Pulse shortening in the passive Q-switched lasers with intracavity stimulated Raman scattering

Transcription

1 Optics Communications 218(2003) Pulse shortening in the passive Q-switched lasers with intracavity stimulated Raman scattering V.L. Kalashnikov *,1 Institut f ur Photonik, Technische Universitaet Wien, Gusshausstrasse 27/387, A-1040 Vienna, Austria Received 18April 2002 received in revised form 19 September 2002 accepted 30 January 2003 Abstract The model of passive Q-switching is expanded on the Raman self-active lasers. The numerical optimization demonstrates a possibility of the essential pulse shortening at fundamental as well as at Stokes wavelengths. This is achieved by the control of an initial saturable absorption and a resonatorõs Q-factor at both wavelengths. The achievable pulse widths are in an order of magnitude shorter than those in the absence of Raman self-scattering even in the presence of the excited-state absorption in the saturable absorber. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Q-switching Solid-state lasers PACS: G P R 1. Introduction The diode-pumped Q-switched lasers allowing nano- and subnanosecond pulsing are of interest due to their low cost and high-peak output powers. Their applications range from spectroscopy and environment sensing to medicine. There are three main techniques of the passive Q-switching, viz., * Tel.: fax: address: v.kalashnikov@tuwien.ac.at (V.L. Kalashnikov). 1 The author is Lise Meitner Fellow at Technical University of Vienna and appreciates the support from the Austrian Science Fund (FWF, Grant No. M611). URL: an use of the modulators based on: the semiconductor-saturable absorber mirrors (SESAM) (see, for example [1 3]) the semiconductor quantum dots dispersed in the glass bulk [4] and the doped bulk crystals such as, for example, Cr 4þ :YAG [5]. The last technique is of interest due to low cost, simplicity and high damage threshold. Yb 3þ -doped active crystals excel the variety of the Q-switched laser media operating around 1 lm in smaller quantum defect, larger gain relaxation time and wider tunability region. The prominent example is Yb 3þ :KGdðWO 4 Þ 2 [6,7], which is known as the Raman active medium [8,9]. As a result, there is possible the simultaneous two-color pulsing at 1033 and 1139 nm without the need of any additional frequency conversion. To date the /03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi: /s (03)01191-x

2 148 V.L. Kalashnikov / Optics Communications 218 (2003) generation of 85 and 20 ns pulses at fundamental and Stokes wavelengths, respectively, was reported in [7]. Such two-color oscillation extends the tunability region and can find the numerous applications. In this article we present the model of Q- switching, which takes into account the stimulated Raman scattering (SRS) within the laser resonator, and report the results of the numerical optimization of the two-color oscillation. The intracavity SRS can be induced by an additional nonlinear element as well as by an active medium. As a model system, the Yb 3þ :KGdðWO 4 Þ 2 laser Q- switched by the Cr 4þ :YAG saturable absorber will be analyzed. We point the main trends of the pulse shortening at fundamental and first Stokes wavelengths, which can be useful for an elaboration of the compact sources of the subnanosecond tunable infrared pulses. 