**2. Fundamentals and operating principles**

A conventional laser diode generates light by a single photon that is generated from an electron interband transition; this means that a high-energy electron in the conduction band recombines with a hole in the valence band, being the energy of the photon determined by the band-gap energy of the material system used. However, QCLs do not use bulk semiconductor materials in their optically active region, but a periodic series of thin layers of varying material composition forming a superlattice, which leads to an electric potential that changes across the length of the device (one-dimensional multiple quantum well confinement), splitting the bandpermitted energies into a number of discrete electronic subbands, making electrons cascade down a series of identical energy steps built into the material during crystal growth, and emitting a photon at every step, unlike diode lasers, which emit only one photon over the equivalent cycle. With an appropriate design of the thickness of these layers, population inversion is achieved between discrete conduction band-excited states in the coupled quantum wells by the control of tunnelling, making laser emission possible. Therefore, the position of energy levels is mainly determined by the thickness of the layers, rather than the material, and thus allowing tuning the emission wavelength of QCLs over a wide range in the same material system. Thus, one electron emits a photon during every intersubband transition within the quantum well (QW) in the superlattice, and then can tunnel into the next period of the structure where another electron can be emitted, leading QCLs to outperform diode lasers operating at

the same wavelength by a factor even greater than 1000 in terms of power.

by tunnelling.

4 Quantum Cascade Lasers

**Figure 1.** Typical conduction band structure of a QCL [4].

Classically, a QCL is made of a periodic repetition of active sections, which consist of tunnelcoupled quantum wells and injector, where a miniband is formed. As **Figure 1** shows [4], from the injector miniband the electrons are injected into the upper laser energy level (4) of the active section, where the laser transition takes place. Afterwards, the lower laser energy level (3) is emptied by longitudinal optical emissions (LO emissions) and the electrons enter the next step

> The main particularity in QCLs is the fact that instead of using bulk semiconductor materials in their optically active region, a periodic series of thin layers are used, that is, superlattices, consisting of a number of quantum well—barrier system equally spaced, introducing a multiple quantum well (MQW) leading to one-dimensional confinement allowing an electric potential variation (band splitting) that results in a number of discrete electronic subbands. In order to achieve the population inversion required for laser emission, it is necessary for a

proper thickness and composition layer design. These confined energy levels depend on the layer thickness, so the tunability of the emission within the same material relies, in principle, on thickness variation, although multiwavelength QCLs emit by means of different materials within the same structure and multiple resonators [7, 8].

In classical semiconductor laser diodes, electrons and holes recombine across the band gap, thus, generating one photon per e− -h+ pair recombined. However, in QCLs this is not the case, as an electron in the conduction band within a QW emits one photon whenever it undergoes an intersubband transition, that is, thermalization into lower-energy levels, emitting one photon. This electron could tunnel into the next period of the structure, where the mentioned transition happens again emitting another photon. This phenomenon of a single electron emitting multiple photons as it passes through different periods of the structure is called cascade, making the quantum efficiency of QCLs greater than unity and leading to their high optical output power.

**Figure 3.** Three-level intersubband transition and scatterings considered in most QCLs.

QCLs are usually three-level lasers, whose level transitions are depicted in **Figure 3**. In these lasers, the wave function formation is faster than scattering between states; hence, the timeindependent solutions of the Schrödinger equation can be applied, so the system can be modelled using rate equations. Each subband, *i*, is considered to have *ni* electrons, and a scatter between the initial and the final subband, *f*, will have a scattering rate, *Wif*, and lifetime, *τif*. If no other subbands are populated, the rate equations are described as extraction and injection of electrons, respectively (depicted in **Figure 3**), whose values are both equal in steady-state conditions, where time derivatives are zero. Hence, the general rate equation for electrons in a subband *i* within an *N* level system is where *I* = *I*ee = *I*ie. At low temperature, absorption is near zero: hence, *τ*32 > *τ*21, so *W*21 > *W*32, and *n*3 > *n*2 leading to the existence of population inversion.

$$\text{dn}\_{\text{l}}/\text{dt} = n\_2/\tau\_{21} + n\_3/\tau\_{31} - n\_1/\tau\_{13} - n\_1/\tau\_{12} - I\_{\text{ee}} \tag{1}$$

$$dn\_2 \mid \text{dt} = n\_3 \mid \tau\_{32} + n\_1 \mid \tau\_{12} - n\_2 \mid \tau\_{21} - n\_1 \mid \tau\_{23} \tag{2}$$

$$\text{dn}\_3 / \text{dt} = I\_{\text{ie}} + \text{n}\_1 / \text{\tau}\_{13} + \text{n}\_2 / \tau\_{23} - \text{n}\_3 / \tau\_{31} - \text{n}\_3 / \tau\_{23} \tag{3}$$

