Femtosecond mode locking based on adiabatic excitation of quadratic solitons

We demonstrate a new approach for pulse formation in mode-locked lasers, based on exciting intracavity solitons in a two-dimensionally patterned quasi-phase-matching (QPM) grating. Through an adiabatic following process enabled by an apodized QPM crystal, we transiently excite multicolor nonlinear states within the crystal, utilize their advantageous properties for pulse formation and stabilization, and then convert the energy back to the resonating laser pulse before the end of the crystal in order to suppress losses. This idea gives access to large nonlinearities that would otherwise be too lossy for use intracavity. In our case, the states accessed are self-defocusing Kerr-like nonlinearities based on phase-mismatched second-harmonic generation. The QPM device has an additional transverse gradient, for tuning the nonlinearity and to aid in laser self-starting. We demonstrate the technique in a semiconductor saturable absorber mirror mode-locked laser with Yb:CALGO as the gain medium, producing 100 fs pulses at 540 MHz repetition rate, with 760 mW of average output power. We present comprehensive theoretical and numerical modeling of the laser to understand the new mode-locking regime. Our approach offers a flexible and compact route to managing nonlinearities inside laser cavities while suppressing the losses that could otherwise prevent or deteriorate mode-locked operation, and is particularly interesting for highly compact bulk, fiber, and waveguide lasers with gigahertz repetition rates and operating wavelengths from the nearto mid-infrared spectral regions. © 2015 Optical Society of America


INTRODUCTION
Soliton dynamics play a critical role in many optical systems and are of fundamental interest in several areas of physics.In the context of mode-locked lasers, temporal solitons support ultrashort pulse formation in a wide variety of laser architectures, including solid-state lasers and fiber lasers [1,2], and more recently even optical microresonators [3].When combined with semiconductor saturable absorber mirrors (SESAMs) to start and stabilize the mode locking [4], soliton shaping allows the pulse to be much shorter than the slow response time of the SESAM [5].Another prominent example of soliton dynamics in the context of ultrafast optics is supercontinuum generation in optical fibers and waveguides [6].The continued development of ultrafast sources including frequency combs is thus connected with our ability to create solitons and control their dynamics.This development is of great interest for numerous application areas [7], including spectroscopy, metrology, optical clocks, and high-speed communication and information processing.In this paper, we experimentally demonstrate a new type of two-dimensionally patterned quasi-phase-matching (2D-QPM) device designed to adiabatically excite intracavity solitons based on the second-order nonlinearity χ 2 .By allowing much greater control over the nonlinear properties of the device, this 2D-QPM technique has the potential to overcome several fundamental limitations in conventional soliton formation techniques, thus opening up many new opportunities for ultrashort pulse generation in near-and midinfrared mode-locked lasers, particularly compact high repetition rate and waveguide lasers.
To understand this potential, we first recount some important properties of solitons based on the widely used Kerr effect.The Kerr effect yields a positive nonlinear refractive index n 2 when the laser wavelength is far enough from the bandgap to avoid multiphoton absorption.Using wide bandgap materials to satisfy this constraint means only moderate values of n 2 are typically accessible, since n 2 scales inversely with the fourth power of the bandgap [8].Thus, achieving sufficient nonlinearity for soliton pulse shaping becomes increasingly difficult for lower energy lasers.Negative group delay dispersion (GDD) is also required when n 2 > 0, which constrains the dispersion management due to the positive dispersion typically offered by bulk materials at nearinfrared wavelengths.These issues become increasingly challenging when scaling to high laser repetition rates, because of the lower pulse energies, greater susceptibility to damage from Q-switching instabilities [9], and more restricted cavity design space.
An alternative to the Kerr nonlinearity is offered by cascaded quadratic nonlinearities (CQNs) [10,11].By tuning a second-harmonic generation (SHG) medium far from phasematching, a large and negative effective nonlinear refractive index can be achieved, supporting a Kerr-like soliton with negative n 2 .CQNs offer numerous potential advantages for mode locking due to the large, adjustable, and negative (self-defocusing) nonlinear refractive indices that can be obtained.Such cascaded χ 2 nonlinearities have been exploited for various applications, including pulse compression and frequency shifting [12][13][14][15], and supercontinuum generation [16,17].The corresponding physics can also play an important role in efficient chirped QPM interactions, including adiabatic frequency conversion and parametric amplification [18,19], a connection that we discuss further in this paper.CQNs have been exploited for mode locking [20][21][22][23][24][25][26][27][28], with the SPM yielding either soliton formation, or a Kerr-like lens to favor pulsed operation, or both.
