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The theoretical framework of the Haus master equation of passive mode-locking is revisited. Reformulating the equation in the frequency domain as coupled ordinary differential equations, the complete set of fundamental soliton solutions is surveyed. For large values of anomalous dispersion, this leads to the well known bell-shaped solutions originally found by inverse scattering. Closer to zero dispersion, mode-locked spectra are affected by the available gain bandwidth, and solitons with Bessel-like temporal profiles are found. These spectrally caged solitons match previously unexplained pulse characterization measurements of few-cycle oscillators and mode-locked fiber lasers in the normal dispersion regime. Moreover, the frequency domain formalism suggests that a phase lock between the modes can even be established in the absence of saturable absorption. This finding may explain numerous mysterious experimental reports of mode-locking or comb formation in passive microring resonators and semiconductor lasers. Therefore our frequency-domain approach sheds new light into soliton physics from a completely different perspective.

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A completely analytical approach is presented that provides deep insight into the pulse formation processes in a hollow fiber compressor by disentangling spatial soliton dynamics.

Today the generation of frequency combs nearly exclusively relies on passive mode-locking, requiring a phase lock between the longitudinal modes of a laser. In order to overcome the non-equidistance of the cold cavity modes, it is generally considered mandatory to include an effective saturable absorption mechanism in the laser cavity. However, there exist a number of experimental demonstrations of mode-locking or comb formation in which saturable absorption was clearly absent. Here we show that four-wave mixing may equally well lead to a mode-locking effect. However, the resulting pulse trains are only partially coherent, and the comb structures lack perfect equidistance. Operation of lasers in the pseudo mode-locked regime can easily be confused with traditional mode-locking. We discuss indications and characterization approaches for unveiling pseudo mode-locking as well as limitations for application of pseudo combs.

We discuss an efficient all-optical switching scheme with ultrafast switching times and high switching contrast which satisfies fan-out [1] , one difficult challenge to be met in all optical computing. To this end, a nonlinear interaction mechanism provides a means to control light by light [2] , [3] , realizing switching between different parts of soliton-fission induced supercontinuum spectra. The experimental setup employed a NL-PM-750 photonic crystal fiber (NKT Photonics), and the complementing numerical simulations were performed using a generalized nonlinear Schrodinger equation [1] , [3] .

Since the advent of mode-locked laser frequency combs, metrology has experienced a rapid improvement of the precision in frequency measurements. Best reported microwave-based measurements with cesium fountain clocks reached a fractional uncertainty of 10 -15 in 2005 [1] . In 2010 optical frequency measurements took the lead, and more recent experiments with optical lattice clocks reached 10 -18 values. Frequency transfer experiments even demonstrated sub 10 -20 levels [2] . Fractional uncertainities currently decrease at the pace of one order of magnitude every 4 years. If this Moore’s Law of frequency metrology continues at the same rate, one expects to see 10 -22 uncertainties at the end of this decade. Similar to Moore’s Law, however, the question arises for how long this trend can further continue before some fundamental physical mechanism eventually causes stagnation. And one such fundamental barrier arises due to Schawlow-Townes noise in the mode-locked laser, causing a 1/ f -like drift mechanism between two independent frequency combs.

During the last decade, self mode-locking and resulting comb formation have emerged from a controversially discussed anomaly to a hot topic in semiconductor laser physics. Initial experimental reports claimed mode-locking in a cavity lacking the stabilizing action of a saturable absorber [1] . These reports were heavily disputed as coherent artifacts [2] , [3] , i.e., an erroneous interpretation of autocorrelation measurements of a multimode continuous wave laser. While this incorrect interpretation may explain some peculiar reports, there are numerous indications that there really is a mode-locking effect in some other cases. In particular, some experimental reports indicated a threshold-like collapse of the intermode beat linewidth [4] , similar to what is seen in traditional laser mode-locking. Despite numerous attempts to tackle the underlying physical mechanism, it is still not understood how mode-locking can be stabilized without a saturable absorber, defying generally accepted conclusions from the Haus Master Equation [5] . In order to analyze self mode-locking, we rewrote the Master Equation formalism in the frequency domain [6] , converting a partial differential equation into a coupled system of ordinary differential equations.

