Nanoscale active tuning of the second harmonic generation efficiency in semiconductors from super-low to gigantic values

We will initially investigate the problem for the most general case which deals with an interaction material with multiple resonance frequencies. The first part of the problem involves the modeling of the wave dynamics. Consider a high-amplitude electric field that is propagating through a material with M resonance frequencies ω = {ω₁, ω₂, ..., ω_M }, and M corresponding polarization decay rates γ = {γ₁, γ₂, ..., γ_M }. The total polarization density that is induced by the electric field can be expressed as the sum of the individual polarization densities that are induced via each resonance, through the wave equation and the Lorentz dispersion equations

$$\begin{equation}{\nabla }^{2}E-{\mu }_{0}{\varepsilon }_{\infty }\frac{{\partial }^{2}E}{\partial {t}^{2}}={\mu }_{0}\sigma \frac{\partial E}{\partial t}+{\mu }_{0}\frac{{\mathrm{d}}^{2}P}{\mathrm{d}{t}^{2}}\quad (\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\enspace \mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d}\enspace \mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\enspace \mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})\end{equation} \tag{ 2 }$$

$$\begin{equation}\frac{{\mathrm{d}}^{2}{P}_{m}}{\mathrm{d}{t}^{2}}+{\gamma }_{m}\frac{\mathrm{d}{P}_{m}}{\mathrm{d}t}+{{\omega }_{m}}^{2}{P}_{m}-\frac{{{\omega }_{m}}^{2}{{P}_{m}}^{2}}{{N}_{m}ed}+\frac{{{\omega }_{m}}^{2}{{P}_{m}}^{3}}{{{N}_{m}}^{2}{e}^{2}{d}^{2}}=\frac{{N}_{m}{e}^{2}E}{{m}_{\mathrm{e}}}\quad (\mathrm{L}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{z}\enspace \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\enspace \mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})\end{equation} \tag{ 3 }$$

$$\begin{equation}P=\sum\limits _{m=1}^{M}{P}_{m},\qquad N=\sum\limits _{m=1}^{M}{N}_{m}=\sum\limits _{m=1}^{M}N{\xi }_{m},\qquad \sum\limits _{m=1}^{M}{\xi }_{m}=1\end{equation} \tag{ 4 }$$

E: electric field, μ₀: free space permeability, ε_∞: background permittivity, t: time, σ: conductivity, P: polarization density, γ: polarization damping rate, ΔT: simulation duration, ω: angular resonance frequency, N: electron density, e: elementary charge, d: atom diameter, m_e: electron mass, ξ_m : oscillation strength for the mth resonance.

After solving for the electric field through equations (2)–(4), the SHGE can be evaluated from the intensity ratio of the second and the first harmonic

$$\begin{equation}{\eta }_{\text{numerical}}=\frac{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}\enspace \mathrm{o}\mathrm{f}\enspace \mathrm{t}\mathrm{h}\mathrm{e}\enspace \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\enspace \mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{c}\enspace \mathrm{a}\mathrm{t}\enspace x={x}_{\text{output}}}{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}\enspace \mathrm{o}\mathrm{f}\enspace \mathrm{t}\mathrm{h}\mathrm{e}\enspace \mathrm{fi}\mathrm{r}\mathrm{s}\mathrm{t}\enspace \mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{c}\enspace \mathrm{a}\mathrm{t}\enspace x={x}_{\text{input}}}=\frac{{\left\vert {\int }_{0}^{{\Delta}T}\left\{E(x={x}_{\text{output}},t){\mathrm{e}}^{-\mathrm{i}\left(2\omega \right)t}\right\}\mathrm{d}t\right\vert }^{2}}{{\left\vert {\int }_{0}^{{\Delta}T}\left\{E(x={x}_{\text{input}},t){\mathrm{e}}^{-\mathrm{i}\omega t}\right\}\mathrm{d}t\right\vert }^{2}}.\end{equation} \tag{ 5 }$$

The value that is obtained by solving equation (5), can be compared with the empirical finding in equation (1) for verifying the accuracy of the computation.

