Rate equations

Subband populations are determined by the intersubband scattering rates and the injection/extraction current.

QCLs are typically based upon a three-level system.^[3] Assuming the formation of the wavefunctions is a fast process compared to the scattering between states, the time independent solutions to the Schrödinger equation may be applied and the system can be modelled using rate equations. Each subband contains a number of electrons ${\displaystyle n_{i}}$ (where ${\displaystyle i}$ is the subband index) which scatter between levels with a lifetime ${\displaystyle \tau _{if}}$ (reciprocal of the average intersubband scattering rate ${\displaystyle W_{if}}$), where ${\displaystyle i}$ and ${\displaystyle f}$ are the initial and final subband indices. Assuming that no other subbands are populated, the rate equations for the three level lasers are given by:

${\displaystyle {\frac {\mathrm {d} n_{3}}{\mathrm {d} t}}=I_{\mathrm {in} }+{\frac {n_{1}}{\tau _{13}}}+{\frac {n_{2}}{\tau _{23}}}-{\frac {n_{3}}{\tau _{31}}}-{\frac {n_{3}}{\tau _{32}}}}$

${\displaystyle {\frac {\mathrm {d} n_{2}}{\mathrm {d} t}}={\frac {n_{3}}{\tau _{32}}}+{\frac {n_{1}}{\tau _{12}}}-{\frac {n_{2}}{\tau _{21}}}-{\frac {n_{2}}{\tau _{23}}}}$

${\displaystyle {\frac {\mathrm {d} n_{1}}{\mathrm {d} t}}={\frac {n_{2}}{\tau _{21}}}+{\frac {n_{3}}{\tau _{31}}}-{\frac {n_{1}}{\tau _{13}}}-{\frac {n_{1}}{\tau _{12}}}-I_{\mathrm {out} }}$

In the steady state, the time derivatives are equal to zero and ${\displaystyle I_{\mathrm {in} }=I_{\mathrm {out} }=I}$. The general rate equation for electrons in subband i of an N level system is therefore:

${\displaystyle {\frac {\mathrm {d} n_{i}}{\mathrm {d} t}}=\sum \limits _{j=1}^{N}{\frac {n_{j}}{\tau _{ji}}}-n_{i}\sum \limits _{j=1}^{N}{\frac {1}{\tau _{ij}}}+I(\delta _{iN}-\delta _{i1})}$,

Under the assumption that absorption processes can be ignored, (i.e. ${\displaystyle {\frac {n_{1}}{\tau _{12}}}={\frac {n_{2}}{\tau _{23}}}=0}$, valid at low temperatures) the middle rate equation gives

${\displaystyle {\frac {n_{3}}{\tau _{32}}}={\frac {n_{2}}{\tau _{21}}}}$

Therefore, if ${\displaystyle \tau _{32}>\tau _{21}}$ (i.e. ${\displaystyle W_{21}>W_{32}}$) then ${\displaystyle n_{3}>n_{2}}$ and a population inversion will exist. The population ratio is defined as

${\displaystyle {\frac {n_{3}}{n_{2}}}={\frac {\tau _{32}}{\tau _{21}}}={\frac {W_{21}}{W_{32}}}}$

If all N steady-state rate equations are summed, the right hand side becomes zero, meaning that the system is underdetermined, and it is possible only to find the relative population of each subband. If the total sheet density of carriers ${\displaystyle N_{\mathrm {2D} }}$ in the system is also known, then the absolute population of carriers in each subband may be determined using:

${\displaystyle \sum \limits _{i=1}^{N}n_{i}=N_{\mathrm {2D} }}$.

As an approximation, it can be assumed that all the carriers in the system are supplied by doping. If the dopant species has a negligible ionisation energy then ${\displaystyle N_{\mathrm {2D} }}$ is approximately equal to the doping density.

Electron wave functions are repeated in each period of a three quantum well QCL active region. The upper laser level is shown in bold.

Active region designs

The scattering rates are tailored by suitable design of the layer thicknesses in the superlattice which determine the electron wave functions of the subbands. The scattering rate between two subbands is heavily dependent upon the overlap of the wave functions and energy spacing between the subbands. The figure shows the wave functions in a three quantum well (3QW) QCL active region and injector.

In order to decrease ${\displaystyle W_{32}}$, the overlap of the upper and lower laser levels is reduced. This is often achieved through designing the layer thicknesses such that the upper laser level is mostly localised in the left-hand well of the 3QW active region, while the lower laser level wave function is made to mostly reside in the central and right-hand wells. This is known as a diagonal transition. A vertical transition is one in which the upper laser level is localised in mainly the central and right-hand wells. This increases the overlap and hence ${\displaystyle W_{32}}$ which reduces the population inversion, but it increases the strength of the radiative transition and therefore the gain.

In order to increase ${\displaystyle W_{21}}$, the lower laser level and the ground level wave functions are designed such that they have a good overlap and to increase ${\displaystyle W_{21}}$ further, the energy spacing between the subbands is designed such that it is equal to the longitudinal optical (LO) phonon energy (~36 meV in GaAs) so that resonant LO phonon-electron scattering can quickly depopulate the lower laser level.

Fabry–Perot lasers

This is the simplest of the quantum cascade lasers. An optical waveguide is first fabricated out of the quantum cascade material to form the gain medium. The ends of the crystalline semiconductor device are then cleaved to form two parallel mirrors on either end of the waveguide, thus forming a Fabry–Pérot resonator. The residual reflectivity on the cleaved facets from the semiconductor-to-air interface is sufficient to create a resonator. Fabry–Pérot quantum cascade lasers are capable of producing high powers,^[23] but are typically multi-mode at higher operating currents. The wavelength can be changed chiefly by changing the temperature of the QC device.

Distributed feedback lasers

A distributed feedback (DFB) quantum cascade laser^[24] is similar to a Fabry–Pérot laser, except for a distributed Bragg reflector (DBR) built on top of the waveguide to prevent it from emitting at other than the desired wavelength. This forces single mode operation of the laser, even at higher operating currents. DFB lasers can be tuned chiefly by changing the temperature, although an interesting variant on tuning can be obtained by pulsing a DFB laser. In this mode, the wavelength of the laser is rapidly "chirped" during the course of the pulse, allowing rapid scanning of a spectral region.^[25]

External cavity lasers

Schematic of QC device in external cavity with frequency selective optical feedback provided by diffraction grating in Littrow configuration.

In an external cavity (EC) quantum cascade laser, the quantum cascade device serves as the laser gain medium. One, or both, of the waveguide facets has an anti-reflection coating that defeats the optical cavity action of the cleaved facets. Mirrors are then arranged in a configuration external to the QC device to create the optical cavity.

If a frequency-selective element is included in the external cavity, it is possible to reduce the laser emission to a single wavelength, and even tune the radiation. For example, diffraction gratings have been used to create^[26] a tunable laser that can tune over 15% of its center wavelength.

Extended tuning devices

There exists several methods to extend the tuning range of quantum cascade lasers using only monolithically integrated elements. Integrated heaters can extend the tuning range at fixed operation temperature to 0.7% of the central wavelength^[27] and superstructure gratings operating through the Vernier effect can extend it to 4% of the central wavelength,^[28] compared to <0.1% for a standard DFB device.