Examples of paraxial

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The slowly varying amplitudes of these two circularly polarized electromagnetic beams satisfy separate and different paraxial wave equations as given by (36) and (38).
Here an expression for the equation gover ning the beam width parameter s of the two laser beams in the paraxial approximation has been derived.
Numerical comparisons of the oscillations are made between the solutions of the linear and nonlinear paraxial equation due to laser-plasma interactions.
They are formed by surface ectoderm and pharyngeal endoderm and are populated by paraxial mesoderm and cranial neural crest cells.
This nonlinear evolution has been studied by using the paraxial ray approximation and the plasma wave intensity has been evaluated with and without coupling.
Solutions of the nonlinear paraxial equation due to laser plasma-interactions.
Using the paraxial-ray approximation, one can solve these coupled equations (7) and (8).
In geometrical optics a paraxial approximation often signifies another simplification, which can be made when there is one preferred coordinate direction.
The propagation of the two beams is governed by two separate and different paraxial wave equations.
An appropriate expression for the nonlinear dielectric constant has been used in the analysis of laser-beam propagation in the paraxial approximation.
The laser field is treated in a standard paraxial approximation in three dimensions.
In this case, paraxial approximations are insufficient for the description of the wave beams propagation.
The solution of pump laser beam has been obtained within the paraxial ray approximation.
We relax the paraxial-ray approximation, and retain the cross space-time derivatives of the laser amplitude, allowing for temporal self-distortion of the laser intensity profile.
A numerical study of the semiclassical limit of linear and nonlinear paraxial equation, in both focusing and defocusing cases is explored.
In fact, all branches of the complete multivalued solution will be found in this way, if x is small enough and the paraxial approximation holds.
In this paper a paraxial theory of interaction between two initially parallel electromagnetic beams in an ionic collision dominated plasma is presented.
Numerical comparisons of the oscillations are made between the solutions of linear and nonlinear paraxial equation due to laser plasma interactions.

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