An inherent property in cascading devices (i.e., those involving CQNs), and more generally solitons based on χ 2 media, is the need for multicolor pulses, since the soliton requires an interplay between the fundamental or first harmonic (FH) and the second harmonic (SH).Indeed, a key feature of a CQN interaction using ultrashort pulses is the presence of a weak SH pulse propagating with a stronger FH pulse.This presents a disparity with many practical laser systems, where we have only the FH available: in this case, energy is lost when exciting the SH pulse.Since these losses are intensity dependent, they can destabilize the modelocked laser.These SHG losses can be reduced, for a given amount of self-phase modulation (SPM), by using a longer CQN crystal and operating further from phase-matching, which we exploited in [28].However, this design procedure has several drawbacks, since it introduces additional dispersion, linear losses, and optical delay to the pulses and requires a long crystal, which prohibits ultracompact devices.For low-loss optical cavities utilizing CQNs, or χ 2 solitons in general, we are thus presented with a fundamental trade-off between SPM and nonlinear losses.
Here, we overcome this limitation via a new type of microstructured cascaded-χ 2 device based on aperiodic QPM.The device concept is illustrated in Fig. 1.The QPM period varies with respect to longitudinal position (propagation direction): the regions near the edges (input and output sides) have a long period, while the region in the middle has a short period.This "apodized" longitudinal QPM profile is designed to adiabatically excite ("turn on") and later de-excite ("turn off ") the SH wave in a CQN-like interaction, allowing a large SPM in the middle region where the SH is strong, while minimizing the energy lost from the fundamental pulse.This approach is similar in spirit to the concept of soliton amplification in chirped QPM media [29,30], as well as the concept of nonlinearity management inside modelocked lasers [31].In addition to the longitudinal chirp, there is a fan-out structure [32], with the period changing with transverse position.This transverse variation allows continuous tuning of the nonlinearity, and helps facilitate self-starting mode locking.With this 2D-QPM device, we thus have access to all the advantageous properties of CQNs, and overcome their fundamental drawback to enable a multitude of new applications.
In terms of laser physics, this device concept opens up the possibility of accessing new mode-locking regimes in low-loss optical cavities by adiabatically exciting the relevant state, even if the fraction of energy in the SH is substantial, as will be the case for many χ 2 solitons [33].Such strong-SH states are now accessible intracavity, because the SH wave is excited from zero and is de-excited after the interaction region, before it can exit the crystal and constitute a loss for the laser.Furthermore, the concept has immediate practical importance for lasers with gigahertz repetition rates, because the features offered by adiabatic cascading simultaneously address many of the challenges faced when scaling femtosecond diode-pumped solid-state lasers to high repetition rates.These features include (1) low-loss soliton formation at low intracavity fluences in a highly compact device, enabled by the large self-defocusing n 2 , of order −1.5 × 10 −5 cm 2 ∕GW in our case; (2) straightforward dispersion management, even in restrictive cavity geometries, since the dispersion provided by bulk materials leads to soliton formation; and (3) suppression of damage to optics from Q-switching instabilities through the variable and nonabsorptive losses from SHG, which allow the laser to be operated at the rollover point of the nonlinear reflectivity to suppress Q-switched mode locking [34].
To demonstrate the concept, we implemented the QPM device depicted in Fig. 1 in a mode-locked laser, using Yb:CaAlGdO 4 (CALGO) as the gain medium [35].The laser produced 100 fs pulses at a repetition rate of 0.54 GHz, with 760 mW average output power, and exhibited reliable self-starting mode locking.Our results prove the viability of adiabatic soliton excitation inside mode-locked lasers, enabled by the versatility of two-dimensional QPM structures.
In this paper, we first present the theory of adiabatic SH excitation (Section 2), our experimental results (Sections 3 and 5), numerical simulation of the laser (Section 4), and concluding remarks (Section 6).

