To understand the behaviour of dielectric Fibre Guides we can begin with the Gaussian Beam equation in free space. Using a similar argument we can say that, in a medium of refractive index, 
, the 
beam mode would be
The phase rate of this mode would therefore be
Near the beam waist plain, 
, we can re-arrange 10.21 to say that
and, by looking in a good maths book, we can find that
We can therefore say that, near the waist plain the beam's phasefront radius is very large — i.e. the beam is almost a ‘plane wave’ — and that the phase rate is
When using a structure such as a fibre to guide a wave we would like to arrange for the field distribution to move along the guide without diffraction effects causing the field distribution to expand and move away from the guide. To do this we can seek to obtain a plane phasefront — i.e. we require a phase rate which is independent of the cross beam distance, r. Expression 10.30 indicates that 
depends upon r. However we can ‘counteract’ this dependency by arranging that the refractive index, 
, should also depend upon r such that 
. We can represent this by modifying 10.30 to write
where 
represents the refractive index at the distance, r, from the beam axis. This can be rearranged into
where
For the paraxial approximation to be valid we require 
. This implies that 
, so we can approximate 10.33 with
and expect that 
. Since 
when 
this means we can simplify 10.32 into
This result tells us that a dielectric medium whose refractive index varies with r according to 10.35 is able to carry a Gaussian Beam Mode maintaining a plane phase front (and hence a uniform width, 
) along the fibre. Such a fibre therefore acts as a waveguide and the mode field is a fundamental Gaussian pattern. The same analysis can be performed for higher Gaussian Modes and yields a similar result although the details differ because, for higher modes, 
so we have to modify expression 10.30 etc.
Fibres of this type are called graded index or square law fibres. In effect, by having a lower refractive index off-axis they ‘speed up’ the local field, straighten out any tendency for the mode to develop a curved phase front and diffract away from the fibre axis. The graded index medium is a sort of ‘continuous lens’ which continually refocuses the beam, preventing it from spreading.
Summary
You should now understand the basic properties of modes. That we can describe a guided or propagating EM field as one or more modes, each having it's own phase rate or propagation constant, 
. That the modes in metallic guide are of two basic types — TE and TM — and that each of these modes are cut off below a given frequency whose value depends upon the guide dimensions and the mode numbers, m & n. That the lowest mode which propagates is called the Fundamental Mode. That it is often advisable to ensure single mode operation to avoid the problems which arise due to different modes propagating at differing rates. You should now also see how the fields in free space and in light fibres can be described in terms of Gaussian Beam Modes. That, in free space, these modes have widths and curvatures which are mode-number-independent. That free space Gaussian modes have a width which varies along the beam, but fibre modes have a width which can be kept constant by using a graded index material.
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