Abstract

Femtosecond power scaling in anomalous dispersion waveguides like telecom fiber is limited by runaway pulse collapse. We achieve an order of magnitude increase over previous femtosecond erbium fiber lasers by using phase-only pulse shaping in a stretcher fiber Bragg grating and identify a truncated Jacobi elliptic function pulse shape with better nonlinear compression characteristics than standard pulses. We generate 1560 nm, 340 nJ, 63 fs, 2.4 MW peak power pulses at 25 MHz repetition rate with standard single-mode fiber and a chirped mirror pair rather than bulk gratings. At a higher 100 MHz repetition rate, we verify frequency comb stability with 110 nJ, 62 fs pulses at 11 W average power. This record output can combine comb precision with strong field physics, and the method broadly applies to improving ultrafast laser sources.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Femtosecond fiber lasers are robust, compact sources for high average power and repetition rate applications such as frequency combs [1]. Higher peak intensities are needed to enable comb implementations for nonlinear optical and strong field interactions. Intense frequency combs are of such interest that resonant enhancement cavities are used to boost the intensity of fiber combs enough to generate high harmonics [2].

Fiber laser power is limited by uncontrolled nonlinearity at high pulse intensities. To compensate, the pulse can be stretched in time with chirped pulse amplification, in space with specialty fibers with larger mode size, or by exiting the fiber and completing compression in free-space bulk optics [3]. The shape of the pulse is a key parameter for maximizing the pulse energy of a given system. For example, in normal dispersion Yb fiber lasers at 1 µm, parabolic pulse shapes exhibit self-similar propagation [4], enabling µJ level amplification [5] while maintaining their functional pulse shape. Past the gain-shaping limit of self-similar amplification is the gain-managed nonlinearity regime with its own characteristic asymmetric pulse shape [6,7].

For longer wavelength Er lasers at 1.5 µm, and Tm and Ho at 2 µm, silica glass fibers have anomalous dispersion (with the exception of small core fibers that limit pulse energy to about 10 nJ [8,9]) so parabolic pulses are no longer self-similar. Intense pulses in anomalous dispersion fiber instead undergo nonlinear self-compression, conveniently generating spectral bandwidth and short pulses, but also making them sensitive and prone to pulse breakup.

Large-mode Tm amplifiers, aided by the lower nonlinearity of their long wavelengths [10], have produced strong output, for example 1 µJ, 24 fs [11] with external compression. For Er fiber lasers, 0.9 mJ pulses with large-mode fiber at 485 fs was shown at kHz repetition rates [12], but below 100 fs, pulse energy has been limited to 34 nJ [13], even with large-mode fiber [14], or 60 nJ if including coherent combination [15]. A recent system reached 0.54 MW peak power with 15 nJ, 9.4 fs pulses with a fiber compression stage [16]. Pulse energies of published Er fiber lasers below 100 fs are plotted in Fig. 1.

 figure: Fig. 1.

Fig. 1. Published high-energy Er fiber lasers by pulse duration and energy for pulse durations below 100 fs [1316] with the work of Luo et al. in [15] being a coherently combined system. Phase shaping lets us use nonlinear compression at much higher pulse energies than before.

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Compared to Yb and Tm, intense Er sources have been lacking [17]. While nonlinearity in the gain fiber can be managed by stretching, increasing Er output requires better dispersion control to prevent nonlinear collapse at high intensities near the end of the fiber where the pulse is short and intense. Standard fiber laser construction adjusts second-order dispersion by changing fiber length. Third-order control is possible if the lengths of two different fiber types are simultaneously adjusted [18]. To gain much finer and more flexible control, we add lossless and economical phase shaping.

Phase shaping has been used in a few fiber lasers. These lasers used external grating compression so that the pulse was still stretched while in fiber [5,19]. Our system, illustrated in Fig. 2, is between such external grating systems and monolithic fiber systems [13]. It uses a fiber Bragg grating (FBG) stretcher and compressor pair for chirped pulse amplification, with nonlinear spectral broadening and temporal compression in the compressor FBG and the remaining 2 cm of fiber labeled “exit fiber.” The stretcher is mounted on a heater array [20], providing computer control of the stretcher group delay dispersion. A chirped mirror pair provides a small amount of final compression. They are not required, but can improve stability by reducing maximum intensity within fiber, without adding the size or pulse front tilt of bulk gratings. Full spectral amplitude and phase shaping with spatial light modulators is also possible [21,22], but we avoid the significant complexity and cost, and find that phase-only shaping already allows us to reach our component power limits.

 figure: Fig. 2.

Fig. 2. Er fiber frequency comb system and pulse shapes. Dispersion tuning of the stretcher fiber grating adjusts the pulse shape for improved nonlinear compression in the exit fiber. This allows power scaling to 340 nJ, 63 fs pulses in standard PM1550 single mode, anomalous dispersion fiber.

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With phase shaping, our Er frequency comb improved from about 20 nJ, 100 fs pulses to reach record 340 nJ, 63 fs, 2.4 MW peak power pulses at a 25 MHz repetition rate. At a 100 MHz repetition rate suitable for frequency combs, we produced 62 fs, 110 nJ, 1.4 MW peak power pulses for 11 W average power. We verified coherent amplification for frequency comb applications by measuring and stabilizing the f-2f interferometry signal of the amplified output with 0.4 rad in-loop integrated phase noise, and locking an amplified comb line to a narrow-linewidth reference laser. An immediate application of such a high power Er comb would be supercontinuum generation with enough power for simultaneous precision timing transfer to multiple wavelengths and f-2f interferometry in a single branch, rather than multiple branches that add relative phase noise [23].

Our calculations show that, for our case of nonlinear compression in a short fiber length with chirped mirrors, standard Gaussian or ${\sec}{{\rm{h}}^2}$ pulses are not optimum. We have found a better pulse shape that compresses in short fibers to higher intensities with high stability, consistent with our improved output. These pulses, which we call Jacobi pulses after the Jacobi sn elliptic function, can be formed from a Gaussian spectrum by phase shaping, resulting in a rounder peak and shorter tails. Better understanding of nonlinear compression is especially important for compression to few cycles, which pairs well with frequency comb stabilization for carrier envelope phase-dependent applications such as high harmonic generation in solids [24] and nanoelectronics [25].

Our approach greatly improves nonlinear waveguide compression in the anomalous dispersion regime. This includes most femtosecond fiber systems at wavelengths longer than 1.3 µm, such as Er, Tm, and Ho fiber amplifiers, as well as compression and broadening systems in fluoride, chalcogenide, and gas-filled, hollow-core fibers. The availability of high intensity fiber combs will accelerate mid-IR and strong field frequency comb development and applications.

2. LASER SYSTEM

Our laser is outlined in Fig. 2. The system is seeded by 35 pJ, 100 fs pulses at a 100 MHz repetition rate from a conventional erbium fiber frequency comb [26,27] based on a nonlinear amplifying loop mirror [28], or a similar 25 MHz oscillator for higher pulse energy. The pulse is stretched in a 5 cm long chirped fiber Bragg grating for chirped pulse amplification. The stretcher grating is contacted to 32 individually addressable resistive heaters. One heating element can increase the group delay by up to about 1 ps with thermal spread within about 5 nm of bandwidth. Several pixels can increase the local group delay by more than 2 ps, as calibrated by frequency-resolved optical gating (FROG) measurements at low pulse energy. Heater settings are found by computer optimization of the second harmonic autocorrelation peak (equivalent to maximizing frequency doubling) of the compressed output pulses. A simple algorithm of optimizing pixels individually in random order compresses the pulse well within tens of minutes, with each temperature change taking several seconds to settle.

