Analysis of a large-mode neodymium laser passively Q switched with a saturable absorber and a stimulated-brillouin-scattering mirror

Kir yanov et al. Vol. 17, No. 1/January 000/J. Opt. Soc. Am. B 11 Analysis of a large-mode neodymium laser passively Q switched with a saturable absorber and a stimulated-brillouin-scattering mirror Alexander V. Kir yanov* and Vicente Aboites Centro de Investigaciones en Optica A.C., Apartado Postal 948 León, 37000, Guanajuato Mexico Nikolai N. Il ichev General Physics Institute of Russian Academy of Sciences, Vavilov Street 38, Moscow 11794, Russian Federation Received April 16, 1999 A neodymium laser passively Q switched with a saturable absorber and a stimulated-brillouin-scattering mirror is numerically studied. An explanation is given for the laser beam spot-size widening. It is shown that this phenomenon is, most probably, due to nonperfect phase conjugation that occurs during the stimulated- Brillouin-scattering process through switching by intracavity radiation. The results of numerical simulation of the laser that account for this phenomenon are compared with experiment. It is demonstrated that the calculated output energy and pulse duration are in good agreement with those measured experimentally. 000 Optical Society of America [S0740-34(99)0091-1] OCIS codes: 140.3530, 140.3540, 140.3410, 90.5900, 190.5940. 1. INTRODUCTION In recent years, the technique of phase conjugation (PC) based on stimulated Brillouin scattering (SBS) has been widely used in solid-state laser engineering for radiation divergency enhancement and output power increase. 1 In particular, a few schemes have been developed that exploit the SBS process in powerful Q-switched neodymium (1.06-m-wavelength) lasers either as a single switching mechanism or as an additional one, when Q switching is started by a passive saturable absorber (SA). 3,4 Despite a general understanding of the processes taking place in lasers of this type and the availability of wellreproducible experimental data, there are no up-to-date models describing lasing in the cavity that contains a passive Q switch and a SBS cell. In addition, it is not clear why such a laser oscillates in the large-size mode regime when diffraction-limited radiation fills virtually the entire aperture of the active medium (AM). The present work is an attempt to give an explanation for this phenomenon. Presented below are the results of numerical simulations of the process in which a Gaussian beam (GB) widens after reflection from a SBS mirror (Section ), a model of a passively Q-switched laser with a SA and a SBS mirror (the model considers the laser beam s widening to be due to SBS reflection; Section 3), and a comparison of the model results with experiments (Section 4).. GAUSSIAN BEAM WIDENING AFTER REFLECTION FROM A STIMULATED- BRILLOUIN-SCATTERING MIRROR Schematically, the PC phenomenon is sketched in Fig. 1(a). Assuming an incident plane wave, after reflection from an ideal phase-conjugating mirror the Gaussian beam s curvature radius R r R i, and its spot size W r W i (the subscripts i and r correspond to the incident and the reflected waves, respectively). However, in a nonideal case, for the reflection of a GB, the reflected beam s curvature radius R r R i, whereas for its spot size we have W r W i, where 1 (nonperfect PC). This effect was first analyzed in Ref. 5, which contains an analytical solution only for the linear scattering regime. Reference 6 gives an approximate evaluation for this relative beam-waist parameter, 0., and reveals the output beam divergency to be close to that of the incidence. Up until now, various evaluations for the parameter have been made, such as 0., 6,7 0.5 1, 8 and 1. 9 One of our aims was to obtain a numerical solution to the problem of laser beam widening after reflection from a SBS mirror. We solved the transcendental equation 10 GIL D lnr sbs1 r sbs expd 1 r sbs D lnr sbs1 r sbs 1 r sbs (1) for calculating the SBS reflection coefficient r sbs, assuming GB incidence: I I 0 exp x, () W i 0740-34/000/010011-07$15.00 000 Optical Society of America

