Optical frequency combs are typically generated in the near-infrared wavelength range, where many mode-locked lasers operate. Nonlinear frequency conversion can then be used to extend optical frequency metrology to other spectral regions, such as the extreme ultraviolet (XUV). High-power frequency combs at the fundamental wavelength can efficiently drive nonlinear frequency conversions. Low phase noise is an important requirement because the frequency conversion process not only multiplies the carrier frequency but also the phase fluctuations. In this work, we have developed a low-noise frequency comb centered at 1030 nm with an average power of 230 W, a temporal pulse duration of 59 fs, and a peak power of 32 MW. One of the modes of the seed laser is phase-locked to a continuous wave reference laser stabilized to an ultra-stable high-finesse cavity. The residual integrated phase noise from 10 Hz to 10 MHz is 41 mrad, which is sufficiently low to address narrow transitions with kHz-level linewidths after the frequency conversion to XUV wavelengths.

Many mode-locked lasers operate in the infrared (IR) wavelength range. Generating optical frequency combs outside of this range often requires nonlinear frequency conversion. For example, a high-power infrared frequency comb can be focused into a gaseous or solid medium to coherently extend its spectrum into the UV to mid-IR region using nonlinear responses such as self-phase modulation (SPM) and difference frequency generation.1,2

Frequency conversion to the extreme ultraviolet (XUV) region is possible using the process of high-order harmonic generation (HHG).3–5 In single-pass geometry, it is challenging to achieve sufficiently high peak intensity to drive the HHG process efficiently at a high repetition rate. This is because many pulses per unit of time share the available average power (see dashed lines in Fig. 1). Tightly focused, high average power lasers have been used for this aim.6,7 However, the demonstrated power levels are insufficient for many frequency metrology applications. More powerful XUV frequency combs may be generated using the intracavity HHG scheme, where high average power laser pulses build up in an enhancement cavity and HHG takes place at the cavity focus.8–12 The HHG process involves partial ionization of the gas target that serves as the nonlinear medium. The resulting plasma buildup during the pulse causes a blue shift of the circulating field and limits the enhancement (intensity clamping). Furthermore, the steady-state plasma in the focal volume affects the active cavity length stabilization, requiring an offset in the servo lock-point to avoid instability.13–15 Both effects can be mitigated at the same time by reducing the cavity finesse and using a higher average power impinging frequency comb.13 Therefore, high-power combs are also essential for the intracavity HHG scheme.

FIG. 1.

Peak power of laser systems for driving the HHG process at repetition rates higher than 50 kHz. Blue squares and circles represent HHG experiments with single-pass geometry based on Ti:sapphire and ytterbium laser systems, respectively. Solid red triangles show the peak powers of laser systems used for intracavity HHG. Open red triangles represent the peak powers within enhancement cavities. Our laser system, based on a Yb:YAG Innoslab amplifier and pulse compression using the multi-pass cell spectral broadening (MPCSB) scheme, is given by the green triangles. Only [Sau19] and our work use the MPCSB pulse compression scheme, which can be employed to achieve higher pulse energy than nonlinear pulse compression by solid core fibers. Our circulating peak power (1.3 GW, open green triangle) is similar to that of [Por19], but is achieved with a smaller enhancement factor and, therefore, mitigates the limitations due to ionization of the HHG medium.16 References: [Car16],17 [Cor18],18 [Mill19],19 [Por19],20 [Sau19],21 [Yos11],22 [Zha22],23 and [Oza15].9 

FIG. 1.

Peak power of laser systems for driving the HHG process at repetition rates higher than 50 kHz. Blue squares and circles represent HHG experiments with single-pass geometry based on Ti:sapphire and ytterbium laser systems, respectively. Solid red triangles show the peak powers of laser systems used for intracavity HHG. Open red triangles represent the peak powers within enhancement cavities. Our laser system, based on a Yb:YAG Innoslab amplifier and pulse compression using the multi-pass cell spectral broadening (MPCSB) scheme, is given by the green triangles. Only [Sau19] and our work use the MPCSB pulse compression scheme, which can be employed to achieve higher pulse energy than nonlinear pulse compression by solid core fibers. Our circulating peak power (1.3 GW, open green triangle) is similar to that of [Por19], but is achieved with a smaller enhancement factor and, therefore, mitigates the limitations due to ionization of the HHG medium.16 References: [Car16],17 [Cor18],18 [Mill19],19 [Por19],20 [Sau19],21 [Yos11],22 [Zha22],23 and [Oza15].9 

Close modal

Figure 1 shows the parameters of the laser systems used for previous studies on HHG at repetition rates above 50 kHz. Both single-pass HHG and intracavity HHG experiments are shown in this figure. Many experiments use frequency combs with average powers exceeding 10 W. With a power enhancement factor typically on the order of 102, the peak intracavity power exceeds 1 GW, comparable to conventional single-pass HHG experiments using low repetition-rate laser systems.

An important application of XUV frequency combs could be high-precision spectroscopy of simple atoms/ions and highly charged ions, as well as laser cooling at XUV wavelengths. To address narrow transitions in the XUV region, the phase noise of the XUV frequency comb must be sufficiently small. In an ideal coherent frequency conversion process, the carrier phase of the qth harmonic ϕq(t) is given by ϕq(t) = 1(t), where ϕ1(t) = ωct + Δϕ(t) is the carrier phase of the fundamental radiation with the carrier frequency ωc and the phase noise Δϕ(t). If the laser is sufficiently low-noise, its spectrum can be described as a narrow carrier at ωc surrounded by a broad phase noise “pedestal” (see Sec. III). The observational transition linewidth is then not affected by this noise; only the excitation probability is reduced. After the conversion, the optical power contained in the narrow carrier is reduced to expq2Δϕrms2, where Δϕrms is the integrated fast phase noise of the fundamental radiation (see Sec. III). Δϕrms should, therefore, be kept small to efficiently drive a narrow transition. The large reduction in the carrier power due to phase noise is known as “carrier collapse.”24 

We plan to perform high-precision spectroscopy of the 1S–2S two-photon transition in trapped He+ ions, which can be driven with an XUV frequency comb at 60.8 nm.16,25 The energy levels of this hydrogen-like ion can be accurately described by the theory of quantum electrodynamics, and the measurement will serve as a precise test of the theory. We will use the intracavity HHG process to produce the XUV frequency comb as the 17th harmonic of an infrared frequency comb centered at 1033 nm. The infrared frequency comb should have a high average power to efficiently drive the HHG process. To avoid a significant reduction of the excitation rate due to phase noise at the 2 × 17 = 34th harmonic, the phase noise integrated from the observational transition linewidth (expected to be a few kHz16) to half the repetition rate should be kept well below (1 rad)/34 = 29 mrad.16 Note that the effective harmonic order is 34 because the 17th harmonic is used to drive a two-photon transition.

