Tightly locked optical frequency comb from a semiconductor disk laser

Ultrafast semiconductor disk lasers (SDLs) passively modelocked using semiconductor saturable absorbers mirrors (SESAMs) generate optical frequency combs (OFCs) with gigahertz line spacings a regime where solid-state and fiber lasers struggle with geometrical and Q-switching limitations. We stabilized both the frequency comb spacing and the offset without any additional external optical amplification or pulse compression. The overall noise performance is competitive with other gigahertz OFCs. A SESAM-modelocked vertical external-cavity surface-emitting laser (VECSEL) at a center wavelength around 1 μm generates 122-fs pulses with 160 mW average output power and we only needed 17-pJ pulse energy coupled into a silicon nitride (Si3N4) waveguide for supercontinuum generation (SCG) and OFC offset stabilization. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Optically pumped SDLs have demonstrated significant progress during the past decade [1,2], both for continuous wave [3] and modelocked [4,5] operation.They provide excellent beam quality and large flexibility to cover wavelengths from the ultraviolet (UV) to the midinfrared (MIR).Compared to solid-state lasers with upper-state lifetimes of up to milliseconds, the carrier lifetime of SDLs only amounts to a few hundreds of picoseconds [6].Modelocked SSDLs are thus naturally suited to operate at (multi)-gigahertz pulse repetition rates [7], since the short lifetime combined with a large gain cross section prevents the buildup of Q-switching instabilities [8].High repetition rate SESAM-modelocked solid-state lasers on the other hand require additional measures to prevent Q-switching such as for example advanced cavity designs [9] or additional intracavity nonlinear components [10].Pulse repetition rates in the gigahertz regime are particularly interesting for applications such as ultrastable microwave generation [11] and photonic computing [12].The resulting large comb line spacing allows for easier access to individual lines and results in more power per comb line for the same average power.These features are particularly useful for instance to calibrate astronomical spectrographs [13], generate arbitrary waveforms [14] or to perform comb line resolved spectroscopy [15][16][17].
In order to obtain a stable OFC, both the line spacing (given by the pulse repetition rate f rep ) and the carrier envelope-offset (CEO) frequency f CEO have to be detected and actively controlled [18][19][20].Extremely low noise of the f rep has been demonstrated with ultrafast SDLs [21], but their f CEO has remained an inaccessible parameter for a long time.The combination of high repetition rates, moderate average output powers in the 100 mW-range, and pulse durations of several hundred femtoseconds prevented octave-spanning supercontinuum generation (SCG) in commercial photonic crystal fibers [22] and therefore self-referenced f CEO detection using f-to-2f interferometry [18].Thus, more complex setups including external pulse amplification and compression were necessary to reach the required pulse parameter regime with sub-100-fs duration and multi-kilowatt peak power, leading to the first selfreferenced f CEO detection [23] and f CEO stabilization [24] of an ultrafast SDL.
Here, we present for the first time a fully stabilized, self-referenced SDL comb with excellent noise properties which does not require additional external pulse amplification or pulse compression.This result became possible with two major advances: the development of an ultrafast SDL approaching the kilowatt peak power regime with near 100-fs pulse duration [4] and a Si 3 N 4 waveguide [25][26][27] with a novel design that supports coherent octavespanning SCG with much lower pulse energy and comparatively long pulse durations.

