Optical frequency combs based on quantum cascade lasers have recently been demonstrated in the mid- and far-infrared spectral regions, opening the possibility for broadband, compact spectrometers. The successful operation of these systems will depend on understanding the frequency noise of these lasers, whose mode-locking dynamics leads to an almost constant optical power rather than pulse generation. We demonstrate that the four-wave mixing process—responsible for comb formation—effectively correlates the quantum frequency noise of the individual comb modes. The plateau observed in the high-frequency portion of the noise spectrum is attributed to the quantum noise limit. This result proves that four-wave mixing introduces no additional frequency noise, showing that quantum cascade laser combs are well suited for high-resolution spectroscopy applications.
© 2015 Optical Society of America
In recent years, optical frequency combs (OFCs) have become fundamental tools for near-infrared (NIR) spectroscopy and metrology [1,2]. Thanks to their wide spectral coverage, high coherence, and absolute traceability, they are used for high-resolution and precision atomic and molecular spectroscopy in this spectral region . Since the fundamental ro-vibrational transitions of simple molecules fall in the mid-infrared (MIR), it is of particular interest to have OFCs operating in this spectral region. Until now, a well-established approach to satisfy this need consisted of directly transferring NIR OFCs’ emission to the MIR region through nonlinear processes, e.g., difference frequency generation using intense fiber-based NIR OFCs [4–6] or optical parametric oscillators [7,8]. This approach guarantees good spectral coverage and coherence, but requires complex and delicate experimental setups. Comb generation has also been achieved by parametric oscillation in high-Q microresonators , with spectral coverage recently extended to the MIR region [10–12].
Quantum cascade lasers (QCLs)  are current-driven semiconductor lasers based on intersubband transitions in quantum wells, emitting MIR or terahertz (THz) radiation. In devices designed with low group delay dispersion, it has been shown that comb operation can be achieved [14,15] thanks to the four-wave mixing (FWM) process taking place in the gain medium itself . For MIR-operating devices, the upper state lifetime, inherent to the intersubband transition of the active region, is very short (subpicosecond range). This is responsible not only for the broadband nature of the FWM that enhances the mode locking, but also for the tendency to operate with a nearly constant output power. For these reasons, the phase relation between the modes is similar to that of frequency-modulated lasers , as theoretically predicted [17,18], and no pulses are emitted.
Quantum cascade laser frequency combs (QCL-combs) have been initially characterized by measuring the autocorrelation of the intermode beat note at the cavity round-trip frequency (7.5 GHz), performing a so-called beat note spectroscopy . A more sensitive technique is provided by comparing two QCL-combs in a heterodyne beat experiment. Recent experiments on QCLs in a dual-comb spectroscopy setup demonstrated a mode equidistance fractional accuracy of relative to the carrier optical frequency , a value close to those measured for microresonator-based combs .
In this paper we investigate the frequency noise of such combs. The frequency noise of single-mode QCLs has been studied in the MIR as well as in the THz [19–21]. However, a detailed study on the frequency noise of QCL-combs has never been reported. This characterization is essential both for spectroscopy applications as well as for a better understanding of the fundamental properties of these devices with such a unique comb formation mechanism . The generation of the comb of frequencies is interpreted within the framework of supermodes. A high-finesse optical cavity is used as a multimode frequency-to-amplitude (FA) converter to retrieve the QCL-comb intrinsic linewidth. A comparison between the linewidth obtained in the comb regime operation and in the single-mode operation is also given, demonstrating that FWM effectively correlates the quantum noise of the comb modes.
2. FREQUENCY NOISE
What distinguishes a frequency comb from a simple array of perfectly equally spaced single-frequency optical sources is the correlation of the frequency noise. While the heterodyne beat of two independent single-frequency laser sources always yields a linewidth wider than that of the individual lasers, this is not the case if the two single frequencies are extracted from a frequency comb source. These considerations are equally true for technical as well as for quantum noise. The intrinsic linewidth of a laser is given by the Schawlow–Townes formula  and can be interpreted as the ratio of the number of photons emitted in the cavity by spontaneous emission over the total number of photons circulating in the cavity. As compared to a single-frequency device, we observe that the only effect of comb operation is the redistribution of the stimulated photons into equally spaced modes with negligible additional frequency noise. This is in contrast to amplifiers and single-mode lasers, where FWM increases the signal wave noise . For this reason, the intrinsic linewidth of single comb modes is expected to be unchanged and can be expressed by the Schawlow–Townes formula considering the total optical power of all comb modes.
