1. INTRODUCTION

Ultrashort physics in the mid-infrared is an underdeveloped field with exciting prospects [1]. High-peak-power sources at these wavelengths find application in nonlinear spectroscopy [2,3], supercontinuum generation [4], and standoff detection [5,6], for example. For more energetic mid-IR pulses, favorable wavelength scaling of electron rescattering processes make these interesting sources for soft x-ray generation and related applications [7–10]. A number of approaches, chiefly based on difference-frequency generation [3,11–14], and cascaded optical parametric amplifiers and oscillators [15–20] have been used to generate pulses in the mid-infrared, with microjoule sub two optical cycle pulses having been demonstrated [21]. The common element to these solutions is that they are tabletop scale; to be able to directly generate mid-infrared pulses in a monolithic platform could significantly broaden the applicability of such systems, facilitating the study of new physics.

As proven stable [22,23], compact, monolithic comb sources [24,25], quantum cascade lasers (QCLs) would make for an interesting system to produce pulses in the mid-infrared and terahertz. However, the very mechanism responsible for the phase locking, the short sub-picosecond upper state lifetime [26] which drives the broadband four-wave mixing, is that which makes it extremely difficult to internally form singular, isolated pulses intracavity [26,27]. Direct phase measurements confirm the laser output to exhibit a continuous-wave character with a strong frequency modulation, where the laser periodically sweeps its bandwidth [28]; this well supports previous conclusions drawn from indirect measurements [24,29].

Nonetheless, some techniques have succeeded in producing pulses in QCLs. A QCL lasing at 6.3 μm, with an active region specially engineered for a long gain recovery time ($\sim 50\mathrm{ps}$), and an electrically isolated short section, used to modulate the loss at the cavity round-trip time, was able to produce transform-limited pulses of the order of 3 ps [30]. However, the device had to be operated at cryogenic temperatures (77 K) close to the lasing threshold, limiting the fractional bandwidth to about 1.5% at $1585{\mathrm{cm}}^{-1}$, and the peak power to less than 10 mW (0.5 pJ per pulse). In the terahertz, the situation is somewhat more favorable, with the gain recovery time naturally in the tens of picoseconds [31,32], comparable to a short cavity round-trip time. One approach taken by Oustinov and co-workers [33] was based on synchronizing the radio-frequency (RF) modulation to a terahertz seed pulse injected directly into the cavity, made possible by the closely matched effective indices for the RF mode and the terahertz mode. In this way, they were able to amplify these naturally transform-limited pulses over several round trips, with pulse widths measured at around 11 ps FWHM.

An alternative approach to direct pulse generation is to compensate for the intermodal phase differences in the laser output field externally. This has already been performed for microresonator combs at 1550 nm using the technique line-by-line pulse shaping [34–37], where liquid-crystal waveshapers modify the phase of individual spectral components. An optimization routine stepwise changes the applied phase with a view to maximizing the second-harmonic signal, thereby ultimately producing near-transform-limited pulses, and characterizing the natural comb state [38,39]. External pulse compression is advantageous in that the device can be operated under normal conditions, hence making the most effective use of the gain medium. As such, the bandwidth will be naturally wide, and the average power in the watt range, which, in the case of a perfect phase correction, would translate to pulses of widths of the order of 300 fs and peak powers of several hundred watts. Certain properties of the QCL comb states, namely their stability, repeatability, and mainly quadratic phase profile [23,28], lend themselves to a fixed and potentially simple setup to correct the phase excursions. In this work we demonstrate the phase compensation to the first order using a grating compressor.

2. EXPERIMENT

Our sample is a 6 mm mid-IR QCL lasing at 8.2 μm, featuring a plasmon-enhanced waveguide for dispersion compensation [40]. The device operates as a comb, lasing on hundreds of modes and showing a singular sharp peak in the radio frequency spectrum at the inverse cavity round-trip time. This device was previously shown to exhibit the reproducible linear chirped state [28], where the spectral phases follow a mainly quadratic profile, quantified by the field group delay dispersion (GDD):

Our experimental setup shown in Fig. 1 comprises the QCL, a compressor to compensate the second-order dispersion, and a Fourier transform spectroscopy measurement based on shifted wave interference Fourier transform spectroscopy (SWIFTS) [25,41] (see Appendix A). The latter is a RF technique which gives direct access to the intermodal phase differences, and has been used to characterize the temporal behavior of QCLs [28,41]. The QCL light is collimated using a high numerical aperture collimation lens (focal length 1.173 mm) and is passed through an optical isolator to mitigate feedback. A beam splitter then directs the beam to the compressor, which modifies only the spectral phases, and reflects the light directly back to the beam splitter. The modified light is then sent through a Michelson interferometer and shined onto a sensitive mercury cadmium telluride detector (MCT) and fast quantum well infrared photodetector (QWIP) (splitting ratio $99:1$), meaning the DC autocorrelation trace and phase information containing correlation (SWIFTS) trace can be recorded simultaneously.

