Diffusion-broadened lineshape under a strong field

The Bennett hole is analysed for a medium of two-level particles with weak collisions beyond the scope of the perturbation theory. In the limit of small homogeneous width, the analytical expression for the hole shape is of the form of a Bessel function. The result is compared with data obtained by the variational approximation and numerical calculation. It is shown that the approximation gives the correct width, but an incorrect shape. A strong monochromatic field resonant with a dipole transition between two excited states of an ion equalizes their population. If the ions execute a thermal motion then only part of them hit the resonance due to the Doppler shift. Bennett holes appear in the velocity distribution of the upper and lower energy level populations [1], with the width of the Bennett hole being determined by different processes. The relaxation of field-induced polarization leads to a homogeneous width determined by the lifetime of the polarization. Effects of saturation under a strong light field give us an additional width proportional to the square root of the intensity of the light. This happens because the amplitude of the Bennett hole on any of the two levels grows, while the intensity of field is increasing for all ion velocities, but as the populations become closer this growth slows down. Moreover, during the lifetime of the excited level the ions interact with the field of other charged particles and change their velocities, then there is a diffusion width determined by the ratio of the transport collision frequency and the rate of the decay to underlying levels [2]. The theory of the diffusion or Coulomb broadening summarized in [4] predicts strong broadening when either of the two states is long-lived; however, the theory was mainly concerned with the case of a weak field and ignored field broadening of the holes due to saturation. Under the field of a relatively weak travelling electromagnetic wave the Bennett hole shape for a long-lived level is cusp-like [2]. In the case of strong saturation the problem becomes more complicated. Even when one level is short-lived the exact solution has the form of integrals with spheroidal functions, then some approximations are applied for experimental data processing. The two-parametric variational approximation [3] gives the width of the resonance with an error of less than 2%. However, this approximation gives the wrong shape, especially in the limit of a weak field. The aim of this paper is to calculate the shape of the Bennett hole in a two-level system with strongly distinct level widths when diffusion and field broadening are comparable. The long-lived upper level is typical for continuous ion lasers, the metastable lower state often occurs in the absorbing transition of an anti-Stokes Raman laser—a source of tunable short-wave coherent radiation, so the problem posed above is of interest for experiments [5]. 0953-4075/97/110377+05$19.50 c © 1997 IOP Publishing Ltd L377 L378 Letter to the Editor For definiteness let us consider a two-level system with an upper level m and lower level n, where the lower level is long-lived. To take into account the Coulomb diffusion in the velocity space one should use the density matrix formalism. Let us consider the case when the diffusion width of the Bennett hole on the upper level wmD = vT (ν/20m) is small in comparison with the homogeneous width wH = 0/k. Here 0j is the relaxation constant of level j ; 0 is the relaxation constant of polarization ρmn, k is the wavenumber of the light field; vT is thermal velocity of the ions, vT = √ 2T/m, T ,m are the temperature in units of energy and the mass of the ions, respectively; ν is the Coulomb effective transport collision frequency. This case is realistic for the upper level of a Raman ionic laser [5]. Then one can neglect the diffusion operators in the Fokker–Planck equations for the density matrix elements ρmm, ρmn [4], and the distribution of the lower state population ρn ≡ ρnn over v‖ satisfies the second-order differential equation ( 0n + 20|G| 2 T 2 + (− kv‖) ) ρn = νv 2 T 2 d2 dv2 ‖ ρn + qn + 20|G| 2 T 2 + (− kv‖) qm 0m . (1) Here qj is the excitation rate of level j , G = Edmn/2h̄  = ω − ωmn where E and ω are the amplitude and the frequency of the field, respectively, dmn is the matrix element of the dipole moment, ωmn is the Bohr frequency of the m–n transition; T is an auxiliary width T 2 = 02 + 20|G|/0m. We also neglect the frictional force which is possible while the concentration of charge carriers in the plasma is moderate. If ρn(v‖) has a width w, then for an estimate one can substitute 1/w for each derivative d/dv‖. The diffusion width wnD appears from the condition of compensation of the term 0nρn and the diffusion term. Then the friction force term νv‖(dρn/dv‖) has a term of order (ν0j )v‖ρn and we can neglect it while ν 0n (the velocity change during the lifetime 0−1 j due to the frictional force is of the order of v‖ν/0j which is small compared with the diffusional change vT (ν/0j )1/2). When |− kv‖| T we can neglect T 2 in the denominators of (1), so its solution has the form ρn = qn 0n + ( qm 0m − qn 0n ) A √ iX ( CKα(X)− S−3/2,α(iX) ) (2) where X = |−kv‖|/kwnD,A = (wF /wnD) and wF is the field width of the Bennett hole due to saturation w2 F = 20|G|/0nk, α = (A + 4 )1/2, Sμ,ν(z) is Lommel’s function [6] and C is the constant defined from joining together (2) and the solution for ρn at small X. Since the upper level is much wider than the lower one, 0n 0m, in the case of a strong field, |G|2 00n, we have wF T /k. If X ' T we can neglect 0n in the brackets in the left-hand side of (1). Then ρn has the form of a sum of some constant and two hypergeometric functions 2F1 whose parameters depend only on α. It gives us a hope that it is possible to join these two asymptotics for ρn(v‖). But there is one rougher but simpler way of doing this—under these conditions the width of the Bennett hole is much greater than T , so we can define a constant C from the condition of nonsingular behaviour of (2) at X = 0. Then C = 2 −3/2