Orientation of Nd 3+ dipoles in yttrium aluminum garnet: Experiment and model

Transcription

1 PHYSICAL REVIEW A 79, Orientation of Nd + dipole in yttrium aluminum garnet: Experiment and model Sylvain Schwartz,, * Gille Feugnet, Maxence Rebut, Fabien Bretenaker, 2 and Jean-Paul Pocholle Thale Reearch and Technology France, Campu Polytechnique, Avenue Augutin Frenel, F-9767 Palaieau Cedex, France 2 Laboratoire Aimé Cotton, CNRS-Univerité Pari Sud, Campu d Oray, F-9405 Oray Cedex, France Received 8 December 2008; revied manucript received 7 April 2009; publihed 9 June 2009 We report an experimental tudy of the.064 m tranition dipole in neodymium-doped yttrium aluminum garnet Nd:YAG by meauring the coupling contant between two orthogonal mode of a laer cavity for different cut of the YAG gain crytal. We propoe a theoretical model in which the tranition dipole, lightly elliptic, are oriented along the crytallographic axe. Our experimental meaurement how a very good quantitative agreement with thi model and predict a dipole ellipticity between 2% and %. Thi work provide an experimental evidence for the imple decription in which tranition dipole and crytallographic axe are collinear in Nd:YAG with an accuracy better than, a point that ha been dicued for year. DOI: 0.0/PhyRevA PACS number: H, Rz While neodymium-doped yttrium aluminum garnet Nd:YAG i one of the mot if not the mot commonly ued olid-tate laer crytal, the exact orientation of tranition dipole within it i till, paradoxically, an unreolved problem. Thi i probably owing to the fact that for mot application it i ufficient to conider the Nd:YAG crytal uually grown along the crytallographic axi a iotropic; although it ha been known for long,2 that Nd + ion in thi configuration rather ee a D 2 ymmetry, with ix poible dodecahedral orientation. The influence of crytal ymmetry on dipole orientation ha been previouly tudied in aturable aborber uch a Cr:YAG 5 and Tm:YAG 6,7. In the firt cae, it ha been clearly etablihed that tranition dipole were aligned with the crytal axe labeled 00, 00, and 00 4,5, while in the econd cae it ha been found that they were rather collinear with the 0, 0, and 0 direction 7. In the cae of Nd:YAG, the anwer to thi quetion i till unclear in pite of everal previou tudie involving, in particular, dynamical polarization effect in Nd:YAG laer 8. In thi paper, we propoe another approach to probe the orientation of tranition dipole in Nd:YAG, by meauring the coupling contant between two linearly polarized orthogonal mode of a laer cavity for different cut of the gain crytal, uing a teady-tate method imilar to the one decribed in 2. The meaured coupling contant i a dimenionle ratio between cro-aturation and elf-aturation coefficient, which i relatively independent of mot laer parameter pumping rate, birefringence, etc., hence a good indicator for teting the validity of theoretical model. Our tudy deal for the mot part with the nm emiion line, ometime referred to a R2, between the upper doublet of 4 F /2 and the Y level of 4 I /2. A a matter of fact, it i known from previou tudie,4 that the R line at nm between the lower doublet of 4 F /2 and the Y2 level of 4 I /2 ha a very mall contribution to the overall gain, epecially at low pumping rate. The paper i organized a follow. We firt propoe a theoretical model for calculating the coupling contant between *ylvain.chwartz@thalegroup.com two orthogonal mode of a Nd:YAG laer cavity on the aumption that tranition dipole are collinear with YAG crytallographic axe. Starting from the very imple cae of linear tranition dipole, we then generalize the model to the cae of lightly elliptical dipole, howing a very good agreement with the experimental value of the coupling contant publihed in 2 with -cut Nd:YAG. We then apply thi model to the decription of our own experimental configuration, namely, a two-mode laer cavity uing a a gain medium a Nd:YAG crytal either or 00 cut. In both cae, the meaured value of the coupling contant i compared with the correponding