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Journal of the Physical Society of Japan

Self-Induced Transparency Due to Two-Photon Transition and Area Theorem

Eiichi Hanamura

Published Dec 1, 1974• 18 citations
Publisher

Abstract

The effect of self-induced transparency due to two-photon transition in a system of three-level atoms is discussed. The area theorem of two-photon transition is derived both with and without taking the account of the inhomogeneous broadening of active levels. The metastable and distortionless propagation of the radiation wave is possible only for the pulse with Lorentzian form for the radiation intensity.

DOI
10.1143/jpsj.37.1598

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We report on computer-generated solutions to the coupled matter-field equations which provide the first evidence for distortionless optical pulse propagation in an absorbing medium of three-level atoms. The associated phenomenon represents the three-level equivalent of self-induced transparency in two-level atoms. Results have application to the propagation of short, coherent optical pulses in a medium in which the atoms have two distinct transition frequencies in near resonance with the pulse carrier frequency. We relate in particular to the case of pulse propagation in a magnetically perturbed sodium vapor where the pulse carrier frequency is tuned near the frequencies of the well-known sodium $D$ lines. Here we obtain particular solutions corresponding to three and five peak pulses with "areas" of approximately $6\ensuremath{\pi}$ and $10\ensuremath{\pi}$, respectively. The solutions are strongly supported by a number of analytical results. In particular, simple algebraic relations are found which give the populations of the levels in terms of the field variables.

Physical Review A•Nov 1, 1977•4 citations
The effect of self-induced transparency due to two-photon transition in a system of three-level atoms is discussed. The area theorem of two-photon transition is derived both with and without taking the account of the inhomogeneous broadening of active levels. The metastable and distortionless propagation of the radiation wave is possible only for the pulse with Lorentzian form for the radiation intensity.
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.
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.
The propagation of two pulses in a three-level absorber is treated by using Maxwell's equations and an effective "two-photon Bloch equations" which are analogous to the one-photon Bloch equations, except for two-photon coupling and frequency shifts. I derive relaxation-dependent shifts in the equilibrium $2n\ensuremath{\pi}$ Lorentzian pulses. Part of these shifts depends on weighted time integrals of harmonics of the area and hence remove the propagation degeneracy of pulses having equal areas but different shapes. Absorption of small-area pulses is studied as a function of absorption length, input energies, and relaxation, and it is found that a local enhancement in the transmission occurs as a function of the absorption length.
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.
In the limit of pulse durations that are shorter than the characteristic times of atomic decays, the dynamics of the propagation of a probe light pulse under the action of a driving pulse in the form of a self-induced transparency soliton applied to the adjacent transition of the V -scheme is considered. For a three-level absorber, it is shown that, in the absence of two-photon inhomogeneous broadening, the effect of soliton-induced transparency (SOIT) for the probe field takes place, which consists of the probe pulse undergoing forced localization by the driving pulse and propagates further without a change in the energy or shape. This type of transparency has no threshold with respect to both the energy and the area and is observed for arbitrarily weak pulses. The SOIT effect disappears in the presence of a two-photon inhomogeneous broad-ening. The effect of Raman amplification in a three-level amplifier with the Raman population inversion is considered. It is demonstrated that, under the action of the two-photon inhomogeneous broadening, the effect of Raman amplification becomes a threshold effect. The broader the two-photon inhomogeneous contour, the larger the Raman inversion required to initiate generation.
We report on computer-generated solutions to the coupled matter-field equations which provide the first evidence for distortionless optical pulse propagation in an absorbing medium of three-level atoms. The associated phenomenon represents the three-level equivalent of self-induced transparency in two-level atoms. Results have application to the propagation of short, coherent optical pulses in a medium in which the atoms have two distinct transition frequencies in near resonance with the pulse carrier frequency. We relate in particular to the case of pulse propagation in a magnetically perturbed sodium vapor where the pulse carrier frequency is tuned near the frequencies of the well-known sodium DDD lines. Here we obtain particular solutions corresponding to three and five peak pulses with "areas" of approximately $6\ensuremath{\pi}$ and $10\ensuremath{\pi}$, respectively. The solutions are strongly supported by a number of analytical results. In particular, simple algebraic relations are found which give the populations of the levels in terms of the field variables.