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Sander et al. Vol. 26, No. 4 /April 2009 /J. Opt. Soc. Am. B 743

Dynamics of dispersion managed octave-spanningtitanium:sapphire lasers

Michelle Y. Sander,1,* Jonathan Birge,1 Andrew Benedick,1 Helder M. Crespo,1,2 and Franz X. Kärtner1

1Department of Electrical Engineering and Computer Science and Research Laboratory of Electronics,Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139-4307, USA

2Departamento de Física, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal*Corresponding author: sanderm@mit.edu

Received October 27, 2008; revised January 29, 2009; accepted January 30, 2009;posted February 2, 2009 (Doc. ID 103340); published March 18, 2009

An extensive one-dimensional laser model based on dispersion managed mode locking is presented that accu-rately describes the pulse dynamics of octave-spanning titanium:sapphire lasers generating sub-two-cyclepulses. By including detailed characteristics for the intracavity elements (mirrors and output coupler), it isdemonstrated that the spectral output and temporal pulse shape of these lasers can be predicted quantitativelyin very good agreement with experimental results. © 2009 Optical Society of America

OCIS codes: 140.4050, 140.7090, 320.5550, 320.7090.

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. INTRODUCTIONince the discovery of Kerr-lens mode locking (KLM) in ti-anium:sapphire (TiSa) lasers in 1991 [1], femtosecondode-locked lasers underwent rapid progress, which led

o the generation of octave-spanning lasers with sub-two-ycle optical pulses [2–4] that also operate at high repeti-ion rates [5]. To better understand the pulse formation inemtosecond mode-locked lasers much advancement inhe modeling and understanding of the pulse formationas made: for special cases with small changes per round

rip, analytical solutions of solitary waves with secant hy-erbolic functions are formed through the distributed bal-nced action of self-phase modulation (SPM) and groupelay dispersion (GDD). The average dynamics are cap-ured by the master equation [6] as a continuous way ofescribing intracavity effects based on average dispersionalues. The importance and limiting effects of dispersionn such solitary systems, especially higher order disper-ion, as a perturbative effect on mode locking were exam-ned in [7–10]. Alternative approaches that combine tem-oral and spatial pulse dynamics based on Fresnelntegrals were pursued as well [11,12].

The temporal pulse evolution in Kerr-lens mode-lockediSa lasers is well-described by dispersion managed mode

ocking (DMM) [13,14]: in systems where the GDD peri-dically changes its sign, with small average GDD peround trip when compared to the absolute GDD in eachection (as is the case in femtosecond laser cavities), dis-ersion managed solitary pulses are formed. Based on thencountered GDD and SPM, together with the localizedmpact and order of the intracavity elements, the pulsesndergo temporal and spectral broadening and recom-ression during each cavity round trip [13,14]. When ap-lied to standard non-octave-spanning mode-locked la-ers, simulations based on DMM have been used toescribe their behavior accurately, thus relating pulse en-rgy to pulse duration [15] or including mirror character-

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stics for detailed output characteristics [16]. Numericalnalysis of the spatiotemporal pulse dynamics, in termsf Gaussian beam propagation combined with DMM ef-ects [17], have been compared with analytical resultsased on the variational principle, including second-orderispersion, self-focusing, diffraction, and second-orderispersion of the mirrors [18,19].The focus of this work is to extend the one-dimensionalodel, which describes the temporal dynamics, to includeigher order material dispersion as well as the detailedirror and output coupler characteristics, and to make a

uantitative comparison between the DMM model and ex-erimental results from sub-two-cycle octave-spanning la-ers. To our knowledge, the full spectral characteristics ofispersive mirrors and output couplers (OCs) have nevereen taken into account to such an extent for sub-two-ycle laser systems. In addition, a comparison betweenhe simulated output pulse and the retrieved pulse fromwo-dimensional spectral shearing interferometry (2DSI)20] is performed for the first time to the best of ournowledge. It is shown that the model can capture thectave-spanning output spectrum and the generated sub-wo-cycle pulses in great detail, thereby allowing for a di-ect and precise comparison with experimental measure-ents. Thus, laser dynamics, stability, and steady-state

pectral output can be predicted and optimized based onealistic parameters.