ADIABATIC HARMONIC EXCITATION
To explain and motivate the longitudinal variation of the QPM structure in Fig. 1 in more detail, we utilize the existence of eigenmodes associated with the coupled wave equations in a uniform χ 2 nonlinear medium.These are solutions where all of the waves propagate indefinitely with no exchange of energy between them.We quantify these eigenmodes for the case of SHG in Section 1 of Supplement 1, with an approach analogous to that Fig. 1.Illustration of the inverted domains in our fan-out, apodized QPM device implemented in MgO:LiNbO 3 .For clarity, the transverse profile of every 15th domain is shown (i.e., domains 1, 16, 31, …), and the thickness of the lines is not to scale (each domain is actually ∼3.5 μm long).The illustration shows the two primary features.First, the fan-out profile, with period decreasing for larger transverse positions.Second, the apodization profile, where there is a large period at the input and output sides of the device (regions 1 and 3), and a smooth but rapid change to a shorter period within the middle part (region 2).To the right of the domain pattern, the time-dependent intensity of the laser and its SH are illustrated, indicating the small SH intensity in regions 1 and 3, and the moderate intensity in region 2.
of [19].The resulting properties of one of these eigenmodes is illustrated [Fig.2(a)] as a function of phase-mismatch Δk.The figure is normalized to a length-scale L NL defining the strength of the SHG process.This length is , where subscripts correspond to the FH (j 1) and SH ( j 2) waves, ω j are angular frequencies, n j are refractive indices, A j are electric field envelopes, and d eff is the effective nonlinear coefficient.The phasemismatch is given by Δk k 2 − 2k 1 − K g for wave vectors k j ω j n j ∕c and QPM grating k-vector K g .
The blue curves in Fig. 2(a) show the fraction of total intensity in the SH part of the eigenmode, while the red curves show the rate γ FH at which the FH part of the eigenmode accumulates SPM.From the dashed red curve, at large Δk, the SPM accumulated by the FH over a distance L is ϕ SPM γ FH L ≈ −L∕ΔkL 2 NL .Similarly, from the dashed blue curve, the fraction of energy in the SH is a NL ≈ 1∕ΔkL NL 2 .Both ϕ SPM and a NL are thus proportional to the input FH intensity.These features closely correspond to those of CQNs.
The utility of the eigenmode description becomes apparent when considering nonuniform phase-matching media.If Δk is varied via a chirped QPM period, we can consider a local eigenmode, associated with the position-dependent value of Δkz k 2 − 2k 1 − K g z.Then, if the QPM period varies slowly enough, the fields can adiabatically follow the eigenmode as it evolves with Δkz.This capability has enabled adiabatic frequency conversion [18], in which Δkz is swept slowly through phase-matching in order to achieve efficient frequency conversion.This concept has been applied to efficient and broadband sum-and difference-frequency generation [18], optical parametric amplification [36], and SHG [37].
In this work, we use a new type of aperiodic QPM structure that bridges the gap between adiabatic frequency conversion and cascaded quadratic nonlinearities: we begin far from phasematching to efficiently couple the fields into an eigenmode, then adiabatically reduce Δk so that the FH and SH evolve into an eigenmode with a large rate of SPM for the FH, and then, after an appropriate distance, we de-excite the SH part of the eigenmode by returning to a large Δk.A typical example Δk profile is shown in Fig. 2(b), designed by adapting the apodization techniques from [19].The resulting evolution for continuous plane waves is shown in Fig. 2(c), blue curve.The SH component of the eigenmode is shown by the green curve: it starts small (region 1, with∕ΔkL NL ≈ 150), is larger in the middle region (region 2, with ΔkL NL ≈ 17), and is small again at the output (region 3, with ΔkL NL ≈ 150).The blue curve shows how the SH field follows this eigenmode.Because Δk0L NL ≈ 150 is large but still finite, launching of the fields into the eigenmode is not perfect, which leads to the oscillations in the blue curve.For comparison, the red curve shows the fields in the case of an unapodized grating (constant Δk): the oscillations are much more pronounced in this case and they persist through the whole device.The chosen range of ΔkL NL in this example implies that the large-Δk approximations of Fig. 2(a) are valid.
