Domain wall characterization in ferroelectrics by using localized nonlinearities

In this paper, a method of domain wall characterization in ferroelectrics through Cherenkov second harmonic generation by localized nonlinearities is proposed. By this method, domain wall width is estimated to be less than 10nm. High spatial angular resolution of about 10mrad in the experiment reveals the fine structures of the domain walls. Combined with scanning techniques, this method can reconstruct domain wall patterns with high resolution. This method has advantages of being nondestructive, noncontact, in situ as well as of high resolution. ©2010 Optical Society of America OCIS codes: (160.2260) Materials: Ferroelectrics; (190.1900) Nonlinear optics: Diagnostic applications of nonlinear optics; (190.2620) Harmonic generation and mixing; (190.4400) Nonlinear optics: Nonlinear optics, materials. References and links 1. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). 2. K. Nassau, H. J. Levinstein, and G. M. Loiacono, “The domain structure and etching of freeoelectric lithium niobate,” Appl. Phys. Lett. 6(11), 228–229 (1965). 3. M. Müller, E. Soergel, and K. Buse, “Visualization of ferroelectric domains with coherent light,” Opt. Lett. 28(24), 2515–2517 (2003). 4. S. I. Bozhevolnyi, J. M. Hvam, K. Pedersen, F. Laurell, H. Karlsson, T. Skettrup, and M. Belmonte, “Second harmonic imaging of ferroelectric domain walls,” Appl. Phys. Lett. 73(13), 1814–1816 (1998). 5. G. Fogarty, B. Steiner, M. Cronin-Golomb, U. Laor, M. H. Garrett, J. Martin, and R. Uhrin, “Antiparallel ferroelectric domains in photorefractive barium titanate and strontium barium niobate observed by high-resolution x-ray diffraction imaging,” J. Opt. Soc. Am. B 13(11), 2636–2643 (1996). 6. S. Zhu, and W. Cao, “Direct observation of ferroelectric domains in LiTaO3 using environmental scanning electron microscopy,” Phys. Rev. Lett. 79(13), 2558–2561 (1997). 7. F. Augereau, G. Despaux, and P. Saint-Gr’egoire, “Imaging ferroic domain structures with an acoustic microscope: example of PPLN,” Ferroelectrics 290(1), 29–38 (2003). 8. O. Tikhomirov, B. Red’kin, A. Trivelli, and J. Levy, “Visualization of 180° domain structures in uniaxial ferroelectrics using confocal scanning optical microscopy,” J. Appl. Phys. 87(4), 1932–1936 (2000). 9. M. Flörsheimer, R. Paschotta, U. Kubitscheck, C. Brillert, D. Hofmann, L. Heuer, G. Schreiber, C. Verbeek, W. Sohler, and H. Fuchs, “Second-harmonic imaging of ferroelectric domains in LiNbO3 with micron resolution in lateral and axial directions,” Appl. Phys. B 67(5), 593–599 (1998). 10. P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Cerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17(10), 447–450 (1970). 11. A. Zembrod, H. Puell, and J. Giordmaine, “Surface radiation from nonlinear optical polarization,” Opt. Quantum Electron. 1(1), 64–66 (1969). 12. A. Fragemann, V. Pasiskevicius, and F. Laurell, “Second-order nonlinearities in domain walls of periodically poled KTiOPO4,” Appl. Phys. Lett. 85(3), 375–377 (2004). 13. D. A. Scrymgeour, and V. Gopalan, “Nanoscale piezoelectric response across a single antiparallel ferroelectric domain wall,” Phys. Rev. B 72(2), 024103 (2005). 14. J. Wittborn, C. Canalias, K. V. Rao, R. Clemens, H. Karlsson, and F. Laurell, “Nanoscale imaging of domains and domain walls in periodically poled ferroelectrics using atomic force microscopy,” Appl. Phys. Lett. 80(9),