 

Fig. 1. Experimental setup. Quantum well infrared photodetector (QWIP); mercury cadmium telluride detector (MCT); a pair of matched ruled diffraction gratings (G1, G2); a pair of lenses ($F=10\mathrm{cm}$); and the retroreflecting mirror MR form the compressor. G2 and MR are mounted on a common translation stage, directed along $z$. The optical axis $z$ is positive in the right-hand direction, and so a displacement of G2 left indicates a positive GDD, and vice versa. To extend the travel of G2, the relay pair are shifted 5 cm toward the first grating as compared to the standard $4F$ configuration.

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The stretcher-compressor is the Martinez type [42], made up of a pair of antiparallel gratings (G1, G2: 150 lines/mm, blazed 35°) and a pair of identical achromatic doublet lenses (focal length $F=10\mathrm{cm}$) separated by twice the focal length to form a relay pair. A mirror retroreflects the light directly back through the system, which both doubles the effective dispersion induced on the field and reverses the spatial chirp. The second grating G2 and retroreflecting mirror MR are mounted on a motorized translation stage, allowing their position to be adjusted along the optical axis $z$, while keeping their relative positions fixed. The effective group delay dispersion per unit displacement is set by the angle of incidence for a fixed central wavelength, which for our system is about 54.5 deg ($\sim 0.53{\mathrm{ps}}^2/\mathrm{cm}$). A larger dispersion would be obtained for a smaller incident angle, but at the price of a more difficult geometry and, in the absence of larger optics, spectral clipping. At the origin $z=0$, where the two gratings are separated by exactly $4F$, the system is nondispersive, and the field is returned simply attenuated. By moving the second grating toward the first grating ($z<0$), a positive GDD is added corresponding to $0.53{\mathrm{ps}}^2$ per centimeter displacement.

The measurement proceeds as follows. The delay stage is set to around $-14\mathrm{cm}$, which is the farthest possible position without clipping of the beam. It is then stepped away in 2 mm increments such that the separation between the two gratings increases. At each grating displacement along $z$, the standard autocorrelation and SWIFTS traces are recorded (resolution $0.12{\mathrm{cm}}^{-1}$), giving complete spectral amplitude $A_n(z)$ and phase information ${\varphi }_n(z)$ for each comb mode $n$. We can therefore reconstruct the instantaneous intensity by the inverse Fourier transform:

3. RESULTS

In Fig. 2(a) we show how the steady-state instantaneous intensity evolves as a function of the grating displacement [Eq. (2)]. It can be seen that, as the displacement becomes more negative, the energy becomes more compressed within the period and a pulse is shown to emerge from the background. Conversely when the grating moves in the positive direction, interferences between the adjacent comb periods can be seen in the form of high-frequency oscillations.

 

Fig. 2. (a) Instantaneous intensity as a function of the grating position, calculated from the measured complex spectra by Eq. (2). As the grating G2 approaches in the direction of G1, indicated by the negative displacement, the pulse narrows and the peak power increases. (b) The field GDD, calculated from fits to each of the measured complex spectra. The red line at $-11.86\mathrm{cm}$ indicates the position at which the quadratic part of the phase has been completely eliminated. $-6.45{\mathrm{ps}}^2$ marks the dispersion at $z=0$. (c) Pulse width (Gaussian FWHM) and (d) peak intensity, normalized to the average, as a function of grating position. The orange curves are calculated by applying the relevant phase shifts given by the grating pair to the reference measurement at $z=0$. (e) Comparison of the waveform at zero displacement and (f) position with highest peak power.

