In-situ monitoring of domain reversal in congruent lithium niobate by a digital holographic technique is described. While the ferroelectric polarization is reversed by electric field poling, the two-dimensional distribution of the phase shift, due mainly to the linear electro-optic and piezoelectric effects, is measured and visualized. Digital holography is used to reconstruct both amplitude and phase of the wavefield transmitted by the sample to reveal the phase shift induced by adjacent reversed domains during the poling. The resulting movies of both amplitude and phase maps, for in-situ visualization of domain pattern formation, are shown. The possibility of using the technique as tool for monitoring in real-time the periodic poling of patterned samples is discussed.
©2004 Optical Society of America
Ferroelectric crystals, such as lithium niobate (LN) and lithium tantalate, find many photonic applications, related to their peculiar nonlinear and electro-optic properties, ranging from fabrication of periodically poled crystals for nonlinear frequency generation by quasi-phase-matching (QPM) [1–2], to complex structures used for photonic bandgap devices . All of the phenomena employed in these devices are strictly dependent on the existence and kinetics of domain structure. As a consequence, the ability to microengineer ferroelectric domains is central to all of these applications, and techniques for visualizing domain structure and dynamics are important for the characterization of ferroelectric materials and devices . For example, the fundamental prerequisite for using the QPM technique, in nonlinear devices, is the ability to fabricate crystals with periodic domain structures of good quality, typically with periods in the 3μm to 30μm range. Various techniques have been used for this purpose, most successfully the technique of electric field poling [5,6]. It is based on the application of high external electric field pulses to periodically reverse the spontaneous polarization using the in-situ monitoring method which measures the displacement current flowing in the external electrical circuit . The major limitation of this technique is the minimal achievable period length, limited by electric field inhomogeneities. In fact, it is well known  that domain growth under the electrodes can be simplified into four main stages: domain nucleation at the electrode edges, domain tip propagation towards the opposite crystal face, the merging of the domains under the electrodes and the domain walls propagation away from the electrodes. A 50% duty cycle between regions of opposite domain orientation is desirable for QPM applications and the last stage makes it difficult to obtain. Hence, visualization of the domain growth and its subsequent morphological evolution, through the domain wall motion, would be desirable to control in real-time and in-situ the domain pattern fabrication. Unambiguous ex-situ identification of domain structures is usually achieved by the well-established technique based on selective chemical etching . Domain polarity is determined by transforming the reversed domain pattern into a topographic structure on the crystal surface, observable by a conventional microscope. This technique provides high resolution and comparatively ease of use but it is destructive and it is not able to give in-situ information about the structure of reversed domains.
In the past few years, it has been reported that the 180° domain walls in ferroelectric materials can be imaged  at room temperature by ordinary light microscopes under polarized or unpolarized light. This discovery has given rise to a wide variety of monitoring techniques. At the beginning much effort has been addressed for studying the domain kinetics by partly ex-situ techniques. Constant pulses were applied to the crystal to nucleate and grow domains and the evolution process was separately monitored, after every pulse, by optical observation under a light microscope . The electric field application and the optical observation steps were separate. Afterwards, other monitoring imaging techniques have been developed, such as confocal frequency doubling , near-field optical microscopy , atomic force microscopy , interferometry  and electro-optic imaging microscopy [15–18]. These techniques provide amplitude-contrast images of the ferroelectric domain pattern during formation, giving information about nucleation and growth kinetics of the reversed domains. Recently, illumination with coherent light along the z axis of electro-optic crystals has been demonstrated to contain information about the domain structures. It comes from an analysis of the near field interference pattern generated by waves travelling through domains of opposite direction and phase shifted by the combined electro-optic and piezoelectric effects . However, to the best of our knowledge, a quantitative measurement of the two-dimensional phase shift distribution experienced by a plane wave, transmitted along the z crystal axis during the domain reversal process, is not presented in literature.
