As in studies [35, 36], the aforementioned laser’s cavity stability was analyzed in [34] in a paraxial approximation involving the use of the ray matrix technique. It was assumed that the axis of a system with two-dimensional transverse inhomoge­neities is located in a single plane. In this instance, as is generally known (see, e. g., study [29]), the conversion of the projections of light rays in the mutually perpen­dicular meridional planes can be independently examined within the framework of Gaussian optics. For the two-mirror linear cavity of the laser type under consider-

which the fission fragment energy is deposited the matrix of beam propagation in the active volume (M2); the matrix of propagation in a homogeneous medium between the other active volume boundary and the second mirror (M3); the matrix of beam reflection on the second mirror (M4); then the M3, M2, and M1 matrices again; and, finally, the M0 matrix, which describes beam reflection on the first mirror. A cavity is stable if |m| < 1, where m = (A + D)/2.

For an active volume with a one-dimensional quadratic transverse refractive index distribution,

the stability parameter takes the form [35, 36]:

my = C1 cos2/ — C2( cos 2/ + 1) — (C3y2 + C4y—2)( cos2/ — 1) — -(C5Y + C6r—1) sin 2y,

where

It is not difficult to demonstrate that for a laser in the active medium of which the refractive index distribution takes the form

the stability parameter is

mx = Cich2<9 — C2(ch20 + 1) — (C302 + C40-2) (ch20 — 1)+
+ (C50 — C60-1 )sh20,

where

while the C1-6 coefficients are in agreement with the coefficients in equation (8.39).

Experiments [2] aimed at studying optical inhomogeneities in a laser with planar uranium layers (see Chap. 7, Sect. 7.4, and Sect. 8.2 of this chapter) revealed that the refractive index distribution in the x = const planes is well described by parabola (8.38), while in the y = const planes, it is fairly well depicted by parabola (8.41). An analysis of the experiments and calculations using the procedure set forth in Sect. 8.2 of this chapter demonstrated that the refractive index distribution in the transverse cross-section of the laser cell’s active volume can be described with an accuracy of up to a few percents by the dependence

n(x, y, t) = n0(t) + 2p(t)x2 — 2a( ()y2. (8.44)

In a paraxial approximation, geometric optics equations (8.17) take the form [30, 33]

For gases, Дn = n — n0 << 1, which when taking dependence (8.44) into account, makes it possible to present these equations in the form of two independent equations for the x — and y-components,

d2x dAn(x)

0 dz2 dx ’

d2y dAn(y)

n0 dZ2 = ~~dT ’

i. e., to independently examine the passage of the light beams and the fulfillment of the stability criterion in the x, z, and у, z planes. It is obvious that a cavity of this type will be stable if two conditions are simultaneously observed: Шх| < 1 and Шу| < 1.

If the cavity geometry remains unchanged, the stability parameters my and mx (see (8.39) and (8.42)) will then only be dependent upon the corresponding dimen­sionless inhomogeneity parameters у and 0, which are determined by equations (8.40) and (8.43). Thus, if the у values vary over time, then my will periodically pass through the I my I = 1 values, which is explained by the presence of the trigono­metric sines and cosines in expression (8.39). In this instance, if the 0 values vary over time, then mx can also intersect the | mx| = 1 values. During the transition of the my (or mx = 1) values to | my| = 1 (or | mx| = 1), the alteration of cavity stability is observed. The у and 0 values at which such a transition occurs are critical values of the dimensionless inhomogeneity parameters. They are designated below as ycr and 0cr.

Over the course of pumping, the inhomogeneities characterized by the dimen­sionless у and 0 parameters monotonically increase [2, 23, 35, 36, 45]. The depen­dence of the my stability parameter upon the value of the у dimensionless inhomogeneity parameter over the variation range of real interest is presented in Fig. 8.17. The corresponding dependence of the mx stability parameter upon the 0

dimensionless inhomogeneity parameter is shown in Fig. 8.18. During irradiation time, the my parameter periodically intersects the | my| = 1 value in making the transition from the stable region to the instable one, then again returns to a stable state. The instability region sections in Fig. 8.17 are numbered in order of growth inhomogeneity parameter values: the first-order instability region comes first, then the second-order region, etc.

Changes in the stability of a laser with one-dimensional inhomogeneities of the type described in (8.38) were calculated in [35, 36], which revealed the cavity stability losses and restorations occurring over the course of a pulse that correspond in time to the moments of the total cessation and restoration of lasing observed during the experiments. Over the course of these calculations, the refractive index distribution was approximated by parabola (8.38), which was common over the entire volume; therefore, the loss of stability corresponded to the total cessation of lasing.

It was previously mentioned that during a number of experiments involving planar and cylindrical uranium layers, both fluctuations in laser power and their total cessation were observed. The reason for the fluctuations can be sought in the temporary departures of only a portion of the cavity’s volume from the stable state. So, direct calculations demonstrated that during the approximation of inhomoge­neities using correlation (8.44) based on their values in the peripheral regions of the transverse cross-section of the cell’s active volume, higher a and p coefficients are generally obtained than during an approximation based on the central regions. Apparently, stability is first lost in the peripheral sections, while the central sections remain stable. An artificial technique was used in study [34] to verify this proposition. During the investigation of stability in the y and z planes, the active section of the cavity was evenly split into separate regions along the x axis that fully
occupied the gap between the uranium layers, d. An a(t) parabola coefficient was selected for each region, i, at a relevant moment in time based on the results of density distribution calculations and it was used to compute the myi stability parameter. The transverse areas of the regions for which | myi | < 1 were then added. If the result of this addition is assigned to the overall area of the cell’s transverse cross-section, we then obtain a characteristic that is dependent upon time, Sy(t), which is found in an approximation of the total absence of a relationship between the individual regions.

We note that during this approach, the stability status of each individual region is examined outside its relationship with the status of the other regions, i. e., as if each region is isolated from the remaining ones. This technique a priori precludes the possibility of the origination of a stable (or quasistable) state between different regions, which is far from obvious. The use of this technique can be justified by the fact that during passage through the inhomogeneous media under consideration, each light beam is deflected by a slight angle so that when the transverse dimensions of a region are not too small, the bulk of the beams remain within the confines of this cavity section if its status is stable. Thus, the region Дxi dimension must be limited from below, because it is apparent that the smaller the transverse dimension, the more numbers of the beams can travel outside the confines of this region to the neighboring ones. As a criterion, it was assumed that a region’s size must not be smaller than a beam’s displacement after one Cavity pass: Дхг — > Lф, where L is the distance between the mirrors and ф is the beam deflection angle. The deflection angle was estimated from the correlation ф ~L (3n3x)~Lpmx0, where x0 is half the dimension of the uranium layer in the direction of the х axis, and pm is the maximum p coefficient in the region most far removed from the center at the time of pumping pulse termination.

A similar approach was also used when calculating the stability in the x and z planes. The maximum number of splits in all the calculations presented below did not exceed N = 30. During the calculations for a cylindrical cell, splitting into coaxial cylindrical regions was carried out. The presence of axial symmetry was assumed.

Once again, we stress that the technique used is artificial and cannot be rigor­ously substantiated. It a priori rules out the possibility of the origination of a stable state between different regions, which is far from obvious.