# Quantum Cascade Lasers: High Performance Mid-infrared Sources

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## Abstract

This chapter will present the principle of operation and theory of QCLs. The \(\alpha \) -factor, key parameter of semiconductor lasers but not well-known in QCLs, will then be defined, before studying in details a specific QCL structure, from which both Fabry-Perot (FP) and distributed feedback (DFB) devices emitting around 5.6 \(\upmu \mathrm {m}\) were fabricated.

QCLs emitting in the terahertz (THz) have also been realized [7], i.e. in the wavelength range from 30 \(\upmu \)m to 1 mm, although they do not operate at room-temperature yet, because of the appearance of thermal relaxation mechanisms between upper and lower laser levels through optical phonons preventing from population inversion when increasing the temperature. Some solutions exist to operate THz QCLs at room temperature, such as the application of a strong magnetic field above 16 T, in order to suppress the inter-Landau-level non-radiative scattering [8]. Another technique based on difference frequency generation in a mid-IR QCL to obtain room-temperature monolithic THz devices has been proposed [9] and has led to the best output power performances at these wavelengths.

Figure 2.1 presents the QCL performances reported in the literature, over the mid-infrared and THz ranges from 3 to 250 \(\upmu \)m and their operating temperature. The graph shows the existence of a wavelength range from 28 to 50 \(\upmu \)m where no QCLs were realized: it corresponds to the Restrahlen band, where the III–V materials are absorbant due to optical phonon absorption. In the following, we will focus on mid-infrared QCLs, since room-temperature operation is more convenient for experimental work.

This chapter will present the principle of operation and theory of QCLs. The \(\alpha \)-factor, key parameter of semiconductor lasers but not well-known in QCLs, will then be defined, before studying in details a specific QCL structure, from which both Fabry-Perot (FP) and distributed feedback (DFB) devices emitting around 5.6 \(\upmu \)m were fabricated.

## 2.1 Principle of Operation

In interband laser diodes, the laser transition occurs between the conduction band and the valence band of the semiconductor material, and the laser frequency \(\nu \) is determined by the energy gap \(E_g\) between these two bands, with the relation \(E_g \le h \nu \le F_c - F_v\), with \(F_c\) and \(F_v\) the quasi-Fermi levels associated with the conduction and valence bands, respectively. The laser diode wavelengths are therefore limited below 2 \(\upmu \)m, because the energy gaps of the III–V materials are in this range. On the other hand, QCL operation is based on intersubband transitions within the conduction band of the III–V semiconductor, as shown in Fig. 2.2. Therefore, the emission wavelength is no longer limited by the gap of the material but by the energy spacing between the subbands, which is determined by quantum engineering of the active area. The only limitation in wavelength is the thickness of the quantum well where the laser transition takes place, which rules the spacing between the subbands: if the quantum well is too narrow, the upper subband will be too close to the continuum, the electron will no longer be confined and no photon will be emitted. There are therefore no QCLs emitting below 3 \(\upmu \)m at room temperature, and the ones emitting between 3 and 4 \(\upmu \)m are mostly based on newly developed structures containing antimony [10]. On the contrary, if the quantum well is too wide, the subbands will be very close from one another and thermal relaxation will compete with the radiative transitions, hence limiting the operation of THz QCLs to cryogenic temperatures.

As shown in Fig. 2.3, the actual design of a QCL structure is much more complex than previously described. The laser transition indeed occurs in a multi-quantum well active region that is a 3-level laser [11]: the photon is emitted during the transition from an upper level \(\vert 3\rangle \) to a lower level \(\vert 2\rangle \). Then the electron relaxes through an optical phonon in level \(\vert 1\rangle \), from which it will tunnel into a injector region, i.e. a succession of narrow quantum wells called minibands that will lead the electron to the upper level \(\vert 3\rangle \) of the next active region.

