Lidar sounding of iodine molecules in atmosphere
It was shown in the papers [1, 2] that the heavy isotope of iodine 131I2 can serve as the indicator of radioactive contamination. This indicator is important during the evaluation of structure of process gases and during the monitoring of atmosphere around nuclear power plants (NPP) or other radiochemical enterprises. It is stipulated by the interest in laser methods of measurement of molecular iodine concentration in gas media. Remote measurement of iodine molecule concentration on a real time basis is a complex task. In the papers [2, 3], the estimations of potential capabilities of the method of differential absorption were performed [1, 4, 5]. The assumption was made that the line of laser generation can be described by delta function, therefore its spectral width does not influence on sounding results. However, radiation spectrums of actual lasers used in lidar measurements have finite width. In the paper  on the basis of numerical computations of the lidar equation for differential absorption and scattering by gas molecules in atmosphere using the genetic algorithm it was shown that it is necessary to take into account the finite width of laser generation line in order to determine very low concentrations of detectable molecules in medium.
The absorption bands of iodine molecules within optical range, which correspond to the transition from the ground state X to the first electronic state B, occupy the interval from 499 to 670 nm. Authors of the paper  recorded approximately 47 000 absorption lines within this range, and the wave numbers of about 23 000 lines are given in their atlas. Nowadays, these numbers are significantly higher. The results of studies carried out previously [8–10] allow assuming that in order to detect the concentrations of molecular iodine of about 1010 cm–3 in atmosphere it is more preferable to use the methods of differential absorption. In such case the effective cross-section of iodine molecule absorption has the highest value in comparison with the cross-section of fluorescence taking into account quenching and cross-section of elastic scattering . Using the attenuation of laser radiation with the relevant selection of radiation wavelength it is possible to develop the sensitive method of measurement of iodine molecule concentration in atmosphere .
The lidar for differential absorption and scattering implies the use of two-wave emitter: one wavelength of laser radiation gets into the center of absorption band of iodine molecules and other wavelength is outside this band. In our variant, the laser radiation at the second harmonic of laser based on YAG: Nd at the wavelength of 532 nm (or frequency = 5.639 THz) is used for sounding of iodine molecules; it gets into the absorption band of iodine molecules within the wavelength range of 499–670 nm with the maximum about 645 nm (or frequencies 4.478–6.012 THz with the maximum about 5.245 THz and half-width G3 = 767 GHz). Laser radiation outside the iodine absorption band should be given at the wavelength longer than 589.5 nm, which corresponds to the maximum of fluorescence of I2 molecules . The laser radiation of the fundamental harmonic of laser based on YAG: Nd at the wavelength of 1064 nm (or frequency = 2.8195 THz) was selected as the reference channel in the scheme of differential absorption. Let us consider the optical layout of such lidar for differential absorption and scattering (Fig. 1).
The lidar transmitting system consists of two lasers 8, the energy of the first laser radiation in the pulse with duration of 10 ns at the wavelength of 532 nm () is equal to 25 mJ, and the energy of the second laser radiation is equal to 40 mJ at the wavelength of 1064 nm (). The area of collecting aperture of telescope 10 is S0 = 0.125 m 2. The intensity of laser radiation at the wavelength of 532 nm, which propagates through the atmosphere layer with iodine molecules of set concentration there and back, is decreased at the expense of absorption in iodine [3, 7], and then it is directed to photodetector 1 through glass light filter 2.
LIDAR EQUATION OF DIFFERENTIAL ABSORPTION AND SCATTERING
The optical axes of lasers and receiving telescope are directed along the axis Z. As it was previously , the directivity diagrams of emitter θL and collecting telescope θT are low θL ≈ θT ≈ 10–4–10–3, therefore the solid angle of divergence of laser radiation will be determined as ΩL = πθL2 and the field of view of receiving telescope – ΩT = πθT2.
Let us designate the optical characteristics of atmosphere at sounding route z through the attenuation coefficient , and the properties of topographic target – through the reflection coefficient or cumulative coefficient of Mie elastic scattering and Rayleigh molecular scattering . Each of two lasers of lidar transmitting system shall be characterized by the power of laser radiation P1 or P2 sent into atmosphere with the laser pulse duration of or , and the line of laser radiation shall be considered as Gaussian line with maximum at the frequency or and half-widths or . Then, the power of laser radiation P1 (or P2) at lidar photodetector can be represented in accordance with  integrating at the full width of generation line from () to () in the form:
where T (ν, z) is the transmission at the frequency of route section from lidar to studied volume, which is equal to the following expression in accordance with 
and is the coefficient of attenuation of laser radiation in atmosphere; – is the spectral coefficient of transmission of lidar receiving system [1, 11] or its instrument function; is the coefficient of backward scattering of topographic target  and G (z) is lidar geometric function [4, 11], which is located within the limits 0 < G (z) < 1.
Using the formula (2) in the expression (1) we will finally obtain the lidar equation for elastic scattering in backward direction for laser generation line with finite half-width in the following form:
(3)where the lidar constant is introduced.