The stretched pulses are preamplified in core pumped Er fiber before amplification in single-mode Er/Yb fiber, cladding pumped by up to 36 W of 976 nm light. The stretched amplified pulses then enter a compact micro-optics circulator which sends the beam first into the compressor grating, and then directs the reflected beam out of the circulator as a free-space beam. The circulator includes lenses for free-space coupling to the two fibers, a polarizing beam splitter, and a Faraday rotator to separate the oppositely directed beams. The compressor grating has the same nominal dispersion as the stretcher.

The compressor grating does not begin immediately at the circulator, but has 2 cm of standard PM1550, anomalous dispersion fiber between the lens and the grating (exit fiber in Fig. 2). The pulse undergoes most nonlinearity during compression by the FBG and its return through the exit fiber, determining the output spectrum and phase. A few meters of passive anomalous and normal dispersion fiber before the stretcher provides coarse dispersion control, with roughly unchirped pulses emerging from the circulator at low power. After leaving fiber, a pair of anomalous dispersion chirped mirrors, equivalent to about 16 cm of linear transmission in PM1550 fiber, performs the final pulse compression. More experimental details can be found in Supplement 1.

A. Spectral Phase Contributions

In terms of phase contributions to the pulse, the system has two main regions: before and after the compressor grating. Spectral phases accumulated by the pulse before the compressor grating can simply be added linearly, while phase after the compressor is determined by the nonlinear propagation of the pulse shape that enters the final length of fiber after the compressor. In this view, the shaper’s task is to control the pulse that emerges from the compressor fiber and ensure that it has the optimal shape for nonlinear propagation in the remaining fiber.

In the region before the compressor, there are spectral phase contributions from fiber dispersion, stretcher dispersion including phase shaping, and self-phase modulation (SPM) in the amplifier. Usually, SPM would change the pulse shape and alter future SPM. However, in analogy to the Fraunhofer approximation [29], the linear stretcher dispersion is high enough that the pulse shape is simply a mapping of the spectral shape to time. Phases added by shaping or SPM are much smaller than the stretching phase, and have minimal effect on the pulse shape, and consequently SPM, so that the various phases gained before the compressor can be added together linearly. After coarse compensation by fiber lengths, the phase shaper can compensate unwanted phase from SPM and higher-order fiber dispersion, as well as add a custom phase that further improves the pulse shape. Practically, this is handled automatically by the shaper optimization algorithm.

B. Measured Pulses—30 nJ

We start at 30 nJ, which is about the limit of this type of system without pulse shaping. At this range, SPM in the amplifier becomes strong enough that the nonlinear compression becomes poor and the output pulse has a significant pedestal. Figure 3 shows second harmonic FROG reconstructions of the pulse (bar plot with a cyclic colormap by phase; the ambiguous phase sign and temporal direction of FROG has not been determined) for three different shaper settings (illustrated in the inset as heating versus wavelength). The top row is with no shaping. The center row is with an optimized linear chirp, which is roughly equivalent to a fiber length adjustment. The pulse is fairly short at 94 fs, but with significant pedestal. The bottom row is with a computer-optimized group delay dispersion. The pulse is shorter at 74 fs, the pedestal is greatly reduced, and the spectral wings are broader and share a common phase (line plot in Supplement 1).

 figure: Fig. 3.

Fig. 3. Er laser at 30 nJ pulse energy with FROG temporal reconstruction (left), and spectrum (right), with phase mapped to color. Insets show the heater settings as power versus wavelength. Top: No pulse shaping. Center: Linear chirp only. Bottom: Computer-optimized pulse. Linear chirp can generate a low-quality pulse. Optimized shaping generates a noticeably cleaner pulse, which is visible in not just the shorter pulse, but also the phase and spectrum.

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C. Measured Pulses—110 nJ

Increasing pumping, we get to our component limits at 110 nJ for a 100 MHz repetition rate. Figures 4(a) and 4(b) show the FROG temporal reconstruction and spectrum of the 110 nJ, 62 fs FWHM pulse (bar plot with cyclic colomap by phase) and a 113 nJ simulation (solid line), including full numerical propagation of a Jacobi shaped pulse in the FBG compressor, which is described later. The peak power is estimated by scaling the temporal intensity to the pulse energy. Note the clean phase in the main pulse, and the small spectral phase range. Figure 4(c) shows the logarithm of the clean and symmetric experimental FROG trace with a 31 dB range colorscale. The calculated pulse matches the experimental pulse quite well, including some detailed spectral features. The experimental pulse is actually cleaner than the calculation, likely from the greater degrees of freedom and faster iteration time available from pulse shaping than was in the simulation.

 figure: Fig. 4.

Fig. 4. (a)–(b) Temporal and spectral profiles of 110 nJ, 1.4 MW peak power, 62 fs, 11 W average power frequency comb pulses from a single-mode Er fiber laser. Experiment is colored by phase, and the simulation is the solid line with 113 nJ pulse energy. Spectra are normalized to have the same total power. Inset is heater power with wavelength. (c) Experimental FROG trace with logarithmic color scale showing clean pulse structure.

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D. Measured Pulses—340 nJ

To further increase the pulse energy in this system, we reduced the repetition rate by seeding with a 25 MHz oscillator. We were able to maintain short pulse generation up to 340 nJ at full pumping, with 63 fs duration and up to 2.4 MW peak power, as shown in Fig. 5. Spectral bandwidth increased, indicating greater nonlinearity, leading to more difficulty suppressing satellite pulses, and some pulse instability. The spectral oscillations are from interference with the satellite pulses. The pulse is still remarkably well controlled for a pulse with 0.6 MW peak power in fiber. This system was designed for 100 MHz repetition rate; at 25 MHz, the nonlinearity would preferably be reduced closer to the levels of the 110 nJ pulses by reducing the length of the exit fiber, and using more chirped mirror bounces. Another option is to add intensity noise stabilization, which would increase the system tolerance to high nonlinearity.

 figure: Fig. 5.

Fig. 5. Temporal and spectral profiles of 340 nJ, 2.4 MW peak power, 63 fs, pulses at 25 MHz repetition rate. Phase is mapped to color by phase, and the spectrum is normalized. Temporal plot inset is a zoom of the main peak. Spectral plot inset is the heater setting versus wavelength.

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E. Frequency Comb

Frequency combs would be an immediate application of this type of high average power, high repetition rate system. With the significant nonlinearity, it is possible that the system is too sensitive to generate a frequency comb. Frequency combs usually have repetition rates on the order of 100 MHz due to limitations of radio frequency electronics, so we test our system seeded with the 100 MHz frequency comb oscillator.

We verified that this system is a frequency comb by performing f-2f interferometry [1] on a 5% sample of the free-space output beam. The f-2f beat frequency was strong at about 40 dB at 100 kHz resolution bandwidth, and could be phase locked to a reference frequency indicating stable carrier-envelope offset (CEO) frequency. We could also lock an amplified comb line to a narrow linewidth telecom reference laser with the oscillator’s repetition rate actuator, indicating that both frequency comb aspects can be stabilized. Both locks had less than half a radian of integrated phase noise from 3 MHz to 0.1 Hz on the in-loop beat note.

