We describe implementation and demonstration of a polarization technique adapted for lidar to measure all unique elements of the volume backscatter phase matrix. This capability allows for detection of preferential orientation within a scattering volume, and may improve scattering inversions on oriented ice crystals. The technique is enabled using a Mueller formalism commonly employed in polarimetry, which does not require the lidar instrument be polarization preserving. Instead, the accuracy of the polarization measurements are limited by the accuracy of the instrument characterization. A high spectral resolution lidar at the National Center for Atmospheric Research was modified to demonstrate this polarization technique. Two observations where the instrument is tilted off zenith are presented. In the first case, the lidar detects flattened large raindrops oriented along the same direction due to drag forces from falling. The second case is an ice cloud approximately 5 km above lidar base that contains preferentially oriented ice crystals in a narrow altitude band.
© 2012 OSA
Interpretation of polarization lidar data generally assumes that the scattering volume consists of randomly oriented particles. This assumption, however, is invalid in some cases. Large raindrops deform from perfect spheroids due to drag; resulting in what is approximated as horizontally oriented oblate spheroids [1, 2]. Ice crystals also orient under certain conditions which sometimes result in optical phenomena such as sun dogs, pillars and arcs .
It has been suggested that ice crystal orientation may have an impact on radiative transfer through increased cloud albedo  and cloud lifetime through fall speed [5, 6]. In  it was suggested that horizontal orientation is the natural state of ice crystals in the atmosphere and  estimated that 40% of all high clouds observed by the Polarization and Directionality of Earth Reflectances (POLDER) satellite mission contained oriented ice crystals. Still, it is not clear if oriented ice crystals occur in sufficient concentration and frequency to significantly impact Earth’s radiation budget.
The occurrence of preferential orientation also presents an issue when interpreting remote sensing data. Satellite retrievals of ice cloud optical depth and particle shape can vary significantly depending on the assumed orientation of the scatterers . Furthermore, polarization parameters retrieved by lidar systems can have ambiguous meaning when oriented ice crystals are present [10, 11]. The core problem is that measurement methods capable of detecting preferential orientation are rarely employed.
Horizontally oriented plates are often identified by directing a lidar along zenith or nadir. The presence of horizontally oriented plates will result in observation of a high backscatter, low depolarization specular reflection [12–16]. CALIPSO used this technique for a 17 month period, but was subsequently tilted off nadir, because the specular signals tend to overwhelm other cloud and aerosol observations . Scanning capability was used in  to allow for better characterization of the orientation distribution and randomly oriented ice crystal properties.
Specular reflections produce robust signatures of oriented plates, but they generally cannot contribute additional information about the oriented crystals (i.e. habit, size, etc.). In one instance, however, a Doppler lidar was used to determine the fall speed of the oriented crystals . Using microphysical growth and aerodynamic properties of plates, the ice crystal size and shape could then be inferred [19, 20].
Off-zenith polarization measurements present a means for detecting and characterizing ice crystals through their unique polarization signatures without sacrificing other cloud and aerosol data. In , the backscatter Stokes vector was measured to detect the presence of oriented ice crystals and the Cloud Aerosol Backscatter and Depolarization Lidar (CAPABL) measures backscatter phase matrix diattenuation, a polarization effect only attributable to oriented scatterers [22, 23].
By measuring the full backscatter phase matrix, all polarization properties of an arbitrary scattering volume may be characterized. Such a measurement of oriented ice crystals provides five unique polarization parameters, which would likely improve scattering inversions of the oriented population. However, to our knowledge, this measurement has only performed by . Perhaps this is because the design, hardware, and processing complexity commonly thought to be necessary for such a measurement create a practical deterrent. However, through the use of a vectorized Mueller matrix formalism that is commonly employed in polarimetry [25–27], we have developed a simple methodology for measuring the full backscattered matrix. The method does not require a highly polarization preserving instrument or elaborate polarization hardware. Its accuracy is defined by the accuracy of instrument characterization, which is necessary to process the data for scattering matrix retrieval. Through the use of a single rotating quarter wave-plate (QWP), we modified the high spectral resolution lidar (HSRL) at the National Center for Atmospheric Research (NCAR) to measure the full backscatter phase matrix without loss of conventional HSRL data products [28, 29]. We will describe the methodology employed in this measurement and system characterization and show observations of scattering matrices for rain and an ice cloud containing oriented ice crystals.
