Modeling and Evaluation of the Systematic Errors for the Polarization-Sensitive Imaging Lidar Technique

: Polarization lidar plays a signiﬁcant role in characterizing the properties of cirrus clouds, classifying aerosol types, retrieving aerosol microphysical properties, etc. However, the retrieval reliability and accuracy of the linear volume depolarization ratio (LVDR) of atmospheric particles rely on many system factors, requiring intensive attention and massive e ﬀ orts on system calibrations and error evaluations, etc. In this work, a theoretical model based on the Stokes–Mueller formalism has been established for the newly developed polarization-sensitive imaging lidar (PSI-Lidar) technique. The systematic errors introduced by the degree of linear polarization (DoLP) of the emitted laser beam, the o ﬀ set angle, and the quantum e ﬃ ciencies (QEs) and polarization extinction ratios (PERs) of the polarization-sensitive image sensor, were evaluated in detail for the PSI-Lidar at 450, 520, and 808 nm. Although the DoLP of typical multimode laser diodes is not very high, the inﬂuence of non-ideal polarized laser beam on the LVDR can be reduced to less than 1% by employing a high-PER linear polarizer to improve the DoLP of the transmitted laser beam. Laboratory measurements have revealed that the relative QEs of the image sensor with four polarized directions are independent of the total illumination intensity and indicate a good consistency with the factory relative QEs (less than 2% deviation). Meanwhile, the inﬂuence of the relative QEs on the LVDR can be well-calibrated from either experimental or factory relative QEs. Owing to the non-ideal PER of the polarization-sensitive image sensor, e.g., ≈ 74 at 808 nm, ≈ 470 at 450 nm, the crosstalk between received signals with di ﬀ erent polarization states can signiﬁcantly deteriorate the measurement accuracy for small LVDRs. A relative error of the LVDR less than 4% can be achieved at 450 and 520 nm with the LVDR varying from 0.004 to 0.3 for a PER uncertainty of ± 5%, by taking the polarization crosstalk e ﬀ ect into account. However, in order to achieve a relative error of less than 10% for a small atmospheric LVDR of 0.004 at 808 nm, the uncertainty of the PER should be less than ± 2.5%. The o ﬀ set angle can be calculated based on the four polarized lidar signals and the PER values at the four polarization angles. It was found out that the retrieval error of the o ﬀ set angle is less than 0.15 ◦ even with a large PER uncertainty ( ± 20%), giving a negligible systematic error on the LVDR (less than 1%).


Introduction
Lidar, as a powerful remote sensing tool featuring high spatial and temporal resolution, greatly contributes to our knowledge of atmospheric radiation [1,2]. Meanwhile, lidar techniques have attracted a large number of worldwide researchers and have made substantial progress in measurements of aerosol, clouds, temperature, humidity, greenhouse gases, wind, and so on during the past decades [3][4][5][6][7][8][9][10][11][12][13][14]. As an Figure 1c, can capture a four-directional polarized image of the transmitted laser beam in one shot in Figure 1b. Four individual images with different polarization angles can be decoupled from the measured four-directional polarized image. Thus, lidar signals with four different polarization angles of 0 • , 45 • , 90 • , and 135 • can be simultaneously obtained from the corresponding polarization image after pixel binning and pixel-distance transform, from which the LVDR can be evaluated. In order to suppress the background signal, a band-pass interference filter can be installed in the optical path of the receiver. The intensity-modulation of the laser diode is synchronized with the exposure of the image sensor. The background image with the laser diode turned off can also be recorded, allowing real-time subtraction of the background signal. The pixel-range relationship is calibrated according to geometrical optics by measuring the backscattering signal from a tall building with a known distance from the lidar system [63]. Owing to the wide wavelength availability of the high-power multimode laser diodes and broadband responsivity of the polarization-sensitive image sensor, the PSI-Lidar technique can be implemented at various wavelengths from the visible to the near-infrared region, e.g., 450, 520, and 808 nm.
Remote Sens. 2020, 12, x FOR PEER REVIEW 4 of 21 Figure 1c, can capture a four-directional polarized image of the transmitted laser beam in one shot in Figure 1b. Four individual images with different polarization angles can be decoupled from the measured four-directional polarized image. Thus, lidar signals with four different polarization angles of 0°, 45°, 90°, and 135° can be simultaneously obtained from the corresponding polarization image after pixel binning and pixel-distance transform, from which the LVDR can be evaluated. In order to suppress the background signal, a band-pass interference filter can be installed in the optical path of the receiver. The intensity-modulation of the laser diode is synchronized with the exposure of the image sensor. The background image with the laser diode turned off can also be recorded, allowing real-time subtraction of the background signal. The pixel-range relationship is calibrated according to geometrical optics by measuring the backscattering signal from a tall building with a known distance from the lidar system [63]. Owing to the wide wavelength availability of the high-power multimode laser diodes and broadband responsivity of the polarization-sensitive image sensor, the PSI-Lidar technique can be implemented at various wavelengths from the visible to the near-infrared region, e.g., 450, 520, and 808 nm. In previous work, a 450-nm PSI-Lidar prototype has been developed for atmospheric depolarization studies and a promising result has been achieved [62]. We hereby briefly introduce the PSI-Lidar system as an example. The 450-nm laser diode utilized has an output power of about 3.5 W. The laser beam is collimated by an achromatic lens (f = 600 mm, Φ = 100 mm), and the backscattering signal is collected by an achromatic lens (f = 100 mm, Φ = 50 mm). The polarization-sensitive image sensor is tilted for 7° with respect to the optical axis of the receiving lens. The receiver and the transmitter are mounted with an approximately 806 mm separation to each other to satisfy the Scheimpflug principle [56,57,64]. Figure 1d shows the lidar signals at different polarization angles measured by the 450-nm PSI-Lidar system as well as the corresponding LVDR profiles. In previous work, a 450-nm PSI-Lidar prototype has been developed for atmospheric depolarization studies and a promising result has been achieved [62]. We hereby briefly introduce the PSI-Lidar system as an example. The 450-nm laser diode utilized has an output power of about 3.5 W. The laser beam is collimated by an achromatic lens (f = 600 mm, Φ = 100 mm), and the backscattering signal is collected by an achromatic lens (f = 100 mm, Φ = 50 mm). The polarization-sensitive image sensor is tilted for 7 • with respect to the optical axis of the receiving lens. The receiver and the transmitter are mounted with an approximately 806 mm separation to each other to satisfy the Scheimpflug principle [56,57,64]. Figure 1d shows the lidar signals at different polarization angles measured by the 450-nm PSI-Lidar system as well as the corresponding LVDR profiles.

