The volume depolarization ratio of the molecular backscatter signal detected with polarization lidar varies by a factor of nearly 4 depending on whether the rotational Raman bands are included in the detected signals of the individual system or not. If the rotational Raman spectrum is included partially in the signals, this calibration factor depends on the temperature of the atmosphere. This dependency is studied for different spectral widths of the receiving channels. In addition, the sensitivity to differences between the laser wavelength and the center wavelength of the receiver are discussed.
© 2002 Optical Society of America
Polarization lidar was among those lidar techniques which could be realized first [1–3] and, since then, became a powerful, widely used tool for the remote study of clouds and atmospheric aerosols. The common concept consists of two receiving channels which detect the backscattered light of a linear polarized laser source parallel and perpendicular to the polarization of the emitted light. The technique is sensitive to the shape of the backscattering particles: Spherical particles reflect light in 180-degree backscattering direction without change of polarization whereas aspherical particles, depending on shape, size, and composition, do. Thus polarization lidar allows to distinguish between different particle types in the atmosphere and is very sensitive to detect layers containing aspherical particles, such as, e.g., cirrus clouds [4–6] or certain types of polar stratospheric clouds [7–9], even if these layers are optical thin.
Though systems that employ the polarization lidar technique to detect depolarizing particles are widely used at date, the measurement of the exact value of depolarization is still connected with many difficulties. Different techniques for dealing with multiple scattering effects which cause depolarization also for spherical particle ensembles have been investigated [10–12]. A basic difficulty, however, lies in the calibration of polarization lidar, i.e., in the fact that the depolarization ratio of purely molecular backscattered signal is needed as input parameter for deriving the depolarization profile. Its value depends highly on the spectral parameters of the individual lidar receiver and varies by nearly a factor of 4 between 3.63∙10-3 and 1.43∙10-2 depending on whether the rotational Raman bands are included in the detected signals or not. Hence, as worst case scenario, the measurement results show an error of the same factor if the calibration is done falsely. In practice, the signals of most of today’s polarization lidar systems include the rotational Raman bands partly because of the conflicting requirements of narrow receiver bandwidth to keep the background signals low and high receiver transmission which is much easier (and cheaper) to achieve with a wider bandwidth.
Instrumental cross-talk which arises of imperfections of either the laser source or the alignment of the receiving channels can be corrected , but only when the molecular depolarization is known. For determining this value which is characteristic for the individual instrument, however, so far only one technique has been described , a method utilizing as reference the backscatter signal of non-polarizing polar stratospheric clouds. Hence, this experimental technique can be used only for systems operating in polar regions. Other systems rely on theoretical calculations.
The properties of atmospheric backscatter signals that are relevant for calculating the molecular depolarization ratio depending on the receiver bandwidth of a lidar system can be derived from other published work [15–22]. However, so far as we know, this information has not been collected yet in a form convenient for such calculations. Therefore we give first a summary of the formula necessary for determining the correct value of this calibration factor.
As it will be shown, care has also to be taken for a possible temperature dependency of the molecular depolarization ratio when the rotational Raman spectrum is included partially in the signals. Objective of this study is to investigate theoretically this dependency on atmospheric temperature for different spectral widths of the receiving channels. In addition, the sensitivity to differences between the laser wavelength and the center wavelength of the receiver are discussed.
As proposed by Young , we use in the following the term ‘Cabannes line’ for the central line of the backscattered spectrum (no change of the rotational or the vibrational state of the scattering molecule). This line is surrounded by the pure rotational Raman spectrum which is due to a change of the rotational state of the scattering molecule. The Cabannes line and the pure rotational Raman spectrum together are called ‘Rayleigh spectrum’.
2. Theoretical background
In this section, a summary of the equations which describe the intensity of the rotational Raman lines and of formulas for calculating the relative intensity of the Cabannes and the rotational Raman backscatter signal depending on polarization are given. Then a formula to calculate the depolarization ratio of the molecular backscatter signal, which depends on the receiver bandwidth of the individual lidar system used for the detection and may depend atmospheric temperature, is derived.