theoretical prediction. The reult, implication, and perpective of thi work are finally dicued. I. EXPRESSION OF THE COUPLING CONSTANT IN THE CASE OF LINEAR DIPOLES In the uual emiclaical decription of laer, the tranition dipole i often modeled by an operator dˆ aociated with the linear vector d=du, whoe interaction with a linearly polarized electric field E = Ex i decribed quantum mechanically by the dipolar Hamiltonian dˆ E. Auming furthermore that dipole coherence lifetime i much horter than photon lifetime and population inverion lifetime T which i indeed the cae in Nd:YAG laer 5, the dipolar interaction can be decribed by the following term in the rate equation for the population inverion denity N and field intenity I 5: dn = N I co 2 x,u, int T I di = cni co 2 x,u int where I i the aturation intenity, i the interaction cro ection, c i the peed of light in vacuum, and where u and x are aumed to be unit vector. The overall rate equation for I and N in a imple Lamb laer model then read 5 a /2009/796/ The American Phyical Society

2 SCHWARTZ et al. dn = W N T + dn, int di = I di + int, 2 where W i the pumping rate. A regard to Nd:YAG laer, the latter equation are ufficient to decribe atifactorily mot of the experimentally oberved phenomena uch a relaxation ocillation 6 or piking during laer turn on 7. However, they do not allow an accurate decription of mode coupling in a Nd:YAG laer cavity 2. A. General expreion of the coupling contant To thi end, one mut take into account, in addition with the two laer mode, the exitence of everal poible orientation for the tranition dipole. In the following model, we hall aume three poible orientation correponding to the unitary vector u, u 2, and u and aociated with population inverion denitie N, N 2, and N. We furthermore conider, in keeping with 2 and with the experiment decribed later on in thi paper, the cae of a laer with two mode linearly polarized along the x and x 2 unitary vector and aociated with intenitie I and. The emiclaical equation for the dipolar interaction can be generalized a follow: dn i int = N i I T =,2 I co2 x,,u i di = ci int N i co 2 x,,u i where we have introduced two poibly different aturation intenitie I and I 2. The overall rate 2 for N i and I become in thi cae, dn i = W N i T + dn i, int di = I + di, 4 int where we have introduced different lo coefficient for each mode =/. In Eq. 4, W ha been choen to be independent of i, which correpond to the cae of iotropic pumping thi ha been checked experimentally; ee further in thi paper. In the teady-tate regime and for near-threhold operation i.e., I /I, Eq. and 4 can be rewritten a, 5 and N i = WT I =,2 I co2 x,u i = c N i co 2 x.,u i The coupling contant C, initially defined by Lamb a the ratio between cro-aturation coefficient and elf-aturation coefficient 8 in the cae of laer with hort population inverion lifetime T, can be generalized to other kind of laer uch a Nd:YAG by the following more general definition, involving mall variation from the teady-tate regime 2: 6 C = I / 2 / I / / 2. Uing thi definition, a traightforward calculation tarting from Eq. 6 and uing Eq. 5 for the expreion of N i lead to the following formula: C = PHYSICAL REVIEW A 79, co 2 x co,u 2 i x 2 2,u i co 4 x,u i co 4 x 2,u i 7. 8 One important point i that thi expreion depend only on the overall geometry, making the coupling contant a ueful tool for tudying dipole orientation. B. Application to the cae of the Š crytal The cae of the crytal i by far the mot common for Nd:YAG laer. In thi configuration, the laer wave vector k i along the axi, while the laer electric field lie in the tranvere plane. Let u aume that the two laer mode are linearly polarized along the two following tranvere unitary vector: x = 2, x 2 = 6 0 2, where the coordinate are expreed in the bae of the crytallographic axe. Although thi orthogonal bae of the tranvere plane ha been arbitrarily choen, it hould be pointed out that thi choice doe not affect the final expreion for the coupling contant calculation hown in