## 2.2 Theory

### 2.2.1 Heterostructure

*m*is one of the band taken into account for the calculation, \(u_{m,\mathbf {k}=0}(\mathbf {r})\) is the Bloch function and \(\varphi _m (\mathbf {r})\) is a slowly varying envelope. The Bloch functions are assumed to be similar in all the layers of the heterostructure. Therefore, due to the translation invariance in the plane of the layers:

*A*the area of the laser, \(\mathbf {k_{\Vert }}\mathbf {r_{\Vert }} = k_x x + k_y y\) and \(\chi _m\) an envelope function for the band

*m*.

*k*are given by:

*l*the quantum well width. The energy of the produced photon \(E_{phot} = h \nu \), \(\nu \) being the frequency, corresponds to the energy difference between two consecutive subbands of the conduction band. For instance, for \(k=2\):

### 2.2.2 Spontaneous Emission and Material Gain Calculation

*i*and

*f*of the conduction band. The stimulated emission rate \(W_{i\rightarrow f}^{st}\) can be expressed as a function of the spontaneous emission rate \(W_{i\rightarrow f}^{sp}\) as [14]:

*c*is the light velocity, \(E(\nu )\) the energy density of the wave at frequency \(\nu \) and

*n*the refractive index.

*i*and

*f*, \(\Delta N = N_i - N_f\).

*q*is the elementary electron charge, \(\epsilon _0\) the vacuum permittivity and \(\vert z_{if}\vert \) the dipole matrix element, which is inversely proportional to \(\nu \) [13, 15]. Therefore, the material gain is directly proportional to \(L(\nu )\) and can be written as:

### 2.2.3 QCL Rate Equations

*j*can be expressed as [16]:

*j*. \(\eta \) is the conversion efficiency, \(\tau _{kl}\) corresponds to the carrier lifetime from level

*k*to level

*l*, \(\tau _{sp}\) is the spontaneous emission lifetime, \(\tau _{p}\) is the photon lifetime inside the laser cavity and \(\beta \) the spontaneous emission factor, which represents the fraction of spontaneous emission coupled into the lasing mode. \(G_{0}\) corresponds to the net modal gain over one period normalized by the group velocity \(v_g\), expressed in \(s^{-1}\), and can be defined as [16]:

*g*the gain in \(cm^{-1}\) and

*A*the area of the laser cavity. Let us stress that the rate Eqs. 2.12–2.15 correspond to the single-mode scenario. In case of multimode operation, the photon rate equation for each mode can be obtained by adding to the single-mode photon density a term \(S_m\), corresponding to the photon population for the \(m^{th}\) longitudinal mode oscillating at the frequency \(\omega _m\) [17].

This simplified model is most of the time sufficient and gives accurate results, as shown in Fig. 2.5. However, for some calculations such as small-signal analysis, it is better to consider the full model, since key parameters such as the time for the electrons to pass through the successive active regions and injectors are not taken into account in the simplified model.

### 2.2.4 QCL Modulation Response

*j*the period number, and \(S = S_{st}+\delta S\). Under external perturbation \(I=I_0+\delta i\), using the full set of rate equations, the modulation response of a QCL can be written as [18]:

## 2.3 Linewidth Enhancement Factor

### 2.3.1 Definition

The \(\alpha \)-factor also impacts many important aspects of the semiconductor lasers, such as brightness, modulation properties or filamentation in broad-area semiconductor lasers [24]. Furthermore the LEF significantly influences the nonlinear dynamics of a semiconductor laser subject to optical injection or optical feedback, and nonlinear dynamics can only be observed in lasers for which \(\alpha >0\) [25].

### 2.3.2 Measurements Methods

As seen in the previous paragraph, the \(\alpha \)-factor can be retrieved directly from linewidth measurements. However, this method can be complex to implement, and other measurements techniques have been proposed.

*m*the modulation indices in frequency and amplitude, respectively, the \(2\beta /m\) coefficient reaches a plateau where:

Typically, the LEF values reported for quantum well lasers range from 1 to 3, whereas for more complex structures, such as quantum dot lasers, the \(\alpha \)-factor is higher, between 3 and 10 [33]. A record value as high as \(\alpha =57\) in InAs quantum dot lasers emitting both on ground state and excited state has even been reported [34].

