Numerical Analysis of Soft-Aperture Kerr-Lens Mode Locking in Ti:Sapphire Laser Cavities by Using Nonlinear ABCD Matrices

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1 Journal of the Korean Physical Society, Vol. 46, No. 5, May 2005, pp Numerical Analysis of Soft-Aperture Kerr-Lens Mode Locking in Ti:Sapphire Laser Cavities by Using Nonlinear ABCD Matrices Yong Woo Lee and Jong Hoon Yi Department of Physics, Yeungnam University, Gyeongsan Yong Ho Cha, Yong Joo Rhee, Byoung Chul Lee and Byung Duk Yoo Laboratory for Quantum Optics, Korea Atomic Energy Research Institute, Daejeon (Received 20 January 2005, in final form 9 March 2005) We have numerically analyzed soft-aperture Kerr-lens mode locking in Ti:sapphire laser cavities. The intensity-dependent Kerr-lens effect was simulated by using nonlinear ABCD matrices, and the power-dependent Gaussian beam mode was calculated for typical linear x-folded laser cavities used for soft-aperture Kerr-lens mode locking. The Kerr-lens effect was found to depend strongly on cavity parameters such as the separation of the two curved mirrors, the crystal position, and the lengths of cavity arms. Based on the numerical analysis, we derived the optimal cavity condition for soft-aperture Kerr-lens mode locking, and we then experimentally demonstrated a Kerr-lens mode-locked Ti:sapphire laser. PACS numbers: By, Jx Keywords: Kerr-lens mode locking, Ti:sapphire laser, ABCD matrices I. INTRODUCTION Since the first Kerr-lens mode locking (KLM) in a Ti:sapphire laser was demonstrated in 1991 [1], KLM has contributed to the generation of picosecond and femtosecond laser pulses in many kinds of solid-state lasers, such as Nd:YLF, Cr:Forsterite, Cr:LiSAF, Yb:YAG, and Yb:glass [2 4] lasers. Especially, Ti:sapphire lasers aided by KLM have been very successful in generating sub-20- fs pulses thanks to the broad gain profile and the excellent physical properties of Ti:sapphire [5], and even a sub-5-fs region has been explored [6]. KLM utilizes the power-dependent beam profile caused in a cavity by the Kerr-lens effect. In a carefully controlled cavity, the beam profile at a specific position in the cavity can be reduced at higher intra-cavity power. If an appropriate aperture is installed at that position, then an effective saturable absorption, a higher transmission at a higher power, can be achieved. Such effective saturable absorption prefers the short pulse generation to cw lasing and leads to stable mode locking (hardaperture KLM) [7]. If the position where the beam profile is reduced at higher power is inside the gain medium, KLM can be achieved even without any aperture. In a longitudinal pumping configuration, the pumping beam profile is usually smaller than the lasing beam profile in the gain medium, and the smaller lasing beam profile yhcha@kaeri.re.kr; Fax: leads to better mode matching with the pumping beam. Therefore, pulsed lasing, which has a higher peak power and a smaller lasing beam profile in the gain medium due to the Kerr-lens effect, experiences a higher gain than cw lasing. Such gain discrimination between pulsed and cw lasing leads to short-pulse generation, so stable mode locking (soft-aperture KLM) [8] can be realized. Because the Kerr-lens effect is nearly insensitive to the laser wavelength and has very fast response time in the fs region, KLM can be applied to the generation of femtosecond and picosecond pulses in various kinds of solidstate lasers. Moreover, the simple configuration for KLM makes it one of the most widely used mode-locking techniques. The most significant drawback of KLM is that KLM is achieved only under limited cavity conditions. Experimental and theoretical studies have shown that KLM is realized near the stability limits and is sensitive to the cavity alignment. This means that the optimal cavity condition for efficient KLM is significantly different from that for the cw lasing. Therefore, in developing a Kerr-lens mode-locked Ti:sapphire laser, it is important to theoretically investigate the optimal cavity conditions for efficient KLM. For a rigorous numerical simulation of the Kerr-lensing effect, a complicated calculation of beam propagation is required, leading to excessive time consumption and inconvenience. Some groups have introduced relatively simple ways for the numerical analysis of KLM by using modified q parameters [9]. Such

