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Zeitschrift für Naturforschung A

Doppler Line Broadening from the Asymptotic Cauchy Distribution

Bernard H. Lavenda

Published Apr 1, 1993• 2 citations
Publisher

Abstract

Abstract The half-width of Doppler broadening is derived without assuming Maxwell's distribution for the velocities of the radiating atoms. In the absence of Doppler broadening there is only natural radiation broadening, and by thermalizing the asymptotic Cauchy distribution one obtains asymp-totic double exponential distributions for the largest (blue) and smallest (red) values of the frequency shift. These are the relevant extreme value distributions for largest and smallest values. Since the mean values are opposite and equal there is a zero shift in the line, while the standard deviation gives an expression for the half-width to within 2% of the Doppler half-width obtained from the Maxwellian when the damping coefficient is taken as the ratio of the natural radiation damping coefficient to the fine structure constant.

DOI
10.1515/zna-1993-0403

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Journal of Physics B Atomic and Molecular Physics•Jun 1, 1976•41 citations
Abstract The half-width of Doppler broadening is derived without assuming Maxwell's distribution for the velocities of the radiating atoms. In the absence of Doppler broadening there is only natural radiation broadening, and by thermalizing the asymptotic Cauchy distribution one obtains asymp-totic double exponential distributions for the largest (blue) and smallest (red) values of the frequency shift. These are the relevant extreme value distributions for largest and smallest values. Since the mean values are opposite and equal there is a zero shift in the line, while the standard deviation gives an expression for the half-width to within 2% of the Doppler half-width obtained from the Maxwellian when the damping coefficient is taken as the ratio of the natural radiation damping coefficient to the fine structure constant.
We generalize the two-photon probe-absorption coefficient to include any amount of Doppler broadening and an arbitrarily intense saturator wave. By analytically performing integrals over a Lorentzian velocity distribution we are able to examine the transition from homogeneous broadening to extreme Doppler broadening for both zero and nonzero Stark shifts. For Doppler widths much larger than the saturator Rabi flopping frequency, the probe-absorption spectrum approaches a Doppler-free Lorentzian with width equal to the homogeneous linewidth and center displaced by the dynamic Stark shift.
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 By means of a direct recording Fabry-Perot interferometer, the contours of several singlet lines of helium were studied over a wide range of pressure and temperature. The source was a microwave discharge cooled with liquid nitrogen or liquid helium. Analysis of the tracings allowed the Doppler and Lorentz components to be separated quantitatively. A detailed study was carried out for the line λ 7281 Å (21P - 31S). By extrapolation of the Lorentz width to zero pressure the radiation width of the level 2 was determined and led to an ƒ-value 0·377 + 0·035 of the resonance line λ 584 Å (11S— 21P) The resonance broadening is found to be independent of the speed of the atoms, over a range of a factor five in the speed, in agreement with theories of optical collision by Weisskopf and others. The absolute value of the resonance broadening constant is compared with various theories. The strong line λ 6678 Å (21P -31D) showed self-absorption due to imprisonment of resonance radiation.
A previously developed theory of spontaneous emission from driven systems has been applied to the case of a Doppler broadened gas of two level atoms radiating into free space. The power spectrum of those photons radiated nearly parallel to the propagation vector of the driving field is predicted to differ from that of the undriven system. The case of zero attenuation coefficient (population equality in the two resonance levels) and the driving field tuned to the center of the Doppler line, is investigated in particular. When the saturation parameter is large there is strong suppression, in an interval about the central frequency comparable to the saturated width, of the spontaneous emission whose spectral components are phased for amplitude modulation relative to the driving field. An amount of power equal to that suppressed, but with spectral components phased for phase modulation relative to the driving field, is radiated in an interval about the central frequency comparable to the homogeneous width.
For pt.I see ibid., vol.8, p.1795 (1975) where the theory of Doppler-free two-photon transitions was discussed for the case of one single laser frequency. Here, the effects of two different frequencies are considered. When the residual Doppler width of the two-photon transition is less than the natural linewidth, the previous results are essentially regained. For large residual Doppler shifts, the absorption profile is a power-broadened Lorentzian if the frequency of the probe field is larger than the frequency of the strong field. In the opposite case, the absorption peak splits because of the AC Stark effect. The high-intensity distortions are minimized when equal frequencies can be used. Power-dependent shifts do not ordinarily occur with a resonantly tuned intermediate level. Some modifications of the spectra are introduced by phase-disturbing collisions.