Measuring Nonlinear Susceptibility in Crystalline Solids: Ellipse Rotation Observation, Notas de estudo de Física

Baixe Measuring Nonlinear Susceptibility in Crystalline Solids: Ellipse Rotation Observation e outras Notas de estudo em PDF para Física, somente na Docsity! I064 I E E E JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-9, NO. 11, NOVEMBER 1973 Ellipse Rotation Studies in Laser Host Materials ADELBERT OWYOUNG Abstracf-Using a TEM,,, near Gaussian mode ruby laser system we report the first experimental measurements of intensity induced changes of optical polarization (ellipse rotation) in a cubic crystalline medium, YAG, for which we obtain the nonlinear susceptibilities x31221 ( - w, w. w, -.w) = 6.34 X ESlJ and (x31111 + x31221 - 2 x 3 1212 = 7.18 x 10-15 ESU, accurate to better than 1 7 percent relative to x31221 ( - w,w,w, -a) for liquid CS,. These values are compared with further results obtained for fused quartz and two laser glasses. Moreover, by time resolving the ellipse rotation data we demonstrate the capability to plot ellipse rotation versus in- put power on a single laser shot, thus increasing the practical feasibility of the technique and introducing the possibility of resolving transient con- tributions to the measurement. I. INTRODUCTION T HE NONLINEAR rotation of an elliptically polarized beam in isotropic media has been well documented in liquids [l] , [2] and glasses [ 3 ] where it has been employed as a means of determining the direct elec- tronic contribution to the nonlinear refractive index nz [4]. In this paper we report the observation of ellipse rotation in a cubic crystalline solid, YAG (Y3Al&), and demonstrate that the ellipse rotation technique may be employed to measure the anisotropy of the third-order nonlinear susceptibility tensor. Values for the nonlinear refractive index n, are inferred for several crystal orien- tations and the results for YAG are compared with measurements in fused quartz and several laser glasses. By time resolving the 9-11s full width at half maximum (FWHM) ellipse rotation signal and comparing it with the 13-ns input laser pulse we obtain on a single laser shot all of the information necessary to plot ellipse rotation versus power for a given material. Hence, the practicability of the ellipse rotation technique is greatly enhanced and tran- sient phenomena such as electrostriction, which might cause erroneous interpretation of the measurement, may be readily detected by comparing data obtained from the leading and trailing edges of the laser pulse. Also, by employing focused Gaussian beams whose focal volumes are completely contained within the samples of interest, we demonstrate ellipse rotation to be entirely independent of focusing parameters. In Section I1 ellipse rotation in cubic media is con- sidered and it is demonstrated that this technique may be employed to measure the anisotropy of the nonlinear susceptibility x ~ ~ ' ~ ~ . Section I11 is addressed to the problem of ellipse rotation with focused beams, and in Section IV the xperimental apparatus is described. Finally, the results obtained and some implications of these results are discussed in Sections V and VI. Manuscript received May 7, 1973. This work was supported by the The author is with Sandia Laboratories, Albuquerque, N. Mex. 871 15. U.S. Atomic Energy Commission. II. ELLIPSE ROTATION IN CUBIC MEDIA As in the case of other intensity dependent refractive in- dex effects, it is convenient to express ellipse rotation in terms of a nonlinear polarization at frequency w, P""(W), which is cubic in the electric field strength. Following Maker and Terhune [5] we write the ith component of P""(w) in the form P,""(w) = 0, w1, wz, wl)EI(wI)ER(w2)E1(W3) X exp{l(kl + k , + k,)z} (1)- where a sum is taken over all repeated indices. In (1) w = w1 + w z + w3; k l , z , 3 are the wave vectors corresponding to w ~ , ~ , ~ ; D is a degeneracy factor which is 6, 3 , or 1 depending upon whether wl, wz, and w3 are nondegenerate, partially degenerate, or totally degenerate; E i ( w ) are the complex amplitudes of the electric field [E i * (w) = E,( - w ) ] ; and x3,jkL( -w, wl, w,, w3) are the components of the fourth rank nonlinear susceptibility tensor [5] . For "self-induced" nonlinear refractive index changes, the relevant parameter is the nonlinear susceptibility xQLjkL ( - w , w , w , - w ) at the driving frequency w. In particular, for cubic crystals of point