Abstract

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

1. INTRODUCTION

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 [3]. 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 [46] 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 [9], with spectral coverage recently extended to the MIR region [1012].

Quantum cascade lasers (QCLs) [13] 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 [16]. 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 [14], 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 [14]. 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 7.5×1016 relative to the carrier optical frequency [18], a value close to those measured for microresonator-based combs [9].

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 [1921]. 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 [14]. 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 [22] 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 [23]. 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 [24]. This permits the retrieval of the Langevin equation for the photon annihilation operator related to the nth QCL-comb mode:

a^˙n=(Gn12τc+iDn)a^nGn2τck,lCklBkla^ka^ma^lκn,k,l,m+1τcV^n,
where
Gn=iγ122πnfrep+iγ12g0,Dn=πδn2fn2πδn,Ckl=γ22γ222πi(lk)frep,Bkl=γ122i(1iγ122πlfrep1iγ122πkfrep)
with κn,k,l,m the spatial superposition integral among the modes involved in the FWM, τc the photon lifetime in the laser cavity, γ22 the scattering rate out of the excited laser state, γ12 the loss of coherence of the laser transition, frep the comb repetition frequency (cavity round trip, without dispersion), g0 the peak gain, and δn the difference between the frequency of the nth mode of the ideal laser cavity fn and the frequency of the nth mode of the laser cavity with dispersion. The coefficients Gn and Dn represent the complex gain coefficient for the mode n and the modal dispersion, respectively. The coefficient Ckl represents the amplitude of the coherent population oscillations. The coefficient Bkl represents the bandwidth of the FWM gain and determines how many modes contribute to the FWM process. V^n are the vacuum Langevin noise operators related to the optical loss processes (waveguide and mirrors) [24,25], characterized by the following statistical properties:
V^n(t)=0,V^n(t)V^n(t)=δ(tt),V^n(t)V^n(t)=0.
The second term in Eq. (1) refers to the FWM and is responsible for the coupling among all the laser modes. The product of the terms Gn, Ckl, and Bkl can be linked to the third-order nonlinear susceptibility χ(3) [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 [26]. The new annihilation operators are given by
b^q=nUqna^n,
where Uqn is the element of a unitary matrix such that U1=U [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,
V^q=nUqnV^n,
will have the same correlation properties [Eqs. (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.

3. EXPERIMENT

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 [14]. 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 frep7.5GHz (multi-longitudinal-mode emission). The comb repetition frequency frep 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 [28]. 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 frep 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 [29]. 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, F6000) 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 T=2.9dB, extinction E=33dB) 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 (BW=010MHz) is employed. A 12-bit vertical resolution, 1 GHz analog bandwidth, 2.5GS/s 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 frep. In order to resolve the laser spectrum, a Vernier ratio Vr=FSR/frep slightly different from 1 is chosen, and a piezoelectric actuator is used to scan the cavity length over one FSR [30]. A schematic representation is depicted in Fig. 2(a).

 figure: Fig. 1.

Fig. 1. Top: Experimental setup used to measure the FNPSD of the laser. The main optical components include the laser (a multistack InGaAs/InAlAs QCL), the optical isolator, the high-finesse optical cavity, and the high-sensitivity MCT detector. The signal is processed by a high-sampling rate oscilloscope. Bottom: Power-versus-current curve of the QCL at fixed temperature. Two operating regimes are observed in this device, a single-mode regime and a comb regime.

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 figure: Fig. 2.

Fig. 2. Schematic of the optical cavity and of the comb spectra with (a) Vr1.0 (comb regime: one mode) and (b) Vr=1.0 (comb regime: all modes). (c) Cavity transmissions acquired in the three conditions: single-mode QCL, QCL in comb regime with only one mode in resonance with the cavity, and QCL in comb regime with all the modes in resonance with the cavity. These acquisitions are obtained by scanning the cavity length. The cavity detuning is the variation of the resonance frequency (FSR) with the length. The comb is composed of many modes, but the most intense ones are six. Essentially only these main modes contribute significantly to the orange transmission peak. On the other hand, only the most intense mode (the one in resonance) contributes to the blue peak. These peaks are used for the calibration of the FA converter (see section 1 of Supplement 1).