Similar to microresonator-based combs, QCL-combs are generated through FWM. For this reason, according to a semiclassical approach to QCL-combs [17,18], it makes sense to compare the quantities with the quantum formalism developed for microresonator-based combs . This permits the retrieval of the Langevin equation for the photon annihilation operator related to the th QCL-comb mode:24,25], characterized by the following statistical properties: 1) refers to the FWM and is responsible for the coupling among all the laser modes. The product of the terms , , and can be linked to the third-order nonlinear susceptibility [17,18]. Solving Eq. (1) is beyond the scope of this paper. However, we note that in comb operation the average relative phases of the modes are fixed (within fluctuations). Therefore, through a unitary transformation, it is possible to select a new basis for the cavity modes, the supermodes basis, such that one of these supermodes corresponds to that selected by the comb operation . The new annihilation operators are given by 27]. In this way, the equation is reduced to that of a single-mode laser (with only one mode excited). In particular, the Langevin operators for the new modes, 2)] because of the unitary nature of the transformation. The resulting frequency noise is expected to be the same as that of a single-mode laser.
The laser used for these experiments is a QCL-comb based on an InGaAs/InAlAs broadband design with multiple active regions (multistack), previously reported in . It operates in continuous wave at room temperature, emitting several milliwatts of power at 7.10 μm on a single transverse mode. The device length is 6 mm, corresponding to (multi-longitudinal-mode emission). The comb repetition frequency can be measured as a radio-frequency (RF) modulation arising directly on the laser-biasing current and extracted from the device through a bias-tee . Therefore the laser is always driven through a bias-tee in order to observe any RF modulation on the laser current. Two main operation regimes are observed in this device. Just above the laser threshold, the device emits single-mode radiation and we do not observe any RF beat note. Above a second current threshold, a comb regime is observed for a significant part of the device working range (see Fig. 1, bottom), where a narrow RF beat note on the laser current corresponding to is observed. In the comb regime, the laser emits a single coherent comb of frequencies, and, as opposed to Kerr combs, the formation of a set of different subcombs is not observed . The presence of these two operating regimes allows the study of the frequency noise in both regimes using the same device. In order to investigate the frequency noise power spectral density (FNPSD), a high-finesse optical cavity (Fabry–Perot, ) is used to resolve the laser spectrum and to detect the frequency fluctuations of the laser, acting as a FA converter (Fig. 1, top). A pair of high-reflectivity mirrors (CRD Optics, 99.96% of declared reflectivity) coated with dielectric layers is used to build a self-made cavity suitable for our experiments. In the setup, an optical isolator (transmission , extinction ) is employed to avoid the instabilities induced on the laser by the backreflection from the input mirror of the cavity. To collect the signal transmitted by the cavity, a high-sensitivity nitrogen-cooled HgCdTe (MCT) detector () is employed. A 12-bit vertical resolution, 1 GHz analog bandwidth, sampling rate oscilloscope is used to acquire the signal and to compute its Fourier transform. The distance between the two mirrors is chosen in order to set the free spectral range (FSR) of the cavity close to . In order to resolve the laser spectrum, a Vernier ratio slightly different from 1 is chosen, and a piezoelectric actuator is used to scan the cavity length over one FSR . A schematic representation is depicted in Fig. 2(a).