2 Journal of the Korean Physical Society, Vol. 46, No. 5, May 2005 methods, however, are still difficult to use for systematic numerical analyses of KLM in various cavity conditions. In this paper, we numerically analyze soft-aperture KLM in Ti:sapphire lasers by simulating the Kerr-lens effect with nonlinear ABCD matrices. The power-dependent beam mode in a cavity is calculated as the cavity condition is varied, and the optimal condition for soft-aperture KLM in Ti:sapphire lasers is derived. II. BEAM PROFILE CALCULATION BY USING ABCD MATRICES ABCD matrices are very convenient and useful tools in calculating Gaussian beam profiles in a cavity containing linear optical elements. The Gaussian beam size at any position in a cavity can be calculated from a following equation: q = Aq + B Cq + D. (1) Here, A, B, C, and D are the ABCD matrix elements of a complete round-trip through a cavity, and q is the beam parameter defined by 1 q = 1 R i λ πw 2, (2) where R is the wavefront curvature, w the Gaussian beam size (1/e 2 radius of the intensity), and λ the lasing wavelength. For a cavity containing only linear optical elements, the round-trip matrix elements are all fixed values, and the beam size, w, can be easily obtained by solving Eq. (1). Because the self-focusing by the Kerr effect is a nonlinear phenomenon that is highly dependent on the power and the beam size, the cavity for soft-aperture KLM cannot be simply described by linear ABCD matrices only. To simulate the Kerr-lensing effect, therefore, one needs to derive nonlinear ABCD matrices for the Kerrlensing plates where the Gaussian beam is self-focused by the Kerr effect. The intensity-dependent refractive index caused by the Kerr effect can be described as n = n L n 2E 2 = n L + n 2 n L c 0 ε 0 I = n L + γ 2 I, (3) where n L is the linear part of refractive index, n 2 and γ 2 are the electric field and the intensity coefficients of the nonlinear refractive index respectively, E is the amplitude of the electric field, I the laser intensity, ε 0 the electric permittivity in vacuum, and c 0 the speed of light in vacuum. The Gaussian laser intensity distribution can be approximated by the first two terms of its Taylor expansion as follows: [ ( r ) ] [ 2 ( r ) ] 2 I = I 0 exp 2 I (4) w w Then, the intensity-dependent refractive index has a parabolic spatial variation: n = n L + n [ ( 2I 0 r ) ] = n I (1 1 ) n L c 0 ε 0 w 2 γ2 r 2,(5) where n I = n L + n 2I 0, n L c 0 ε 0 ( 4n2 I 0 γ = n L c 0 ε 0 n I P = πw 2 I 0 /2. ) 1/2 1 w = ( ) 1/2 8n2 P 1 n L c 0 ε 0 n I π w 2, The ABCD matrix, A Kerr, for a plate which has a parabolic refractive index, according to Eq. (5), is ( ) cos(γl) sin(γl)/(n A Kerr = I γ), (6) n I γ sin(γl) cos(γl) where l is the plate thickness. Because n I and γ depend on the power and the beam size, the elements of A Kerr themselves depend on the beam size. Therefore, the round-trip ABCD matrix through a cavity including Kerr-lens plates is a function of the beam sizes at the plates, and Eq. (1) cannot be solved analytically unless the correct beam size at each plate is determined. In our numerical calculation, the beam sizes at the Kerr-lens plates are iteratively determined by starting with zero intra-cavity power. Then, the initial ABCD matrices for the Kerr-lens plates can be defined at a given intra-cavity power with these beam sizes. From these initial matrices, a new set of beam sizes can be calculated by solving Eq. (1) at each Kerr-lens plate. Then, the ABCD matrices for the Kerr-lens plates are newly defined with the new set of beam sizes. In such a