Electrooptical effects in glass forming liquids of dipolar nano-clusters embedded in a paraelectric environment

Alexander Gumennik; Yael Kurzweil-Segev; Aharon J. Agranat

doi:10.1364/OME.1.000332

1. Introduction

Potassium tantalate niobate (KTN) is a perovskite crystal which at the paraelectric (PE) phase exhibits an exceptionally strong quadratic electrooptical (EO) effect. At temperatures slightly above the ferroelectric (FE) phase transition temperature T_c, KTN manifests an electrically induced change in the refractive index of ≈10⁻². And yet this exceptionally strong electrooptical effect has hardly been exploited for applications. The reason for this is the fact that in the vicinity of the ferroelectric phase transition the electrooptical effect is accompanied with random scattering due to the formation of dipolar clusters that fluctuate in time and space.

In general, the EO effect is regarded as a deterministic phenomenon treated within the framework of crystal optics. In particular, it is normally assumed that in KTN at the paraelectric phase the EO effect is quadratic due to the centro-symmetric structure of the crystal. It is known, however, that the dielectric behavior of KTN at the PE phase has the characteristics of soft condensed matter [1]. More specifically it was shown that KTN behaves as a dipolar ‘glass forming liquid’ in which dipolar clusters are embedded in a PE medium and fluctuate at random in size and orientation.

These dipolar clusters that in close proximity to the ferroelectric phase transition are metastable were shown to influence significantly the electroholographic behavior [2–4], and the formation of light induced spatial solitons-like waveguiding structures in this region [5]. In addition, it has recently been shown that quenching these clusters by fast cooling from elevated temperatures to the phase transition region causes the induction of waveguiding structures with minute dimensions which open the way to “scale free optics” [6].

In this paper we report direct observation of the influence of these clusters on the electrooptic behavior of KTN at the paraelectric phase in the immediate vicinity of the phase transition. In particular, we observed that the application of a uniform electric field causes a linearly polarized light beam to depolarize as it traverses the crystal. It is also shown that in KTN doped with Li (KLTN) these effects are inhibited which allows full exploitation of the strong EO effect in this material.

Consider a plane wave propagating in a crystal. For a given direction of propagation, the wave experiences refractive indices which depend on its polarization. These are derived from the index ellipsoid given by

\frac{x_{1}^{2}}{n_{11}^{2}} + \frac{x_{2}^{2}}{n_{22}^{2}} + \frac{x_{3}^{2}}{n_{33}^{2}} = 1,

where n₁₁, n₂₂, and n₃₃ are the refractive indices for a plane wave polarized along the principal dielectric axes defined by their unit vectors

{\hat{x}}_{1}

,

{\hat{x}}_{2}

, and

{\hat{x}}_{3}

respectively. For a crystal at the paraelectric phase, where the electrooptic effect is quadratic, the application of a static (low frequency) electric field will cause deviations in the index ellipsoid given by

Δ (\frac{1}{n_{i j}^{2}}) = \sum_{k, l = 1}^{3} g_{i j k l} P_{k} P_{l},

where g_ijkl are the elements of the quadratic EO tensor, and P_k is the static polarization induced by the applied electric field. In particular, in the centro-symmetric KTN crystal the induced polarization is given by

P_{k} = ε E_{k},

where ε is the (scalar) dielectric constant, and E_k is the electric field applied along the

{\hat{x}}_{k}

axis. It is assumed in (3) that ε_r>>1, so that the static dielectric constant is given by ε = ε_o(ε_r−1)≈ε_oε_r, where ε_o is the vacuum permittivity and ε_r is the relative static dielectric constant. The non-zero electrooptic coefficients of KTN are given by: g₁₁ = g_iiii = 0.16 m⁴/C²; g₁₂ = g_ijij = −0.02 m⁴/C² i≠j; and g₄₄ = g_iijj = 0.08 m⁴/C² i≠j [7].

