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Pulse shortening in the passive Q-switched lasers with intracavity stimulated Raman scattering
Optics Communications 218 (2003) 147–153 www.elsevier.com/locate/optcom
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 18 April 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: 42.60.G; 42.55.P; 42.55.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.: +43-158-8013-8723; fax: +43-158-8013-8799. E-mail address: [email protected] (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: http://www.geocities.com/optomaplev.
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, Cr4þ :YAG [5]. The last technique is of interest due to low cost, simplicity and high damage threshold. Yb3þ -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 Yb3þ :KGdðWO4 Þ2 [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
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01191-X
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V.L. Kalashnikov / Optics Communications 218 (2003) 147–153
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 Qswitching, 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 Yb3þ :KGdðWO4 Þ2 laser Qswitched by the Cr4þ :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 ng ðtÞ and of the ground-level population density in the saturable absorber na ðtÞ obey the following rate equations: n_ g ðtÞ ¼ crg c/p ðtÞng ðtÞ; n_ a ðtÞ ¼ qrgsa c/p ðtÞna ðtÞ;
ð1Þ
where a dot denotes the partial derivative with respect to the time t; c is the light velocity; rg and rgsa 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
na ðng Þ ¼ na;i
ng ng;i
ðfqÞ ;
ð2Þ
where f ¼ rgsa =ðcrg Þ, na;i and ng;i 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, as and ap 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) a0s ¼ iQ ap ; a0p ¼ iQas ;
ð3Þ
€ þ 2 Q_ þ X2 Q ¼ lap a ; Q s T which result in g 2 a0s ¼ jap j as ; 2 g 2 a0p ¼ jas j ap ; 2
ð4Þ
where g ¼ lT =X is the SRS gain coefficient (g ¼ 4:8 cm/GW for Yb3þ :KGdðWO4 Þ2 [15]) and jap;s j2 denotes the intensities. The integration of Eqs. (4) and the subsequent transition to the photon densities result in /s;f ¼
/s;i ðmp /p;i þ ms /s;i Þ ; mp /p;i exp½lg gchðmp /p;i þ ms /s;i Þ þ ms /s;i
/p;f ¼
/p;i ðmp /p;i þ ms /s;i Þ : ms /s;i exp½lg gchðmp /p;i þ ms /s;i Þ þ mp /p;i ð5Þ
V.L. Kalashnikov / Optics Communications 218 (2003) 147–153
Here /s;i and /s;f (/p;i and /p;f ) are the photon densities before and, respectively, after propagation within the active crystal at the Stokes (fundamental) wavelength, ms and mp are the Stokes and fundamental wavelengths, respectively, lg 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:
149
Up 1 Ny N þ ð1 dp Þ ln ð1 y aq Þ U_ p ¼ N T02 Up þ N Us þ 1 ; Up þ N Us exp½GðUp þ N Us Þ
Us 1 U_ s ¼ ð1 y aq Þ Ns þ ð1 ds Þj ln T02 N Up þ N Us þ 1 ; Up exp½GðUp þ N Us Þ þ N Us clcav Up ; y_ ¼ lg N ð7Þ
Up;s ¼ 2rg lg /p;s ;
rg lg ng;i t s¼2 ; tcav
ms lcav ; v¼ ; mp lg 1 gchmp G¼ ; 2 rg 1 1 N ¼ ln þ ln þ Lp ; 2 Rp T0 1 1 Ns ¼ j ln þ ln þ Ls ; 2 T0 Rs ng 1 N y¼ ; ng;i ¼ : 2 rg l g ng;i N¼
ð6Þ
Here T0 ¼ expðrgsa na;i la Þ is the initial transmission of the saturable absorber defining the maximum modulation depth of passive Qswitching; the initial population na;i is equal to the concentration of the absorption ions, la is the absorber width, Rp and Rs are the reflection coefficients of the output mirror at fundamental and Stokes wavelengths, respectively, Lp and Ls are the linear intracavity losses at the fundamental and Stokes wavelengths, respectively, j ¼ rgsa ðms Þ=rgsa ðmp Þ (this is 0.38 for the Yb3þ :KGdðWO4 Þ2 laser) and lcav and tcav are the cavity length and period, respectively. It is evident, that the value of the initial inversion ng;i 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]:
where dp ¼ resa ðmp Þ=rgsa ðmp Þ, ds ¼ resa ðms Þ=rgsa ðmp Þ, resa ðmp Þ and resa ðms 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 rgsa ðmp Þ=crg 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.
