Electric Field Envelope Research Articles

Analytical method for calculating the electric field envelope of ultrashort pulses by approximating the wavenumber up to third order

For optical pulses shorter than 20 fs duration or highly dispersive materials in the visible range of the spectrum, high-order terms in the Taylor expansion for the wave vector, around the carrier frequency, should be considered. By expanding the wave vector near the center of optical frequency ω0 in a Taylor series up to the third-order approximation, we present an analytical method for calculating the electric field envelope of a pulse after it has propagated through a medium that contributes second- and third-order group velocity dispersion. To verify the method we present some examples for both 20 and 15 fs pulses propagating through pieces of glass made of low and high dispersive material. Limitations of the method are discussed.

Higher-order effects and ultrashort solitons in left-handed metamaterials

Starting from Maxwell's equations, we use the reductive perturbation method to derive a second-order and a third-order nonlinear Schrödinger equation, describing ultrashort solitons in nonlinear left-handed metamaterials. We find necessary conditions and derive exact bright and dark soliton solutions of these equations for the electric and magnetic field envelopes.

Localized plasmons in quantum plasmas

We consider the nonlinear interactions between finite amplitude electron and ion plasma oscillations in a fermionic quantum plasma. Accounting for the quantum statistical electron pressure and the quantum Bohm potential, we derive a set of coupled nonlinear equations that govern the dynamics of modulated electron plasma oscillations (EPOs) in the presence of the nonlinear ion oscillations (NLIOs). We numerically study stationary solutions of our coupled nonlinear equations. We find that the quantum parameter H (equal to the ratio between the plasmonic and electron Fermi energy densities) introduces new features to the electron density and electric potential humps of localized NLIOs in the absence of EPOs. Furthermore, the nonlinear coupling between the EPOs and NLIOs gives rise to a new class of envelope solitons composed of bell shaped electric field envelope of the EPOs, which are trapped in the electron density hole (and an associated negative oscillatory electric potential) that is produced by the ponderomotive force of the EPOs. The knowledge of the localized plasmonic structures is of immense value for interpreting experimental observations in dense quantum plasmas.

A MODEL FOR TRANSITION BETWEEN OUTDOOR AND INDOOR PROPAGATION

We present a novel outdoor-indoor radio wave propagation model. It predicts the electric field envelope Cumulative Distribution Function (CDF) in a room placed near a radio communication emitter. The experimental CDF obtained from the simulation, fits the experimental CDF obtained from a measurement campaign carried out over 19200 sampling points inside the room. The maximum deviation found between these CDFs is less than 1%. Kolmogorov-Smirnov test is employed to analyze the goodness of fit. P-values around 99% are reached. A comparison is made with other classical methods reported in the literature as ray-tracing (RT) and a hybrid method employing finite-difference time-domain (FDTD) together with RT. The proposed model significantly improves the results achieved in those previous investigations. Although we study the problem in three dimensions, the repetitive nature of the algorithm allows us to parallelize the computation process speeding the calculations.

Finite amplitude envelope solitons in a pair-ion plasma

The nonlinear coupling between finite amplitude ion thermal waves (ITWs) and quasistationary density perturbations in a pair-ion plasma is considered. A generalized nonlinear Schrödinger equation is derived for the ITW electric field envelope, accounting for large amplitude quasistationary plasma slow motion describing the ITW ponderomotive force. The present theory accounts for the trapping of ITWs in a large amplitude ion density hole. The small amplitude limit is considered and exact analytical solutions are obtained. Finite amplitude solutions are obtained numerically and their characteristics are discussed.

