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arXiv:1407.0187 · Physical Review A

Fourier theory of linear gain media

Hans Olaf Hågenvik, Markus Edward Malema, Johannes Skaar

Published Apr 15, 2015• 9 citations
arXivHTMLPDF

Abstract

The analysis of wave propagation in linear, passive media is usually done by considering a single real frequency (the monochromatic limit) and also often a single plane-wave component (plane-wave limit). For gain media, we demonstrate that these two limits generally do not commute; for example, one order may lead to a diverging field, while the other order leads to a finite field. Moreover, the plane-wave limit may be dependent on whether it is realized with a finite-support excitation or Gaussian excitation, eventually of infinite widths. We consider wave propagation in gain media by a Fourier-Laplace integral in space and time, and demonstrate how the correct monochromatic limit or plane-wave limit can be taken, by deforming the integration surface in complex frequency--complex wave-number space. We also give the most general criterion for absolute instabilities. The general theory is applied in several cases, and is used to predict media with novel properties. In particular, we show the existence of isotropic media which in principle exhibit simultaneous refraction, meaning that they refract positively and negatively at the same time.

DOI
10.1103/physreva.91.043826

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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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The analysis of wave propagation in linear, passive media is usually done by considering a single real frequency (the monochromatic limit) and also often a single plane-wave component (plane-wave limit). For gain media, we demonstrate that these two limits generally do not commute; for example, one order may lead to a diverging field, while the other order leads to a finite field. Moreover, the plane-wave limit may be dependent on whether it is realized with a finite-support excitation or Gaussian excitation, eventually of infinite widths. We consider wave propagation in gain media by a Fourier-Laplace integral in space and time, and demonstrate how the correct monochromatic limit or plane-wave limit can be taken, by deforming the integration surface in complex frequency--complex wave-number space. We also give the most general criterion for absolute instabilities. The general theory is applied in several cases, and is used to predict media with novel properties. In particular, we show the existence of isotropic media which in principle exhibit simultaneous refraction, meaning that they refract positively and negatively at the same time.
The stability of a monochromatic constant-intensity laser beam propagating through a resonant medium is considered. It is found that many absorbers should exhibit a region of negative conductivity and, consequently, amplify perturbations. The associated gain is dependent on a number of variable parameters, such as laser-beam intensity, perturbation frequency, relaxation times, degeneracy, transverse mode, inhomogeneous broadening, etc. Putting absorbers inside Fabry-Perot interferometers allows the construction of plane-wave devices with a bistable output.
General overview In many areas in the physical sciences, the propagation of waves in complex mediaplays a central role. Acoustics, applied mathematics, elastics, environmental sciences, mechanics, marine sciences, medical sciences, microwaves, and seismology are just a few examples, and of course nanophotonics [6, 429, 448, 449]. Here, complex media are understood to exhibit a strongly inhomogeneous spatial structure, which determines to a large extent their linear and nonlinear properties. Examples of complex media are random, heterogeneous, porous, and fractal media. The challenges that researchers in these areas must surmount to describe, understand, and ultimately predict wave propagation are formidable. Right from the start of this field – in the early 1900s – it was clear that new concepts and major approximations had to be introduced. Famous examples of such concepts are the effective medium theory [55, 60, 321] and the radiative transport theory [81, 491]. These concepts are very much alive, even today. The fundamental challenge with these approximate concepts is that often the length scales of the inhomogeneities in the complex medium are comparable with the wavelength, whereas the range of validity of these approximations is restricted to situations where these length scales are much larger than the wavelength. Consequently, many relevant situations arise where either the effective medium theory, or radiative transport theory, or both, fail dramatically. Examples of such situations are given in this introduction and throughout this book. The complexity of the medium that supports wave propagation can be classified in a number of ways. Many different types of spatial inhomogeneities in a host matrix can be envisioned, varying from completely random, via aperiodic and waveguide structures, to quasi-crystalline and even structures with long-range periodic order. The in homogeneities could be of a self-organized form, or the fruit of precise engineering. An additional classification is whether or not the complexity is confined to the surface (in two dimensions (2D)) or is present throughout the volume of the medium (in three dimensions (3D)), as in porous media.
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 ${({\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2})}^{\ensuremath{-}m}$.
We investigate the time evolution of waves in evanescent media generated by a source within this medium and observed at some distance away from the location of the source. The aim is to find a velocity which describes a causal process and is thus, for a medium with relativistic dispersion, limited by the velocity of light. For a source with a sharp onset in time, the wave function consists of a forerunner generated by the onset of the source, and of a monochromatic front. The forerunner is dominated by a frequency which decreases with time, and the monochromatic front carries the oscillation frequency of the source into the evanescent medium. For a medium with Schrödinger-like dispersion the velocity of the front is infinite and the monochromatic front propagates with a velocity which is in agreement with the traversal time for tunneling. In the relativistic case the forerunners travel with the velocity of light and the velocity the monochromatic front is smaller than the velocity of light and only for special energies equal to the velocity of light. For sources with a sharp onset, the forerunners are not attenuated and in magnitude far exceed the monochromatic front. This renders the detection of the monochromatic front difficult. To avoid the different behavior of forerunners and monochromatic fronts, sources which are frequency-band limited can be considered or a short-time Fourier transform of the field at the observation point can be taken. Both discussions suggest that the traversal time can be determined only up to a factor of 𝒪(1).