the Appendix. Differently peaking, the coupling contant i independent, in the cae, of the crytal orientation. Following the author of 9, we conider tranition dipole du, du 2, and du collinear with the crytal axe, namely, = 0 0 u , u 2 = 0, u =. Thi hypothei will be elf-conitently confirmed by our experimental reult later on in thi paper. Expreion 8 immediately lead in thi cae to the coupling-contant value C=/90.. Thi value can be compared with the experimental meaurement C from Ref. 2. The dicrepancy i attributed to the fact that dipole from each ite are not perfectly linear but rather lightly elliptic, a will be decribed in what follow. II. TAKING INTO ACCOUNT CROSS COUPLINGS BETWEEN DIPOLES FROM DIFFERENT CRYSTAL SITES In order to account for poible cro coupling between dipole from different crytal ite, we hall aume a mall

3 ORIENTATION OF Nd + DIPOLES IN YTTRIUM PHYSICAL REVIEW A 79, u i E x u i E x C = Baed on the experimental reult C of Ref. 2, the etimate 2.75%0.5% can be deduced from expreion. A poible explanation for thi ellipticity i the exitence of arbitrarily oriented Nd + ion reiding in defect ite of the YAG matrix that would induce energy tranfer between dipole from different intrinic crytal ite. FIG.. Comparion between uual linear dipole coupling left and elliptical dipole coupling right. The latter ha been phenomenologically introduced to allow for cro coupling between dipole from different ite. dipole ellipticity. In thi phenomenological decription illutrated on Fig., the field-dipole interaction take the new following form: dn i = N 2 i I int T I co2 x,u i + in 2 x,,u i = di = ci int N i co 2 x,u i + in 2 x.,u i Thi model will be ued to calculate a new expreion for the coupling contant in both the and 00 cae. A. Cae of the Š crytal The latter equation can be applied to the previouly tudied cae of the Nd:YAG crytal, uing the vector defined in Eq. 9 and 0. Thi lead to the following rate equation for the population inverion denitie: dn = N int T I I I , 6 dn 2 = N 2 int T I dn = N int T I I I 2 and for the laer mode intenitie, , I + 2 +, di = c N + + N 2+ + N I, int 2 2 d = c N + N N. int 6 We eventually obtain the following expreion for the coupling contant in the preence of mall elliptical dipolar coupling, up to the firt order in : I 2 B. Cae of the Š00 crytal We now turn to the cae of a 00 crytal, on the reaonable aumption that dipole ellipticity i independent of crytal cut. In thi new configuration, the laer wave vector k i aligned with one crytallographic axi ay u and the two orthogonal laer mode are linearly polarized along two tranvere axe defined by x = co = in, x 2 0 in co, 0 where the coordinate have been expreed in the bae u,u 2,u of the crytallographic axe. With the latter definition, the polarization direction of the two laer mode x and x 2 make an angle with the crytal or dipole axe u and u 2. Thi lead, up to the firt order in, to the following rate equation for the population inverion denitie: dn = N int T I I A + 2 I B, dn 2 = N 2 int T I I B + 2 I A, where the following notation have been introduced: A = co 2 + in 2, B = in 2 + co 2. Similarly, one get the following rate equation for the mode intenitie: di = cn A + N 2 BI, int d = cn B + N 2 A. int It hould be mentioned that the influence of N ha been neglected in thi analyi ince it would involve only term on the order of 2 or maller. One finally obtain C =4A 2 B 2 /A 2 +B 2 2, which reduce, up to the firt order in 2 and, to the following expreion: C = Thi expreion will be ued later on in thi paper to compare thi theoretical model with data from our experiment. III. EXPERIMENT We now turn to the decription of our experimental etup ketched in Fig. 2. We ue a 8-cm-long linear laer cavity 0684-

