Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Optics Express

Saturation of radiation trapping and lifetime measurements in three-level laser crystals

Ching-Hsu Chen, Yue-Heng Wu, Cheng-Ping Fan, Tai-Hei Wei

Published Oct 26, 2012• 10 citations
Publisher

Abstract

In this study, we take the pump rate into consideration for the first time to give a theoretical description of radiation trapping in three-level systems. We numerically verify that under strong pumping, the population of the ground state is depleted, which leads to saturation of the radiation trapping within the pumped region. This saturation inevitably clamps the lifetime lengthening that is experimentally verified on a 0.05 at% thin ruby crystal based on the axial pinhole method. Our model is confirmed to be valid in lifetime measurement when the ruby fluorescence is collected from both the pumped and the unpumped regions.

DOI
10.1364/oe.20.025613

Related papers

Measurement of upper level lifetime in presence of radiation trapping: A revisitation of the pinhole method

Guido Toci, Angela Pirri, Daniele Alderighi +1 more

In this work we expose a recently developed theoretical model for the analysis of the measurements obtained with the so-called pinhole method, for the determination of the upper level lifetime in materials where the superimposition between fluorescence and absorption spectra determines significant radiation trapping effect. Under fairly general conditions the fluorescence decay curve after a pulsed excitation of the sample obtained with this experimental set-up can be conveniently described by a double exponential functional form. The faster of the two decay times is close to the intrinsic fluorescence lifetime, with a difference that can be calculated with geometrical considerations. The theoretical results were tested with a suitably designed experiment, where the upper level lifetime of the laser transition of some Yb doped samples were investigated, obtaining results that are consistent with literature data and in good agreement with the theoretical results as for the temporal dependence of the fluorescence decay.

Laser Physics•Nov 29, 2012•3 citations

Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials

Stéphan Guy

Radiation trapping causes significant lengthening of the measured fluorescence lifetimes in solid samples, which leads to overestimating them. Self-absorption inside solid-state samples is considerably enhanced by successive total internal reflections at the sample/air interface. A simple method for quantitative estimation of radiation trapping in solid-state materials within the frame of the Holstein-Biberman equations is presented. This method is based on the assumption that the radiation propagation follows paths with many total internal reflections; it can be applied to many resonant two-level systems such as ${\mathrm{Er}}^{3+}$-, ${\mathrm{Yb}}^{3+}$-, or ${\mathrm{Ho}}^{3+}$-doped materials. The equations are solved for a pulsed excitation localized in a finite volume inside the sample. In our modeling approach, we take into account the initial geometric population distribution and the imaging setup. We derive analytical expressions of the measured decays, which give the deformation of the measured curves compared with the ideal exponential decay. We investigate the effect of the fraction of self-trapped light on the decay modification. We then show that the ratio between the pumping volume, the collected volume, and the sample dimension changes the measured decay significantly.

Physical Review B•Apr 4, 2006•32 citations

Analytical approach to dynamics of reverse saturable absorbers

Andrey E. Kobyakov, David J. Hagan, Eric W. Van Stryland

We develop an analytical approach to the dynamics of band populations of reverse saturable absorbers modeled by the three-level approximation of the five-level rate equations. We find high-accuracy approximate solutions to these rate equations, taking into account the temporal shape of the incident laser pulse for different regimes of excitation. The results obtained are confirmed by direct numerical integration of the rate equations and are verified by solution of the full system of the rate and the propagation equations. The validity ranges of the approximations are determined. We also prove that for input pulses that are much longer than the lifetime of the first excited state the ground-state depletion obeys the same functional dependence on the input fluence as in the case of rectangular input pulses. The dynamics of the excited states, however, explicitly depends on the pulse shape. We quantitatively estimate the effect of various parameters on the nonlinear absorption coefficient and discuss implementation of the approach by a beam propagation method to reduce the computational time.

Journal of the Optical Society of America B•Nov 1, 2000•36 citations

Nonlinear radiation trapping in an atomic vapor excited by a strong laser pulse

Nikolai N. Bezuglov, Andrey N. Klyucharev, A. F. Molisch +3 more

We investigate radiation trapping in an atomic vapor which has been excited by a strong short laser pulse. Since the saturation of the vapor by the pulse leads to a decrease in the effective absorption coefficient, the radiation trapping becomes nonlinear. We derive approximate analytical expressions for the excited-state density in the directly excited region, the fluorescence-excited region, and the density averaged over the whole cell. Starting out from fairly simple approximate expressions based on a prescribed distribution of excited atoms, we then develop physically motivated correction factors that drastically improve the accuracy. All these expressions are given for three important cell geometries: the plane-parallel slab, the infinite cylinder, and the sphere. We compare our results to accurate numerical solutions, and find agreement within 5--10%. We then derive the decay time of the emergent radiation, and find that it can be smaller than the natural lifetime of the excited atoms, in agreement with recent experimental results obtained for sodium vapors.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Mar 1, 1997•17 citations

A theoretical study of saturation effects in three-level laser systems

Yakir Reuven, Michael Baer

Exact numerical solution of a time-space (one dimensional) three-level lasing system (homogeneously broadened) is used to study the relevance of steady-state models to such a system. This study revealed: 1) the existence of two disconnected (quasi) steady-state regions-the small-signal gain region (slow pumping rate) and the high saturation region (large pumping rate), and 2) the spatially averaged equations for the high saturation region have a form similar to those usually employed in the small-signal gain region, except that a constant which in the latter case is unity becomes smaller than 1 in the high saturation region.

