# 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 Δ*n_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?