The end-to-end relationship from laser to measured optical signal is given by the Stokes Vector Lidar Equation (SVLE) where polarization states are represented as Stokes vectors and each term along the optical path is a Mueller matrix . The equation is written
In the case of polarization measurements, atmospheric extinction is not expected to have a significant impact on the transmitted polarization state [15, 30] so the scattering matrix F(k⃗i, k⃗s, R) is the only polarization term in the SVLE that contains information about the atmosphere. However, additional terms in the instrument matrices generally present an issue necessitating correction to produce an accurate characterization of the atmosphere.
The polarization properties of the backscatter phase matrix can be used to distinguish randomly oriented and preferentially oriented scatterers. In cases where the volume consists of randomly oriented particles, the backscatter phase matrix takes the form [31–34]
The backscatter phase matrix for preferentially oriented scatterers is distinctively different, exhibiting all three polarization effects (depolarization, diattenuation and retardance) . The fundamental form of this matrix is [24, 31]35]
The scattering matrix in Eq. (4) is given the superscript (0) to indicate it is in the scatterer’s coordinate basis. In this work we do not assume to know what this basis is. The matrix under interrogation is assumed to be rotated relative to the instrument’s polarization basis by some angle φ so that the actual matrix under interrogation is10].
2.1. Measuring the scattering matrix
An imperfect polarization lidar may contain depolarizing, diattenuating and retarding effects in MTX and MRX, non-ideal polarization analyzers, and elliptical laser polarization states with degree of polarization less than one. The scattering volume, however, does not see the behavior of the instrument directly, and we can reduce Eq. (1) to reflect the perspective of FEq. (1), is N. The incident Stokes vector is given from Eq. (1) as Eq. (1) have been omitted.
The process for measuring a Mueller matrix described by Eq. (7) can be changed to a form that reflects the desire to measure the elements of F. The scattering matrix is rewritten as a vector, f⃗, of all independent variables with elements [25, 26]
The signal recorded on a detector channel is then given by Eq. (7) and is given for a general sixteen element Mueller matrix as
Because the fundamental forms of the scattering matrices have redundant terms, we can further reduce the dimensionality of f⃗ by eliminating elements where i > j. This is reflected in the measurement vector a⃗ which is then written for the ten elements in Eq. (6) asEq. (3), f⃗ and a⃗ are further reduced to have three elements.
If the system makes multiple linearly independent measurements such that rank(A) ≥ Dim(f⃗) the scattering matrix can be determined byEq. (17) is over-defined and the inverse operation on A is replaced with a pseudo-inverse. In this way, additional temporal integration may be achieved in the scattering matrix retrievals.
The uncertainty in this measurement is then given by the covariance matrix of f⃗ which is related to the statistical uncertainty of the detected vector N⃗ through 
The HSRL uses temporally varying measurement matrices to resolve the full backscatter matrix. This technique of substituting temporally changing polarization measurements for simultaneous measurements is not an uncommon lidar technique [33, 37]. Spacing measurements in time means the measured signal will contain both uncertainty from shot noise and atmospheric variability. As a result, the covariance of the measured signals isEq. (20) into (19) and substituting that result into (18) gives
Eq. (20) and therefore Eq. (21) account for atmospheric variability on time scales greater than or equal to the measurement integration time. For HSRL’s 0.5 sec integration time, there is also likely variation at sub-integration time scales. Numerical simulations of HSRL have shown that higher frequency variations result in small covariance terms between between all the scattering matrix elements in . As a result, some zero mean noise is imparted on all scattering matrix elements due to high frequency atmospheric backscatter, extinction and depolarization variability.
