Parametrization of optimum filter passbands for rotational Raman temperature measurements

Eva Hammann; Andreas Behrendt

doi:10.1364/OE.23.030767

1. Introduction

Atmospheric temperature measurements with rotational Raman lidar (RRL [1–4],) or non-range-resolved temperature measurements, e.g., temperature measurements for aircraft control using rotational Raman backscatter [5, 6], are based on the fact that the relative intensity of rotational Raman lines is temperature-dependent. Q, being the ratio between two signals extracted from the spectrum with opposite temperature dependence, can be related to absolute temperature after calibration [7].

The backscatter cross section of the strongest rotational Raman lines is about a factor 1/1500 weaker than the one of elastic molecular scattering [3]. Therefore, the solar background plays an important role for the measurement uncertainties of rotational Raman temperature lidar. Daytime measurements are more challenging than nighttime measurements. All of today’s rotational Raman systems use Nd:YAG lasers as light sources and operate either at the second or third harmonic wavelength near 532 nm or 355 nm, respectively. For daytime measurements, UV systems are generally preferred due to the higher backscatter cross section and lower solar background [3, 8]. Examples for rotational Raman lidars operating in the UV and with interference filter based polychromators are the systems of NASA Goddard Space Flight Center [9], of University of Hohenheim [10], of University of Basilicata [11], of Xi’an University [12] and of Hampton University [13]. Rotational Raman lidars at 532 nm reach larger range at night than UV systems due to the higher laser power available at 532 nm compared to 355 nm, higher efficiency in signal separation, and lower atmospheric extinction. Therefore 532 nm is often used for stratospheric measurements [14–16] and ranges of up to 40 km have been reported [15]. Some of the systems at 532 nm are also based on interference filters [7, 9, 10, 14, 16, 17] and some employ double-grating polychromators [18, 19].

The measurement uncertainty depends critically on which parts of the rotational Raman spectrum are extracted. With interference filters, typically not just one but a few Raman lines are within the filter transmission bands. As the central wavelength of an interference filter is related to the angle of incidence, the central wavelength is adjustable by changing the angle. Depending on temperature, lines with high rotational quantum states J are weak and maybe hardly indistinguishable from the background. On the other hand, these weak high-J lines are more sensitive to temperature changes than low-J lines with higher intensity. Therefore, the optimum combination of the receiver passbands of both rotational Raman channels depends on temperature and signal-to-background ratio. The latter depends then on solar elevation angle, receiver transmission, filter bandwidth, filter out-of-band blocking and laser power.

For the low-J channel, the lowest statistical error can be achieved with a filter passband near the laser wavelength. However, a further demand on the rotational Raman channels is sufficiently high blocking of the elastically scattered light. Thus, low transmission at the laser wavelength is required. But a filter passband close to the laser wavelength shows lower blocking than a filter placed further away. In consequence, both these selection criteria are in conflict to each other and a practical compromise must be found.

Elastic signal leakage in the first rotational Raman channel can be corrected [7, 13]. But leakage correction increases the statistical uncertainty and decreases the accuracy because the correction factor cannot be determined exactly. We therefore recommend putting priority on minimizing the systematic error due to leakage. Experimental results prove that this is indeed possible by positioning the filter passband in sufficient distance to the laser wavelength [14, 17]. After minimizing this systematic error, we recommend minimizing the statistical error with the method presented in this paper.

In order to meet the measurement requirements at daytime, the University of Hohenheim (UHOH) RRL was optimized for high temperature measurement performance in daytime in the convective boundary layer [10]. The data of the UHOH RRL have been used for studies on the characterization of transport and optical properties of aerosol particles near their sources [20, 21], on the initiation of convection [22, 23], and on atmospheric stability indices [24, 25]. Simulations for identifying the optimum pair of central wavelengths for one specific system, the RRL of UHOH, and different signal-to-background ratios were presented already in [25]. It was shown that for the given filter characteristics, a configuration for low and high background levels can be identified, which decreased the relative measurement uncertainty by up to 70%. We present here a concept which explains how to find optimum central wavelengths or shifts of the passband for other lidar systems. This concept avoids the tedious effort of repeating the whole simulations. We investigated whether a parametrization is possible which takes into account all or at least some of the relevant parameters (signal-to-background ratio, temperature range of interest, filter bandwidth, filter shape, etc.). The results are useful for lidar applications but also beyond, as the rotational Raman technique can also be applied e.g. for temperature sensors for aircraft control [5, 6].

In section 2, the basics of the rotational Raman technique for temperature measurements are recapitulated and the optimization procedure is described. In section 3, the results are discussed. Section 4 contains a summary.