### 2.3.3 \(\alpha \)-Factor of QCLs

Typically for mid-infrared QCLs, the measured sub-threshold \(\alpha \)-factor using this Hakki-Paoli technique varies between −0.6 and 0.3 [35, 36, 37]. This value is low, but definitely non-zero. It is also important to point out, that most of these measurements were performed at cryogenic temperature. When increasing the temperature up to 300 K, thermal agitation of phonons will lead to broader linewidth, and hence to higher linewidth broadening factor values.

Furthermore, the spatial hole burning is very large in QCLs compared to laser diodes [38] and the above-threshold \(\alpha \)-factor is expected to be significantly different from that measured below threshold. There are very few reports of above-threshold linewidth broadening factor measurements at room temperature for a mid-infrared QCL. Using the fit of the L-I curves while controlling the internal laser temperature, Hangauer et al. [39] reported values between 0.167 and 0.483 close to threshold. Moreover, von Staden et al. [40] deduced the \(\alpha \)-factor from the self-mixing interferometers and obtained values between 0.26 and 2.4, strongly increasing with the bias current. One measurement of the FM-AM ratio using optical heterodyning led to \(\alpha \)-factor values of 0.02 ± 0.2 at 243 K [41]. This measurement has the advantage at high frequency to be independent of the thermal effects, but it might lead to some issues when considering structures with complex carrier dynamics, such as quantum dot lasers or QCLs. It would be interesting to apply techniques such as optical injection far from the maximum gain mode to measure the LEF of a QCL, to obtain temperature independent values.

## 2.4 Detailed Study of a QCL Design

In this thesis, we will focus mainly on a specific QCL structure that produced performant lasers, both Fabry-Perot and DFB, emitting around 5.6 \(\upmu \)m.

### 2.4.1 Fabrication of QCL Devices

^{-3}) InP cladding. The upper InP cladding is then grown by metal organic chemical vapor epitaxy. In the case of the DFB QCL, the upper cladding was designed following [43] to enable single-mode emission. A top metal grating was added, with a coupling efficiency of \(\kappa ~\approx \) 4 cm

^{-1}, leading to a \(\kappa L\) is close to unity. Contrary to buried gratings or conventional top gratings with a highly doped dielectric layer between the cladding and the grating, which are based on gain-guiding, this technology is based on index-guiding. The modulation of the refractive index originates from the coupling between the guided modes in the active region and the surface mode, also called plasmon-polariton, which is confined at the interface between metal and upper cladding, two materials with permittivities of opposite signs [14].

The wafer is then processed using double-trench technology, in order to reduce the lateral current spreading in the device, and therefore to reduce the self-heating of the laser [15, 44].

To improve the performances, a high-reflectivity (HR) coating (\(R>95\%\)) on the back facet reduces mirror losses, while the front facet is leaved as cleaved (\(R=0.3\)). Finally, for efficient heat extraction, the QCL is most of the time episide-down mounted with gold-tin soldering on AlN submount. Figure 2.10 shows a schematic and a scanning electron microscopy (SEM) picture of the DFB device under study.

### 2.4.2 QCL Internal Parameters

Furthermore, the repartition of the modes inside the QCL can be simulated using COMSOL, as shown in Fig. 2.11b and c, presenting the simulations of the fondamental mode TM0 and first order mode TM1, respectively. The group refractive index and the confinement factor can be retrieved.