Thus, statistical variability of the atmosphere adds uncertainty to the scattering matrix retrieval. It is important to note that it does not bias the measurement in a way that would generate false positive identification of oriented scatterers. We have found, however, that nonlinear detector response, such as pulse pile-up, can bias measurement of off diagonal elements. For this reason, low altitude, high backscatter, water clouds can have false, non-zero off diagonal elements. These false terms generally do not satisfy the criteria for Eq. (5) or the matrix form in Eq. (6). This and our knowledge of the data acquisition system’s linear dynamic range make such cases easy to identify.
The polarization formalism shown here provides a simple processing technique for bias free polarization measurements regardless of the lidar’s polarization effects. The accuracy of this measurement is limited by the accuracy to which D⃗ and S⃗ are known instead of the polarization purity of the system. Imperfections in the measurement such as system retardance, depolarization and diattenuation, non-ideal polarizers and differences in detector gain are folded into D⃗ and S⃗ and are fully removed from the scatterer characterization. While we apply this formalism to measuring a full backscatter phase matrix here, it can also be applied to any conventional polarization lidar measurements where optical system effects reduce the accuracy of the measurement.
3. System description
The high spectral resolution lidar at NCAR was designed and built at the University of Wisconsin to directly measure cloud and aerosol backscatter, extinction and circular depolarization [28, 29]. A simplified instrument layout is shown in Fig. 1. The laser is vertically polarized and passes through the transceiver thin film polarizer (TFP). The vertical polarization is then transformed by a quarter wave-plate (QWP) to right hand circular (RHC) polarization. The beam is expanded by an afocal Mersenne Cassegrain telescope. Upon backscattering, the polarization becomes left hand circular (LHC) due to the change in propagation direction and the QWP transforms this polarization to horizontal. This polarization reflects off the TFP as the parallel channel. Part of parallel polarized signal is picked off by a beam sampler (BS) and passed through an iodine absorption cell to measure the molecular backscatter signal, and the rest passes onto high backscatter (10% port on a beam sampler) and low backscatter (90% port on a beam sampler) parallel polarized total molecular and aerosol detectors (not depicted). The total molecular and aerosol cross polarized signal is picked off behind the TFP with a beam splitter (BS). This signal is then passed to the cross polarized detector. Not shown is an energy monitor in the transmitter that is used to normalize fluctuations in transmitted laser power (less than 5% shot to shot variation).
Some of the NCAR HSRL specifications are listed in Table 1. The narrow field-of-view serves to suppress multiple scattering , and even optically thick water clouds have very little depolarization. For this reason, multiple scattering is not likely to be a significant contributer to detected signals.
3.1. Modified operation
To measure the full scattering matrix of oriented scatterers we must obtain a measurement matrix of rank greater than ten. Conventional polarization operation only allows for rank(A) = 2. The QWP mount was replaced with a motorized mount to provide more linearly independent polarization measurements. As this QWP rotates, it changes both the transmitted and received polarizations. Both D⃗ and S⃗ must change to characterize the full scattering matrix, so systems that do not use a shared telescope configuration would require a rotating wave-plate in both the transmitter and receiver.
The QWP is rotated by a high speed rotation mount with better than 0.1° accuracy of motor rotation angle. This rotation is performed continuously, so the QWP position is not constant over the course of an integrated data point. The measurement vector corresponding to a time integrated data point becomes
The data acquisition hardware on the HSRL has a minimum integration time of 0.5 seconds. Thus, 2000 laser shots are recorded over a QWP rotation of 37°. This rotation rate was selected because it minimized the measured depolarization and backscatter uncertainty (uncertainty in other matrix terms were not considered) for all possible QWP angles. The fact that 37 and 360 have no common multiples, forgoes the need to slave the QWP rotation to the HSRL timing unit. Instead, we only record the start and stop position of the QWP for each data point and determine the measurement vector using Eq. (22).
Though only 2.5 seconds of data (5 data points from each channel) are needed to invert Eq. (17), typical scattering matrix calculations use 40 seconds of data per matrix retrieval (80 data points from each channel) to suppress shot noise at altitudes up to 14 km.
