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Few- and Single-Cycle Coherent Pulse Propagation in Multi-Level Media

Robert Marskar

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

Recent experimental realizations of single-cycle pulses in the terahertz, infrared, and optical regions provide promising light sources for applications in telecommunication, medical imaging, spectroscopy, and coherent control. In the resonant regime, the short duration of such pulses imply that they may escape certain effects that are detrimental to coherence; spontaneous emission is one such example. In combination with an octave-spanning spectral bandwidth, an adequate theoretical approach necessitates a semiclassical coherence model outside the conventional few-level, rotating wave, and slowly varying envelope approximations. The evolution equations that govern the underlying dynamics of such systems are inherently nonlinear and possess solutions only in certain idealized, limiting cases. In this thesis we investigate the viability of obtaining general-purpose numerical solutions to such problems.\n\nA computational approach based on Maxwell’s equations coupled with a multi-level quantum mechanical model for homogeneously and inhomogeneously broadened media is reported. The formulation leads to a numerical algorithm applicable to bi-directional propagation of pulses of arbitrary duration along all three spatial coordinates. For sufficiently dense inhomogeneously broadened materials we show, numerically, that when the forward propagation approximation of Maxwell’s wave equation is not valid, certain types of optical solitons may lose a significant amount of energy with increasing propagation distance through the co-excitation of a backward mode.\n\nA second set of computational routines is based on neglection of optical backpropagation in Maxwell’s wave equation. The resulting propagation equation generalizes the paraxial wave equation to the single-cycle pulse regime and is shown to be susceptible to numerical integration in conjunction with quantum theory. Our computer code is applied to femtosecond conical emission in atomic sodium. We find that cone emission occurs together with spatiotemporal pulse breakup analogous to the soliton-type splitting observed in self-induced transparency. We furthermore support these results by presenting complementary numerical calculations of the same phenomenon in the picosecond pulse regime. The angular structure of the conical emission observed numerically agrees with expermental measurements performed by others. Our computational results resolve some of the underlying questions surrounding conical emission in alkali metal vapors.\n\nConsideration of optical selection rules in rigid-rotor molecules is shown to greatly simplify the numerical evaluation of single-cycle pulse propagation in multi-level molecular systems. Numerical application of linearly polarized single-cycle terahertz pulses in rigid rotor molecular media, incorporating a full rotational manifold, shows that coherent bleaching effects ultimately lead to self-steepening of the driving pulse. A possible practical consequence of these results is that coherent, nonlinear saturation effects may partially cancel dispersion-induced temporal broadening of single-cycle pulses, opening up the possibility of increased energy transmission of terahertz pulses through molecular media.

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Single-cycle coherent terahertz-pulse propagation in rigid-rotor molecular media

Robert Marskar, Ulf L. Österberg

We theoretically analyze linear and nonlinear coherent propagation of linearly polarized, plane-wave, resonant single-cycle terahertz pulses through spatially extended rigid-rotor molecular media. Our model incorporates mixed state medium preparation, nonperturbative nonlinearities, saturation, coherence, memory effects, and propagation, but ignores the effects of damping. Explicit solutions are reported in the linear propagation regime. These solutions are the multilevel superposition of linear, single-cycle $0\ensuremath{\pi}$ pulses, and appear as temporal beats in the time domain. For media initially in thermal equilibrium, the pulse and molecular beats are dispersive and broaden temporally with increased propagation distance. In the simplified limit of equal rotational line strength (an idealized situation), the emitted impulses are exact temporal copies of the input pulse. An efficient, scalable computational method for solving the reduced multilevel Maxwell-Bloch equations for molecular media is reported. This method is based on a standard differential method for the propagation equation together with an operator splitting method for the Bloch equations. It invokes neither the slowly varying envelope (SVEA) or rotating wave approximations (RWA), and incorporates a large number of possible energy eigenstates (we solve for 7744 levels). Case studies of nonlinear single-cycle pulse propagation are then provided by means of computer solutions. In the nonlinear regime, we observe strong molecular orientations and suppression of the pulse and orientational revivals predicted by linear theory. For sufficiently strong pulses, coherent bleaching effects lead to increased transmission of the driving pulse, which also bears signs of self-modulation and carrier-shock formation.