Consider a KTN sample installed in a “crossed polarizers” set-up (CPS) illustrated schematically in Fig. 1 [8].The setup is arranged so that the $\hat{z}$ axis is co-aligned with the optical axis, the input polarizer is oriented in the $\hat{x} + \hat{y}$ direction, and the output polarizer is oriented in the $\hat{x} - \hat{y}$ direction. The output intensity $I$ of a plane wave of input intensity $I_{0}$ propagating through the set-up in the $\hat{z}$ direction is given by

I = I_{0} \sin^{2} (\frac{Δ Φ}{2}),

where ΔΦ is the relative phase shift between the two optical field components with polarizations along

\hat{x}

and

\hat{y}

respectively. ΔΦ is accumulated during the propagation of the wave through the crystal due to the electrooptically induced birefringence (BR). For a KTN sample cut to form a rectangular prism so that its edges are co-aligned with the crystallographic axes, and the CPS axes

\hat{x}

,

\hat{y}

and

\hat{z}

, and an electric field E is applied along the

\hat{x}

axis, ΔΦ is given by

Δ Φ = - \frac{π n_{0}^{3} (g_{11} - g_{12}) ε^{2} E^{2} L}{λ},

where L is the length of the crystal along the

\hat{z}

axis, λ is the wavelength of the propagating wave, n_o and ε_r are the index of refraction and the static dielectric constant at the paraelectric phase respectively, and g₁₁ and g₁₂ are the appropriate electrooptic coefficients.

Fig. 1 Schematics of crossed-polarizer setup.

Download Full Size | PDF

2. Experimental observation

The electrooptical behavior was investigated in two crystals: KTN and KTN doped with Li (KLTN). The crystals were grown by us using the top seeding solution growth method [9]. The Nb/Ta concentrations ratios in the crystals were determined by electron microprobe analysis. The Li concentration in the KLTN crystal was evaluated based on the Li concentration in the flux from which the crystal was grown [9]. The paraelectric-ferroelectric phase transition temperature [10] of each sample was determined from a measurement of the dielectric constant dependence on the temperature. The characteristics of the samples are summarized in Table 1 .

Table 1. Sample Properties

View Table | View all tables in this article

The experimental results are shown in Fig. 2 in which the output power of the CPS vs. the applied electric field is presented. Consider first Fig. 2(a) in which the behavior of the KTN crystal is presented. The measurements were done at T = 29.4°C = T_c + 4°C. It can be seen that the output power of the CPS deviate significantly from the strict sin²(aE²) behavior predicted by (4) and (5). In fact it was observed that as the applied field is strengthened the modulation depth of the output signal decays to zero and the signal converges to half of its maximum level. Furthermore, it was found that when the modulation of the output light is completely suppressed, it is also completely depolarized. (i.e. it does not depend on the orientation of the output polarizer). In addition it was observed that when the input light was polarized along one of the principal dielectric axes ( $\hat{x}$ or $\hat{y}$ ), the output signal was not depolarized and remained at its initial polarization as predicted by crystal optics considerations. (See Eqs. (2) and (3) above).

Fig. 2 Experimental CPS output for a) KTN at Tc + 4.0°C and b) KLTN at Tc + 4.4°C.

Download Full Size | PDF

Similar effects were observed in the KLTN crystal (Fig. 2(b)). The measurements were done at T = 30.9°C = T_c + 4.4°C. However, as can be seen in the KLTN crystal the depolarization envelope was substantially weaker. Moreover, the depolarization was at its maximum at E around 3kV and then gradually disappeared as the applied field was increased further.

