Bifurcation of fundamental Gaussian modes in Kerr-lens mode-locked lasers

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1 1 Septemer 1999 Ž. Optics Communications Bifurcation of fundamental Gaussian modes in Kerr-lens mode-locked lasers Ming-Dar Wei, Wen-Feng Hsieh ) Institute of Electro-Optical Engineering, National Chiao-Tung UniÕersity, Ta-Hsueh Rd. 1001, Hsinchu 30050, Taiwan Received 16 March 1999; received in revised form 17 June 1999; accepted 17 June 1999 Astract We propose that more than one spatial fundamental Gaussian mode may occur in concentric and confocal resonators under persisting nonlinear effect. Extensively studying the influence of the resonator s parameters in a Kerr-lens mode-locked resonator, pitchfork ifurcation results in more symmetrical configurations, and saddle-node ifurcation appears as the symmetry eing roken. From the properties of ifurcation, we suggest that the equal-arm and near-confocal resonator is suitale for the emergence of istaility in Kerr-lens mode-locked lasers. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Bifurcation; Kerr effect and resonator 1. Introduction Recently, istaility in Kerr-lens mode-locking Ž KLM. resonators has een studied y many researchers since the KLM laser was uilt w1 3 x. Bistaility was first predicted in a near-concentric unstale KLM resonator y considering saturale Gaussian gain and Kerr nonlinearity when the pumping rate is modulated aout the threshold for laser operation wx 1. When oth spatial and temporal effects are simultaneously taken into account, the S-shaped istale ehavior of spot size, pulse width and pulse energy with varying pump power was found at a specific near-confocal configuration wx 2. In these two papers, istaility results mainly from the mixing effect of asorptive Ž gain saturation. and dispersive ) Corresponding author. Tel.: q ext 56316; fax: q ; wfhsieh@cc.nctu.edu.tw Ž optical Kerr effect. nonlinearity. Excluding gain effect, istale ehavior was also studied in Ref. wx 3. By applying the reduced self-focusing ABCD matrix for a Kerr medium and considering the nonlinear coupling of two transversal directions in the elliptical eam, they found that more than one TEM 00 resonator mode exists around concentric configuration. Summarizing the previous results, multiple Gaussian modes are numerically otained in some specific configurations with the different source of nonlinearity. However, why these configurations are sensitive to nonlinear effects and whether this istale character is configuration-dependent is not known. In this paper, we show that the istaility depends upon resonator configuration, and we propose a reasonale illumination from studying the dynamics of Gaussian eam propagation. In previous research, the iterative map was used to study Gaussian eam propagation in a general r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. Ž. PII: S

2 162 ( ) M.-D. Wei, W.-F. HsiehrOptics Communications resonator wx 4. Analyzing the map with the help of Greene s residue theorem wx 5, we propose that ifurcation can easily occur around the confocal and concentric configurations whilst keeping a nonlinear perturation. This result will e numerically verified in KLM resonators. For the sake of discussing the ifurcation of configuration dependence, we will only focus on the Gaussian eam propagation in a cold KLM resonator having pure self-focusing. The multiple fundamental Gaussian modes indeed exist in the aove-mentioned configurations, even just considering the self-focusing. Moreover, we extensively explored the ifurcation ehaviors under the influence of the resonator s parameters, such as crystal position and astigmatism angle of curved mirrors in equal- and unequal-arm resonators. These resonators have the different classifications of ifurcation that mainly result from the degree of symmetry. These results offer a useful analysis for istaility resonator design in KLM experiment, ecause self-focusing governs the character of the KLM resonator. 2. Theoretical prediction In the paraxial approximation, the fundamental propagation of Gaussian eam follows the ABCD law for the q parameter. The q parameter of Gaussian eam is defined as 1rqs1rRyilrŽpw 2. 