Experimental and theoretical study of monomode vectorial lasers passively Q switched byacr 4 : yttrium aluminum garnet absorber

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1 PHYSICAL REVIEW A VOLUME 60, NUMBER 5 NOVEMBER 1999 Experimental and theoretical study of monomode vectorial lasers passively Q switched byacr 4 : yttrium aluminum garnet absorber Marc Brunel, Olivier Emile, Marc Vallet, Fabien Bretenaker, and Albert Le Floch Laboratoire d Electronique Quantique Physique des Lasers, Unité Mixte de Recherche du Centre National de la Recherche Scientifique 667, Université de Rennes I, Campus de Beaulieu, F-3504 Rennes Cedex, France Laurent Fulbert, Jean Marty, Bernard Ferrand, and Engin Molva Laboratoire d Electronique, de Technologie et d Instrumentation, Département Optronique, Commissariat à l Energie Atomique, 17 rue des Martyrs, F Grenoble Cedex 9, France Received 17 May 1999 The dynamics of a solid-state laser sustaining the oscillation of two orthogonally polarized eigenstates and passively Q switched by a Cr 4 :YAG YAG denotes yttrium aluminum garnet saturable absorber is experimentally investigated. A set of rate equations in which the orientation-dependent interactions between the intensities and the saturable absorber have been taken into account is developed to describe theoretically the evolution of such a laser. When a 001-cut Cr 4 :YAG plate is used, orthogonally polarized pulsed eigenstates are emitted either successively or simultaneously, depending on the orientations of the eigenstates with respect to the crystallographic axes. Namely, in the latter case, beat note carrying pulses are emitted, whose beat frequency is continuously tunable. On the other hand, when a 111-cut Cr 4 :YAG plate is used, the simultaneity regime is never observed. All experimental results, obtained using a diode pumped Nd:YAG laser, are well confirmed by the theoretical model. S PACS numbers: 4.55.Rz, 4.60.Gd I. INTRODUCTION Passively Q-switched diode laser-pumped solid-state lasers are the subject of a growing interest due to their potential applications in Doppler Lidars, sources for global altimetry systems, injection seeds for power amplifiers, and, generally, whenever a compact single-frequency pulsed source is needed. Recent developments have shown that diode-pumped Nd:YAG YAG denotes yttrium aluminum garnet lasers passively Q switched by a Cr 4 :YAG saturable absorber can radiate transversally and longitudinally monomode pulses at the millijoule level with kilohertz repetition rates 1,. Besides its spectroscopic properties, which make it a good saturable absorber for the ubiquitous neodymiumdoped solid-state lasers, Cr 4 :YAG exhibits interesting properties with respect to the polarization of the absorbed radiation. Indeed, its transmission is isotropic in the small-signal regime, but electric dipoles responsible for the absorption in the 1-m wavelength range have been shown to be aligned along the YAG crystal axes, thus making the saturated transmission anisotropic 3 5. This effect often leads to a linearly polarized output when microchip lasers are considered 6 8, and to the appearance of spurious losses in higher power lasers when crossed anisotropies, such as Brewster windows, are inserted inside the laser cavity 9. However, peculiar regimes where unpolarized or alternate polarization states are emitted in the pulse train have been reported and hardly controlled Moreover, due to the low value of the coupling constant between linearly polarized eigenstates, quasi-isotropic solid-state lasers in the cw regime usually oscillate in dual polarization eigenstates with a tunable frequency difference 13,14. One can hence wonder whether two-frequency Q-switched pulses with a tunable frequency difference could also be emitted in two orthogonal polarization eigenstates. In addition, owing to the orientation of the absorber s dipoles along the matrix crystal axes, one may question the influence of the orientation of these axes with respect to the residual anisotropies inside the cavity on the occurrence of one- or two-polarization state Q-switched emission. Consequently, the aim of this paper is to investigate both experimentally and theoretically vectorial lasers in the sense that two polarization eigenstates may oscillate Q switched by Cr 4 :YAG. In particular, we aim to show that the regime commonly observed in which only one eigenstate oscillates is not the only possible