1 Theory

The type of optical fibre used in this project, is the well known step-index fibre which consists of a core of constant refractive index , surrounded by a cladding of a lower refractive index .

Figure 1: Schematic diagram of a simple step-index optical fibre.

Figure 1 illustrates a simple step-index fibre whose core-cladding interface is at a radius from the centre of the fibre. When the fields in a fibre interfere constructively we obtain a stable field distribution in the transverse direction with a periodic dependence, which is known as a mode. Optical fibres only support a finite number of modes which depend on the core radius and the relative refractive index difference between the core and the cladding. The field guided in the fibre can be written as a superposition of bound modes in the fibre. The modes in an optical fibre are solutions of the wave equation,

$$\left( \nabla ^{2}-\frac{n^{2}}{c^{2}}\frac{\partial ^{2}}{\partial t^{2}}\right) \vec{E}=0$$ (1)

which in cylindrical coordinates is given by

$$\left( \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{... ...}}{c^{2}}\frac{\partial ^{2}}{\partial t^{2}}\right) \vec{E}=0$$ (2)

and consist of Bessel functions. Much of the work done in this project mainly focuses on the TE01, TM01 and HE11 modes. The TE (Transverse Electric) mode is one where the longitudinal component of the electric field is zero (ie. ). Similarly the TM (Transverse Magnetic) mode is one where the longitudinal component of the magnetic field is zero (ie. ). In addition to these modes, we also deal with hybrid modes, such as the HE and EH modes.

Note that working in a cylindrical polar coordinate system is much preferred due to the cylindrical geometry of an optical fibre. Cartesian vectors (ie. with , , components) can be transformed to cylindrical polar vectors (ie. with , , components) using the following equations

$$\vec{E}_{r} = \vec{E}_{x}\cos \phi +\vec{E}_{y}\sin \phi$$ (3)

$$\vec{E}_{\phi } = -\vec{E}_{x}\sin \phi +\vec{E}_{y}\cos \phi$$ (4)

and similarly, cylindrical polar vectors can be transformed to Cartesian vectors using the following equations

$$\vec{E}_{x} = \vec{E}_{r}\cos \phi -\vec{E}_{\phi }\sin \phi$$ (5)

$$\vec{E}_{y} = \vec{E}_{r}\sin \phi +\vec{E}_{\phi }\cos \phi$$ (6)

. It is useful to define the normalised frequency, given by

$$V=\frac{2\pi }{\lambda }a\sqrt{n_{core}-n_{cladding}}$$ (7)

which is a dimensionless parameter and shall later to be referred to instead of the wavelength . Shown below is a typical example of a dispersion curve, which shows the relationship between the effective refractive index () and for various modes.

Figure 2: An example dispersion curve.