To utilize the cavity as a FA converter, we act on the piezoelectric actuator and on the temperature controller of the laser to set and to let the comb offset frequency be equal to that of the optical cavity. In this way, the comb modes and the optical cavity resonances are perfectly matched [see Fig. 2(b) for a schematic and Fig. 2(c) for the measured cavity transmission profile]. As a consequence, in these conditions and only in these conditions of temperature and driving current of the laser, all the comb modes are transmitted by the cavity. The cavity can thus be used as a multimode FA converter to collect the frequency fluctuations of all the modes at the same time  (see section 2 of Supplement 1 for a demonstration). Since , an accurate value of the optical cavity FSR can be obtained by measuring as described above. Such an accurate FSR value is needed for the calibration of the FA converter (see section 1 of Supplement 1). The laser emits a power of when the comb modes are exactly matched to the cavity resonances. Thanks to the high finesse of the cavity, it is also possible to collect the frequency fluctuations of an individual comb mode by slightly varying the FSR. A spectrum retrieved with the laser in single-mode operating conditions () was also acquired. The FNPSDs measured on the single-mode and comb regimes are reported in Fig. 3(a). The spectra are compensated for the FA converter cutoff (see section 1 of Supplement 1). We observe that the frequency noise on the comb regime is below the frequency noise on the single-mode regime. Moreover, the frequency noise of an individual comb mode is also equivalent to that acquired on all comb modes together. By integrating the FNPSD using Elliott’s formula , the full width at half-maximum (FWHM) of a laser mode can be retrieved. In our case, we obtain a FWHM of about 600 kHz in a 1 s timescale, which is consistent with the linewidth shown by distributed-feedback (DFB) QCLs [19,20,32]. Moreover, the contributions of the current driver noise and the laser intensity noise as well as the detection noise floor are reported. Taking into account the detection noise floor shape, the spectra are reliable up to 2 MHz. Around 1 MHz, a flattening can be observed. Figure 3(b) shows a portion of the same FNPSDs (from 100 kHz to 3 MHz). This flattening, characteristic of white frequency noise, corresponds to the intrinsic quantum noise level due to the spontaneous emission, the so-called Schawlow–Townes limit .
At this point it is interesting to compare these levels to those expected for single-mode emissions with the same characteristics, given by the Schawlow–Townes limit Supplement 1), we can compute the Schawlow–Townes limit relative to the single-mode emission () and to the comb emission (). The two values are and , respectively. These values are consistent with those obtained from the spectra , which are for the single-mode emission and for the comb emission [see Fig. 3(b)]. The fact that the measured Schawlow–Townes limit for the comb emission corresponds to the one computed using Eq. (5) justifies the theoretical framework introduced in Section 1.
More importantly, the measurement of the FNPSD in the comb regime shows that the quantum fluctuations of the different modes are correlated. In fact, we observe that the FNPSD—in particular the portion limited by the quantum noise—is identical when measured with one comb mode and with all comb modes simultaneously. This quantum limit—a value that is given by the Schawlow–Townes expression—would be at least a factor of 6 larger than the one shown in Fig. 3(b), assuming that the quantum fluctuations of each comb mode are uncorrelated. This factor is outside the uncertainty of the measurement.
With this work, we have demonstrated that in QCL-combs, the FWM process—at the origin of the comb operation—correlates the frequency fluctuations between the modes until the quantum limit. As a result, the linewidth is shown to be limited by the Schawlow–Townes formula, as it is for single-mode lasers of the same total power. More importantly, QCL-combs do not suffer from additional frequency noise and are indeed suitable for high-resolution spectroscopy applications. As a consequence, instruments using the spectral multiplexing of dual-combs or multi-heterodyne spectrometers hold an inherent noise advantage compared to similar systems using arrays of single-mode lasers. Finally, the same technique used to retrieve the FNPSD could be used to implement an active stabilization, for locking these combs to high-finesse ultra-stable optical cavities.
ETH Pioneer Fellowship programme; European Laboratory for Non-linear Spectroscopy (Florence); Italian National Institute of Optics (CNR-INO); Swiss National Science Foundation (SNF200020–152962).
We thank Dr. Paolo De Natale for useful scientific discussions and Lauren Clack for editorial help.
See Supplement 1 for supporting content.