3. Mathematical formulation of theoretical framework explaining the experimental observation

It is well established that in KTN at the paraelectric phase, well above T_c, the Nb ions emerge from the center of inversion of the unit cell and shift at random between eight equivalent dipolar states [11]. As the crystal temperature approaches T_c a correlation between the elementary Nb dipoles starts to form, forcing them to team up in creating dipolar regions, or nano-clusters which fluctuate at random in size and orientation. Thus, although these clusters exhibit momentarily non-zero polarization, the net macroscopic spontaneous polarization averaged over long time scales is zero. In KTN with Nb concentration exceeding 30% per mole (T_c>225K) the phase transition is of the first order [12]. In these crystals, in the immediate vicinity of the phase transition temperature, there is an intermediate state in which the dipolar regions combine, forming large domains [13], or “superclusters”, that exhibit non-zero polarization over longer time scales [1]. As the phase transition temperature is approached these superclusters grow in size until at T_c they percolate into one ferroelectric domain. Under the application of an electric field, the superclusters tend to maintain the direction of the applied field. In terms of the Gibbs free energy this means that the system will reside more in the lowest local minimum with non-zero polarization occurring in the immediate vicinity of a first order phase transition as shown in Fig. 3 [4,14].

Fig. 3 qualitative picture of the mean-field approximation Gibbs free energy in centrosymmetric crystal as function of polarization without field applied (left) and when electric field applied (right) with the parameters of the double well model used to derive the probability χ (for the definition of χ see the paragraph below).

Download Full Size | PDF

The spatial distribution of the superclusters at a given time can be derived by assuming: (i) the average size of the supercluster depends on the temperature; and (ii) the concentration of the superclusters embedded in the paraelectric surrounding lattice at a given temperature is a function of the applied field. A crystal at a given temperature can schematically be divided into elementary cells. Each cell is of length ΔL, and is either paraelectric or ferroelectric with probabilities (1-χ) and χ respectively, and where ΔL is the average linear dimension of a ferroelectric supercluster at the given temperature T.

This is illustrated schematically in Fig. 4 . Assume the crystal is divided into columns of cells that are spanned between the two ends of the crystal in parallel to the optical axis. The binomial distribution of the ferroelectric cells can be treated as a normal distribution if the number of cells along the optical path in the crystal is large, i.e. N≡L/ΔL >>1, where L is the length of the crystal. In these terms the probability P(n) of finding n ferroelectric cells along one column of cells is given by

P (n) = \frac{1}{{(2 π σ^{2})}^{\frac{1}{2}}} \cdot \exp (- \frac{{(n - < n >)}^{2}}{2 σ^{2}}),

where <n> is the average number of superclusters in one column at a given temperature T, and σ is the width of the probability distribution. In terms of the relative concentration of the ferroelectric superclusters χ, <n> = Nχ and σ² = Nχ(1-χ), so that P(n) can be written as

P (n) = \frac{1}{{(2 π \frac{L}{Δ L} χ (1 - χ))}^{\frac{1}{2}}} \cdot \exp (- \frac{{(n - \frac{L}{Δ L} χ)}^{2}}{2 \frac{L}{Δ L} χ (1 - χ)}) .

Consider a linearly polarized plane wave propagating through a column of cells spanned along the

\hat{z}

principal axis of the crystal with its polarization oriented at 45° with respect to the

\hat{x}

and

\hat{y}

principal axes. Assume the birefringence in the paraelectric and ferroelectric cells are Δn_P and Δn_F respectively, and the respective static dielectric constants are ε_P and ε_F. In these terms the phase difference ΔΦ accumulated between the

\hat{x}

and

\hat{y}

polarization components of the propagating wave as it traverses a column for which n cells are ferroelectric is given by

\begin{array}{l} {Δ Φ |}_{n} = 2 π \cdot \frac{Δ n_{e f f} L}{λ} = 2 π \frac{Δ n_{f} n Δ L + Δ n_{p} (N - n) Δ L}{λ} = 2 π \frac{Δ L}{λ} (Δ n_{f} n + Δ n_{p} (\frac{L}{Δ L} - n)) \\ = 2 π \frac{Δ L}{λ} ((Δ n_{f} - Δ n_{p}) n + Δ n_{p} \frac{L}{Δ L}) = \frac{2 π L}{λ} Δ n_{p} + \frac{2 π Δ L n}{λ} (Δ n_{f} - Δ n_{p}), \end{array}