In the case of a pulsed interaction with finite group velocity mismatch (GVM) between the FH and SH envelopes, the oscillations in Fig. 2(c) manifest as two pulse components.This behavior is evident in Fig. 2(d), which shows a plane-wave simulation of the SH in an apodized QPM grating similar to Fig. 1.The figure uses a reference velocity of v 1 vω 1 for group velocity vω d k∕d ω −1 .We assume a 150 fs FH with ≈1.5 GW∕cm 2 peak intensity, and choose Δk 60 mm −1 , yielding n 2 ≈ −1.5 × 10 −5 cm 2 ∕GW.In this pulsed case, the SH component coupled into an eigenmode propagates with the FH, at velocity ≈v 1 , and therefore stays at ≈0 ps delay.This "non-delayed" component follows the eigenmode as it is "turned on" in the input-apodization region and "turned off " in the output-apodization region.In contrast, the SH component not coupled into the eigenmode propagates with constant energy at the normal group velocity v 2 vω 2 and is therefore delayed with respect to the FH. Figure 2(e) shows the SH intensity at the output of the crystal, to show how much weaker the remaining pulse components are in an apodized grating compared with an unapodized grating (peak of ≈0.01%instead of ≈0.25% ).In the mode-locked laser this SH can constitute a significant loss, since four such pulses are generated [two for each pass through the crystal, as seen in Fig. 2(e), and two passes through the crystal for a linear cavity].
Given the above, for a QPM grating with an adiabatic chirp profile such that the fields follow the relevant eigenmode [19], two SH components are generated: one into the eigenmode, and another that propagates linearly.We can therefore estimate the energy lost to the SH in a single pass as a NL ≈ Δk0L NL −2 ΔkLL NL −2 ; (1) where the first term is from the energy lost into the "delayed" pulse component of Fig. 2(d), and the second term is from energy still remaining in the eigenmode at the output of the device.In contrast, the SPM accumulated by the FH in a single pass corresponds to an integral over the whole device: where these equations assume Δkz remains far enough from phase-matching that (i) the approximate dashed curves in Fig. 2(a) can be applied instead of the solid curves, and (ii) the bandwidth of the cascading process is large compared to the pulse bandwidth; this bandwidth can be quantified via the response function theory introduced in Supplement 1, Section 2. The sign of ϕ SPM determines the sign of n 2 : ϕ SPM < 0 means n 2 < 0, i.e., a self-defocusing nonlinearity.
To estimate the cavity losses from Eq. ( 1), we can first calculate L NL using the peak of the electric fields, and then multiply a NL in Eq. ( 1) by 2∕3 to account for integration over space and time (assuming a Gaussian spatial profile and a sech 2 temporal profile), and two passes through the crystal per round trip.
Equations ( 1) and ( 2) show how we can obtain a large SPM without SHG losses as long as the QPM grating is chirped adiabatically [19], and we impose the appropriate boundary conditions on Δkz.Moreover, in comparing the blue and red curves in Fig. 2(c) and identifying the two pulse components in Fig. 2(d), we see that the oscillations in intensity associated with conventional CQN interactions are not an inherent aspect of the desired SPM process, but originate from poor launching of the input fields into the relevant eigenmode that gives rise to SPM.These oscillations are strongly reduced in the apodized case.
The QPM design used for our experiments is illustrated in Fig. 3.The transverse variation in the phase-mismatch is also shown.Figure 3(b) shows the resulting QPM period profile at the three transverse positions x j , assuming a nominal wavelength of 1045 nm.When converted into a ferroelectric domain pattern, this design yields Fig. 1.

EXPERIMENTAL SETUP
To demonstrate the device concept and its applicability to shortpulse mode locking, we built a laser based on Yb:CALGO, shown in Fig. 4. Yb:CALGO was chosen since it supports short pulses and has favorable thermal properties [35].Pump light at ≈980 nm is directed to the antireflection-coated 3-mm-long Yb:CALGO crystal through the end mirror M1.The CALGO and aperiodically poled lithium niobate (APPLN) crystals are both placed close to M1; M1 also provides −400 fs 2 GDD.The APPLN crystal had a thickness of 0.5 mm, and was designed according to Fig. 1.We configure mirror M2 as either a high reflector (HR) or a GTI-type mirror with ≈ − 500 fs 2 per reflection.We estimate total round-trip GDD values of 1237 and 237 fs 2 for the HR and GTI cases, respectively.There are uncertainties of order 100 fs 2 from the lengths of the materials involved and the properties of the dispersion-compensating mirrors.Mirror M2 is a 2% output coupler and is used as a turning mirror.The gain crystal and the SESAM are the same as in [28]: the SESAM has a 2.8% modulation depth and a saturation fluence of 5.8 μJ∕cm 2 .The beam 1∕e 2 radius at the Yb:CALGO, APPLN crystal, and mirror M1 is ≈150 μm.The beam radius at the SESAM is ≈150 μm.
