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Abstract
The molecular depolarization ratio (MDR) is of great significance for polarization lidar techniques in terms of validating the measurement accuracy, etc. However, previous studies mainly focused on cases with narrowband laser linewidths, and the transmittance of the Cabannes line in the receiver has been assumed constant. In this work, the narrowband theoretical model of MDR has been re-examined by taking the transmittance of the Cabannes line into account. A large relative deviation of beyond 200% has been found if the wavelength-shift reaches up to 0.5 nm for a receiving bandwidth of 0.5 nm at 532 nm, which is much larger than the case without considering the transmittance of the Cabannes line, i.e., only 15%, reported in previous studies. Besides, a broadband theoretical model has been proposed to evaluate the MDR for polarization lidar using high-power multimode laser diodes as light sources. Simulation studies have revealed that the MDR is highly related to the laser linewidth, the receiving bandwidth, as well as the wavelength-shift between the laser wavelength and the center wavelength of the receiver. The MDR at 520 nm calculated by the broadband theoretical model is about 21% larger than the value evaluated without considering the laser linewidth, when the receiving bandwidth is equivalent to the laser linewidth (e.g., 2 nm). Validation measurements, employing a 520-nm imaging-based polarization lidar with a 3.4-nm laser linewidth and a 10-nm receiving bandwidth, illustrated that the volume depolarization ratio in a clean atmospheric region (0.129±0.0025) was highly consistent with the theoretical MDR (0.132). The good agreement between theoretical and experimental results demonstrated a high measurement accuracy of the imaging-based polarization lidar and excellent feasibility of the broadband theoretical model.
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1. Introduction
Lidar, featuring high spatial and temporal resolution, has attracted much interest from researchers worldwide and has been widely developed for atmospheric remote sensing of aerosol, clouds, gases, wind and so on over the past few decades [1–4]. As a branch of the atmospheric lidar techniques, polarization lidar plays a crucial role in studying dust transportation, characterizing optical and microphysical properties of aerosol/clouds, as well as classifying aerosol types, since the first demonstration of the polarization lidar in 1971 [5–10]. In traditional pulsed polarization lidar, a linearly polarized light is transmitted into the atmosphere, and the corresponding backscattering lights collected by a telescope were split into a cross-polarization component and a parallel-polarization component by utilizing a polarization beam splitter (PBS). The ratio of the cross- to the parallel-polarized backscatter coefficient is referred to as the linear volume depolarization ratio, which is a key parameter related to particle shapes [11]. However, the measurement accuracy of the depolarization ratio is affected by the polarization purity of the laser, the gain ratio, the polarization crosstalk of the PBS, the misalignment (offset) angle between the polarization plane of the laser and the polarization plane of the PBS in the receiver, etc. [12]. Among these factors, the gain ratio, sensitive to measurement environments, is the most difficult one for calibration and has become a great challenge in the development history of polarization lidar [13–16]. Various calibration approaches have been proposed, such as the clean atmosphere method [17], the unpolarized light method [18], the rotating half-wave plate (HWP) method [11,19–21], the pseudo-depolarizer method [22], etc. The clean atmosphere method is an absolute calibration method, which compares the measured depolarization ratio and the theoretical value of the molecular depolarization ratio (MDR) in the region, where only atmospheric molecules exist [17]. Besides, the MDR is also a crucial input parameter for retrieving the linear particle depolarization ratio (LPDR).
The MDR is mainly due to the Rayleigh scattering from atmospheric molecules including nitrogen (N2) and oxygen (O2) when radiated by the transmitted laser beam. In the Rayleigh scattering spectrum, the term “Cabannes line”, which was proposed by Young [23], is referred to as the central line of the scattering spectrum. The central Cabannes line is surrounded by the pure rotational Raman spectrum (PRRS), which is caused by a change in the rotational states of scattering molecules. In 1997, Chance et al. improved parameters for the description of Rayleigh scattering and rotational Raman scattering in the wavelength range of 200 -1000 nm, allowing more accurate calculations of molecular scattering for a variety of atmospheric radiative transfer and constituent retrieval applications [24]. In 2001, She et al. reviewed the spectral structure of the molecular scattering (strength and bandwidth) and its constituent spectra associated with Rayleigh and vibrational Raman scattering. Moreover, She et al. also carried out a systematic comparison of relative scattering strengths and receiver bandwidths among various lidar techniques [25]. In 2002, Behrendt et al. studied the MDR for different receiving bandwidths as well as its dependency on temperature. Besides, the influence of the wavelength-shift between the laser wavelength and the center wavelength of the optical filter in the receiver has also been investigated [26]. However, the transmittance of the Cabannes line was assumed constant in these early studies, which may lead to large uncertainties of the MDR if the wavelength-shift is not trivial. Moreover, the laser linewidth has not been taken into account in these studies, considering the narrow linewidth (1 cm-1 or even narrower) of Nd: YAG lasers typically used in pulsed polarization lidar systems.