We propose in this paper a digital holography (DH) based technique [20–22] for in-situ interferometric analysis of domain reversal process in congruent LN providing quantitative spatially resolved phase shift data. The phase shift distribution is generated simultaneously by linear electro-optic and piezoelectric effect along the z crystal axis. Such phase shift distribution is numerically reconstructed by the DH method and it is denoted here as the phase-map of the sample. Differently from the conventional phase-contrast imaging performed with a microscope, in which an optical mechanism is used to translate phase variations into corresponding amplitude changes, the technique presented here provides direct measurement of the phase shift distribution of the object wavefield. A reflective grating interferometer (RGI)  set-up generates an interference fringe pattern obtained by the recombination of a reference wavefield with a wavefield transmitted by the sample during the poling process. While an external electric field that reverses the spontaneous polarization of the material is applied, the fringe pattern changes due to electro-optic and piezoelectric effects. A charge-coupled device (CCD) continuously records the evolving fringe pattern. The digital holograms acquired during the poling process are used to numerically reconstruct both the amplitude and the phase of the wavefield transmitted in quasi real-time by the sample. Sequences of amplitude- and phase-maps of the domain walls moving under the effect of the external electric field are obtained and collected into movies. Quasi real-time in-situ qualitative information about the spatial evolution of the reversed domain pattern is given by the reconstructed amplitude-map movie, while quantitative information is provided by the phase-map movie.
Results for congruent LN samples, spontaneous polarization of which is reversed by application of a single external electric field pulse exceeding the coercive field of the material, are presented and discussed. The technique provides full-field simultaneous information about the optical amplitude and phase deformation induced by domain reversal process during poling. In principle, this technique could be used as a new non-invasive tool for in-situ monitoring of the electric field periodic poling in ferroelectric materials, by a relatively simple procedure. Moreover, by building the set-up in a microscope configuration, the technique can be used for studying the optical phase shift effects induced by the point defect complexes due to lithium nonstoichiometry and arising in the vicinity of domain boundaries. The behaviour of independently growing and merging domains would be studied by means of the phase shift analysis in order to investigate interaction between domains which arise from independent nucleation events.
2.1 The experimental set-up
Congruently melting z-cut LN samples (25×25×0.5)mm sized were obtained by dicing single domain 3-inch diameter crystal wafers polished on both sides. Figure 1 shows schematically the sample holder and the external electric circuit used for the poling process at room temperature. Figure 2 shows a picture of the sample holder. The structure of the sample holder is inspired by that used by Wengler et al. . Domain inversion is achieved by applying one positive high voltage pulse, slightly exceeding the coercive field of the material (~21kV/mm for LN), to the z + face of the crystal sample . Electrical contact on the sample surfaces is obtained by a liquid electrode configuration consisting of two electrolyte containing chambers which squeeze the sample between two O-ring gaskets. Tap water is used as liquid electrolyte. This configuration insures both the homogeneity of the external electric field within the sample, due to the uniform adhesion of the electrolyte to the crystal surface, and transparency along the z direction, which allows illumination of the sample through the quartz windows during the poling process.
A conventional Signal Generator (SG) drives an High Voltage Amplifier (HVA - 2000×) to deliver +12kV/and a series resistor Rs = 100MΩ is used to limit the current flowing in the circuit. Because the applied voltage exceeds the coercive field of LN, a displacement current Ipol flows in the external circuit due to the charge redistribution within the crystal structure. This current is measured by acquiring the voltage drop across the resistor Rm through the oscilloscope OSC. A computer-controlled technique is used  to apply the voltage for a duration T such that T = Q/Ipol = 2PsA/Ipol where Q is the delivered charge, Ps the spontaneous polarization of the crystal and A the area to be polarization reversed.
The poling system is positioned into the RGI set-up  as shown in Fig. 3. It is made of two components, a mirror and a reflective grating (RG) with 1200lines/mm and (44×44)mm sized. A He-Ne laser emitting at 632.8nm is launched into a single mode optical fiber and expanded to a spherical wave which impinges onto the off-axis parabolic mirror POM to obtain a plane wavefront beam. The collimated wavefront w(x,y) is spatially divided in two half wavefronts w 1(x,y) and w 2(x,y) by the mirror and the grating while the wavefront w 1 (x, y) is reflected onto the grating by the mirror. The angle of incidence of the two half wavefronts on the grating is such that both of them are diffracted along the normal to the reflective grating. The wavefront w 1(x, y) is folded on the other wavefront w 2(x,y) and interferes with it giving place to a non-localized fringe pattern in front of the grating. The fringe pattern is digitized by a CCD camera with 512×512 square pixels 11.7μm sized. The field of view captured by the CCD array is around (5×5)mm sized.
The poling set-up is positioned in front of the reflective grating RG before the recombination occurs, so that the half wavefront w 2(x,y) is that transmitted by the crystal. A careful alignment is necessary to position the wave vector of the incident light parallel to the z crystal axis and the laser beam is linearly polarized with the polarization direction along the x-axis of the crystal.