The QPM crystal design had several gratings, including simple period gratings, apodized but non-fan-out gratings, as well as the apodized fan-out gratings illustrated in Figs. 1 and 3. When using an apodized, fan-out QPM grating, the laser exhibits reliable selfstarting mode locking in both M2 configurations (HR and GTI).Both cases correspond to a net positive cavity GDD, and the main source of SPM is from the negative contribution of the APPLN crystal.The fan-out apodized QPM gratings almost always supported reliable self-starting mode locking in a variety of cavity configurations and over a large range of QPM periods.We sometimes observed mode locking when using the non-fan-out apodized gratings, but never observed mode locking with the "control" gratings (non-fan-out, non-apodized).These results indicate not only the expected importance of apodization (longitudinal variation of QPM period) for achieving mode locking, but also the importance of the fan-out design on achieving self-starting.In Supplement 1, Section 5, we show how the transverse variation of the absolute phase of the QPM structure can explain this initially surprising property of fan-out gratings.
The largest mode-locking range was obtained in the case with M2 as an HR, while the shortest pulses were obtained with M2 as a GTI.The parameters in this latter case were 760 mW average power (total from both output beams), 0.54 GHz repetition rate,   3. (a) Phase mismatch Δkx; z Δk 0 ω 0 − K g x; z in the device used, assuming a wavelength λ 0 2πc∕ω 0 1045 nm.The apodization profile is designed according to [19], with additional buffer regions of constant and large Δk at the ends (regions outside the vertical dashed lines).(b) QPM profile 2π∕K g x j ; z for three different transverse positions x j .The longitudinal coordinate is centered at 0 mm in this case to emphasize the symmetry.The grating k-vector required for phasematched SHG at λ 0 1045 nm is Δk 0 ≈ 953.5 mm −1 , corresponding to the dashed horizontal line.The slope in grating k-vector ∂K g ∕∂x 40 mm −2 .For the middle part of the grating ("Pos.2"), the minimum grating k-vector is ≈894 mm −1 .To test the expected performance with respect to SHG suppression, we show in Fig. 6 the spectrum of the SH outside the crystal.As discussed in Section 2, in a normal cascading interaction, the temporal profile of the SH field corresponds to a double pulse, with each pulse having equal energy (although slightly different dispersion and nonlinear effects).The harmonic spectrum therefore resembles a double-pulse spectrum, whose envelope scales as F A 1 t 2 , and delay between the pulses given by the group velocity walk-off between the FH and SH, which is vω 2 −1 − vω 1 −1 L ≈ 1.5 ps in our case.In apodized cascading devices, we strongly suppress this normal SH response [see Fig. 2(d)].However, the SH can also be generated as a consequence of random duty cycle (RDC) errors in the QPM grating.Such RDC errors lead to a phase-matching pedestal [38], for which individual SH spectra are strongly modulated.For an apodized device having small but finite RDC errors, we thus expect the SH spectrum to be strongly modulated and to bear little resemblance to the FH from which it is generated, since the normal double-pulse response is almost completely suppressed by apodization, but the RDC contribution remains.
We confirmed the expected qualitative behavior of the SH, as shown in Fig. 6, which indicates that there is no signature of the normal cascading-type response, with most of the SH spectrum attributable to imperfect poling instead.Furthermore, for a fanout structure, we find that the beam profile of the SH is modified, as well as its spectrum.While the beam quality for the FH is always excellent (M 2 ≈ 1), the beam profile of the discarded SH is poor, and it exhibits significant spatiotemporal coupling, as illustrated in Fig. 6(b).For this measurement, the SH beam was scanned spatially over the fiber-collimating lens before the optical spectrum analyzer.The properties of the SH and the role of fan-out gratings in modulating the SH spectrum and beam are discussed further in Section 5.

