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.