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Annalen der Physik

Quantenmechanische Behandlung des Lasers mit nichtresonanter Rückkopplung. III Berechnung des Frequenzspektrums

W. Brunner, Henryk Paul

Published Jan 1, 1969• 0 citations
Publisher

Abstract

Abstract Neglecting intensity fluctuations of the radiation the mean photon number per mode, nλ, is calculated for a laser with nonresonant feedback in case of a homogenously as well as an inhomogenously broadened atomic line. The dependence of nλ on the frequency ωλ displays the frequency spectrum of the radiation which is found to be narrowed with respect to the spontaneous line. The linewidth, nevertheless, is much greater than in the case of a 1‐mode‐laser (with resonant feedback).

DOI
10.1002/andp.19694790105

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Acta Optica Sinica•Jan 1, 1985•0 citations
Abstract Neglecting intensity fluctuations of the radiation the mean photon number per mode, nλ, is calculated for a laser with nonresonant feedback in case of a homogenously as well as an inhomogenously broadened atomic line. The dependence of nλ on the frequency ωλ displays the frequency spectrum of the radiation which is found to be narrowed with respect to the spontaneous line. The linewidth, nevertheless, is much greater than in the case of a 1‐mode‐laser (with resonant feedback).
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.
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 Previous calculations [1] yielding the frequency spectrum of a one‐mode radiation field interacting with a single atom are extended to the case of M atoms. Under the assumption M „N (N mean photon number) the total Hamiltonian is diagonalized, and the correlation function 〈𝔈̂(t1 𝔈̂(t2)〉 revealing the frequency spectrum of the interacting field calculated quite generally. The following special initial conditions are discussed in detail: a) Any atom is either in the upper or the lower state, and b) the electric dipole moment of the ensemble of atoms is oscillating in phase or with a phase difference of π with respect to the electric field strength. In case a) the initial monochromatic line is broadened, and in case b) a frequency shift appears.
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 homogeneous line-broadening. The ratio of the number of atoms or 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 γ. E0 is the amplitude of the electric field in the incident radiation and the dipole moment matrix element for the transition. Exact expressions for N2/N as a function of time are given for the situation when the frequency ω/2π of the radiation is exactly equal to that of the transition ω0/2π. Numerical calculations are made for the situation when the ‘detuning’ δ ═ ω — ω0 is not equal to zero. Approximate expressions are also derived in terms of δ for use in a later paper which deals with mixed homogeneous and inhomogeneous line-broadening.