Question #42878

A monochromatic laser beam of initial intensity $I_0$ (in $\text{W m}^{-2}$) propagates along the positive $x$-axis into a thick semiconductor crystal starting at $x=0$. As the beam propagates, photon absorption increases the local temperature of the crystal, which in turn decreases the energy band gap. Assume the local steady-state temperature rise is directly proportional to the local beam intensity $I(x)$, such that the effective absorption coefficient becomes $\alpha(I) = \alpha_0 + \lambda I(x)$. Here, $\alpha_0$ (in $\text{m}^{-1}$) is the intrinsic absorption coefficient and $\lambda$ (in $\text{m W}^{-1}$) is a positive thermal-coupling constant. Determine the depth $x_{1/2}$ (in $\text{m}$) at which the beam intensity drops to half of its initial value ($I_0/2$).