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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Numerical simulation of laser pulse propagation in rare-earth-doped materials

Karl W. Beeson, Evgueni Parilov, Mary J. Potasek

Published Feb 1, 2012• 2 citations
Publisher

Abstract

We describe a general numerical method for calculating short-pulse laser propagation in rare-earth-doped materials, which are very important as gain materials for solid-state lasers, fiber lasers and optical amplifiers. The split-step, finite difference method simultaneously calculates changes in the laser pulse as it propagates through the material and calculates the dynamic populations of the rare-earth energy levels at any position within the material and for times during and after the laser pulse has passed through the material. Many traditional theoretical and numerical analyses of laser pulse propagation involve approximations and assumptions that limit their applicability to a narrow range of problems. Our numerical method, however, is more comprehensive and includes the processes of single- and multi-photon absorption, excited state absorption (ESA), energy transfer, upconversion, stimulated emission, cross relaxation, radiative relaxation and non-radiative relaxation. In the models, the rare-earth dopants can have an arbitrary number of energy levels. We are able to calculate the electron population density of every electronic level as a function of, for example, pulse energy, dopant concentration and sample thickness. We compare our theoretical results to published experimental results for rare-earth ions such as Er3+, Yb3+, Tm3+ and Ho3+.

DOI
10.1117/12.909065

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We describe a general numerical method for calculating short-pulse laser propagation in rare-earth-doped materials, which are very important as gain materials for solid-state lasers, fiber lasers and optical amplifiers. The split-step, finite difference method simultaneously calculates changes in the laser pulse as it propagates through the material and calculates the dynamic populations of the rare-earth energy levels at any position within the material and for times during and after the laser pulse has passed through the material. Many traditional theoretical and numerical analyses of laser pulse propagation involve approximations and assumptions that limit their applicability to a narrow range of problems. Our numerical method, however, is more comprehensive and includes the processes of single- and multi-photon absorption, excited state absorption (ESA), energy transfer, upconversion, stimulated emission, cross relaxation, radiative relaxation and non-radiative relaxation. In the models, the rare-earth dopants can have an arbitrary number of energy levels. We are able to calculate the electron population density of every electronic level as a function of, for example, pulse energy, dopant concentration and sample thickness. We compare our theoretical results to published experimental results for rare-earth ions such as Er3+, Yb3+, Tm3+ and Ho3+.
Many spectroscopic techniques today rely on time-resolved measurements under short excitation pulses. Instead of using a chopped pump excitation, or ultrafast optical pulses, we expand on and apply the previously developed set of frequency domain methods to analyze the population level dynamics in rare-earth-doped media. By identifying the full frequency response of the gain medium, this method can accurately yield excited state lifetimes and can also be used to estimate transition cross-sections. The accuracy of the frequency domain methods are verified with Er3+-and Tm3+-doped fibers, and an Al2O3:Tm3+waveguide, recovering similar results as reported by time-resolved techniques. The complete frequency domain model presented here can be used in characterization of novel optical gain media, and can provide insights into population dynamics in solid state amplifiers and lasers.
The enhanced laser cooling performance of rare-earth-ion-doped nanocrystalline powders is predicted, compared to the bulk material, using Yb3+:Y2O3{\mathrm{Yb}}^{3+}:{\mathrm{Y}}_{2}{\mathrm{O}}_{3}Yb3+:Y2​O3​ as the model material. This is achieved by enhancing the off-resonance, phonon-assisted absorption, which is proportional to the three factors considered in this paper: the dopant concentration, the pumping field energy, and the excitation coefficient. Using the energy transfer theory for concentration quenching, the optimum concentration corresponding to the maximum cooling power is found to be considerably larger than the currently used value, suggesting noticeable enhancement effects for laser cooling. The pumping field energy is enhanced in random nanopowders compared with bulk crystals under the same irradiation, due to the multiple scattering of photons. Photons are thus localized in the medium and do not propagate through, increasing the photon absorption of the pumping beam (and it is shown that the reabsorption of the fluorescence is negligible). Using molecular dynamics simulations, the phonon density of states (DOS) of the nanopowder is calculated, and found to have broadened modes, and extended small tails at low and high frequencies. The second-order electronic transition rate for the anti-Stokes luminescence is calculated using the Fermi golden rule, which includes the influence of this phonon DOS, and is shown to have enhancement effects on the laser cooling efficiency using nanopowders. It is finally concluded that these three enhancement mechanisms are essentially to increase the population of the three participating carriers (electron, photon, and phonon) in the interacting volume, and this also points out directions for enhancing laser cooling performance in bulk materials.