Nitrogen and oxygen molecules may be treated here as simple linear molecules, i.e., linear molecules with no electronic momentum coupled to the scattering. The backscatter coefficient of a rotational Raman line, which results from such a molecule of atmospheric constituent i in a state with rotational-angular-momentum quantum number J, is described for Stokes lines by 
with J = 0, 1, 2,...
and for anti-Stokes lines by
with J = 2, 3, 4 ...
where gi (J) denotes the statistical weight factor which depends on the nuclear spin Ii , ν 0 the frequency of the incident light, γi the anisotropy of the molecular-polarizability tensor, h Planck’s constant, c the velocity of light, k Boltzmann’s constant and T temperature. B 0,i and D 0,i are the rotational constant and centrifugal distortion constant for the ground state vibrational level, respectively. The rotational energy E rot,i(J) of a homonuclear diatomic molecule like molecular nitrogen and oxygen in quantum state J is 
For all wavelengths of the incident light, the shift of the rotational Raman lines is constant on a frequency scale and is, for the Stokes branch, given by
and for the anti-Stokes branch by
The splitting of the O2 lines in triplets  can be neglected in this context because the two satellite lines are located close to the main lines with distances of 0.056 nm on either side and their intensities are about a factor 1/30 weaker than the intensity of each central line. Pressure broadening of the rotational Raman lines  can also be disregarded as the filters transmission bands discussed here are very broad compared to the width of the PRRS lines.
For linear molecules and linear polarized incident light, the differential backscatter cross-sections for the Cabannes line (Cab) and for the total rotational Raman wings (RR) for an atmospheric constituent i can be calculated with Placzek’s scattering theory  (compare also Ref. 20, 22) and are given by
where p and s denote polarization parallel and perpendicular to the polarization of the incident light, respectively, and αi is the trace of the molecular polarizability tensor. κ is a common factor with
where λ s is the wavelength of the scattered light, ν s its frequency, and ν 0 the frequency of the incident light. Note that the relative intensities of the Cabannes lines and the rotational Raman wings do not depend on ν 0 provided that αi and γi are wavelength independent, which is a very good approximation .
The volume depolarization ratio δ(z) is commonly defined as the ratio of the backscatter coefficients for scattering perpendicular and parallel relative to the polarization of the transmitted laser beam, β s(z) and β p(z), respectively,
with z for height. Superscripts “mol” and “par” indicate molecular and particle backscatter coefficients, respectively. Care has to be taken as other definitions for the depolarization ratio are also in use [26, 14]. The molecular backscatter coefficients depend on the receiver characteristics of the lidar system used. They may depend on atmospheric temperature and thus on height. δ(z) is measured with polarization lidar by
where P s(z) and P p(z) are the lidar signals with polarization perpendicular and parallel to the polarization of the transmitted laser light, respectively, and k is a calibration factor equal to the inverse of the ratio of the channel efficiencies. Generally, the efficiency of the lidar receiver is not the same for the parallel and the perpendicular channel and k can be determined by two methods: (1) experimentally by comparing the signals of both lidar channels when coupling unpolarized light into the receiver while no laser light is emitted. The light source can either be artificial (lamp, torch) or natural (e.g., multiple scattering of sunlight provides a source of unpolarized light when the lidar receiver field-of-view is pointed at thick clouds in daytime). (2) Another method is to normalize δ(z) to the value of the molecular depolarization ratio
in an altitude region z 0 where particle scattering can be neglected, which gives
For the second technique the importance to use the correct value for δ mol(z 0) is obvious. Both techniques, however, assume a perfect system without imperfections of either the laser source or the alignment of the receiving channels. To check for imperfections, method (1) can be employed and the value found for k can be compared with measurements using the calculated value for δ mol(z 0) and Eq. 14.
With a polarization lidar that includes Raman channels, the particle depolarization ratio
can be measured. R s(z) and R p(z) are the measured backscatter ratio for perpendicular and parallel polarization, respectively. Also here, δ mol(z) scales the measurement result directly and relative errors when using wrong values for δmol(z) cause the same relative errors of δ par(z).
With Eq. 8 and 9 the volume depolarization ratio for the rotational Raman wings becomes δ RR = 0.75 and independent of the species of the scattering molecule, whereas for the Cabannes line (Eq. 6 and 7) and therefore also for the whole Rayleigh spectrum it is not, but
with εi = (γi /αi )2.
εi can be derived best from depolarization measurements of Cabannes or Rayleigh scattering. For comparing literature values, one has to be aware of the scattering geometry (usually 90-degree scattering) and type of incident light (linear polarized or unpolarized) . The values for εi used for this study were derived from Ref. 20 and are shown in Table 1. They are in close agreement, e.g., with the data given in Ref. 22. Using the data of Table 1 one gets with Eq. 16 δ Cab = 0.0027 (0.0077) and δ Ray = 0.0106 (0.0299) for N2 (O2) for linear polarized incident light and 180-degree backscattering.
As the polarization of Cabannes and pure rotational Raman scattering differ by two orders of magnitude, the depolarization ratio of the molecular signal detected by a lidar system depends highly on the fraction of the rotational Raman wings transmitted by the lidar receiver and is given by
where ci is the relative concentration of atmospheric species i and xi the relative part of the intensity of the rotational Raman wings of this species detected by the lidar. (If the filter widths for the parallel and the perpendicular lidar channel are not the same, this would be taken into account by different factors for the transmitted part of the rotational Raman spectrum in nominator and denominator.) Eq. 17 yields with Eq. 6 – 9
Because the intensity of each rotational Raman line depends on the temperature of the atmosphere, also xi , and therefore δ mol may be temperature dependent. As already mentioned above, γi, and εi can be regarded as constant with temperature in this context as within the range of atmospheric temperatures their temperature dependency is very weak .