Recently, atmospheric lidar techniques based on the optical imaging principle have been developed for atmospheric pollution monitoring, gas concentration measurements, aerosol polarization studies, etc. [27–31]. In 2021, an imaging-based polarization lidar technique, employing a linearly polarized continuous-wave (CW) 450-nm multimode laser diode and a polarization camera, has been proposed for all-day accurate retrieval of the atmospheric depolarization ratio [32,33]. Later on, a visible, near-infrared (VIS-NIR) dual-polarization lidar technique has been implemented for investigating the aerosol spectral characteristics and identifying the aerosol types [34]. In spite of these successful measurements of atmospheric depolarization ratio, absolute calibration with the clean atmosphere method is still highly desirable to further verify the measurement accuracy of these imaging-based polarization lidar systems. However, the laser linewidths of high-power multimode laser diodes are much broader, typically in the order of 1 ∼ 4 nm (full-width at half maximum, FWHM). Thus, the previous narrowband theoretical model of the MDR is not suitable for characterizing the performance of the imaging-based polarization lidar technique [26]. Besides, the dependency of the MDR on the temperature, the wavelength-shift and the receiving bandwidth is still unclear in the case of a broadband laser linewidth.
In view of above challenges, this paper re-examines the narrowband theoretical model of MDR by taking the transmittance of the Cabannes line into account, and then proposes a broadband theoretical model for calculating the MDR, which is suitable for the polarization lidar using high-power multimode laser diodes as light sources. The dependency of the MDR on the temperature, the bandwidths of the receiving channels, and the wavelength-shift has been assessed for broadband laser linewidths. Moreover, atmospheric validation measurements have been carried out by employing an imaging-based polarization lidar with a broadband laser diode as the light source.
2. Theoretical modeling of the MDR
The linear volume depolarization ratio (LVDR, ${\delta _\textrm{v}}$), arising from the depolarization effect of air molecules and aerosols, can be defined by the ratio between the backscattering coefficients of the cross- and parallel-polarization components measured by the polarization lidar
Here z is the measurement distance. Superscripts s and p refer to the cross- and the parallel-polarization in respect to the polarization state of the incident light, respectively. The cross- or parallel-polarization backscattering coefficient ($\beta $) consists of the molecular (subscript mol) and aerosol (subscript aer) components. ${P^\textrm{s}}(z)$ and ${P^\textrm{p}}(z)$ are the recorded lidar profiles of the cross- and parallel-polarization channels, respectively. The gain ratio k refers to the ratio of optoelectronic responses between the two orthogonally polarized channels. The gain ratio can be calibrated and determined experimentally by employing various approaches. However, the clean atmosphere method is an absolute calibration method, which can evaluate the gain ratio by comparing the measured depolarization ratio and the theoretical MDR in the clean atmosphere region [15,16]. On the other hand, the performance of polarization lidars after calibration of the gain ratio can be verified and the instrumental polarization cross-talk can be corrected by the clean atmosphere method, by comparing the theoretical MDR and the measured depolarization ratio at an altitude only containing molecules. As a key input parameter, the MDR (${\delta _{\textrm{mol}}}$) should be determined in priority to obtain the LPDR (${\delta _{\textrm{aer}}}$), which is given by
Here the backscatter ratio R is defined as the ratio of the total backscattering coefficient to the molecular component.
When detecting the MDR using lidar techniques with a certain receiving bandwidth, the backscattering signals could not only contain the Cabannes line, but also the PRRS of atmospheric molecules. In order to model the MDR measured by polarization lidar techniques, the PRRS of N2 and O2 molecules should be taken into account, while the influence of other trace gases such as carbon dioxide, neon and hydrogen is ignored. N2 and O2 molecules may be treated as simple linear molecules with no electronic momentum coupled into the scattering. The shift of the PRRS is a constant on a frequency scale. For linear molecules and linearly polarized light, the differential backscatter cross-sections (DBCS) of the Cabannes line and the total PRRS for atmospheric molecules can be calculated with the Placzek's scattering theory [35]. The depolarization ratio, corresponding to extraction of pure PRRS signals, is independent of molecule species and found to be 0.75 by calculating the ratio of cross-DBCS to parallel-DBCS of the PRRS. The depolarization ratios of the Cabannes line and the whole Rayleigh spectrum are given by
Here ${\varepsilon _i}$ is the anisotropy of the molecular polarizability tensor and the subscript i refers to different atmospheric components (N2 and O2), which is given by ${\varepsilon _i} = 4.5({{F_i} - 1} )$. Fi is the King correction factor related to wavelength ($\lambda $), estimated by [25,36,37]
For linearly polarized light, the depolarization ratio of the Cabannes lines originating from the backscattering of N2 and O2 are 0.0026 and 0.0076, respectively. The depolarization ratio of the Cabannes lines of air is 0.0036 at 532 nm. The depolarization ratio of the whole Rayleigh spectrum (including Cabannes lines and PRRS) of N2 and O2 are 0.01039 and 0.02959, respectively. The volume depolarization ratio of the whole Rayleigh spectrum of air is 0.01396 at 532 nm. As can be seen, the depolarization ratio of Cabannes lines is lower by two orders of magnitude than that of the PRRS. Thus, the measured value of the MDR strongly depends on the fraction of the PRRS, which is related to temperature, spectral bandwidths of the receiving channels, etc.
