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.

Last Modified: February 26, 2018
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