regime. Thus, we investigate the laser dynamics with respect to the relative orientations of the cavity phase anisotropies and the Cr 4 :YAG matrix orientation. We focus on the possibility for these lasers to emit two orthogonally polarized eigenstates in alternate or simultaneity regimes. Namely, pulses carrying a beat note at a continuously adjustable frequency are investigated. Moreover, in order to describe theoretically the laser dynamics, we derive a rate equations model which includes the anisotropic properties of both the cavity and the saturable absorber. This paper is organized as follows. In Sec. II, we first derive a theoretical model consisting in rate equations for the photon number and population inversion densities in the active medium and in the absorber. The vectorial nature of the electromagnetic field-atom interaction is taken into account. In Sec. III, we then test experimentally the predictions of this model. We first describe the experimental apparatus used for this study. Second, we emphasize the peculiar dynamics obtained with a 001-cut Cr 4 :YAG plate. These results are then compared to the ones obtained with the more usual 111-cut Cr 4 :YAG epitaxially grown on a YAG substrate /99/605/4057/$15.00 PRA The American Physical Society

2 PRA 60 EXPERIMENTAL AND THEORETICAL STUDY OF Our purpose is to derive a model predicting the behavior of solid-state lasers which are transversally and longitudinally monomode, vectorial, and containing a passive saturable absorber having anisotropic intensity-dependent properties, e.g., Cr 4 :YAG. Usual monomode solid-state lasers are well described by two coupled rate equations for the population inversion in the active medium and the photon number in the laser resonator 15. When a passive Q switch is inserted inside the cavity, a third equation governing the evolution of the population inversion in the absorber permits us to predict the behavior of such scalar oscillating in only one polarization eigenstate lasers 16,17. These equations, however, do not take into account the vectorial nature of light. When the passive Q switch consists in a Cr 4 :YAG crystal, we know that the saturation is anisotropic, as a result of the preferential orientation of the dipole moments responsible for the absorption in the vicinity of 1 m along the crystallographic axes of the YAG matrix 3. Il ichev et al. have modified the rate equations including an angulardependent field-atom coupling coefficient in the absorbing medium 18. But these authors experimentally investigated lasers containing also partial polarizers, e.g., Brewster windows, which enforce the oscillation of one eigenstate only. Up to now, no theoretical model has taken the possible oscillation of two polarization eigenstates in Cr 4 :YAG Q-switched lasers into account. To this aim, we follow previous works describing dual-polarization lasers in the continuous-wave regime. For the sake of simplicity, we start from the phenomenological model described in Ref. 19 and introduce three additional equations of evolution for the saturable absorption, with special attention paid to the orientation of the dipole moments responsible for this absorption with respect to the eigenstate directions. Let us consider the laser cavity of Fig. 1, whose propagation axis is z. It is closed with mirrors M 1 and M and contains an isotropic active medium and a linear phase anisotropy creating a retardance xy between the x and y polarizations. It also contains a Cr 4 :YAG saturable absorber whose 100, 010, and 001 crystallographic axes are labeled u, v, and w, respectively. In the linear regime small signal the absorption is isotropic 3. Then a Jones matrix analysis for the cold cavity yields two linearly polarized eigenstates aligned along the x and y axes with a frequency difference vv y v x c/l xy /, where c is the velocity of light and L the optical length of the cavity. Following Ref. 19, the rate equations governing the evolutions of the intensities and population inversions of these laser eigenstates are then İ x x I x n x n y I x x a x I x, İ y y I y n y n y I y y a y I y, 1a 1b FIG. 1. Laser cavity setup. M 1,M : mirrors. xy : xy phase anisotropy. Finally, the results, as well as the perspectives of this work, are summarized in Sec. IV. II. THEORETICAL MODEL ṅ x P x n x I x I y n x. ṅ y P y n y I y I x n y. 1c 1d where I x, I y, n x, and n y are the intensities and population inversions