We also tested f-2f stabilization with an acousto-optic frequency shifter in the fundamental beam to shift the beat frequency, allowing us to stabilize the CEO frequency at 0 and small kHz level frequencies. This means that the carrier envelope phase can be constant or slowly changing, as would be useful in measuring phase-sensitive processes [24,25]. More details of the comb tests are in Supplement 1.

3. SIMULATION

There are many practical limits to what we can measure; for example, the stretched pulse is too long for our FROG, and the double pass from the compressor grating prevents measurements inside the exit fiber. We use numerical simulation to further understand this new operating regime, and to verify that the results are realistic. From these simulations, we find that phase control is key for both compensating the nonlinear phase added by the power amplifier, and for forming a Jacobi pulse which greatly improves the nonlinear compression.

As done earlier, we conceptually divide the laser at the FBG compressor. Before the compressor, spectral phases such as from SPM add linearly to the stretched pulse, and can be precisely compensated and manipulated by phase shaping. There is an ambiguous region within the FBG compressor itself as the pulse compresses. While we have done simulations of the full FBG with nonlinearity, just one calculation, such as in Fig. 4, takes many hours. We simplify the system by approximating the FBG compressor as an infinitely thin element that linearly adds the FBG phase, and represent the nonlinearity within the fiber grating as propagation in standard fiber with about half the length of the actual grating after the thin FBG. This makes the post-compressor region a PM1550 fiber with a length of 4 cm (approximately half the FBG compressor length added to the exit fiber length) followed by a chirped mirror pair. In the simpler pre-compressor region, the shaper cancels phases such as amplifier SPM and fiber dispersion, and the phase of the pulse leaving this region can be set as desired. The problem then reduces to choosing the optimum pulse for nonlinear propagation in the short length of fiber after the compressor FBG.

As a first approximation, we modeled nonlinear propagation based on the standard nonlinear Schrödinger equation,

$$i\frac{{\partial A}}{{\partial z}} = \frac{{{\beta _2}}}{2}\frac{{{\partial ^2}A}}{{\partial {t^2}}} - \gamma |A{|^2}A,$$
where $A$ is the slowly varying complex pulse envelope, $z$ is the propagation distance, ${\beta _2}$ is the fiber dispersion, and $\gamma$ is the nonlinearity parameter of the fiber. For the standard fibers used here we have ${\beta _2} = 2.6 \times {10^{- 26}}\;{{\rm{s}}^2}/{\rm{m}}$ and $\gamma = 0.0011\;({\rm{W}}\;{\rm{m}}{)^{- 1}}$. We ignored gain narrowing in the power amplifier, which we verified experimentally to be small.

When leaving the exit fiber, the phase has contributions from: the original phase of the pulse leaving the FBG compressor (effectively set by the shaper); fiber dispersion; and the phase from nonlinear propagation. The phase from nonlinear propagation dominates, and is governed primarily by the temporal pulse power profile $P(t) = |{A_{\rm{in}}}{|^2}$ launched into the exit fiber, which can be controlled by the pulse shaper. In equation form, the output of the exit fiber is roughly ${A_{\rm{out}}}(t) = {A_{\rm{in}}}\exp (i\gamma {z_{\textit{nl}}}{P_{\rm{in}}}(t))$, for a fiber length ${z_{\textit{nl}}}$, before including the spectral phase of the chirped mirror pair.

More accurately, propagating the nonlinear Schrödinger equation lets us calculate ${A_{\rm{out}}}(t)$ as a function of the input pulse ${P_{\rm{in}}}(t)$. This means we can iteratively optimize the input to improve the output pulse. As detailed in Supplement 1, we used a stochastic parallel gradient descent algorithm to optimize a fitness parameter $F = ({\rm{peak}}\;{\rm{power}}) \times ({\rm{Strehl}}\;{\rm{ratio}})$, with peak power for the output pulse. This temporal version of the Strehl ratio measures pulse compression quality, calculated as the peak power of the compressed output pulse divided by the transform-limited (uniformly zero spectral phase) peak power in time. This ranges from 0 for a very long pulse, to 1 when transform-limited.

We start with ${P_{\rm{in}}}(t)$ of transform limited Gaussian pulses based on our experimental parameters (Gaussian spectrum with 105 nJ pulse energy and 14 nm spectral bandwidth entering a 4 cm long fiber). The algorithm optimizes the fitness function by applying phase, or phase and amplitude modifications in the spectral domain. We emulated the FBG shaper with 15 sections of individual group delay elements, with group delay ranging from 0–2 ps with a bandwidth of around 5 nm as in the actual laser system, while also having the option to give the virtual elements amplitude-shaping capabilities.

The standard pulses are clearly not optimal, as the algorithm easily finds modified pulses with better peak powers and Strehl ratios, in either case of phase, or phase and amplitude shaping. This was also true for initial ${\sec}{{\rm{h}}^2}$ pulses, while parabolic pulses had good peak power, but poor Strehl ratio from their large bandwidth. The improved pulse form had a mostly parabolic main pulse and short tails. To conveniently work with this improved pulse shape, we fit the optimal shape as the square of the Jacobi elliptic function sn(x, 0.61), truncated between the two zeros at ${\rm{x}} = {{0}}$ and 3.92, scaling x to time, with zero power outside these points, which we call a Jacobi pulse.

Figure 6 summarizes these calculations, showing the pulse input to the fiber, the simulated pulse that leaves the fiber, and the pulse after compression by the chirped mirrors. The pulses here are: a Gaussian, a phase-optimized Gaussian, a Jacobi pulse matched to the optimized Gaussian, and a phase-optimized Jacobi pulse. The Jacobi pulse here matches the main peak of the shaped Gaussian temporal power, so it actually has a lower total pulse energy from the lack of wings. A pure Gaussian input generates a reasonable output, with some wings (noting that a system without shaping would not have a pure Gaussian at this point because of SPM and fiber dispersion). After phase optimization, the dashed blue set shows much higher peak power and smaller wings when compressed. The Jacobi version of this shaped Gaussian input (having flat phase and non-Gaussian spectrum) is shown by the orange set. The dashed green set is the slightly improved version found by optimizing the Jacobi pulse with phase shaping. The Jacobi and optimized pulses are almost identical, verifying that the Jacobi shape accurately represents the new optimized pulse form.

 figure: Fig. 6.

Fig. 6. Simulation of propagation through 4 cm of single-mode fiber, followed by chirped mirrors. The three groups of curves are for three points in the system, from top to bottom: Input to fiber, output from fiber (shifted down), and pulse after chirped mirrors. The four input shapes entering the fiber are: Gaussian, phase-shaped Gaussian, matching Jacobi, and phase-shaped Jacobi, the last three of which are very similar. The three Jacobi-like pulses compress to much higher peak power with cleaner shape than a pure Gaussian.

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This particular Jacobi pulse is an optimum for our system. Different system parameters, or desired pulse characteristics will change the exact shape of the optimum, but overall, the Jacobi pulse shape clearly provides much better nonlinear compression than Gaussian or ${\sec}{{\rm{h}}^2}$ pulses, with up to seven times the peak power in our calculations, depending on SPM in the fiber segment, up to the onset of Raman scattering. Relations for scaling the laser system for Jacobi pulses at different powers are given in Supplement 1.