2. Model Our model is based on the rate equations approach developed in [10 12]. When the pulse duration is small enough in order to neglect the gain and loss relaxations and the inversion change induced by a pump during the pulse generation stage is negligible, the evolutions of the population inversion density in the active medium n g ðtþ and of the ground-level population density in the saturable absorber n a ðtþ obey the following rate equations: _n g ðtþ ¼ cr g c/ p ðtþn g ðtþ ð1þ _n a ðtþ ¼ qr gsa c/ p ðtþn a ðtþ where a dot denotes the partial derivative with respect to the time t c is the light velocity r g and r gsa are the gain cross-section and the absorption cross-section in the saturable absorber, respectively / p is the intracavity photon density at the fundamental wavelength c is the inversion reduction factor in the gain medium, which is equal to 2 or 1 for the three- or four-levels media, respectively q is the ratio of the beam area within active medium to that within saturable absorber. Eqs. (1) allow the solution in the form n a ðn g Þ¼n ai ðfqþ ð2þ n g n gi where f ¼ r gsa =ðcr g Þ, n ai and n gi are the initial ground-state population density in the saturable absorber and the initial inversion density in the gain medium, respectively. To describe the field evolution we have to take into account the SRS induced by some intracavity nonlinear element (for example, by the active medium) that is the main difference of our model from [10 12]. In the nanosecond time region it is possible to neglect the pulse groupdelay effects, to use the adiabatic approximation for the phonon oscillations, and to assume the exact photon phonon resonance. Then the wellknown propagation equations for the Raman medium are (see, for example [13,14] a prime denotes the partial derivative with respect to the propagation distance, a s and a p are the field amplitudes at Stokes and fundamental wavelengths, respectively Q is the phonon oscillation amplitude X is the phonon frequency T is the phonon relaxation time l is the coupling constant) a 0 s ¼ iq a p a 0 p ¼ iqa s Q þ 2 ð3þ Q T _ þ X 2 Q ¼ la p a s which result in a 0 s ¼ g 2 ja pj 2 a s a 0 p ¼ g 2 ja sj 2 a p ð4þ where g ¼ lt =X is the SRS gain coefficient (g ¼ 4:8cm/GW for Yb 3þ :KGdðWO 4 Þ 2 [15]) and ja ps j 2 denotes the intensities. The integration of Eqs. (4) and the subsequent transition to the photon densities result in / sf ¼ / pf ¼ / si ðm p / pi þ m s / si Þ m p / pi exp½ l g gchðm p / pi þ m s / si ÞŠ þ m s / si / pi ðm p / pi þ m s / si Þ m s / si exp½l g gchðm p / pi þ m s / si ÞŠ þ m p / pi : ð5þ

3 Here / si and / sf (/ pi and / pf ) are the photon densities before and, respectively, after propagation within the active crystal at the Stokes (fundamental) wavelength, m s and m p are the Stokes and fundamental wavelengths, respectively, l g is the active crystal length and h is the PlanckÕs constant. Eqs. (5) describe the nonlinear Raman gain (loss) for the Stokes (fundamental) component. Later on, it is convenient to use the following dimensionless quantities: V.L. Kalashnikov / Optics Communications 218 (2003) _U p ¼ U p N _U s ¼ U s N Ny 1 N þð1 d p Þ ln ð1 y aq Þ T0 2 U p þ NU s þ U p þ NU s exp½gðu p þ NU s ÞŠ 1 1 N s þð1 d s Þj ln ð1 y aq Þ T0 2 U p þ NU s þ 1 U p exp½ GðU p þ NU s ÞŠ þ NU s _y ¼ cl cavu p l g N ð7þ U ps ¼ 2r g l g / ps s ¼ 2 r gl g n gi t t cav N ¼ m s v ¼ l cav m p l g G ¼ 1 gchm p 2 r g 1 1 N ¼ ln þ ln þ L T0 2 p R p 1 1 N s ¼ j ln þ ln þ L s T 2 0 R s y ¼ n g n gi ¼ 1 N : n gi 2 r g l g ð6þ where d p ¼ r esa ðm p Þ=r gsa ðm p Þ, d s ¼ r esa ðm s Þ=r gsa ðm p Þ, r esa ðm p Þ and r esa ðm s Þ are the excited-state absorption cross-sections at the fundamental and Stokes wavelengths, respectively. A dot denotes the partial derivative with respect to the dimensionless time parameter s, the dimensionless densities of the photons and of the inversion in the active medium are functions of s. a ¼ r gsa ðm p Þ=cr g and we suppose the absence of the Stokes componentõs contribution to the loss saturation. The system (7) can be solved only numerically on the basis of the fourth-order Runge Kutta method, for example. The obtained numerical results are presented in Section 3. Here T 0 ¼ expð r gsa n ai l a Þ is the initial transmission of the saturable absorber defining the maximum modulation depth of passive Q- switching the initial population n ai is equal to the concentration of the absorption ions, l a is the