group symmetry 432, 43m, or m3m, xaijhL is reduced by intrinsic permutation symmetry [6] and crystal symmetry [7] to three independent elements which are denoted x3111'(-w, w , w , -w), X ~ ' ~ ~ ~ ( - W , w , W , -w), and x31221(-w, w, w, -w). Here "1" or "2" denotes x, y , or z and 1 # 2. Hence, for cubic crystals of the above symmetry, the nonlinear polarization may be written in the form P,""(w)= {3x311'1EL(w)Ei(w)E,*(w) + 3X3'""',(W)E,(W)El*(W) + ~x~'~~~E~(w)E,*(w)E~(w)) exp ( ikz) . (2) The nonlinear propagation of an elliptically polarized beam in a cubic crystal is most easily understood by ex- pressing the incident field E(t) in the form of circularly polarized components, E(t) = Re ((E,,+e'm?+ +E,-&) exp (i(kz - ut))}. (3) Here i., = (a, 5 iay)/\/2 are the right and left circularly polarized unit vectors, E,, are the real amplitudes of the circularly polarized components of E(t) , and 4/2 is the angle at which the major axis of the polarization ellipse rests with respect to the x axis at the input to the medium. Substituting in (2) a trial solution similar in form to that given in ( 3 ) , and expressing the results in terms of circular modes for a beam propagating along a (100) crystal axis, z , one finds Authorized licensed use limited to: Universidade Federal de Alagoas. Downloaded on February 1, 2010 at 12:58 from IEEE Xplore. Restrictions apply. where E , are the complex amplitudes of the field at z, 1 El ' = Eo+2 + E,_' is fixed for all z , and higher order terms resulting in changes to E,,' are neglected. Substituting (4) into Maxwell's equations and assuming that the resultant refractive index change, 6n, is much less than n, the linear refractive index of the medium, one sees immediately that 6n is well defined for the two cases, 4 = 0 [ellipse oriented along (100) crystal axis] and 4 = n/2 [ellipse oriented along (1 lo)]. For 4 = 0 67r n an, = - { (x31111 + 2 x 3 1 2 1 s - X312a' 1 P I 1 + (j31111 - 2 ~ ~ ~ ' ' ' + x ~ ~ ~ ~ ~ ) E ~ , ~ ) (5a) and for d = 7r/2 an, = - { ( x 3 6~ 1111 + 2 X 3 1 a ~ a - X 3 ~ a a l n 1 P I 1 + 2 ~ 8 " ~ ' E 3 ~ ' ) . (5b) Hence, for an elliptically polarized beam oriented along these crystal axes, the induced birefringence expressed by (5) will cause the polarization ellipse to rotate. For small angles of rotation 0 one may then write [ 5 ] - = 2~ ( a n , - an-) de w dz where B = 3 ( x 3 1 1 1 1 - 2 x 3 1 2 1 2 + x3lZz1 ) for 4 = 0 and B = 6 ~ 3 " ~ ' for 4 = ~ / 2 . We note here that for isotropic media where it will be recalled that xal"'= 2x31212 + x31221, B becomes identically equal to 6 ~ 3 ~ ~ ~ ~ . It should be emphasized that ( 5 ) and (6) are valid only for small values of 8. For larger index changes I E,I ' # E,' and the coupling of energy between right and left circular modes cannot be neglected. 111. ELLIPSE ROTATION WITH FOCUSED BEAMS Rather than performing the experimental measurement of ellipse rotation with unfocused beams as implied by (6), it is advantageous to focus the beam into the sample so as to minimize the intensity at the sample surface. This is par- 1065 ticularly true in the case of solids which generally exhibit a low nonlinear index, thus making surface breakdown a potential problem at he high intensities required for measuring index changes. Also focusing minimizes the volume of strain-free material required in the study and thus relaxes the requirements on material quality. This is especially helpful when examining crystalline samples where imperfections may be avoided by proper position- ing of the sample. We consider the case of a focused Gaussian beam in the limit of sufficiently low power such t at the spatial distortion of the intensity distribution due to self-focusingis negligible. This assumption is quite reasonable since ellipse rotation measurements are generally made in an intensity range where the resultant induced phase shifts amount to only several degrees. Hence, we write the right and left circularly polarized components of the field in the form where Eo*(r, z) are of constant amplitude and change with position only in that they specify the phase shift induced by the nonlinear index changes; and U(r, z) is the Gaussian beam function given by U ( t , z) = - exp w o W(Z> Here w(z) = wo{l + (Xz/(~w,'n))'}''' is the l/e spot size, h being the free-space wavelength, R(z) = z(l + [mvo2n/(hz)]~) is the radius of the wavefront, and the phase factor y given by y = tan-' [Xz/(~w,'n)]. Since a Gaussian beam exhibits a Gaussian radial pro- file for all z , the energy flow in the beam may be traced by a set of rays defined by the hyperboloids, for various values of the parameter K . Equation (6) may be integrated along each set of rays to estimate the total amount of power rotated into a polarization orthogonal to that of the elliptically polarized input beam. For the case of a beam focused centrally into a sample of length L, the rotation angle OK for the set of rays parameterized by K is given by Authorized licensed use limited to: Universidade Federal de Alagoas. Downloaded on February 1, 2010 at 12:58 from IEEE Xplore. Restrictions apply. 