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The direction of the principal axes of the residual birefringence cannot be controlled and it changes randomly along the fiber at a length scale of ∼ 1 m [51]. When fibers shorter than 10 m are used, fiber birefringence can be ignored, and the fiber can be assumed to be isotropic. For fibers much longer than the correlation length of residual birefringence, effects of randomly varying birefringence cannot be ignored. Dependence of FOPA gain on the polarizations of the pumps, signal and idler is investigated in the short and long length scales separately.

Solid curves show the theoretically expected results. 2 Dual-Pump Parametric Amplifiers In the previous Section the simpler case of single-pump FOPAs is used to explain the basic features of FOPA design. This Section focuses on dual-pump FOPAs that employ the nondegenerate FWM process using two pumps with different frequencies [10]– [12], [45]–[50]. Because of the extra degrees of freedom presented by the second pump in terms of its frequency and SOP, dual-pump FOPAs have several advantages over single-pump FOPAs.

Similar to the single-pump case, one can expand the linear phase mismatch ∆β = β (ω3 ) + β (ω4 ) − β (ω1 ) − β (ω2 ) in a Taylor series as [46]: ∞ ∆β = 2 1 ∑ (2m)! m=1 d 2m β dω 2m (ω3 − ωc )2m − ωd2m . 15) ω=ωc where ωc = (ω1 +ω2 )/2 is the mean frequency of the two pumps and ωd = (ω1 −ω2 )/2 is the half of their frequency difference. This equation differs from the phase-mismatch term in the case of single-pump FOPA Eq. 10) by the last term. The ωd term contributes only when two pumps are used and is independent of the signal and idler frequencies.

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