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Physical Review A

Self-Induced Transparency in Two-Pass Attenuators and Related Phenomena

Frederic A. Hopf

Published Jul 1, 1970• 5 citations
Publisher

Abstract

A theory describing two sequential electromagnetic pulses interacting with a resonant, inhomogeneously broadened attenuator is presented. The resulting equations are solved on a computer. Limiting cases are considered which indicate the conditions under which the second pulse is expected to increase in energy. The self-induced transparency effect in a two-pass attenuator is treated using this theory. It is shown that there are qualitative as well as quantitative differences between the single-pass and double-pass cases. For example, curves of output versus input energy in the two-pass case may have observable structure that is sensitive to the presence or absence of atomic coherence. It is found that under a wide range of circumstances, the output for the second pulse is independent of whether the pulse is going in the same or in the opposite direction from the first. This result is seen to be related to more general considerations involving the interaction of a pulse with spatially inhomogeneous systems such as a chain of amplifiers and attenuators. Also discussed are the measurement of ${T}_{1}$ using two-pass techniques, and the possibility of using these phenomena in a practical application to optical logic circuitry.

DOI
10.1103/physreva.2.195

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Two-photon self-induced transparency in a resonant medium. II. Transient pulse behaviour

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For pt.I see ibid., vol.8, no.3, 339 (1975). The transient behaviour of two short optical pulses in two-photon coherent propagation is obtained by calculating numerically the coherent propagation equations which couple the two electromagnetic pulses and the atomic system in self-consistent fashion. The coherent interaction of the two different frequency pulses gives rise to lossless propagation in the absorptive medium. This description is characterized by a two-photon area theorem. Novel coherent propagation effects of peak amplification and multiple pulse breakup are found. These effects are ascribed to two-photon self-induced transparency analogous to that of a one-photon transition in a two-level system. A difference of input intensity between the two different frequency pulses yields curious modulation effects involving sharp peak amplification and marked pulse breakup for the low-intensity input. In contrast, the high-intensity input pulse displays a smooth variation with distance and time. Low-loss pulse transmission above threshold, and pulse advance and delay are found for the present case in contrast to that for one-photon self-induced transparency.

