## Abstract

Using an atomic/molecular vapor as an aerosol blocking filter for atmospheric temperature measurements with a Cabannes lidar is revisited. Different problems in previously used barium and iodine filters prevented them from delivering the 78 times signal advantage (8.8 times less uncertainty) over rotational Raman lidar. We conclude that, despite the sensitivity optimization in rotational Raman lidar, a proposed Cabannes lidar utilizing potassium vapor filters can have 6.1 times less temperature uncertainty. By tuning the laser frequency cyclically to above and below the potassium D_{1} transition, the lidar system can measure temperature and wind simultaneously.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fiocco et al. [1] in 1971 first used a scanning high resolution Fabry–Perot interferometer to obtain a spectrum of backscattered light from a CW argon-ion laser at 488 nm resulting in a central peak with pedestals on both sides due respectively to aerosol and molecular Cabannes scattering [2,3]. They suggested that an analysis of such a spectrum, could determine atmospheric temperature and the aerosol to molecular scattering ratio in the troposphere. A decade later, Schwiesow and Lading in 1981 [4] proposed measuring the atmospheric temperature profile by Rayleigh (now termed Cabannes) scattering using a stabilized Michelson interferometer. Two years later, Shimizu et al. [5] proposed to do the same with atomic vapor filters (AVFs) for a better blocking of aerosol scattering. Both papers predicted tropospheric temperature measurements with an accuracy of ${\pm} 1\;K$.

The concept of temperature measurement by comparing Cabannes scattering passing through a high-temperature (HT) AVF or a low-temperature (LT) AVF is straight forward. As is well known, the fraction of Cabannes scattered light passing through the HT AVF increases more rapidly than that through the LT AVF as atmospheric temperature increases. Thus, the ratio of the former to the latter is an increasing function of atmospheric temperature. To achieve this proposed temperature measurement in practice turned out to be a challenge. Nonetheless, it was first demonstrated with observational data in 1992-93 [6,7] using barium filters with a tunable dye laser at wavelength 553.7 nm. Owing to the need of very high filter temperature (∼ 800 K) to produce sufficient barium vapor, it was not pursued further. Realization that using iodine filters at much lower temperatures (∼ 350 K) with the widely available frequency-doubled YAG laser at wavelength 532 nm occurred nearly a decade later. The measurements of Hair et al. [8] with 11 nights of temperature data in 2001 covered the entire troposphere despite the presence of aerosols; it still appears to be the only tropospheric atmospheric temperature measurements based on Cabannes scattering at 532 nm using iodine filters. Cabannes scattering for atmospheric temperature measurements was carried out at 355 nm by Hua et al. [9] in 2004. There, the backscattering signal is detected through 3 Fabry-Perot etalons (FPEs) with different frequency shifts and FWHMs, denoted as Channel-1 (1 GHz, 0.3 GHz), Channel-2 (3 GHz, 0.6 GHz) and Mie-Channel (0 GHz, 0.2 GHz); atmospheric temperature is retrieved from the signal ratio of Channel-2 to Channel-1. To achieve the required sensitivity, the temperature of these FPEs must be stabilized to within 0.001°C which may have discouraged further deployment and field experimentation.

Even though the pure rotational backscattering coefficient ${\beta _R}$ and its anti-Stokes (S-branch) coefficient ${\beta _{aRR}} \approx 0.375{\beta _R}$ [3] are weaker than the associated Cabannes backscattering coefficient ${\beta _C}$ by about 29 and 78 times respectively, the atmospheric temperature measurements by rotational Raman lidar technique appears to enjoy more success partially because of its less stringent requirement in laser linewidth and stability. Such a lidar was first deployed and reported along with field observational data in 1993 [10] and continued with improvements in 2000 [11] and in 2004 [12], producing impressive atmospheric temperature profiles.

The reason that a molecular filter does not perform to the anticipated level of excellence is the interference of the (photo-dissociation) continuum absorption. In the iodine case, the culprit lies in the strength of the bound-free transitions $^1{\Pi _{1u}} \leftarrow X{(^1}{\Sigma _{{o^ + }g}})$ continuum transition which increases relative to the bound-bound transitions $B({^3{\Pi _{{o^ + }u}}} )\leftarrow X{(^1}{\Sigma _{{o^ + }g}})$[13] of interest to lidar [14,15] as the vapor density increases. This leads to the off-resonance transmittance (excluding the passive loss of windows) for the LT and HT iodine vapor filters (IVFs) for the required bandwidths to be respectively 0.5 and 0.1 as shown in Fig. 6 of [8]. Thus, to achieve the full potential of AVFs for atmospheric temperature measurements, a workable atomic vapor, without the interfering continuum absorption in the infrared and visible wavelengths is required. We thus propose the use of a potassium vapor filter (KVF) as a better alternative and a new potassium backscattering temperature lidar in Section 4 of this paper after a brief theoretical discussion of AVF-based measurement sensitivity and uncertainty in section 2 and a brief account of the three existing scattering temperature lidars in Section 3. Then, we proceed to the first objective of this paper to compare the figure of merits between four similar atmospheric temperature measurement methods, all using the ratio between two filtered scattering signals for temperature measurements, utilizing (1) barium vapor filters (BVFs) and Cabannes scattering (CS) at 553.7 nm, (2) iodine vapor filters (IVFs) and CS at 532 nm, (3) custom interference filters (CIFs) and rotational Raman scattering (RRS) at 532 nm and (4) the proposed potassium vapor filters (KVFs) and CS at 770 nm. This will be done in Section 5.

Although IVF as blocking filter did not deliver the promise for the anticipated atmospheric measurements, it has been very successfully used for measurements of aerosol optical properties [16,17] and atmospheric wind [18,19]. For these purposes, only one filter at a lower temperature is used, thus minimizing the continuum absorption; for example, Piironen and Eloranta [16] were able to use a 43-cm long cell at 27°C to limit the off-resonance transmission to be around ∼0.8 with an aerosol rejection ratio of 1000:1. The second objective of this paper is to propose (a) a potassium lidar utilizing two KVFs for temperature only measurement by locking the laser frequency at KD_{1} transition at 770 nm, i.e., at 0 GHz relative to the resonance KD_{1} transition in Section 4, and (b) for simultaneous measurements of temperature and wind by tuning the laser frequencies back and forth between +0.208 GHz and -0.180 GHz from the resonance transition frequency in Section 6. The measurement sensitivities and uncertainties for both temperature and wind will be discussed along with the procedure for data processing, followed by discussion and conclusion in Section 7.

## 2. Temperature lidars based on the ratio of two differently filtered scattering signals: *sensitivity and single-photon uncertainty of a measurement*

We first consider a vertically pointing lidar where we can ignore the small vertical component of the air velocity, a lidar located at altitude *z*_{0} transmitting a laser pulse with energy ${E_L}$ at frequency ${\nu _L}$ into the atmosphere. The number of photons $N({\nu _L},z;{r_B})$ quasi-elastically scattered from scatterers at an altitude z in the range-bin ${z_B} = {r_B}\cos \Theta $ for a beam pointing at an angle $\Theta $ off zenith into a ground-based telescope with area ${A_R}$ is given by the lidar equation as:

The collection efficiencies of the two channels are ${\eta _1} = {\rho _2}\;\textrm{and}\;{\eta _2} = {\tau _2}$, with ${\eta _1} + {\eta _2} = 1$ nominally for Cabannes lidar using AVF. In this case, the attenuation factor of the AVF ${f_i}[T(z),P(z)]$ depends on the filter function (frequency dependent AVF transmission) ${F_i}(\nu )$, with $i = 1$ for HT filter and $i = 2$ for LT filter. The normalized Cabannes scattering spectrum $R(T,P,\nu )$[8], which depends on atmospheric temperature T and pressure P (as well as on LOS wind to a lesser extent) is given as:

Here, the filter transmission ${F_i}(\nu )$ is either a measured or a theoretical notch function of frequency. Since the transmission function of most AVFs approximates that of a notch filter, they can often be approximated by the product of off-resonance transmittance ${\tau _{ioff}}$ and an ideal blocking filter (to be defined below) with bandwidth $\Delta {\nu _B} \approx $ FWHM of the AVF in question.

To retrieve the atmospheric temperature, we define and express the temperature ratio $R_T^{Obs}(z)$ as the ratio of received signal counts in channel 1 to those in channel 2 from (2b), and the associated essential temperature ratio ${R_T}(z)$ by setting ${\eta _1} = {\eta _2}$ as given in (4) below:

Since the channel efficiencies ${\eta _1}\;\textrm{and}\;{\eta _2}$ are given, the quantity on the right-hand-side of the second identity in (4), depending only on the attenuation factors, is known or measured; this equation is basically identical to that of (2) in [6] or (1) in [7], suggesting that the procedures for temperature retrieval would be identical to those described. We now estimate the temperature measurement sensitivity and uncertainty of an AVF-based temperature lidar.

