*6.2. Triple-Frequency Tuning for Gain Factor Optimization*

Assume that a 300 THz (λ*f ree space* =1 μm) infra-red stimulus wave *Est* and a high-power pump wave *Ehp* that is composed of three ultrashort pulses (frequencies are to be determined), are generated to propagate in a micro-resonator with two reflecting walls. The wall on the left side can be thought as an optical isolator, which fully transmits from its left side and almost fully reflects from its right side. The wall on the right side represents an optical band-pass filter with a frequency-dependent reflection coefficient Γ(*f*)(see Figure 7). Both waves are generated at x = 0 μm and at the time instant t = 0 ps. The waves and the parameters of the gain medium are as given below:

$$E\_{hp}(\mathbf{x} = 0 | \mu m, t) = \sum\_{i=1}^{3} A\_i \cos(2\pi f\_i t + \psi\_i) (u(t) - u(t - \Delta T\_i)) (u(t) : \mathbf{U} \text{in} t \text{ step function}) \; \; \; \; \quad (1)$$

where *<sup>A</sup>*<sup>1</sup> = 1.5 <sup>×</sup> <sup>10</sup>8, *<sup>A</sup>*<sup>2</sup> = <sup>2</sup> <sup>×</sup> <sup>10</sup>8, *<sup>A</sup>*<sup>3</sup> = 2.5 <sup>×</sup> 108, <sup>Δ</sup>*T*<sup>1</sup> = <sup>4</sup>*ps*, <sup>Δ</sup>*T*<sup>2</sup> = <sup>2</sup>*ps*, <sup>Δ</sup>*T*<sup>2</sup> = <sup>1</sup>*ps*

*Est*(*<sup>x</sup>* <sup>=</sup> <sup>0</sup>μ*m*, *<sup>t</sup>*) <sup>=</sup> <sup>1</sup> <sup>×</sup> sin 2π <sup>3</sup> <sup>×</sup> <sup>10</sup>14 *t V*/*m*, *f or* 0 ≤ *t* ≤ 40*ps*

*Dielectric constant o f the gain medium* = ε<sup>∞</sup> = 1 + χ = 10 (μ*<sup>r</sup>* = 1)

*Resonance f requency o f the gain medium* : *f*<sup>0</sup> = 600*THz*

*Damping rate o f the gain medium* : <sup>γ</sup> = <sup>1</sup> <sup>×</sup> 1010*Hz*

*Duration o f simulation* : 0 ≤ *t* ≤ 40*ps*; *Range o f the gain medium* : 0μ*m* < *x* < 10μ*m*

*Right cavity wall location* : *x* = 10μ*m*; *Le f t cavity wall location* : *x* = 0μ*m*

*Electron density o f the medium* : *<sup>N</sup>* = 3.5 <sup>×</sup> <sup>10</sup>28/*m*3; *Atomic diameter* : *<sup>d</sup>* = 0.3 *nm*

**Figure 7.** Configuration of the cavity and the parameters for Section 6.2.

**Our problem:** Find the optimal pump wave pulse frequencies *fp*1, *fp*2, *fp*<sup>3</sup> that maximize the magnitude of the monochromatic stimulus wave in the cavity ( *Est*( *fst* <sup>=</sup> <sup>300</sup>*THz*) ), for 50 THz < *fp*1, *fp*2, *fp*<sup>3</sup> < 500 *THz*, 0μ*m* < *x* < 10μ*m*, 0 ≤ *t* ≤ 40 *ps*, such that

