The and Y^2(.),respectively. The wave which is

The divergence angle is directly proportional to the wavelength ? and inversely proportional to the spot size 2W0 .Squeezing the spot size 2W0 (beam-waist diameter) so that beam divergence increased. So it is clear that a highly directional beam is made by making use of a small wavelength and a thick beam waist diameter.
The optical intensity I( r)= ?|U(r)| ?^2 is a function of the radial and axial positions , ?=?((x^2+y^2)) an Z respectively .
The Gaussian beam is not the only the solution of the paraxial Helmholtz equation. There are other solutions that exhibit non-Gaussian intensity distributions but share the paraboloidal wave fronts of the Gaussian beam. It is possible to break a coherent paraxial beam using the orthogonal set of so-called Hermite-Gaussian modes, which are given by the product of x and y factor. Such solutions are possible to separate in x and y in the paraxial wave equations written in Cartesian coordinates. This mode are given in order ( l , m) referring to the x and y directions.
1. The phase of hermite gaussian beam is the same as that of the Gaussian beam, except for an excess phase Z (z) that is not dependent of x and y. If Z(z) is a slowly varying function of z, both hermite and gaussian beam have paraboloidal wave fronts with the same radius of curvature R( z).
2. This hermite distribution is as Gaussian function which is modulated in the x and y directions by the functions X^2(.) and Y^2(.),respectively. The wave which is modulated in x, y direction, therefore represents a beam of non-Gaussian intensity distribution.


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