In traditional polarization lidar techniques with narrowband lasers, the MDR can be calculated by
Here $c_i$ is the relative concentration of atmospheric component (N2 or O2). $(\mathrm{d} \sigma / \mathrm{d} \Omega)_\pi^{\mathrm{Cab} \text { or RR,s},i}$ and $(\mathrm{d} \sigma / \mathrm{d} \Omega)_\pi^{\mathrm{Cab} \text { or RR,p},i}$ are the cross-DBCS and parallel-DBCS of the Cabannes lines or PRRS for atmospheric component i, respectively. $x_i^{\textrm{Cab}}$ and $x_i^{\textrm{RR}}$ are the transmittances of the Cabannes lines and the PRRS of atmospheric component i, respectively, after passing through the receiving filter. The transmittance of the Cabannes line of N2 and O2 can be considered equal. In the case of a perfect match between the laser wavelength and the center wavelength of the receiver, the transmittance of the Cabannes lines of N2 and O2 can be considered as 100%. The transmittance of the rotational Raman scattering is given by
Here $\lambda $ is the wavelength of the emitted laser, $(\mathrm{d} \sigma / \mathrm{d} \Omega)_\pi^{\mathrm{RR}, \mathrm{St}, i}$ and $(\mathrm{d} \sigma / \mathrm{d} \Omega)_\pi^{\mathrm{RR}, \mathrm{ASt}, i}$ are the DBCS of Stokes lines and anti-Stokes lines for atmospheric component i in a state with rotational-angular-momentum quantum number J, respectively. ${\eta _{\textrm{filter}}}(J)$ is the transmittance of the filter for PRRS. Equation (7) can be rewritten as
Here ${\gamma _i}$ is the anisotropy of the molecular-polarizability tensor. ${\gamma _i}$ and ${\varepsilon _i}$ can be considered as constants with temperature since their temperature dependencies are very weak within the range of atmospheric temperatures [35]. Values of ${\gamma _{{\textrm{O}_{2}}}}$ and ${\gamma _{{\textrm{N}_{2}}}}$ are estimated as [24]
Here v is the wavenumber, which is defined as the number of the wavelength per unit length. v can be expressed by ${{\nu \textrm{ = }1} / \lambda }$ with the unit of µm-1. It should be noted that above equations are only valid in the range of 0.2 to 1 µm.
Figure 1 (a) shows the Cabannes line and PRRS of the air molecule at 532 nm with an infinitely narrow laser linewidth. Since above equations do not take the laser linewidth into account, they are not suitable for the calculation of the MDR with a broadband laser linewidth, e.g., 1 to 4 nm FWHM for high-power multimode laser diodes.
Fig. 1. The Cabannes line and pure rotational Raman spectrum (PRRS) of air molecules at 532 nm with (a) a narrowband laser linewidth and (b) a broadband laser linewidth.
In this work, a broadband theoretical model of MDR is proposed by taking the laser linewidth, temperature, bandwidths of the receiving channels and the wavelength-shift into account. The typical broadband laser spectrum with a 2-nm FWHM is simulated by a Gaussian function. In order to simplify the calculation, the normalized laser spectrum was divided into n equal parts (n = 300 in simulation), and each part can be utilized to calculate the parallel- and cross-components of a certain atmospheric component. The corresponding Cabannes line and the PRRS of air molecules are shown in Fig. 1 (b). As a result, the MDR can be obtained through spectral integration from ${\lambda _{\textrm{lower}}}$ to ${\lambda _{\textrm{upper}}}$. The MDR with a broadband laser linewidth is thus given by
Here $\eta_{\small{\textrm{LD}}}(\lambda)$ is the relative intensity of the emitted laser spectrum at wavelength $\lambda $.