absorber width, R p and R s are the reflection coefficients of the output mirror at fundamental and Stokes wavelengths, respectively, L p and L s are the linear intracavity losses at the fundamental and Stokes wavelengths, respectively, j ¼ r gsa ðm s Þ=r gsa ðm p Þ (this is 0.38for the Yb 3þ :KGdðWO 4 Þ 2 laser) and l cav and t cav are the cavity length and period, respectively. It is evident, that the value of the initial inversion n gi is defined by the small signal netloss at the fundamental wavelength (see the last equation from Eqs. (6)). The resulting dynamical system, which takes into account the excited-state absorption [12], has the following form [16]: 3. Numerical results and discussion The principle of pulse shortening due to the SRS is quite simple (Fig. 1). The growth of the field power at the fundamental wavelength (solid curve) resulting from the loss saturation amplifies the scattered field at the Stokes wavelength (dashed curve). The exponentially growing scattered field depletes the field at the fundamental wavelength and thereby shortens the fundamental pulse tail. As a result, the duration of the pulses at both wavelengths can be essentially shorter than the pulse duration in the absence of the SRS. Our goal is the numerical analysis of the pulse shortening at fundamental as well as Stokes wavelengths. As an example, we shall consider the Yb 3þ :KGdðWO 4 Þ 2 laser with the Cr 4þ :YAG saturable absorber. The fundamental and Stokes wavelengths in this case are 1033 and 1139 nm,

4 150 V.L. Kalashnikov / Optics Communications 218 (2003) (a) (b) Fig. 1. The logarithms of the dimensionless photon densities at the fundamental (solid curve) and Stokes (dashed curve) wavelengths in the dependence on the dimensionless time parameter. The initial photon density is v ¼ 7, q ¼ 1, T 0 ¼ 0:81, R p ¼ 0:82, R s ¼ 0:33, L p ¼ L s ¼ 0:05, a ¼ 178:57, N ¼ 0:91, c ¼ 1, d p ¼ d s ¼ 0 and G ¼ 512:68. respectively. It is natural, that the optimization procedures differ for these wavelengths. Therefore, at first, we consider the fundamental pulse shortening. Fig. 2 shows the dependencies of the dimensionless minimal pulse width and the corresponding R p and R s on the initial transmission of the saturable absorber at the fundamental wavelength. We consider q and v as the varied parameters. The former can be increased by the field focusing in the absorber. The latter decreases with the diminishing l cav =l g -ratio. The solid curve demonstrates the hypothetic situation of the vanishing SRS. For the chosen q ¼ 1 and v ¼ 15 the dimensionless pulse width increases from 8.5 to 21 for T 0 growing from 0.5 to 0.95 (Fig. 2(a)). R p producing the minimum pulse width for the given T 0 is shown in Fig. 2(b). The transition to the dimensional pulse width results from the multiplication by t cav =N. For generality sake, we do not fix the laser configuration and suppose that the transition to the dimensional quantities can be easily made by a reader. The SRS changes the situation dramatically. When the initial transmission of the absorber is (c) Fig. 2. The minimum dimensionless pulse width at the fundamental wavelength (a) and the corresponding output coupler reflectivity at the fundamental (b) and Stokes (c) wavelengths versus the initial transmission of the saturable absorber. q ¼ 1 (solid, dashed and dashed-dotted curves), 10 (dotted) v ¼ 29 (dashed, dotted), 15 (solid, dashed-dotted). G ¼ 0 for the solid curves. Other parameters correspond to Fig. 1. small (this requires a large pump to exceed the lasing threshold), the SRS increases the pulse width (Fig. 2(a) broken curves in the comparison with the solid one). This effect has a simple explanation. The saturation of the large initial loss produces the extremely high power at the fundamental wavelength (Q-switching for the large modulation depths). As a result, the SRS begins already on the pulse front and depletes the laser field in the vicinity of the pulse maximum. This increases the pulse width like the action of the passive negative feedback. Hence the fundamental pulse width minimization can be achieved only by the SRS suppression due to R s! 0 (Fig. 2(c)). Additionally, the effect of the nonlinear depletion can be reduced by the R p increase (Fig. 2(b), the