We present a spectral phase measurement and correction method utilizing a pulse shaper. Positive and negative π/2 spectral phase steps are scanned across the spectrum of the pulse while collecting second harmonic spectra. Second and third order phase distortions show characteristic features in the difference between the spectra. Computer simulations show the amplitude and sign of these features quantitatively determine the magnitude and sign of the second and third order dispersion with milliradian accuracy. Experimental dispersion measurements are benchmarked by measuring the group velocity dispersion of air as well as fused silica windows.

Amplified spontaneous emission (ASE) causes fluctuations of pulse energy, of the optical phase and of the timing of the pulse intensity envelope in a mode-locked laser or frequency comb. Starting from the assumption of one ASE photon per longitudinal laser mode and roundtrip, we rederive analytic equations for the three fundamental types of quantum noise in a laser. To this end, we analyze the interference of the coherent intracavity field and a spectrally localized ASE photon as a function of wavelength and phase of the latter. Performing an integration over all wavelengths and phases and taking stochastic noise into account, we compute ASE-induced jitters for all quantities considered. Continuing this approach, we then derive an expression for the resulting carrier-envelope phase noise of the comb, for which so far only numerical estimates exist. We further compute analytical estimates for ASE induced pulse chirp and duration variations and address the issue of resulting pulse contrast in a mode-locked laser and the resulting coherence properties. Considering three example cases, we finally compute estimates for all quantities analyzed. Taken together, our analysis provides a comprehensive view of ASE effects in a mode-locked laser, which unites numerous scattered reports across the literature.

Microresonator combs have revolutionized the field of frequency metrology, replacing bulky mode-locked laser setups with chip-scale devices. While these novel comb sources have demonstrated their usefulness in numerous applications the exact nature of the mode-locking process in these devices appears to be not fully understood as it is commonly believed that synchronization between laser modes can only be achieved in the presence of an effective saturable absorber inside cavity. In the absence of saturable absorption, one would therefore expect that soliton solutions of the Haus master equation are not unconditionally stable against residual third-order dispersion. Consequently, deviations from a perfect equidistance may arise. Here we show that four-wave mixing processes can, to some extent, take over the role of saturable absorption and lead to a synchronization of modes. Within a certain range of dispersions, stable soliton solutions can be found not only in the anomalous dispersion regime, but also for zero, normal, or third-order dispersion. However, if the soliton conditions are not exactly matched, breather solutions form, and the resulting combs show deviations from equidistance.

We experimentally demonstrate how the optical Doppler effect can be exploited to modulate the carrier-envelope frequency of a mode-locked laser without causing detrimental side-effects and without any intervention into the laser.

We present carrier-envelope phase stabilization of an Er:Yb:glass laser at 1.55 pm via the feed-forward method with 2.9 as (1 Hz - 3 MHz) timing jitter and continuous stabilization over 24 hours.

We demonstrate a novel algorithm for pulse recovery from FROG traces contaminated with coherent artifacts. It separately retrieves the fields corresponding to the coherent artifact and the average features of unstable pulses from a single trace.

Self-mode-locking of semiconductor lasers has been frequently reported, defying all known solutions of the Haus Master Equation approach. Here we show that four-wave mixing all by itself suffices to mode-lock a laser.

Controlling the CEP in mode-locked lasers over long timescales is crucial for real-world applications. We present a hybrid stabilization system in an Er: Yb:glass mode-locked laser with 75 hours of stabilization and IPN below 14 mrad.

Controlling the CEP in mode-locked lasers over long timescales is crucial for real-world applications. We present a hybrid stabilization system in an Er:Yb:glass mode-locked laser with 75 hours of stabilization and IPN below 14 mrad.

The coherent artifact hinders accurate inversion of pulse measurements. Here we demonstrate a FROG algorithm to accurately retrieve not only the average pulse shape but also the artifact itself, providing access to the underlying coherence properties.