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 Yb3þ :KGdðWO4 Þ2 laser with the Cr4þ :YAG saturable absorber. The fundamental and Stokes wavelengths in this case are 1033 and 1139 nm,
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(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 107 . v ¼ 7, q ¼ 1, T0 ¼ 0:81, Rp ¼ 0:82, Rs ¼ 0:33, Lp ¼ Ls ¼ 0:05, a ¼ 178:57, N ¼ 0:91, c ¼ 1, dp ¼ ds 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 Rp and Rs 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 lcav =lg -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 T0 growing from 0.5 to 0.95 (Fig. 2(a)). Rp producing the minimum pulse width for the given T0 is shown in Fig. 2(b). The transition to the dimensional pulse width results from the multiplication by tcav =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 Rs ! 0 (Fig. 2(c)). Additionally, the effect of the nonlinear depletion can be reduced by the Rp increase (Fig. 2(b), the
V.L. Kalashnikov / Optics Communications 218 (2003) 147–153
region in the vicinity of T0 ¼ 0:2). In any case, the pulse width exceeds that in the absence of SRS. The T0 growth decreases the fundamental field power due to the modulation depth decrease. As a result, the stimulated scattering efficiency, which 2 depends on jap j , 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 T0 0:4 the delay of the SRS pulse relatively the fundamental pulse can be achieved efficiently by the power reduction due to the Rp decrease (Fig. 2(b)). However, the further power decrease produced by the T0 growth needs the Rs increase (Fig. 2(c), T0 > 0:45) in order to enlarge the SRS efficiency. The stimulated scattering efficiency in2 creases due to the jas j growth for Rs ! 1. So, the low threshold operation (T0 ! 1) needs Rs ! 1 and Rp ! 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 T0 0:45–0:85. The pulse width here is smaller than that in the absence of SRS. The abrupt pulse width growth after T0 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 (T0 1). The shift of the minimum pulse width region toward T0 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 T0 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 lcav < 12 mm. The control of the pulse shortening can be realized by the variation of Rs (Fig. 3(c)). When the delay between pulses at fundamental and Stokes
151
(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 Rs ! 0 for small T0 . The T0 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 Rs increase (i.e., by the jas j2 increase). The optimal value of Rp in the wide region of T0 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.
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V.L. Kalashnikov / Optics Communications 218 (2003) 147–153
(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, dp ¼ ds ¼ 0:44. G ¼ 0 for the solid curves. Other parameters correspond to Fig. 1.
As a rule, the measured value of resa for Cr4þ :YAG is scattered within the wide range. We use the data presented in [17,18]. resa increases within 920–1064 nm and amounts to 2 1018 cm2 [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 dp ¼ ds 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 Rp (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 T0 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 Rp decrease for the small T0 (dashed curve in Fig. 4(b)). As it was in the absence of the excited-state absorption for T0 > 0:7 (dashed-dotted curve in Fig. 2(b)), in our case of resa 6¼ 0 the fundamental pulse power decrease causing the SRS suppression can be compensated by the additional Rp increase for T0 > 0:5 (dashed curve in Fig. 4(b)). To enhance the SRS for T0 > 0:3 (when jap j2 decreases due to the modulation depth decrease) the Rs 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 T0 (compare the dotted curves in Figs. 4(a) and 3(a)). The optimal Rp 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 Rs as a consequence of the T0 growth (dotted curve in Fig. 4(c)). This switching takes a place for the smaller T0 (compare with dotted curve in Fig. 3(c)). The common feature of the considered Qswitching regimes is the contraction of the SRS generation region for the growing T0 as a result of 2 the jap j decrease. The SRS generation becomes possible for Rp 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 Rp 1 (dotted line in Fig. 4(b)) but also in the vicinity of this minimal Rp (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
V.L. Kalashnikov / Optics Communications 218 (2003) 147–153
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 Rs ! 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.
153
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. 141. [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. 187. [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. 322. [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) 2220. [12] X. Zhang, S. Zhao, Q. Wang, Q. Zhang, L. Sun, S. Zhang, IEEE J. Quantum Electron. 33 (1997) 2286. [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) 1732. [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, http://xxx.lanl.gov/physics/0009056. [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 72–74 (1997) 222.
