Optical Parametric Amplifiers (OPA) have been of wide interest for the past decades due to their potential for low noise amplification and generation of squeezed light. However, the existing OPAs for fiber applications are based on Kerr effect and require from few centimeters to kilometers of fiber for significant gain.

In this thesis, I review the principles of phase sensitive amplification and derive the expression for gain of a lossless Kerr medium based Nonlinear Mach-Zehnder Interferometer (NMZI OPA) using a classical physics model . Using quantum optics, I calculate the noise of a lossless Kerr medium based OPA and show that the noise figure can be close to zero.

Since in real life a Kerr medium is lossy, using quantum electrodynamics, I derive equations for the evolution of a wave propagating in a lossy Kerr medium such as an optical fiber. I integrate these equations in order to obtain the parametric gain, the noise and the noise figure. I demonstrate that the noise figure has a simple expression as a function of loss coefficient and length of the Kerr medium and that the previously published results by a another research group are incorrect. I also develop a simple expression for the noise figure for high gain parametric amplifiers with distributed loss or gain.

In order to enable construction of compact parametric amplifiers I consider using different nonlinear media, in particular a Saturable Absorber (SA) and a Semiconductor Optical Amplifier (SOA). Using published results on the noise from SOA I conclude that that such device would be prohibitively noisy. Therefore, I perform a detailed analysis of

noise properties of a SA based parametric amplifier. Using a quantum mechanical model of an atomic 3 level system and the Heisenberg's equations, I analyze the evolution in time of a single mode coherent optical wave interacting with a saturable absorber. I solve the simultaneous differential equations and find the expression for the noise figure of the SA based NMZI OPA. The results show that noise figure is still undesirably high. The source of the noise is identified. A new approach for low noise parametric amplifier operating with short pulses is proposed.