way, the beam size at each Kerr-lens plate can be calculated iteratively until the beam sizes effectively do not change. Once the beam size at each Kerr-lens plate is determined at the given intra-cavity power, the beam size at any position in the cavity can be calculated by using Eq. (1). The linear Ti:sapphire laser cavity for soft-aperture KLM is typically composed of two curved mirrors, two flat end mirrors, and one Ti:sapphire crystal, and the x-folded cavity configuration is used. Because an actual x-folded laser cavity contains a Brewster-cut Ti:sapphire crystal and curved mirrors which are appropriately tilted for astigmatism compensation, the ray tracing should be performed separately on the sagittal and the tangential planes. In our calculation, however, two ideal lenses are used instead of two curved mirrors for simple calculation, as shown in Fig. 1. The Kerr-lens effect takes places in the Ti:sapphire crystal, so we divide the crystal into many Kerr-lens plates to ensure negligible change in the beam size through each plate. The refractive index, n L, and the electric field coefficient of the nonlinear refractive index, n 2, of Ti:sapphire are 1.76 and m 2 V 2 [10], respectively.

3 Numerical Analysis of Soft-Aperture Kerr-Lens Yong Woo Lee et al Fig. 1. Schematic of the linear cavity for the simulation of the soft-aperture KLM. M1 and M2 are flat mirrors, L1 and L2 are lenses with 50-mm focal lengths, C is a Ti:sapphire crystal with normal ends, d1 and d2 are the lengths of the short and the long arms, l1 and l2 are the lengths from the crystal ends to L1 and L2. III. NUMERICAL RESULTS Because KLM utilizes the power-dependent beam profile in a cavity, it is convenient to define a following KLM strength, δ, as a measure of the gain modulation caused by Kerr lensing [11]: δ = 1 w w P P 0, where P is the intra-cavity power. δ must be positive and large enough for stable KLM because the beam size at a specific position should become smaller as the intracavity power increases. For soft-aperture KLM, we average δ over the Ti:sapphire crystal to take into account the contribution of the whole region, not a specific position, inside the crystal. Experimentally KLM is well known to be generally achieved only around the boundary of the stable cavity region, and we verify that in Fig. 2 by calculating δ over the whole stable cavity region. Here, d1 and d2 are 600 mm and 1000 mm, respectively, and the crystal length is chosen as 5 mm. When d1 d2, two separate stable regions exist [12], and δ is calculated for both regions. The stable cavity region is scanned by moving L2, and the displacement, l 2, is defined by the displacement of L2 from the boundary of each stable region ( l 2 = 0 at the inner boundary of each region). The Ti:sapphire crystal is positioned at the beam waist generated between L1 and L2, and l 1 is the displacement of the crystal from the beam waist; positive (negative) l 1 means a shift of the crystal toward L2 (L1). As Fig. 2 shows, δ is enhanced near the inner limits in both stable regions, indicating that KLM can be achieved in both stable regions. In the first (inner) stable region, however, the enhancement of δ is confined to a very narrow region close to the inner boundary regardless of the crystal position; the range of l 2 for δ > 2 is less than 0.05 mm. This means that KLM is achieved only in the undesirable region where the laser output power is reduced and the cavity becomes sensitive to misalignment. In the second Fig. 2. δ variation over (a) the first and (b) the second stable cavity regions. d1 and d2 are 600 mm and 1000 mm, respectively, and the crystal length is 5 mm. The crystal position varied slightly around the beam waist. Fig. 3. (a) δ variation around the inner boundary of the second stable region. The crystal position was varied around the beam waist. (b) δ variation with respect to the crystal position. The numbers in the figure are the values of l 2 in the second stable region.

