Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Optical Society of America B

Formation of an optical breather

R. Michalska-Trautman

Published Jan 1, 1989• 4 citations
Publisher

Abstract

The generation of an optical breather in coherent, resonant pulse propagation is studied. The initial field envelope is assumed to be in the form of a hyperbolic secant, with one half inverted so that the total area of the pulse is equal to zero. The Maxwell–Bloch equations for an inhomogeneously broadened atomic line are solved by the inverse-scattering method for all values of the independent variables x and t. The scattering amplitudes are found for any value θ of the absolute area of the initial pulse. For θ = 2πn the reflection coefficients are rational functions of the spectral parameter, and the inverse problem can be solved exactly. It is shown that the x-dependent poles of the reflection coefficients are responsible for the transient pulse shape. Such poles appear only in the presence of inhomogeneous broadening. The threshold for transparency is found at θ = 2π. At this value of the area, a breatherlike decaying solution is obtained. At θ = 4π a breather appears. Its energy amounts to approximately 70% of the initial pulse energy. For θ = 2π and 0 = 4π the solutions are analyzed in detail.

DOI
10.1364/josab.6.000036

Related papers

Analytical properties of the initial-value problem for zero-area pulses

Sami A. Shakir

The inverse scattering technique is used to show that the phase term in the Maxwell-Bloch equations should be set to zero when the slowly varying envelope approximation is assumed for self-induced transparency. A solvable model of two square pulses shifted and inverted with respect to each other is considered, and the threshold for "breather" generation for this profile is shown to correspond to an initial absolute area of $\ensuremath{\surd}2\ensuremath{\pi}$ when the separation between the two pulses is zero, and approaches $\ensuremath{\pi}$ as the separation is increased. When the two pulses are identical, only breathers are found. Upper bounds on the energy of the $0\ensuremath{\pi}$ pulses are derived, along with threshold conditions for soliton generation. For an absolute area less than $\ensuremath{\pi}$ there are no bound solutions.

Physical Review A•Oct 1, 1979•2 citations

Exact solution of the initial eigenvalue problem for coherent pulse propagation

R. Michalska-Trautman

The inverse scattering method is used to investigate the resonant propagation of an optical pulse assumed to be initially unchirped and to have a hyperbolic-secant shape. An exact solution is found for the two-component initial eigenvalue problem. The discrete spectrum consists of a finite set of purely imaginary, equidistant values. The number of discrete eigenvalues (number of solitons) is determined by the initial pulse area $A$. The scattering amplitudes are expressed in terms of Euler gamma functions and hyperbolic functions. In addition to poles in the upper half-plane, the function $\frac{\overline{b}}{a}$ has a pole on the real axis for $A=(2n+1)\ensuremath{\pi}$.

Physical Review A•Jan 1, 1981•8 citations

Polarisation effects in the formation of optical breathers at the inhomogeneously broadenedJ= 0 →J= 1 quantum transition

А. В. Волков, Oleg M. Parshkov

The formation and collisions of breathers excited by laser radiation at the inhomogeneously broadened J = 0 → J = 1 quantum transition are studied by numerical simulations in the slowly varying envelope approximation. Conditions are obtained under which laser pulses with the initial shape quite simply realised in experiments can be transformed into elliptically polarised breathers in the medium, each of the components of their field being a breather described by the theory of self-induced transparency at a nondegenerate quantum transition. It is shown that the collision of such breathers is not elastic in the general case and leads to the appearance of more general types of resonance breather-like pulses. Taking into account relaxation processes, the possibility of the formation of a breather at the 6p 2 3 P 0 → 6p7s 3 P 1 transition in the 208 Pb isotope is investigated. It is found that relaxation in some case not only causes the pulse decay but also changes the eccentricity of its polarisation ellipse.

Quantum Electronics•Sep 30, 2008•0 citations

Numerical simulation of the evolution of an intense 0πpulse and formation of an optical breather at the inhomogeneously broadened resonance quantum transition

Oleg M. Parshkov

The interaction of a sufficiently intense 0 π pulse with the inhomogeneously broadened resonance quantum transition is studied numerically in the slowly varying envelope approximation. The formation of a single optical breather and evolution of the population of energy levels related to it are described. It is found that in the quasi-resonance case, two 2 π pulses of the same duration with different frequencies appear instead of the breather at a large distance. The frequency modulation also prevents the formation of the breather, giving rise to one 2 π pulse or two 2 π pulses of different frequencies and different durations. The results of numerical analysis of the known experiment on the observation of the 0 π pulse in a ruby crystal are presented. It is shown that due to the presence of irreversible relaxation in this experiment, an optical breather was detected at the stage of its transformation to a weak 0 π pulse.

