June 1, 1994 / Vol. 19, No. 11 / OPTICS LETTERS 807

Resonators for Kerr-lens mode-lockedfemtosecond Ti:sapphire lasers

G. Cerullo, S. De Silvestri, V. Magni, and L. Pallaro

Dipartimento di Fisica del Politecnico, Centro di Elettronica Quantistica e Strumentazione Elettronica delConsiglio Nazionale delle Ricerche, Piazza L. da Vinci 32, 20133 Milano, Italy

Received December 6, 1993

Resonators for Kerr-lens mode-locked lasers are studied with the help of an analytical model, and a new designprocedure is presented. The experiments performed with a femtosecond Ti:sapphire laser demonstrate that thismodel permits an effective resonator design and offers reliable guidelines for the final experimental optimization.With a suitable resonator configuration, self-starting of the Kerr-lens mode-locking regime has been achieved.

Kerr-lens mode-locking (KLM) is a powerful tech-nique for ultrashort-pulse generation from a varietyof solid-state lasers."~ This technique is based onthe self-focusing effect in a resonator containing amaterial with Kerr nonlinearity that modifies thetransverse mode and results in power-dependentloss or gain modulation. This modulation isprovided either by a suitable intracavity aperture(hard-aperture KLM) or by the transverse gainprofile of the active material (soft-aperture KLM).Hard-aperture KLM has the advantage of beingmore reproducible and easier to specify. A stablemode-locking regime critically depends, in bothcases, on a careful adjustment of the resonator: inparticular, the optical stability of the resonator andthe focusing conditions in the Kerr medium must beproperly controlled. This observation agrees withseveral analyses of the models of resonators for KLMlasers.' 3 Although these models suggest resonatordesign guidelines, until now a detailed comparison ofthe theoretical predictions with experimental resultshas been lacking, and the resonator implementationhas been based mostly on empirical rules.

The purpose of this Letter is to identify the res-onator configurations most suitable for hard-apertureKLM and to provide design criteria for resonatorsthat allow one to obtain KLM with high reliabilityand reproducibility. The predictions of a theoreticalanalysis, based on a formalism previously publishedby the authors,12" 3 are compared with the experimen-tal results obtained with a femtosecond Ti:sapphirelaser. The experiments clearly demonstrate that ourmodel can be successfully employed for practical con-struction and optimization of KLM lasers.

We consider here a standard astigmatically com-pensated resonator configuration such as that shownin Fig. 1(a), in which the Ti:sapphire rod simulta-neously acts as a Kerr medium and a gain medium,but the obtained results can be directly extendedto more-complex structures. To take into accountthe astigmatism of the Brewster-cut Ti:sapphire rodand of the tilted focusing mirrors, we can separatethe resonator into two equivalent one-dimensionalresonators corresponding to the tangential (6t) andsagittal (77') planes. A matrix representation of

these resonators is shown in Fig. 1(b). As a functionof the distance between the folding mirrors, z, eachof these two resonators presents two stability regionsof the same width. In one of the two stabilityregions the folding mirrors are arranged in analmost confocal configuration, and the resonator hashigh misalignment sensitivity (HMS) region, sincethe C element of the one-way matrix for propagationbetween the two end mirrors vanishes for theconfocal arrangement. 1415 In the second region,because of the greater distance between foldingmirrors the end mirrors lie in nearly conjugatedplanes, and the resonator has a low misalignmentsensitivity (LMS region). To analyze resonatorscontaining a Kerr medium within the frameworkof the aberrationless theory of self-focusing, wepreviously introduced12'13 a formalism based on anonlinear ABCD matrix that allows one to calculatein closed form the self-consistent TEMoo mode

