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Journal of the Optical Society of America B

Photon echo formed by exciting pulses with smooth envelopes

Igor V. Yevseyev, Mikhail Vladimirovich Fedorov, V. N. Tsikunov

Published Jul 1, 1989• 1 citation
Publisher

Abstract

A photon echo formed by exciting pulses with smooth envelopes is considered. The echo signal is found analytically. The result is compared with known results for rectangular exciting pulses. It is shown that if the echo is formed on wide spectral lines there is an essential difference between the cases of smooth and rectangular exciting pulses.

DOI
10.1364/josab.6.001314

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A photon echo formed by exciting pulses with smooth envelopes is considered. The echo signal is found analytically. The result is compared with known results for rectangular exciting pulses. It is shown that if the echo is formed on wide spectral lines there is an essential difference between the cases of smooth and rectangular exciting pulses.
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.
An expression for the electric field intensity of the photon echo formed in a gaseous medium on transition between two degenerate levels under the action of two linearly polarized exciting pulses is derived. Using the obtained relationships the effect of photon-echo shape locking is analysed for a square shaped second (‘locking’) exciting pulse as well as for the case of a smooth envelope of the hyperbolic secant form. It is shown that in degenerate systems as well as in non-degenerate ones, photon-echo shape locking takes place if the shape of the ‘locking’ pulse is square and does not take place in the case of a smooth envelope.
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 excitation of the primary and stimulated photon echoes by pulses modulated in phase with Barker codes is modeled. The shape of the complex envelopes of the primary and stimulated echoes is found by solving the optical Bloch equations. Linear and nonlinear excitation modes are considered. In linear modes, it is possible to realize a time delay and form the correlation function of modulated pulses. The results obtained can be used in modeling and analyzing the corresponding algorithms of the photon echo excitation and in calculating the characteristics of signal processors based on the phenomena of photon and spin echoes.
Multiple photon echo signals were found to be the reflection of the oscillation structure of the primary echo signals formed by excitation of two-level active centers. Amplitude pulses are of the same length, but their amplitude is different. The length of the pulse is comparable with the delay between them. The primary photon echo of the oscillation structure is shown to exist in the intervals (-t1,t1) and (-2t1,2t1). It was established that under certain correlation between the amplitudes of exciting pulses the oscillation signal structure becomes asymmetric. That is why in the regime of square-law detection of the primary photon echo signal its asymmetry oscillation structure is observed in the form of frequent of isolated signals (multiple photon echo), the distance between them being equal to the pulse length.
The excitation of photon echo by noise pulses that are formed by modulation of the carrying frequency with Gauss noise is modeled. The modeling is based on optical Bloch equations the solution of which for noise pulse realizations is constructed by their stepwise approximation. In terms of the formalism of state transfer matrices, the two- and three-pulse excitation modes are analyzed. The complex envelopes of the primary and stimulated echo responses are determined. In the linear (low-level-signal) mode, the shape of the two-pulse echo corresponds to that of the time delayed and inverted noise pulse. The boundary of the linear mode, upon exceeding of which distortions of the shape of the noise pulse become noticeable, is determined. The shape of the stimulated (three-pulse) echo in the linear mode corresponds to that of the autocorrelation function of the noise pulse realization. Upon passage beyond the boundary of the linear mode, the shape of the three-pulse echo corresponds either to the cross-correlation function of distorted noise pulses (with different intensities) or to the autocorrelation function of distorted pulses (with the same intensities). The modeled photon echo excitation modes can be used in photon echo processors to process signals in the light range.