**Example 1.** *Sum frequency generation via non-linear wave mixing (frequency up-conversion)*

*This example is about the generation of a higher frequency component*(ω3)*, by mixing of two monochromatic waves with frequencies* ω1 *and* ω2 *via non-linear coupling, such that* ω3 = ω2 + ω1.

The pump wave *E***2** is originated at x = 2.5 μm, with an amplitude of *A*2 V/m and a frequency of 180 THz.

$$E\_2(\mathbf{x} = 2.5 \mu m, t) = A\_2 \times \sin\left(2\pi (1.8 \times 10^{14})t + q\varphi\right) \text{ V/m}^2$$

The input wave *E***1** is originated at x = 2.5 μm, with an amplitude of *A*1 V/m and a frequency of 120 THz (see Figure 8).

$$E\_1(\mathbf{x} = 2.5 \mu m, t) = A\_1 \times \sin \left( 2\pi (1.2 \times 10^{14})t + \varphi\_1 \right) \text{ V/m} \ (assume \ \varphi\_1 = 0, \ \varphi\_2 = 0)$$

*Spatial and temporal simulation parameters* 0 ≤ *x* ≤ 10μ*m*, 0 ≤ *t* ≤ 60*ps*

*Resonance f requency o f the medium* : *f*0 = 1.1 × 1015*Hz*

*Damping rate o f the medium* : γ = 1 × 1012*Hz*

*Dielectric coe f ficient o f the medium* (ε∞) = 1 + χ = 10 (μ*r* = 1)

*Le f t PML* (*le f t absorption layer*) *is f rom x* = 0 *to x* = 2.25μ*m*

*Right PML* (*right absorption layer*) *is f rom x* = 7.75μ*m to x* = 10μ*m*

**Figure 8.** Configuration for the frequency up-conversion in example 1.

The experimentally verified theoretical formula for frequency up-conversion efficiency is given as [1,4]

$$\begin{aligned} \eta\_{\text{thermal}} &= \frac{\omega\_3}{\omega\_2} (\sin \sqrt{2d^2n^3} w\_3^2 (\pi \alpha \varepsilon\_0 A\_2^{-2}) L^2) \ &= \frac{\omega\_3}{\omega\_2} (\sin \sqrt{2d^2n^4} w\_3^2 \varepsilon \varepsilon\_0 A\_2^{-2} L^2) \ & \text{(20)}\\ \omega\_2 &= \text{Frequency of the pump wave} \quad , \quad \omega\_1 = \text{Frequency of the input wave} \\ \mathbf{d} &= \text{Material non}-\text{linearity coefficient, } \mathbf{u} = \text{ Refractive index} \\ A\_2 &= \text{Pump wave amplitude } A\_1 = \text{Input wave amplitude, } \mathbf{L} = \text{Length of the non}-\text{linear media} \\ \omega\_3 &= \omega\_1 + \omega\_2 = \text{Frequency of the upward wave} \end{aligned} \tag{20}$$

*Appl. Sci.* **2020**, *10*, 1770

Our computational model, which is based on the non-linear electron motion equation, is implemented via FDTD discretization. Coupled with the wave equation, the total wave *E* = *E*1 + *E*2 is computed from:

$$\begin{array}{c} \frac{\mathbb{E}(i+1,j) - 2\mathbb{E}(i,j) + \mathbb{E}(i-1,j)}{\Delta x^2} - \mu\_0 \varepsilon\_{\cos} \left( i,j \right) \frac{\mathbb{E}(i,j+1) - 2\mathbb{E}(i,j) + \mathbb{E}(i,j-1)}{\Delta t^2} \\ = \mu\_0 \sigma(i,j) \frac{\mathbb{E}(i,j) - \mathbb{E}(i,j-1)}{\Delta t} + \mu\_0 \frac{\mathbb{P}(i,j+1) - 2\mathbb{P}(i,j) + \mathbb{P}(i,j-1)}{\Delta t^2} . \end{array} \tag{21a}$$

$$\begin{split} \frac{P(i,j+1) - 2P(i,j) + P(i,j-1)}{\Lambda t^2} + \gamma \frac{P(i,j) - P(i,j-1)}{\Lambda t} + a\alpha^2 (P(i,j)) - \frac{a\chi^2}{\text{N} \text{cl}} (P(i,j))^2 \\ - \frac{a\psi^2}{\Lambda^2 c^2 d^2} (P(i,j))^3 = \frac{\Lambda c^2}{m} (E(i,j)). \end{split} \tag{21b}$$

For a time interval of 0 ≤ *t* ≤ *tmax*, the computational formula for frequency up-conversion efficiency is:

$$\eta\_{\text{compitudinal}} = \frac{\text{Intensity of the } \omega\_3 \text{ frequency component of the total wave at } t = t\_{\text{max}}}{\text{Intensity of the } \omega\_2 \text{ frequency component of the total wave at } t = 0} \tag{22}$$

In this example, we have used the following values for each efficiency formula:

$$\alpha\_2 = (2\pi \times 180) T H z\_\prime\\\alpha\_1 = (2\pi \times 120) T H z\_\prime$$

L = length of the interaction media = 3.33 μ*m* (from x = 3.33 μ*m* to 6.66 μ*m*)

ω3 = *f requency o f the upconverted wave* = 2π × 300*THz*, n= *Re f ractive index* = √ 10

*d* = material non-linearity coefficient = 6.3 × 10−22 (The theoretical and the computational results agree for this value of *d* for a sample pump wave amplitude of *A*2 = 109*V*/*m* (see Figure 9). Our aim is to see if the results also agree for all the other pump wave amplitudes for this value of d)

**Figure 9.** Comparison of the frequency up-conversion efficiencies for *f*3 = 300 THz and d = 6.3 × 10−22, versus the pump wave amplitude.