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The field group delay dispersion, estimated by a fit to the measured phase traces at each position, is shown in Fig. 2(b) to change linearly with the grating position, as expected. We also show in Fig. 2(c) the (fitted) Gaussian equivalent pulse width and in Fig. 2(d) the peak intensity evolving with the grating position, both compared to the values predicted when applying the expected phase correction to the phase spectrum measured at 0 cm. The calculated points were deliberately extended past the geometric constraint of the setup to highlight that this is not the limiting factor, with the performance in terms of pulse width and peak power expected to quickly degrade past $-14\mathrm{cm}$. The red line in Figs. 2(b)–2(d) indicates the position at which the quadratic part of the phase has gone to zero. Note that this coincides with neither the maximum peak power, nor the shortest pulse. The discrepancy arises predominantly due to the stronger negative dispersion in the redmost lobe of the spectrum, which contains a substantial part of the power; the GDD evaluated at a central wavelength is an overly simplistic metric of the spectral phase profile for these lasers, especially in the context of compression.

Cuts of the instantaneous intensity at the $z=0$ and $z=-13.6\mathrm{cm}$ are shown in Figs. 2(e) and 2(f), respectively. A pulse of equivalent Gaussian FWHM 12 ps is observed to emerge, with a peak to average ratio of 40.7. In the ideal, transform-limited case, with the DC spectrum as is and the phases all set identical, a pulse with peak to average ratio in excess of 320 would be expected, with a pulse width less than 300 fs. At an estimated optical bandwidth of 2.79 THz at 10 dB, the Fourier-limited pulse exhibits a time-bandwidth product of 0.8, limited by the rectangular spectral shape, which gives rise to a quasi-sinc pulse. An equivalent Gaussian spectrum would give a pulse length of $\sim 0.44/2.79\mathrm{THz}=157\mathrm{fs}$, though to shape the full amplitude spectrum would strongly reduce the integrated pulse energy.

Though we start with 600 mW, accounting for the loss in the final beam splitter, only about 14 mW is observed to emerge from the system. This means our pulse has a peak power $P_0=P_{\mathrm{avg}}{I}_0/I_{\mathrm{avg}}$ of about 574 mW, with a pulse energy $P_{\mathrm{avg}}/f_{\mathrm{rep}}\approx E$ 1.9 pJ. Fifty-two percent of the intensity is lost at the optical isolator, and the rest can be accounted for through the beam splitter, the nonperpendicularity of the incident light with respect to the grating rulings, and lossy coatings present on some of the mirrors.

In addition to the fundamental beat note, which we use to characterize the phase spectrum, we also detect the second, third, and fourth radio frequency harmonics on the QWIP. In Fig. 3 we measure the power in the first four harmonics using a spectrum analyzer (Rohde and Schwarz FSW) as a function of grating displacement, and compare their evolution to that predicted. For each beating order $b$, the expected power is given by

 

Fig. 3. Beat-note power at the fundamental, second, third, and fourth electronic harmonics measured as a function of grating position, with the calculated traces superimposed (orange lines).

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As expected for an FM comb [24,29], the beatings are shown to be strongly suppressed at $z=0$, increasing as a general trend with negative grating displacement. Strikingly, at the position of maximum peak intensity ($-13.6\mathrm{cm}$), the beatings are shown to be simultaneously maximized, a key signature for an AM pulse train where all the phases have the same sign and similar magnitude. These behaviors are reflected in the calculated beat-note traces, which show good qualitative agreement for all measured grating displacements.

Traditionally, the fast noise dynamics of mode-locked lasers was characterized by the timing jitter of the pulse train. Indeed, the timing jitter, described either in terms of its integrated value ${\sigma }_T$ or its pulse-to-pulse value ${\sigma }_{pp}$, can be conveniently extracted from an integral of the electrical noise power spectral density on a fast detector [45], without relying on nonlinear detection methods (e.g., [46]). This technique is especially well-suited for the characterization of mode-locked lasers with cavities in the meter range, such that the optical modes are tightly spaced in the frequency domain and the electronic beat notes easily accessible, being typically in the megahertz. While the approach is most appropriate for actively mode-locked lasers, with some care similar inferences can be made about passively mode-locked lasers such as ours [47,48]. In contrast with the normal QCL emission with its strong FM characteristics, the pulsed mode-locked emission of our device after correction enables a straightforward application of Von der Linde’s analysis [45].

