Maxwell-Bloch Model Research Articles

Self-consistent Maxwell–Bloch model for high-order harmonic generation in nanostructured semiconductors

In pursuit of efficient high-order harmonic conversion in semiconductor devices, modeling insights into the complex interplay among ultrafast microscopic electron–hole dynamics, nonlinear pulse propagation, and field confinement in nanostructured materials are urgently needed. Here, a self-consistent approach coupling semiconductor Bloch and Maxwell equations is applied to compute transmission and reflection high-order harmonic spectra for finite slab and sub-wavelength nanoparticle geometries. An increase in the generated high harmonics by several orders of magnitude is predicted for gallium arsenide nanoparticles with a size maximizing the magnetic dipole resonance. Serving as a conceptual and predictive tool for ultrafast spatiotemporal nonlinear optical responses of nanostructures with arbitrary geometry, our approach is anticipated to deliver new strategies for optimal harmonic manipulation in semiconductor metadevices.

Optical pump assisted broadband terahertz frequency comb

A broadband terahertz (THz) frequency comb assisted by an optical pump in THz quantum cascade lasers (QCLs) is investigated theoretically and numerically through a Maxwell–Bloch model combined with the coupled wave theory. When an optical pump is injected into the laser cavity with dispersion, the intrinsic four-wave-mixing nonlinear process becomes not only an important elementary phase-locking mechanism during the mode proliferating process, but also the bandwidth of the frequency comb is increased and the power is amplified through the nonlinear parametric process. The relative shift between the frequency of the optical pump and the zero-dispersion frequency of THz QCLs tremendously affects the conversion efficiency of the nonlinear parametric process. The simulation results show that appropriately optical pumping could assist in generating the broadband THz frequency comb with over 1 THz and more than 80 lines, which may open many potential applications in designing and optimizing high resolution THz spectroscopy sources.

Propagation of electric field generated by periodic pumping in a stable medium of two-level atoms of the Maxwell–Bloch model

We consider the problem of the propagation of an electric field generated by periodic pumping in a stable medium of two-level atoms as the mixed problem for the Maxwell–Bloch equations without spectrum broadening. An approach to the study of such a problem is proposed. We use the inverse scattering transform method in the form of the matrix Riemann–Hilbert (RH) problem, using simultaneous spectral analysis of both the Lax equations. The proposed matrix RH problem solves the problem of the propagation of a sinusoidal signal in an unperturbed stable medium (attenuator). It is proved that this RH problem provides the causality principle for the region t < x, and for the region of the light cone, 0 < x < t allows us to find the asymptotics of the transmitted signal. First, we study the asymptotics of the RH problem for large times, and then, we obtain asymptotic formulas for the mixed problem solution of the Maxwell–Bloch equations when the attenuator is long enough. Three sectors are obtained in the light cone where the asymptotics have essentially different behaviors.

The numerical investigation of colliding optical solitons as an all-optical-gate using the method of Lines

The bidirectional propagation and collision of counter-propagating optical solitons in a two-level resonant medium is analyzed. For sufficiently short duration of the pulses and sufficiently high power levels, self-induced transparency allows the pulses to propagate without losses at anomalously low velocities. Here, we use an extended semiclassical approach, called Maxwell–Bloch model. The quantization of the electromagnetic field is neglected, however, a full quantum mechanical formulation is used to describe the behavior of the matter. The numerical simulations are performed by using the method of lines. For improved accuracy, spline interpolation of the fields is introduced. Results are presented to demonstrate the use of soliton collision for all-optical switching, in particular, as OR- and XOR-gates.