Since ideally the essential temperature ratio ${R_T}(z)$ defined in (4) depends on the ratio of two attenuation factors, ${f_i}[T(z),P(z)]:\;i = 1,2$ which in turn depends on atmospheric temperature and pressure, we can express the fractional uncertainty of the observed temperature ratio $\delta R_T^{Obs}(z)/R_T^{Obs}(z)$, which equals the fractional uncertainty of the essential temperature ratio $\delta {R_T}/{R_T}$ as the channel efficiencies are assumed to be constant, as in (5a) below. This then leads to the expression for the uncertainty of the measured temperature $\delta {T_{\textrm{air}}}$ in (5b) in terms of temperature and pressure sensitivities, ${S_{T12}}\;\textrm{and}\;{S_{P12}}$ defined in (5c) as the fractional change in temperature ratio ${R_T}$ per 1 K temperature change and per 1 Pa pressure change. The temperature measurement sensitivity ${S_{T12}}$ is then the difference between one-channel sensitivity ${S_{Ti}} = ({\delta {f_i}/{f_i}} )\;per\;1\;K$ temperature change for channel 1 ($i = 1$) and that for channel 2 ($i = 2$), each defined as the fractional change of the respective temperature ratio (or fraction attenuation factor $\delta {f_i}/{f_i}$ per 1 K). Thus, to the extent possible (to be explained below), one would choose the vapor cell temperatures to maximize the difference between the two one-channel sensitivities, ${S_{T12}} = {S_{T1}} - {S_{T2}}$.

Since for a typical vertical resolution, say 1 km, the pressure sensitivity is negligible [21], as given in the second expression in (5b), the observed 1-σ temperature uncertainty $\delta {T_{air}}$ is understandably the ratio of fractional signal photon count uncertainty to the temperature sensitivity of the measurement technique. To facilitate the comparison between various temperature measurement techniques, we express the fractional signal photon uncertainty $\delta {R_T}/{R_T}$ in terms of the photon counts received at point B for a measurement, ${N_0}$, and the attenuation factors of the two AVFs, ${f_1}\;\textrm{and}\;{f_2}$. By noting ${N_i} = {\eta _i}{f_i}{N_0}$ and assuming the fluctuations in photo-electrons are shot-noise-limited with $\delta {N_i} = \sqrt {{\eta _i}{f_i}{N_0}} $, the fractional signal photon uncertainty becomes

For figure of merit comparisons of the 4 temperature measuring techniques, all based on the ratio between two filtered scattering signals, we will assume the achievement of optimum conditions when possible, i.e., for all 3 techniques based on Cabannes scattering, but not for RRS as explained below.

## 3. Characteristics of an AVF and performance of three existing scattering temperature lidars

Since both sensitivity and single-photon uncertainty depend on the attenuation factor ${f_i}[T(z),P(z)]:\;i = 1,2$, it plays a fundamental role for the resulting uncertainty of an atmospheric temperature measurement. To develop a strategy for choosing and comparing the effect of the LT and HT blocking AVFs, we consider an ideal blocking filter, defined as a filter with zero transmission within ($- {\nu _B} < \nu < {\nu _B}$) and unity transmission outside the blocking frequency ${\pm} {\nu _B}$. The fraction of signal transmitted through (or filtered signal) is of fundamental importance for an AVF-based and rotational Raman lidar in question here. The FWHM of an ideal blocking filter by definition equals its blocking bandwidth $\Delta {\nu _B}$, i.e., $FWHM = \Delta {\nu _B} = 2{\nu _B}$. Unlike a Fabry-Perot interferometer, the transmission function of an AVF approaches that of an ideal filter with sharp cutoffs whose $FWHM \approx 2{\nu _B}$. It is instructive to consider the fraction of the Cabannes signal transmitted through an ideal filter with blocking bandwidth $\Delta {\nu _B} = 2{\nu _B}$ and its associated one-channel sensitivity as a function of filter blocking bandwidth. In Fig. 2 we replot these quantities for Cabannes scattering at 532 nm from an atmosphere at 275 K and 0.75 atm, taken from Fig. 5 of Hair et al. [8].

The one-channel sensitivity ${S_{Ti}}$ of an ideal filter increases sharply between bandwidth 2 to 4.5 GHz. Ideally, one would set the vapor pressures of the vapor cell to realize these bandwidths for LT and HT filters respectively. A bandwidth of 2 GHz is wide enough to block aerosol scattering and 4.5 GHz is narrow enough to have significant Cabannes scattering passing through the filter. Unfortunately, as explained below, it is not possible to do so for many atom/molecular species for different reasons. In addition to the fundamental species mass limitation, the problem for systems using AVFs usually lies on the inability to choose a desirable HT filter. For barium, to increase bandwidth beyond 3 GHz requires vapor temperature greater 800 K. For iodine filters, this problem is circumvented by the combined use of two neighboring lines by paying a high price in filtered signal due to continuum absorption. After a discussion of vapor mass as a fundamental limit for AVF performance, we utilize the associated ideal filters and briefly describe three temperature lidars reported in the literature, two based on Cabannes scattering with iodine or barium vapor filters, termed CS(532 nm)/IVF and CS(553.7 nm)/BVF lidars, and one based on rotational Raman scattering with custom-built dielectric interference filters (CIFs), termed RR(532 nm)/CIF lidar.

#### 3.1. Molecular mass is a fundamental limit on vapor filter performance

Briefly, we consider the transmission function of a typical cylindrical vapor cell with length $L$ and Pyrex windows along with a side finger of the reservoir partially filled with a metal of interest. The saturated vapor density $n(\# {m^{ - 3}})$ is controlled by the cell temperature and vapor pressure which is regulated by controlling the temperature of the finger [8]. The transmittance of the AVF $\textrm{T}(\nu )$ is an exponential function of the product $({n{\sigma^A}(\nu )L} )$ as given in (7.a), with ${\sigma ^A}(\nu )$ being the absorption cross-section of a resonance transition at frequency ${\nu _0}$ between ground state |1> and excited state |2> with degeneracies, respectively ${g_1}$ and ${g_2}$; it is proportional to the Einstein A-coefficient of the transition ${A_0}$ and a Gaussian function $G(\nu - {\nu _0})$, representing Doppler broadening of the absorption line. The Doppler-broadened line-shape function $G(\nu - {\nu _0})$ and its variance $\sigma _\nu ^A$ or HWHM width $\Delta {\nu _{HWHM}}$ which is proportional to the square-root of the ratio of temperature to molecular weight of the vapor $\left( {\sqrt {T/M} } \right)$ as given in (7.b), where ${k_B}$ is the Boltzmann constant, are given by:

#### 3.2. Iodine vapor for high- and low-temperature blocking filters

Since iodine ($M = 253.8\;amu$) begins to sublime at about 300 K, the Doppler-broadened FWHM is calculated to be 0.437 GHz at 532 nm. This is much narrower than the FWHM of atmospheric ($M = 28.97\;amu$) backscattered Cabannes spectrum of FWHM=2.59 GHz. Thus, it is impossible for the blocking bandwidth of a single iodine absorption line to be much wider than 2 GHz as required by a HT filter before significant continuum absorption sets in.

Hair et al. [8] got around this problem by using a combination of iodine 1107 and 1108 lines with a cell 15.42 cm long with two temperature controllers (on the body and the finger of the cell) to respectively set the vapor temperature and pressure. The cell-finger temperatures used were (82.19 °C -72.03 °C) and (56.18 °C - 47.74 °C) for high-temperature (HT) filter (filter 1) and low-temperature (LT) filter (filter 2), respectively. To ensure near zero transmission between the two lines where the frequency of the laser is located to reject aerosol scattering, the FWHM ($2{\Delta _{HW\textrm{T}}}$) of the LT filter is set to 3.0 GHz, paying the price of 50% off-resonance transmission. The FWHM of the HT filter is set to 4.3 GHz to ensure significant light (3%) transmission. This combination resulted in temperature sensitivity ${S_{T12}} = {S_{T1}} - {S_{T2}}$ of 0.42%K^{-1} but cost dearly in received signal. Ignoring the reduction of the off-resonance transmission, these filters are very close to two ideal filters with $\Delta {\nu _B}$ = 3.0 and 4.3 GHz. Here, the light transmitted through these two ideal filters are about 0.15 and 0.03 (circles in Fig. 2) and the one-channel sensitivities are 0.59%K^{-1} and 1.04%K^{-1} (squares in Fig. 2) leading to the measurement sensitivity of 0.45%K^{-1}, comparable to 0.42%K^{-1} of the iodine filters. Though the one-channel sensitivity calculated from ideal filters may be used approximately for the corresponding iodine filters, to use the calculated fraction signal from ideal filters for the real AVFs, the off-resonance transmittance (due to continuum absorption) must be included. Thus for the iodine filters in question, the attenuation factors are $({{f_1} = 0.1\ast 0.033 = 0.0033} )$ for H.T. channel and $({{f_2} = 0.5\ast 0.148 = 0.074} )$ for L.T. channel. The optimum single-photon uncertainty is then ${\xi _{opt}} = 1/\sqrt {{f_2}} + 1/\sqrt {{f_1}} \,\, = \,\,21$.