Introduction
Ferroelectric is one kind of the most important materials in nonlinear optics.In the past decade, domain engineering, which is mainly based on ferroelectrics, has attracted great attention and been widely studied, such as the fabrication and application of periodically inverted domain structures [1].Meanwhile, a number of detection and visualization methods have been developed to observe the domain structures, like chemical etching method [2], optical imaging [3][4][5] and scanning microscopy techniques [6][7][8][9].Chemical etching is the most widely utilized method because of its high resolution and comparative ease of use.The disadvantage is also obvious that it is destructive.Optical imaging is comparatively simple and can be used to acquire real-time or in situ information about the domain structures, but diffraction limits the resolution.Scanning microscopy is non-destructive and of high resolution.However, it cannot give real-time information and strongly depends on the surface patterns.
In this paper, we propose a new optical method that can characterize domain walls through Cherenkov second harmonic generation (CSHG) by using localized nonlinearities.This method is not only non-destructive and in situ, but also gives domain wall information with high precision, such as wall width and orientation.Moreover, this method can be used to acquire the information of domain structures buried inside a crystal other than on the surface since it has strong response to domain walls anywhere in a transparent crystal.

Cherenkov second harmonic generation by localized nonlinearities in domain wall
In medium with normal dispersions, nonlinear polarization P 2ω forced by fundamental beam (FB) propagates faster than a free second harmonic (SH) beam in the same medium.Then CSHG should be generated by such polarization and emit at Cherenkov angle to the FB [10].Although typical setup for CSHG is always a waveguide with nonlinear substrate [10], conical CSHG with rather low efficiency in single domain nonlinear crystals has been reported in very early days [11].Recently, it has been shown that new localized nonlinearities exist in domain wall regions [12], which can generate enhanced CSHG.Compared with the conical CSHG in single domain crystals [11], CSHGs generated in domain wall regions are a pair of well-collimated beams symmetrical to domain wall.This is because in single domain medium, nonlinear polarizations forced by FB can be treated as isotropic in transverse directions, while nonlinear polarizations in domain wall regions are closely related to the domain wall width, where new nonlinearities only exist in the vicinity of the domain walls.It is reported in previous study that the effective domain wall width is about 100nm [13,14], much smaller than the micrometer-scale FB spot.So the polarization in domain wall region has planar geometry.Considering the coherence of the polarization over the full beam width, the generated CSHGs should be a pair of well collimated beams symmetrical to the domain wall.Here, we define a pair of well collimated beams as each of the CSHG beam is less than 1° in its angular distribution, i.e. total angular distribution of the CSHG is less than 2°. Figure 1(a) is a schematic of CSHG patterns generated by the FB-illuminated domain wall.When the width of domain wall is larger than or equal to the FB spot, conical CSHG will be generated obviously.With the increasing of the FB spot size, the conical CSHG will degenerate to a pair of arcs, and finally to a pair of well-collimated beams.In order to get the angular distribution of the CSHG arc, we need to calculate the electrical field by counting all the contributions of the area illuminated by FB spot in domain wall.The CSHG electrical field at distance r is then determined by 0 ( ) where ds is the element of radiating source driven by the FB in domain wall, r 0 is the distance between this element and the zero point, k is the wave vector of the harmonic beam, and A is the integrating area illuminated by FB spot in domain wall, which can be changed when the ratio of beam width over domain wall width r b /r dw changes.From Eq. ( 1  When rb/rdw = 1, the CSHG should be conical, so its angular distribution is 2π, as indicated as 360°.When rb/rdw>100, this pair of CSHGs are well-collimated.

