Slowly Varying Envelope Approximation Research Articles

Delayed optical feedback-regulated artificial soliton molecule in a femtosecond optical parametric oscillator

Soliton molecules (SMs) play a crucial role in nonlinear optical systems, enriching our understanding of nonlinear science through the study of their interaction dynamics. While passively mode-locked fiber lasers offer an efficient platform for generating diverse types of SMs, the complex internal dynamics of the laser often pose challenges in achieving predetermined temporal separations between SMs. Here, we implement a delayed optical feedback technique within a femtosecond optical parametric oscillator, enabling the generation of SMs with precise and controllable temporal separations. A theoretical model, which models the intracavity iterations of the signal with a simplified Ikeda map, is proposed to study the impact of parametric gain, intracavity feedback delay, and cavity length on the internal separations of the SMs. Our experimental results confirm that adjusting the cavity length allows for producing desired temporal separations within SMs. To reveal the evolution dynamics of the SMs, we further develop a rigorous numerical model using the carrier-resolved Forward Maxwell Equation, which is capable of modeling ultra-broadband complex dynamics based on a single equation without relying on the slowly-varying envelope approximation. The numerical model unveils the rich formation dynamics of the SMs at various separations, which confirms the critical role of the gain window provided by the pump. This work opens up new opportunities for the on-demand generation of SMs and provides valuable insights into the complex dynamics in femtosecond optical parametric oscillator systems with optical delayed feedback.

Stability of nonparaxial gap-soliton bullets in waveguide gratings

We investigate the formation and propagation of gap-soliton bullets in nonlinear periodic waveguides at frequencies close to the gap for Bragg reflection beyond the paraxial approximation. Using a multiple-scales analysis, we derive a two-dimensional (2D) nonlinear Schrödinger equation with higher-order correction terms that consider the nonparaxial regimes in the slowly-varying envelope approximation. In addition, a fully numerical simulation of the newly derived model equation demonstrates that the mutual balancing between Kerr, dimensionality, higher-order dispersions and nonparaxiality allows shape-preserving propagation of gap-soliton bullets in a grating waveguide.

Propagation and Transformation of Vortexes in Linear and Nonlinear Radio-Photon Systems

The article is devoted to issues related to the propagation and transformation of vortexes in the optical range of frequency. Within the framework of the traditional and modified model of slowly varying envelope approximation (SVEA), the process of converting vortex beams of the optical domain into vortex beams of the terahertz radio range based on nonlinear generation of a difference frequency in a medium with a second-order susceptibility is considered. The modified SVEA splits a slowly varying amplitude into two factors, which makes it possible to more accurately describe the three-wave mixing process. The theoretical substantiation of the rule of vortex beams topological charges conversion is given—the topological charge of the output radio-vortex beam is equal to the difference between the topological charges of the input optical vortex beams. A numerical simulation model of the processes under consideration has been implemented and analyzed.

New precise results on dynamics of weak-pulse excitation of mirrorless slab of two-level atoms

We consider by precise calculations the result of passing a very short coherent electromagnetic pulse through a gaseous slab of resonant (“two-level”) atoms bounded by vacuum on both sides. By “very short” we mean that at time t=0 (just after the passage) the system is left in an initial condition in which (except in a negligibly small region next to the exit face) the atoms have not yet felt reflection from the exit boundary; we assume only weak excitation from the ground state. A large rôle is played in our calculations by what we call ENR (exact numerical results) which are obtained by summing over a very great number of eigenmodes in the modal expansion of our formulas. Among other things, we show that, for thicknesses exceeding 5 resonant wavelengths λ, the forward flux of radiation is almost precisely inverse to the thickness, contrary to Beer’s law. We also introduce a generalized definition of frequency shift by regarding the forward flux spectrum as a mass distribution and finding the shift in its center of mass. With this definition we are able to exhibit oscillations of extremely small amplitude but extremely well defined period in the shift as a function of thickness, starting around 5λ and losing no quality even at 80λ. We have given a new discussion of the landmark 1969 paper of Burnham and Chiao in which the slowly varying envelope approximation (SVEA) was used to solve the atomic excitation problem exactly subject to neglect of the backward wave. We have elucidated the distinction between the thick-slab approximation (decay of excitation faster than return from back face), used there, and our medium-thickness problem (return from back face faster than local decay). We have shown how the method of Burnham and Chiao can be adapted to the medium-thickness problem, yielding an SVEA result almost identical to the one they presented. Other new results are summarized in the Introduction.