3. Results and Discussion
Relative atmospheric mixing ratios for nitrogen and oxygen are 79 % and 21 %, respectively. Thus with Eq. 18 and the data given in Table 1 the depolarization ratios of air can be computed easily for the Cabannes line (x O2 = x N2 = 0) and the Rayleigh spectrum (x O2 = x N2 = 1), yielding δ Cab, air = 3.63∙10-3 and δ Ray, air = 1.43∙10-2 for linear polarized incident light and 180-degree backscattering. These values are consistent with an effective ε of dry air of 0.218 (calculated with Eq. 16), which is very close to the value of 0.216 which can be deduced from the data quoted in Ref. 20 for direct measurements of the Rayleigh depolarization of air for unpolarized incident light and 90-degree scattering geometry. Strictly, these results are valid for dry air only, but the water vapor mixing ratio is even for saturation so small (< 1 %) that it can be neglected here.
For different widths of Gaussian filter transmission curves we calculated x O2 (T) and x N2 (T) using Eq. 1 to 5. First we assume that the center of the receiver transmission band is equal to the wavelength of the transmitted laser light. The results are shown for N2 in Fig. 2.
For full widths at half maximum (FWHM) of the receiver of about 0.3 nm at 532 nm, i.e., 10.6 cm-1, or smaller the extracted signal is nearly purely due to Cabannes scattering (the two closest rotational Raman lines lie in distances of ±11.9 cm-1 (±14.4 cm-1) for N2 (O2)), while for FWHM = 15 nm at 532 nm, i.e., 530 cm-1, about 95 % of the rotational Raman signal are extracted. For the calculation, a primary wavelength of 532 nm was taken, because this is the wavelength of the frequency-doubled radiation of a Nd:YAG laser, which is at present the most widely used light source for polarization lidar. However, as the pure rotational Raman spectrum is invariant on a frequency scale for any primary wavelength (with intensity relative to the intensity of the Cabannes line), the results are the same for any primary wavelength when the filter width is taken in frequency units (as long as αi and γi can be regarded as wavelength independent as already mentioned above).
Fig. 3 shows x O2 (T) and x N2 (T) in comparison for FWHM = 1.0 nm as an example. As it can be seen, x O2 (T) and x N2 (T) cannot be considered as equal, but have to be treated separately: For this FWHM, x O2 (T) is up to ~50 % larger (relatively) than x N2 (T). Fig. 4 shows the molecular volume depolarization ratio δ mol(T) calculated from x O2 (T) and x N2 (T) by use of Eq. 18. Absolute values for δ mol(T) range nearly from δ Cab, air =3.63∙10-3 to δ Ray, air = 1.43∙10-2. With rising temperature the pure rotational Raman spectrum becomes broader. Thus, for a certain FWHM, δ mol(T) is generally smaller at higher temperatures than at lower temperatures, because at higher temperatures a smaller fraction of the rotational Raman wings is included in the detected signals. The temperature dependency of δ mol(T) is largest for FWHM ≈ 2.0 nm and small when the signals are nearly only due to Cabannes scattering or nearly the whole rotational Raman wings are included in the detected signal (Fig. 5). For FWHM ≈ 2.0 nm , δ mol(T) rises from 300 K to 180 K by about 17 % (relatively), whereas for FWHM ≤ 0.5 nm or FWHM ≥ 15.0 nm the relative changes between 180 and 300 K are δ mol(T)/δ mol(T = 240 K) ≲ ±1 %. Table 2 summarizes the results and gives the exact values for x O2 (T), x N2 (T), and δ mol(T) for T = 240 K and the relative variation of δ mol(T) between 200 and 280 K defined as
A good compromise between reasonable small but not too small spectral width of the receiver (in order to compress the background signal, but to achieve also good receiver transmittance) might be FWHM = 0.5 nm which yields δ mol(T = 240K) = 3.76∙10-3 (which is 3.6 % larger than δ Cab, air) and Δδ mol = 1.2 %.