The master oscillator power amplifier (MOPA) scheme with chirped pulse amplification (CPA) enables the generation of ultra-short-pulse lasers with high pulse energy. In the design of a MOPA-based CPA system, special attention is required to maintain pulse-to-pulse phase coherence and achieve a low-noise frequency comb. Using a Yb-doped fiber oscillator and a large-mode-area Yb-doped fiber amplifier, Ruehl and co-authors generated an infrared frequency comb with 80 W average power and 120 fs pulse duration.26 Luo and co-authors demonstrated chirped pulse amplification up to 130 W average power.27 Ye and co-authors reported a 200 W average power frequency comb based on an electro-optic comb generator operating at a GHz repetition rate.28 High-power frequency combs based on Yb-doped fiber lasers are also realized using only standard off-the-shelf components to achieve 80 W average power with a 155 fs pulse duration.29 By seeding a Yb-doped fiber amplifier with a Ti:sapphire oscillator, Saule and co-authors demonstrated a frequency comb with an average power of 80 W and a pulse duration of 30 fs.30 By appropriately controlling the chirp of the seed pulses, the nonlinear spectral broadening that occurs in Yb-doped fiber amplifiers can be exploited to generate short pulses. Using this scheme, Luo and co-authors demonstrated an optical frequency comb with a pulse duration of 42 fs and an average power of 109 W,31 and Liu and co-authors achieved a peak power of 51 MW.32 By coherently combining the output of multiple Yb-doped fiber amplifiers, a carrier-envelope-offset-stable pulse train with 1 kW average power was demonstrated.33 By stabilizing the comb modes with respect to an ultra-stable optical reference, a low-noise optical frequency comb has been demonstrated with a 10 W average power at 75 fs pulse duration.34 Thulium-doped fiber lasers and amplifiers are an interesting option for high-power frequency comb generation at longer wavelengths of ∼2 μm.35,36 Ultrafast thin-disk oscillators can produce optical frequency combs with high average power directly from the oscillator without using the MOPA scheme.37,38

In this work, we demonstrate a low-noise and high-power infrared frequency comb that uses Yb-based solid-state laser technology for the oscillator and amplifiers. Compared to widely used fiber amplifiers, solid-state amplifiers operate with a shorter interaction length and a larger beam cross section.39 Therefore, the nonlinear response of the gain medium is smaller, and no CPA scheme is required. A laser system without CPA avoids a possible effect on the phase noise of the frequency comb. For the spectrally dispersed beams in the stretcher and in the compressor, phase fluctuations caused, e.g., by moving air, could have a different effect on the different wavelengths. Such an effect could not be compensated by beam-pointing and path-length stabilization. Using an Yb:YAG Innoslab femtosecond amplifier,39 we obtain an output power of 360 W, of which we retain 270 W after the optical isolator and spatial mode filter. The spectral narrowing due to the limited gain bandwidth of the Yb:YAG gain medium restricts the pulse duration to around 1 ps. Nonlinear spectral broadening of pulses using conventional nonlinear fibers is limited to about 4 MW of peak power due to catastrophic self-focusing. We employ the multi-pass cell spectral broadening (MPCSB) scheme40,41 and compress the pulse duration to 59 fs.

We use a cavity-stabilized ultra-stable continuous wave (cw) laser42 as an optical reference to phase-lock one of the comb modes at around 1033 nm. High fidelity phase stabilization of this mode is achieved by using multi-stage feedback loops acting on the oscillator and external frequency shifters. The phase noise of the stabilized comb mode integrated from 10 Hz to 10 MHz is estimated to be 41 mrad. The phase noise is measured with respect to the cw reference laser, which was independently characterized to have 10 mrad of phase noise integrated from 10 kHz to 10 MHz.42 When the intracavity HHG scheme is used to generate an XUV frequency comb, the enhancement cavity acts as a narrow spectral filter, and the phase noise is further reduced. In our setup designed for He+ spectroscopy, the phase noise between the comb mode and the reference laser filtered by the enhancement cavity is estimated to be 14 mrad. This is small enough to efficiently drive the target transition after wavelength conversion to 60.8 nm, provided that the reference laser is stable enough.

The laser system reported in this paper is represented by the green triangle in Fig. 1. By combining the Yb:YAG Innoslab amplifier with the MPCSB pulse compression scheme, we reach a peak power of 32 MW, which is suitable for driving an enhancement cavity for HHG. The frequency comb developed in this work has sufficiently low phase noise to produce an XUV frequency comb suitable for precision spectroscopy.

A schematic of our high-power frequency comb system is shown in Fig. 2. It starts with a soft-aperture Kerr-lens mode-locked Yb:KYW oscillator, which is depicted in more detail in Fig. 3. The laser is pumped by a single-mode polarization-maintaining fiber-coupled cw diode laser at 981 nm with a maximum power of 500 mW. This wavelength corresponds to a sharp absorption peak in Yb:KYW and allows for emission in a broad spectrum around 1020–1060 nm when the pump polarization is parallel to the Nm-axis of the gain crystal. The crystal has a 2-mm thickness, and its 3 × 4 mm2 surfaces are anti-reflection coated. The dopant concentration is 10%. To ensure emission at 1030 nm, a low-pass filter (Alluxa) is introduced into the laser cavity. The filter has a cutoff wavelength of 1075 nm at normal incidence, and the transition interval from 10 to 90% transmission is specified to be about 5 nm. The center wavelength is tuned by adjusting the angle of incidence of the beam onto the filter. This is necessary to match the output spectrum to the gain peak of the subsequent amplifiers. The pump beam is collimated by a triplet-lens collimator (Thorlabs TC25APC-980) and then focused into the laser crystal by a doublet lens (f = 60 mm) to minimize spherical aberrations. The cavity is arranged in a linear geometry that is folded by a set of seven flat mirrors. Some of the mirrors are negatively chirped for intracavity dispersion compensation. The total group-delay dispersion (GDD) given by the chirped mirrors is about −2000 fs2. The total cavity length is 3.75 m, corresponding to a repetition rate of 40 MHz. The cavity length is stabilized by acting on two independent mirrors. The short-arm end mirror (M1 in Fig. 3) is a highly reflective mirror actuated by a lead zirconium titanate piezoelectric element (PZT) mounted on a high-damping alloy. The long-term drifts are taken care of by the long-arm outcoupling-mirror (M2 in Fig. 3), actuated by a flexure guided linear stage driven by a PZT. The output coupler’s reflectivity is 99%. The concave focusing mirrors (M3 and M4 in Fig. 3) have a radius of curvature of 100 mm and are transmissive for the pump. All components are placed on a custom machined monolithic aluminum breadboard with a massive airtight lid and temperature stabilization.

