# Electromagnetic Waves in Optical Fibers

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Light or other electromagnetic waves can propagate through optical fibers. This Demonstration visualizes the field, energy distribution, and propagation of light in step index fibers. A step index fiber is composed of a core and cladding that have different refractive indices and , respectively. Many modes are possible, including the transverse magnetic (TM) and transverse electric (TE) waves, of which the zeroth modes and are chosen here.

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In this Demonstration, the instantaneous fields, energy density, and power density are displayed for the designated time for a given mode number , core diameter , and the frequency . The maximum electric field is fixed at 1000 V/m for all conditions. The frequency has to be higher than the cut-off frequency determined by and . Using cylindrical coordinates with the wave propagation direction as the axis, the fields are a function of , , and . Energy flows along the channel (along the positive direction). The electric and magnetic fields are shown on a perpendicular plane by red and blue arrows, respectively. The energy density is displayed on two planes by coloring. The energy transport or power density ( component) is shown. The fields proportional to the magnitudes are displayed, with their relations listed in the table.

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Contributed by: Y. Shibuya (December 2012)
Open content licensed under CC BY-NC-SA

## 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.