## Abstract

The frequency-noise power spectral density of a room-temperature distributed-feedback quantum cascade laser emitting at *λ* = 4.36 *μ*m has been measured. An intrinsic linewidth value of 260 Hz is retrieved, in reasonable agreement with theoretical calculations. A noise reduction of about a factor 200 in most of the frequency interval is also found, with respect to a cryogenic laser at the same wavelength. A quantitative treatment shows that it can be explained by a temperature-dependent mechanism governing the transport processes in resonant tunnelling devices. This confirms the predominant effect of the heterostructure in determining shape and magnitude of the frequency noise spectrum in QCLs.

©2011 Optical Society of America

## 1. Introduction

The interest of the scientific community in frequency-noise properties of quantum-cascade lasers (QCLs) is growing in parallel with increasing demand of such sources for high-resolution spectroscopy and frequency metrology.

Cryogenically-operated QCLs have demonstrated a very low intrinsic linewidth [1], related to the white-noise component of their frequency noise spectrum. This peculiar feature, predicted since the very first demonstration of QCLs [2], relies on the very small Henry’s linewidth enhancement factor and on the huge increase of the number of photons coupled to the lasing mode occurring just above threshold. This latter effect, that is already taken into account by the Schawlow-Townes formula [3], has been deeply investigated from both the theoretical and experimental point of view [4] and marks a big difference with respect to conventional bipolar lasers. However, a small intrinsic linewidth, for its own sake, does not necessarily mean that the QCL emission can be easily narrowed: when its frequency noise spectrum is also well behaved at lower frequencies, then locking loops with only moderate gains and bandwidths are needed to achieve narrow linewidths and low energy content in the wings of the emission spectrum.

As is well known, the most complete characterization of the spectral purity of a laser is the measurement of its frequency-noise power spectral density (FNPSD), since its allows the emission spectrum to be derived over any accessible timescale [5]. Besides the interest related to the implementation of ultra-narrow mid-infrared sources, the study of FNPSD in QCLs can help understanding the physics of the transport processes inside the laser heterostructure. In particular, the operating temperature of the device was already predicted [6] to play a key role for the frequency noise of the laser, since several mechanisms contributing to noise (such as carrier transport processes in resonant tunnelling heterostructures) are expected to depend on frequency, as well as on the specific heat of the QCL. This latter quantity, in particular, is expected to affect FNPSD, as it eventually sets a thermal cut-off for the dissipative current-to-frequency-conversion mechanisms occurring in the laser, thus affecting the current-related portion of the noise spectrum. From this point of view, the comparative measurements of the FNPSD of two QCLs operating at similar wavelengths but at two different temperatures (80 K and 290 K) offers a unique opportunity for testing the hypotheses outlined above.

In the present work, following the methods described in Ref. [1], we measure the FNPSD of a room-temperature (RT) QCL working at 4.36 *μ*m and we compare it with the same quantity already measured for a cryogenic QCL at 4.33 *μ*m. The much lower overall FNPSD, shown by the RT device, is discussed, as well as the noise contribution from the current driver. For frequencies higher than 30 MHz a white-noise region is observed, corresponding to an intrinsic linewidth of about 260 Hz. The comparison with the expected value is discussed in depth, evidencing a good agreement with the theoretical model. The dependence on temperature of both the intrinsic linewidth and the low-frequency 1/*f*–type FNPSD is also explained. In the last part of the paper we briefly discuss some perspectives concerning narrowing of QCL linewidth, showing that the reduced FNPSD enables to achieve sub-kHz linewidth with standard frequency-locking techniques.

## 2. Experimental setup

In the experiment, the frequency noise of the QCL is retrieved from a measurement of the intensity fluctuations of the laser beam after interaction with a frequency-to-amplitude converter. Following a well established method [7, 8], the adopted discriminator is the side of the direct-absorption profile of a molecular transition. The spectrum of the transmitted intensity, when the laser frequency *ν*
_{0} is stabilized at the half-height position, reproduces the spectrum of the laser frequency fluctuations, scaled by the slope of the absorption profile. Fig. 1 shows the concept scheme of the experimental set-up.