In this paper, we describe the numerical model basedn the nonlinear Schrödinger equation (NLS) for the crys-al and then derive the different cavity parameters. Next,mportant intracavity elements and the experimentaletup are explained. In Section 4, the pulse dynamics inteady state for one round trip are presented. Experimen-al output spectra for three different laser cavity configu-ations and laser parameters are compared with numeri-al results, and the impact of dispersion compensationnd mirror ripples on laser performance is evaluated.

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744 J. Opt. Soc. Am. B/Vol. 26, No. 4 /April 2009 Sander et al.

. MODELo follow the femtosecond pulse generation in sub-two-ycle octave-spanning laser systems we conduct a numeri-al analysis based on a one-dimensional model and thenompare the results with experimental data. We applyhis analysis to three different cavity configurations, thusnderlining the validity of the chosen approach, and dem-nstrate its capability to predict important laser charac-eristics such as pulse duration and shape, bandwidth,nd spectral output.The schematic setups of the considered femtosecond la-

er cavities are given in Fig. 1(a) for a ring and in Fig. 1(c)or a linear laser: in this work, we focus on prismless TiSaaser cavities that directly generate octave-spanning out-ut spectra at low and high repetition rates [21–25]. Allasers developed in our group use custom-designed,ouble-chirped mirror (DCM) pairs, thereby enablingroadband dispersion compensation over almost one oc-ave. For dispersion compensation, either BaF2 or twoedges of fused silica (FS) and BaF2 are inserted. To ob-

ain the best agreement with the experimental setup, thearious elements within the cavity are modeled as accu-ately as possible in terms of their physical characteris-ics.

The pulse dynamics for any TiSa laser are determinedy the nonlinear propagation through the TiSa crystalnd the linear evolution through the remaining passiveavity elements such as air path, mirrors, output coupler,nd wedges. When taking into account the discrete im-

ig. 1. (Color online) (a) Experimental setup of 500 MHz laser.b) Schematic model for ring laser used in numerical analysis.he numbering in the ring laser denotes the cavity position used

n Fig. 3. (c) Model of linear laser cavity.

act, spatial location, and order of each intracavity ele-ent, the temporal evolution of the pulse and its spectral

nvelope for the cavity position z can be followed. Withinhe crystal, the pulse dynamics are described by Eq. (1),he NLS [26]: during each pass through the crystal, theulse encounters gain saturation together with Lorentz-an filtering g�z , t�, dispersion of the TiSa crystal, SPMnd the crucial KLM, which is approximated by a fastaturable absorber q�z , t�. By assuming a slowly varyingulse envelope, A�z , t� in units of �W, Eq. (1) describes theulse dynamics with regard to a retarded time frame toving with the group velocity of the pulse.

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The complex interplay between the different effectsaptured by Eq. (1) can be analyzed by applying the splittep Fourier method [26] to slices of propagation distancez within the crystal [see Fig. 1(b)]. To represent thetudied laser system appropriately, the parameters werehosen as follows: the SPM coefficient �=2�n2,L / ��Aeff�ith n2,L=3�10−7 �cm2/W� and �=800 nm is determinedy the effective beam cross-section Aeff in the crystal. Be-ause there is some experimental uncertainty associatedith the value of the beam cross section (the beam radiusas estimated to be 10–20 �m), we assumed the cross

ection so that �=4�10−7 �W mm�−1, which agrees withalues in [15,17]. Once a specific Aeff for the beam wasxed, the small signal gain and average gain saturationnergy were calculated for a ring laser configuration:

Sat,Ring=hfLAeff / �LfRep�=3.24 nJ and g0=RPL /2Aeff�=1.9 (with h Planck’s constant, fL=3.75�1014 Hz,=3�10−19 cm2, L=3 �s, fRep=500 MHz, RP=PP / �hfL�6.2 W/ �hfL� for the 500 MHz laser, ��L=0.24 rad PHz).o obtain stable steady-state pulses with the desiredulse energies, the parameters were slightly adjusted toalues of g0=1.5 and WSat,Ring=3.35 nJ in order to simu-ate the 500 MHz laser.