2. Methods

2.1 Rotational Raman temperature measurements

For lidar applications, elastic scattering by particles and molecules can be employed as well as inelastic scattering by molecules. For inelastic scattering, so-called Raman scattering, the vibrational and/or rotational quantum state of the scattering molecule changes and accordingly the energy of the scattered photon differs from the energy of the incident photon. The rotational energy level E_rot,i(J) of quantum state J for an atmospheric constituent i is given by

E_{r o t, i} (J) = [B_{0, i} J (J + 1) - D_{0, i} J^{2} {(J + 1)}^{2}] h c, J = 0, 1, 2, \dots

with Planck’s constant h and velocity of light c [3]. B₀_,i and D₀_,i are the rotational constant and centrifugal distortion constant for the ground state vibrational level. The selection rule for energy transitions of diatomic molecules like nitrogen and oxygen is J ± 2. As changes of J thus cause both increases and decreases of the energy, the elastic backscatter signal at the excitation frequency is surrounded by a spectrum of rotational Raman lines on both sides. The Stokes branch is found at lower frequencies, corresponding to J + 2, with shifts to the excitation frequency of

Δ ν_{S t, i} (J) = - B_{0, i} 2 (2 J + 3) + D_{0, i} [3 (2 J + 3) + {(2 J + 3)}^{3}], J = 0, 1, 2, \dots .

J is here and in the following the rotational quantum number of the molecule before scattering takes place.

The anti-Stokes branch at higher frequencies, corresponding to J - 2, follows

Δ ν_{A S t, i} (J) = B_{0, i} 2 (2 J - 1) + D_{0, i} [3 (2 J - 1) + {(2 J - 1)}^{3}], J = 2, 3, 4, \dots .

The differential backscatter cross section for a certain rotational quantum state J is temperature-dependent [26] and is given by:

{(\frac{d σ}{d Ω})}_{π}^{R R, i} (J, T) = \frac{112 π^{4}}{15} \frac{g_{i} (J) h c B_{0, i} {(ν_{0} + Δ ν_{i} (J))}^{4} γ_{i}^{2}}{{(2 I_{i} + 1)}^{2} k T} X (J) \exp (- \frac{E_{r o t, i} (J)}{k T})

where g_i is a statistical weight factor depending on the nuclear spin I_i. ν₀ stands for the frequency of the exciting light, γ_i is the anisotropy of the molecular polarizability and k the Boltzmann constant. X(J) differs between the two branches. For the Stokes branch, one must apply

X (J) = \frac{(J + 1) (J + 2)}{2 J + 3}, J = 0, 1, 2, \dots

and for the anti-Stokes branch

X (J) = \frac{J (J - 1)}{2 J - 1}, J = 2, 3, 4, \dots .

Table 1 lists the values used for all constants in Eq. (4) [3, 27]. Only the rotational Raman spectra of nitrogen and oxygen are calculated as these molecules account for more than 95% even for wet air and the contribution of other molecules (mainly water vapor) is negligible [5]. Figure 1 shows an example of the spectrum calculated for two different temperatures.

Table 1. Values used for the constants in Eqs. (1)-(4)

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Fig. 1 Shift of pure rotational Raman lines of nitrogen and oxygen relative to the excitation frequency ν₀ and relative intensities of these lines for temperatures of 180 and 300 K.

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Lidar systems transmit laser pulses into the atmosphere and detect the backscattered light with high temporal resolution. From the run time of the signals, the distance to the scatterer is derived. The lidar equation for inelastic scattering describes the portion P_λR of the transmitted laser light P₀, which is detected from distance z

P_{λ R} (z) = P_{o} \frac{c Δ t}{2} \frac{O (z)}{z^{2}} η_{λ R} A_{T} β_{λ R} (z) \exp [- \int_{0}^{z} [α_{o} (z') + α_{R} (z')] d z'] .

In this equation, Δt is the laser pulse duration, O(z) the overlap function, η_λR the detection efficiency, A_T the telescope area, β_λR the backscatter coefficient, and α₀ and a_R are the extinction coefficient for the excitation wavelength and the Raman wavelength, respectively. For rotational Raman lidar, two rotational Raman signals are used. Typically, the one closer to the excitation wavelength is called P_RR1 and the one further away P_RR2. The ratio of rotational Raman signals is then [26]

Q (T) = \frac{P_{R R 2}}{P_{R R 1}} = \frac{\sum_{O_{2} N_{2}} \sum_{J_{i}} τ_{R R 2} (J_{i}) η_{i} {(\frac{d σ}{d Ω})}_{π}^{R R, i} (J_{i})}{\sum_{O_{2} N_{2}} \sum_{J_{i}} τ_{R R 1} (J {}_{i}) η_{i} {(\frac{d σ}{d Ω})}_{π}^{R R, i} (J_{i})}

with P_RRn for the background-corrected signal in detection channel n, τ_RRn is the transmission for the wavelength of channel n and η_i the number density of the molecule.

Calibration is usually performed through comparison with an independently measured temperature profile as the uncertainties by analyzing the instrumental parameters and calculating a calibration with these are typically larger [28]. There are several approaches to derive temperature T(Q), depending on the number of detected lines and the temperature range. Details and examples for calibration functions can be found in [3]. In the optimization calculations described here, we apply the exact formulas and not the fit determined by calibration.