where Δn_eff denotes the average BR along the column. The optical power at the output of the CPS can be derived by calculating the weighted sum of the contributions from an ensemble of columns for which the columns with n ferroelectric cells are distributed according to Eq. (7). In these terms the output power is given by

\begin{array}{l} I_{o u t} = I_{i n} \sum_{n} P (n) \cdot \sin^{2} ({\frac{Δ Φ}{2} |}_{n}) = \\ = I_{i n} \int_{- \infty}^{\infty} (\frac{1}{{(2 π \frac{L}{Δ L} χ (1 - χ))}^{\frac{1}{2}}} \cdot \exp (- \frac{{(n - \frac{L}{Δ L} χ)}^{2}}{2 \frac{L}{Δ L} χ (1 - χ)})) \cdot \sin^{2} (\frac{π L}{λ} (Δ n_{p} + \frac{Δ L n}{L} (Δ n_{f} - Δ n_{p}))) \cdot d n, \end{array}

where it is assumed that the number of parallel columns in the sample is large, and that the effect of interference between neighboring cells cancels out. Solving the integral in Eq. (9) finally yields the expression for the output power of the CPS.

Fig. 4 A modulator setup with a crystal in a two phase mixture state.

Download Full Size | PDF

I_{o u t} = I_{i n} \cdot \frac{1}{2} [1 - \exp (- \frac{8 π^{2} Δ L \cdot L}{λ^{2}} {(Δ n_{f} - Δ n_{p})}^{2} \cdot χ (1 - χ)) \cdot \cos [\frac{2 π L}{λ} (Δ n_{p} + χ (Δ n_{f} - Δ n_{p}))]] .

Note that for the extreme cases χ = 0 and χ = 1 the expression (10) converges to the deterministic cases when Eq. (5) is plugged into Eq. (4) with Δn_P and Δn_F respectively.

In order to test the agreement of Eq. (10) with the experimental data, it is necessary to derive the electric field dependencies of χ, Δn_P, and Δn_F, and ΔL.

Regarding χ(E): the electric field dependence of χ can be derived by assuming that immediately above T_c, the elementary cells reside in either minima (P(E = 0) = 0 and P(E = 0)≠0) of the Gibbs free energy function (Fig. 3). Assuming Maxwell statistics, the relaxation times of the elementary cells at each minimum are given by

τ_{P} = τ_{0} \exp (V / k_{B} T)

τ_{F} = τ_{0} \exp ([V + Δ] / k_{B} T),

where V is the barrier height and Δ is the energy difference between the minima (Fig. 3). In these terms the exchange rate equations are given by

\frac{d N_{F}}{d t} = - \frac{N_{F}}{τ_{0} \exp (V / k_{B} T)} + \frac{N_{P}}{τ_{0} \exp ([V + Δ] / k_{B} T)}

\frac{d N_{P}}{d t} = + \frac{N_{F}}{τ_{0} \exp (V / k_{B} T)} - \frac{N_{P}}{τ_{0} \exp ([V + Δ] / k_{B} T)},

where N_F = N∙χ is the number of elementary cells in the FE phase, N_P = N∙(1−χ) is the number of elementary cells in the PE phase (note: N_F + N_P = N). At the steady state the population of the phases is constant so that

\frac{N_{F}}{\exp (V / k_{B} T)} - \frac{N_{P}}{\exp ([V + Δ] / k_{B} T)} = 0,

By substituting N_F = N∙χ, and N_P = N∙(1−χ) we can rewrite Eq. (13) to obtain χ given by

χ = \frac{1}{1 + \exp (- Δ / k_{B} T)},

The energy asymmetry of a double well potential is known to be a linear function of the applied electric field. Hence, Δ = Δ_o−AE where Δ_o = Δ(E = 0) and A = A(T). Defining

E_{0}^{χ} = Δ_{0} / A

and

σ = k_{B} T / A

we finally get

χ (E) = \frac{1}{1 + \exp [- \frac{E - E_{0}^{χ}}{σ}]} .