2.2 Amplitude-map and phase-map reconstruction for in-situ visualization of reversed domains
LN being an electro-optic material, the refractive index n increases from n to n + Δn under a uniform external electric field, in one domain, while in the oppositely oriented one it decreases from n to n-Δn, thus providing an index contrast across the domain wall . The refractive-index change, due to the linear electro-optic effect along the z crystal axis, depends on the domain orientation according to Δn ∝ r 13 E 3, where E 3 is the external electric field parallel to the z crystal axis. The index difference across a domain wall is equal to 2Δn and causes phase retardation of the transmitted beam in the crystal. This effect is widely used for in-situ electro-optic imaging of domain reversal in ferroelectric materials [16,17], through the crystal thickness, without any polarizers , thus avoiding the ex-situ invasive chemical etching process . The phase retardation of a plane wave incident along the domain boundaries is also affected by the piezoelectric effect, which induces a negative or positive sample thickness variation Δd in reversed domain regions. Therefore, during the electric field poling, an incident plane wave experiences a phase shift Δϕ, mainly due to the linear electro-optic and piezoelectric effects along the z crystal axis, according to
where the piezoelectric thickness change Δd is dependent on k 3, the ratio between the linear piezoelectric and the stiffness tensor (k 3=7.57×10-12 m/V) , nw =1.33 is the refractive index of water and U is the applied voltage. A DH technique [20–22] is used here for in-situ visualization and optical analysis of domain reversal in LN during the electric field poling process. DH is an imaging method in which the hologram resulting from the interference between the reference and the object complex fields, w 1(x,y) and w 2(x,y) respectively, is recorded with a CCD camera and numerically reconstructed. The hologram is multiplied by the reference wavefield in the hologram plane, namely the CCD plane, to calculate the diffraction pattern in the image plane. The reconstructed field Γ(ν,μ) in the image plane, namely the plane of the object, at a distance d from the CCD plane, is obtained by using the Fresnel approximation of the Rayleigh-Sommerfeld diffraction formula
where r(ξ, η) ≡ w 1 (ξ, η) is the reference wave which, in the case of a plane wave, is simply given by a constant value, h(ξ, η) = |w 1 (ξ, η) + w 2 (ξ, η)|2 is the hologram function, λ is the laser source wavelength, the factor cos α, with α ≅ 49°, is introduced for correcting the anamorphism effect  and d is the reconstruction distance, namely the distance measured between the object and the CCD plane along the beam path. The coordinates (,μ) are related to the image plane coordinates (x’, y’) by ν = x'/λd and μ = y'/λd . The reconstructed field Γ(ν,μ) is obtained by the Fast Fourier Transform (FFT) algorithm applied to the hologram h(ξ,η) multiplied by the reference wave r(ξ,η) and the chirp function exp[(iπ/λd)(ξ 2 + η 2)]. The discrete finite form of Eq. (2) is obtained through the pixel size (Δx', Δy') of the CCD array , which is different from that (Δξ, Δη) in the image plane and related as follows:
where N is the pixel number of the CCD array. The wavefields impinging on the CCD surface are digitized at 8-bit rate and stored as numerical array data corresponding to the hologram patterns which are processed by Matlab program according to the discrete finite form of Eq. (2). The great advantage of this technique is the possibility to numerically reconstruct the complex field of the object beam. The two-dimensional amplitude A[x',y') and phase ϕ(x',y') distributions of the object wavefield can be re-imaged by using one hologram acquisition and performing simple calculations on the object wavefield Γ(ν,μ) reconstructed from the numerical solution of diffraction equations:
The DH technique is a novel tool for microscopic visualization and optical analysis of ferroelectric domain boundaries. In fact, conventional microscope imaging suffers from small depths of focus , because of the high numerical apertures of the lenses and the high magnification ratios, so that mechanical motion along the optical axis is required to check the focus of the image. In contrast, DH allows 3D simultaneous calculation of the complex wave front, in amplitude and phase, by numerically solving diffraction equations . In comparison with standard interference techniques, which also provide amplitude and phase map imaging, DH provides many advantages. Recording of only one hologram is necessary while recording of four or more interferograms is required in phase-shifting interferometry, such that the required stability for the acquisition of multiple images is very much relaxed and, in addition, dynamic events can be recorded. Numerical reconstruction of the back propagated beam does not suffer for spreading of diffraction since the amplitude and the phase of the object beam are reconstructed at right focusing distance. This allows a correct exact correlation between the spatial distribution of the phase-map and the area of the sample under investigation. Furthermore, DH provides the possibility to digitally correct the aberrations [21,26] due to the objective lens.
