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Cascaded second order and piezoelectric contributions to degenerate four wave mixing in non-centrosymmetric materials are analyzed in detail. The effective third order susceptibility measured in degenerate four-wave mixing becomes strongly dependent on experimental parameters that do not normally influence the third order response in centrosymmetric materials. This introduces new important requirements for a reliable reporting of experimental results. A new technique that allows to experimentally relate the third order susceptibility to the high-frequency electro-optic and dielectric properties is introduced and demonstrated in BaTiO3 and KNbO3. PACS Numbers: 78.20.Bh, 42.65.Hw, 77.65.-j, 77.84.-s

Pulsed Degenerate Four Wave Mixing (DFWM) is a wide-spread tool for characterizing third order susceptibilities of candidate materials for nonlinear optical applications. We discuss important second order effects that cause unexpected geometrical and pulse-length dependencies of the DFWM signal. They are due to the concurrent processes of optical rectification and linear electro-optic effect, and to piezoelectric elastic relaxation. If neglected, these "cascaded" second order effects can lead to misleading DFWM results in non-centrosymmetric materials. In this work we give the first complete expressions to calculate their contributions to the DFWM signal, and show how they can be used to relate experimentally the third order susceptibility to the linear electro-optic and dielectric properties.

We first point out the origin of second order contributions and give general equations for calculating them for the most important DFWM set-ups, 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 electro-optic and dielectric tensors to be used when relaxation to a new elastic configuration is possible. Lastly, we demonstrate experimentally, in the well known electro-optic materials BaTiO₃ and KNbO₃, how these results can be used to obtain absolute third order susceptibilities by comparing to independently determined electro-optic 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:  

(1)

where the E_n(w,k_n) are the complex amplitudes of the three input waves, which can be distinguished by their wavevectors. In case of pulsed experiments, (1) describes the field in a time interval much shorter than the laser pulses, which we consider so long that the an "instantaneous" response can be safely assumed.

The field of (1) induces a time-dependent third order polarization [2,3]:  

(2)

where c⁽³⁾_ijkl is the third order nonlinear optical susceptibility in the time-domain, e₀ is the permittivity of vacuum, and we use S.I. units. Summation over repeated indices is understood.

Figure 1 shows two ways of arranging the three input waves so that the nonlinear polarization at frequency w and wavevector k₄=k₁ + k₂ - k₃, with complex amplitude P(w,k₄), 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 k₄, one finds:  

(3)

c⁽³⁾_ijkl(w,w,- w,- w,k₁,k₂,-k₃,-k₄) is the third order optical susceptibility tensor that describes DFWM [3].  

Figure 1: Two common DFWM experimental geometries. Three beams with wavevectors k1, k2, and k3 interact in a sample to generate a fourth beam with wavevector k4. The interacting beams and the sample are drawn in the x-y plane in the first line, and the coordinates of the beam wavevectors are plotted in the second line. For clarity, the x and y axes are not in scale. (a) Beams 1 and 2, and beams 4 (signal) and 3, are counterpropagating [3]. (b) All input input beams travel towards the positive x direction. The beams are distinguished by a slightly different z direction [11].

Our S.I. expressions use the same convention as in Ref. 4. They go over to the ones in electro-static 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  

(4)

where c is the speed of light in vacuum in m/s. This takes into account the additional factor (1/4) that was included in the definition of the third order susceptibilities c_ijkl [2].

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  

(5)

and  

(6)

These two polarizations are induced by two pairs of input waves. They interact with the remaining input wave in (1) to generate a nonlinear polarization of exactly the same form as (3), with frequency w and wavevector k₄:  

(7)

The factor e_ii + 2 has been discussed by Flytzanis and Bloembergen [5]. It is obtained in the Lorenz local field approximation by relating to the microscopic (local) electric field induced by the polarization. E^OR(k) is the macroscopic electric field that can be induced by the polarization P^OR(k)=P^OR(w=0,k) in (5) and (6). For laser pulses so long that time varying magnetic fields can be neglected, E^OR( k) must be curl-free. Through the linear susceptibility, it induces a polarization that must be added to P^OR to get the displacement field D_i=e₀ e_ij E^OR_j + P^OR_i, which must be divergence free in the absence of free charges. From rot E^OR=0 and div D=0 we obtain  

(8)

It is useful to separate P^OR in a transversal part with zero divergence and P^OR(k) perpendicular k, and a longitudinal part with zero curl and P^OR(k) parallel k. E^OR=0 for a transversal polarization, and E^OR_i = - P^OR_i/(e₀ e_ii) for a longitudinal polarization oriented parallel to a main axis of the dielectric tensor. In a DFWM experiment, the contributions from (3) and (7) add. By combining (7) with (5) and (6), and comparing to (3), we can define an effective susceptibility that must replace c⁽³⁾_ijkl in (3) whenever a non-centrosymmetric material is used:  

(9)

ccasc, kaijkl and ccasc, kbijkl describe the cascaded contributions and depend on the wavevector differences ka = k1 - k3 and kb = k2 - k3, via (5)-(8). They can be calculated in a more compact form by relating the second order susceptibilities appearing in (5)-(7) to the electro optic coefficients rijk=rjik describing the change of the optical indicatrix by an electric field E as D(1/e)ij = rijk Ek:  

(10)

where the n_i are the refractive indices at frequency w, and c⁽²⁾_kij(0,w,-w)=c⁽²⁾_ijk(w,-w,0).