Equations (2)–(4) can be solved through the use of the finite difference time domain method, based on which the discretization is given as follows

$$\begin{align}\hfill & \frac{E\left(i+1,j\right)-2E\left(i,j\right)+E\left(i-1,j\right)}{{\Delta}{x}^{2}}-{\mu }_{0}{\varepsilon }_{\infty }\frac{E\left(i,j+1\right)-2E\left(i,j\right)+E\left(i,j-1\right)}{{\Delta}{t}^{2}}\hfill \\ \hfill & \quad ={\mu }_{0}\sigma \frac{E\left(i,j+1\right)-E\left(i,j-1\right)}{2{\Delta}t}+{\mu }_{0}\frac{P\left(i,j+1\right)-2P\left(i,j\right)+P\left(i,j-1\right)}{{\Delta}{t}^{2}}\hfill \end{align} \tag{ 6 }$$

$$\begin{align}\hfill & \frac{{P}_{m}\left(j+1\right)-2{P}_{m}\left(j\right)+{P}_{m}\left(j-1\right)}{{\Delta}{t}^{2}}+{\gamma }_{m}\frac{{P}_{m}\left(j+1\right)-{P}_{m}\left(j-1\right)}{2{\Delta}t}+4{\pi }^{2}{{f}_{m}}^{2}{P}_{m}\left(j\right)\hfill \\ \hfill & \qquad -\frac{4{\pi }^{2}{{f}_{m}}^{2}}{{N}_{m}(j)ed}{\left({P}_{m}\left(j\right)\right)}^{2}+\frac{4{\pi }^{2}{{f}_{m}}^{2}}{{({N}_{m}(j))}^{2}{e}^{2}{d}^{2}}{\left({P}_{m}\left(j\right)\right)}^{3}=\frac{{N}_{m}(j){e}^{2}}{{m}_{\mathrm{e}}}\left(E\left(i,j\right)\right).\hfill \end{align} \tag{ 7 }$$

$$\begin{equation}P={P}_{1}+{P}_{2}+\cdots +{P}_{M},N={N}_{1}+{N}_{2}+\cdots +{N}_{M}.\end{equation} \tag{ 8 }$$

Here one should pay attention when choosing the spatial and temporal sampling rates (Δx⁻¹, Δt⁻¹), as they should be chosen according to the Courant stability condition to prevent numerical instabilities [31].

Since we are interested in the active tuning of the SHGE, the electron density for each transition (resonance) should be adjusted for SHGE maximization. The electron density for each transition can be set or swept to investigate the SHGE directly using equations (2)–(5). But in order to find the required optical pumping rate for each transition, the corresponding injection-rate equations must be solved. Considering an interaction medium with M resonances, and assuming that the electronic transition rates are much lower than the non-radiative decay rates, the mathematical description for the electron dynamics can be stated by the following coupled differential equations

$$\begin{equation}\frac{\mathrm{d}{N}_{0}}{\mathrm{d}t}=-[{{\Gamma}}_{1}+{{\Gamma}}_{2}+\cdots +{{\Gamma}}_{M}]{N}_{0}+\sum\limits _{i=1}^{M}\frac{{N}_{i}}{{\tau }_{i}}\end{equation} \tag{ 9 }$$

$$\begin{equation} \frac{\mathrm{d}{N}_{1}}{\mathrm{d}t}={{\Gamma}}_{1}{N}_{0}+\sum\limits _{i=2}^{M}\frac{{N}_{i}}{{\tau }_{i}}-\frac{{N}_{1}}{{\tau }_{1}}\end{equation} \tag{ 10 }$$

$$\begin{equation} \frac{\mathrm{d}{N}_{M-1}}{\mathrm{d}t}={{\Gamma}}_{M-1}{N}_{0}+\frac{{N}_{M}}{{\tau }_{M}}-\frac{{N}_{M-1}}{{\tau }_{M-1}}\end{equation} \tag{ 11 }$$

$$\begin{equation} \frac{\mathrm{d}{N}_{M}}{\mathrm{d}t}={{\Gamma}}_{M}{N}_{0}-\frac{{N}_{M}}{{\tau }_{M}}\end{equation} \tag{ 12 }$$

N₀: density of electrons at the ground level, N_M : electron density for the Mth resonance, Γ: optical pumping rate for the Mth resonance, τ_M : electron lifetime for the Mth resonance.