NUMERICAL MODELING OF THE LASER
The results of Section 3 showed features of the laser that warrant further study, including the asymmetric laser spectrum [Fig.5(a)] and the modulated SH spectrum [Fig.6(b)].
To understand the behavior of the laser and its trade-offs in more detail, we need to consider the effect of the new QPM device on broadband pulses.For example, the presence of GVM, as evident in Fig. 2(d), leads to a slope in the frequency-dependent phase mismatch Δkω.Operation closer to phase matching enables a greater rate of SPM, but also increases the relative variation of Δkω across the optical spectrum.Moreover, imperfections in the QPM domains, namely, RDC errors [38], can modify the behavior of the SH compared with Fig. 2(d).To model the resulting dynamical effects, we derive in Sections 2 and 3 of Supplement 1 a perturbative model of the nonlinearity experienced by FH pulses in nonuniform QPM media.The result is a Raman-like response function, with examples shown in Figs.S1 and S2 (Supplement 1).In this section, we incorporate that response function into general numerical simulations of the laser.
These simulations describe the evolution of the pulse through the cavity, including nonlinear processes in all the intracavity elements, as well as the population dynamics of the gain medium.Certain important assumptions are made to speed up the calculations, including modal overlaps to obtain a (1 1D) model; approximating the single-pass propagation in APPLN via the above-mentioned response function; and artificially accelerating the laser population dynamics by numerically scaling the cross section, lifetime, and doping concentration, such that the gain and pulse-shaping dynamics remain the same but relaxation oscillations are damped.Further details are given in Supplement 1 (Section 4).
In Fig. 7, we show numerical results, which are in good agreement with the experimental results from Fig. 5.The simulation predicts ≈100 fs pulses with comparable power to the actual laser and given physically realistic values for all of the parameters of the simulation.Figure 7(a) shows the spectrum and temporal profile (inset).Remarkably, the asymmetry in the spectrum is extremely similar to the experimentally measured spectrum of the same laser [Fig.5(a)].Figure 7(b) shows the evolution of the population dynamics over time and through the length of the Yb:CALGO crystal.Figure 7(c) shows the evolution of the laser spectrum, which exhibits a frequency shift after the initial buildup from noise. Figure 7(d) shows the corresponding pulse profile on a logarithmic scale, to show the more gradual evolution of the laser into a single clean pulse.
The trend in Fig. 7(c) suggests a χ 2 self-frequency-shift (SFS) effect originating from the APPLN crystal.This SFS can be explained via the phase curve of the response function in Fig. S1(b) (Supplement 1).Such an SFS is reminiscent of Raman nonlinearities, where the loss-related part of the corresponding "response function" directly scatters energy from the high-frequency to the low-frequency part of the spectrum [39].The case of our APPLN medium is different, since the loss-related part of the response is suppressed via apodization, leaving mainly the phaserelated part that gives rise to SPM.Nonetheless, CQNs also support SFS effects, which have been discussed in the context of single-pass pulse compression devices [14].In the configuration involved here, there is a blueshift, which shifts the center frequency away from phase matching and causes the asymmetric laser spectrum observed.We checked this by turning off the frequency dependence of the response function numerically: in that case, there is no frequency shift effect, and the predicted laser spectrum is symmetric.
These results show that by modeling each intracavity element we reproduce the observed experimental results.The noninstantaneous nonlinearity associated with the response function is important in APPLN devices when generating short pulses, and can be incorporated efficiently into simulations.

STUDY OF MODE-LOCKED OPERATION
Having identified important pulse-shaping effects, in this section we further experimentally study the nonlinear properties of the laser.For this we use M2 as an HR mirror instead of a GTI-type mirror: in this case there is more positive intracavity dispersion, yielding a wider range of mode-locked operation.For example, we can continuously scan from one side of a 2 mm fan-out QPM grating to the other by translating the crystal.We calculate the total round-trip GDD as 1237 fs 2 , after replacing the ≈ − 500 fs 2 GTI used for Fig. 5 with an HR.A typical result is power and a repetition rate of 1.45 W and 544 MHz, respectively, with slightly longer 149 fs pulses.