Physical Review A•Aug 24, 2015•2 citations

Computational Modeling of Ultrashort Pulse Propagation in Nonlinear Optical Materials

Peter M. Goorjian, Govind Prasad Agrawal

Recent progress in ultrafast technology has led to the generation of optical pulses as short as 8 fs. The propagation of such short pulses in nonlinear optical media requires new computational techniques in which the standard approximations such as the slowly-varying envelope approximation (SVEA) and rotating-wave approximation (RWA) are not made while using the Maxwell-Bloch equations [1]. Previous calculations, which used the finite-difference time-domain (FDTD) method, found that for a a Kerr-like instantaneous nonlinearity [2] significant differences can occur when the full vector Maxwell equations are used instead of the nonlinear Schrodinger equation. Recently, the FDTD method has been extended to the case of a homogeneously broadened two-level atomic system [3].

Nonlinear Optics•Jan 1, 1996•0 citations

Conical emission as a result of pulse breakup into solitary waves

Michael E. Crenshaw, Cyrus D. Cantrell

We report computational results that show that the conical emission observed when a nearly resonant laser pulse propagates through an atomic vapor is the result of the breakup of the pulse into solitary waves. For an incident pulse with a radial as well as a temporal profile, the crests and troughs of the two-dimensional solitary waves are curved in a time-radius plane. The temporal modulation associated with pulse breakup appears as Rabi sidebands in the spectrum, while the curvature of the solitary waves in the time-radius plane results in a transverse spatial modulation that leads to conical emission in the far field. These results were obtained by performing detailed numerical calculations of the time-dependent paraxial propagation of a cylindrically symmetric laser pulse through a vapor of two-level atoms under the rotating-wave and slowly-varying-envelope approximations in the limit of no collisional damping or Doppler broadening. Our results can be interpreted readily in terms of optical nutation on the Bloch sphere and in terms of noncollinear phase matching through the curvature of the solitary waves rather than through additional parametrically generated waves. We find that self-focusing is not required for the generation of conical emission, although it would be difficult to separate these effects experimentally.

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Generation of optical sub-cycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects

Vladimir P. Kalosha, Joachım Herrmann

Summary form only given. Recent advances in ultrafast laser technology have made possible the generation of extremely short and intense pulses with only two optical periods of less than 5 fs in duration in the visible region. Efforts for generation of still shorter pulses is motivated by the possibility to study light matter interactions under unique conditions as the pulse duration approaches the duration of a single optical cycle. We study the propagation of a 5-fs pulse through a layer of a dense medium of two-level systems. As a result we demonstrate the formation of single-cycle and half-cycle pulses in the sub-fs region and novel types of spectral changes during propagation as large blue and red shifts in the transmitted and reflected pulses, respectively, explained by intra-pulse frequency-difference mixing. Solitary pulse propagation phenomena beyond the limits of slowly-varying envelope (SVEA) and rotating-wave approximation (RWA) are illustrated as the formation of full Maxwell-Bloch half-cycle solitons and two-soliton pulses.

Conference on Lasers and Electro-Optics•Jan 20, 2003•0 citations

Resonant pulse propagation in dense atomic vapours

Thomas Peter Ogden

This thesis presents theoretical models and results of numerical simulations describing the propagation of optical pulses through dense, thermal atomic vapours. In particular we investigate the nonlinear effects of optical solitons due to self-induced transparency (SIT) in two-level systems, optical simultons in V-type three-level systems and electromagnetically induced transparency (EIT) in Λ-type systems, including the storage and retrieval of dark-state polaritons. An investigation is made into two-photon excitation of the 5D states of rubidium in a high-intensity beam including the hyperfine structure of the relevant atomic levels. Decay from these states to the 6P manifolds is ruled out as a cause of experimentally observed fluorescence due to the amount of power broadening associated with intensities necessary to provide any significant level of population in these highly excited states. We combine the nonlinear effects of optical solitons and EIT to explain experimentally observed steepened pulses in a V-type system in a micron-length cell. We explain the behaviour as the early formation of a simulton pulse drawn from a CW probe field by a strong coupling pulse, due to coherent population trapping. We predict that in a longer cell it may be possible to facilitate propagation of matched pulses, even when the transitions in the system have different propagation coefficients, as long as decoherence from collision broadening can be controlled. The fact that weak pulses can propagate with this scheme suggests an approach to achieving transparent propagation of single or few photon pulses distinct from, but related to, both SIT and EIT.

•Jan 1, 2016•0 citations

Circularly polarized few-optical-cycle solitons in the short-wave-approximation regime

Hervé Leblond, Houria Triki, DUMITRU BOGDAN MIHALACHE

We consider the propagation of few-cycle pulses (FCPs) beyond the slowly varying envelope approximation in media in which the dynamics of constituent atoms is described by a two-level Hamiltonian by taking into account the wave polarization. We consider the short-wave approximation, assuming that the resonance frequency of the two-level atoms is well below the inverse of the characteristic duration of the optical pulse. By using the reductive perturbation method (multiscale analysis), we derive from the Maxwell-Bloch-Heisenberg equations the governing evolution equations for the two polarization components of the electric field in the first order of the perturbation approach. We show that propagation of circularly polarized (CP) few-optical-cycle solitons is described by a system of coupled nonlinear equations, which reduces in the scalar case to the standard sine Gordon equation describing the dynamics of linearly polarized FCPs in the short-wave-approximation regime. By direct numerical simulations, we calculate the lifetime of CP FCPs, and we study the transition to two orthogonally polarized single-humped pulses as a generic route of their instability.

