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Physical Review A

Small-signal gain in lethargic and conventional laser amplifiers

F. A. Hopf, Stig Stenholm

Published Aug 1, 1983• 2 citations
Publisher

Abstract

The lethargic regime of a swept-gain laser amplifier is investigated theoretically to determine the procedure for passing over to the conventional limit ${T}_{1}(\mathrm{level}\mathrm{lifetime}) \ensuremath{\gg}{T}_{2} (\mathrm{phase}\mathrm{memory}\mathrm{time})$ of laser amplifiers. The calculation is done with a model in which the transit time of the atoms ${t}_{0}$, is less than ${T}_{1}$, which is soluble in closed form. The conventional limit is taken as ${t}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$. The general and conventional limits involve contradictory conditions of operation; nonetheless, both agree with experiment. This apparent contradiction is resolved by our finding that the conventional limit is a singular case.

DOI
10.1103/physreva.28.1176

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The lethargic regime of a swept-gain laser amplifier is investigated theoretically to determine the procedure for passing over to the conventional limit T1(levellifetime)\ensuremath≫T2(phasememorytime){T}_{1}(\mathrm{level}\mathrm{lifetime}) \ensuremath{\gg}{T}_{2} (\mathrm{phase}\mathrm{memory}\mathrm{time})T1​(levellifetime)\ensuremath≫T2​(phasememorytime) of laser amplifiers. The calculation is done with a model in which the transit time of the atoms t0{t}_{0}t0​, is less than T1{T}_{1}T1​, which is soluble in closed form. The conventional limit is taken as t0\ensuremath→\ensuremath∞{t}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}t0​\ensuremath→\ensuremath∞. The general and conventional limits involve contradictory conditions of operation; nonetheless, both agree with experiment. This apparent contradiction is resolved by our finding that the conventional limit is a singular case.
The buildup of radiation from noise is considered in the case of the small-signal regime of a swept-gain amplifier with small Doppler width, of the sort indicated in recent schemes for an x-ray laser. The treatment uses the fully quantized electromagnetic field in order to take into account the spontaneous emission. The pulse is seen to grow nonexponentially with a rate that vanishes in the limit of large distances. The gain is generally much smaller than would be expected on the basis of the usual development. Many of the features normally associated with the small-signal regime of an amplifier, such as threshold conditions and spectral narrowing, are not seen. Instead we find spectral broadening, saturation, and the formation of steady states.
The steady-state behavior of amplifiers in which the excitation is swept at the speed of light is discussed in the semiclassical approximation. In the present work we examine the case where the decay time of the population is comparable to that of the polarization. Pulse propagation is shown to obey a generalized sine-Gordon equation which contains the effects of atomic relaxations. The analytical expression of the steady-state pulses (SSP) gives two threshold conditions. In the region of limited gain the SSP is a broad pulse with small area which can be obtained by small signal theory. In the second region of high gain the SSP is the superradiant $\ensuremath{\pi}$ pulse. Its pulse power is not limited as in usual superradiant theory because, as we show, for a swept excitation the cooperation-length limit does not exist.
Pulse propagation in maser-type traveling-wave amplifiers with a homogeneously broadened transition is treated by a formalism analogous to the Bloch equations. Phenomenological dephasing (T2) and recovery (T1) times are defined, and a linear (nonsaturable) loss mechanism is included. In the numerical calculations, only the case of negligible excitation during the time it takes a pulse to pass is considered. When pulses are allowed to grow until the amplifier is ``saturated,'' steady-state pulses having a unique shape and intensity independent of the initial pulse are found. The parameters of these steady-state pulses depend only on the ratio of the linear loss and gain coefficients. Steady-state pulses have a peak intensity that decreases monotonically from infinity to zero as the linear loss coefficient varies from zero to the gain coefficient, while the pulse width correspondingly varies from zero to infinity, and the pulse energy varies from a finite value to zero. Steady-state pulses propogate at a velocity less than that of the small signal velocity in the medium.
The theory of small-signal laser amplification is developed for homogeneously broadened systems in which the three main decay rates, for the collective atomic dipole moment, the population inversion, and the field in the laser cavity, have arbitrary relative magnitudes. The calculations extend previous work on class-AAA and -BBB lasers, where the dipole decay rate greatly exceeds the other two rates, to class-CCC lasers, where the dipole decay rate is comparable to the others. The free-running laser is assumed to excite a single longitudinal mode of the cavity, whose frequency generally differs from that of the coupled atomic transition. The linear gains of the laser are derived for input signals whose frequencies are close to that of the lasing mode or to one of its adjacent nonlasing longitudinal modes. Divergences in the gain that occur for these two arrangements are associated with the previously studied single-mode and multimode instabilities of the free-running laser, respectively.