Quantum Cascade Laser

Abstract

A quantum cascade laser contains active regions and conducting regions in turn. An active region contains three electron subbands.

Problems

29.1

Determine the frequency (and the wavelength) of laser radiation at which cooling of the active medium is favorable.

29.2

Estimate the gain (gain coefficient and gain factor per round trip) of radiation propagating in the active medium of a quantum cascade laser. [Hint: assume that the Einstein coefficient of stimulated emission is the same as for interband transitions.]

29.3

A quantum cascade laser cannot be realized if the laser transition frequency $v_{1}$ coincides with the longitudinal optic frequency of the semiconductor material. Then, fast nonradiative relaxation of the upper laser level, by emission of a longitudinal optic phonon (frequency $v_{\text {LO}}$) near the Brillouin zone center, makes population inversion almost impossible. Relaxation is still strong if $v_{1}$ lies in the vicinity $v_{\text {LO}}$ (=8.7 THz for GaAs).

• (a) Determine the frequency range for which the nonradiative lifetime of the upper laser level is shorter than $10^{-6}$ s, assuming that the nonradiative lifetime is $10^{-12}$ s at resonance ($v_{1}= v_{\text {LO}}$) and that the relaxation rate decreases for laser frequencies around $v_{\text {LO}}$ according to a Lorentz resonance function.

• (b) Determine the power density in the laser medium that is necessary for reaching population inversion $(B_{21} = 4 \times 10^{21} \,\text {m}^{3}\,\text {J}^{-1}\,\text {s}^{-2})$.

• (c) Discuss the influence due to thermal broadening of the electron distribution in the upper laser subband.

29.4

Strong absorption at the transversal optical frequency (=8.0 THz for GaAs) at the zone center results in damping of optical waves. In which frequency range is the optical thickness of the laser material (length 1 mm) of a quantum cascade laser larger than 0.02)? [Hint: assume that the absorption coefficient varies according to a Lorentz resonance function and that the absorption coefficient has a maximum value of $10^{5}$ cm${}^{-1}$.]