A. Jacobi Pulse Shape

The Jacobi elliptic functions are general doubly periodic functions which can describe systems like the nonlinear large angle motion of a simple pendulum, periodic pulse train solutions of the nonlinear Schrödinger equation [30], and solitons in a ring microresonator [31]. Feeding the Jacobi pulse back into the optimization routine finds only modest improvement (plotted in Supplement 1), even when increasing the spectral resolution of pulse shaping by a factor of two and adding more fiber dispersion terms to the propagation calculation.

Various analytic pulse forms are shown in Fig. 7. Gaussian and ${\sec}{{\rm{h}}^2}$ pulses have a large fraction of their energy in the tails, which do not participate effectively in the nonlinear compression process, so simply confining more power to the peak already improves performance. Unlike a ${\sec}{{\rm{h}}^2}$ soliton, or a self-similar parabolic pulse, the Jacobi pulse here is not an exact solution to a steady-state problem, but rather a fit to a numerical optimization of a system with variable parameters and variable output criteria. However, the Jacobi shape does have a clear physical meaning when considering a simpler system.

 figure: Fig. 7.

Fig. 7. Comparison of analytic Jacobi and standard pulse shapes. The Jacobi pulse has a parabolic peak with more realistic wings. Gaussian and ${\sec}{{\rm{h}}^2}$ pulses have sharp peaks and intense wings. The pulses have the same energy and FWHM.

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First, we note that the fiber is followed by mainly linearly chirped mirrors, so the best case would be a linearly chirped output from the fiber. In SPM, the frequency is modified proportionally to the time derivative of the intensity, so a parabolic intensity profile (as most pulses approximately are near the peak) produces a linear chirp. Second, Fig. 6 shows us the important point that the Jacobi pulse does not strongly change shape after 4 cm of fiber, so we ignore the shape change during propagation here, and consider the spectral broadening to be determined just from the time derivative of the initial shape. That makes a parabolic pulse shape ideal, as all generated spectrum will have the same linear chirp, as shown on the top of Fig. 8. However, unstretched parabolic pulses are impractical, as the cutoff at the edges requires very broad spectra.

 figure: Fig. 8.

Fig. 8. Calculations illustrating the Jacobi pulse shape viewed as phase shaping a Gaussian spectrum toward a parabolic shape. The dashed curves are phase-shaped Gaussian pulses with transform limited pulse durations of 40, 65, or 90% of the plotted parabolic pulse they are trying to emulate. With more bandwidth, the shaped Gaussian is able to trace the parabola closer to the sharp edges. The intensity time derivative (top, arbitrary units) is proportional to the chirp added by self-phase modulation. The broader the linear region in the center, the more spectrum and energy that can be compressed by linearly chirped mirrors.

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The problem then becomes trying to emulate a parabolic pulse using a Gaussian spectrum and phase shaping. This is a straightforward numerical calculation, with calculated pulses and their first time derivatives shown in Fig. 8. The plotted parabolic pulse is the target shape. The dashed curves are shaped Gaussians with different spectral bandwidths. When transform-limited, they have temporal FWHM of 40, 65, and 90% of the FWHM of the target parabolic pulse. A standard Powell optimization algorithm has adjusted the phases pixel by pixel as well as the total pulse energy to minimize the square of the difference to the parabolic shape. A Jacobi pulse with the same FWHM as the parabola is plotted for reference.

The specific FWHM and bandwidth values here are artificial; it is the relative FWHM that is important. Practically, there is a given Gaussian bandwidth from the amplifier, and it would be the FWHM of the target parabola that expands. We use a constant parabola here for the direct graphical comparison. For the 90% FWHM Gaussian, there is little leeway to change, so the shaped pulse still has a sharp peak with large tails, and has a linearly chirped region of about 200 fs. At 65%, the greater bandwidth can provide a closer fit, and the shaped pulse has a linear derivative over about 300 fs. The fitted shape is quite similar to the Jacobi pulse. At 40%, the greater bandwidth allows it to follow the parabola more closely as it approaches zero. This tail region has little energy, but also the largest spectral excursion, making it important for compression.

In this view, the Jacobi pulse arises from phase shaping a Gaussian spectrum to approximate a parabolic temporal shape. This assumes a short fiber, such that the chirp from SPM dominates fiber dispersion, and the pulse shape does not significantly change within the fiber, followed by a linear compressor. The Jacobi pulse is then an instance within a continuum of pulses from a Gaussian to a nearly parabolic shape as the pulse duration and chirp increases. The optimum will depend on system parameters like initial spectrum, fiber length, and chirped mirror dispersion.

B. Measured Jacobi Pulse

Experimentally, we cannot measure the Jacobi pulse inside the FBG compressor, and given the finite FBG length, a Jacobi pulse will not exactly occur as it does in our thin-FBG approximation. We can however measure Jacobi pulse formation at low powers without chirped mirrors, where the measured output is approximately the same as the pulse immediately after the FBG compressor.

Figure 9 shows a measured Jacobi pulse synthesized from a Gaussian spectrum, both experimentally with a simple shaping of 3 stepped group delays, and in a matching simulation. Pulse energy was 4 nJ to avoid nonlinearity. Synthesizing a Jacobi pulse from a Gaussian spectrum leads to small wings that are too weak to interfere with the following nonlinear compression. In this calculation, the pulse shaper was simulated as 15 elements with a 2 ps group delay range, with thermal broadening of 4 THz FWHM, and was optimized to match the experimental pulse shape.

 figure: Fig. 9.

Fig. 9. Analytic Jacobi pulse (dashed green), and measured (solid orange) and simulated (dashed blue) synthesized Jacobi pulses showing that it is possible to generate a Jacobi pulse from a Gaussian spectrum by phase-only shaping in the fiber Bragg grating stretcher. Inset shows the three-step heating profile with wavelength.

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C. Full Simulation

In addition to the faster approximation, we have also performed numerical simulations including the full length of FBG compressor, such as shown in Fig. 4. In the full simulation, linearly chirped, stretched Jacobi sn shaped pulses were propagated in a nonlinear chirped FBG. Time-dependent nonlinear coupled-mode equations [32] were generalized to include a linear chirp and tanh taper that closely matched the FBG reflection apodization. The compressed reflected pulses were then nonlinearly propagated through 20 mm of fiber, followed by compression by the chirped mirrors.

Both the full and approximate calculations generate similar overall characteristics such as spectral bandwidth or pulse duration that are comparable to the experimental results. The full simulation is better at matching the temporal and spectral asymmetry of the experiment. The full simulation includes the earlier long wavelength side compressing and experiencing more nonlinearity than the trailing short wavelength side that compresses last, and which is not part of the thin FBG approximation.

4. CONCLUSIONS

We have generated record 340 nJ, 63 fs pulses, and a 110 nJ, 62 fs, frequency comb from a single-mode Er fiber laser using phase shaping in a chirped fiber Bragg grating. Adapting the pulse shape to the propagation conditions allows nonlinear compression in anomalous dispersion fiber lasers to operate stably at much higher intensities, resulting in a higher pulse energy, average power, and peak intensity. We have identified the Jacobi pulse, an optimal pulse shape for nonlinear compression in short fibers that has the good compression characteristics of a parabolic pulse with improved tails and spectrum. It can be formed from a Gaussian spectrum by phase shaping for efficient and clean nonlinear compression.

Our results have several implications. For laser technology, while our results were in standard 9 µm core fiber, similar intensity threshold improvements should also apply to large mode area fibers, Tm fiber lasers, and anomalous waveguide pulse compression such as hollow core, chalcogenide and fluoride fibers, and external compression systems.