The statistical error of a temperature measurement with the rotational Raman technique is mainly determined by shot noise [29]. Thus, ΔT_stat can be simulated from the photon numbers by applying Poisson statistics giving [7]

Δ T_{s t a t} = \frac{d T}{d Q} Δ Q = \frac{d T}{d Q} \sqrt{\frac{P_{R R 1} + P_{B 1}}{P_{R R 1}^{2}} + \frac{P_{R R 2} + P_{B 2}}{P_{R R 2}^{2}}}

with P_Bx for the background in rotational Raman channel x . Equation (9) is only valid if the background was determined over a high number of range bins and its statistical error is negligible.

2.2 Simulation concept and parameters

For the simulation, the anti-Stokes lines of nitrogen and oxygen were first calculated for two temperatures in order to approximate the differential in Eq. (9) by

\frac{d T}{d Q} = \frac{T_{1} - T_{2}}{Q_{1} - Q_{2}} .

We found that a suitable difference of the two temperatures in the simulation is 5 K. For much larger temperature ranges, non-linear effects of T(Q) may become significant [3]. Then, the simulated photon count rates in the detection for the first and second rotational Raman channel with the passband centers SPC1 and SPC2 respectively, were calculated using different shapes and widths for the transmission bands and different background levels.

The uncertainties for the different combinations of SPCs were calculated for temperatures between 180 and 300 K, which cover the range of typical temperatures in the troposphere and stratosphere, i.e., those altitudes which can be covered with rotational Raman lidar. We varied the SPCs in such a way that the optimum for all of these temperatures is always found. The relative intensities and energy shifts of the rotational Raman spectrum are the same independent of the frequency of the excitation radiation (see Fig. 1). Consequently, we express SPCs and widths in units of wavenumbers, in order to discuss the results in a generalized way independent of the laser wavelength. SPC1 was varied from 0 to 74 cm⁻¹ and SPC2 from 66 cm⁻¹ to 194 cm⁻¹, both in 0.8 cm⁻¹ steps. The corresponding filter parameters in wavelength units for the second to fourth harmonic of Nd:YAG lasers are listed in Table 2. With these ranges, a total of 93 and 161 values were used in the simulation for SPC1 and SPC2, respectively. For each of the value pairs, the statistical uncertainty was calculated resulting in an array of 93 ˟ 161 = 14 973 values for each combination of temperature, filter shape and bandwidth as well as background level. As the aim of this study is to find optimum filter parameters for a given configuration, only the relative errors d_rT in each array have then been investigated further, which means that all the values in each array were divided by the smallest value of the array.

Table 2. Wavenumber ranges covered by our simulations with corresponding wavelength ranges for the second to fourth harmonic of a Nd:YAG laser. SPC1 and SPC2 are the shifts of the passband centers of the filters for the low-J and high-J rotational Raman channel, respectively, relative to the frequency of the initial radiation.

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The bandwidths of the filters, which extract the two rotational Raman signals, must fulfil the following criteria. First, the passbands of both channels should not overlap. Second, they must not include the frequency of the laser. Furthermore, both signals should show an as large as possible sensitivity to temperature. This sensitivity should be opposite for the two channels. Figure 2 shows as example the temperature sensitivity of the extracted signals

\frac{d P}{d T} \approx \frac{P (T_{2}) - P (T_{1})}{T_{2} - T_{1}}

as a function of the filter bandwidth for rectangular shaped filter transmission curves at 270 K. One can see that for the RR1 channel with negative temperature sensitivity (decreasing signal intensity with increasing temperature) only bandwidths smaller than about 100 cm⁻¹ avoid overlap with the excitation frequency. But it should be noted that typically a very high blocking (transmission less than 10⁻⁷) is needed at the excitation frequency [14] in order to avoid contamination of the temperature measurements by elastic backscattering of particles in the atmosphere. These requirements are relaxed for 355 nm compared to 532 nm because of the wavelength dependencies for particle scattering (typically 1/λ) and Raman scattering (1/λ⁴). Thus, in practice the optimum SPC1 cannot be used for RR1 and a larger shift is necessary.

Fig. 2 Temperature sensitivity dP/dT of the extracted rotational Raman signals depending on the wavenumber shift of the center of the filter passband and its bandwidth. Only values for which the excitation frequency is outside of the transmission band are shown.

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But, it is not sufficient to study only temperature sensitivity when selecting optimum filter parameters, the signal intensities relative to the background are also important for the measurement uncertainty like already outlined in the introduction.

A full width at half maximum (FWHM) of 23.8 cm⁻¹ for the first and 39.6 cm⁻¹ for the second channel was used in the first simulations. These bandwidth values were successfully tested in previous experiments [10, 25]. These widths are located in the lower region of Fig. 2, where it is indeed possible to extract signals with opposite temperature sensitivity at 270 K.

