Nonlinear dynamics of the counter-propagating waves in
superradiance
Yu.A. Avetisyan 
and E.D. Trifonov 
Blagonravov Mechanical Engineering Institute
of the Russian
Academy of Sciences, Saratov, Russia;
Russian State Pedagogical University, St.Petersburg, Russia
Superradiance (SR) is the cooperative spontaneous emission of the identical excited atoms or molecules. In the present paper we analyze the counter-propagating waves correlation (observed in SR experiments [1]) using the semiclassical Maxwel-Bloch equations for two-level atoms population inversion W(x,t) and slowly varying in the time t (but not in space x) amplitudes of the density matrix nondiagonal element R(x,t) and the electric field E(x,t)
c.c., 
where 
is the polarization phase relaxation constant, 
is the wavenumber, other constants have the same definitions as
in [2]. By the numerical solution of these Eqs. for initially
completely inverted state W(x, 0) = 1 with 
- correlated
polarization ( 
), L and N are
the sample length and the total number of atoms
correspondingly) we calculated the counter-propagating
waves correlation coefficients for delay times and for
maximal intensities using standard technique [3]. Our
calculations (performed for the sample length in the region
) demonstrates anticorrelation of the counter-
waves maximal intensities and the increasing of these waves
synchronization with increasing of N, and with decreasing
of L that is in agreement with the results reported
earlier [2,3]. However, our calculations revealed a strong
dependence of the correlation coefficients and variances on
times repeated half of wavelength 
in L. If L is integer
-multiple of then
there is the minimum of correlation coefficient and the
maximum of variance and vice versa when L is half integer
multiple of . The explanation of this feature, based on
the conception of resonant reflection from the transversal
boundaries of active volume [4] has been done.