The deviation of light by domain-walls, first mentioned in barium titanate in 1955 [1], has been widely reported in ferroelastics (FEL) [2–7]. This phenomenon has been named deflection by Tsukamoto et al. and explained in Ref [3]. More recently, a similar deflection phenomenon has been evidenced in ferroelectric crystals, lithium niobate (LN) and lithium tantalate (LT), when submitted to high electric fields [8,9]. These crystals are now widely used for nonlinear optical applications, especially periodically-poled lithium niobate (PPLN) that is usually fabricated by poling techniques. The deflection phenomenon is here particularly interesting because the deflected spots can be used as real-time tracers of domain reversal during the poling process [8–10].

Up to now, deflection in ferroelectrics has been interpreted in terms of diffraction and attributed to the existence of a refractive index gradient through a finite domain-wall thickness of a few micrometers [8]. The present paper suggests a quite different interpretation.

In FEL crystals domain-walls are natural optical discontinuities that induce refractive transmission and reflection because of the crystal anisotropy. In non-ferroelastic ferroelectrics (FE), such as lithium niobate (LN) or lithium tantalate (LT), domain-walls are, in principle, optically continuous. However, in the presence of an electric field, the refractive indices are modified by the electro-optic (EO) effect. Field-induced optical changes in neighbor domains are always opposite, whatever the direction of the applied field, because LN and LT are full-FE crystals in which all EO coefficients go to zero in the paraelectric phase, and correlatively change sign from domain to domain in the FE phase. Any electric field applied in multidomain LN or LT gives optical contrast to the FE domain structure, which thus behaves similarly to a FEL one.

The case studied by Müller et al. in LN and LT is particularly simple because both the electric field and the incident wave-vector are parallel to the polar axis. In this configuration, the crystal remains uniaxial. Both the ordinary index n_o and the extraordinary index n_e are modified by the electric field, as follows:

 

Fig.1. . Construction of the deflected beams on the slowness curves. Electric field and incident beam parallel to z-axis. For clarity, the angles and the birefringence have been widely exaggerated.

Download Full Size | PDF

In these equations the sign ± refers to the orientation of the domain: positive (resp. negative) if the spontaneous polarization is parallel (resp. antiparallel) to the applied electric field. Since both electro-optic coefficients r ₁₃ and r ₃₃ are positive, both refractive indices are enlarged in negative domains. At grazing incidence on a domain wall, refraction thus occurs from positive domains to negative domains, as shown in Fig. 1. If not polarized, the incident beam gives in principle two refracted beams for each domain-wall orientation: one ordinary beam (polarized perpendicularly to the plane of incidence), one extraordinary beam (polarized in the plane of incidence). Their wave-vectors can be constructed on the slowness curves. The internal angle θ_o of the ordinary wave-vector with respect to the domain-wall is given by:

or, using Eq. (1):

Taking account of refraction at the output face of the sample, one finds the corresponding external angle α_o:

A similar calculation gives the following formulas for the extraordinary wave (θ_e internal angle, α_e external angle):

At ‘low’ fields, approximate formula are valid:

Since the Pockels coefficient r ₁₃ in LN and LT does not exceed 20 pm/V, these approximations are in fact always valid (up to 4 MV/cm, a value which largely exceeds the coercive field). Crystal optics hence predicts deflection angles α_o and α_e to be proportional to the square root of the applied field. Since n_e~n_o, Eqs. (8) & (9) do not differ significantly: the ordinary beam and the extraordinary beam are deflected at so close angles that it is most probably difficult to resolve them in experimental observations, unless a polarizer is used to extinguish one or the other.

Figure 2(a) compares the deflection angle measured by Müller et al. in LN at 351 nm versus electric field and the calculations given by Eqs (8) & (9), taking LN data at this wavelength, n_o=2.5410, n_e=2.4160 [11], r ₁₃=17.7 pm/V [12]. The agreement is excellent, provided that the applied field is corrected for an internal field of 1.4 kV/mm. In Ref [8] Müller et al. have estimated this internal field to 2 kV/mm. Figure 2(b) shows the spectral dependence α(λ) at E=14 kV/mm. The solid line is calculated by Eq. (8) using LN data from Refs [11,12]. According to this equation, α(λ) follows in principle the same dispersion law as the product n ²(λ)×r ^1/2 ₁₃(λ). The agreement with experimental results of Müller et al. is again fairly good.

 

Fig. 2. Variations of the deflection angle in lithium niobate: (a) versus electric field at wavelength 351 nm, (b) versus wavelength at field -14 kV/mm. Theoretical curves are calculated from Eqs (8) & (9), for the ordinary beam (solid line) and the extraordinary beam (dash line), assuming an internal field of 1.4 kV/mm in the crystal.

Download Full Size | PDF

It should be stressed that, contrarily to the natural deflection exhibited by FEL domain structures, the field-induced deflection at FE domain-walls in LN and LT consists here – in the configuration chosen by Müller et al. – in refractive transmissions only. There is no reflected beam symmetrical to the transmitted one, since the wall is not here a mirror plane for the optical indicatrices of neighbor domains. Because of this asymmetry, triangular positive domains in LT do not give a six-fold deflection pattern but a three-fold one. However, triangular negative domains can give a six-fold pattern, owing to wave-guiding by internal reflections inside the domains, as correctly explained in Ref [9].

To conclude, the field-induced deflection reported by Müller et al. in LN and LT is a consequence of refraction at domain-walls. It is not fundamentally different from classical deflection earlier observed in FEL crystals. Contrarily to what has been suggested in Refs. [8,9], the finite domain-wall thickness is not involved in deflection. It is involved only in diffraction, which gives the star-like branches observed by Müller et al. at lower angles than the deflection spots (α<α_o). This new interpretation does not weaken the interest of the method for studying the nucleation and growth of domains during the poling process, but forbids using it to estimate the domain-wall thickness, which anyway ranges far below the optical wavelength. Other configurations are worth being studied, especially the one in which both the electric field and the propagation direction are perpendicular to the polar axis. In this case, it can be predicted that the deflection angle is constant (related to natural birefringence) whereas the deflected intensity is proportional to E ², as will be shown in another paper [13].