Shown in Fig. 4 is the power spectral density of the beat note at the fundamental round-trip frequency compared to the power of the carrier at the center of the band (dBc), showing a dynamical range of almost 80 dB at the point of maximum compression. We find an integrated timing jitter 20 kHz–100 MHz of 335 fs, and a pulse-to-pulse timing jitter of 2.0 fs, with reasonable agreement between the different orders (see Appendix D). The same analysis applied to the measured beat note of a QCL whose FM output was converted to an AM one using an optical discriminator yielded an integrated timing jitter of 342 fs, which is very close to that found for our device. In general, our values compare well with those measured on quantum dot semiconductor passively mode-locked lasers, which typically feature broader beat notes on the order of several kilohertz. As an example, [47] reports an integrated jitter value of 1.6 ps for such a device over the same frequency range of 20 kHz to 100 MHz.

 

Fig. 4. Radio frequency spectra as recorded on a fast detector (QWIP) with a spectrum analyzer, with the grating at the position where the field shows the strongest pulses. The resolution bandwidth (RBW) is 1 Hz for all measurements (a) Comparison of the first four harmonics, showing the expected broadening. (b) Radio frequency spectrum of the fundamental beat note as measured at a 1 Hz resolution bandwidth, scaled to the total power in the carrier, evaluated at a 200 Hz RBW. The blue line is a $1/f^2$ fit to the tail, from which the integrated timing jitter is inferred. (c) Third-harmonic electronic beat note, both free-running and when the laser has been RF injection locked at the fundamental beat frequency (7.42 GHz).

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Jitter performance can be drastically improved by injecting RF from a stable source at approximately the fundamental beat frequency, which acts to lock the comb repetition rate. QCLs have an intrinsically high electronic cutoff frequency, owing to the lack of relaxation peak [49], and remain responsive to bias modulations above 10 GHz [50]; this makes them particularly amenable candidates for RF injection locking, and this has been successfully demonstrated [51,52]. Importantly, all lasing modes can be coherently locked to the external oscillator, and this has been shown not to perturb the comb state [53]. We show in Fig. 4 the dramatic difference between the third-order beat note for the free-running QCL, and that observed when the QCL has been injected with RF at the fundamental of around 7.42 GHz.

4. CONCLUSION

We have demonstrated the compression of a free-running continuous-wave QCL comb, of period $\sim 134\mathrm{ps}$, into a train of well-defined pulses of around 12 ps FWHM. With this pulse train, we were able to characterize the timing jitter of our comb. That we are able to achieve this correction with a grating compressor, a staple of the ultrafast community, gives an additional strong confirmation of the intrinsic stability and reproducibility of the intermodal phases.

While the peak to average ratio increases to 40.7 at the optimal location, a 10-fold increase from the uncompressed output, losses in the experiment mean that there is no overall increase in peak power. Some of the responsible issues can be mitigated through optimization of the setup, but we are fundamentally limited by the higher-order dispersion present in the field. To compensate for this, a more sophisticated scheme would have to be used. For example, prisms could be placed in the focal plane between the two lenses: this would be of most use on the wings of the spectrum, which have a stronger dispersion quite dissimilar from the central portion. One could also take advantage of the various aberrations to tailor the dispersion up to the fourth order [54], at the price of the greater complexity that comes with the additional degrees of freedom. In the other direction, as the spectrum is deterministic, a fixed phase mask in the Fourier plane, or indeed a tunable thermal slab [55], could in principle compensate the dispersion almost entirely for a given state.

This proof-of-concept experiment demonstrates that external compensation schemes are a feasible approach to controlling the phases of the QCL comb. A spatial light modulator, for example, could be used to not only negate the field dispersion, but would allow for total control of the individual amplitudes and phases [35], enabling arbitrary tailoring of the QCL waveform, limited only by the intrinsic period, bandwidth, and power of the laser. In the context of compression, it should be possible to convert the typical QCL comb emission into a train of stable pulses with favorable time-bandwidth products, and potentially peak powers in the hundreds of watts.

APPENDIX A: SWIFTS

The standard autocorrelation is usually measured with a Michelson interferometer and a slow detector, with a scanning mirror inducing a delay $\tau$ between two paths, and the interference measured as a function of this delay. Such a measurement contains no phase information. With SWIFTS, the fundamental beat note is instead measured. As it is the sum of all heterodyne tones between neighboring modes, a measurement of the beat note will be sensitive to the phase relationship between these modes. The beat note is demodulated in quadrature as a function of mirror delay:

(A1)$$x(\tau )=⟨E(t+\tau )E^{*}(t)\cos ({\omega }_{r}t)⟩,$$

(A2)$$y(\tau )=⟨E(t+\tau )E^{*}(t)\sin ({\omega }_{r}t)⟩.$$

It can be shown that, for a comb source, and a sufficient total path delay, these quadratures give access to the intermodal phase differences after a Fourier transform:

(A3)$$X(\omega )-iY(\omega )=\sum _n|A_n||A_{n-1}|e^{i({\varphi }_n-{\varphi }_{n-1})}\delta [\omega -({\omega }_0+n{\omega }_r)].$$

This is a direct measurement of the field group delay. The phase profile is then reconstructed by summing across these phase differences.

