Markus Pollnau

The Boltzmann distribution is commonly interpreted as a classical approximation of the quantum-mechanical Fermi-Dirac and Bose-Einstein distributions. We arrive at a different conclusion by directly investigating the three equilibrium distributions without considering statistical variations. A general differential equation of thermal equilibrium is obtained that applies to any particles. Simple integration yields the Boltzmann distribution. Furthermore, quantum conditions are established, ruling the occupation numbers of (i) fermions by utilizing Pauli's exclusion principle and (ii) bosons by utilizing Einstein's rate-equation approach to Planck's law of blackbody radiation. By exploiting the respective quantum condition, integration of the same differential equation of thermal equilibrium yields either the Fermi-Dirac or the Bose-Einstein distribution. It suggests that the Boltzmann distribution is a general distribution of thermal equilibrium, whereas the Fermi-Dirac and Bose-Einstein distributions are special cases of the Boltzmann distribution, ruled by specific conditions. This suggestion is further underlined by the fact that inserting the respective quantum condition directly into the Boltzmann distribution delivers either the Fermi-Dirac or the Bose-Einstein distribution. This finding implies that fermions and bosons simultaneously obey both their own specific and the general Boltzmann distribution.

Rare-earth-doped solid-state lasers utilize distributed-feedback (DFB) resonators to generate ultra-narrow-linewidth emission down to a few kHz. The longitudinal modes of a single Fabry-Pérot and a DFB resonator are influenced by the structure, the optical excitation configuration, and external feedback, as has been observed experimentally. Here we investigate the influence of bi-directional launching of light and external feedback on the longitudinal modes of single Fabry-Pérot, double Fabry-Pérot, and DFB resonators. These results can be exploited for the technological advancement of DFB lasers.

The compound SrAl₂O₄:Eu²⁺,Dy³⁺ is currently one of the best‐performing persistent phosphors and numerous studies have been performed to understand the mechanisms involved in its afterglow process. One aspect which so far has received only limited attention is the dependence of loading curve characteristics on dopant concentrations. Herein, a detailed study of the loading curves of SrAl₂O₄:Eu²⁺,Dy³⁺ as a function of composition, sample temperature, and pump intensity is presented, completed by emission decay and quantum yield measurements. The observed emission decays can be described using the Inokuti−Hirayama equation for a Dexter energy transfer (ET) process. As the Dexter ET rate and the electron transfer rate have a similar radial dependence, the Inokuti−Hirayama equation can also describe the electron transfer process involved for the trapping process. These observations indicate that in this persistent phosphor, the trapping process is a local process; however, different types of traps appear to present different “Dexter” critical radii R₀. This electron transfer is also temperature dependent, which requires further investigations.

SrAl₂O₄ that is optically activated by Eu²⁺, often additionally co-doped with Dy³⁺, is a non-radioactive persistent phosphor which is known for its excellent afterglow properties. It has found various applications, e.g. in the watch industry, for security signs, in medical diagnostics, and in photovoltaics. The monoclinic SrAl₂O₄ was synthesized in polycrystalline form and structurally characterized. Its luminescence and afterglow properties were studied. Wavelength-dependent thermoluminescence experiments were performed on SrAl₂O₄:Eu and SrAl₂O₄:Eu,Dy polycrystalline samples. Substitution of Sr²⁺ by Eu²⁺ on two different Sr sites in the crystal is associated with blue and green Eu²⁺ emission. Excitation at 445 nm allows to selectively excite one of the two different Eu²⁺ ions, whereas excitation at 375 nm excites both Eu²⁺ ions. Incorporation of dysprosium increases significantly (by a factor of about 4 to 8) the total number of traps involved in the afterglow of this persistent phosphor. Increasing the temperature at which the samples are irradiated (loaded) from 173 K to 248 K reveals that many new traps can only be occupied or activated at higher temperatures, leading to a strong increase of the integrated thermoluminescence intensity, in particular for the Dy-codoped samples. The results of this study reveal that the diversity of traps leading to the long afterglow is much larger than previously reported in the literature. We propose that the presence of dysprosium induces an excitation-induced charge-transfer reaction Eu²⁺ + Dy³⁺ → Eu³⁺ + Dy²⁺. However, the principal traps responsible for the efficient afterglow are temperature-activated and appear to be associated with the green-emitting Eu²⁺ ion on the Sr2 site coupled to a nearby dysprosium ion.