5 V.L. Kalashnikov / Optics Communications 218 (2003) region in the vicinity of T 0 ¼ 0:2). In any case, the pulse width exceeds that in the absence of SRS. The T 0 growth decreases the fundamental field power due to the modulation depth decrease. As a result, the stimulated scattering efficiency, which depends on ja p j 2, decreases, too. Hence, the front of the exponentially growing Stokes field moves to the pulse tail (Fig. 1). This sharpens the fundamental pulse and decreases its width (see Fig. 2(a)). For T 0 0:4 the delay of the SRS pulse relatively the fundamental pulse can be achieved efficiently by the power reduction due to the R p decrease (Fig. 2(b)). However, the further power decrease produced by the T 0 growth needs the R s increase (Fig. 2(c), T 0 > 0:45) in order to enlarge the SRS efficiency. The stimulated scattering efficiency increases due to the ja s j 2 growth for R s! 1. So, the low threshold operation (T 0! 1) needs R s! 1 and R p! 1 for the pulse width minimization. The characteristic feature of the considered regime is the existence of a wide plateau in the pulse width dependence within the interval of T 0 0:45 0:85. The pulse width here is smaller than that in the absence of SRS. The abrupt pulse width growth after T 0 0:9 is caused by the SRS reduction due to the power decrease at the fundamental wavelength. The latter is a result of the vanishing modulation depth (T 0 1). The shift of the minimum pulse width region toward T 0 1 can be achieved by the mode focusing within the absorber (the q growth, transition from dashed to dotted curves in Fig. 2) or by the resonator shortening (the v decrease, transition from dashed to dash-dotted curves in Fig. 2). The most essential pulse shortening can be achieved for the Stokes pulse (Fig. 3(a)). As it is seen, there is the deep pulse width minimum for T 0 0:7 0:8, which tends toward the larger initial saturable absorber transmissions as a result of the q increase or the v decrease (transition from solid to dashed and dotted curves, respectively). The achievable pulse durations are in an order of magnitude shorter than those in the absence of the SRS and can be made shorter 100 ps already for l cav < 12 mm. The control of the pulse shortening can be realized by the variation of R s (Fig. 3(c)). When the delay between pulses at fundamental and Stokes (a) (b) (c) Fig. 3. The minimum dimensionless pulse width at the Stokes wavelength (a) and the corresponding output coupler reflectivity at the fundamental (b) and Stokes (c) wavelengths versus the initial transmission of the saturable absorber. q ¼ 1 (solid and dotted curves), 10 (dashed) v ¼ 29 (solid, dashed), 15 (dotted). Other parameters correspond to Fig. 1. wavelengths is small, the fundamental field amplifies the Stokes pulseõs tail. As a result, the width of the latter is large. However, the loss growth for the Stokes component retards the Stokes pulse formation and thereby reduces its tail and decreases its duration. This is achieved by R s! 0 for small T 0. The T 0 growth reduces the fundamental pulse power (see above) and, as a result, retards the Stokes pulse formation. This requires the stimulated scattering efficiency growth in order to reduce the pulse duration. Such growth is caused by the R s increase (i.e., by the ja s j 2 increase). The optimal value of R p in the wide region of T 0 is close to 1 (Fig. 3(b)). Now let us briefly consider the contribution of the excited-state absorption. This contribution can reduce the pulse power and thereby suppress the SRS.