For a full treatment of SWIFTS, see for example [25,41,56]. The setup used in this paper (marked “Acquisition” in Fig. 1) is described in more detail in Supplement 1 of [28].

APPENDIX B: DEVICE CHARACTERISTICS

The power spectrum, measured at low frequency on the MCT detector, and field group delay $\partial \varphi /\partial \omega$, measured at the device round-trip frequency of 7.42 GHz, are plotted in Figs. 5(a) and 5(c). The latter is calculated by normalizing relative phases between the modes ${\varphi }_n-{\varphi }_{n-1}$, extracted from the SWIFTS traces, to the circular cavity round-trip frequency ${\omega }_r$. By summing across along this trace, the phase spectrum can be recovered. This is plotted in Fig. 5(b), where at 0 cm the parabolic profile is very clear. Also plotted are the predicted and measured phase traces at $-13.6\mathrm{cm}$, showing good agreement, and the curve with no group delay dispersion for comparison. An 11th-order polynomial fit is made to the 0 cm trace and its derivative with respect to $\omega$, that is to say the group delay dispersion Eq. (1), is taken and plotted in Fig. 5(d). Qualitatively, this highlights that the field GDD of the comb possesses third- and higher-order dispersion, which neither the grating compressor, nor any other mainly quadratic compressor, e.g., a fiber, can compensate for alone.

 

Fig. 5. Device characteristics, measured at 20°C, 1.1688 A. (a) Optical spectrum, normalized to show the power per mode, as measured directly in front of the laser. (b) Phase spectrum as found by integrating the measured phase difference spectrum in (c). (c) Laser output field group delay measured at the nondispersive grating offset 0 cm. The orange curve is an 11th-order polynomial fit. (d) Field group delay dispersion, calculated as the numerical derivative of the black curve with respect to the circular frequency (${\delta }^2\varphi /\delta {\omega }^2$)

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APPENDIX C: BEAT-NOTE INTERFEROGRAMS

The action of the grating on the field can be seen directly on the beat-note interferogram, which shows the RF power in the beat note as a function of the interferometric path difference [24]. In Fig. 6, we plot at five different grating displacements ${|x(\tau )+iy(\tau )|}^2$, a quantity closely related to the beat-note interferogram, measured at the centerburst with a resolution of $16{\mathrm{cm}}^{-1}$. The reference point is indicated at a displacement of zero, where the grating sits at exactly the focal spot, and the system is dispersionless. Here, the FM character of the field is demonstrated very clearly, with a minimum at the zero path difference [24,29], and a maximum at approximately $1/2B$, where $B$ is the optical bandwidth. As the second grating is moved closer to the first, the correlation starts to become strongest at the zero path difference. The beat-note interferogram can be seen to more and more resemble the standard autocorrelation trace, as the spectral components are moved in phase. In the case of an AM signal, the ratio of the central fringe to that of the maximum would have a value of 1 by definition, since placing the light through an unbalanced interferometer should not lead to an enhancement of the beat-note strength. For more FM signals the opposite is true, with large enhancements expected.

 

Fig. 6. Unnormalized beat-note interferograms measured at $16{\mathrm{cm}}^{-1}$ as a function of the grating displacement (left) with the matching intermodal phase difference trace (right), where 0.0 cm is the reference (unmodified) position, and $-13.2\mathrm{cm}$ represents a strongly AM signal. The grating position is noted in the left-hand column. Fits to the phase difference spectra are plotted in orange, and the estimated GDD noted. A green DC spectrum is provided for reference.

