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 of$\text{ }n$ -$\text{ }n\left(\lambda \right)$ can be modeled by Sellmeier Eqn.; measured by material dispersion$\text{ }{D}_{material}$
Freq dep of$\text{ }k$ - nonlinear relation between$\text{ }k$ and$\text{ }\omega$ 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.