#### 3.3. Atmospheric temperature lidar using barium filters at 553.7 nm

The setup of this lidar at the barium absorption line at 553.7 nm is like that using iodine filters at one of the many absorption lines of iodine near 532 nm. The advantage of the barium system lies in its ability to tune on and off resonance easily. Thus, it can perform total scattering (aerosol plus molecular) without a separate total scattering channel (channel 3/Detector 3 in Fig. 1). Atmospheric optical properties may be obtained by tuning the lidar off-resonance in addition to on-resonance for temperature retrieval. By prolonging the duty cycle of observation, an added advantage comes from the instantaneous determination of efficiency ratio ${\eta _1}/{\eta _2}$ by the ratio of off-resonance signals, i.e., ${\eta _1}/{\eta _2} = {N_{\textrm{1off}}}/{N_{\textrm{2off}}}$, leading to an essential temperature ratio ${R_T}(z)$ of (4) in terms of 4 measured photocounts ${R_T}(z) = ({{N_1}/{N_{\textrm{1off}}}} )/({{N_2}/{N_{\textrm{2off}}}} )$ without requiring the channel efficiencies, see Eq. (20) of [6] or Eq. (1) of [7]. The problem of the barium system lies in its low vapor pressure, requiring very high temperatures to obtain sufficient vapor density.

The atmospheric temperature measurements with the barium system have been carried out and published in [6,7] and thoroughly analyzed in [21]. The key information that is relevant to the analysis here may be found in Table 2 of [21] with experimentally measured parameters used for field measurements. The bandwidth, fractional transmission and one-channel sensitivity for the LT filter are 2.04 GHz, 0.4644 and 0.41%K^{-1}, and those for the HT filter are 2.99 GHz, 0.1951 and 0.59%K^{-1}. Although a wider filter is possible respectively with 3.25 GHz, 0.1534 and 0.63%K^{-1}, it was not used because a filter temperature exceeding 820 K is necessary. There was a narrower filter with 1.82 GHz, 0.5582 and 0.37%K^{-1}, which could have been used. Since there is no continuum absorption to worry about, the fractional transmissions through the 2.04 and 2.99 GHz barium filter pairs are respectively ${f_1} = 0.1951\;\textrm{and}\;{f_2} = 0.4644$, along with sensitivity ${S_{T12}} = 0.18\%{K^{ - 1}}$. Using (6b), the optimal single-photon uncertainty is ${\xi _{opt}} = 3.73$. Fundamentally, the problem of this lidar is the vapor mass restricting the range of vapor density that limits the temperature sensitivity to be lower than that of the iodine system by more than a factor of 2.

#### 3.4. Atmospheric temperature measurement by rotational Raman lidar at 532 nm using narrowband custom interference filters

The pure rotational Raman (PRR) scattering spectrum, which consists of the O-branch or Stokes (${\nu _S} < {\nu _L}$) lines with $\Delta J ={+} 2$, the Q-branch of unshifted lines with $\Delta J = 0$ and the S-branch, or anti-Stokes lines with (${\nu _S} > {\nu _L}$) with $\Delta J ={-} 2$ is well known. The relative strengths (probabilities) of the three branches are given by the well-known Placzek-Teller coefficients that sum to unity for a given *J* value. The ratio of the total PRR cross-section $\sigma (PRR)$ to the CS cross-section $\sigma (CS)$ may be calculated since the atmosphere consists of 78.1% nitrogen and 20.9% oxygen, and $\sigma (PRR)/\sigma (CS) = 7{\gamma ^2}/45{a^2}$ [3] with the square of isotropic (trace) and anisotropy polarizability, ${a^2}$ and ${\gamma ^2}$, for Rayleigh scattering [3] for nitrogen and oxygen given in Table 1, taken from Table 9.3 of [24]. Thus $\sigma (PRR)/\sigma (CS)$ for atmosphere consisting of 78.1% nitrogen and 20.9% oxygen is (7/45)* (0.781*0.52 + 0.209*1.26) / (0.781*3.17 + 0.209*2.66) = 0.03435. The cross-section of the shifted PRR (sum of O- and S-branches) $\sigma (RR) \approx 0.75\ast \sigma (PRR)$ is then $0.02576\;\sigma (CS)$ consistent with the relation among backscattering coefficients,${\beta _{RR}} = 0.75{\beta _R}$ and ${\beta _{PRR}} = 0.22\ast 7{\beta _C}/45 = 0.0342{\beta _C}$ [3], or ${\beta _C} = 29.2{\beta _{PRR}} = 78.0{\beta _{aRR}}$.

With aerosol scattering filtered, a realistic temperature lidar may probe the intensity ratio of two specified spectral bands of the anti-Stokes (S-) branch of the PRR spectrum, exploring its temperature dependence. This was done with considerable field observations in 1993 by Neldejevoc et al. [10], and later in 2000 for the GKSS lidar and in 2004 for the RASC lidar by Behrendt et al. [11,12], using CIFs. Our discussion follows the last two references along with the approximate equivalent idealized filter transmission functions for simplicity. Though RASC lidar considered a system that combines rotational Raman and the integration technique to profile atmospheric temperature from ground to 80 km, we only consider the rotational Raman component for measuring troposphere and stratosphere temperatures with a single-frequency laser at 532.25 nm. Because the anti-Stokes spectrum covers a wide wavelength band between 528 nm and 532 nm, unlike Cabannes scattering, there is considerable freedom in placing the location of HT and LT CIFs for a rotational Raman lidar. In fact, the RASC rotational Raman temperature lidar selects its CIFs with an algorithm that minimizes its temperature uncertainty around 240 K [12]. To facilitate discussion, we recall the spectral strengths when summing over all lines in all 3 branches of the PRR is unity. Thus, the rotational Raman lines form the normalized spectrum of PRR.

The optical properties of the filter polychromator of the RASC lidar are shown in Table 2 of [12]. To make real dielectric filters look more like the ideal ones with sharp cutoffs, two complementary filters BS4a + BS4b were used to perform the job of the LT filter; together, they behave as a filter centered at 531.1 nm with FWHM of 0.65 nm with a broad peak transmission of ${\tau _2} = 0.72$. The HT filter BS5 is centered at 528.64 nm with FWHM of 0.80 nm with a peak transmission of ${\tau _1} = 0.87$. For simplicity, in our performance estimate, we represent the transmission function of the two filters as passband filters with location, bandwidth and top-hat peak transmission indicated below. Since there is no spectral overlap between LT and HT passbands for the rotational Raman spectrum, the constraint of ${\eta _1} + {\eta _2} = 1$ is voided; in fact, ${\eta _1} = {\eta _2} = 1$ for the system deployed at RASC [12]. The peak transmission here plays the role of off-resonance transmittance in KVF. For LT filter, BS4a + BS4b, ${\tau _2} = 0.72$ at 531.14 nm and, because the reflectivity of BS4a + BS4b at 528.5 nm is 0.96, the peak transmission for the HT filter is ${\tau _1} = 0.96\ast 0.87 = 0.835$. So, the single-photon uncertainty of this system $\xi $ is not optimum; it is rather $\xi = \sqrt {1/({{\eta_1}{f_1}} )+ 1/({{\eta_2}{f_2}} )} = \sqrt {1/{f_1} + 1/{f_2}}$. The filter attenuation factors may be calculated by (11a) via $R_{Air}^{RR}(\Delta {\nu _i};T)$ defined in (11b) below.

