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Physical Review A

Rotation of the Plane of Polarization of a Pulse Undergoing Self-Induced Transparency

B. D. Silverman, James C. Suits

Published Aug 1, 1972• 5 citations
Publisher

Abstract

The rotation of the plane of polarization during the propagation of a pulse undergoing self-induced transparency is reinvestigated. The rotation per unit length, which is constant along the pulse, is shown to be exactly equal to the linear result obtained far out on the trailing edge of the hyperbolic-secant pulse. If the two circular components have their center frequencies shifted with respect to each other, it is pointed out that the rotation will not be stationary in the laboratory frame, but will propagate with a characteristic velocity. The existence of such propagation of a plane of polarization could be useful to test experimentally the existence of frequency pulling.

DOI
10.1103/physreva.6.847

Related papers

Self-Induced Transparency

S. L. McCall, Erwin Louis Hahn

Above a critical power threshold for a given pulse width, a short pulse of coherent traveling-wave optical radiation is observed to propagate with anomalously low energy loss while at resonance with a two-quantum-level system of absorbers. The line shape of the resonant system is determined by inhomogeneous broadening, and the pulse width is short compared to dissipative relaxation times. A new mechanism of self-induced transparency, which accounts for the low energy loss, is analyzed in the ideal limit of a plane wave which excites a resonant medium with no damping present. The stable condition of transparency results after the traversal of the pulse through a few classical absorption lengths into the medium. This condition exists when the initial pulse has evolved into a symmetric hyperbolic-secant pulse function of time and distance, and has the area characteristic of a "$2\ensuremath{\pi}$ pulse." Ideal transparency then persists when coherent induced absorption of pulse energy during the first half of the pulse is followed by coherent induced emission of the same amount of energy back into the beam direction during the second half of the pulse. The effects of dissipative relaxation times upon pulse energy, pulse area, and pulse delay time are analyzed to first order in the ratio of short pulse width to long damping time. The analysis shows that the $2\ensuremath{\pi}$ pulse condition can be maintained if losses caused by damping are compensated by beam focusing. In an amplifying inhomogeneously broadened medium an analytic "$\ensuremath{\pi}$ pulse area" solution is presented in the limit of a sharp leading edge of the pulse. The dynamics of self-induced transparency are studied for the particular effects of Doppler velocities upon a resonant gas. The analysis of transparency for random orientations of dipole moments associated with degenerate rotational states yields modified forms of self-induced transparency behavior, which indicates a finite pulse energy loss as a function of distance in some cases. The effect of self-induced transparency on the photon echo is considered. Experimental observations of self-induced transparency have been made in a ruby sample at resonance with a pulsed ruby-laser beam. Single and multiple $2\ensuremath{\pi}$ pulse outputs have been observed, and pulse areas measured in the range of $2\ensuremath{\pi}$. The experimental results are compared with the predictions of the ideal-plane-wave theory. Deviations from the ideal-plane-wave theory are discussed. An analysis is made of the effect of a transverse mode of the propagating beam upon the transparency properties of the pulse.

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Self-induced transparency in a dispersive and nonlinear Kerr host medium

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A steady-state solution that describes pulse propagation in inhomogeneously broadening two-level atoms embedded in a dispersive and nonlinear Kerr lossless host medium has been found. The solution is of an unchirped hyperbolic-secant form with a linear spatial phase delay. The features of the solution are (i) the group velocity, the amplitude, and the width of the pulse are uniquely determined for given parameters of the medium and the carrier frequency; (ii) the group velocity is independent of the density of the resonant atoms; and (iii) the solution can exist in the positive dispersion region.

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Self-induced transparency in the presence of the Kerr nonlinearity

R. Michalska-Trautman

The self-induced transparency solution of the Bloch-Maxwell equations is found in the case of inhomogeneously broadened resonant atomic lines and a host medium with a Kerr-type nonlinearity in the index of refraction. The characteristic feature of the solution is that the atomic functions are not factorized. The time dependence of the pulse phase is different from that in the case of a sharp atomic line. In addition to the known term quadratic in the pulse envelope, it contains a term increasing in time, attaining a maximum at the trailing edge of the pulse, and rapidly decreasing afterwards. The average carrier frequency is proportional to the square of the spectral width of the atomic line divided by the squared width of the pulse. A satisfactory solution is found by modulation of the center of the Lorentz distribution function of the purely resonant solutions. It is shown that in the presence of the inhomogeneous broadening the problem cannot be solved by a linear perturbation expansion with respect to the Kerr constant. This method leads to divergencies near the trailing edge of the pulse.

