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Quantum Electronics

Spectral perturbations in a semiconductor laser: II. Nonlinear interaction of modes

P G Eliseev

Published Sep 30, 2005• 7 citations
Publisher

Abstract

The spectral perturbation is considered in a semiconductor active medium in the vicinity of the frequency of a strong electromagnetic wave. The nonlinear interaction of waves via the dynamic interference grating, or nonlinear scattering by population waves leads to the shift of resonance frequencies, which is manifested, in particular, in a change of the mode-beating spectrum of the laser. The spectral profile of this perturbation depends on the amplitude—phase coupling factor α. For large α, the profile becomes symmetrical in detuning from the strong-mode frequency. A change in the group refractive index corresponding to the 'slowing' of light is also calculated.

DOI
10.1070/qe2005v035n09abeh009187

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The spectral perturbation is considered in a semiconductor active medium in the vicinity of the frequency of a strong electromagnetic wave. The nonlinear interaction of waves via the dynamic interference grating, or nonlinear scattering by population waves leads to the shift of resonance frequencies, which is manifested, in particular, in a change of the mode-beating spectrum of the laser. The spectral profile of this perturbation depends on the amplitude—phase coupling factor α. For large α, the profile becomes symmetrical in detuning from the strong-mode frequency. A change in the group refractive index corresponding to the 'slowing' of light is also calculated.
Nonlinear optics normally involves the change in the susceptibility of a material due to the presence of optical radiation. In many cases, the light-matter interaction produces a change in the refractive index of the medium, as occurs in photorefractive media, as well as in Kerr-like media, Brillouin-active media and absorbing media with thermallyinduced refractive index changes. Such nonlinear refractive index variation corresponds to a change in the real part of the optical susceptibility %. In resonant materials, the optical radiation induces stimulated emission (or absorption) causing transfer of population between different energy states of the medium. This can lead to saturation of the amplification (or absorption) corresponding to changes in the imaginary part of the susceptibility x= i2odk where the amplitude gain coefficient a(I) is a saturable function of the intensity /, and k is the magnitude of the wavevector of the optical radiation. The interference of two (or more) coherent beams I(x) in the laser amplifier can cause a modulation of the population inversion N(x). This modulation of the inversion is commonly known as spatial hole burning in lasers and leads to formation of a gain-grating a(x) = 1/2GN(X) where G is the stimulated emission cross-section. The gain grating can act as a very efficient diffractive optical element exhibiting high diffraction efficiency that may be even greater than unity [16] and also very high phase conjugate reflectivity using four-wave mixing (FWM) geometries [5-7].
A theory is given of the stimulated scattering of light as a result of interaction of a strong electromagnetic wave with a dynamic grating (wave) of the population N of an excited state if N affects the refractive index of the medium. The medium is regarded as an assembly of two-level systems and the dependence of the permittivity ε on N is assumed to be linearized. If δε / δ N <0, the amplification due to the stimulated scattering is expected in the anti-Stokes region on illumination of an unexcited medium. Under lasing conditions the amplification should occur in the Stokes region of the strong electromagnetic field (wave). This mechanism may be the cause of the mutual coupling between laser modes.
The theory of nondegenerate four-wave mixing (NDFWM) in semiconductor lasers and amplifiers is presented with particular emphasis on the physical processes that lead to population pulsations. In the case of nearly degenerate four-wave mixing, modulation of the carrier density at the beat frequency Ω of the pump and probe waves creates a dynamic population grating whose effectiveness is governed by the spontaneous carrier lifetime τs. Such a grating affects both the gain and the refractive index of the probe wave. In particular, the probe gain exhibits features analogous to those observed in a detuned atomic system arising from the optical Stark effect. Both the gain grating and the index grating contribute to NDFWM, with the dominant contribution coming from the index grating. For detunings such that Ωτs ≫ 1, population pulsations correspond to modulation of the intraband population arising from spectral hole burning. Our results show that NDFWM is then limited by the phase-mismatch effects governed by the transit time τ rather than by the intraband population-relaxation time T1. Significant NDFWM is expected to occur for detunings up to about 300 GHz for typical transit-time values of 3 psec in semiconductor lasers.
I recent years, significant attention has been given to nonlinear optical phenomena, such as nonlinear optical absorption and dispersion, derived from a driven quantum transition of semiconductor nanostructures, such as semiconductor quantum wells, quantum disks and mainly quantum dots. An important issue that is widely studied is the effect of probe electromagnetic field intensity on the total (nonlinear) absorption coefficient and on the total (nonlinear) index of refraction. Some of the phenomena that are commonly found for large electromagnetic field intensities are the creation of a strong dip in the absorption spectrum near or at resonance (the so-called bleaching effect), negative absorption near or at resonance that leads to optical gain and the change of the slope of the total index of refraction near or at resonance from negative to positive that changes the behavior of the system from fast to slow light. Unfortunately, these phenomena do not exist and are artifacts of a common abuse of the theoretical methodology for the study of the total absorption coefficient and the total index of refraction. The main error is that the total absorption coefficient and the total index of refraction was initially derived from density matrix equations using perturbation theory and includes first-order (linear) as well as third-order (nonlinear) terms. This method gives correct results for small electromagnetic field intensities and it leads to saturation of absorption and dispersion in this regime. However, for larger intensities it gives the misleading results mentioned above due to either failure of the order of perturbation theory considered in the derivation of the total absorption coefficient and the total index of refraction or even failure of perturbation theory at all in the treatment of the problem. In this work, we revisit the problem of the total absorption coefficient and the total index of refraction in a symmetric semiconductor quantum dot structure (we choose the example of a core-shell quantum dot) under a strong probe field excitation. We use the two-level model, solve the relevant density matrix equations under the rotating wave approximation and under steady state conditions, and obtain the correct form of the nonlinear optical susceptibility that is then used for the derivation of the formulae of the total absorption coefficient and the total index of refraction under the interaction with a strong probe field. Actually, the commonly used perturbation theory formulae can be obtained by our formulae using a series expansion and keeping terms up to second order in electric field amplitude. We also derive the electronic structure of the quantum dot system by solving numerically the time-independent Schrodinger equation, under the effective mass approximation, using the potential morphing method and obtain the energy difference and the electric dipole matrix element for the relevant transitions of the quantum dot system. Then, for the specific quantum dot system we compare the results of the total absorption coefficient and the total index of refraction for the two methodologies (the perturbation result and our result) for different electromagnetic field intensities.
A study was made of the modulation of the coherent emission spectrum by spatial gratings of the refractive index, produced by the dye laser field in an absorbing medium inside the resonator. It was found that the frequencies of the reflection maxima of a nonlinear mirror resulting from the inhomogeneity of the refractive index along the direction of propagation of the light wave were in agreement with the frequencies satisfying the conditions for the maxima of interference reflection in the presence of an additional plane in the resonator.