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 18 April 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: 42.60.G; 42.55.P; 42.55.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.: +43-158-8013-8723; fax: +43-158-8013-8799. E-mail address: [email protected] (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: http://www.geocities.com/optomaplev.
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, Cr4þ :YAG [5]. The last technique is of interest due to low cost, simplicity and high damage threshold. Yb3þ -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 Yb3þ :KGdðWO4 Þ2 [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
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01191-X
148
V.L. Kalashnikov / Optics Communications 218 (2003) 147–153
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 Qswitching, 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 Yb3þ :KGdðWO4 Þ2 laser Qswitched by the Cr4þ :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 ng ðtÞ and of the ground-level population density in the saturable absorber na ðtÞ obey the following rate equations: n_ g ðtÞ ¼ crg c/p ðtÞng ðtÞ; n_ a ðtÞ ¼ qrgsa c/p ðtÞna ðtÞ;
ð1Þ
where a dot denotes the partial derivative with respect to the time t; c is the light velocity; rg and rgsa 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
na ðng Þ ¼ na;i
ng ng;i
ðfqÞ ;
ð2Þ
where f ¼ rgsa =ðcrg Þ, na;i and ng;i 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, as and ap 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) a0s ¼ iQ ap ; a0p ¼ iQas ;
ð3Þ
€ þ 2 Q_ þ X2 Q ¼ lap a ; Q s T which result in g 2 a0s ¼ jap j as ; 2 g 2 a0p ¼ jas j ap ; 2
ð4Þ
where g ¼ lT =X is the SRS gain coefficient (g ¼ 4:8 cm/GW for Yb3þ :KGdðWO4 Þ2 [15]) and jap;s j2 denotes the intensities. The integration of Eqs. (4) and the subsequent transition to the photon densities result in /s;f ¼
/s;i ðmp /p;i þ ms /s;i Þ ; mp /p;i exp½lg gchðmp /p;i þ ms /s;i Þ þ ms /s;i
/p;f ¼
/p;i ðmp /p;i þ ms /s;i Þ : ms /s;i exp½lg gchðmp /p;i þ ms /s;i Þ þ mp /p;i ð5Þ
V.L. Kalashnikov / Optics Communications 218 (2003) 147–153
Here /s;i and /s;f (/p;i and /p;f ) are the photon densities before and, respectively, after propagation within the active crystal at the Stokes (fundamental) wavelength, ms and mp are the Stokes and fundamental wavelengths, respectively, lg 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:
149
Up 1 Ny N þ ð1 dp Þ ln ð1 y aq Þ U_ p ¼ N T02 Up þ N Us þ 1 ; Up þ N Us exp½GðUp þ N Us Þ
Us 1 U_ s ¼ ð1 y aq Þ Ns þ ð1 ds Þj ln T02 N Up þ N Us þ 1 ; Up exp½GðUp þ N Us Þ þ N Us clcav Up ; y_ ¼ lg N ð7Þ
Up;s ¼ 2rg lg /p;s ;
rg lg ng;i t s¼2 ; tcav
ms lcav ; v¼ ; mp lg 1 gchmp G¼ ; 2 rg 1 1 N ¼ ln þ ln þ Lp ; 2 Rp T0 1 1 Ns ¼ j ln þ ln þ Ls ; 2 T0 Rs ng 1 N y¼ ; ng;i ¼ : 2 rg l g ng;i N¼
ð6Þ
Here T0 ¼ expðrgsa na;i la Þ is the initial transmission of the saturable absorber defining the maximum modulation depth of passive Qswitching; the initial population na;i is equal to the concentration of the absorption ions, la is the absorber width, Rp and Rs are the reflection coefficients of the output mirror at fundamental and Stokes wavelengths, respectively, Lp and Ls are the linear intracavity losses at the fundamental and Stokes wavelengths, respectively, j ¼ rgsa ðms Þ=rgsa ðmp Þ (this is 0.38 for the Yb3þ :KGdðWO4 Þ2 laser) and lcav and tcav are the cavity length and period, respectively. It is evident, that the value of the initial inversion ng;i 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]:
where dp ¼ resa ðmp Þ=rgsa ðmp Þ, ds ¼ resa ðms Þ=rgsa ðmp Þ, resa ðmp Þ and resa ðms 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 rgsa ðmp Þ=crg 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.