The transmission curves of multi-cavity interference filters have Gaussian-like shapes. Thus, we used for the simulation idealized Gaussian curves which were compared with real filter curves. The filter of the UHOH RRL (manufactured by Materion Precision Optics and Thin Film Coating Inc.) has an optical thickness of 6 at the excitation wavelength and a peak transmission of more than 0.5. (The additional blocking at the laser wavelength by one order of magnitude is achieved by a sequential mount of the rotational Raman channels and the elastic channel [10, 25].) The filter transmission curve F can be approximated by the following equation:

F (ν) = A \exp [\frac{- B {(ν - ν_{0})}^{4}}{F W H M}]

Parameters A and B are chosen to model the actual curve like described in [10]. In section 2.3 the impact of other transmission curves is discussed. Equation (12) describes the filter passband for a beam divergence of 0°. In practice, however, the beam divergence is not exactly 0°. Therefore a beam divergence of 0.5° was used in the simulation and later it was studied how other values for the divergence influence the results.

For the determination of optimum SPCs, the value P_Bx for the background is crucial. It is given here in relation to the intensity $P_{J}^{\max}$ of the strongest rotational Raman line in order to generalize the discussion. The background takes the filter bandwidth $Δ ν_{F W H M}$ into account by a factor which is 1 for a filter bandwidth of 8 cm⁻¹ (corresponding to 0.10 nm for an excitation wavelength of 355 nm and 0.23 nm for 532 nm) and is correspondingly adjusted for other bandwidths via

P_{B} = S (z) \frac{Δ ν_{F W H M}}{8 {cm}^{- 1}} P_{J}^{\max} (z)

The factor S is then varied for different background levels. At night, values between S = 0 and S = 0.1 for lower altitudes (up to 2-3 km) are typical for the UHOH RRL. Figure 3 is the result of a derivation of S from experimental data collected with the UHOH RRL (200 mJ pulse energy, laser wavelength of 354.8 nm). At daytime, S is larger and increases quickly with altitude as the backscatter signal decreases exponentially. Typically, S is larger than 1 for altitudes above 2 km at noon. The result of the simulation does not change significantly with higher values of S [30]. Therefore, S = 1.0 can be used as a worst-case value at daytime. In the simulation, S was thus varied between 0.0 and 1.0 – also, as larger values will result in very large uncertainties of the temperature measurements anyway and are therefore outside the preferable set of parameters.

Fig. 3 Background-level S for the two rotational Raman signals RR1 and RR2 at different times of the day (local noon is at 11:40 UTC). The profiles were measured on May 19, 2013 under cloud-free conditions. Total incoming short wave radiation measured by a nearby EC station was 0 W/m² at 0 UTC, 128 W/m² at 5 UTC and 890 W/m² at 11 UTC.

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P_J^max can be estimated from the specific filter passband and the relative intensity of the lines within the passband in comparison with the intensity of the strongest anti-Stokes Raman line. The telescope diameter and further detection efficiencies have no influence on the ratio, as they influence both the signal and the background in the same way.

How can the results presented here now be transferred to other systems? The most relevant parameter is laser pulse energy E_pulse. For systems with excitation wavelengths near 355 nm and the same filter passband widths as ours (24 cm⁻¹ and 40 cm⁻¹ for the first and second RR channel, respectively), one will find S´ values which scale with the one shown in Fig. 3 with the pulse power of the other system E’_pulse relative to the pulse power E_pulse used here according to

S^{'} = \frac{E_{p u l s e}}{{E^{'}}_{p u l s e}} S

At 532 nm, the daylight background per nm bandwidth is a factor of about 1.6 larger than at 355 nm (see Appendix). Thus S is a factor of (1.5)²x 1.6 = 3.6 larger for the same filter passband widths in wavenumber units. The simulation was performed like described in section 2.2 resulting in one array for each pair of temperature and background level. Figure 4 shows as an example the simulation results for 270 K and S = 0, i.e. no background radiance. The optimum pair of SPC1 and SPC2 (with d_rT = 1.0, minimum error per each array) is found at 23 cm⁻¹ and 132 cm⁻¹, respectively. It can be seen that a small shift of 8 cm⁻¹ (corresponding to 0.1 nm at 355 nm) may result in significant increase of the measurement uncertainty.

Fig. 4 Example for the simulated temperature measurement uncertainties d_rT (relative to the minimum uncertainty of the array shown in black) depending on SPC1 and SPC2, respectively. This case is for a temperature of 270 K and zero background (S = 0). The star denotes the largest possible value for SPC1 for which d_rT ≤ 1.2; this point is of practical importance as it marks a setting at which the required blocking of the elastically scattered light at the excitation frequency can be reached.

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A limiting factor for SPC1 is the minimum shift which is sufficient to block the elastic backscattered signal completely. Without this constraint, the optimum is approximately at 25 cm⁻¹ for SPC1, which can be found indeed in Fig. 2 in the area of negative correlation. However, current interference filters require a larger shift to the elastic line to guarantee sufficient blocking. Therefore we decided to study not only the parametrization of the shifts for the case d_rT = 1.0, but to extend the analysis for the SPC pair which provides a relative uncertainty of 1.2 with the largest possible value for SPC1. There are already interference filter which show the required blocking with the resulting SPCs. The performance will be even better if it is possible to choose a smaller value for SPC1, i.e. a passband which is nearer to the frequency of the laser. If a smaller SPC1is chosen, the already determined SPC2 can be applied without significant increase of the measurement uncertainty.

