Fiber Nonlinearity

Feb. 22, 2018

• Fiber nonlinearity: 2nd order nonlinearity `chi^{(2)} ee` occurs only in anisotropic crystal, not in fiber.
3rd order nonlinearity `chi^{(3)} eee` occurs in fiber.
• Effective area (`A_e`) and length (`L_e`) (pp. 77-79 2nd Ed., pp. 79-81 3rd Ed.): Since nonlinear effects are much dependent on power and intensity, the propagation distance `/L_e` and the fiber cross section `A_e` where high power and high intensity are maintained.
Effective length `L_e = (1 - e^{-alpha L})/alpha`; for ` alpha L ">>" 1`, `L_e ~~ 1 / alpha` where `alpha` is the attenuation coefficient of the fiber.
Effective area `A_e = {| int I ds |^2}/{int I^2 ds}`; effective intensity `I_e = P/{A_e}`
• Stimulated Raman Scattering (SRS) (pp. 80-81, 326-329 2nd Ed, pp. 82-83, 332-334 3rd Ed.): Input high freq `->` output low freq.
It has broadband (100’s of nm) output and provide forward & backward scattering.
It requires power above a threshold `P_{th} = {16 A_e}/{g_R L_e}` where `A_e` is almost equal to core area, `g_R` is the Raman gain coef., typical `P_{th} ~~ 1 W`.
Application: optical amplifier, laser.
Problems: Generate cross talk and limits power per channel, e.g.
considering Raman gain profile as a triangle with linewidth `Delta lambda_c` and `W` equally spaced channels `0, 1, ..., W-1` with spacing `Delta lambda _s` & equal power `P`.
The fraction of power coupled from Channel 0 (with shortest wavelength) to all other channels is ` P_0 = [{g_R Delta lambda_s P L_e} /{2 Delta lambda_c A_e}] {W(W-1)}/2`
Example on calculating the threshold power

• Stimulated Brillouin Scattering (SBS) (pp. 79-80 2nd Ed., pp. 81-82 3rd Ed): SBS causes backward scattering and down shifted by 11 GHz at 1550nm.
It requires power above a threshold `P_{th} = {21 alpha A}/g_B` where `A` is core area, `g_B` is the Brillouin gain coef., typical `P_{th} ~~ 0.01W`
SBS is narrowband 100´S of MHz. As bandwidth increases, `P_{th}` increases. (pp. 325-326 2nd Ed, pp. 331-332 3rd Ed)
• Four-wave mixing (FWM): Mixing 3 beams with different frequencies `->` dominant freq. `f_{FWM} = f_1 + f_2 - f_3`, `f_1 + f_3 - f_2` and `f_3 + f_2 - f_1`.
Mixing 2 beam with different frequencies `->` freq. `f_{FWM} = 2f_1 -f_3`, `2 f_3 - f_1`.
FWM increases when a) channel spacing small and even distribution (`lambda` match), b) power / channel large, c) Small chromatic dispersion, d) short fiber distance (pulses overlapping), e) very close refractive indices for channels, f) large `chi^{(3)}`
• Temporal FWM: Near end, pulses overlap `->` strong FWM
Far end, pulses overlapping decrease `->` less FWM.

Last Modified: February 22, 2018
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