FIG. 2.

Schematic of the main optical components in our high-power IR frequency comb system. The Yb:KYW oscillator is mode-locked by Kerr-lensing and emits a spectrum centered at 1030 nm. Two Yb:LuAG amplifiers in combination with a Yb:YAG Innoslab amplifier provide an average power of 270 W. Two multi-pass spectral broadening stages, along with two Gires–Tournois-Interferometer (GTI) mirrors and a grating pair for pulse compression, provide ultra-short pulses of 59 fs. The average output power is 230 W. Central wavelengths are denoted by λ, bandwidths by Δλ, and pulse durations by τ.

FIG. 2.

Schematic of the main optical components in our high-power IR frequency comb system. The Yb:KYW oscillator is mode-locked by Kerr-lensing and emits a spectrum centered at 1030 nm. Two Yb:LuAG amplifiers in combination with a Yb:YAG Innoslab amplifier provide an average power of 270 W. Two multi-pass spectral broadening stages, along with two Gires–Tournois-Interferometer (GTI) mirrors and a grating pair for pulse compression, provide ultra-short pulses of 59 fs. The average output power is 230 W. Central wavelengths are denoted by λ, bandwidths by Δλ, and pulse durations by τ.

Close modal
FIG. 3.

Layout of our mode-locked Yb:KYW oscillator. The mirrors M1 and M2 form the ends of the linear cavity, where M2 is the output coupler. Mirrors M3 and M4 focus the resonant beam at the Yb:KYW crystal (green). The pump is a laser diode with an emission centered at 981 nm and a maximum output power of 500 mW (blue). A low-pass filter (LPF) centers the resonant spectrum at 1030 nm and provides a secondary output from its reflection. The setup is enclosed in an airtight housing to minimize disturbances from pressure and temperature fluctuations.

FIG. 3.

Layout of our mode-locked Yb:KYW oscillator. The mirrors M1 and M2 form the ends of the linear cavity, where M2 is the output coupler. Mirrors M3 and M4 focus the resonant beam at the Yb:KYW crystal (green). The pump is a laser diode with an emission centered at 981 nm and a maximum output power of 500 mW (blue). A low-pass filter (LPF) centers the resonant spectrum at 1030 nm and provides a secondary output from its reflection. The setup is enclosed in an airtight housing to minimize disturbances from pressure and temperature fluctuations.

Close modal

The laser stays mode-locked for several months without the need for realignment. Figure 4 shows the repetition rate of the oscillator measured over 100 days. Such long-term stability is essential because accidental termination of mode locking results in the abrupt shutdown of the subsequent high-power amplifier by an interlock system. This may degrade the solder contact of the slab laser crystal with its heat sink due to rapid temperature changes.

FIG. 4.

Repetition rate of our Yb:KYW oscillator measured over 100 days. The oscillator can stay mode-locked for months without realignment or manual intervention.

FIG. 4.

Repetition rate of our Yb:KYW oscillator measured over 100 days. The oscillator can stay mode-locked for months without realignment or manual intervention.

Close modal

The main output of the oscillator has an average power of 10 mW and emits 105-fs pulses at 1030 nm with a 10 nm FWHM bandwidth. There is also a secondary output from the reflection of the intracavity low-pass filter with an average power of 60 mW. The pulses from this output are not Fourier-limited due to the spectral phase distortion caused by the reflection from the filter. Nevertheless, it is used to generate a beat note with the reference cw laser for frequency stabilization (see Sec. III).

The main output of the Yb:KYW oscillator then goes through three amplification stages. First, two Yb:LuAG amplifiers are used for preamplification. The two preamplifiers are almost identical, and their layout is shown in Fig. 5. The beams are focused into the Yb:LuAG crystals that act as the gain media. The 3 × 1 × 6 mm3 rectangular-shaped crystals are clamped in water-cooled heat sinks using indium sheets for thermal contact. The crystals are pumped by high-power multi-mode laser diodes emitting at 940 nm. The fiber-coupled diodes are collimated by achromat doublet lenses with f = 45 mm for the first stage and f = 40 mm for the second stage, and then focused into the crystal by a f = 60 mm lens. The pump and seed beam waist radii are set to be 140 and 160 μm within the crystals for the first and second stages, respectively. The diodes are attached to a copper block that is water cooled at a constant temperature of 22 °C. The pump and signal beams at the crystals are imaged onto CCD cameras to ensure a good spatial overlap between the beams. The preamplifiers are operated in a double-pass geometry by using λ/4-plates and polarizing beam splitters (PBSs) (see Fig. 5).

FIG. 5.

Two-stage Yb:LuAG preamplification setup, including double-pulse generation. Each gain crystal is pumped by a multi-mode laser diode (LD), and the system is operated in a double-pass configuration by using a λ/4-plate and a polarizing beamsplitter (PBS). Faraday isolators (FI) are used to prevent parasitic lasing by eliminating the back reflections that can be amplified by the previous stages. A double-pulse generation setup is needed for our two-photon spectroscopy scheme for He+.16 The acousto-optic modulators (AOM1 and AOM2) are used for phase-stabilization (see Sec. III B and Fig. 13).

FIG. 5.

