Analytical theory of frequency-modulated combs: generalized mean-field theory, complex cavities, and harmonic states.

Frequency-modulated (FM) combs with a linearly-chirped frequency and nearly constant intensity occur naturally in certain laser systems; they can be most succinctly described by a nonlinear Schrödinger equation with a phase potential. In this work, we perform a comprehensive analytical study of FM combs in order to calculate their salient properties. We develop a general procedure that allows mean-field theories to be constructed for arbitrary sets of master equations, and as an example consider the case of reflective defects. We derive an expression for the FM chirp of arbitrary Fabry-Perot cavities-important for most realistic lasers-and use perturbation theory to show how they are affected by finite gain bandwidth and linewidth enhancment in fast gain media. Lastly, we show that an eigenvalue formulation of the laser's dynamics can be useful for characterizing all of the stable states of the laser: the fundamental comb, the continuous-wave solution, and the harmonic states.

Sensitivity of SWIFT spectroscopy.

SWIFT spectroscopy (Shifted Wave Interference Fourier Transform Spectroscopy) is a coherent beatnote technique that can be used to measure the temporal profiles of periodic optical signals. While it has been essential in understanding the physics of various mid-infrared and terahertz frequency combs, its ultimate limits have not been discussed. We show that the envelope of a SWIFTS interferogram is physically meaningful and is directly related to autocorrelation. We derive analytical expressions for the SWIFTS signals of two prototypical cases-chirped pulses from a mode-locked laser and a frequency-modulated comb-and derive scaling laws for the noise of these measurements, showing how it can be mitigated. Finally, we confirm this analysis by performing the first SWIFTS measurements of near-infrared pulses from femtosecond lasers, establishing the validity of the technique for highly-dispersed sub-picojoule pulses.

Mitigating offset frequency drift in frequency combs using a customized power law dispersion.

We introduce a new, to the best of our knowledge, passive technique of mitigating the phase noise in optical frequency combs (FC) by reducing the drift of offset frequency. This can be achieved by customizing the dispersion to attain a power law dependence of the wave vector on frequency, k(ω)∼ωα, ensuring a constant ratio between group and phase velocities. When this condition is maintained, the drift offset frequency is passively mitigated, and phase noise is reduced. Using quantum cascade laser (QCL) FCs as an example, we demonstrate, analytically and numerically, that the desired dispersion can be easily engineered by properly adjusting the thickness of the QCL active region and that stable offset frequency can be combined with low residual group dispersion.

Generalized method for the computational phase correction of arbitrary dual comb signals.

We demonstrate a computational phase correction algorithm that is able to correct for phase and timing fluctuations of arbitrary dual comb spectra. By augmenting a Kalman filter with a global search and decoupling the interferogram estimation, we show that dual comb signals having a wide range of structures can be predicted and corrected. Furthermore, we derive an upper bound for the accuracy of any self-correction technique and show that the augmented filter is capable of reaching this bound when the phase and frequency noise are bandlimited. Finally, we show how expectation maximization can be used to learn the statistical parameters of a system without any free parameters. This approach is hands-off, robust, and accurate for a wide range of dual comb systems. Demonstration code is provided.

Compact and high-precision wavemeters using the Talbot effect and signal processing.

Precise knowledge of a laser's wavelength is crucial for applications from spectroscopy to telecommunications. Here, we present a wavemeter that operates on the Talbot effect. Tone parameter extraction algorithms are used to retrieve the frequency of the periodic signal obtained in the sampled Talbot interferogram. Theoretical performance analysis based on the Cramér-Rao lower bound as well as experimental results are presented and discussed. With this scheme, we experimentally demonstrate a compact and high-precision wavemeter with below 10 pm single-shot estimation uncertainty under the 3-σ criterion around 780 nm.

Temporal characteristics of quantum cascade laser frequency modulated combs in long wave infrared and THz regions.

We consider here a time domain model representing the dynamics of quantum cascade lasers (QCLs) generating frequency combs (FCs) in both THz and long wave infrared (LWIR λ = 8-12µm) spectral ranges. Using common specifications for these QCLs we confirm that the free running laser enters a regime of operation yielding a pseudo-randomly frequency modulated (FM) radiation in the time domain corresponding to FCs with stable phase relations in the frequency domain. We provide an explanation for this unusual behavior as a consequence of competition for the most efficient regime of operation. Expanding the model previously developed in [Opt. Eng. 57(1), 011009 (2017)] we analyze the performance of realistic THz and LWIR QCLs and show, despite the vastly different scale of many parameters, that both types of lasers offer very similar characteristics, namely FM operation with an FM period commensurate with the gain recovery time and an FM amplitude comparable with the gain bandwidth. We also identify the true culprit behind pseudo-random dynamics of the FM comb to be spatial hole burning, rather than the more pervasive spectral hole burning.

Achieving comb formation over the entire lasing range of quantum cascade lasers.

Frequency combs based on quantum cascade lasers (QCLs) are finding promising applications in high-speed broadband spectroscopy in the terahertz regime, where many molecules have their "fingerprints." To form stable combs in QCLs, an effective control of group velocity dispersion plays a critical role. The dispersion of the QCL cavity has two main parts: a static part from the material and a dynamic part from the intersubband transitions. Unlike the gain, which is clamped to a fixed value above the lasing threshold, dispersion associated with the intersubband transitions changes with bias, even above the threshold, and this reduces the dynamic range of comb formation. Here, by incorporating tunability into the dispersion compensator, we demonstrate a QCL device exhibiting comb operation from Ith to Imax, which greatly expands the operation range of the frequency combs.

