Albert et al.

Vol. 13, No. 1 / January 1996 / J. Opt. Soc. Am. B

29

Time-resolved nonlinearities from nonsymmetric polar phonons in PbTiO3 and KNbO3 perovskites O. Albert, M. Duijser, J. C. Loulergue, and J. Etchepare ´ Unite´ de Recherche Associ´ee 1406, Laboratoire d’Optique Appliquee, Centre National de la Recherche Scientifique, Ecole Polytechnique, Ecole Nationale ´ ´ Superieure de Techniques Avancees, Centre de l’Yvette, 91120 Palaiseau, France Received February 16, 1995; revised manuscript received August 28, 1995 We present time-resolved measurements of low-frequency nonsymmetric phonon polaritons driven impulsively in perovskite crystals. The definition of the phonon wave vector as a function of the excitation wavelength is addressed for the case of one and two pump beams. Results obtained with KNbO3 and PbTiO3 for B2 and E symmetry modes, respectively, are presented, and the value of the lowest wave vector that can be reached for a specific wavelength of the pump beams is discussed.  1996 Optical Society of America

1.

INTRODUCTION

There has been renewed interest in recent years in the analysis of phonon modes by nonlinear spectroscopy. In particular, the development of ultrashort temporal laser pulses has permitted the direct impulsive excitation of low-frequency phonons; moreover, the excitation process of Raman resonances is increased in the case of polar phonons as a result of the generation of infrared radiation by x s2d nonlinearities. It has been demonstrated that a three-beam scheme is ideally suited for the study of phonon – polariton dispersion near the Brillouin zone center.1 – 5 Pump – probe experiments have also been performed to characterize polar and nonpolar phonons in dielectrics, semiconductors, and metals.6 – 10 In all cases in which the signal is at the same wavelength as the probe, the physical process addressed during the detection procedure involved the variation of the refractive index of the material under study. In fact these nonlinearities have been analyzed in the time domain mainly through the creation of a transient index grating or electro-optic characteristics and by reflectivity or spatial profile changes experienced by a probe beam. To our knowledge, most of the studies published on impulsively driven phonons were devoted to symmetric motions. Moreover, it has been shown that in several cases only A1 symmetric phonons were concerned, no matter what the polarization direction of the pump beam was with respect to the crystal axis.8,9 Only Dougherty et al.3 recently published a transient grating analysis of KNbO3 (and BaTiO3 ) in which B2 (and E) modes were driven by ultrashort pulses. Unfortunately, in the case of KNbO3 the high dispersion between the frequencydependent ordinary and extraordinary refractive indices prevented Dougherty et al. from access to low values of the wave vector in a region where the phonon – polariton characteristics are mostly of interest. We present examples of excitation of nonsymmetric phonon modes and discuss several of the specific problems related to the use of ultrashort temporal pulses. We focus first on the exact definition of the phonon wave 0740-3224/96/010029-05$06.00

vectors involved, in the general case of a three-beam experimental setup. The parameters that are analyzed are the angle between the two driving pump beams, their spectral width, and the value(s) of the Raman resonance(s) involved. The discussion is afterward specialized to a pump – probe experiment. Examples of measurements in which three and two beams are used are described. Two materials, characterized by high (KNbO3 ) and low (PbTiO3 ) birefringence, are measured.

2. PHONON WAVE-VECTOR CHARACTERISTICS The easiest way to drive wave-vector-dependent nonsymmetric phonon polaritons lies in the use of a twopump-beam arrangement with two linear polarizations perpendicular to each other. For the excitation of true polaritons, as opposed to quasi-mode behavior, crystal anisotropy requires a geometry in which the induced polarization is perpendicular to the polariton propagation wave vector. We present in Fig. 1 the case of PbTiO3 , 1 for which the spatial symmetry is 4mm (or C4v ); the use of polarization components along the y and z directions leads to the excitation of the c coefficient of the Raman polarizability tensor, whereas the induced polarization is along the y axis. As the driven polariton is propagating in the xz plane, one is concerned with true TO phonons. For KNbO3 , whose spatial symmetry is mm2, polarization configurations and symmetry rules are similar, as far as one is concerned with B2 modes. The refractive indices that the two pump beams p1 and p2 experience, namely, n1 (ordinary) and n2 (extraordinary), are different. Therefore the excited phonon q vector can be described by two components, qk and q' , respectively, along and perpendicular to the direction of propagation of the beams: q' ­ 2vL sinsuy2d , qk ­ vL hfn1 2 2 sin2 suy2dg1/2 2 fn2 2 2 sin2 suy2dg1/2 j , where u is the angle between the pump beams, external  1996 Optical Society of America

