Femtosecond mode locking based on adiabaticexcitation of quadratic solitonsC. R. PHILLIPS,* A. S. MAYER, A. KLENNER, AND U. KELLER

Department of Physics, Institute of Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland*Corresponding author: cphillips@phys.ethz.ch

Received 1 April 2015; revised 8 July 2015; accepted 8 July 2015 (Doc. ID 237268); published 30 July 2015

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

OCIS codes: (140.4050) Mode-locked lasers; (190.5530) Pulse propagation and temporal solitons; (190.4360) Nonlinear optics, devices.

http://dx.doi.org/10.1364/OPTICA.2.000667

1. INTRODUCTION

Soliton dynamics play a critical role in many optical systems andare of fundamental interest in several areas of physics. In the con-text of mode-locked lasers, temporal solitons support ultrashortpulse formation in a wide variety of laser architectures, includingsolid-state lasers and fiber lasers [1,2], and more recently evenoptical microresonators [3]. When combined with semiconductorsaturable absorber mirrors (SESAMs) to start and stabilize themode locking [4], soliton shaping allows the pulse to be muchshorter than the slow response time of the SESAM [5].Another prominent example of soliton dynamics in the contextof ultrafast optics is supercontinuum generation in optical fibersand waveguides [6]. The continued development of ultrafastsources including frequency combs is thus connected with ourability to create solitons and control their dynamics. This develop-ment is of great interest for numerous application areas [7], in-cluding spectroscopy, metrology, optical clocks, and high-speedcommunication and information processing. In this paper, we ex-perimentally demonstrate a new type of two-dimensionally pat-terned quasi-phase-matching (2D-QPM) device designed toadiabatically excite intracavity solitons based on the second-ordernonlinearity χ�2�. By allowing much greater control over the non-linear properties of the device, this 2D-QPM technique has the

potential to overcome several fundamental limitations in conven-tional soliton formation techniques, thus opening up many newopportunities for ultrashort pulse generation in near- and mid-infrared mode-locked lasers, particularly compact high repetitionrate and waveguide lasers.

To understand this potential, we first recount some importantproperties of solitons based on the widely used Kerr effect. TheKerr effect yields a positive nonlinear refractive index n2 whenthe laser wavelength is far enough from the bandgap to avoid mul-tiphoton absorption. Using wide bandgap materials to satisfy thisconstraint means only moderate values of n2 are typically acces-sible, since n2 scales inversely with the fourth power of thebandgap [8]. Thus, achieving sufficient nonlinearity for solitonpulse shaping becomes increasingly difficult for lower energy la-sers. Negative group delay dispersion (GDD) is also required whenn2 > 0, which constrains the dispersion management due to thepositive dispersion typically offered by bulk materials at near-infrared wavelengths. These issues become increasingly challeng-ing when scaling to high laser repetition rates, because of the lowerpulse energies, greater susceptibility to damage from Q-switchinginstabilities [9], and more restricted cavity design space.

An alternative to the Kerr nonlinearity is offered by cas-caded quadratic nonlinearities (CQNs) [10,11]. By tuning a

2334-2536/15/080667-08$15/0$15.00 © 2015 Optical Society of America

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second-harmonic generation (SHG) medium far from phase-matching, a large and negative effective nonlinear refractive indexcan be achieved, supporting a Kerr-like soliton with negative n2.CQNs offer numerous potential advantages for mode locking dueto the large, adjustable, and negative (self-defocusing) nonlinearrefractive indices that can be obtained. Such cascaded χ�2� non-linearities have been exploited for various applications, includingpulse compression and frequency shifting [12–15], and supercon-tinuum generation [16,17]. The corresponding physics can alsoplay an important role in efficient chirped QPM interactions,including adiabatic frequency conversion and parametric ampli-fication [18,19], a connection that we discuss further in thispaper. CQNs have been exploited for mode locking [20–28], withthe SPM yielding either soliton formation, or a Kerr-like lens tofavor pulsed operation, or both.