Figure 5 shows the simulation results for S = 0 and temperatures between 180 and 300 K. The wavenumber pairs, which result in d_rT = 1.2 are marked as spheres to visualize the temperature dependency (for a temperature of 270 K, they correspond to the outer boundary of the yellow area of Fig. 4).

Fig. 5 Spheres mark pairs SPC1 and SPC2 for which the relative temperature measurement uncertainty d_rT = 1.2. These simulation results are for zero background (S = 0). It is interesting to note that the range of both SPC1 and SPC2 for d_rT ≤ 1.2 increases with increasing temperature. Furthermore, theses ranges shift to larger values. Gray areas mark the projections of the data points to the parameter planes.

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The optimum wavenumber shifts decrease with decreasing temperature due to the weakening high-J lines. It is also noteworthy that the area within the spheres becomes smaller with decreasing temperatures. The reason for this characteristic is that the rotational Raman spectrum becomes broader with increasing temperature. In consequence, it is more demanding to optimize for measurements at lower than at higher atmospheric temperatures.

2.3 Influence of temperature and background

To clarify the dependency, Figs. (6a) and 6(b) show the minimum values of the temperature measurement uncertainty of given SPC1 and SPC2, respectively. For all values of the other SPC (if not only d_rT = 1.2 would be shown in Fig. 5 but all values, Fig. 6a and b would correspond to a projection of the minimum data of Fig. 5 to the T-SPC1 plane and T-SPC2-plane, respectively). Interestingly, the optimum SPC1 changes only by 2 cm⁻¹ over the whole temperature range. The SPC1 range with d_rT < 1.2 becomes larger with higher temperatures. It increases from 42 cm⁻¹ at 180 K to 57 cm⁻¹ at 300 K. It can be concluded that the optimum SPC1 is not very sensitive to temperature and that the same values can be applied in practice for temperature ranges of several tenths of K. On the other hand, SPC2 shows for the optimum, as well as for the range with d_rT <1.2, a quite strong temperature dependency. The optimum position increases by more than 25 cm⁻¹ between 180 and 300 K and the d_rT = 1.2 isolines follow almost parallel. Here the effect of higher relative intensity of high-J lines with higher temperatures becomes significant.

Fig. 6 (a) Minimum values of d_rT for a combination of SPC1 and temperature T for all values of SPC2 for zero background (S = 0). (b) Same as (a) but for pairs of SPC2 and T for all values of SPC1. The need of taking the temperature measurement range of interest into account for the filter selection becomes evident.

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As was done for 270 K above, we investigate now the maximum possible SPC1 for uncertainties d_rT < 1.2 and non-zero background. Figure 7 shows the results for different background levels and temperatures. Like expected, the SPCs decrease the lower the temperature. The optimum shifts change nearly linearly by 15 cm⁻¹ for temperatures between 180 and 300 K (with fixed background level) corresponding to a slope of 0.125 cm⁻¹/K.

Fig. 7 SPC1 and SPC2 from the excitation wavenumber for d_rT = 1.2 (marked with the star in Fig. 4 for T = 270 K and S = 0). Points are results of the simulation; thin lines show the best linear fit for each case. These data form the input for the suggested parametrization (see Table 3). Error bars show the uncertainties due to the discretization steps used for SPC1 and SPC2.

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Furthermore, the shifts are smaller the higher the background. When changing the background, the whole line shifts approximately by a constant value while the slope is nearly preserved. This shift is 2.9 cm⁻¹, 4.5 cm⁻¹, and 5.5 cm⁻¹ from zero background to S = 0.1, 0.5 and 1.0, respectively. Same applies for SPC2, where the shift is about 30 cm⁻¹ over the whole temperature range. Here the slope is 0.250 cm⁻¹/K with shifts of 11.5 cm⁻¹, 20 cm⁻¹, and 23.5 cm⁻¹ for S = 0.1, 0.5 and 1.0, respectively.

The next step is to investigate how large the temperature dependence of the shift of SPC1 and SPC2 is. Therefore the SPC difference d_rSPC_{S, 240K} to the SPCs for 240 K and the same background were calculated with

d_{r} S P C_{S, 240 K} (T) = \frac{S P C (S = 0, T) - S P C (S = 0, 240 K)}{S P C (S = 0, 240 K)}

The results for S = 0 and S = 1 are shown in Fig. 8a. Of course, d_rSPC_{S, 240K} becomes larger the larger the temperature difference to the reference value of 240 K. Interestingly, all curves are almost linear and nearly coincide. d_rSPC_S,240K varies by about 30% over the temperature range of 180 to 300 K giving a mean change of 0.25%/K.

Fig. 8 (a) d_rSPC_{S, 240K} for SPC1 and SPC2 with S = 0 and S = 1 respectively (see Eq. (15)).(b) Same as (a), but at same temperatures and relative to zero background (S = 0).