Optical frequency comb (OFC) stabilization
The gigahertz OFC was generated with a SESAM modelocked VECSEL which is a member of the ultrafast SDL technology platform [2,28].Here we used an ultrafast VECSEL generating 122-fs-pulses at a pulse repetition rate of 1.6 GHz with an average output power of 160 mW at a center wavelength of 1025 nm (see Section 3.1 for more details).The Si 3 N 4 waveguide has been optimized to handle the comparatively low peak power and long pulses of the SDL.It features two zero-dispersion crossings on either side of the anomalous groupvelocity dispersion region that is centered around the pump wavelength [Fig.1(b)].The new type of waveguide design enables the generation of two dispersive waves one octave apart.Instead of aiming for a spectrally flat supercontinuum (SC), the novel design focusses on transferring more spectral power into these dispersive waves.Hence, only ≈17 pJ of coupled pulse energy is needed to generate a coherent octave-spanning SC suitable for f-to-2f interferometry.This is a factor of two less than in previous Si 3 N 4 -waveguide results where a flatter SC was generated [25].
The compact stabilization setup is depicted in Fig. 1(a).We use a reflection of the laser output beam from a glass wedge to measure the f rep with a fast photodiode.The remaining output power is coupled into the Si 3 N 4 waveguide with a coupling efficiency of 28%.The generated SC features two coherent dispersive waves centered around 700 nm and 1440 nm [Fig.1(c)], which are used to detect the f CEO in an f-to-2f interferometer.The microwave spectrum of the f-to-2f photodiode signal shows the two CEO beat frequencies between DC and the f rep [Fig.2(a)].The CEO beat signal is centered around 780 MHz and exhibits a signal-to-noise ratio (SNR) of 25 dB measured with a resolution bandwidth (RBW) of 10 kHz.The f CEO can be shifted over 100 MHz by modulating the pump power [Fig.2(b)].Note that the SNR of the CEO signal cannot simply be increased by increasing the pulse energy coupled into the waveguide.For coupled pulse energies exceeding 17 pJ, the CEO signal power decreases again while the noise floor rises at the same time, which indicates the onset of incoherence in the SCG [Fig.2(c)].Simulations of the SCG in the waveguide confirm the loss of coherence for high pulse energies (see Section 3.2).In order to maintain highest coherence, the length of the waveguide must be chosen to match the desired pulse energy; it needs to be sufficiently long to allow for the generation of the two dispersive waves but should not significantly exceed the soliton fission length.
The free-running CEO beat has a full width at half maximum (FWHM) linewidth of only ≈20 kHz [Fig.2(d)].The narrow linewidth represents a substantial improvement compared to the megahertz CEO linewidths [23,24] detected via external pulse amplification and compression, indicating that additional amplifier noise and amplitude-to-phase noise conversion in more complex lasers-amplifier systems can have a detrimental impact on the noise performance.Furthermore, it is also interesting to compare with the f CEO of a 1-GHz diode pumped solid-state laser (DPSSL) based on Yb:CALGO [29], which was detected using a previous generation of Si 3 N 4 waveguides and showed a linewidth of 170 kHz.The Yb:CALGO laser material exhibits a comparatively long carrier lifetime of 420 µs [30], which acts like a low-pass filter for pump power fluctuations.However, its narrow absorption spectrum [30] increases the sensitivity to pump frequency fluctuations.The SDL on the other hand has a very broad absorption spectrum which is insensitive to pump frequency fluctuations but has a much shorter carrier lifetime (on the order of 100-200 ps) [6], which is below the cavity roundtrip time.The ability to directly measure the f CEO fluctuations opens up interesting possibilities to study the complex carrier-photon coupling dynamics within an SDL gain medium, which is the topic of ongoing work.
The f rep and f CEO are stabilized using an ultra-high frequency digital lock-in amplifier (UHFLI) (see Sections 3.3-3.5).To stabilize the f rep , the cavity length is modulated with a piezo-mounted output coupler and the f CEO is stabilized by pump current modulation.The f CEO   not to the linewidth of the signal, indicating a tight lock characterized by the presence of a coherent peak.There is negligible influence of the f rep stabilization loop on the f CEO stabilization.The residual phase noise integrated from 1 Hz to 1 MHz is 456 mrad when the f rep stabilization is off and 448 mrad when it is on.Taking into account the linear scaling with the f rep , these residual noise values are comparable to the best results obtained by stabilized gigahertz solid-state lasers [29] which are not monolithic [32], including commercial Ti:sapphire combs [33].The f rep noise measurement above 1 kHz is limited by the detector shot noise [Fig.4(a)].The timing jitter of the individually stabilized f rep amounts to 52 fs, which is close to the best values for SDLs [21].With a simultaneously stabilized f CEO , the timing jitter increases to 257 fs.The additionally introduced noise can be explained by a coupling of the pump modulation used for the f CEO stabilization to the timing jitter.The coupling seems to be stronger compared to typical solid state laser systems presumably due to the different upper level lifetimes.We showing that the frequency deviation can be reduced with longer average times without running out of the stabilization loop.Even though the f rep is more stable, the frequency stability of the optical frequencies ν opt = f CEO + nf rep is limited by the f rep noise, which is multiplied by a factor n ≈182'300 (corresponding to 1025 nm), and not by the f CEO stabilization.

Laser setup and performance
In our experiment, we use a diode-pumped VECSEL passively modelocked with a SESAM.The laser cavity is V-shaped with the SESAM and the output coupler (OC, 100 mm radius of curvature, 1% transmission) as end-mirrors and the VECSEL chip as the folding mirror [Fig.
1 indicate clean fundamental modelocking.To obtain the highest signal-to-noise ratio (SNR) of the detected CEO beat, the laser was not operated at the maximum average output power of 160 mW, but at a reduced power of 120 mW with a pump power of 14.2 W.