Efficient topology optimization of optical waveguide using finite element method based on slowly varying envelope approximation

Abstract We propose a novel finite element method (FEM) for efficient optical waveguide analysis to improve the computational efficiency in inverse design of optical waveguide devices. In our formulation, applying slowly varying envelope approximation (SVEA) to FEM, an electromagnetic field expression is split into a slowly varying complex envelope amplitude and a steady-state oscillating term. Then, field profile can be obtained by just solving the slowly varying complex envelope amplitude and the number of unknown variables along the propagation direction can be greatly reduced. The proposed method can achieve efficient optical waveguide analysis. Furthermore, by employing this analysis method in the inverse design of optical waveguide devices, automatic optimization of device structure can be carried out with reasonable computational costs.

Analysis of ultra-short bunches in free-electron lasers

Free-electron lasers (FELs) operate at wavelengths from millimeter waves through hard x-rays. At x-ray wavelengths, FELs typically rely on self-amplified spontaneous emission (SASE). Typical SASE emission contains multiple temporal ‘spikes’ which limit the longitudinal coherence of the optical output; hence, alternate schemes that improve on the longitudinal coherence of the SASE emission are of interest. In this paper, we consider electron bunches that are shorter than the SASE spike separation. In such cases, the spontaneously generated radiation consists of a single optical pulse with better longitudinal coherence than is found in typical SASE FELs. To investigate this regime, we use two FEL simulation codes. One (MINERVA) uses the slowly-varying envelope approximation (SVEA) which breaks down for extremely short pulses. The second (PUFFIN) is a particle-in-cell simulation code that is considered to be a more complete model of the underlying physics and which is able to simulate very short pulses. We first anchor these codes by showing that there is substantial agreement between the codes in simulation of the SPARC SASE FEL experiment at ENEA Frascati. We then compare the two codes for simulations using electron bunch lengths that are shorter than the SASE slice separation. The comparisons between the two codes for short bunch simulations elucidate the limitations of the SVEA in this regime but indicate that the SVEA can treat short bunches that are comparable to the cooperation length.

Impact of Transient Population Grating on the Superradiant Phase Transition in Semiconductor Laser Structures

We study theoretically and experimentally superradiant dynamics in multiple-section GaAs/AlGaAs laser structures. We develop a model based on the one-dimensional Maxwell curl equations without making use of the slowly-varying envelope approximation and demonstrate that strong transient periodic modulation of both e – h density and refractive index drastically affect the dynamics of emission generation in semiconductor media. We show that the coherent λ/2 transient population and the relative permittivity gratings facilitate e – h self-ordering and the onset of the superradiant phase transition in the e – h system.

Effects of few optical cycles approximation to spectral broadening generated in two-zero dispersion wavelengths fibers

It is both experimentally and numerically confirmed that Supercontinuum can be generated by pumping a few-tens-femtoseconds short pulse in anomalous dispersion region of a two zero-dispersion wavelengths (ZDW) fiber, such as, photonic crystal fiber. In numerical method, a well-known slowly varying envelope approximation (SVEA) has always been used to illustrate how the short pulse expands to a broadband spectrum in two-ZDWs fiber. But, the limit of SVEA is that it can only be applied for the short pulse, which its temporal envelope changes slowly comparing with its optical cycle. However, a few optical cycles short pulse is not fit with the condition of the SVEA approximation. In this report, another model, namely slowly evolving wave approximation (SEWA), is applied for numerical demonstration of supercontinuum generated by pumping the few optical cycles short pulse in two-ZDWs fiber. According to our simulation, under SEWA the final spectral width of the very short input pulse in the two-ZDWs fiber is between about 500 - 2000 nm, even the center wavelength of input pulse is near second-ZDW of fiber. Moreover, other effects of these two approximations since the center wavelength of the very short input pulse is close to both upper and lower ZDWs of the fiber in the same fiber’s length are also compared in our presentation.

On the Asymmetric Spectral Broadening of a Hydrodynamic Modulated Wave Train in the Optical Regime

Amplitude modulation of a propagating wave train has been observed in various media including hydrodynamics and optical fibers. The notable difference of the propagating wave trains in these media is the magnitude of the nonlinearity and the associated spectral bandwidth. The nonlinearity and dispersion parameters of optical fibers are two orders of magnitude smaller than the hydrodynamic counterparts, and therefore, considered to better assure the slowly varying envelope approximation (SVEA) of the nonlinear Schrödinger equations (NLSE). While most optics experiment demonstrate an NLSE-like symmetric solutions, experimental studies by Dudley et al. (Optics Express, 2009, 17, 21497–21508) show an asymmetric spectral evolution in the dynamics of unstable electromagnetic waves with high intensities. Motivated by this result, the hydrodynamic Euler equation is numerically solved to study the long-term evolution of a water-wave modulated wave train in the optical regime, i.e., at small steepness and spectral bandwidth. As the initial steepness is increased, retaining the initial spectral bandwidth thereby increasing the Benjamin–Feir Index, the modulation localizes, and the asymmetric and broad spectrum appears. While the deviation of the evolution from the NLSE solution is a result of broadband dynamics of free wave interaction, the resulting asymmetry of the spectrum is a consequence of the violation of the SVEA.