The molecular ro-vibrational transition used in this case is the (00^{0}1–00^{0}0) P(56) line of CO_{2}, at a constant pressure of 1 mbar. At this pressure, the conversion coefficient of 22 mV/MHz is maximum, providing the highest sensitivity. Considering the 30 nV/
$\sqrt{\text{Hz}}$ output voltage noise of the 200-MHz-bandwidth HgCdTe photovoltaic detector used, an ultimate sensitivity of about 2 Hz^{2}/Hz for the FNPSD measurement is obtained. The transfer function of the discriminator is calculated for the experimental absorption profile according to the method described in [1].

The source is a distributed-feedback (DFB) QCL provided by Hamamatsu Photonics, with single-mode continuous-wave operation at temperatures in the range between 0 and 20 °C. The compact casing of the QCL also includes a thermo-electric cooler for efficient temperature stabilization. The beam, collimated by an aspherical ZnSe lens, is sent to a 10-cm-long cell for direct-absorption spectroscopy of CO_{2} gas. The absorption signal is acquired by a HgCdTe detector and processed by a real-time FFT spectrum analyzer. The chosen operating QCL temperature during the measurement is *T* = 15 °C. At this temperature, the threshold current is *I _{th}* = 674 mA, while the molecular resonance is found at an operating current

*I*= 776 mA, giving a ratio

_{o}*I*/

_{o}*I*= 1.15. In these conditions, the laser output power is about 20 mW.

_{th}The device used in the present work is a strain-compensated In_{0.72}Ga_{0.28}As/In_{0.31}Al_{0.69}As DFB QCL [9] with 25 cascade stages, 12 *μ*m bare ridge, 3 mm length, and an injector doping density of ∼ 2 × 10^{11} cm^{−2}. The active region has a bound-to-bound quantum design, based on single phonon-continuum (SPC) depopulation [10, 11]. The layered structure was grown by metal organic vapour-phase epitaxy and the laser chip was mounted epi-side-down on a heat sink.

## 3. Measurements

We measured the FNPSD of the RT QCL with two different current drivers: in Fig. 2a the QCL is powered by the same home-made current driver used in our previous work [1], which is based on the well-known Hall–Libbrecht design [12], while in Fig. 2b a commercial current driver (Wavelength Electronics, model QCL1000) is used. Both plots also report the noise contribution from the corresponding current driver. Fig. 2a shows, in addition, the FNPSD measured in our previous work for the cryogenic DFB QCL. The first remarkable result is that the FNPSD of the RT QCL is considerably below that of the cryogenic QCL in most of the investigated spectral range. For frequencies lower than 10 kHz, where both trends follow a 1/*f* behavior, the ratio between the two noise spectra is constant at about 200. In the 10–200 kHz range, the noise contribution from the current driver is dominant and hinders precise measurement of the QCL frequency noise. This was not observed with the cryogenic QCL, due to its larger FNPSD. Between 1 MHz and 10 MHz the RT QCL shows a noise feature not originating from the current noise and resulting from an envelope of adjacent spikes. We think that this technical noise could arise from a non-optimized electrical grounding of the RT QCL. Finally, above 10 MHz, both the current noise of the home-made driver and the technical noise are negligible, allowing the pure QCL frequency noise to emerge. Despite the spurious contributions described above, we can state that all the FNPSD of the QCL (before the white-noise flattening) is consistent with an 1/*f* behavior.

The situation is slightly different when the commercial current driver is used (see Fig. 2b). Here, the larger contribution from the current noise is even more dominant below 10 MHz, and prevents any useful interpretation. However, above 10 MHz, the commercial driver current noise drastically drops down, allowing a clean measurement of the QCL FNPSD, in agreement with what is measured with the home-made driver. A zoom of this last portion of both spectra is shown in Fig.3: the flattening of the FNPSD down to a white-noise level (*N _{w}*) is evident, and yields an intrinsic linewidth

*δν*=

_{exp}*πN*= 260±90 Hz, about four times smaller than that measured for the cryogenic QCL at the same value of

_{w}*I*/

_{o}*I*(1.15). The uncertainty is calculated by taking into account both the scattering of the data in a single trace and the slope calibration error.