Coherent scattering of near-resonant light by a dense, microscopic cloud of cold two-level atoms: Experiment versus theory

We measure the coherent scattering of low-intensity, near-resonant light by a cloud of laser-cooled two-level rubidium atoms with a size comparable to the wavelength of light. We isolate a two-level atomic structure by applying a 300G magnetic field. We measure both the temporal and the steady-state coherent optical response of the cloud for various detunings of the laser and for atom numbers ranging from 5 to 100. We compare our results to a microscopic coupled-dipole model and to a multi-mode, paraxial Maxwell-Bloch model. In the low-intensity regime, both models are in excellent agreement, thus validating the Maxwell-Bloch model. Comparing to the data, the models are found in very good agreement for relatively low densities ($n/k^3\lesssim 0.1$), while significant deviations start to occur at higher density. This disagreement indicates that light scattering in dense, cold atomic ensembles is still not quantitatively understood, even in pristine experimental conditions.

Self-adjustment of a nonlinear lasing mode to a pumped area in a two-dimensional microcavity [Invited

We numerically performed wave dynamical simulations based on the Maxwell-Bloch (MB) model for a quadrupole-deformed microcavity laser with spatially selective pumping. We demonstrate the appearance of an asymmetric lasing mode whose spatial pattern violates both the x- and y-axes mirror symmetries of the cavity. Dynamical simulations revealed that a lasing mode consisting of a clockwise or counterclockwise rotating-wave component is a stable stationary solution of the MB model. From the results of a passive-cavity mode analysis, we interpret these asymmetric rotating-wave lasing modes by the locking of four nearly degenerate passive-cavity modes. For comparison, we carried out simulations for a uniform pumping case and found a different locking rule for the nearly degenerate modes. Our results demonstrate a nonlinear dynamical mechanism for the formation of a lasing mode that adjusts its pattern to a pumped area.

Nonadiabaticity of cavity-free neutral nitrogen lasing

We report on a theoretical study of cavity-free lasing of neutral nitrogen molecules in femtosecond laser filaments with a nonadiabatic Maxwell-Bloch model, compared with recent pump-seed experiments. The nonadiabaticity of the lasing process is revealed and it is found that electron-neutral collisions dominate the dipole dephasing rate. Moreover, we show that the asymmetry between forward and backward lasing not only depends on the different amplification lengths but also on the temporal dynamics of electron-neutral collisions. The comparison of the nonadiabatic model with simulations based on the adiabatic approximation (such as radiative transfer equations) explicitly sets a bound on the validity of the latter model for cavity-free nitrogen lasing phenomenon, which holds a unique potential in optical remote sensing applications.

Self-consistent Maxwell-Bloch model of quantum-dot photonic-crystal-cavity lasers

We present a powerful computational approach to simulate the threshold behavior of photonic-crystal quantum-dot (QD) lasers. Using a finite-difference time-domain (FDTD) technique, Maxwell-Bloch equations representing a system of thousands of statistically independent and randomly positioned two-level emitters are solved numerically. Phenomenological pure dephasing and incoherent pumping is added to the optical Bloch equations to allow for a dynamical lasing regime, but the cavity-mediated radiative dynamics and gain coupling of each QD dipole (artificial atom) is contained self-consistently within the model. These Maxwell-Bloch equations are implemented by using Lumerical's flexible material plug-in tool, which allows a user to define additional equations of motion for the nonlinear polarization. We implement the gain ensemble within triangular-lattice photonic-crystal cavities of various length $N$ (where $N$ refers to the number of missing holes), and investigate the cavity mode characteristics and the threshold regime as a function of cavity length. We develop effective two-dimensional model simulations which are derived after studying the full three-dimensional passive material structures by matching the cavity quality factors and resonance properties. We also demonstrate how to obtain the correct point-dipole radiative decay rate from Fermi's golden rule, which is captured naturally by the FDTD method. Our numerical simulations predict that the pump threshold plateaus around cavity lengths greater than $N=9$, which we identify as a consequence of the complex spatial dynamics and gain coupling from the inhomogeneous QD ensemble. This behavior is not expected from simple rate-equation analysis commonly adopted in the literature, but is in qualitative agreement with recent experiments. Single-mode to multimode lasing is also observed, depending on the spectral peak frequency of the QD ensemble. Using a statistical modal analysis of the average decay rates, we also show how the average radiative decay rate decreases as a function of cavity size. In addition, we investigate the role of structural disorder on both the passive cavity and active lasers, where the latter show a general increase in the pump threshold for cavity lengths greater than $N=7$, and a reduction in the nominal cavity mode volume for increasing amounts of disorder.