6 152 V.L. Kalashnikov / Optics Communications 218 (2003) (a) (b) (c) Fig. 4. The minimum dimensionless pulse width at the fundamental (solid and dashed curves) and Stokes (dotted) wavelengths (a) and the corresponding output coupler reflectivity at the fundamental (b) and Stokes (c) wavelengths versus the initial transmission of the saturable absorber. q ¼ 1, v ¼ 15, d p ¼ d s ¼ 0:44. G ¼ 0 for the solid curves. Other parameters correspond to Fig. 1. As a rule, the measured value of r esa for Cr 4þ :YAG is scattered within the wide range. We use the data presented in [17,18]. r esa increases within nm and amounts to cm 2 [17]. Then there is the pronounced decrease of the excited-state absorption in the region >1100 nm [18], but we will use its highest value to stress the main tendencies. Then d p ¼ d s ¼ 0:44 and Fig. 4 presents the results of the pulse width optimization. There exists the pronounced increase of the pulse width in the absence of the SRS (solid curve in Fig. 4(a) in the comparison with solid curve in Fig. 2(a)), which is accompanied with some increase of the optimal R p (solid curve in Fig. 4(b) in the comparison with solid curve in Fig. 2(b)). The SRS allows the pulse shortening up to 70% at the fundamental wavelength, but the minimum pulse width can be achieved by the cost of the T 0 decrease (dashed curve in Fig. 4(a)). The delay of the Stokes pulse relatively the fundamental pulse, which is necessary for the pulse shortening (see above), results from the R p decrease for the small T 0 (dashed curve in Fig. 4(b)). As it was in the absence of the excited-state absorption for T 0 > 0:7 (dashed-dotted curve in Fig. 2(b)), in our case of r esa 6¼ 0 the fundamental pulse power decrease causing the SRS suppression can be compensated by the additional R p increase for T 0 > 0:5 (dashed curve in Fig. 4(b)). To enhance the SRS for T 0 > 0:3 (when ja p j 2 decreases due to the modulation depth decrease) the R s growth is necessary (dashed curve in Fig. 4(c)). This is smaller value than that in the absence of excited-state absorption (dashed-dotted curve in Fig. 2(c)). However, in spite of the moderate decrease of the pulse width at the fundamental wavelength, the minimum Stokes pulse width is noticeably small (dotted curve in Fig. 4(a)). Nevertheless the pulse width minimum corresponds to the smaller T 0 (compare the dotted curves in Figs. 4(a) and 3(a)). The optimal R p is close to 1 (dotted line in Fig. 4(b)) and, as it was in the absence of the excitedstate absorption, there is a switching between small and large optimal R s as a consequence of the T 0 growth (dotted curve in Fig. 4(c)). This switching takes a place for the smaller T 0 (compare with dotted curve in Fig. 3(c)). The common feature of the considered Q- switching regimes is the contraction of the SRS generation region for the growing T 0 as a result of the ja p j 2 decrease. The SRS generation becomes possible for R p approaching to 1, i.e., there exists some minimum output coupler reflectivity at the fundamental wavelength providing the SRS. The short pulses at the Stokes wavelength (dasheddotted curve in Fig. 4(a)) can be obtained not only for R p 1 (dotted line in Fig. 4(b)) but also in the vicinity of this minimal R p (dashed-dotted curve in Fig. 4(b)). This regime can be usable because it provides also the output at the fundamental wavelength. So, we can see that the SRS allows the pronounced pulse shortening at the fundamental

