10 Generation of High Order Modes
Several methods have been proposed to generate LG modes. We discuss here a particular method based on
fiber optics. It is known that cylindrical fibers having three layers (core plus two claddings) can exhibit
modes having an annular structure more or less analogous to an LG mode. Consider a fiber having a
core of radius
, a cladding confined to the zone
and an extra cladding for
. The latter can be assumed infinite for the guided modes, which have an evanescent
behavior in that region, so that the external radius is never reached by the light. The refractive
indices are
in the core,
in the first cladding and
in the external cladding. We
assume
. Assuming a wave of the form (
is any integer and
)
where
is any component of the optical wave and
is a parameter depending on the fiber geometry
(radii and indices). The wave equation reduces to
There exist families of modes depending on the value of
compared to
and
. Modes
such that
are called core modes. Modes such that
are called
cladding modes. We are interested in cladding modes because the central part of the beam
is vanishing in this case (as in an
mode). A further (realistic) assumption is
that the indices are slightly different. In this case, the weak guidance model holds, leading
to linearly polarized modes called LP. Solving Equation (10.2) leads to a wave of the form
with the following notation:
and
are Bessel functions of the first and second kind, respectively.
is a modified
Bessel function of the second kind. The structure of the solution was dictated by the following
considerations: The solution must be regular at
, (no
contribution in the core), the
solution must be evanescent in the external cladding (no
contribution there). Now the
arbitrary constants
and
can be reduced to one after taking into account the boundary
conditions. The boundary conditions require continuity of the components of the field tangential to
the cylindrical interfaces at
and
. In the weak guidance model, this is equivalent to
requiring a smooth solution at the interfaces and smooth derivatives. Only discrete values of
make it possible, and these discrete values determine families of modes. The central issue of the
guide theory is thus to find these values. If we adopt the following notation (
):
then the solutions
are determined by the dispersion equation
Solutions of Equation (10.9) may or may not exist, depending on the parameters. Their (finite)
number depends also on these parameters. Standard numerical procedures allow one to extract
the number of solutions and the effective indices corresponding to each of the modes. For a
given mode,
being known, the quantities
and
are known, and the constants
can be computed from
, which can be used for normalization. Specifically, we have
It is possible to find parameters such that an LP mode has a structure comparable to an LG mode. We
show a specific (but somewhat arbitrary) example in Figure 67, for which the Hermitian scalar product of
the fiber (
) mode with an LG mode is about 75% in power. The parameters of the LP mode are
The
mode has w = 4.472 cm. It is surely possible to have a better matching after an
optimization study that we are planning. The last step is to couple a
laser beam
with such a fiber and then the resulting fundamental LP mode to the
via a Bragg
coupler.