_{th}## 4. Discussion

The experimental value of the intrinsic linewidth can be compared with theory by evaluating the Schawlow-Townes-Henry formula. Alternatively, we prefer to use the approach adopted in our previous work [1], inspired by the theoretical treatment of Ref. [4]. The intrinsic linewidth, referring to a 3-level model, can be written as:

The parameters involved in the formula above are provided and discussed below, and we refer to Ref. [4] and equations therein (in particular Equations A6, A11 and A14) for their numerical estimation. For the sake of brevity, we will also refer to the RT QCL measured in the present work as QCL_{1}, and to the cryogenic QCL used in our previous work [1] as QCL_{2}. Due to the CW-operation, all the parameters must be calculated at active region temperatures. At threshold, *I _{th}* = 674 mA for QCL

_{1}, the active region temperature is determined to be the temperature giving, in pulsed operation, the same threshold current as in CW-operation, namely ∼ 340 K, about 50 K higher than the heat-sink temperature

*T*∼290 K (see Ref. [13]). For QCL

_{1}, it results an active region temperature

*T*

_{1}∼350 K at the bias conditions

*I*

_{0}= 776 mA and

*V*= 10 V (with a thermal resistance

_{bias}*R*

_{therm}_{1}∼ 7.42 K/W). With a different manner, the active region temperature of QCL

_{2}(at its experimental bias conditions

*I*

_{0}= 219 mA and

*V*= 10 V) is

_{bias}*T*

_{2}∼ 182 K, about 97 K higher than the heat-sink temperature

*T*= 85 K (see discussion for the higher thermal resistance

*R*

_{therm}_{2}∼ 44.5 K/W of QCL

_{2}, below Eq. 3).

Focusing on QCL_{1} at the active region temperature *T*
_{1}, the non-radiative relaxation times due to LO-phonons and temperature-insensitive elastic (alloy-disorder) scattering processes [14, 15] are *τ _{t}* ≈

*τ*

_{3}∼ 1 ps,

*τ*

_{31}∼ 1.7 ps and

*τ*

_{21}∼ 0.15 ps, while the radiative lifetime of the upper laser level, due to spontaneous emission,

*τ*∼ 10 ns, is four orders of magnitude longer than

_{r}*τ*. Similarly, the key parameter

_{t}*β*=

_{eff}*βτ*/

_{t}*τ*∼ 5 × 10

_{r}^{−9}, called “effective coupling efficiency”, is four orders of magnitude smaller than the spontaneous emission coupling efficiency,

*β*∼ 5 × 10

^{−5}. With an injection efficiency

*η*∼ 0.7, a photon decay rate

*γ*∼ 1 × 10

^{11}s

^{−1}and

*α*∼ 0, an intrinsic linewidth

_{e}*δν*∼ 340 Hz at

*I*

_{0}/

*I*= 1.15 and a threshold current

_{th}*I*∼ 430 mA are obtained, the latter estimated by equations (11) and (12) in Ref. [4], assuming an energy separation

_{th}*E*

_{21}∼ 130 meV (value from design [9]). By slightly adjusting the parameters related to optical processes

*γ*∼ 1.2 × 10

^{11}s

^{−1}, and

*β*∼ 3 × 10

_{eff}^{−9}, while leaving unchanged the relaxation times and the injection efficiency

*η*, we obtain

*δν*∼ 245 Hz at

*I*

_{0}/

*I*= 1.15 and

_{th}*I*∼ 634 mA. The excellent agreement with the experimental results of the two independent quantities (

_{th}*δν*∼ 260 Hz and

*I*∼ 674 mA) justifies the use of these latter parameter values.

_{th}A second interesting finding is obtained from the comparison between the measurements on RT and cryogenic QCLs: the intrinsic linewidth is larger at lower temperatures. A definitive test would be measuring the linewidth of QCL_{1} at different heat-sink temperatures (for example 300 K and 80 K). Unfortunately, we have not been able to perform this experiment yet, because the QCL case does not fit our cryostat. Nevertheless, the theoretical model just used for predicting the values of the intrinsic linewidth can also be applied for predicting its dependence on temperature. In order to do this, we figure out the expected temperature-dependence of the intrinsic linewidth of QCL_{1} in which, due to the temperature change, the pitch of the DFB grating is virtually tuned to match the lasing wavelength with peak-gain wavelength, thus keeping *α _{e}* ∼ 0. The ratio of the relaxation times at the two different temperatures is estimated to be