To avoid back reflections from the transmitted laser into the receiver, the original HSRL design tilted the QWP in the plane of the optical bread board by approximately 10° relative to the optic axis. As a result, the device is not a true QWP, because the crystal cut designed to produce a phase shift between ordinary and extraordinary axes assumes normal incidence. Additionally, the phase shift of the QWP is not constant with rotation angle . Through both modeling [40, 41] and experimentation, we have found that the QWP phase shift can be reliably approximated at tilt angles less than 20° as
The valid range for this approximation generally depends on the wave-plate material, thickness and design. Materials with greater birefringence, thicker wave-plates, and compound zero order wave-plates generally have a narrower field-of-view and therefore shrink the valid range of Eq. (23).
The value of each term in Eq. (23) was determined by rotating the wave-plate between a horizontal polarizer and an analyzer at 0°, 30°, 60° and 90°. We fit the Fourier coefficients from the recorded signal to Eq. (23) to obtain an initial QWP characterization.
Tilting the QWP has an effect of tilting the trajectory of the QWP’s output polarization on the Poincaré sphere (see Fig. 2). Large tilt angles also tend to introduce additional lobes to the path, which are of little benefit, but the expanded lateral coverage from the tilted trajectory improves our ability to resolve angular features, such as the orientation angle φ in Eq. (6).
In addition to the QWP coefficients in Eq. (23), we also must determine the laser polarized state, the diattenuation vector of all detector channels (total high backscatter, total low backscatter, molecular and cross polarized), and the Mueller matrix of the remaining transmitter optics (folding mirror and a telescope). Each of these terms are known to some reasonable approximation. The laser is designed to be vertically polarized with a high DOP through use of a Glan-Taylor polarizer before the TFP. The total high and low and molecular channels are designed to have horizontal diattenuation vectors with magnitudes near one. Likewise the cross polarized channel is designed to have a vertical diattenuation vector also with magnitude near one. The 45° laser line mirror used to fold the outgoing beam into the telescope was characterized during the lidar modification (diattenuation near zero, phase shift of approximately ). The low angles of incidence on the cassigrain telescope’s mirrors should make the device nearly polarization preserving, but allowances were made for small amounts of depolarization, retardance and diattenuation in the device.
The estimated polarization effects of each of these terms provides the initial conditions for a sequence of optimizations that allows us to determine the measurement vectors of the instrument for any QWP rotation angle. Intensity on each channel is measured as a function of QWP angle in the following setups.
- A retro-reflection mirror is placed immediately after the QWP at (1.) in Fig. 1.
- A retro-reflection mirror is placed immediately after the folding mirror at (2.) in Fig. 1.
- A polarizer, then retro-reflecting mirror are placed immediately after the folding mirror at (2.) in Fig 1. Data is recorded with the polarizer set at 0°, 30° and 60°.
Running a simultaneous Fourier coefficient fit to all the above setups provides the descriptions for all HSRL terms except the telescope. A final fit, including the telescope is performed in the process of data collection off the sky. It is unlikely that any of these polarization terms will change rapidly over time. Thus, the instrument is recharacterized periodically using depolarizing sky returns to guarantee the performance of the instrument. Also, when the HSRL’s transmit tilt angle is changed, the only significant change is in the orientation of the folding mirror. The mirror Mueller matrix is rotated by the new tilt angle and the system is characterized using sky returns.
After determining their polarization properties, each channel is well characterized to within a scalar constant. This scalar is the optical efficiency of each channel and must be properly accounted for to use both cross and parallel channels in the full scattering matrix calculation. The HSRL’s configuration produces a measurement matrix of rank 15 when both channels are combined and 9 in each individual channel. Typically we use the combination of both polarization channels to determine the volume scattering matrix
The value of K(R) can be determined when we can assume only randomly oriented scatterers are present. In that case each channel can independently calculate the reduced scattering matrix of rank three using three or more successive data points.
Fig. 3 shows the initially recorded backscatter profiles, f⃗‖(R) and f⃗⊥(R). The perpendicular channel is then corrected by a factor and the resulting profile very nearly agrees with the parallel channel.