0, where R is the radius of curvature, w0 the spot size and l the wavelength of the eam. After using the q parameter to construct the iterative map, the character of fundamental resonator mode can e determined from the ehavior of the map at the period-1 fixed point wx 4. Analyzing the staility of the fixed point with the Greene s residue theorem, we otained the residue in a linear system as 2 Ž 1 2. Ž. Ress1y 2G G y1. 1 Here we have defined G1sayrr1 and G2sdy rr2 as the G-parameters for general optical resonators, where r1 and r2 are their radii of curvature a for the two end mirrors and is the transfer c d matrix of one-way pass etween these two end mirrors. If a fixed point of a map without multiplier q1 is isolated, there are no other fixed points within its proximity wx 5. In contrast, another periodic orit could e created or destroyed when the fixed point has a multiplier q1. Such saddle-node or pitchfork ifurcation may occur under persisting nonlinear perturation. Since the system with residue equal to zero has multiplier q1 wx 5, the confocal Ž GG s or concentric Ž GG s configuration could have the aove-mentioned ifurcation with the help of Eq. Ž. 1. Thus, a general resonator at these configurations has the intrinsic character that it may have multiple Gaussian resonator modes under the nonlinear effect. Based on the previous discussion, we will analyze the properties of ifurcation in a KLM resonator. The nonlinear self-focusing of Gaussian eam in the Kerr medium can e descried y using the renorwx 6. After renormalizing the q malized q parameter parameter to e X sre yj Im ' 1yK, Ž 2. q q q q X will follow the free-space propagation in the Kerr medium. Here the Kerr parameter K is the cavity eam power over the critical power of self-trapping and the Re and Im represent the real and imaginary parts of a complex numer. From self-consistency of the q parameters in a resonator, a simple analytical approach was proposed to design the four-mirror folded KLM laser wx 7. The four-mirror folded KLM resonator, shown in Fig. 1, consists of two flat end mirrors Ž M and M. 1 4 and a pair of curved mirrors Ž M and M. 2 3 of focal length f. The distance etween M1 and M 2 is d 1, etween M 3 and M 4 is d 2, etween M and the end face I of the Kerr medium 2 Fig. 1. The KLM resonator. The four-mirror folded KLM resonator consists of two flat end mirrors Ž M and M. 1 4 and a pair of curved mirrors Ž M and M. 2 3 with the same focal length f. The distance etween M1 and M 2 is d 1, M 3 and M 4 is d 2, M 2 and the end face I of Kerr medium is r1 and two curved mirrors is z, respectively.

3 ( ) M.-D. Wei, W.-F. HsiehrOptics Communications is r1 and etween two curved mirrors is z, respectively. Since the curvature of Gaussian resonator mode must match those of the end mirrors in lossless system, the curvature of parameter q at output flat mirror M1 is zero. Thus, we can assume that the q parameter at M1 is q1sjysjpw 2 rl. Using the renormalized q parameter concept to transform q 1 through the Kerr medium and the self-consistence at end face II, one can otain the spot size of fundamental mode at M1 to satisfy a quartic equation of 2 y as wx Ž. Ž. Ž. Ž. h w, K, G sa y qa y qa y Ž. qa y 2 a s0. Ž Here G represents a configuration variale that depends on d 1, d 2, r 1, r 2, z, f and the Kerr medium length L. The coefficients a 4, a 3, a 2, a 1, and a0 are functions of K and G. Owing to a quartic equation having analytic solutions, one can extensively study the ifurcation ehavior of spatial fundamental Gaussian mode in the KLM resonator with various configurations G s and Ks. This approach is used in the following numerical calculation. numerical simulation. The Kerr parameter offers the nonlinear effect to e the ifurcation parameter. Fig. 2Ž. a shows spot size at M1 versus Kerr parameter for z s mm, r1 s r2 s mm and u s in the symmetrical stale resonator. The solid line in this figure indicates the stale solutions of self-consistent Gaussian eam and the dashed line represents the unstale one. The configuration is near confocal, since the confocal configuration for linear Ž cw. resonator corresponds to a separation of the curved mirrors