∇<sup>2</sup>*Ehp fp*1, *fp*2, *fp*<sup>3</sup> − μ0ε<sup>∞</sup> ∂<sup>2</sup>*Ehp fp*1, *fp*2, *fp*<sup>3</sup> <sup>∂</sup>*t*<sup>2</sup> <sup>=</sup> <sup>μ</sup>0<sup>σ</sup> ∂*Ehp fp*1, *fp*2, *fp*<sup>3</sup> <sup>∂</sup>*<sup>t</sup>* <sup>+</sup> <sup>μ</sup><sup>0</sup> ∂<sup>2</sup>*Php* ∂*t*<sup>2</sup> ∂<sup>2</sup>*Php* <sup>∂</sup>*t*<sup>2</sup> <sup>+</sup> <sup>γ</sup> ∂*Php* <sup>∂</sup>*<sup>t</sup>* <sup>+</sup> <sup>ω</sup><sup>0</sup> 2 *Php* <sup>−</sup> <sup>ω</sup><sup>0</sup> 2 *Ned Php*<sup>2</sup> <sup>−</sup> <sup>ω</sup><sup>0</sup> 2 *N*2*e*2*d*<sup>2</sup> *Php*<sup>3</sup> <sup>=</sup> *Ne*<sup>2</sup> *<sup>m</sup> Ehp fp*1, *fp*2, *fp*<sup>3</sup> . ∇2*Est fp*1, *fp*2, *fp*<sup>3</sup> − μ0ε<sup>∞</sup> ∂2*Est fp*1, *fp*2, *fp*<sup>3</sup> <sup>∂</sup>*t*<sup>2</sup> <sup>=</sup> <sup>μ</sup>0<sup>σ</sup> ∂*Est fp*1, *fp*2, *fp*<sup>3</sup> <sup>∂</sup>*<sup>t</sup>* <sup>+</sup> <sup>μ</sup><sup>0</sup> ∂2*Pst* ∂*t*<sup>2</sup> ∂2(*Pst*) <sup>∂</sup>*t*<sup>2</sup> + <sup>γ</sup> ∂(*Pst*) <sup>∂</sup>*<sup>t</sup>* + ω<sup>0</sup> <sup>2</sup>(*Pst*) <sup>−</sup> <sup>ω</sup><sup>0</sup> 2 *Ned Pst*<sup>2</sup> + 2*PstPhp* <sup>−</sup> <sup>ω</sup><sup>0</sup> 2 *N*2*e*2*d*<sup>2</sup> *Pst*<sup>3</sup> + 3*Pst*<sup>2</sup>*Php* + 3*PstPhp*<sup>2</sup> = *Ne*<sup>2</sup> *<sup>m</sup> Est fp*1, *fp*2, *fp*<sup>3</sup> 

It is important to emphasize that our aim is to maximize the magnitude of the stimulus wave at its original frequency *fst* = 300 *THz* (monochromatic form). This is a precaution against any degree of spectral broadening that the stimulus wave may go through while being amplified. A plain attempt to maximize the magnitude of the stimulus wave (|*Est*|) independent of the original excitation frequency (*fst*), may result in an amplified stimulus wave with different frequency components. In fact, these different frequency components might be even much more dominant than the original excitation frequency of the stimulus wave. Therefore, our cost function is chosen as (at any spatial point *x* = *x* )

$$\begin{array}{lcl} Q = & \left| E\_{\rm sf} (f\_{\rm sf} = 300 THz) \right| \\ &= \left| \int\_{3 \times 10^{14} - \Delta f}^{3 \times 10^{14} + \Delta f} \left\{ \int\_{0}^{\Delta T} \langle E\_{\rm sf} (\mathbf{x} = \mathbf{x}', t) e^{-i (2\pi \Omega) t} \rangle dt \right\} e^{i (2\pi \Omega) t} d\Omega \right| \\ & & (\cdot \quad \cdot \quad \cdot) \qquad (\cdot \quad \cdot) \end{array}$$

*where* Δ*T* = 40*ps*, 0 ≤ *t* ≤ 40*ps*, <sup>3</sup> <sup>×</sup> 1014 <sup>−</sup> <sup>Δ</sup>*<sup>f</sup>* < Ω < <sup>3</sup> <sup>×</sup> 1014 + <sup>Δ</sup>*<sup>f</sup>* , Δ*f* = 0.5*THz* **Initial conditions:**

#### *Php*(*x*, 0) = *Php* (*x*, 0) = *Ehp*(*x*, 0) = *Ehp* (*x*, 0) = *Pst*(*x*, 0) = *Pst* (*x*, 0) = *Est*(*x*, 0) = *Est* (*x*, 0) = 0