Two-stage Yb:LuAG preamplification setup, including double-pulse generation. Each gain crystal is pumped by a multi-mode laser diode (LD), and the system is operated in a double-pass configuration by using a λ/4-plate and a polarizing beamsplitter (PBS). Faraday isolators (FI) are used to prevent parasitic lasing by eliminating the back reflections that can be amplified by the previous stages. A double-pulse generation setup is needed for our two-photon spectroscopy scheme for He+.16 The acousto-optic modulators (AOM1 and AOM2) are used for phase-stabilization (see Sec. III B and Fig. 13).

Close modal

In the first preamplifier, the seed bandwidth is considerably larger than the gain bandwidth. Therefore, the spectral overlap limits the spectrally averaged small-signal gain to around 10. The small-signal gain at 1030 nm is measured to be around 100. The first preamplifier has a maximum output power of 145 mW when pumped with 7.5 W and a seed input power of 9.4 mW.

After the first preamplifier, the beam is sent through an acousto-optic modulator (AOM), which is used for phase stabilization of the frequency comb, as explained in Sec. III B. This laser system is designed for our envisioned He+ spectroscopy, as described in Ref. 16. To achieve Doppler-free two-photon spectroscopy, it is required that two counter-propagating excitation pulses meet at the target He+ ions. For this aim, we produce pulse pairs that have a temporal interval of about 3 ns by introducing a Mach–Zehnder-type delay line in between the first and second preamplifier stages.

After this, the beam entering the second preamplification stage has 40 mW of average power. Whereas the first preamplifier is operating in the small signal regime, the second one operates close to saturation because its seed power is larger and the seed spectrum overlaps better with the gain spectrum. 3 W of output power is obtained from the second preamplifier with 20 W of pump power. The beam quality is measured to be M2 = 1.05 × 1.05. Another AOM is placed at the output of the second preamplifier and is used for compensating path-length fluctuations introduced in the rest of the setup (see Sec. III C). The average power after the second AOM is 2.7 W. The B-integral of the preamplifiers is estimated to be a few mrad. A beam pointing stabilization system (MRC Systems) fixes the alignment of the beam sent to the main amplifier and reduces beam pointing fluctuations.

After the preamplifiers, beam pointing stabilization, and mode-matching optics, the beam with 2.5 W of average power is delivered to an Innoslab amplifier.39 This Yb:YAG solid-state amplifier was developed by Fraunhofer ILT and delivers 270 W of average power after a Faraday isolator and a spatial filter. The spatial filter acts only on the slow-axis of the Innoslab gain crystal, where the beam quality is worse.43 With this, we obtain a beam quality of M2 = 1.07 × 1.20. A second beam pointing stabilization system (TEM Messtechnik) compensates for beam pointing fluctuations in the output from the Innoslab amplifier. Due to the limited gain bandwidth of the Yb:YAG crystal, the FWHM bandwidth of the spectrum narrows from 2.3 nm at the input to 1.6 nm at the output, both centered at 1030 nm. A comparison of the output spectra from the Yb:KYW oscillator, the preamplifiers, and the Innoslab amplifier can be seen in Fig. 6. The output pulse duration is 1.1 ps.

FIG. 6.

Spectrum at different positions in the Yb-based laser system. The green trace shows the output spectrum of the Yb:KYW oscillator. The orange trace shows the spectrum at the output of the Yb:LuAG preamplifers, which has a 2.3 nm FWHM bandwidth due to gain narrowing. The blue trace shows the spectrum after the Innoslab amplifier with a FWHM bandwidth of 1.6 nm due to further gain narrowing.

FIG. 6.

Spectrum at different positions in the Yb-based laser system. The green trace shows the output spectrum of the Yb:KYW oscillator. The orange trace shows the spectrum at the output of the Yb:LuAG preamplifers, which has a 2.3 nm FWHM bandwidth due to gain narrowing. The blue trace shows the spectrum after the Innoslab amplifier with a FWHM bandwidth of 1.6 nm due to further gain narrowing.

Close modal

To drive the intracavity HHG process more efficiently, the pulses are nonlinearly compressed. We employ a multi-pass cell spectral broadening (MPCSB) scheme, which consists of spectral broadening by self-phase modulation in a fused silica medium placed in a multi-pass cell and chirp removal by GTI mirrors and a grating compressor. The MPCSB relies on the accumulation of nonlinear phases in multiple small steps in a Herriott-type cell while avoiding catastrophic self-focusing. The scheme is suited for high average power, imperfect beam quality, and pulse energies of a few μJ, i.e., above the critical power for self-focusing where a solid-core fiber for spectral broadening is not applicable.40,41 Note that this problem cannot be circumvented by lowering the intensity at a given laser power.

After two multi-pass cells (MPCs), we obtain a spectrum that covers wavelengths from 1010 to 1060 nm (see Fig. 7). Each MPC consists of a pair of 2 in.-diameter focusing mirrors and an anti-reflection coated fused-silica plate placed in the middle, acting as the nonlinear medium. The first MPC uses 45 passes through a 20 mm thick plate, while the second one uses 21 passes through a 13 mm thick plate. The focusing mirrors have a dispersive coating that partially compensates for the dispersion introduced by the silica plate. The designed group-delay dispersion (GDD) at 1030 nm of the mirrors in the first MPC is −350 fs2, and that of the mirrors in the second MPC is −250 fs2. The GDD introduced by the fused-silica elements is 380 and 250 fs2 for the first and second MPCs, respectively.

FIG. 7.

Experimental and simulated spectra after the multi-pass cells for spectral broadening. The experimental spectrum (red trace) is obtained at 230 W of power measured after the compression. It has a significant spectral intensity ranging from 1010 to 1060 nm. The simulated spectrum (blue trace) is obtained assuming third-order dispersion TOD = −0.51 ps3 is present in the pulses before spectral broadening. The red trace in the inset shows the spectrum measured before spectral broadening, and the blue trace shows a Gaussian spectrum with a FWHM of 1.6 nm, which is used as the input spectrum for the simulations of the spectral broadening in the multi-pass-cell spectral broadening scheme.

FIG. 7.

Experimental and simulated spectra after the multi-pass cells for spectral broadening. The experimental spectrum (red trace) is obtained at 230 W of power measured after the compression. It has a significant spectral intensity ranging from 1010 to 1060 nm. The simulated spectrum (blue trace) is obtained assuming third-order dispersion TOD = −0.51 ps3 is present in the pulses before spectral broadening. The red trace in the inset shows the spectrum measured before spectral broadening, and the blue trace shows a Gaussian spectrum with a FWHM of 1.6 nm, which is used as the input spectrum for the simulations of the spectral broadening in the multi-pass-cell spectral broadening scheme.

