Copyright The American Physical Society 1997. All rights reserved. Except as provided under U.S. copyright law, this work may not be reproduced, resold, distributed or modified without the express permission of The American Physical Society. The archivial version of this work is published in Phys. Rev. Lett. 82, 193 (1997).
PRL online  Relevant Issue, 82(1)  This paper NonLocal Contributions to Degenerate Four Wave Mixing in Noncentrosymmetric MaterialsIvan Biaggio
(Received 11 September 1998)
We first point out the origin of second order contributions and give general equations for calculating them for the most important DFWM setups, correcting and completing some expressions given in Ref. 1. We then discuss the piezoelectric relaxation of the crystal as it is constrained by the boundary conditions defined by the DFWM experimental geometry, and calculate the effective electrooptic and dielectric tensors to be used when relaxation to a new elastic configuration is possible. Lastly, we demonstrate experimentally, in the well known electrooptic materials BaTiO_{3} and KNbO_{3}, how these results can be used to obtain absolute third order susceptibilities by comparing to independently determined electrooptic and dielectric properties, without using a reference material. We start with the definition of the third order nonlinear optical susceptibilities. Consider an electric field that is a sum of three plane waves with wave vectors k_{i} and frequency w:
The field of (1) induces a timedependent third order polarization [2,3]:
Figure 1 shows two ways of arranging the three input waves so that the nonlinear polarization at frequency w and wavevector k_{4}=k_{1} + k_{2}  k_{3}, with complex amplitude P(w,k_{4}), radiates the signal wave in a phase matched way over the whole sample thickness. Inserting (1) into (2), and collecting the terms with frequency w and wavevector k_{4}, one finds:
Our S.I. expressions use the same convention as in Ref. 4. They go over to the ones in electrostatic units (e.s.u.) of Refs. 2,3, used in a relevant part of the literature, with the substitution ^{}, while numerical values are converted using the rule
In a noncentrosymmetric material, the field of (1) also leads to a second order polarization. The only part that gives a large phase matched contribution to DFWM is induced by optical rectification, and it consists of the sum of two components oscillating in space like a plane wave. Their complex amplitudes are
We can now write c^{casc, ka}_{ijkl} and c^{casc, kb}_{ijkl} for k_{a} and k_{b} parallel to a main axis of the dielectric tensor. For P^{(OR)}(w=0,k) perpendicular k (transversal polarization):
In (11)(14), r_{ijk} and e_{ij} are electrooptic and dielectric tensors at constant strain when short enough pulses are used, but they are effective tensors including acoustic phonon contributions when the spatial periods 2p/k_{a} or 2p/k_{b} are so small that elastic deformations can build up during the laser pulse length. This can already be the case for 100 ps pulses for one of the contributions of Fig.1a: 2p/k_{b} can reach 0.1 µm inside the crystal for a light wavelength of ~0.5 µm, and an acoustic wave with a speed of 5 km/s travels that distance in 20 ps. These effective tensors are not simply the directly measurable ones at constant stress (unclamped) because only certain acoustic phonon contributions are allowed. The electric and strain fields in the crystal must have in our case a planewave spatial dependence E(k)= E exp(i k x) and u(k)= u exp(i k x) with the complex (vectorial) amplitudes E and u. This boundary condition breaks the symmetry of the crystal, leading to modified, wavevector dependent r_{ijk} and e_{ij} tensors with a lower symmetry. A similar problem has been treated in Ref. 6 assuming an electric field always parallel to the modulation wavevector. In our case the rectified polarization can also be perpendicular to the wavevectors k_{a} and k_{b} and we need to treat the problem in general by looking at the general relationships between electric field, strain, and dielectric tensor. The electrooptic tensor r^{S}_{ijk} at constant strain and the elastooptic tensor p^{E}_{ijkl} at constant electric field determine the change in the dielectric tensor induced by the spatially sinusoidal electric and strain fields:
By calculating u(k) from E(k), we can relate the amplitude of the dielectric tensor modulation (15) to the amplitude of the electric field modulation E with an effective electrooptic tensor r_{ijk} defined by D e^{1}_{ij}= r_{ijk}E_{k}. The stress tensor T_{ij} can be expressed as a function of the strain tensor S_{kl}=(u_{kl}+u_{lk})/2 and the electric field by means of the elastic stiffness tensor at constant electric field C^{E}_{ijkl} and the piezoelectric tensor e_{ijk}:
Defining ^{} and ^{}, and substituting in (17), reveals the system of three linear inhomogenous equations A_{ik} u_{k} = b_{i}. The matrix A_{ik} is symmetric and can be inverted [7] to obtain the solution u_{k}=A_{ki}^{1}b_{i}, which is then inserted in (15) to get the effective electrooptic coefficient we were looking for:
In centrosymmetric materials, any coefficient c^{(3)}_{ijkl} defined in (3) can be measured in DFWM by appropriately choosing the polarizations of the 3 input beams and the signal beam in the sample referenceframe. This is not true for noncentrosymmetric materials. Even when the polarizations are kept constant in the sample referenceframe, the second order nonlocal contributions depend on the wavevector differences k_{a} and k_{b}, and change with sample orientation, DFWM setup, and pulselength. To fix the ideas, we calculate the influence of these parameters for tetragonal BaTiO_{3} and orthorhombic KNbO_{3}. We use the r^{S}_{ijk} extrapolated at 1.06 µm [8], and Refs. 6 and 9 for the other material constants. The separate contributions from P^{OR}(k_{a}) and P^{OR}(k_{b}) are shown in Table 1 for the two DFWM geometries of Fig.1 and two orientations of the polar 3axis of the crystals (labelled c). All light polarizations are kept constant in the sample referenceframe and the direct third order contributions do not change with crystal orientation or experimental set up. The cascaded contributions, on the other hand, vary considerably. As an example, c^{(3)}_{3333} can be measured with (i) c parallel y and all beams polarized along y, or (ii) c parallel z and all beams polarized along z (the angles between the beams can always be chosen so small that the xcomponents of the optical electric field are negligible). For c^{(3)}_{3333} only r_{333} contributes to P^{OR}, which is then parallel to c. For the setup in Fig.1a, both rectified polarizations are transversal for c parallel z, while P^{OR}(k_{a}) becomes longitudinal for c parallel y. For the Fig.1b setup, P^{OR}(k_{a}) is transversal and P^{OR}(k_{b}) longitudinal for c parallel z, and viceversa for c parallel y. To give an example of its effect, we include piezoelectric elastic relaxation for the contribution with the large wavevector k_{b} in Fig.1a. All other cascaded contributions were calculated using the clamped (strainfree) coefficients, and are valid up to laser pulse lengths of several nanoseconds, depending on the angle between the beams. The same reasoning can be applied to c^{(3)}_{1133} and c^{(3)}_{2233}, but here the piezoelectric contribution for k_{a,b} perpendicular c always vanishes by symmetry, as can be demonstrated with (18). Despite the fact that direct third order contributions are identical for every pair of rows of Table 1, the total cascaded contributions can differ by a factor of ~ 2. They are different for the two experimental setups of Fig.1 and change with sample orientation in the setup of Fig.1a. Interestingly, in the setup of Fig.1b, the total cascaded contribution does not depend on the orientation of the sample. Note also that, in the setup of Fig.1a, acoustic phonons contribute about 50 % to c^{(3), EFF}_{3333} (c parallel y) of KNbO_{3} (which corresponds to more than a factor of 2 for the DFWM signal). The total energy in the signal pulse in DFWM is ^{}, where z is an unknown calibration factor that depends on difficult to control parameters such as beam profiles and overlap in the sample, h collects all known experimental quantities that affect the measurement, and c^{(3), EFF} is the active effective susceptibility coefficient. For noncentrosymmetric materials the geometry dependencies showcased in Table 1 can be exploited, in place of a reference material with well known susceptibility, to determine the calibration factor z:
We applied this technique in the experimental geometry of Fig.1a for BaTiO_{3} and KNbO_{3}. The measurements were performed using c^{(3)}_{2233} for the calibration and comparing the other coefficients to the c^{(3)}_{2233} setup by taking into account the different light beam polarizations. c^{(3)}_{2233} was chosen because of its large cascaded contributions and the absence of piezoelectric relaxation (r_{ijk}=r^{S}_{ijk}), which minimizes the influence of experimental errors and the number of material parameters that must be known. The experimental susceptibilties values we measured at 1.06 µm and for 100 ps pulses are, in units of 10^{22} m^{2}/V^{2}, c^{(3) EFF}_{1133}(c parallel y)=560± 80, c^{(3) EFF}_{1133}(c parallel z)=340± 80 for BaTiO_{3}, and c^{(3) EFF}_{2233}(c parallel y)=330 ± 70, c^{(3) EFF}_{2233}(c parallel z)=190 ± 50 for KNbO_{3}. Note that the direct third order contributions are only c^{(3)}_{1133} ~ 110 for BaTiO_{3} and c^{(3)}_{2233} ~ 60 for KNbO_{3}. For the other coefficients, c^{(3)}_{1111}~ 160, c^{(3)}_{3333} ~ 100 for BaTiO_{3}, and c^{(3)}_{2222} ~ 180, c^{(3)}_{3333} ~ 60 for KNbO_{3}. A more detailed discussion of these experimental results will be given elsewhere. The results above where checked with a classical reference measurement. Comparing the experimental signal to the one observed with a 1 mm thick cell filled with CS_{2}, and using c^{(3) CS2}_{1111} =263 ± 30 [10], we got c^{(3)}_{1133}(c parallel y)=550 ± 100 for BaTiO_{3} and c^{(3)}_{2233}(c parallel y)=290 ± 60 for KNbO_{3}. This is in excellent agreement with the results obtained above using the cascading contributions in Table 1 as a reference, confirming the validity of the expressions and theoretical interpretations given in this paper. In conclusion, we have demonstrated that DFWM in noncentrosymmetric materials requires special care in selecting experimental geometries in order to deliver reliable results, and we have shown how to calculate the geometry dependent second order contributions for different pulse lengths. The second order contributions can be exploited to calibrate a DFWM experiment and relate third order susceptibilities to second order dielectric and electrooptic properties experimentally. References