Equation (9) describes the rate of change in the electron density of the ground state under optical pumping for M different transitions (each with a rate of Γ_i ), and the corresponding electron lifetimes τ_i for each transition which indicate the rate that the excited electrons are recombined to the ground state. In equations (9)–(12), electrons are pumped from the ground level to the upper levels at different rates (the first term in each equation indicates the pumping), from where they fall/recombine to lower levels/transitions, increasing the electron population in lower levels. As an example, the first term in equation (10) represents the rate of injection for the first transition, the second term indicates the rate of electron flow (recombination) to the first electronic-transition from all the upper transitions, and the last term signifies the rate of recombination from the first electronic-transition to the ground-level (hence the minus sign, which indicates the decrease of electron population in level 1). In this study, it is assumed that the electron excitation/recombination follows the same pattern for all the other transitions (i = 2, 3, ..., M − 2). It should be noted that the SHGE is computed at the steady state, that is when the electron density associated with each transition (resonance) stabilizes around a certain value in time.

For materials that have only a single resonance frequency, the previously described set of equations (equations (2)–(4) and (9)–(12)) simplify to the following set of coupled equations

$$\begin{equation}{\nabla }^{2}E-{\mu }_{0}{\varepsilon }_{\infty }\frac{{\partial }^{2}E}{\partial {t}^{2}}={\mu }_{0}\sigma \frac{\partial E}{\partial t}+{\mu }_{0}\frac{{\mathrm{d}}^{2}P}{\mathrm{d}{t}^{2}}\end{equation} \tag{ 13 }$$

$$\begin{equation}\frac{{\mathrm{d}}^{2}P}{\mathrm{d}{t}^{2}}+\gamma \frac{\mathrm{d}P}{\mathrm{d}t}+{{\omega }_{0}}^{2}P-\frac{{{\omega }_{0}}^{2}{P}^{2}}{Ned}+\frac{{{\omega }_{0}}^{2}{P}^{3}}{{N}^{2}{e}^{2}{d}^{2}}=\frac{N{e}^{2}E}{{m}_{\mathrm{e}}}\end{equation} \tag{ 14 }$$

$$\begin{equation}\frac{\mathrm{d}N}{\mathrm{d}t}={\Gamma}N-\frac{N}{{\tau }_{\mathrm{e}}},\quad {\tau }_{\mathrm{e}}=\frac{1}{A+BN+C{N}^{2}}\end{equation} \tag{ 15 }$$

τ_e: electron lifetime, Γ: optical pumping rate, A: non-radiative recombination constant, B: radiative recombination constant, C: Auger recombination constant.

In this case, the following discretized equations are implemented to solve for the required optical pumping rate that corresponds to the optimum transition electron density N, which maximizes the SHGE via equations (5) and (13)–(15)

$$\begin{align}\hfill & \frac{E\left(i+1,j\right)-2E\left(i,j\right)+E\left(i-1,j\right)}{{\Delta}{x}^{2}}-{\mu }_{0}{\varepsilon }_{\infty }\frac{E\left(i,j+1\right)-2E\left(i,j\right)+E\left(i,j-1\right)}{{\Delta}{t}^{2}}\hfill \\ \hfill & \qquad ={\mu }_{0}\sigma \frac{E\left(i,j+1\right)-E\left(i,j-1\right)}{2{\Delta}t}+{\mu }_{0}\frac{P\left(j+1\right)-2P\left(j\right)+P\left(j-1\right)}{{\Delta}{t}^{2}}\hfill \end{align} \tag{ 16 }$$

$$\begin{align}\hfill & \frac{P\left(j+1\right)-2P\left(j\right)+P\left(j-1\right)}{{\Delta}{t}^{2}}+\gamma \frac{P\left(j+1\right)-P\left(j-1\right)}{2{\Delta}t}+{{\omega }_{0}}^{2}P\left(j\right)\hfill \\ \hfill & \qquad -\frac{{{\omega }_{0}}^{2}}{N(j)ed}{\left(P\left(j\right)\right)}^{2}+\frac{{{\omega }_{0}}^{2}}{{(N(j))}^{2}{e}^{2}{d}^{2}}{\left(P\left(j\right)\right)}^{3}=\frac{N(j){e}^{2}}{m}E\left(i,j\right).\hfill \end{align} \tag{ 17 }$$

Once the optimal transition electron density N that maximizes the magnitude of the electric field E is found, the corresponding optical pumping rate can be evaluated based on the discretization of equation (15) as

$$\begin{equation}{\Gamma}\left(j\right)=\frac{N\left(j+1\right)-N\left(j-1\right)}{2{\Delta}t}+A\left(N\left(j\right)\right)+B{(N(j))}^{2}+C{(N(j))}^{3}\quad \left(\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{d}\enspace \mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\enspace \mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{g}\enspace \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\right)\end{equation} \tag{ 18 }$$

x: spatial coordinate, t: time, $E\left(x,t\right)=E\left(i{\Delta}x,j{\Delta}t\right)\to E\left(i,j\right),i=1,2,\dots N,j=1,2,\dots ,M$.