Multimode nonlinear optics is used to overcome a long-standing limitation of fiber optics, tightly phase locking several spatial modes and enabling the coherent transport of a wave packet through a multimode fiber. A similar problem is encountered in the temporal compression of multimillijoule pulses to few-cycle duration in hollow gas-filled fibers. Scaling the fiber length to up to 6 m, hollow fibers have recently reached 1 TW of peak power. Despite the remarkable utility of the hollow fiber compressor and its widespread application, however, no analytical model exists to enable insight into the scaling behavior of maximum compressibility and peak power. Here we extend a recently introduced formalism for describing mode locking to the analog scenario of locking spatial fiber modes together. Our formalism unveils the coexistence of two soliton branches for anomalous modal dispersion and indicates the formation of stable spatiotemporal light bullets that would be unstable in free space, similar to the temporal cage solitons in mode-locking theory. Our model enables deeper understanding of the physical processes behind the formation of such light bullets and predicts the existence of multimode solitons in a much wider range of fiber types than previously considered possible.

International audience; Few-cycle pulse characterization methods face a serious challenge in providing sufficient signal-to-noise ratios together with superior spectral fidelity, as imposed by phase-matching conditions and linear dispersion effects. Here we investigate the effect of linear dispersion inside the nonlinear medium inherently present in such arrangements. We demonstrate that pulse characterization using a cross-polarized wave generation dispersion scan is surprisingly insensitive to the group-velocity dispersion itself. We characterize sub-4 fs pulses at 780 nm center the wavelength utilizing crystals of different thicknesses, yielding nearly identical pulse shapes. Numerical simulations shed light on this behavior indicating practical limits of usable medium lengths.

Frequency comb measurements have enormously progressed in the last decade, with 10−18 precisions coming into reach [1]. Similar to Moore's law, the precision in frequency metrology measurements has improved by about 3 orders of magnitude in a little bit more than a decade. Along these lines, the question may arise whether there will be continuous improvements for decades to come — or whether, at one point, an ultimate limitation to further improvement arises. Moreover, frequency metrology has proven superior to other methods to evince a drift of fundamental physical constants, e.g., the fine-structure constants, yet no such drift has ever been convincingly proven using either optical [2] or cosmological methods [3]. So the question is: how low can we go before quantum effects impose an ultimate limit for precision frequency metrology?

The wave-particle dualism is one of the most important paradigms of modern physics, often attributing a particle character to objects that are in principle waves. One such example is a soliton, representing a wave that can propagate over extended distances without changing its shape, resembling the trajectories of rigid mechanical bodies in Newtonian physics, even in a collision process [1]. However, this analogy has strong restrictions as still wave properties are required and more importantly only the trivial solution of the momentum conservation equation (exactly equal solitions) can be addressed. Here we take the analogy one decisive step further, discussing completely Newtonian soliton collisions, detached from the wave description. Moreover, we show that the solitons act like extended massive objects, which deform in accord with the continuum mechanical concept of classical theory of elasticity. The interaction offers a variety of possibilities for controlled exchange of energy and momentum, opening a perspective for new applications of solitons in optical switching or trapping.

Carrier envelope phase (CEP) stabilization has proven an indispensable tool in attosecond science [1, 2] and precision frequency metrology [3]. Although fully CEP stabilized mode-locked fiber lasers and Ti:sapphire lasers are commercially available, CEP stabilization and all-optical frequency synthesis are still far away from general applicability to any given mode-locked laser. Stabilizability ultimately relies on the availability of at least 10 photons per pulse in the detection scheme [4]. Astro-combs, e.g., are rather demanding in this regard as they require ultra-high repetition rates. Moreover, despite their high pulse energies, thin-disk lasers have proven notoriously difficult to stabilize. Here we demonstrate that introduction of active optical gain into the infrared arm of an f-to-2f interferometer [5] can boost the signal by up to 20 dB, making stabilization of notoriously unstabilizable mode-locked lasers possible, e.g., high repetition-rate oscillators or thin-disk lasers.

When light sources approach the single cycle limit, both their dispersion compensation as well as pulse characterization become increasingly challenging. As all pulse characterization methods rely on some optical non-linearity, concomitant phase matching issues or dispersive broadening inside the nonlinear medium may severely corrupt the measurement result. One of the best suited methods for characterizing sub-2-cycle pulses is the dispersion scan (d-scan) technique that traditionally relies on second harmonic generation (SHG) as the nonlinear interaction [1]. However, using frequency conversion as a nonlinearity phase matching is the dominant limitation obscuring smaller propagation effects. By using cross-polarized wave (XPW) generation [2] in the d-scan arrangement [3], where no frequency conversion takes place, the phase matching issues are eliminated, uncovering the dispersion and self-phase modulation (SPM), inherently present in the propagation.