*A*2 = pump wave amplitude (varied from 5 × 107*V*/*mto* 2 × 109*V*/*m* ) *A*1 = input wave amplitude = *A*2/100 (*A*1 *A*2)

**Example 2.** *Optical amplification via non-linear wave mixing (parametric amplification).*

*Appl. Sci.* **2020**, *10*, 1770

The high-intensity pump wave *E***2** is generated at x = 2.5 μm. It has an amplitude of *A*2 V/m and a frequency of 220 THz.

$$E\_2(\mathbf{x} = 2.5 \mu m, t) = A\_2 \times \sin\left(2\pi (2.2 \times 10^{14})t + q\_2\right) \text{ V/m}$$

The input wave *E***1** is generated at x = 2.5 μm. It has an amplitude of *A*1 V/m and a frequency of 140 THz (see Figure 10)).

$$E\_1(\mathbf{x} = 2.5 \mu m, t) = A\_1 \times \sin\left(2\pi (1.4 \times 10^{14})t + \varphi\_1\right) \text{ V/m} \text{ (assume that } \varphi\_1 = 0, \,\varphi\_2 = 0)$$

**Figure 10.** Configuration for optical parametric amplification in example 2.

 The computational results are obtained by solving (21a,b) and by computing the ratio of the spectral intensity of the input wave for ω = ω1 at t = *tmax* to the input wave intensity for ω = ω1 at t = 0:

$$\eta\_{\text{computed}} = \frac{\text{Intensity of the } \omega\_1 \text{ frequency component of the total wave at } t = t\_{\text{max}}}{\text{Intensity of the } \omega\_1 \text{ frequency component of the total wave at } t = 0} \tag{23}$$

The experimentally verified theoretical formula for parametric amplification, which is derived from the solution of the non-linear wave equation that is based on material non-linearity coefficient, is given as [1,4]:

$$\eta\_{\text{theoretical}} = \cos \text{h}^2 \left( \text{L}d \sqrt{\omega\_1 (\omega\_2 - \omega\_1) \eta^3} \sqrt{0.5 \text{cm} \varepsilon\_0 E\_{\text{pump}}}^2 \right) \tag{24}$$

*L* : *Length o f the interaction medium* = 5μ*m*, *d* : *Nonlinearity coe f ficient* = 1.2 × 10−21*C*/*V*2

$$η =
Intrinsic\
implies = 119.2Ω\ (olm)$$

The resulting amplification of the input wave (gain factor) is plotted with respect to the pump wave amplitude based on both the theoretical and the computational formulations (see Figure 11). Notice that the gain factor increases quadratically with the pump wave amplitude. The resulting gain is quite small, as parametric amplification practically requires an interaction medium length on the order of centimeters [16–20], whereas in this example we have an interaction medium length of 5 μm.

**Figure 11.** Comparison of the parametric amplification rates with respect to the pump wave amplitude.

#### **8. Discussion**

It is important to note that although the presented algorithm is very accurate in obtaining the highest gain factor and determining the optimal frequencies for the ultrashort pump wave pulses, the parameters of the gain medium may limit the maximum achievable gain factor. For example, in our presented numerical experiments, the background permittivity values were set as ε∞ = 10 and ε∞ = 12 respectively, these values correspond to the background permittivity of many solid dielectric media in the near infra-red and the visible frequency range. A very large gain factor is also achievable for smaller values of the background permittivity at the expense of a slight increase in ultrashort pump wave intensity to compensate for the decreased stored energy in the micro-resonator. Similarly, for a gain material with a higher background permittivity, the required pump wave intensity can be lowered in order to achieve the same gain factor. Another important parameter that heavily influences the gain factor is the polarization damping rate or simply the damping rate γ. A damping rate that is greater than γ = 1011 slows down the rate of amplification via non-linear wave mixing and results in a lower gain factor [21].

The frequencies of the ultrashort pump wave pulses can be practically tuned from the far IR (infrared) range to the near UV (ultraviolet) range. Commercial IR laser sources such as the Neodymium-YAG laser, or the Helium-Neon laser can be used in the near IR range and also in the visible range via frequency doubling. A visible laser light can be used in combination with a near IR laser light to generate a UV laser light via the process of sum frequency generation. A far IR laser light and even a THz laser light can be generated using two near IR laser lights of slightly different frequencies, through difference frequency generation process.

Optical amplification via non-linear wave mixing offers the advantage of wide-band amplification of a monochromatic stimulus wave, practically in a frequency interval ranging from the far IR to the near UV part of the spectrum [1,4,19]. Hence, the numerical algorithm presented in Figure 4, can be used to amplify a monochromatic stimulus wave of any frequency that lies between the far IR to the near UV part of the spectrum.

Our main aim in this article is to show that super-gain optical parametric amplification can be achieved in a simple fabry-perot type micro-resonator. For more complicated resonators including microring and microdisc resonators, the one-dimensional model presented in this article must be extended to a two-dimensional model. Our algorithm, which is presented in Figure 4, can be extended to a two-dimensional or a three-dimensional micro-resonator analysis using the technique that is presented in the article referenced in [22]. This article explains how, using FDTD, Maxwell's 2-D and 3-D curl equations in vector form can be effectively stepped in time simultaneously with a system of auxiliary differential equations, in order to create an integrated model that is capable of accurately simulating a medium involving instantaneous non-linearity, dispersive non-linearity, and multiple linear dispersions. Concerning implementation, the Greene-Taflove algorithm given in [22] can easily be incorporated into our algorithm during the discretization of the equations via finite difference time domain method. Hence the algorithm presented in this article can be employed in the analysis of arbitrary-shaped 2-D and 3-D micro-resonators.