Here, $R_{Air}^{aRR}(\Delta {\nu _i};T)$ symbolically represents a collection of normalized anti-Stokes spectral line-strengths of nitrogen and oxygen that fall within the bandwidth $\Delta {\nu _i}$ of the HT $({i = 1} )$ and LT $({i = 2} )$ filters. It is well known that the scattering from both even- and odd-J states are allowed in nitrogen, i.e., $R_{{N_2}}^{aRR}(\nu ;T) = R_{{N_2} - Even}^{PRR}(\nu ;T) + R_{{N_2} - Odd}^{PRR}(\nu ;T)$, while only odd-J states $R_{{O_2} - odd}^{aRR}(\Delta {\nu _i};T)$ are allowed in oxygen. Here, we also express the filter ${F_i}(\nu )$ as the product of its peak transmission and an ideal passband filter with passband width $\Delta {\nu _i}$, or ${\tau _i}F_i^{idael}({\Delta {\nu_{iBP}}} )$. The filtered anti-Stokes spectrum $R_{Air}^{aRR}(\Delta {\nu _i};T)$ in (11b) is constructed based on the knowledge of the nitrogen and oxygen species concentration, and the PRR scatter cross-section difference between the two major species in Table 1. Writing out the filter function more explicitly with the peak transmission of the respective filters ${\tau _{HT}}\;\textrm{and}\;{\tau _{LT}}$, we have ${f_1}(T,\Delta {\nu _1}) \equiv {\tau _i}R_{Air}^{aRR}(\Delta {\nu _i};T)$ with $i = 1\;for\;HT$, i.e., ${\tau _1} = {\tau _{HT}}\;and\;\Delta {\nu _1} = \Delta {\nu _{HT}}$ and $i = 2\;for\;LT$, i.e., ${\tau _2} = {\tau _{LT}}\;and\;\Delta {\nu _2} = \Delta {\nu _{LT}}$. In summary, the filtered spectral strength of rotational Raman scattering $R_{Air}^{RR}(\Delta {\nu _i};T)$ is the sum of all lines within the filter passband weighted by fractional concentration and relative PRR cross-section of the molecular species, nitrogen and oxygen, and it may be calculated by (11b). These results along with one-channel uncertainty ${S_{Ti}}$, calculated by (5c) are shown in Table 2, leading to temperature sensitivity, ${S_{T12}}{ = }{S_{T1}} - {S_{T2}}$ = 0.93%K^{-1}. With the values of attenuation factors, we use (6a) to calculate the single photon uncertainty $\xi = \sqrt {1/({{f_1}} )+ 1/({{f_2}} )} = \sqrt {1/(0.0416) + 1/(0.0124)} = 10.2$. Had we chosen the LT and HT filters by educated guess to be near the spectral peak instead of minimization of temperature uncertainty, the resulting temperature sensitivity could be easily a factor of 2 smaller.

## 4. Choosing a preferred vapor and a proposed potassium temperature lidar

With the unusual usage of two neighboring iodine lines to increase filter’s bandwidth (thus temperature sensitivity), the system still suffers low off-resonance transmittance suggesting a need to search for better atomic vapors, where the continuous absorption is not a problem. The alkali earth metals are attractive because their singlet ${({\textrm{x}s} )^2} - ({\textrm{x}s\textrm{x}p} )$ transition forms a two-level system. Unfortunately, their low vapor pressure requires too high a finger temperature to produce enough vapor density. Alkali metals can produce enough vapor at reasonable finger temperature, but their transitions are in the form of a doublet with more complex transition structures. Since most alkali’s ground state splitting is relatively small, i.e., the difference between doublet transitions much smaller than the width of Cabannes scattering of air, they could be good candidates for AVF vapors. For example, the ground state splittings are $\Delta \nu = 1.772\;\textrm{GHz}$ for Na and $\Delta \nu = 0.4618\;\textrm{GHz}$ for K; their respective D-doublet transitions are 16956.172 cm^{-1} (NaD_{1}) and 16973.368 cm^{-1} (NaD_{2}), and 12985.186 cm^{-1} (KD_{1}) and 13042.896 cm^{-1} (KD_{2}). What makes these vapors particularly attractive is that the relevant lasers are now becoming available. Because of the ongoing interest in Na laser guided stars, there exist high power single-frequency CW tunable lasers at 589 nm. Even the YAG-based, laser-diode-seeded all-solid-state Q-switched system based on sum-frequency-generation has made Na LIF lidar at 589 nm practically a maintenance-free operation [25]. In addition, a diode-pumped, Q-switched single-longitudinal mode Alexandrite ring laser has very recently been successfully developed, and the lidar operation at 770 nm demonstrated [26]. Therefore, it is timely to consider the use of Na and K AVFs for lidar aerosol and state-parameter measurements in the troposphere based on Mie-Cabannes scattering. Obviously, one has a choice of using D_{1} or D_{2} lines of sodium (589 nm) or of potassium (770 nm). The analysis and performance of these possibilities are expected to be similar. Below, for simplicity, we discuss the performance of one such lidar operated at the D_{1} transition of potassium at 770 nm.

Using the parameters for the D_{1} transitions of potassium atom (its Einstein A-coefficient ${A_0}$ and the transition frequencies of each of the 4 hyperfine transitions), and assuming each transition lineshape is Doppler-broadened at a specified temperature, its absorption cross-section for the KD_{1} transition ${\sigma ^A}(\nu )$ may be calculated with Eq. (21a) of She et al. [27]. We plot in Fig. 4 the absorption cross-section spectrum for atoms in a vapor cell cool enough (50 K for example) to resolve the doublet (in dash-dots on right scale); its higher peak is noted at – 0.18 GHz. Also shown in the figure are two normalized Cabannes scattering spectra of air near ground (at 275 K and 0.75 atm) centered at zero (grey solid) and at -0.18 GHz (grey dash). The three potassium vapor filters could be constructed from a 10 cm long Pyrex cell with a tip for metal reservoir with the temperature of the reservoir and cell controlled separately; the temperatures for the reservoir and cell $({{T_{res}}/{T_{cell}}} )$ are set at (300 K/305 K), (350 K/355 K) and (400 K/405 K) for LTB, LTA, and HT filters, respectively. To calculate a filter transmission function, we first obtain the vapor pressure ${p_K}$ for ${T_{res}}$ with metal in solid and liquid phases in the finger using Eq. (22) of [28],

We then determine vapor density ${\textrm{N}_K}$ for $({{T_{res}}/{T_{cell}}} )$ with ideal gas law and calculate the transmission function $\textrm{T}(\nu )$ using (7a) and (7b) by substituting ${\textrm{N}_K} = {p_K}/{k_B}{T_{cell}}$ for $n$ and the absorption ${\sigma ^A}(\nu )$ with Doppler-broadening at the cell temperature ${T_{cell}}$. The resulting transmission functions for the three KVFs are also shown in dashed (LTB), in solid (LTA) and in dots (HT), respectively in Fig. 4. At the center (0 GHz) of the absorption cell, the filters can block scattered light at a level of 21.4, 43.8 and 763 dB, respectively for LTB, LTA and HT. At – 0.18 GHz, they respectively block at a level of 20.4, 41.9 and 701 dB.

#### 4.1. Comparing two lidars at different frequencies

Since the resonance of KD_{1} transition (770 nm) is different from that of the iodine absorption in question (532 nm) and Rayleigh scattering cross section is wavelength dependent, we need to discuss the wavelength dependence in scattered signal and filter bandwidth. We note in (6), measured temperature uncertainty depends on three factors, one of which is the total photon counts received at point B in a measurement, ${N_0}$. In addition to the power-aperture product (${P_L}{A_R}$), the photon counts depend not only on the scattering process (CS vs. RRS) but also on the laser wavelength. For a given incident photon flux, the ${\lambda ^{ - 4}}$ dependence of photon scattering cross-sections is well known. Since for a given incident power P, the number of scattered photons is proportional to the wavelength, the wavelength dependence on scattered photon becomes ${\lambda ^{ - 3}}$ i.e., the received photon ${N_0}$ becomes ${N_0}{({\lambda _0}/\lambda )^3}$ when scaled to a system with the same ${P_L}{A_R}$ at wavelength ${\lambda _0} = 532nm$. To consider the impact of wavelength dependence on the filtered light through an ideal filter bandwidth $\Delta {\nu _B}$, we revisit the nature of the normalized Cabannes spectrum $R(\nu ;T,P)$, which is related to a model-dependent theoretical function ${C_0}(x,y)$ of two dimensionless parameters $x$ and $y$ defined in Tenti et al. [29] with a $\sqrt 2 $ difference in the definition of ${\textrm{v}_0}$ as:

*x*is proportional to the product of $\nu$ and $\lambda $. Since the spectrum ${C_0}(x,y)$ is normalized in the $x - \textrm{axis}$, to compare the Cabannes spectrum with the same ${\rm{V}}_0 $ (or atmospheric temperature) resulting from scattering at two different wavelengths $\lambda \;and\;{\lambda _0}$, we need to compare them as a function of the dimensionless frequency $x$, or investigate the quantities of interest in terms of a scaled frequency ${\nu _{\textrm{sca}}} = (\lambda /{\lambda _0})\nu$. For example, to compare a potassium lidar at 770 nm to a lidar at 532 nm, we could plot quantities in the scaled frequency ${\nu _{\textrm{sca}}} = (770/532)\nu = 1.45\nu$.