Experiments and analysis
In our experiment, a Ti: Sapphire oscillator producing about 100 fs pulses at 84 MHz repetition rate at wavelength 800 nm with averaged power 200mW is used as the detecting light source.We fabricate different domain inversions with different wall patterns and qualities in 1mm z-cut LiTaO 3 samples, as shown in Figs.2(d), we can estimate that r b /r dw > 100 here, which means the domain wall width should be less than 100nm.This result is in good agreement with previous reports [13,14].In subsequent steps, we focus the FB tightly to the smallest of about 1µm, limited by the experimental setup.Also, a pair of well-collimated CSHGs with intensive divergence is observed and no conical CSHG is found.So we can improve the estimation of the domain wall width to be less than 10nm.More precise measurement can be done by decreasing the FB spot to smaller size.Although it seems that our estimation of domain wall width is much smaller than previously reported ones of about 100nm [13,14], we believe there is no conflict.It is very important to define precisely what one means by a domain wall width: whether in terms of a polarization gradient across the wall or of a range of material properties that might be influenced by the presence of the wall.In our research, domain wall width represents the region where enhanced nonlinearities exist.It is different from previously reported works [13,14] where domain wall width represents the range in which the material has good response to PFM and AFM techniques.By utilizing CSHG method, we can measure the orientation of domain wall with high precision as well as the width of domain wall.Figure 3 gives a pair of CSHGs which contains the information of the fine structures of the domain wall.Since smaller numerical aperture of the focused FB can provide higher angular resolution, here the FB is still loosely focused to about 10µm on the input facet, whose small numerical aperture will provide higher angular resolution.The domain wall covered by the FB spot has tiny difference in its orientation, which can generate CSHGs with tiny angular difference.The spatial angular resolution of the CSHGs is about 10mrad.Such tiny difference of the domain wall orientation is hard to detect by other optical methods, but it can be easily visualized by CSHG method with high precision.As well, with the decreasing of the FB spot size and its divergence, the precision of measurement can be further improved.This high spatial angular resolution may be meaningful in domain engineering of fine structures.When the domain walls' scale is smaller than the FB spot size and the domain walls have complex patterns, CSHG method can still be used to determine the orientations of the domain walls and show a first glance at the geometry of the domain wall structures.Due to the point-group structure and the nonstoichiometry, domain inversions in LiTaO 3 crystals are trigonal [15].We fabricate such trigonal micro domain inversions with averaged size of 7~8µm in the 1mm z-cut LiTaO 3 sample as shown in Fig. 4(a) and the 10µm FB spot can coves it totally.The generated CSHGs are hexagonally distributed as shown in Fig. 4(b).They are obviously three pairs of CSHGs symmetrical to the trigonal walls respectively.Since the FB spot size does not impact the CSHG, this method is valid even when domain inversions are smaller than the FB spot.For a more complex domain wall pattern, more complex CSHGs are expected, through whose intensities we can also determine the relative importance of the domain walls.Since CSHGs are quite sensitive to domain walls, it can also be used to monitor domain inversions inside any ferroelectrics.Different from typical optical method, CSHG method gives the information of the domain wall instead of visualizing the illuminated domain wall directly.In order to get the information of domain walls in a large area, we need to use the scanning technique.In this step, the FB is tightly focused to the smallest of about 1µm to increase the resolution.A stepping motor is used to move the sample with step length 1µm, so a point by point scanning can be realized.The CSHG patterns are captured by a CCD camera set behind the sample and stored by a computer.
All the stored information is used to reconstruct the domain wall patterns.Figure 5(a) is the domain wall pattern after etching and Fig. 5(b) is the reconstructed one by scanning CSHG method.We get good domain wall pattern at the resolution of 1µm.A more precise resolution is also expected with a smaller FB spot size.Since the 1µm FB spot nearly reaches the diffraction limit, it is hard to focus the FB to a smaller size.A proper method here to decrease the FB spot size is the near-field optical method.For example, a tapered fiber can squeeze down the light spot to sub-wavelength or even smaller, which can improve the resolution greatly.A useful feature of this method lies in the possibility of focusing slightly below the surface, therefore reducing the influence of the roughness of the surface on the measurement.Furthermore, we can perform a time-resolved measurement by using the pulsed laser source to reveal the growth process of domain structures.Conventional optical time-resolved measurement, such as visualization with coherent light, can provide overall status of the domain inversions.But it cannot give the precise information of the domain walls and is hard to make depth-resolved measurement.By CSHG method, it is convenient to focus the FB where we want to detect.Compared with previous optical method, this scanning CSHG method has no diffraction limit and can provide more precise measurement.Information provided by it should be of major interest for domain engineering in fine structures as well as over large crystals.CSHG by localized nonlinearities has been demonstrated in a variety of ferroelectrics, such as PPKTP [12], PPLN and PPLT [16], SBN [17] and so on.Since the localized nonlinearities should exist in domain wall regions of all ferroelectrics, this CSHG method can be used in all ferroelectrics and even other nonlinear materials with domain structures.