3. Experimental results
Figure 4 shows the in-situ video of the interferograms acquired by the CCD during the electric field poling of a LN sample. The video frames have a temporal resolution of 10frame/s. The z + crystal face, photoresist patterned to have a total electrode area about (5×5)mm sized, is imaged. The polarization axis is normal to the image and the field of view is slightly larger than the electrode area. The diffraction pattern of the photoresist window is visible. The out of focus diffraction image of two principal domain walls which grows from the upper and lower regions is clearly superimposed on the interference fringe pattern. They move towards the center until the spontaneous polarization of the crystal under the electrode area is reversed. The visibility of the domain wall under transmitted light is determined by the opposite phase shift induced on the reversed domains by the linear electro-optic and piezoelectric effects along the z crystal axis, as mentioned in Section 2.2.
The DH technique illustrated in Section 2.2 is applied to all of the recorded interferograms for reconstructing the amplitude and the phase-maps of the domain pattern during the electric field poling. The domain switching of the crystal polarization under the electrode area took about 5s. Figure 5(a) shows the movie obtained by collecting the two-dimensional distribution of the object wavefield amplitude numerically reconstructed from the interferograms recorded during the poling process.
A full-field visualization of the domain wall, able to provide in-situ information about the spatial evolution of the reversed domain pattern, is obtained. It is clearly visible that domain nucleation preferentially starts at the edges of the electrode area. Then, domains merge to form a few dominant domain wall fronts. The lateral resolution achieved in these images is around 55μm, according to Eq. (3). The resolution can be easily improved by using a smaller distance between the sample and the CCD array. In fact, the reconstruction pixel in the image plane is lower for smaller reconstruction distances . Without any magnification, about 11.7μm (CCD pixel size) spatial resolution can be achieved by using the so-called convolution numerical procedure for resolving the diffraction problem . Otherwise, by using a DH set-up in a microscope configuration, it is possible to obtain lateral resolutions down to 0.5μm .
A reference interferogram of the sample at its initial virgin state is acquired when no voltage is applied. With respect to this, the phase shift experienced by the object wavefield during poling is calculated. The DH reconstruction is performed for both the reference hologram and the nth hologram recorded during the domain switching, to obtain the corresponding object wavefield phase distributions ϕ 0(x',y') and ϕn(x',y'). The two-dimensional map of the phase shift Δϕ(x', y')= ϕn(x', y') - ϕ 0(x', y') is calculated for each hologram and the corresponding images are collected into the movie presented in Fig. 5b. In Figs. 5(a–b) the reconstruction distance is d=540mm. Both the out of focus real image and the zero-order diffraction term, generated by the DH numerical procedure , are filtered out for clarity. Moreover, the anamorphism effect, visible in Fig. 4 by the deformed shape of the square resist window, was numerically corrected in the reconstruction process by inserting the cos α factor in the diffraction formula (see Eq. (2)) . A surface plot representation of the unwrapped phase-map is performed for the central region of the more representative frames, to reveal the profile of the phase shift occurring at domain boundaries. Figure 6 shows the resulting movie, which shows the evolution of the object wavefield phase-map profile during the poling process. The mean value of the phase-shift occurring across the domain boundary is calculated to be around 3.6rad. Such value is in good agreement with the theoretical value calculated by Eq. (1) (no=2.29; nw=1.33; r 13=10pm/V; k 3=7.57pm/V ) apart from a modulo 2π shift. Mean value and variance of the two-dimensional distribution of both the amplitude Aref (x', y') and the phase Δϕref(x', y') reconstructed for the reference hologram were calculated to estimate the accuracy of the method. The mean value of the reference amplitude is (4.7×10-4)a.u. with a variance of 0.09a.u., while the mean value of the refernce phase is 0.82rad with a variance of 0.11rad corresponding to about λ/60.
The in-situ reconstruction of the object wavefield phase distribution is useful basically for two reasons: it gives information about the local orientation of the ferroelectric axis by means of the phase shift profile and it provides quasi real-time evaluation of the amount of optical path difference in adjacent reversed domains.