We can now write ccasc, kaijkl and ccasc, kbijkl for ka and kb parallel to a main axis of the dielectric tensor. For P(OR)(w=0,k) perpendicular k (transversal polarization):  

(11)

 

(12)

For P^(OR)(w=0,k) parallel k (longitudinal polarization):  

(13)

 

(14)

For high dielectric constants these last "longitudinal" contributions are negligible. But in low-dielectric constant materials such as molecular crystals they can remain comparable to the direct third order susceptibility.

In (11)-(14), r_ijk and e_ij are electro-optic 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 plane-wave 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 electro-optic tensor r^S_ijk at constant strain and the elasto-optic tensor p^E_ijkl at constant electric field determine the change in the dielectric tensor induced by the spatially sinusoidal electric and strain fields:  

(15)

where u_kl(k)=du_k(k)/dx_l.

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 electro-optic tensor r_ijk defined by D e^-1_ij= r_ijkE_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:  

(16)

For a static deformation the divergence of the stress tensor d T_ij/dx_j must vanish [7]. Since all spatial dependencies are in the form of a plane-wave, taking the divergence leads to an algebraic equation relating the amplitudes of the strain field and the electric field:  

(17)

with k the modulus of the wavevector k and .

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^-1b_i, which is then inserted in (15) to get the effective electro-optic coefficient we were looking for:  

(18)

The dielectric tensor at constant strain e^S_ij and the piezoelectric tensor e_ijk determine the displacement field D_i(k) = e₀ e^S_ijE_j(k) + e_ijkS_jk(k). Inserting u_k=A_ki^-1b_i from (17) we obtain D_i=e₀ e_ijE_j for the modulation amplitudes, with the effective dielectric tensor  

(19)

Equations (18) and (19) must be used in (11)-(14) whenever elastic deformations can be established during the laser pulse length. Note the symmetry breaking induced by the presence of the wavevector k. The effective tensors (18) and (19) belong in general to a lower symmetry, e.g. orthorhombic instead of tetragonal for k along the 1 axis in BaTiO₃ (where the 1 and 2 axis are otherwise equivalent).

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 reference-frame.

This is not true for non-centrosymmetric materials. Even when the polarizations are kept constant in the sample reference-frame, the second order non-local contributions depend on the wavevector differences k_a and k_b, and change with sample orientation, DFWM set-up, and pulse-length.

To fix the ideas, we calculate the influence of these parameters for tetragonal BaTiO₃ and orthorhombic KNbO₃. 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 3-axis of the crystals (labelled c). All light polarizations are kept constant in the sample reference-frame 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 x-components of the optical electric field are negligible). For c⁽³⁾₃₃₃₃ only r₃₃₃ contributes to P^OR, which is then parallel to c. For the set-up 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 set-up, P^OR(k_a) is transversal and P^OR(k_b) longitudinal for c parallel z, and vice-versa 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 (strain-free) 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 set-ups of Fig.1 and change with sample orientation in the set-up of Fig.1a. Interestingly, in the set-up of Fig.1b, the total cascaded contribution does not depend on the orientation of the sample. Note also that, in the set-up of Fig.1a, acoustic phonons contribute about 50 % to c^{(3), EFF}₃₃₃₃ (c parallel y) of KNbO₃ (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:  

(20)

where the subscripts z and y indicate two orientations of the sample with the same direct third order contribution, and c^casc is the sum of the two cascaded contributions in (9). This equation relates the variations in DFWM signal strength to the calculated cascaded contributions, and therefore the experimental values of the third order susceptibilities to the linear electro optic properties.

We applied this technique in the experimental geometry of Fig.1a for BaTiO₃ and KNbO₃. The measurements were performed using c⁽³⁾₂₂₃₃ for the calibration and comparing the other coefficients to the c⁽³⁾₂₂₃₃ set-up 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 m²/V², c^{(3) EFF}₁₁₃₃(c parallel y)=560± 80, c^{(3) EFF}₁₁₃₃(c parallel z)=340± 80 for BaTiO₃, and c^{(3) EFF}₂₂₃₃(c parallel y)=330 ± 70, c^{(3) EFF}₂₂₃₃(c parallel z)=190 ± 50 for KNbO₃. Note that the direct third order contributions are only c⁽³⁾₁₁₃₃ ~ 110 for BaTiO₃ and c⁽³⁾₂₂₃₃ ~ 60 for KNbO₃. For the other coefficients, c⁽³⁾₁₁₁₁~ 160, c⁽³⁾₃₃₃₃ ~ 100 for BaTiO₃, and c⁽³⁾₂₂₂₂ ~ 180, c⁽³⁾₃₃₃₃ ~ 60 for KNbO₃. 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 CS₂, and using c^{(3) CS2}₁₁₁₁ =263 ± 30 [10], we got c⁽³⁾₁₁₃₃(c parallel y)=550 ± 100 for BaTiO₃ and c⁽³⁾₂₂₃₃(c parallel y)=290 ± 60 for KNbO₃.

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 non-centrosymmetric 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 electro-optic properties experimentally.

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