When I expand the phase mismatch (for three signals interacting nonlinearly in optical fiber delta-beta with dispersion slope (dB = B(f1) + B(f2) - B(f3) - B(f1+f2-f3) , where B (Propagation Constant) will expanded using Taylor series around f0. The result will depend on f0, which seem unlogical because the phase mismatch will change according to the point of your expansion!
Did anybody face the same problem before?
When I plot the FWM mixing with f0 and without f0, I find a big difference.
There should be a unique value for the phase mismatch, but as Michael Reilly points out, you need to include sufficient terms in the Taylor expansion.
The calculation is simplified if the Taylor series is expanded about the mid frequency fm = (f1 + f2) / 2
In this case the odd orders of the propagation constant (group velocity and dispersion slope) cancel, as do the zero order terms. The second derivative of the propagation constant (chromatic dispersion) will dominate in most cases. For very widely spaced frequencies close to the zero dispersion wavelength, it may be desirable to include the 4th derivative (dispersion curvature term).
Are you working directly with the Taylor coefficients, or using a published formula including dispersion and dispersion slope? Some widely cited papers from 30 years ago present formulas with sign errors in the dispersion slope term, and are inaccurate near the dispersion zero wavelength.
Thanks a lot, Michael, and Alan.
I drive the formula myself and it is equal to
delta B = - (2 pi)^2 beta2 (f1-f3)(f2-f3)(1+pi*beta3/beta2[f1-f0+f2-f0])
where f0 the point of expansion for the propagation constant.
Abdallah,
You can re-write your equation as
Δβ = - 4π2 (f1-f3)(f2-f3) { β2(f0) + 2 π (fm-f0) β3(f0) }
where βn(f) = (dnβ / dωn) evaluated at ω = 2πf, and fm = (f1 + f2) / 2
which becomes: Δβ = - 4π2 (f1-f3)(f2-f3) β2(fm)
provided that β3 (dispersion slope) is constant over the frequencies of interest, the results from both expressions should be independent of the frequency chosen for the Taylor expansion. If the dispersion slope varies significantly, then you need to include higher order terms.
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