For nonlinear optics and strong-field physics, our frequency comb provides the necessary ${10^{12}}\;{\rm{W/c}}{{\rm{m}}^2}$ intensity level at 1.5 µm wavelengths, which can be particularly useful for semiconductors and molecules, where 1 µm Yb wavelengths are too close to electronic transitions, but 2 µm Tm wavelengths are too far. High intensities also mean efficient harmonic conversion; for example, replacing several-watt Ti:sapphire lasers with much more robust frequency doubled Er fiber lasers.

For real-world applications, we have shown that fiber Bragg grating systems can outperform external grating compressor systems. Concerns about FBG environmental stability can be overcome by using the adaptability of pulse shaping and simple engineering. High-dispersion bulk grating compressors are large and have pulse-tilt alignment sensitivity, requiring careful and stable alignment and positioning. In contrast, our FBG system is fully fiberized other than two chirped mirrors that are not alignment-dependent. A box of all the optics could easily be carried with one hand. For applications of intense frequency combs to become common instruments, compact and robust approaches like this one will be needed.

Funding

IMRA America, Inc.

Acknowledgment

The authors are grateful to S. T. Cundiff for stimulating discussions.

Disclosures

Authors Kevin F. Lee, Yu Yun, Jie Jiang, and Martin E. Fermann: IMRA America, Inc. (F, E, P).

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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6. P. Sidorenko, W. Fu, and F. Wise, “Nonlinear ultrafast fiber amplifiers beyond the gain-narrowing limit,” Optica 6, 1328–1333 (2019). [CrossRef]  

7. P. Sidorenko and F. Wise, “Generation of 1 µJ and 40 fs pulses from a large mode area gain-managed nonlinear amplifier,” Opt. Lett. 45, 4084–4087 (2020). [CrossRef]  

8. J. W. Nicholson, A. D. Yablon, P. S. Westbrook, K. S. Feder, and M. F. Yan, “High power, single mode, all-fiber source of femtosecond pulses at 1550 nm and its use in supercontinuum generation,” Opt. Express 12, 3025–3034 (2004). [CrossRef]  

9. Y. Tang, Z. Liu, W. Fu, and F. W. Wise, “Self-similar pulse evolution in a fiber laser with a comb-like dispersion-decreasing fiber,” Opt. Lett. 41, 2290–2293 (2016). [CrossRef]  

10. C. Gaida, M. Gebhardt, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Thulium-doped fiber chirped-pulse amplification system with 2 GW of peak power,” Opt. Lett. 41, 4130–4133 (2016). [CrossRef]  

11. C. Gaida, M. Gebhardt, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Self-compression in a solid fiber to 24 mw peak power with few-cycle pulses at 2 µm wavelength,” Opt. Lett. 40, 5160–5163 (2015). [CrossRef]  

12. X. Peng, K. Kim, M. Mielke, S. Jennings, G. Masor, D. Stohl, A. Chavez-Pirson, D. T. Nguyen, D. Rhonehouse, J. Zong, D. Churin, and N. Peyghambarian, “Monolithic fiber chirped pulse amplification system for millijoule femtosecond pulse generation at 1.55 µm,” Opt. Express 22, 2459–2464 (2014). [CrossRef]  

13. M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018). [CrossRef]  

14. J. W. Nicholson, A. Desantolo, W. Kaenders, and A. Zach, “Self-frequency-shifted solitons in a polarization-maintaining, very-large-mode area, Er-doped fiber amplifier,” Opt. Express 24, 23396–23402 (2016). [CrossRef]  

15. P. Luo, Q. Hao, H. Fu, and H. Zeng, “60 nJ, 95 fs environmentally stable Er-doped fiber laser system,” Chin. Opt. Lett. 17, 061401 (2019). [CrossRef]  

16. D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021). [CrossRef]  

17. G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013). [CrossRef]  

18. R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020). [CrossRef]  

19. J. Želudevičius, R. Danilevičius, and K. Regelskis, “Optimization of pulse compression in a fiber chirped pulse amplification system by adjusting dispersion parameters of a temperature-tuned chirped fiber Bragg grating stretcher,” J. Opt. Soc. Am. B 32, 812–817 (2015). [CrossRef]  

20. S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004). [CrossRef]  

21. T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19, 255–260 (2011). [CrossRef]  

22. F. He, H. S. S. Hung, J. H. V. Price, N. K. Daga, N. Naz, J. Prawiharjo, D. C. Hanna, D. P. Shepherd, D. J. Richardson, J. W. Dawson, C. W. Siders, and C. P. J. Barty, “High energy femtosecond fiber chirped pulse amplification system with adaptive phase control,” Opt. Express 16, 5813–5821 (2008). [CrossRef]  

23. A. Rolland, P. Li, N. Kuse, J. Jiang, M. Cassinerio, C. Langrock, and M. E. Fermann, “Ultra-broadband dual-branch optical frequency comb with 10–18 instability,” Optica 5, 1070–1077 (2018). [CrossRef]  

24. S. Ghimire and D. A. Reis, “High-harmonic generation from solids,” Nat. Phys. 15, 10–16 (2019). [CrossRef]  

25. M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019). [CrossRef]  

26. N. Kuse, J. Jiang, C.-C. Lee, T. Schibli, and M. Fermann, “All polarization-maintaining Er fiber-based optical frequency combs with nonlinear amplifying loop mirror,” Opt. Express 24, 3095–3102 (2016). [CrossRef]  

27. IMRA America, Inc., https://www.imra.com/products/imra-scientific/.

28. M. E. Fermann, F. Haberl, M. Hofer, and H. Hochreiter, “Nonlinear amplifying loop mirror,” Opt. Lett. 15, 752–754 (1990). [CrossRef]  

29. D. N. Schimpf, J. Limpert, and A. Tünnermann, “Controlling the influence of SPM in fiber-based chirped-pulse amplification systems by using an actively shaped parabolic spectrum,” Opt. Express 15, 16945–16953 (2007). [CrossRef]  

30. J. M. Soto-Crespo, N. Devine, and N. Akhmediev, “Adiabatic transformation of continuous waves into trains of pulses,” Phys. Rev. A 96, 023825 (2017). [CrossRef]  

31. R. D. D. Bitha and A. M. Dikandé, “Elliptic-type soliton combs in optical ring microresonators,” Phys. Rev. A 97, 033813 (2018). [CrossRef]  