1 J. Opt. Soc. Am. B/Vol. 17, No. 1/January 000 Kir yanov et al. Fig. 1. (a) Schematic showing widening of a GB after reflection from a SBS mirror. W i and W r are the beam spot sizes in the caustics, W 0 and W 1 are those at the focusing lens, and b is the GB confocal parameter. (b) Equivalent scheme for a nonideal phase-conjugating mirror. where W i is determined as above; G is the SBS gain; I is the intensity of incidence, with I 0 being its maximum value; L is the effective interaction length; D 5 is the initial Stokes noise level; and x is the coordinate across the beam spot size. Note that we assumed the steady state for SBS reflection. The results, for some particular incident intensities (i.e., for some values of the GIL parameter), are shown in Fig. (a) for determining the SBS reflection coefficient r sbs (x) at each point of the Gaussian profile of incidence. Figure (b) shows the reflected beam profiles that are obtained when an incident GB is reflected by the SBS mirror for certain GIL values (curves 1 4). Note that the reflected beam distributions do not greatly differ from the Gaussian distributions (curves 1 4), which are obtained by application of the optimal approximation method. 11 These results allow us to introduce, for the reflected beam (now approximated by the Gaussian function), the beam spot size W r and the SBS reflection coefficient in the maximum of the output beam r max sbs. It is also now suitable to operate with the relative beam-waist parameter for the output beam (see text above). The dependences of the parameters r max sbs and versus GIL are shown in Fig. max 3(a). It can be seen that the reflection coefficient r sbs depends on GIL in the traditional manner according to measurements made in several experimental studies: It Fig.. (a) Transversal distributions of the SBS reflection coefficient. Relative intensity of incidence: GIL 1 (curve 1), 5 (curve ), 50 (curve 3), 300 (curve 4). (b) Transversal distributions of the reflected beam amplitude: Curves 1 4 are the real shapes of the reflected beam, whereas curves 1 4 are their approximations by the Gaussian function; GIL 1 (curves 1, 1); 5 (curves, ); 50 (curves 3, 3); 300 (curves 4, 4). Fig. 3. (a) Results of numerical calculations performed with relations (1) and (): SBS reflection coefficient in maximum of distribution r max sbs (x) (curve 1), and relative beam-waist parameter in the lens caustic (curve ) versus parameter GIL. (b) Relative beam waist at the focusing lens 1/ versus parameter GIL [results of numerical calculations performed with relations (1) and () and by application of Eq. (8)].

Kir yanov et al. Vol. 17, No. 1/January 000/J. Opt. Soc. Am. B 13 increases sharply in the proximity of the SBS threshold and asymptotically approaches unity with an increase in GIL. The parameter actually runs all the values from 1 to 0. and back with an increase in GIL; it is characteristic that the most dramatic changes (significant narrowing) in the relative beam waist occur when GIL values are close to the SBS threshold. Because of the narrowing in the vicinity of the SBSmirror region (the lens caustic), the spot size of the reflected beam has to increase at the lens, as shown schematically in Fig. 1(a). One can obtain the corresponding widening by simply applying the ABCD formalism to the propagation of a GB in the forward and the backward directions through the system described in Fig. 1(a). Operating a nonperfect SBS mirror in terms of an ideal phase-conjugating mirror that has an aperture just behind it (hence taking into account the relative beam-waist parameter ), one can reduce the system shown in Fig. 1(a) to the equivalent one sketched in Fig. 1(b). Its corresponding ABCD matrix for an entire pass of a GB is A B 1 i f /d C D K, (3) 0 1 where is the wavelength, f is the lens focal length, d is the aperture diameter its amplitude transmittance is proportional to the function exp(x /d ), and K is the complex conjugation operator. 