1068 IEEE JOURNAL OF QUANTUM ELECTRONICS, NOVEMBER 1973 6.0 5.0 Fig. 3. (a) Oscillator (01) and amplifier (02) output monitor 20 ns/div. (b) Ellipse rotated signal (03) 5 ns/div. (c) Transmitted signal (04) 10 nx/div. Fig. 4. Graph of F versus (P)av for Owens-Illinois ED-4 glass ob- tained on one time-resolved laser shot. The data on the leading and trailing edges of the pulse are taken in 2-11s increments. The scaling fac- tor Fo e 8.7 X lo-' and (P)av = 1 is approximately 14.5 kW. ED-4 glass, and American Optical LSO glass are tabulated in Table 11. The absolute error in the measurement arising from data reproducibility, filter calibration, and error in the calibration standard is assessed to be 7 percent. The importance of using a calibration standard is underlined by the fact that the absolute values of (P)," given in Figs. 4 and 5 are approximately a factor of 2 larger than would be expected from (12). Such a discrepancy could easily occur Fig. 5. Graph of F versus (P)av for YAG with 6 = 7r/2 taking data only from the peaks of each laser shot. The scaling factor Fo 8.7 X and (F ' )av = 1 is approximately 15 kW. TABLE 11 RESULTS OF ELLIPSE ROTATION MEASUREMENTS Material S / O x 10" esu n at 6943 '*"+J 7.18 1.829 7.6 YAGpi,rr/2 a,b 6.34 1.829 Refractive Index Lecgth of Saqoles L ( m ) a,b 7.6 Fused Qcartz 1.23 1.455 11.0 Owens-Illinois ED-4 Glass 2.39 1.557 9.7 Ameri.can Optical LSO Glass 1.92 1.505 10.3 c s, 378 1.623 9.0 a For the purpose of comparing n, for YAG in various orientations we measure the anisotropy 2x312~21/(~31111 + x31221 - 2x8l2I2) = 0.88 f 0.04. The YAG samples are grown by Crystal Optics Research Inc. Dimen- sions are 9.5 X 9.5 X 76.2-mm cut with (100) faces and X-ray oriented to better than 0.5". The results for fused quartz are approximately 15 percent lower than those quoted in [3]. This discrepancy is presumably the result of a small error of 0.07 density in the attenuating filters employed in [3]. due to a reduction in focal intensity arising from beam truncation by the oscillator aperture or the input polariz- ing optics of the experiment [9]. Although all of the results cited on Table I1 are obtained without damage to the samples, stimulated backward Brillouin scattering is observed in the glass samples at higher input powers. Also we observe that the data are in- deed independent of focal parameters as predicted by (12) for the variety of focal length lenses used in this study. Since electrostrictive processes would be transient to a varying degree under this variety of experimental con- ditions, the consistency of our results provides an ad- ditional piece of evidence which, in conjunction with our time-resolved data, strongly supports the assertion that electrostriction is not causing an erroneous interpretation of the results. Authorized licensed use limited to: Universidade Federal de Alagoas. Downloaded on February 1, 2010 at 12:58 from IEEE Xplore. Restrictions apply. OWYOUNG: ELLIPSE ROTATION STUDIES Since two independent measurements are necessary to completely specify nonlinear refractive index changes in isotropic media (three in cubic crystalline media), it is im- possible to conclusively determine values of the nonlinear refractive index, n,, for a linearly polarized beam from ellipse rotation data alone. A supplementary measure- ment giving accurate ac Kerr, self-focusing, or spon- taneous Raman scattering results would provide the ad- ditional information necessary to resolve this problem [IO]. Until sufficient information becomes available to ascer- tain the physical mechanisms responsible for nonlinear in- dex changes in these materials, it is useful to estimate the nonlinear index n2 by using our ellipse rotation data and assuming a purely electronic mechanism. This is not an unreasonable assumption since