Ray and Dispersion in Fibers |
Feb. 15, 2018 |
• Ray in fibers: based on internal
reflection.
`->`
zig-zap abruptly
`->`
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 -
`->`
aka - { multipath, intermode, modal } dispersion; measured
by `D_{modal} = {Delta tau}/L "(ns / km)"`.
`->`
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.
• :
2 phenomena determine it
Freq dep
of$\text{\hspace{0.5em}}n$
-$\text{\hspace{0.5em}}n(\lambda )$
can be modeled by Sellmeier Eqn.; measured by material
dispersion$\text{\hspace{0.5em}}{D}_{material}$
Freq dep
of$\text{\hspace{0.5em}}k$
- nonlinear relation
between$\text{\hspace{0.5em}}k$
and$\text{\hspace{0.5em}}\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.