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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Simulation model of accumulated programming in an optically thick spectral hole burning material

Yoann Attal, Perrine Berger, T. Chanelière, Anne Louchet-Chauvet

Published Jun 1, 2017• 1 citation
Publisher

Abstract

In this work, we present a model that describes the atomic reaction to a periodic excitation in a general case. Starting with a holeburning material, we aim at computing the steady state of the atoms of interest within the whole bulk material. This work is composed of three steps. In the first one, we generalize the optical Bloch equations to any atomic structure, keeping as long as possible the Bloch sphere picture that can provide good intuition and a better understanding of the atomic response than a huge set of equations.

DOI
10.1109/cleoe-eqec.2017.8087359

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In this work, we present a model that describes the atomic reaction to a periodic excitation in a general case. Starting with a holeburning material, we aim at computing the steady state of the atoms of interest within the whole bulk material. This work is composed of three steps. In the first one, we generalize the optical Bloch equations to any atomic structure, keeping as long as possible the Bloch sphere picture that can provide good intuition and a better understanding of the atomic response than a huge set of equations.
A large number of signal-processing protocols are based on recording a spectral pattern via spectral hole-burning in an inhomogeneously broadened absorption profile. We present a simulation method specifically designed for periodic excitation sequences leading to the creation of a spectral pattern. This method is applicable to any multilevel atomic structure. The atomic variables’ coherent dynamics are solved for a single temporal excitation step. The result is expressed as an equivalent population transfer rate. This way, the whole sequence is described as a matrix product and the steady state of the system under periodic excitation is easily derived. The propagation through the atomic medium is fully decoupled from the temporal evolution. We apply this method to the engraving of a spectral grating in a large-absorption Tm:YAG sample for wideband spectral analysis.
There have recently been several studies of the performance of laser frequency stabilization using spectral holes in solids, instead of an external cavity, as a frequency reference. Here an analytical theory for Pound-Drever-Hall laser frequency stabilization using spectral hole-burning is developed. The interaction between the atomic medium and the phase modulated light is described using a linearized model of the Maxwell-Bloch equations. The interplay between the carrier and modulation sidebands reveals significant differences from the case of locking to a cavity. These include a different optimum modulation index, an optimum sample absorption, and the possibility to lock the laser in an inherent linear frequency drift mode. Spectral holes in solids can be permanent or transient. For the materials normally used, the dynamics and time scales of transient holes often depend on population relaxation processes between ground state hyperfine levels. These relaxation rates can be very different for different solid state materials. We demonstrate, using radio-frequency pumping, that the hyperfine population dynamics may be controlled and tailored to give optimum frequency stabilization performance. In this way also materials with initially non-optimum performance can be used for stabilization. The theoretical predictions regarding the inherent linear frequency drift is compared to experimental data from a dye laser stabilized to a spectral hole in a Pr(3+)3+:Y(2)SiO(5) crystal.
The theoretical understanding of such laser stabilization exists for the case of cavities, while it is not very well developed for the case using spectral holes. Our contribution is to provide such a theory in the case of using spectral holes. The main difficulty is the requirement to calculate the coherent response of the atomic reference material to the incoming laser radiation. These calculations are inherently non-linear, but we develop a linearized, analytical theory, which is valid in the frequency stabilized condition.
A many-body analysis of spectral hole burning and light-induced band renormalization in the gain region of an initially inverted semiconductor is presented. The results are based on the numerical solution of the semiconductor Bloch equations, specialized to the case of noncopropagating pump-and-probe light fields. These equations account for coherent pump-probe scattering and include the Coulomb correlation between charge carriers. They also contain optical dephasing and carrier-carrier scattering rates that are computed within the random-phase approximation.
We study the Maxwell–Bloch model, which describes the propagation of a laser through a material and the associated interaction between laser and matter (polarization of the atoms through light propagation, photon emission and absorption, etc.). The laser field is described through Maxwell's equations, a classical equation, while matter is represented at a quantum level and satisfies a quantum Liouville equation known as the Bloch model. Coupling between laser and matter is described through a quadratic source term in both equations. The model also takes into account partial relaxation effects, namely the trend of matter to return to its natural thermodynamic equilibrium. The whole system involves 6 + N(N + 1)/2 unknowns, the six-dimensional electromagnetic field plus the N(N + 1)/2 unknowns describing the state of matter, where N is the number of atomic energy levels of the considered material. We consider at once a high frequency and weak coupling situation, in the general case of anisotropic electromagnetic fields that are subject to diffraction. Degenerate energy levels are allowed. The whole system is stiff and involves strong nonlinearities. We show the convergence to a nonstiff, nonlinear, coupled Schrödinger-rate model, involving 3 + N unknowns. The electromagnetic field is eventually described through its envelope, one unknown vector in ℂ3. It satisfies a Schrödinger equation that takes into account propagation and diffraction of light inside the material. Matter on the other hand is described through a N-dimensional vector describing the occupation numbers of each atomic level. It satisfies Einstein's rate equation that describes the jumps of the electrons between the various atomic energy levels, as induced by the interaction with light. The rate of exchange between the atomic levels is proportional to the intensity of the laser field. The whole system is the physically natural nonlinear model. In order to provide an important and explicit example, we completely analyze the specific (two-dimensional) Transverse Magnetic case, for which formulae turn out to be simpler. Technically speaking, our analysis does not enter the usual mathematical framework of geometric optics: it is more singular, and requires an ad hoc Ansatz.
So far, persistent spectral hole-burning (PSHB) phenomenon has been observed in a number of materials. They are molecules in glasses, crystals and polymers, color centers in alkali halide crystals, and rare earth and transition metal ions in glasses and crystals. They are molecular or ionic guests embedded in glass, crystal or polymer hosts. Basic requirements for the PSHB are summarized as follows: (1) The optical absorption of guests embedded in hosts must be broadened inhomogeneously. (2) There must exist more than one ground state configurations of the total system consisting of guests and hosts. (3) The relaxation among the ground states must be slower than the decay rate of any excited state. In fact, this set of requirements is satisfied in many kinds of molecular or ionic guests in hosts, such as glasses, crystals and polymers, at low temperatures. We can naturally understand that guests consisting of small number of atoms are greatly influenced by the surrounding hosts and that the energy levels of the guests are inhomogeneously broadened.