32. H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985). [CrossRef]  

References

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  1. T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Commun. Phys. 2, 1–16 (2019).
    [Crossref]
  2. C. Zhang, S. B. Schoun, C. M. Heyl, G. Porat, M. B. Gaarde, and J. Ye, “Noncollinear enhancement cavity for record-high out-coupling efficiency of an extreme-UV frequency comb,” Phys. Rev. Lett. 125, 093902 (2020).
    [Crossref]
  3. J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
    [Crossref]
  4. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
    [Crossref]
  5. H. Chang, Z. Cheng, R. Sun, Z. Peng, M. Yu, Y. You, M. Wang, and P. Wang, “172-fs, 27-µJ, yb-doped all-fiber-integrated chirped pulse amplification system based on parabolic evolution by passive spectral amplitude shaping,” Opt. Express 27, 34103–34112 (2019).
    [Crossref]
  6. P. Sidorenko, W. Fu, and F. Wise, “Nonlinear ultrafast fiber amplifiers beyond the gain-narrowing limit,” Optica 6, 1328–1333 (2019).
    [Crossref]
  7. P. Sidorenko and F. Wise, “Generation of 1 µJ and 40 fs pulses from a large mode area gain-managed nonlinear amplifier,” Opt. Lett. 45, 4084–4087 (2020).
    [Crossref]
  8. J. W. Nicholson, A. D. Yablon, P. S. Westbrook, K. S. Feder, and M. F. Yan, “High power, single mode, all-fiber source of femtosecond pulses at 1550 nm and its use in supercontinuum generation,” Opt. Express 12, 3025–3034 (2004).
    [Crossref]
  9. Y. Tang, Z. Liu, W. Fu, and F. W. Wise, “Self-similar pulse evolution in a fiber laser with a comb-like dispersion-decreasing fiber,” Opt. Lett. 41, 2290–2293 (2016).
    [Crossref]
  10. C. Gaida, M. Gebhardt, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Thulium-doped fiber chirped-pulse amplification system with 2 GW of peak power,” Opt. Lett. 41, 4130–4133 (2016).
    [Crossref]
  11. C. Gaida, M. Gebhardt, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Self-compression in a solid fiber to 24 mw peak power with few-cycle pulses at 2 µm wavelength,” Opt. Lett. 40, 5160–5163 (2015).
    [Crossref]
  12. X. Peng, K. Kim, M. Mielke, S. Jennings, G. Masor, D. Stohl, A. Chavez-Pirson, D. T. Nguyen, D. Rhonehouse, J. Zong, D. Churin, and N. Peyghambarian, “Monolithic fiber chirped pulse amplification system for millijoule femtosecond pulse generation at 1.55 µm,” Opt. Express 22, 2459–2464 (2014).
    [Crossref]
  13. M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
    [Crossref]
  14. J. W. Nicholson, A. Desantolo, W. Kaenders, and A. Zach, “Self-frequency-shifted solitons in a polarization-maintaining, very-large-mode area, Er-doped fiber amplifier,” Opt. Express 24, 23396–23402 (2016).
    [Crossref]
  15. P. Luo, Q. Hao, H. Fu, and H. Zeng, “60 nJ, 95 fs environmentally stable Er-doped fiber laser system,” Chin. Opt. Lett. 17, 061401 (2019).
    [Crossref]
  16. D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021).
    [Crossref]
  17. G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
    [Crossref]
  18. R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
    [Crossref]
  19. J. Želudevičius, R. Danilevičius, and K. Regelskis, “Optimization of pulse compression in a fiber chirped pulse amplification system by adjusting dispersion parameters of a temperature-tuned chirped fiber Bragg grating stretcher,” J. Opt. Soc. Am. B 32, 812–817 (2015).
    [Crossref]
  20. S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004).
    [Crossref]
  21. T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19, 255–260 (2011).
    [Crossref]
  22. F. He, H. S. S. Hung, J. H. V. Price, N. K. Daga, N. Naz, J. Prawiharjo, D. C. Hanna, D. P. Shepherd, D. J. Richardson, J. W. Dawson, C. W. Siders, and C. P. J. Barty, “High energy femtosecond fiber chirped pulse amplification system with adaptive phase control,” Opt. Express 16, 5813–5821 (2008).
    [Crossref]
  23. A. Rolland, P. Li, N. Kuse, J. Jiang, M. Cassinerio, C. Langrock, and M. E. Fermann, “Ultra-broadband dual-branch optical frequency comb with 10–18 instability,” Optica 5, 1070–1077 (2018).
    [Crossref]
  24. S. Ghimire and D. A. Reis, “High-harmonic generation from solids,” Nat. Phys. 15, 10–16 (2019).
    [Crossref]
  25. M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
    [Crossref]
  26. N. Kuse, J. Jiang, C.-C. Lee, T. Schibli, and M. Fermann, “All polarization-maintaining Er fiber-based optical frequency combs with nonlinear amplifying loop mirror,” Opt. Express 24, 3095–3102 (2016).
    [Crossref]
  27. IMRA America, Inc., https://www.imra.com/products/imra-scientific/ .
  28. M. E. Fermann, F. Haberl, M. Hofer, and H. Hochreiter, “Nonlinear amplifying loop mirror,” Opt. Lett. 15, 752–754 (1990).
    [Crossref]
  29. D. N. Schimpf, J. Limpert, and A. Tünnermann, “Controlling the influence of SPM in fiber-based chirped-pulse amplification systems by using an actively shaped parabolic spectrum,” Opt. Express 15, 16945–16953 (2007).
    [Crossref]
  30. J. M. Soto-Crespo, N. Devine, and N. Akhmediev, “Adiabatic transformation of continuous waves into trains of pulses,” Phys. Rev. A 96, 023825 (2017).
    [Crossref]
  31. R. D. D. Bitha and A. M. Dikandé, “Elliptic-type soliton combs in optical ring microresonators,” Phys. Rev. A 97, 033813 (2018).
    [Crossref]
  32. H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
    [Crossref]

2021 (1)

D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021).
[Crossref]

2020 (3)

P. Sidorenko and F. Wise, “Generation of 1 µJ and 40 fs pulses from a large mode area gain-managed nonlinear amplifier,” Opt. Lett. 45, 4084–4087 (2020).
[Crossref]

C. Zhang, S. B. Schoun, C. M. Heyl, G. Porat, M. B. Gaarde, and J. Ye, “Noncollinear enhancement cavity for record-high out-coupling efficiency of an extreme-UV frequency comb,” Phys. Rev. Lett. 125, 093902 (2020).
[Crossref]

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
[Crossref]

2019 (6)

S. Ghimire and D. A. Reis, “High-harmonic generation from solids,” Nat. Phys. 15, 10–16 (2019).
[Crossref]

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

H. Chang, Z. Cheng, R. Sun, Z. Peng, M. Yu, Y. You, M. Wang, and P. Wang, “172-fs, 27-µJ, yb-doped all-fiber-integrated chirped pulse amplification system based on parabolic evolution by passive spectral amplitude shaping,” Opt. Express 27, 34103–34112 (2019).
[Crossref]

P. Sidorenko, W. Fu, and F. Wise, “Nonlinear ultrafast fiber amplifiers beyond the gain-narrowing limit,” Optica 6, 1328–1333 (2019).
[Crossref]

T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Commun. Phys. 2, 1–16 (2019).
[Crossref]

P. Luo, Q. Hao, H. Fu, and H. Zeng, “60 nJ, 95 fs environmentally stable Er-doped fiber laser system,” Chin. Opt. Lett. 17, 061401 (2019).
[Crossref]

2018 (3)

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
[Crossref]

A. Rolland, P. Li, N. Kuse, J. Jiang, M. Cassinerio, C. Langrock, and M. E. Fermann, “Ultra-broadband dual-branch optical frequency comb with 10–18 instability,” Optica 5, 1070–1077 (2018).
[Crossref]

R. D. D. Bitha and A. M. Dikandé, “Elliptic-type soliton combs in optical ring microresonators,” Phys. Rev. A 97, 033813 (2018).
[Crossref]

2017 (1)

J. M. Soto-Crespo, N. Devine, and N. Akhmediev, “Adiabatic transformation of continuous waves into trains of pulses,” Phys. Rev. A 96, 023825 (2017).
[Crossref]

2016 (4)

2015 (2)

2014 (1)

2013 (1)

G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
[Crossref]

2011 (1)

2009 (1)

J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
[Crossref]

2008 (1)

2007 (1)

2004 (2)

S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004).
[Crossref]

J. W. Nicholson, A. D. Yablon, P. S. Westbrook, K. S. Feder, and M. F. Yan, “High power, single mode, all-fiber source of femtosecond pulses at 1550 nm and its use in supercontinuum generation,” Opt. Express 12, 3025–3034 (2004).
[Crossref]

2000 (1)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref]

1990 (1)

1985 (1)

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[Crossref]

Abdollahi, M.