*τ*

_{3}(300 K)/

*τ*

_{3}(80 K)≃0.6 (a relatively small change even for such a large temperature change, because of the presence of alloy disorder scatterings). The coupling efficiency

*β*is inversely proportional to the spectral width

*w*of spontaneous emission, since this is the only temperature-dependent term involved in

_{s}*β*(see equation (A14) of Ref. [4]). From the experimental result of the spectral width of the present device we obtain the ratio of the coupling efficiencies

*β*(300 K)/

*β*(80 K)=

*w*(80 K)/

_{s}*w*(300 K)≃ 0.5. On the other hand, the spontaneous emission lifetime

_{s}*τ*and the photon decay rate

_{r}*γ*are independent of temperature variations, as well as the small correction factor

*ɛ*≃ 0.2. Consequently, for

*I*

_{0}/

*I*≃ 1.15, the ratio of the intrinsic linewidths (∝

_{th}*β*(

*τ*

_{3}/

*τ*)

_{r}*γ*) is predicted to be

*δν*(300 K)/

*δν*(80 K)≃ 0.3. This can be considered a first theoretical confirmation of the dependence of the intrinsic linewidth on the operating temperature. Finally, we want to remark that the Schawlow-Townes-Henry formula [3] does not allow to evaluate the temperature dependence of the intrinsic linewidth.

Besides the comments on the intrinsic linewidth, the comparison between the low-frequency portion of QCL_{1} and QCL_{2} FNPSDs deserves further comments. The most relevant fact needing an explanation is the drastic reduction (*δ f*
_{2}/*δ f*
_{1})^{2} ∼ 200 of the 1/*f* –type frequency noise for the RT QCL (QCL_{1}). In Ref. [6], some experimental evidences were given to support the hypothesis that the flicker frequency noise originates from temperature fluctuations induced by an internal tunnelling-current flicker noise *δ I*: *δ f* ∝ *δ T* ∼ *R _{therm}δ P* ∝

*R*. As a consequence, the frequency noise reduction is given by:

_{therm}δ I*f*-type frequency noise of QCL

_{1}and QCL

_{2}. This is further validated by the fact that all other key parameters involved in the comparison, such as injection barrier thickness, number of periods, thickness of one period and interface density are only sligthly different between the two devices (respectively 3.5 nm, 25, 51.7 nm and 425.53

*μ*m

^{−1}for QCL

_{1}[9] and 4.6 nm, 22, 57.7 nm and ∼ 390

*μ*m

^{−1}for double-phonon design [16] QCL

_{2}with a ∼ 11

*μ*m bare ridge waveguide, a shorter cavity length of 1.5 mm, a comparable injector doping density of ∼ 3 × 10

^{11}cm

^{−2}and epi-side-up mounting; see Refs. [17, 18] for the definition of the interface density). On the contrary, by considering the temperature-dependent cross-plane thermal resistivity [19], and scaling for the different device length, the mounting geometry (epi-side up or down) and the active region temperature, the thermal resistance

*R*

_{therm}_{2}of QCL

_{2}is convincingly inferred to be about 6 times higher than that of QCL

_{1}:

*R*

_{therm}_{2}∼ 6

*R*

_{therm}_{1}∼ 44.5 K/W. This higher thermal resistance well explains the observed thermal cut-off at 200 kHz for QCL

_{2}, as shown in Fig. 2(a) and also in Ref. [6], though the thermal cut-off for QCL

_{1}, expected at around 1 MHz, is unfortunately masked by spurious noise. However the scaling of thermal resistance can only partially explain the huge reduction of the flicker FNPSD of Eq. 3 (a factor 36 vs. 200). An additional effect involving the internal current squared fluctuations

*δ*

*I*

^{2}must be assumed for explaining the remaining factor 200/36 ≃ 5.5. Switching from current to current-density squared fluctuations

*δ*

*J*

^{2}leads to an expected reduction of (

*δ*

*J*

_{2}/

*δ*

*J*

_{1})

^{2}≃ 22, since the active region dimensions only differ for a factor two in length. This behavior could eventually involve the alternate capture and emission of electrons at individual quantum sites (defect sites in case of homogeneous semiconductors) which is known to generate discrete switching in the device resistance and, consequently, current fluctuations. A single occurrence of this effect is referred to as a random telegraph signal (RTS), and the summation of many RTSs in structured devices like QCLs may result in 1/

