Question #42854

Two coherent electromagnetic waves, having vacuum intensities $I_1 = 4 \text{ W/m}^2$ and $I_2 = 9 \text{ W/m}^2$, are superposed in phase at $x = 0$ to form a single resultant wave. This resultant wave propagates in the $+x$ direction through a semiconductor medium of length $L = \ln(2) \text{ m}$. Inside the medium, local energy dissipation alters the band gap, creating a feedback loop between the wave intensity and the medium's properties. The local steady-state refractive index $n(x)$ depends on the local intensity $I(x)$ as $n(x) = n_0 + \mu I(x)$. Consequently, the local absorption coefficient $\alpha(x)$ of the medium becomes intensity-dependent and is given by $\alpha(x) = \alpha_0 + \sigma (n(x) - n_0)$, where $\alpha_0 = 1 \text{ m}^{-1}$ is the linear absorption coefficient. Given that the product $\sigma \mu = 0.2 \text{ m/W}$, and the intensity decays according to $\frac{dI}{dx} = -\alpha(x) I(x)$, determine the intensity of the wave as it exits the medium at $x = L$.