where *I*ee and *I*ie are the extraction and injection of electrons, respectively (depicted in **Fig‐ ure 3**), whose values are both equal in steady-state conditions, where time derivatives are zero. Hence, the general rate equation for electrons in a subband *i* within an *N* level system is

$$dn\_i \mid \mathbf{dt} = \mathbf{S}\_{f=1}^N n\_j \mid \boldsymbol{\tau}\_{j\boldsymbol{t}} - n\_i \mathbf{S}\_{f=1}^N \boldsymbol{\tau}\_{i\boldsymbol{f}}^{-1} + \mathbf{I}(d\_{i0} - d\_{iN}) \tag{4}$$

where *I* = *I*ee = *I*ie. At low temperature, absorption is near zero:

proper thickness and composition layer design. These confined energy levels depend on the layer thickness, so the tunability of the emission within the same material relies, in principle, on thickness variation, although multiwavelength QCLs emit by means of different materials

In classical semiconductor laser diodes, electrons and holes recombine across the band gap,

as an electron in the conduction band within a QW emits one photon whenever it undergoes an intersubband transition, that is, thermalization into lower-energy levels, emitting one photon. This electron could tunnel into the next period of the structure, where the mentioned transition happens again emitting another photon. This phenomenon of a single electron emitting multiple photons as it passes through different periods of the structure is called cascade, making the quantum efficiency of QCLs greater than unity and leading to their high

pair recombined. However, in QCLs this is not the case,

electrons, and a scatter

within the same structure and multiple resonators [7, 8].


**Figure 3.** Three-level intersubband transition and scatterings considered in most QCLs.

modelled using rate equations. Each subband, *i*, is considered to have *ni*

QCLs are usually three-level lasers, whose level transitions are depicted in **Figure 3**. In these lasers, the wave function formation is faster than scattering between states; hence, the timeindependent solutions of the Schrödinger equation can be applied, so the system can be

between the initial and the final subband, *f*, will have a scattering rate, *Wif*, and lifetime, *τif*. If no other subbands are populated, the rate equations are described as extraction and injection of electrons, respectively (depicted in **Figure 3**), whose values are both equal in steady-state conditions, where time derivatives are zero. Hence, the general rate equation for electrons in a subband *i* within an *N* level system is where *I* = *I*ee = *I*ie. At low temperature, absorption is near zero: hence, *τ*32 > *τ*21, so *W*21 > *W*32, and *n*3 > *n*2 leading to the existence of population

thus, generating one photon per e−

optical output power.

6 Quantum Cascade Lasers

inversion.

$$
\mathfrak{n}\_3 \mid \mathfrak{r}\_{32} = \mathfrak{n}\_2 \mid \mathfrak{r}\_{21} \tag{5}
$$

hence, *τ*32 > *τ*21, so *W*21 > *W*32, and *n*3 > *n*2 leading to the existence of population inversion.

The scattering rate between two subbands strongly depends upon the overlap of the wave functions and energy spacing between the subbands. In order to decrease *W*32, the overlap of the upper and lower laser levels is reduced. This is often achieved through designing the layer thickness such that the upper laser level is mostly localized in the left part of the well of the 3QWs active region, while the lower laser level wave function is made to mostly reside in the central and right part of the wells, leading to the so-called diagonal transition. A vertical transition is one which the upper laser level is rather localized in the central and right part of the wells, increasing the overlap and, therefore, *W*32, reducing the population inversion but increasing the strength of the radiative transition and hence the gain. On the other hand, in order to increase *W*21, the lower laser level and the ground level wave functions are designed to obtain a good overlap, and the energy spacing between the subbands is designed such that it is equal to the longitudinal optical (LO) phonon energy so that the resonant LO phononelectron scattering can depopulate the lower laser level. For instance, the LO-phonon energy is around 36 meV in GaAs (which is comparable to the room temperature *kBT* value of around 26 meV) and around 91 meV in GaN [9].

Tunable semiconductor lasers can be produced by using multiple resonators or multisection injection devices [10–12]. In fact, tunable QCLs have been demonstrated [7] some time ago. Moreover, a QCL with a heterogeneous cascade containing two substacks previously optimized to emit at 5.2 and 8.0 μm, respectively, was presented by Gmachl et al. [13]. On the other hand, single-mode tunable QC distributed feedback lasers emitting between 4.6 and 4.7μm wavelength have been reported [14]. These lasers were pulsed, continuously tunable singlemode emission and were achieved from 90 to 300 K with a tuning range of 65 nm and a peak output power of approximately 100 mW at room temperature—so the lasers described by Pellandini et al. [12], Gmachl et al. [13] and Köhler et al. [14] were laser sources for the midinfrared region. In order to realize near-infrared QCLs, optical non-linearity in intersubband lasers has been used to design such lasers emitting at 4.76 μm, with third harmonic and second harmonic generation at 1.59 and 2.38 μm, respectively [15].