associated with the x and y eigenstates. These rate equations are obtained using the following approximations: i we eliminate the atomic coherences adiabatically and ii we neglect the phase-sensitive nonlinear interactions in the active medium 0, since the beat frequency v is much larger than the system relaxation rates. In these equations, x and y are the intensity loss coefficients along the x and y directions, given by x,y c/l lnr 1 R (1 x,y ), where R 1 and R are the intensity reflection coefficients of mirrors M 1 and M, respectively, and x and y are the single-pass intensity loss coefficients experienced by x and y eigenstates through propagation inside the cavity. is a coupling constant accounting for cross-saturation in the gain medium. This constant may be related to the experimentally measured coupling constant C between laser eigenstates 13, as pointed out in the Appendix. x and y are small quantities which hold for spontaneous emission. is the decay time of the population inversion in the gain medium. P x and P y are the pumping rates of the x and y eigenstates. and are constant field-atom coupling coefficients. Finally, a x and a y are the saturable loss coefficients along the x and y directions. These coefficients are related to the absorptions by the three species of dipoles a u, a v, and a w aligned along the crystallographic axes through a x x u a u x v a v x w a w, a y y u a u y v a v y w a w, a b where x, y, u, v, and w are unit vectors along the x,y,u,v, and w directions, respectively. The equations of evolution of the absorptions are given by ȧ u a a 0 a u ux I x uy I y a u, ȧ v a a 0 a v vx I x vy I y a v, ȧ w a a 0 a w wx I x wy I y a w, 3a 3b 3c where a is the decay time of the excited state of the absorbing transition, a 0 c/l ln T a (T a is the small-signal intensity absorption, and ij are coupling coefficients defined by ux x u C a x v x w, uy y u C a y v y w, vx x v C a x u x w, vy y v C a y u y w, 4a 4b 4c 4d

3 4054 MARC BRUNEL et al. PRA 60 FIG.. Orientation of the Cr 4 :YAG crystallographic axes u,v,w with respect to the cavity eigenaxes x,y,z. a111 sample: A1/&(wv) and B1/6(uvw) are in the xy plane. b001 sample. wx x w C a x u x v, wy y w C a y u y v, 4e 4f where C a is a coupling constant accounting for crosssaturation in the absorber, and a a a g g. a and a 1/ a are the absorption cross section and the upper level lifetime of the absorbing transition, respectively. g and g 1/ are the stimulated emission cross section and upper level lifetime of the laser transition in the gain medium, respectively. In the following, two particular situations are emphasized, corresponding to two different Cr 4 :YAG samples see Fig. : a111-cut sample used with light propagating along this 111 direction, and a 001-cut sample used with light propagating along this 001 direction. The saturation of the three absorber species depends strongly on the orientation of the x and y directions of the eigenstates with respect to the crystallographic axes of the absorber, as evidenced by Eqs. 4. In the usual case of the 111 orientation, the scalar products between x and y and u, v, and w are given in the second column of Table I, where is the angle made by A 1/&(wv) with the eigenstate direction x see Fig. a. TABLE I. Scalar products between the eigenstate unit vectors x and y and the crystallographic axes unit vectors u, v, and w for the considered orientations of the Cr 4 :YAG crystal. Crystal orientation z//111 z//001 (x u) (x v) (x w) (y u) (y v) (y w) sin 3 cos sin & 6 cos sin & 6 cos 3 sin cos & 6 sin cos & 6 cos sin 0 sin cos 0 5 In the 001 orientation, however, these products are given in the last column of Table I, where is the angle made by the direction u with the eigenstate direction x see Fig. b. These two orientations lead to different behaviors for the eigenstate dynamics, as is now going to be investigated experimentally. In the following, we use Eqs. 1 4 to simulate the behavior of the laser. Depending on the sample used to Q switch the laser, we will use the scalar products given either in the second or third column of Table I. This set of coupled nonlinear differential equations is numerically integrated using a fourth-order Runge-Kutta algorithm 1 and using the parameters given by the experiments. III. EXPERIMENTAL RESULTS We consider the longitudinally pumped laser schematized in Fig. 1. The active medium is a 1.1-mm-long crystal of 1% at. doped Nd:YAG. One of