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APPENDIX D: TIMING JITTER

We evaluate the integrated timing jitter by following the procedure of Von der Linde [45,47]:

(D1)$${\sigma }_T=\frac{1}{2\pi m{f}_r}\sqrt{{\int }_{f_d}^{f_u}S(f)\mathrm{d}f},$$

and the pulse-to-pulse timing jitter:

(D2)$${\sigma }_{pp}=\frac{1}{m\pi f_r}\sqrt{{\int }_0^{+f_r/2}S(f){\sin }^2\left(\frac{\pi f}{m{f}_r}\right)},$$

where $S(f)$ is the measured single-sided phase noise power spectral density, $m$ is the electronic beat-note harmonic index (fundamental $m=1$), and $f_r$ is the fundamental beat frequency.

Beat notes up to the fourth order were acquired at the point of maximum compression (see Fig. 4, main article), and Gaussian, Lorentz, $1/f^2$ and pseudo-Voigt fits performed on them. The results are tabulated in Table 1. As can be seen, the Gaussian FWHM increases approximately linearly with $m$, indicating that the correlation time of the jitter noise is much longer than the pulse repetition time [48]. For ${\sigma }_T$, we evaluate Eq. (D1) from the $1/f^2$ fit, and the pulse-to-pulse jitter (D2), which must also include the noise up to 0 Hz, from the pseudo-Voigt. Note that this tends to overestimate the pulse-to-pulse jitter.

Table 1. Beat-Note Parameters and Timing Jitter Extracted from Fits to Beat-Note Orders $m=1,2,3,4\dots {\delta }_G$ and $2\gamma$ Are the FWHM for the Gaussian and Lorentzian Fits, Respectively^a

View Table

Funding

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (SNF) (SNF200020-165639).

Acknowledgment

We thank Dmitry Kazakov for stimulating discussions, and Martin Franckié, Mehran Shahmohammadi, and Zhixin Wang for proofreading the manuscript and for constructive feedback.

REFERENCES

1. H. Pires, M. Baudisch, D. Sanchez, M. Hemmer, and J. Biegert, “Ultrashort pulse generation in the mid-IR,” Prog. Quantum Electron. 43, 1–30 (2015). [CrossRef]  

2. S. A. Meek, A. Hipke, G. Guelachvili, T. W. Hänsch, and N. Picqué, “Doppler-free Fourier transform spectroscopy,” Opt. letters 43, 162–165 (2018). [CrossRef]  

3. A. Sugiharto, C. M. Johnson, H. De Aguiar, L. Alloatti, and S. Roke, “Generation and application of high power femtosecond pulses in the vibrational fingerprint region,” Appl. Phys. B 91, 315–318 (2008). [CrossRef]  

4. I. Kubat, C. S. Agger, U. Møller, A. B. Seddon, Z. Tang, S. Sujecki, T. M. Benson, D. Furniss, S. Lamrini, K. Scholle, P. Fuhrberg, B. Napier, M. Farries, J. Ward, P. M. Moselund, and O. Bang, “Mid-infrared supercontinuum generation to 12.5 μm in large NA chalcogenide step-index fibres pumped at 4.5 μm,” Opt. Express 22, 19169–19182 (2014). [CrossRef]  

5. O. Katz, A. Natan, Y. Silberberg, and S. Rosenwaks, “Standoff detection of trace amounts of solids by nonlinear Raman spectroscopy using shaped femtosecond pulses,” Appl. Phys. Lett. 92, 171116 (2008). [CrossRef]  

6. A. Ting, I. Alexeev, D. Gordon, E. Briscoe, J. Peñano, R. Hubbard, P. Sprangle, and G. Rubel, “Remote atmospheric breakdown for standoff detection by using an intense short laser pulse,” Appl. Opt. 44, 5315–5320 (2005). [CrossRef]  

7. V. S. Yakovlev, M. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft X-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15, 15351–15364 (2007). [CrossRef]  

8. S. Cousin, F. Silva, S. Teichmann, M. Hemmer, B. Buades, and J. Biegert, “High-flux table–top soft x-ray source driven by sub-2-cycle, CEP stable, 1.85-μm 1-kHz pulses for carbon K-edge spectroscopy,” Opt. Lett. 39, 5383–5386 (2014). [CrossRef]  

9. P. Colosimo, G. Doumy, C. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. March, G. Paulus, and H. G. Muller, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4, 386–389 (2008). [CrossRef]  

10. C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature 483, 194–197 (2012). [CrossRef]  

11. R. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phase-matched difference frequency mixing of femtosecond pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999). [CrossRef]  

12. P. Krogen, H. Suchowski, H. Liang, N. Flemens, K.-H. Hong, F. X. Kärtner, and J. Moses, “Generation and multi-octave shaping of mid-infrared intense single-cycle pulses,” Nat. Photonics 11, 222–226 (2017). [CrossRef]  