Carrier-envelope phase (CEP) stabilization plays an important role in the field of attosecond pulse generation and high-field physics [1]. Low-jitter CEP stabilization of ultrashort pulses typically relies on active stabilization schemes, either employing feedback or feed-forward control. However, both established methods come with a detrimental side effect. Feedback control directly modulates pump power or intracavity loss, which causes unwanted power fluctuations of the laser output. While feed-forward control does not corrupt the output stability, it nevertheless induces beam pointing variations in the acousto-optic frequency shifter employed. Here we demonstrate a novel method for modulating the CEP of a laser oscillator, which is based on the Doppler effect and displays neither of the above-mentioned side-effects.

Frequency combs have revolutionized frequency metrology, enabling optical clockworks with unprecedended precisions approaching the 10−18 range [1]. Early frequency comb measurements exclusively relied on mode-locked lasers, which provide a periodic waveform as a matter of working principle. In turn, the resulting frequency comb is guaranteed to be absolutely equidistant. While this expectation is ultimately based on theoretical mode-locking models, experimental tests have scrutinized deviations from equidistance of mode-locked laser combs down to the 10−15 level. Experimental tests of equidistance are rather elaborate and typically require beating of different spectral parts of the comb with a reference comb. Such tests have also been performed with microresonator combs [2], which are not a priori guaranteed to be equidistant from theoretical considerations. Finally, there are rather recent reports of quantum cascade laser combs; yet careful investigations of the equidistance of such sources still appear to be outstanding. Here we propose and demonstrate a novel experimental test for the equidistance of frequency combs.

Pulse train instabilities have often given rise to confusion and misinterpretation in ultrafast pulse characterization measurements. Most prominently known as the coherent artifact, a partially mode-locked laser with a non-periodic waveform may still produce an autocorrelation that has often been misinterpreted as indication of a coherent pulse train. Some modern pulse characterization methods easily miss the presence of a coherent artifact, too. Here, we address the particularly difficult situation of a pulse train with chirp-only instability. This instability is shown to be virtually invisible to autocorrelation measurements, but can be detected with frequency-resolved optical gating, spectral phase interferometry for direct electric-field reconstruction, and dispersion scan. Our findings clearly show that great care is necessary to rule out a chirp instability in lasers with an unclear mode-locking mechanism and in compression experiments in the single-cycle regime. Among all dynamical pulse train instabilities analyzed so far, this instability appears to be the best-hidden incoherence and is most difficult to detect.

Few-cycle pulsed laser technology highlights the need for control and stabilization of the carrier-envelope phase (CEP) for applications requiring shot-to-shot timing and phase consistency. This general requirement has been achieved successfully in a number of free space and fiber lasers via feedback and feed-forward methods. Expanding upon existing results, we demonstrate CEP stabilization through the feed-forward method applied to a SESAM mode-locked Er:Yb:glass laser at 1.55 um with a measured ultralow timing jitter of 2.9 as (1 Hz - 3 MHz) and long-term stabilization over a duration of eight hours. Single-digit attosecond stabilization at telecom wavelengths opens a new direction in applications requiring ultra-stable frequency and time precision such as high-resolution spectroscopy and fiber timing networks., Comment: 4 pages, 4 figures, submitted to Optics Letters

Optics express 29(10), 14314 (2021). doi:10.1364/OE.422739, A pulse-shaper-based method for spectral phase measurement and compression with milliradian precision is proposed and tested experimentally. Measurements of chirp and third-order dispersion are performed and compared to theoretical predictions. The single-digit milliradian accuracy is benchmarked by a group velocity dispersion measurement of fused silica., Published by Soc., Washington, DC

Nonlinear dynamics of continuous-wave pumped regenerative amplifiers operating at 2 μm are investigated. At repetition rates near 1 kHz, three different operation regimes are observed, including stable regular, chaotic, and subharmonic dynamics. Numerical simulations reproduce this behavior in a quantitative way. In particular, we find stable periodic doubling regimes in which every other seed pulse experiences high gain. Exploiting a narrow parameter window beyond the onset of chaos enables operation of a high-gain picosecond Ho:YLF regenerative amplifier which delivers up to 16 mJ picosecond pulses at 2050 nm. Energy fluctuations of the 700 Hz pulse train are as low as 0.9% rms.