3. Simulation studies on the MDR
In the following sections, simulation studies on the MDR have been carried out for the cases of narrowband and broadband laser linewidths, by taking the transmittance of the Cabannes line into account. Meanwhile, the dependencies of the MDR on the temperature, the bandwidths of the receiving channels, and the wavelength shift have been assessed. The operation wavelength of the imaging-based polarization lidar system can be chosen in a wide range (375-1000 nm) owing to the wide spectral selectivity of laser diodes. Among these wavelengths, 520 nm is a favorite wavelength for the imaging-based polarization lidar, which is also close to the typical operation wavelength (532 nm) of the conventional pulsed polarization lidar. Meanwhile, the MDR at 532 nm has been thoroughly investigated in previous polarization lidar studies using Nd: YAG lasers as light sources. Thus, simulation studies are focused on the wavelengths of 520 nm and 532 nm.
A main focus of the present work is to investigate the mutual influence of different factors on the MDR for practical atmospheric lidar systems. This implies that the laser linewidth, the receiving bandwidth, as well as the wavelength-shift cannot vary arbitrarily in simulation studies. The variations of these parameters should be physically meaningful. For example, the laser linewidth and the wavelength-shift are generally smaller than the receiving bandwidth to optimize the signal-to-noise ratio (SNR) performance of a practical lidar system.
For the laser linewidth, there are mainly two cases in atmospheric polarization lidar techniques, narrowband laser linewidth (single frequency) much smaller than the receiving bandwidth and a relatively broadband laser linewidth in the order of nm. Thus, the simulation studies have been classified into two categories, which are discussed below.
3.1 Simulation of the MDR with a narrowband laser linewidth
In previous studies, the 532-nm MDR has been evaluated theoretically and experimentally [17,20]. In this work, the difference of MDRs between 532 nm and 520 nm have been examined in the case of a narrowband laser linewidth. It has been found out that the 532-nm MDR at 273 K varies between 3.54 × 10−3 and 12.83 × 10−3, as the receiving bandwidth increases from 0.1 nm to 10 nm (FWHM), while the corresponding 520-nm MDRs are in the range of 3.55 × 10−3-12.95 × 10−3. As can be seen, the relative deviations between the MDRs at the two wavelengths are lower than 4%. Such a small deviation implies that we can directly compare the MDR values at 520 nm and 532 nm without considering the wavelength dependency. It should be noted that the transmittance curve is simulated by a Gaussian function, which is a widely used filter type for lidar systems. However, if the actual transmittance curve of the receiver, e.g., Lorentzian-shape, flat-top shape, has a large discrepancy with the Gaussian-shape transmittance model, it is advised to re-evaluate the MDR.
In practical atmospheric measurements, a mismatch between the center wavelength in the receiver and the laser wavelength (wavelength-shift) might occur due to the deviation of the center wavelength of the receiving filter or the variation of the laser emission wavelength. Figure 2 shows the transmittance curve of a Gaussian-shape filter with a bandwidth of 0.5 nm for different wavelength-shifts (e.g., -0.5 nm, 0 nm, 0.5 nm) in respect to the laser wavelength of 532 nm. As can be seen, the transmittance of the Cabannes line after passing the filter decreases, while the transmittance of the PRRS increases, as the wavelength-shift increases. Clearly, the transmittance of the Cabannes line, which is assumed to be a constant in previous studies [26], should be considered.
Fig. 2. Transmittance curve of a Gaussian-shape filter with a receiving bandwidth of 0.5 nm (FWHM) for different wavelength-shifts (e.g., -0.5 nm, 0 nm, 0.5 nm) in respect to the laser wavelength of 532 nm.
Figure 3 shows the relationship between the MDR and the temperature with different wavelength-shifts at a receiving bandwidth of 0.5 nm. As shown in Fig. 3 (a), if the transmittance of the Cabannes line is not taken into account, the relative deviation of the MDR is about 15% with a wavelength-shift of ±0.5 nm for a receiving bandwidth of 0.5 nm (240 K), which is in good agreement with the results reported by Behrendt et al. [26]. However, a wavelength-shift of ±0.5 nm can result in an increment of the MDR beyond 200% considering the transmittance of the Cabannes line, as shown in Fig. 3 (b), which is significantly larger than the values evaluated without considering the transmittance of the Cabannes line. Meanwhile, the MDR value become more sensitive to the ambient temperature and its relative deviation can reach up to 40.1% for a wavelength-shift of ±0.5 nm as the temperature varies between 200 K and 350 K. Besides, the variation of the MDR for the red shift of the receiver is slightly larger than that for the blue shift. This can be attributed to the fact that the intensity of the Stokes PRRS is stronger than that of the anti-Stokes PRRS.