4 Journal of the Korean Physical Society, Vol. 46, No. 5, May 2005 Fig. 4. δ variation with respect to the crystal position. The numbers in the figure are the values of l 2. The crystal length is (a) 7 mm and (b) 3 mm. Fig. 5. δ variation with respect to the crystal position. The numbers in the figure are the values of l 2. d1 is (a) 400 mm and (b) 900 mm. The crystal length and d2 are 5 mm and 1000 mm, respectively. (outer) stable region, on the other hand, the condition for the enhancement of δ is noticeably relaxed, depending on the crystal position; at l 1 = 1.6 mm, the range of l 2 for δ > 2 is twice as wide (> 0.1 mm) as that in the first stable region. Therefore, the second stable region is a better candidate for soft-aperture KLM, and its inner boundary region deserves closer investigation. Figure 3(a) shows the dependence of the δ variation around the inner boundary on the crystal position. The figure clearly shows that the range of δ enhancement is significantly expanded only when the crystal is properly positioned. When l 1 0, i.e., the crystal is shifted toward L2 from the beam waist, the range of δ enhancement is tightly confined to just near the inner boundary. When l 1 < 1.2 mm, on the other hand, the range of δ enhancement is expanded away from the boundary, and, at l 1 = 1.6 mm, the δ enhancement is optimized. For an excessive shift of the crystal from the beam waist ( l 1 < 2.0 mm), the peak value of δ at the boundary is significantly lowered. Such a dependence of δ on the crystal position is very important in adjusting the cavity alignment for the soft-aperture KLM. Because one starts with cw operation in aligning the laser cavity, the cw output power is usually optimized first, which means that the Ti:sapphire crystal is positioned near the beam waist. With the crystal near the beam waist, however, the softaperture KLM is hardly achieved because the cavity is too close to the stable limit for the δ enhancement, as explained above. For efficient soft-aperture KLM, therefore, one needs to sacrifice the cw output power by shifting the crystal from the beam waist toward the curved mirror on the short-arm side (L1 in Fig. 1) to expand the range of δ enhancement sufficiently. The importance of the crystal position is illustrated more clearly in Fig. 3 (b) showing the δ variation with respect to the crystal position ( l 1 ) for 0.02 mm l mm. The crystal position where δ is maximized is significantly shifted ( l mm) from the beam waist, and one can have a better chance for soft-aperture KLM when the crystal is pushed toward L1 from the beam waist. The crystal length can be an important parameter in achieving efficient soft-aperture KLM because the length of the Kerr-lens medium directly influences self-focusing. To investigate the effect of the crystal length on the δ enhancement, we calculate the δ variation with respect to the crystal position for crystal lengths of 7 mm and 3 mm, and the results are shown in Fig. 4. As we intuitively expected, Fig. 4 and Fig. 3 (b) show that a longer crystal leads to a higher δ enhancement and a better chance for soft-aperture KLM. Here, it should be noted that the optimal crystal position where δ is maximized depends on the crystal length; for the 7-mm crystal, δ is maximized at l mm, and for the 3-mm crystal, δ is maximized at l mm, indicating that a longer crystal requires a bigger shift to achieve the max-

5 Numerical Analysis of Soft-Aperture Kerr-Lens Yong Woo Lee et al imal value of δ. In spite of the higher δ enhancement, a long Ti:sapphire crystal is not always good for femtosecond pulse generation because a longer crystal leads to a longer pulse duration [13 16] and the mode matching with pumping laser beam is more difficult for a longer crystal. Figure 5 shows the δ variation with respect to the crystal position for two different lengths of the short arm: d1 = 400 mm and 900 mm. The crystal length and the length of the long arm are fixed to 5 mm and 1000 mm, respectively. As Fig. 5 shows, a longer short-arm