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Physical Review A

Slowly varying envelope approximation in a laser with optical feedback

Alejandro A. Duarte, Hernán Gustavo Solari

Published Aug 8, 2001• 6 citations
Publisher

Abstract

We examine the application of the slowly varying envelope approximation (SVEA) to the study of the laser with optical feedback, comparing the monochromatic modes obtained using a SVEA approximation with those emerging from a full treatment of the Maxwell-Bloch equations in a coupled cavity formulation of the laser with optical feedback [A. A. Duarte and H. G. Solari Phys. Rev. A 58, 614 (1998)]. While the SVEA approximation in the body of laser produces reliable results, the same approximation applied to the boundary conditions completely distorts the metamorphosis of the spectrum present in the original model, yet far from the transition region the SVEA gives acceptable results. The failure of the SVEA at the metamorphosis is related to the high sensitivity of the dispersion relations $k(w)$ with respect to frequency changes.

DOI
10.1103/physreva.64.033803

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We examine the application of the slowly varying envelope approximation (SVEA) to the study of the laser with optical feedback, comparing the monochromatic modes obtained using a SVEA approximation with those emerging from a full treatment of the Maxwell-Bloch equations in a coupled cavity formulation of the laser with optical feedback [A. A. Duarte and H. G. Solari Phys. Rev. A 58, 614 (1998)]. While the SVEA approximation in the body of laser produces reliable results, the same approximation applied to the boundary conditions completely distorts the metamorphosis of the spectrum present in the original model, yet far from the transition region the SVEA gives acceptable results. The failure of the SVEA at the metamorphosis is related to the high sensitivity of the dispersion relations k(w)k(w)k(w) with respect to frequency changes.
By solving the Maxwell-Bloch equations beyond the slowly varying envelope approximation (SVEA), we investigate the interaction of ultrashort pulse with two-level medium. When the pulse width decreases to the order of the optical period, the evolutions of the 2π, 4π and 6π ultrashort pulses deviate from the results obtained using the SVEA and present some new features, such as the frequency shift occurring on the trailing edge of the pulse. Moreover, a factor is defined to quantify the envelope deviation between the cases of the SVEA and the non-SVEA in order to quantitatively evaluate the error caused by the SVEA, then we discuss the applicable condition of the 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.
The propagation of laser light a few wavelengths inside a saturable absorber is theoretically studied. Computed steady-state solutions of the Maxwell–Bloch equations that describe the unidimensional propagation of a plane monochromatic wave, without introducing the slowly varying envelope approximation, show how saturation effects can influence this propagation. This can even lead to the appearance of a self-reflected wave inside the absorber because of the refractive-index changes that are induced by nonlinear coupling. One finds a close relation between the failure of the slowly varying envelope approximation and the presence of the self-reflected wave.
We investigate the propagation of laser pulses in media with non-stationary optical and magnetic response. A new amplitude integro-differential equation, governing the evolution of wave packets with one and few cycles under the envelope, is derived. The application of slowly varying amplitude approximation to this equation reduces it to a nonlinear Slowly Varying Equation of Amplitudes (SVEA) in a second approximation of the linear dispersion. It is important to point out that the SVEA obtained in media with non-stationary optical and magnetic response is similar to the SVEA equation with stationary magnetic response. The only difference is in the expressions for the group velocity and for the dispersion parameters. In particular, in paramagnetic media near to the magnetic resonances and also in experiments with a strong DC longitudinal magnetic field, it is possible to attain negative group velocity and group velocity, higher than the velocity of light in vacuum.
Resonant absorption in a laser pulse can be reduced to zero if the trailing part induces stimulated emission and returns the absorbed energy back to the pulse. This kind of effect is called Self-Induced Transparency (SIT). Analytic descriptions of short pulse propagation in a resonant medium, based on Slowly Varying Envelope Approximation (SVEA) and in linear media, have been extended in studies of SIT along two principal directions: First, to improve the SVEA by introducing Short Pulse Approximation (SPA); and second, to consider the effect of nonlinear refractive index.
The superradiance (SR) of initially inverted two-level active medium in a cavity is considered. We discuss the results specific for the rigorous simulation of the SR pulse (obtained on the basis of 1D Maxwell-Bloch equations solution without slowly varying envelope approximation (SVEA) in the spatial variable): the strong dependencies of the counter-propagating waves correlation and pulse development acceleration not only on the module of the cavity mirrors reflection coefficient, but on the integer- factor of the ratio of the sample length and half of the resonant wavelength. The simple modification of the SVEA model approximately describing the above features is considered.