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Quantum Electronics

On the normalisation of the observed spectral gain line profile with increasing optical thickness of a substance layer

Nikolai S. Bukhman

Published Sep 30, 2000• 0 citations
Publisher

Abstract

It is shown that a spectral gain line narrows down and its profile is normalised with increasing optical thickness of a layer of the amplifying medium. As a result, the gain line profile always becomes Gaussian, independently of the true form factor of the line, when the thickness of the active medium layer is sufficiently large. The normalisation of the line profile is demonstrated for the lines with Lorentzian, Gaussian, and 'time-of-flight' profiles.

DOI
10.1070/qe2000v030n09abeh001815

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It is shown that a spectral gain line narrows down and its profile is normalised with increasing optical thickness of a layer of the amplifying medium. As a result, the gain line profile always becomes Gaussian, independently of the true form factor of the line, when the thickness of the active medium layer is sufficiently large. The normalisation of the line profile is demonstrated for the lines with Lorentzian, Gaussian, and 'time-of-flight' profiles.
The propagation of a narrow-band signal (wave packet) in a medium with population inversion is considered. It is shown that in an optically thick medium layer the signal decomposes into the initial and amplified signals propagating at different velocities and having different durations. The initial signal propagates without distortions at the velocity c / n 0 ( n 0 is the refractive index of the medium away from the resonance frequency) and plays the role of a precursor of the amplified signal. The amplified signal moves at a lower velocity. It lags behind the initial signal during propagation and acquires a universal Gaussian shape (irrespective of the shape of the initial signal and the spectral line profile). The appearance of the amplified signal, which substantially differs from the initial signal in all the parameters, as well as the interference between the amplified and initial signals looks like a 'extra-distortion' of the initial signal.
General relations are obtained that govern the evolution of the leading edge of a signal propagating in a selectively amplifying medium. The propagation of a pulse with a rectangular envelope, the carrier frequency of which is close to the frequency of one of the spectral amplification lines of a selective (in frequency) amplifying medium, is considered. The time dependence of the envelope of the initially rectangular signal is studied for various values of the optical thickness of the substance layer for the Gaussian and Lorentzian contours of the spectral line. It is shown that in the case of a shift of the carrier frequency of the signal relative to the center of the spectral line, the interference of the primary and amplified signal can lead to very significant oscillations of the total signal in time.
Introduction In this chapter we will present the theory of spectral gain in optical amplifiers and the related optical power distribution. The concepts are general, as they apply to every optical amplifier configuration, independent of the specific architecture and implementation technology. We will configure the optical amplifying medium as a traveling-wave device, where the optical field undergoes optical amplification as it propagates from the input to the output sections. Generally speaking, optical amplification is achieved by a net stimulated emission process. Referring to a generic section z of infinitesimal thickness dz , optical gain is determined by the imbalance between net stimulated emission and cavity losses, including extrinsic absorption, reflection and radiated fields. As we know from the theory of light amplification in the three-level atomic system presented in Chapter 5, stimulated absorption of photons is governed by the same transition rate as stimulated emission, and is included in the net stimulated emission process. With this in mind, we note the identity between stimulated emission and the stimulated absorption rates, Γ 12 = Γ 21 , and the net stimulated photon term Γ[ j 2 ( t ) – j 1 ( t )] p ( t ) in the rate equation. In order to develop the traveling-wave model, the photon density becomes a function of the longitudinal position 0 ≤ z ≤ L along the amplifier medium. Accordingly, we will set p ( t , z ) as the relative density of stimulated photons per unit volume (0 ≤ p ( t , z ) ≤ 1) at time t and section z . When photons with appropriate energy proceed along the amplifying medium, they are amplified by means of net stimulated emission. At some section z = z 0 , the steady-state photon density P ( z ) can approach the saturation density P sat ( z ).
Propagation of a short quasi-monochromatic electromagnetic (light or radio) signal (wave packet) is considered. The carrier frequency of the signal is close to the frequency of the absorption spectral line (or group of lines) of a medium, and the spectrum width is large relative to the width of this spectral line (or group of lines). Analytic formulas for the temporal dependence of the signal are derived and numerically confirmed for paths of various lengths. It is shown that the signal is substantially distorted long before it noticeably decays owing to absorption. The signal’s temporal dependence turns out to be rather universal with respect to the form factor of the spectral line, the optical thickness of the substance layer, and the initial temporal dependence of the signal.
We analytically study the linear propagation of arbitrarily shaped light pulses through an absorbing medium with a narrow transparency window or through a resonant amplifying medium. We point out that, under certain general conditions, the pulse acquires a nearly Gaussian shape, irrespective of its initial shape and of the spectral profile of the line. We explicitly derive in this case the pulse parameters, including its skewness, responsible for a deviation of the delay of the pulse maximum from the group delay. We illustrate our general results by analyzing the slow-light experiments having demonstrated the largest fractional pulse delays.