An inherent property in cascading devices (i.e., those involvingCQNs), and more generally solitons based on χ�2� media, is theneed for multicolor pulses, since the soliton requires an interplaybetween the fundamental or first harmonic (FH) and the secondharmonic (SH). Indeed, a key feature of a CQN interaction usingultrashort pulses is the presence of a weak SH pulse propagatingwith a stronger FH pulse. This presents a disparity with manypractical laser systems, where we have only the FH available:in this case, energy is lost when exciting the SH pulse. Since theselosses are intensity dependent, they can destabilize the mode-locked laser. These SHG losses can be reduced, for a givenamount of self-phase modulation (SPM), by using a longerCQN crystal and operating further from phase-matching, whichwe exploited in [28]. However, this design procedure has severaldrawbacks, since it introduces additional dispersion, linear losses,and optical delay to the pulses and requires a long crystal, whichprohibits ultracompact devices. For low-loss optical cavitiesutilizing CQNs, or χ�2� solitons in general, we are thus presentedwith a fundamental trade-off between SPM and nonlinear losses.

Here, we overcome this limitation via a new type of micro-structured cascaded-χ�2� device based on aperiodic QPM. Thedevice 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 longperiod, while the region in the middle has a short period. This“apodized” longitudinal QPM profile is designed to adiabaticallyexcite (“turn on”) and later de-excite (“turn off”) the SH wave in aCQN-like interaction, allowing a large SPM in the middle regionwhere the SH is strong, while minimizing the energy lost from thefundamental pulse. This approach is similar in spirit to the con-cept of soliton amplification in chirped QPM media [29,30], aswell as the concept of nonlinearity management inside mode-locked lasers [31]. In addition to the longitudinal chirp, thereis a fan-out structure [32], with the period changing with trans-verse position. This transverse variation allows continuous tuningof the nonlinearity, and helps facilitate self-starting mode locking.With this 2D-QPM device, we thus have access to all the advanta-geous properties of CQNs, and overcome their fundamentaldrawback to enable a multitude of new applications.

In terms of laser physics, this device concept opens up the pos-sibility of accessing new mode-locking regimes in low-loss opticalcavities by adiabatically exciting the relevant state, even if the frac-tion of energy in the SH is substantial, as will be the case for manyχ�2� solitons [33]. Such strong-SH states are now accessible intra-cavity, because the SH wave is excited from zero and is de-excitedafter the interaction region, before it can exit the crystal and con-stitute a loss for the laser. Furthermore, the concept has imme-diate practical importance for lasers with gigahertz repetitionrates, because the features offered by adiabatic cascading simulta-neously address many of the challenges faced when scaling femto-second diode-pumped solid-state lasers to high repetition rates.These features include (1) low-loss soliton formation at low intra-cavity fluences in a highly compact device, enabled by the largeself-defocusing n2, of order −1.5 × 10−5 cm2∕GW in our case;(2) straightforward dispersion management, even in restrictivecavity geometries, since the dispersion provided by bulk materialsleads to soliton formation; and (3) suppression of damage tooptics from Q-switching instabilities through the variable andnonabsorptive losses from SHG, which allow the laser to be op-erated at the rollover point of the nonlinear reflectivity to suppressQ-switched mode locking [34].

To demonstrate the concept, we implemented the QPMdevice depicted in Fig. 1 in a mode-locked laser, usingYb:CaAlGdO4 (CALGO) as the gain medium [35]. The laserproduced 100 fs pulses at a repetition rate of 0.54 GHz, with760 mW average output power, and exhibited reliable self-startingmode locking. Our results prove the viability of adiabatic solitonexcitation inside mode-locked lasers, enabled by the versatility oftwo-dimensional QPM structures.

In this paper, we first present the theory of adiabatic SH ex-citation (Section 2), our experimental results (Sections 3 and 5),numerical simulation of the laser (Section 4), and concludingremarks (Section 6).