(a) S 2

pum bL2 ea

S, SLin pump beam

(b) (A, B,1 j (A 2 8r2\

(Cri Ar/) j jCr2Arc7~ ,. d

Fig. 1. (a) Resonator configuration: folding mirrors M3and M4 have radii of curvature R = 100 mm and incidenceangles 0/2 = 14.5°; end mirrors M1 and M2 are flat; theTi:sapphire rod length is d = 20 mm (refractive indexn = 1.76). Sl, S2, slits. (b) Matrix representation of theresonator: the matrices describe the propagation froma face of the Ti:sapphire rod to the corresponding endmirrors and back to the rod face; Ti is a propagation cor-responding to the Ti:sapphire rod. In the tangential (6)plane the equivalent length of the rod is de = din 3 andthe folding mirrors correspond to a lens with focal lengthf = [R cos(0/2)]/2; in the sagittal (-q4) plane de = d/nand f = R/[2 cos(0/2)].

0146-9592/94/110807-03$6.00/0 © 1994 Optical Society of America

808 OPTICS LETTERS / Vol. 19, No. 11 / June 1, 1994

in resonators with cylindrical symmetry. Thefirst-order loss variation caused by an aperture isproportional to the small-signal relative spot sizevariation, defined as9"3

( w dp )p=0 (1)

where w is the spot size at the aperture plane and pis the power normalized to the critical power for self-focusing.12 If 8 is negative and sufficiently large,an aperture gives rise to KLM because the lossesdecrease as the power increases. In particular, atthe plane of end mirror Ml, 8 can be expressed as

81 =- al + a2S (2)2(ai + a2 + 2ala 2S)

with

ai = Bri/de + AniS = AoDo + BoCo,

(i = 1, 2),

where BR and Ani are the matrix elements de-fined in Fig. 1(b), de is the equivalent length of theTi:sapphire rod, and AO, Bo, C0 , and Do are the ele-ments of the one-way matrix relevant to propagationfrom mirror Ml to M2 . Note that the resonator isoptically stable if - 1 < S < 1. As a first approxima-tion, Eq. (2) can be applied to the two equivalent one-dimensional resonators separately. One thereforeobtains a value, Ofe, for the tangential plane and adifferent one, 6,1, for the sagittal plane. Since, how-ever, Eq. (2) was derived for resonators with cylindri-cal symmetry and does not strictly hold for astigmaticbeams, this approximation was compared with more-accurate iterative numerical calculations. The re-sults, in agreement with those of Ref. 11, have shownthat for most of the resonators commonly used theeffect of astigmatism is to increase 61f and reduce 81,with respect to the predictions of Eq. (2). Our ap-proach therefore also describes reasonably well, apartfrom a scale factor, the mode behavior also in astig-matic resonators. It should be noted that Eq. (2)allows one to obtain in a simple way most of the re-sults previously reported in the literature and oftenobtained with more-complicated calculations.5k"1

To design and optimize a resonator that best ex-ploits the nonlinearity for KLM, one can start fromEq. (2) and find the resonator parameters that max-imize 161. The most critical parameters turn out tobe the distance between the folding mirrors, z, whichessentially determines the optical stability, and thedistance between mirror M3 and the closer face of theTi:sapphire rod, x, which controls the focusing condi-tion without affecting the resonator stability. There-fore a practical way to optimize the resonator is toplot 5 as a function of x and z and find the regionsin which the best trade-off between large values of181 and acceptable tolerances can be achieved. Todemonstrate experimentally the validity of this pro-cedure we applied it to various resonators of the typeshown in Fig. 1(a), each characterized by differentdistances Ll and L2 and operating in both the HMS

and LMS regions. For each resonator we systemat-ically changed the value of 8 by varying both the po-sition of the Ti:sapphire rod and the folding distancewhile checking whether KLM could be achieved.

