An improved calibration method for lidar depolarization measurement is described. With this method the system constants including the electronic gain ratio of the parallel and perpendicular channels, the optical reflectance and transmission parameters of the polarizing beam splitter, and the linear polarization ratio of the emitting laser beam can be determined conveniently by using lidar measurements with a half-wave plate oriented at selected angles.
©2013 Optical Society of America
The polarization lidar technique first described by Schotland et al in the early 1970s  is a powerful tool for distinguishing ice clouds from water clouds and identifying non-spherical aerosol particle. This technique is based on the principle that backscattered radiation from spherical particles does not significantly differ from its original polarization state. Partial depolarization of backscattered light is caused by non-spherical particle scatters, or by multiple scattering effects, which allow non-backscatted light scatters back into the instrument filed-of-view. Traditional linear polarization measurements determine the ratio of the parallel and perpendicular polarized (with respect to the linearly polarized source) return signals. Unfortunately, other factors contributing to observed depolarized signals, such as an incomplete polarization of the laser source, non-ideal behavior of the polarizing beam splitter, and different gain factors between the parallel and perpendicular channels, have to be considered to determine depolarization due to atmospheric particles and molecules. Therefore, calibration of a polarization lidar is important to provide accurate atmosphere measurements.
Usually, the instrument gain ratio of the two polarization channels is determined through observations in aerosol and cloud free regions, by assuming that the observed ratio in these cases is equal to the molecular depolarization ratio obtained with theoretical calculations  or with observational data . The drawback of this method is low concentrations of undetected aerosols can cause significant errors. Furthermore, molecular depolarization must be known with high accuracy. Considering that lidar measured molecular depolarization ratio can range from 0.36% to 1.4% depending on lidar receiver spectral width and may exhibit temperature dependency , errors can be significant due to assumed molecular depolarization ratio. Another calibration method uses unpolarized light to generate equal signals on both channels [4, 5]. But unpolarized light is not easily to obtain, especially in the field. A single detector technique can also be used, switching optics to detect the two polarizations for alternate laser pulses [6, 7]. However, one must accordingly know the optical reflectance and transmission parameters of a polarizing beam splitter (PBS) in this method. These properties are not easily determined and could change significantly, not only by misalignment of the polarizing cube but also by non-ideal collimation of the light beam, especially for broad spectral band prisms [8–10].
Another calibration method is to generate a balanced signal on both detectors by insertion of a half-wave plate into the optical path of either the transmitter  or receiver . Freudenthaler et al. proposed a ‘ ± 45° calibration’ method  to improve the calibration accuracy. But it is still limited by previous knowledge of the optical reflectance and transmission parameters of the polarizing beam splitter. In fact, almost all of the calibration methods described consider the source laser beam perfectly linearly polarized, which is not always true due to the laser source itself and possible depolarization by the beam transmitting optics. This will also introduce errors in lidar depolarization measurements. Here we propose an improved ‘ ± 45°calibration’ method to calibrate the system constants of a polarization lidar.
2. Polarization lidar calibration method
Assuming that there is no multiple scattering (which is true for observations with optical depth smaller than 0.5 ), the total backscattered power P(r) from distance r is described with the single-scattering lidar equation:
For polarization lidars, atmospheric backscatter signals are separated into parallel and perpendicular polarization components by using a PBS. The angle between the plane of polarization of the laser and the incident plane of the PBS(φ) can be adjusted by rotating a half-wavelength-plate(HWP) which is inserted in the optical path of the transmitting laser beam(as shown in Fig. 1). Assuming the laser beam is perfectly linearly polarized, for φ = 0°, the received power components before the PBS with respect to parallel (PP) and perpendicular (PS) to the incident plane of the PBS (see Fig. 1) can be written respectively as
Here, we denote the power measured in the reflected and transmitted channels after PBS with the subscripts R and T, respectively. The total reflected (PR) and transmitted (PT) power components actually recorded by the data acquisition can be written asFigure 1 should be considered as a schematic with key components to represent the effects of other optical components on lidar transmitting and receiving system.