its ends M 1 is highly transmitting (T95%) at 809 nm and highly reflecting (R99.5%) at 1064 nm. The resonator is L97 mm long and is closed with a 100-mm radius of curvature output coupler M with transmission T1% at 1064 nm. The pump laser is a laser diode emitting up to 750 mw at 809 nm in a 00-m-core diameter fiber. The end of the fiber is simply butted against the Nd:YAG crystal. In order to control the cavity phase anisotropies, we introduce two antireflection coated quarterwave plates L 1 and L between the saturable absorber and M. The neutral axes of L 1 are rotated by 45 with respect to x and y. The fast axis of L is rotated by an angle with respect to the fast axis of L 1. With these elements, the laser eigenstates are then linearly polarized along the x and y axes in the active medium, helicoidally polarized between the two quarter-wave plates, and linear again at the output of the laser, making angles of 45 to the fast axis of L,3. The self-consistency for the phase of the electric field over one round trip inside the resonator yields the eigenfrequencies v x and v y, whose difference is vv y v x /c/l. Thus, the quarter-wave plates create a linear phase retardance xy between the x and y polarization directions, which is continuously adjustable by simply rotating one of the quarter-wave plates. The laser output is then passed through an optical isolator to avoid spurious feedbacks and detected by a fast photodiode. A half-wave plate inserted between the laser output coupler and the isolator permits us to analyze the laser output polarization. We now focus on the Q-switch regimes observed when a Cr 4 :YAG plate is inserted inside the laser cavity. A. 111 Cr 4 :YAG In a first step, we introduce inside the laser cavity an antireflection coated Cr 4 :YAG plate grown by liquid phase epitaxy on a 111 YAG substrate 8. This 111 crystallographic axis is set parallel to the optical axis of the cavity z see Fig. a, which corresponds to the usual YAG orientation. The small-signal intensity transmission of this sample is 84% at 1064 nm. At an excitation rate x y 1.45 ( x and y are the ratios of the unsaturated gain by the unsaturated losses for the x and y eigenstates, respectively; see the Appendix, weset15. The results of the numerical

4 PRA 60 EXPERIMENTAL AND THEORETICAL STUDY OF FIG. 3. Theoretical time evolutions of a the total intensity and b the population inversions n x full line and n y dashed line, in the case of a 111 sample. The values of the parameters used in the simulations are L0.097 m, R 1 1, R 0.99, 1, a 1/ a 4 s, g 1/ 30 s,.875, T a 0.85, 1 x 1 y 0.983, x y 10 0, 0.1, C a 0.03, 15, and x y FIG. 4. Zoom on the two first pulses of Fig. 3. Theoretical time evolutions of a,b the intensities I x thick full line and I y thick dashed line and population inversions n x full line and n y dashed line, and c,d the same intensities together with absorptions a u, a v, and a w. Note that the x- and y-polarized pulses are emitted alternately delay 36 s. The values of the parameters used in the simulations are the same as in Fig. 3. integration of Eqs. 1 4 with the use of the second column of Table I are given in Figs. 3 and 4. Figure 3 shows that Q-switched pulses are obtained at a repetition rate equal to 4.3 khz. A closer examination of the pulses leads to the results of Fig. 4. The pulses have a full width at half maximum equal to 50 ns. Moreover, it appears that the two eigenstates are emitted successively, as evidenced by Figs. 4a and 4b. The time delays between the x and y pulses and between the y and x pulses are found to be identical. The population inversions in the gain medium and the absorptions of the three absorbing species are also given in Fig. 4, showing that the population inversion is depleted successively by the two orthogonal eigenstates. Moreover, this regime is obtained for any value of. The values of the parameters used in the simulation are L0.097 m, R 1 1, R 0.99, 1, a 1/ a 4 s, g 1/ 30 s, and.875. Besides, among the 84% transmission of the absorber, we take T a 0.85 due to saturable absorption. The extra absorber losses, which correspond to a transmission coefficient equal to 0.84/ , are background nonsaturable losses which are included in the parameters x and y. Moreover, the imperfections of the other intracavity elements lead to an extra 0.5% losses per pass. This leads to 1 x 1 y For x and y, we take a value (10 0 ), which is small enough to have