Details

Snapshot 1: fields for mode for a fiber of 10 m diameter at

Snapshot 2: fields of mode for a fiber of 10 m diameter at

Snapshot 3: fields of mode for a fiber of 10 m diameter at

The periodic solution of the wave equation for the modes and has to be axisymmetrical. In the case of , for example, it takes the form

.

Here is the core radius and is the angular frequency. Considering the connection conditions for other field components leads to transcendental equations:

.

The values of and are the roots of the characteristic equations. There is a cut-off frequency determined by the fiber property and the value . The frequency has to be higher than in order for there to be a solution for and . The cut-off frequency is the same for the TM and TE modes, but the values of and are different because of a slight difference in the characteristic equations. Once and are determined, is obtained from the relation The constant is set so that the maximum electric field is 1000 V/m. Other field components can then be readily calculated.

For the sake of simplicity, the electrical and magnetic fields are shown in the areas where the power flow is at a maximum. The energy density is calculated by , where and are the instantaneous field values. The average Poynting vector is given by , for which the component is evaluated and shown in the graph. The cladding takes a fraction of the power in the vicinity of core.

The step index fibers are known as a multi-mode fibers. More modes are possible beside the simple TM and EM modes shown here.

Reference

[1] T. Okoshi, et al., Optical Fibers (in Japanese), Tokyo: Ohm Publishing Co., 1983.