Theoretical Modeling of the PSI-Lidar
The PSI-Lidar system, from the emitted laser beam to the image sensor, can be interpreted as multiple functional modules with the corresponding Stokes vectors and Mueller matrices, as depicted in Figure 2. The Mueller matrices describe how the optical elements and the atmospheric constituents change the Stokes vectors of the laser beam. A polarization element can be described by a Mueller matrix with a combination of diattenuation, retardance, and the depolarization in general [65]. According to the principle of the polarization lidar technique and the Stokes-Mueller formalism, the Mueller matrix of the PSI-Lidar can be described as Here I L is the Stokes vector describing the polarization state of the emitted laser beam from the multimode laser diode, R(α) and R(θ) are the rotation Mueller matrices, α is the rotation angle of the polarization plane of the linear polarizer with respect to the polarization plane of the laser beam, θ is the rotation angle between the polarization plane of the laser beam after passing through the linear polarizer and the 0 • -directional on-chip polarizer of the polarization-sensitive image sensor, M LP is the Mueller matrix of the linear polarizer, M T is the Mueller matrix of the transmitter (e.g., the refractor telescope), F(ψ) is the atmospheric scattering matrix with a scattering angle of ψ, M R is the Mueller matrix of the receiver including the receiving telescope and the interference filters, the Mueller matrix of the polarization-sensitive image sensor can be expressed as the product between the Mueller matrix of the linear polarizer (M x ) and the QE of the pixel below the on-chip polarizer (η x ), where the subscript x refers to 0 • , 45 • , 90 • , and 135 • polarization angles. In the following sections, each functional block will be described by means of Stokes-Mueller formalism.

Theoretical Modeling of the PSI-Lidar
The PSI-Lidar system, from the emitted laser beam to the image sensor, can be interpreted as multiple functional modules with the corresponding Stokes vectors and Mueller matrices, as depicted in Figure 2. The Mueller matrices describe how the optical elements and the atmospheric constituents change the Stokes vectors of the laser beam. A polarization element can be described by a Mueller matrix with a combination of diattenuation, retardance, and the depolarization in general [65]. According to the principle of the polarization lidar technique and the Stokes-Mueller formalism, the Mueller matrix of the PSI-Lidar can be described as Here L I is the Stokes vector describing the polarization state of the emitted laser beam from the multimode laser diode, ()   The emitted light of the laser diode is often linearly polarized with a non-ideal PER varying in a wide range. In order to be consistent with the defined PER of the linear polarizer [65], the PER of the laser beam is defined by the ratio between the light intensity at the supposed polarization state (referred to as the parallel polarization) and the light intensity with an orthogonal polarization state (perpendicular polarization), namely The emitted light of the laser diode is often linearly polarized with a non-ideal PER varying in a wide range. In order to be consistent with the defined PER of the linear polarizer [65], the PER of the laser beam is defined by the ratio between the light intensity at the supposed polarization state (referred to as the parallel polarization) and the light intensity with an orthogonal polarization state (perpendicular polarization), namely Here, I 0 and I ⊥ 0 are the parallel and perpendicular polarized light intensities. The total radiation intensity (I 0 ) of the laser diode can be regarded as the sum of the parallel and perpendicular polarized lights The Stokes vector, describing the flux and the polarization state of a laser beam, can be generally written as S = [S 0 , S 1 , S 2 , S 3 ] T . The four elements in the vector describe the total intensity (S 0 ), the intensity difference on the x (parallel) and y (perpendicular) axes (S 1 ), the intensity difference on Remote Sens. 2020, 12, 3309 6 of 21 the +45 • and −45 • axes (S 2 ), and the intensity that is right-hand circular (RHC) or left-hand circular (LHC) (S 3 ), respectively [51,65]. Another parameter describing the polarization property of a laser beam is the DoLP, which is mathematically given by [66] p Laser = For the laser beam emitted by a laser diode, the Stokes vector can be written by In the PSI-Lidar technique, a linear polarizer with a high-PER may be used to improve the DoLP of the transmitted laser beam. However, most likely the polarization plane of the laser beam is not perfectly aligned with the polarization axis of the linear polarizer. If the angle between the polarization axis of the linear polarizer and the polarization plane of the emitted laser beam from the multimode laser diode is denoted as α, the rotation Mueller matrix of the linear polarizer can be written as [54] A non-idealized linear polarizer in practice has diattenuation and retardance, which can be described by the combination of the diattenuator and retarder through the Mueller matrix [65] Here, T LP is the intensity transmittance, T max LP and T min LP are the maximum and minimum intensity transmittances of the linear polarizer with respect to the incident plane, D LP is the diattenuation, ∆ LP is the phase retardance.
The polarization plane of the laser beam passing through the linear polarizer may have an offset angle of θ with respect to the 0 • -directional on-chip polarizer of the polarization image sensor, which can be described by a rotation Mueller matrix Remote Sens. 2020, 12, 3309 7 of 21 In the PSI-Lidar technique, the laser beam emitted from the high-power laser diode is often collimated by an achromatic lens (or refractor telescope with achromatic lens) in order to reduce the divergence of the transmitted laser beam. The Mueller matrix of achromatic lenses is given by Here, T T is the intensity transmittance of the achromatic lens, D T and ∆ T are the diattenuation and the phase retardance of the achromatic lens, respectively.
The atmospheric scattering volume can be described by the size, shape, composition, orientation, and symmetry of the particles [67]. In this work, the atmospheric scattering volume is assumed as randomly oriented, axially symmetric scatterers [68], and the scattering angle (ψ) is approximately equal to 180 • for typical PSI-Lidar systems that can be considered as constant. The corresponding scattering matrix can be written by [69] Here, F 11 is the volume backscatter coefficient (β) and d is the depolarization parameter related to the propensity of the scattering medium to preserve the incident polarization with a range of 0-1 [69]. δ is the atmospheric LVDR, which is the primary parameter obtained from the PSI-Lidar.
Since the receiver module consists of the receiving telescope and narrowband interference filter, the Mueller matrix of the receiver module is the matrix product of the two matrices as Here, T R , D R , Z R and ∆ R are given by Here, T Tel. and T Filter are the intensity transmittances of the receiving telescope and narrowband interference filter, respectively. D R and ∆ R are the diattenuation and the phase retardance of the receiving module, respectively. The polarization-sensitive image sensor consists of pixel (photo-detector) arrays and on-chip linear polarizers that are fabricated between the micro-lens and the photodetector. The linear polarizer for a specific polarization angle can be described by the Mueller matrix Here, the subscript x refers to 0 • , 45 • , 90 • , and 135 • polarization angle, T x is the intensity transmittances of the linear polarizer, respectively, D x and ∆ x are the diattenuation and the phase retardance of the linear polarizer, respectively.
According to the above equations, the relationship between the detected lidar signals (I x ) at four different polarization angles and the LVDR can be established by the Stokes vectors and the Mueller matrices. The measurement errors introduced by the DoLP of the laser beam, the offset angle between the polarization plane of the transmitted laser beam and the 0 • -directional polarizer of the image sensor, the QE and the PER of the image sensor can thus be evaluated through the Stokes-Mueller formalism.

