## 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.