30

J. Opt. Soc. Am. B / Vol. 13, No. 1 / January 1996

Albert et al.

a function of three parameters, say, v, V, and u: qk a,b sv, V, ud ­

Z

1`

2`

dv

I sv 1 Vy2dIsv 2 Vy2d I svL 1 Vy2dIsvL 2 Vy2d

3 hsv 6 Vy2dfn1 2 sv 6 Vy2d 2 sin2 suy2dg1/2 2 sv 7 Vy2dfn2 2 sv 7 Vy2d 2 sin2 suy2dg1/2 j ,

Fig. 1. Polarization configuration: pump pulses p1 and p2 have their linear polarization directed along the y and z axes. 1 Upper left: Polarizability tensor for PbTiO3 sC4v d and KNbO3 14 sC2v d. Upper right: Index ellipsoid in the yz plane. Lower right: Components of the phonon – polariton wave vector.

to the crystal, and vL is the central frequency of the laser. A more general formulation has to be built up to take into account the dispersion of the refractive indices as a function of the actual wavelengths used to drive a specific phonon of frequency V. If we ascertain that the maximum efficiency of the excitation process is obtained by choice of the frequency pair svL 6 Vy2d, two equations result: qk a ­ k1 a 2 k2 a and qk b ­ k1 b 2 k2 b , where qk a ­ hsvL 1 Vy2dfn1 2 svL 1 Vy2d 2 sin2 suy2dg1/2 2 svL 2 Vy2dfn2 2 svL 2 Vy2d 2 sin2 suy2dg1/2 j , qk b ­ hsvL 2 Vy2dfn1 2 svL 2 Vy2d 2 sin2 suy2dg1/2 2 svL 1 Vy2dfn2 2 svL 1 Vy2d 2 sin2 suy2dg1/2 j . These equations lead to two different numerical values. In fact these two waves interact in the crystal to create a resultant wave with a definite wave-vector value. These waves can be described as P1 ­ A expfisqk a x 1 q' zdg and P2 ­ A expfisqk b x 1 q' zdg, and the result of the interaction as P ­ A expfisqk x 1 q' zdgcos Dqk x, with qk ­ 1y2sqk a 1 qk b d and Dqk ­ 1y2sqk a 2 qk b d. More generally, there exists an infinity of frequency pairs inside the spectral width of the pulses, which can drive a specific phonon of frequency V. As a result, the qk -vector component is

where I svd corresponds to the intensity spectral profile of the pulses. Integration over the spectral width of the pulses leads to a spread in wave-vector values. As a consequence, several phonon–polariton frequencies are driven in the same excitation geometry, which leads to the appearance of damping. We applied numerical calculations to the lowest E (and B2 ) polariton branches of PbTiO3 (and KNbO3 ), using Fourier-transform-limited sech2 styT d pulses with T ­ 0.45 ps centered at 620 nm. Several numbers are given in Table 1 for illustration. Dq values are roughly independent of u and V. Therefore the polaritons from various branches, which are driven at a same u value, are characterized by a same wave vector, in contrast to what can be observed in a spectral-domain measurement. The relaxation time, which results from the spread in the driven polariton frequencies, is larger and larger as u and q values increase. Its influence on the measured relaxation time of the polariton is always negligible for the case of PbTiO3 ; on the contrary, at the lowest q value accessible for KNbO3 , which corresponds to the phononlike regime, its contribution to the effective phonon damping measurement reaches 10%. The above expressions describe the creation of a polariton with wave vector q ­ k1 2 k2 . An equally intense polariton is also created in the opposite direction: 2q ­ 2sk1 2 k2 d. The two counterpropagating polaritons lead, by interference, to a stationary wave that will be probed by the test pulse. This approximation is in fact valid only when the probe beam size is smaller than the pump beam size in the interaction region. The smallest value of the wave vector is indeed reached in a one-pump-beam experiment. In this case the exciting beam is directed along the x axis and its polarization at 45± from z (the c axis of the crystal). Therefore only the longitudinal (along the x axis) component of the wave vector has a nonzero value. As the induced polarization is directed along y, we are still concerned with a true TO mode. The actual wave-vector value depends on the dispersion of the refractive indices11,12 at the central wavelength of the laser that is used. At 620 nm it is 163 cm–1 for PbTiO3 and 2622 cm–1 for KNbO3 . With a Ti:sapphire laser the latter value will be 1843 cm–1 inside the dispersive portion of the polariton curve. We can nevertheless conclude that, in the case of highly dispersive