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Figure 8b shows in a similar fashion the dependency on the background in relative values d_rSPC_S _{= 0, T} to S = 0 at the same temperature following

d_{r} S P C_{S = 0, T} (T) = \frac{S P C (S, T) - S P C (S = 0, T)}{S P C (S = 0, T)}

d_rSPC_{S = 0, T} is smaller than about 10% for SPC1 and 20% for SPC2.

In order to illustrate the significance of these results, it is interesting to investigate how much the temperature measurement uncertainties d_rT increase if these dependencies of the optimum SPCs on temperature and background are neglected. In comparison with Fig. 5 and Fig. 6, the same value of d_rSPC_{S, 240K} or d_rSPC_{S = 0,} _T leads to larger increases in d_rT the smaller T. This is caused by the narrower rotational Raman spectrum. The same is true for both parameters if S increases because the region of the rotational Raman spectrum, which differs significantly from the background, becomes narrower with increasing background. Consequently, the largest increase of d_rT for given values of d_rSPC_{S, 240K} or d_rSPC_S _{= 0,}_T is found in our study at 180 K and S = 1. Here, – as worst case – d_rT increases drastically by about 170% if SPC1 and SCP2 have been optimized for 240 K and S = 0, but measurement are made at 180 K and S = 1 with this pair.

2.4 Influence of filter transmission curve and passband widths

There are rotational Raman lidars which detect only single Raman lines (e.g [31].) but this is not the approach here as the error is definitely larger than using several lines – also for high background, unless the filter bandwidth are narrower than the spectral distance of the rotational Raman lines. As this cannot be realized with interference filters now and in the foreseeable future (and since the maximum transmission becomes also smaller the smaller the bandwidth), we focus the discussions here on filter passbands, which include more than one line.

In section 2.2 and 2.3, the simulations were based on a shape of the filter transmission curve SFT similar to the one of the filters used in the UHOH RRL [25]. As the transmitted passband shifts when changing the angle of incidence on the filter (see [14]), the resulting shape of the filter transmission depends on the divergence of the light passing the filter: The larger the divergence, the broader the transmission band becomes. Figure 9 shows different forms of the transmission band of filter 1.

Fig. 9 Filter transmission curves used in the simulation to study the effect of their shape on the optimum filter setting. SFT1 is a Gaussion (see Eq. (12)). SFT2 has the same width as SFT1 but a slightly narrower peak and broader tails. SFT3 and SFT4 are for the same filter as SFT1 but with smaller and higher beam divergence in the receiver, respectively.

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SFT1 corresponds to the UHOH RRL filters and was used for the simulations above. As the beam divergence in the receiver is about 0.5°, this value was used to derive realistic shapes of the transmission bands (see section 2.2). SFT2 has the same width but a narrower peak and broader tails. This transmission curves corresponds to an older filter we had in use. SFT3 and SFT4 are for beam divergences of 0.1° and 1.0°, respectively, and thus narrower and broader than SFT1. The passband of filter 2 was varied in the same way as filter 1 in the simulations. The difference in optimum SPC for different transmission curve shapes d_rSPC_S,T,SFT1 is calculated (similar to Eq. (15) and (16)) with

d_{r} S P C_{S, T, S F T 1} (T, S) = \frac{S P C (S, T, S F T) - S P C (S, T, S F T 1)}{S P C (S, T, S F T 1)}

Figure 10 shows how SFT2, SFT3 and SFT4 change the results shown for SFT1 in Fig. 7. Temperatures of 200, 250 and 300 K and background value S = 0 and S = 1.0 were chosen for this test. SPC1 varies within −5.5 and + 1.8%, when comparing with the values for same T and S. SPC2 changes only within −2.2 and 1.3%. These changes are very small in comparison to the changes when T and S are varied for the same filter SFT. In addition to these variations, we studied also the effects of keeping SFT1 but changing the bandwidth (see Eq. (12)). For this, we also found small effects (changes of less than ± 6%). Thus, we conclude that the results obtained with SFT1 and discussed in section 2.3 are representative also for other realistic SFTs and widths which are similar (Gaussian-like, effective widths between 15 and 55 cm⁻¹ corresponding to 0.2 and 0.7 nm at 355 nm). As a result it is recommended to focus on expected temperature and background conditions for determining optimum positions of the filter passbands.

Fig. 10 (a) Relative change d_rSPC_S,T,SFT1 of selected SPC1s shown in Fig. 7a if the SFT2 to SFT4 are used instead of SFT1. (b) Same as (a) but for SPC2 and Fig. 7b.

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3. Resulting parametrization for passband shifts

As discussed in the previous sections, a complete parametrization of the filter parameters taking all effects into account is difficult and also not required in practice. An optimum filter setting can be determined with sufficient accuracy when focusing on the temperature range of interest and on the background level. Therefore, we have investigated whether the presented simulation results can be parametrized with respect to temperature and background only.

