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Electromagnetic Waves in Optical Fibers

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.
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.

SNAPSHOTS

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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.
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