# Homework #4

Due 5pm Friday of the 4th week

## Stimulated emission in a laser diode (10pts)

a) What is the peak spectral energy density inside the recombination region of a 1 mW laser diode that emits at 1 micron wavelength, has a cavity length of 1 mm, cavity cross-sectional area of 100 square microns, and a quality factor of 50? (Note, spectral energy density has units of energy per unit volume per unit frequency.)

b) Compare your answer in part a) to the spectral energy density of quantum zero-point fluctuations in the same cavity mode at 1 micron wavelength. For example, 100 fold?, 1,000,000-fold?

## Transitions in a Quantum Cascade Laser (15 pts)

Consider a quantum cascade laser with GaAs/AlGaAs layers stacked in the z-direction. The lattice constant describing the two-atom unit cell of GaAs is a. A photon is emitted when an electron in the n_z = 2 state transitions to the n_z = 1 state of the quantum well. The width of the quantum well is 10 nm. The electron effective mass is 0.067 times the free electron mass.

ψ_1(z) & ψ_2(z) can be accurately described by a linear combinations of atomic orbitals, as illustrated below (not drawn to scale).

a) With an accuracy of approximately ±10%, find the z-component of the transition dipole moment

P_z = q<ψ_2|z|ψ_1>

• Don't use an explicit form for the atomic orbitals - the key properties are that the atomic orbitals are normalized and the overlap between neighboring atomic orbitals is small.
• A reasonable approximation for the boundary conditions is ψ_1 = ψ_2 = 0 at z = ±5 nm.
• You may prefer to set the boundaries at 0 & 10 nm.

b) By considering the parity of the states, prove that the only allowed optical transitions have odd values of Δn_z.

c) Compare the relative strengths of the 1 → 2 transition and the 1 → 4 transition. What is the wavelength of the 1 → 2 transition?