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

Optical Materials Express, Vol. 1, Issue 3, pp. 332-343 (2011)

http://dx.doi.org/10.1364/OME.1.000332

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### Abstract

Studies of the electrooptic effect in potassium tantalate niobate (KTN) and Li doped KTN in the vicinity of the ferroelectric phase transition are reported. It was observed that in KTN the standard electrooptic behavior is accompanied by electrically induced depolarization of the light traversing through the crystal. This behavior is attributed to the influence of the fluctuating dipolar clusters that are formed in KTN above the ferroelectric phase transition due to the emergence of the Nb ions out of the center of inversion of the unit cell. It was shown in addition that this behavior is inhibited in Li doped KTN, which enables exploiting the large electrooptic effect in these crystals.

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## 1. Introduction

_{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.

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]

4. G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K_{1-x}Li_{x}Ta_{1-y}Nb_{y}O_{3}: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]

_{11}, n

_{22}, and n

_{33}are the refractive indices for a plane wave polarized along the principal dielectric axes defined by their unit vectors

_{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 bywhere ε is the (scalar) dielectric constant, and E

_{k}is the electric field applied along the

_{r}>>1, so that the static dielectric constant is given by ε = ε

_{o}(ε

_{r}−1)≈ε

_{o}ε

_{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: g

_{11}= g

_{iiii}= 0.16 m

^{4}/C

^{2}; g

_{12}= g

_{ijij}= −0.02 m

^{4}/C

^{2}i≠j; and g

_{44}= g

_{iijj}= 0.08 m

^{4}/C

^{2}i≠j [7

7. A. Bitman, N. Sapiens, L. Secundo, A. J. Agranat, G. Bartal, and M. Segev, “Electroholographic tunable volume grating in the g_{44} configuration,” Opt. Lett. **31**(19), 2849–2851 (2006). [CrossRef] [PubMed]

_{o}and ε

_{r}are the index of refraction and the static dielectric constant at the paraelectric phase respectively, and g

_{11}and g

_{12}are the appropriate electrooptic coefficients.

## 2. Experimental observation

9. R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K_{1-y}Li_{y}Ta_{1-x}Nb_{x}O_{3}:Cu,” J. Cryst. Growth **131**(3-4), 486–494 (1993). [CrossRef]

9. R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K_{1-y}Li_{y}Ta_{1-x}Nb_{x}O_{3}:Cu,” J. Cryst. Growth **131**(3-4), 486–494 (1993). [CrossRef]

_{c}+ 4°C. It can be seen that the output power of the CPS deviate significantly from the strict sin

^{2}(aE

^{2}) 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 (

_{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

_{c}, the Nb ions emerge from the center of inversion of the unit cell and shift at random between eight equivalent dipolar states [11

11. Y. Girshberg and Y. Yacoby, “Off-centre displacements and ferroelectric phase transition in dilute KTa_{1}_{−}_{x}Nb_{x}O_{3},” J. Phys. Condens. Matter **13**(39), 8817–8830 (2001). [CrossRef]

_{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

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]

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]

_{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

4. G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K_{1-x}Li_{x}Ta_{1-y}Nb_{y}O_{3}:Cu,V at the paraelectric phase,” Phys. Rev. B **58**(9), 5282–5286 (1998). [CrossRef]

^{2}= Nχ(1-χ), so that P(n) can be written asConsider a linearly polarized plane wave propagating through a column of cells spanned along the

_{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

_{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

_{P}and Δn

_{F}respectively.

_{P}, and Δn

_{F}, and ΔL.