In section 2.3, it was shown that the optimum values of SPC1 and SPC2 can be calculated with the linear approximation

S P C x (T, S) = m_{S P C x} T + b_{S P C x} (S) with x = 1, 2

In this equation m_SPCx is the slope depending on the d_RT and the SPCx temperature dependency and b_SPCx is a shift which depends on the background. Some parameters were already given in section 2.3 for d_rT = 1.2. Table 3 lists these parameters and adds the corresponding values for d_rT = 1.0. As discussed above, m_SPCx does not depend on the background. The values of b_SPCx are given for selected values of S in Table 3. For other values of S, an approximation formula was derived (see Fig. (11)). The values were retrieved by fitting SPCx(T) and determining of the mean slope of the linear fits and the respective background-related shift.

Table 3. Parameters for the suggested parametrization which determines the optimum filter central passbands depending on background S, for d_rT = 1.0 and d_rT = 1.2. m_SPCx: Slope of the linear fit, b_SPCx: y-intercept of the linear fit.

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Fig. 11 Mean y-intercept b_SPCx of the suggested linear parametrization for SPC1 and SPC2, respectively, versus background parameter S.

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The fit functions are for SPC1

b_{S P C 1} (S) = 5.52 ​ {cm}^{- 1} \exp (\frac{S}{- 0.25}) + 19.59 {cm}^{- 1}

and for SPC2

b_{S P C 2} (S) = 28.20 {cm}^{- 1} \exp (\frac{S}{- 0.36}) + 41.28 {cm}^{- 1}

In consequence, one obtains the following parametrizations for the passband centers SPC1 of the low-J rotational Raman channel, for which the measurement error is not more than 20% higher than the optimum (d_rT = 1.2) and required blocking of the elastic backscatter is feasible:

S P C 1 (T, S) = 0.120 \frac{{cm}^{- 1}}{K} T + 5.52 ​ {cm}^{- 1} \exp (\frac{S}{- 0.25}) + 19.59 {cm}^{- 1}

The corresponding equation for SPC2, the passband center of the high-J channel for d_rT = 1.2, reads

S P C 2 (T, S) = 0.238 \frac{{cm}^{- 1}}{K} T + 8.20 {cm}^{- 1} \exp (\frac{S}{- 0.36}) + 41.28 {cm}^{- 1} .

The relative fit error (RFE) between the exact SPCs determined by detailed simulation (SPC_Sim) and by the parametrized equations (SPC_Fit) were calculated for both approaches with

R F E = \frac{S P C_{F i t} - S P C_{S i m}}{S P C_{S i m}} .

The results for the linear equation are shown in Fig. 12. The fit error RFE is smaller than ± 2% for temperatures between 200 and 300 K. For 180 K, the error is larger but still smaller than ± 4%. A second-order polynomial fit would decrease these errors only marginally by up to 1% (not shown).

Fig. 12 Performance of the suggested parametrization. (a) Relative fit errors (RFE) for SPC1 for temperatures between 180 and 300 K and background parameters S from 0 to 1. (b) Same as (a) but for SPC2. All deviations are smaller than 2% with the exception of temperatures lower than 200 K for which they are smaller than 4%.

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What are the related changes in the measurement uncertainty d_rT due to the fit errors RFE? A difference of 2% in wavenumber shift results in 1 cm⁻¹ for SPC1 of 50 cm⁻¹ (corresponding to approximately 0.015 nm for a laser wavelength of 354.83 nm) relative uncertainty values d_rT change by only 0.9% for this shift of 1 cm⁻¹. For SPC2, e.g., 2% change of 120 cm⁻¹ correspond to 2.4 cm⁻¹. For an excitation wavelength of 354.83 nm this results in a shift of 0.03 nm and in a change in relative uncertainty d_rT of only 0.2%. In summary, the errors of the parametrization are indeed small.

4. Conclusions

The statistical measurement uncertainty of atmospheric temperature measurements with the rotational Raman technique depends critically on the choice of the filter passbands for the two rotational Raman channels. We have calculated the relative temperature measurement uncertainties for a range of suitable frequency shifts of the passband centers relative to the excitation frequency for temperatures between 180 and 300 K. Furthermore, different realistic values for the background have been investigated as well different widths and shapes of the passbands.

We found that the effects of differences between realistic filter transmission curves and bandwidths are small (only changes the SPCs by 2 to 5.5%). Consequently, we suggest to neglect these dependencies and to focus the optimization on temperature and background condition because both are responsible for much larger differences.

Then, we have investigated the feasibility of a parametrization for the passband centers, which takes temperature and background into account. We found that two simple parametrizations, one for the SPC of each rotational Raman channel, approximate the optimum shifts well (only differences <2% between 200 and 300 K; <4% at 180 K). We suggest using a linear function for the temperature dependency. The crossings with the y-axis depend on the background level. We provide the fit parameters not only for the optimum pair of passband centers but also for the pair with largest shift of the low-J rotational Raman channel for which the measurement error is not more than 20% higher than the optimum. This information is important, because, in practice, the optimum setting cannot be chosen for these filters as it is otherwise too close to the excitation frequency and thus the blocking of the elastic backscatter light is not sufficient.