Fig. 3. The relationship between the MDR and the temperature with different wavelength-shifts at a receiving bandwidth of 0.5 nm. (a) Without considering the transmittance of the Cabannes line, see also in [26]; (b) considering the transmittance of the Cabannes line (the present narrowband theoretical model).
Table 1 summarizes the transmittances of the Cabannes line and the PRRS with a receiving bandwidth of 0.5 nm for different wavelength-shifts, respectively. In an extreme case, the lidar system only receives the PRRS signal and rejects the Cabannes line when the wavelength-shift is beyond 2 nm. The corresponding MDR is about 0.75. Thus, the MDR detected by a lidar system depends highly on the relative proportion of the Cabannes lines in respect to the PRRS measured by the receiver. The wavelength-shift should be less than ± 0.1 nm to achieve a small deviation of the MDR (<2%). Thus, the wavelength-shift between the center wavelength in the receiver and the laser wavelength should be considered carefully for the calculation of the MDR in practical applications of polarization lidar.
Table 1. Simulation results of the MDR with different wavelength-shifts considering the transmittance of the Cabannes line.a
Besides, the mutual influence of the wavelength-shift and the receiving bandwidth on the MDR has also been investigated, as shown in Fig. 4. It should be emphasized that the maximum wavelength-shifts are not constants for different receiving bandwidths. In general, the maximum wavelength-shift should be less than the receiving bandwidth. Otherwise, it implies that the present optical filter is an incorrect choice for the lidar system. In the following simulation analysis, the maximum wavelength-shift is set to a half of the receiving bandwidth, e.g., a receiving bandwidth of 1 nm corresponds to a maximum wavelength-shift of ±0.5 nm.
Fig. 4. The MDR at 532 nm with a narrowband laser linewidth under the different wavelength-shifts for different receiving bandwidths at the atmospheric temperature of 273 K. The maximum wavelength-shift is set to a half of the receiving bandwidth.
As can be seen from Fig. 4, the value of the MDR generally increases with the increasing of the receiving bandwidths. The MDR with a 10-nm receiving bandwidth is about 4 times larger than that with a 0.3-nm receiving bandwidth. However, the MDR is also influenced by the wavelength-shift. The relative deviation introduced by the maximum wavelength-shift firstly increases and then decreases with the increasing of the receiving bandwidth, e.g., 3.2% at 0.3 nm, 41.3% at 1 nm, 74.8% at 2 nm and 18.8% at 10 nm.
3.2 Simulation of the MDR with a broadband laser linewidth
3.2.1 Dependency of the MDR on the laser linewidth and receiving bandwidth
Since broadband light sources are mainly employed in imaging-based polarization lidar techniques, the MDR with a broadband laser linewidth has been investigated at 520 nm. As shown in Fig. 5, the MDR value can change substantially with the increasing of the laser linewidth and the receiving bandwidth. In the case of infinitely narrowband laser linewidth, the transmittance of the PRRS increases with the broadening of the receiving bandwidth, leading to the increasing of the MDR. The entire PRRS lines are almost covered by the receiver with a bandwidth of 15 nm, and the MDR is close to the depolarization ratio of the whole Rayleigh scattering spectrum (∼0.14). On the contrary, only the Cabannes lines can be received with a receiving bandwidth of 0.2 nm, and the MDR is about 0.0036.
Fig. 5. The MDR values with different laser linewidths and receiving bandwidths at 520 nm. The atmospheric temperature is set to 273 K.
When the receiving bandwidth is less than e.g., 3 nm, the MDR increases sharply with the increasing of the laser linewidth. This is because more PRRS passes through the receiving filter as the laser linewidth increases. For a broader receiving bandwidth, e.g., ≥ 5 nm, the dependence of the MDR on the laser linewidth will be greatly reduced.
On the other hand, if the laser linewidth is fixed, the MDR generally increases with the increasing of the receiving bandwidth. However, when the laser linewidth is larger, e.g., 8-10 nm, the MDR value is insensitive to the receiving bandwidth, as the whole Rayleigh scattering spectrum may be detected. In practical measurement, the laser linewidth of the high-power laser diode is typically less than 4 nm, as illustrated by the pink region in Fig. 5. Thus, the following simulation studies are mainly focused on 2-nm and 4-nm laser linewidths.
3.2.2 Dependency of the MDR on the temperature, receiving bandwidth and wavelength-shift
Figures 6(a) and (b) show the temperature and bandwidth dependency of the MDR with a fixed laser linewidth of 2 nm. As can be seen, a larger receiving bandwidth corresponds to a larger MDR, e.g., an MDR of ∼0.14 with 10-nm bandwidth. The MDR would generally become smaller with the increasing of the temperature, as the PRRS becomes broader with the temperature. The dependency of the MDR on the temperature is reduced for a larger receiving bandwidth, and the temperature-induced relative deviation is less than ±2% with a receiving bandwidth of 10 nm. Moreover, the temperature and bandwidth dependencies of the MDR with a laser linewidth of 4 nm are also shown in Fig. 6 (c) and (d). In general, a similar temperature dependency has been found. However, in the case of the same receiving bandwidth, the temperature-induced relative deviation with the laser linewidth of 4 nm is smaller than that with the laser linewidth of 2 nm.