The single-resonance modeling is a valid approximation in many cases, as many well-investigated materials have a single dominant resonance. In this case, it is much easier to control the sweeping of the associated electron density through the tuning of the pumping rate Γ. Such active tuning of the transition electron density leads to a huge variation in the ratio of the intensities of the first and the second harmonics. This variation will be investigated in the next section.

Figure 4 illustrates the practical mechanism for sweeping the transition electron density through the tuning of the optical pumping rate for the investigation of the variation of SHGE. The basic setup involves an optical pump source, which provides the pump beam that is used for tuning the transition electron density, a beam-splitter for combining the input beam with the pump beam, a lens for focusing the combined-beam onto the semiconductor surface, and an isolator for directing the combined-beam pathway towards the semiconductor surface with the help of a mirror. In this study, it is assumed that all the pump-light is absorbed by the semiconductor. Although the suggested setup is simple, one complicacy that might arise in practice is the overheating of the semiconductor due to the high pump intensity (∼10¹³ W m⁻²), which can lead to the generation of thermal-noise in the output signal, and also the modification of the electrical properties of the semiconductor that may lead to deviations from the expected SHGE. Such a high-intensity can also lead to thermal damage for relatively long optical pulses. However, since the pulse lengths are chosen to be on the order of a few picoseconds in our computations, and since most semiconductors incur thermal damage for optical intensities above ∼10¹⁵ W m⁻², we believe that thermal damage is unlikely for the given parameters in this study.

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Using numerical simulations, we have initially investigated the variation of the SHGE via tuning the electron density for a multi-resonant media of three resonances, with spectral locations f = {400 THz, 700 THz, 1200 THz}. The corresponding polarization damping rates for each resonance are given as γ = {10⁹ Hz, 4 × 10⁹ Hz, 1.2 × 10¹⁰ Hz}. In this section, we will only deal with the sweeping of the total transition electron density N, and not the optical pumping rate. The wave and the material parameters of the simulation are set as follows

$$\begin{equation*}E(x=3\enspace \mu \mathrm{m},t)=1{0}^{8}\times \mathrm{sin}\left(2\pi \left(1.5\times {10}^{14}\right)t\right)\enspace \mathrm{V}\enspace {\mathrm{m}}^{-1}\quad (\mathrm{a}\mathrm{n}\enspace \mathrm{i}\mathrm{n}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}\enspace \mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e})\end{equation*}$$

$$\begin{equation*}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\enspace \mathrm{a}\mathrm{n}\mathrm{d}\enspace \mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\enspace \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\enspace \mathrm{f}\mathrm{o}\mathrm{r}\enspace \mathrm{t}\mathrm{h}\mathrm{e}\enspace \mathrm{s}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:0\leqslant x\leqslant 10\enspace \mu \mathrm{m},0\leqslant t\leqslant 5\enspace \mathrm{p}\mathrm{s}\end{equation*}$$

$$\begin{equation*}\mathrm{R}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\enspace \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}:\left({\varepsilon }_{\mathrm{r}}\right)=13\enspace ({\mu }_{\mathrm{r}}=1),\qquad \mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\enspace \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}:d=0.3\enspace \mathrm{n}\mathrm{m}\end{equation*}$$

$$\begin{equation*}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace \mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m}:5\enspace \mu \mathrm{m}< x< 5.3\enspace \mu \mathrm{m},\quad \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\enspace \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{s}:{\boldsymbol{\xi }}_{\boldsymbol{m}}=\left\{0.3,0.4,0.3\right\}\quad (\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{r}\mathrm{y}\enspace \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{s})\end{equation*}$$

$$\begin{equation*}5\times 1{0}^{26}\enspace {\mathrm{m}}^{-3}< N< 1\times 1{0}^{29}\enspace {\mathrm{m}}^{-3},\qquad {N}_{1}=N{\xi }_{1},\qquad {N}_{2}=N{\xi }_{2},\qquad {N}_{3}=N{\xi }_{3}.\end{equation*}$$