Then, according to the ‘ ± 45°calibration’ method, the relative amplification factor ratio V*(V* = VR / VT) can be calculated from two subsequent measurements by setting φ at ± 45° by ,
For φ = 90°, the polarization plane of the laser has been rotated by 90°, which mean, , then Eq. (4) change toEqs. (6) and (8) the followings can be derived:
Thus, we can set initial values for RP, TP, RS, TS and iterate following Eqs. (5), (9), and (10) until the relative residual difference between two iterations is less than 0.1%. For calibration, the value of δv in Eq. (9) can be calculated theoretically for clean air . We can then calculate the linear volume depolarization ratio δv from Eq. (6) as,
Figure 2 shows the iterated solutions of RP, TP, RS, and TS based on simulated lidar signals generated with RP = 0.04, TP = 0.96, RS = 0.98, TS = 0.02, and V* = 1.67. The initial values for the iteration were arbitrarily set to: RP = 0.01, TP = 0.99, RS = 0.99, TS = 0.01. The errors in determined V* and the other optical parameters are not sensitive to the errors of orientation angle φ, as shown in Fig. 3. The errors of the solutions are also insensitive to the assumption error of δv. As illustrated in Fig. 4, the relative errors in derived RP, TP, RS, and TS are only ~0.3% of the assumption error of δv.
For Eq. (2), perfect linearly polarized laser beam is assumed. We next consider a more realistic situation. The outgoing laser power can also be split up to the parallel (P||) and orthogonally polarized (P⊥) components relative to the incident plane of the receiver PBS, and Eq. (2) is then rewritten as
Here, we introduce an outgoing laser power linear depolarization ratio δL .
P|| and P⊥ are components of the outgoing laser power, thus δL is a factor including all the transmitting system effects. Such as the laser source is not perfect linearly polarized, possible depolarization/diattenuation by the beam transmitting optics and the HWPs rarely are able to provide a perfect π phase shift.
In this circumstance, Eq. (3) is rewritten as
Then, we haveEq. (11). For an aerosol-free lidar range in the free troposphere, δA = δm, the linear depolarization ratio δm of air molecules can be calculated theoretically . We then solve all the system constants V*, RP, TP, RS, TS and δL in a manner consistent with the iteration process described above, which can then be used for regular measurements. Finally, the linear volume depolarization ratio δv can be calculated by:
3. Calibration of the UW Airborne Raman lidar
The University of Wyoming (UW) compact airborne Raman lidar was developed for aerosol and short range (within 1km) water vapor measurements from the UW King Air. The transmitter is a flashlamp-pumped Nd: YAG laser which emits about 50 mJ per pulse at 355 nm with a repetition rate of 30 Hz. A HWP is installed close to the laser emitting window to adjust the polarization direction of the outgoing laser. The receiver is a Cassegrain telescope with aperture of 300 mm, which is mounted on a rigid optical table together with the laser. The lidar has four receiving channels; two channels for elastic scattering parallel and perpendicular respectively, and the other two for Raman scattering from water vapor and nitrogen respectively. All of the channels use narrow band filters with FWHM (full width at half maximum) of 0.3 nm. A four-channel 12-bit data acquisition card with 100M sampling frequency is used to record the signals. A simplified instrument layout is shown in Fig. 5.
A calibration exercise was carried out under a clear sky condition after snowing event on Feb 3, 2010. The lidar was aligned for zenith observation from the ground. First, we rotated the HWP to + 22.5°to set the angle φ (see Fig. 1) to be + 45°. After 900 shots of accumulating (30 seconds), the HWP was rotated to −22.5°to set the angle φ to be −45°, then φ = 0° and φ = 90°. Thus, we collected measurements of δ*( ± 45), δ*(0) and δ*(90) to use the method described in Section 2 to calibrate the system. The signals at an aerosol-free range with good signal to noise ratio at about 4 km altitude were chosen to for the calculation. The initial values were set as RP = 0.01, TP = 0.99, RS = 0.99, TS = 0.01 (based on the factory specifications of the PBS) and δV = 0.0045 (calculated based on 355nm and 0.3 nm filter FWHM for clear air), and the solutions are achieved after 9 iteration loops. The solutions are RP = 0.077, TP = 0.923, RS = 0.957, TS = 0.043, δL = 0.0031, V* = 1.745.
An improved depolarization lidar calibration method, which just requires a HWP in the emitting optical path, is described in detail. The method is based on lidar measurements at φ(see Fig. 1) = 0°, ± 45° and 90° respectively. With an iteration based processing algorithm and the observations within an aerosol-free range, the optical reflectance and transmission parameters of the polarizing beam splitter and the gain factor ratio of the parallel and perpendicular channels can be determined. The linear depolarization ratio of emitting laser power can also be determined. The lidar can be well calibrated for atmospheric depolarization measurement with these system parameters, which has been demonstrated on the University of Wyoming airborne Raman lidar. This calibration method can be applied to all of the polarization-sensitive lidars if there is a HWP in the emitting optical path, which is true for most systems.
In this method, the assumption of molecular depolarization δv is used in an iteration process rather than directly used for the calibration. The calibration errors in the relative amplification factor ratio and the other optical parameters are only ~0.3% of the error in assumed δv which makes the calibration method more accurate.
This work is supported by National Science Foundation (NSF) under Award AGS-0645644.
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