2. ADIABATIC HARMONIC EXCITATION

To explain and motivate the longitudinal variation of the QPMstructure in Fig. 1 in more detail, we utilize the existence of ei-genmodes associated with the coupled wave equations in a uni-form χ�2� nonlinear medium. These are solutions where all of thewaves propagate indefinitely with no exchange of energy betweenthem. We quantify these eigenmodes for the case of SHG inSection 1 of Supplement 1, with an approach analogous to that

Fig. 1. Illustration of the inverted domains in our fan-out, apodizedQPM device implemented in MgO:LiNbO3. For clarity, the transverseprofile of every 15th domain is shown (i.e., domains 1, 16, 31, …), andthe thickness of the lines is not to scale (each domain is actually ∼3.5 μmlong). The illustration shows the two primary features. First, the fan-outprofile, with period decreasing for larger transverse positions. Second, theapodization profile, where there is a large period at the input and outputsides of the device (regions 1 and 3), and a smooth but rapid change to ashorter period within the middle part (region 2). To the right of the do-main pattern, the time-dependent intensity of the laser and its SH areillustrated, indicating the small SH intensity in regions 1 and 3, and themoderate intensity in region 2.

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of [19]. The resulting properties of one of these eigenmodes isillustrated [Fig. 2(a)] as a function of phase-mismatch Δk.The figure is normalized to a length-scale LNL defining thestrength of the SHG process. This length is L−1NL � ω1

d eff

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1jA1j2 � n2jA2j2

p∕�n1 ffiffiffiffiffi

n2p

c�, where subscripts corre-spond to the FH (j � 1) and SH ( j � 2) waves, ωj are angularfrequencies, nj are refractive indices, Aj are electric field enve-lopes, and d eff is the effective nonlinear coefficient. The phase-mismatch is given by Δk � k2 − 2k1 − K g for wave vectorskj � ωjnj∕c and QPM grating k-vector K g.

The blue curves in Fig. 2(a) show the fraction of total intensityin the SH part of the eigenmode, while the red curves show the rateγFH at which the FH part of the eigenmode accumulates SPM.From the dashed red curve, at large Δk, the SPM accumulatedby the FH over a distance L is ϕSPM � γFHL ≈ −L∕�ΔkL2NL�.Similarly, from the dashed blue curve, the fraction of energy inthe SH is aNL ≈ 1∕�ΔkLNL�2. Both ϕSPM and aNL are thus pro-portional to the input FH intensity. These features closely corre-spond to those of CQNs.

The utility of the eigenmode description becomes apparentwhen considering nonuniform phase-matching media. If Δk isvaried via a chirped QPM period, we can consider a localeigenmode, associated with the position-dependent value of

Δk�z� � k2 − 2k1 − K g�z�. Then, if the QPM period variesslowly enough, the fields can adiabatically follow the eigenmodeas it evolves with Δk�z�. This capability has enabled adiabaticfrequency conversion [18], in which Δk�z� is swept slowlythrough phase-matching in order to achieve efficient frequencyconversion. This concept has been applied to efficient and broad-band sum- and difference-frequency generation [18], opticalparametric amplification [36], and SHG [37].

In this work, we use a new type of aperiodic QPM structurethat bridges the gap between adiabatic frequency conversionand cascaded quadratic nonlinearities: we begin far from phase-matching to efficiently couple the fields into an eigenmode, thenadiabatically reduce Δk so that the FH and SH evolve into aneigenmode with a large rate of SPM for the FH, and then, afteran appropriate distance, we de-excite the SH part of the eigen-mode by returning to a large Δk. A typical example Δk profileis shown in Fig. 2(b), designed by adapting the apodization tech-niques from [19]. The resulting evolution for continuous planewaves is shown in Fig. 2(c), blue curve. The SH component of theeigenmode is shown by the green curve: it starts small (region 1,with∕ΔkLNL ≈ 150), is larger in the middle region (region 2,with ΔkLNL ≈ 17), and is small again at the output (region 3,with ΔkLNL ≈ 150). The blue curve shows how the SH field fol-lows this eigenmode. Because Δk�0�LNL ≈ 150 is large but stillfinite, 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 thiscase and they persist through the whole device. The chosen rangeof ΔkLNL in this example implies that the large-Δk approxima-tions of Fig. 2(a) are valid.