1. T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, “Absolute optical frequency measurement of the cesium D1 line with a mode-locked laser,” Phys. Rev. Lett. 82, 3568–3571 (1999). [CrossRef]
2. S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hänsch, “Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb,” Phys. Rev. Lett. 84, 5102–5105 (2000). [CrossRef]
3. P. Maddaloni, P. Cancio, and P. De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20, 052001 (2009). [CrossRef]
4. A. Ruehl, A. Gambetta, I. Hartl, M. E. Fermann, K. S. E. Eikema, and M. Marangoni, “Widely-tunable mid-infrared frequency comb source based on difference frequency generation,” Opt. Lett. 37, 2232–2234 (2012). [CrossRef]
5. F. Zhu, H. Hundertmark, A. A. Kolomenskii, J. Strohaber, R. Holzwarth, and H. A. Schuessler, “High-power mid-infrared frequency comb source based on a femtosecond Er:fiber oscillator,” Opt. Lett. 38, 2360–2362 (2013). [CrossRef]
6. I. Galli, F. Cappelli, P. Cancio, G. Giusfredi, D. Mazzotti, S. Bartalini, and P. De Natale, “High-coherence mid-infrared frequency comb,” Opt. Express 21, 28877–28885 (2013). [CrossRef]
7. F. Adler, K. C. Cossel, M. J. Thorpe, I. Hartl, M. E. Fermann, and J. Ye, “Phase-stabilized, 1.5 W frequency comb at 2.8–4.8 μm,” Opt. Lett. 34, 1330–1332 (2009). [CrossRef]
8. K. L. Vodopyanov, E. Sorokin, I. T. Sorokina, and P. G. Schunemann, “Mid-IR frequency comb source spanning 4.4–5.4 μm based on subharmonic GaAs optical parametric oscillator,” Opt. Lett. 36, 2275–2277 (2011). [CrossRef]
9. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011). [CrossRef]
10. C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5 μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013). [CrossRef]
11. A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015). [CrossRef]
12. C. Lecaplain, C. Javerzac-Galy, E. Lucas, J. D. Jost, and T. J. Kippenberg, “Quantum cascade laser Kerr frequency comb,” arXiv:1506.00626 (2015).
13. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994). [CrossRef]
14. A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012). [CrossRef]
15. D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014). [CrossRef]
16. P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102, 222104 (2013). [CrossRef]
17. J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104, 081118 (2014). [CrossRef]
18. G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015). [CrossRef]
19. S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19, 17996–18003 (2011). [CrossRef]
20. L. Tombez, J. D. Francesco, S. Schilt, G. D. Domenico, J. Faist, P. Thomann, and D. Hofstetter, “Frequency noise of free-running 4.6 μm distributed feedback quantum cascade lasers near room temperature,” Opt. Lett. 36, 3109–3111 (2011). [CrossRef]
21. M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics 6, 525–528 (2012). [CrossRef]
22. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958). [CrossRef]
23. R. Hui and A. Mecozzi, “Phase noise of four-wave mixing in semiconductor lasers,” Appl. Phys. Lett. 60, 2454–2456 (1992). [CrossRef]
24. Y. K. Chembo, “Quantum correlations, entanglement, and squeezed states of light in Kerr optical frequency combs,” arXiv:1412.5700 (2014).
25. C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, “Role of pumping statistics in laser dynamics: quantum Langevin approach,” Phys. Rev. A 41, 2756–2765 (1990). [CrossRef]
26. H. Haken and M. Pauthier, “Nonlinear theory of multimode action in loss modulated lasers,” IEEE J. Quantum Electron. 4, 454–459 (1968). [CrossRef]
27. G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics, from the Semi-Classical Approach to Quantized Light (Cambridge University, 2010).
28. G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014). [CrossRef]
29. T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]
30. I. Galli, S. Bartalini, P. Cancio, F. Cappelli, G. Giusfredi, D. Mazzotti, N. Akikusa, M. Yamanishi, and P. De Natale, “Mid-infrared frequency comb for broadband high precision and sensitivity molecular spectroscopy,” Opt. Lett. 39, 5050–5053 (2014). [CrossRef]
31. D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982). [CrossRef]
32. F. Cappelli, I. Galli, S. Borri, G. Giusfredi, P. Cancio, D. Mazzotti, A. Montori, N. Akikusa, M. Yamanishi, S. Bartalini, and P. De Natale, “Sub-kilohertz linewidth room-temperature mid-IR quantum cascade laser using a molecular sub-Doppler reference,” Opt. Lett. 37, 4811–4813 (2012). [CrossRef]
33. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]