Bessel function solution for the gain of one-pump fiber optical parametric amplifier

The 1-pump fiber optical parametric amplifier (OPA) with dispersion and loss, but no pump depletion, has recently been characterized by a 2×2 transfer matrix, with each term expressed in terms of four Hankel functions, H_iu of imaginary order iu. Unfortunately, these functions are not commonly available in mathematical packages such as Mathematica, and their use in practical applications thus requires the development of special software. The evolution of the related modulation instability (MI) has also been studied in terms of modified Bessel functions I_iv, of imaginary order iv. However, these results do not directly provide expressions for the complex output fields of an OPA; furthermore, they assume a simplified form for fiber dispersion. In the paper we present an alternate approach, which yields the complex OPA idler gain in terms of just four Bessel functions J_v, which have the advantage of being commonly available in mathematical packages. In addition, fiber dispersion is handled in a most general manner