Fig. 6. (a) The relationship between the MDR and the temperature for different receiving bandwidths and (b) the corresponding relative deviation of the MDR introduced by temperature variations with a laser linewidth of 2 nm. (c) The relationship between the MDR and the temperature for different receiving bandwidths and (d) the corresponding relative deviation of the MDR introduced by temperature variations with a laser linewidth of 4 nm.
The mismatch between the center wavelength of the emitted laser and the center wavelength of the optical filter in the receiver can also introduce a systemic error on the MDR. Thus, the influence of the wavelength-shift between the receiver center wavelength and the laser wavelength on the MDR should be carefully studied. Figure 7 (a) and (b) show the MDR at the wavelengths of 520 nm under different wavelength-shifts and receiving bandwidths with the laser linewidths of 2 nm and 4 nm, respectively. In the simulation analysis, the maximum wavelength-shift is set to a half of the receiving bandwidth. As can be seen, the wavelength-shift also has a large impact on the MDR for a broadband laser linewidth, as the transmittance of the Cabannes line would decrease with the increasing of the wavelength-shift. The relative deviations of the MDR introduced by the maximum wavelength-shift are 13.4% at 1-nm bandwidth, 33.3% at 2-nm bandwidth, 35.5% at 5-nm bandwidth and 16.6% at 10-nm bandwidth for a laser linewidth of 2 nm. Besides, under the same receiving bandwidth, the relative deviation caused by the wavelength-shift with the laser linewidth of 4 nm is generally smaller than that with the laser linewidth of 2 nm.
Fig. 7. The MDR at the wavelengths of 520 nm under different wavelength-shifts and receiving bandwidths at 273 K. (a) The laser linewidth is 2 nm. (b) The laser linewidth is 4 nm.
The dependency of the MDR on the temperature and the wavelength-shift has been evaluated with different receiving bandwidths and laser linewidths, as shown in Fig. 8. A common feature is that the relative deviation of the MDR induced by the temperature variation decreases as the increasing of the wavelength-shifts. It can also be concluded that the relative deviation introduced by the temperature variation generally decreases with the increasing of the receiving bandwidth and the laser linewidth. However, a wide receiving bandwidth would also introduce relatively larger solar background and thus noise in practical measurements, leading to the deterioration of the SNR. Thus, the receiving bandwidth should be comparable to the laser linewidth in practical measurements. For a laser linewidth of 2 nm and a receiving bandwidth of 2 nm, the relative deviation introduced by the wavelength-shift is less than 4% if the wavelength-shift is less than ±0.3 nm, requiring precise alignment between the receiver center wavelength and the emission wavelength of the laser diode that can be tuned by injection current and case temperature.
Fig. 8. The dependency of the MDR on the temperature for different wavelength-shifts. (a) Laser linewidth: 2 nm, receiving bandwidth: 2 nm, the MDR without wavelength shift is 0.0079. (b) Laser linewidth: 2 nm, receiving bandwidth: 10 nm, the MDR without wavelength shift is 0.013. (c) Laser linewidth: 4 nm, receiving bandwidth: 4 nm, the MDR without wavelength shift is 0.011. (d) Laser linewidth: 4 nm, receiving bandwidth: 10 nm, the MDR without wavelength shift is 0.013.
3.2.3 MDR in typical polarization lidar systems
In practical measurement, various receiving bandwidths would be employed for the polarization lidar technique, resulting in different values of the MDR. The determination of the theoretical value of the MDR is significant for validating the performance of a polarization lidar system.
Table 2 shows the theoretical MDR values at typical operation wavelengths of polarization lidar systems under different receiving bandwidths at 273 K. Since the multimode laser diodes (e.g., 405 nm, 450 nm, 520 nm, 808 nm) used in the imaging-based polarization lidar have a relatively broader linewidth (e.g., 1-4 nm), the MDR values evaluated with a receiving bandwidth of less than 1 nm most likely cannot be observed in practical atmospheric measurements. As can be seen from $\delta _{520\textrm{nm}}^{2\textrm{nm}}$ and $\delta _{520\textrm{nm}}^{narrow}$, the discrepancy of the MDR values between different laser linewidths gradually decreases with the increasing of the receiving bandwidth. However, when the receiving bandwidth is equivalent to the laser linewidth, e.g., FWHM = 2 nm, which is the case in most practical measurements, the MDR ($\delta _{520\textrm{nm}}^{2\textrm{nm}}$) at 520 nm calculated by the broadband theoretical model is about 21% larger than the value ($\delta _{520\textrm{nm}}^{narrow}$) evaluated without considering the laser linewidth. The large discrepancy implies that the broadband theoretical model is necessary for evaluating the MDR in the imaging-based polarization lidar, which utilizes multimode laser diodes as light sources.
