While single pump configurations are often preferred for low distortion signal amplification [16], or in general, in scenarios where the signal gain is not a key performance metric, a dual pump configuration is preferred in applications requiring a large PSA signal gain, ideally with polarisation insensitive operation. Single pump optical parametric amplification with fibers having a longitudinally oscillating dispersion profile, or in other words dispersion oscillating fibers (DOF) were initially investigated in the context of dispersion management in long-haul optical communication links [17]. These investigations, that predicted the presence of high-order modulation instability sidebands in a DOF gain spectrum made it an interesting candidate for other applications such as frequency conversion [13] and wide band optical parametric amplification [18]. However, the use of DOFs as a single pump PSA is still an open area of research to the best of our knowledge. Furthermore, with the continual development of newer all optical signal processing capabilities of PSA based systems [19; 20; 21; 22], albeit using multiple pumps, we plan to investigate a single pump DOF based PSA system, where the dispersion variation provides us with an extra degree of freedom for tailoring the system functionalities. To this aim we investigate analytically and numerically the physics of the signal amplification in a DOF acting as a single pump PSA and unravel the relationship between the dispersion profile with various system parameters \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Nonlinear Medium** & **Advantage** & **Disadvantage** \\ \hline Highly nonlinear fiber (HNLF) & High gain and low noise figure; Typical nonlinear coefficient is high, \(\sim\)10 (W  km)\({}^{-1}\) & High power required to invoke nonlinearities, low stimulated Brillouin scattering (SBS) threshold \\ \hline Periodically poled Lithium Niobate (PPLN) & Moderate gain and compact & High power required to invoke nonlinearities \\ \hline Semiconductor optical amplifier (SOA) & Compact & High noise figure \\ \hline Dispersion oscillating fiber (DOF) & High SBS threshold, low figure, moderate (potentially large) gain & High power required to invoke nonlinearities since experimental nonlinear coefficient in this work is 3.5 (W km)\({}^{-1}\). However this value can be increased by using a preform of a HNLF \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table for comparison of advantages and disadvantages of different nonlinear media. and investigate its potential for developing wavelength selective wide band tunable filters for optical signal processing applications. Contrary to the single pump configuration, the dual pump configuration enjoys the advantage of not requiring a copier stage for achieving PSA functionality when operated in a degenerate signal-idler mode [5; 23]. Recently, the case of a dual-pump PSA with a dispersion oscillating fiber was investigated analytically in [24]. However, the question of whether a sinusoidal dispersion profile is the optimal profile for attaining a large PSA gain is still an open one. Therefore, in this article we also address the question, what is the optimal non-constant dispersion profile for achieving a large PSA gain for practical applications? Although FOPA-based dual pump PSAs provide an exquisite combination of low noise and high gain modality, achieving a high gain in practice is often challenging due to energy loss from stimulated Brillouin scattering (SBS) within the fiber. SBS prevents transmission of high power, highly coherent pump waves through long lengths of nonlinear fiber due to the coherent build up of Stokes wave in the counter-propagating direction. In order to get around the power scaling inhibition posed by SBS within a nonlinear fiber, several approaches had been adopted, such as applying a distributed strain [25; 26], applying a temperature gradient along the fiber length [27; 28], broadening the pump linewidth through phase modulation [29; 30; 31; 32; 33], etc.