The electric field dependence of both Δn_P and Δn_F is implicit in their respective dependence on the induced (static) polarization

P_{P, F} (E)

in the paraelectric and ferroelectric regions respectively.

The BR in the PE regions is derived from Eq. (2) and is given by

Δ n_{P} (E) = - (\frac{1}{2}) n_{0}^{3} g_{P} {(P_{P} (E))}^{2}

and the BR in the FE regions is given by

Δ n_{F} (E) = - (\frac{1}{2}) n_{0}^{3} [g_{F} {(P_{F} (E))}^{2} + r_{F} P_{F} (E) + C_{0}],

where n_o is the refractive index in the crystal, g_P,F are the quadratic EO coefficients in the PE and FE regions respectively, r_F is the linear EO coefficient in the FE regions, and

(\frac{1}{2}) n_{0}^{3} C_{0}

is the BR induced by the spontaneous polarization in the FE regions.

In the vicinity of T_c the expression (3) is no longer valid, and the induced polarization is given by

P_{P, F} (E) = \int_{0}^{E} ε_{s s}^{P, F} (E') d E',

where

P_{P, F} (E)

are the induced polarizations, and

ε_{s s}^{P, F} = (d P_{F, P} / d E)

are the small signal dielectric constants in the paraelectric and ferroelectric regions respectively. In terms of

ε_{s s}^{P, F}

and χ, the effective small signal dielectric constant is given by

ε_{s s} (E) = ε_{s s}^{P} [1 - χ (E)] + ε_{s s}^{F} χ (E) .

For

ε_{s s}^{P}

and

ε_{s s}^{F}

we used the following phenomenological expressions:

ε_{s s}^{P} = a + \frac{b}{s^{2} + {(E - E_{0}^{P})}^{2}}

ε_{s s}^{F} = c,

where a, b, c, s, and

E_{0}^{P}

are temperature dependent parameters.

The expressions (20a) and (20b) are phenomenological expressions that were chosen in order to avoid using the common expression derived from the Gibbs free energy [14] that does not hold in close proximity of the phase transition, and in addition is non-integrable thus cannot be used in Eq. (18).

4. Approval of the theoretical framework validity through fitting of the mathematical model to the experimental data

$ε_{s s} (E)$ was measured directly in the KTN sample at T = 29.4°C = T_c + 4°C, and in the KLTN sample at T = 35.3°C = T_c + 4.4°C. The results are shown in Fig. 5 together with a fitting to the model (19) using the expressions (20a), (20b) and (15). The parameters a, b, c, s, $E_{0}^{P}$ ,σ and $E_{0}^{χ}$ that were derived from these fittings are summarized in Table 2 .

Fig. 5 Birefringence measurements in modulator configuration (right) with respective dielectric constants (left) of pure KTN 4°C above T_c (top) and KLTN 4.4°C above T_c (bottom) with Eq. (10) fitted to the birefringence measurement (solid red) and Eq. (19) fitted to the dielectric constant (dashed red). The experimental data is represented by solid black lines.

Download Full Size | PDF

Table 2. Fitting Parameters of the Dielectric Constant Part of the Model

View Table | View all tables in this article

These parameters were used to fit the BR measurements presented in Fig. 2 to the model (10) so that the only free parameters in the fitting were ΔL, g_P, g_F, r_F, C_o and I_in. The fitting was done for both the KTN and the KLTN samples. As shown in Fig. 5, the model produced a very good fit to the experimental data for both cases. The values for ΔL, g_P, g_F, r_F, C_o and I_in are presented in Table 3 . (See Appendix I for a detailed description of the model fitting procedure).