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To utilize the cavity as a FA converter, we act on the piezoelectric actuator and on the temperature controller of the laser to set Vr=1.0 and to let the comb offset frequency fo 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 [6] (see section 2 of Supplement 1 for a demonstration). Since Vr=1.0, an accurate value of the optical cavity FSR can be obtained by measuring frep 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 P=25mW 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 (P=15mW) 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 [31], 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 Dδν due to the spontaneous emission, the so-called Schawlow–Townes limit [22].

 figure: Fig. 3.

Fig. 3. (a) FNPSD of the QCL-comb taken in three different conditions. The spectra are compensated for the FA converter cutoff. The technical contributions to the noise are also reported: taking into account the detection noise floor shape, the spectra are reliable up to 2 MHz; the two contributions, one given by the current driver and the other one related to the intensity noise, are negligible. (b) Zoom of the flattening portion of the spectra around 1 MHz, corresponding to the Schawlow–Townes limit. The spectra are related to the two operating conditions of the laser: single-mode with P=15mW (blue) and comb regime (with all the modes in resonance with the cavity) with P=25mW (orange).

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4. DISCUSSION

At this point it is interesting to compare these levels Dδν to those expected for single-mode emissions with the same characteristics, given by the Schawlow–Townes limit [33]

δν=hνPαtotc24πng2αmnsp(1+αe2).
Taking ν=42.2THz as the central frequency, αm=2.2cm1 as mirror losses, αtot=7.2cm1 as total losses (the relatively high waveguide losses are due to the residual cross absorption given by the multistack structure), ng=3.4, nsp=2 as spontaneous emission factors, and αe2=0.0023 as the squared Henry linewidth enhancement factor averaged over the laser spectrum (see section 3 of Supplement 1), we can compute the Schawlow–Townes limit relative to the single-mode emission (P=15mW) and to the comb emission (P=25mW). The two values are δν=383Hz and δν=230Hz, respectively. These values are consistent with those obtained from the spectra δν=πDδν, which are (474±100)Hz for the single-mode emission and (292±79)Hz 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.

5. CONCLUSIONS

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.

Funding

ETH Pioneer Fellowship programme; European Laboratory for Non-linear Spectroscopy (Florence); Italian National Institute of Optics (CNR-INO); Swiss National Science Foundation (SNF200020–152962).

Acknowledgment

We thank Dr. Paolo De Natale for useful scientific discussions and Lauren Clack for editorial help.

 

See Supplement 1 for supporting content.

REFERENCES

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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]  

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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]  

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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]  

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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]  

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References

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  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]

2015 (2)

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]

G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
[Crossref]

2014 (4)

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]

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]

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]

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]

2013 (4)

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]

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]

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]

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]

2012 (5)

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]

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]

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]

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]

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]

2011 (4)

2009 (2)

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]

P. Maddaloni, P. Cancio, and P. De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20, 052001 (2009).
[Crossref]

2000 (1)

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]

1999 (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]

1994 (1)

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]

1992 (1)

R. Hui and A. Mecozzi, “Phase noise of four-wave mixing in semiconductor lasers,” Appl. Phys. Lett. 60, 2454–2456 (1992).
[Crossref]

1990 (1)

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]

1982 (2)

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[Crossref]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[Crossref]

1968 (1)

H. Haken and M. Pauthier, “Nonlinear theory of multimode action in loss modulated lasers,” IEEE J. Quantum Electron. 4, 454–459 (1968).
[Crossref]

1958 (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

Adler, F.

Akikusa, N.

Aspect, A.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics, from the Semi-Classical Approach to Quantized Light (Cambridge University, 2010).

Bartalini, S.

Beck, M.

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]

Benkert, C.

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]

Bergou, J.

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]

Blaser, S.

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]

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]

Borri, S.

Burghoff, D.

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]

Cai, X.

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]

Cancio, P.

Capasso, F.

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]

Cappelli, F.

Cardenas, J.

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]

Chan, C. W. I.

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]

Chembo, Y. K.

Y. K. Chembo, “Quantum correlations, entanglement, and squeezed states of light in Kerr optical frequency combs,” arXiv:1412.5700 (2014).

Cho, A. Y.

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]

Consolino, L.

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]

Cossel, K. C.

Cundiff, S. T.

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]

Davidovich, L.

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]

De Natale, P.

Del’Haye, P.

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]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

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]

Dikmelik, Y.

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]

Domenico, G. D.

Edamura, T.

Eikema, K. S. E.

Elliott, D. S.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[Crossref]

Fabre, C.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics, from the Semi-Classical Approach to Quantized Light (Cambridge University, 2010).

Fain, R.

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]

Faist, J.

