Special Fibers and Attenuation |
Feb. 20, 2018 |
• Chromatic dispersion away
from `lambda_0`: `D_{"intra"} = {lambda S_0}/4 [1 - (lambda_0/lambda)^4]`
where `S_0`
is the zero dispersion slope,
i.e. `k prime prime prime` or `beta_3`
at `lambda_0`
• Chromatic dispersion
near `lambda_0`
: E.g. dispersion shift fibers, we need to consider high
order dispersion (i.e. 3rd term in Taylor series
or `k prime prime prime`
term) and use
the
equation `D= S_0 ( lambda - lambda_0 )`
•
:
use optical elements (e.g.
dispersion compensated fiber or fiber grating) to negate the
effect of `D_{"intra"}`
so that on average chromatic dispersion is zero and even the
dispersion slope can be reduced (see Sec 5.7.3 pp. 314-319
2nd Ed., pp. 320-325 3rd Ed.)
•
:
the single mode
is composed of 2 orthogonal polarizations. PMD introduces
random delays for pulses in different
polarizations.
`Delta tau = L | n_{gx} - n_{gy} | /c`
(`=> {Delta tau}/L = | n_{gx} - n_{gy} | /c`)
A slight difference in delay for high speed (>2.5Gb/s)
channel widens pulses. The dispersion
parameter `D_{PMD} = {Delta tau }/ sqrt{L}`
too larger than 10% of a bit period, power in the slow
polarization is considered lost. (see Sec 5.7.4 pp. 320-323
2nd Ed., pp. 325-328 3rd Ed.)
• Fiber Attenuation: nonuniform refractive
index `->`
Rayleigh scattering, attenuation
coef. `alpha(lambda) = C_1 / lambda^4`
Imperfection of fiber (e.g.
non-circular), `alpha = C_2`
Absorption from
impurities, `alpha(lambda) = A(lambda)`
• Transmission windows: 3rd window (C band)
around `1.55 mu m`
has the lowest absorption
(`~~0.2 {dB}/{km}`)
1st window around 800 nm.
2nd window (S band)
around `1.3 mu m`
4th window (L band)
> `1.55 mu m`
5th window (pending on removing impurities) between 1.55 and
1.3`mu m`
• Definition
of `alpha`: In terms of
power, `P_o = P_i e^{-alpha L}`
In terms of
field, `E_o = E_i e^{-alpha_e L}`
In terms of decibel
(dB), `P_o = P_i 10^{-{alpha_{dB}L}/10 }`
Conversion
`- alpha_{dB} = 8.685 alpha_e`
and `alpha_{dB} = 4.343 alpha`
Also `dBm = 10 log_{10} (P /{1mW})`
• Power
budgeting: `(P_{rec})_{dB} = (P_{tx})_{dB} - |losses|_{dB}`
Note
- `|losses|_{dB}`
is degradation factors from components in dB, e.g. 1dB loss
in connector means 79.4% of power surviving after the
connector.
• Cut-off
wavelength `lambda_c`: In theory for step index fiber, we can
obtain `lambda_c`
from `V = 2.405`
which means 2nd and higher order mode occurring
if `lambda < lambda_c`.
In experiment, the single (fundamental) mode decreases to
less than 0.1 dB while the 2nd mode attenuated by 19.3dB.
• Birefringence: Birefringence in fiber causes
polarization evoluting along the fiber. A complete cycle is
reached at the beat
length `lambda / B`
with `B = |n_x - n_y |`