third harmonic [ l l ] and three wave mixing [5], [12] studies in glasses and a variety of crystalline solids all indicate that the electronic contribu- tion is of the order of magnitude which is observed in this study. Yet the results of these studies are not extensive or accurate enough to conclude that the mechanism is purely electronic [ 101. Under the electronic assumption xs‘22‘= x31212(see [5] and [lo]), hence allowing us to solve x31111 in YAG and also to estimate x31111 = 3x3lZz1 in the glasses. If a nuclear non- linearity were found to yield an additional contribution to the nonlinear index change, this assumption would lead us to an estimate of n, which would be high or low, depen- ding upon whether the mechanism responsible for the nuclear nonlinearity is of a type which gives rise to a depolarized or polarized light scattering process [lo]. Since the nonlinear efractive index n2 for a linearly polarized beam is given by [13] x31111(- n, = - m, w , w , -0) (1 3) n one finds under the electronic assumption the values of n, X 1013 = 1.00, 1.73, 1.44, and 4.27 ESU for fused quartz, ED-4 glass, LSO glass, and YAGloo, respectively. The sub- script 100 denotes the fact that the electric field of the beam is directed along a (100) crystallographic direction. For propagation along the (100) direction of YAG with the field directed along a (1 10) axis one finds 67r n n2’ = - iX31111 + x31221 + 2X31212} (14) for which we obtain the value for YAG of n,’ = 4.1 X ESU, again assuming that apurely electronic mechanism is involved. It is to be noted that (14) is also valid for the propagation of a linearly polarized beam along the (1 11) direction of YAG which is the orienta- tion common to laser operation. In order to verify the assertion indicated by our results that n,’/n, = 0.96 f 0.015 we have made some preliminary measurements of damage in YAG using a linearly polarized beam and Rayleigh limit lenses ranging from 1069 24-38 cm. These data exhibit no observable.anisotropy as the field is oriented along the (100) or (1 10) crystal axes and are thus quite consistent with our ellipse rotation measurements. VI. CONCLUSIONS In this paper we have reported the first observation of ellipse rotation in a cubic crystalline medium and com- pared these results with measurements in several glasses. As typified by Fig. 4, the capability of time resolving the ellipse rotation data suggests a practical means of ob- taining rough (f 15-percent) measurements of ellipse rota- tion on a single laser shot, and the similarity of data between leading and trailing edges of the laser pulse es- tablishes that electrostrictive effects may be neglected in the present measurements. This conclusion is also sup- ported by the independence of ellipse rotation on focusing parameters. Moreover, extremely reproducible ellipse rotation measurements (5 5 percent) are obtained by averaging over many shots, thus permitting the resolution of a small but definite anisotropy in the third-order non- linear susceptibility of YAG. ACKNOWLEDGMENT The author wishes to thank R. W. Willey for technical assistance in the experimental phases of this work. REFERENCES P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity- dependent changes in the refractive index of liquids,” Phys. Reu. Left . , vol. 12,pp. 507-509, 1964. P. D. McWane and D. A. Sealer, “New measurements of intensity- dependent changes in the refractive index of liquids,” Appl. Phys. Lett.. vol. 8, pp. 278-279, 1966. A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Reu. B. , vol. 5, pp. 628-633, 1972. R. W. Hellwarth, A. Owyoung, and N . 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Owyoung, “The origins of the nonlinear refractive indices of li- guids and glasses,” M. s. thesis, California Institute of Technology, Pasadena, Dec. 1971. C . C. Wang and E. L. Baardsen, “Study of optical third harmonic generation in reflection,” Phys. Reu., vol. 185, pp, 1079-1082, 1969; also errata in Phys. Reu., vol. B1, p. 2827, 1970. H. Hauchecorne, F. Kerherve, and G. Mayer, “Measures Des Interactions Entre Ondes Lumineuses Dans Diverses Substances,” J . Phys. (Paris), vol. 32, pp. 47-62, 1971. C. C. Wang, “Nonlinear susceptibility constants and self-focusing of optical beams in liquids,” Phys. Rev., vol. 152, pp. 149-156, 1966. Authorized licensed use limited to: Universidade Federal de Alagoas. Downloaded on February 1, 2010 at 12:58 from IEEE Xplore. Restrictions apply.