G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
[Crossref]

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]

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]

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]

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]

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]

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]

Fermann, M. E.

Francesco, J. D.

Friedli, P.

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]

Gaeta, A. L.

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]

Galli, I.

Gambetta, A.

Gao, J.-R.

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]

Gavartin, E.

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]

Giusfredi, G.

Gorodetsky, M. L.

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]

Griffith, A. G.

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]

Grynberg, G.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics, from the Semi-Classical Approach to Quantized Light (Cambridge University, 2010).

Haken, H.

H. Haken and M. Pauthier, “Nonlinear theory of multimode action in loss modulated lasers,” IEEE J. Quantum Electron. 4, 454–459 (1968).
[Crossref]

Hall, J. L.

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]

Han, N.

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]

Hänsch, T. W.

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]

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]

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]

Hartinger, K.

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]

Hartl, I.

Hayton, D. J.

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]

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[Crossref]

Herr, T.

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]

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]

Hillery, M.

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]

Hinkov, B.

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]

Hofer, J.

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]

Hofstetter, D.

Holzwarth, R.

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]

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]

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]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

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]

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]

Hu, Q.

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]

Hugi, A.

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]

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]

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]

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]

Hui, R.

R. Hui and A. Mecozzi, “Phase noise of four-wave mixing in semiconductor lasers,” Appl. Phys. Lett. 60, 2454–2456 (1992).
[Crossref]

Hundertmark, H.

Hutchinson, A. L.

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]

Inguscio, M.

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]

Javerzac-Galy, C.

C. Lecaplain, C. Javerzac-Galy, E. Lucas, J. D. Jost, and T. J. Kippenberg, “Quantum cascade laser Kerr frequency comb,” arXiv:1506.00626 (2015).

Jones, D. J.

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]

Jost, J. D.

C. Lecaplain, C. Javerzac-Galy, E. Lucas, J. D. Jost, and T. J. Kippenberg, “Quantum cascade laser Kerr frequency comb,” arXiv:1506.00626 (2015).

Kao, T.-Y.

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]

Khurgin, J. B.

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]

Kippenberg, T. J.

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]

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]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

C. Lecaplain, C. Javerzac-Galy, E. Lucas, J. D. Jost, and T. J. Kippenberg, “Quantum cascade laser Kerr frequency comb,” arXiv:1506.00626 (2015).

Kolomenskii, A. A.

Lau, R. K. W.

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]

Lecaplain, C.

C. Lecaplain, C. Javerzac-Galy, E. Lucas, J. D. Jost, and T. J. Kippenberg, “Quantum cascade laser Kerr frequency comb,” arXiv:1506.00626 (2015).

Lee, Y. H. D.

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]

Lipson, M.

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]

Liu, H. C.

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]

Lucas, E.

C. Lecaplain, C. Javerzac-Galy, E. Lucas, J. D. Jost, and T. J. Kippenberg, “Quantum cascade laser Kerr frequency comb,” arXiv:1506.00626 (2015).

Maddaloni, P.

P. Maddaloni, P. Cancio, and P. De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20, 052001 (2009).
[Crossref]

Marangoni, M.

Mazzotti, D.

Mecozzi, A.

R. Hui and A. Mecozzi, “Phase noise of four-wave mixing in semiconductor lasers,” Appl. Phys. Lett. 60, 2454–2456 (1992).
[Crossref]

Mohanty, A.

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]

Montori, A.

Okawachi, Y.

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]

Orszag, M.

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]

Pauthier, M.

H. Haken and M. Pauthier, “Nonlinear theory of multimode action in loss modulated lasers,” IEEE J. Quantum Electron. 4, 454–459 (1968).
[Crossref]

Phare, C. T.

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]

Picqué, N.

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]

Poitras, C. B.

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]

Ranka, J. K.

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]

Reichert, J.

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]

Reno, J. L.

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]

Riedi, S.

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]

Riemensberger, J.

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]

Roy, R.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[Crossref]

Ruehl, A.

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

Schilt, S.

Schliesser, A.

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]

Schuessler, H. A.

Schunemann, P. G.

Scully, M. O.

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]

Sigg, H.

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]

Sirtori, C.

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]

Sivco, D. L.

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]

Smith, S. J.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[Crossref]

Sorokin, E.

Sorokina, I. T.

Strohaber, J.

Taschin, A.

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]

Thomann, P.

Thorpe, M. J.

Tombez, L.

Townes, C. H.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

Tredicucci, A.