Silicon nitride waveguide
The silicon nitride waveguide is compatible with complementary metal-oxide-semiconductor (CMOS) fabrication techniques.It has a sub-wavelength cross-section (height: 750 nm, width: 780 nm) with a silicon oxide cladding.The waveguide dispersion is simulated using a finite element software and exhibits two zero-dispersion wavelengths [Fig.1(b)].The 5-cmlong waveguide is spiraled to fit on the 1 x 1 mm writing area of the electron-beam Fig. 6.Simulation of the generated supercontinuum.The simulation of the generated supercontinuum by the silicon nitride waveguide for 8 pJ (green) and 14 pJ (orange) of coupled pulse energy, respectively.For the pulse energy of 14 pJ, the coherence of the spectral parts used for f-to-2f interferometry is drastically reduced.The coupled pulse energies in the simulation are smaller than the pulse energies in the experiment, since in the experiment, not all the power is coupled into the desired TM 0 mode.lithography tool [Fig.1(b)].For the free-space coupling into the waveguide, we use a telescope to enlarge the optical beam and an aspheric lens to focus onto the waveguide facet [Fig.1(a)].The observed loss of coherence in the generated SC for increased pulse energies [Fig.2(c)] is confirmed in our simulations (Fig. 6).The propagation of the spectrum inside the waveguide is described by the generalized nonlinear Schrödinger equations including shot noise and spontaneous Raman scattering.The coupled pulse energies used in the simulation are smaller than the experimental values, since in the experiment, a fraction of the power is not coupled into the desired TM 0 .The incoherence observed for increased pulse energies results from amplified noise inside the waveguide and can be avoided by using a shorter waveguide.

Pulse repetition rate stabilization
The pulse repetition rate f rep is detected with a beam reflection from a glass wedge and measured with a fast fiber-coupled photodiode (Model 1434, Newport).For the f rep stabilization, the 12th harmonic at 19.2 GHz is used which increases the sensitivity of the stabilization.To pre-process the signal, the 12th harmonic is mixed down to 70 MHz (MM93PG-40, Remec Inc.) with an ultra-stable microwave source (83650A, Hewlett-Packard), amplified by a low noise amplifier (DUPVA-1-70, Femto), and bandpass filtered (BBP-70 + , Mini-Circuits) [Fig.1(a)].The pre-processed signal is then used as input to the ultra-high frequency digital lock-in amplifier (UHFLI, Zurich Instruments), where the input signal is demodulated with the internal ultra-stable oven-controlled oscillator.The resulting error signal is then sent through the UHFLI's proportional-integral (PI) loop filter to generate the correction signal.The correction signal modulates a piezo driver (SQV 1/150, Piezomechanik GmbH), which controls a piezo-mounted output coupler, similar to the one used by Mangold et al. [21].The −3 dB operation bandwidth of the actuator used for the pulse repetition rate is around 1 kHz.

CEO frequency stabilization
The CEO frequency f CEO is measured by f-to-2f interferometry [Fig.1(a)].The octavespanning SC is split into a short and a long wavelength part by a dichroic beam splitter for temporal overlap tuning.After recombination, the long wavelength part is frequency-doubled in a 3-mm long, magnesium-doped periodically poled lithium niobate (PPLN) crystal.Finally, an optical bandpass filter centered around 720 nm selects the desired spectral components before they interfere on a high-speed avalanche photodetector (APD 210, Menlo Systems).The photodetector has a bandwidth of 1 GHz visible in the drop of signal strength in the microwave spectrum [Fig.2(a)].For pre-processing, the photodiode signal is mixed down to 70 MHz (ZX05-30W-S + , Mini Circuits) with an ultra-stable microwave source (83650B, Agilent), amplified by a low noise amplifier (DHPVA-100, Femto), and bandpass filtered (BBP-70 + , Mini-Circuits).The corresponding correction signal is generated with the UHFLI in the same way as for the f rep .The correction signal drives a custom-made voltage-to-current converter, which is connected in parallel to the noise-filtered electronic power supply of the pump diode to modulated the pump current.The modulation bandwidth is estimated to be around 100 kHz.