We conclude that quite a simple parametrization is possible and it is thus not necessary to to repeat tedious optimization calculations for each individual rotational Raman receiver. The parameters of the suggested parametrization are given for temperatures between 180 and 300 K and different relative values of the background. These conditions cover the practical range of atmospheric temperature measurements with the rotational Raman technique for both lidar and non-range-resolved measurements. The suggested parametrization can be used to adapt existing receivers to different temperature and background conditions and has the potential to reduce the statistical measurement uncertainties significantly.

Appendix

A.1 Background values

The background count rate P_B is related to the solar radiance Ψ [32]

P {}_{B}= η_{λ R} A_{\det} Ψ ω_{r e c} λ_{R R}

with the detection efficiency k_RR, the telescope surface A_det, the angle area of view ω_rec and the transmission of the filter λ_RR. Most of the factors cancel in comparison with the lidar equation (Eq. (7)), as they are equal for the backscattered signal. It follows for a range bin z in the full overlap region

Ψ = S \frac{P_{0}}{8 {cm}^{- 1}} \frac{c Δ t}{2} β_{λ J \max} \exp [- \int_{0}^{z} (α_{λ_{0}} (z') + α_{R} (z')) d z']

Solar radiance Ψ is approximately 250 mW/(m² nm sr) for 355 nm and 400 mW/(m² nm sr) for 532 nm for clear sky and up to a factor 1.25 higher in case of clouds. The extinction and backscatter coefficients of 355 nm and 532 nm can be estimated with [3]

P_{R R} (532 nm, J) = 0.6 {(\frac{τ (532 nm) (z)}{τ (355 nm) (z)})}^{2} P_{R R} (355 nm, J)

with the optical transmission τ of both wavelengths at a certain altitude range. Therefore one should solve for S´ for the case of high daytime background (250 mW)

S^{'} = S \frac{200 mJ}{E_{0 x}} \frac{P_{R R} (x)}{P_{R R} (355 nm)} \frac{250 mW}{Ψ_{x}}

with the values of the laser system x and the level of expected radiance.

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15. M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and Rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004). [CrossRef]

16. P. Achtert, M. Khaplanov, F. Khosrawi, and J. Gumbel, “Pure rotational-Raman channels of the Esrange lidar for temperature and particle extinction measurements in the troposphere and lower stratosphere,” Atmos. Meas. Tech. 6, 91–98 (2013).

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Molecule	B₀ (cm⁻¹)	D₀ (cm⁻¹)	g_J (J odd)	g_J (J even)	I	γ² (10⁻⁴⁸ cm⁶)
N₂	1.989500	5.48x10⁻⁶	6	3	1	0.509
O₂	1.437682	4.85x10⁻⁶	0	1	0	1.27

	Wavenumber (cm⁻¹)	For 532.0 nm (nm)	For 354.8 nm (nm)	For 266.0 nm (nm)
SPC1 max	74.0	529.91	353.90	265.48
SPC1 min	0.0	532.00	354.80	266.00
SPC2 max	194.0	526.60	352.40	264.63
SPC2 min	66.0	530.14	354.00	265.53
Step size	0.8	0.022	0.010	0.0055

	m_SPCx (cm⁻¹/K)	b_SPCx (cm⁻¹)
	all S	S = 0	S = 0.1	S = 0.5	S = 1
SPC1, d_rT = 1.2	0.120	25.5	22.6	21.0	20.0
SPC2, d_rT = 1.2	0.238	72.0	60.5	52.0	48.5
SPC1, d_rT = 1.0	0.051	10.0	11.0	12.0	12.5
SPC2, d_rT = 1.0	0.223	71.0	61.5	53.0	49.5

Molecule	B₀ (cm⁻¹)	D₀ (cm⁻¹)	g_J (J odd)	g_J (J even)	I	γ² (10⁻⁴⁸ cm⁶)
N₂	1.989500	5.48x10⁻⁶	6	3	1	0.509
O₂	1.437682	4.85x10⁻⁶	0	1	0	1.27

	Wavenumber (cm⁻¹)	For 532.0 nm (nm)	For 354.8 nm (nm)	For 266.0 nm (nm)
SPC1 max	74.0	529.91	353.90	265.48
SPC1 min	0.0	532.00	354.80	266.00
SPC2 max	194.0	526.60	352.40	264.63
SPC2 min	66.0	530.14	354.00	265.53
Step size	0.8	0.022	0.010	0.0055

Parametrization of optimum filter passbands for rotational Raman temperature measurements

Abstract

1. Introduction

2. Methods

2.1 Rotational Raman temperature measurements

2.2 Simulation concept and parameters

2.3 Influence of temperature and background

2.4 Influence of filter transmission curve and passband widths

3. Resulting parametrization for passband shifts

4. Conclusions

Appendix

A.1 Background values

References and links

Cited By

Figures (12)

Tables (3)

Equations (27)

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