Laser parameters

Parameter | Value (ps) | Parameter | Value |
---|---|---|---|

Carrier lifetime 3-2 \(\tau _{32}\) | 2.27 | Group index \(n_{g}\) | 3.2 |

Carrier lifetime 3-1 \(\tau _{31}\) | 2.30 | Confinement factor \(\Gamma _{opt}\) | 68% |

Carrier lifetime 2-1 \(\tau _{21}\) | 0.37 | Net modal gain \(G_{0}\) | 1.2 \(\times \) 10 |

Carrier escape time \(\tau _{out}\) | 0.54 | Photon lifetime \(\tau _{p}\) | 4.74 ps |

### 2.4.3 Laser Static Properties

The DFB QCLs are 2 mm long and 9 \(\upmu \)m wide, there are epi-side down mounted with a high-reflectivity coating on the back facet. The lasers can be operated both in continuous-wave and pulsed mode. The L-I-V characteristic curves of such a QCL in continuous-wave operation at 10\(^\circ \), 20\(^\circ \), 30\(^\circ \) and 40 \(^\circ \)C are represented in Fig. 2.12a. For instance, at 20 \(^\circ \)C, the laser threshold is at 421 mA (current density of \(J_{th}=2.34\) kA/cm\(^2\)), and 9.22 V and the maximum emitted power is 140.4 mW, but these characteristics may slightly vary from one laser to the other and depending on the current source and detection optics used in the experimental setup. The dip that appears sometimes in the L-I curves, for instance at 581 mA at 10 \(^\circ \)C, is a measurement artifact due to the strong water absorption on the path between laser and detector at this wavelength.

As shown in Fig. 2.12b, the DFB QCLs are perfectly single-mode all along the L-I curve, and the wavelength red-shifts from 1769.5 cm\(^{-1}\) (5.651 \(\upmu \)m) to 1764.4 cm\(^{-1}\) (5.668 \(\upmu \)m) when increasing the bias current. Finally, the far-field of the DFB QCL is drawn in Fig. 2.12c, presenting a relatively round beam. The full width at half maximum of the far-field is 47\(^\circ \) horizontally and 59\(^\circ \) vertically.

Figure 2.12d presents the electroluminescence spectra of the DFB QCL, measured in pulsed mode, with a pulse length of 300 ns and a repetition rate of 100 kHz, using a lock-in amplifier and sensitive mercury-cadmium-telluride (MCT) photodetector at cryogenic temperature. The red curve corresponds to a measurement far below threshold, where the electroluminescence spectrum follows the gain shape. It is centered around 1782 cm\(^{-1}\) (5.61 \(\upmu \)m) and its full width at half maximum (FWHM) is 138 cm\(^{-1}\) (FWHM \(=44~\upmu \)m expressed in wavelength). Furthermore, it is worth noticing that the spectrum is not perfectly symmetrical, which suggests a non-symmetrical gain, and hence a non-zero \(\alpha \)-factor. The blue curve was measured just below threshold. In this case, the electroluminescence spectrum is much narrower, with a FWHM of 15 cm\(^{-1}\) (FWHM \(=5~\upmu \)m in terms of wavelength), showing clear gain saturation.

The optical spectra of the epi-up FP QCL at 15 \(^\circ \)C, with pulses of 300 ns and a repetition rate of 50 kHz, are represented in Fig. 2.12f for several bias currents, clearly showing the broadening of the gain and hence of the FP spectrum when increasing the pump current. In this case, the center frequency is around 1820 cm\(^{-1}\) (5.45 \(\upmu \)m), but it can vary depending on the laser geometry.

### 2.4.4 QCL Gain Measurements

*L*the laser length and \(k = \sqrt{I_{max}/I_{min}}\), where \(I_{max}\) and \(I_{min}\) are two consecutive maximum and minimum of the DFB optical intensity transmitted through the Fabry-Perot laser. The mirror losses are \(\alpha _{M} = 4\) cm\(^{-1}\). The waveguide losses \(\alpha _W\) can be estimated by studying the transverse electrical (TE) transmission of the DFB through the Fabry-Perot. However, it was not possible in our setup to rotate the Fabry-Perot QCL, and the value was taken at \(\alpha _{W} \approx 10\) cm\(^{-1}\), as measured in a previous work for a similar DFB QCL. However, this value varies significantly from one device to the other, especially with the laser length, and must therefore be considered carefully. Finally, the intersubband losses \(\alpha _{ISB}\) could be extracted from a measurement far below threshold, where the gain is negligible. However, no signal was detected at such a low bias current, and these losses could not be extracted for this QCL. The evolution of \(G -\alpha _{ISB} - \alpha _W\) with the Fabry-Perot current density is presented in Fig. 2.14b, when the DFB QCL is operated at 589 mA and 9.8 V.