no influence on the results. From the calculation given in the Appendix, we take 0.1. Finally, we deduce the value C a 0.03 from Ref. 4. Experimentally, we also observe that, at an excitation rate x y 1.45, the laser emits a train of Q-switched pulses at 5-kHz repetition rate. The pulses have a full width at half maximum equal to 55 ns, in fairly good agreement with the simulation. Each pulse has a 3-J energy. Using a confocal 7.5-GHz Fabry-Perot interferometer, we check that the laser is longitudinally monomode. Moreover, owing to the high losses experienced by higher order transverse modes in the saturable absorber, the laser emits in the fundamental TEM 00 mode. We also check that, for every value of, x- and y-polarized pulses are emitted alternately. In order to find out how the two eigenstates can be emitted simultaneously, we thus have to turn to other Cr 4 :YAG orientations. B. 001 Cr 4 :YAG In a second step, we introduce inside the laser cavity a 1-mm-long antireflection coated Cr 4 :YAG plate whose 001 crystallographic axis is set parallel to the optical axis of the cavity z, and whose small-signal intensity transmission at 1064 nm is 90%. Depending on the value of the angle made by the 100 axis of the Cr 4 :YAG labeled u in the equations and in Fig. with respect to the x axis, two different regimes are theoretically predicted. First, we set 0. In this case, the absorbing dipoles are aligned with the cavity eigenaxes. We numerically integrate the rate equations 1 4 using the last column of Table I. The pulses have a full width at half maximum equal to 55 ns, as shown in Fig. 5. It appears that, as with the 111 sample, the two eigenstates are emitted successively, as evidenced by Figs. 5a and 5b, at a 4.7-kHz repetition rate. The population inversion in the gain medium and the absorptions of the three absorbing species are also given in Fig. 5, showing that, again, the population inversion is depleted successively by the two orthogonal eigenstates. The values of the parameters used in the simulation are the same as before, except for the excitation rate ( x y 1.7), small-signal absorption (T a 0.9) of this Cr 4 :YAG sample, and the extra intracavity losses x y 510 3, which correspond to the imperfections of the intracavity elements only. The experimentally observed pulse train is shown in Fig. 6a when the neutral axes of the half-wave plate are rotated by a few degrees with respect to x and y. At the excitation rate x y 1.7, the laser emits a train of Q-switched pulses, whose full width at half maximum is 50 ns see Fig. 6b, in good agreement with the simulation. One can see

5 4056 MARC BRUNEL et al. PRA 60 FIG. 5. Theoretical time evolutions of a,b the intensities I x thick full line and I y thick dashed line and population inversions n x full line and n y dashed line, and c,d the same intensities together with absorptions a u, a v, and a w, in the case of a 001 sample with 0. Note that the x- and y-polarized pulses are emitted alternately delay 09 s. The values of the parameters used in the simulations are the same as in Fig. 3 except T a 0.9, 1 x 1 y 0.995, and x y 1.7. FIG. 6. Experimental time evolution of the laser intensity in the case of a 001 sample with 0 showing the alternate emission of x- and y-polarized pulses. a A polarizer permits us to distinguish the two polarization states. b Zoom on one of the pulses. that the two orthogonally polarized eigenstates are emitted alternately, as expected. We check again that the laser is longitudinally and transversally monomode. We also verify that these two eigenstates are linearly polarized aligned along x and y, respectively. The repetition rate is then 4.6 khz the repetition rate corresponding to one eigenstate emission only is.3 khz. By rotating the quarter-wave plate L, we observe that this regime is obtained for any value of the frequency difference v. We now set 45. The cavity eigenstates are now expected to interact equally with both absorbing dipole sets, along the 100 and 010 axes see Fig. b. We numeri- FIG. 7. Theoretical time evolution of a the intensities I x and I y and population inversions n x and n y, and b the same intensities together with absorptions a u, a v, and a w, in the case of a 001 sample with 45. Note that both eigenstates are emitted simultaneously. The values of the parameters used in the simulations are the same as in Fig. 5. cally integrate the rate equations 1 4. The pulses have a full