A technique for the derivation of the electric field strength of varieties of photon-echo signals formed by exciting pulses with smooth envelopes is developed. This technique is based on joint solution of the d'Alembert and Schrödinger equations. The matrix of the evolution operator is constructed, which permits a description of the state of an individual atom (molecule) resulting from the action of an arbitrary exciting pulse sequence. The special form of exciting pulses with smooth envelope considered here is, together with the model of rectangular exciting pulses, one of the models that permits an analytical solution when the areas of the exciting pulses are not small. As an example, the technique is applied to the calculation of the electric field strength of signals of the modified three-level photon echo and the three-level photon echo formed by exciting pulses with smooth envelopes. The signal characteristics of photon-echo varieties formed by rectangular and non-rectangular exciting pulses are compared. It is shown that if the spectral lines of the resonant transitions are broad then the shapes and the peaks of the echo signals depend essentially on the shapes of the exciting pulses. The appearance of the fine structure of the modified three-level photon-echo and three-level photon-echo signal shapes is predicted. The necessity of setting up experiments aimed at observing the theoretically predicted special features of these photon-echo varieties is pointed out.
The behavior of the photon echo produced by an inhomogeneously broadened spectral line is analyzed, with particular emphasis on the effect of spatial degeneracy of the molecular energy levels. The theoretical results are used to aid in the identification of the states involved in the echoes produced from gaseous S
F6 excited by pulses of C
O2 laser radiation near 10.6 \ensuremath{\mu}. The analysis demonstrates that for the $|{\ensuremath{\Delta}}_{j}|=1$ transitions (
P and
R branches) the echo induced by two linearly polarized pulses propagating along the same direction, with an angle $\ensuremath{\psi}$ between their electric vectors, is polarized at an angle greater than $\ensuremath{\psi}$ with respect to the first pulse, and the echo intensity decreases to a minimum, but not zero, as $\ensuremath{\psi}$ approaches $\frac{\ensuremath{\pi}}{2}$. An exception is the $j=1\ensuremath{\leftrightarrow}j=0$ transition, for which the echo is always polarized along the second pulse and the intensity varies as ${cos}^{2}\ensuremath{\psi}$. For ${\ensuremath{\Delta}}_{j}=0$ transitions (
Q branch) the echo polarization generally lies at an angle smaller than $\ensuremath{\psi}$, i.e., between the first two pulses, and its intensity also decreases as $|\ensuremath{\psi}|$ approaches $\frac{\ensuremath{\pi}}{2}$. The exceptions for the ${\ensuremath{\Delta}}_{j}=0$ transitions are the cases $j=\frac{1}{2}\ensuremath{\leftrightarrow}j=\frac{1}{2}$ and $j=1\ensuremath{\leftrightarrow}j=1$. In the former case, the echo is polarized at an angle $2\ensuremath{\psi}$ with respect to the electric field vector of the first pulse, and the echo intensity is independent of $\ensuremath{\psi}$; whereas in the latter case, the echo behaves like that for $j=1\ensuremath{\leftrightarrow}j=0$. The dependence of the echo behavior on the angular momentum of states involved in the transition can serve to identify the transitions when measurements of the echo polarization and intensity are possible.
We have studied the information locking effect and the effect of correlation of the shape of an object laser pulse with the shape of a stimulated photon echo response in the presence of external spatially inhomogeneous electric fields. We have shown that, for the transition 3 H 4 - 3 P 0 in a LaF 3 :Pr 3+ crystal, one can observe the effect of the correlation of the shape of an object laser pulse with the shape of a stimulated photon echo response and, depending on the scheme of the action of external spatially inhomogeneous electric fields, either the information locking effect or the information destroying effect.
In the limit of small exciting light-pulse areas the amplitude and polarization of the tri-level photon echo in gases formed by the exciting pulses with arbitrary shapes are obtained. It is shown that the dependence of the tri-level photon echo signal on the mutual orientation of the exciting pulse polarization vectors may be used to determine the optical coherence-matrix relaxation characteristics for an optically forbidden transition. The conditions under which a tri-level photon echo pulse repeats the shape of each of the exciting pulses are defined. It is proved that the tri-level photon-echo polarization properties do not depend on the shapes of the exciting pulses, either for small light-pulse areas or for the tri-level echo formation on narrow spectral lines.
The polarization properties of photon echoes in strong magnetic field, when the action of this field during the time of propagation of the exciting pulses and the echo signal through the resonant gaseous medium cannot be neglected, are studied theoretically. The photon echo polarization in strong magnetic field cannot be described by the polarization vector but rather by the polarization matrix. The degree of the photon echo polarization decreases with the increase of the magnetic field strength. The dependence of the photon echo intensity on the magnetic field strength was also analyzed numerically. In strong magnetic field the fast oscillations of this intensity appear to be damping and modulated by the slower ones. The spectrum of these oscillations depend also on the spectra of the exciting pulses, so that in case of small area of the first exciting pulse its temporal shape may be reconstructed from the oscillations of the echo intensity versus the magnetic field strength.