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Annalen der Physik

Quantenmechanische Berechnung der Photonenzahl und Amplitudenfluktuation einer Laserstrahlung bei inhomogen verbreiterter Atomlinie

W. Brunner

Published Jan 1, 1968• 2 citations
Publisher

Abstract

Abstract Using the quantum mechanical equations describing the laser process we calculate the photon number and the amplitude fluctuations of the laser radiation for the stationary case, if the atomic frequencies are assumed to be distributed inhomogeneously in form of a LORENTZ line. It is shown, that, on principle, the results in comparision with the case of homogeneously broadened atomic line are unchanged. However, the photon number and therefore especially the threshold condition for the laser effect as well as the amplitude fluctuations are changed quantitatively in general due to the dependence of the line width 2ϵ of the LORENTZ line.

DOI
10.1002/andp.19684770109

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Inhomogeneous broadening

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Up to this point we have always assumed that all the atoms which interact with the radiation field have the same atomic transition frequency, with an atomic linewidth given by γ ⊥ . This configuration is called homogeneous broadening. However, in the great majority of lasers the linewidth is inhomogeneously broadened , because the atomic frequency follows a distribution. For example, in solid-state lasers such as the ruby laser and fiber lasers the inhomogeneity is due to the imperfections in the crystalline structure of the material. In gas lasers, where the atoms or molecules are free to move, the inhomogeneous broadening is due to the Doppler effect, for which a molecule moving with velocity v in a field of frequency ω and wave vector k experiences an effective frequency ω ′ = ω − k · v [8]. Since what matters is the difference between the atomic transition frequency and the field frequency, we can keep the field frequency fixed and imagine that the field interacts with different groups of atoms that move with different velocities and therefore have different transition frequencies. The realistic velocity distribution is a Gaussian, for which the corresponding distribution for the modulus of the velocity is Maxwellian. However, in the following we will mainly analyze the case of a Lorentzian distribution for the atomic transition frequency, because this allows an analytical treatment. Again for the sake of simplicity, we will limit our analysis to the case of a ring cavity. In Section 15.1 we will generalize to the inhomogeneous-broadening configuration the modal equations for the case of homogeneous broadening, which were formulated in Chapter 12. In Section 15.2 we will specialize to the single-mode model and discuss its stationary solutions for the laser case. Multimode dynamical equations The atomic transition frequency of the generic group of atoms is given by ω a + δ D , where ω a is the central atomic frequency and δ D takes positive and negative values. We call ω c the frequency of the cavity mode nearest to ω a and denote by δ ac the difference ω a − ω c .

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Optica Acta International Journal of Optics•Apr 1, 1981•12 citations
Abstract Using the quantum mechanical equations describing the laser process we calculate the photon number and the amplitude fluctuations of the laser radiation for the stationary case, if the atomic frequencies are assumed to be distributed inhomogeneously in form of a LORENTZ line. It is shown, that, on principle, the results in comparision with the case of homogeneously broadened atomic line are unchanged. However, the photon number and therefore especially the threshold condition for the laser effect as well as the amplitude fluctuations are changed quantitatively in general due to the dependence of the line width 2ϵ of the LORENTZ line.
Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly inhomogeneous line broadening due to such causes as doppler shift arising from translational velocity. The ratio of the number of molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width ∆, where E0 is the amplitude of the electric field in the incident radiation and is the dipole moment matrix element for the transition. Excitation functions obtained in a previous paper (I) for various values of the ratio γ/β, where γ is the half line-width in the absence of inhomo­geneous broadening, are used to obtain the variation of N2/N with time under conditions of mixed broadening for various values of the ratio γ/∆ and γ/β, when the exciting radiation is in exact resonance with the central frequency of the transition.
Up to this point we have always assumed that all the atoms which interact with the radiation field have the same atomic transition frequency, with an atomic linewidth given by γ ⊥ . This configuration is called homogeneous broadening. However, in the great majority of lasers the linewidth is inhomogeneously broadened , because the atomic frequency follows a distribution. For example, in solid-state lasers such as the ruby laser and fiber lasers the inhomogeneity is due to the imperfections in the crystalline structure of the material. In gas lasers, where the atoms or molecules are free to move, the inhomogeneous broadening is due to the Doppler effect, for which a molecule moving with velocity v in a field of frequency ω and wave vector k experiences an effective frequency ω ′ = ω − k · v [8]. Since what matters is the difference between the atomic transition frequency and the field frequency, we can keep the field frequency fixed and imagine that the field interacts with different groups of atoms that move with different velocities and therefore have different transition frequencies. The realistic velocity distribution is a Gaussian, for which the corresponding distribution for the modulus of the velocity is Maxwellian. However, in the following we will mainly analyze the case of a Lorentzian distribution for the atomic transition frequency, because this allows an analytical treatment. Again for the sake of simplicity, we will limit our analysis to the case of a ring cavity. In Section 15.1 we will generalize to the inhomogeneous-broadening configuration the modal equations for the case of homogeneous broadening, which were formulated in Chapter 12. In Section 15.2 we will specialize to the single-mode model and discuss its stationary solutions for the laser case. Multimode dynamical equations The atomic transition frequency of the generic group of atoms is given by ω a + δ D , where ω a is the central atomic frequency and δ D takes positive and negative values. We call ω c the frequency of the cavity mode nearest to ω a and denote by δ ac the difference ω a − ω c .
Abstract Extending the method developed in a previous paper [1] the correlation functions both for electric field strength and intensity are calculated. In the latter case the result is obtained quite easily by solving a differential equation. In the first case, on the contrary, a detailed knowledge of the photon creation operator b+ is necessary, especially of the different temporal behaviour with respect to phase and amplitude. In the main, the results are in agreement with those published by other authors. Of special interest is the result that the damping constant for intensity correlations tends to the value corresponding to the case of thermal radiation, in the region far from threshold. In addition, the ansatz b+ = (ro + ϱ) exp{i χ(t)} (ro c‐number) introduced by HAKEN is discussed from a quantum mechanical point of view.
A quantum-statistical analysis of multi-photon lasers including the effect of detuning and atomic motion is presented. The equations are solved to fourth order in the atom-field coupling constant. Systems in detailed balance (with multi-photon loss mechanism) as well as systems without detailed balance (with single-photon loss mechanism) are considered separately. Treatments for the single-mode and multi-mode processes are given. Expressions for threshold conditions, peak photon number and the half width of photon distribution are obtained. The variation of mean output photon number, the effect of detuning, the coherence characteristics, etc. are studied and results are illustrated for two- and three-photon lasers as examples.