We will now examine the influence of turbulent pulses on the optical quality of a medium without allowance for the potential effect of turbulent mixing in the presence of an inhomogeneous averaged temperature field. It is obvious that taking this effect into account [10] will not improve the optical characteristics.

The refractive index dependence, n, for the gases under consideration is quite accurately imparted by the equation

n = 1 + Cnp, (9.20)

where Cn is the constant for a gas of this type. Using Eq. (9.13) for the refractive index pulses in the smallest-scale inhomogeneities, we obtain

-3/2

An0 — Cnp(v/Л)1/2 — (9.21)

u2

Correlation (9.21) corresponds to definition (9.10) for pressure pulses. However, if dependence (9.15) is used instead of Eq. (9.10), the corresponding expression for estimating the Дп0 value takes the form

Дп0 — Cnp(v/Л)1/4 —r. (9.22)

u2

According to Eqs. (9.11) and (9.20), the expression for refractive index pulses in the largest inhomogeneities with a typical scale of Л can be written as

Дnm — CnP-2/u2- (9.23)

Estimates for helium based on dependence (9.21) using the numerical parameter values previously assigned at U = 10 m/s yield Дп01~10~10, while based on dependence (9.22), Дn02~10~9. According to Eq. (9.23), Дnm ~6 x 10~9; at a velocity of U = 100 m/s, we get Дп01~5 x 10~9, Дn02~6x 10~8, and Дnm ~6 x 10-7.

Refractive index pulses cause energy dissipation in a coherent light beam passing through a turbulent medium. In an optically active medium, the dissipated energy lost during diffraction on inhomogeneities can be viewed as a coherent beam loss, i. e., as a kind of absorption [14, 15]. In order to calculate the absorption constant (see refs. [15—17] for example), use was made of the results that Sutton

[14] obtained under the following assumptions: a turbulent medium with a dimen­sion of L, is homogeneous and isotropic; the light wave front at the medium inlet is planar; and consideration is given to the effective value averaged over time intervals that exceed the turbulence time scales.

The absorption constant that reflects the effect of turbulent pulses is determined by the formula [14]

aT = 2k{ An2) Л, (9.24)

where kx is the wave number, and (An2) is the mean-square pulse of refractive index.

The aT coefficient can be estimated by replacing (An2) with (Anm)2. Because, in point of fact (An2) < (Anm)2, this replacement can only lead to the overstatement of the absorption coefficient. We assume that Л ~ 1 cm. Placing kx = 2л/Х into Eq. (9.24) (for certainty’s sake, we assume that X = 1.73 qm [18, 19]), together with the Anm values found above, we obtain aT~2 x 10-7 cm-1 at U = 10 m/s and aT~2 x 10-3 cm-1 at U = 100 m/s. For comparison: the gain in NPLs based on rare gas mixtures is вк~ 10-3-10-2 cm-1 [20].

According to ref. [14], the dependence of coherent beam intensity upon the distance traversed, x, is determined by the law

I(x) = l0exp{(Pk — aT)xg.

Here, if D/Л ~ 1 (where D is the transverse dimension of the beam), and

aTL, < < 1, (9.25)

the diffraction pattern in the far-field zone then virtually replicates the diffraction pattern from the inlet following the traversal of a homogeneous medium that does not contain turbulent pulses, and the noticeable attenuation of the laser beam is absent. Thus, if condition (9.25) is satisfied, turbulent pulses in density should not exert a direct influence on the shaping of laser beam power and angular parameters. Only statistically averaged density inhomogeneities identical to those originating in sealed laser cells, or in the presence of laminar flows, will affect these character­istics (see ref. [10] for an example of the determination of this average).