13. C. Phillips, J. Jiang, C. Mohr, A. Lin, C. Langrock, M. Snure, D. Bliss, M. Zhu, I. Hartl, J. Harris, M. E. Fermann, and M. M. Fejer, “Widely tunable midinfrared difference frequency generation in orientation-patterned GaAs pumped with a femtosecond Tm-fiber system,” Opt. Lett. 37, 2928–2930 (2012). [CrossRef]  

14. B. Golubovic and M. Reed, “All-solid-state generation of 100-kHz tunable mid-infrared 50-fs pulses in type I and type II AgGaS 2,” Opt. Lett. 23, 1760–1762 (1998). [CrossRef]  

15. C. McGowan, D. Reid, M. Ebrahimzadeh, and W. Sibbett, “Femtosecond pulses tunable beyond 4 μm from a KTA-based optical parametric oscillator,” Opt. Commun. 134, 186–190 (1997). [CrossRef]  

16. M. Hemmer, A. Thai, M. Baudisch, H. Ishizuki, T. Taira, and J. Biegert, “18-μJ energy, 160-kHz repetition rate, 250-MW peak power mid-IR OPCPA,” Chin. Opt. Lett. 11, 013202 (2013). [CrossRef]  

17. H. Liang, P. Krogen, Z. Wang, H. Park, T. Kroh, K. Zawilski, P. Schunemann, J. Moses, L. F. DiMauro, F. X. Kärtner, and K. H. Hong, “High-energy mid-infrared sub-cycle pulse synthesis from a parametric amplifier,” Nat. Commun. 8, 141 (2017). [CrossRef]  

18. B. Mayer, C. Phillips, L. Gallmann, M. Fejer, and U. Keller, “Sub-four-cycle laser pulses directly from a high-repetition-rate optical parametric chirped-pulse amplifier at 3.4 μm,” Opt. Lett. 38, 4265–4268 (2013). [CrossRef]  

19. I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, T. Paasch-Colberg, I. Znakovskaya, M. Pescher, W. Schweinberger, and V. Pervak, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9, 721–724 (2015). [CrossRef]  

20. T. Kanai, P. Malevich, S. S. Kangaparambil, K. Ishida, M. Mizui, K. Yamanouchi, H. Hoogland, R. Holzwarth, A. Pugzlys, and A. Baltuska, “Parametric amplification of 100 fs mid-infrared pulses in ZnGeP 2 driven by a Ho: YAG chirped-pulse amplifier,” Opt. Lett. 42, 683–686 (2017). [CrossRef]  

21. S. Chaitanya Kumar, A. Esteban-Martin, T. Ideguchi, M. Yan, S. Holzner, T. W. Hänsch, N. Picqué, and M. Ebrahim-Zadeh, “Few-cycle, broadband, mid-infrared optical parametric oscillator pumped by a 20-fs Ti: sapphire laser,” Laser Photonics Rev. 8, L86–L91 (2014). [CrossRef]  

22. F. Cappelli, G. Campo, I. Galli, G. Giusfredi, S. Bartalini, D. Mazzotti, P. Cancio, S. Borri, B. Hinkov, J. Faist, and P. De Natale, “Frequency stability characterization of a quantum cascade laser frequency comb: frequency stability characterization of a quantum cascade laser frequency comb,” Laser Photonics Rev. 10, 623–630 (2016). [CrossRef]  

23. S. Bartalini, L. Consolino, F. Cappelli, G. Campo, I. Galli, D. Mazzotti, P. Cancio, G. Scalari, J. Faist, and P. De Natale, “Direct measurement of the phase coherence of quantum cascade comb sources,” in CLEO: Applications and Technology (Optical Society of America, 2018), pp. ATh4Q-6.

