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Why Modes? 

Transmission patterns are named after so-called modes not only because of plain amount of incident ray angles. The reason is that when light enters the optical fiber, energy oscillations are trapped between walls of the cladding, and light becomes a standing wave. Similar thing happens in all waveguides (electromagnetic, sound).

Two components of the standing wave: electric and magnetic, are oscillating perpendicularly to each other. Wavefront of standing waves propagates to crosswise direction (perpendicular to both EM oscillations) along the fiber core with speed near to speed of light. This means that light in the fiber consists from transverse standing waves. Modes are distinguished by number of peaks and nodes present in a standing wave pattern:

• Fundamental mode has no nodes and oscillation pattern is half-wavelength. This mode is main mode for single-mode optical fibers. It corresponds to the direct perpendicular angle of light incidence into the cable, so that single ray of light propagates in the center of the core, as most of the energy is concentrated in the center (where is the biggest standing wave amplitude, l=0,m=1 on the picture)
• Higher order modes describe behavior of light, when it enters core at an angle (acceptance cone). Standing waves with bigger amount of peaks are considered to be of higher order. Peaks are distributed along cross section according to the mode indexes: one for angular distribution and other for circular distribution.
  Higher the order of the mode, more oscillation happens partially in the cladding (see 1st picture) due to partial refraction of light into the cladding. In angular distributed modes, energy tends to "stick" to the outer layers of the core. This behavior indicates larger angle of incidence, when in optical ray representation, it bounces off the core walls.

How then multiple light rays are propagating together in the fiber?

When light becomes trapped in the cable, it can be represented by rays (left picture) or by standing waves (right picture). In case of standing waves, different incident angles are establishing different standing wave patterns, which propagate in superposition with each other (right picture demonstrates superposition of modes 01e, 31o and 31e).

For graded-index fibers, mode energy distribution pattern is spread evenly, such that oscillation amplitude is the same at all cross-sectional area.