In the case of a pulsed interaction with finite group velocitymismatch (GVM) between the FH and SH envelopes, theoscillations in Fig. 2(c) manifest as two pulse components.This behavior is evident in Fig. 2(d), which shows a plane-wavesimulation of the SH in an apodized QPM grating similar toFig. 1. The figure uses a reference velocity of v1 � v�ω1� forgroup velocity v�ω� � �dk∕dω�−1. We assume a 150 fs FH with≈1.5 GW∕cm2 peak intensity, and choose Δk � 60 mm−1,yielding n2 ≈ −1.5 × 10−5 cm2∕GW. In this pulsed case, theSH component coupled into an eigenmode propagates withthe FH, at velocity ≈v1, 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 componentnot coupled into the eigenmode propagates with constant energyat the normal group velocity v2 � v�ω2� and is therefore delayedwith respect to the FH. Figure 2(e) shows the SH intensity at theoutput of the crystal, to show how much weaker the remainingpulse components are in an apodized grating compared with anunapodized grating (peak of ≈0.01% instead of ≈0.25% ). In themode-locked laser this SH can constitute a significant loss, sincefour such pulses are generated [two for each pass through the crys-tal, as seen in Fig. 2(e), and two passes through the crystal for alinear cavity].

Given the above, for a QPM grating with an adiabatic chirpprofile 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 estimatethe energy lost to the SH in a single pass as

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

Research Article Vol. 2, No. 8 / August 2015 / Optica 669

aNL ≈ �Δk�0�LNL�−2 � �Δk�L�LNL�−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 energystill remaining in the eigenmode at the output of the device. Incontrast, the SPM accumulated by the FH in a single passcorresponds to an integral over the whole device:

ϕSPM ≈ −

ZL

0

�Δk�z 0�L2NL�−1dz 0; (2)

where these equations assume Δk�z� remains far enough fromphase-matching that (i) the approximate dashed curves inFig. 2(a) can be applied instead of the solid curves, and (ii) thebandwidth of the cascading process is large compared to the pulsebandwidth; this bandwidth can be quantified via the responsefunction theory introduced in Supplement 1, Section 2. The signof ϕSPM determines the sign of n2: ϕSPM < 0 means n2 < 0, i.e.,a self-defocusing nonlinearity.

To estimate the cavity losses from Eq. (1), we can first calculateLNL using the peak of the electric fields, and then multiply aNL inEq. (1) by 2∕3 to account for integration over space and time(assuming a Gaussian spatial profile and a sech2 temporal profile),and two passes through the crystal per round trip.

Equations (1) and (2) show how we can obtain a large SPMwithout SHG losses as long as the QPM grating is chirped adia-batically [19], and we impose the appropriate boundary condi-tions on Δk�z�. Moreover, in comparing the blue and redcurves in Fig. 2(c) and identifying the two pulse componentsin Fig. 2(d), we see that the oscillations in intensity associatedwith conventional CQN interactions are not an inherent aspectof the desired SPM process, but originate from poor launching ofthe input fields into the relevant eigenmode that gives rise toSPM. These oscillations are strongly reduced in the apodized case.

The QPM design used for our experiments is illustrated inFig. 3. The transverse variation in the phase-mismatch is alsoshown. Figure 3(b) shows the resulting QPM period profile atthe three transverse positions xj, assuming a nominal wavelengthof 1045 nm. When converted into a ferroelectric domain pattern,this design yields Fig. 1.

3. EXPERIMENTAL SETUP

To demonstrate the device concept and its applicability to short-pulse mode locking, we built a laser based on Yb:CALGO, shownin Fig. 4. Yb:CALGO was chosen since it supports short pulsesand has favorable thermal properties [35]. Pump light at≈980 nm is directed to the antireflection-coated 3-mm-longYb:CALGO crystal through the end mirror M1. The CALGOand aperiodically poled lithium niobate (APPLN) crystals areboth placed close to M1; M1 also provides −400 fs2 GDD.The APPLN crystal had a thickness of 0.5 mm, and was designedaccording to Fig. 1. We configure mirror M2 as either a highreflector (HR) or a GTI-type mirror with ≈ − 500 fs2 per reflec-tion. We estimate total round-trip GDD values of �1237 and�237 fs2 for the HR and GTI cases, respectively. There areuncertainties of order 100 fs2 from the lengths of the materialsinvolved and the properties of the dispersion-compensating mir-rors. Mirror M2 is a 2% output coupler and is used as a turningmirror. The gain crystal and the SESAM are the same as in [28]:the SESAM has a 2.8% modulation depth and a saturation flu-ence of 5.8 μJ∕cm2. The beam 1∕e2 radius at the Yb:CALGO,APPLN crystal, and mirror M1 is ≈150 μm. The beam radius atthe SESAM is ≈150 μm.

