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Piezoelectric Contributions to Pulsed Degenerate Four-Wave Mixing
Swiss Federal Institute of Technology, ETH-Hönggerberg,
CH-8093 Zürich, Switzerland
(Received 6 November 2000)
In solid state materials lacking a center of symmetry, cascaded second order nonlinear optical effects [1,2,3], especially the combination of optical rectification and the linear electro-optic effect, are an important contribution to Degenerate Four-Wave Mixing (DFWM) [4,5], an experimental tool widely used to measure the third-order nonlinear optical susceptibility. This quantity determines the magnitude of several interesting phenomena , among which the optical Kerr effect, self-phase modulation, soliton formation, and the interaction of different optical waves, with applications like optical limiting and all-optical switching.
In order to extract an accurate value of the third-order susceptibility of noncentrosymmetric materials from a DFWM experiment, it is necessary to take into account the peculiar geometry dependences introduced by second order effects [4,5,7]. They can both lead to misleading results, if ignored, or to a alternative possibility for calibrating DFWM experimental set-ups, if properly taken into account . Some predicted effects of the electro-optic and optical rectification contributions to DFWM have been observed in Refs. 4,5,7. This letter now reports the experimental observation of another, associated effect: the nanosecond fast piezoelectric relaxation whose contributions to DFWM was calculated in Ref. 5. This is important because the possibility of piezoelectric relaxation introduces additional dependencies of the DFWM signal from the angle between the interacting beams and from the pulse-length. This must be taken into account in order to be able to compare experimental results obtained with different DFWM set-ups.
At the origin of the piezoelectric contributions discussed in this work are two space-modulated rectified polarizations induced by two pairs of interacting waves in DFWM. For the experimental configuration shown schematically in Fig. , the rectified polarizations can be described by two static (on the time scale of the pulse length used in DFWM) plane waves with wavevectors ka and kb. [4,5] Approximating the interacting beams and the generated polarizations by plane waves with real amplitude (1/2)A(w, k) exp[i ( t - k x)] +c.c. and complex amplitude A(w, k), one can write the complex amplitude of the rectified polarization with wavevector ka as
where e0 is the permittivity of vacuum, En( w, k) is the component of the electric field amplitude along n, En(- w, - k) is its complex conjugate, and cpjl(2)(0, - w, w) is the second order susceptibility describing optical rectification and the linear electro-optic effect. It is related to the standard electro optic coefficients rijk by ckij(2)(0, - w, w) = - (1/2) ni2 nj2 rijk, where ni is the refractive index for light polarized along the i-axis. An analogous expression gives the rectified polarization with wavevector kb. 
The electro-optic interaction of one of the input beams with these optical rectification gratings contributes to DFWM, and can be represented by two effective nonlinear optical susceptibility tensors, c kaijkl and c kbijkl, which must be added to the genuine third-order susceptibility c(3)ijkl to get an effective susceptibility tensor c(3), EFFijkl that describes DFWM in noncentrosymmetric materials. More details about the nomenclature used here for these tensors can be found in Ref. 5. The results of Ref. 5 can be cast in the form:
where rijk and eij are effective electro-optic and dielectric tensors to be discussed below, dqp=0, q<>p;dqp=1,q = p, and zqp(k) is
Piezoelectric effects become important when the propagation time of an acoustic elastic wave over a distance equal to the modulation period of the rectified polarization (L = 2 p/ka for Eq. (1) is shorter than the laser pulse length used. In this case a strain pattern can be established in the crystal while the interacting pulses are still present, and it leads to additional contributions to the DFWM signal. These additional contributions can be included in the limit of long wavevectors by using, in Eqs. (2)-(4), effective electro-optic and dielectric tensors that take into account that the crystal textemdash\ because of the sinusoidal spatial modulation of the rectified polarizations textemdash\ is neither in a completely strain-free case, nor in a completely stress-free case. These effective tensors can be calculated by 
where k is the wavevector of the corresponding optical rectification grating (see Eq. (1)), rSijk is the electro-optic tensor at constant strain, eSij is the dielectric tensor at constant strain, ki=ki/k, pijkl is the elasto-optic tensor, eijk is the piezo-electric tensor, Aik = CEijkl kj kl, and CEijkl is the elastic stiffness tensor. For short wavevectors the crystal has no time to elastically relax, and rijk(k)=rSijk, eij(k)=eSij.
This leads to a dependence of the third-order susceptibility determined in DFWM on the length of the wavevector of the rectified polarization grating, a fact that influences DFWM measurements in various ways. As an example, in the arrangement of Fig. 1 the wavevector kb is generally quite long, and piezoelectric relaxation can already become possible in some picoseconds. The kb contribution will thus be different for experiments performed, e.g., with 100 ps pulses and for experiments performed with 1 ps pulses. However, to directly observe such an effect, one would need to tune the pulse length, which is impractical.
To experimentally demonstrate the existence of the piezoelectric contribution, and to relate its magnitude to the predicted theoretical value, it is better to focus on the rectified polarization grating with the shorter wavevector ka. When the duration of the laser pulse is similar to the acoustic wave propagation time discussed above, the effect of piezoelectric relaxation can be observed by using a nanosecond laser in the experimental configuration of Fig. 2 and measuring the DFWM signal as a function of the angle b between the interacting beams (which is related to the spatial period L = 2 p/ka of the optical rectification grating by L = l/[2 sin (b/2)], with l the laser wavelength in vacuum).
