In this tutorial we calculate the optical gain upon optical irradiation. The irradiation parameters are the

• photon energy of the irradiation and the
• line width.

The transition rate per volume element can be expressed with the following sum, \[ R = R_{ab} - R_{ba} = \frac{2}{V} \sum_{k_a} \sum_{k_b} \frac{2 \pi}{ \hbar} |H_{ba}| ^2 \delta(E_b - E_a -\hbar \omega)(f_a-f_b). \]

In order to evaluate the sum much faster we calculate the $H_{ba}$ matrix element at $k_a = 0; k_b = 0$ (Remark: $k_a = k_b$), and we neglect the $k$ dependence of it. Then we can simplify the sum as follows, \[ R(E, w) = C_0(E) \int \gamma(E_a-E, w) \cdot H(E_a-E) \cdot [n(E_a) - p(E_b)] {\rm d}E_a {\rm d}E_b, \] where $E$ is the irradiation energy, $w$ is the line width and we assume that the irradiation has the $\gamma(E, w)$ broadening function.

Here $C_0(E)$ is an energy dependent constant, \[ C_0 = \frac{\pi e^2 \hbar}{n_{\rm r} c \epsilon_0 m_0^2 E}. \]

A new keyword has been introduced to handle an optical device, opticaldevice{}.

opticaldevice{
	name = "quantum_region_name"
	line_broadening = 1            # Line broadening model (1: Lorentzian)
	photon_energy   = 1.0          # Photon energy in (eV)
	line_width      = 1.0          # Line width in (eV)
}

The run keyword requires solve_optical_device{} to be included.