Nonlinear Refraction |
Feb. 27, 2018 |
•
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.
• 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.
`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"}`
, `D_{"intra"} < 0 -> {partial tau_g}/{partial lambda} <0`, i.e. short wavelength slower.
, `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.
• : New type of fibers that may
allow more flexibility in adjusting dispersion.