Ridge waveguides in amorphous Al2O3:Yb³⁺ are produced by reactive co-sputtering and reactive-ion etching. Their spectroscopic properties, optical gain, and cooperative upconversion are studied and explained based on a model of distinct ion classes.

We perform a systematic spectroscopic study in channel waveguides of potassium gadolinium lutetium double tungstate doped with different Er³⁺ concentrations. Transition cross sections of ground-state absorption (GSA) and excited-state absorption (ESA), as well as stimulated emission (SE) at the pump wavelength around 980 nm are determined. ESA is directly measured by the pump-probe technique. Evaluation of GSA and ESA spectra indicates that ESA may be diminished by an appropriate choice of pump wavelength near 980 nm. Besides, GSA and SE at the signal wavelength around 1.5 µm are measured and the wavelength-dependent gain cross section as a function of excitation density is determined. Non-exponential luminescence decay curves from the ⁴I_13/2 and ⁴I_11/2 levels are analyzed and the probabilities of the energy-transfer-upconversion (ETU) processes (⁴I_13/2, ⁴I_13/2) → (⁴I_15/2, ⁴I_9/2) and (⁴I_11/2, ⁴I_11/2) → (⁴I_15/2, ⁴F_7/2) are quantified. Despite the large interionic distance between neighboring rare-earth sites in potassium double tungstates, the probability of ETU is comparatively large because of the large cross sections of the involved transitions. A rate-equation analysis of the influence of ETU and ESA on gain at ∼1.5 µm is performed, revealing that ETU from the ⁴I_13/2 amplifier level strongly limits the gain when the doping concentration increases above ∼6at.%. The calculated maximum achievable internal net gain per unit length amounts to ∼15 dB/cm for an optimized Er³⁺ concentration of ∼4 × 10²⁰ cm⁻³ and a launched pump power of 300 mW at a pump wavelength of 984.5 nm, in reasonable agreement with recent experimental results.

We explore the arguably most fundamental aspect of energy-transfer upconversion (ETU), namely the dependence of upconversion luminescence from a higher-energy level, following ETU excitation from a metastable lower-energy level, on direct luminescence from that metastable level. We investigate ETU among neighboring Nd³⁺ ions in single crystals of GdVO₄ and LaSc₃(BO₃)₄ with different doping concentrations by measuring, after short-pulse laser excitation with different pump energies, the infrared luminescence decay from the metastable ⁴F_3/2 level and the yellow upconversion luminescence decay from the ⁴G_7/2 level. We observe a highly super-quadratic dependence of upconversion on direct luminescence intensity. We conclude that the commonly assumed quadratic law of ETU, as proposed by Grant’s model and frequently employed in rate-equation simulations, is inadequate to the description of ETU processes. Whereas Zubenko’s model, which considers a finite migration rate, provides significantly better fits to the experimental luminescence-decay curves, also this model cannot accurately reproduce the measured decay curves, partly because it does not take the non-homogeneous distribution of active ions into account.

The laser linewidth manifests the spectral coherence of a lasing resonator mode. It is of fundamental importance, theoretically as much as practically. For more than half a century we have been told that the laser linewidth is a result of quantum fluctuations due to spontaneous emission, which induce amplitude and phase fluctuations, whereas stimulated emission occurs in phase with the incident field. Unfortunately, in such a scenario each emission process would violate the law of energy conservation. We show that a (spontaneous or stimulated) emission process occurs 90 degrees in lead of the incident field, converts one atomic excitation into one photon in the lasing mode and, therefore, conserves the energy, whereas the mentioned fluctuations are the result of vacuum energy and occur at an orders-of-magnitude faster time scale than an emission process.