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 1.3mu 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)