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
[Crossref]

Abramski, K. M.

G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
[Crossref]

Aguirregabiria, G.

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

Aizpurua, J.

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

Akhmediev, N.

J. M. Soto-Crespo, N. Devine, and N. Akhmediev, “Adiabatic transformation of continuous waves into trains of pulses,” Phys. Rev. A 96, 023825 (2017).
[Crossref]

Barty, C. P. J.

Baumann, E.

T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Commun. Phys. 2, 1–16 (2019).
[Crossref]

Bitha, R. D. D.

R. D. D. Bitha and A. M. Dikandé, “Elliptic-type soliton combs in optical ring microresonators,” Phys. Rev. A 97, 033813 (2018).
[Crossref]

Borisov, A. G.

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

Brida, D.

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

Carstens, H.

Cassinerio, M.

Chang, H.

Chavez-Pirson, A.

Cheng, Z.

Churin, D.

Daga, N. K.

Danilevicius, R.

Dawson, J. W.

Desantolo, A.

Devine, N.

J. M. Soto-Crespo, N. Devine, and N. Akhmediev, “Adiabatic transformation of continuous waves into trains of pulses,” Phys. Rev. A 96, 023825 (2017).
[Crossref]

Diddams, S. A.

D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021).
[Crossref]

Dikandé, A. M.

R. D. D. Bitha and A. M. Dikandé, “Elliptic-type soliton combs in optical ring microresonators,” Phys. Rev. A 97, 033813 (2018).
[Crossref]

Dudley, J. M.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref]

Ebrahimi, A.

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
[Crossref]

Eidam, T.

T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19, 255–260 (2011).
[Crossref]

J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
[Crossref]

Feder, K. S.

Fermann, M.

Fermann, M. E.

Fortier, T.

T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Commun. Phys. 2, 1–16 (2019).
[Crossref]

Fu, H.

Fu, W.

Gaarde, M. B.

C. Zhang, S. B. Schoun, C. M. Heyl, G. Porat, M. B. Gaarde, and J. Ye, “Noncollinear enhancement cavity for record-high out-coupling efficiency of an extreme-UV frequency comb,” Phys. Rev. Lett. 125, 093902 (2020).
[Crossref]

Gaida, C.

Gebhardt, M.

Ghimire, S.

S. Ghimire and D. A. Reis, “High-harmonic generation from solids,” Nat. Phys. 15, 10–16 (2019).
[Crossref]

Gholami, A.

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
[Crossref]

Haberl, F.

Hadrich, S.

J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
[Crossref]

Hädrich, S.

Hamidnejad, E.

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
[Crossref]

Hanna, D. C.

Hao, Q.

Harvey, J. D.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref]

Hasan, T.

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
[Crossref]

He, F.

Hekmat, M.

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
[Crossref]

Heyl, C. M.

C. Zhang, S. B. Schoun, C. M. Heyl, G. Porat, M. B. Gaarde, and J. Ye, “Noncollinear enhancement cavity for record-high out-coupling efficiency of an extreme-UV frequency comb,” Phys. Rev. Lett. 125, 093902 (2020).
[Crossref]

Hochreiter, H.

Hofer, M.

Hung, H. S. S.

Jansen, F.

Jauregui, C.

Jennings, S.

Jiang, J.

Kaczmarek, P.

G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
[Crossref]

Kaenders, W.

Kim, K.

Kowligy, A.

D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021).
[Crossref]

Kruglov, V. I.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref]

Krzempek, K.

G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
[Crossref]

Kubota, F.

S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004).
[Crossref]

Kuse, N.

Lai, W.

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
[Crossref]

Langrock, C.

Lee, C.-C.

Leitenstorfer, A.

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

Lesko, D. M.

D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021).
[Crossref]

Li, P.

Limpert, J.

Lind, A. J.

D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021).
[Crossref]

Liu, Z.

Ludwig, M.

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

Luo, P.

Marinica, D. C.

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

Masor, G.

Matsumoto, S.

S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004).
[Crossref]

Mielke, M.

Misas, C. J.

J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
[Crossref]

Miyazaki, T.

S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004).
[Crossref]

Naz, N.

Nguyen, D. T.

Nicholson, J. W.

Normohamadi, H.

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
[Crossref]

Omoomi, M.

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
[Crossref]

Pasternak, I.

G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
[Crossref]

Peng, X.

Peng, Z.

Peyghambarian, N.

Porat, G.

C. Zhang, S. B. Schoun, C. M. Heyl, G. Porat, M. B. Gaarde, and J. Ye, “Noncollinear enhancement cavity for record-high out-coupling efficiency of an extreme-UV frequency comb,” Phys. Rev. Lett. 125, 093902 (2020).
[Crossref]

Prawiharjo, J.

Price, J. H. V.

Regelskis, K.

Reis, D. A.

S. Ghimire and D. A. Reis, “High-harmonic generation from solids,” Nat. Phys. 15, 10–16 (2019).
[Crossref]

Rhonehouse, D.

Richardson, D. J.

Ritzkowsky, F.

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

Rolland, A.

Roser, F.

J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
[Crossref]

Rothhardt, J.

T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19, 255–260 (2011).
[Crossref]

J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
[Crossref]

Ruan, S.

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
[Crossref]

Rybka, T.

M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
[Crossref]

Schibli, T.

Schimpf, D. N.

J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
[Crossref]

D. N. Schimpf, J. Limpert, and A. Tünnermann, “Controlling the influence of SPM in fiber-based chirped-pulse amplification systems by using an actively shaped parabolic spectrum,” Opt. Express 15, 16945–16953 (2007).
[Crossref]

Schoun, S. B.

C. Zhang, S. B. Schoun, C. M. Heyl, G. Porat, M. B. Gaarde, and J. Ye, “Noncollinear enhancement cavity for record-high out-coupling efficiency of an extreme-UV frequency comb,” Phys. Rev. Lett. 125, 093902 (2020).
[Crossref]

Seise, E.

J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
[Crossref]

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Siders, C. W.

Sidorenko, P.

Sobon, G.

G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
[Crossref]

Soto-Crespo, J. M.

J. M. Soto-Crespo, N. Devine, and N. Akhmediev, “Adiabatic transformation of continuous waves into trains of pulses,” Phys. Rev. A 96, 023825 (2017).
[Crossref]

Sotor, J.

G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
[Crossref]

Stohl, D.

Strupinski, W.

G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
[Crossref]

Stutzki, F.

Sugihara, T.

S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004).
[Crossref]

Sun, R.

Sun, Z.

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
[Crossref]

Takabayashi, M.

S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004).
[Crossref]

Tang, Y.

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref]

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D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021).
[Crossref]

Tunnermann, A.

J. Limpert, F. Roser, D. N. Schimpf, E. Seise, T. Eidam, S. Hadrich, J. Rothhardt, C. J. Misas, and A. Tunnermann, “High repetition rate gigawatt peak power fiber laser systems: challenges, design, and experiment,” IEEE J. Sel. Top. Quantum Electron. 15, 159–169 (2009).
[Crossref]

Tünnermann, A.