Close modal

Four reflections on two GTI mirrors placed in between the MPCs compensate for the chirp imprinted during the spectral broadening, with a total GDD of −20 000 fs2. After the second MPC, a transmission grating pair removes the chirp acquired in this last stage. The transmission gratings (Gitterwerk GmbH) have a groove density of 1000 lines/mm, and the diffraction efficiency exceeds 99% at 1030 nm. The grating distance is set to be less than 2 mm, and the spatial chromatic dispersion after the grating pair is only a small fraction of the beam diameter. The reduction in spatial mode matching to the enhancement cavity due to the spatial chromatic dispersion is estimated to be less than 0.3%. Both gratings were aligned with great care to ensure their grooves were precisely parallel in order to prevent angular chromatic dispersion. This is needed for an efficient coupling into our enhancement cavity. A high beam quality (M2 = 1.1 × 1.05) is maintained without compensating for the spatial chromatic dispersion using a second grating pair.

A third beam pointing stabilization system from TEM Messtechnik is used to stabilize the output of the MPCs. The temporal pulse duration is measured to be 59 fs using the frequency-resolved optical gating (FROG) method, as shown in Fig. 8. The pulse consists of a main peak on top of a relatively broad pedestal structure that extends for about 500 fs. A fraction of 70% of the power is contained in the main peak. After the grating pair, the peak power is increased by more than tenfold compared to the pulses sent to the MPCSB setup. The fluctuation and drift of the average output power are less than 0.7% over a few hours, as shown in Fig. 9. In principle, one can reach even shorter pulses with a different configuration of the MPCs, but we designed the parameters of the MPCSB such that the broadened spectrum still fits within the acceptance bandwidth of our enhancement cavity.

FIG. 8.

Pulse characterization after pulse compression from frequency-resolved optical gating (FROG). (a) Measured and (b) retrieved FROG traces, showing good agreement and a FROG trace error of 0.52%. (c) and (d) show the retrieved pulse intensity and phase in the time and frequency domain, respectively (blue and green traces), as well as the directly measured spectrum for comparison (red trace, same as in Fig. 7). The pulse duration is 59 fs.

FIG. 8.

Pulse characterization after pulse compression from frequency-resolved optical gating (FROG). (a) Measured and (b) retrieved FROG traces, showing good agreement and a FROG trace error of 0.52%. (c) and (d) show the retrieved pulse intensity and phase in the time and frequency domain, respectively (blue and green traces), as well as the directly measured spectrum for comparison (red trace, same as in Fig. 7). The pulse duration is 59 fs.

Close modal
FIG. 9.

Output power of the laser system over time, measured after pulse compression using a water-cooled power meter. The measurement is performed every 30 s over 270 min. The fluctuation and drift of the power are less than 0.7%.

FIG. 9.

Output power of the laser system over time, measured after pulse compression using a water-cooled power meter. The measurement is performed every 30 s over 270 min. The fluctuation and drift of the power are less than 0.7%.

Close modal
The primary nonlinear process in MPCSB is self-phase modulation (SPM), which is expected to broaden the spectrum symmetrically if the initial temporal shape of the pulse is symmetric. However, a pronounced asymmetric broadening is observed in our setup. Some previous studies attribute the asymmetry to self-steepening and Raman shift.44–46 Both effects would extend the spectrum toward longer wavelengths and, hence, do not explain the observed spectrum. However, the complex interplay between SPM, GDD, and higher-order dispersion may broaden the spectrum asymmetrically. In particular, third-order dispersion (TOD) can be expected to be responsible for asymmetric spectral broadening.47 This may be caused by the dispersion of the MPCSB components or may be present in the pulses entering the MPC stages. To study this, we have numerically simulated the pulse propagation dynamics in our MPCSB setup by solving the nonlinear Schrödinger equation using the split-step Fourier method. Our model includes SPM, GDD, and TOD introduced at the MPCSB setup but ignores the self-steepening and the Raman shifting effects. With the pulse amplitude A(z,t), where z and t are the propagation axis and time, the propagation equation is48 
Az=(D+N)A,
(1)
where
D=iβ22!2t2+β33!3t3
(2)
and
N(t)=iγ|A(t)|2.
(3)

β2 and β3 are the GDD and TOD per unit length at the center frequency ω0, corresponding to 1030 nm. The spectral phase of the pulses entering the MPCSB stages may also have an initial chirp due to GDD and TOD. The nonlinear term N(t) accounts only for SPM through the γ parameter.

The initial spectrum is assumed to have a Gaussian shape with a 1.6 nm width (FWHM), which is chosen to reproduce the spectrum coming from the Innoslab amplifier (see the inset of Fig. 7). The pulse propagation through the two MPCSB setups and the GTI mirrors in between is simulated. The β2 parameter of fused silica at 1030 nm is 19 fs2/mm.49 Since the beam diameter in the MPCSB is not precisely known, the γ parameter is adjusted to best reproduce the observed broadening and modulation of the spectrum. The simulation was repeated for different values of the TOD contribution to the spectral phase of the initial pulses before entering the MPCSB. An initial TOD = −0.51 ps3 with γ = 4 × 10−9 (W m)−1 resulted in the simulated spectrum shown in Fig. 7, which is in reasonable agreement with the experimentally obtained spectrum. Introducing only the GDD contribution in the initial spectral phase, ignoring TOD, failed to reproduce the observed spectral asymmetry. Adding the TOD contribution of the fused silica medium hardly changed the results. This suggests that a strong TOD contribution to the spectral phase present in the pulses before entering the MPCSB is responsible for the observed asymmetry in the broadened spectrum. Recently, we found that the large amount of TOD can be attributed to our preamplifier stages, where the pulses experience a strong gain narrowing effect.50 

The modes of a frequency comb can, in principle, be infinitely narrow, limited only by the laser’s on-time. Meanwhile, a real laser system has phase noise, which leads to a broadening of the modes. An atomic transition can only be excited efficiently if a significant fraction of the mode power is contained within the observational transition linewidth. In our case, the 1S–2S transition in He+ has a natural linewidth of only 84 Hz.51 We expect that the observational transition linewidth will be broadened to a few kHz due to ionization and quenching of the 2S state.16,52