Characterization of fiber Bragg gratings using spectral interferometry based on minimum-phase functions

The focus of this paper is on interferometric systems that utilize spectral interferometry based on minimum-phase functions (MPFs) to fully characterize any fiber-Bragg-grating (FBG) spectra without exceptions. The approach presented involves sending a broadband light source (e.g., a short laser pulse) into the FBG of interest and using an optical spectrum analyzer (OSA) to record the spectrum of the interference between the reflected pulse from the grating and the time-delayed version(s) of the original short pulse. The square root of this measured spectrum, which yields the Fourier-transform (FT) magnitude of the input-pulse-sequence electric-field envelope, is then processed to uniquely recover both the phase and the amplitude of the FBG spectrum. The underlying principle for this unique recovery is that by construction, the input pulse sequence sent to the OSA is close to an MPF; thus, it is possible to recover its FT phase spectrum using only the measurement of its FT magnitude spectrum. This is an important result since, by merely measuring the FT magnitude, the entire complex spectrum of the grating can be recovered. Furthermore, this technique can conveniently be used to simultaneously characterize more than one FBG, using a single FT magnitude measurement. This technique has important advantages over existing techniques: a higher resolution and the ability to use longer duration laser pulses.

Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect

A numerical approach called Fourier direct method (FDM) is applied to nonlinear propagation of optical pulses with the central wavelength 800 nm, the width 2.67-12.00 fs, and the peak power 25-6870 kW in a fused-silica fiber. Bidirectional propagation, delayed Raman response, nonlinear dispersion (self-steepening, core dispersion), as well as correct linear dispersion are incorporated into 'bidirectional propagation equations' which are derived directly from Maxwell's equations. These equations are solved for forward and backward waves, instead of the electric-field envelope as in the nonlinear Schroedinger equation (NLSE). They are integrated as multidimensional simultaneous evolution equations evolved in space. We investigate, both theoretically and numerically, the validity and the limitation of assumptions and approximations used for deriving the NLSE. Also, the accuracy and the efficiency of the FDM are compared quantitatively with those of the finite-difference time-domain numerical approach. The time-domain size 500 fs and the number of grid points in time 2048 are chosen to investigate numerically intensity spectra, spectral phases, and temporal electric-field profiles up to the propagation distance 1.0 mm. On the intensity spectrum of a few-optical-cycle pulses, the self-steepening, core dispersion, and the delayed Raman response appear as dominant, middle, and slight effects, respectively. The delayed Raman responsemore » and the core dispersion reduce the effective nonlinearity. Correct linear dispersion is important since it affects the intensity spectrum sensitively. For the compression of femtosecond optical pulses by the complete phase compensation, the shortness and the pulse quality of compressed pulses are remarkably improved by the intense initial peak power rather than by the short initial pulse width or by the propagation distance longer than 0.1 mm. They will be compressed as short as 0.3 fs below the damage threshold of fused-silica fiber 6 MW. It is demonstrated that the carrier envelope phase (CEP) causes the difference on the temporal electric-field profile and the intensity spectrum for the initial peak power of the order of megawatts. At the propagation distance longer than the coherence length for third-order harmonics, the difference grows in the spectral components around the third-order and higher-order harmonics. The CEP can be a sensitive marker to monitor the evolution of nonlinear optical process by a few-optical-cycle electric-field wave-packet source.« less

Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials

We investigate the nonlinear propagation of electromagnetic waves in left-handed materials. For this purpose, we consider a set of coupled nonlinear Schrödinger (CNLS) equations, which govern the dynamics of coupled electric and magnetic field envelopes. The CNLS equations are used to obtain a nonlinear dispersion, which depicts the modulational stability profile of the coupled plane-wave solutions in left-handed materials. An exact (in)stability criterion for modulational interactions is derived, and analytical expressions for the instability growth rate are obtained.

Diffraction managed solitons: asymptotic validity and excitation thresholds

Solitons in discrete optical systems have been a topic of keen interest over the past decade. In a diffraction managed waveguide array, where the waveguide's diffraction profile is varied periodically in the direction of propagation, the slow evolution of the electric field envelope is modelled by a nonlocal discrete equation that exhibits soliton solutions numerically. In this paper, we verify the validity of the asymptotic approximation, demonstrating that solutions of the averaged equation are close in the l2 sense to those of the original model for long time scales. Moreover, we prove that the averaged equation has stable ground state solutions, complementing previous experimental and numerical observations. Finally, we show that for a generalization of the averaged equation, ground states of an associated Hamiltonian exist if and only if their l2 norm exceeds a certain excitation threshold, which depends on the system's dimension and degree of nonlinearity.

