# Basic Equations of Forward SRS in the Atmosphere

The excitation of SRS at vibrational transitions (Av *=* ±1, where *v* is the vibrational quantum number) of molecular nitrogen and oxygen is considered. A radiation source installed on board an aircraft flying at an altitude of *h** _{0}* = 10 km operates along slant paths. The laser radiation is assumed to be monochromatic and spatially coherent, the pulse duration

*t*

*satisfies the inequality*

_{p}*t*

*^*

_{p}*T*

_{2}*,*where

*T*

*is the characteristic relaxation time of molecular vibrations, which corresponds to the stationary SRS conditions. In the parabolic approximation, the system of SRS equations for the slowly varying complex amplitudes*

_{2}*A*

_{n}(r_{±},*z*

*)*of main radiation and the Stokes-anti-Stokes components with allowance for diffraction, linear absorption at the path, electronic Kerr effect, and parametric four-photon interaction effects has the following form [3,51]:

where *E _{n}*(r;

*t*) =

*A*(

^{r}±,z)exp{i(o>

_{n}t —

*k*

*„*

_{z}*z*)} is the complex strength of electric field of the n-th component

*E*

*z);*

_{n}(t_{±},*u*

*= w*

_{n}_{0}+

*nO*

*,*

_{R},*n*= 0, ±1, ±2, ...;

*i*

*the frequency of the Raman transition;*

_{R }*k*

_{n}*= n*

_{g}(*w*

_{n})*w*

_{n}/*c*the absolute value of the wave vector of the component kn

*k*

*„*is the projection of the wave vector k onto the z-axis. Characteristics

_{z}*a*

*and*

_{ab}*n*

*are the linear absorption coefficient and the refractive index of the gas medium, respectively;*

_{g}*g*

*is the stationary Raman gain coefficient connected with the imaginary part of the cubic susceptibility of the medium;*

_{R}*Y*

*n*

*m*

*ip*

*= ^*

_{0}*y[*

*^n*are the coefficients responsible for the condition of conservation of the photon number at molecular transitions (Manley-Rowe relations); n

_{2}the nonlinear part of the refractive index associated with the Kerr effect;

*c* is the speed of light in vacuum; *z* and *r _{±}* are, respectively, the longitudinal and transverse moving coordinates of radiation propagation. The pulse duration of the main radiation is assumed to be short, and the effects of group delay of interacting waves and the distortion of their envelope due to the time dispersion can be neglected.

To determine the phase factors A_{nm}p at the parametric terms in Equation 4.47, it is necessary to take into account the conditions of spatial synchronism of interacting waves, which depend on the dispersion conditions of the medium. Within the framework of the collinear geometry of parametric interaction and the normal dispersion of air described by the Cauchy formula [3]:

where *a* = 2.23 X 10^{-4}; *b* = 4.73 • 10^{-37} Hz^{-2}; *p _{g}* is the air density in kg/m

^{3}, and the frequency is in Hz, it can be easily shown that frequency detuning A

_{nm}p can be determined as follows:

The effect of purely Raman scattering, when the interacting waves are always spatially synchronized along the path *(A _{nm}p* = 0), corresponds to the situation with

*n = l, m = p = n*± 1. In this case, the cascade excitation of the Stokes component occurs. These components propagate simultaneously and collinearly with the initial radiation [48], and generally have the wider angular divergence than the radiation at the main frequency owing to diffraction. All the other admissible combinations of the indices n, m, l,

*p*describe the four-wave parametric generation of combination frequencies. The angular structure of this radiation usually looks like a system of concentric rings with a center at the axis of the main beam and diverges much more strongly in comparison with axial SRS [54].

For quantitative comparison of the contributions of combination and parametric terms in Equation 4.47, the following parameter is usually used [55]:

where *I*_{0}* = cn _{g}A_{0}(z =* 0)

^{2}/8n is the initial peak intensity of the power beam. In fact, this parameter compares the characteristic length

*L*at which the weak first Stokes component is amplified in

_{R}= 1/g_{R}I_{0},*e*times, and the length

*L*= A

_{P}_{100-1}/n, at which the phase factor of the first anti-Stokes component alternates the sign. At the length

*L*and under conditions

_{R},*r) <*1, the interacting waves are always phase- matched and, consequently, the role of the parametric effects in SRS is important. The condition

*r) >*1, contrary, points to the fast change in the phase of parametric terms and to the predominant effect of the Raman interaction.

Using Equations 4.48 and 4.49, we can estimate the parameter *n* for the radiation of three laser sources operating in different spectral ranges (according to data of Reference 48):

At the peak intensity of radiation at the path start *I** _{0}* = 1 kW/cm

^{2}and

*h*

*= 10 km, we have n ~ 10*

_{0}^{3}(OIL), 50 (CO), and 2 X 10

^{5}(UV). As can be seen, in all the cases

*r)*^ 1 and the parametric generation of the SRS can be neglected.

It should be noted that the air density entering into Equation 4.48 indicates the decrease of the air dispersion with an increase of the height above the ground level. Consequently, we can expect the increasing role of the parametric interaction at high altitudes, as was predicted in Reference 51. However, as is shown in the following subsection, the Raman gain coefficient *g _{R}* also changes with height, and its decrease also obeys the barometric formula.

The situation can change dramatically either at an increase of the radiation intensity or at excitation of SRS at rotational sublevels of atmospheric gases (RSRS). In the last case, for molecular nitrogen the frequency shift of the Stokes component is too small (U_{R} ^ 76 cm^{-1}) to decrease significantly the wave detuning |Д_{100-1}| and to increase the length of coherent parametric interaction. In addition, the higher RSRS gain coefficient in comparison with vibrational SRS also favors the decrease of the parameter n. From the practical point of view, for energy conversion from the main radiation into the Stokes components, the alternation of the predominant type of interaction from Raman to parametric leads to the suppression of the exponential gain of the Stokes wave and to its growth by the more slowly, power law [56] up to the almost complete termination of the pump depletion due to SRS [55,57].

So, in view of the obtained estimations and the experimental results [3], we can ignore RSRS and the parametric relation between components and restrict our consideration to only purely Raman interaction.

The influence of the Kerr effect on the laser beam propagation through the atmosphere is determined by the value of the parameter *n _{2}.* In the IR region of laser radiation wavelengths, the typical values of the nonlinear addition n

_{2}for the atmospheric air are Th.2 X 10

^{-23}m

^{2}/W [3]. This gives the effective length of Kerr nonlinearity

*L*(k

_{K}=_{0}n

_{2}I

_{0})

^{-1}

*>*6.5 X 10

^{4}km at

*I*

_{0}*<*10 kW/cm

^{2}, which falls far beyond actual optical paths considered in this chapter. Consequently, SRS is the strongest nonlinear effect within the framework of the considered model of radiation propagation in the middle atmosphere.