width at half maximum equal to 49 ns, as shown in Fig. 7, and are emitted at a repetition rate of 4.7 khz. Contrary to the preceding results, it appears that the two eigenstates are now emitted simultaneously, as evidenced by Figs. 7a and 7b. The population inversion in the gain medium and the absorptions of the three absorbing species are also given in Fig. 7, showing that, now, the population inversion is depleted concurrently by the two orthogonal eigenstates. Moreover, this simultaneity regime is observed for angles in the range of a few degrees around 45, depending on the value of the loss anisotropy x / y 1. We now turn to the corresponding experiment with 45, and we analyze the polarization of the pulses by rotating the external half-wave plate. As expected from the simulation, we observe that all the pulses are identical and contain both eigenstates. This is evidenced in Fig. 8. The repetition rate is then 5.1 khz. Figure 8b depicts one of these Q-switched pulses full width at half maximum: 5 ns on a shorter time scale, with the analyzer s angle set at 45 to the x and y directions. One can observe here the simultaneous oscillation of both eigenstates within the pulse, resulting in a beat note observable behind the polarizer. Moreover, we check that this beat frequency is continuously adjustable up to c/4l730 MHz by rotating the quarterwave plate from 0 to 45, as shown in Fig. 9. It is striking to

6 PRA 60 EXPERIMENTAL AND THEORETICAL STUDY OF FIG. 8. Experimental evolution of the laser intensity in the case of a 001 sample with 45 showing the simultaneous emission of both x- and y-polarized pulses. Observation behind a polarizer: a All pulses contain both eigenstates; b zoom on one of the pulses exhibiting a beat note at 150 MHz. notice that this two-frequency oscillation obeys the same eigenstate physics in the Q-switch regime as in the usual cw regime 14. IV. DISCUSSION AND CONCLUSION In order to give a simple physical picture of these behaviors, let us recall that the absorbing dipole sets are oriented along the u, v, and w directions. We first consider the 001 orientation for the Cr 4 :YAG sample. When 0, x(y) is parallel to u(v). Thus, each eigenstate interacts preferentially with one dipole set only. Let us suppose that one of the eigenstates say x reaches threshold first. It then bleaches the absorption of its own dipole set, thus enhancing the loss anisotropy. As a result, a one-eigenstate x-polarized pulse is emitted. The population inversion accounting for this eigenstate is then fully depleted. If the loss anisotropy is weak which is the case in our experiment, the y-polarized eigenstate has consequently a significant advance to reach the threshold first. It then bleaches the absorption of its own dipole set, generating a y-polarized eigenstate. As a result, we obtain this alternate sequence of x- and y-polarized pulses. Conversely, when 45, each eigenstate interacts with both dipole sets. When one eigenstate reaches threshold, it then saturates both dipole sets equally. Both dipole sets are then bleached and the x- and y-polarized eigenstates are emitted concomitantly. To some extent, the nonlinearities of the saturable absorber can be said to lead to some kind of anticoupling or cooperative coupling, contrary to crosssaturation in the active medium. Finally, in the 111 oriented Cr 4 :YAG case, one can notice that no angle can be found for which the three absorber sets are saturated equally by the x and y polarizations. Consequently, the simultaneous emission of both eigenstates in the same pulse is never obtained. In conclusion, we have shown both experimentally and theoretically that two different dynamical regimes may be obtained in vectorial solid-state lasers passively Q switched byacr 4 :YAG saturable absorber. When a 001 oriented Cr 4 :YAG plate is used, orthogonally polarized pulsed eigenstates are emitted either successively or simultaneously, depending on the orientations of the eigenstates with respect to the crystal axes. Namely, to obtain beat-note carrying pulses with a continuously tunable beat frequency, one must choose i the 001 orientation for the Cr 4 :YAG sample and ii a 45 orientation within a few degrees of the eigenstate directions, as defined by the phase anisotropy of the cavity, with respect to the crystallographic axes of the saturable absorber. On the contrary, when a 111 oriented Cr 4 :YAG plate is used, the simultaneity regime