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]

Udem, T.

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]

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]

Villares, G.

G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
[Crossref]

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]

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]

Vitiello, M. S.

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]

Vodopyanov, K. L.

Wang, C. Y.

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]

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]

Windeler, R. S.

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]

Yamanishi, M.

Yang, Y.

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]

Ye, J.

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]

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]

Yu, M.

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]

Zhu, F.

Appl. Phys. Lett. (3)

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]

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]

R. Hui and A. Mecozzi, “Phase noise of four-wave mixing in semiconductor lasers,” Appl. Phys. Lett. 60, 2454–2456 (1992).
[Crossref]

IEEE J. Quantum Electron. (2)

H. Haken and M. Pauthier, “Nonlinear theory of multimode action in loss modulated lasers,” IEEE J. Quantum Electron. 4, 454–459 (1968).
[Crossref]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[Crossref]

Meas. Sci. Technol. (1)

P. Maddaloni, P. Cancio, and P. De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20, 052001 (2009).
[Crossref]

Nat. Commun. (3)

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]

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]

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]

Nat. Photonics (3)

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]

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]

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]

Nature (1)

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]

Opt. Express (3)

Opt. Lett. (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]

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]

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]

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]

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]

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]

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]

Phys. Rev. (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

Phys. Rev. A (2)

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]

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[Crossref]

Phys. Rev. Lett. (2)

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]

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]

Science (2)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

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]

Other (3)

C. Lecaplain, C. Javerzac-Galy, E. Lucas, J. D. Jost, and T. J. Kippenberg, “Quantum cascade laser Kerr frequency comb,” arXiv:1506.00626 (2015).

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics, from the Semi-Classical Approach to Quantized Light (Cambridge University, 2010).

Y. K. Chembo, “Quantum correlations, entanglement, and squeezed states of light in Kerr optical frequency combs,” arXiv:1412.5700 (2014).

Supplementary Material (1)

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» Supplement 1: PDF (1116 KB)     

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Figures (3)

Fig. 1.
Fig. 1. Top: Experimental setup used to measure the FNPSD of the laser. The main optical components include the laser (a multistack InGaAs/InAlAs QCL), the optical isolator, the high-finesse optical cavity, and the high-sensitivity MCT detector. The signal is processed by a high-sampling rate oscilloscope. Bottom: Power-versus-current curve of the QCL at fixed temperature. Two operating regimes are observed in this device, a single-mode regime and a comb regime.
Fig. 2.
Fig. 2. Schematic of the optical cavity and of the comb spectra with (a) Vr1.0 (comb regime: one mode) and (b) Vr=1.0 (comb regime: all modes). (c) Cavity transmissions acquired in the three conditions: single-mode QCL, QCL in comb regime with only one mode in resonance with the cavity, and QCL in comb regime with all the modes in resonance with the cavity. These acquisitions are obtained by scanning the cavity length. The cavity detuning is the variation of the resonance frequency (FSR) with the length. The comb is composed of many modes, but the most intense ones are six. Essentially only these main modes contribute significantly to the orange transmission peak. On the other hand, only the most intense mode (the one in resonance) contributes to the blue peak. These peaks are used for the calibration of the FA converter (see section 1 of Supplement 1).
Fig. 3.
Fig. 3. (a) FNPSD of the QCL-comb taken in three different conditions. The spectra are compensated for the FA converter cutoff. The technical contributions to the noise are also reported: taking into account the detection noise floor shape, the spectra are reliable up to 2 MHz; the two contributions, one given by the current driver and the other one related to the intensity noise, are negligible. (b) Zoom of the flattening portion of the spectra around 1 MHz, corresponding to the Schawlow–Townes limit. The spectra are related to the two operating conditions of the laser: single-mode with P=15mW (blue) and comb regime (with all the modes in resonance with the cavity) with P=25mW (orange).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

a^˙n=(Gn12τc+iDn)a^nGn2τck,lCklBkla^ka^ma^lκn,k,l,m+1τcV^n,
Gn=iγ122πnfrep+iγ12g0,Dn=πδn2fn2πδn,Ckl=γ22γ222πi(lk)frep,Bkl=γ122i(1iγ122πlfrep1iγ122πkfrep)
V^n(t)=0,V^n(t)V^n(t)=δ(tt),V^n(t)V^n(t)=0.
b^q=nUqna^n,
V^q=nUqnV^n,
δν=hνPαtotc24πng2αmnsp(1+αe2).

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