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CRC Press eBooks

GAS LASER THEORY

Murray, Iii Sargent

Published Mar 5, 2018• 0 citations
Publisher

Abstract

In Chaps. 8 and 9, we considered a laser medium consisting of homo­ geneously broadened, stationary atoms. When the lasing atoms move as in a gas, they see an electric field with shifted frequency due to the Doppler effect, as shown in Fig. 10-1. One might argue that it is merely necessary to average the complex polarization or the susceptibility Xn over the frequency range corresponding to the velocity distribution, that is, calculate a new ^ n , for which Here the frequency co = coo + Kv, where coo is the frequency at atomic line (1) center, v is the component of velocity along the laser (z) axis, and the frequen­ cy distribution is determined by a Maxwell-Boltzmann velocity distribution W(y) = ( w)-1 exp [ - (v/w)2]. (2) Here u is the most probable speed of the atom, and K is the wave number. The simple recipe (1) is, in fact, valid for an inhomogeneously broadened mediumt consisting of stationary atoms such as ruby at low temperatures. However, in the standing-wave laser, the atoms not only see Doppler-shifted frequencies, but also move through the standing-wave electric field, effectively seeing an amplitude-modulated field. Equivalently, each atom sees two frequencies, for the standing-wave field is the sum of oppositely directed running waves, one of which appears to be Doppler upshifted, one down­ shifted. Hence it is necessary to consider both the frequency shift and the time dependence of the atomic z coordinate when calculating the polarization of a gaseous medium.

DOI
10.1201/9780429493515-17

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In the present work, an analytical calculation of the value of xo is presented for the same physical conditions as in [5, 6], with one additional approximation. In a gas laser, movement of the atoms has the result that the periodic structure of the standing wave field of the laser modes is weakly reflected in the spatial distribution of the population inversion [7]. On this basis we shall carry out the calculation of xo assuming that the intensity of the fields ml and ma is constant along the axis of the laser. We thus exclude from consideration a certain range of phenomena related to the effect of a spatial displacement between the standing waves m~ and ma on the mode competition. We consider this reasonable, since those phenomena were analyzed in sufficien.t detail in [3], where the basic attention of the work was directed at analysis of the competition effect for various values of q. Following Evseev et al. [5,6] we shall use the fact that, near the boundaries of the two-frequency and slngle-frequency regimes (x =  the intensity of one of the modes is much larger than the intensity of the other, almost to the point of suppression. We shall assume at the outset that lasing takes place in one strong mode, for example m,, caused by saturation of the active medium. We shall initially assume the mode ma to be suppressed and '~eak, i.e., not saturating the active medium. In the steady state, the saturated amplification gt of the mode m~ in the active medium is equal to the losses T. The saturated amplification ga of the weak mode may be larger than, smaller than, or equal to T, depending on x. In the first case, the field of the mode ma will grow and two-frequency operation, not slngle-frequency, will be stable, while the third is realized for x =  Based on the results of [8] it is possible to obtain the following expressions for gt and ga:

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In Chaps. 8 and 9, we considered a laser medium consisting of homo­ geneously broadened, stationary atoms. When the lasing atoms move as in a gas, they see an electric field with shifted frequency due to the Doppler effect, as shown in Fig. 10-1. One might argue that it is merely necessary to average the complex polarization or the susceptibility Xn over the frequency range corresponding to the velocity distribution, that is, calculate a new ^ n , for which Here the frequency co = coo + Kv, where coo is the frequency at atomic line (1) center, v is the component of velocity along the laser (z) axis, and the frequen­ cy distribution is determined by a Maxwell-Boltzmann velocity distribution W(y) = ( w)-1 exp [ - (v/w)2]. (2) Here u is the most probable speed of the atom, and K is the wave number. The simple recipe (1) is, in fact, valid for an inhomogeneously broadened mediumt consisting of stationary atoms such as ruby at low temperatures. However, in the standing-wave laser, the atoms not only see Doppler-shifted frequencies, but also move through the standing-wave electric field, effectively seeing an amplitude-modulated field. Equivalently, each atom sees two frequencies, for the standing-wave field is the sum of oppositely directed running waves, one of which appears to be Doppler upshifted, one down­ shifted. Hence it is necessary to consider both the frequency shift and the time dependence of the atomic z coordinate when calculating the polarization of a gaseous medium.
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 .
In the present work, an analytical calculation of the value of xo is presented for the same physical conditions as in [5, 6], with one additional approximation. In a gas laser, movement of the atoms has the result that the periodic structure of the standing wave field of the laser modes is weakly reflected in the spatial distribution of the population inversion [7]. On this basis we shall carry out the calculation of xo assuming that the intensity of the fields ml and ma is constant along the axis of the laser. We thus exclude from consideration a certain range of phenomena related to the effect of a spatial displacement between the standing waves m~ and ma on the mode competition. We consider this reasonable, since those phenomena were analyzed in sufficien.t detail in [3], where the basic attention of the work was directed at analysis of the competition effect for various values of q. Following Evseev et al. [5,6] we shall use the fact that, near the boundaries of the two-frequency and slngle-frequency regimes (x =  the intensity of one of the modes is much larger than the intensity of the other, almost to the point of suppression. We shall assume at the outset that lasing takes place in one strong mode, for example m,, caused by saturation of the active medium. We shall initially assume the mode ma to be suppressed and '~eak, i.e., not saturating the active medium. In the steady state, the saturated amplification gt of the mode m~ in the active medium is equal to the losses T. The saturated amplification ga of the weak mode may be larger than, smaller than, or equal to T, depending on x. In the first case, the field of the mode ma will grow and two-frequency operation, not slngle-frequency, will be stable, while the third is realized for x =  Based on the results of [8] it is possible to obtain the following expressions for gt and ga:
In this chapter we treat single-mode laser operation in which both the atoms and the field obey the laws of quantum mechanics. For simplicity, we suppose for most of the discussion that the field is tuned to atomic line center and that the active atoms are two-level systems which pass through the cavity in a time r, as in Sec. 16-1. To simulate a homogeneously broadened medium, we suppose that the transit times are distributed with the probability y exp(— yr). The more realistic (and complicated) calculation underlying the theory of Chap. 8 is given in Appendix I. We also suppose that excitation occurs only to the upper level; Probs. 17-13 and 17-14 consider excitation to the lower level as well. We describe cavity losses by an ensemble of nonresonant atoms injected into the lower level, as discussed in Sec. 16-1. The response of these “loss” atoms does not saturate (i.e., small coupling constant), as does that of the active atoms. The loss atoms here play the same role as the fictitious con­ ductivity in the semiclassical laser theory of Chap. 8. As shown in Sec. 16-5, an equivalent description is obtained by a reservoir of simple harmonic os­ cillators with a spread of frequencies. Our discussion depends on the reservoir theory of Sec. 16-1, in which the reduced density operator represents the ra­ diation field. In Chap. 20 we give a related development in which the Langevin (Heisenberg picture) approach is used.
The advent of the laser opened the way to the development of powerful methods for producing and manipulating cold atomic gases. To set the stage we describe a typical experiment, which is shown schematically in Fig. 4.1. A beam of sodium atoms emerges from an oven at a temperature of about 600 K, corresponding to a speed of about 800 m s −1 , and is then passed through a so-called Zeeman slower, in which the velocity of the atoms is reduced to about 30 m s −1 , corresponding to a temperature of about 1 K. In the Zeeman slower, a laser beam propagates in the direction opposite that of the atomic beam, and the radiation force produced by absorption of photons retards the atoms. Due to the Doppler effect, the frequency of the atomic transition in the laboratory frame is not generally constant, since the atomic velocity varies. However, by applying an inhomogeneous magnetic field designed so that the Doppler and Zeeman effects cancel, the frequency of the transition in the rest frame of the atom may be held fixed. On emerging from the Zeeman slower the atoms are slow enough to be captured by a magneto-optical trap (MOT), where they are further cooled by interactions with laser light to temperatures of order 100 μK. Another way of compensating for the changing Doppler shift is to increase the laser frequency in time, which is referred to as ‘chirping’.
A formal treatment of a traveling-wave optical maser is presented in which an assumed electromagnetic field in a rotating cavity, obeying Maxwell's equations, nonlinearly polarizes the moving gaseous atoms. The interaction is treated quantum mechanically in the frame of the moving atom. The resultant polarization, statistically summed over all velocity ensembles, is used as a source term in Maxwell's equations. The self-consistency gives a set of equations which determine the amplitudes and frequencies of oscillation of the modes of the independent oppositely directed traveling waves in terms of the parameters of the system. The results reduce to those obtained by Lamb for the case of a standing-wave optical maser with a stationary cavity. In addition, stability conditions on the oppositely directed waves are obtained for the cases where the active medium is a single isotope and a mixture of two isotopes.