Based on the solution of Equations (2)–(4), the corresponding induced electric energy density is computed as

$$\begin{equation}{W}_{\mathrm{e}}=\frac{1}{2}{\varepsilon }_{\mathrm{\infty }}{E}^{2}+\frac{1}{2}EP.\end{equation} \tag{ 19 }$$

Using equations (2)–(4), the electron density is tuned from 5 × 10²⁶ m⁻³ to 10²⁹ m⁻³ in increments of 5 × 10²⁶ m⁻³. The resulting variation of the SHGE is shown in figure 5. The SHGE reaches to a maximum value of 0.27 for an electron density of 4.2 × 10²⁸ m⁻³, which is 10⁷ times higher than for mediums with low nonlinearity, and 10⁵ times higher than for mediums with high nonlinearity, considering a 300 nm-long interaction medium based on figures 2 and 3. This finding emphasizes the importance of active tuning for increasing the SHGE. At the optimal electron density where peak SHGE occurs, the material exhibits an ultra-nonlinear electrical response, such that the generated second harmonic intensity is comparable with that of the first harmonic. At the nanoscale, such a high SHGE cannot be attained using any well-known nonlinear optical crystal with passive tuning as illustrated by figures 2 and 3.

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The reason behind the attainment of such high nanoscale SHGE can be uncovered by investigating the intra-material electric energy density and the polarization density. Figure 6 shows that, around an electron density of 4.2 × 10²⁸ m⁻³, both the electric energy density and the polarization density reaches to a maximum value. This means that for this electron density, the material allows the generation and coupling of maximum energy density to the incident wave, such that even the second and the third order polarization terms become significantly high. Figure 7 shows the spectrum of the output signal with the newly generated harmonics. Notice that not only the second harmonic, but also the third harmonic is generated via the induced ultra-nonlinear response (though at a lower efficiency).

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Interestingly, such high SHGE peak is not observed when the frequency of the first harmonic is set as 200 THz instead of 150 THz. Figure 8 shows the SHGE versus electron density for f = 200 THz. This time, the highest observed SHGE is on the order of 10⁻⁴, which still indicates a strong nonlinear response, but much less than the case for f = 150 THz. From this observation we understand that there is a complex mathematical relationship among the frequency of the first harmonic, the electron density, and the SHGE, which necessitates the use of elaborate fitting algorithms after making thousands of simulations. In this study we will not go into uncovering the exact relationship between these parameters but instead we will focus on how to tune the electron density for well-known semiconductors such as silicon or gallium-arsenide under commonly used frequencies of optical excitation for maximizing the SHGE.

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Silicon and gallium-arsenide are well-known semiconductors with relatively high nonlinearity. In the nanoscale, their SHGE is on the order of 10⁻⁴ in the near-infrared frequency range. Using equations (13)–(15), initially, the SHGE is actively tuned for silicon under excitation by an arbitrary near-IR laser source (220 THz). The corresponding wave and material parameters are given as [30–32]

$$\begin{equation*}E(x=3\enspace \mu \mathrm{m},t)=6\times 1{0}^{7}\times \mathrm{sin}\left(2\pi \left(2.2\times {10}^{14}\right)t\right)\enspace \mathrm{V}\enspace {\mathrm{m}}^{-1}\end{equation*}$$

$$\begin{equation*}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\enspace \mathrm{a}\mathrm{n}\mathrm{d}\enspace \mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\enspace \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\enspace \mathrm{f}\mathrm{o}\mathrm{r}\enspace \mathrm{t}\mathrm{h}\mathrm{e}\enspace \mathrm{s}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:0\leqslant x\leqslant 10\enspace \mu \mathrm{m},0\leqslant t\leqslant 5\enspace \mathrm{p}\mathrm{s}\end{equation*}$$

$$\begin{equation*}\mathrm{R}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\enspace \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}:\left({\varepsilon }_{\mathrm{r}}\right)=12\enspace ({\mu }_{\mathrm{r}}=1),\qquad \mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\enspace \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}:d=0.3\enspace \mathrm{n}\mathrm{m}\end{equation*}$$

$$\begin{equation*}\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}\enspace \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}{:=}1{0}^{10}\enspace \mathrm{H}\mathrm{z},\quad \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace (\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{g}\mathrm{a}\mathrm{p})\enspace \mathrm{f}\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}=\mathrm{272}\enspace \mathrm{T}\mathrm{H}\mathrm{z}\end{equation*}$$