is never observed. We have developed a set of rate equations to describe theoretically the evolution of such a laser, in which the orientation-dependent interactions between the intensities and the saturable absorber have been taken into account. All experimental results, obtained using a diode pumped Nd:YAG laser, are well confirmed by the theoretical model. This model could find applications in other anisotropic passively Q-switched systems 5. Finally, the new twofrequency regime may find applications in pulsed detection systems at modulation frequencies where external modulators are not currently available, e.g., in the 10-GHz range with, of course, a shorter cavity. ACKNOWLEDGMENTS We thank F. Leplingard and P. Pierrard for fruitful discussions. This work was partially supported by the Conseil Régional de Bretagne and was performed in the framework of the Center Laser et Applications à la Chimie. APPENDIX FIG. 9. Experimental beat frequency observed in the case of a 001 sample with 45 vs angle. The frequency difference is continuously tunable up to c/4l730 MHz. may be related to the coupling constant C between the two linearly polarized eigenstates, which has been experimentally measured to be equal to 0.16 in Nd:YAG lasers using a differential measurement of the intensity variations of two orthogonally polarized eigenstates 13. In this experiment 13, we consider a cw Nd:YAG laser emitting two x and y linearly polarized eigenstates. We then modify the losses x of the x eigenstate only and measure the induced

7 4058 MARC BRUNEL et al. PRA 60 variations I x and I y of the intensities of these two eigenstates. The coupling constant C is then given by CI y /I x. A1 To find out a relation between and C, we derive the stationary solutions for the differential equations 1 without the absorber cw laser regime, i.e., when a x a y a 0 0. Setting all the derivatives to zero in Eqs. 1 yields the stationary values for the population inversions: n 0 1 x 1 x y, A n 0 1 y 1 y x. A3 We introduce the relative excitation rates x P x /n 0 x and y P y /n 0 y. The intensities are then given by I 0 x 1 x1 y 1, A4 I 0 y 1 y1 x 1, A5 Setting x y and P x P y yields n x 0 n y 0 and I x 0 I y 0. We now introduce a small variation of the losses x ( x ) and calculate the resulting variations I x and I y of the eigenstate intensities. It yields I y I x 1. Combining Eq. 6 with Eq. 9 then gives 1 C 11C. A6 A7 With C0.16 given by the experiment for linearly polarized eigenstates 13, we obtain the value 0.1 taken in our simulations. 1 H. Liu, S.-H. Zhou, and Y. C. Chen, Opt. Lett. 3, R. S. Afzal, A. W. Yu, J. J. Zayhowski, and T. Y. Fan, Opt. Lett., H. Eilers, K. R. Hoffman, W. M. Dennis, S. M. Jacobsen, and W. M. Yen, Appl. Phys. Lett. 61, A. Brignon, J. Opt. Soc. Am. B 13, M. J. Damzen, S. Camacho-Lopez, and R. M. P. Green, Phys. Rev. Lett. 76, Y. C. Chen, S. Li, K. K. Lee, and S. Zhou, Opt. Lett. 18, J. J. Zayhowski and C. Dill III, Opt. Lett. 19, L. Fulbert, J. Marty, B. Ferrand, and E. Molva in Conference on Lasers and Electro-Optics, Vol. 15 of 1995 Technical Digest Series Optical Society of America, Washington, D.C., 1995, p N. N. Il ichev, A. V. Kir yanov, E. S. Gulyamova, and P. P. Pashinin, Quantum Electron. 8, Kvant. Elektron. 5, M. Lukac, S. Trost, and M. Kazic, IEEE J. Quantum Electron. QE-8, P. Yankov, J. Phys. D 7, H. Liu, O. Hornia, Y. C. Chen, and S.-H. Zhou, IEEE J. Sel. Top. Quantum Electron. 3, M. Brunel, M. Vallet, A. Le Floch, and F. Bretenaker, Appl. Phys. Lett. 70, M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, Opt. Rev. 4, A. E. Siegman, Lasers University Science, Mill Valley, CA, A. Szabo and R. A. Stein, J. Appl. Phys. 36, H. T. Powell and G. J. Wolga, IEEE J. Quantum Electron. QE-7, N. N. Il ichev, A. V. Kir yanov, and P. P. Pashinin, Quantum Electron. 8, Kvant. Elektron. 5, K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, P. Mandel, C. Etrich, and K. Otsuka, IEEE J. Quantum Electron. QE-9, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Wetterling, Numerical Recipes in Pascal Cambridge University Press, Cambridge, England, 1989, Chap. 15. V. Evtuhov and A. E. Siegman, Appl. Opt. 4, A. Kastler, C. R. Acad. Sci. B Paris 71, S. Camacho-Lopez, R. P. M. Green, G. J. Crofts, and M. J. Damzen, J. Mod. Opt. 44, P. Thony, 1.55 m Passively Q-switched Microchip Laser, in Advanced Solid-State Lasers, OSA Technical Digest Optical Society of America, Washington, D.C., 1998.