Yttrium Aluminum Garnet (YAG, Y_3 〖Al〗_5 O_12) crystals doped with Erbium have posed an interesting position in the field of rare earth solid state lasers as they possess the property of self-saturation, in which the upper energy level in a laser has a much shorter lifetime than that of the lower level. This is the case of the 2.94μm transition in Er:YAG. That property at first seems to make this material unfit for use as a laser gain medium, however further research into these classes of rare earth materials revealed interesting non-liner energy transfer mechanisms that allow it to be a useful mid wave infra-red (MWIR) coherent source in the continuous wave (CW) and pulsed regimes. This came to fruition due to decades of modeling and spectroscopic research investigating the non-linear energy transfer mechanisms of excited state absorption (ESA), energy transfer up-conversion (ETU), and cross relaxation (XR). These effects show up in the rise and fall times of the levels observed through fluorescence. Resulting in the non-exponential rise and fall characteristics and having a squared or even cubed relation to the population. The population evolution for each of the lasing levels is now affected by ions recycling energy and in turn cause the lifetimes of the levels to “effectively” change to a point where simple linear models do not adequately describe the system and its performance. The non-linear energy transfer dynamics of Er:YAG are modeled under high resonant pump conditions. By pumping with selective resonant pumps, the interaction dynamics of the (_ ^4)I_(11/2) and the (_ ^4)I_(13/2) levels in the Erbium ion reveal the contributions from generally ignored non-linear energy transfer mechanisms. Specifically, the multi-photon effect known as excited state absorption (ESA) is modeled by measuring and characterizing the cross section in single crystal Er:YAG samples utilizing a pump and probe technique coupled with transient fluorescence measurements. The measured ESA cross section is then included in the rate equation modeling.
Laser cooling of rare-earth doped solids has been demonstrated across a wide range of material platforms, inspiring the development of simple phenomenological models such as the four-level model to elucidate the universal properties of laser cooling under various operating conditions. However, these models usually require the input of full absorption spectra that must be provided experimentally or by additional complicated atomic modeling. In this Letter, we propose that a four-level model, when extended to admit effective energy levels adaptive to the pumping photon energy, can accurately predict the cooling efficiency as a function of temperature and pumping frequency using only few inputs such as the absorption coefficient measured at a single frequency and temperature. Our model exploits the quasi-equilibrium properties of the excitation of rare-earth ions for the determination of the effective four energy levels. The model is validated against published experimental results for a span of materials including ytterbium/thulium-doped glass and crystals. With the verified model, we derive explicit expressions for the optimal frequency and the operating bandwidth of pumping lasers. Our model significantly simplifies the modeling process of laser cooling and is expected to stimulate further development of optical refrigeration.
In this work, we have investigated experimentally and theoretically the process of frequency upconversion at low temperatures in rare earth doped vitreous samples. A sample of chalcogenide glass doped with erbium ions (Er3+) was pumped with a laser source at 1.54 μm. We have observed, at room temperature, three different emission bands around 530 nm, 555 nm and 670 nm, originating from the excited state erbium levels 2H11/2, 4S3/2 and 4F9/2 to the 4I15/2 ground-state, respectively. A cryogenic system cooled the sample down to 4 K, and we have recorded many spectra in this range of temperature. We have observed relative increase in the intensity of the green fluorescence, which comes from the level 4S3/2, and also the vanishing emission of the 530 nm line around 210 K. For all emission lines, the frequency upconversion process has been accounted for with three photons absorbed from the pumping source, but saturation effects were observed above 10 mW of incident power. We have made a theoretical treatment of the experimental data and a numeric simulation, which has revealed that the observed behaviors are related to processes that involve step-wise photon absorption and phonon assisted nonradiative decay, both dependent on the temperature and phonon energy of the sample. We also present, in the third chapter, an analytic treatment of pulse propagation in nonlinear media. We have used a variational approach to predict the behavior of an optical pulse propagating in the vicinity of a two-photon resonance, where the third order nonlinearities may be enhanced and limits optical devices development. A possible solution for the problem is the propagation of a pulse with suitable profile and frequency chirp resulting in a balancing of the nonlinear effects.
Two pumping schemes are considered for optical amplifiers based on rare-earth-doped optical fibers in which pump light is fed either along with the signal light (copropagating) or through the opposite end of the fiber (counterpropagating), including the effects of the excited-state absorption (ESA) process. A rate-equation analysis shows that when ESA is absent the signal gain is the same in both pumping schemes, even under gain saturation, whereas when ESA is present the saturated signal gain is larger in the counterpropagating scheme. Independently of ESA the optical signal-to-noise ratio at the amplifier output is the same in both pumping schemes if the dopant behaves as a four-level laser system, and it is larger in the copropagating scheme for a three-level dopant.