It is well established that the phase transition temperature TC of ferroelectric oxygen perovskites is strongly affected by the presence of impurities and defects. For example, the replacement of a Ta ion in potassium tantalate niobate (KTN) by an Nb ion will cause a change in TC of magnitude: δTC≈8.5 K° /1% per mole of Nb. A similar effect can be achieved by replacing a K ion in KTN by Li. Here the effect is more dramatic and results in certain cases in δTC≈50 K° /1% per mole of Li [KLTN]. Thus, at the paraelectric state where the dielectric constant is given by the Curie law, a spatial variation in the composition that causes a correlated modulation in the Curie temperature (designated by δTC(x)) will yield a change in the dielectric constant given by:
where is the relative static (and low frequency) dielectric constant, C is the Curie-Weiss constant, and T is the temperature. Applying a uniform electric field E to a crystal in which δTC is modulated will generate a modulation in the induced polarization that is spatially correlated with δε(x) and is given by:
where it is assumed that the crystal is slightly above TC so that εr>>1. At the paraelectric phase where the electrooptic effect is quadratic, the electric field induced birefringence is given by:
where n0 is the index of refraction at the paraelectric phase, geff is the effective quadratic electrooptic coefficient, and P is the induced (low frequency) polarization. Note that for a KTN crystal at T=TC+7K where εr=104, geff=0.2m²/C² , and applied field of E = 3 kV/cm, equation (3) yields induced birefringence of Δn≈0.01!
Finally, combining (1), (2) and (3) will yield the spatially modulated induced birefringence:
In summary: a permanent spatial modulation in TC is created by periodic changes in the composition introduced during the growth of the crystal. This generates a correlated modulation in the static dielectric constant. Under the application of an electric field the latter induces a correlated modulation of the refractive index. This feature, namely a refractive index grating activated by the application of an electric field is the dielectric electrooptic effect [REF16].