The past 30 years have seen spectacular progress in the development of techniques for measuring the complete temporal field, and even the complete spatiotemporal field, of ultrashort laser pulses. The challenge has been to measure a pulse without the use of a shorter event or an independent known reference pulse, neither of which is typically available. We begin with autocorrelation, the first such “self-referenced” pulse-measurement method ever proposed, which measures only a rough pulse length, and we describe its limitations. One such limitation is the presence of a somewhat unintuitive “coherent artifact,” which occurs for complicated pulses and also when averaging over a train of pulses whose shapes vary from pulse to pulse. We then describe the most important modern techniques capable of measuring the complete temporal intensity and phase of even complicated ultrashort pulses, as well as their ability (or inability) to measure such unstable pulse trains. A pulse reliably measured with such a device can then be used as a reference pulse in conjunction with another technique, such as spectral interferometry or holography, to measure pulses otherwise unmeasurable by a self-referenced technique. Examples include techniques for measuring low-intensity pulse(s) and for measuring the complete spatiotemporal intensity and phase of arbitrary pulse(s). This Tutorial is limited to well-established, proven methods, but other methods whose description proves instructive will be discussed.

Controlling the carrier envelope phase (CEP) in mode-locked lasers over practically long timescales is crucial for real-world applications in ultrafast optics and precision metrology. We present a hybrid solution that combines a feed-forward technique to stabilize the phase offset in fast timescales and a feedback technique that addresses slowly varying sources of interference and locking bandwidth limitations associated with gain media with long upper-state lifetimes. We experimentally realize the hybrid stabilization system in an Er:Yb:glass mode-locked laser and demonstrate 75 hours of stabilization with integrated phase noise of 14 mrad (1 Hz to 3 MHz), corresponding to around 11 as of carrier to envelope jitter. Additionally, we examine the impact of environmental factors, such as humidity and pressure, on the long-term stability and performance of the system., 11 pages, 6 figures, 1 table; to be published in Optics Express; revisions include adjustment of some citations, additions to section 2, and added information in table 1

Spectral coherence is an important prerequisite for many applications of femtosecond supercontinua, including precision frequency metrology, attosecond science, and high-field physics. These applications often critically depend on measurement of the carrier-envelope phase via f − 2 f interferometry. As mode-locked laser sources cannot directly provide octave-spanning spectra, one typically resorts to spectral broadening in highly nonlinear optical fibers to generate the necessary spectral coverage. This process comes with a caveat, as coherence can be severely degraded in the broadening process. This degradation can be mitigated by suitable choice of a fiber dispersion profile. Here we numerically investigate prototypical fiber designs and analyze their susceptibility to coherence degradation. It is found that the generally favored soliton fission process provides the best broadening efficiency at the expense of rather scarce coherence conditions. Sufficient pulse energy provided, all-normally dispersive fiber designs fare far better in terms of spectral coherence. Finally, fiber designs with flattened normal dispersion provide the best compromise between efficiency and resulting spectral coherence. Our study indicates that the spectral width of a supercontinuum may be deceiving. Even when compressible into a short pulse, this does not automatically qualify for sufficient spectral coherence to conduct carrier-envelope-phase-sensitive experiments. The numerical simulations in this study provide a guideline for coherent spectral broadening. In turn, these guidelines may help to improve carrier-envelope phase stabilization and, in particular, to enable stabilization of laser oscillators that have previously proven difficult to stabilize.

Mode locking of a 1.34 μm vertical external cavity surface emitting laser is demonstrated using a GaSb-based semiconductor saturable absorber mirror (SESAM). The SESAM includes six AlGaSb quantum wells (QWs) with an absorption edge at ∼1.37 μm. The proposed approach has two key benefits: the QWs can be grown lattice matched, and only a small number of Bragg reflector layers is required to provide high reflectivity. Pump–probe measurements also reveal that the AlGaSb/GaSb structure exhibits an intrinsically fast absorption recovery on a picosecond timescale. The mode-locked laser pulse train had a fundamental repetition rate of 1.03 GHz, a pulse duration of ∼5 ps, and a peak power of ∼1.67 W. The demonstration paves the way for exploiting GaSb-based SESAMs for mode locking in the 1.3–2 μm wavelength range, which is not sufficiently addressed by GaAs and InP material systems.