$$\begin{equation*}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace \mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m}\enspace \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}:5\enspace \mu \mathrm{m}< x< 5.3\enspace \mu \mathrm{m},\qquad \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}=1{0}^{-3}\enspace \mathrm{S}\enspace {\mathrm{m}}^{-1}\end{equation*}$$

$$\begin{equation*}A=1.35\times 1{0}^{9}\enspace {\mathrm{s}}^{-1},\qquad B=5.6\times 1{0}^{-16}\enspace {\mathrm{m}}^{3}\enspace {\mathrm{s}}^{-1},\quad C=1.5\times 1{0}^{-40}\enspace {\mathrm{m}}^{6}\enspace {\mathrm{s}}^{-1}.\end{equation*}$$

The variation of the SHGE is shown in figure 9. For a transition electron density of N = 1.9 × 10²⁸ m⁻³, the SHGE reaches 10.8, which is an extremely high value compared to the values observed in the nanoscale without active tuning. This means that the intensity of the generated second harmonic is 10.8 times greater than the intensity of the first harmonic of the input wave. Figure 10 shows the corresponding amplitude spectrum of the input and output waves. Notice that there is also a simultaneous generation of a strong third harmonic whose spectral amplitude is comparable to that of the first harmonic of the input wave.

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To attain a transition electron density of N = 1.9 × 10²⁸ m⁻³, one should first find the corresponding optical pumping rate Γ using equation (15). Doing so, a steady state optical pumping rate of 2.5 × 10³⁸ m⁻³ s⁻¹ is found using the given parameters. Such an optical pumping rate requires a beam intensity of 10¹³ W m⁻², which corresponds to an electric field amplitude of 10⁸ V m⁻¹ (equation (20)). In the small-scale, this intensity level can be easily attained by a precise focusing of the pump beam through a lens, onto the interaction material cross-section. If we assume a typical small-scale material cross section of 100 μm², this requires a pumping power of 1000 W.

$$\begin{equation}{\Gamma}=\frac{I}{hf{\Delta}}=\frac{0.5c\sqrt{{\varepsilon }_{\infty }}{\varepsilon }_{0}{{E}_{\mathrm{p}}}^{2}}{4\times {10}^{-26}}=\frac{{10}^{13}\enspace \mathrm{W}\enspace {\mathrm{m}}^{-2}}{(6.62\times {10}^{-34}\enspace \mathrm{J}\times \mathrm{s})\times (2.2\times {10}^{14}\enspace \mathrm{H}\mathrm{z})\times (3\times {10}^{-7}\enspace \mathrm{m})}\end{equation} \tag{ 20 }$$

h: Planck's constant, f: pump frequency, Δ: propogation distance, I: intensity, c: speed of light, E_p: pump wave electric field amplitude.

Using equations (13)–(15), the computations are repeated for gallium arsenide under an arbitrary near-IR laser excitation at 290 THz. This time the wave and the material parameters are taken as [30–32]

$$\begin{equation*}E\left(x=3\enspace \mu \mathrm{m},t\right)=1\times 1{0}^{8}\times \mathrm{sin}\left(2\pi \left(2.9\times {10}^{14}\right)t\right)\enspace \mathrm{V}\enspace {\mathrm{m}}^{-1},\quad A=1.35\times 1{0}^{9}\enspace {\mathrm{s}}^{-1},\quad B=5.6\times 1{0}^{-16}\enspace {\mathrm{m}}^{3}\enspace {\mathrm{s}}^{-1}\end{equation*}$$

$$\begin{equation*}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\enspace \mathrm{a}\mathrm{n}\mathrm{d}\enspace \mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\enspace \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\enspace \mathrm{f}\mathrm{o}\mathrm{r}\enspace \mathrm{t}\mathrm{h}\mathrm{e}\enspace \mathrm{s}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:0\leqslant x\leqslant 10\enspace \mu \mathrm{m},0\leqslant t\leqslant 5\enspace \mathrm{p}\mathrm{s}\end{equation*}$$

$$\begin{equation*}\mathrm{R}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\enspace \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}:\left({\varepsilon }_{\mathrm{r}}\right)=13\enspace ({\mu }_{\mathrm{r}}=1),\qquad \mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\enspace \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}:d=0.3\enspace \mathrm{n}\mathrm{m}\end{equation*}$$