The saturation power for the saturable absorber washosen as PSat=3 MW. This is equal to about one-third ofhe pulse peak power in the crystal, so that the saturablebsorber is best exploited to form stable pulses (cf. [27]).he modulation depth q0 of the saturable absorber waset to 5% and represents two to three times the estimatedntracavity losses of the laser. This number is justified byhe observation that, in order to obtain mode locking, theaser cavity is initially misaligned with a laser thresholdhat is two to three times higher than under optimumontinuous-wave lasing conditions. Since the thresholdor lasing is given by g0�1+q0 (see [28]), this conditionives an approximate value for the equivalent absorberodulation depth; assuming that, when mode locked, the

aser cavity realigns to maximize output power. As theajor pulse shaping effects in the crystal are due to dis-

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Sander et al. Vol. 26, No. 4 /April 2009 /J. Opt. Soc. Am. B 745

ersion and SPM [15], gain filtering and saturable ab-orption largely stabilize the pulse against backgroundadiation and scale the final pulse energy through theombined adjustment of the parameters. Therefore, slighteviations in the latter set of parameters have less effectn the pulse dynamics.

The material dispersion was determined by the refrac-ive index n���, which was obtained from the Sellmeierquations for the different materials [29]. An exceptionas the refractive index of air, which was calculated by

he Edlén formula [30] for a better agreement with theaboratory environment. The spectral phase term ���� forach material incorporates second-order and higher dis-ersive effects, and is proportional to the actual optical

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ig. 2. (Color online) (a) DCM reflectivity and output coupler re-ectivity for the 500 MHz ring laser. (b) Designed and measuredroup delay for each type of DCM. The measured data is deter-ined by white-light interferometry at normal incidence angle.

c) Amplitude and phase of linear passive cavity transfer functionersus wavelength.

ath length of that particular element. To include thesehase contributions in the free-space section of the laserogether with the spectral shaping of the DCMs, a spec-ral transfer function with amplitude S��� and dispersivehase P��� for the linear passive cavity elements wassed. For the ring laser cavity configuration in Fig. 1(a)he pulse propagation outside the crystal is given by Eq.2), with R1/2��� denoting the power reflectivity of theCMs M1 and M2, and TOC��� the transmission of theC [compare Fig. 2(a) for the 500 MHz ring laser]. A2�t�nd A9�t� represent the pulse amplitude exiting the crys-al and after one pass through the remaining passive cav-ty elements before re-entering the TiSa crystal, respec-ively. The dispersive phase �i��� captures the impact ofach mirror [subscript 1 or 2, according to the type of mir-or; see Fig. 1(a)] or intracavity material (denoted withhe respective material as a subscript).

A9��� = S���e−jP���A2���,

S��� = �R2���R1���TOC���R2���R1���,

P��� = �2��� + �1��� + �FS��� + �BaF2��� + �2���

+ �1��� + �Air���. �2�

Figure 2(c) shows the magnitude S and phase P of theinear passive cavity transfer function versus wavelengthor the 500 MHz laser cavity and exemplifies the overallhape of the transfer functions for the examined laser sys-ems. It can be seen that the magnitude S supports theuildup of a broad intracavity spectrum between 640 and050 nm, while the phase P noticeably incorporatesigher order dispersion effects.

. EXPERIMENTAL SETUPhe numerical model is applied to laser configurations

ound in the literature, which are referenced for details: ainear 200 MHz laser [22], a 500 MHz ring laser21,23,24], and a 1 GHz ring laser [25]. Using the00 MHz ring laser as an example, the important cavitylements in this experimental setup are explained.

The experimental setup of an octave-spanning TiSaing laser with a 2 mm long crystal and a repetition ratef 500 MHz is presented [see Fig. 1(a)]. This configurationroduces two outputs, a main, carrier-envelope (CE)hase stabilized pulse from the reflective OC and a 1f–2futput coupled out through the flat mirror M3, which issed for the detection of the CE offset frequency in af–2f interferometer (for details see [24]).This laser encompasses new, custom-designed DCMs,

hich were optimized to provide exact compensation ofhe material dispersion in the cavity while maintaining amooth phase over a broad wavelength range [31,32]. Theirror characteristics are shown in Fig. 2(a): the mirrors

abeled M1 and M3 feature a pump window around32 nm and 50% transmission around 1160 and 580 nm,hich corresponds to the 1f–2f frequency componentssed for CE phase stabilization of the laser [33]. Mirrors2 and M4 are both of the same type and exhibit a high

eflectivity over a broader wavelength window to support

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746 J. Opt. Soc. Am. B/Vol. 26, No. 4 /April 2009 Sander et al.