Table 1. Numerical Values of Dqk for the Lowest E and B2 Nonsymmetric Phonon – Polariton Modes in PbTiO3 and KNbO3 a uy2 ­ 0±

uy2 ­ 0.25±

uy2 ­ 0.8±

Material

TO1 scm21 d

Dqk scm21 d

q scm21 d

T2 (ps)

q scm21 d

T2 (ps)

q scm21 d

T2 (ps)

PbTiO3 KNbO3

88 50

9 60

168 2646

31 9

219 2650

52 26

480 2684

155 26

a Additional relaxation times that are due to the spread in the number of driven frequencies have been calculated for two-beam and three-beam experiments with pump pulses at 620 nm.

Albert et al.

Vol. 13, No. 1 / January 1996 / J. Opt. Soc. Am. B

materials, one can still describe the wave-vector dependence of the phonon polariton, using a pump – probe experiment, by tuning the pump’s central wavelength from the visible to the infrared region.

3. DETERMINATION OF THE REFRACTIVE-INDEX CHANGE Three-beam experiments have been performed in the wellknown transient grating scheme, and we will not go into further details here. Two-beam experiments have also been performed to reach the smallest value of q compatible with the dispersion of the refractive indices. As we stated above, the pump beam polarization direction is 45± to the direction of the c axis of the crystal (in the yz plane) to drive nonsymmetric phonons. On the action of two pump fields Ep1 and Ep2 p with components along y and z, the index ellipsoid will rotate by an angle f about the x axis in the yz plane. The value of the ellipsoid is13 tan 2f ­

31

rise to a problem only in the detection process, when the probe polarization is not directed along a principal axis of the ellipsoid index, a situation that occurs when symmetric phonons are driven; the dynamics of the signal depends there on the efficiency of the compensator.

4.

EXPERIMENTAL RESULTS

We present results obtained with two ferroelectrics whose birefringence and dispersion parameters are drastically different at the pump wavelength used. The characteristics of the laser are as follows: the central wavelength is 620 nm, the temporal pulse width is 80 fs, and the energy density of the pump beams impinging upon the sample is of the order of 10 GWycm2 . Figure 2 shows two different measurements performed on KNbO3 crystal. The first [Fig. 2(a)] was obtained in a fourwave mixing scheme with an angle uy2 ­ 0.8± between

2r44 Ep1 Ep2 p , s1yn1 d2 2 s1yn2 d2

where r44 is the symmetry-allowed quadratic electrooptic (or Kerr) coefficient. This effect holds at least for nonresonant electronic third-order nonlinearities and nonpolar phonon modes. In the case of polar modes the transient field E sVd associated with the driven phonon polariton at frequency V creates a refractive-index change by means of the linear electro-optic effect. According to selection rules, the index ellipsoid will still rotate about the x axis in the yz plane by an angle f, whose value is now given by tan 2f ­

2r42 E sVd , s1yn1 d2 2 s1yn2 d2

where r42 is the contribution of the V phonon to the linear electro-optic coefficient. To measure this refractive-index change we direct the probe beam polarization along the c axis and measured the signal with an analyzer tuned at 90± along the b axis. A configuration rotated by 90± will give rise in fact to comparable results. This crossed polarization scheme for the probe and signal beams is reminiscent of Kerrgate experiments, in which the anisotropy induced in a material is measured through the polarization change encountered by the probe beam. We recall that, as in a transient grating experiment, the measured signal in this scheme is proportional to the square of the induced phase change. Finally we briefly compare this experimental setup with techniques developed by Auston and Nuss6 and Cheung and Auston.7 Although the first of them is devoted to the excitation of nonsymmetric polar phonons, and the other to symmetric ones, both can be used to derive symmetric or nonsymmetric phonons. In each case the analysis of the signal is performed through a polarization change measurement. The use of homodyne or heterodyne detection, the choice of which depends only on the characteristics of the laser source, leads to a signal that is quadratic or linear with respect to the refractive-index changes. In each case, also, the static birefringence gives

