Optimization of gain in intersubband quantum well lasers by supersymmetry

A systematic method is described for the optimized design of quantum-well structures, in respect to maximizing the stimulated gain in optically pumped intersubband lasers. It relies on applying supersymmetric quantum mechanics to an initial Hamiltonian, in order to (i) add one bound state below the spectral range of the initial Hamiltonian, and (ii) to generate a parameter-controlled family of isospectral Hamiltonians with the desired energy spectrum. By changing the value of the control parameter one changes the potential shape and thus the values of dipole matrix elements and electron-phonon scattering matrix elements. The use of this procedure is demonstrated by designing an appropriate smooth or stepwise-constant graded ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ ternary alloy quantum well.