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Optics and Spectroscopy

A transient two-photon–one-photon double resonance on interband transitions in crystals: I. General relations

M. A. Bondarev, E. Yu. Perlin

Published Apr 1, 2017• 2 citations
Publisher

Abstract

A theory of the transient absorption of femtosecond light pulses at a two-photon–one-photon double resonance on adjacent interband transitions in the electronic system has been developed. Approximate expressions for the reactive component of the nonlinear polarization of the three-level system, P s, which determines the absorbed power in the medium, have been obtained, and the dependences of P s on the light intensity and detunings of resonances have been calculated.

DOI
10.1134/s0030400x17040063

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A theory of the transient absorption of femtosecond light pulses at a two-photon–one-photon double resonance on adjacent interband transitions in the electronic system has been developed. Approximate expressions for the reactive component of the nonlinear polarization of the three-level system, P s, which determines the absorbed power in the medium, have been obtained, and the dependences of P s on the light intensity and detunings of resonances have been calculated.
We present solutions for two-photon optical transients from a three-level system which include the effects of (a) direct spontaneous emission to and from the nonresonant intermediate state and (b) phase-interrupting collisions. These effects are, in general, expected to be important for electronic transitions in the visible or ultraviolet region relevant for many two-photon transient experiments on free atoms. The steady-state absorption profile shows power narrowing and desaturation due to spontaneous relaxation. The two-photon delayed optical nutation signal is obtained explicitly. A perturbative calculation of optical nutation shows intensity-dependent damping. We find that for a particular case, namely, when the upper level spontaneous decay rate is twice that of the intermediate level, transient signals have especially simple forms and can be usefully employed for experimental determination of decay constants. The paper is a direct but nontrivial generalization of Schenzle and Brewer's corresponding investigation on one-photon transients.
An unconventional perturbation treatment of the theory of two-photon absorption in a four-level atomic system close to two-photon resonance is developed on the basis of the optical Bloch equations. The results for the longtime populations of the intermediate and upper states are valid in low- and medium-intensity regimes. In both regimes the upper state population shows a lorentzian resonance centred on zero two-photon detuning with a width equal to the natural linewidth of the upper state. The intermediate state population shows a similar behaviour in the intermediate regime only, with no significant dependence on two-photon detuning in the low-intensity regime. However, in both regimes the population of the virtual intermediate state is at least comparable to that of the resonant upper state. The perturbation theory results have been checked by direct numerical calculations based on the Bloch equations. Numerical calculations confirm the presence of a high-intensity regime characterized by power shifts, broadening and saturation effects.
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 expression for the absorption coefficient in two-photon resonance transitions is deduced under general conditions. No particular restrictions are imposed on the level system or on the frequencies and intensities of the fields. The following cases are considered: (i) a two-level system (with and without singularities) irradiated by a single wave and (ii) a three-level system irradiated by two waves. Some theoretical and experimental disagreements reported by authors of previous papers are examined and clarified by the present theory.
Based on five-level rate-equation theory, we develop the laser-pulse-duration dependence of two-photon-absorption-induced singlet and triplet excited-state absorptions (ESAs). We present analytical expressions for the effective three-photon absorption coefficients caused by both singlet and triplet ESAs under the pulsed excitation on time scales from femtoseconds to microseconds. We demonstrate that the triplet ESA is predominant with longer laser pulses (microseconds to tens of nanoseconds) and that the resultant nonlinear absorption (NLA) can be adequately interpreted by a simplified four-level model. Under the excitation of picosecond laser pulses, generally speaking, the competition between singlet and triplet ESAs is observable. In this instance, the photodynamics of the system can be understood by a five-level model. In the femtosecond regime, however, a three-level model is validated in the prediction of NLA, because the triplet ESA becomes negligible.
For the interaction of coherent radiation with a nondegenerate, homogeneously broadened multilevel system the nonlinear susceptibilities and excitations and their time dependence are evaluated using a perturbation approach to the density matrix up to fourth order. This is done for the "off-resonance" case, ${(\ensuremath{\Delta}{\ensuremath{\omega}}^{2}+{\ensuremath{\Gamma}}^{2})}^{\frac{1}{2}}\ensuremath{\gg}\frac{1}{\ensuremath{\tau}}$, as well as for the "on-resonance" case, ${(\ensuremath{\Delta}{\ensuremath{\omega}}^{2}+{\ensuremath{\Gamma}}^{2})}^{\frac{1}{2}}\ensuremath{\ll}\frac{1}{\ensuremath{\tau}}$, with $\ensuremath{\tau}$ being the pulse duration, by taking into account the temporal change of the electric field amplitude and the phase and energy relaxation times, T2{T}_{2}T2​ and T1{T}_{1}T1​. It is shown that the transient one-photon excitation known as adiabatic following, is not only important for a pulse duration $\ensuremath{\tau}<{T}_{2}$, but may also be important for ${T}_{2}\ensuremath{\lesssim}\ensuremath{\tau}<{T}_{1}$. In addition, expressions for the transient two-photon excitation in a multilevel system are derived. Finally, the two-level limit is evaluated to all orders of the perturbation including damping effects, which gives results valid also for large incident light intensities. For the "off-resonance" case an expression is obtained which describes not only the transient adiabatic following process, but also the usual one-photon absorption including saturation effects giving rise to power broadening.