#### 4.2. Filtered Cabannes scattering and one-channel temperature sensitivity of KVF

Shown in Fig. 5 are the fractional transmission of Cabannes scattering ${f_i}$ of air (laser locked at 0 GHz) at 275 K and 0.75 Atm (thin solid) through an ideal blocking filter, along with the one-filter temperature sensitivity ${S_{Ti}}$(thin dashed) at 770 nm plotted as a function of the bandwidth of the ideal filter. The FWHM widths of the 3 potassium filters (KVFs) can be measured from Fig. 4 respectively to be 1.7, 2.0 and 2.9 GHz. Their fractional (or filtered) signal transmitted and one-channel sensitivity may be numerically calculated, and are also shown in Fig. 5, respectively as 29.0%, 18.6% and 3.74% (solid circles), and 0.295%K^{-1}, 0.466%K^{-1} and 1.012%K^{-1} (open squares). As can be seen, they fall almost exactly (identical) on the curves for ideal filters. The temperature sensitivities for the two pair of filters are ${S_{T12A}} = 0.55\%{K^{ - 1}}$ and ${S_{T12B}} = 0.72\%{K^{ - 1}}$. Had we locked the laser at the K-D_{1} transition peak at -0.18 GHz, the corresponding one-filter sensitivities would be somewhat smaller, with 0.256%K^{-1}, 0.387%K^{-1}, and 0.883%K^{-1}, giving rise to ${S_{T12A}} = 0.50\%{K^{ - 1}}$ and ${S_{T12B}} = 0.63\%{K^{ - 1}}$. For temperature only measurements, we would lock the laser at 0 GHz. For the comparison to be discussed in Section 4(c), we use KVF-1 and KVF-2B for HT and LT filters respectively, i.e., ${S_{T12B}} = 0.72\%{K^{ - 1}}$, and ${\xi _{opt}} = 1/\sqrt {{f_2}} + 1/\sqrt {{f_1}} = 1/\sqrt {0.2902} + 1/\sqrt {0.0374} = 7.028$. We also analyzed these for an atmosphere at 240 K and 40 kPa. With the laser locked at 0 GHz, ${f_i} = \;25.4\%,\;15.6\%\;\textrm{and}\;2.95\%$ and ${S_{Ti}} = 0.57\%,\;0.37\;and\;1.17\%{K^{ - 1}}$, leading to ${S_{12A}} = 0.60\%{K^{ - 1}}$ and ${S_{T12B}} = 0.80\%{K^{ - 1}}$.

It is instructive to compare the use of KVFs (Fig. 5) at 770 nm against that of iodine filters at 532 nm (Fig. 2) by employing the wavelength scaling discussion in Section 4(a). This can be done by replotting the two curves in Fig. 5 in the scaled frequency and transforming the thin curves into the corresponding thick curves. Now, we can see an ideal filter, say of $\Delta {\nu _B} = 2GHz$ for scattering at 770 nm is equivalent to a filter of $\Delta {\nu _B} = 2.9GHz$ wide at 532 nm, because, under otherwise the same (atmospheric) conditions, the Cabannes spectrum at 770 nm in the frequency space is much narrower. Indeed, the thick curves in Fig. 5 look very similar to those in Fig. 2. The very small difference between the two sets of curves may be accounted for by their difference in the y value, larger in 770 nm by a factor of (770/532), reflecting more collective scattering contributions. This bit of physical understanding increases confidence in these plots and their applications.

For data processing of the proposed lidar, we need to return to (4), and relate the measured photocount ratio $R_T^{Obs}(z)$ to the ratio of filter attenuation factors ${f_1}[T(z),P(z)]/{f_2}[T(z),P(z)]$, or the essential temperature ratio ${R_T}(z)$ as:

Here, we assume an optimum division of the scattered photons according to the KVF attenuation factors of air at a standard point with 275 K and 0.75 kPa.

To this point we have focused on calculation of lidar signals from the air at known temperature and pressure. For the inverse problem of finding temperatures from lidar signals, we write the (essential) temperature ratio, ${R_T}(T,P)$, in Taylor series around a standard point with $({T_{std}},{P_{std}})$ through second order:

*μ*as {std, T, P, TT, TP, and PP}

*,*are respectively the function, the first and second-order partial derivatives of ${R_T}(T,P)$ with respect to T and/or P all evaluated at the standard point. The coefficients, ${R^\mu }$ are found by fitting calculated values of ${R_T}$ around the standard point. We have used ${T_{std}}(z)$ and ${P_{std}}(z)$ values given by the 1976 U S Standard Atmosphere [http://www.pdas.com/atmosTable1SI.html] and find, for example at 10 km altitude, ${T_{std}}$ = 223.15 K, ${P_{std}}$ = 26,436 Pa, using KVF1 for HT and KVF2B for LT filter respectively, R

^{μ}={ 1.0885E-01, -6.2930E-07, 9.4778E-04, 1.0421E-06, 5.2945E-12, -2.5571E-09 } in MKS units. Over a range of 12 K and 5,000 Pa near the standard point, using the full expansion in (10a) in place of the exact result will give temperature errors defined as (R

_{T}-R

_{Tapprox})/R

^{T}of less than 0.02 K, whereas the temperature error when using just R

^{std}, R

^{T}and R

^{P}is less than 0.1 K. To analyze an experiment, we use the observed value of ${R_T}$ at an altitude where the pressure is P and the first three terms of (10a) giving the temperature at that altitude as,

This expression will give temperatures accurate to better than 0.5 K over a range of altitudes from 0 to 60 km if T is within T_{std} ± 20 K and if P is within 0.75 P_{std} and P_{std}. The atmospheric pressure at the neighboring height (range bin $\Delta z$) may be calculated by assuming hydrostatic equilibrium and the ideal gas law. Iteration of the hydrostatic equation to obtain the pressure at z_{i+1} from the pressure at z_{i} can be implemented using P(z_{i+1}) = P(z_{i})exp[–M_{air}gΔz/k_{B}T_{L}(z_{i}) ]. Alternatively, Witschas [30] noted that, since the atmospheric pressure at a given altitude has limited variation, it may be sufficient to replace the hydrostatic equation by assuming a standard atmospheric pressure profile in the analysis. Some of our model calculations show that this assumption will change the temperatures by less than 0.1 K if the assumed pressure, P_{o}, is known accurately. However, a 3% error in P_{o} leads to a temperature error of about 1.5 K near the surface, independent of whether iteration goes from low-to-high altitudes or high-to-low altitudes. We have found no evidence of instability in iterations either upward or downward. Iterations in altitude yield vertical profiles of atmospheric temperature and pressure and thus density profiles. Once the temperature and pressure profiles are obtained along with ${f_i}(T,P,z)$, the signal from added total scattering channel (channel 3 in Fig. 1 with aerosol plus molecular) can be used to determine the profiles of extinction aerosol coefficient and lidar ratio, see for example Eqs.3(a)-3(e) of [7].

## 5. Figure of merit comparison between four atmospheric temperature lidars

In this Section, we compare four atmospheric temperature measurement methods, all using ratios between two filtered scattering signals: (1) Cabannes scattering (CS) at 553.7 nm filtered by barium filters, (2) CS at 532 nm by iodine filters, (3) rotational Raman scattering (RRS) at 532 nm by custom-built dielectric interference filters (CIFs), and (4) CS at 770 nm by the proposed potassium filters. For three of them that are based on Cabannes scattering, we first list their respective single-photon uncertainty ${\xi _{opt}}$ and ${S_{T12}}$, and then calculate the respective temperature uncertainty $\delta {T_{air}}(K)$ based on ${N_0} = {10^8}$ photon counts arriving at point B for temperature measurements, along with $\delta {T_{air}}(K)$ scaled to 532 nm, i.e., normalized to the same ${P_L}{A_R}$ product. For the RRS lidar, since we could optimize the signal for each channel to the extent possible as the two channels are spectrally independent, we list $\xi$ calculated from (6a) instead. The photon counts arriving at point B would be those of PRR instead, and the effective photocounts become $N_0^{PRR} = 0.0342{N_0} = 3,420,000$.