The values given above were calculated for Gaussian-shape receiver transmission curves. This shape is a suitable model for interference filters made up of two or three cavities, which is the most widely used filter type for polarization lidar today. However, note that significant differences to the above data may result if the actual receiver transmission differs significantly from this model. E.g., as extreme case, no rotational Raman lines are within FWHM = 0.5 nm for the model of a perfectly rectangular transmission curve, which yields δ mol(T)= δ Cab, air = 3.63∙10-3 for all temperatures. On the other hand, for a Lorentzian-shape transmission curve (which is a good approximation for single-cavity interference filters and has very shallow cut-off slopes) of FWHM = 0.5 nm gives δ mol(T=240K)=4.16∙10-3 and Δδ mol = 2.9 % . Thus in case the transmission of a given receiver differs significantly from the Gaussian model individual calculations are recommended.
For practical purposes, it is important to know how sensitive a system is to small misalignments. To study this question, we examined the effect of small shifts (Shift) between the receiver central wavelength and the laser wavelength for a Gaussian-shape receiver transmission curve with FWHM = 0.5 nm at 532 nm, i.e., 17.7 cm-1. These shifts can either be caused by small misalignments of the angle of incidence of the interference filters used (already an error of ±1° might cause a shift of ∓0.5 nm; see, e.g., Ref. 28) or when the emission of the laser in use is not exactly known, e.g., for Nd:YAG laser the frequency doubled radiation may vary by tenths of a nm .
Fig. 6 and Fig. 7 show δ mol(T) for shifts up to ±0.5 nm. Compared with the unshifted receiver a shift of +0.5 nm results in a relative increase of δ mol of ~15 %. Also the temperature dependency rises from Δδ mol =1.2% to Δδ mol = 5.3%. (δ mol and Δδ mol are a little bit larger when the receiver transmission band is shifted to the Stokes side because the rotational Raman lines are stronger here than on the anti-Stokes side.)
Table 3 gives the exact values for x O2 (T = 240 K), x N2 (T = 240 K), δ mol(T = 240 K), Δδ mol and the systematic measurement error at T = 240 K if instead of the correct value for δ mol, which takes the shift into consideration, falsely the value without shift would be used:
To keep Δ(Shift) smaller than, e.g., 1 % the relative position of the receiver bandwidth has to be known better than ±0.1 nm. Whereas such accuracy is common for measuring laser wavelengths, the alignment of the lidar receiver in such an accuracy requires some effort. But also with an accuracy of Shift ≤ ±0.1 nm, temperature effects might become significant: If Shift = 0.1 nm is neglected and δ mol(Shift = 0, T = 300 K) = 3.73∙10-3 is taken falsely, the measurement errors add up to 3.5 % at T = 180 K, the condensation temperature regime of polar stratospheric clouds, as δ mol(Shift = 0.1 nm, T = 180K) = 3.86∙10-3. For Shift = 0.5 nm an error of 22 % would be made under the same conditions as δ mol(Shift = 0.5 nm, T = 180 K) = 4.54∙10-3. Even for a relatively narrow receiver bandwidth of FWHM = 0.5 nm at 532 nm, it is therefore beneficial to take the temperature dependency of δ mol into account.
4. Conclusions and outlook
We calculated the volume depolarization ratio of the molecular backscatter signal for different bandwidths of a lidar receiver and studied the dependency of this parameter on atmospheric temperature. Findings show that significant temperature dependency may result if the pure rotational Raman signals are included partly in the detected signals. For a laser wavelength of 532 nm and Gaussian-shape receiver bandwidths centered to this, a bandwidth of FWHM ≈ 2.0 nm causes maximum temperature dependency and δ mol(T) rises from 300 K to 180 K by about 17 % (relatively). Small temperature effects occur for FWHM ≤ 0.5 nm or FWHM ≥ 15.0 nm and relative changes between 180 and 300 K are δ mol(T)/δ mol(T =240K)≲ ±1%. However, care has to be taken to determine accurately both the spectral bandwidth and shape of the receiver and the center wavelength of the receiver bandwidth relative to the wavelength of the laser. A good compromise for the receiver width (for both reasonable transmission of the lidar signals and high suppression of the background) is, e.g., FWHM = 0.5 nm at 532 nm. For such a Gaussian-shape receiver bandwidth, δ mol(T =240 K) is 3.6 % larger (relatively) than the volume depolarization of pure Cabannes backscattering and the laser wavelength and center wavelength of the receiver have to be determined with an accuracy better than ±0.1 nm to keep errors below ~3.5 % at atmospheric temperatures between 180 and 300 K if the temperature dependency of δ mol(T) is neglected. Thus, for lidar systems which extract the rotational Raman signal partially, i.e., the large majority of today’s systems, the measurement accuracy may be improved significantly by taking the atmospheric temperature into account and correcting cross-talk effects. The improvement which can be achieved depends on the characteristics of the individual lidar system.
This work was supported by the Japanese Society for the Promotion of Science (JSPS) with a postdoctoral fellowship for A. Behrendt (P-00765) and a Grant-in-Aid for JSPS Fellows (12440127).
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