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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.

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 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.

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.

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.

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.

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.

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.

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.

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

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.

The concept of coherence is of fundamental importance for describing the physical characteristics of light and for evaluating the suitability for experimental application. In the case of pulsed laser sources, the pulse-to-pulse coherence is usually considered for a judgment of the compressibility of the pulse train. This type of coherence is often lost during propagation through a highly nonlinear medium, and pulses prove incompressible despite multioctave spectral coverage. Notwithstanding the apparent loss of interpulse coherence, however, supercontinua enable applications in precision frequency metrology that rely on coherence between different spectral components within a laser pulse. To judge the suitability of a light source for the latter application, we define an alternative criterion, which we term intrapulse coherence. This definition plays a limiting role in the carrier-envelope phase measurement and stabilization of ultrashort pulses. It is shown by numerical simulation and further corroborated by experimental data that filamentation-based supercontinuum generation may lead to a loss of intrapulse coherence despite near-perfect compressibility of the pulse train. This loss of coherence may severely limit active and passive carrier-envelope phase stabilization schemes and applications in optical high-field physics.

A precise control of the electric field nodes relative to the maximum of the intensity envelope is a prerequisite for attosecond pulse generation and experiments in high-field physics [1]. Many carrier-envelope phase (CEP) stabilization schemes start with stabilization of an oscillator, which is nevertheless often difficult to obtain as the CEP has proven highly sensitive to environmental parameters, including temperature, air pressure, and humidity. This very high susceptibility to environmental changes is considered more of a nuisance than anywhere useful. Here, we show that the extreme sensitivity of the CEP can be exploited for quantitative measurements of the electro-optic Kerr effect in atmospheric air, i.e., an extremely weak effect that has previously only been measurable by combining effective path lengths of 100m with tens of kilovolt voltages. In contrast, we can obtain a measurable effect by placing an 8 cm long high voltage capacitor into the intracavity beam path of a mode-locked few-cycle Ti: sapphire oscillator, and we only require rather moderate driver voltages of less than 5 kV.

Stabilizing the carrier-envelope phase (CEP) of mode-locked laser sources has made tremendous progress in recent years, enabling the generation of isolated attosecond pulses and precision frequency metrology at the 10−18 uncertainty level. While some laser types can be very well stabilized, other interesting laser materials have proven notoriously difficult or even impossible to be CEP stabilized. The exact mechanisms behind these problems are poorly understood. Here, we present a novel diagnostic method that provides unprecedented insight into the inner workings of noise generating mechanisms in laser oscillators.

A novel pulse characterization method is presented, favorably combining interferometric frequency-resolved optical gating (FROG) and time-domain ptychography. This new variant is named ptychographic-interferometric frequency-resolved optical gating (πFROG). The measurement device is simple, bearing similarity to standard second-harmonic FROG, yet with a collinear beam geometry and an added bandpass filter in one of the correlator arms. The collinear beam geometry allows tight focusing and circumvents possible geometrical distortion effects of noncollinear methods, making πFROG especially suitable for the characterization of unamplified few-cycle pulses. Moreover, the direction-of-time ambiguity afflicting most second-order FROG variants is eliminated. Possible group delay dispersion of pulses leads to a characteristic tilt in the πFROG traces, allowing the detection of uncompensated dispersion without a retrieval. Using nanojoule, three-cycle pulses at 800 nm, the πFROG method is tested, and the results are compared with spectral phase interferometry for direct electric field reconstruction measurements. Measured pulse durations agree within a fraction of a femtosecond. As a further test, the πFROG measurements are repeated with added group delay dispersion, and found to accurately reproduce the dispersion computed with Sellmeier equations.

Attosecond spectroscopy and precision frequency metrology depend on the stabilization of the carrier-envelope phase (CEP) of mode-locked lasers. Unfortunately, the phase of only a few types of lasers can be stabilized to jitters in the few-hundred millirad range. In a comparative experimental study, we analyze a femtosecond Ti:sapphire laser and three mode-locked fiber lasers. We numerically demodulate recorded time series of the free-running carrier-envelope beat note. Our analysis indicates a correlation between amplitude and frequency fluctuations at low Fourier frequencies for essentially all lasers investigated. While this correlation typically rolls off at frequencies beyond 100 kHz, we see clear indications for a broadband coupling mechanism in one of the fiber lasers. We suspect that the observed coupling mechanism acts to transfer intracavity power fluctuations into excess phase noise. This coupling mechanism is related to the mode-locking mechanism employed and not to the gain medium itself. We further verify this hypothesis by numerical simulations, which identify resonances of the saturable absorber mirror as a possible explanation for the coupling mechanism. Finally, we discuss how to avoid a detrimental influence of such resonances.