zcs mm; therefore, as predicted in Section 2, the ifurcation appears as a pitchfork ifurcation with the ifurcation point at Ž K, w. sž , mm., where K and w represent the critical ifurcation Kerr parameter and spot size, respectively. Furthermore, we numerically otain hshwshwwshks0 at the ifurcation point, where the suscripts of 3. Numerical results We will concentrate initially on the symmetric confocal configuration in the KLM laser. This resonator has equal arms with d sd s850 mm and 1 2 the crystal is placed at the center of two curved mirrors with r1sr 2. The parameters of the optical elements referred to the experimental ones wx 8 in which the radii of curvature of the curved mirrors M 2 and M 3 are oth 100 mm and the length of Brewster-cut Ti:sapphire rod is L s 20 mm. The curved mirrors have een tilted y an angle u for the astigmatism compensation aout Brewster-cut laser rod. Thus, the resonator could e divided into two astigmatic optical systems associated with the sagittal and tangential planes. These two planes are considered to e orthogonal. Because similar ehavior can also e found in oth planes, we focused the following numerical simulation on the sagittal plane. Since the eam s q-parameter contains only the spot size with zero curvature at the output mirror M, the 1 spot size is chosen as the scalar measure in the Fig. 2. Pitchfork ifurcation. The spot size at M1 versus Kerr parameter in equal-arm resonator with d1s d2s85 cm is shown for zs115.3 mm and Ž. a r s47.65 mm, Ž. 1 r1s46.5 mm and Ž. c r1s49 mm. The solid line corresponds to the stale solutions of self-consistent Gaussian eam and the dashed line represents the unstale one.

4 164 ( ) M.-D. Wei, W.-F. HsiehrOptics Communications function hw, Ž K, G. represent the various order partial derivatives with respect to the variales indicated as suscripts. According to the singularity theory wx 9, this result also verifies that the classification elongs to the pitchfork ifurcation. When the crystal is located away from the center of two curved mirrors, the characteristics of ifurcation will e changed. By constraining zs115.3 mm, the ifurcation diagram with r1 s46.5 mm is shown in Fig. 2Ž., note that r1 s47.65 mm for symmetry resonator. It is no longer a standard pitchfork ifurcation ut a pertured one. As pitchfork ifurcation is not generic, it usually results from some peculiar symmetry or the inadequacy of the idealization, in which some small effects are neglected wx 9. Therefore, moving the crystal away from the center of two curved mirrors had roken the symmetry and induced the pertured variation of ifurcation. In Fig. 2Ž., the upper ranch of this typical pertured ifurcation corresponds to the continuous evolution ranch from efore to after ifurcation. On the contrary, the other typical pertured pitchfork ifurcation takes place at r1 greater than mm. For example, r s49 mm in Fig. 2Ž. 1 c, the lower ranch in this type is a continuous evolution one. When a slit is inserted at M1 in a hard aperturing KLM resonator, the KLM strength d is determined y the rate of change of the spot size as increasing the laser power w7,10 x. Experimentally, for achieving the larger self-amplitude modulation, in general the slit is inserted vertical to the tangential plane, which has the larger KLM strength. If the configuration has the pitchfork ifurcation in the sagittal plane with d-0 in the tangential plane, it is suitale for oserving the istaility in hard-aperturing KLM laser. Under well-matching astigmatism, d has the same sign in oth planes, thus the configuration with the type r s49 mm in Fig. 2Ž. 1 c prefers to operate at KLM due to d- 0 in oth planes. However, the configuration with the type r1s 46.5 mm in Fig. 2Ž. will not e a preferred KLM one. Further calculating the ifurcation points y varying the configuration-parameters z and r 1, we otain the contour of critical ifurcation parameter K, shown in Fig. 3 with progressively increasing K from curves Ž. a to Ž. e. The dashed line represents the symmetrical configuration that the crystal is located at the center of two curved mirrors. Clearly, the K Fig. 3. Contour of the critical ifurcation parameter in equal-arm resonator. The