Optical Parameters of the PSI-Lidar
The polarization-sensitive image sensor can capture a four-directional polarized image in one shot. Thus, lidar signals at four different polarization angles can be simultaneously obtained from the polarization-sensitive image sensor, from which the LVDR can be evaluated according to the Stokes-Mueller formulas. Nevertheless, as discussed in Section 2, the retrieval of the LVDR depends on many system factors of the PSI-Lidar system, such as the DoLP of the laser beam, the QE and the PER of the polarization-sensitive image sensor, etc. The systematic errors of the PSI-Lidar technique thus arise from the uncertainties of these system parameters, which should be determined. Table 1 gives the PER as well as the corresponding DoLP of multimode laser diodes at different wavelengths, which are obtained from datasheets provided by the suppliers. As can be seen, the PER of multimode laser diodes at 450 and 520 nm are typically in the region of 100-400 leading to a DoLP beyond 0.98. However, the PER value of the 808-nm laser diode is seldom provided by manufacturers. Thus, an experimental setup was developed to measure the PER (DoLP) of the laser beam emitted from the 808-nm laser diode. As shown in Figure 3, the laser beam emitted by the laser diode first passed through an 1-mm aperture and then a high-PER linear polarizer (>10,000:1), which can be considered as an ideal polarizer. A second aperture was installed in front of the amplified photodetector to suppress the stray light. The transmission intensity of the polarized laser beam through the linear polarizer was detected by the photodetector and then acquired by a data acquisition (DAQ) card for data analysis. The linear polarizer was manually rotated through a rotational polarizer mount from 0 • to 360 • with a step of 2 • , and the transmission intensity through the linear polarizer is recorded for 8 s at each rotation angle of the linear polarizer. Figure 4 shows the relationship between the detected intensities and the rotation angles of the linear polarizer. The PER can be obtained by fitting the polarization response curve according to a cosine function. It has been found out that the PER of the 808 nm laser diode is about 68 ± 8, which is much smaller than those at shorter wavelengths.

The Relative Quantum Efficiencies (QEs) of the Polarization-Sensitive Image Sensor
The relative QEs of the four polarization channels of the image sensor also has a direct impact on the measurement results, as indicated by Equation (1). Although the relative QEs of the image sensor can be obtained directly from its datasheet, it is worthwhile to further investigate the dependency of the relative QEs on the total illumination intensity of the incoming light, which can vary substantially during atmospheric measurements. The experimental setup for measuring the relative QE of the four-directional polarization channels is shown in Figure 5a. An integrating sphere with a halogen light source is utilized to create a near-ideal unpolarized light. A convex lens with a diameter of 50 mm and a focal length of 100 mm can focus the unpolarized light emitted by the integrating sphere on the polarization-sensitive image sensor. As the halogen light source has a wide emission spectrum from 400-1000 nm, different interference filters were utilized to select the emission spectrum, e.g., 450 nm interference filter (FWHM = 10 nm), 520 nm interference filter (FWHM = 10 nm), and 808 nm interference filter (FWHM = 3.1 nm) with an RG780 long-pass filter, which were placed in front of the image sensor. The cone half-angle (CHA) of the incident light to the image sensor was about ±5°.