Nonlinear optical effects during femtosecond superradiant emission generation in semiconductor laser structures.

This paper presents theoretical and experimental studies of ultrabright internal second harmonic during femtosecond superradiant emission generation in multiple sections GaAs/AlGaAs laser structures at room temperature. Experimentally measured conversion efficiencies are by 1-2 orders of magnitude greater than expected. To explain this fact, a model based on one-dimensional nonlinear Maxwell curl equations without taking into consideration the slowly-varying envelope approximation has been developed. It has been demonstrated that strong transient periodic modulation of e-h density and refraction index dramatically affects the process of superradiance in semiconductor media and can explain the ultrastrong internal second harmonic generation.

Propagation of ultrashort pulses in a two-level medium beyond the slowly varying envelope approximation

The semiclassical Maxwell–Bloch equations beyond the slowly varying envelope approximation (SVEA) are derived in order to study the propagation of the ultrashort pulse in a two-level medium. The evolutions of the pulse shape, the group velocity and the pulse peak of 2[Formula: see text] ultrashort pulse are investigated as well as the pulse splitting of 4[Formula: see text] ultrashort pulse. By comparing the numerical results with those obtained using the SVEA, the error introduced by the approximation is quantitatively evaluated with respect to the pulse width and the applicable condition of the SVEA is discussed consequently. The envelope solutions presented by us allow clear observation on the evolution of the envelope.

Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion

We study linear and nonlinear pulse propagation models whose linear dispersion relations present bands of forbidden frequencies or forbidden wavenumbers. These bands are due to the interplay between higher-order dispersion and one of the terms (a second-order derivative with respect to the propagation direction) which appears when we abandon the slowly varying envelope approximation. We show that as a consequence of these forbidden bands, narrow pulses radiate in a novel and peculiar way. We also show that the nonlinear equations studied in this paper have exact soliton-like solutions of different forms, some of them being embedded solitons. The solutions obtained (of the linear as well as the nonlinear equations) are interesting since several arguments suggest that the Cauchy problems for these equations are ill-posed, and therefore the specification of the initial conditions is a delicate issue. It is also shown that some of these equations are related to elliptic curves, thus suggesting that these equations might be related to other fields where these curves appear, such as the theory of modular forms and Weierstrass ℘ functions, or the design of cryptographic protocols.

Fast Nonlinear Ultrasound Propagation Simulation Using a Slowly Varying Envelope Approximation.

Medical systems usually consider linear propagation of ultrasound, an approximation of reality. However, numerous studies have attempted to accurately simulate the nonlinear pressure wave distortion and to evaluate the contribution of harmonic frequencies. In such simulations, the computation time is very large, except for the method based on the angular spectrum scheme where the derivative order is reduced using the Fourier transform. However, the harmonic computation is usually limited to the second harmonic because of quasi-linear approximation. In this paper, a slowly varying envelope approximation (SVEA) is used in the Fourier domain to compute the entire nonlinear distortion induced, including high harmonics and nonlinear mixing frequencies. The simulation by SVEA is evaluated by comparison with other simulation tools. The obtained deviation and difference remain low enough to fully validate such an approximation. Moreover, the simulator is implemented on a GPU to obtain a very fast tool, where the full nonlinear distorted [Formula: see text] field is computed in less than 10 s.

Few-cycle solitons in supercontinuum generation dynamics

We review several propagation models that do not rely on the slowly-varying-envelope approximation (SVEA), and can thus be considered as fundamental models addressing the formation and propagation of few-cycle pulsed field structures and solitary waves arising in the course of intense ultrashort optical pulse evolution in nonlinear media and beyond octave-bandwidth optical spectrum broadening. These generic models are: the modified-Korteweg-de Vries (mKdV), the sine-Gordon (sG), and the mixed mKdV-sG equations. To include wave polarization dynamics, the vector extensions of both mKdV and sG equations are introduced. Multi-octave-spanning supercontinuum generation and few-cycle soliton structures are highlighted from numerical simulations.

