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Physical Review A

Enhancing nonlinear frequency conversion using spatially dependent coherence

Z. Kis, Emmanuel Paspalakis

Published Oct 13, 2003• 12 citations
Publisher

Abstract

We consider pulse propagation in a $\ensuremath{\Lambda}$-type medium with spatially dependent coherence. It has been shown in previous works that it is impossible to get complete nonlinear energy conversion between an injected pulse and a generated pulse for homogenous coherence distribution. The aim of our work is to achieve unity conversion efficiency. We show by analytic considerations and numerical simulations that this can be achieved only if the propagation satisfies the conditions of adiabaticity in the local frame on the position domain. We also derive an exact analytic model for pulse propagation in our $\ensuremath{\Lambda}$-type medium, which is valid even if adiabaticity is not satisfied.

DOI
10.1103/physreva.68.043817

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We consider pulse propagation in a \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-type medium with spatially dependent coherence. It has been shown in previous works that it is impossible to get complete nonlinear energy conversion between an injected pulse and a generated pulse for homogenous coherence distribution. The aim of our work is to achieve unity conversion efficiency. We show by analytic considerations and numerical simulations that this can be achieved only if the propagation satisfies the conditions of adiabaticity in the local frame on the position domain. We also derive an exact analytic model for pulse propagation in our \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-type medium, which is valid even if adiabaticity is not satisfied.
We show that nonadiabatic, resonant amplitude- and phase-modulated pulses can be frequency converted with greater efficiency than adiabatic resonant pulses in a coherently prepared $\ensuremath{\Lambda}$ system. Indeed, conversion efficiencies close to unity, similar to those achieved using highly detuned pulses, can been obtained by using highly nonadiabatic resonant pulses. Moreover, by solving the Maxwell-Bloch equations using Fourier transforms, we derive analytical expressions for the probe and the generated four-wave mixing (FWM) pulses as a function of time and propagation distance. From these expressions, which are valid for either adiabatic or nonadiabatic pulses, we derive the result that starting from a nonadiabatic probe pulse, an asymptotically matched probe-FWM pulse pair with the same shape as the initial probe pulse is obtained. In addition, we show that, starting with a nonadiabatic matched pulse pair or a pair of matched pulse trains, we obtain propagation of these pulses without either deformation or losses.
We develop a general method allowing one to construct the consistent theory of light pulse propagation through an atomic medium in arbitrary nonlinear regime with respect to the field strength, taking into account the light polarization, temporal (frequency), and spatial dispersions. The method is based on the reduced Maxwell equation, atomic density matrix formalism, and adiabatic approximation. In order to demonstrate the efficiency of our method we investigate in detail the case of propagation of the pulses under conditions of coherent population trapping in a medium of two-level atoms with degenerate energy levels. Equations describing the evolution of various field parameters are derived. It is shown that pulses of ellipticity and spatial orientation of polarization ellipse propagate with slowing. Analytical expressions for the slowing factor are obtained. Previously unknown effects of the stimulated phase modulation and the generation of a phase pilot pulse by the variation of spatial orientation of the polarization ellipse are predicted.
The propagation of a weak-field pulse through a medium of three-level atoms is considered. Each atom has a $\ensuremath{\Lambda}$-type level scheme in which the two lower levels are stable. A strong control field drives one of the electronic transitions while the signal field drives the coupled transition under conditions where EIT (electromagnetically induced transparency) is usually operative. The input pulse is slowed and compressed as it enters the medium, adiabatically following the EIT solution. However, if the control field is changed suddenly when the pulse is in the medium, the signal field is transformed into two pulses, one of which propagates as a normal EIT pulse and the other with a different speed and an amplitude that oscillates in time. The temporal oscillations are transformed into both spatial and temporal oscillations as the pulse exits the medium. An analytic expression is derived for the pulse intensity which provides a good approximation to the exact result at all times. It is shown that the oscillating component of the exiting pulse can be spatially compressed in comparison with the input pulse.
We derive an analytic solution for pulsed frequency conversion based on electromagnetically induced transparency or maximum coherence in resonant atomic vapors. In particular, drive-field and coherence depletion are taken into account. The solutions are obtained with the help of a Hamiltonian approach, which in the adiabatic limit allows one to reduce the full set of Maxwell-Bloch equations to simple canonical equations of Hamiltonian mechanics for the field variables. Adiabatic integrals of motion can be obtained and general expressions for the spatiotemporal evolution of field intensities derived. Optimum conditions for maximum conversion efficiency are identified and the physical mechanism of nonlinear conversion in the limit of drive-field and coherence depletion discussed.
The spatial and temporal dynamics of two short pulses propagating in an optically dense medium of resonant three-level Λ-atoms is investigated numerically and analytically. The maximum coherence for the Raman transition due to coherent population trapping. It is shown that, at the initial stage of propagation, the waveforms of such pulses only slightly change along the length of the medium, which may considerably exceed the length of linear absorption for a single weak pulse. As the length of the absorbing medium increases, the energy of the probe (first) pulse is completely transferred into the second (control) pulse.