contour values are Ž. a K s0.01, Ž. K s0.05, Ž. c K s0.1, Ž. d K s0.2, and Ž. e K s0.3, respectively. The dashed line represents the crystal locating at the center of two curved mirrors in which corresponds to the symmetric configuration. increases as the crystal is located away from the center of two curved mirrors or as the configuration far from the confocal structure. Similar ehavior can also e found in the tangential plane, ut the ifurcation region will reduce to less than half of that in the sagittal plane. The smaller region results from shorter effective rod length and effective focal length of the curved mirrors in the tangential plane. The ifurcation region on the right hand side of dashed line elongs to the pertured pitchfork ifurcation having the type of Fig. 2Ž. c. Compared with the preferale KLM region in the experiments in Ref. w10x and the theory in Ref. w11 x, we find that the ifurcation region is located within the preferred KLM region though much smaller than the preferred one. Thus, we elieve that istaility may e oserved in such a ifurcation region. If we consider the asymmetric resonator with d /d ut d qd s170 cm, a different classifica tion of ifurcation will take place. In Fig. 4 with d1s70 cm, d2s100 cm, zs mm and r1s 41 mm, saddle-node ifurcation occurs at critical ifurcation parameter K f There is no real solution of spot size as K-K, and two self-con sistent spot sizes exist as K)K. The lower ranch with solid line indicates stale solutions and the upper ranch with dashed line represents unstale solutions. When the configuration varies close to the confocal one, the K decreases. In general, pitchfork

5 ( ) M.-D. Wei, W.-F. HsiehrOptics Communications ifurcation will occur in more symmetrical configurations and saddle-node ifurcation exists as this symmetry eing roken wx 5. Therefore, we think the emergence of different classification mainly results from the configurations having the unequal arms, which have roken the symmetry of resonator. However, the symmetry roken from the slightly unequal arms can e compensated y the tilted angle of the curved mirrors. The critical ifurcation parameter against the tilted angle is shown in Fig. 5 with d s83 cm, d s87 cm, zs115 mm and r s mm which corresponds to near-confocal configuration. We constrain our discussions on K that the associated power can e otained from general experiments. As u- u s , the configuraa tion has only one real solution of spot size and no ifurcation. Increasing the tilted angle to u)u a,we find the saddle-node ifurcation takes place. The upper ranch corresponds to the stale solution and the lower ranch corresponds to the unstale solution. Moreover, K increases as the tilted angle is increased, ut the staility reverses and the K de- creases as increasing the tilted angle to u) u s When u is greater than uc s , the ifurcation transits to the pertured pitchfork ifurcation and K still decreases as increasing u. The minimum K is at uds corre- sponding to the standard pitchfork ifurcation. The other type of pertured pitchfork ifurcation exists as ud-u-ues , in which K increases as increasing u. The configuration returns to having one reasonale spot size as u)ue if one constrains K In fact, dynamical characteristics in this region still elongs to the pitchfork ifurcation for K )0.4. Because such a K value is not easy to reach in general KLM lasers, the phenomenon of Fig. 4. Saddle-node ifurcation. The ifurcation diagram is shown as d s70 cm, d s100 cm, zs mm and r s41 mm Fig. 5. The critical ifurcation parameter versus the tilted angle. It is shown the critical ifurcation parameter against the tilted angle with d s83 cm, d s87 cm, zs115 mm and r s40 mm. The value of specific tilted angle are uas , u s , u s , u s and u s These tilted anc d e gles are transition point for the changing the character of ifurcation diagram. ifurcation may not e oserved in experiments. It is worth noting that the aove regions all have d-0 in the tangential plane, i.e., these regions prefer KLM operation with hard aperture. Although the tilted angle can compensate the roken symmetry resulting from unequal arms, the latter one governs the emergence of ifurcation. If we constrain d qd s cm and d1fd 2, the region of pitchfork ifurcation is aout hundreds of mm for z translation in equalarm resonator. The region quickly shrinks as increas- ing < d y d < 1 2 and ecomes less than 10 mm as < d yd < 1 2 G4 cm, even having the tilted angle compensation. 