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Perturbation theory in self-induced transparency for short pulses

Zofia Białynicka-Birula

A systematic perturbation expansion with respect to the small parameter ${(\ensuremath{\Delta}\ensuremath{\tau}\ensuremath{\omega})}^{\ensuremath{-}1}$, $\ensuremath{\Delta}\ensuremath{\tau}$ being the duration of the pulse and $\ensuremath{\omega}$ the optical frequency, is introduced. This method is applied to the problem of self-induced transparency for sharp-line absorbers at exact resonance. Corrections to the McCall and Hahn solution are obtained which describe the propagation of the form-preserving pulse with modulated frequency and slightly changed shape, group velocity, and wave vector. Earlier results by Marth, Holmes, and Eberly on phase modulation and wave-vector shift are confirmed.

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Tunable optical pulses of given duration by polarization technique

Ishkhan G. Harutyunyan, Toros A. Papazian

The main regularities of induced change of light polarization in the field of powerful ultrashort pulses in the resonant gaseous medium are investigated. Frequency- tunable ultrashort pulses allows to observe the polarization plane rotation at large values of detuning, in the different schemes of single-photon and two-photon interaction. On the other hand, the high intensity of these pulses allows to investigate the nonresonant interaction in the crystalline and solid transparent media. Our experiments carried out permit to propose the polarization technique for forming the given parameters pulses or train of pulses. In the observing medium placed between crossed polarizers, at counter propagation, powerful pump and linearly polarized probe pulses interact, because of which the probe pulse polarization changes. The polarization rotation angle as a time function is determined by the intensity of ultrashort pulse, the parameters of medium and frequencies of interacting waves. At the definite choice of the experimental conditions, requiring a probe wave after a crossed polarizer we can obtain the pulse which duration is determined by the length of medium, where interaction occurs. Using a nonresonant transparent media we have the possibility of forming the pulses of given duration tunable in a wide spectral region.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Oct 9, 1998•0 citations

Theory of self-induced transparency in a Kerr host medium beyond the slowly-varying-envelope approximation

Sien Chi, Tsun-Yee Wang, Senfar Wen

The self-induced transparency in the Kerr host medium has been studied beyond the slowly-varying-envelope approximation. An analytic solution is obtained. There exists a hyperbolic-secant soliton whose power, pulse width, group velocity, and propagation constant are uniquely determined for given parameters of the medium. The reductions of the group velocities of the solitons by the self-induced-transparency effect in the resonant medium with homogeneous and inhomogeneous broadening are also studied. It is found that there is negative dispersion induced by the self-induced transparency, which is not predicted by the theory with the slowly-varying-envelope approximation. The amount of the induced dispersion is determined by the total dispersion of the system required to be compensated by the Kerr effect. Numerical examples of the self-induced transparency in an erbium-doped fiber are shown. In a typical erbium-doped fiber, the soliton solution has not been found because the inhomogeneous-broadening linewidth of the medium is too large.