he buildup of a wide spectrum in the cavity. The groupelay for the DCMs in the 500 MHz cavity is illustrated inig. 2(b) for the optimized design [31] and the measuredirrors, as determined by white-light interferometry for a

ormal incidence angle.In this laser setup an OC in reflection is used that is

oated onto the FS wedge. The OC has a 2% reflectivityor the main part of the spectrum [Fig. 2(a)]. It allows theeneration of a broad intracavity spectrum while reflec-ivities higher than 50% below 590 nm and above040 nm effectively enhance and couple out the spectralings. With such a reflective OC, the dispersive phase im-act of the OC on the intracavity dispersion is less signifi-ant, since the phase in transmission is constrained byramers–Kronig relations [34]. Another special compo-ent of this laser cavity is the combination of two wedgesonsisting of FS and BaF2. To fine-tune the intracavityispersion, the total inserted material thickness ishanged, while at the same time allowing adjustment ofhe CE phase independently of the intracavity dispersiony interchanging the two materials [35].

. RESULTS AND DISCUSSION. Pulse Dynamicsccording to the dispersion managed mode-locking model,ach pulse undergoes significant pulse broadening andompression during each cavity round trip, depending onhe material dispersion encountered in the respective sec-ion of the laser. While it is experimentally challenging toetermine the actual pulse stretching within the laseravity, simulations of the intracavity pulse propagationlearly capture the impact of each element on the pulseormation.

The numerical analysis follows the temporal and spec-ral breathing, as demonstrated in Fig. 3, for the00 MHz laser: starting at the geometric middle of theiSa crystal [label 1 in Fig. 1(b)], the pulse evolution inteady state for one cavity round trip is displayed in Fig.(a): the pulse duration reaches its relative minimumulse full width at half-maximum (FWHM) in the crystalith 9 fs, which is slightly offset from the geometriciddle of the crystal and mostly due to gain filtering and

igher order dispersion effects. This minimum pulse du-ation with its corresponding spectral envelope is deter-ined by interplay among the gain filtering, the chirp of

he pulse, and the SPM, which generates new frequencyomponents, filling up and even exceeding the gain band-idth. Thus, the strength of the SPM encountered in the

rystal is a major factor that influences the femtosecondulse generation as the spectrum breathes within therystal from a FWHM of 50 THz near the center (position) to 75 THz (position 2) in Fig. 3(b). While the linear partf the cavity, mainly the mirrors, support the full spec-rum, the pulse stretches and compresses according to theispersion encountered from each cavity element. Underptimized dispersion compensation achieved through theedge insertion, the pulse assumes a minimum pulse du-

ation with a peak instantaneous power of 9 MW at theC [position 5 in Fig. 3(a)], as well as after bouncing off1 (position 8) before re-entering the crystal again. From

minimum pulse duration of 5.7 fs, it stretches to 53 fsnd is recompressed to 32 fs before re-entering the TiSarystal.

. Output Spectrumfter analyzing the intracavity pulse dynamics, the out-ut spectrum for different laser configurations was exam-ned and the impact of different types of OCs evaluated:he linear 200 MHz laser has an OC in transmission inne of its arms, the 500 MHz laser has an OC in reflec-ion, coated on a FS wedge, while the 1 GHz laser essen-ially uses a flat output coupler by reflecting off a BaF2edge. The goal of the simulations was to predict the

pectral bandwidth and shape of the output for compa-able pulse energies and average output power (e.g., forhe 500 MHz ring laser the average output power fromhe numerical analysis was 380 mW, which is in accor-ance with the measured output power of 350 mW [24]).ach displayed power spectral density (PSD) curve inigs. 4–6, whether from the experiment or the numericalnalysis, is normalized to its maximum amplitude.In Fig. 4(a) the output spectrum for the linear 200 MHz

aser from the experimental setup and simulation is pre-ented, and the numerical analysis predicts the outputpectrum shape accurately. The ZnSe/MgF2 OC enhanceshe wings relative to the main output, and the powerransmission for the center wavelengths is 1% (see [22]).

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ig. 3. (Color online) (a) Pulse breathing for the 500 MHz ringaser for one cavity round trip starting in the geometric middle ofhe TiSa crystal. (b) Spectral breathing of the power spectral den-ity (PSD) for the 500 MHz ring laser. The position in the cavityabeling refers to Fig. 1(b).

