Ray and Dispersion in Fibers

Feb. 15, 2018

• Ray in fibers: based on internal reflection.
Step index `->` zig-zap abruptly
Grade index `->` gradual oscillation like sin and cos.
Note: Path lengths of rays are different.
• Numerical aperture (NA): `NA = n_a sin alpha_m` where `n_a` is the index outside the fiber and `alpha_m` is the maximum acceptance angle.
Working out the relation with critical angle `theta_c` and `n_1 ~~ n_2`, `NA = sqrt{n_1^2 - n_2^2} ~~sqrt{2 n_1 Delta n} = n_1 sqrt{2 Delta}` where `Delta n = n_1 - n_2` and percentage change in index `Delta = {Delta n}/n_1`
Assuming `a` > `w` of the beam, fraction of power coupled into fiber `prop NA^2`
Note: For coupling light into fiber, we need to consider core area and `alpha_m`
• Dispersion: difference in delays for signals originating from a source in reaching a certain receiving point.
2 parameters affect dispersion -
Path length `->` aka - { multipath, intermode, modal } dispersion; measured by `D_{modal} = {Delta tau}/L "(ns / km)"`.
Freq dep. speed `->` aka - { intramode, chromatic } dispersion; measured by `{Delta tau} /{L Delta lambda}"(ns / km-nm)"`.
• Group velocity: `v_g = {d omega}/{dk} = 1/{k prime}` (or `beta_1` in text); group index `n_g -= c / {v_g}`
• Max. data rate: `R_b = 1/{4 Delta tau}`, conservative estimate.
• Modal dispersion: step index `D_{modal} = {n_{1g} Delta}/c` where `n_{1g}` is group index of the core and `Delta = {n_1 - n_2}/n_1 ~~{n_1 - n_2}/n_2`
grade index `D_{modal} = {n_{1g} Delta^2}/{8c}`
* Reduce dispersion - shortest path has slower speed than that of longest path.
Chromatic dispersion : 2 phenomena determine it
Freq dep ofn -n(λ) can be modeled by Sellmeier Eqn.; measured by material dispersionDmaterial
Freq dep ofk - nonlinear relation betweenk andω which is determined by the eigen-solution of Maxwell’s eqn.; measured by waveguide (wavelength) dispersion `D_{waveguide}`
Overall, `D_{"intra"} = D_{waveguide} = D_{material}`
• Definition of `D_{"intra"}`: `D_{"intra"}= 1/L {partial tau_g}/{partial lambda}=-k primeprime {2 pi c}/ {lambda_o^2}` where `tau_g` is the group delay and `k primeprime = {d^2 k}/{d omega^2}` (`beta_2` in the text pp. 731-733 2nd Ed., pp. 769-771 3rd Ed.).
• Zero dispersion wavelength: `D_{material} =0` at `lambda prime_0 = 1.276 mu m`
Since the negative `D_{waveguide}` cancels ` D_{material}`, `D_{"intra"} =0` at `lambda_0 = 1.3 mu m`
Using this idea, special single mode fibers like dispersion-shifted fiber and dispersion-flattened fiber are designed.
Typically, `D_{"intra"} = (lambda S_0)/4 [ 1 - ({lambda_0}/lambda)^4]` where `S_0` is the zero dispersion slope.


Last Modified: February 11, 2018
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