<title>X-ray laser coherence in the presence of density fluctuations</title>

Abstract

A ray-tracing technique based on the radiation transfer equation is used to describe the spontaneous emission gain in active media. Using this approach analytical solutions for the intensity distribution and coherence function in the output plane of an active medium with parabolic transverse profiles of dielectric constant and gain coefficient are presented. Applicability of the approximation when contribution into output emission is made by only spontaneous sources adjacent to the far face region of an active medium is analyzed. This approximation is seen not to be applicable for many real situations and it is necessary take into account the sources in the whole active medium. The effect of dielectric constant fluctuations on output coherence is treated by using the phase approximation of Huygens-Kirchhoff method.Keywords: x-ray laser, inhomogeneous active media, fluctuations, degree of coherence, ray-tracing technique. INTRODUCTIONThere are several approaches for theoretical description of the output emission of the lasers of interest. All of them are based on paraxial version of the wave equation. The approach based on expansion of the wave field into infmite series over transverse modes"2 is widely used. This approach is similar to conventional methods used in the laser optics. However, it can be practically used for a limited number of dielectric constant and gain coefficient profiles in an active medium for which transverse modes can be obtained analytically.The second approach3 is based on the numerical solution of the paraxial wave equation using the fast Fourier-transform split-operator method. The spontaneous emission is treated with the random-phase Monte-Carlo method. The coherence function is calculated by averaging over 1 00-200 temporal steps.The third approach is based on solving the equation for the transverse coherence function.4'5 Using this approach authors succeeded in analytical solving for a number of profiles of dielectric constant distribution (but for homogeneous distribution of gain coefficient) and for the case of saturation of emission gain.In this paper, we present an approach based on the solution of radiation transfer equation. This equation is an approximating the equation for transverse coherence function and allows not only to realize effective numerical algorithms for solution of the given problem but also obtain analytical solutions for inhomogeneous distribution of dielectric constant and gain coefficient in an active medium.3. Our starting point is the paraxial wave equationwhere k is the wave number, ic is the relative perturbation of the complex dielectric constant, P, is the spontaneous polarization , and r (z, p). Let us consider an active medium with relative dielectric constant distribution As(z, p) = s(z, p) + ioz,p) ,where s is the real component and is the imaginary part of the dielectric constant which is connected with the gain coefficient of the medium g by a(z,p) = -k'g(z,p).The form of the functions and c is determined by the spatial distribution of inverse pop...

<title>Adaptive focusing of laser radiation onto a rough reflecting surface through the turbulent and nonlinear atmosphere</title>

“…It is also smaller than the characteristic scales of the spatial variations of functions (r,t) and S(r,t) in eqn (21). For this case, in eqns (19) and (21) Eqn (18) describing evolution of the mutual coherence function is difficult to analyze because it represents the second order partial differential equation (PDE) for a complex function that depends on two vector (p and R) and two scalar (t and z) variables. The equation (18) can be simplified by considering propagation of a returned wave in a medium with relatively smooth refractive index fluctuations, so that the characteristic scale l (z)for MCF falloff over the difference coordinate (coherence length) p in eqn (18) is smaller than the characteristic spatial scales 4, for refractive index fluctuations -the smooth-refractive-index (SRI) approximation16'9.…”

“…The parameter l in eqn (22) defines the characteristic spatial scale for the fall-off in the returned field MCF (19) at the target surface and can be referred to as the returned-wave coherence length at the target plane. The distance l is inversely proportional to the characteristic angle of surface roughness slopes o /.…”

“…By taking into account these approximations, the boundary condition for the MCF(19) in the sum and differencecoordinates reads: F(p, R, L, t) = y2 (R, t) exp[ikVRS(R, t)]exp(-p2 I 2l )I(R, L, t) . (25) For Xr,t)=1 and S(r,t)const expression (25) coincides with the well-known Gaussian-Schell model beam describing a partially coherent laser source14'19.…”