer	Union Optic, SHP1025	Extinction ratio >2000:1 at 458 nm
Linear polarizer	Thorlabs, LPNIRE100-B	Extinction ratio >1500:1 at 808 nm
Dichroic Mirror	Thorlabs, DMLP567R	T >97% at 808 nm; R >99% at 458nm,
Cylindrical lens	Thorlabs, LJ1227L2-A&LK1426L1-A	Convex lens: f = 6.35 mm, height: 6 mm
Cylindrical lens	Thorlabs, LJ1227L2-A&LK1426L1-A	Concave lens: f=-25 mm, height: 10 mm
Elliptical Mirror	Edmund Optics	R >98% at 458nm and at 808nm
Receiving telescope	Bosma Corp., Maksutov-Cassegrain	Focal length: 1000 mm; Diameter: 105 mm
Detectors	Sony, Polarization camera	CMOS: IMX250MZR; Area: 2048×2448 Pixels; 10-bit; 3.45 µm×3.45 µm
Filters	Alluxa & Edmund Optics (458 nm)	458 nm:2.3nm FWHM; T > 90%
Filters	Alluxa & Edmund Optics (458 nm)	452 nm:51nm FWHM; T > 93%
	Edmund Optics (808 nm)	RG715: Long pass 715 nm; T: >96% 808 nm: 3 nm FWHM; T: >90%
Optical windows	BOROFLOAT 33	90 mm (transmitter) and 150 mm (receiver)

Visible, near-infrared dual-polarization lidar based on polarization cameras: system design, evaluation and atmospheric measurements

Abstract

1. Introduction

2. VIS-NIR dual-polarization imaging lidar

2.1 Optical setup

2.2 Lidar signal acquisition, processing and inversion

3. Evaluation of the polarization crosstalk and the offset angle

3.1 Evaluation of the polarization crosstalk for the polarization camera

3.2 Online evaluation of the offset angle

4. Atmospheric aerosol measurements

4.1 Dust observation – a case study

4.2 Observation and evaluation of the depolarization ratio and color ratio

5. Conclusion

Appendix A: Derivation of the linear volume depolarization ratio and the offset angle

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (1)

Equations (40)

Optics Express