Table 3. Other Fitting Parameters—The Birefringence Part of the Model

View Table | View all tables in this article

5. Conclusions

In conclusion, we showed that the EO behavior of KTN at the PE phase deviates strongly from that of an ideal isotropic crystalline medium. This deviation was attributed to the fact that above the FE phase transition temperature, KTN is constituted of an ensemble of FE clusters that are embedded in a PE medium. A priori it is expected that a plane wave propagating through a KTN medium will be affected by the random nature (in size and orientation) of these clusters. However, their fast fluctuations cause this randomness to be canceled out. The application of an electric field, in addition to inducing uniform birefringence through the quadratic EO effect, causes these fluctuations to slow down thus allowing the propagating wave to accumulate random BR across its wave-front, and to become depolarized. The scattering from the dipolar clusters limits considerably the applicability of KTN although it possesses an exceptionally strong electrooptical effect.

We have shown however that the scattering exhibited in KTN is strongly inhibited in KLTN crystals. Thus, in KLTN, the application of the electric field causes some depolarization to occur. But, as the field is strengthened further the depolarization is completely inhibited, and the crystalline nature of the medium is completely restored.

In the proposed model, the depolarization was attributed to the variance in the optical path lengths between the two polarization components aligned along the principal axes. This variance is accumulated in the wavefront of the beam as it propagates along different columns (Fig. 4). The variance stems from the typical cluster size and the contrast in BR between the paraelectric and ferroelectric regions in the column. The model fitting indicates that the clusters size is similar in KTN and KLTN. However the contrast of the BR is much stronger in KTN than in KLTN. Thus the difference in the depolarization between KTN and KLTN must be based on the difference in the contrast of the BR.

A possible explanation of the difference in the BR contrast between KTN and KLTN can be found by considering the fact that the dipolar clusters that are created in KLTN by the emergence of the Nb ions are formed around the small number of Li ions in this crystal. It was found in copper doped KTN crystals that the dielectric relaxation of the copper ions deviated from the anticipated Arrhenius behavior, and manifests a Volger-Fulcher-Tammann behavior [15]. This was attributed to cooperative behavior that occurs between the dipolar clusters that were formed around the copper ions by the Nb ions that emerge outside the center of inversion of their respective unit cells. The Li ions respond fast to the application of an electric field since they are much smaller than their respective site in the unit cell. In addition they reside in the K site which differs in symmetry then the Nb/Ta site. Thus, the polarization of the clusters in KLTN is mediated by the fact that its directions are in parallel to the Li ions which are not the natural directions for the Nb ions. This can account for the difference in the BR and lead to the small depolarization that occurs in KLTN. Nevertheless, the microscopic mechanism that causes the different behavior in KTN and KLTN merit further research.

This makes KLTN to be an attractive electrooptical medium functional in both the visible and mid IR wavelengths. If doped with non-stoichiometric ions such as copper it exhibits a strong photorefractive effect [16]. This makes KLTN an attractive medium for electroholographic applications for both wavelength selective photonic switching [7], and the creation of minute metastable waveguiding structures that were recently proposed as the platform for scale-free optics [6].

Appendix I

A priori it may seem that the large number of parameters in the model facilitates the latter to fit the experimental data. It should be noted however, that there are two independent sets of the parameters, the dielectric parameters and the BR parameters. Each set of parameters is extracted from an independent set of experimental data: the dielectric parameters are extracted from electrical measurements of the dielectric constant whereas the BR parameters are extracted from optical measurements of the birefringence. Moreover, the value of each parameter is evaluated in a one-to one procedure from a different region of the respective set of experimental data.

The dielectric parameters, a, b, c, s, $E_{0}^{P}$ , $σ$ and $E_{0}^{χ}$ , are evaluated by fitting the general expression (19) to the measurements of the dielectric constant $ε_{s s}^{}$ as function of the applied electric field using the expressions for phase-specific dielectric constants - $ε_{s s}^{P}$ and $ε_{s s}^{F}$ given in Eq. (20). This is shown in Fig. 6 .

Fig. 6 Determination of the dielectric constant parameters on the example of KLTN by fitting Eq. (19) to the experimental data collected at 4.4°C above T_c.