*f*noise. The emission and trapping rates are expected to be proportional to the exponential form exp(−

*E*/

_{a}*k*

_{B}*T*), which means that the rate drastically increases with temperature. Here, the experimental results suggest the following hypothesis: at higher rates the contributions of individual RTSs may cancel out more effectively, resulting in a lower 1/f noise. Basing on these considerations we assume, for the squared current-density flicker noise, an inverse exponential form

*δ*

*J*

^{2}∝ 1/exp(−2

*E*/

_{a}*k*

_{B}*T*). Consequently, the factor 22 in the ratio of squared current-density fluctuations (

*δ*

*J*

_{2}/

*δ*

*J*

_{1})

^{2}= exp(2

*E*/

_{a}*k*

_{B}*T*

_{2})/exp(2

*E*/

_{a}*k*

_{B}*T*

_{1}) leads to an empirical estimation of the activation energy

*E*∼ 50 meV, which is comparable to thermal energies in the considered temperature range. A deeper investigation is required in order to give a more reliable and quantitative treatment. Nevertheless, the discussion presented above adds valuable elements supporting the correlation between internal current fluctuations and flicker fluctuations in the electron populations of the QCL quantum structure.

_{a}Finally, we want to discuss some perspectives concerning linewidth reduction of our RT QCL, based on the experimental data shown above. First, we must conclude that a further improvement of the laser driver current noise is required in order to fully exploit the drastic reduction of the QCL frequency noise. The noise level, currently in the range of 1–2 nA/ $\sqrt{\text{Hz}}$, needs to be lowered down of at least one order of magnitude. Alternatively, a low-pass filtering on the current output should be implemented, at the expense of the modulation bandwidth of the driver. In this frame, it is possible to implement a frequency-locking loop acting on the driving current for a reduction of the laser linewidth. Following Ref. [5], from the experimental FNPSD of the RT QCL we calculate a free-running linewidth of about 400 kHz (FWHM) over an observation time of 10 ms. This is not a suitable linewidth for demanding spectroscopic experiments in the mid-infrared range. As an example, the present broad free-running linewidth of our QCL limits its application to the case of ultra-high-sensitivity Saturated-Absorption Cavity-Ring-down spectroscopy (see Ref. [20]), since it requires laser radiation coupling to a high-finesse cavity mode with a few kHz width. Indeed, mid-IR QCLs can be combined with well-established techniques [21] to efficiently narrow their linewidth even below the intrinsic levels, as demonstrated in a pioneering work by Taubman et al. [22]. Following the recent work by Di Domenico et al. [23], we can conclude that with moderate bandwidth (less than 100 kHz) and gain (about 50 dB at 10 Hz) a narrowing of the QCL source down to the kHz level can be easily achieved. Comparable narrowing of the cryogenic QCL would clearly require a larger servo loop bandwidth, given its measured FNPSD.

## 5. Conclusions

In this work the frequency-noise power spectral density of a room-temperature QCL at 4.3 *μ*m is completely characterized. An intrinsic linewidth of 260 Hz, about four times smaller than in a cryogenic QCL, is measured, in reasonable agreement with the theoretical value. The flicker 1/*f* –type frequency noise, occurring at lower frequencies, is also drastically reduced (about a factor 200) with respect to the case of a cryogenically-operated QCL at the same wavelength. A physical interpretation of the measured behaviour is proposed, based on the hypothesis that the flicker frequency noise originates from temperature fluctuations induced by an internal tunnelling-current flicker noise. The temperature dependence of both the thermal resistance and the current flicker noise well explains the experimental results, assigning a predominant effect of the heterostructure in determining the shape and amplitude of the FNPSD curve. In conclusion, if an important element for the success of QCLs has undoubtedly been the tailoring of their emitted frequency, we believe that tailoring of QCLs spectral features is now possible and it can be the key for their next most demanding applications.

## Acknowledgments

We gratefully acknowledge Dr. Pablo Cancio Pastor for his critical reading of the manuscript and M. Giuntini, A. Montori and M. De Pas from LENS for their help on driving electronics. This work was partially supported by Ente Cassa di Risparmio di Firenze and by Regione Toscana through the projects CTOTUS and SIMPAS, in the framework of POR–CReO FESR 2007–2013.

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