The model given in the preceding paper is used to calculate the nonlinear gain and phase of a laser amplifier operating at field intensities that may be too large for the perterbational results to be valid. The induced polarization is calculated exactly by integrating the macroscopic density-matrix differential equations. The incremental gain and phase shift are found to be complicated functions of the circularly polarized field intensity components. Simple physical interpretations of these functions are given. The results are used to compute the change in the ellipticity of a single-frequency signal in a high-gain amplifier.
The model given in the preceding paper is used to calculate the nonlinear gain and phase of a laser amplifier operating at field intensities that may be too large for the perterbational results to be valid. The induced polarization is calculated exactly by integrating the macroscopic density-matrix differential equations. The incremental gain and phase shift are found to be complicated functions of the circularly polarized field intensity components. Simple physical interpretations of these functions are given. The results are used to compute the change in the ellipticity of a single-frequency signal in a high-gain amplifier.
Using classical fields and the density-matrix formalism, we analyze the nonlinear characteristics of a gas laser amplifier subject to an axial magnetic field, operating with two optical-frequency signals of arbitrary polarization. Perturbational solutions, valid for relatively weak intensities, are obtained which describe the saturation interaction of the signals and the generation of combination tones. An arbitrary amount of Doppler broadening is considered throughout. The simple $J=1\ensuremath{\rightarrow}J=0$ model is found convenient to describe the physical processes taking place. In general, however, the interactions are found to depend on the J values of the levels. We find that only saturation and no combination-tone generation occurs for two opposite circularly polarized signals, while both effects are, in general, present for two linearly or elliptically polarized fields. The limiting case of zero magnetic field is examined separately. It is also found that in many cases the medium is made effectively anisotropic by the nonlinear interactions. Applications to measurements of atomic parameters such as J values, g values, and lifetimes are discussed.
Theoretical and experimental studies of a gas laser amplifier are presented, assuming the amplifier is operating with a saturating optical frequency signal. The analysis is primarily concerned with the effects of the gas pressure and the presence of an axial magnetic field on the characteristics of the amplifying medium. Semiclassical radiation theory is used, along with a density matrix description of the atomic medium which relates the motion of single atoms to the macroscopic observables. A two-level description of the atom, using phenomenological source rates and decay rates, forms the basis of our analysis of the gas laser medium. Pressure effects are taken into account to a large extent through suitable choices of decay rate parameters. Two methods for calculating the induced polarization of the atomic medium are used. The first method utilizes a perturbation expansion which is valid for signal intensities which barely reach saturation strength, and it is quite general in applicability. The second method is valid for arbitrarily strong signals, but it yields tractable solutions only for zero magnetic field or for axial magnetic fields large enough such that the Zeeman splitting is much larger than the power broadened homogeneous linewidth of the laser transition. The effects of pressure broadening of the homogeneous spectral linewidth are included in both the weak-signal and strong-signal theories; however the effects of Zeeman sublevel-mixing collisions are taken into account only in the weak-signal theory. The behavior of a He-Ne gas laser amplifier in the presence of an axial magnetic field has been studied experimentally by measuring gain and Faraday rotation of linearly polarized resonant laser signals for various values of input signal intensity, and by measuring nonlinearity - induced anisotropy for elliptically polarized resonant laser signals of various input intensities. Two high-gain transitions in the 3.39-μ region were used for study: a J = 1 to J = 2 (3s2 → 3p4) transition and a J = 1 to J = 1 (3s2 → 3p2) transition. The input signals were tuned to the centers of their respective resonant gain lines. The experimental results agree quite well with corresponding theoretical expressions which have been developed to include the nonlinear effects of saturation strength signals. The experimental results clearly show saturation of Faraday rotation, and for the J = 1 t o J = 1 transition a Faraday rotation reversal and a traveling wave gain dip are seen for small values of axial magnetic field. The nonlinearity induced anisotropy shows a marked dependence on the gas pressure in the amplifier tube for the J = 1 to J = 2 transition; this dependence agrees with the predictions of the general perturbational or weak signal theory when allowances are made for the effects of Zeeman sublevel-mixing collisions. The results provide a method for measuring the upper (neon 3s2) level quadrupole moment decay rate, the dipole moment decay rates for the 3s2 → 3p4 and 3s2 → 3p2 transitions, and the effects of various types of collision processes on these decay rates.
This paper presents a semiclassical theory of a single-mode gas laser valid for essentially arbitrary values of field intensity. The theory is formulated in terms of an ensemble-averaged form of the density-matrix equations of motion involving a single set of coordinates. The polarization and the population-inversion density are expanded in spatial Fourier components. One obtains a set of coupled difference equations in the Fourier coefficients yn, which may be solved subject to appropriate boundary conditions on yn for large n. The solutions may be expressed in terms of continued fractions in the general case, and in closed form in important special cases. Quantities of interest are evaluated by computer. The steady-state laser intensity and frequency are obtained from the coupled electromagnetic field equations as functions of the cavity parameters and the linear gain of the amplifying medium. At moderate values of field intensity, the velocity distribution of the space-averaged population inversion density exhibits a fine structure for low-velocity atoms superimposed on the usual broader depletion. This structure is a manifestation of the coherent ringing of the amplifying medium. The results are compared with those obtained in the rate-equation approximation, and with the recent work of Stenholm and Lamb. Discrepancies with the latter paper are explained.