Based on inequality Eq. (9.25), we will estimate the permissible gas flow velocities at which turbulent pulses in density do not necessarily exert a direct influence on the optical quality of a medium. Deliberately reducing the permissible gas-flow velocity values, we will replace (An2) with (Anm)2, then from Eqs. (9.24) and (9.25), we obtain

(Дnm)2 << {2kjALt)

Placing Eqs. (9.23) into (9.26), we obtain

We note that in inequality Eq. (9.26), in which the U velocity value enters the biquadrate, must be strictly executed; therefore, in order to estimate the permissible U values, it is not obligatory to require that the “much less” condition in Eq. (9.27) be satisfied: it is sufficient that its left side be roughly two to three times smaller than its right side. In order to facilitate estimates of the upper limit of the gas-flow velocities at which the effect of turbulent pulses on the optical quality of a medium is small, we will transform Eq. (9.27) into the inequality U < U*, where

During estimates of permissible gas-flow velocity based on formula (9.28) in an amplification regime, the Lt quantity is used (the geometric dimension of the medium along the optical axis, which equals the distance between the amplifier’s end windows.) During estimates of gas-flow velocity in the lasing regime, L, should be replaced with Lp (the effective photon path in the resonator, Lp = ctp, where c is the light velocity in the medium under consideration and tp is the photon lifetime in the laser cavity). The latter can be determined by means of the formula [21]:

tp = clnrm(1 — Rd)] . (9:29)

Here, La is the length of the active amplifying section of the gas medium that fills the cavity (which can be taken to equal the size of the uranium layer in the direction of the optical axis); RD is the share of diffraction and other losses; and rm is the reflectivity of the output mirror (the reflectivity of the second mirror is taken to equal unity).

The permissible gas-flow velocities, U*, for mixtures based on He (P0 = 2 atm) and Ar (P0 = 0.5 atm) at LA = 1 m, L, = 1.4 m, Л = 1 cm, RD = 0, and rm = 0.9 obtained using formulas (9.28) and (9.29) are presented in Table 9.1.

In addition to the direct influence examined, turbulence can have a strong indirect effect. Actually, medium turbulization, as is generally known, leads to an increase in heat conduction process efficiency. Effective turbulent thermal diffu — sivity coefficient noticeably exceeds conventional thermal diffusivity coefficient in a stationary medium. As a result, the averaged temperature and density profile

Gas	Amplification regime	Lasing regime
He	60	30
Ar	20	10

Table 9.1 Estimated U* velocity values, in m/s

equalization rate increases. When conditions are present that favor the origination of a passive region in a gas within which lasing is absent, this phenomenon can lead to a perceptible increase in the subject region’s dimensions as compared to the laminar flow mode.

An approximate estimate of turbulent thermal diffusivity coefficient can be derived from the following considerations. Temperature equalization time in the scales characterized by linear dimensions on the order of the maximum turbulent pulse scale is determined by correlation Eq. (9.18). On the other hand, assuming that the heat exchange process within macroinhomogeneities between individual smaller inhomogeneities has the same mechanism as during conventional molecu­lar heat exchange in a stationary medium (i. e., individual migrating inhomogene­ities play the role of molecules during the turbulent mixing process), the temperature equalization time can be estimated from the correlation

TTm — Л2/ат, (9.30)

where aT is the effective turbulent thermal diffusivity coefficient. Comparing Eqs. (9.18) and (9.30), we obtain

ат — AU. (9.31)

This expression is commensurate with the well-known correlation ат ~ Л x Дu

[4] , where Ди is the typical variation of large-scale pulse velocity at distances on the order of Л (under specific conditions, it can be comparable to U).

The average time required for a portion of the gas to traverse a section of a laser channel with a length of x comes to t — x/U. Assuming that the dependence of passive zone development upon gas residence time in the channel comes in accordance with dependence (8.3), then for the size of this zone within the channel part involved in the turbulence at a medium thermal diffusivity coefficient, aT, that conforms to Eq. (9.31), we get I — л/Ax. Thus, the passive zone will involve almost the entire width of the channel, d, after the gas traverses a turbulent section with a length of x~ d 2/Л, which, assuming that Л~d, yields x~d <<bL.

Of course, these are only approximate estimates; however, they illustrate some of the difficulties that would possibly be faced during the development of NPLs with longitudinal gas flowing and that it would require considerable effort to overcome. This situation was one of the reasons for seeking alternative solutions.