4 SCHWARTZ et al. PHYSICAL REVIEW A 79, FIG. 2. Color online Sketch of our experimental laer cavity etup. Each razor blade can be tranlatory moved perpendicularly to the laer axi, in order to create additional loe to the correponding laer mode. M and M2 are the cavity mirror. with a 2.5-cm-long Nd:YAG crytal a a gain medium. The cavity alo contain a 0-mm-long uniaxial birefringent crytal YVO 4 cut at 45 of it optical axi, in order to patially eparate the two orthogonal mode of the cavity. We have checked that the two perpendicular ordinary and extraordinary polarization correpond to the two patially eparated mode inide the cavity and that no ignificant cro coupling reulting in forked eigentate operation 9 wa induced inide the Nd:YAG crytal which impoe a pecific crytal orientation in the 00 cae, ee further. We have alo checked, uing a Fabry-Perot analyzer, that each one of the two perpendicular laer mode wa longitudinally ingle mode. Thi i probably due to the fact that the YVO 4 crytal act a an etalon, in conunction with the near-threhold operation of the laer. The razor blade placed in the vicinity of the two eparated beam are intended for creating additional loe to the correponding laer mode i.e., changing and 2.We limit the loe introduced by the razor blade to about %, allowing u to neglect the beam truncation and to keep a good overlap of the beam in the active medium. The intenitie of the two orthogonal mode are then monitored on two photodiode after being eparated by a polarizing beam plitter. Typically, the poition of one razor blade i changed periodically in time uing a piezoelectric tranducer, and the relative dependence of I and i monitored on an ocillocope. We ucceively introduce razor blade loe to the and 2 mode, thu obtaining experimental value for / /I / and I / 2 / / 2. The coupling contant i eventually deduced from expreion 7. It i worth noticing that thi reult i independent at leat in the framework of our theoretical model of mot laer parameter, in particular, aturation intenitie I and I 2, pumping rate, and calibration of photodiode. The dominant error ource in thi meaurement i intenity elf-modulation, which introduce an uncertainty in the lope meaurement. Another error ource i reidual pumping aniotropy. In order to ae the contribution of the latter effect, we have checked experimentally that pumping light had no preferential polarization axi with an accuracy better than a few percent, reulting in a relative error of a few percent on the meaurement of C. The overall meaurement error i etimated to C/C0%. A. Cae of the Š crytal In a firt experiment, we ue a typical Nd:YAG crytal a a gain medium, with arbitrary orientation. The raw I FIG.. Experimental curve howing veru I in the cae when the lo rate left curve and 2 right curve are changed. Slope meaurement provide the following value: / /I / = left curve and I / 2 / / 2 = , leading to C= output light from the fibered laer diode i focued on the crytal for optical pumping. Following the method decribed above for the meaurement of the coupling contant, we have obtained the data reported on Fig.. Thi lead to the following value C , in very good agreement with the previouly publihed value C Thi agreement i probably owing to the fact that the coupling contant i relatively independent of mot laer parameter, a pointed out previouly, provided the two cavity mode are properly eparated by the YVO 4 crytal and patial hole burning can be neglected which can be hown to be the cae here and in the work of Ref. 2. Taking into account thi new experimental value with maller error bar than in 2, the following finer etimate of the ellipticity can be deduced from Eq. : 2.2%0.2%. B. Cae of the Š00 crytal In a econd experiment, the gain medium i replaced by a 00 Nd:YAG crytal manufactured by the German company FEE GmbH. The crytallographic axe are known preciely by the x-ray analyi. A mall birefringence i oberved for thi crytal, probably due to the mechanical tre from the mount. In order to avoid the appearance of forked mode in the laer cavity 9, it i neceary to properly align the birefringence axe of the YAG crytal with the ordinary and extraordinary polarization axe of the YVO 4 crytal. A illutrated on Fig. 4, uch a configuration i experimentally obtained, on our etup, with an angle =0 between the crytal axe and the YVO 4 axe. With uch an alignment, the cavity eigenmode are linearly polarized and coincide with the ordinary and extraordinary polar- YVO 4 axe YVO 4 crytal axe <00> YAG I birefringence axe FIG. 4. Relative orientation, around the longitudinal cavity axi, of the YVO 4 crytal left and the 00 Nd:YAG crytal right. A can be een on thi ketch, the YAG crytal ha been oriented in order to align it reidual birefringence axe with the YVO 4 crytal axe, in order to enure that the cavity eigenmode are linearly polarized along the YVO 4 axe intead of being forked mode

