Localized light waves: Paraxial and exact solutions of the wave equation (a review)

Linear and Nonlinear Femtosecond Optics in Isotropic Media - Ionization-Free Filamentation

“…A systematic study on the different kinds of exact solutions and methods for solving wave equation (6) was performed recently in (29). Here, as in (26) we suggest another method: Starting with the ansatz E (x, y, z, t) = V (x, y, z, t) exp (i(k 0 z − ω 0 t)), we separate the main phase and reduce the wave equation to 3D + 1 parabolic type one (2).…”

Photon Localization Revisited

“…The answer is: yes, they are indeed, since browsing various integral transform tables reveals rather few examples where both the real and imaginary part of a wave function and of its time derivative have simultaneously an exponential or stronger localization in conjunction with other requisite properties. Fortunately, the list of proper wave functions with an extraordinary strong localization is growing-in addition to an optically feasible version (Reivelt & Saari, 2002;2004) of the FWM various new interesting solutions with Gaussian falloff can be derived (Kiselev, 2007). Yet, it could be argued that the well-known Gaussian beam pulse has the same quadratically exponential radial profile in the waist region.…”

Wavelet-based integral representation for solutions of the wave equation

“…There are several practical methods that allow one to obtain an exact solution of the wave equation directly in the position space, without integrating over the whole R 3 . These methods were originated in papers [26]- [30] (see [31] for a review of such methods). The aim of this section is to look at some of these methods from the point of view of physical wavelets.…”