This thesis deals with the propagation of optical waves in Kerr nonlinear media, with emphasis on optical beam propagation beyond the slowly varying envelope approximation (SVEA).
The first topic in this thesis is the propagation of a continuous-wave in one-dimensional (1D) nonlinear grating structures. For this purpose we develop a numerical scheme based on a variational method. It directly implements the nonlinear Helmholtz (NLH) equation and transparent-influx boundary conditions (TIBC) without introducing any approximation except the finite element discretization. This is different from nonlinear transfer matrix formalisms that are based on the SVEA and other approximations. Therefore our method can also be used to study the validity of the nonlinear transfer matrix methods. To illustrate our method, we study the optical response of linear and nonlinear quarter-wavelength reflectors and show that the method performs well, even for large nonlinear effects. The method is also found to be able to deal with the optical bistable behavior of ideal periodic structures or gratings with defect as a function of either the frequency or the intensity of the input light. We predict that a relatively low threshold of bistability can be achieved in a defect structure which has good optical features (in our case are large field enhancement and narrow resonance) by selecting the frequency of the incident light in vicinity of the defect mode frequency.
A numerical and analytical investigation of the deformation of bichromatic waves (or equivalently bi-plane waves in a spatial domain) is also presented. Within the paraxial approximation, it is shown that an optical pulse that is initially linearly bichromatic may deform substantially, resulting in large variations in amplitude and phase. Such deformations may lead to a train of soliton-like waves. The strong deformation of a bichromatic pulse is found to depend on exceeding a critical value of the quotient of amplitude and frequency difference. This behavior holds equally well for the spatial analog.
Using the SVEA, we derive a beam propagation model that includes a transverse linear refractive index variation. Based on this model we show that a stationary spatial soliton placed in a triangular waveguide will always oscillate inside the waveguide, and that the period of the oscillation depends on the soliton amplitude. Therefore, if a bound-N-soliton, which consists of N solitons of different amplitudes but with the same velocity, is excited in a triangular waveguide, it will break up into N individual solitons. After break up, solitons produced by the break up may exit from the waveguide or at least may have a perturbed oscillation path.
The abovementioned study on the bi-plane wave distortion and the propagation of spatial solitons is based on the SVEA. To study nonparaxial effects related to these phenomena, we derive a nonparaxial beam propagation model which is called the nonparaxial nonlinear Schrodinger equation. The accuracy of this model exceeds the standard SVEA. From several numerical experiments, we find that in cases where the degree of nonparaxiality is small, the paraxial equation is in good agreement with the nonparaxial model, as expected. However, in cases where rapid changes of the envelope occur, e.g. in the break up of a bound-N-soliton or in the propagation of bi-plane waves where the product of the amplitude and the modulation period is relatively big, the paraxial model may not describe the correct physical phenomena for sufficiently large degrees of nonparaxiality.