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Journal of the Optical Society of America B

Numerical calculations of spatially localized wave emission from a nonlinear waveguide: two-level saturable media

David R. Heatley, Ewan M. Wright, G. I. Stegeman

Published Jun 1, 1990• 4 citations
Publisher

Abstract

We investigate parameter trade-offs in spatially localized wave emission from a nonlinear waveguide in which the cladding is modeled as a saturable two-level medium. This model is the simplest example of a resonant medium that displays both saturable refractive index and absorption. Numerical results are presented that show the features of the wave emission, and an approximate scaling law is obtained that greatly reduces the parameter space.

DOI
10.1364/josab.7.000990

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We investigate parameter trade-offs in spatially localized wave emission from a nonlinear waveguide in which the cladding is modeled as a saturable two-level medium. This model is the simplest example of a resonant medium that displays both saturable refractive index and absorption. Numerical results are presented that show the features of the wave emission, and an approximate scaling law is obtained that greatly reduces the parameter space.
Only for optically thin layers and plane waves may the intensity of a light wave be unambiguously expressed in terms of the nonlinear susceptibility of the medium. In aperture limited light beams the saturation parameter and the Stark shifts are local variables, while the output intensity of the pump and probe beams is an integral characteristic of the transmission process. The output spectrum is affected by diffraction, nonlinear refraction, and saturated absorption. The nonlinear refraction is a cause of a spectral line asymmetry which is well- known in saturation spectroscopy. In addition to usual line broadening, the saturated absorption causes specific beam effects, related to the formation of a self-induced transparency channel. The diffraction of the beam due to this self-induced aperture may result in the enhancement of the on-axis intensity, which is not a result of the self-induced lens effects. All these factors are taken into account by the mathematical model suggested in the present work. The propagation of the near-resonance saturating pump and non-saturating probe beams of various initial profiles through a homogeneously broadened two-level gas is simulated numerically. The calculations carried out make it possible to analyze the contribution of the propagation phenomena, mentioned above, to the output spectra of the beams.
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].
We study the self-action of an amplitude-modulated beam in a two-level saturable absorbing medium. We also consider the radial quadratic dependence of the linear refraction index to apply the results to doped waveguides. As the modulation period approaches the relaxation times, the medium response is no more instantaneous, so that one should solve the full set of Maxwell-Bloch equations. We propose a second-order scheme with the Gauss-Laguerre transformation of the transverse field pattern. A simplified approach based on the synchronous interaction approximation is used for thin layers. We analyze the transient behavior of the medium response and its manifestation in the modulation of the beam diffracting after a thin saturable absorber. Nonlinear distortions of the modulation signal passed through a doped waveguide appear to be unexpectedly small compared with those of the local polarization and population difference.
A novel method is reported for calculating the nonlinear resonant change in refractive index of a doped saturable medium at the short-pulse excitation. The derived system of equations for energy density and phase of a short pulse propagating in the doped medium allows one to address a rather general case of the resonant index non-linearity caused by the dope's population perturbation (bleaching) effect. As an example, the energy-dependent changes in the refractive index and transmission coefficient of a Cr4+:YAG crystal under the ns-range (wavelength 1.06 μm) excitation are modeled.
The four-level model is used to study the nonlinear media with coincident absorption and emission bands in both principal (S0S1) and excited (T1T2) channels.The excitation of molecules occurred by two light elds: radiations with intensity I12 at frequency ω0 tuned at the centre of absorption band of principal singletsinglet channel and radiations with intensity I34 at frequency ω tuned at the centre of absorption band of excited triplettriplet channel.The monotonic dependence of phase response of media in both principal and excited channels and the saturation processes are studied theoretically and numerically for four-level model.