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 Yb3þ :KGdðWO4 Þ2 laser with the Cr4þ :YAG saturable absorber. The fundamental and Stokes wavelengths in this case are 1033 and 1139 nm,
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(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 107 . v ¼ 7, q ¼ 1, T0 ¼ 0:81, Rp ¼ 0:82, Rs ¼ 0:33, Lp ¼ Ls ¼ 0:05, a ¼ 178:57, N ¼ 0:91, c ¼ 1, dp ¼ ds 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 Rp and Rs 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 lcav =lg -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 T0 growing from 0.5 to 0.95 (Fig. 2(a)). Rp producing the minimum pulse width for the given T0 is shown in Fig. 2(b). The transition to the dimensional pulse width results from the multiplication by tcav =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 Rs ! 0 (Fig. 2(c)). Additionally, the effect of the nonlinear depletion can be reduced by the Rp increase (Fig. 2(b), the
V.L. Kalashnikov / Optics Communications 218 (2003) 147–153
region in the vicinity of T0 ¼ 0:2). In any case, the pulse width exceeds that in the absence of SRS. The T0 growth decreases the fundamental field power due to the modulation depth decrease. As a result, the stimulated scattering efficiency, which 2 depends on jap j , 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 T0 0:4 the delay of the SRS pulse relatively the fundamental pulse can be achieved efficiently by the power reduction due to the Rp decrease (Fig. 2(b)). However, the further power decrease produced by the T0 growth needs the Rs increase (Fig. 2(c), T0 > 0:45) in order to enlarge the SRS efficiency. The stimulated scattering efficiency in2 creases due to the jas j growth for Rs ! 1. So, the low threshold operation (T0 ! 1) needs Rs ! 1 and Rp ! 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 T0 0:45–0:85. The pulse width here is smaller than that in the absence of SRS. The abrupt pulse width growth after T0 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 (T0 1). The shift of the minimum pulse width region toward T0 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 T0 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 lcav < 12 mm. The control of the pulse shortening can be realized by the variation of Rs (Fig. 3(c)). When the delay between pulses at fundamental and Stokes
151
(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 Rs ! 0 for small T0 . The T0 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 Rs increase (i.e., by the jas j2 increase). The optimal value of Rp in the wide region of T0 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.
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(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, dp ¼ ds ¼ 0:44. G ¼ 0 for the solid curves. Other parameters correspond to Fig. 1.
As a rule, the measured value of resa for Cr4þ :YAG is scattered within the wide range. We use the data presented in [17,18]. resa increases within 920–1064 nm and amounts to 2 1018 cm2 [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 dp ¼ ds 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 Rp (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 T0 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 Rp decrease for the small T0 (dashed curve in Fig. 4(b)). As it was in the absence of the excited-state absorption for T0 > 0:7 (dashed-dotted curve in Fig. 2(b)), in our case of resa 6¼ 0 the fundamental pulse power decrease causing the SRS suppression can be compensated by the additional Rp increase for T0 > 0:5 (dashed curve in Fig. 4(b)). To enhance the SRS for T0 > 0:3 (when jap j2 decreases due to the modulation depth decrease) the Rs 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 T0 (compare the dotted curves in Figs. 4(a) and 3(a)). The optimal Rp 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 Rs as a consequence of the T0 growth (dotted curve in Fig. 4(c)). This switching takes a place for the smaller T0 (compare with dotted curve in Fig. 3(c)). The common feature of the considered Qswitching regimes is the contraction of the SRS generation region for the growing T0 as a result of 2 the jap j decrease. The SRS generation becomes possible for Rp 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 Rp 1 (dotted line in Fig. 4(b)) but also in the vicinity of this minimal Rp (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
V.L. Kalashnikov / Optics Communications 218 (2003) 147–153
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 Rs ! 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.
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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. 141. [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. 187. [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. 322. [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) 2220. [12] X. Zhang, S. Zhao, Q. Wang, Q. Zhang, L. Sun, S. Zhang, IEEE J. Quantum Electron. 33 (1997) 2286. [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) 1732. [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, http://xxx.lanl.gov/physics/0009056. [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 72–74 (1997) 222.