Precision frequency metrology and attosecond pulse generation critically rely on stabilization of the carrier-envelope phase (CEP) of mode-locked lasers. So far, only a relatively small class of lasers has been successfully stabilized to warrant phase jitters of a few hundred milliradians as they are required for the generation of an isolated attosecond pulse. For stabilizing certain laser types, the exact reasons for the observed difficulties (or the lack thereof) is only poorly understood. Here we compare the free-running CEP noise of four different lasers, including a femtosecond Ti:sapphire laser and three mode-locked fiber lasers. This study indicates a correlation between amplitude and frequency fluctuations at low Fourier frequencies for essentially all lasers investigated. This finding is well explained with technical noise sources and thermal coupling mechanisms below the upperstate lifetime of the laser gain material. However, for one of the lasers under test, we observe a broadband amplitude-to-phase coupling mechanism well above the upperstate lifetime. This coupling mechanism is related to a dynamical loss modulation. We verify our explanation by numerical simulations, which identify resonances of the saturable absorber mirror as a possible explanation for the coupling mechanism. In case of high modulation depth and resonantly enhanced saturation characteristics, such a saturable absorber can give rise to broadband conversion of spontaneous emission amplitude noise into phase noise, which may cause, in turn, extremely broadband noise signatures, exceeding a megahertz bandwidth.

We demonstrate a novel carrier-envelope frequency noise power spectral density characterization method with >70 dB dynamic range across 8-decade Fourier frequency range.

This paper represents a systematic investigation of detection shot noise in carrier-envelope phase (CEP) stabilization. Numerical simulations are conducted to calculate the influence of shot noise in laser oscillators. These results are compared with experimental results for Ti:sapphire lasers. It is found that shot noise imposes a limitation for obtaining sub-100 mrad CEP jitters. Careful interferometer design is necessary to push this limit toward 10 mrad. In contrast to oscillator stabilization, shot noise appears to play a much more restrictive role in amplifier stabilization. Using spectral interferometry together with spectral broadening in sapphire, it already appears practically challenging to reach sub-100 mrad jitters. Adaption of the optical nonlinearity in the broadening step appears key to further improvements of the CEP jitter of amplified systems. We believe that these improvements open a perspective for currently unfeasible applications of CEP stabilized pulses. Moreover, our considerations can be easily adapted to CEP stabilization of other laser types beyond Ti:sapphire.

Focusing short CO2 laser pulses into air reveals rich ionization physics that is best explained by solid-state theories and results in centimetre-scale-diameter megafilaments that transport joules of energy.

We investigate the influence of the optical Doppler effect on the carrier-envelope frequency (CEF) of a mode-locked pulse train. The laser pulses are Doppler-shifted in frequency during reflection off a periodically moving mirror that is driven by an electro-dynamical exciter inside an 𝑓-2𝑓 interferometer. Depending on the relative movement of the mirror at the instant of reflection, we experimentally demonstrate a CEF shift of the laser pulses up to ±69 kHz, which is sufficient for the carrier-envelope phase control of laser amplifiers with repetition rates of 10 kHz and beyond. Using piezoelectric thick films, we show that the scheme also appears to be adaptable to the megahertz repetition rates of typical oscillators. As the phase control is exerted extra-cavity, Doppler-induced CEF modulation is virtually free of any side effects of traditional stabilization schemes that typically act on the pump power. Finally, the Doppler scheme may overcome servo loop bandwidth limitations associated with pump power control.

When confronted with a pulse train whose intensity and/or phase versus time varies from pulse to pulse, multi-shot pulse-measurement techniques usually exhibit a coherent artifact (CA), which substantially complicates the interpretation of the measurement. In frequency-resolved optical gating (FROG), such instabilities are indicated by discrepancies between the measured and retrieved FROG traces. Here we consider the simultaneous retrieval of the CA and the average pulse characteristics from a single FROG trace in the limit of significant fluctuations. We use a modified generalized projections algorithm. Two electric fields are simultaneously retrieved, while the data constraint is updated as the algorithm progresses using only the assumption that the trace can be modeled as the sum of two spectrograms, one corresponding to the pulse and the other corresponding to the CA. An additional flat-spectral-phase constraint is added to one of the fields to ensure that it only reacts to the presence of the CA. Using this novel retrieval method, the complete retrieval of the characteristics of pulses in an unstable train from FROG traces is demonstrated.