Lasing dynamics of neutral nitrogen molecules in femtosecond filaments

We investigate the dynamics of cavity-free lasing from neutral nitrogen molecules in femtosecond laser filaments. An important difference of intensity and temporal duration is observed between backward and forward lasing, in both amplified spontaneous emission and seed-amplification regimes. Numerical simulations based on a nonadiabatic Maxwell-Bloch model reproduce well the observations, and attribute these differences to the finite gain lifetime and the traveling excitation nature of this gas laser.

Controllable bright and dark rogue waves in inhomogeneous erbium doped fibers

We report bright and dark rogue waves for a generalized nonlinear Schrödinger–Maxwell–Bloch model which describes the wave propagation in an inhomogeneous erbium doped fiber. This is achieved by reducing the inhomogeneous dynamical system to homogeneous dynamical model under certain parametric conditions (integrable conditions). We discuss the recurrence and annihilation of controllable bright and dark rogue waves as they propagate through a periodically distributed fiber-amplifier system. Furthermore, we investigate the nonlinear tunneling of these rogue waves as they propagate through a dispersion barrier or well.

Characteristics of spectral-hole burning in Tm3+:YAG based on the perturbation theory

In this paper, the physical mechanism of the interaction between electromagnetic wave and spectral-hole burning crystal material is investigated in detail. In the small signal regime, a perturbation theory model is used to analyze the mechanism of spectral-hole burning. By solving the Liouville equation, three-order perturbation results are obtained. From the theoretic analysis, spectral-hole burning can be interpreted as a photon echo of the zero-order diffraction echo when the first optical pulse and the second optical pulse are overlapped in time. According to the model, the spectral-hole width is dependent on the chirp rate of the reading laser. When the chirp rate is slow with respect to the spectral features of interest, the spectral hole is closely mapped into time domain. For a fast chirp rate, distortions are observed. The results follow Maxwell—Bloch model and they are also in good agreement with the experimental results.

A proposal for multi-tens of GW fully coherent femtosecond soft X-ray lasers

X-ray free-electron lasers1,2 delivering up to 1 × 1013 coherent photons in femtosecond pulses are bringing about a revolution in X-ray science3,4,5. However, some plasma-based soft X-ray lasers6 are attractive because they spontaneously emit an even higher number of photons (1 × 1015), but these are emitted in incoherent and long (hundreds of picoseconds) pulses7 as a consequence of the amplification of stochastic incoherent self-emission. Previous experimental attempts to seed such amplifiers with coherent femtosecond soft X-rays resulted in as yet unexplained weak amplification of the seed and strong amplification of incoherent spontaneous emission8. Using a time-dependent Maxwell–Bloch model describing the amplification of both coherent and incoherent soft X-rays in plasma, we explain the observed inefficiency and propose a new amplification scheme based on the seeding of stretched high harmonics using a transposition of chirped pulse amplification to soft X-rays. This scheme is able to deliver 5 × 1014 fully coherent soft X-ray photons in 200 fs pulses and with a peak power of 20 GW. Researchers describe a path towards 5 × 1014 fully coherent soft X-ray photons in 200 fs pulses reaching 20 GW peak power. The proposed amplification scheme is based on seeding stretched high harmonics using a transposition of Chirped Pulse Amplification to soft x-rays.