Barium Titanate (BaTiO3) is a good candidate to observe these effects because its electro-optic, piezoelectric, and elastic properties are well characterized, and because it has large electro-optic coefficients. All the relevant material parameters have been tabulated in Ref. 8 and the dispersion of the electro-optic coefficients is given in Ref. 9. BaTiO3 has tetragonal symmetry with point group mm4, with the c-axis parallel to the ferroelectric polarization. The piezoelectric contribution is expected to be particularly strong for c ka || c1133, i.e. for the contribution of the optical rectification grating with wavevector ka when the c-axis of the crystal is parallel to ka. Numerical values are given below. They were all calculated using the material parameters in Ref. 8 and in Ref. 9 at a wavelength of 1.06 mum.
The DFWM experiments were realized with a Q-switched laser producing 7 ns long pulses at a wavelength of 1.064 mum and the experimental set-up shown in Fig. 2. This set-up has been especially constructed to allow a relatively easy adjustment of the angle between the beams while keeping the path lengths of all beams, and therefore the relative coherence of the interacting pulses, constant. To measure the effective susceptibility c(3), EFF1133, beams 1 and 2 were polarized in the plane of incidence, while beams 3 and 4 were polarized perpendicular to it (by the half-wave plate marked ``HW'' in Fig. 2.
For the BaTiO3 crystal oriented as in Fig. 2, with its c axis in the beam incidence plane, c ka || c1133 contributes to the signal, and it is proportional to (r131)2/(e11+2) (see Eq. (1)). It can be calculated to be c ka || c1133 = (224 ± 45) x 10-22 m2/V2 when the strain-free tensor components are used . When elastic relaxation is allowed, the effective electro-optic and dielectric coefficients of Eqs. (5) and (6) must be used, resulting in c ka || c1133 = (355 ± 70) x 10-22 m2/V2. The errors are estimated on the basis of a ~10% uncertainty in the dielectric and electro-optic tensors . The inclusion of piezoelectric effects does not lead to larger uncertainties because textemdash\ in this particular case in BaTiO3 textemdash\ only a shear deformation can be induced, and the effective tensors calculated from Eqs.(5)-(6) correspond to the stress-free values, which can be taken directly from Ref. 9. The magnitude of the total piezoelectric contribution in this case is thus expected to be (130 ± 60) x 10-22 m2/V2 (the error takes into account that this value is calculated from a difference, but also that any errors in the strain-free and stress-free values of the material tensors are not necessarily uncorrelated). The contribution from the rectification grating with wavevector kb is fixed at c kb perp c1133 = (224 ± 45) x 10-22 m2/V2 because there any piezoelectric contribution vanishes by symmetry, as follows from Eq. (5). The experimental value of the effective DFWM third-order susceptibility should thus increase by approximately (25 ± 13)% over its (560 ± 80) x 10-22 m2/V2 strain-free value  when the wavevector of the rectified polarization increases to a magnitude that allows for elastic relaxation during the pulse duration.
The beam intersection angle when this increase takes place is given by the modulation period L = 2 p/ka of the rectified polarization grating for which the propagation time of a transverse acoustic wave is similar to the pulse duration of 7 ns. From the relevant speed of sound of a transverse wave, v=[CE1313/r]1/2=3.18 mu m/ns (r is the mass density), one gets L ~ 20 mm. This corresponds to a beam intersection angle b between 50 and 60 mrad.
An important fact that can be exploited for the measurements is that when the BaTiO3 crystal is oriented with its c axis perpendicular to the beam incidence plane, c ka perp c3311 vanishes almost completely . For this crystal orientation no dependence of the DFWM signal from the beam intersection angle b is expected. In addition, for both c || ka and c perp ka, c kb perp c1133 is the same and independent of the angle between the interacting beams.
It follows that the effects of possible variations of mirror reflectivity and/or beam overlap conditions when changing the beam intersection angle can be avoided by measuring the DFWM signal for both c || ka and c perp ka and taking the ratio between the two values. Because of the expected constancy of the signal for c perp ka, the data measured in such a way should directly reflect the dependence of c ka || c1133 on the magnitude of ka, or the beam intersection angle b.
The results are shown in Fig. 3. The data points are averages over several measurements. The effect of acoustic phonons for longer wavevectors can clearly be seen as the increase of the third-order susceptibility when the angle between the beams increases. The transition from a lower to a higher susceptibility value does indeed take place for a period of the rectified polarization grating around 20 mum, as expected from the estimation above. The ratio between the two susceptibility values at short and long wavevectors is higher than the value calculated above, but is within the expected uncertainties. The value of c ka || c1133/\c ka perp c3311 measured at short wavevectors is somewhat lower than what is expected from the calculated values of the second order contributions  but this small discrepancy is also within the experimental error.
The piezoelectric effect whose signature is visible in Fig. 3 is peculiar to DFWM in noncentrosymmetric materials. In contrast to the example of thermal gratings, where textemdash\ given short enough pulses textemdash\ acoustic phonons can be induced and detected (as an oscillation of the signal intensity as a function of the delay time of the probe pulse) no matter what the magnitude of the wavevector ka is, in the process observed here acoustic phonons only contribute to the signal if their characteristic frequency (given by the the speed of sound divided by the spatial period of the rectification grating) is higher than the frequency content of the laser pulses. Their effect can only be seen as a dependence of the DFWM signal from the pulse length, or from the angle between the beams.
It should be noted that in this experiment the beams inducing the rectified polarization gratings are always polarized perpendicular to each other, so that there is no light intensity modulation with wavevector ka or kb. A generation of acoustic phonons by transient temperature gratings can therefore be excluded, together with any other effects relying on spatially inhomogeneous energy deposition.
In conclusion, the measurements presented above give a clear demonstration of the existence of piezoelectric contributions to pulsed DFWM, which must be taken into account in order to determine reliable values of third-order susceptibilities in noncentrosymmetric materials, and to be able to compare DFWM experiments performed with different pulse lengths or experimental setups.