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Physical Review Letters

Spectral response and propagational instability for pump-probe propagation in a two-level medium

Michael E. Crenshaw, C. M. Bowden

Published Sep 2, 1991• 4 citations
Publisher

Abstract

The interaction of a strong nearly resonant cw pump laser and a weak cw probe laser in a medium consisting of two-level atoms was studied by numerical integration of the Maxwell-Bloch equations. With propagation into the optically thick medium, the numerically calculated probe gain spectrum deviates substantially from one calculated by a four-wave-mixing formulation of coupled-wave theory with an undepleted pump. For probe frequencies inside the modified gain region, the laser field, initially in an unstable steady-state condition, evolves to a stable solution of the sine-Gordon equation.

DOI
10.1103/physrevlett.67.1226

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Optics Letters•Sep 1, 1982•45 citations
The interaction of a strong nearly resonant cw pump laser and a weak cw probe laser in a medium consisting of two-level atoms was studied by numerical integration of the Maxwell-Bloch equations. With propagation into the optically thick medium, the numerically calculated probe gain spectrum deviates substantially from one calculated by a four-wave-mixing formulation of coupled-wave theory with an undepleted pump. For probe frequencies inside the modified gain region, the laser field, initially in an unstable steady-state condition, evolves to a stable solution of the sine-Gordon equation.
We investigated the sideband instability by numerical integration of the Maxwell-Bloch equations for propagation of a nearly resonant pump laser and a weak probe laser in a system of two-level atoms. In an optically thick medium, the probe gain (absorption) spectrum becomes a function of propagation distance. We find that the well-known gain resonance at the generalized Rabi frequency Ω' is suppressed and a broad asymmetric gain region forms on each side of the laser frequency. For probe-pump detunings inside these gain regions, the field envelope breaks up into periodic traveling waves that are compared to the polycnoidal solutions of the sine-Gordon equation with periodic boundary conditions.
We examine the effect of propagation on the resonance Rabi sideband of the four-wave mixing (FWM) spectrum, obtained when short temporally displaced pump and probe pulses interact with an optically thick medium of two-level atoms. We find that the dependence of the time-integrated FWM signal on the pump-probe delay is considerably altered by propagation. In particular, the logarithm of the FWM signal, for the case where the probe precedes the pump, deviates from linearity and may even increase over a range of values. An explanation is given in terms of the overlap of the pump envelope with the coherent response of the atomic system to the probe, both of which are modified on propagation.
The propagation of a strong coherent "pump" wave traveling through a resonant medium is discussed. The medium is assumed to consist of identical stationary atoms, and the pump-wave frequency is assumed to coincide exactly with the resonance frequency for transitions between a particular pair of atomic states. A previously developed theory of the response of a strongly driven atom to a weak nearly resonant "signal" field is used to deduce the existence of a coupling between any two waves travelling parallel to the pump wave whose (nearly equal) frequencies sum to twice the pump-wave frequency. In the limit of very intense (highly saturating) pump fields, the coupling between the two waves of nearby frequencies leads to amplification of both waves if their frequencies lie within an interval about the pump-wave frequency equal to the Rabi frequency of population inversion. The basic process, which is roughly described by travelling-wave parametric-amplifier equations of motion, consists of the absorption of two pump-field photons followed by their emission at different nearby frequencies, and thus implies a frequency instability of the initially coherent pump wave.
Experimental and numerical investigation of single-beam and pump-probe interaction with a resonantly absorbing dense extended medium under strong and weak field-matter coupling is presented. Significant probe beam amplification and conical emission were observed. Under relatively weak pumping and high medium density, when the condition of strong coupling between field and resonant matter is fulfilled, the probe amplification spectrum has a form of spectral doublet. Stronger pumping leads to the appearance of a single peak of the probe beam amplification at the transition frequency. The greater probe intensity results in an asymmetrical transmission spectrum with amplification at the blue wing of the absorption line and attenuation at the red one. Under high medium density, a broadband of amplification appears. The theoretical model is based on the solution of the Maxwell-Bloch equations for a two-level system. Different types of probe transmission spectra obtained are attributed to complex dynamics of a coherent medium response to broadband polychromatic radiation of a multimode dye laser.
Analytic and numerical calculations for the propagational dependence of Rabi-Split resulting from a strong resonant pump and a coupled weak off-resonance probe are presented for a Lambda three-level atomic configuration. A semi-classical Maxwell-Bloch formulism was adopted for both the perturbational (using stationary phase and steepest descent asymptotic techniques) and the rigorous computational treatments. This calculation can be considered as a Double Self-Induced Transparency SIT' where the strong pump (multiple of 2n on-axis area) experiences coherent pulse break up, coherent self-focusing 2-4 and depletion as the weak probe (multiple of 0.02n area) cooperatively builds up. Different transition gain ratios (i.e., oscillator strength: p2w) are considered to insure that the weak probe does not get (a) delayed with respect to the pump or (b) get out of synchronization, and cease to overlap. The probe detuning can be as large as the input on-axis pump Rabi frequency. The interplay of nonlinear Raman gain action, dispersion and diffraction give rise concommitantly to a number of effects previously studied independently such as self-phase modulation, wave front encoding, transverse ring formation, self-focusing, quasi-trapping (when pump detuning is also allowed), and asymptotic three-level solitary waves.
A self-consistent description is given for the propagation of a weak optical probe beam through a homogeneously broadened two-level medium in the presence of a strong, collinear, near-resonant pump beam. The propagation of the probe beam is given in terms of bichromatic natural modes, which contain frequency components symmetrically displaced from that of the pump beam. The two frequency components have a definite relative amplitude and phase, which are determined solely by the frequency detunings of the optical fields from the atomic resonance and by the intensity of the pump field. A simple method for calculating these natural modes is derived, and their importance in describing nearly degenerate four-wave mixing, optical bistability, and the stability of homogeneously broadened ring lasers is discussed.