**Özüm Emre A¸sırım \* and Mustafa Kuzuo ˘glu**

Department of Electrical and Electronics Engineering, Middle East Technical University, 06800 Ankara, Turkey; kuzuoglu@metu.edu.tr

**\*** Correspondence: e176154@metu.edu.tr; Tel.: +90-5323329262

Received: 6 February 2020; Accepted: 28 February 2020; Published: 4 March 2020

**Abstract:** The goal of this paper is to show that super-gain optical parametric amplification can be achieved even in a small micro-resonator using high-intensity ultrashort pump waves, provided that the frequencies of the ultrashort pulses are tuned to maximize the intracavity magnitude of the wave to be amplified, which we call the stimulus wave. In order to accomplish this, we have performed a dispersion analysis via computational modeling of the electric polarization density in terms of the non-linear electron cloud motion and we have concurrently solved the electric polarization density and the wave equation for the electric field. Based on a series of non-linear programming-integrated finite difference time-domain simulations, we have identified the optimal pump wave frequencies that simultaneously maximize the stored electric energy density and the polarization density inside a micro-resonator by using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization algorithm. When the intracavity energy and the polarization density (which acts as an energy coupling coefficient) are simultaneously high, an input wave can be strongly amplified by efficiently drawing energy from a highly energized cavity. Therefore, we propose that micrometer-scale achievement of super-gain optical parametric amplification is possible in a micro-resonator via high-intensity ultrashort "pump wave" pulses, by determining the optimal frequencies that concurrently maximize the stored electric energy density and the polarization density in a dielectric interaction medium.

**Keywords:** optical parametric amplification; non-linear wave mixing; micro-resonator; optimization