leads to a higher δ enhancement over a wider range of crystal positions. This means that the soft-aperture KLM can be more easily achieved when the lengths of the short and the long arms are comparable to each other. IV. EXPERIMENTAL DEMONSTRATION Based on the above analysis of soft-aperture KLM, we demonstrated soft-aperture KLM in a Ti:sapphire laser. The Ti:sapphire laser was composed of a 6-mmlong Brewster-cut Ti:sapphire crystal, two curved mirrors with 50-mm focal lengths, a 10 % output coupler, two fused silica prisms for dispersion compensation, and four flat mirrors. The Ti:sapphire crystal was 0.15-wt.% doped and was installed between the two curved mirrors. The folding angles at the curved mirrors were adjusted to 17 to compensate for the astigmatism caused by the Brewster-cut crystal, and the lengths of the short and the long arms were 600 and 900 mm, respectively. The two curved and the four flat mirrors (Layertec GmbH) were all chirped mirrors with a negative group-delay dispersion of 70 fs 2 over nm, and the separation between the fused silica prisms was 250 mm. We first maximized the cw output power around the inner boundary of the second stability region and shifted the Ti:sapphire crystal from the position of the maxi- mal cw output power toward the curved mirror on the short-arm side by 1.5 mm. Then, we could achieve KLM by tapping mirrors or moving prisms as pushing the cavity closer to the inner boundary of the second stability region. The cavity parameters, such as the crystal position, the separation between the curved mirrors, and the insertion of fused silica prisms, were further optimized for stable mode locking and maximal output power. The mode-locked output power was 350 mw at a 3.5-W pumping. Figure 6 shows that the output spectrum of our mode-locked Ti:sapphire laser has a spectral width of more than 170 nm. The transform-limited pulse duration calculated from the spectrum is 8.5 fs, and the pulse duration measured by using an interferometric autocorrelator was 14 fs, which is much longer than the transform-limited pulse duration due to incomplete dispersion compensation during the measurement. V. CONCLUSION We numerically analyzed soft-aperture KLM in Ti:sapphire laser cavities by using nonlinear ABCD matrices. The nonlinear ABCD matrices describing the Kerr-lensing plates where the Gaussian beam was selffocused by the Kerr effect were derived, and the powerdependent Gaussian beam mode was analyzed in linear Ti:sapphire laser cavities for soft-aperture KLM. We verified that soft-aperture KLM had a better chance of occurrence when the cavity was around the inner boundary of the second stability region. The position of the Ti:sapphire crystal was found to be a critical parameter; when the Ti:sapphire crystal was slightly shifted from the beam waist toward the curved mirror on the shortarm side, the KLM strength was significantly enhanced. The effect of the crystal length and the short-arm length on the soft-aperture KLM was analyzed, too. Based on the numerical analysis of the soft-aperture KLM, we were able to demonstrate successfully soft-aperture KLM in a Ti:sapphire laser. REFERENCES Fig. 6. Output spectrum of the soft-aperture Kerr-lens mode-locked Ti:sapphire laser. The inset is the measured interferometric autocorrelation with a pulse duration of 14 fs. [1] D. E. Spence, P. N. Kean and W. Sibbett, Opt. Lett. 16, 42 (1991). [2] M. Ramaswamy, A. S. Gouveia-Neto, D. K. Negus, J. A. Izatt and J. G. Fujimoto, Opt. Lett. 18, 1825 (1993). [3] A. Sennaroglu, C. R. Pollock and H. Nathel, Opt. Lett. 18, 826 (1993). [4] A. Miller, P. LiKamWa, B. H. T. Chai and E. W. VanStryland, Opt. Lett. 17, 195 (1992). [5] P. F. Moulton, J. Opt. Soc. Am. B 3, 125 (1986). [6] A. Baltuska, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma and R. Szipocs, Appl. Phys. B 65, 175 (1997). [7] T. Brabec, P. F. Curley, Ch. Spielmann, E. Wintner and A. J. Schmidt, J. Opt. Soc. Am. B 10, 1029 (1993).

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