HIGH FREQUENCY BEHAVIOR OF THE MAXWELL–BLOCH MODEL WITH RELAXATION: CONVERGENCE TO THE SCHRÖDINGER-RATE SYSTEM

We study the Maxwell–Bloch model, which describes the propagation of a laser through a material and the associated interaction between laser and matter (polarization of the atoms through light propagation, photon emission and absorption, etc.). The laser field is described through Maxwell's equations, a classical equation, while matter is represented at a quantum level and satisfies a quantum Liouville equation known as the Bloch model. Coupling between laser and matter is described through a quadratic source term in both equations. The model also takes into account partial relaxation effects, namely the trend of matter to return to its natural thermodynamic equilibrium. The whole system involves 6 + N(N + 1)/2 unknowns, the six-dimensional electromagnetic field plus the N(N + 1)/2 unknowns describing the state of matter, where N is the number of atomic energy levels of the considered material. We consider at once a high frequency and weak coupling situation, in the general case of anisotropic electromagnetic fields that are subject to diffraction. Degenerate energy levels are allowed. The whole system is stiff and involves strong nonlinearities. We show the convergence to a nonstiff, nonlinear, coupled Schrödinger-rate model, involving 3 + N unknowns. The electromagnetic field is eventually described through its envelope, one unknown vector in ℂ3. It satisfies a Schrödinger equation that takes into account propagation and diffraction of light inside the material. Matter on the other hand is described through a N-dimensional vector describing the occupation numbers of each atomic level. It satisfies Einstein's rate equation that describes the jumps of the electrons between the various atomic energy levels, as induced by the interaction with light. The rate of exchange between the atomic levels is proportional to the intensity of the laser field. The whole system is the physically natural nonlinear model. In order to provide an important and explicit example, we completely analyze the specific (two-dimensional) Transverse Magnetic case, for which formulae turn out to be simpler. Technically speaking, our analysis does not enter the usual mathematical framework of geometric optics: it is more singular, and requires an ad hoc Ansatz.

Stochastic pulse switching in a degenerate resonant optical medium

Using the idealized integrable Maxwell-Bloch model, we describe random optical-pulse polarization switching along an active optical medium in the $\ensuremath{\Lambda}$ configuration with disordered occupation numbers of its lower-energy sublevel pair. The description combines complete integrability and stochastic dynamics. For the single-soliton pulse, we derive the statistics of the electric-field polarization ellipse at a given point along the medium in closed form. If the average initial population difference of the two lower sublevels vanishes, we show that the pulse polarization will switch intermittently between the two circular polarizations as it travels along the medium. If this difference does not vanish, the pulse will eventually forever remain in the circular polarization determined by which sublevel is more occupied on average. We also derive the exact expressions for the statistics of the polarization-switching dynamics, such as the probability distribution of the distance between two consecutive switches and the percentage of the distance along the medium the pulse spends in the elliptical polarization of a given orientation in the case of vanishing average initial population difference. We find that the latter distribution is given in terms of the well-known arc sine law.

Propagation of dispersion–nonlinearity-managed solitons in an inhomogeneous erbium-doped fiber system

In this paper, a generalized nonlinear Schrödinger–Maxwell–Bloch model with variable dispersion and nonlinearity management functions, which describes the propagation of optical pulses in an inhomogeneous erbium-doped fiber system under certain restrictive conditions, is under investigation. We derive the Lax pair with a variable spectral parameter and the exact soliton solution is generated from the Bäcklund transformation. It is observed that stable solitons are possible only under a very restrictive condition for the spectral parameter and other inhomogeneous functions. For various forms of the inhomogeneous dispersion, nonlinearity and gain/loss functions, construction of different types of solitary waves like classical solitons, breathers, etc is discussed.

Propagation d'impulsions laser ultracourtes dans un cristal non linéaire.

In this paper, we present a mathematical model for ultrashort pulses propagation in nonlinear optical crystals. For ultrashort pulses, classical models based on the slowly varying envelope approximation are no longer relevant. The microscopic model based on the Maxwell-Bloch model, that we present, can accurately describe the wave-propagation while accounting for dispersive nonlinearities. We will describe this model and its numerical discretization in two dimensions of space. Finally, we perform two numerical experiments.