Table 2. The theoretical MDR values at typical operation wavelengths of polarization lidar systems under different receiving bandwidths at 273 K.a
4. Experimental investigation of the MDR
Atmospheric depolarization measurements have been carried out by an imaging-based polarization lidar for cross validation of the MDR values obtained from the broadband theoretical model. The polarization lidar system utilizes a multimode 520 nm laser diode as the light source and a polarized image sensor as the detector [33,34]. A linear polarizer is placed in front of the laser diode to improve the degree of linear polarization (>99.99%). The backscattering signal is collected by a Maksutov-Cassegrain telescope (f = 1000 mm, ϕ=105 mm) and then detected by a polarization camera (2048 × 2448, 3.45 µm × 3.45 µm) equipped with four micro-polarizer arrays. An interference filter with 10 nm FWHM is used to reject the solar background. The baseline between the transmitted laser beam and the optical axis of the Maksutov-Cassegrain telescope is 1 m. Four polarized lidar profiles can be simultaneously obtained by the polarization camera, from which the depolarization ratio can be evaluated.
Atmospheric vertical measurements have been carried out from 22:00 on 11th October to 05:30 on 12th October 2022 (UTC +8) in Dalian University of Technology (DUT), China. The ambient temperature and pressure at different altitudes were provided by a nearby radiosonde at around 00:00 (local time) on 12th October 2022, which can be used for calculating the molecular extinction and the backscattering coefficient at different altitudes. Consequently, the molecular lidar signal at different altitudes can be simulated by the atmospheric model. The aerosol backscattering/extinction coefficients are retrieved by using the Fernald method with two key parameters as inputs, namely the boundary value and the lidar ratio. The boundary value can be determined at a certain altitude, where no aerosol exists. Although “clean” atmosphere generally exists at an altitude of more than 10 km, it might occasionally appear at lower altitudes, e.g., in the range of 3 to 5 km. The aerosol lidar ratio, which is defined by the ratio of the aerosol extinction coefficient to the aerosol backscattering coefficient, is set to 50 sr [38,39].
Figure 9 shows the time-space map of parallel- and cross-polarized backscattering signals, the extinction coefficient, the ratio of 45° to 135° polarized backscattering signals, the LVDR and the LPDR. Figure 10 shows the total backscattering profiles, the extinction coefficient profiles, the LVDR profiles and the LPDR profiles in different hours. As can be seen, the boundary layer was slightly less than 2 km before 0:00 on 12th October 2022 and then increased over time. In most cases, the boundary layer height during nighttime is less than 2 km. However, it can reach up to about 2 km in some occasions [40,41]. According to Fig. 10 (a), the lidar profile at the altitude of 2.5-5 km highly coincided with the simulated molecular lidar profile, indicating that a clean atmosphere containing only molecules appears in this region. As can be seen from Fig. 9 (c) and Fig. 10 (b), the extinction coefficient is close to zero at the altitude of 2.5-4 km. Figure 9 (d) illustrates the time-space map of the ratio of the 45°-polarized lidar signal to the 135°-polarized lidar signal, from which the offset angle between the polarization plane of the transmitted laser beam and the receiver polarization plane can be obtained. The offset angle was 0.28° with a standard deviation of ± 0.009° during the whole measurement period, implying good system alignment.
Fig. 9. Atmospheric vertical measurements have been carried out from 22:00 on 11th October to 05:30 on 12th October 2022 (UTC +8) in Dalian University of Technology (DUT), China. The time-space map of (a) parallel- and (b) cross-polarized backscattering signals, (c) the extinction coefficient, (d) the ratio of 45° to 135° polarized signals, (e) the LVDR and (f) the LPDR.
Fig. 10. (a) The total backscattering profiles (b) the extinction coefficient profiles (c) the LVDR profiles (d) the LPDR profiles in different hours.