For the experiments we set up an argon-laser-pumped Ti:sapphire laser in the configuration ofFig. 1(a). The 20-mm-long Ti:sapphire rod and thefolding mirrors were mounted on precision transla-tors. A pair of SF10 prisms placed 35 cm apart com-pensates the dispersion. Slit S5 was placed close tomirror Ml and was used to sustain KLM, whereasslit S2 was used for tuning and to avoid wavelengthinstabilities. In the cw regime, with 5% coupling,the typical output power was 850 mW at 4.5 W ofpump power. KLM was initiated with the help ofpiezoactuator that vibrates mirror Ml at 300 Hz withan amplitude of <1.5 1tum. The position of the fold-ing mirrors was directly read on the micrometers ofthe mirror translation stages. To obtain the mirrordistance, z, we initially performed a calibration pro-cedure. The resonator was aligned to work in theHMS stability region, and the distance z was pro-gressively reduced until laser action stopped. Weassumed that in this condition the mirrors were inthe confocal arrangement corresponding to the sta-bility limit. The folding distances of the other threestability limits read on the micrometers correspondedto the theoretical values within 0.2 mm. The zeroreference of the micrometer of the rod translationstage was calibrated with the help of a precisionruler placed between mirror M3 and the closer rodface. For each resonator and for each value of x andz, first the laser was aligned for maximum outputpower; then slit Si, oriented to cut the beam in the4 direction, was progressively closed until pure KIMoperation, without any cw component in the spec-trum, was achieved. By varying the distance of thepump focusing mirror, we adjusted the pump beamwaist position in such a way that KLM could not beobtained without slit Si as a result of gain-guidingeffects in the active material. The pulse durationranged between 50 and 60 fs FWHM with a spec-trum of =15 nm FWHM centered at 800 nm.

We first considered an asymmetric resonator, withLl = 500 mm and L2 = 1100 mm, operating in eitherthe HMS or the LMS region. Slit S1 was insertedinto the short arm of the cavity because our calcula-tions, in agreement with Refs. 8 and 11, showed that,as a function of x and z, broader regions with nega-tive 816 could be achieved. Figure 2 shows the con-tour curves of 31e as a function of x and z. On thesame plot we also show, with filled circles, the pointsfor which KLM could be experimentally obtained.The results, in good agreement with the theoreticalmodel, demonstrate that KLM could be achieved onlyfor 8lf <0. Qualitatively, for I81 I > 0.5, KLM re-quired only slight cavity perturbations to be initiatedand remained stable for hours; on the other hand,for lower 181fl, KLM required careful adjustment ofthe aperture size and was sensitive to perturbations.The same experiments were performed with a nearlysymmetric resonator, with Ll = 700 and L2 = 900:Fig. 3 shows that similar results in agreement withthe theory have also been obtained. Because of the

June 1, 1994 / Vol. 19, No. 11 / OPTICS LETTERS 809

1 9 1

20 0 _ _.LMS region -0.2

-0. . ..

16

15 - _

;;3 X0A 2 HMS region

41 43 45 47 49 51 53 55 57x (mm)

Fig. 2. Contour curves of the small-signal relative spotsize variation 81e (parameter of the curves) in the tan-gential plane of the resonator with L1 = 500 mm andL2 = 1100 mm. Only curves of negative b16 are shown.The filled circles mark the points where KLM has beenexperimentally achieved. The dashed horizontal linesare the stability limits.

120

119

118

E?117E

N 1 16

115

114

Fig. 3. Same as in Fig. 2 for the resonator withLI = 700 mm and L2 = 900 mm.

astigmatism effects, KLM could not be achieved withthe aperture oriented to cut the beam in the ij direc-tions. In the HMS region both resonators were, asexpected, sensitive to misalignment, and KLM wasin general more difficult to achieve and less stable.For this reason there are fewer experimental datapoints in Figs. 2 and 3 in the HMS regions than inthe LMS regions. Although, as already mentionedin the literature,7 -' 3 large nonlinear loss modulationscan be achieved by bringing the resonator close tothe stability limits, where S = +1, approaching thestability limit too closely degrades the laser perfor-mance. Our results indicate that the best trade-offin terms of output power, stability, and tolerance tothe exact positions of the components is achieved byoperating the laser in the LMS region at a distanceof Az = 0.5-1 mm from the lower stability limit.