Compared to the barium and iodine system, the proposed potassium (M=39.1 amu) performs better for different reasons. Iodine (M=253.8 amu) and barium (M=173.3 amu) have the problem of very narrow Doppler width, thus making it difficult to have sufficient blocking bandwidth required by the HT filter. Coupled with the need for unreasonably high temperature to produce sufficient Ba vapor, the temperature sensitivity ended up being about 4 times lower than the potassium case. With the creative use of two neighboring absorption lines, the iodine system has achieved sensitivity only a factor of 1.7 less, but its single-photon uncertainty becomes 3 times larger than potassium due to the presence of continuum absorption. The signal strength of anti-Stokes PRR scattering is about 78 times weaker than that of CS, suggesting a potential temperature uncertainty of 8.8 times higher. However, the resulting uncertainty of the $\delta {T_{Air}}$ of the RR lidar system with custom made interference filters (CIFs) is only a bit lower than the CS/iodine system because it is possible to employ custom-built filters with a bandwidth more than 100 times broader. Since the two spectral regions are independent in this case, we can make ${\eta _1} = {\eta _2} = 1$. Furthermore, the beauty of being able to choose its center wavelength and bandwidth, tailored to the application at hand, made it possible to optimize the temperature sensitivity. For example, the center wavelength of the HT filter, BS5, has moved from near 529.5 nm for the GKSS lidar to 528.76 nm for the RASC lidar because the emphasis of the former was the condensation of polar stratospheric clouds (with temperature ∼190 K). Alternatively, the interest of RASC lidar was temperature profiling from troposphere to stratosphere, thus choosing filters to minimize temperature uncertainties at ∼ 240 K [12]. In fact, this optimization has made the one-channel sensitivity negative for the LT filter and positive for the HT filter (see last column in Table 2), making the temperature sensitivity 0.93%K^{-1}, 1.3 times higher than the proposed CS/KVF (2.2 times higher than CS/IVF). Of course, the strength of Cabannes scattering over anti-Stokes PRR (37.5% of PRR) is still an advantage for CS/KVF; it alone should reduce the temperature uncertainty by 8.8 times. This leads to a single-photon uncertainty of 10.2 for RR lidar in comparison to 7.03 for CS/KVF lidar, partially negating the gain in sensitivity for RR/CIF (0.93% as compared to 0.72%). The net effect is a factor of 6.1 times smaller in uncertainty for CS/K lidar. As can be seen in Table 3, the temperature uncertainty of the proposed potassium lidar is a factor of 5.2 and 6.1 times smaller than CS/IVF lidar and RR/CIF lidar, or 3.0 and 3.5 times smaller when scaled to 532 nm respectively.

We next briefly discuss an estimation of the system parameters that are needed to deliver the assumed 100,000,000 Cabannes scattering photons employed in Table 3 above for temperature measurements. For this, we return to (2a) and derive from it the received photon counts of the molecular channel ${N_m}({\nu _S},z;{r_B})$ in terms of ${P_L}{A_R}$ product and atmospheric density $\textrm{N}(z)$ as:

^{8}photons from the standard atmosphere at 1 km, 5 km and 10 km with $\textrm{N}(z)$ of 2.31 × 10

^{25}, 1.53 × 10

^{25}, 5.42 × 10

^{24}m

^{-3}. Using $d{\sigma ^C}({\nu _L})/d\Omega = 5.96\textrm{x}{10^{ - 32}}{m^2}/sr$, we determine from (12) the required measurement time for the tabulated temperature error for 532 nm at 1 km, 5 km and 10 km to be about 0.90, 6.83, and 38.5 min, respectively. For these same integration times and vertical resolutions, or ${N_m} = 1.0\textrm{x}{10^8}$, the temperature uncertainty for the potassium system scaled to 532 nm is estimated to be 0.17 K and that for iodine system is 0.5 K, see Table 3. The calculated $({P{A_R}} ){\tau _{{\mathop{\rm int}} }}({\Delta z} )$ product for the hypothetical system at 532 nm at $z = 1\;km$ turns out to be 0.045 Wm

^{2}-min-km; this may be compared to the same for the experimental barium system (3 mW, A

_{R}= 0.0324 m

^{2}, 20 min and 375 m resolution) and iodine (6W, A

_{R}= 0.0324 m

^{2}, 1 hr and 300 m resolution) system of 7.3 × 10

^{−4}and 3.5 Wm

^{2}-m-km, respectively. At 1 km, the measured temperature error from the barium system at 553.7 nm [6,7] is 8 K (8.5 K when scaled to 532 nm) which becomes 1.08 K when scaled to the $({P{A_R}} ){\tau _{{\mathop{\rm int}} }}({\Delta z} )$ of the hypothetical reference system which is 6.4x larger than the 0.17 K for the CS(770 nm)/KVF system in Table 3. The measured error of the experimental iodine system [8] at 1 km is 0.25 K, which becomes 0.7 K when $({P{A_R}} ){\tau _{{\mathop{\rm int}} }}({\Delta z} )$ is scaled; it is 1.4x larger than the 0.5 K for the iodine system in Table 3. Owing to the possible difference in the total collection efficiency ${\eta _m}\exp [\ldots ]$ and the real single-photon uncertainty ${\xi _{opt}}$ of the experimental systems from those given in Table 3, it is gratifying that with educated assumptions, the results of both measurements at 532 nm and at 553.7 nm appear to be consistent (both larger than a factor of less than 10) with the hypothetical system. These $P{A_R}$ values needed to deliver 10

^{8}photons will not affect the relative figures of merit between the 4 lidars as outlined in Table 3.

Since the estimated measurement times for the modest hypothetical lidar system appear to be acceptable, the proposed potassium system in Table 3 could lead to a realistic lidar if implemented. It may be also beneficial to consider other pairs of (HT/LT) filters. Similar analysis can be made for a system at 589 nm using sodium filters, which will enjoy a more favorable vapor mass limitation and wavelength scaling for Cabannes scattering from 532 nm, but its larger ground state hyperfine splitting, $\Delta \nu = 1.772\;\textrm{GHz}$ for Na compared to $\Delta \nu = 0.4618\;\textrm{GHz}$ for K will set a limit for the bandwidth of the LT filter, thus reducing the filtered signal and sensitivity.

## 6. Proposed temperature/wind lidar with Cabannes scattering and KVFs

For lidars pointing off-zenith, the radial component resulting from the horizontal wind cannot be ignored. Since 1 m/s line-of-sight (LOS) wind gives rise to a Doppler-shift in backward scattering of -3.76 MHz at 532 nm and of -2.60 MHz at 770 nm, only Cabannes scattering can be used to monitor molecular motion (or wind). The flexibility in CIF optimization in RR lidar unfortunately cannot give a high enough spectral resolution for measuring wind. Indeed, the backward Cabannes scattering at 532 nm with iodine filter have been employed for LOS wind measurements more than 2 decades [18,19,31,32]. Despite the continuum absorption, the iodine filter performs well in this case because only one filter with moderate width is needed. It turns out the temperature-only potassium lidar system can be used for this objective, if, instead of locking the laser frequency at the center of mass of the filter absorption line ${\nu _0} = 0\;GHz$, we shift the laser frequency between two fixed frequencies, above at ${\nu _ + }$ and below it at ${\nu _ - }$. Due to the ground-state hyperfine splitting of 0.4618 GHz, the KD_{1} absorption spectrum and its associated KVF transmission is slightly asymmetric. For this reason, we select two locations, ${\nu _ - } ={-} 0.180\;GHz$ and ${\nu _ + } ={+} 0.208\;GHz$, away from the KD_{1} transition ${\nu _0}$ but within the edge of the filter to reject aerosol scattering. We next discuss the methodology, temperature and wind uncertainties for ${N_0}$ photons at point B in the lidar receiver (Fig. 1), as well as the procedure of data processing.

To compare with the temperature measurements discussed previously (lock ${\nu _0} = 0\,GHz$), we still consider the same temporal resolution, i.e., the time interval for the lidar to collect ${N_0}$ photons at point B. For temperature measurements, Eq. (4) still applies, except the photocounts and attenuation factors should be taken as the average of those collected between setting laser at ${\nu _ - } ={-} 0.180\;GHz$ and at ${\nu _ + } ={+} 0.208\;GHz$. In other words, at a given altitude z, with the same two KVF channels, Eq. (4) becomes,

Notice that the mean values of both $N\;\textrm{and}\;f$ will be a bit smaller than (but comparable to) those when laser was locked at ${\nu _0}$ as in the temperature only measurement. The factors ${\eta _1}\;\textrm{and}\;{\eta _2}$ are detection efficiencies of the setup (after point B in Fig. 1). Theoretically, they may be set to the optimized values, $\eta _1^{opt}\;\textrm{and}\;\eta _2^{opt}$ as is done in the estimates below. With these values substituted, the same formulae, (5c), (6a) and (6b) may be utilized to determine ${\xi _{opt}},\;{S_{T12}}\;and\;\delta {T_{Air}}$.