Information on iodine molecule concentration is contained in cofactor in the expression (2), and the attenuation coefficient in atmosphere is determined on the basis of correlation in the form [1, 10]
here the first summand is the attenuation coefficient of atmosphere at the laser radiation wavelength after deduction of studied molecules, and the second summand is the product of cross-section of resonance absorption of iodine molecules and their concentration.
In order to implement the method of differential absorption and scattering , let us take two lidar equations of the type (3) for two frequencies of laser radiation and . Let us believe that the second wavelength is outside the absorption band of iodine molecule; let us substitute in the first equation for the expression (4) and divide one expression by another one. As a result, we will obtain the equation for the most general case of differential absorption and scattering with the assumption on the difference of all cofactors depending on laser radiation frequency as in the papers [10, 11].
Let us approximate the instrument function by Gaussian curve and contour of iodine molecule absorption band – by Lorentz function . Let us believe that atmosphere is homogeneous and coefficient of laser radiation attenuation in atmosphere is the function of frequency only .
Using the error integral erf (Γ)  and having proceeded to the optical density, as previously in the paper , we can write the final equation in the form:
SOLUTION OF LIDAR EQUATION
Let us consider the parameters of this task for our experimental case. Let us believe that for our lidar G1 (z) = G2 (z), and reflection coefficients are approximately identical for both channels and equal to 0.15 – for diffuse target and 10–7 – for total elastic aerosol scattering in atmosphere . The energy E = τP in the pulse with duration of 10 ns of laser radiation at the wavelength of 532 nm () is equal to 25 mJ, at the wavelength of 1064 nm () – 40 mJ. The half-width of generation line Г2 will vary within the range of 1–5 GHz and Г1 – within the range of 2–10 GHz respectively. Let us believe that the half-width of instrument function is higher by order than the half-width of laser generation line, and this assumption is quite acceptable for industrial lasers and interference light filters used in the capacity of lidar spectrum analyzers . Let us assume that it is equal to 100 GHz. The values of coefficients of attenuation in atmosphere were taken from the paper  and are equal to 0.16 km–1, 0.12 km–1 respectively. The maximum value of absorption cross-section at the wavelength of 532 nm for iodine absorption band is = 4.6 10–18 cm 2, and half-width of this band – Г3 = 767 GHz  according to the data given in the paper .
Let us perform the numerical solution of equation (5) for the values of half-widths 1 within the range of 1–5 GHz and 2 – with the values of 2–10 GHz, sounding distances within the range of 100–5 000 m, range of concentrations from 1010 cm–3 to 1017 cm–3 and remaining abovementioned parameters of the task. Results of the solution of equation (5) are given in Fig. 2 for aforementioned parameters, range of concentrations of 1010–1013 cm–3 and sounding distance up to 5 km.
This restriction by the values of concentrations and distances is associated with the fact that the method of spectroscopy of differential absorption [11, 13] has lower and upper limits by the range of possible values of products of concentrations and layer thicknesses, which are determined on the basis of lidar optical layout and photodetector.
For the values of optical density D, which are higher than 4.6, the calculations were not performed because the photodetector with dynamic range of 104 was selected in accordance with data in the papers [1, 10].
Results of the solution of equation (5) are given in Fig. 3 for the same case as in Fig. 2 but for different values (half-width 1 = 1 and 5 GHz and two values of half-width 2 = 2 and 10 GHz). Analysis of graphical representation of solution (Fig. 3) showed that taking into account the width of laser generation line allows decreasing the value of optical density in calculations. For the concentration of 1010 cm–3 such decrease is noticeable: from 1.66 for the distances of 100 m to 1.13 for the distance of 5 km, and for the concentration of 1012 cm–3 – by 1.6 times for the distance of 100 m and 1.07 times – for the distance of 5 km.
Let us further check obtained results. In order to check the results of numerical simulation experimentally, let us find the value of cross-section of iodine molecule absorption for laboratory lidar of differential absorption. The diagram of dependence of optical density D on concentration of iodine molecules N in the layers with thicknesses of 10 and 20 cm and two values of half-width 1 = 1 and 5 GHz and two values of half-width 2 = 2 and 10 GHz (Fig. 4) reflects the experimental results. Their processing allowed obtaining the value of cross-section of iodine molecules at the wavelength of 532 nm, which is equal to = (4.1±0.5) 10–18 cm 2. It is in agreement with the values obtained in the paper [1, 8] σ = (1.88±0.37) 10–18 cm 2 and in the paper  σ = 4.6 · 10–18 cm 2 to satisfactory extent.
Thus, it was shown for the first time that taking into account the finite width of laser generation line during sounding of iodine molecules in atmosphere by the lidar of differential absorption and scattering noticeably demonstrates the decrease of optical density value with the growth of sounding distance and increase of iodine molecule concentration. Thus, for the concentration of iodine molecules of 1010 cm-3 this decrease is equal to 1.66 times for the distance of 100 m and 1.13 times for the distance of 5 km.