The QPM crystal design had several gratings, including simpleperiod gratings, apodized but non-fan-out gratings, as well as theapodized fan-out gratings illustrated in Figs. 1 and 3. When usingan apodized, fan-out QPM grating, the laser exhibits reliable self-starting mode locking in both M2 configurations (HR and GTI).Both cases correspond to a net positive cavity GDD, and the mainsource of SPM is from the negative contribution of the APPLNcrystal. The fan-out apodized QPM gratings almost always sup-ported reliable self-starting mode locking in a variety of cavityconfigurations and over a large range of QPM periods. We some-times observed mode locking when using the non-fan-outapodized gratings, but never observed mode locking with the“control” gratings (non-fan-out, non-apodized). These results in-dicate not only the expected importance of apodization (longi-tudinal variation of QPM period) for achieving mode locking,but also the importance of the fan-out design on achievingself-starting. In Supplement 1, Section 5, we show how the trans-verse variation of the absolute phase of the QPM structure canexplain this initially surprising property of fan-out gratings.

The largest mode-locking range was obtained in the case withM2 as an HR, while the shortest pulses were obtained with M2 asa GTI. The parameters in this latter case were 760 mW averagepower (total from both output beams), 0.54 GHz repetition rate,

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Fig. 3. (a) Phase mismatch Δk�x; z� � Δk0�ω0� − K g �x; z� in the de-vice used, assuming a wavelength λ0 � 2πc∕ω0 � 1045 nm. The apod-ization profile is designed according to [19], with additional bufferregions of constant and large Δk at the ends (regions outside the verticaldashed lines). (b) QPM profile 2π∕K g �xj; z� for three different transversepositions xj . The longitudinal coordinate is centered at 0 mm in this caseto emphasize the symmetry. The grating k-vector required for phase-matched SHG at λ0 � 1045 nm is Δk0 ≈ 953.5 mm−1, correspondingto 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.

Pump

M2: HR or GTI

M3: Outputcoupler

Fused silica BPM1: Inputcoupler

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Apodized, fan-out APPLN

M4: SESAM

Fig. 4. Soliton mode-locked Yb:CALGO laser setup: 3 mmYb:CALGO crystal, a ≈2-mm-long fan-out apodized APPLN crystal,a 2-mm-thick fused silica Brewster plate (BP), a 200 mm radius mirror(M2), a 500 mm radius, 2% output coupling mirror (M3), and a SESAM(M4). M1–M4: laser cavity mirrors.

Research Article Vol. 2, No. 8 / August 2015 / Optica 670

and 100 fs pulse duration measured by an autocorrelation. Theoptical and microwave spectra are shown in Figs. 5(a) and 5(b),respectively. The inset of Fig. 5(a) shows the autocorrelation. Thebeam quality was M 2 ≈ 1 for the fundamental, within theaccuracy of the measurement device.

To test the expected performance with respect to SHG sup-pression, we show in Fig. 6 the spectrum of the SH outside thecrystal. As discussed in Section 2, in a normal cascading interac-tion, the temporal profile of the SH field corresponds to a doublepulse, with each pulse having equal energy (although slightlydifferent dispersion and nonlinear effects). The harmonicspectrum therefore resembles a double-pulse spectrum, whoseenvelope scales as F �A1�t�2�, and delay between the pulses givenby the group velocity walk-off between the FH and SH, which is�v�ω2�−1 − v�ω1�−1�L ≈ 1.5 ps in our case. In apodized cascadingdevices, we strongly suppress this normal SH response [seeFig. 2(d)]. However, the SH can also be generated as a conse-quence of random duty cycle (RDC) errors in the QPM grating.Such RDC errors lead to a phase-matching pedestal [38], forwhich individual SH spectra are strongly modulated. For an apo-dized device having small but finite RDC errors, we thus expectthe SH spectrum to be strongly modulated and to bear littleresemblance to the FH from which it is generated, since the nor-mal double-pulse response is almost completely suppressed byapodization, but the RDC contribution remains.

