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Nonlinear effects in traveling wave laser amplifiers

Donald H. Close

Published Jan 1, 1965• 5 citations
Publisher

Abstract

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.

DOI
10.7907/y5fc-gv56.

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Strong field saturation effects in laser media

Donald H. Close

Using Lamb's model, we analyze the effects of gain saturation by strong traveling-wave fields in dilute laser media. Using approximate solutions of an integral equation for the population inversion density (PID), the index of refraction and incremental gain are studied for arbitrarily strong fields. Effects of atomic motion are included for a Maxwellian velocity distribution, but pressure effects are neglected. The case of a monochromatic field leads to the saturation results of Gordon, White, and Rigden, which are studied as a function of frequency. For a small ratio of natural to Doppler linewidths, there is a transition from inhomogeneous to homogeneous broadening for sufficiently strong fields. An effect of particular interest is the generation of waves at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, $2{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$, and higher order sidebands by two strong input signals at ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$. The source of the parametric gain at these frequencies is the time-dependent gain saturation due to the presence of multiple strong fields. For ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ small compared to the decay rates ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ of the laser levels, the gain at these intermodulation sidebands is computed as a function of the field strengths. The limiting cases of homogeneous and inhomogeneous (due to atomic motion) broadening are studied in detail. These two cases give essentially the same results, for a given unsaturated gain. Numerical results indicate that a first-order side-band intensity at least 10% as large as that of the inducing fields can be easily observed in practice. The integral equation for the PID is converted into an infinite set of linear algebraic equations for a typical solid-state laser. The conditions under which this set of equations can be limited to a finite number is discussed, and the dependence of the sideband gain on ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ is calculated. For large ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, the gain at the $m\mathrm{th}$ sideband decreases as ${({\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2})}^{\ensuremath{-}m}$.

IEEE Journal of Quantum Electronics•Apr 1, 1966•2 citations

Strong-Field Saturation Effects in Laser Media

Donald H. Close

Using Lamb's model, we analyze the effects of gain saturation by strong traveling-wave fields in dilute laser media. Using approximate solutions of an integral equation for the population inversion density (PID), the index of refraction and incremental gain are studied for arbitrarily strong fields. Effects of atomic motion are included for a Maxwellian velocity distribution, but pressure effects are neglected. The case of a monochromatic field leads to the saturation results of Gordon, White, and Rigden, which are studied as a function of frequency. For a small ratio of natural to Doppler linewidths, there is a transition from inhomogeneous to homogeneous broadening for sufficiently strong fields. An effect of particular interest is the generation of waves at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, $2{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$, and higher order sidebands by two strong input signals at ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$. The source of the parametric gain at these frequencies is the time-dependent gain saturation due to the presence of multiple strong fields. For ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ small compared to the decay rates ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ of the laser levels, the gain at these intermodulation sidebands is computed as a function of the field strengths. The limiting cases of homogeneous and inhomogeneous (due to atomic motion) broadening are studied in detail. These two cases give essentially the same results, for a given unsaturated gain. Numerical results indicate that a first-order side-band intensity at least 10% as large as that of the inducing fields can be easily observed in practice. The integral equation for the PID is converted into an infinite set of linear algebraic equations for a typical solid-state laser. The conditions under which this set of equations can be limited to a finite number is discussed, and the dependence of the sideband gain on ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ is calculated. For large ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, the gain at the $m\mathrm{th}$ sideband decreases as ${({\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2})}^{\ensuremath{-}m}$.

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Theory of Nonlinear Effects in a Gas Laser Amplifier. I. Weak Signals

A. Dienes

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.