For LOS wind measurements, we realize that only one channel is necessary to compare the photocounts shift between ${\nu _ + }$ and ${\nu _ - }$ laser lock, i.e., each channel gives an independent LOS wind measurement when the plus and minus cycle is completed. At a given altitude z, the LOS wind is then determined by the wind ratios defined as

Note the absence of an efficiency ratio because we compare ${N_i}({\nu _ + })$ and ${N_i}({\nu _ + })$ from the same channel and ${R_{Wi}}(z)$ represents both observed and essential wind ratios. Since one cycle (two acquisitions) must be completed for a wind measurement, we can employ the same temporal resolution if we assume only ${N_0}/2$ photons at point B. On the other hand, wind measurements from both channels may be averaged to reduce the photon noise error. With these differences kept in mind, it is not difficult to derive single-photon efficiencies and sensitivities for LOS wind measurements for the average, at a given z (or T and P). They are:

To provide numerical estimates, we analyze the simultaneous temperature/wind measurement of this proposed temperature/wind lidar for an atmosphere at the standard point, 275 K and 0.75 atm and the results are summarized in Table 4.

It is expected that the temperature uncertainty in the simultaneous temperature/wind measurement to be a bit larger than the temperature only measurement, i.e., $\delta {T_{Air}} = 0.101\;K$ as compared to $\delta {T_{Air}} = 0.097\;K$. It is perhaps somewhat surprising to see the wind measurement uncertainty with HT KVF $\delta {W_1}$ to be more than a factor of 2 less than with the LT KVF $\delta {W_2}$. Apparently, the frequency dependence of the normalized Cabannes spectrum made the loss in signal with the HT KVF over-compensated by its gain in sensitivity. Unlike with iodine filter, one can take the advantage of this characteristics due to the absence of accompanying continuum absorption. Since both wind measurements (${W_1}\;\textrm{and}\;{W_2}$) are carried out by default in this scheme, the estimated uncertainty of the mean wind if deduced using (14b) is $\delta {W_{Air}} = 0.128\;m/s$.

The simultaneous atmospheric temperature and LOS wind may be retrieved from lidar data this way. To facilitate this discussion, we relate as in (4) the essential temperature ratio ${R_T}$ to the observed temperature ratio $R_T^{Obs}$ as,

Both the essential temperature ratio in (15) and the wind ratio in (13b) can be expressed as the ratio of two (different) theoretical attenuation factors. Since these attenuation factors are functions of (T,P,W), and assuming ${\eta _2}/{\eta _1}$ is known, we can retrieve atmospheric $T\;and\;W$ from measured temperature and wind ratios $({R_T^{Obs}\;\textrm{and}\;{R_{Wi}}} )$, once the atmospheric pressure P is known. To shed some light on the relationships between $({T,\;{W_i}} )$ and $({{R_T},\;{R_{Wi}}} )$, we plot their calibration curves for two atmospheric pressures and two detection channels in Fig. 6 below. For comparison, the calibration curves using the Doppler-broadened spectrum (in dash), G(ν) from (7b), independent of *P*, and using the normalized CS spectrum (in solid) $R(T,P,\nu )$ in (3) are both shown.

As expected, at lower pressures and higher temperatures (a) and (b), there is little difference between the Cabannes and Doppler contours. At higher pressures, the Cabannes temperature contours deviate significantly from the Doppler contours, although the zero velocity contours overlap significantly. Spanning the velocity from - 50 to 50 m/s at 240 K changes R_{W1} by 1.2 whereas R_{W2} changes by only 0.3 suggesting that, globally, R_{W1} is more sensitive to wind than R_{W2}, as is shown in Table 4.

To retrieve the atmospheric temperature and wind profiles, we first deduce the essential temperature ratio profile ${R_T}(z)$ from the measured temperature photocount profile $R_T^{Obs}(z) = \overline {{N_1}} (z)/\overline {{N_2}} (z) = {\eta _1}{R_T}/{\eta _2}$, and start the data processing at a reference point ${z_0}$ where the pressure P_{o} is measured or estimated by independent means. The temperature and LOS wind at this point can be determined from the measured ratio pairs via essential ratios, $({{R_T},{R_{W1}}} )$ and $({{R_T},{R_{W2}}} )$, from a calibration curve for pressure P_{o}. With the temperature T_{o} at this point known, using hydrostatic equilibrium and the ideal gas law, we can calculate the density at z_{o} and the pressure P_{1} at the next lower altitude z_{1} = z_{o} - Δz, with which we can calculate T_{1} and W_{1} from the measured ratio pairs at z_{1}. The process is continued yielding 3 profiles, T(z), W_{1}(z), and W_{2}(z). The simple average of the two wind profiles yields the wind profile $W(z)$. The measured profiles of atmospheric/molecular parameters are then obtained, along with their estimated photon uncertainties. More specifically, prior to the experiment, R_{T}(T,W,P) and R_{W}(T,W,P) will be calculated for given HT and LT filter pair for a range of values of {T,W} and a range of calibration pressures {P}. To find temperature and wind from R_{T} and R_{W} values at a given pressure, an efficient method is to first calculate bilinear forms of T(R_{T},R_{W},P), W_{1}(R_{T},R_{W},P) and W_{2}(R_{T},R_{W},P) in powers of (R_{T}-R_{To}) and (R_{W}-R_{Wo}) obtaining, for example, for the KVF1 as,

T(R_{T},R_{W1},P_{o}) = 274.0 + 1021ΔR_{T} + 10.41 ΔR_{W1} + 218.7ΔR_{T}^{2} -56.04ΔR_{W1}^{2} +106.8ΔR_{T}ΔR_{W1}

W_{1}(R_{T},R_{W1},P_{o}) = -3.741 -47.62 ΔR_{T}+ 103.9 ΔR_{W1} -399.3 ΔR_{T}^{2} -110.5 ΔR_{W1}^{2}+569.4

ΔR_{T}ΔR_{W1}

where P_{o} = 76 kPa, ΔR_{T0 }= R_{T} -0.1508 and ΔR_{W10} = R_{W1} - 0.9090 which give T and W_{1} with an accuracy of 0.05 K and 0.1 m/s, respectively, over a range of about ±6 K and ±5 m/s. These expansions were calculated using 12 × 12 regions in the R_{T}-R_{W1} plane covering temperatures from 150 to 300 K and LOS wind speeds from - 52 to 52 m/s. For other pressures, T and W would be calculated by interpolation between calibration pressure levels. The pressure at level z_{i+1} may be obtained from the hydrostatic equation or by assuming the pressure of a standard atmosphere as discussed earlier.

Witschas [30] also gives an analytic approximation for the Cabannes spectrum which we have compared with our Cabannes spectrum for velocities, -50 m/s < V < 50 m/s, and temperatures 200 < T < 280 K for P = 76,000 Pa (∼2.35 km), and for 0.277 kPa (∼40 km). For measurements of temperatures near 275 K using the analytic approximation in the data analysis gives temperatures within a few degrees of the Cabannes results. For measurements of temperatures at 0.277 kPa using the analytic line shape, the Gaussian line shape, or the Cabannes line shape in the data analysis all result in temperatures within a degree or so. Near the surface the analytic line shape would give temperatures too high by about 8, 6 and 4 K for actual temperatures of 200, 240 and 280 K respectively. These results, of course, depend upon the particular vapor filter and laser frequencies being used and must be evaluated for future experiments.

We note if Channel 1 and 2 are used independently for single-channel wind measurement, then we would have $\eta = 1$ and ${N_0} = 50,000,000$ for each channel, and in this case, $\xi /\sqrt {{N_0}} (\%)$ will be 0.093 and 0.036 respectively for channel 1 and 2, leading to $\delta {W_1} = 0.080\;m/s$ and $\delta {W_2} = 0.125\;m/s$, smaller than the corresponding ones listed in Table 4 as expected. We also note the uncertainty of horizontal wind velocity when computing from a LOS wind measurement increases as the pointing angle (off zenith) $\Theta $ decreases. This is much more of a concern at higher altitudes as photon noise increases. In addition, we note a similar temperature/wind lidar was proposed [33] and demonstrated [34] in 2009. Piggybacked on the three-frequency sodium fluorescence lidar transmitter [35], this lidar implemented a Na double-edge magneto optic filter to receive the right- and left-circularly polarized signal separately, utilizing four out of six received signals to form two independent ratios for simultaneous temperature and wind measurements. In addition, based on the double-edge technique [36] of a pair of Fabry-Perot interferometers (FPI), the wind measurement of the latter was improved in 2012 by incorporating a system-level optical frequency control method [37]. Continuing the use of an interferometer for atmospheric temperature measurements [1,9] by scanning either the laser [38] or the FPI [39] in 2014, the received signal is retrieved to simultaneously probe atmospheric temperature. Since these measurements also depend on Cabannes scattering, when implemented properly for aerosol rejection and modeling, the temperature and wind measurement uncertainties are expected not to be very different from the proposed potassium lidar. Owing to the ease of alignment, excellent rejection of aerosol contamination, simplicity in implementation, and newly available maintenance-free lidar transmitter, we hope the proposed potassium lidar will become an attractive choice for future atmospheric temperature and wind observations.