The theory of the laser amplifier is developed for conditions in which the strength of the input signal is increased from small values, where the amplification is linear, to larger values, where the amplification becomes nonlinear. The below-threshold laser amplifier oscillates at a single frequency equal to that of the input signal, and its properties are found by solution of the nonlinear equation of motion for the single excitation amplitude. For the above-threshold laser amplifier, the effects of the nonlinear behavior are to shift the laser frequency from its free-running value and to transfer intensity from the laser line to the signal frequency and to a range of satellite lines, whose frequency detunings are integer multiples of the signal detuning. The intensities of the various emission lines of the laser are calculated by power-series expansions of the field amplitudes up to terms of fourth order in the input signal strength. The onset of injection locking is determined by the conditions for which the intensity at the shifted free-running laser frequency falls to zero. The injection-locked state is characterized by a single excitation frequency equal to that of the input signal, and its properties are found by solution of the same nonlinear equation of motion as for the below-threshold amplifier. The ranges of input signal strength and detuning are determined for which the injection-locked state is stable. The energy conservation properties of the laser amplifier are considered for each of its operating states. \textcopyright{} 1996 The American Physical Society.
Using semiclassical radiation theory, a formalism similar to that used by Lamb in his Theory of an Optical Maser is developed for studying the amplification of vector traveling waves in a laser-type medium. The effect of the medium on the waves is given in terms of space (or time) dependent field amplitudes and phases and a nonlinear index of refraction. With particular emphasis on typical gaseous media, the effects of Doppler broadening are treated in detail for arbitrary ratios of natural to Doppler linewidths. Polarization and propagation vectors in various directions are considered, and the nonlinear effects are found to make an isotropic medium effectively anisotropic.
Lowest order nonlinear effects (due to a polarization cubic in the field amplitudes) are studied extensively, and the frequency dependence of several of these processes is presented in graphical form. In particular, the introduction of fields at new frequencies and polarization effects are considered. The characteristics of these nonlinear processes peculiar to Doppler broadened lines are discussed, and the processes are interpreted in terms of saturation and coherent modulation of the population inversion density.
Strong nonlinear effects are considered in a more approximate way and are found to consist of saturation of the various linear and nonlinear processes previously considered. These strong nonlinear effects should occur at low enough intensities to be easily observed in practice on a CW basis. With the present formalism, the analytical results of Gordon, White and Rigden regarding gain saturation in laser amplifiers are obtained, and the extension is made to include frequencies away from line center and the effects of multiple spectral components. Again, the introduction of fields at new frequencies is considered in detail. These results are also discussed in terms of saturation and coherent modulation of the populations and hole burning.
Using semiclassical radiation theory and a density matrix
formalism we analyze the nonlinear characteristics of a gas laser
amplifier operating with two optical frequency signals of arbitrary
polarization and having an axial magnetic field. Both perturbational
solutions, valid for relatively weak intensities and solutions valid
for arbitrarily strong fields are obtained for two nonlinear effects:
the saturation interaction of the electromagnetic waves, and the
generation of combination tones. An arbitrary amount of Doppler
broadening is considered throughout. The detailed treatment of J = 1 to J = 0 model yields the
frequency, magnetic field and polarization dependence of the nonlinear
effects. The results are presented analytically and graphically
and are discussed using physical arguments. It is found that only
saturation but no combination tone generation occurs for two opposite
circularly polarized input signals while both are, in general, present
for two arbitrary linearly or elliptically polarized fields. For
two opposite circular waves the interaction is found to comprise three
parts, each with a different behavior: self saturation, common level
mutual saturation and a coherent double quantum interaction. The
total interaction (coupling) between the two fields is always weak.
The limiting case of a single linearly polarized field is considered
separately, the zero magnetic field dip and the nonlinear behavior
of the Faraday rotation is discussed. For two linearly (or elliptically) polarized waves the three
nonlinear processes listed above take place between opposite circular
components. In addition a modulation of the population inversion
densities occurs due to the presence of two different frequencies with
the same circular polarization. This results in the generation of new
frequencies and also contributes to the coupling between the input
fields. The coupling depends on the magnetic field, and on the frequency
separation and the polarization states of the signals. The
limiting case of zero magnetic field is examined. It is found that the
medium is made effectively anisotropic by the nonlinear interactions.
The polarization vectors of two linearly polarized fields rotate apart
unless the angle between them is zero or 90 degrees. The results are extended to the general Ja to Jb transition.
In zero magnetic field the nonlinear effects are found to depend on
ΔJ, while for nonzero magnetic field resonances in the interaction
occur whenever the frequency difference between two opposite circularly
polarized transitions that have common level equals the frequency
separation of the input fields. Combination tone generation takes
place for all but two opposite circularly polarized signals.