Additional scaling and cross talk between the molecular and total channels are covered in [28, 29]. The new polarization operation of the HSRL does not change this calibration except that the individual channel retrievals of the backscatter coefficient (F11) should be used instead of background subtracted photon counts.
With a complete system description we calculate the condition number of the measurement matrix A is approximately 25. This relatively high value indicates that the system parameters are not optimized for full scattering matrix retrieval. It may be possible to improve the system performance by further tilting the rotating QWP or replacing it with a custom retarder as suggested by . Slower QWP rotation rates improve the condition of A. However, further investigation into uncertainty contributions by atmospheric variability and system time resolution need to be investigated in further detail to perform a proper system optimization. Never-the-less, in its present configuration, the HSRL is able to retrieve the full backscatter matrix of oriented scatterers.
The NCAR HSRL began collecting full backscatter phase matrix data over Boulder, CO, USA starting in May 2012. On July 16, 2012 at 2PM local time (1600 UT), the lidar was tilted 20° off zenith when a heavy rain storm passed over. During this time the cloud overhead was ice or mixed phase. The time resolved backscatter ratio, depolarization and diattenuation of the event are shown in Fig. 4. Note the use of a single depolarization term to describe the oriented rain is not technically correct, but the product is shown to provide a conventional lidar context for oriented scatterer observations. We presently use diattenuation defined as to identify the presence of oriented scatterers. The time integrated backscatter phase matrix profile is shown in Fig. 5. The cloud above 2.5 km is strictly depolarizing where there are no off diagonal terms in the cloud scattering matrix. However, the altitudes where rain is present have significant off diagonal terms due to the preferential orientation of the larger raindrops. This profile shows that the instrument is able to characterize oriented and randomly oriented scatterers within the same profile.
In another observation on July 2, 2012 at 5AM local time (1100 UT), the HSRL was operating at a 30° tilt angle when an ice cloud passed over at approximately 5 km. Horizontally oriented ice crystals were present. Fig. 6 shows the time resolved backscatter ratio depolarization and diattenuation of the ice cloud. Again the use of depolarization where ice crystals are oriented is not technically correct, but the profile shows that the randomly oriented volumes, and therefore by extension oriented volumes, are composed of ice. The observed cloud consists entirely of ice, but preferential orientation, indicated by the diattenuation profile, only exists in a relatively narrow altitude range. There is also some temporal variability in the orientation. It is not clear if this is because of a change in the ice crystal characteristics, advection or other environmental conditions that might disrupt the crystal orientation behavior (such as turbulence).
Figure 7 shows the time integrated scattering matrix parameters of the oriented ice cloud from Fig. 6. Off diagonal terms in the matrix are non-zero only within a narrow altitude band of between 5 and 5.5 km. Note also that where oriented ice crystals are present, the magnitudes of diagonal terms F22, F33 and F44 are larger than the rest of the randomly oriented cloud. The depolarizing effect where the oriented ice crystals are present is less than that of fully randomly oriented volumes. Depolarization is often an indicator of randomness within the volume. The fact that the depolarization does not become zero may be the result of ice crystal properties (such as roughness), broad size distributions or a significant randomly oriented sub-population.
We have demonstrated that the capability to measure the full backscatter phase matrix can be integrated into a lidar instrument through a relatively simple hardware modification. The processing of the instrument’s data requires some basic matrix operations to retrieve the full backscatter phase matrix. Unlike conventional polarization lidar data products, this quantity is unambiguous when oriented scatterers are present and in such cases provides additional polarization data that may be used to improve the accuracy of scattering inversions.
In performing these modifications to NCAR’s HSRL, we demonstrate that the measurement can be performed on an instrument with significant polarization effects. The modification did not sacrifice the already existing capabilities of the instrument and further improved our ability to calibrate the two polarization channels.
While polarization lidar measurements of the atmosphere have been made for decades, there are very few experimental studies demonstrating specific values of backscatter phase matrices. The significance of oriented ice crystals for remote sensing inversions and radiative transfer inevitably depends on these scattering terms. Through this expanded polarization capability, NCAR’s HSRL will begin to further develop the relationship between orientation and polarization and other physical parameters of the atmosphere.
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