Journal of Physics B Atomic and Molecular Physics•Feb 21, 1975•10 citations
A theory describing two sequential electromagnetic pulses interacting with a resonant, inhomogeneously broadened attenuator is presented. The resulting equations are solved on a computer. Limiting cases are considered which indicate the conditions under which the second pulse is expected to increase in energy. The self-induced transparency effect in a two-pass attenuator is treated using this theory. It is shown that there are qualitative as well as quantitative differences between the single-pass and double-pass cases. For example, curves of output versus input energy in the two-pass case may have observable structure that is sensitive to the presence or absence of atomic coherence. It is found that under a wide range of circumstances, the output for the second pulse is independent of whether the pulse is going in the same or in the opposite direction from the first. This result is seen to be related to more general considerations involving the interaction of a pulse with spatially inhomogeneous systems such as a chain of amplifiers and attenuators. Also discussed are the measurement of T1{T}_{1}T1​ using two-pass techniques, and the possibility of using these phenomena in a practical application to optical logic circuitry.
The influence of degeneracy on coherent pulse propagation in an inhomogeneously broadened attenuator is examined by computer calculations, and the connection with self-induced transparency in a nondegenerate system is described. Specific modifications that arise in a degenerate system are the development of pulse shapes characteristic of the degeneracy, the association of a finite loss with propagation, a reduction of pulse delay times, and the suppression of pulse separation of multi-$2\ensuremath{\pi}$ pulses. Additionally, results of calculations pertaining to experiments on pulse propagation in SF6{\mathrm{F}}_{6}F6​ are presented which indicate two basically different phenomena. One corresponds to delay phenomena qualitatively similar to nondegenerate self-induced transparency, while the other is a coherent saturation effect generally characteristic of degenerate media. The computed results fully substantiate our earlier experimental findings.
Above a critical power threshold for a given pulse width, a short pulse of coherent traveling-wave optical radiation is observed to propagate with anomalously low energy loss while at resonance with a two-quantum-level system of absorbers. The line shape of the resonant system is determined by inhomogeneous broadening, and the pulse width is short compared to dissipative relaxation times. A new mechanism of self-induced transparency, which accounts for the low energy loss, is analyzed in the ideal limit of a plane wave which excites a resonant medium with no damping present. The stable condition of transparency results after the traversal of the pulse through a few classical absorption lengths into the medium. This condition exists when the initial pulse has evolved into a symmetric hyperbolic-secant pulse function of time and distance, and has the area characteristic of a "$2\ensuremath{\pi}$ pulse." Ideal transparency then persists when coherent induced absorption of pulse energy during the first half of the pulse is followed by coherent induced emission of the same amount of energy back into the beam direction during the second half of the pulse. The effects of dissipative relaxation times upon pulse energy, pulse area, and pulse delay time are analyzed to first order in the ratio of short pulse width to long damping time. The analysis shows that the $2\ensuremath{\pi}$ pulse condition can be maintained if losses caused by damping are compensated by beam focusing. In an amplifying inhomogeneously broadened medium an analytic "$\ensuremath{\pi}$ pulse area" solution is presented in the limit of a sharp leading edge of the pulse. The dynamics of self-induced transparency are studied for the particular effects of Doppler velocities upon a resonant gas. The analysis of transparency for random orientations of dipole moments associated with degenerate rotational states yields modified forms of self-induced transparency behavior, which indicates a finite pulse energy loss as a function of distance in some cases. The effect of self-induced transparency on the photon echo is considered. Experimental observations of self-induced transparency have been made in a ruby sample at resonance with a pulsed ruby-laser beam. Single and multiple $2\ensuremath{\pi}$ pulse outputs have been observed, and pulse areas measured in the range of $2\ensuremath{\pi}$. The experimental results are compared with the predictions of the ideal-plane-wave theory. Deviations from the ideal-plane-wave theory are discussed. An analysis is made of the effect of a transverse mode of the propagating beam upon the transparency properties of the pulse.
The authors develop a theoretical analysis which describes coherent nonlinear resonant effects where two optical pulses of different frequencies undergo two-photon self-induced transparency (two-photon SIT) in an inhomogeneously broadened three-level system. The field (Maxwell) and atomic (Schrodinger; density matrix representation) equations are coupled in a self-consistent manner by virtue of the resultant macroscopic polarizations at the two frequencies. This treatment is accomplished under circumstances where atomic coherence plays a predominant role. The resulting equations yield a new two-photon area theorem which gives first the intensity dependent, two-photon absorption law analogous to Beer's law for the one-photon transition. Secondly, a 2 pi Lorentzian pulse solution is derived which propagates without loss at the modified pulse velocity.
The dynamics of self-induced transparency in an inhomogeneously broadened attenuator is investigated with special emphasis on delay times. One finds a strong dependence of delay times on the input value of the area parameter $\ensuremath{\theta}$ with pulses of ${\ensuremath{\theta}}_{\mathrm{in}}\ensuremath{\approx}\ensuremath{\pi}$ exhibiting much larger delay times than for ${\ensuremath{\theta}}_{\mathrm{in}}=2\ensuremath{\pi}$. This investigation is extended to include the effects of an atomic phase memory time T2{T}_{2}T2​ which is of the order of ten times the pulse width (i.e., in accord with current experimental circumstances). The variation of other parameters is considered. A method of measuring $\ensuremath{\theta}$ directly is presented.
A review is given of the current status of the self-induced transparency effect. The McCall–Hahn theory (2π pulses) is considered and its generalization to the case of phase modulation of radiation pulses, degeneracy of transitions, and two-photon resonance is discussed. The relationship with the theory of nonlinear waves, obeying the Korteweg–de Vries equation, is pointed out. The possibility of similar coherent effects in direct interband transitions in semiconductors is discussed. A summary is given of the experimental results demonstrating a strong fall in the absorption and a considerable slowing down of light pulses, typical of the self-induced transparency. Possible physical applications of the effect, based on these properties, are discussed.
For pt.I see ibid., vol.8, no.3, 339 (1975). The transient behaviour of two short optical pulses in two-photon coherent propagation is obtained by calculating numerically the coherent propagation equations which couple the two electromagnetic pulses and the atomic system in self-consistent fashion. The coherent interaction of the two different frequency pulses gives rise to lossless propagation in the absorptive medium. This description is characterized by a two-photon area theorem. Novel coherent propagation effects of peak amplification and multiple pulse breakup are found. These effects are ascribed to two-photon self-induced transparency analogous to that of a one-photon transition in a two-level system. A difference of input intensity between the two different frequency pulses yields curious modulation effects involving sharp peak amplification and marked pulse breakup for the low-intensity input. In contrast, the high-intensity input pulse displays a smooth variation with distance and time. Low-loss pulse transmission above threshold, and pulse advance and delay are found for the present case in contrast to that for one-photon self-induced transparency.