Quantum Electronics•Sep 30, 2007•0 citations

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.

Physical Review•Jul 10, 1969•1,394 citations

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.

Physical Review A•May 1, 1978•6 citations
The generation of an optical breather in coherent, resonant pulse propagation is studied. The initial field envelope is assumed to be in the form of a hyperbolic secant, with one half inverted so that the total area of the pulse is equal to zero. The Maxwell–Bloch equations for an inhomogeneously broadened atomic line are solved by the inverse-scattering method for all values of the independent variables x and t. The scattering amplitudes are found for any value θ of the absolute area of the initial pulse. For θ = 2πn the reflection coefficients are rational functions of the spectral parameter, and the inverse problem can be solved exactly. It is shown that the x-dependent poles of the reflection coefficients are responsible for the transient pulse shape. Such poles appear only in the presence of inhomogeneous broadening. The threshold for transparency is found at θ = 2π. At this value of the area, a breatherlike decaying solution is obtained. At θ = 4π a breather appears. Its energy amounts to approximately 70% of the initial pulse energy. For θ = 2π and 0 = 4π the solutions are analyzed in detail.
The inverse scattering technique is used to show that the phase term in the Maxwell-Bloch equations should be set to zero when the slowly varying envelope approximation is assumed for self-induced transparency. A solvable model of two square pulses shifted and inverted with respect to each other is considered, and the threshold for "breather" generation for this profile is shown to correspond to an initial absolute area of $\ensuremath{\surd}2\ensuremath{\pi}$ when the separation between the two pulses is zero, and approaches $\ensuremath{\pi}$ as the separation is increased. When the two pulses are identical, only breathers are found. Upper bounds on the energy of the $0\ensuremath{\pi}$ pulses are derived, along with threshold conditions for soliton generation. For an absolute area less than $\ensuremath{\pi}$ there are no bound solutions.
The inverse scattering method is used to investigate the resonant propagation of an optical pulse assumed to be initially unchirped and to have a hyperbolic-secant shape. An exact solution is found for the two-component initial eigenvalue problem. The discrete spectrum consists of a finite set of purely imaginary, equidistant values. The number of discrete eigenvalues (number of solitons) is determined by the initial pulse area AAA. The scattering amplitudes are expressed in terms of Euler gamma functions and hyperbolic functions. In addition to poles in the upper half-plane, the function b‾a\frac{\overline{b}}{a}ab​ has a pole on the real axis for $A=(2n+1)\ensuremath{\pi}$.
The formation and collisions of breathers excited by laser radiation at the inhomogeneously broadened J = 0 → J = 1 quantum transition are studied by numerical simulations in the slowly varying envelope approximation. Conditions are obtained under which laser pulses with the initial shape quite simply realised in experiments can be transformed into elliptically polarised breathers in the medium, each of the components of their field being a breather described by the theory of self-induced transparency at a nondegenerate quantum transition. It is shown that the collision of such breathers is not elastic in the general case and leads to the appearance of more general types of resonance breather-like pulses. Taking into account relaxation processes, the possibility of the formation of a breather at the 6p 2 3 P 0 → 6p7s 3 P 1 transition in the 208 Pb isotope is investigated. It is found that relaxation in some case not only causes the pulse decay but also changes the eccentricity of its polarisation ellipse.
The interaction of a sufficiently intense 0 π pulse with the inhomogeneously broadened resonance quantum transition is studied numerically in the slowly varying envelope approximation. The formation of a single optical breather and evolution of the population of energy levels related to it are described. It is found that in the quasi-resonance case, two 2 π pulses of the same duration with different frequencies appear instead of the breather at a large distance. The frequency modulation also prevents the formation of the breather, giving rise to one 2 π pulse or two 2 π pulses of different frequencies and different durations. The results of numerical analysis of the known experiment on the observation of the 0 π pulse in a ruby crystal are presented. It is shown that due to the presence of irreversible relaxation in this experiment, an optical breather was detected at the stage of its transformation to a weak 0 π pulse.
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.
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.