IEEE Journal of Quantum Electronics•Oct 1, 1980•7 citations

Free of reabsorption upper-state lifetime measurements by the method of transient gratings

I. G. Kisialiou

Light reabsorption and radiation trapping can noticeably distort fluorescence decay in solid-state laser gain media with a partial overlap of absorption and emission bands. In this paper, the upper-state lifetime measurements in laser materials by the method of transient gratings are discussed. A theoretical model for description of reabsorption in a thin layer is developed. It is shown that under certain conditions reabsorption has no effect on the transient time of refractive index grating and thus the true upper-state lifetime can be measured. Some experimental results confirming effectiveness of the technique are presented.

Applied Optics•Jul 26, 2012•5 citations
In this study, we take the pump rate into consideration for the first time to give a theoretical description of radiation trapping in three-level systems. We numerically verify that under strong pumping, the population of the ground state is depleted, which leads to saturation of the radiation trapping within the pumped region. This saturation inevitably clamps the lifetime lengthening that is experimentally verified on a 0.05 at% thin ruby crystal based on the axial pinhole method. Our model is confirmed to be valid in lifetime measurement when the ruby fluorescence is collected from both the pumped and the unpumped regions.
In this work we expose a recently developed theoretical model for the analysis of the measurements obtained with the so-called pinhole method, for the determination of the upper level lifetime in materials where the superimposition between fluorescence and absorption spectra determines significant radiation trapping effect. Under fairly general conditions the fluorescence decay curve after a pulsed excitation of the sample obtained with this experimental set-up can be conveniently described by a double exponential functional form. The faster of the two decay times is close to the intrinsic fluorescence lifetime, with a difference that can be calculated with geometrical considerations. The theoretical results were tested with a suitably designed experiment, where the upper level lifetime of the laser transition of some Yb doped samples were investigated, obtaining results that are consistent with literature data and in good agreement with the theoretical results as for the temporal dependence of the fluorescence decay.
Radiation trapping causes significant lengthening of the measured fluorescence lifetimes in solid samples, which leads to overestimating them. Self-absorption inside solid-state samples is considerably enhanced by successive total internal reflections at the sample/air interface. A simple method for quantitative estimation of radiation trapping in solid-state materials within the frame of the Holstein-Biberman equations is presented. This method is based on the assumption that the radiation propagation follows paths with many total internal reflections; it can be applied to many resonant two-level systems such as Er3+{\mathrm{Er}}^{3+}Er3+-, Yb3+{\mathrm{Yb}}^{3+}Yb3+-, or Ho3+{\mathrm{Ho}}^{3+}Ho3+-doped materials. The equations are solved for a pulsed excitation localized in a finite volume inside the sample. In our modeling approach, we take into account the initial geometric population distribution and the imaging setup. We derive analytical expressions of the measured decays, which give the deformation of the measured curves compared with the ideal exponential decay. We investigate the effect of the fraction of self-trapped light on the decay modification. We then show that the ratio between the pumping volume, the collected volume, and the sample dimension changes the measured decay significantly.
We develop an analytical approach to the dynamics of band populations of reverse saturable absorbers modeled by the three-level approximation of the five-level rate equations. We find high-accuracy approximate solutions to these rate equations, taking into account the temporal shape of the incident laser pulse for different regimes of excitation. The results obtained are confirmed by direct numerical integration of the rate equations and are verified by solution of the full system of the rate and the propagation equations. The validity ranges of the approximations are determined. We also prove that for input pulses that are much longer than the lifetime of the first excited state the ground-state depletion obeys the same functional dependence on the input fluence as in the case of rectangular input pulses. The dynamics of the excited states, however, explicitly depends on the pulse shape. We quantitatively estimate the effect of various parameters on the nonlinear absorption coefficient and discuss implementation of the approach by a beam propagation method to reduce the computational time.
We investigate radiation trapping in an atomic vapor which has been excited by a strong short laser pulse. Since the saturation of the vapor by the pulse leads to a decrease in the effective absorption coefficient, the radiation trapping becomes nonlinear. We derive approximate analytical expressions for the excited-state density in the directly excited region, the fluorescence-excited region, and the density averaged over the whole cell. Starting out from fairly simple approximate expressions based on a prescribed distribution of excited atoms, we then develop physically motivated correction factors that drastically improve the accuracy. All these expressions are given for three important cell geometries: the plane-parallel slab, the infinite cylinder, and the sphere. We compare our results to accurate numerical solutions, and find agreement within 5--10%. We then derive the decay time of the emergent radiation, and find that it can be smaller than the natural lifetime of the excited atoms, in agreement with recent experimental results obtained for sodium vapors.
Exact numerical solution of a time-space (one dimensional) three-level lasing system (homogeneously broadened) is used to study the relevance of steady-state models to such a system. This study revealed: 1) the existence of two disconnected (quasi) steady-state regions-the small-signal gain region (slow pumping rate) and the high saturation region (large pumping rate), and 2) the spatially averaged equations for the high saturation region have a form similar to those usually employed in the small-signal gain region, except that a constant which in the latter case is unity becomes smaller than 1 in the high saturation region.
Light reabsorption and radiation trapping can noticeably distort fluorescence decay in solid-state laser gain media with a partial overlap of absorption and emission bands. In this paper, the upper-state lifetime measurements in laser materials by the method of transient gratings are discussed. A theoretical model for description of reabsorption in a thin layer is developed. It is shown that under certain conditions reabsorption has no effect on the transient time of refractive index grating and thus the true upper-state lifetime can be measured. Some experimental results confirming effectiveness of the technique are presented.