The f-to-2f interferometer plays a key role for carrier-envelope phase (CEP) measurement and subsequent stabilization. The CEP measurement typically relies on the application of two optical nonlinearities, namely supercontinuum generation and second-harmonic generation. Then the cascadation of these nonlinearities often leads to signal levels on the order of a few photons per pulse. We experimentally demonstrate that the introduction of optical gain into the infrared arm of an f-to-2f interferometer can mitigate this detection bottleneck and improve signal-to-noise ratios by 20 dB compared to purely passive schemes. We further show that this measure allows for residual phase jitters between the carrier and envelope of about 10 mrad, corresponding to record-breaking timing jitters in the single attosecond regime. Moreover, the method appears generally applicable to a wide range of oscillators in the near-infrared and may enable stable CEP locking of mode-locked oscillators that so far have resisted stabilization. Finally, we propose a parametric variant of the active f-to-2f interferometer that can be used for laser amplifiers with kilohertz repetition rates and below.

Rogue waves, by definition, are rare events of extreme amplitude. At the same time, they are surprisingly ubiquitous, in the sense that they can exist in a wide range of physical contexts and possess probability distributions that exhibit heavier tails than the normal Gaussian distribution. While many mechanisms have been demonstrated to explain the appearance of rogue waves in various specific systems, there is no known generic mechanism or general set of criteria shown to rule their appearance. Presupposing only the existence of a nonlinear Schrodinger-type equation together with a concave dispersion profile around a zero-dispersion wavelength, we demonstrate that solitons may experience acceleration and strong reshaping due to the interaction with continuum radiation, giving rise to extreme-value phenomena. The mechanism appears to be widely independent from interactions specific to the optical context, e.g., the Raman effect or other scattering processes that have no equivalent in other wave-supporting physical systems. In our system, a strong increase in the peak power may appear via reshaping while the pulse energy is nearly conserved. The conservative nature of the proposed reshaping-induced appearance of rogue waves makes this mechanism particularly robust.

Multi-shot pulse-shape measurements of trains of ultrashort pulses with unstable pulse shapes are studied. Measurement techniques considered include spectral-phase interferometry for direct electric-field reconstruction (SPIDER), second harmonic generation frequency-resolved optical gating (FROG), polarization gate FROG, and cross-correlation FROG. An analytical calculation and simulations show that SPIDER cannot see unstable pulse-shape components and only measures the coherent artifact. Further, the presence of this instability cannot be distinguished from benign misalignment effects in SPIDER. FROG methods yield a better, although necessarily rough, estimate of the pulse shape and also indicate instability by exhibiting disagreement between measured and retrieved traces. Only good agreement between measured and retrieved FROG traces or 100% SPIDER fringe visibility guarantees a stable pulse train.

Fiber-optical rogue waves appear as rare but extreme events during optical supercontinuum generation in photonic crystal fibers. This process is typically initiated by the decay of a high-order fundamental soliton into fundamental solitons. Collisions between these solitons as well as with dispersive radiation affect the soliton trajectory in frequency and time upon further propagation. Launching an additional dispersive wave at carefully chosen delay and wavelength enables statistical manipulation of the soliton trajectory in such a way that the probability of rogue wave formation is either enhanced or reduced. To enable efficient control, parameters of the dispersive wave have to be chosen to allow trapping of dispersive radiation in the nonlinear index depression created by the soliton. Under certain conditions, direct manipulation of soliton properties is possible by the dispersive wave. In other more complex scenarios, control is possible via increasing or decreasing the number of intersoliton collisions. The control mechanism reaches a remarkable efficiency, enabling control of relatively large soliton energies. This scenario appears promising for highly dynamic all-optical control of supercontinua.

We demonstrate an extremely accurate method for measuring ultrabroadband, sub-10 fs pulses even if they exhibit a highly modulated spectrum, space-time coupling, or both. The method uses a spatially encoded arrangement for spectral phase interferometry for direct electric field reconstruction, which allows a zero additional phase measurement to be performed with a relatively low signal-to-noise ratio in real time and single shot.

Rogue waves are extremely large waves that exceed any expectation based on long-term observation and Gaussian statistics. Ocean rogue waves exceed the significant wave height in the ocean by a factor 2. Similar phenomena have been observed in a multiplicity of optical systems. While the optical systems show a much higher frequency of rogue events than the ocean, it appears nevertheless questionable what conclusions can be drawn for the prediction of ocean rogue waves. Here we tackle the problem from a different perspective and analyze the predictability of rogue events in two optical systems as well as in the ocean using nonlinear time-series analysis. Our analysis is exclusively based on experimental data. The results appear rather surprising as the optical rogue wave scenario of fiber-based supercontinuum generation does not allow any prediction whereas real ocean rogue waves and a multifilament scenario do bear a considerable amount of determinism, which allows, at least in principle, the prediction of extreme events. It becomes further clear that there exist two fundamentally different types of rogue-wave supporting systems. One class of rogue waves is obviously seeded by quantum fluctuations whereas in the other class, linear random interference of waves seems to prevail.

