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`
Example for dispersion calculation

• 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 )`
Dispersion compensation: 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.)
Polarization mode dispersion (PMD): 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.
Example of dB calculation

• 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 |`

Application: Polarization maintaining fiber (PMF)

Last Modified: February 20, 2018
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