1 The ABCD law for a system possessing a phase-conjugating element is q 1 W 1, R 1 Aq 0* B Cq 0 * D, (4) where 1/q 0 and 1/q 1 are the complex curvatures of the wave front at the input and the output of the system, respectively. Hence, for the case of Fig. 1(b), we obtain W 1 W 0 f d, (5) where W 0 and W 1 are the beam spot sizes at the lens for the input and the output (note that curvature radius R 1 R 0 ), respectively. It is easy to show, assuming the curvature radius to be infinite in the lens caustic, that d W i 1, (6) Figure 3(b) shows the dependence of the beam spot-size output versus input calculated by use of Eq. (8). We found that, in the approximation accepted above, the spot size of the Stokes signal increases at the focusing lens [Fig. 3(b)], especially for GIL values close to the SBS threshold. Note that the relative beam waist does not depend, within the limits of approximation (7), on the lens focal length. We conclude that (i) the process of a GB reflection from a SBS cell can be treated as a reflection from a perfect phase conjugating mirror and a further pass through a Gaussian aperture; (ii) this process is described by the SBS reflection coefficient r sbs and by the relative beamwaist parameter, determined from the solution to relations (1) and (); and (iii) the output beam waist at the focusing lens is related to the input beam waist by the simple relationship given in Eq. (8). Note that the result of Eq. (8) is not trivial: That the initial assumption W r W i can be extended and is also valid at the focusing lens only after a nonideal phase-conjugating mirror is operated as an ideal one with a Gaussian aperture just behind it has been demonstrated. 3. LARGE-MODE LASER Q SWITCHED WITH A STIMULATED-BRILLOUIN- SCATTERING MIRROR AND A SATURABLE ABSORBER The above results are important for understanding the phenomenon of mode widening in passively Q-switched lasers that use SBS cells 3,4 (see Fig. 4). When evaluating the output parameters of such lasers, one has to take into account the changes in the beam cross section during a giant pulse formation. The system of differential equations that describe a laser with a SA and a SBS cell as part of a rear mirror, which also account for the nonlinear changes in the laser beam spot size that is due to nonperfect PC (see Section ), is written as 13 dn d t N t R am a S dm a d t s M s S am a t R S 1 ln L r 1 R* 0, (9.1) ds m a l a d t, (9.) where the parameter as a function of GIL is found from the above-described numerical calculations [see Fig. 3(a), curve ]. Finally, since for a thin lens W i f W 0 1 f 1 W 0 f W, (7) 0 we obtain, for the output beam spot size at the lens, W 1 W 0. (8) Fig. 4. Schematic of laser with self-switched SBS mirror. 1, : output (M1) and rear (M) mirrors, respectively; 3: AM; 4: SA; 5: focusing lens; 6: SBS cell. t R and t R * correspond to the round-trip times in the initial and switched resonators, respectively.

14 J. Opt. Soc. Am. B/Vol. 17, No. 1/January 000 Kir yanov et al. dm s d t sm s t R S M 0 M s s m s l s ds d t, (9.3) ds d t S 1 1 S t, t R For calculating the reflection coefficient R* of the composed rear mirror (shown by components, 5, and 6 in Fig. 4), one can write 14 S S AM, (9.4) 0, S S AM where (see also Fig. 4) Eq. (9.1) describes the cavity photon number (N) evolution and Eqs. (9.) and (9.3) are the rate equations for the AM population inversion (M a ) and the SA ground-state population (M s ), respectively (M s0 is the total number of absorbing centers in the SA). Finally, Eq. (9.4) has been added to describe the beam spotsize cross-section (S) temporal evolution that is due to the SBS process in a phase-conjugating mirror (for simplicity, the spot-size cross sections in the AM and in the SA are assumed to be equal); S AM is the AM cross section. As in Ref. 13, the coupling parameters are a,s /t R S a,s are the lasing (AM) and the resonant absorption (SA) cross sections, respectively]. The parameter in Eq. (9.) represents the reduction factor in the population inversion in the AM. The losses in Eq. (9.1) are the sum of the unsaturated nonresonant losses in all the cavity components, L 0, and the useful losses on the cavity mirrors, ln(1/r 1 R*), where r 1 is the reflection coefficient of the