A method for analyzing the statistical nature of laser noise is demonstrated. By application to CEP noise characteristics, the method is shown to enable discriminating between deterministic technical and stochastic quantum noise contributions.

The noise performance of a feed-forward scheme for carrier-envelope phase stabilization is discussed. This scheme uses an acousto-optic frequency shifter to directly correct for fluctuations of the carrier-envelope phase in a pulse train emitted by a mode-locked laser without manipulating the intracavity dispersion. Generation of zero-offset frequency combs is demonstrated. Furthermore, it is shown that pump laser noise has only a minor effect on the achievable performance. Limited only by the travel time of the acoustic wave in the shifter, pump laser noise can be corrected up to near-megahertz frequencies, which yields superior noise performance compared to traditional feedback operation. Residual phase jitters down to 45 mrad are experimentally verified.

The introduction of a dressing beam to extend the reach of filaments promises a wealth of new applications that extend beyond the laboratory scale.

Broadband characterization of the carrier-envelope phase (CEP) noise spectral density of free-running mode-locked lasers is essential for advanced low-noise optical frequency comb designs. Here we present a direct method that utilizes an optical heterodyne beat between a pair of repetition-rate-locked mode-locked lasers for CEP noise characterization, without requiring an f-2f interferometer or nonlinear optical conversion steps. A proof-of-principle experiment in a femtosecond Yb-fiber laser achieves CEP noise spectral density characterization with >270 dB dynamic range over a Fourier frequency range from 5 mHz to 8 MHz. The measurement noise floor is well below 1 μrad/√Hz, enabling dependable detection down to a quantum-limited noise floor. The method can resolve various noise mechanisms that cause specific CEP noise spectral shapes. The underlying mechanisms are further analyzed in terms of spurious temporal correlation to distinguish between technical and stochastic noise signatures. Moreover, a Hadamard deviation analysis reveals a varying degree of frequency stability in the measured CEP time series.

Scientists demonstrate a simple self-referenced feed-forward approach for stabilizing the carrier–envelope phase of femtosecond light pulses. Twelve attoseconds of residual timing jitter below the atomic unit of time is achieved, surpassing the precision of previous methods by more than a factor of five.

A plasma induced temporal break-up in filamentary propagation has recently been identified as one of the key events in the temporal self-compression of femtosecond laser pulses. An analysis of the Nonlinear Schrodinger Equation coupled to a noninstantaneous plasma response yields a set of stationary states. This analysis clearly indicates that the emergence of double-hump, characteristically asymmetric temporal on-axis intensity profiles in regimes where plasma defocusing saturates the optical collapse caused by Kerr self-focusing is an inherent property of the underlying dynamical model.

The carrier-envelope offset frequency of a laser oscillator is determined from the visibility of spectrally resolved fringes in a combined two-path multiple-path interferometer. At maximum visibility the pulses have zero carrier-envelope phase drift, while the visibility becomes zero for uncorrelated pulses. The method is widely independent of bandwidth and pulse energy. The effects of carrier-envelope offset phase noise, finite detection time, and dispersion are also discussed.

Performance and noise immunity of different interferometer set-ups for carrier-envelope phase detection are compared. The frequently used Mach–Zehnder interferometer is found to be easily corrupted by acoustic noise contributions and air streaks, whereas a quasi-common-path variant of the f-to-2f interferometer exhibits a 40% reduction of residual noise. This comparative analysis also provides deeper insight into additional mechanisms that are currently limiting the performance of carrier-envelope phase stabilization schemes.

Digital holographic shape measurements using femtosecond laser pulses are reported. For contouring of very fast moving objects, the simultaneous generation of at least two spectrally separated ultrashort pulses is required. To deliver this particular spectral signature at high pulse energies, a chirped-pulse Ti:sapphire laser amplifier was modified to emit two spectrally separated pulses with energies above 1 μJ each. The wavelength separation of these pulses was adjustable within the 50 nm gain bandwidth, cutting out two distinct wavelength peaks by a variable double-slit assembly in a prism sequence. A Michelson-type interferometer was employed to perform the two-wavelength contouring. The phases of the holograms and the phase differences are calculated numerically, which allow us to deduce the contour lines of the topology of the object. The suitability of the light source for digital holography is demonstrated with contouring of stationary objects and the potential for high-speed applications is indicated.