Download Full Size | PDF

a - is the low field baseline of $ε_{s s}^{}$ ; b - is relative to the a height of the maximal $ε_{s s}^{P}$ ; c - is the high field baseline of $ε_{s s}^{}$ , which is $ε_{s s}^{F}$ ; s - is the width of the peak of $ε_{s s}^{P}$ ; $E_{0}^{P}$ - is the field where the $ε_{s s}^{P}$ gets its maximal value; $E_{0}^{χ}$ - is the field at which the volume ratio between the paraelectric and pseudo-ferroelectric phases is 1:1; and $σ$ is the width of the range of fields at which the material is in the mixed-phase state. All the above result from the measurement of the dielectric constant $ε_{s s}^{}$ .

The BR parameters are evaluated from fitting of the model (10) to the measurements of the output of the CPS as presented in Fig. 7 .

Fig. 7 Determination of the birefringence parameters on the example of KLTN by fitting Eq. (10) to the experimental data collected at 4.4°C above T_c.

Download Full Size | PDF

The values of the dielectric parameters used in this work are evaluated independently in the fitting procedure of the dielectric data. The quadratic electro-optic coefficient at the paraelectric phase g_P is extracted from the low field behavior of $I_{o u t}$ ; the quadratic and linear electro-optic coefficients of pseudo-ferroelectric phase, g_F and r_F respectively, are extracted from the high field behavior of $I_{o u t}$ ; the natural BR C_o of the pseudo-ferroelectric phase is extracted from the $I_{o u t}$ modulation frequency in terms of electric field in the mixed-phase range; the super-cluster size ΔL is extracted from the maximal depolarization rate in the mixed-phase range; and $I_{i n}$ is determined by the maximal amplitude of $I_{o u t}$ .

Thus the fitting procedure does not allow for interdependence of the parameters in the fitting process.

Acknowledgments

This research was supported by the Israel Ministry of Science The “Eshkol Scholarship” of Israel Ministry of Science (grant No. 3-4341), the Israel Science Foundation (grant No. 1519/08), and the Brojde Center for Innovative Engineering and Computer Science.

References and links

1. S. E. Lerner, P. Ben Ishai, A. J. Agranat, and Yu. Feldman, “Percolation of polar nanoregions: a dynamic approach to the ferroelectric phase transition,” J. Non-Cryst. Solids 353(47-51), 4422–4427 (2007). [CrossRef]

2. A. J. Agranat, M. Razvag, M. Balberg, and V. Leyva, “Dipolar holographic gratings induced by the photorefractive process in potassium lithium tantalate niobate at the paraelectric phase,” J. Opt. Soc. Am. B 14(8), 2043–2048 (1997). [CrossRef]

3. A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K_1-xLi_xTa_1-yNb_yO₃:Cu,V,” Phys. Rev. B 55(19), 12818–12821 (1997). [CrossRef]

4. G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K_1-xLi_xTa_1-yNb_yO₃:Cu,V at the paraelectric phase,” Phys. Rev. B 58(9), 5282–5286 (1998). [CrossRef]

5. E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83(10), 1954–1957 (1999). [CrossRef]

6. E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nano-disordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011). [CrossRef]

7. A. Bitman, N. Sapiens, L. Secundo, A. J. Agranat, G. Bartal, and M. Segev, “Electroholographic tunable volume grating in the g₄₄ configuration,” Opt. Lett. 31(19), 2849–2851 (2006). [CrossRef] [PubMed]

8. A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, 1984), Chapter 7.3.1.

9. R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K_1-yLi_yTa_1-xNb_xO₃:Cu,” J. Cryst. Growth 131(3-4), 486–494 (1993). [CrossRef]

10. M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, 1977), Chapter 4.

11. Y. Girshberg and Y. Yacoby, “Off-centre displacements and ferroelectric phase transition in dilute KTa₁₋_xNb_xO₃,” J. Phys. Condens. Matter 13(39), 8817–8830 (2001). [CrossRef]