Physical Review A•Aug 18, 2011•12 citations
Recent experimental realizations of single-cycle pulses in the terahertz, infrared, and optical regions provide promising light sources for applications in telecommunication, medical imaging, spectroscopy, and coherent control. In the resonant regime, the short duration of such pulses imply that they may escape certain effects that are detrimental to coherence; spontaneous emission is one such example. In combination with an octave-spanning spectral bandwidth, an adequate theoretical approach necessitates a semiclassical coherence model outside the conventional few-level, rotating wave, and slowly varying envelope approximations. The evolution equations that govern the underlying dynamics of such systems are inherently nonlinear and possess solutions only in certain idealized, limiting cases. In this thesis we investigate the viability of obtaining general-purpose numerical solutions to such problems.\n\nA computational approach based on Maxwell’s equations coupled with a multi-level quantum mechanical model for homogeneously and inhomogeneously broadened media is reported. The formulation leads to a numerical algorithm applicable to bi-directional propagation of pulses of arbitrary duration along all three spatial coordinates. For sufficiently dense inhomogeneously broadened materials we show, numerically, that when the forward propagation approximation of Maxwell’s wave equation is not valid, certain types of optical solitons may lose a significant amount of energy with increasing propagation distance through the co-excitation of a backward mode.\n\nA second set of computational routines is based on neglection of optical backpropagation in Maxwell’s wave equation. The resulting propagation equation generalizes the paraxial wave equation to the single-cycle pulse regime and is shown to be susceptible to numerical integration in conjunction with quantum theory. Our computer code is applied to femtosecond conical emission in atomic sodium. We find that cone emission occurs together with spatiotemporal pulse breakup analogous to the soliton-type splitting observed in self-induced transparency. We furthermore support these results by presenting complementary numerical calculations of the same phenomenon in the picosecond pulse regime. The angular structure of the conical emission observed numerically agrees with expermental measurements performed by others. Our computational results resolve some of the underlying questions surrounding conical emission in alkali metal vapors.\n\nConsideration of optical selection rules in rigid-rotor molecules is shown to greatly simplify the numerical evaluation of single-cycle pulse propagation in multi-level molecular systems. Numerical application of linearly polarized single-cycle terahertz pulses in rigid rotor molecular media, incorporating a full rotational manifold, shows that coherent bleaching effects ultimately lead to self-steepening of the driving pulse. A possible practical consequence of these results is that coherent, nonlinear saturation effects may partially cancel dispersion-induced temporal broadening of single-cycle pulses, opening up the possibility of increased energy transmission of terahertz pulses through molecular media.
We theoretically analyze linear and nonlinear coherent propagation of linearly polarized, plane-wave, resonant single-cycle terahertz pulses through spatially extended rigid-rotor molecular media. Our model incorporates mixed state medium preparation, nonperturbative nonlinearities, saturation, coherence, memory effects, and propagation, but ignores the effects of damping. Explicit solutions are reported in the linear propagation regime. These solutions are the multilevel superposition of linear, single-cycle $0\ensuremath{\pi}$ pulses, and appear as temporal beats in the time domain. For media initially in thermal equilibrium, the pulse and molecular beats are dispersive and broaden temporally with increased propagation distance. In the simplified limit of equal rotational line strength (an idealized situation), the emitted impulses are exact temporal copies of the input pulse. An efficient, scalable computational method for solving the reduced multilevel Maxwell-Bloch equations for molecular media is reported. This method is based on a standard differential method for the propagation equation together with an operator splitting method for the Bloch equations. It invokes neither the slowly varying envelope (SVEA) or rotating wave approximations (RWA), and incorporates a large number of possible energy eigenstates (we solve for 7744 levels). Case studies of nonlinear single-cycle pulse propagation are then provided by means of computer solutions. In the nonlinear regime, we observe strong molecular orientations and suppression of the pulse and orientational revivals predicted by linear theory. For sufficiently strong pulses, coherent bleaching effects lead to increased transmission of the driving pulse, which also bears signs of self-modulation and carrier-shock formation.
Recent progress in ultrafast technology has led to the generation of optical pulses as short as 8 fs. The propagation of such short pulses in nonlinear optical media requires new computational techniques in which the standard approximations such as the slowly-varying envelope approximation (SVEA) and rotating-wave approximation (RWA) are not made while using the Maxwell-Bloch equations [1]. Previous calculations, which used the finite-difference time-domain (FDTD) method, found that for a a Kerr-like instantaneous nonlinearity [2] significant differences can occur when the full vector Maxwell equations are used instead of the nonlinear Schrodinger equation. Recently, the FDTD method has been extended to the case of a homogeneously broadened two-level atomic system [3].