On the optical bistability of ferroelectrics via two-wave mixing: Beyond the slowly varying envelope approximation

In this paper, we investigate the Optical Bistability (OB) resulting from a degenerate two-wave mixing process in a single layer of Kerr nonlinear ferroelectric material. The nonlinear process is mathematically described by a nonlinear coupled-wave (CW) equations propagating across the nonlinear medium. The system is solved numerically with and without employing the Slowly Varying Envelope Approximation (SVEA). The effects of layer thickness, operating frequency, and the intensity of the probe beam on the bistable response are investigated in both cases with and without SVEA. We have found that there is a significant difference between both solutions (with and without SVEA) in the analysis of the CW theory of two-wave mixing for OB. The SVEA solution fails to predict the bistable response for various combinations of the input parameters. Furthermore, the solution with SVEA overlooks the detection of several bistable response periods, reduces the switching contrast of the detected OB, and shows only higher threshold values of bistable response. A comparison between our theoretical results and some available experimental data is also shown.

Passive Lossless Huygens Metasurfaces for Conversion of Arbitrary Source Field to Directive Radiation

We present a semi-analytical formulation of the interaction between a given source field and a scalar Huygens metasurface (HMS), a recently introduced promising concept for wavefront manipulation based on a sheet of orthogonal electric and magnetic dipoles. Utilizing the equivalent surface impedance representation of these metasurfaces, we establish that an arbitrary source field can be converted into directive radiation via a passive lossless HMS if two physical conditions are met: local power conservation and local impedance equalization. Expressing the fields via their plane-wave spectrum and harnessing the slowly-varying envelope approximation we obtain semi-analytical formulae for the scattered fields, and prescribe the surface reactance required for the metasurface implementation. The resultant design procedure indicates that the local impedance equalization induces a Fresnel-like reflection, while local power conservation forms a radiating virtual aperture which follows the total excitation field magnitude. The semi-analytical predictions are verified by finite-element simulations of HMSs designed for different source configurations. Besides serving as a flexible design procedure for HMS radiators, the proposed formulation also provides a robust mechanism to incorporate a variety of source configurations into general HMS models, as well as physical insight on the conditions enabling purely reactive implementation of this novel type of metasurfaces.

A Novel Finite Element Method Based Beam Propagation Method without Using Slow Varying Envelope Approximation

A novel finite element method (FEM) based wide-angle (WA) beam propagation method (BPM) was presented whereby the scalar Helmholtz equation was solved without using the slow varying envelope approximation (SVEA) and was more general and precise than the previous well-known WA-BPM proposed by Hadley. The accuracy of this novel scheme was demonstrated in comparison with the existing approach.

Simulations of a FIR Oscillator with Large Slippage parameter at Jefferson Lab for FIR/UV pump-probe experiments

We previously proposed a dual FEL configuration on the UV Demo FEL at Jefferson Lab that would allow simultaneous lasing at FIR and UV wavelengths [1]. The FIR source would be an FEL oscillator with a short wiggler providing diffraction-limited pulses with pulse energy exceeding 50 microJoules, using the exhaust beam from a UVFEL as the input electron beam. Since the UV FEL requires very short pulses, the input to the FIR FEL is extremely short compared to a slippage length and the usual Slowly Varying Envelope Approximation (SVEA) does not apply. We use a non-SVEA code [2] to simulate this system both with a small energy spread (UV laser off) and with large energy spread (UV laser on).

Interaction between optical fields and their conjugates in nonlinear media

Motivated by recent experimental results, we demonstrate that the ubiquitous pulse propagation equation based on a single generalized nonlinear Schrödinger equation is incomplete and inadequate to explain the formation of the so called negative-frequency resonant radiation emitted by optical solitons. The origin of this deficiency is due to the absence of a peculiar nonlinear coupling between the positive and negative frequency components of the pulse spectrum during propagation, a feature that the slowly-varying envelope approximation is unable to capture. We therefore introduce a conceptually new model, based on the envelope of the analytic signal, that takes into account the full spectral dynamics of all frequency components, is prone to analytical treatment and retains the simulation efficiency of the nonlinear Schrödinger equation. We use our new equation to derive from first principles the phase-matching condition of the negative-frequency resonant radiation observed in previously reported experiments.

Theory of third-harmonic generation using double half-Gaussian hollow beams and self-phase-matching

We first (to best of out knowledge) use the slowly varying envelope approximation (SVEA) to obtain the equation of third harmonic (TH) generated by focusing the double half-Gaussian hollow beams into an atomic medium. The two-dimension intensity distribution is achieved by simulating, and it is similar to that generated by the 0th order Bessel beams. In addition to the similarities between the experimental setups, we demonstrate the accuracy of the equation. Compared with that generated by the Bessel beams, the pattern of the TH intensity generated by the double half-Gaussian hollow beams has salient features.