We confirmed the expected qualitative behavior of the SH, asshown in Fig. 6, which indicates that there is no signature of thenormal cascading-type response, with most of the SH spectrum

attributable to imperfect poling instead. Furthermore, for a fan-out structure, we find that the beam profile of the SH is modified,as well as its spectrum. While the beam quality for the FH isalways excellent (M 2 ≈ 1), the beam profile of the discardedSH is poor, and it exhibits significant spatiotemporal coupling,as illustrated in Fig. 6(b). For this measurement, the SH beamwas scanned spatially over the fiber-collimating lens before theoptical spectrum analyzer. The properties of the SH and the roleof fan-out gratings in modulating the SH spectrum and beam arediscussed further in Section 5.

4. NUMERICAL MODELING OF THE LASER

The results of Section 3 showed features of the laser that warrantfurther 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 inmore detail, we need to consider the effect of the new QPM de-vice on broadband pulses. For example, the presence of GVM, asevident in Fig. 2(d), leads to a slope in the frequency-dependentphase mismatch Δk�ω�. Operation closer to phase matching en-ables a greater rate of SPM, but also increases the relative variationof Δk�ω� across the optical spectrum. Moreover, imperfections inthe QPM domains, namely, RDC errors [38], can modify thebehavior of the SH compared with Fig. 2(d). To model theresulting dynamical effects, we derive in Sections 2 and 3 ofSupplement 1 a perturbative model of the nonlinearity experi-enced by FH pulses in nonuniform QPM media. The result isa Raman-like response function, with examples shown inFigs. S1 and S2 (Supplement 1). In this section, we incorporatethat response function into general numerical simulations ofthe laser.

These simulations describe the evolution of the pulse throughthe cavity, including nonlinear processes in all the intracavityelements, as well as the population dynamics of the gain medium.Certain important assumptions are made to speed up the calcu-lations, including modal overlaps to obtain a (1� 1D) model;approximating the single-pass propagation in APPLN via theabove-mentioned response function; and artificially acceleratingthe laser population dynamics by numerically scaling the crosssection, lifetime, and doping concentration, such that the gainand pulse-shaping dynamics remain the same but relaxation os-cillations are damped. Further details are given in Supplement 1(Section 4).

In Fig. 7, we show numerical results, which are in good agree-ment with the experimental results from Fig. 5. The simulationpredicts ≈100 fs pulses with comparable power to the actual laserand given physically realistic values for all of the parameters of thesimulation. Figure 7(a) shows the spectrum and temporal profile(inset). Remarkably, the asymmetry in the spectrum is extremelysimilar to the experimentally measured spectrum of the same laser[Fig. 5(a)]. Figure 7(b) shows the evolution of the populationdynamics over time and through the length of the Yb:CALGOcrystal. Figure 7(c) shows the evolution of the laser spectrum,which exhibits a frequency shift after the initial buildup fromnoise. Figure 7(d) shows the corresponding pulse profile on a log-arithmic scale, to show the more gradual evolution of the laserinto 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 ex-plained via the phase curve of the response function in Fig. S1(b)

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(b)Wavelength (nm)515 520 525 530 535

10-4

10-3

10-2

10-1

100

SH

spe

ctru

m (

arb.

u.)

(a)

Fig. 6. Measured SH, indicating suppression of the normal cascadingresponse in our APPLN device. (a) Spectrum from part of the beamcoupled into an optical fiber. (b) Spectrum when translating the SH beamacross the fiber collimator, to couple different parts of the beam; thisindicates the presence of some spatiospectral coupling.

Research Article Vol. 2, No. 8 / August 2015 / Optica 671

(Supplement 1). Such an SFS is reminiscent of Raman nonlinear-ities, where the loss-related part of the corresponding “responsefunction” directly scatters energy from the high-frequency tothe low-frequency part of the spectrum [39]. The case of ourAPPLN medium is different, since the loss-related part of theresponse is suppressed via apodization, leaving mainly the phase-related part that gives rise to SPM. Nonetheless, CQNs alsosupport SFS effects, which have been discussed in the contextof single-pass pulse compression devices [14]. In the configura-tion involved here, there is a blueshift, which shifts the centerfrequency away from phase matching and causes the asymmetriclaser spectrum observed. We checked this by turning off the fre-quency dependence of the response function numerically: in thatcase, there is no frequency shift effect, and the predicted laserspectrum is symmetric.

