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We analyze the effect of transverse inhomogeneity on nonlinear beam propagation in a dielectric medium. Specifically, we consider the propagation of both CW beams and pulses in optical waveguides possessing a real nonlinear refractive index of the form n=n1 ( ̄r,?) + n2|E|2. The CW problem is treated within the paraxial approximation, for the case of a Gaussian beam incident on-axis. For powers lower than the homogeneous medium critical power, waveguiding dominates, and the beam focal parameter, although altered quantitatively, continues to vary sinusoidally as a function of distance as in the linear waveguide case, with a spectral period independent of the nonlinearity. Above the critical power, however, waveguiding is superceded and nonlinearity dominates. The beam becomes unstable, and displays oscillatory focussing in a fashion which is very similar to self-focussing in homogeneous media. Our pulse propagation studies employ a rather different starting point, based on the slowly varying envelope approximation and involving an averaging over the transverse coordinates. Our principal objective is to determine the conditions for undistorted pulse propagation, i.e., the existence of optical solitons. We obtain the equations governing the existence of solitons and find that they differ significantly from those for the homogeneous medium case. In particular, while "bright" soliton propagation is restricted to the anomolous dispersion regime in homogeneous media, in waveguides it is possible to propagate "bright" solitons in regimes of normal dispersion as well.
It is ironic that the ideas ofNewton, which described a beam of light as a stream ofparticles made it difficult for him to explain things like thin film interference. Yet these particles, called 'photons', have caused the adjective 'photonic' to gain common usage, when referring to optical phenomena. The purist might argue that only when we are confronted by the particle nature of light should we use the word photonics. Equally, the argument goes on, only when we are face-to face with an integrable system, i. e. one that possesses an infinite number of conserved quantities, should we say soliton rather than solitary wave. Scientists and engineers are pragmatic, however, and they are happy to use the word 'soliton' to describe what appears to be an excitation that is humped, multi humped, or localised long enough for some use to be made of it. The fact that such 'solitons' may stick to each other (fuse) upon collision is often something to celebrate for an application, rather than just evidence that, after all, these are not really solitons, in the classic sense. 'Soliton', therefore, is a widely used term with the qualification that we are constantly looking out for deviant behaviour that draws our attention to its solitary wave character. In the same spirit, 'photonics' is a useful generic cover-all noun, even when 'electromagnetic theory' or 'optics' would suffice.
This book provides researchers at the forefront of nonlinear optical technologies with robust procedures and software for the systematic investigation of the fundamental phenomena in nonlinear optical waveguide structures. A full vectorial electromagnetic formulation is adopted and the conditions under which simplification to a scalar formulation is possible are clearly indicated. The need to model the dielectric saturation properly is identified, and improved algorithms are presented for obtaining the complete power dispersion curve of structures exhibiting bistability. As the stability analysis of nonlinear modes is crucial to the development of nonlinear model methods, an effective procedure to investigate the propagation of the scalar nonlinear waves in 3D is another important feature of the book. All of the procedures described, as well as an automatic mesh generator for the finite element method, are incorporated into a software package which is included with this book.