The Relative Quantum Efficiencies (QEs) of the Polarization-Sensitive Image Sensor
The relative QEs of the four polarization channels of the image sensor also has a direct impact on the measurement results, as indicated by Equation (1). Although the relative QEs of the image sensor can be obtained directly from its datasheet, it is worthwhile to further investigate the dependency of the relative QEs on the total illumination intensity of the incoming light, which can vary substantially during atmospheric measurements. The experimental setup for measuring the relative QE of the four-directional polarization channels is shown in Figure 5a. An integrating sphere with a halogen light source is utilized to create a near-ideal unpolarized light. A convex lens with a diameter of 50 mm and a focal length of 100 mm can focus the unpolarized light emitted by the integrating sphere on the polarization-sensitive image sensor. As the halogen light source has a wide emission spectrum from 400-1000 nm, different interference filters were utilized to select the emission spectrum, e.g., 450 nm interference filter (FWHM = 10 nm), 520 nm interference filter (FWHM = 10 nm), and 808 nm interference filter (FWHM = 3.1 nm) with an RG780 long-pass filter, which were placed in front of the image sensor. The cone half-angle (CHA) of the incident light to the image sensor was about ±5°.

The Relative Quantum Efficiencies (QEs) of the Polarization-Sensitive Image Sensor
The relative QEs of the four polarization channels of the image sensor also has a direct impact on the measurement results, as indicated by Equation (1). Although the relative QEs of the image sensor can be obtained directly from its datasheet, it is worthwhile to further investigate the dependency of the relative QEs on the total illumination intensity of the incoming light, which can vary substantially during atmospheric measurements. The experimental setup for measuring the relative QE of the four-directional polarization channels is shown in Figure 5a. An integrating sphere with a halogen light source is utilized to create a near-ideal unpolarized light. A convex lens with a diameter of 50 mm and a focal length of 100 mm can focus the unpolarized light emitted by the integrating sphere on the polarization-sensitive image sensor. As the halogen light source has a wide emission spectrum from 400-1000 nm, different interference filters were utilized to select the emission spectrum, e.g., 450 nm interference filter (FWHM = 10 nm), 520 nm interference filter (FWHM = 10 nm), and 808 nm interference filter (FWHM = 3.1 nm) with an RG780 long-pass filter, which were placed in front of the image sensor. The cone half-angle (CHA) of the incident light to the image sensor was about ±5 • . The image of the unpolarized light was recorded with different exposure times ranging from 20 to 400 ms to investigate the influence of different illumination intensities. Meanwhile, 20 frames were acquired at each exposure time to perform image averaging and thus to improve the signal-to-noise ratio (SNR). Figure 5b shows a typical image recorded by the image sensor. The uniform part of the image, marked with a rectangle, referred to as the region of interest (ROI), was chosen to evaluate the relative QEs. The image in the ROI was decoupled into four directional polarized images. The relative QE for each individual pixel in the ROI was then evaluated from the ratio between the light intensity for a specific polarization angle and the mean light intensity of all polarization angles. The relative QEs for all pixels in the ROI may deviate slightly owing to the photon-response nonuniformity (PRNU) of the image sensor, which is also susceptive to the illumination intensity [70]. The measurement result revealed that the standard deviation of the relative QEs for the pixels in the ROI is less than 0.8% for high light-level condition (incoming light intensity is more than 10% of the full well capacity of each pixel), while it could increase to 1.3% for low light level condition. Nevertheless, the pixel-to-pixel variation of the relative QE is considered as random noise for lidar signal. Meanwhile, the image of the laser beam is vertically binned to obtain the lidar profile in practical lidar measurements, which can also greatly suppress the influence of the PRNU. Thus, the mean value of the relative QE for all pixels in the ROI is more appropriate for the evaluation of the depolarization ratio. The final relative QE is taken from the average value of the six-time measurements with different exposure times, and the standard deviation indicates the variation of the relative QE with the illumination intensity.
The measured relative QEs as well as the values from the datasheet at three different wavelengths are shown in Table 2. In general, the measured relative QEs were in good agreement with those provided by the datasheet. The deviations of the relative QEs between the measured values and the factory values at 450, 520, and 808 nm are less than 0.2%, 0.31%, and 2%, respectively. Meanwhile, the fluctuation of the measured relative QE is less than 0.04% when the total illumination intensity varied from 5% to 80% of the full-well capacity (FWC) of the image sensor. This implies that the relative QE is independent of the illumination intensity, which is of great benefit for employing the polarization-sensitive image sensor for depolarization studies. The image of the unpolarized light was recorded with different exposure times ranging from 20 to 400 ms to investigate the influence of different illumination intensities. Meanwhile, 20 frames were acquired at each exposure time to perform image averaging and thus to improve the signal-to-noise ratio (SNR). Figure 5b shows a typical image recorded by the image sensor. The uniform part of the image, marked with a rectangle, referred to as the region of interest (ROI), was chosen to evaluate the relative QEs. The image in the ROI was decoupled into four directional polarized images. The relative QE for each individual pixel in the ROI was then evaluated from the ratio between the light intensity for a specific polarization angle and the mean light intensity of all polarization angles. The relative QEs for all pixels in the ROI may deviate slightly owing to the photon-response nonuniformity (PRNU) of the image sensor, which is also susceptive to the illumination intensity [70]. The measurement result revealed that the standard deviation of the relative QEs for the pixels in the ROI is less than 0.8% for high light-level condition (incoming light intensity is more than 10% of the full well capacity of each pixel), while it could increase to 1.3% for low light level condition. Nevertheless, the pixel-to-pixel variation of the relative QE is considered as random noise for lidar signal. Meanwhile, the image of the laser beam is vertically binned to obtain the lidar profile in practical lidar measurements, which can also greatly suppress the influence of the PRNU. Thus, the mean value of the relative QE for all pixels in the ROI is more appropriate for the evaluation of the depolarization ratio. The final relative QE is taken from the average value of the six-time measurements with different exposure times, and the standard deviation indicates the variation of the relative QE with the illumination intensity.
The measured relative QEs as well as the values from the datasheet at three different wavelengths are shown in Table 2. In general, the measured relative QEs were in good agreement with those provided by the datasheet. The deviations of the relative QEs between the measured values and the factory values at 450, 520, and 808 nm are less than 0.2%, 0.31%, and 2%, respectively. Meanwhile, the fluctuation of the measured relative QE is less than 0.04% when the total illumination intensity varied from 5% to 80% of the full-well capacity (FWC) of the image sensor. This implies that the relative QE is independent of the illumination intensity, which is of great benefit for employing the polarization-sensitive image sensor for depolarization studies.