We report computational results that show that the conical emission observed when a nearly resonant laser pulse propagates through an atomic vapor is the result of the breakup of the pulse into solitary waves. For an incident pulse with a radial as well as a temporal profile, the crests and troughs of the two-dimensional solitary waves are curved in a time-radius plane. The temporal modulation associated with pulse breakup appears as Rabi sidebands in the spectrum, while the curvature of the solitary waves in the time-radius plane results in a transverse spatial modulation that leads to conical emission in the far field. These results were obtained by performing detailed numerical calculations of the time-dependent paraxial propagation of a cylindrically symmetric laser pulse through a vapor of two-level atoms under the rotating-wave and slowly-varying-envelope approximations in the limit of no collisional damping or Doppler broadening. Our results can be interpreted readily in terms of optical nutation on the Bloch sphere and in terms of noncollinear phase matching through the curvature of the solitary waves rather than through additional parametrically generated waves. We find that self-focusing is not required for the generation of conical emission, although it would be difficult to separate these effects experimentally.
Summary form only given. Recent advances in ultrafast laser technology have made possible the generation of extremely short and intense pulses with only two optical periods of less than 5 fs in duration in the visible region. Efforts for generation of still shorter pulses is motivated by the possibility to study light matter interactions under unique conditions as the pulse duration approaches the duration of a single optical cycle. We study the propagation of a 5-fs pulse through a layer of a dense medium of two-level systems. As a result we demonstrate the formation of single-cycle and half-cycle pulses in the sub-fs region and novel types of spectral changes during propagation as large blue and red shifts in the transmitted and reflected pulses, respectively, explained by intra-pulse frequency-difference mixing. Solitary pulse propagation phenomena beyond the limits of slowly-varying envelope (SVEA) and rotating-wave approximation (RWA) are illustrated as the formation of full Maxwell-Bloch half-cycle solitons and two-soliton pulses.
This thesis presents theoretical models and results of numerical simulations describing the propagation of optical pulses through dense, thermal atomic vapours. In particular we investigate the nonlinear effects of optical solitons due to self-induced transparency (SIT) in two-level systems, optical simultons in V-type three-level systems and electromagnetically induced transparency (EIT) in Λ-type systems, including the storage and retrieval of dark-state polaritons. An investigation is made into two-photon excitation of the 5D states of rubidium in a high-intensity beam including the hyperfine structure of the relevant atomic levels. Decay from these states to the 6P manifolds is ruled out as a cause of experimentally observed fluorescence due to the amount of power broadening associated with intensities necessary to provide any significant level of population in these highly excited states. We combine the nonlinear effects of optical solitons and EIT to explain experimentally observed steepened pulses in a V-type system in a micron-length cell. We explain the behaviour as the early formation of a simulton pulse drawn from a CW probe field by a strong coupling pulse, due to coherent population trapping. We predict that in a longer cell it may be possible to facilitate propagation of matched pulses, even when the transitions in the system have different propagation coefficients, as long as decoherence from collision broadening can be controlled. The fact that weak pulses can propagate with this scheme suggests an approach to achieving transparent propagation of single or few photon pulses distinct from, but related to, both SIT and EIT.
We consider the propagation of few-cycle pulses (FCPs) beyond the slowly varying envelope approximation in media in which the dynamics of constituent atoms is described by a two-level Hamiltonian by taking into account the wave polarization. We consider the short-wave approximation, assuming that the resonance frequency of the two-level atoms is well below the inverse of the characteristic duration of the optical pulse. By using the reductive perturbation method (multiscale analysis), we derive from the Maxwell-Bloch-Heisenberg equations the governing evolution equations for the two polarization components of the electric field in the first order of the perturbation approach. We show that propagation of circularly polarized (CP) few-optical-cycle solitons is described by a system of coupled nonlinear equations, which reduces in the scalar case to the standard sine Gordon equation describing the dynamics of linearly polarized FCPs in the short-wave-approximation regime. By direct numerical simulations, we calculate the lifetime of CP FCPs, and we study the transition to two orthogonally polarized single-humped pulses as a generic route of their instability.