7 V.L. Kalashnikov / Optics Communications 218 (2003) wavelength and the essential pulse shortening at the Stokes wavelength in the presence of the excitedstate absorption as well. 4. Conclusion The intracavity SRS plays a role of the strong negative passive feedback for the Q-switched laser. For the relatively small initial transmission of the saturable absorber providing the highest intracavity powers, the SRS begins already on the front of the laser pulse. The resulting depletion of the fundamental field in the vicinity of the pulse peak results in its widening. In this case, the pulse width minimization is provided by the decrease of the resonatorõs Q-factor at the Stokes wavelength due to R s! 0. However, the increase of the initial absorberõs transmission resulting in the power decrease shifts the SRS pulse toward the tail of the fundamental pulse. As a result of the fundamental pulse tail depletion, its width shortens. The Stokes pulse width can be shortened in an order of magnitude. This can be achieved by the appropriate increase of the resonatorõs Q-factors at fundamental and Stokes wavelengths. The additional pulse shortening for the relatively large absorberõs transmission can be obtained due to the field focusing in the absorber and the cavity shortening relatively the active medium length. The excitedstate absorption hinders from the pulse shortening and shifts the minimum of the pulse width toward the smaller initial transmission of the saturable absorber. However, the essential pulse shortening at the Raman wavelength is possible in this case as well. We suppose that the presented model and the numerical results can be useful for an elaboration of the compact passively Q-switched two-color lasers. References [1] G.J. Spuhler, R. Paschotta, R. Fluck, B. Braun, M. Moser, G. Zhang, E. Gini, U. Keller, J. Opt. Soc. Am. B 16 (1999) 376. [2] R. Fluck, B. Braun, U. Keller, E. Gini, H. Melchior, in: C.R. Pollock, W.R. Bosenberg (Eds.), Advanced Solid-State Lasers, OSA Trends in Optics and Photonic Series, vol. 10, Optical Society of America, Washington, DC, 1997, p [3] G.J. Spuhler, R. Pashotta, M.P. Kullberg, M. Graf, M. Moser, U. Keller, L.R. Brovelli, C. Harder, E. Mix, G. Huber, in: M.M. Fejer, H. Injeyan, U. Keller (Eds.), Advanced Solid-State Lasers, OSA Trends in Optics and Photonic Series, vol. 26, Optical Society of America, Washington, DC, 1999, p [4] A.M. Malyarevich, V.G. Savitski, P.V. Prokoshin, N.N. Posnov, K.V. Yumashev, E. Raaben, A.A. Zhilin, J. Opt. Soc. Am. B 19 (2002) 28. [5] I.J. Miller, A.J. Alcock, J.E. Bernard, in: OSA Proc. Advanced Solid-State Lasers, vol. 13, 1992, p [6] N.V. Kuleshov, A.A. Lagatsky, V.G. Shcherbitsky, V.P. Mikhailov, E. Heumann, T. Jensen, A. Diening, G. Huber, Appl. Phys. B 64 (1997) 409. [7] A.A. Lagatsky, A. Abdolvand, N.V. Kuleshov, Opt. Lett. 25 (2000) 616. [8] A.M. Ivanyuk, M.A. Ter-Pogosyan, P.A. Shaverdov, V.D. Belyaev, V.L. Ermolaev, N.P. Tikhonov, Opt. Spectrosc. 59 (1985) 572. [9] K. Andryunas, Y. Vishchakas, V. Kabelka, I.V. Mochalov, A.A. Pavlyuk, G.T. Petrovskij, V. Syrus, JETP Lett. 42 (1985) 410. [10] J.J. Degnan, IEEE J. Quantum Electron. 25 (1989) 214. [11] J.J. Zaykowski, P.L. Kelly, IEEE J. Quantum Electron. 27 (1991) [12] X. Zhang, S. Zhao, Q. Wang, Q. Zhang, L. Sun, S. Zhang, IEEE J. Quantum Electron. 33 (1997) [13] H.A. Haus, I. Sorokina, E. Sorokin, J. Opt. Soc. Am. B 15 (1998) 223. [14] V.L. Kalashnikov, E. Sorokin, I.T. Sorokina, J. Opt. Soc. Am. B 18(2001) [15] P.G. Zverev, T.T. Basiev, A.A. Sobol, V.V. Skornyakov, L.I. Ivleva, N.M. Polozkov, V.V. Osiko, Quantum Electron. 30 (2000) 55. [16] V.L. Kalashnikov, [17] Z. Burshtein, P. Blau, Y. Kalisky, Y. Shimony, M.R. Kokta, IEEE J. Quantum Electron. 34 (1998) 292. [18] S. K uck, K.L. Schepler, S. Hartung, K. Petermann, G. Huber, J. Luninescence (1997) 222.