Noise characterization
The noise of the f rep and f CEO is analyzed with a signal source analyzer (SSA; E5052B, Agilent).The noise is measured inside the feedback loop using the pre-processed signals before the input of the UHFLI.For the f rep characterization, the "reference source/phaselocked loop" (PLL) method is applied.This method uses a PLL to lock the internal reference source of the SSA to the carrier frequency of the device under test (DUT) in phase quadrature.The signal of the DUT and the internal reference source are compared in a double-balanced mixer and analyzed with baseband spectrum analyzer.The "reference source/PLL" method has a very good sensitivity and is insensitive to amplitude noise.The frequency noise PSD was measured at the 12th harmonic of f rep and then subsequently divided by factor of 12 2 to account for the quadratic scaling of the frequency noise with the harmonic number.
The stabilized f CEO is also characterized with the "reference source/PLL" method.However, the higher drift rate of free-running f CEO together with the limited SNR do not allow to measure the phase noise with the "reference source/PLL" method.Thus, we use the "heterodyne (digital) discriminator" method of the SSA, which is modified digital version of the analog delay-line discriminator method.Whereby instead of the internal reference source, the signal of the DUT is compared to a delayed copy of itself.The discriminator method is able to track the free-running f CEO but it degrades the measurement sensitivity, especially with our limited SNR.Consequently, the measured phase noise of the free-running f CEO is only an upper estimate.This explains, why free-running phase noise is not compatible with the measured CEO linewidth of ≈20 kHz and why the noise spectrum of the stabilized f CEO does not reach the level of the free-running case above the modulation feedback bandwidth >1 MHz.For the discriminator measurement, the f-to-2f photodiode signal is mixed to 300 MHz and bandpass filtered (3TF-250/500-5S, Lorch).

Frequency stability
The frequency time series is calculated from the measured phase deviation.The phase deviation is recorded by the UHFLI while simultaneously calculating the PI-loop correction signal.The input signal is demodulated with the internal reference oscillator at a fixed frequency, to which the signal is stabilized.We then record the phase of the demodulated signal, which allows us to calculate the corresponding frequency offset.From this, we calculated the overlapping relative Allan deviation estimator (ADEV) [34,35]: where τ is a multiple of the measurement interval τ 0 , N is the number of measurements, and x i is the i th measurement of the time-error function x(t) = φ(t)/(2πν 0 ).The instantaneous frequency ν(t) = ν 0 + d t φ(t)/(2π) is then given by the nominal frequency ν 0 the measured phase deviation φ(t).For the f CEO characterization, the nominal frequency is given by ν 0 = f CEO = 776 MHz.Since the Allan deviation is not expected to scale with the CEO frequency, the absolute Allan deviation given by σ y (t) f CEO is shown in Fig. 3(c).For the f rep characterization, the nominal frequency is given by the 12th harmonic: ν 0 = 12f rep = 19.2GHz.The measurement interval τ 0 = 1.14 μs leads to confidence intervals below visibility in Fig. 3(c), 4(c), which thus are omitted.
To distinguish between white phase noise and flicker phase noise, we calculated the modified Allan deviation.The τ -3/2 -slope of the modified Allan deviance (MDEV, not shown) [35] implies that white phase noise is dominant for the f CEO and f rep .The recording time of 15 min was not limited by the laser stabilization, but by the available storage space.The Allan deviation is calculated from the in-loop phase deviation, thus drifts of the electronic reference oscillators do not contribute.
For validation of the measurement, the Allan deviation is also calculated from the measured one-sided frequency noise PSD S y (f) [36]: ) , which is in good agreement with the overlapping Allan deviation calculated from the timeseries with a discrepancy factor of 2.0 and 1.2 for the f CEO and f rep , respectively.

Conclusions
In conclusion, we have presented the first fully stabilized OFC from an ultrafast SDL without external pulse amplification or pulse compression.A novel Si 3 N 4 waveguide design allowed for energy-efficient self-referenced detection of the f CEO .Using an all-digital stabilization setup, a noise performance comparable to the best results achieved with other non-monolithic gigahertz laser sources was achieved already with this first demonstration.This result opens up a new observable to study the underlying semiconductor laser physics and provides a new compact OFC source based on semiconductor lasers with a great flexibility in operation wavelength ranging from UV to MIR potentially enabling many future applications.