12. G. Bitton, Yu. Feldman, and A. J. Agranat, “Relaxation processes of off-center impurities in KTN:Li crystals,” J. Non-Cryst. Solids 305(1-3), 362–367 (2002). [CrossRef]

13. J. Toulouse, “The three characteristic temperatures of relaxor dynamics and their meaning,” Ferroelectrics 369(1), 203–213 (2008). [CrossRef]

14. R. Blinc and B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics (Elsevier, 1974).

15. P. Ishai, C. de Oliveira, Y. Ryabov, Y. Feldman, and A. Agranat, “Glass forming liquid kinetics manifested in a KTN:Cu crystal,” Phys. Rev. B 70(13), 132104 (2004). [CrossRef]

16. A. J. Agranat, “Optical lambda-switching at telecom wavelengths based on electroholography,” Top. Appl. Phys. 86, 133–161 (2003). [CrossRef]

Sample	Distance between the contacts [mm]	Optical axes length [mm]	Solid composition K_1-yLi _yTa_1-xNb_xO₃		T _measurement	T_c[°C]
Sample	Distance between the contacts [mm]	Optical axes length [mm]	x	y
KTN	1.60	2.40	0.395	-	29.4 ± 0.1	25.4 ± 0.1
KLTN	2.17	2.33	0.360	0.011	30.9 ± 0.1	26.5 ± 0.1

Sample	Epsilon parameters
Sample	a [10⁴]	b [10¹³ V²/m²]	c [10³]	s[10⁴ V/m]	$E_{0}^{p}$ [10⁵ V/m]	$σ$ [10⁴ V/m]	$E_{0}^{χ}$ [10⁵ V/m]
KTN	1.943	5.210	7.264	9.795	2.203	7.823	3.613
KLTN	1.870	7.234	8.556	6.116	2.435	3.567	3.509

Sample	$I_{i n}$ [a.u.]	$Δ L$ [10⁻⁶ m]	Birefringence parameters
Sample	$I_{i n}$ [a.u.]	$Δ L$ [10⁻⁶ m]	$g_{p}$ [m⁴/(F²•V²)]	$g_{f}$ [ m⁴/(F²•V²)]	$r_{f}$ [ m²/(F•V)]	$C_{0}$
KTN	0.089	1.710	0.1061	2.2149	0.0768	4.627·10⁻⁴
KLTN	0.206	3.014	0.0947	3.1570	−0.1698	3.258·10⁻³

Sample	Distance between the contacts [mm]	Optical axes length [mm]	Solid composition K_1-yLi _yTa_1-xNb_xO₃		T _measurement	T_c[°C]
Sample	Distance between the contacts [mm]	Optical axes length [mm]	x	y
KTN	1.60	2.40	0.395	-	29.4 ± 0.1	25.4 ± 0.1
KLTN	2.17	2.33	0.360	0.011	30.9 ± 0.1	26.5 ± 0.1

Sample	Epsilon parameters
Sample	a [10⁴]	b [10¹³ V²/m²]	c [10³]	s[10⁴ V/m]	$E_{0}^{p}$ [10⁵ V/m]	$σ$ [10⁴ V/m]	$E_{0}^{χ}$ [10⁵ V/m]
KTN	1.943	5.210	7.264	9.795	2.203	7.823	3.613
KLTN	1.870	7.234	8.556	6.116	2.435	3.567	3.509

Electrooptical effects in glass forming liquids of dipolar nano-clusters embedded in a paraelectric environment

Abstract

1. Introduction

2. Experimental observation

3. Mathematical formulation of theoretical framework explaining the experimental observation

4. Approval of the theoretical framework validity through fitting of the mathematical model to the experimental data

5. Conclusions

Appendix I

Acknowledgments

References and links

Cited By

Figures (7)

Tables (3)

Equations (23)

Optical Materials Express