Blue five-level frequency-upconversion system in rubidium

We demonstrate production of continuous coherent blue laser light by using a five-level system in rubidium vapor. Two low-power lasers, at 780 and 776 nm, induce strong atomic coherence in the 5S-5P-5D states. The atoms decay to the 6P excited state, from which stimulated emission produces a coherent blue (420 nm) beam. We have coupled both ground-state hyperfine levels, effecting coherence between four levels. The coherent blue output is enhanced by several mechanisms, including stronger coupling to a larger fraction of the atomic population, operation at a detuning such that the vapor is nominally transparent to the 780 nm pump field, reduced losses owing to optical pumping, and optimal phase matching. We report experimental findings and compare them with results from a semiclassical Maxwell-Bloch model.

Double-resonant optical materials with embedded metal nanostructures

We derive equations modeling the resonant interaction of electric and magnetic components of light fields with metal nanostructures. This paired resonance was recently shown to produce negative refractive index. The model equations are a generalization of the well-known Maxwell-Lorentz model. We demonstrate that in the case of nonlinear polarization and linear magnetization, these equations are equivalent to a system of equations describing the resonant interaction of light with plasmonic oscillations in metal nanospheres. A family of solitary wave solutions is found that is similar to pulses associated with self-induced transparency in the framework of the Maxwell-Bloch model. The evolution of incident optical pulses is studied numerically, as are the collision dynamics of the solitary waves. These simulations reveal that the collision dynamics vary from near perfectly elastic to highly radiative, depending on the relative phase of the initial pulses.

Numerical methods for the bidimensional Maxwell–Bloch equations in nonlinear crystals

Two numerical schemes are developed for solutions of the bidimensional Maxwell–Bloch equations in nonlinear optical crystals. The Maxwell–Bloch model was recently extended [C. Besse, B. Bidégaray, A. Bourgeade, P. Degond, O. Saut, A Maxwell–Bloch model with discrete symmetries for wave propagation in nonlinear crystals: an application to KDP, M2AN Math. Model. Numer. Anal. 38 (2) (2004) 321–344] to treat anisotropic materials like nonlinear crystals. This semi-classical model seems to be adequate to describe the wave–matter interaction of ultrashort pulses in nonlinear crystals [A. Bourgeade, O. Saut, Comparison between the Maxwell–Bloch and two nonlinear maxwell models for ultrashort pulses propagation in nonlinear crystals, submitted (2004)] as it is closer to the physics than most macroscopic models. A bidimensional finite-difference-time-domain scheme, adapted from Yee [IEEE Trans. Antennas Propag. AP-14 (1966) 302–307], was already developed in [O. Saut, Bidimensional study of the Maxwell–Bloch model in a nonlinear crystal, submitted (2004)]. This scheme yields very expensive computations. In this paper, we present two numerical schemes much more efficient with their relative advantages and drawbacks.

Frequency-chirped readout of spatial-spectral absorption features

This paper examines the physical mechanisms of reading out spatial-spectral absorption features in an inhomogeneously broadened medium using linear frequency-chirped electric fields. A Maxwell-Bloch model using numerical calculation for angled beams with arbitrary phase modulation is used to simulate the chirped field readout process. The simulation results indicate that any spatial-spectral absorption feature can be read out with a chirped field with the appropriate bandwidth, duration, and intensity. Mapping spectral absorption features into temporal intensity modulations depends on the chirp rate of the field. However, when probing a spatial-spectral grating with a chirped field, a beat signal representing the grating period can be created by interfering the emitted photon echo chirped field with a reference chirped field, regardless of the chirp rate. Comparisons are made between collinear and angled readout configurations. Readout signal strength and spurious signal distortions are investigated as functions of the grating strength and the Rabi frequency of the readout pulse. Using a collinear readout geometry, distortions from optical nutation on the transmitted field and higher-order harmonics are observed, both of which are avoided in an angled beam geometry.