Time domain modeling of terahertz quantum cascade lasers for frequency comb generation.

The generation of frequency combs in the mid-infrared and terahertz regimes from compact and potentially cheap sources could have a strong impact on spectroscopy, as many molecules have their rotovibrational bands in this spectral range. Thus, quantum cascade lasers (QCLs) are the perfect candidates for comb generation in these portions of the electromagnetic spectrum. Here we present a theoretical model based on a full numerical solution of Maxwell-Bloch equations suitable for the simulation of such devices. We show that our approach captures the intricate interplay between four wave mixing, spatial hole burning, coherent tunneling and chromatic dispersion which are present in free running QCLs. We investigate the premises for the generation of QCL based terahertz combs. The simulated comb spectrum is in good agreement with experiment, and also the observed temporal pulse switching between high and low frequency components is reproduced. Furthermore, non-comb operation resulting in a complex multimode dynamics is investigated.

Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs.

Recently, much attention has been focused on the generation of optical frequency combs from quantum cascade lasers. We discuss how fast detectors can be used to demonstrate the mutual coherence of such combs, and present an inequality that can be used to quantitatively evaluate their performance. We discuss several technical issues related to shifted wave interference Fourier Transform spectroscopy (SWIFTS), and show how such measurements can be used to elucidate the time-domain properties of such combs, showing that they can possess signatures of both frequency-modulation and amplitude-modulation.

Broadband all-electronically tunable MEMS terahertz quantum cascade lasers.

In this work, we demonstrate all-electronically tunable terahertz quantum cascade lasers (THz QCLs) with MEMS tuner structures. A two-stage MEMS tuner device is fabricated by a commercial open-foundry process performed by the company MEMSCAP. This provides an inexpensive, rapid, and reliable approach for MEMS tuner fabrication for THz QCLs with a high-precision alignment scheme. In order to electronically actuate the MEMS tuner device, an open-loop cryogenic piezo nanopositioning stage is integrated with the device chip. Our experimental result shows that at least 240 GHz of single-mode continuous electronic tuning can be achieved in cryogenic environments (∼4 K) without mode hopping. This provides an important step toward realizing turn-key bench-top tunable THz coherent sources for spectroscopic and coherent tomography applications.

Terahertz tomography using quantum-cascade lasers.

The interfaces of a dielectric sample are resolved in reflection geometry using light from a frequency agile array of terahertz quantum-cascade lasers. The terahertz source is a 10-element linear array of third-order distributed-feedback QCLs emitting at discrete frequencies from 2.08 to 2.4 THz. Emission from the array is collimated and sent through a Michelson interferometer, with the sample placed in one of the arms. Interference signals collected at each frequency are used to reconstruct an interferogram and detect the interfaces in the sample. Because of the long coherence length of the source, the interferometer arms need not be adjusted to the zero-path delay. A depth resolution of 360 µm in the dielectric is achieved with further potential improvement through improved frequency coverage of the array. The entire experiment footprint is <1 m × 1 m with the source operated in a compact, closed-cycle cryocooler.

High-quality microresonators in the longwave infrared based on native germanium.

The longwave infrared (LWIR) region of the spectrum spans 8 to 14 µm and enables high-performance sensing and imaging for detection, ranging, and monitoring. Chip-scale LWIR photonics has enormous potential for real-time environmental monitoring, explosive detection, and biomedicine. However, realizing technologies such as precision sensors and broadband frequency combs requires ultra low-loss and low-dispersion components, which have so far remained elusive in this regime. Here, we use native germanium to demonstrate the first high-quality microresonators in the LWIR. These microresonators are coupled to partially-suspended Ge waveguides on a separate glass chip, allowing for the first unambiguous measurements of isolated linewidths. At 8 µm, we measured losses of 0.5 dB/cm and intrinsic quality (Q) factors of 2.5 × 105, nearly two orders of magnitude higher than prior LWIR resonators. Our work portends the development of novel sensing and nonlinear photonics in the LWIR regime.

Frequency comb ptychoscopy.

Multiheterodyne techniques using frequency combs-radiation sources whose lines are perfectly evenly-spaced-have revolutionized science. By beating sources with the many lines of a comb, their spectra are recovered. Even so, these approaches are fundamentally limited to probing coherent sources, such as lasers. They are unable to measure most spectra that occur in nature. Here we present frequency comb ptychoscopy, a technique that allows for the spectrum of any complex broadband source to be retrieved using a comb. In this approach, the spectrum is reconstructed by unfolding the simultaneous beating of a source with each comb line. We demonstrate this both theoretically and experimentally, at microwave frequencies. This approach can reconstruct the spectrum of nearly any complex source to high resolution, and the speed, resolution, and generality of this technique will allow chip-scale frequency combs to have an impact in a wide swath of new applications, such as remote sensing and passive spectral imaging.

Computational multiheterodyne spectroscopy.

Dual-comb spectroscopy allows for high-resolution spectra to be measured over broad bandwidths, but an essential requirement for coherent integration is the availability of a phase reference. Usually, this means that the combs' phase and timing errors must be measured and either minimized by stabilization or removed by correction, limiting the technique's applicability. We demonstrate that it is possible to extract the phase and timing signals of a multiheterodyne spectrum completely computationally, without any extra measurements or optical elements. These techniques are viable even when the relative linewidth exceeds the repetition rate difference and can tremendously simplify any dual-comb system. By reconceptualizing frequency combs in terms of the temporal structure of their phase noise, not their frequency stability, we can greatly expand the scope of multiheterodyne techniques.