The Polarization Extinction Ratios (PERs) of the Polarization-Sensitive Image Sensor
According to the datasheet of the polarization-sensitive image sensor, the PER varies from 40 to 470 in the region of 400-900 nm and it could also deviate at different polarization angles. Table 3 shows the PERs of different polarization angles at three different wavelengths, obtained from the datasheet of the polarization-sensitive image sensor. As can be seen, the PER at 808 nm is significantly smaller than those in the blue-green region. Due to the non-ideal PER, crosstalk between different polarization states could occur. For instance, the 90 • polarization channel (or pixel) may detect the parallel polarized backscattering signal, leading to the overestimation of the 90 • polarization signal. The overestimation can result in a measurement error comparable to the molecular depolarization ratio (typically about 0.004). Thus, it is of great importance to investigate the polarization crosstalk effect on the polarized lidar signals.

Evaluation and Discussion of Systematic Errors for the PSI-Lidar
In this section, systematic errors introduced by various factors, e.g., the DoLP of the emitted laser beam, the offset angle, the QEs and the PERs of the polarization-sensitive image sensor, were analyzed and discussed. The systematic errors introduced by the transmitter, e.g., the small depolarizing effect of the achromatic lens, are neglected as the uncertainty introduced by possible birefringence is very difficult to analyze and estimate [54]. Furthermore, the polarization effects of the receiver, e.g., telescope, are neglected in this work [71,72]. Even so, the analysis framework presented below is feasible to include such effects. It should be noted that the polarization crosstalk of a Cassegrain telescope with small incidence angles of the incoming light in lidar is negligible, guiding us to choose an optimal receiving telescope to reduce the systematic error [52].

The Systematic Error Introduced by the DoLP of the Transmitted Laser Beam
The LVDR can be significantly influenced by the crosstalk between lights with different polarization states originating from the non-ideal polarization property (DoLP < 1) of the laser beam that has been transmitted into the atmosphere. According to Stokes-Mueller formalism, the DoLP-related relative error of the LVDR can be deduced by neglecting the influence from other factors Here, p Trans. is the DoLP of the laser beam that is transmitted into the atmosphere, δ 0 is the true value of the LVDR, which is only dependent on the optical properties of the scatters (see Equation (12)). Figure 6 illustrates the relationship between the relative error and the DoLP of the transmitted laser beam at typical atmospheric LVDRs for molecules (0.004), anthropogenic aerosol (0.05), and dust (0.3) [35,73,74]. As can be seen, the relative error decreases with the increasing of the DoLP, and increases when the LVDR becomes smaller. It can be concluded that the relative error of the LVDR would be limited to 1% even in the case of molecular depolarization ratio (≈0.004) if the DoLP of the transmitted laser beam can reach up to 99.992%.
p is the DoLP of the laser beam that is transmitted into the atmosphere, 0  is the true value of the LVDR, which is only dependent on the optical properties of the scatters (see Equation (12)). Figure 6 illustrates the relationship between the relative error and the DoLP of the transmitted laser beam at typical atmospheric LVDRs for molecules (0.004), anthropogenic aerosol (0.05), and dust (0.3) [35,73,74]. As can be seen, the relative error decreases with the increasing of the DoLP, and increases when the LVDR becomes smaller. It can be concluded that the relative error of the LVDR would be limited to 1% even in the case of molecular depolarization ratio (≈0.004) if the DoLP of the transmitted laser beam can reach up to 99.992%. According to the PERs of the multimode laser diode shown in Table 1, the relative error can reach up to 250% at 450 nm (3.2 W), 83% at 520 nm, and 417% at 808 nm with a molecular depolarization ratio of about 0.004. Thus, a linear polarizer has to be utilized to improve the DoLP of the transmitted laser beam as well as the systematic error. The DoLP of the laser beam after passing through a linear polarizer is given by Here, Laser p is the DoLP of the laser beam emitted from the multimode laser diode. As can be seen, the DoLP of the transmitted laser beam is primarily determined by the PER of the linear polarizer ( LP ER ), the rotation angle (  ) as well as the original DoLP of the emitted laser beam from the multimode laser diode. The rotation angle can be readily controlled to ±2° through manual alignment with a fine dial. For a laser diode with a small PER of 68 (e.g., 808-nm laser diode), the DoLP of the transmitted laser beam can reach up to 99.992% by using a linear polarizer with a PER of 500:1, which is widely available. In summary, the systematic error introduced by the non-ideal polarized laser beam of multimode laser diodes can be neglected by employing a relatively high-PER linear polarizer to improve the DoLP of the transmitted laser. According to the PERs of the multimode laser diode shown in Table 1, the relative error can reach up to 250% at 450 nm (3.2 W), 83% at 520 nm, and 417% at 808 nm with a molecular depolarization ratio of about 0.004. Thus, a linear polarizer has to be utilized to improve the DoLP of the transmitted laser beam as well as the systematic error. The DoLP of the laser beam after passing through a linear polarizer is given by p Trans. = (ER LP − 1) + (ER LP + 1)p Laser cos(2α) (ER LP + 1) + (ER LP − 1)p Laser cos(2α) (21) Here, p Laser is the DoLP of the laser beam emitted from the multimode laser diode. As can be seen, the DoLP of the transmitted laser beam is primarily determined by the PER of the linear polarizer (ER LP ), the rotation angle (α) as well as the original DoLP of the emitted laser beam from the multimode laser diode. The rotation angle can be readily controlled to ±2 • through manual alignment with a fine dial. For a laser diode with a small PER of 68 (e.g., 808-nm laser diode), the DoLP of the transmitted laser beam can reach up to 99.992% by using a linear polarizer with a PER of 500:1, which is widely available. In summary, the systematic error introduced by the non-ideal polarized laser beam of multimode laser diodes can be neglected by employing a relatively high-PER linear polarizer to improve the DoLP of the transmitted laser.