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Sander et al. Vol. 26, No. 4 /April 2009 /J. Opt. Soc. Am. B 747

The reflection off the wedge, which generates the mainutput beam for the 1 GHz ring laser [25], essentially cor-esponds to the intracavity spectrum, since no spectralhaping occurs by the OC. Figure 4(b) shows the outputpectrum and demonstrates how an ultrabroadband spec-rum can be supported within the cavity because of thisat OC. A very good agreement between the simulatednd measured spectrum on average can be noted, andmall substructures occur with the same magnitude inoth PSDs. The offset of 10 nm between certain featuresn the experiment and the simulation suggests slight dif-erences between the fabricated and the designed mirrorata, since the 200 MHz and 1 GHz laser setups featurehe same mirrors (which are different from the 500 MHzaser). In addition, deviations in the wavelength calibra-ion of the measurement equipment used to characterizehis particular laser setup also explain the slight wave-ength offset.

To emphasize the importance of the mirror characteris-ics, the numerical analysis for the 500 MHz ring laser23,24] is conducted with the designed and the experi-entally measured mirror data. The difference between

he two mirror descriptions is shown in the group delayGD) of the DCMs, as illustrated in Fig. 2(b). The mea-ured GD follows the designed GD closely in the wave-ength range of interest (the maximum deviation betweenoth curves for the main part of the spectrum is ±2 fs),ut it shows somewhat reduced oscillations in the long-avelength regime when compared to the design.

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ig. 4. (Color online) Comparison of the simulated and mea-ured main output spectra: (a) linear 200 MHz laser, (b) 1 GHzing laser. Each PSD curve is normalized to its maximummplitude.

The impact of these slight differences in the GD on thepectra can be seen in Figs. 5(a) and 5(b), where the00 MHz ring laser output spectra for the main outputnd the 1f–2f referencing beam are shown for three dif-erent cases: for simulations using the measured and de-igned mirror characteristics (i.e., reflectivity and disper-ion), and for the experimentally obtained results. Theimulated spectra in Figs. 5(a) and 5(b) show good agree-ent between each other and, on average, good agree-ent with the measured data. The main features in the

pectral wings are accurately reproduced and we can pre-ict the optimized power levels achievable for the 1f andf output [with 1160 and 580 nm, as 1f and 2f frequen-ies, illustrated in Fig. 5(b)] on the chosen normalizedcale. When comparing the two numerical results for bothhe 1f–2f output and the main output, the PSD with theesigned mirrors falls off slightly steeper in the long-avelength regime (larger than 1160 nm). As the longavelengths penetrate deeper into the mirror stacks, theyre more prone to fabrication tolerances, which can ex-lain the slight deviations. In addition, the oscillations inhe GD in Fig. 2(b) are more pronounced for the designedCMs compared to the smoothed measured GD for longavelengths. Considering these slight deviations between

he designed and measured mirror data, the numericalnalysis under two GD conditions predicts stable steady-

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ig. 5. (Color online) (a) Main output spectrum for the 500 MHzing laser. (b) 1f–2f output spectrum for the 500 MHz ring laser.n both cases the measured spectrum is compared with the simu-ated output spectra using the designed and experimentally de-ermined DCM data. Each PSD curve is normalized to its maxi-um amplitude in the center part of the spectrum.

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748 J. Opt. Soc. Am. B/Vol. 26, No. 4 /April 2009 Sander et al.

tate outputs, which in both cases compare well with thexperimental results. At the same time, incorporating theeasured mirror data enhances the agreement between

xperiment and simulation even further.

. Dispersion Compensationn important factor in optimizing the output spectrum is

he overall dispersion compensation in the cavity: fine ad-ustment of the wedge thickness changes the intracavityDD by a few fs2 in a very precise fashion and relative

hanges in the wedge insertion can be determined veryccurately in the experimental setup. To examine the de-endence of the spectral wings on the intracavity disper-ion, the spectra for small dispersion variations related tohanges in the wedge insertion were numerically evalu-ted. The impact of dispersion fine adjustment is shownn Fig. 6, where the main output spectrum of the00 MHz laser is examined for different BaF2 materialhicknesses, while keeping all other parameters constant.he simulated 500 MHz laser was optimized to its broad-st, steady-state intracavity spectrum if the cavity con-ained insertion values of 2.05 mm BaF2 and 1.84 mm FS,hich compare well to the estimated experimental valuesf 2.07 mm BaF2 and 1.95 mm FS. For reduced wedgehicknesses, the wings in the spectrum are notably de-reased (approximately −12 dB at 1160 nm for smallhanges in BaF2 material insertion of only 100 �m, whichorresponds to a GDD of 4 fs2 at 800 nm for the00 MHz ring laser). This is in good agreement with theequired experimental fine-tuning of the wedges. At theame time, the sensitivity to the dispersion compensationepends on the actual laser configuration, since the00 MHz linear laser exhibits approximately −7 dB de-rease at 1160 nm for the same relative change in BaF2aterial insertion.