Fig. 2. Time-resolved nonlinearities for KNbO3 on a semilogarithmic scale. (a) Diffraction efficiency obtained in a three-beam experiment with q ­ 2684 cm21 . (b) Transmission of a probe beam through an analyzer in a two-beam experiment for which q ­ 2646 cm21 .

32

J. Opt. Soc. Am. B / Vol. 13, No. 1 / January 1996

Fig. 3. Time-resolved nonlinearities for PbTiO3 on a semilogarithmic scale. (a) Diffraction efficiency obtained in a three-beam experiment with q ­ 215 cm21 . (b) Transmission of a probe beam through an analyzer in a two-beam experiment for which q ­ 163 cm21 .

the pump beams. The associated wave-vector value, 2684 cm–1 , stands in a region where the photon contribution to the polariton is of no account. The values of the frequency V and the relaxation time T, determined from the data, are 43.8 cm–1 and 0.25 ps, respectively, in agreement with already reported time-domain experiments conducted under nearly the same conditions.3 Figure 2(b) corresponds to a one-pump beam with a probe in a Kerr-gate configuration. We obtain, for the driven polariton, a value close to that of Fig. 2(a), a fact that confirms that we are in the phononlike low-dispersion region. The behavior of PbTiO3 is drastically different. The smallest angle between the pump beams that was feasible in our transient grating apparatus, uy2 ­ 0.246±, was used. It corresponds to a wave vector of 215 cm–1 . The complicated signal observed [Fig. 3(a)] has a beating behavior. We will comment here only on the origin of these beats. An exhaustive analysis of the q-dependent polariton behavior will be published elsewhere.14 We demonstrate there that, unlike in previously published results,15 the lowest phonon – polariton frequency tends toward small values with no resonance in the damping. Such a characteristic signal was observed previously in time-resolved experiments, particularly for LiTaO3 and LiNbO3 and for KNbO3 . We address these two cases to enlighten the discussion on the origin of the signal observed in E mode PbTiO3 phonon polaritons. For the

Albert et al.

first case, Bakker et al. found5 that the existence of resonances located inside the spectral region that can be driven by the pump pulses is a consequence of an extreme anharmonicity of the potential associated with the same lattice vibration; this potential consists of several wells. Because of heterodyne detection of the setup, the existence of a signal at vi 2 vj is therefore a signature of polariton beats. For the second case16 the use of subpicosecond pulses for the excitation of A1 modes in KNbO3 permitted the two lowest (v1 and v2 ) lattice phonon modes to be driven simultaneously. The appearance of a signal at sv1 2 v2 d is only a consequence of the quadratic nature of the signal inherent in our technique of measurement. We have indeed demonstrated that, in a transient grating experiment, four peaks could be detected in the Fourier transform of the temporal signal, at frequencies 2v1 , 2v2 , v1 2 v2 , and v1 1 v2 . For the case with which we are concerned here, the behavior of the temporal signal is due to the same interference from the two lowest phonon – polariton modes, located, respectively, at 21.8 and 133.5 cm–1 . The first one corresponds to the lowest branch, whose TO frequency is 88 cm–1 ; it has a pronounced photonlike character. The second polariton is situated on the still phononlike part of the upper branch, close to the first LO frequency at 128 cm–1 . We specify that previously published1,17 temporal-domain experiments performed on the lowest A1 -symmetry phononpolariton branch in PbTiO3 did not present any beating behavior, although the anharmonicity of this A1 phonon potential was stressed recently.18 Figure 3(b) represents results obtained in a two-beam experiment (Kerr-gate configuration). The corresponding q value, which reduces to the qk component, is 163 cm–1 , very close to the Brillouin zone center. The phonon – polariton frequencies obtained, 18 and 132 cm–1 , are close to their respective limits: 0 cm–1 and vLO .