Domenico and co-authors have numerically studied the effect of phase noise on laser line shapes.53 It turns out that the noise spectrum can be separated into two different regimes. The first regime corresponds to large phase excursions at low Fourier frequencies (slow phase noise). This noise is responsible for the linewidth and often leads to a Gaussian line shape. If a stable reference, for example, a high-finesse cavity, is available, this noise can be efficiently suppressed using a feedback loop. By using cavities manufactured from ultra-low expansion glass, linewidths of around 1 Hz are routinely achieved.54,55 A linewidth below 10 mHz has been demonstrated by employing single-crystal silicon cavities.56 

The second regime corresponds to small phase excursions at high Fourier frequencies (fast phase noise). This noise contributes to the wings of the spectrum and often leads to a broad phase noise “pedestal.”42 The fast phase noise redistributes power from the center of the spectrum to the wings but has no significant effect on the linewidth. For typical laser sources, the wings contain at most a few percent of the total power and can often be neglected.42 However, the situation changes if a harmonic of the laser is being generated. At the qth harmonic, the phase noise Δϕ(t) increases by a factor q.57 The fractional power remaining in the narrow carrier is given by expq2Δϕrms2, where Δϕrms is the integrated fast phase noise.57 Once qΔϕrms exceeds around 1 rad, the carrier starts to collapse.24 Since we plan to excite the 1S–2S two-photon transition in He+ using the 17th harmonic of our infrared frequency comb, the effective multiplication factor is q = 34. It is, therefore, important to minimize the fast phase noise by using lasers that are intrinsically low-noise and by employing feedback loops with high bandwidth.

The stabilization principle used in our setup is depicted in Fig. 10. We use a laser stabilized to an ultra-stable high-finesse cavity operating at 1033 nm as the reference. This laser has a linewidth of around 1 Hz. By analyzing the heterodyne beat note between the laser and the transmission through its high-finesse reference cavity, a small integrated phase noise of 10.2 mrad was measured between 10 kHz and 10 MHz.42 It is tuned such that its output frequency fcw is close to the 34th subharmonic of the 1S–2S transition frequency f1S−2S. Here, f1S−2S is defined as the frequency that corresponds to the energy difference between the 1S and 2S states. The closest mode of the frequency comb is then tightly phase locked to the cw laser by producing a beat note between the lasers and giving feedback to the frequency comb. This mode then determines the fixed point of the frequency comb, and the residual frequency noise of the other comb modes is antisymmetric with respect to the stabilized one.58 Since the two-photon excitation is driven by the pairwise interaction of the comb modes,59 the noise then drops out.

FIG. 10.

Frequency comb stabilization principle for exciting the 1S–2S two-photon transition in He+. It is sufficient to stabilize the comb mode at the 34th subharmonic of the transition frequency since the residual noise of the other modes is canceled by the pairwise excitation.

FIG. 10.

Frequency comb stabilization principle for exciting the 1S–2S two-photon transition in He+. It is sufficient to stabilize the comb mode at the 34th subharmonic of the transition frequency since the residual noise of the other modes is canceled by the pairwise excitation.

Close modal

The absolute frequency of the stabilized comb mode fcw is measured relative to a GPS disciplined hydrogen maser using a self-referenced frequency comb (Menlo Systems FC1500-ULN). One straightforward extension would be to stabilize the carrier-envelope offset frequency fceo of the seed frequency comb by measuring it with an f − 2f interferometer and giving feedback to the pump diode current of the Yb:KYW oscillator. This would enable applications where all comb modes need to be stabilized. The frequency of the nth comb mode is then given by nfrep + fceo, where frep is the repetition rate.

The performance of the stabilization is ultimately limited by the signal-to-noise ratio (SNR) of the beat detection between the frequency comb mode and the reference cw laser. In this section, we describe a low-noise balanced beat detection setup that is used throughout the stabilization system.

If the technical noise is sufficiently small and the power of the cw laser significantly exceeds that of the frequency comb at the detector, the SNR is only limited by the shot noise of the comb mode and is given by60 
SN=ηPnhνBw,
(4)
where η is the photodetector quantum efficiency, Pn is the power of the comb mode, is the photon energy, and Bw is the detection bandwidth. The design goal of the beat detection setup is to reach the shot-noise limit by minimizing technical noise while maximizing the comb mode power that reaches the photodetector.

A schematic is shown in Fig. 11. Only one frequency comb mode contributes to the beat note, while the other modes add shot noise and technical amplitude noise. Furthermore, we found that short pulses with high peak power can saturate photodiodes and subsequent amplifiers at significantly lower average power levels compared to cw lasers. This limits how much frequency comb light can be sent into the setup. We, therefore, spectrally filter the frequency comb light with an interference filter with a 0.4 nm FWHM bandwidth (Layertec 157313) followed by an etalon filter with a free spectral range of 207 GHz (0.74 nm) and an FWHM linewidth of 6.9 GHz (LightMachinery OP-6204-M). The filtered light is then overlapped with light from the cw laser on a PBS. A second PBS is used to project the laser beams onto common polarization axes. The two outputs of the second PBS are sent into a differential photodetector, which consists of a pair of photodiodes and a transimpedance amplifier (see Fig. 11). The idea behind differential detection is that technical noise is correlated and, hence, effectively canceled if the signal strengths are balanced.61 This is achieved by fine-tuning the angle of the half-wave plate before the second PBS. Ideally, this detector even reduces the shot noise limited SNR by 3 dB as compared to a single detector with a 50% beamsplitter since this uses only half the optical power.62 

FIG. 11.

Balanced beat detection setup. The light is sent to the differential photodetector using single-mode fibers, which ensures spatial mode matching. IF: interference filter, ET: etalon filter, PBS: polarizing beam splitter, PD: photodiode, and TIA: transimpedance amplifier.

FIG. 11.

Balanced beat detection setup. The light is sent to the differential photodetector using single-mode fibers, which ensures spatial mode matching. IF: interference filter, ET: etalon filter, PBS: polarizing beam splitter, PD: photodiode, and TIA: transimpedance amplifier.

Close modal

We typically send around 1 mW of cw laser light and a few ten μW of comb light (after spectral filtering) onto the photodetector. A typical beat note spectrum is shown in Fig. 12. The SNR is around 60 dB in the 100 kHz bandwidth, which is sufficient for our application.