output mirror (M1) and R* is that of the complex mirror composed of the rear mirror (M) and the SBS mirror [the lens along with the SBS cell (components 5 and 6, respectively, in Fig. 4)]. t R is the round-trip time for a cavity of length l t R l/c, where c is the velocity of light; for simplicity, we assume that t R t R * (see Fig. 4)], and s is the relaxation time of the SA. Note that the last terms in Eqs. (9.) and (9.3) have been introduced to take into account the participation, during a giant pulse formation, of additional areas of the AM and the SA because of generation beam spot-size increase resulting from the SBS process in the phaseconjugating mirror (m a,s and l a,s are the concentrations and the lengths of the AM and the SA, respectively). Note also that the change in the spot-size cross section S is governed by the widening of the spot size at the lens (see Fig. 1), which is numerically calculated, as explained in Section. It is easy to derive from Eq. (8) that S S St S Fund S 1 1, (10) and only the first Stokes generation is taken at each reflection of the fundamental beam from the SBS mirror. Also, it is necessary to bear in mind that the process of the laser mode s widening as a result of the SBS process is accounted for in our analysis in a formal way. Actually, we do not take into consideration the radiation frequency shift after each reflection from the SBS cell. This approximation, rather rough, is nevertheless valid for a laser like ours, where the laser beam remains in the laser cavity for only a few round trips. R* r SBS* r, (11) 1 r SBS *r again remembering that the validity of neglecting the spectral shift of additionally injected Stokes frequencies during the nondegenerate PC is only approximate. It should be recalled that the parameters r SBS and, which are necessary to solve the system of Eqs. (9), are themselves dependent on the cavity photon number N, because the SBS reflection coefficient in a GB maximum (and, consequently, the relative beam-waist parameter ) are intensity (GIL) dependent and can be determined from the numerical solution to relations (1) and (). Since the intracavity radiation is I hc N Sl (1) (h is the radiation photon energy), then, taking for the effective fundamental Stokes interaction length L the confocal parameter b [see Fig. 1(a) (Ref. 13)]: we obtain where L b W 0, (13) GIL N, (14) 4Gh t R. (15) In the last derivation we noted that the spot-size cross section of a GB is W /. Hence the system of Eqs. (9) should be solved with intermediate calculations of the functions r SBS f(n) and g(n). Finally, geometrical losses of the cavity have to be accounted for, because these arise during the increase of the generation spot-size cross section S [we can see from Fig. 3(b) that this increase can be very high, especially in the vicinity of the SBS-mirror threshold, even for a single reflection]; hence the appearance of aperturing properties of the AM. These losses are accounted for by the transmission factor, T h(n), when writing the expression for the SBS reflection coefficient: where r SBS *N r SBS max NTN, (16) exp 1 S S TN AM N,. (17) 1, S S AM We found numerically the dynamics of intracavity intensity along a giant pulse formation; the dynamics of population inversion in the AM and of ground-state popu-

Kir yanov et al. Vol. 17, No. 1/January 000/J. Opt. Soc. Am. B 15 Fig. 5. Results of numerical calculations performed with relations (9) for a Nd:YAG laser passively Q switched with a LiF:F crystal, without consideration of SBS-mirror self-switching (curve 1) and with it (curve ): (a) intracavity intensity, (b) beam cross section, (c) population inversion in AM, (d) relative beam-waist factor, (e) doped centers number in SA, (f) reflection coefficient R* of the rear complex mirror. All dependences are versus cavity round-trip number. Table 1. Parameters of the Laser Used in This Study AM: Nd:YAG SA: LiF:F SBS Mirror: Acetone Cavity a 6.6 10 19 cm s.5 10 16 cm G 1.9 10 8 cm/w L 0 0.01 m a 9.5 10 17 m s 1.85 10 15 D 5 L 110 cm l a 6.0 cm l s 0.8 cm W 0 0.08 cm a 0.6 s 80 ns a W 0 is the initial spot size in calculations. lation in the SA; and the dynamics of the laser beam spot size and of the reflected coefficient of the composed rear mirror. In addition, we found the integral output parameters of the laser (see Fig. 4) output energy and pulse duration by using the traditional approach for their evaluation. 15 In particular, we used the formula