The last point of Fig. 2.14b was measured just below threshold. However, at threshold we expect \(G - \alpha _{M} -\alpha _{ISB} - \alpha _W = 0\), which is not the case in our measurement. This could be explained by the existence of a shift between the maximum gain peak and the DFB wavelength, leading to gain measurement that does not correspond to the maximum gain equal to the total losses at threshold [46]. Indeed, from the electroluminescence spectrum (Fig. 2.12d), we deduce that the laser hits the gain at about 92% of its maximum and we expect a maximum gain of \(G_{max}=0.92 \times (\alpha _m + \alpha _W + \alpha _{ISB})\), although it can not be verified here.

### 2.4.5 Intensity Noise Measurements

*i*(

*t*) the temporal fluctuations of the emitted signal, and \(\sigma _{i\, laser}^{2}\) the variance of the laser noise, as defined in Fig. 2.15.

This laser noise originates mainly from the beating between stimulated and spontaneous emissions [47]. The existence of spontaneous emission indeed leads to photons with random polarization, direction and phase, that will compete with the coherent light from stimulated emission and generate noise.

*t*the integration time.

In this paragraph, the RIN of a DFB QCL is measured both in continuous-wave and pulsed operation, in order to conclude whether the studied QCL structure can be used for spectroscopic applications.

*B*of the ESA, here 200 Hz. The center frequency \(\nu \) can be taken in the frequency range of the detector, by carefully avoiding the range where the signal is dominated by the 1 /

*f*noise of the detector, in our case between 10 kHz and 1 MHz. The RIN at a given frequency can be expressed as:

The RIN of the laser using the ARMEXEL source at a repetition rate of 100 kHz is measured under several operating conditions. In pulsed mode, the RIN does not decrease exponentially with the bias current as in continuous-wave operation, but oscillates around \(-110\) dB/Hz after a short decrease for low bias currents. Figure 2.17a presents the RIN at different center frequencies within the detector range, showing little dependence of the RIN with the measurement frequency. However, the study of the RIN evolution with the pulse width at a fixed center frequency of 20 kHz shows that a longer pulse duration results in a lower RIN (Fig. 2.17b).

The RIN measurements are repeated with the PICOLAS source at a repetition rate of 100 kHz. The RIN evolution with the bias current has the same tendency as with the ARMEXEL source, and oscillates around the same value of -110 dB/Hz after a short decrease for low bias currents. The study of the RIN at different center frequencies (Fig. 2.17c) shows that this source is optimized at higher frequency, with a RIN decreasing as the measurement frequency increases. However, the RIN remains almost constant with the pulse duration at a center frequency of 20 kHz (Fig. 2.17d).

The measured RIN values are higher than the typical ones for single-mode interband laser diodes (around \(-160\) dB/Hz, see [47]), but are consistent with other measurements realized on QCLs [48]. Typical RIN values acceptable for spectroscopic applications are below \(-150\) dB/Hz. The QCL under study can therefore be used for spectroscopy easily in continuous-wave, but also in pulsed mode by averaging over a longer acquisition time *t*, since the key parameter remains the SNR, which is proportional to \(\sqrt{t}\). This can be a drawback for some applications, but several techniques can be implemented to improve the SNR, such as matched filter, synchronous detection or use of a reference path.

## 2.5 Conclusions

In this chapter, the QCL technology has been studied. Thanks to their intersubband transitions, the QCLs can operate from 3 \(\upmu \)m up to 250 \(\upmu \)m, depending on the active region design. They are compact sources with high output power, and have therefore become favored lasers sources for mid-infrared applications.

The specific QCL design that will be used in the following chapters has been characterized in details. Both Fabry-Perot and DFB QCLs emitting around 5.6 \(\upmu \)m are available, with output power as high as 255 mW in continuous wave operation at 20 \(^\circ \)C.

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