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Physical Review

Theory of an Inhomogeneously Broadened Laser Amplifier

Frederic A. Hope, Marlan O. Scully

Published Mar 10, 1969• 166 citations
Publisher

Abstract

Recent advances in laser technology have led to the production of ultrashort electromagnetic pulses of very high peak power. In this paper we develop a theory which describes the transient behavior of such pulses in a laser amplifier. The effects of atomic coherence and inhomogeneous broadening, necessary for a complete treatment of the problem, are included. In the present development, the degree of homogeneous and inhomogeneous broadening is variable. The two-level active atoms are characterized by a phase memory time ${T}_{2}$ and an inhomogeneous frequency distribution (which leads to a reversible decay time $T_{2}^{}{}_{}{}^{*}$). The field (Maxwell) and atomic (Schr\"odinger) equations are coupled in a self-consistent manner as in Lamb's theory of an optical maser. The resulting equations are solved analytically in certain limits, while the general case is integrated numerically. The emphasis is placed on the situation in which $T_{2}^{}{}_{}{}^{*}$ and the pulse width are much shorter than ${T}_{2}$. Thus our main effort is devoted to the physics of the inhomogeneously broadened system. The theory itself, however, contains the effects of both forms of broadening. The basic difference between inhomogeneous and homogeneous broadening is discussed. In the former case, the decay of radiation is caused by the dephasing of dipoles, and is reversible. The atomic memory is retained during the dephasing, and an internal reflection of the atomic coordinates, as is exemplified in the "photon echo" process, will cause the dipoles to rephase. The phonon interruptions or atomic collisions that bring about the homogeneous broadening are random processes which lead to an irreversible destruction of the atomic phase memory. The implications of the reversible versus irreversible decay processes are seen to have important consequences in the problem of pulse amplification.