As Figs. 2 and 3 show, both asymmetric and nearlysymmetric cavities are suitable for KLM, providedthat the rod position x and the foldihg distance z areproperly adjusted. However, these figures also showthat the region in the xz plane for which high valuesof 181 can be achieved progressively broadens as theresonator approaches the symmetric configuration, incontradiction to Ref. 11. As an example, for 81 = -1to be reached, the maximum allowed distance from

the stability limit is 0.22 mm for the asymmetric cav-ity of Fig. 2 and 0.44 mm for the cavity of Fig. 3,which is closer to a symmetric one. The highest tol-erance is reached with a perfectly symmetric cavity(L1 = L2). Moreover, in a symmetric resonator theHMS and the LMS regions become joined' 4 and re-sult in a single stability region in terms of z. In thecenter of this wider region the resonator becomesequivalent to a confocal one with a well-behavedtransverse mode and no power reduction even if thestability parameter S is -1. Therefore, if the rodis suitably positioned and the distance z is set closeto the center of the stability region, 51 can even di-verge to +o without sacrificing the laser stability.Experimentally, with a symmetric cavity (L1 = L2 =

850 mm), in a range of ±0.6 mm around z = 116 mm(corresponding to the center of the stability region)KLM did not require external perturbations to startbut was completely self-starting. This regime wasstable for hours and easily reproducible since only mi-nor alignments were required from day to day. Theamplitude fluctuations on a millisecond time scalewere <1%. The perfect symmetry of the resonatorwas not necessary for self-starting: in fact, we couldchange LI by as much as 50 mm and still keep thelaser in the self-starting regime. Also, the toleranceon the exact position of the rod was as much as 2 mm.On the basis of these results and of the above consid-erations we attributed the self-starting behavior tothe very high value of Al achieved with these res-onators, but further studies of the phenomenon arein progress.

The authors thank I. Piffari and S. Stagira for theirhelp with the experiments.

References

1. D. E. Spence, P. N. Kean, and W. Sibbett, Opt. Lett.16, 42 (1991).

2. F. Krausz, Ch. Spielmann, T. Brabec, E. Wintner, andA. J. Schmidt, Opt. Lett. 17, 204 (1992).

3. B. E. Lemoff and C. P. Barty, Opt. Lett. 17, 1367(1992).

4. M. T. Asaki, C. Huang, D. Garvey, J. Zhou, H. C.Kapteyn, and M. M. Murnane, Opt. Lett. 18, 977(1993).

5. M. Pich6, Opt. Commun. 86, 156 (1991).6. G. W. Pearson, C. Radzewicz, and J. S. Krasinski, Opt.

Commun. 94, 221 (1992).7. M. Pich6 and F. Salin, Opt. Lett. 18, 1041 (1993).8. T. Brabec, Ch. Spielmann, P. F. Curley, and F. Krausz,

Opt. Lett. 17, 1292 (1992).9. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, IEEE J.

Quantum Electron. 28, 2086 (1992).10. D. Georgiev, J. Herrmann, and U. Stamm, Opt. Com-

mun. 92, 368 (1992).11. T. Brabec, P. F. Curley, C. Spielmann, E. Wintner,

and A. J. Schmidt, J. Opt. Soc. Am. B 10, 1029 (1993).12. V. Magni, G. Cerullo, and S. De Silvestri, Opt. Com-

mun. 96, 348 (1993).13. V. Magni, G. Cerullo, and S. De Silvestri, Opt. Com-

mun. 101, 365 (1993).14. V. Magni, S. De Silvestri, and A. Cybo-Ottone, Opt.

Commun. 82, 137 (1991).15. A. Cybo-Ottone, M. Nisoli, V. Magni, and S. De Sil-

vestri, IEEE J. Quantum Electron. 28, 1825 (1992).

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