Ray and Dispersion in Fibers Feb. 15, 2018

+ Ray in fibers: based on internal reflection.
Step index [equation] zig-zap abruptly
Grade index [equation] gradual oscillation like sin and cos.
Note: Path lengths of rays are different.

+ Numerical aperture (NA): [equation] where [equation] is the index outside the fiber and [equation] is the maximum acceptance angle.
Working out the relation with critical angle [equation] and [equation], [equation] where [equation] and percentage change in index [equation].
Assuming a > w of the beam, fraction of power coupled into fiber [equation].
Note: For coupling light into fiber, we need to consider core area and [equation].

+ Dispersion: difference in delays for signals originating from a source in reaching a certain receiving point.
2 parameters affect dispersion -
Path length [equation] aka - { multipath, intermode, modal } dispersion; measured by [equation] (ns / km).
Freq dep. speed [equation] aka - { intramode, chromatic } dispersion; measured by [equation] (ns / km-nm).

+ Group velocity: [equation] ( or [equation] in text); group index [equation].

+ Max. data rate: [equation], conservative estimate.

+ Modal dispersion: step index [equation] where [equation] is group index of the core and [equation].
grade index [equation]
* Reduce dispersion - shortest path has slower speed than that of longest path.

+ Chromatic dispersion : 2 phenomena determine it
Freq dep of [equation] - [equation] can be modeled by Sellmeier Eqn.; measured by material dispersion [equation].
Freq dep of [equation] - nonlinear relation between [equation] and [equation] which is determined by the eigen-solution of Maxwell's eqn.; measured by waveguide (wavelength) dispersion [equation].

+ Definition of [equation]: [equation] where [equation] is the group delay and [equation] ([equation] in the text pp. 731-733 2nd Ed, pp. 769-771 3rd Ed.).

+ Zero dispersion wavelength: [equation] at [equation]
[equation] at [equation] owing to the the negative [equation] cancelling [equation].
Using this idea, special single mode fibers like dispersion-shifted fiber and dispersion-flattened fiber are designed.
[equation] where [equation] is the zero dispersion slope.

Last Modified: February 11, 2018
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