Gap solitons in metamaterials

Electromagnetic localization and the existence of gap solitons in nonlinear metamaterials, which exhibit a stop band in their linear spectral response, is theoretically investigated. For a self-focusing Kerr nonlinearity, the equation for the electric field envelope with carrier frequency in the stop band—where the magnetic permeability µ(?) is positive and the dielectric permittivity ϵ(?) is negative—is described by a nonlinear Klein–Gordon equation with a dispersive nonlinear term. A family of standing and moving localized waves for both electric and magnetic fields is found, and the role played by the nonlinear dispersive term on solitary wave stability is discussed.

Optimisation approach to modelling nonlinear optical absorption in thick media

It is shown that an optimisation procedure with Gaussian trial functions may be utilised to model optical beam propagation within optically thick nonlinear absorbers. Approximate expressions for the transmitted electric field and phase envelopes have been obtained. The resultant prediction for the attenuation of transmitted optical power with propagation distance is found to agree well with the corresponding result obtained from a numerical solution of the wave equation.

Propagation of laser radiation in anisotropic media

Weakly nonparaxial master equations are obtained for transverse components of the electric field envelope of laser beams and pulses propagating in an anisotropic medium with frequency dispersion and absorption (dichroism). It is demonstrated that, in the special case of a linear regime, the field is separated into ordinary and extraordinary waves and the propagation of the latter waves occurs through a mixed mechanism of diffraction and diffusion. It is shown that the derived equations are applicable to nonlinear-optical problems, including the description of the polarization structure of a soliton field in media with quadratic optical nonlinearity.

Dynamics of nonlinear dust lattice waves in the presence of ion plasma waves

The parametric interaction between nonlinear dust lattice waves and ion plasma waves in a plasma sheath is considered. It is found that the interaction is governed by a pair of equations comprising a modified Kortweg–de Vries equation for the driven (by the ion plasma wave ponderomotive force) nonlinear dust lattice waves and a nonlinear Schrödinger equation for the modulated ion plasma wave envelops. The pair admits stationary solutions in the form of a dust density cavity in which localized ion plasma wave electric field envelope is trapped.

Computer simulations of nonlinear propagation of an optical pulse using a finite-difference in the frequency-domain method

To simulate propagation of an optical pulse in a nonlinear medium, a finite-difference in frequency-domain (FDFD) method was developed. In this method, Maxwell's equations were solved rigorously without introducing an electric-field envelope function commonly used in conventional methods. This method was used to calculate the propagation of an optical soliton in a fused-silica-like material, and results were compared with those of a finite-difference in time-domain (FDTD) method. It was found that the FDFD method was efficient and more robust than the FDTD method. Another advantage of the FDFD method is the case of incorporating arbitrary linear dispersion relations into the calculations.

Launched pulse-shape dependence of the power spectrum of the spontaneous Brillouin backscattered light in an optical fiber

We theoretically analyze the relationship between the electric field envelope shape of an optical pulse launched into an optical fiber and the power spectrum of the spontaneous Brillouin backscattered light it produces. The electric field envelope is characterized by the pulse width, leading-trailing time, and steepness. The peak power of the launched, pulsed-light power spectrum is proportional to the square of the pulse width regardless of the pulse leading-trailing time and steepness, and the power spectrum broadens in inverse proportion to the pulse width. The peak power of the spontaneous Brillouin backscattered light produced by the launched, pulsed light is proportional to the pulse width when it is above approximately 100 ns and is proportional to the square of the pulse width when it is below approximately 1 ns. The power spectrum of the spontaneous Brillouin backscattered light also broadens rapidly corresponding to the pulse width, when the pulse width falls below approximately 30 ns. As the pulse leading-trailing time is shortened or the pulse leading-trailing part becomes steep, the Brillouin backscattered-light power spectrum broadens greatly, even if the launched pulse width remains constant. Our analysis showed that an optical pulse with a triangular-shaped electric field envelope forms the Brillouin backscattered-light power spectrum with the narrowest profile and consequently gives the minimum error in measuring the peak-power frequency, when the pulse width is below approximately 50 ns. The measurement error with the triangular-shaped pulsed light is 1/square root(2) times smaller than that for a rectangular-shaped pulsed light, when the pulse width falls below several nanoseconds. By contrast, the rectangular-shaped envelope gives the minimum error when the pulse width exceeds approximately 50 ns.