16 J. Opt. Soc. Am. B/Vol. 17, No. 1/January 000 Kir yanov et al. Table. Comparison of the Basic Characteristics of Three Lasers, Each with a Different AM AM Output Pulse Energy (mj) Pulse Duration (0.1 of Intensity Amplitude; ns) Experiment Simulation Experiment Simulation Nd:YAP 180 00 189 30 4 Nd:YAG 160 180 177 4 1 Nd, Cr:GSGG 100 10 1-19 Fig. 6. Experimental normalized transversal distributions of the output beam of a Nd:YAG laser Q switched with a LiF:F crystal in the cases of a blocked (curve 1) and an unblocked (curve ) SBS mirror. evident that all these changes appear to be due to switching of the SBS mirror by intracavity radiation [compare curves 1 and in Figs. 5(b), 5(d), and 5(f)]. The results of numerical simulations have been verified experimentally. The layout of the experiment is the same as in Refs. 3 and 4 and corresponds to the configuration sketched in Fig. 4. Table allows us to compare the basic characteristics of lasers that are based on Nd:YAG (this study), Nd:YAP (6.3-mm-diameter, 80-mmlong rod), and Nd, Cr:GSGG (3.5-mm-diameter, 44-mmlong rod). 3,4 Good agreement is observed between the theory and the experiments. It is necessary to bear in mind that the laser beam, despite filling virtually the entire aperture of the AM (Fig. 6), remained diffraction limited (this fact was first observed experimentally in Ref. 3 and, it must be remembered, was analyzed theoretically in Ref. 6). Finally, Fig. 7 depicts a comparison between the numerical calculations and the experimental data for the laser based on Nd:YAG for different regimes of the laser operation. Curve 1 corresponds to the case in which a screen is put in front of the SBS mirror [i.e., between the mirror (M) and the lens (component 5 in Fig. 4)]; curve 3 corresponds to the case in which the screen is removed. Also depicted are the results of the calculations performed when widening of the laser beam spot size during a giant pulse formation is not taken into consideration (curve ). out hsfin 1 a ln R av * ln M max a M a min (18) for giant pulse output energy, where M a max and M a min are the maximum and the minimum population inversions, respectively, in the AM; S fin is the final generation cross section; and R av * is the average reflectivity of the composed rear mirror during the giant pulse oscillation. 4. NUMERICAL RESULTS AND COMPARISON WITH EXPERIMENTS The results of simulations (see Fig. 5) were obtained for a laser that utilized a Nd:YAG rod (diameter, 6.3 mm; length, 60 mm) as the AM, a LiF:F crystal (T init 60%, T fin 100%) as the SA, and an acetone-filled cuvette along with a 1-cm focusing lens as the SBS mirror (see Fig. 4). The other parameters of the laser are listed in Table 1. As is shown in Refs. 3 and 4, the most apparent peculiarities of a laser with a SBS mirror and a SA are those expressed during operation with small reflectivities of both the rear and the output mirrors. So we analyzed only the case in which r 1 r 0.04. One can see that the spot size of the laser beam [curve in Fig. 5(b)] increases sharply when it is very close, in the time domain, to the SBS-mirror switching; as a result, additional areas of the AM and the SA begin to participate in laser generation, resulting, in turn, in an increase of giant pulse energy and a decrease of the pulse duration [compare curves 1 and in Figs. 5(a), 5(c), and 5(e)]. It is Fig. 7. Numerical data (curves) and experimental data (squares and circles) for a Nd:YAG laser s output parameters [(a) giant pulse energy, (b) duration] versus initial transmission of a SA. 1: SBS cell is blocked; : SBS cell is unblocked, and laser beam spot-size widening owing to SBS is not accounted for; 3: SBS cell is unblocked, and laser beam spot-size widening owing to SBS is accounted for.

