Nonlinear Refraction Feb. 27, 2018

Freq Chirp and Gaussian pulses (pp. 734-735 2nd Ed, pp. 772-773 3rd Ed.): at z=0, G(t) = A_o e^{(-t/{sqrt{2}T_o})^2) cos (omega_o t + 0.5 kappa (t/{T_o})^2 ) where kappa is the chirp factor. kappa > 0 means high frequency having longer delay while kappa < 0 is just opposite.
With dispersion, pulse width at propagation distance z T_z^2 = T_o^2 [(1+ (k prime prime z kappa )/{T_o^2})^2 + ( (k prime prime z)/{T_o^2})^2 ]
If z > L_D where dispersion length L_D = (T_o^2) /{|k prime prime |}, pulse width widens significantly.
If k prime prime kappa < 0 , pulse can be compressed.
Model equation for Gaussian pulse propagation

• Spectral broadening (pp. 81-93 2nd Ed., pp. 83-95 3rd Ed.): Intensity modulation can causes change in refractive index, i.e. phase modulation, -> widen the signal spectrum.
Recall E ~ exp(j( omega t - kz)) and phase phi = -kz.
In nonlinear media,  phi - -{2 pi z (n_o + Delta n )}/lambda_o
Nonlinear phase shift Delta phi = -{2 pi L Delta n }/lambda_o
Notice that Delta n = n_2 I is a function of time where n_2 ( cm^2 "/W") is the nonlinear refractive index coefficient and I (W"/" cm^2 ) is intensity.
Freq chirping delta omega = {d Delta phi}/{dt} = -{2 pi}/lambda_o L n_2 {dI}/{dt}
Note: Leading edge of a pulse has longer wavelength relative to trailing edge.
Compare to D_{"intra"}
Normal dispersion, D_{"intra"} < 0 -> {partial tau_g}/{partial lambda} <0, i.e. short wavelength slower.
Anomalous dispersion, D_{"intra"} > 0 -> {partial tau_g}/{partial lambda} >0, i.e. short wavelength faster.
• Self phase modulation (SPM) and cross phase modulation (XPM):
A special case of FWM, f_FWM = f_1 + f_1 -f_1, i.e, f_1 = f_2 = f_3
Recall n = n_o + n_2 I
n_2 < 0

positive (self-focusing) nonlinearity

n_2 < 0

negative (self-defocusing) nonlinearity
Define power P = IA_e where A_e = {| int I dS |^2} /{int I^2 dS} is the effective area.
E.g. for a 3-channels-system.
Nonlinear phase in channel 1 Delta phi_1 = gamma L_e (P_1 + 2 (P_2 + P_3))
Nonlinear phase in channel 2 Delta phi_2 = gamma L_e (P_2 + 2 (P_1 + P_3))
where gamma = {2 pi n_2} / {lambda_o A_e} in rad/W-m and L_e = P_0^(-1) int_0^L P dz = int_0^L e^{-alpha z} dz is the effective length.
SPM - gamma L_e P_1 term
XPM -  gamma L_e 2 (P_2+P_3)
If the intensity and pulse width are satisfied soliton condition, the pulse is self-trapping in time without dispersion -> so called soliton.
• Modulation instability: For pulse propagation in a positive dispersion fiber (D_{"intra"} > 0), if the intensity is too high, the beam will break up in time (side lobes) and spatial (filaments) owing to interaction with SPM.
Under appropriate intensity, SPM and positive dispersion can preserve pulse width and shape, i.e. generation of solitons.
• Countering FWM: unevenly freq spacing, increase freq spacing, reduce power, dispersion compensation or management.
• Fiber type considerations (pp. 93-98 2nd Ed., pp. 95-99 3rd Ed.): Non-zero dispersion fibers (NZ-DSF) with small D_{"intra"} (to avoid FWM), small dispersion slope (to enable dispersion compensation) and large A_e are preferred.
Positive dispersion fiber is used in terrestrial systems to enable future expansion to L band.
Negative dispersion fiber is used in undersea systems for avoiding modulation instability.
Photonic crystal fibers: New type of fibers that may allow more flexibility in adjusting dispersion.