_{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 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 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 thatBy substituting N

_{F}= N∙χ, and N

_{P}= N∙(1−χ) we can rewrite Eq. (13) to obtain χ given byThe 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

_{P}and Δn

_{F}is implicit in their respective dependence on the induced (static) polarization

_{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

_{c}the expression (3) is no longer valid, and the induced polarization is given bywhere

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

_{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,

_{P}, g

_{F}, r

_{F}, C

_{o}and I

*. 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*

_{in}_{P}, g

_{F}, r

_{F}, C

_{o}and I

*are presented in Table 3 . (See Appendix I for a detailed description of the model fitting procedure).*

_{in}## 5. Conclusions

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]

7. A. Bitman, N. Sapiens, L. Secundo, A. J. Agranat, G. Bartal, and M. Segev, “Electroholographic tunable volume grating in the g_{44} configuration,” Opt. Lett. **31**(19), 2849–2851 (2006). [CrossRef] [PubMed]

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]

## Appendix I

_{P}is extracted from the low field behavior of

_{F}and r

_{F}respectively, are extracted from the high field behavior of

_{o}of the pseudo-ferroelectric phase is extracted from the

## Acknowledgments

## 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 |

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 |

3. | A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K |

4. | G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K |

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. |

6. | E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nano-disordered ferroelectrics,” Nat. Photonics |

7. | A. Bitman, N. Sapiens, L. Secundo, A. J. Agranat, G. Bartal, and M. Segev, “Electroholographic tunable volume grating in the g |

8. | A. Yariv and P. Yeh, |

9. | R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K |

10. | M. E. Lines and A. M. Glass, |

11. | Y. Girshberg and Y. Yacoby, “Off-centre displacements and ferroelectric phase transition in dilute KTa |

12. | G. Bitton, Yu. Feldman, and A. J. Agranat, “Relaxation processes of off-center impurities in KTN:Li crystals,” J. Non-Cryst. Solids |

13. | J. Toulouse, “The three characteristic temperatures of relaxor dynamics and their meaning,” Ferroelectrics |

14. | R. Blinc and B. Zeks, |

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 |

16. | A. J. Agranat, “Optical lambda-switching at telecom wavelengths based on electroholography,” Top. Appl. Phys. |

**OCIS Codes**

(160.2100) Materials : Electro-optical materials

(250.4110) Optoelectronics : Modulators

(160.2710) Materials : Inhomogeneous optical media

**ToC Category:**

Crystalline Materials

**History**

Original Manuscript: April 6, 2011

Revised Manuscript: May 26, 2011

Manuscript Accepted: May 26, 2011

Published: June 2, 2011

**Citation**

Alexander Gumennik, Yael Kurzweil-Segev, and Aharon J. Agranat, "Electrooptical effects in glass forming liquids of dipolar nano-clusters embedded in a paraelectric environment," Opt. Mater. Express **1**, 332-343 (2011)

http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-1-3-332

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### References

- 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]
- 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]
- A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K1-xLixTa1-yNbyO3:Cu,V,” Phys. Rev. B 55(19), 12818–12821 (1997). [CrossRef]
- G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K1-xLixTa1-yNbyO3:Cu,V at the paraelectric phase,” Phys. Rev. B 58(9), 5282–5286 (1998). [CrossRef]
- 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]
- 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]
- 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]
- A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, 1984), Chapter 7.3.1.
- R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K1-yLiyTa1-xNbxO3:Cu,” J. Cryst. Growth 131(3-4), 486–494 (1993). [CrossRef]
- M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, 1977), Chapter 4.
- Y. Girshberg and Y. Yacoby, “Off-centre displacements and ferroelectric phase transition in dilute KTa1−xNbxO3,” J. Phys. Condens. Matter 13(39), 8817–8830 (2001). [CrossRef]
- 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]
- J. Toulouse, “The three characteristic temperatures of relaxor dynamics and their meaning,” Ferroelectrics 369(1), 203–213 (2008). [CrossRef]
- R. Blinc and B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics (Elsevier, 1974).
- 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]
- A. J. Agranat, “Optical lambda-switching at telecom wavelengths based on electroholography,” Top. Appl. Phys. 86, 133–161 (2003). [CrossRef]

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