Derivation of the Paraxial Wave Equation:
In pointtopoint communication, we may think of the electromagnetic field as propagating in a kind of "searchlight" mode  i.e. a beam of finite width that propagates in some particular direction. In analyzing this mode of wave propagation, we make use of an important solution to the so call paraxial approximation of the electromagnetic wave equation (or, more precisely, the paraxial approximation of the Helmholz equation).To that end, we first derive the paraxial approximation and then examine the freespace Gaussian Beamsolution(s). We start with the homogeneous Helmholz equation for the vector potential in the form  see Equation [ I13a ]
[ III1 ]We are looking for a wave propagating in, say, the zdirection, so we write a particular component of the potential in the form
[ III2 ]In Equation [ III2 ], the function represents a spatial modulation or "masking" of a plane wave propagating in the zdirection. The zdirection is obviously special and it is useful to appropriately parse the differential operators. For the gradoperator we may write
[ III3 ]where, for example,so that
[ III4 ]
[ III5 ]For the Laplacian operator we may write
[ III6 ]where, for example,
[ III7 ]Therefore,
[ III8 ]and the parsed Helmholz equation (without approximation) becomes
[ III9 ]The paraxial approximation is precisely defined by the condition
[ III10 ]
which means that the longitudinal variation in the modulation function, , changes very little in the wavelength associated with beam  i.e. . In this approximation, we neglect the third term and obtain the equation
[ III11 ]

which is called the paraxial approximation of the wave equation.[1]
Solutions of the Paraxial Wave Equation:
For an introductory reference see Newport's Gaussian Beam Tutorial
The Gaussian beamTo inform or motivate our next step, we consider the paraxial approximation of a known solution of the Helmholz equation  i.e. a spherical wave
[ III12 ]

Reflecting on the "quadratic" form of this approximate expression, it is reasonable to look for an axially symmetric solution of the paraxial wave equation in the following form  i.e.a Gaussian beam:
[ III13 ]

where .We test our conjecture by substituting the Gaussian beam function  i.e. Equation [ III13 ]  into the paraxial wave equation  i.e. Equation [ III11 ]  to wit
[ III14 ]

Executing the indicated operations, we obtain
[ III15a ]

or simplifying
[ III15b ]

Hence, for an arbitrary r this equation is separable into two parts  viz.
[ III15c ]

and
[ III15d ]

which are satisfied by the simple solutions
[ III16a ]

[ III16b ]

On comparison with the paraxial approximation of a spherical wave  i.e Equation [ III12 ]  we may write in terms of a radius of curvature and a width  viz.
[ III17 ]

To standardize the constants of integration we assume a plane wavefront at an arbitrary reference point  i.e. we take . Thus,
[ III18a ]

and
[ III18b ]

where is the critical Gaussian beam scaling parameter which is called variously the Fresnel length, the diffraction length, or the confocal parameter. In terms of this parameter, Equation [ III17 ] may be written
[ III19 ]

Equating real and imaginary parts, we obtain
or, finally, in standardized form
[ III20 ]

Now since Equation [ III16b ] may be written
[ III16b' ]

we may write
to obtain the usual, officially approved form of the Gaussian Beam
[ III21 ]

The following kind of picture is sometimes found to be helpful in understanding the propagation of a Gaussian Beam (the bold curve depicts the spatial variation of the beam width and the light curve the beam curvature at particular points in space):
Higher order HermiteGaussian beams
In order to study the propagation of higher order beams, we substitute the following trial solution:
[ III22 ]

into the paraxial wave equation  viz. Equation [ III11 ]  and obtain
[ III23 ]

Since the sum of the first and fifth terms already satisfies the paraxial wave equation, Equation [ III23 ] reduces to
[ III24 ]

From Equations [ III19 ] and [ III20 ] we see thatso that the reduced equation  i.e. Equation [ III24 ]  becomes
[ III25 ]

where and .
A Hermite polynomial of order n [2] has the following differential equation:
[ III26 ]

With the simple change in variables and , Equation [ III25 ] may be written
[ III27 ]

Thus, it is apparent that we can write the functions and as Hermite polynomials  viz.
if we require that . Henceor .
Finally we may write a general solution for the paraxial equation as
[ III28 ]

[0, 0] HermiteGaussian 
[0, 1] HermiteGaussian

[1, 1] HermiteGaussian 
[2, 2] HermiteGaussian 
Gaussian Beam Transformation Matrices:
What we have shown above is that a given HermiteGaussian beam is essentially completely specified or defined by the complex function . In propagating through an optical system, the beams are transformed by various optical components. The amazing fact is that the transformation produced by a given component follows a simple ABCD transformation law  viz.
[ III29 ]

where A, B, C, D are the matrix elements found in our analysis of geometric optics!!
To "prove" this, we argue by example. For example, the transformation through a uniform dielectric region is given by
so that and the transformation through a thin lens is given by
so that .
Further "justification" of this transformation lawmay be found in terms of the so called "r / r' argument"  viz
From geometric optics and, in particular, Equation [ II11 ], we may write
or
which is identical to transformation equation [ III29 ] if we interpret as the wave optics generalization of . QED
A MATLAB Gaussian Beam Plotter:
To download the MATLAB program "gbprop" click here
[2] The Hermite polynomials have the generator .