Discussions

The role played by SRS in the propagation of a power beam in the atmosphere is indicated by some parameters, namely, the length of nonlinear interaction at SRS Lr, the length of free diffraction of the initial beam LD, and the path length L.

The effect of SRS on the intensity of the main radiation is usually characterized by the integral gain factor

Under conditions of well-developed effect, when the ratio of the intensity of the first Stokes component I_1 to the pump intensity achieves the level I_1/I0 ^ 1%, the gain factor amounts to ^25 ^ 30, which corresponds to the initial level of the Stokes seed в = (U_1/U0)| и 10_8. Correspondingly, the characteristic length

LR, at which the effect achieves the given level GR, at small values of GR, can be found as LR = GR /((gR )I0). It is obvious that with an increase of the gain factor Gr, the effect of pump depletion should be taken into account, and LR can be determined only numerically.

Another important factor is beam diffraction, which can affect significantly the efficiency of Raman interaction of the main radiation and the Stokes components.

Consider the simplified model of propagation of the power radiation at short paths, when the condition L < LD is valid and the excitation of only the first Stokes component is taken into account. Under these conditions, we can neglect the beam diffraction and obtain the analytical solution of system (4.54) and (4.55). Thus, the initial problem can be reduced to the following form:

If the absorption coefficient is constant all over the path (an = const), the solution of Equation 4.61 is known to be a modified analog of SRS for plane waves [60]:

Here,

The structure of Equation 4.62 allows us to introduce the combined parameter G*, characterizing the degree of SRS manifestation at the optical path

where Tn (L) = exp {— /L an (z)dz} is the linear transmittance of the optical path for radiation with the frequency is peak pump intensity.

Figure 4.5 shows the coefficient of conversion of the main radiation power P0 into power of Raman components PR:

as a function of this parameter for different initial beam profiles (A0 = 1.315 pm, an = 0). Since the super-Gaussian beam has the smallest effective radius (RJyS-gauss) = 0.79), while the ring and Gaussian beams are somewhat wider

Coefficient of SRS conversion of radiation power n as a function of the parameter G* for beams with different initial profiles

Figure 4.5 Coefficient of SRS conversion of radiation power nR as a function of the parameter G* for beams with different initial profiles: Gaussian (1), ring (2), super-Gaussian (3). Calculation without linear absorption and diffraction (A0 — 1.315 pm).

(Re(ring) = 1.2, Re(gauss) = 1), the beam with the plateau-like profile has the higher peak intensity, the initial beam power being the same.

Consequently, SRS in this case progresses faster, which can be seen from Figure 4.5. Correspondingly, the conversion coefficient r/R at fixed G* has the smallest values for beams with shadowing. The maximum achievable level of SRS conversion can be determined from the Manley—Rowe relations and, for the given conditions, is nit™ < 71%. The threshold value of the parameter G*, at which the SRS becomes noticeable, is G1 и 30 ^ 35.

What happens if the condition L < LD is no longer valid and it becomes necessary to take into account the wave diffraction in addition to the nonlinear interaction of waves? This question is answered in Figures 4.6—4.8. The calculation

The same as in Figure 4.5 but for L = 1.05

Figure 4.6 The same as in Figure 4.5 but for L = 1.05.

Power conversion coefficient of the Gaussian beam with neglected diffraction (1), and at L = 0,02 (2); 0,17 (3); 0,72 (4); 1,05 (5)

Figure 4.7 Power conversion coefficient of the Gaussian beam with neglected diffraction (1), and at L = 0,02 (2); 0,17 (3); 0,72 (4); 1,05 (5).

Coefficient n as a function of the dimensionless path length at G* = 50 for the Gaussian (1), ring (2), and super-Gaussian (3) beam profiles

Figure 4.8 Coefficient nR as a function of the dimensionless path length at G* = 50 for the Gaussian (1), ring (2), and super-Gaussian (3) beam profiles.

parameters here are the same as in Figure 4.5, but Equations 4.61 are supplemented with the transverse Laplacian.

Figure 4.6 shows the dependence of the conversion coefficient of the power beam on the combined SRS gain parameter for different initial beam profiles and the dimensionless path length L = L/LD = 1.05.

