The crossflow piezo-electrooptic effect in crystals . Example of lithium tantalate

In the present paper the thermodynamic and phenomenological descriptions of the crossflow piezo-electrooptic effect in crystals have been made. For example the necessary experimental measurements of this effect have been carried out in the lithium tantalate crystals. For these crystals at first the P33113−P11113 = −4.6·10 m/ N·V absolute coefficients difference of crossflow piezo-electrooptic effect were determined by two measurement methods.


Introduction
The crossflow piezo-electrooptic effect, resulting from mutual interaction of piezoand electrooptic effects in crystal materials, is poorly investigated because of its small values [1] on the background of a direct action of the piezo-or electrooptic effects.
The aim of the present work is to carry out thermodynamic and phenomenological description of the crossflow piezo-electrooptic effect and make necessary experimental measurements for its verification.

Thermodynamic description
According to the general thermodynamic theory [2][3][4], a crystal appears as a thermodynamic system the monocrystal state of which is determined by certain numbers of variables.Mechanical stress σ ij , electrical field strength E i and temperature T are chosen as independent variables.Herein Gibbs free energy G is assumed as thermodynamic function for which [1,5]: where U is the inner energy.Then, the components of the mechanical deformation ε ij , electrical field induction D i and entropy S will be functions of independent variables.Let's spread in the MacLaurin's series the ε ij and D i up to the third order of smallness (no thermal effects are taken into account) because their meanings of the second and the third order of smallness are insignificant [6]).Setting for convenience δ m = D m /4π, we will have: Besides, the following equations are valid [5]: Considerations of the physical sense of the partial derivatives are given more in full in [1].We will take into consideration only components describing crossflow effects.They are: It is a correction part to the elastic compliance constant S klij , which is connected with the electrical field E n acting in the crystal, or the same, correction part to the reverse piezoelectric effect constant d nij under the action of the mechanical stress σ ij .It is fifth-rank tensor, which describes a crossflow piezo-electrostrictive effect caused in the crystal by mutual interaction of the electrical field and the mechanical stress.Furthermore: Here k mn -permittivity tensor and B mnijo -a tensor of crossflow piezo-electrooptic effect, which determine the change of the piezooptic module µ mnij = ∂k mn /∂σ ij under the action of the electrical field E o , or the same, the change of the liner electrooptic effect tensor ρ mno = ∂k mn /∂E o under the action of the mechanical stress σ ij .
Then, as parts of the second order of smallness are neglected, the simplified correlations ( 2) and (3) are: where d mij is a tensor of piezoelectric effect.

Phenomenological description
In the given work, the phenomenological description of a crossflow effect needed for the explanation of our experimental measurements is carried out.
As is generally known, the external action (be it the mechanical stress σ mn or the external electric field E l ) exerted on the crystal sample results in the change of birefringence δ(∆n k ) = δn i − δn j (or refractive indices δn i , δn j ) of the sample as well as its length δt k in the direction k of light propagation, which are registered by polarization-optic method through the change of an optical path difference δ∆ k of this sample: The value δ(∆n k ) or δn i , δn j can be determined from the tensor for polarization constants a ij .Beside the coefficients of piezo-π ijmn and linear electrooptic r ijl effects, this tensor in the first approximation for acentric crystals also contains the coefficients of their crossflow effect [1], analogous to (8): where P ijmnl = ∂π ijmn /∂E l = ∂r ijl /∂σ mn is the tensor of crossflow piezo-electrooptic effect (tensor of absolute coefficients), σ mn E l = g mnl is the 3-rank tensor, which equals the product of the 2-rank tensor σ mn and vector E l .From a thermodynamic description of crossflow piezo-electrooptic effect P mnijl (see formula ( 6)), one can describe symmetrical properties of this tensor while replacing the indexes P ijmnl = P jimnl = P ijnml = P ijlmn .The complete form of this tensor is given in [7].Analogously, the value δt k is determined from the deformation tensor ε kl (see formula (7)), which beside the coefficients of elastic S klij and piezoelectric d nkl effects also contains the coefficients of their crossflow A klijn effect.This crossflow A klijn effect is named "false" effect in our case.
To derive the necessary operating correlations we use the matrix interpretation of the processed tensors.For a crystal with small initial birefringence the correlation (9) will be simplified to δ∆ k = t k δ(∆n k ), where the δ(∆n k ) value is equal to [5,8]: Here π * km = π im n 3 i − π jm n 3 j is a known piezooptic coefficient of the induced birefringence.For half-wave stresses σ o m , when δ∆ k = λ/2 (λ is a length of light wave), one can obtain: After applying the electric field E l , for repeatedly measured half-wave stresses σ o ′ m , we obtain: Analogously P * kml = P iml n 3 i −P jml n 3 j is the crossflow piezo-electrooptic coefficient of the induced birefringence.Mutual solution of ( 12) and ( 13) gives us the value of this coefficient to be searched for: A similar formula can be obtained at measuring the magnitudes of half-wave electric fields E o l and E o ′ l for σ m = 0 and σ m = 0 accordingly.The measurements of crossflow piezoelectrooptic effect were carried out using the polarisation-interferometrical technique by Senarmont method and half-wave stresses method [5].

Experimental results
To exclude a possible error, connected with the measurement of "false" crossflow effect, the LiTaO 3 crystals with a small initial birefringence (∆n k = 0, 005) were used.For these crystals, the elastic component in (9) and therefore "false" crossflow effect could be neglected.
We have determined the crossflow piezo-electrooptic effect P * 22113 component on the sample of direct cut, when k Y, σ m X and E l Z.In the figures the dependences for the δ(∆n 2 ) value under the normal mechanical stress σ 11 for different magnitudes of E 3 (figure 1) are shown as well as the δ(∆n 2 ) value under the electrical field E 3 for different magnitudes of σ 11 (figure 2).From the changes of angular coefficients for linear interpolation of these behaviours we have calculated the average magnitudes of P *

Conclusions
1.The thermodynamic descriptions of crossflow piezo-electrooptic effect in crystals and the necessary phenomenological descriptions for anisotropic materials with small initial birefringence have been made in the paper.
2. For lithium tantalate crystals, at first the P 33113 −P 11113 = −4.6•10−19 m 3 /N • V absolute coefficients difference of crossflow piezo-electrooptic effect were determined.For these crystals, the change of ∆π * 2211 because of crossflow effect corresponds to ∼ 10% of coefficient π * 2211 at electrical field change of ∆E 3 = 2.3 • 10 5 V/m. 3. The equality (within the limits of experimental error) of the P * 22113 coefficient, obtained from different measurement methods, demonstrated the correctness of our results and showed the possibility of measuring the crossflow piezo-electrooptic effect in anisotropic crystals with small initial birefringence using these methods.

Figure 1 .
Figure 1.The dependences of birefringence change δ(∆n 2 ) for lithium tantalate crystals on the normal mechanical stress σ 11 for different magnitudes of electrical fields E 3 (for light wave λ = 0, 6328 µm and temperature T=293 K).

Figure 2 .
Figure 2. Analogous to figure 1 the dependences of δ(∆n 2 ) on the electrical fields E 3 for different magnitudes of mechanical stress σ 11 .