Physical Review A•Apr 1, 1993•9 citations
The rotation of the plane of polarization during the propagation of a pulse undergoing self-induced transparency is reinvestigated. The rotation per unit length, which is constant along the pulse, is shown to be exactly equal to the linear result obtained far out on the trailing edge of the hyperbolic-secant pulse. If the two circular components have their center frequencies shifted with respect to each other, it is pointed out that the rotation will not be stationary in the laboratory frame, but will propagate with a characteristic velocity. The existence of such propagation of a plane of polarization could be useful to test experimentally the existence of frequency pulling.
Above a critical power threshold for a given pulse width, a short pulse of coherent traveling-wave optical radiation is observed to propagate with anomalously low energy loss while at resonance with a two-quantum-level system of absorbers. The line shape of the resonant system is determined by inhomogeneous broadening, and the pulse width is short compared to dissipative relaxation times. A new mechanism of self-induced transparency, which accounts for the low energy loss, is analyzed in the ideal limit of a plane wave which excites a resonant medium with no damping present. The stable condition of transparency results after the traversal of the pulse through a few classical absorption lengths into the medium. This condition exists when the initial pulse has evolved into a symmetric hyperbolic-secant pulse function of time and distance, and has the area characteristic of a "$2\ensuremath{\pi}$ pulse." Ideal transparency then persists when coherent induced absorption of pulse energy during the first half of the pulse is followed by coherent induced emission of the same amount of energy back into the beam direction during the second half of the pulse. The effects of dissipative relaxation times upon pulse energy, pulse area, and pulse delay time are analyzed to first order in the ratio of short pulse width to long damping time. The analysis shows that the $2\ensuremath{\pi}$ pulse condition can be maintained if losses caused by damping are compensated by beam focusing. In an amplifying inhomogeneously broadened medium an analytic "$\ensuremath{\pi}$ pulse area" solution is presented in the limit of a sharp leading edge of the pulse. The dynamics of self-induced transparency are studied for the particular effects of Doppler velocities upon a resonant gas. The analysis of transparency for random orientations of dipole moments associated with degenerate rotational states yields modified forms of self-induced transparency behavior, which indicates a finite pulse energy loss as a function of distance in some cases. The effect of self-induced transparency on the photon echo is considered. Experimental observations of self-induced transparency have been made in a ruby sample at resonance with a pulsed ruby-laser beam. Single and multiple $2\ensuremath{\pi}$ pulse outputs have been observed, and pulse areas measured in the range of $2\ensuremath{\pi}$. The experimental results are compared with the predictions of the ideal-plane-wave theory. Deviations from the ideal-plane-wave theory are discussed. An analysis is made of the effect of a transverse mode of the propagating beam upon the transparency properties of the pulse.
A steady-state solution that describes pulse propagation in inhomogeneously broadening two-level atoms embedded in a dispersive and nonlinear Kerr lossless host medium has been found. The solution is of an unchirped hyperbolic-secant form with a linear spatial phase delay. The features of the solution are (i) the group velocity, the amplitude, and the width of the pulse are uniquely determined for given parameters of the medium and the carrier frequency; (ii) the group velocity is independent of the density of the resonant atoms; and (iii) the solution can exist in the positive dispersion region.
The self-induced transparency solution of the Bloch-Maxwell equations is found in the case of inhomogeneously broadened resonant atomic lines and a host medium with a Kerr-type nonlinearity in the index of refraction. The characteristic feature of the solution is that the atomic functions are not factorized. The time dependence of the pulse phase is different from that in the case of a sharp atomic line. In addition to the known term quadratic in the pulse envelope, it contains a term increasing in time, attaining a maximum at the trailing edge of the pulse, and rapidly decreasing afterwards. The average carrier frequency is proportional to the square of the spectral width of the atomic line divided by the squared width of the pulse. A satisfactory solution is found by modulation of the center of the Lorentz distribution function of the purely resonant solutions. It is shown that in the presence of the inhomogeneous broadening the problem cannot be solved by a linear perturbation expansion with respect to the Kerr constant. This method leads to divergencies near the trailing edge of the pulse.
A systematic perturbation expansion with respect to the small parameter ${(\ensuremath{\Delta}\ensuremath{\tau}\ensuremath{\omega})}^{\ensuremath{-}1}$, $\ensuremath{\Delta}\ensuremath{\tau}$ being the duration of the pulse and $\ensuremath{\omega}$ the optical frequency, is introduced. This method is applied to the problem of self-induced transparency for sharp-line absorbers at exact resonance. Corrections to the McCall and Hahn solution are obtained which describe the propagation of the form-preserving pulse with modulated frequency and slightly changed shape, group velocity, and wave vector. Earlier results by Marth, Holmes, and Eberly on phase modulation and wave-vector shift are confirmed.
The main regularities of induced change of light polarization in the field of powerful ultrashort pulses in the resonant gaseous medium are investigated. Frequency- tunable ultrashort pulses allows to observe the polarization plane rotation at large values of detuning, in the different schemes of single-photon and two-photon interaction. On the other hand, the high intensity of these pulses allows to investigate the nonresonant interaction in the crystalline and solid transparent media. Our experiments carried out permit to propose the polarization technique for forming the given parameters pulses or train of pulses. In the observing medium placed between crossed polarizers, at counter propagation, powerful pump and linearly polarized probe pulses interact, because of which the probe pulse polarization changes. The polarization rotation angle as a time function is determined by the intensity of ultrashort pulse, the parameters of medium and frequencies of interacting waves. At the definite choice of the experimental conditions, requiring a probe wave after a crossed polarizer we can obtain the pulse which duration is determined by the length of medium, where interaction occurs. Using a nonresonant transparent media we have the possibility of forming the pulses of given duration tunable in a wide spectral region.
The self-induced transparency in the Kerr host medium has been studied beyond the slowly-varying-envelope approximation. An analytic solution is obtained. There exists a hyperbolic-secant soliton whose power, pulse width, group velocity, and propagation constant are uniquely determined for given parameters of the medium. The reductions of the group velocities of the solitons by the self-induced-transparency effect in the resonant medium with homogeneous and inhomogeneous broadening are also studied. It is found that there is negative dispersion induced by the self-induced transparency, which is not predicted by the theory with the slowly-varying-envelope approximation. The amount of the induced dispersion is determined by the total dispersion of the system required to be compensated by the Kerr effect. Numerical examples of the self-induced transparency in an erbium-doped fiber are shown. In a typical erbium-doped fiber, the soliton solution has not been found because the inhomogeneous-broadening linewidth of the medium is too large.