Fig. 1 .
Fig. 1.Laser and stabilization setup.(a) Schematic of the laser and stabilization setup including the laser cavity, pulse repetition rate (f rep ) detection, supercontinuum generation, f-to-2f interferometry, signal pre-processing, and stabilization feedback loop.OC, output coupler; T, transmission; R, radius of curvature; PPLN, periodically poled lithium niobate; BPF, optical band-pass filter; UHFLI, digital ultra-high lock-in amplifier (Zurich Instruments); V-to-A, voltage to current; (b) Silicon nitride waveguide dispersion for a cross-section of 750 x 780 nm simulated for the TM 0 mode featuring two zero-dispersion wavelengths (ZDW).Inset: illustration of the 5 cm long spiraled waveguide.(c) Generated octave-spanning supercontinuum for a coupled pulse energy of 17 pJ.The highlighted wavelength regions are used for f-to-2f interferometry.

Fig. 3 .
Fig. 3. Noise characterization of the CEO frequency.(a) Frequency noise power spectral density (PSD) of the CEO frequency: free-running, CEO stabilized, and both CEO and pulse repetition rate stabilized.The noise of the stabilized CEO frequencies lies below the βseparation line indicating a tight lock.The corresponding integrated phase noise values are shown below.(b) Frequency stability of the CEO frequency recorded over 15 min with an averaging time of 1 s.(c) Allan deviation of the CEO frequency with the corresponding relative Allan deviation with respect to the optical frequencies on the right axis.The Allan deviations are calculated from the recorded time traces and from the frequency noise PSDs.

Fig. 4 .
Fig. 4. Noise characterization of the repetition stabilization.(a) Frequency noise PSD of the pulse repetition rate: free-running, pulse repetition rate stabilized, and both repetition rate and CEO stabilized.The frequency noise PSD was measured at the 12th harmonic of the repetition rate and subsequently divided by factor of 12 2 to account for the quadratic scaling of the noise with the harmonic number.The corresponding timing jitters are shown below.(b) Frequency stability of the pulse repetition rate recorded over 15 min with an averaging time of 1 s.(c) Allan deviation of the pulse repetition rate with the corresponding relative Allan deviation with respect to the optical frequencies on the right axis.The Allan deviations are calculated from the recorded time traces and from the frequency noise PSDs.could not stabilize the f rep noise above 1 kHz because of the limited modulation bandwidth of the piezo actuator.Both the f rep and f CEO were recorded over 15 min in the tight lock condition [Fig.3(b), 4(b)] (limited by data storage capacity).The standard deviation with a 1-s averaging time is 2 mHz and 2 µHz for the f CEO and f rep , respectively.The corresponding Allan deviations (see Section 3.6) are linearly decreasing [Fig.3(c), 4(c)],showing that the frequency deviation can be reduced with longer average times without running out of the stabilization loop.Even though the f rep is more stable, the frequency stability of the optical frequencies ν opt = f CEO + nf rep is limited by the f rep noise, which is multiplied by a factor n ≈182'300 (corresponding to 1025 nm), and not by the f CEO stabilization.

f
f rep = 1.605 662 GHz σ STD, rep = 2 μHz (1 s avg.) rep stab.f rep & f CEO stab.257fs (a)].The VECSEL gain chip is optically pumped by a commercial multimode pump diode at 808 nm at an angle of 45°.The pump diode is wavelength-stabilized with a volume holographic grating (VHG) and we use a custom-made low-pass filter for the electrical power supply of the diode.The VECSEL and SESAM structures are analogous to those presented by Waldburger et al.[4], with the difference being that the VECSEL chip was grown by molecular beam epitaxy (MBE) instead of metalorganic vapor phase epitaxy (MOVPE).The VECSEL and the SESAM are temperature controlled with a water-cooled Peltier element.The laser is boxed in order to reduce the influence of mechanical vibration and air turbulences.The modelocking characterizations are depicted in Fig.5.The VECSEL produces pulses with a pulse duration of 122 fs [Fig.5(a)].The optical spectrum is centered at 1025.3 nm with a full width at half maximum (FWHM) of 11.8 nm [Fig.5(b)] and the pulse repetition rate is 1.599 GHz [Fig.5(c)].The side-peak free microwave spectrum of the pulse repetition rate and the equally powerful harmonics of the pulse repetition rate [Fig.5(d)]

Fig. 5 .
Fig. 5. Modelocking characterization of the SDL.(a) Autocorrelation of the modelocked pulses fitted with a 122 fs sech2-shaped pulse.(b) Optical spectrum centered around 1025.3 nm with a full width at half maximum (FWHM) of 11.8 nm.(c) Microwave spectra of the 1.599 GHz pulse repetition rates showing a 20 MHz span measured with a resolution bandwidth (RBW) of 3 kHz.(d) Microwave spectra of the equally powerful higher harmonics of the repetition rates measured with a RBW of 30 kHz.