4. Discussion The ifurcation phenomenon can also e found in the near-concentric resonator. However, the region having ifurcation is smaller than that of near-confocal resonator, and its classification elongs to the saddle-node ifurcation. Since saddle-node ifurcation has only one stale solution, istaility will not e found in such configuration if we just consider the self-focusing effect. Moreover, no steady spot size in the range with K-K represents that the resonator lacks a steady pulse generating mechanism from cw with Ks0 transiting to KLM with higher K. This system may not e spontaneous in KLM

6 166 ( ) M.-D. Wei, W.-F. HsiehrOptics Communications laser. On the contrary, istaility could e oserved in the configuration with unpertured or pertured pitchfork ifurcation having two stale spot sizes in K)K and a steady one in K-K ; especially, the configurations having d- 0 in tangential plane is achievale to mode-locking operation with hard aperture. Thus, we suggest that the equal-arm and near-confocal resonator is suitale for the emergence of istaility in experiment due to the region with pitchfork ifurcation eing large. When we further consider the spatial and temporal effects in a KLM resonator, a simple quadratic equation is otained to determine the pulse width from space time analogy w12 x. Owing to the spot size variation in Kerr medium couples to temporal self-phase modulation matrix, the ifurcation of spot size will result in the ifurcation of pulse width. We indeed found the same classification of ifurcation in pulse width versus Kerr parameter. It is also to e found in Ref. wx 2 that oth spot size and pulse width appear to have the same istaility ehavior. Howwx 2 is not found ever, the S-shaped ehavior in Ref. and instead of pitchfork ifurcation in this research, this difference may e attriuted to the gain saturation effect. 5. Conclusion Studying the iterative map constructed from the propagation of Gaussian q parameter in a resonator, we found that the confocal and concentric configuration corresponding to the map having multiplier q1. Such a map will induce ifurcation under nonlinear perturation in general. This result gives a reasonale interpretation for the existence of multiple fundamental Gaussian modes, which often occur at the limit of stale region. When the numerical simulations contain only the self-focusing and ignore the gain saturation effect in KLM resonator, istaility takes place in the aove-mentioned configurations. Under extensively studying the influence of different arm-lengths, crystal position and tilted angle, there are two main classification ifurcation corresponding to pitchfork and saddle-node ones. Pitchfork ifurcation occurs in configurations with higher symmetry and saddle-node ifurcation exists as this symmetry eing roken. In addition, we suggest that the equalarm and near-confocal resonator is suitale for the emergence of istaility in experiments. Acknowledgements The research was partially supported y the National Science Council of the Repulic of China under grant NSC M and one of the authors, M.-D.W., would like to thank NSC for a Ph.D. fellowship. References wx 1 M. Piche, Opt. Commun. 86 Ž wx 2 S. Longhi, Opt. Commun. 110 Ž wx 3 R.E. Bridges, R.W. Boyd, G.P. Agrawal, Opt. Lett. 18 Ž wx 4 M.-D. Wei, W.-F. Hsieh, C.C. Sung, Opt. Commun. 146 Ž wx 5 R.S. MacKay, Renormalisation in Area-preserving Maps, World Scientific, Singapore, wx 6 H.A. Haus, J.G. Fujimoto, E.P. Ippen, IEEE J. Quantum Electron. 28 Ž wx 7 K.-H. Lin, W.-F. Hsieh, J. Opt. Soc. Am. B 11 Ž wx 8 D.-G. Juang, Y.-C. Chen, S.-H. Hsu, K.-H. Lin, W.-F. Hsieh, J. Opt. Soc. Am. B 14 Ž wx 9 R. Seysel, Practical Bifurcation and Staility Analysis, Springer-Verlag, New York, 1994 Ž Chapter 8.. w10x G. Cerullo, S. De Silvestri, V. Magni, Opt. Lett. 19 Ž w11x M.-D. Wei, W.-F. Hsieh, C.C. Sung, Opt. Commun. 155 Ž w12x K.-H. Lin, W.-F. Hsieh, J. Opt. Soc. Am. B 13 Ž