Even more than 15 years after their first experimental demonstration, supercontinua have remained an extremely active field of research. In particular, photonic crystal fibers enable the broadening a laser source with a few nanometer initial spectral coverage to a supercontinuum that may encompass several octaves. Unfortunately, this extreme broadening does not come without a caveat: the initially coherent laser light loses most of this favorable property, which makes it impossible to compress the white-light pulse train to the bandwidth-limited single-cycle duration that two octaves could theoretically support. More precisely, this problem arises because of a loss of interpulse coherence, i.e., subsequent white-light pulses exhibit dramatically varying amplitude and phase structures. In fact, these variations are so extreme that they do not even follow a Gaussian distribution anymore, which gives rise to the phenomenon of optical rogue waves. Optical rogue waves are often explained by soliton dynamics. In the light of the above considerations, however, this seems quite paradoxical as solitons are coherent waveforms that evolve in a highly deterministic fashion. Moreover, frequency metrology applications of supercontinua exist which do not seem to be corrupted by a loss of interpulse coherence. In order to resolve this apparent conflict, we propose a new intrapulse coherence definition, which is experimentally verified by fringe contrast measurements in an f-to-2f interferometer. Numerical simulations indicate that intrapulse coherence is typically quite robust in the case of Kerr-dominated spectral broadening whereas it also quickly vanishes in plasma-dominated broadening scenarios, e.g., during filamentation. Interpulse coherence, in contrast, becomes more fragile at the low photon numbers of oscillator sources. As these two types of coherence appear rather independent of each other, situations can arise where interpulse coherence is conserved but intrapulse coherence vanishes and vice versa. Moreover, compressibility into a train of short pulses must not be used to conclude on the robustness of intrapulse coherence. We believe that this new coherence criterion has important implications, both for frequency metrology as well as for carrier-envelope phase stabilization of lasers. Specifically, in frequency metrology, limitations may arise for the obtainable maximum precision of optical frequency measurements. These limitations may impact the redefinition of the fundamental time and length units replacing the current microwave cesium standard by an optical one.

The near-ultraviolet photoluminescence of ZnO nanorods induced by multiphoton absorption of unamplified Ti:sapphire pulses is investigated. Power dependence measurements have been conducted with an adaptation of the ultrashort pulse characterization method of interferometric frequency-resolved optical gating. These measurements enable the separation of second harmonic and photoluminescence bands due to their distinct coherence properties. A detailed analysis yields fractional power dependence exponents in the range of 3-4, indicating the presence of multiple nonlinear processes. The range in measured exponents is attributed to differences in local field enhancement, which is supported by independent photoluminescence and structural measurements. Simulations based on Keldysh theory suggest contributions by three- and four-photon absorption as well as avalanche ionization in agreement with experimental findings.

We present a detailed study of all-optical signal amplification that exploits the extreme sensitivity of supercontinuum generation to input power fluctuations. As useful signal amplification relies on determinism and correlation between input and output signals, the question naturally arises whether such conditions can be maintained in the presence of modulation instability, which is known to strongly affect the coherence of supercontinuum pulse trains. In order to address this question, the effect of supercontinuum coherence on the amplification of a modulation is investigated when injecting weakly modulated 200-fs input pulses into a photonic crystal fiber. Our study, which is performed for multiple spectral channels across the full bandwidth of the supercontinuum, clearly shows that even in the case of partial coherence, the amplification mechanism based on soliton-dispersive wave coupling can be maintained, allowing for the amplification of weakly modulated signals by factors in excess of 40 dB in the normal dispersion regime. acceptedVersion

We demonstrate frequency-doubling of an amplified femtosecond laser with electric-field induced second-harmonic generation in air. The method promises application with unattenuated terawatt lasers and offers wavelength conversion beyond the ultraviolet limit of nonlinear crystals.

Frequency doubling of an amplified laser is experimentally demonstrated with electric-field induced second-harmonic generation in air. The method promises application with unattenuated terawatt laser and offers wavelength conversion beyond the ultraviolet limit of nonlinear crystals.

The intrapulse coherence is introduced as a measure for judging a fixed phase relation between different spectral components within a laser pulse. This new criterion plays an important role for passive CEP stabilization of OPA systems.