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Analytical expressions for saturated signal gain and output power of fiber optical parametric amplifiers

With the increasing availability of high output power EDFA as well as the development of highly nonlinear fibers, more and more fiber optical parametric amplifiers are operated near or in the saturated-gain region rather than small-signal gain region, where the signal gain characteristics are expected to be different from those in the linear-gain region. For the first time, two very pithy and powerful analytical expressions for saturated signal gain and signal output power of fiber optical parametric amplifiers are educed with numerical analysis. Those are realized in three steps, first, a series of numerical values are worked out by numerically solving the couple NLS equations that governed the fiber parameter process. Then the function form for saturated signal gain is figured out by control variates method. At last, adopting least square method, three coefficients are finely obtained with the maximum relative error (in terms of calculated saturated signal gain with numerical integration) no more than 0.46%. At the same time, we worked out the analytical expression for saturated signal output power. From those expressions, we can safely come to a conclusion that saturated signal gain of fiber optical parametric amplifiers is related to pump and signal power while saturated signal output power only rely on pump power, but both have nothing to do with nonlinear coefficient and fiber length, which is quite different from small-signal gain but accord well with law of conservation of energy. Two expressions presented in this paper are very useful in designing and analyzing fiber optical parametric amplifiers. Results of this work agree well with existing experiment results.