These results show that by modeling each intracavity elementwe reproduce the observed experimental results. The noninstan-taneous nonlinearity associated with the response function isimportant in APPLN devices when generating short pulses,and can be incorporated efficiently into simulations.

5. STUDY OF MODE-LOCKED OPERATION

Having identified important pulse-shaping effects, in this sectionwe further experimentally study the nonlinear properties of thelaser. For this we use M2 as an HR mirror instead of a GTI-typemirror: in this case there is more positive intracavity dispersion,yielding a wider range of mode-locked operation. For example, wecan continuously scan from one side of a 2 mm fan-out QPMgrating to the other by translating the crystal. We calculatethe total round-trip GDD as 1237 fs2, after replacing the≈ − 500 fs2 GTI used for Fig. 5 with an HR. A typical resultis power and a repetition rate of 1.45 W and 544 MHz, respec-tively, with slightly longer 149 fs pulses.

We first examine the laser properties as a function of outputpower 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 forsoliton mode locking of a decreasing pulse duration with increas-ing laser power. Figure 8(b) shows that the center wavelength isalso power dependent, which is primarily due to the SFS effectdiscussed in Section 4, although there could also be some influ-ence from shifts in the gain peak in the Yb:CALGO crystal. Theshift to shorter wavelengths brings the wavelength further fromphase matching, thereby reducing the effective n2 of the device.As such, Fig. 8(a) need not follow the τ ∝ 1∕�power� trend ofconventional soliton mode locking. The dashed lines are fullnumerical simulations of the same configuration, showing goodqualitative agreement with experiment; we assumed a realisticΔk ≈ 55 mm−1 (at 1045 nm), 0.5% additional cavity losses,and no other fit parameters.

The SFS effect also limited the achievable pulse durationsin the M2-HR configuration, since SFS effects scale rapidlywith decreasing pulse duration [39]. With M2 as a GTI(Section 3), we reduced the GDD to reduce the requiredround-trip SPM and hence the strength of the SFS at a given pulseduration. This procedure allowed for 100 fs pulses that were nei-ther excessively perturbed by SFS nor by SHG from fabricationerrors.

We next show the properties of the SH. In general this SH isstrongly modulated, likely due to the presence of RDC errors inthe apodized grating. Figures 9(a) and 9(b) show the SH spectrumversus laser power and translation of the APPLN crystal transverseto the laser beam, respectively. Despite the SFS exhibited inFig. 8(b), there is no corresponding shift in the structure ofthe SH spectrum in Fig. 9(a). This can be understood by notingthat the SH spectrum scales with the spatial Fourier transform(FT) of the QPM grating. This FT is power independent, sothe structure of the SH remains unchanged when changingthe power. The situation changes when we keep the pump power

SFS

−100

−80

−60

−40

−20

0

Wavelength (nm)1020 10601040 1080

(c)

0

0.40.2

0.60.8

1

Tim

e (µ

s)

Yb:CALGO position (mm)

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

0.040.060.080.10.120.140.16

Tim

e (µ

s)

(b)

1020 1040 1060 10800

0.2

0.4

0.6

0.8

1

Wavelength (nm)

Spe

ctru

m

−100 0 1000

1

Time (fs)

Inte

nsity

(a)

00.20.40.60.8

1

-100-80-60-40-200

Tim

e (µ

s)

Delay (ps)−15 −10 −5 0 5 10 15

(d)

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

(a) (b)Output power (W)1 1.2 1.4 1.6

Pul

se d

urat

ion(

fs)

120

140

160

180DataSimulation

Output power (W)1 1.2 1.4 1.6

Cen

ter

wav

elen

gth

(nm

)

1044

1045

1046DataSimulation

Fig. 8. Dependence of the (a) pulse duration and (b) center wave-length on output power for the configuration with mirror M2 as anHR. The output power corresponds to the total from both beams passingthrough the output coupler.

