Microscopic modeling of second harmonic generation in quantum cascade lasers

During the last 20 years since the first Quantum Cascade Laser (QCL) [1] there has been substantial development. The long-wavelength limit exploiting intersubband transitions below the reststrahlenband with frequencies in the terahertz range [2] is currently an area of active research but efforts are also being made in the opposite direction towards the short-wavelength limit. As the lasing is based on intersubband transitions with relaxation and thus gain recovery times of picoseconds, they far exceed the solid state lasers when it comes to modulation speed [3]; although the frequencies are limited by the conduction band offset. Possible high speed applications makes it worthwhile to try to extend the wavelength regime towards the telecom frequencies. In this work our implementation of the Keldysh non-equilibrium Green’s function (NEGF) formalism [4] is applied to the phenomena of frequency doubling, i.e. second harmonic generation, which is a straightforward way to achieve shorter wavelengths. The key concept here is that the time dependence of each observable is expressed as a Fourier series with the driving frequency as the fundamental frequency, an approach originally proposed by Brandes [5]. As an example, average current is expressed in this form as