24. 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]  

25. 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]  

26. H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, “Gain recovery dynamics and photon-driven transport in quantum cascade lasers,” Phys. Rev. Lett. 100, 167401 (2008). [CrossRef]  

27. W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, and G. Strasser, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008). [CrossRef]  

28. M. Singleton, P. Jouy, M. Beck, and J. Faist, “Evidence of linear chirp in mid-infrared quantum cascade lasers,” Optica 5, 948–953 (2018). [CrossRef]  

29. 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]  

30. C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, and P. Grant, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009). [CrossRef]  

31. R. P. Green, A. Tredicucci, N. Q. Vinh, B. Murdin, C. Pidgeon, H. E. Beere, and D. A. Ritchie, “Gain recovery dynamics of a terahertz quantum cascade laser,” Phys. Rev. B 80, 075303 (2009). [CrossRef]  

32. D. R. Bacon, J. R. Freeman, R. A. Mohandas, L. Li, E. H. Linfield, A. G. Davies, and P. Dean, “Gain recovery time in a terahertz quantum cascade laser,” Appl. Phys. Lett. 108, 081104 (2016). [CrossRef]  

33. D. Oustinov, N. Jukam, R. Rungsawang, J. Madéo, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, J. Tignon, and S. Dhillon, “Phase seeding of a terahertz quantum cascade laser,” Nat. Commun. 1, 69 (2010). [CrossRef]  

34. A. Weiner, D. Leaird, G. P. Wiederrecht, and K. A. Nelson, “Femtosecond pulse sequences used for optical manipulation of molecular motion,” Science 247, 1317–1319 (1990). [CrossRef]  

35. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. 15, 326–328 (1990). [CrossRef]  

36. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995). [CrossRef]  

37. Z. Jiang, D. E. Leaird, and A. M. Weiner, “Spectral line-by-line pulse shaping,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2005), p. CWB4.

38. F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]  

39. P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015). [CrossRef]  

40. P. Jouy, J. M. Wolf, Y. Bidaux, P. Allmendinger, M. Mangold, M. Beck, and J. Faist, “Dual comb operation of λ∼8.2 μm quantum cascade laser frequency comb with 1 W optical power,” Appl. Phys. Lett. 111, 141102 (2017). [CrossRef]  

41. D. Burghoff, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs,” Opt. Express 23, 1190–1202 (2015). [CrossRef]  

42. O. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3–1.6 μm region,” IEEE J. Quantum Electron. 23, 59–64 (1987). [CrossRef]  

43. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986). [CrossRef]  

44. H. C. Liu, J. Li, M. Buchanan, and Z. R. Wasilewski, “High-frequency quantum-well infrared photodetectors measured by microwave-rectification technique,” IEEE J. Quantum Electron. 32, 1024–1028 (1996). [CrossRef]  

45. D. Von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 201–217 (1986). [CrossRef]  

46. L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38, 1047–1052 (2002). [CrossRef]  

47. F. Kéfélian, S. O’Donoghue, M. T. Todaro, J. G. McInerney, and G. Huyet, “RF linewidth in monolithic passively mode-locked semiconductor laser,” IEEE Photonics Technol. Lett. 20, 1405–1407 (2008). [CrossRef]  

48. D. Eliyahu, R. A. Salvatore, and A. Yariv, “Noise characterization of a pulse train generated by actively mode-locked lasers,” J. Opt. Soc. Am. B 13, 1619–1626 (1996). [CrossRef]  

49. R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79, 2526–2528 (2001). [CrossRef]  

50. B. Hinkov, A. Hugi, M. Beck, and J. Faist, “Rf-modulation of mid-infrared distributed feedback quantum cascade lasers,” Opt. Express 24, 3294–3312 (2016). [CrossRef]  

51. M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, P. Filloux, C. Sirtori, J.-F. Lampin, G. Ferrari, S. P. Khanna, E. H. Linfield, and H. E. Beere, “Stabilization and mode locking of terahertz quantum cascade lasers,” IEEE J. Sel. Top. Quantum Electron. 19, 8501011 (2013). [CrossRef]  

52. M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photonics Rev. 8, 443–449 (2014). [CrossRef]  

53. J. Hillbrand, A. M. Andrews, H. Detz, G. Strasser, and B. Schwarz, “Coherent injection locking of quantum cascade laser frequency combs,” Nat. Photonics 13, 101–104 (2019). [CrossRef]  

54. W. White, F. Patterson, R. L. Combs, D. Price, and R. Shepherd, “Compensation of higher-order frequency-dependent phase terms in chirped-pulse amplification systems,” Opt. Lett. 18, 1343–1345 (1993). [CrossRef]  

55. K. Osvay, K. Varjú, and G. Kurdi, “High order dispersion control for femtosecond CPA lasers,” Appl. Phys. B 89, 565–572 (2007). [CrossRef]  

56. D. Burghoff, Some Notes on Shifted Wave Interference Fourier Transform Spectroscopy (SWIFTS) (2018).