**1. Introduction**

Electromagnetic wave amplification by non-linear wave mixing has been studied for decades with most of the research focusing on the field of non-linear optics. It has been well-established that a low power input wave can be amplified via mixing with a high-intensity pump wave, in a medium that demonstrates a high degree of non-linearity. The fundamental theory of intra-cavity non-linear optical amplification is about the energy transfer from the pump wave-energized resonator to the input wave as a consequence of non-linear optical coupling [1–3]. The more elaborate theory of non-linear wave amplification involves a dispersion analysis that takes into account the frequency dependence of the non-linearity of the interaction medium. However, based on our in-depth investigations, a direct gain factor maximization approach via a non-linear constrained optimization algorithm is lacking. The background research about the theory of non-linear wave mixing has been mostly experimental rather than computational, especially for the purpose of optical amplification. The major reason of this tendency is that this topic is usually studied in the micrometer- or nanometer-wavelength range, but the required interaction medium length to observe a strong non-linear effect is in the millimeter or centimeter range [4]. This requires a tremendous computational cost for acquiring meaningful results, particularly for wave amplification purposes.

Another incentive for the experimental investigation of the topic is the desire to discover new materials that exhibit unusually high non-linearity under excitation. More recently, researchers focus on determining the resonance frequencies of the non-linear electrical response of commonly used materials in photonics such as graphene and gallium nitride. This approach has been mostly successful and resulted in an improvement of efficiencies in many applications in photonics. However, for the purpose of optical amplification, achieving a resonant non-linear optical response usually fails as the dielectric absorption around any non-linear resonance peak is quite strong to prevent significant amplification. Moreover, the well-established concept of optical parametric amplification yields a negligible gain factor in a few micrometers-long resonator. The required interaction medium length for high-gain optical parametric amplification is usually in the order of centimeters. Therefore, optical amplification by non-linear wave mixing is not so feasible for utilization in micro-resonators, optical transistors, micro-modulators, and many other device models in integrated photonics such as those mentioned in [5,6]. However, if one can perform super-gain optical parametric amplification in the micrometer scale, much more powerful macroscale devices can be produced for high-power applications, by engineering the individual micro-scale devices to operate in an array form to maximize optical interference, such as high-power welding machines and super-intensity lasers that are employed in particle accelerators and fusion reactors. A major scientific advancement can be in the field of optical antennas via supercontinuum generation for achieving ultra-wideband operation.

In this article, we have performed a numerical analysis that provides evidence for the high-gain amplification of a low-power stimulus wave, via intense pump waves of ultrashort duration, inside a several micrometers long micro-resonator, by maximizing the electric energy density of the pump wave in the resonator.

Since the order of amplification that can be achieved via non-linear wave mixing depends on the amount of stored electric energy density and the pump wave-initiated polarization density, which acts as a non-linear coupling coefficient, we will first present the energy storage dynamics for a resonator and we will recall the definitions of the electric polarization density and the gain factor of an amplification process. Then we will present the basic background for wave propagation in non-linear dispersive media and we will formulate our energy maximization (optimization) problem based on the partial differential equations given to express wave propagation in non-linear dispersive media. Finally, we will present two numerical experiments and analyze their results.

## **2. Polarization Density and the Cavity Quality (Q) Factor**

The Q factor indicates the amount of the stored energy inside a resonator for a given round trip loss. A high Q value usually signifies that the resonator is low loss, i.e., has a low loss factor, which means that high energy can be trapped efficiently in the resonator. The Q factor depends on the resonator length, the reflectivities of the resonator walls, the frequency of the propagating wave, the total absorption coefficient of the interaction medium between the resonator walls, and any sort of diffraction or scattering loss that may take place inside a resonator. The Q factor of a resonator is defined as:

$$\text{CAVITY QILATION (Q)}\\\text{FACTOR} = 2\pi \frac{\text{Stored energy}}{\text{Energy dissipated per round trip}}\\= f T\_{\text{fl}} \frac{2\pi}{\zeta} = \frac{4f \text{L}\pi}{\zeta \omega}.\tag{1}$$

*Trt* : *Round trip time*, *f* : *Wave f requency*, ζ : *Fractional power loss per round trip c* : *Speed o f light*, *L* : *Cavity length*

The electric energy density in a resonator depends on the magnitude of the electric field and the polarizability of the interaction medium. For a given resonator configuration, the stored electric energy density in a homogenous isotropic interaction medium is given as [7]:

$$\mathcal{W}\_{\varepsilon} = \text{Stored energy density} = \frac{1}{2} ED = \frac{1}{2} E (\varepsilon\_{\text{co}} E + P) = \frac{1}{2} \varepsilon\_{\text{co}} E^2 + \frac{1}{2} EP. \tag{2}$$

*D* : *Electric flux density*, *P* : *Polarization density*, *E* : *Electric field intensity*, ε<sup>∞</sup> : *Background* (*infinite spectral band*)*permittivity*

The electrical polarization density (*P*) in an interaction medium is a microscopic parameter that indicates both the volume density of electrons and the displacement of an electron with respect to the position of the nucleus [3]. For a given electron density of a medium, the more an electron is allowed to displace from its initial position (and the nucleus) the higher the polarization density becomes. It is a measure of the electrical excitability of a material by an incident electromagnetic wave [1]. Polarization density is an important parameter for energy storage in a resonator as more energy can be stored in a cavity with a highly polarizable interaction medium. This is because, as each electron displaces further from the nucleus, their potential energy becomes higher, and an abundance of electrons in the medium will lead to a higher potential energy being stored in the medium [4]. For a highly non-linear medium, the polarization density itself depends on the magnitude of the electric field [1,4]. For these reasons, we will frequently use the term polarization density in our analyses.