Wavelength (nm)

Lase

r po

wer

(W

)

515 525 535

1.11.21.31.41.5

00.20.40.60.81

(a) Wavelength (nm)

Cry

stal

pos

ition

(m

m)

515 525 5357.8

8.0

8.2

8.4

8.6

00.20.40.60.81

(b)

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

Research Article Vol. 2, No. 8 / August 2015 / Optica 672

constant but translate the fan-out crystal. The QPM profile isthen stretched/compressed, leading to a shift in its Fourierspectrum. This expected behavior is manifested in Fig. 9(b):the spatial FT of the grating shifts with crystal position, leadingto a shift in the SH spectrum. The transverse dependence of theSH spectrum also explains Fig. 6(b) and the poor SH beam qual-ity that we observe when using fan-out gratings: different partsof the beam see different parts of the QPM grating, and hencehave their optical spectra shifted. Thus, for each spectral compo-nent, there is transversally varying intensity and phase, leading toa poor beam quality of each spectral component of the SH. Thisalso explains the spatiotemporal coupling of the SH in Fig. 6(b).

6. CONCLUSIONS

In conclusion, we have demonstrated a new type of QPM devicefor soliton formation in mode-locked lasers. The device adiabati-cally excites and subsequently de-excites the SH wave in a solitonbased on the second-order nonlinearity χ�2� of the QPM material.This technique allows for a large SPM, as in conventionalcascaded χ�2� interactions, while suppressing or modifying the as-sociated nonlinear losses. This approach gives access to large self-defocusing nonlinearities, of order n2 ≈ −1.5 × 10−5 cm2∕GWin our case, and the accessible nonlinearity increases with pulseduration. Small residual losses to SHG occur due to imperfectapodization and QPM fabrication errors. The combination ofa large and self-defocusing nonlinearity, relaxed constraints oncavity dispersion, and adjustable nonlinear losses are uniquelyfavorable for supporting soliton formation far into the GHzmode-locking regime.

We demonstrated the concept via a proof of principle laser,which produced 100 fs pulses at 0.54 GHz repetition rate with760 mW average power. Using the same cavity but replacing aGTI 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 residualSH wave leaving the crystal. To support these experiments, weperformed a comprehensive theoretical and numerical analysisof the laser dynamics, showing good agreement with experiments.We also identified the influence of a fan-out QPM structure onthe self-starting characteristics of the laser.

Two important issues we identified were the SFS effect andlosses associated with RDC errors. We designed the laser andAPPLN device so that the influence of the expected RDC errorsbased on previous studies [38] would be manageable. These errorscan be reduced by improved fabrication. The SFS effect is strongbecause of the substantial GVM between the short FH and SHpulses in PPLN at 1 μm. Two ways to reduce the influence of thiseffect 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 reducedGVM coefficient would allow operating at smaller Δk, yieldingan even larger effective n2 prior to the onset of strong SFS-typeeffects. A detailed study of the maximum practically achievablenonlinearity is planned for future work. More generally, adiabaticexcitation combined with a reduced GVM could give access toa broader class of χ�2� solitons [33], supporting previouslyinaccessible mode-locking regimes.

The technique we have demonstrated has fundamental impor-tance, since it gives access to a wide variety of intracavity solitonsbased on second-order nonlinearities (i.e., multi-color solitons)

without incurring losses that would otherwise be associated withexciting them. It also has immediate practical relevance in scalingmode-locked solid-state lasers further into the gigahertz regime,including in the challenging infrared and mid-infrared spectralregions. Moreover, in mode-locked semiconductor disk lasers[40], the moderate intracavity intensities and low output couplingrates typically used provide a strong motivation for low-loss, high-nonlinearity media for achieving robust pulse formation. In thefuture, as well as accessing gigahertz repetition rates and mid-infrared wavelengths, the QPM devices could be implementedin PPLN waveguides rather than bulk, enabling extremelycompact and robust waveguide lasers.

Funding. European Union Seventh Framework Programme,Marie Curie International Incoming Fellowship (PIIF-GA-2012-330562); ETH Zurich (ETH-26 12-1); European ResearchCouncil (ERC) (ERC-2012-ADG_20120216).

See Supplement 1 for supporting content.

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