In order to quantitatively evaluate the precision of LVDR, statistical analysis has been performed at the altitude of 3 ∼ 4 km during the whole night. The temperature dependence of the MDR is very weak with a 10-nm bandwidth interference filter, and thus the temperature-induced variation of the MDR simulated by the broadband theoretical model at the altitude of 3∼4 km is only 2%. Figure 11 shows the hour-averaged depolarization ratio in the altitude of 3-4 km from 22:00 to the next morning measured by the polarization lidar. As can be seen, the measured volume depolarization ratio in the clean region is 0.0129 ± 0.0025 and the deviation is only 3% compared with the theoretical MDR of 0.01324 ± 0.00005 at 273 K, which was calculated based on the practical 520-nm laser spectrum of a laser diode (FWHM = 3.4 nm) and the typical transmittance of a commercial optical filter. The optical wavelength has a center wavelength of 520 nm with an uncertainty of ± 2.0 nm and a bandwidth of 10 nm.
Fig. 11. Hour-averaged depolarization ratio at the altitude of 3∼4 km from 22:00 on 11th October to 05:30 on 12th October, 2022 (blue dot with error bars) measured by the polarization lidar. The red dash-dotted lines show the theoretical MDR for a 2-nm laser linewidth and a 10-nm receiving bandwidth.
5. Conclusion
The MDR is a key parameter for polarization lidar techniques, which cannot only be used for calibration of the gain ratio but also often used to validate the measurement accuracy. Theoretical and experimental investigation on the MDR is of great significance for various applications. Previous studies mainly focused on cases with a narrowband laser linewidth, and the transmittance of the Cabannes line was assumed constant in these early studies. The present work re-examines the narrowband theoretical model of MDR by taking the transmittance of the Cabannes line into account. A large relative deviation of beyond 200% has been found if the wavelength-shift reaches up to 0.5 nm for a receiving bandwidth of 0.5 nm, which is significantly larger than the value (15%) calculated without considering the transmittance of the Cabannes line reported in previous studies. Thus, the wavelength-shift (misalignment) between the center wavelength in the receiver and the laser wavelength should be deliberated in practical applications of the pulsed polarization lidar employing Nd: YAG lasers as light sources.
A broadband theoretical model has been proposed for calculating the MDR, which is suitable for the polarization lidar using high-power multimode laser diodes as light sources. The dependency of the MDR on the temperature, the receiving bandwidth, as well as the wavelength-shift between the laser wavelength and the center wavelength of the receiver has been carefully investigated for the broadband polarization lidar techniques, and a few conclusions can be summarized. The MDR increases sharply with the increasing of the laser linewidth for the receiving bandwidth of less than 3 nm, while the dependence of the MDR on the laser linewidth will be greatly reduced for a broader receiving bandwidth, e.g., ≥ 5 nm. On the other hand, if the laser linewidth is fixed, the MDR generally increases with the increasing of the receiving bandwidth. The MDR would become smaller with the increasing of the temperature, and the dependency of the MDR on the temperature is reduced with the increasing of the wavelength-shift, the receiving bandwidth and the laser linewidth. Besides, the wavelength-shift can result in a significant increasing of the MDR for a broadband laser linewidth as well. The relative deviations introduced by the maximum wavelength-shift are 13.4% at 1-nm bandwidth, 33.3% at 2-nm bandwidth, 35.5% at 5-nm bandwidth and 16.6% at 10-nm bandwidth for a laser linewidth of 2 nm, which would decrease for a larger laser linewidth (e.g., 4 nm).
The theoretical MDR values at typical operation wavelengths of polarization lidar systems under different receiving bandwidths at 273 K are also evaluated based on the broadband theoretical model. If the receiving bandwidth is equivalent to the laser linewidth (e.g., 2 nm), which is the case in most practical measurements, the MDR at 520 nm calculated by the broadband theoretical model is about 21% larger than the value evaluated without considering the laser linewidth. The large discrepancy implies that the broadband theoretical model of the MDR is necessary for the polarization lidar employing broadband light sources.
Atmospheric measurements for the verification of the MDR have been carried out by employing a 520-nm imaging-based polarization lidar on 11th October 2022. It has been found out that the lidar profile at the altitude of 2.5-4 km highly coincides with the simulated molecular signal for the whole night, indicating the presence of a clean atmosphere containing only molecules. In this clean atmospheric region, the volume depolarization ratio is found to be 0.129 ± 0.0025, which is highly consistent with the theoretical MDR at 520 nm with a 10-nm receiving bandwidth. The good agreement between theoretical and experimental results demonstrated a high measurement accuracy of the imaging-based polarization lidar and excellent feasibility of the broadband theoretical model.
Funding
National Natural Science Foundation of China (62075025); Dalian High-Level Talent Innovation Program (2020RQ018); Fundamental Research Funds for the Central Universities (DUT22JC17, DUT22QN246).
Acknowledgments
The authors greatly acknowledge Dr. Zhenping Yin for valuable discussion and Dr. Ioannis Binietoglou for providing the original simulation code of the molecular depolarization ratio.
Disclosures
The authors declare no conflicts of interest.
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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