  We analyze the modulation instability induced by periodic variations of group
velocity dispersion and nonlinearity in optical fibers, which may be
interpreted as an analogue of the well-known parametric resonance in mechanics.
We derive accurate analytical estimates of resonant detuning, maximum gain and
instability margins, significantly improving on previous literature on the
subject. We also design a periodically tapered photonic crystal fiber, in order
to achieve narrow instability sidebands at a detuning of 35 THz, above the
Raman maximum gain peak of fused silica. The wide tunability of the resonant
peaks by variations of the tapering period and depth will allow to implement
sources of correlated photon pairs which are far-detuned from the input pump
wavelength, with important applications in quantum optics.