The Systematic Error Introduced by the Relative QEs of the Four Polarization Channels
The relative QEs directly influence the ratio between lidar signals measured at different polarization angles and thus the value of the LVDR. In order to investigate the potential systematic error introduced by the relative QEs, the LVDRs are evaluated with the measured relative QEs and the factory relative QEs, respectively, while the contributions from other optical components are not taken into account. The deviation of the LVDR evaluated with different relative QEs is given by Remote Sens. 2020, 12, 3309 13 of 21 Here, δ Meas. is the LVDR evaluated with the measured relative QEs (η Meas. 0 • , η Meas. 90 • ), and δ Data is the LVDR calculated with the factory relative QEs (η Data 0 • , η Data 90 • ). The deviation of the LVDR is generally less than 0.3% at 450 and 520 nm, but slightly larger at 808 nm, i.e., about 3%. Nevertheless, these small deviations can be neglected for practical measurements of LVDR. Moreover, the influence of different illumination intensities is also negligible, since the relative QEs is independent of the illumination intensity, as was discussed in Section 3.2. The small deviations also imply that the influence of the relative QEs on the LVDR can be well-calibrated from either experimental or factory relative QEs.

The Systematic Error Introduced by the Non-Ideal PERs of the Image Sensor
Owing to the non-ideal PERs of the image sensor, there would be crosstalk for the lidar signals detected at different polarization angles, leading to considerable systematic errors. However, there are some differences between the crosstalk effect of the conventional polarization pulsed lidar technique based on the PBS and the PSI-Lidar technique, which are worthwhile to elaborate. In conventional polarization lidar technique, the backscattering beam is divided into parallel and perpendicular polarized beams through a PBS. Owing to the non-ideal PER of the PBS, crosstalk can occur, e.g., the parallel polarized laser beam may contain the light with perpendicular polarization state and vice versa. Nevertheless, the sum intensity of the two orthogonally polarized laser beams is equal to the intensity of the backscattering beam before entering the PBS. On the other hand, in the PSI-Lidar technique, the backscattering laser beam is detected simultaneously by many polarized pixels that are spatially spaced as shown in Figure 1c. The incident light on a specific pixel will have a maximum transmission if its polarization state is parallel to the polarization angle of the corresponding on-chip linear polarizer, and a minimum transmission for the light with perpendicular polarization. The primary part of the perpendicular polarized light is rejected by the on-chip linear polarizer, which will not be detected by the photodiode fabricated below the linear polarizer. The crosstalk effect is mainly determined by the PER value of the on-chip linear polarizer. Assuming that other optical components are ideal and the atmospheric particles are randomly oriented, axially symmetric scatterers, the relative error (ε ER ) of the LVDR introduced by ignoring the crosstalk effect can be expressed by Here, δ 0 is the true value of the LVDR, ER 0 • and ER 90 • are the extinction ratios of the 0 • -and 90 • -linear polarizers, described by T max 0 • /T min 0 • and T max 90 • /T min 90 • , respectively. In the following discussions, ER 0 • and ER 90 • are also referred to as 0 • -PER and 90 • -PER, respectively. Table 4 shows the relative error of the LVDR introduced by ignoring the crosstalk effect due to non-ideal PERs. If the value of the LVDR is beyond 0.05, the relative error introduced by ignoring the crosstalk effect is less than 6% in the blue-green region and less than 27% in the near-infrared region. Meanwhile, the relative error at 808 nm considerably decreases with the increasing of the LVDR, and it is about 4% for a LVDR of 0.3. In clean atmospheric conditions dominated by molecules, the value of the LVDR could be smaller than 0.01, e.g., 0.004. In this case, the relative errors for all wavelengths increase significantly, up to about 338% at 808 nm, 76% at 520 nm, and 53% at 450 nm, as the crosstalk effect is comparable to the atmospheric depolarization effect, which should be carefully considered. In fact, the relative error of the LVDR is the tradeoff result between the crosstalk effect and the atmospheric depolarization effect. If the crosstalk effect is taken into account, the measured LVDR can be mathematically given by In principle, as long as the PER of the image sensor can be precisely obtained, the influence of the crosstalk effect on the measured LVDR can be readily eliminated according to Equation (24). In other words, δ Meas. is equal to the true value of the LVDR, namely δ Meas. = δ 0 . Although the PER values can be obtained from the datasheet provided by the manufacturer, the polarization characteristic may not be guaranteed the same for every product due to differences in production batches. The uncertainty or deviation of the PER with respect to the factory or true value can thus introduce a certain amount of error on the retrieved LVDR. The relative error of the LVDR due to the uncertainty of the PERs of the image sensor can thus be expressed by Here, δ Meas. (ER) is the LVDR evaluated with an arbitrary PER in the uncertainty region of the true PER.
Simulation studies were carried out to investigate the influence of the PER uncertainty on the basis of the factory or true PER value. The relative errors of the LVDR due to the uncertainties of the PERs were calculated with a LVDR value varying from 0.004 to 0.3, as shown in Figure 7. For urban aerosols or dust, the value of LVDR is often larger than 0.05 and can reach up to 0.3 [35,74]. Under these circumstances, the relative error can be limited to 1.5% at 450 and 520 nm even if the uncertainty of the PER reaches up ±20%, while the relative error is less than 7% at 808 nm. In order to achieve the above relative error level (e.g., 1.5%) at around 808 nm, the PER uncertainty at 808 nm should be limited to ±5%. If the atmosphere is dominated by molecules, the value of the LVDR is often less than 0.01 (e.g., 0.004) depending on ambient temperature, spectral width, wavelength, etc. [73]. For such a small LVDR, the relative errors at 450 and 520 nm would be less than 4% for a PER uncertainty less than ±5%. However, the relative error at 808 nm varies from 7% to 18% under the same conditions. To achieve a relative error of less than 10% for an LVDR of 0.004, the uncertainty of the PER at 808 nm should not be larger than ±2.5%. Clearly, the relative errors can be substantially reduced by taking the polarization crosstalk effect into account, particularly for the cases with small LVDRs. It can be concluded that precise information regarding the PER value of the polarization-sensitive image sensor is critical for accurate measurements of small LVDRs, particularly at 808 nm.