We first examine the laser properties as a function of output power by changing the pump power.Figures 8(a) and 8(b) show the trends for the pulse duration and center wavelength, respectively.The former shows the typical qualitative trend for soliton mode locking of a decreasing pulse duration with increasing laser power.Figure 8(b) shows that the center wavelength is also power dependent, which is primarily due to the SFS effect discussed in Section 4, although there could also be some influence from shifts in the gain peak in the Yb:CALGO crystal.The shift to shorter wavelengths brings the wavelength further from phase matching, thereby reducing the effective n 2 of the device.As such, Fig. 8(a) need not follow the τ ∝ 1∕power trend of conventional soliton mode locking.The dashed lines are full numerical simulations of the same configuration, showing good qualitative agreement with experiment; we assumed a realistic Δk ≈ 55 mm −1 (at 1045 nm), 0.5% additional cavity losses, and no other fit parameters.
The SFS effect also limited the achievable pulse durations in the M2-HR configuration, since SFS effects scale rapidly with decreasing pulse duration [39].With M2 as a GTI (Section 3), we reduced the GDD to reduce the required round-trip SPM and hence the strength of the SFS at a given pulse duration.This procedure allowed for 100 fs pulses that were neither excessively perturbed by SFS nor by SHG from fabrication errors.
We next show the properties of the SH.In general this SH is strongly modulated, likely due to the presence of RDC errors in the apodized grating.Figures 9(a) and 9(b) show the SH spectrum versus laser power and translation of the APPLN crystal transverse to the laser beam, respectively.Despite the SFS exhibited in Fig. 8(b), there is no corresponding shift in the structure of the SH spectrum in Fig. 9(a).This can be understood by noting that the SH spectrum scales with the spatial Fourier transform (FT) of the QPM grating.This FT is power independent, so the structure of the SH remains unchanged when changing the power.The situation changes when we keep the pump power   constant but translate the fan-out crystal.The QPM profile is then stretched/compressed, leading to a shift in its Fourier spectrum.This expected behavior is manifested in Fig. 9(b): the spatial FT of the grating shifts with crystal position, leading to a shift in the SH spectrum.The transverse dependence of the SH spectrum also explains Fig. 6(b) and the poor SH beam quality that we observe when using fan-out gratings: different parts of the beam see different parts of the QPM grating, and hence have their optical spectra shifted.Thus, for each spectral component, there is transversally varying intensity and phase, leading to a poor beam quality of each spectral component of the SH.This also explains the spatiotemporal coupling of the SH in Fig. 6(b).

CONCLUSIONS
In conclusion, we have demonstrated a new type of QPM device for soliton formation in mode-locked lasers.The device adiabatically excites and subsequently de-excites the SH wave in a soliton based on the second-order nonlinearity χ 2 of the QPM material.This technique allows for a large SPM, as in conventional cascaded χ 2 interactions, while suppressing or modifying the associated nonlinear losses.This approach gives access to large selfdefocusing nonlinearities, of order n 2 ≈ −1.5 × 10 −5 cm 2 ∕GW in our case, and the accessible nonlinearity increases with pulse duration.Small residual losses to SHG occur due to imperfect apodization and QPM fabrication errors.The combination of a large and self-defocusing nonlinearity, relaxed constraints on cavity dispersion, and adjustable nonlinear losses are uniquely favorable for supporting soliton formation far into the GHz mode-locking regime.
We demonstrated the concept via a proof of principle laser, which produced 100 fs pulses at 0.54 GHz repetition rate with 760 mW average power.Using the same cavity but replacing a GTI with an HR mirror, we obtained slightly longer pulses: 149 fs at 0.544 GHz repetition rate and 1.45 W average power.We studied several properties of the laser, including the residual SH wave leaving the crystal.To support these experiments, we performed a comprehensive theoretical and numerical analysis of the laser dynamics, showing good agreement with experiments.We also identified the influence of a fan-out QPM structure on the self-starting characteristics of the laser.