It can be seen that the general character of the dependence rjR(G*) remains the same, but the relative position of the curves for beams with different transverse profiles becomes different: the SRS threshold G1 and the boundary (on the abscissa) of the developed effect G2 shift, when rjR и nRTx.

This behavior is detailed in Figure 4.7, which shows only the Gaussian beam at different values of L . It follows from Figure 4.7 that while G1 increases mono- tonically with an increase of the path length, G2 behaves differently: first increases and then decreases with an increase of L and has a maximum at L и 0.25. The similar dependence is also characteristic for beams of other initial profile (see Figure 4.8).

To understand this effect, it is necessary to consider the evolution of transverse beam intensity profiles under conditions of joint manifestation of SRS ad diffraction. First, as the interaction length increases, the SRS threshold is achieved at the places with maximal values of the transverse intensity distribution of the main wave (Figure 4.9b). The active Raman scattering starting here leads to the distortion of the initial profile of the pump beam and its local depletion. Then other beam zones begin to satisfy the threshold SRS condition, thus providing the continuous increase of the conversion parameter r)R (Figure 4.9c).

The diffraction of radiation leads to the diffusion blooming of the beam in the transverse direction and to decrease of the peak intensity. Consequently, the increasing role of diffraction (increase of the parameter L), on the one hand, should

Transverse intensity profile of the main (1) and the 1-st Stokes (2) beams propagating along the horizontal atmospheric path (h = 10 km) at L = 0.05 (a); 0.44 (b); 0.64 (c). Parameters of calculation

Figure 4.9 Transverse intensity profile of the main (1) and the 1-st Stokes (2) beams propagating along the horizontal atmospheric path (h0 = 10 km) at L = 0.05 (a); 0.44 (b); 0.64 (c). Parameters of calculation: Gaussian beam with l0 = 1.6 • 103 W/cm2, R0 = 20 cm, A0 = 1.315 p,m.

manifest itself in the decrease of the coefficient г/r and the increase of the threshold Gj. In Figure 4.8, this situation is illustrated by the descending branches of the function (L).

This effect can be estimated through the solution of system of Equations 4.54 for two waves with regard for the diffraction of the main radiation and with the neglected depletion, as well as without regard for diffraction of the Stokes wave:

The solution of Equation 4.69 for the Gaussian transverse profile can be expressed as

Upon the substitution of Equation 4.71 into Equation 4.70 at x = y = 0 and path-constant Raman gain coefficient gR, we obtain the law of increase of the Stokes radiation intensity at the beam axis

Equating the exponent in Equation 4.72 to G1 and assuming T0 = T_1 и 1, we obtain the functional dependence of the threshold value of the SRS gain factor on the dimensionless path length G1 ^ arctg(L/LD), which qualitatively describes the data depicted in Figure 4.7.

On the other hand, diffraction by its nature tends to smooth the transverse profile of the beam, leading to diffusion of the wave amplitude from zones with high intensity to zones with the decreased intensity level. This is especially prominent in the zone, where the transverse amplitude gradient is large, that is, in the zone of active SRS manifestation. Thus, the pump depletion, for example, at the beam center due to intensification of the Stokes components is partially compensated by the energy inflow from peripheral zones. In this sense, the diffraction of the main beam favors SRS and intensifies it (ascending branches of the dependence (L)),

as was noted previously in Reference 53.

In the beams with the non-Gaussian initial profile, this effect is even more pronounced, because the transverse intensity gradient here is initially high. Numerical calculations of the intensity at the axis of collimated beams of different profiles along the path at their diffraction in the absence of the SRS effect show that, in contrast to the Gaussian profile, the intensity at the center of the ring and super-Gaussian beams first increases, as in the case of focusing, and then decreases due to the beam blooming.

Thus, the diffraction, redistributing intensity in the transverse profile of the beam, acts in two ways in the Raman active medium: it simultaneously intensifies and suppresses the SRS effect, which causes the ambiguous behavior of the beam power conversion coefficient for different values of the propagation path length. This effect is most pronounced for beams with the non-Gaussian initial spatial profile.

 
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