Optical Propagation Wave Equation Approach

Rigorous treatment of the self-action problem needs the transformation of Eq.(2.1), (2.5) into a system of integro-differential equations. However, if just some orders of group velocity dispersion and nonlinearity are taken into account, an approximate approach can be used based on differential equations solution. When dealing with the ID-i-T problem of optical pulse propagation in a dielectric waveguide, one comes to the wave equation with up to the third order GVD terms taken into account ... 

Numerics has been used intensively in the field of integrated optics (10) since its early days, simply due to the fact that even the basic example of the slab waveguide requires the solution of a transcendental equation in order to calculate the propagation constants of the slab- guided modes. Of course, the focus was directed to analytical methods, primarily, as long as the power of a desktop computer did allow for a few coupled equations and special functions only e.g. to describe the nonlinear directional coupler in a coupled mode theory (CMT) picture. During the years, lots of analytic and semi-analytic approaches to solve the wave equation have been developed in order... 

To understand better the connection between the geometrical optics approach and wave equation solutions, we will discuss in this section the basic equations describing high frequency scalar wavefield propagation. Following Bleistein (1984) and Bleistein et al. (2001), we represent the solution of the scalar wave equation (13.56) outside of the source in the form of the Debye series... 

As we have noted in the introduction, experimental evidence for a kinematic description of excitable wave fronts is rich. Based on hyperbolic wave equations and the Huygens principle, Wiener and Rosenblueth [81] recommend the eikonal approach of geometric optics waves propagate at a constant normal speed... 

The preceding analysis has been appropriate to the study of propagation and to the study of continuous-wave optical processes and devices. For comparison, we will review a simpler approach that can be rather securely applied to the interaction of pulsed light fields, but that had better not to be used for study of propagation problems. The modal approach consists in considering annihilation operators Aj(z), j = 1,2, related to the pulses centered at the frequencies coy, j = 1,2. It is assumed that these operators obey the quasiclassical equations... 

Porous silicon materials are described as a mixture of air, silicon, and, in some cases, silicon dioxide. The optical properties of a porous silicon layer are determined by the thickness, porosity, refractive index, and the shape and size of pores and are obtained from both experimental- and model-based approaches. Porous silicon is a very attractive material for refractive index fabrication because of the ease in changing its refractive index. Many studies have been made on one- and two-dimensional refractive index lattice structures. The refractive index is a complex function of wavelength, i.e., n(X) = n(X) — ik(k), where k is the extinction coefficient and determines how light waves propagate inside a material (Jackson 1975). The square of the refractive index is the dielectric function e(co) = n(co), which contributes to Maxwell s equations. 

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