Kir yanov et al. Vol. 17, No. 1/January 000/J. Opt. Soc. Am. B 17 One can conclude that widening of the generation beam spot size has to be accounted for; otherwise, serious errors arise in estimating the laser output parameters. 5. CONCLUDING REMARKS We have shown that the effect of generation beam spotsize widening that is due to nonperfect phase conjugation of a laser beam after reflection from a SBS mirror should be taken into consideration during optimization of passively Q-switched lasers with a SA and a SBS mirror. Note that the present model is acceptable for evaluating the integral output parameters of the laser: giant pulse output energy and pulse duration. ACKNOWLEDGMENTS A. V. Kir yanov and V. Aboites gratefully acknowledge support from Cathedra Patrimonial de Excelencia and from El Consejo de Ciencia y Tecnología del Estado de Guanajuato, respectively. This work is partially supported by the Russian Fund for Basic Research (project 98-0-17676). Address correspondence to A. V. Kir yanov, who is on leave from the General Physics Institute of the Russian Academy of Sciences. He can be reached at Centro de Investigaciones en Optica, Apartado Postal 948, 37150 León, Guanajuato, Mexico; by telephone at 5-47-17583; by fax at 5-47-17-5000; or by e-mail at kiryanov @foton.cio.mx or kiryanov@kapella.gpi.ru. REFERENCES 1. D. A. Rockwell, Review of phase conjugate solid-state lasers, IEEE J. Quantum Electron. 4, 114 1140 (1988).. N. N. Il ichev, A. A. Malyutin, and P. P. Pashinin, Laser with diffraction-limited divergence and Q switching by stimulated Brillouin scattering, Sov. J. Quantum Electron. 1, 1161 1164 (198). 3. P. P. Pashinin and E. I. Shklovskii, Laser with a stimulated Brillouin scattering mirror switched on by its own priming radiation, Sov. J. Quantum Electron. 18, 1190 119 (1988). 4. P. P. Pashinin and E. I. Shklovskii, Solid-state lasers with stimulated-brillouin-scattering mirrors operating in the repetitive-pulse mode, J. Opt. Soc. Am. B 5, 1957 1961 (1988). 5. G. G. Kochemasov and V. D. Nikolaev, Reproduction of the spatial amplitude and phase distributions of a pump beam in stimulated Brillouin scattering, Sov. J. Quantum Electron. 7, 60 63 (1977). 6. V. M. Rysakov, Yu. V. Aristov, and V. I. Korotkov, Threedimensional stimulated Mandelshtam Brillouin scattering, Opt. Spectrosc. 47, 41 415 (1979). 7. G. Giuliani, M.-M. Denariez-Roberge, and P.-A. Belanger, Transverse modes of a stimulated scattering phaseconjugate resonator, Appl. Opt. 1, 3719 374 (198). 8. P. A. Belanger, Phase conjugation and optical resonators, Opt. Eng. 1, 66 70 (198). 9. M. Ostermeyer, A. Heuer, and R. Menzel, 7-W average output power with 1.*DL beam quality from a single-rod Nd:YAG laser with phase-conjugating SBS mirror, IEEE J. Quantum Electron. 34, 37 377 (1998). 10. A. Agnesi and G. C. Reali, Passive and self-q-switching of phase-conjugation Nd:YAG laser oscillators, Opt. Commun. 89, 41 46 (199). 11. O. O. Silichev, Gaussian optics of resonators containing non-gaussian components, Sov. J. Quantum Electron. 0, 715 719 (1990). 1. B. Ya. Zel dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985), p.. 13. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 669, 104. 14. B. Liby and D. Statman, Phase delay in phase-conjugate external cavity lasers, Opt. Commun. 101, 113 13 (1993). 15. W. Koechner, Solid-State Laser Engineering (Springer- Verlag, Berlin, 1976), p. 401.