$$\begin{align*}\hfill & \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}\enspace \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}{:=}4\times 1{0}^{9}\enspace \mathrm{H}\mathrm{z},\,\hfill \\ \hfill & \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace (\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{g}\mathrm{a}\mathrm{p})\enspace \mathrm{f}\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}=\mathrm{345}\enspace \mathrm{T}\mathrm{H}\mathrm{z},\qquad C=1.5\times 1{0}^{-40}\enspace {\mathrm{m}}^{6}\enspace {\mathrm{s}}^{-1}\hfill \end{align*}$$

$$\begin{equation*}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\enspace \mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m}\enspace \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}:5\enspace \mu \mathrm{m}< x< 5.3\enspace \mu \mathrm{m},\qquad \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}=1{0}^{-4}\enspace \mathrm{S}\enspace {\mathrm{m}}^{-1}.\end{equation*}$$

Although not as high as in the case for silicon, once again, a very high peak SHGE is observed for a transition electron density of N = 3.7 × 10²⁸ m⁻³ (figure 11). Based on equation (15), this corresponds to the steady state optical pumping rate of 4.6 × 10³⁸ m⁻³ s⁻¹. The resulting spectrum for the input and output electric fields is shown in figure 12. Here the investigation is done in a single dimension, therefore this observation indicates the sensitivity of the SHGE in silicon (compared to GaAs) without regarding its crystal-structure. Hence, the investigation that is carried out in this section emphasizes the drastic enhancement/improvement of the SHGE via active tuning, regardless of the crystal-structure of a material. We believe this finding is of particular importance in the sense that weak SHGE in centrosymmetric media can be greatly enhanced via active tuning.

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In this section we will investigate the variation of the SHGE with respect to the spectral distance between the excitation frequency and the resonance frequency. To examine this relation, we have investigated four different cases, in which the resonance frequency is gradually brought closer to the excitation frequency. The excitation and resonance frequency values for each case is provided below. Figures 13–16 illustrate the variation of the SHGE for each case. In the first case, the SHGE is quite weak and does not peak around a certain value of the transition electron density. For the second case, the SHGE is increased for all transition electron densities within the investigated range, although the SHGE still remains relatively weak compared to the values that are observed without any active tuning.

$$\begin{equation*}\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e}\enspace 1:\;\;f=100\enspace \mathrm{T}\mathrm{H}\mathrm{z},\;\;{f}_{0}=1500\enspace \mathrm{T}\mathrm{H}\mathrm{z},\qquad \mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\enspace 2:\;\;f=200\enspace \mathrm{T}\mathrm{H}\mathrm{z},\;\;{f}_{0}=1000\enspace \mathrm{T}\mathrm{H}\mathrm{z}\end{equation*}$$

$$\begin{equation*}\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e}\enspace 3:\;\;f=300\enspace \mathrm{T}\mathrm{H}\mathrm{z},\;\;{f}_{0}=700\enspace \mathrm{T}\mathrm{H}\mathrm{z},\qquad \mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\enspace 4:\;\;f=300\enspace \mathrm{T}\mathrm{H}\mathrm{z},\;\;{f}_{0}=400\enspace \mathrm{T}\mathrm{H}\mathrm{z}\end{equation*}$$

f: excitation frequency, f₀: resonance frequency.

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For the third case, in which the resonance frequency is now only 400 THz beyond the excitation frequency, the SHGE peak attains a very high value for an electron density of N = 2 × 10²⁸ m⁻³, which cannot be practically observed without the involvement of active tuning in the nanoscale. In the final case, the resonance frequency is just 100 THz away from the excitation frequency and the SHGE peak reaches to a super-high value of 3.7 for an electron density of N = 3.3 × 10²⁸ m⁻³. From these four investigations we can see that the proximity of the resonance and the excitation frequencies has a very strong effect on the SHGE. However, regardless of their spectral distance, proper tuning of the transition electron density almost always leads to a sharp increase in the SHGE.

Based on our investigation, the SHGE does not change significantly with respect to the spectral-range of investigation, compared to the dramatic change that occurs when the excitation frequency is near the transition frequency, or the change that occurs when the transition electron density is optimally tuned. Hence, to attain a high SHGE at a particular range of the optical spectrum, one should choose an interaction material such that the resonance frequency of the material is in the close-proximity of the excitation frequency (or the first harmonic) and should make sure that the transition electron density is optimally tuned for SHGE maximization.