Two important issues we identified were the SFS effect and losses associated with RDC errors.We designed the laser and APPLN device so that the influence of the expected RDC errors based on previous studies [38] would be manageable.These errors can be reduced by improved fabrication.The SFS effect is strong because of the substantial GVM between the short FH and SH pulses in PPLN at 1 μm.Two ways to reduce the influence of this effect are reducing the cavity dispersion (as we did experimentally) or operating at longer wavelengths (infrared and mid-infrared) where there the GVM coefficient is smaller.This reduced GVM coefficient would allow operating at smaller Δk, yielding an even larger effective n 2 prior to the onset of strong SFS-type effects.A detailed study of the maximum practically achievable nonlinearity is planned for future work.More generally, adiabatic excitation combined with a reduced GVM could give access to a broader class of χ 2 solitons [33], supporting previously inaccessible mode-locking regimes.
The technique we have demonstrated has fundamental importance, since it gives access to a wide variety of intracavity solitons based on second-order nonlinearities (i.e., multi-color solitons) without incurring losses that would otherwise be associated with exciting them.It also has immediate practical relevance in scaling mode-locked solid-state lasers further into the gigahertz regime, including in the challenging infrared and mid-infrared spectral regions.Moreover, in mode-locked semiconductor disk lasers [40], the moderate intracavity intensities and low output coupling rates typically used provide a strong motivation for low-loss, highnonlinearity media for achieving robust pulse formation.In the future, as well as accessing gigahertz repetition rates and midinfrared wavelengths, the QPM devices could be implemented in PPLN waveguides rather than bulk, enabling extremely compact and robust waveguide lasers.

Fig. 2 .
Fig. 2. Eigenmodes of the coupled wave equations and their excitation.(a) Properties of the relevant eigenmodes for SHG.Blue curves, SH fraction of the eigenmode; red curves, rate of SPM for the first-harmonic part of the eigenmode.Dashed curves: large-Δk limit of the solid curves.(b) Example apodized QPM profile for adiabatic excitation of the desired eigenmode.(c) Green curve: trajectory of the relevant eigenmode given the QPM profile from (b), starting and ending with a very small SH component while having a smaller Δk in the middle to provide SPM.Blue curve: simulation of a plane-and continuous-wave SHG interaction, showing adiabatic following of the eigenmode.Red curve: conventional CQN interaction with Δk constant, exhibiting a larger peak SH intensity and strong oscillations.(d) Simulation of a pulsed plane-wave interaction relevant to our experimental conditions, illustrating how the ripples in (c) manifest as the two pulse components.(e) Output SH pulse profile from apodized and conventional (unapodized) QPM gratings; the apodized case corresponds to the output of (d).

Fig.
Fig. 3. (a) Phase mismatch Δkx;z Δk 0 ω 0 − K g x; z in the device used, assuming a wavelength λ 0 2πc∕ω 0 1045 nm.The apodization profile is designed according to[19], with additional buffer regions of constant and large Δk at the ends (regions outside the vertical dashed lines).(b) QPM profile 2π∕K g x j ; z for three different transverse positions x j .The longitudinal coordinate is centered at 0 mm in this case to emphasize the symmetry.The grating k-vector required for phasematched SHG at λ 0 1045 nm is Δk 0 ≈ 953.5 mm −1 , corresponding to the dashed horizontal line.The slope in grating k-vector ∂K g ∕∂x 40 mm −2 .For the middle part of the grating ("Pos.2"), the minimum grating k-vector is ≈894 mm −1 .

Fig. 7 .
Fig. 7. Numerical simulations of the mode-locked laser, including the dynamics in the APPLN crystals (assuming Δk ≈ 85 mm −1 ) and population dynamics in the Yb:CALGO crystal.The population dynamics are artificially accelerated by a factor of 1000 (see text).(a) Laser spectrum, in agreement with the experimentally measured spectrum shown in Fig. 5(a).Inset: intensity profile.(b) Evolution of the normalized population inversion β in the gain crystal.(c) and (d) Evolution of the laser spectrum and pulse profile over time, respectively (dB scales).The self-frequency shift is indicated in (c) by an arrow.

Fig. 8 .
Fig. 8. Dependence of the (a) pulse duration and (b) center wavelength on output power for the configuration with mirror M2 as an HR.The output power corresponds to the total from both beams passing through the output coupler.

Fig. 9 .
Fig. 9. SH spectra as a function of (a) laser power and (b) APPLN crystal position.The given range of the crystal position was read from the translation stage used to move the crystal.The laser was configured with mirror M2 as an HR.