Philosophical magazine

The modification of the Kronig-Kramers relations under saturation conditions

H. C. Bolton, GORDON J. TROUP

Published Mar 1, 1969• 17 citations

Abstract

When the real and imaginary parts of the susceptibility, χ′ and χ″ respectively, of narrow saturated absorption lines are of Lorentzian form, it is shown that a constant A can always be found such that χ′ and A χ″ are a Kronig-Kramers transform pair. Lines made up of weighted linear super-positions of Lorentzian lines having the same shape and the same saturation behaviour also possess this property, as do certain continuous distributions of narrow Lorentzian lines. The Kronig-Kramers calculation of a saturated absorption lineshape from a saturated dispersion signal is therefore very likely to yield the true lineshape.

DOI: 10.1080/14786436908216306

Related papers

The character of the dispersion of the refractive index near an isolated spectral line

Nikolai S. Bukhman, Anastasia Kulikova

An approximate analog of the Kramers-Kronig relationships for an isolated spectral line is proposed. Expressions are obtained for the real part of the dielectric susceptibility of a substance near a spectral line for collisional, Doppler, and time-of-flight spreading of the spectral line. It is shown that, generally, the imaginary part of the complex form-factor of the spectral line attenuates very slowly, namely, according to the hyperbolic law, as the deviation from the center frequency grows. Therefore, even in the case when practically there is no radiation absorption beyond the spectral line, the influence of this spectral line on the refractive index of the substance manifests itself rather far from it.

Journal of Communications Technology and ElectronicsMay 1, 20154 citations

Saturation broadening by inhomogeneous fields

James C. Camparo

According to the standard Karplus-Schwinger theory of saturation broadening, saturated line shapes are Lorentzians with linewidths that increase linearly with the perturbing field strength. This, however, is not what is observed experimentally when the saturating field is inhomogeneous. If the saturating field strength varies significantly over the experimental signal volume, we find that the saturated line shapes are strikingly non-Lorentzian. The ``sharpness'' of the experimental line shapes is quantified by an ``effective linewidth,'' which is the half width at half maximum of a Lorentzian that approximates the experimental line shape near line center. For certain classes of inhomogeneous fields, we find that this effective linewidth increases approximately as the square root of the saturating field strength, rather than linearly. We show that this class of inhomogeneous fields is distinguished by the presence of a node in the field geometry, and that the effect arises because the line shape near line center is dominated by the atomic population in the vicinity of the node. These results indicate the importance of understanding and accounting for inhomogeneous field effects when extracting physical information from experimental line shapes.

Physical Review AJan 1, 198913 citations

Comments on “Study of the Complex Atomic Susceptibility of Erbium-Doped Fiber Amplifiers”

Jean-Fran�ois Bisson, Dmitrii Kouznetsov

This comment aims to report and correct a mistake in the paper by Desurvire (J. Lightw. Technol., vol. 8, no. 10, pp. 1517-1527, Oct. 1990) with regard to the expression of the complex susceptibility of quasi-three-level laser systems. The derived susceptibility formula contradicts the assumption of thermal equilibrium between sublevels within each manifold and leads to the wrong conclusion that the real and imaginary parts of the susceptibility of saturated materials do not verify the Kramers-Kronig relations (KKR). The corrected formula for the susceptibility is hereby proposed. The poles of the susceptibility obtained with the new formula are all located in the upper half plane of the complex plane, in accordance with the principle of causality. Hence, the KKR should be valid in the saturated regime.

Journal of Lightwave TechnologyJan 1, 20081 citation

Pump–probe model for the Kramers–Kronig relations in a laser

H. Yum, M. S. Shahriar

In this paper, we study theoretically a pump–probe model for the Kramers–Kronig (KK) relations during laser operation. A solution of laser equations reveals that the dispersion profile of a saturated laser gain medium is non-analytical at the boundary between the lasing and the non-lasing spectral region. Such a non-analyticity cannot be explained in terms of the KK relations. In order to interpret this situation, it is important to consider carefully the physical basis of the KK relations. We conclude that the KK relation is expected to apply only to an independent probe applied to the medium, which is under excitation by the pump producing the gain as well as the lasing mode. The absorption/gain and dispersion profiles are then analytical, and satisfy the KK relations. Specifically, these are variants of the so-called Mollow–Ezekiel spectra of probe absorption/gain and dispersion in the presence of a pump, with the exception that in this case the medium is inverted.

arXiv:1003.3686Journal of OpticsSep 24, 201025 citations

Generalized line broadening and saturation behavior

Albert T. Rosenberger

One characteristic normally used to distinguish inhomogeneous broadening from homogeneous broadening is the saturation behavior occurring when irradiation takes place at line center. The usual behavior found is that the absorption coefficient decreases as a negative power of 1 + I/Isat; a power of −1 indicates homogeneous broadening and a power of −½ denotes inhomogeneous broadening. It is shown here that, in the rate equation limit, this behavior is followed in most cases even though the homogeneous and inhomogeneous absorption profiles can have any line shape. The homogeneous line shape, in particular, need not be Lorentzian; its line center saturation behavior will be the same as that of a Lorentzian, however. If the inhomogeneous broadening is due to a distribution of atomic center frequencies, in the inhomogeneous limit the usual line center saturation behavior will be found no matter what the combination of homogeneous and inhomogeneous line shapes. However, if the broadening is inhomogeneous because of a distribution, among the individual atoms, in any property of the absorption profile other than center frequency (for example, the homogeneous linewidth), the saturation properties are not as simple, and in fact the inhomogeneous limit may not even exist.

Annual Meeting Optical Society of AmericaJan 1, 19890 citations

Lineshape dependence of ultrafast saturation of absorption and fluorescence

Anil K. Patnaik, Sukesh Roy, James R. Gord

Ultrafast saturation of signals from an optical process occurs when intense laser pulses, with the duration of the driving laser pulse shorter than the effective decay timescales, interact with the target system. Recently, it has been shown that such saturation is marked by transition from transient-absorption to the transient-gain regime, unlike in continuous-wave or long-duration excitations where saturation occurs once the underdamping regime is achieved. The threshold intensities of the drive pulse that drives the optical signal to saturate can be estimated by calculating the area of the pulse. We demonstrate that depending on the shape of the pulse, the saturation behavior differs. The lineshape dependence of ultrafast saturation of absorption and fluorescence is studied for a variety of pulse shapes. We demonstrate that the saturation threshold is lower for super-Gaussian pulses and almost an order of magnitude lower for sech pulses compared to a Gaussian pulse.

Physica ScriptaOct 1, 20157 citations