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Journal of the Optical Society of America B

Theory of the Faraday effect with weak laser pulses

Betsy M. Harvey, Frank C. Spano

Published Jul 1, 1994• 2 citations
Publisher

Abstract

The propagation of a weak optical pulse along the direction of an applied magnetic field in an atomic or molecular gas is investigated theoretically. When the Zeeman splitting is much smaller that the inhomogeneous linewidth σ, the Faraday rotation of the polarization plane depends linearly on optical depth and is independent of time for long pulses with tp ≫ σ−1 (as in the conventional Faraday effect) but depends linearly on time and is independent of optical depth for short pulses with tp ≪ σ−1; tp is the pulse duration. In the intermediate-pulse-width regime, frequency pushing causes the polarization plane to rotate at greater average speeds as the pulse propagates.

DOI
10.1364/josab.11.001177

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Physical Review A•Dec 1, 1981•1 citation
The propagation of a weak optical pulse along the direction of an applied magnetic field in an atomic or molecular gas is investigated theoretically. When the Zeeman splitting is much smaller that the inhomogeneous linewidth σ, the Faraday rotation of the polarization plane depends linearly on optical depth and is independent of time for long pulses with tp ≫ σ−1 (as in the conventional Faraday effect) but depends linearly on time and is independent of optical depth for short pulses with tp ≪ σ−1; tp is the pulse duration. In the intermediate-pulse-width regime, frequency pushing causes the polarization plane to rotate at greater average speeds as the pulse propagates.
We propose a dynamical model for description of the nonlinear Faraday rotation experienced by a short pulse propagating in a resonant medium subject to an ultra-strong static magnetic field. Under the assumptions of a sufficiently strong external magnetic field, such that the Zeeman splitting of the quantum system energy levels is large compared to the linewidth of the optical transitions involved and the bandwidth of the incident light, the light effectively interacts with a two-level system. Our numerical simulations show that the Faraday effect under these conditions is significantly distinctive from the one caused by weak to moderately strong magnetic field. Nonlinear coherent effects such as inhomogeneous polarization rotation along the pulse duration and an onset of a circularly polarized stimulated emission and coherent ringing have been demonstrated. Some views on the experimental observation of the predicted phenomena are given.
Nonlinear in laser power, near-resonant light-atom interaction leads to a number of spectacular modifications of a well-known effect -polarization plane rotation in a magnetic field applied along the direction of light propagation.In particular, the rotation can be enhanced by up to nine orders of magnitude compared to the linear case.Faraday effect [ 1 ] is the rotation of the plane of linear light polarization by a medium in a magnetic field applied in the direction of light propagation.In 1898, Macaluso and Corbino eJ, studying absorption spectra of the alkali atoms in the presence of magnetic fields, discovered that the Faraday effect in the vicinity of resonance absorption lines also had a distinct resonant character.The essence of the Macaluso-Corbino effect can be illustrated by considering a simple case of a F= 1 ~ F'=O transition (Fig. 1 ).Linearly polarized light incident on the sample can be resolved into two counter-rotating circular components ( cr~.In the absence of a magnetic field, the M=±l sublevels are degenerate and the optical resonance
Results of a fairly straightforward experiment on resonant magneto-optical rotation by rubidium-87 atoms revealed strong time-dependence of the polarization plane of light emerging from atomic vapors following a sudden irradiation with a laser beam. The rotation of the plane appears as a not direct consequence of the influence of the magnetic field on atoms. Reported measurements conducted using a vapor cell without any buffer gas or an anti-relaxation wall coating show that transmitted light has initially the same (linear) polarization as the incident one. Rotation of the polarization plane caused by an axial magnetic field develops in time scales similar to the pace of establishing the optical pumping/relaxation equilibrium in the atomic ensemble. The traditional passive Faraday rotation picture providing working description for the resonant magneto-optical effects in steady-state conditions does not explain the observed sequence of evolution of the polarization. The picture has to be augmented with analysis of time dependence in the spatial distribution of relevant quantum states within the atomic ensemble as well as longitudinal and transversal intensity profiles of the light beam. Isolation of external effects caused by the optical pumping and relaxation processes from consequences of internal atomic coherences should uncover more details of mechanisms of the latter in the presence of magnetic field. Plentitude of data available from time-dependent polarization measurements provides strong incentives for further development and tests of comprehensive models of coherent interactions of atomic ensembles with radiation and possibly a base for additional advance in understanding and applications of these phenomena.
The enhanced Faraday effect in gases for optical frequencies near absorption lines provides a basis for modulating a light beam. In the present study, incident radiation consisting of the Zeeman s components of the D lines of sodium passed through a polarizer to the Faraday cell, which employed saturated sodium vapor at approximately 240°C as an optical medium. Rotation of the plane of polarization of radiation traversing the cell was modulated at frequencies as high as 698 Mc/sec. On the basis of cw modulation studies, there appears to be no reduction in the integrated time-varying Faraday effect for frequencies as high as 265 Mc/sec; this frequency is 1100 times the Larmor frequency in the magnetic fields involved and 26 times the natural linewidth. There was no evidence of any reduction in frequency response for pulsed modulation at 698 Mc/sec.
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 effect of a dc magnetic field on the rate at which a ground-level, Zeeman-degenerate atom is photoionized via a short-lived autoionizing level is calculated. We consider single-step as well as two-step photoionization; in the latter case we allow the light to be either resonant or far from resonance with an intermediate level. In all cases we find that the magnetic field may raise the fraction of the atoms available for photoionization and consequently increase the photoion fraction produced by a light pulse of given fluence. In the resonant case we find that if the magnetic-field Zeeman splitting of the ground and intermediate level is much larger than the rate at which population enters or leaves that level then the Zeeman-state populations are decoupled from one another and there exist simple conditions under which they obey rate equations. In the nonresonant case an additional condition for two-photon rate equations is that the optical Stark shift of the ground state be much smaller than the magnetic Zeeman splitting. These results apply specifically to excitation by lasers whose spectral widths are narrow compared to the homogeneous linewidths of the corresponding atomic transitions, but appear to be generalizable to the opposite limit, in which the laser spectral widths are much broader than the homogeneous linewidths.