. Mirror Rippleshe DCMs are designed in pairs to cancel ripples in theD, so that an overall smooth GD is achieved over aroad bandwidth. The impact of the GD ripples on the00 MHz laser setup was studied, since the overall mirrorscillations are the most pronounced for this cavity con-

ig. 6. (Color online) Main output spectrum for different disper-ion compensation with varying BaF2 insertion for the 500 MHzaser, normalized to its maximum amplitude. A decrease of.1 mm of BaF2 corresponds to a GD of 4 fs and a decrease inhe spectral wings around 1160 nm of −12 dB.

aining 3 DCM pairs: to numerically determine the im-act of the GD ripples on an absolute scale itself, the os-illations from the DCMs were extracted and the averageD ripple for each DCM amounted to 15 fs. These were

mposed with certain strengths on each individual mirrorn the laser setup. With stronger GD oscillations in theinear laser (up to double the original magnitude, i.e.,ipples of 30 fs), but the same overall net GD at 800 nm,ulse generation was still possible at the cost of reducedulse energies and smaller maximum instantaneous in-ensities. For enhanced mirror ripples, the trailing edge ofhe pulse envelope fell off more slowly, thus resulting inostpulses and redistribution of energy into the wings ofhe pulse. Due to the larger perturbations seen at eachirror reflection, the number of round trips until the

teady state was reached increased by a factor of 2.

. Output Pulseor the 500 MHz ring laser the main output pulse was ex-mined after it was externally compressed by four addi-ional octave-spanning DCMs, a BaF2 plate, and two thinS wedges for fine-dispersion compensation. The

ransform-limited FWHM pulse duration extracted fromhe output spectrum could be reduced to 4.2 and 3.7 fs, ac-ording to simulation and experiment. To evaluate the ac-ual output pulse after compression, the numerical resultas compared with the retrieved pulse from 2DSI mea-

urements [20,21,24], as shown in Fig. 7. The pulse shapend pulse duration of 4.9 fs are exactly reproduced. Thisesult shows that sub-two-cycle pulse generation in TiSascillators can be very well captured with the describedne-dimensional temporal model.

. CONCLUSIONSe demonstrate that a one-dimensional numerical analy-

is can accurately describe the laser dynamics of octave-panning lasers that generate sub-two-cycle pulses forifferent cavity configurations by using measured (or de-

ig. 7. (Color online) Output pulse for 500 MHz laser after ex-ernal compression through a delay line with 2 DCM pairs andS/BaF2 material. The measured pulse is retrieved from two-imensional spectral shearing interferometry (2DSI) measure-ents and shows excellent agreement with the simulated output

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Sander et al. Vol. 26, No. 4 /April 2009 /J. Opt. Soc. Am. B 749

igned) mirror dispersion and reflectivity and the disper-ion of the other intracavity elements. Our approachhows that the impact of the DCMs and the OC on theulse dynamics and their spectral details are very wellaptured by this analysis. By optimizing the dispersionompensation and including the experimental deviceharacteristics, we can predict the best achievable perfor-ance in terms of output spectrum, the enhancement of

pectral wings, and achievable output pulse shape anduration in an actual experimental setup. Thus, the pre-ented numerical model is a powerful tool for the analy-is, design, and optimization of laser components and theeneral design of octave-spanning lasers.

CKNOWLEDGMENTShis work was supported in part by Defense Advancedesearch Projects Agency (DARPA) grant HR0011-05-C-155, and United States Air Force Office of Scientific Re-earch (AFOSR) grant FA9550-07-1-0014. H. M. Crespocknowledges grant SFRH/BPD/27127/2006 fromundação para a Ciência e Tecnologia (FCT-MCTES),ortugal.

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