FIG. 12.

Beat note between the cw laser and one mode of the Yb:KYW oscillator (blue trace) recorded with a resolution bandwidth of 100 kHz. The orange trace is the noise floor, which is given by the amplitude noise of the cw laser.

FIG. 12.

Beat note between the cw laser and one mode of the Yb:KYW oscillator (blue trace) recorded with a resolution bandwidth of 100 kHz. The orange trace is the noise floor, which is given by the amplitude noise of the cw laser.

Close modal

The cavity-stabilized cw laser is located in a neighboring lab and is delivered to the frequency comb setup via a 20 m long polarization-maintaining fiber using active fiber noise cancellation.63 It is then amplified from 5 to 100 mW with a Yb-doped fiber amplifier. The seed frequency comb is locked to this laser by using a two-stage feedback system, which is shown in Fig. 13.

FIG. 13.

Frequency comb stabilization setup consisting of a slow and a fast stage. The double-pulse generation is omitted in this sketch. LP filter: low-pass filter, SP: RF splitter, PFD: phase-frequency detector, VCO: voltage-controlled oscillator, and AOM: acousto-optic modulator.

FIG. 13.

Frequency comb stabilization setup consisting of a slow and a fast stage. The double-pulse generation is omitted in this sketch. LP filter: low-pass filter, SP: RF splitter, PFD: phase-frequency detector, VCO: voltage-controlled oscillator, and AOM: acousto-optic modulator.

Close modal

The first stage (slow feedback) pre-stabilizes the frequency of the comb mode using a beat note with the secondary output of the Yb:KYW oscillator. In this way, the full power of the main output remains available for seeding the subsequent amplifiers. The signal is low-pass filtered to isolate the beat note and sent into a digital phase and frequency detector (PFD, onsemi MC100EP140). There, it is compared with a 10 MHz reference from a frequency synthesizer. A home-built loop filter gives feedback to the cavity length of the Yb:KYW oscillator using a piezo-actuated cavity end mirror. When the frequency comb is unlocked and the input frequencies to the PFD are different, it outputs a voltage that is proportional to the frequency difference. This allows the feedback loop to reliably acquire the lock, even when the comb mode frequency is far away from the cw laser frequency. Once the inputs have the same frequency, the PFD outputs a voltage that is proportional to the phase difference. This allows the feedback loop to phase-stabilize the signals. While the feedback loop achieves a bandwidth of a few kHz, the actuator has a small range and cannot compensate for larger drifts in the cavity length. The output of the loop filter is, therefore, used as the error signal for a second loop filter, which controls a piezoelectric stage that moves the other end mirror of the laser cavity. This stage has a very large range of 600 μm, but the bandwidth is only a few Hz. In this way, the piezoelectric actuator is always kept in the center of its travel range.

The bandwidth of the piezo-actuated mirror is too low to achieve a tight phase lock. We, therefore, use an external AOM as a fast actuator in a second feedback stage. The AOM is placed behind the first preamplifier stage (see Fig. 13). It is driven by a voltage-controlled oscillator (VCO, Pasternack PE1V31008). The VCO has a small tuning port capacitance of 82 pF, which allows for fast frequency adjustments. Some of the comb light is picked off after the AOM, and a second beat note is generated with the cw laser. An analog mixer is used to detect the phase difference between the beat note and a reference signal at 9 MHz. The feedback loop is closed with a fast analog loop filter (Vescent Photonics D2-125 Laser Servo), which acts on the tuning port of the VCO. An offset voltage is added to the signal in order to match the VCO frequency to the operation frequency of the AOM.

The bandwidth of a feedback loop is ultimately limited by the time delay of the signal propagating in the loop. The speed of the acoustic wave in an AOM is around five orders of magnitude slower than the propagation speed of electrical signals. It is, therefore, crucial to minimize the distance between the laser beam and the AOM transducer to achieve a large feedback bandwidth. The beam is loosely focused through the AOM with an f = 200 mm lens to minimize the beam size in the AOM medium. In this way, a delay of around 110 ns can be achieved before the beam is clipped by the transducer. The resulting feedback bandwidth is around 600 kHz.

To analyze the performance of the circuit, we recorded the in-loop phase noise with an RF spectrum analyzer (Agilent E4445A) at the monitor port of the fast feedback (see Fig. 13). The results are shown in Fig. 14. The root mean square integrated phase noise from 10 Hz to 10 MHz is 35.5 mrad.

FIG. 14.

In-loop single-sideband phase noise between the cw laser and the mode of the Yb:KYW oscillator. The orange trace is recorded using only the slow feedback, and the blue trace with the slow and fast feedback. The green trace shows the detection noise floor, which is due to the amplitude noise of the cw laser. The dashed blue curve shows the root mean square integrated phase noise calculated from the blue trace starting at 10 Hz (right axis). The power fraction contained in the carrier of the comb mode is estimated to be 99.9%. Note that under this small phase noise condition, the single-sideband phase noise is equivalent to the power spectrum of the comb mode.

FIG. 14.

In-loop single-sideband phase noise between the cw laser and the mode of the Yb:KYW oscillator. The orange trace is recorded using only the slow feedback, and the blue trace with the slow and fast feedback. The green trace shows the detection noise floor, which is due to the amplitude noise of the cw laser. The dashed blue curve shows the root mean square integrated phase noise calculated from the blue trace starting at 10 Hz (right axis). The power fraction contained in the carrier of the comb mode is estimated to be 99.9%. Note that under this small phase noise condition, the single-sideband phase noise is equivalent to the power spectrum of the comb mode.

Close modal

At the 34th harmonic,64 this would reduce the strength of the carrier to 23%. The dashed line in Fig. 14 shows the root mean square integrated phase noise starting at 10 Hz. It is evident that most of the phase noise is contributed by spectral components with Fourier frequencies above 1 MHz, which is beyond the feedback bandwidth. Furthermore, in this frequency range, the phase noise is close to the measurement noise floor, which makes an accurate measurement difficult. We expect that the HHG enhancement cavity passively filters out a significant part of the high-frequency phase noise. We assume a FWHM cavity linewidth of 200 kHz and numerically filter the measured phase noise using the expected noise transfer function.65 The resulting integrated phase noise for the light circulating in the cavity is 7.3 mrad, which is well below the threshold for carrier collapse at the 34th harmonic.