The Systematic Error Introduced by the Offset Angle
In an ideal PSI-Lidar system, the polarization plane of the transmitted laser beam would be aligned perfectly with the 0°-directional on-chip polarizer of the polarization-sensitive image sensor. However, perfect alignment is very difficult to achieve. There could be a misalignment (offset) angle (  ) between the polarization plane of the transmitted laser beam and the polarization plane of the 0°-directional on-chip polarizer of the polarization-sensitive image sensor. The misalignment can cause the leakage of the parallel-polarized signal to the cross-polarized signal, leading to overestimation of the LVDR.
Assuming that the atmospheric particles are randomly oriented axially symmetric scatterers and neglecting the influence of other optical components, the relative error of the LVDR introduced by the offset angle can be deduced as Simulation results of the relative errors introduced by the offset angle with different offset angles and LVDRs are shown in Figure 8. As can be seen, the relative error will increase with the increasing of the offset angle. Meanwhile, the relative error may increase considerably for a small value of the LVDR. The relative error introduced by the offset angle would be limited to 5% for the molecular depolarization of around 0.004, as long as the offset angle is smaller than 0.81°. The relative error can be further reduced to below 1%, if the offset angle is in the range of ±0.37°.

The Systematic Error Introduced by the Offset Angle
In an ideal PSI-Lidar system, the polarization plane of the transmitted laser beam would be aligned perfectly with the 0 • -directional on-chip polarizer of the polarization-sensitive image sensor. However, perfect alignment is very difficult to achieve. There could be a misalignment (offset) angle (θ) between the polarization plane of the transmitted laser beam and the polarization plane of the 0 • -directional on-chip polarizer of the polarization-sensitive image sensor. The misalignment can cause the leakage of the parallel-polarized signal to the cross-polarized signal, leading to overestimation of the LVDR.
Assuming that the atmospheric particles are randomly oriented axially symmetric scatterers and neglecting the influence of other optical components, the relative error of the LVDR introduced by the offset angle can be deduced as Simulation results of the relative errors introduced by the offset angle with different offset angles and LVDRs are shown in Figure 8. As can be seen, the relative error will increase with the increasing of the offset angle. Meanwhile, the relative error may increase considerably for a small value of the LVDR. The relative error introduced by the offset angle would be limited to 5% for the molecular depolarization of around 0.004, as long as the offset angle is smaller than 0.81 • . The relative error can be further reduced to below 1%, if the offset angle is in the range of ±0.37 • . In the PSI-Lidar technique, the offset angle can be determined from the polarized lidar signals at 0°-, 90°-, 45°-, and 135°-polarization angles, which is given by According to Equations (27)- (29), the offset angle can be calculated based on the four polarized lidar signals and the PER values at the four polarization angles. The retrieval error of the offset angle is primarily determined by the uncertainties of the PER values. Numerical simulation on the retrieval error of the offset angle was carried out with the PER values at each polarization angle varying ±20% in respect to the factory or true PER values. In general, the retrieval error of the offset angle increases with the increasing of the PER uncertainty. Table 5 shows the retrieval error of the offset angle with a PER uncertainty of ±20% for different wavelengths and different atmospheric LVDRs. As can be seen, the retrieval error of the offset angle increases with the increasing of the atmospheric LVDR. The retrieval error at 808 nm is much larger than those at the blue-green region, due to the relatively small values of the PER. However, the maximum retrieval error of the offset angle is less than 0.15° for a LVDR value of 0.3. It can be concluded that the retrieval error of the offset angle is not very sensitive to the uncertainty of the PERs at the four polarization angles. According to the simulation results shown in Figure 8, the systematic error of LVDR with an offset-angle measurement error of 0.15° is less than 1% for a LVDR in the region of 0.0040-0.3, which can be neglected during practical polarization studies.  