5 ORIENTATION OF Nd + DIPOLES IN YTTRIUM PHYSICAL REVIEW A 79, olid-tate gain media or aturable aborber, in order to probe the orientation of their active dipole. ACKNOWLEDGMENTS I FIG. 5. Experimental curve howing veru I in the 00 cae angle between laer and crytal axe: 0 when the lo rate left curve and 2 right curve are changed. Slope meaurement provide the following value: / /I / = left curve and I / 2 / / 2 = 0.0.0, leading to C= ization axe of the YVO 4 crytal, making the experiment uitable for comparion with our theoretical model. It i then poible to meaure the coupling contant between both orthogonal mode following the previouly-decribed method, a reported on Fig. 5. The reult i C to be compared with the theoretical value predicted by our model Eq. 2 with =0 and 2.2%0.2%, namely, C Thi remarkable agreement i an experimental evidence for the imple theoretical decription propoed in thi paper. IV. CONCLUSION To ummarize, we have propoed a imple yet accurate theoretical model for decribing the orientation of tranition dipole in a Nd:YAG crytal, by making the following hypothee: firt, dipole are lightly elliptic; econd, their main direction are collinear with the crytallographic axe. Thi model ha been teted by making experimental meaurement of the coupling contant between two orthogonal mode of a Nd:YAG laer for two different crytal cut and 00. A remarkable quantitative agreement between theory and experiment ha been oberved. In particular, thi tudy i an experimental evidence for the fact that tranition dipole are indeed aligned with crytallographic axe of the Nd:YAG crytal, with an accuracy better than. The ignificant reduction in coupling between orthogonally polarized mode in a laer cavity uing a 00-cut gain medium, a demontrated in thi paper, could be ued to ignificantly increae the tability in bifrequency laer, with poible application in the field of lidar or multiocillator ring laer gyrocope. Our work predict that the more favorable ituation for uch application will occur when the crytallographic axe are aligned with the laer cavity axe, with a minimum achievable coupling contant a mall a Furthermore, the imple and original protocol propoed in thi paper could be applied to other kind of I The author are happy to thank Philippe Goldner from Chimie PariTech and Daniel Rytz from FEE GmbH for helpful dicuion. APPENDIX: CALCULATION OF THE COUPLING CONSTANT FOR AN ARBITRARY CRYSTAL ORIENTATION IN THE Š CASE WITHOUT DIPOLE ELLIPTICITY In thi appendix, we will how that in the cae of a crytal with linear dipole oriented along the crytallographic axe, the coupling contant i independent of the orientation between the crytal axe and the direction of the laer mode. To thi end, we conider linear dipole oriented along the crytallographic axe, namely, = 0 0 u 0 0 A 0, u 2 = 0, u =. Any pair of orthogonal direction for the laer mode in the tranvere plane can be obtained by rotating by the appropriate angle the initial bae defined by Eq. 9, which read a x = x 2 = co + in , in + co A2 A traightforward although tediou calculation lead, from Eq. A and A2, to the following expreion: Similarly, we obtain: co 2 x co,u 2 i x 2,u i = 6. A co 4 x,u i = 2, co 4 x 2,u i = 2. A4 It i a remarkable fact that expreion A and A4 are independent of. Uing expreion 8 for the coupling contant eventually lead to C =/ 9, independently, of the orientation between the crytal and the laer mode

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