The stabilization setup tightly phase locks one mode of the frequency comb to the cw laser at the position of the fast feedback beat detection. However, the frequency comb laser beam then travels through the second preamplifier, the high power amplifier, and the MPCs for spectral broadening. We found that path length fluctuations lead to considerable phase noise in the acoustic range that has to be compensated in order to achieve a narrow linewidth in the XUV.

Figure 15 shows the active path length stabilization system we have implemented. A small amount of frequency comb light is split off after the chirp removal using the grating pair. A beat note is generated with light from the cw laser. This light is taken before the fiber amplifier in order to avoid potential phase noise contributions due to path length fluctuations in the gain fiber. A second AOM is used as an actuator for giving feedback on the frequency of the comb modes. The AOM is placed behind the second preamplifier but before the high power amplifier and pulse compression units. In this way, the AOM only has to handle around 3 W of optical power, and the material dispersion is less critical due to the narrow spectrum. However, the light cannot be focused as tightly as in the frequency stabilization setup due to the higher power. This leads to a larger travel time for the acoustic wave in the AOM, which limits the feedback bandwidth to 300 kHz due to the associated phase shift.

FIG. 15.

Path length stabilization setup. LP filter: low-pass filter, SP: RF splitter, VCO: voltage-controlled oscillator, and AOM: acousto-optic modulator.

FIG. 15.

Path length stabilization setup. LP filter: low-pass filter, SP: RF splitter, VCO: voltage-controlled oscillator, and AOM: acousto-optic modulator.

Close modal

The feedback electronics are identical to those of the fast feedback, except that a 1.9 MHz low-pass filter is used for the error signal. This reduces the amount of noise entering the loop filter, while the larger delay of the AOM makes the phase delay introduced by the filter unimportant.

Figure 16 shows the resulting in-loop phase noise measured in the path length stabilization setup. The Innoslab amplifier, MPCSB setup, a part of the preamplifier setup, and the long beam path add 738 mrad of integrated phase noise (orange trace). This is significantly suppressed by the feedback loop, so that the integrated phase noise of the stabilized in-loop signal (blue trace) is 41 mrad. By taking into account the expected noise filtering of the enhancement cavity as described above, we obtain an integrated phase noise for the light circulating in the cavity of 14 mrad. This corresponds to 80% of the power remaining in the carrier at the 34th harmonic.

FIG. 16.

In-loop single-sideband phase noise measured in the path length stabilization setup. The orange trace is the phase noise without active stabilization, and the blue trace with stabilization. For comparison, the in-loop phase noise between the cw laser and the Yb:KYW oscillator (blue trace in Fig. 14) is shown in gray. The green trace shows the detection noise floor, which is due to the amplitude noise of the cw laser. The dashed blue curve shows the root mean square integrated phase noise calculated from the blue trace starting at 10 Hz (right axis).

FIG. 16.

In-loop single-sideband phase noise measured in the path length stabilization setup. The orange trace is the phase noise without active stabilization, and the blue trace with stabilization. For comparison, the in-loop phase noise between the cw laser and the Yb:KYW oscillator (blue trace in Fig. 14) is shown in gray. The green trace shows the detection noise floor, which is due to the amplitude noise of the cw laser. The dashed blue curve shows the root mean square integrated phase noise calculated from the blue trace starting at 10 Hz (right axis).

Close modal

We report on a low-noise and high-power infrared frequency comb at 1 μm wavelength with an average power of 230 W and 59 fs pulse duration. The system consists of Yb-based solid-state lasers as the oscillator and amplifiers. While the Yb:YAG gain medium of the main amplifier provides high gain and output power, the pulse duration of the amplifier output is limited to about 1 ps due to the narrow gain bandwidth. We use the MPCSB scheme and compress the pulse duration to 59 fs. The laser system operates without the CPA scheme.

Using multi-stage feedback loops acting on the oscillator and external frequency shifters, one of the comb modes is stabilized to the cw reference laser at around 1033 nm.

The low-noise and high-power IR frequency comb developed in this work can drive an enhancement cavity for high harmonic generation to generate a narrowband XUV frequency comb. The integrated phase noise of the light circulating in the cavity is estimated to be 14 mrad using in-loop error signals and reasonable assumptions. This is expected to be low enough to prevent carrier collapse even after frequency conversion to 30 nm. Experimentally confirming this would require a heterodyne beat measurement by setting up another identical laser system or performing spectroscopy on a narrow line in the XUV. We will use the generated XUV frequency comb to excite the 1S–2S transition of He+ ions.16 

In our previously reported experiment,16 we obtained an intracavity power of 8 kW, which is indicated by an open green triangle in Fig. 1. The combination of a high-power Yb:YAG Innoslab amplifier and pulse compression using the MPCSB scheme makes it possible to drive the enhancement cavity for HHG with high peak power. The enhancement cavity also provides efficient phase and amplitude noise filtering for Fourier frequencies above half the cavity linewidth.

We expect that low-noise frequency combs with even higher average power or shorter pulse durations are feasible: an average power in excess of 1 kW has been achieved using multi-stage Yb:YAG Innoslab amplifiers66 and pulse durations as short as sub-20 fs have been reported using three-stage MPCSB setups.67 Frequency combs with such high average power or short pulse durations may not be optimal for driving enhancement cavities, as thermal and dispersion management of the cavities becomes more difficult. However, high repetition rate HHG in a single-pass geometry could be an interesting application of such a laser system.

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 742247). This work was supported by the Fraunhofer–Max–Planck cooperation project KORONA. We thank the Fraunhofer ILT for allocating the Innoslab amplifier setup to the Max–Planck-Institute of Quantum Optics. T. W. Hänsch acknowledges support from the Max–Planck Foundation.

The authors have no conflicts to disclose.

Fabian Schmid: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Software (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Jorge Moreno: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Software (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Johannes Weitenberg: Conceptualization (equal); Data curation (equal); Project administration (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Peter Russbüldt: Conceptualization (equal); Writing – review & editing (equal). Theodor W. Hänsch: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Writing – review & editing (equal). Thomas Udem: Conceptualization (equal); Funding acquisition (lead); Project administration (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal). Akira Ozawa: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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As mentioned in Sec. I, the effective harmonic order is 34 because we plan to excite the 1–2S two-photon transition in He + using the 17th harmonic of our infrared frequency comb.

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