The technique has been applied to another sample, previously processed by electric field poling to obtain two adjacent antiparallel domains, in order to investigate the method in presence of a domain wall. Figures 7(a–b) show the corresponding movies obtained by collecting the images of the reconstructed two-dimensional distribution of the amplitude and the phase of the object wavefield. The reconstruction distance is d = 570mm.
The standing horizontal line visible in both amplitude and phase images corresponds to the domain wall previously generated. It is interesting to note in the phase movie (see Fig. 7(b)) how the phase changes in the region between the two merging domains. Such localized phase shift distribution could be due to refractive index stresses occurring in correspondence of reversing domain walls and still present as residuals on domain boundaries after electric field poling patterning [29,30].
In principle, this technique could be used as a tool for real-time monitoring of the electric field periodic or aperiodic poling on resist patterned LN samples. In fact, while the amplitude map is not useful for distinguishing unambiguously the refractive index contrast due to the domain wall from that induced by the photoresist edges, DH provides the advantage to reconstruct the phase-map Δϕ(x',y') of the object wavefield during poling. Here, the phase shift contribution due to the photoresist layer can be numerically eliminated. In this way the domain regions having the polarization reversed by the external electric field, are unequivocally visualized (see Fig. 5(b)–7(b)). Therefore, once the area to be poled is measured, a computer-controlled monitoring technique can be set. It would be able to switch off the input poling voltage when the real-time measured phase shift pattern matches, within acceptable tolerances, the desired a-priori known distribution in the frame. As a consequence, the lateral spreading of reversed domains under the insulating layer could be controlled with high accuracy and the correspondence between the intensity variations of the flowing current and the spatial evolution of the domain walls can be studied. According to some measurements reported for the sideways velocities of 180° ferroelectric domains under steady-state electric field , the accuracy of the method proposed here dramatically depends on the temporal resolution of the interferogram acquisition. This could be improved by making use of high-speed cameras without any other change in the set-up and in the method presented here. This monitoring technique could be an alternative to the usual method based on the measurement of the poling current which only gives a good indication of how much area has been poled, by the evaluation of the amount of charge delivered to the sample, without any information about the spatial evolution of the reversed domain pattern.
As a consequence, the current monitoring technique is not able to provide data on the merging phenomenon that occurs between sideways growing domains.
4. Conclusions and further developments
A technique for in-situ amplitude and phase-map visualization of reversed domain patterns during electric field poling in LN, by making use of the linear electro-optic and piezoelectric effect, has been proposed and demonstrated in this paper. A DH method is used to reconstruct the amplitude and phase distribution of the wavefield transmitted by the specimen during the domain reversal process. The technique is non-invasive and provides important in-situ information about the phase shift distribution induced by the linear electro-optic and piezoelectric effect across domain boundaries. As opposed to the conventional phase-contrast imaging, that uses an optical mechanism to translate the phase variations into amplitude changes, the technique presented here provides a direct measurement of the phase shift distribution of the object wavefield. When used with the conventional monitoring of the poling current while images are recorded, the method is useful for studying the correlation between the spatial evolution of domain walls and the intensity variations of the flowing current. In principle, the technique could be used as a new non-invasive tool for in-situ monitoring of the electric field periodic and aperiodic poling in ferroelectric materials and the possibility to use the technique as an alternative tool for monitoring the poling of patterned samples has been discussed by observing that, while in the amplitude-map of the object wavefield it is not possible to unambiguously distinguish the refractive index contrast due to the domain walls from that induced by the resist edges. The obtained results provide a framework for future investigations, at a microscopic scale, of the optical phase shift effects. They can occur in the proximity of point defect lithium nonstoichiometry in the vicinity of a domain wall, and they are responsible for internal fields, optical birefringence and strains at domain boundaries. Simultaneous polarization imaging of the object wavefield transmitted across domain walls could be obtained by DH , offering a novel and more powerful diagnostic tools for measuring local birefringence effects due to the domain reversal process and related to several phenomena such as electro-optic, piezoelectric, elasto-optic, photorefractive effects. It could thus provide a complete characterization of the crystal properties. Furthermore, the different behaviour of independently growing and merging domains could be interpreted by an optical phase shift analysis, in order to study the interaction between domains which arise from independent nucleation events.
Authors would like to thank Marotta S.r.l. for the fabrication of the sample holder.
This work is supported by the Italian Ministry for Research and University, FIRB project number RBNE01KZ94: ‘Microdispositivi in Niobato di Litio’.
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