On the propagation of electromagnetic pulses under the conditions of quasi-resonance

A theory of quasi-resonance for the model of atoms with an arbitrary number of levels is developed by the method of unitary transformation without recourse to Bloch’s equations. Corrections to the results known from the theory of adiabatic following, which refine these results and take into account dispersion of the nonlinear response of the resonant system, are obtained. The condition for quasi-resonance with a single atomic transition is shown to impose stringent limitations on the form of the nonlinear evolution equation for the electric field envelope of the pulse with dispersion of the nonlinear response taken into account, which prohibit the reduction of the equation indicated to familiar, fully integrable systems.

Electromagnetic shocks on the optical cycle of ultrashort pulses in triple-resonance Lorentz dielectric media with subfemtosecond nonlinear electronic Debye relaxation.

The dynamical evolution of an intense ultrashort sub-10-fs two-cycle optical pulse is considered as it propagates through a transparent third-order dielectric medium characterized by three resonance lines and a finite sub-fs relaxation time of the electronic nonlinearity. Numerical integration of the full Maxwell's equations incorporating triple-resonance Lorentz linear dispersion and Debye nonlinear dispersion, for a linearly polarized electromagnetic pulse centered at lambda(0)=1.24 microm in the normal dispersion region near the zero dispersion wavelength, shows the formation of shocks occurring on the optical cycle due to the generation of optical harmonics. The finite relaxation time of the nonlinear electronic response (sub-fs time scale) (i) slows down the steepening rate of the optical cycle; (ii) does not limit the generation of strongly phase matched optical harmonics, and consequently the development of infinitely sharp edges on the optical cycle producing its breaking when linear dispersion is not included; (iii) reduces the production of phase matched harmonics and consequently the sharpening of the jumps when dispersion is present, compared to the case of an instantaneous nonlinear response; and (iv) reduces the harmonic spectrum spreading and modulation at later times on the appearance of self-steepening of the electric field envelope.

Amplitude and chirp characterization of high-power laser pulses in the 5-fs regime

Frequency-resolved optical gating (FROG) based on second-harmonic generation has been demonstrated to be capable of high-fidelity measurement of the electric-field envelope and of the temporal evolution of the instantaneous carrier frequency of 0.1-TW 5-fs pulses without the need for any correction for systematic experimental errors. At a 1-kHz repetition rate, pulse energies of a few microjoules are sufficient for reliable FROG characterization of pulses with durations down to the single-cycle regime. The results obtained reveal that carefully designed hollow-fiber chirped-mirror compressors are able to deliver high-power sub-10-fs pulses with a smooth Gaussianlike leading edge that has an intensity contrast of approximately 10(-2) .

Intense laser pulse dynamics in dense gases

The results of a theoretical investigation on the interaction between a strong ionizing laser pulse and an initially neutral gas are presented, based on a 3-dimensional numerical simulation of the radiation-gas system. The electromagnetic radiation is described, in the frame of a quasi-optical model, in terms of the complex amplitude of the electric field envelope, while the free-electron density, which determines the laser propagation characteristics, is treated by means of a fluid model, considering different ionization mechanisms. Ions are assumed to be at rest and the electron motion is nonrelativistic.The macroscopic manifestations of the ionization induced scattering of the laser radiation are discussed in detail with reference to specific experimental conditions.

Coupled-mode equations for periodic superstructure Bragg gratings

Periodic superstructure Bragg gratings are gratings in which the parameters vary periodically. Although one can determine their properties by using the standard coupled-mode equations for the electric field envelopes, it is shown that the analysis is simplified if one considers the super envelopes, i.e., the envelopes of the usual envelope functions. These super envelopes satisfy a set of (super) coupled-mode equations, which are formally identical to the standard coupled-mode equations for a uniform grating. This shows that, within certain frequency ranges, any superstructure Bragg grating acts approximately as a uniform grating.