Wang, J.

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
[Crossref]

Wang, M.

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
[Crossref]

H. Chang, Z. Cheng, R. Sun, Z. Peng, M. Yu, Y. You, M. Wang, and P. Wang, “172-fs, 27-µJ, yb-doped all-fiber-integrated chirped pulse amplification system based on parabolic evolution by passive spectral amplitude shaping,” Opt. Express 27, 34103–34112 (2019).
[Crossref]

Wang, P.

Wei, R.

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
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Wise, F. W.

Xing, S.

D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021).
[Crossref]

Yablon, A. D.

Yan, M. F.

Yan, P.

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
[Crossref]

Yazdabadi, A. B.

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
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C. Zhang, S. B. Schoun, C. M. Heyl, G. Porat, M. B. Gaarde, and J. Ye, “Noncollinear enhancement cavity for record-high out-coupling efficiency of an extreme-UV frequency comb,” Phys. Rev. Lett. 125, 093902 (2020).
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S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004).
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Yu, M.

Zach, A.

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C. Zhang, S. B. Schoun, C. M. Heyl, G. Porat, M. B. Gaarde, and J. Ye, “Noncollinear enhancement cavity for record-high out-coupling efficiency of an extreme-UV frequency comb,” Phys. Rev. Lett. 125, 093902 (2020).
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R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
[Crossref]

Zong, J.

Appl. Phys. B (1)

M. Hekmat, M. Omoomi, A. Gholami, A. B. Yazdabadi, M. Abdollahi, E. Hamidnejad, A. Ebrahimi, and H. Normohamadi, “All-fiber high-power monolithic femtosecond laser at 1.59 µm with 63-fs pulse width,” Appl. Phys. B 124, 4 (2018).
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Appl. Phys. Lett. (1)

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Commun. Phys. (1)

T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Commun. Phys. 2, 1–16 (2019).
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IEEE J. Sel. Top. Quantum Electron. (1)

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IEEE Photon J. (1)

R. Wei, M. Wang, Z. Zhu, W. Lai, P. Yan, S. Ruan, J. Wang, Z. Sun, and T. Hasan, “High-power femtosecond pulse generation from an all-fiber Er-doped chirped pulse amplification system,” IEEE Photon J. 12, 1–8 (2020).
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IEEE Photon. Technol. Lett. (1)

S. Matsumoto, M. Takabayashi, K. Yoshiara, T. Sugihara, T. Miyazaki, and F. Kubota, “Tunable dispersion slope compensator with a chirped fiber grating and a divided thin-film heater for 160-Gb/s RZ transmissions,” IEEE Photon. Technol. Lett. 16, 1095–1097 (2004).
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J. Opt. Soc. Am. B (1)

Laser Phys. Lett. (1)

G. Sobon, J. Sotor, I. Pasternak, W. Strupinski, K. Krzempek, P. Kaczmarek, and K. M. Abramski, “Chirped pulse amplification of a femtosecond Er-doped fiber laser mode-locked by a graphene saturable absorber,” Laser Phys. Lett. 10, 035104 (2013).
[Crossref]

Nat. Photonics (1)

D. M. Lesko, H. Timmers, S. Xing, A. Kowligy, A. J. Lind, and S. A. Diddams, “A six-octave optical frequency comb from a scalable few-cycle erbium fibre laser,” Nat. Photonics 15, 281–286 (2021).
[Crossref]

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M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, “Sub-femtosecond electron transport in a nanoscale gap,” Nat. Phys. 16, 341–345 (2019).
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M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
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Other (1)

IMRA America, Inc., https://www.imra.com/products/imra-scientific/ .

Supplementary Material (1)

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» Supplement 1       Supplemental document

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)

Fig. 1.
Fig. 1. Published high-energy Er fiber lasers by pulse duration and energy for pulse durations below 100 fs [1316] with the work of Luo et al. in [15] being a coherently combined system. Phase shaping lets us use nonlinear compression at much higher pulse energies than before.
Fig. 2.
Fig. 2. Er fiber frequency comb system and pulse shapes. Dispersion tuning of the stretcher fiber grating adjusts the pulse shape for improved nonlinear compression in the exit fiber. This allows power scaling to 340 nJ, 63 fs pulses in standard PM1550 single mode, anomalous dispersion fiber.
Fig. 3.
Fig. 3. Er laser at 30 nJ pulse energy with FROG temporal reconstruction (left), and spectrum (right), with phase mapped to color. Insets show the heater settings as power versus wavelength. Top: No pulse shaping. Center: Linear chirp only. Bottom: Computer-optimized pulse. Linear chirp can generate a low-quality pulse. Optimized shaping generates a noticeably cleaner pulse, which is visible in not just the shorter pulse, but also the phase and spectrum.
Fig. 4.
Fig. 4. (a)–(b) Temporal and spectral profiles of 110 nJ, 1.4 MW peak power, 62 fs, 11 W average power frequency comb pulses from a single-mode Er fiber laser. Experiment is colored by phase, and the simulation is the solid line with 113 nJ pulse energy. Spectra are normalized to have the same total power. Inset is heater power with wavelength. (c) Experimental FROG trace with logarithmic color scale showing clean pulse structure.
Fig. 5.
Fig. 5. Temporal and spectral profiles of 340 nJ, 2.4 MW peak power, 63 fs, pulses at 25 MHz repetition rate. Phase is mapped to color by phase, and the spectrum is normalized. Temporal plot inset is a zoom of the main peak. Spectral plot inset is the heater setting versus wavelength.
Fig. 6.
Fig. 6. Simulation of propagation through 4 cm of single-mode fiber, followed by chirped mirrors. The three groups of curves are for three points in the system, from top to bottom: Input to fiber, output from fiber (shifted down), and pulse after chirped mirrors. The four input shapes entering the fiber are: Gaussian, phase-shaped Gaussian, matching Jacobi, and phase-shaped Jacobi, the last three of which are very similar. The three Jacobi-like pulses compress to much higher peak power with cleaner shape than a pure Gaussian.
Fig. 7.
Fig. 7. Comparison of analytic Jacobi and standard pulse shapes. The Jacobi pulse has a parabolic peak with more realistic wings. Gaussian and ${\sec}{{\rm{h}}^2}$ pulses have sharp peaks and intense wings. The pulses have the same energy and FWHM.
Fig. 8.
Fig. 8. Calculations illustrating the Jacobi pulse shape viewed as phase shaping a Gaussian spectrum toward a parabolic shape. The dashed curves are phase-shaped Gaussian pulses with transform limited pulse durations of 40, 65, or 90% of the plotted parabolic pulse they are trying to emulate. With more bandwidth, the shaped Gaussian is able to trace the parabola closer to the sharp edges. The intensity time derivative (top, arbitrary units) is proportional to the chirp added by self-phase modulation. The broader the linear region in the center, the more spectrum and energy that can be compressed by linearly chirped mirrors.
Fig. 9.
Fig. 9. Analytic Jacobi pulse (dashed green), and measured (solid orange) and simulated (dashed blue) synthesized Jacobi pulses showing that it is possible to generate a Jacobi pulse from a Gaussian spectrum by phase-only shaping in the fiber Bragg grating stretcher. Inset shows the three-step heating profile with wavelength.

Equations (1)

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i A z = β 2 2 2 A t 2 γ | A | 2 A ,