DOI
10.1103/physrev.179.399

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Recent advances in laser technology have led to the production of ultrashort electromagnetic pulses of very high peak power. In this paper we develop a theory which describes the transient behavior of such pulses in a laser amplifier. The effects of atomic coherence and inhomogeneous broadening, necessary for a complete treatment of the problem, are included. In the present development, the degree of homogeneous and inhomogeneous broadening is variable. The two-level active atoms are characterized by a phase memory time T2{T}_{2}T2​ and an inhomogeneous frequency distribution (which leads to a reversible decay time T2∗T_{2}^{}{}_{}{}^{*}T2​​∗). The field (Maxwell) and atomic (Schr"odinger) equations are coupled in a self-consistent manner as in Lamb's theory of an optical maser. The resulting equations are solved analytically in certain limits, while the general case is integrated numerically. The emphasis is placed on the situation in which T2∗T_{2}^{}{}_{}{}^{*}T2​​∗ and the pulse width are much shorter than T2{T}_{2}T2​. Thus our main effort is devoted to the physics of the inhomogeneously broadened system. The theory itself, however, contains the effects of both forms of broadening. The basic difference between inhomogeneous and homogeneous broadening is discussed. In the former case, the decay of radiation is caused by the dephasing of dipoles, and is reversible. The atomic memory is retained during the dephasing, and an internal reflection of the atomic coordinates, as is exemplified in the "photon echo" process, will cause the dipoles to rephase. The phonon interruptions or atomic collisions that bring about the homogeneous broadening are random processes which lead to an irreversible destruction of the atomic phase memory. The implications of the reversible versus irreversible decay processes are seen to have important consequences in the problem of pulse amplification.
The problem of a light pulse propagating in a nonlinear laser medium is investigated. The electromagnetic field is treated classically and the active medium consists of thermally moving atoms which have two electronic states with independent decay constants ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ in addition to the decay constant ${\ensuremath{\gamma}}_{\mathrm{ab}}$ describing the phase memory. The self-consistency requirement that the field sustained by the polarized medium be equal to the field inducing the polarization leads to coupled equations of motion for the density matrix, and equations of propagation for the electromagnetic field. Although the theory is developed for a Doppler-broadened gaseous medium, it may also be applied to a solid medium with inhomogeneous broadening. A unified treatment is given encompassing a wide range of pulse durations from cw signals to psec pulses. Continuous pumping is allowed, as well as any amount of detuning of the carrier frequency of the pulse from the atomic resonance frequency. The three independent decay constants ${\ensuremath{\gamma}}_{a}$, ${\ensuremath{\gamma}}_{b}$, and ${\ensuremath{\gamma}}_{\mathrm{ab}}$ provide greater flexibility than that obtained by using 1T1\frac{1}{{T}_{1}}T1​1​ and 1T2\frac{1}{{T}_{2}}T2​1​. The equations are solved analytically in a few specialized cases and numerically in the general case. Flow charts for accomplishing the numerical integration are given. Among the special problems considered is the apparent paradox of pulses propagating faster than the velocity of light under circumstances described by Basov et al. It is shown that this contradiction with relativity arises from the use of an unphysical initial condition.
The increasing interest in the study of the interaction of several resonant fields with multilevel quantum systems is due to a wide range of possible applications of various effects observed in the multi-frequency excitation of quantum objects. Among them, we can note color echo holography, information compression in three-level environments, copying of quantum information, and multi-level quantum gates that perform logical operations. In addition, the recording and reproduction of echo holograms in multi-level systems leads to the possibility, along with logical operations, to perform a change in the real-time scale and the sequence of events in the echo-hologram response, the data on which were embedded into the temporal structure of the object laser pulse. The formation of optical transient processes in multilevel systems occurs when the short laser pulses separated in time and resonant of different frequency transitions at the substance have an effect on the system, so that their interaction with each other becomes possible only through the medium and only if the medium has a sufficiently long phase memory. In this situation, each laser pulse transmits information about its wave characteristics to the medium, and this information is retained here until the arrival of the next laser pulses. The formation of optical transient processes in a multilevel system in a solid depends essentially on the degree of correlation of the inhomogeneous broadening on different frequency transitions and different time intervals, in connection with the possible destruction of the reversible phase memory of the system due to a partial mutual fixation energies of transitions. The influence of external spatially inhomogeneous electric fields on the resonant medium can lead to an additional artificially created inhomogeneous broadening. In this case, it is possible to control the phase memory of the system by changing the parameters of external spatially inhomogeneous electric fields, and the most pronounced effect on phase memory is the case of laser excitation of narrow frequency regions of inhomogeneously broadened lines of the resonance transitions. Moreover, the artificially created inhomogeneous broadening by external spatially inhomogeneous electric fields due to the Stark effect can be comparable with the spread of frequencies due to a partial mutual fixation of the transition energies, which leads to a partial restoration of the coherence of the multilevel system. This effect can lead to an increase in the intensity of the response of the stimulated photon echo (SPE) in the three-level system, which is the opposite effect of photon echo locking.
In a recent paper [Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007)] and in the two preceding papers [Gorshkov et al., this issue, Phys. Rev. A 76, 033804 (2006); 76, 033805 (2006)], we used a universal physical picture to optimize and demonstrate equivalence between a wide range of techniques for storage and retrieval of photon wave packets in homogeneously broadened $\ensuremath{\Lambda}$-type atomic media, including the adiabatic reduction of the photon group velocity, pulse-propagation control via off-resonant Raman techniques, and photon-echo-based techniques. In the present paper, we generalize this treatment to include inhomogeneous broadening. In particular, we consider the case of Doppler-broadened atoms and assume that there is a negligible difference between the Doppler shifts of the two optical transitions. In this situation, we show that, at high enough optical depth, all atoms contribute coherently to the storage process as if the medium were homogeneously broadened. We also discuss the effects of inhomogeneous broadening in solid state samples. In this context, we discuss the advantages and limitations of reversing the inhomogeneous broadening during the storage time, as well as suggest a way for achieving high efficiencies with a nonreversible inhomogeneous profile.
The interaction between an electromagnetic (em) pulse and a maser medium is described by a general set of five equations, under the assumption of a homogeneously broadened electric-dipole transition with two Bloch relaxation times T2and T1and of a linear broadband loss mechanism. When the equations are specialized at resonance, their solutions include the results of the previous treatments on the amplifier problem obtained under particular assumptions. The steady-state pulse (S. S. P.) introduced by Wittke and Warter forT_{2}/T_{1} = 0is here generalized forT_{2}/T_{1} \neq 0and it is shown to propagate at the same velocity of the light in the medium. In the caseT_{2}/T_{1} = 0the steady state is described by exact analytical relations. For times short in comparison to the relaxation times, a solution is given which generalizes the usual interaction formula between an em field and a two-level system by introducing propagation effects. In the general case out of resonance, it is shown that an S. S. P. exists, and that its frequency coincides with the frequency of the atomic transition, independent of the frequency of the input field.
In this paper we develop a theoretical analysis which describes a coherent transient effect of two entering pulses---one laser and one Stokes---in a Raman-active medium. The field (Maxwell) and atomic (Schr\"odinger, density-matrix representation) equations are coupled in a self-consistent manner in time and space by virtue of the resultant second-order nonlinear macroscopic polarizations at the two frequencies. This treatment is accomplished under the circumstances where atomic coherence plays a predominant role. The equations derived here may be referred to as "Raman Bloch-Maxwell equations." The resultant equations yield a new aspect of transient stimulated Raman scattering and thus coherent Raman propagation. The transient-pulse behavior may be understood as self-induced modulation of coherent amplification and absorption. The self-induced modulation for the step-function input pulse is described by a combination of Jacobian elliptic functions depending on the input intensity. Computer calculations of transient-pulse evolution reveal interesting pulse breakup phenomena with peak amplification, energy transmission, pulse advance or delay, and pulse-narrowing effects.
We discuss the propagation of an electromagnetic pulse through an active medium prepared in a state of inversion between two levels of the same parity. Since no electric dipole transition is possible between the chosen atomic levels, we investigate the possibility of amplification of an injected signal having a carrier frequency equal to one-half the atomic-transition frequency. We show that under suitable conditions a nonlinear atomic polarization can be generated which oscillates at the same frequency as the incident electromagnetic pulse. The coupled atom-field evolution is described by the usual self-consistent approach. When atomic relaxation effects are negligible, we derive an equation describing the spatial evolution of the energy of the propagating pulse. From this equation we characterize the threshold condition for power amplification and classify the multiple steady-state solutions of the propagation problem. The evolution of the pulse envelope through the amplifier is analyzed with the help of a hybrid computer simulation. Pulse-envelope modulation and multiple-pulse formation even in the asymptotic limit of long amplifiers are displayed.