In the PSI-Lidar technique, the offset angle can be determined from the polarized lidar signals at 0 • -, 90 • -, 45 • -, and 135 • -polarization angles, which is given by Here, ER 45 • and ER 135 • are referred to as 45 • -PER and 135 • -PER, respectively. V 1 and V 2 are defined as According to Equations (27)- (29), the offset angle can be calculated based on the four polarized lidar signals and the PER values at the four polarization angles. The retrieval error of the offset angle is primarily determined by the uncertainties of the PER values. Numerical simulation on the retrieval error of the offset angle was carried out with the PER values at each polarization angle varying ±20% in respect to the factory or true PER values. In general, the retrieval error of the offset angle increases with the increasing of the PER uncertainty. Table 5 shows the retrieval error of the offset angle with a PER uncertainty of ±20% for different wavelengths and different atmospheric LVDRs. As can be seen, the retrieval error of the offset angle increases with the increasing of the atmospheric LVDR. The retrieval error at 808 nm is much larger than those at the blue-green region, due to the relatively small values of the PER. However, the maximum retrieval error of the offset angle is less than 0.15 • for a LVDR value of 0.3. It can be concluded that the retrieval error of the offset angle is not very sensitive to the uncertainty of the PERs at the four polarization angles. According to the simulation results shown in Figure 8, the systematic error of LVDR with an offset-angle measurement error of 0.15 • is less than 1% for a LVDR in the region of 0.0040-0.3, which can be neglected during practical polarization studies. Since the PERs at the four-directional polarized channels are much larger than 1, Equation (27) can also be approximated by As can be seen, the offset angle is mainly determined by the values of V 1 and V 2 . Although the value of the offset angle can always be retrieved from the four polarization lidar signals, it is more preferable to have a 0 • offset angle in practical depolarization measurements. A simple and efficient approach to achieve a 0 • offset angle is to let the value of V 2 equal to 1. In other words, the offset angle equals 0 • , if signal intensities at 45 • and 135 • polarization angles are identical after normalization by the relative QE.

Conclusions
In this work, a theoretical model based on the Stokes-Mueller formalism was established for the newly developed polarization-sensitive imaging lidar (PSI-Lidar) technique. The systematic errors introduced by the DoLP of the emitted laser beam, the offset angle, the relative QEs, and the PERs of the polarization-sensitive image sensor, were evaluated in detail for the PSI-Lidar at several wavelengths from the visible to the near-infrared region. The following conclusions can be summarized according to experimental as well as simulation studies.
The PERs of multimode laser diodes are often not very high, e.g., varying from about 60 to 400 (DoLP > 0.967) depending on the wavelengths, etc. Directly transmitting the laser beam emitted from the laser diode into the atmosphere can introduce a large measurement error due to the crosstalk in the image sensor for light with different polarization states, particularly for the cases with very small LVDR values. However, the influence of a non-ideal polarized laser beam on the LVDR can be reduced to less than 1% by employing a linear polarizer (PER > 500) before transmitting the laser beam into the atmosphere.
The relative QEs of the image sensor have a direct impact on the measured LVDR. Laboratory measurements have found that the relative QEs are independent of the total illumination intensity and indicates a good consistency with the factory relative QEs (less than 2% deviation). The promising result demonstrated that the influence of the relative QEs on the LVDR can be well-calibrated from either experimental or factory relative QEs. Moreover, as the four polarization channels are on-chip fabricated having good uniformity, the relative QEs of the image sensor are insensitive to ambient environment or measurement conditions, which implies that there is no need for frequent or periodical calibrations.
Owing to the non-ideal PER of the image sensor (about 74 at 808 nm, about 470 at 450 nm), the crosstalk between lights with different polarization states can significantly influence the measured LVDR, e.g., the relative error can reach up to 338% at 808 nm, 76% at 520 nm, and 53% at 450 nm for a LVDR of 0.004. For a LVDR value beyond 0.05, the relative error of LVDR can be limited to 1.5% at 450 nm and 520 nm even if the uncertainty of the PER reaches up ±20%, while the relative error is less than 7% at 808 nm. Meanwhile, the systematic error introduced by the crosstalk effect would be reduced to 4% in various atmospheric conditions at 450 and 520 nm with a PER uncertainty of 5%, which can be neglected for most polarization studies. However, high precision measurement on the LVDR at 808 nm in clean atmosphere dominated by molecules could be very challenging, owing to the much smaller value of the 808-nm PER. This implies that high precision measurement for the PER value of the polarization-sensitive image sensor is of great importance for small LVDR measurements, particularly at 808 nm.
According to numerical studies, it has been found that the offset angle is primarily determined by the four polarized lidar signals. The retrieval error of the offset angle is not sensitive to the PER values at the four polarization angles, which is less than 0.15 • even for a PER uncertainty of ±20% at all wavelengths in various LVDRs. Thus, the relative error of the LVDR due to the retrieval error of the offset angle is less than 1%, which can be ignored. It is also worth mentioning that a 0 • offset angle can be readily achieved during practical measurements if signal intensities at 45 • and 135 • polarization angles are identical after normalization by the relative QEs.