## 7. Discussion and conclusion

We have revisited the use of atomic vapor cells as aerosol blocking filters for backscattering lidars for the purpose of temperature measurements. A basic requirement is that the width of a HT filter be wide enough to reject the low-frequency portion of the Cabannes spectrum of the atmosphere ($M = 28.97\;amu$, FWHM ∼ 3 GHz). Thus, atoms like barium (137.3 amu) and lead (M = 207.2 amu) with high atomic weight, despite their desirable spectral characteristics, are not the best choice. Ironically, both lead [24] and barium vapor filters [6,7] have been attempted for atmospheric temperature measurements. In addition to the difficulty of producing a wide enough blocking band from a single absorption line, it is also difficult for these atoms to produce sufficient vapor density at a reasonable vapor temperature for the intended applications. The barium filters at 553.7 nm [6,7] have been used successfully in field experiments, but were discontinued because of such difficulties.

Iodine ($M = 253.8\;amu$) with high molecular mass has the same problem. However, its high vapor pressure (thus density) at room temperature made it more attractive, as it is possible to combine two neighboring lines [8] to widen the blocking width for the HT filter. As a molecular filter, even with the use of two neighboring lines to increase sensitivity to 0.42%K^{-1}, the high cell temperature needed to produce high enough bandwidth for the HT filter is accompanied by high continuum absorption. This is enough to reduce the transmission at the off-resonance frequencies from unity to 0.1 for a blocking bandwidth of 4.3 GHz which increases the single-photon uncertainty by several times. The performance of the iodine system should still be better than the rotational Raman technique for temperature measurements because the Cabannes scattering (CS) cross-section is about 29 (78) times stronger than that of PRR (anti-Stokes PRR), giving a factor of ∼ 5.4 (8.8) advantage in photon uncertainty. Since RR lidar utilizes anti-Stokes PRR signal for data processing, the numbers in the bracket are more appropriate for the CS and RR lidar comparison. We pointed out a leverage in rotational Raman lidar, which has not been well appreciated. This is the ability to optimize the bandwidth and place the center wavelength of the HT filter relative to that of the LT filter using the customized interference filters (CIFs) with much wider spectral bandwidth. The result is a great increase in the sensitivity of the technique to as high as 0.93%K^{-1}, a factor of 2.2 increase compared to the iodine system. The increase in sensitivity just about cancels the loss due to photon uncertainty, giving rise to a comparable performance between the CS/iodine and RR/CIF. We therefore propose the use of potassium atomic filter at 770 nm with Cabannes lidar (CS/potassium) for atmospheric temperature measurements. Even though the atomic weight of potassium ($M = 39.1\;amu$), not too much heavier than air molecular weight ($M = 28.97\;amu$), with the HT KVF we used, the sensitivity of 0.72%K^{-1} is still a factor of 1.3 lower than that of RR/CIF, resulting in a temperature uncertainty (with 100,000,000 scattered photons) of the proposed system to be 0.097 K, which is about a factor of 5.2 and 6.1 times smaller than CS/Iodine and RR/ CIF respectively. Since CS counts are estimated to be 78 times higher than that of the anti-Stokes PRR, one might anticipate a factor of ∼ 8.8 times smaller in temperature uncertainty. The ability to freely locate the center of the CIFs, thus optimizes measurement sensitivity for RR/CIF leading to a net 6.1 times smaller temperature uncertainty for CS/KVF for the same Cabannes photons received. When scaled to 532 nm (i.e., normalized to the same ${P_L}{A_R}$ product), the temperature uncertainty of CS/KVF in fact is only ∼ 0.592/0.169 = 3.5 times smaller than the RR/CIF system, due mainly to the effectiveness of RR/CIF filter optimization. One could use a lidar 12.3 times smaller in ${P_L}{A_R}$ product to produce the same temperature uncertainty. Note that the factor of 6.1 improvement in temperature uncertainty of CS/KVF over RR/CIF assumes that the same number of photons are received in a measurement. Of course, with the much larger available Nd:YAG laser power one can build a RR/CIF lidar with much higher ${P_L}{A_R}$ product to obtain a more precise measurement. Depending on scientific objectives, a more compact and stable (maintenance-free) lidar system may be preferred for long-term and airborne observations of the atmosphere. At the current 1 W available level power at 770 nm [40], a Diode Pumped Alexandrite Ring Laser is being designed for the VAHCOLI (Vertical And Horizontal COverage by Lidar) network intended for worldwide measurement campaigns by scientists from the Leibniz-Institute of Atmospheric Physics and the Fraunhofer Institute for Laser Technology, Germany [41]. The proposed CS/KVF lidar could possibly be considered and integrated into such an Alexandrite-based system.

Although we stated the aerosol rejection capability of each filter for temperature measurement, i.e., of 21.4, 43.8 and 763 dB respectively for LTB, LTA and HT, we assume complete aerosol rejection in the figure of merit comparison and results presented in Table 3. This is certainly a valid assumption for the mesosphere, but it may be questionable in the troposphere depending on aerosol loading at each altitude. In practice, the level of aerosol contamination can be estimated from the signal received in the total (aerosol plus molecular) scattering channel (DET3 in Fig. 1) provided the linewidth and spectral purity of the transmitting laser are known. Even then, the analysis to incorporate it into the measurement uncertainty is too complicated to be considered here. In all cases of lidar temperature measurements we discussed, the detection efficiency ratio ${\eta _2}/{\eta _1}$ or $\eta _2^{Opt}/\eta _1^{Opt}$ is assumed to be constant. To maintain a constant ratio throughout the course of measurement in practice may however be very challenging. If this ratio, for example, changes from one night to the next because the mount for NBS at position 2 in Fig. 1 rotates slightly thus changing the relative values of ${\tau _2}\;\textrm{and}\;{\rho _2}$ slightly, then the absolute value of the measure temperatures becomes uncertain, although the relative temperature (and the structure of a temperature profile) may remain very good. Hair et al. [8] in 2001 experienced this problem and discussed it in some detail. After an adjusted constant temperature shift, the lidar profiles they measured from ground to the tropopause are seen to yield excellent agreement with collocated balloon-sonde. More careful airborne aerosol optical property measurements in 2008 with a built-in highly accurate real-time calibration system yielded an average change in normalization error of less than 0.3% [20]. This is indeed a non-trivial issue, as a publication in the same year stated that it is not easy to determine the normalization much better than 1% in practice [42]. A remedy is to tune the laser frequency to on- and off-resonance and measure molecular and total scattering from the same and identical setup/channels, thus avoiding the need for the efficiency ratio calibration; this was done with the barium system [6,7]. Unfortunately, it is impossible for the iodine vapor filters because of the existence of too many closely spaced absorptions lines in addition to the continuum absorption, but could be done in the proposed potassium filters, if deemed necessary. The stability problem in channel efficiency and filter function encountered in [8] exists in principle in RRS lidars as well. However, we have not noticed such discussion explicitly in the literature [10,11,12], assuming excellent mechanical stability in the polychromator. On the other hand, published statements do include: “The theoretical calculation of the method led to an analytic calibration function which, once adjusted with a radiosonde, can provide the temperature on successive days of measurement” in [10], “For the temperature determination we calibrated the instrument with data from a local radiosonde” in [11], and “We found the ratio of rotational Raman signal efficiencies experimentally by comparing the theoretical and experimental calibration functions for temperature measurements” in [12]. Thus, it is prudent to test the operational stability of each lidar on a case-by-case basis.

By tuning the laser frequency cyclically to above and below the absorption line of potassium, unlike the RR lidar, the proposed CS/K lidar can be used for simultaneous measurement of atmospheric temperature and LOS wind. The data processing and retrieval methods are discussed with the temperature and LOS wind uncertainty for N_{0} = 100,000,000 estimated to be $\delta {T_{Air}} = 0.101\;m/s$ and $\delta {W_{Air}} = 0.128\;m/s$. In other words, at the signal level of N_{0} = 1,000,000, the uncertainties are respectively ∼ 1 K and 1.5 m/s. In principle, the problem of changing efficiency ratio between two channels though affecting temperature measurements is not a concern for LOS wind measurement, since here we compare scattered light between laser lock at ${\nu _ + }\;\textrm{and}\;{\nu _ - }$ through the same channel within a very short cycling time, and just like in the Na lidar [34], the detection efficiency of a channel cannot change in the time for a cycle of data collection between ${\nu _ + }\;\textrm{and}\;{\nu _ - }$ to be collected.

## Funding

Strategic Priority Research Program of Chinese Academy of Sciences (XDA17010303).

## Acknowledgment

CYS acknowledges helpful discussions with John Hair and Raul Alvarez concerning their work in the 1990s.

## Disclosures

The authors declare no conflicts of interest.

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