Conclusion
In conclusion, we propose a new method to characterize domain walls in ferroelectrics by using localized nonlinearities.By this method, we estimate that the width of the domain walls in LiTaO 3 is less than 10nm.The symmetry of the CSHGs is used to measure the orientation of domain walls and the spatial resolution is as high as 10mrad in our experiments.It is also demonstrated that even when the measured domain inversion is smaller than the FB spot can the CSHGs reveal the information of the domain walls of such tiny domain inversions.Combined with scanning techniques, CSHG method can reconstruct the domain wall patterns with high precision.This method has advantages of nondestructive, noncontact, in situ and of high resolution.Since CSHG by localized nonlinearities exist in domain wall regions of all ferroelectrics, this method should work in general in all ferroelectrics and even some other materials with domain wall structures.
), we can easily get the angular distribution of the CSHG.Calculations with different r b /r dw show that only when the ratio of beam width over domain wall width r b /r dw > 100, can a pair of well collimated CSHGs be observed in the far field (as shown in Fig.1(b)).By this principle, we can estimate the domain wall width by decreasing the diameter of the FB spot on domain wall.Whenever the pair of CSHGs changes to a conical CSHG, it means the diameter of the FB spot on domain wall is comparable to domain wall width.Moreover, the pair of CSHGs symmetrical to the domain wall can be used to acquire the orientation of the domain wall with high precision.

Fig. 1 .
Fig. 1.(a) Schematic of CSHG patterns of different ratio of beam width over domain wall width rb/rdw; and (b) Calculated CSHG's angular distribution changes with the increasing of rb/rdw.When rb/rdw = 1, the CSHG should be conical, so its angular distribution is 2π, as indicated as 360°.When rb/rdw>100, this pair of CSHGs are well-collimated.
2(a), 2(b) and 2(c).We firstly focus the FB loosely to about 10µm on the input facet.The straight domain wall with high quality in Fig.2(a) can generate a pair of well-collimated CSHGs, which are symmetrical to the domain wall, shown in Fig.2(d).The same experiment in low-quality straight domain wall in Fig.2(b) still generates a pair of CSHGs, shown in Fig.2(e).However, each of the pair is distributed in an arc due to the roughness of the domain wall.Two high-quality domain walls with 150° angle between them as in Fig.2(c) can generate two pairs of CSHGs, which can exactly reflect the domain wall orientations as shown in Fig.2(f).In this step, the focused FB spot is about 10µm.From the calculation results of Fig.1(b) and the well-collimated CSHG in Fig.

Fig. 2 .
Fig. 2. (a) and (b) are straight domain walls with high and low qualities respectively; (c) two high-quality domain walls with 150° angle; (d), (e) and (f) are corresponding CSHGs to domain walls in (a), (b) and (c).

Fig. 3 .
Fig. 3. CSHG patterns which reveal the fine structures of the domain wall.The spatial angular resolution is about 10mrad.

Fig. 4 .
Fig. 4. (a) Trigonal domain wall patterns on the + z surface of the sample; (b) corresponding CSHG patterns.