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Physical Review A•Feb 1, 1978•30 citations
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 Lamb's model, we analyze the effects of gain saturation by strong traveling-wave fields in dilute laser media. Using approximate solutions of an integral equation for the population inversion density (PID), the index of refraction and incremental gain are studied for arbitrarily strong fields. Effects of atomic motion are included for a Maxwellian velocity distribution, but pressure effects are neglected. The case of a monochromatic field leads to the saturation results of Gordon, White, and Rigden, which are studied as a function of frequency. For a small ratio of natural to Doppler linewidths, there is a transition from inhomogeneous to homogeneous broadening for sufficiently strong fields. An effect of particular interest is the generation of waves at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, $2{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$, and higher order sidebands by two strong input signals at ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$. The source of the parametric gain at these frequencies is the time-dependent gain saturation due to the presence of multiple strong fields. For ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ small compared to the decay rates ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ of the laser levels, the gain at these intermodulation sidebands is computed as a function of the field strengths. The limiting cases of homogeneous and inhomogeneous (due to atomic motion) broadening are studied in detail. These two cases give essentially the same results, for a given unsaturated gain. Numerical results indicate that a first-order side-band intensity at least 10% as large as that of the inducing fields can be easily observed in practice. The integral equation for the PID is converted into an infinite set of linear algebraic equations for a typical solid-state laser. The conditions under which this set of equations can be limited to a finite number is discussed, and the dependence of the sideband gain on ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ is calculated. For large ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, the gain at the mthm\mathrm{th}mth sideband decreases as ${({\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2})}^{\ensuremath{-}m}$.
Using Lamb's model, we analyze the effects of gain saturation by strong traveling-wave fields in dilute laser media. Using approximate solutions of an integral equation for the population inversion density (PID), the index of refraction and incremental gain are studied for arbitrarily strong fields. Effects of atomic motion are included for a Maxwellian velocity distribution, but pressure effects are neglected. The case of a monochromatic field leads to the saturation results of Gordon, White, and Rigden, which are studied as a function of frequency. For a small ratio of natural to Doppler linewidths, there is a transition from inhomogeneous to homogeneous broadening for sufficiently strong fields. An effect of particular interest is the generation of waves at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, $2{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$, and higher order sidebands by two strong input signals at ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$. The source of the parametric gain at these frequencies is the time-dependent gain saturation due to the presence of multiple strong fields. For ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ small compared to the decay rates ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ of the laser levels, the gain at these intermodulation sidebands is computed as a function of the field strengths. The limiting cases of homogeneous and inhomogeneous (due to atomic motion) broadening are studied in detail. These two cases give essentially the same results, for a given unsaturated gain. Numerical results indicate that a first-order side-band intensity at least 10% as large as that of the inducing fields can be easily observed in practice. The integral equation for the PID is converted into an infinite set of linear algebraic equations for a typical solid-state laser. The conditions under which this set of equations can be limited to a finite number is discussed, and the dependence of the sideband gain on ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ is calculated. For large ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, the gain at the mthm\mathrm{th}mth sideband decreases as ${({\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2})}^{\ensuremath{-}m}$.
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.
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 JJJ 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 JJJ values, ggg 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.
The amplification of noise in a laser amplifier is treated theoretically. The model for the active medium and its description using density-matrix techniques, are taken from the theory of laser operation. The fundamental concern of this investigation is the spectral behavior of the radiation in the nonlinear regime; hence, the formalism is written from the onset in the frequency domain. The statistics of the light are gradually modified by the nonlinear amplification process, and expressions are derived for the rate of change of fluctuations in intensity as a measure of statistical changes. These expressions should be easily susceptible to detailed experimental observations. In addition, the range of validity of Litvak's Gaussian-statistics approximation is discussed with some detail. In the homogeneous-broadening case, the evolution of initially broadband Gaussian radiation toward quasimonochromatic oscillations with laserlike statistics is extensively explored in several numerical examples. The connections of this study with the time-domain work of Risken and Nummedal, on self-pulsing in a ring-laser configuration, are clearly established. Finally, spectral-narrowing and -rebroadening effects in Doppler-broadened media are discussed both analytically and with numerical examples. These examples show the distinct contribution of pulsations in the population ("Raman-type terms"), and saturation phenomena. The predicted narrowing-rebroadening rate is compared with expressions found in previous literature.