5.

CONCLUSION

We have presented the results of time-resolved experiments on the excitation of nonsymmetric phonon polaritons in PbTiO3 and KNbO3 by an impulsive excitation process. A precise study of the wave vector associated with the driven phonons led to the definition of a component whose value depends on the pulse’s spectral width, the phonon frequency, and the geometrical angle between the pump beams. We have seen that a two-beam setup allows the lowest value of the wave vector accessible for a given pump wavelength to be reached. This fact permits the determination of ´svd close to null frequency for the lowest branch of the phonon – polariton dispersion and therefore the determination of the lowest-frequency mode contribution to the static dielectric constant. When several polariton branches are described, this scheme also permits a determination of v values close to vLO phonons. The wavelength used in this experiment led, in the case of PbTiO3 , to the excitation of phonons with very low wave-vector values. We have shown that such phonon polaritons still have low damping values, contrary to reports of previous experiments in which Raman spectroscopy was used. This fact demonstrates the advantages of a temporal-domain nonlinear spectroscopy

Albert et al.

technique. We have seen that 620 nm is not an adequate pump wavelength at which to study phonon – polariton dispersion in KNbO3 because of the large difference between no and ne in this spectral range. One can nevertheless overcome this point and study the q-dependent behavior of highly dispersive materials by merely tuning, in a pump – probe arrangement, the wavelength of the pump beam from the visible to the infrared spectral region. J. C. Loulergue is also with Centre Lorrain d’Optique et d’Electronique du Solide, Universit´e de Metz-Supelec, 2 rue Belin, 57078 Metz, France.

REFERENCES 1. J. Etchepare, G. Grillon, A. Antonetti, J. C. Loulergue, M. D. Fontana, and G. E. Kugel, Phys. Rev. B 41, 12362 (1990). 2. T. P. Dougherty, G. P. Wiederrecht, and K. A. Nelson, Ferroelectrics 120, 79 (1991). 3. T. P. Dougherty, G. P. Wiederrecht, K. A. Nelson, M. H. Garret, H. P. Jensen, and C. Ward, Science 258, 770 (1992); Phys. Rev. B 50, 8996 (1994). 4. P. C. M. Planken, L. D. Noordam, J. T. M. Kennis, and A. Lagendijk, Phys. Rev. B 45, 7106 (1992).

Vol. 13, No. 1 / January 1996 / J. Opt. Soc. Am. B

33

5. H. J. Bakker, S. Hunsche, and H. Kurz, Phys. Rev. B 48, 13524 (1993); B 50, 914 (1994). 6. D. H. Auston and M. Nuss, IEEE J. Quantum Electron. 24, 184 (1988). 7. K. P. Cheung and D. H. Auston, Phys. Rev. Lett. 55, 2152 (1985). ˆ 8. G. G. Cho, W. Kutt, and H. Kurz, Phys. Rev. Lett. 65, 764 (1990). 9. H. J. Zeiger, J. Vidal, T. K. Cheng, E. P. Ippen, and G. Dresselhaus, Phys. Rev. B 45, 768 (1992). 10. O. Albert, D. P. Kien, J. C. Loulergue, and J. Etchepare, Opt. Commun. 114, 315 (1995). 11. S. Singh, J. P. Remeika, and J. R. Potopowicz, Appl. Phys. Lett. 20, 135 (1972). ¨ 12. B. Zysset, I. Biaggio, and P. Gunter, J. Opt. Soc. Am. B 9, 380 (1992). 13. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 230 – 234, 237. 14. J. C. Loulergue, O. Albert, and J. Etchepare, “E-symmetry phonon – polaritons in PbTiO3 ,” Phys. Rev. B (to be published). 15. D. Heiman and S. Ushioda, Phys. Rev. B 17, 3616 (1978). 16. D. P. Kien, J. C. Loulergue, and J. Etchepare, Opt. Commun. 101, 53 (1993). 17. D. P. Kien, J. C. Loulergue, and J. Etchepare, Phys. Rev. B 47, 11027 (1993). 18. C. M. Foster, Z. Li, M. Grimsditch, S. K. Chan, and D. J. Lam, Phys. Rev. B 48, 10160 (1993).