Electro-optical interferometric microscopy of periodic and aperiodic ferroelectric structures

LASER & PHOTONICS REVIEWS Laser Photonics Rev. 9, No. 2, 214–223 (2015) / DOI 10.1002/lpor.201400122 O R IG IN A L P A P ER Abstract The ferroelectric domain structures of periodically poled KTiOPO4 and two-dimensional short range ordered poled LiNbO3 crystals are determined non-invasively by interferomet- ric measurements of the electro-optically induced phase retar- dation. Owing to the sign reversal of the electro-optical coeffi- cients upon domain inversion, a π phase shift is observed for the inverted domains. The microscopic setup provides diffraction- limited spatial resolution allowing us to reveal the nonlinear and electro-optical modulation patterns in ferroelectric crys- tals in a non-destructive manner and to determine the poling period, duty cycle and short-range order as well as detect lo- cal defects in the domain structure. Conversely, knowing the ferroelectric domain structure, one can use electro-optical mi- croscopy so as to infer the distribution of the electric field therein. Electro-optical interferometric microscopy of periodic and aperiodic ferroelectric structures Duc Thien Trinh1,3, Vasyl Shynkar1,∗, Ady Arie2, Yan Sheng4, Wieslaw Krolikowski4,5, and Joseph Zyss1 1. Introduction Nonlinear crystals in which the sign of the nonlinear coeffi- cient is spatially modulated are widely used to obtain quasi- phase-matched nonlinear interactions [1, 2]. The modula- tion of the nonlinear coefficient in ferroelectric crystals can be achieved by the electric field poling technique [3], in which a series of electric pulses that are applied through pat- terned electrodes invert the direction of the electric dipole in the material. Domain inversion reverses the sign of all non-zero elements of the third-rank tensors of the material, including the second-order nonlinear tensor and the Pockels electro-optical tensor [4], but has no effect on the refractive index of the crystal. Various methods have been proposed in order to charac- terize the domain and the defect structures, such as selective etching [5], scanning electron microscopy (SEM) [6] and atomic force microscopy (AFM) [7] and its piezoresponse force microscopy variant (PFM) [8]. However, surface etch- ing is a destructive method, while SEM- and AFM-based measurements provide the surface topography but do not allow us to access the optical properties in the bulk. In 1 Laboratory of Quantum and Molecular Photonics, UMR 8537 CNRS, ´Ecole Normale Supe´rieure de Cachan, France 2 Department of Physical Electronics, School of Electrical Engineering, Tel-Aviv University, Israel 3 Faculty of Physics, Hanoi National University of Education, Vietnam 4 Research School of Physics and Engineering, Australian National University, Canberra, Australia 5 Science Program, Texas A&M University at Qatar, Doha, Qatar ∗Corresponding author: e-mail: vasyl.shynkar@gmail.com addition, the results of surface measurements are not al- ways simple to analyze [8]. Optical techniques based on interferometry [9], the electro-optical effect [10] and non- linear optics [11, 12] permit us to solve many of these drawbacks. In particular, optical methods are non-invasive, allowing for in-situ measurements and three-dimensional (3D) domain-structure characterization. Moreover, they are less time consuming and provide more reliable data analysis. Digital holography [9] was demonstrated for two-dimensional (2D) domain-structure studies with a res- olution of 55 microns, which did not prove sufficient for single-domain investigations. In order to visualize ferro- electric domain boundaries in two-dimensional periodi- cally and quasi-periodically poled structures with small periodicity values, the ˇCerenkov second-harmonic gener- ation (SHG) method appears to be well suited [11, 13, 14]. It provides a high 3D optical resolution but requires an in- tense laser beam to generate a measurable second-harmonic signal and can neither determine the full vector orientation of ferroelectric domains nor variations of the nonlinear or electro-optical response within the domains. In addition, the in-depth scanning range of ˇCerenkov SHG is limited by C© 2015 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ORIGINAL PAPER Laser Photonics Rev. 9, No. 2 (2015) 215 Figure 1 (A) General scheme of the PLEOM setup with He-Ne laser source. PBS, polarization beam splitter; S, sample beam; R, reference beam; two objectives (40×, 0.6 NA); λ/2, half-wave plate; λ/4, quarter-wave plate; P, photodiode; LF, low-frequency signal fed into the feedback loop for phase-compensation control; HF, high-frequency signal. EO signal is measured by lock-in voltage proportional to induced phase shift. (B) Gold electrodes on a glass cover slip. The size of the cover slip is 3 × 6 cm2. the working distance of the microscope objective, which is generally of the order of a few hundreds of microns, thereby limiting the volume to be investigated. We propose here an alternative technique based on inter- ferometric imaging microscopy [15], which was introduced recently for the examination of individual nanoscale ferro- electric crystals with an homogeneous domain structure. We apply this technique for the first time, in order to investi- gate the domain structure in both cases of periodically and aperiodically poled crystalline structures. We experimen- tally demonstrate the ability of this approach to characterize crystals embedding inverted domains down to diffraction- limited resolution. This method enables us to determine the period and duty cycle of a periodically poled KTP crys- tal and the short-range order of a quasi-periodically poled LiNbO3 crystal using a low-power He-Ne continuous-wave (CW) laser. Conversely, in the case of a crystal with known ferroelectric domain structure, this method allows us to ac- cess and measure the electric field inside the crystal. This is of specific importance in cases where the electric field distribution needs to be visualized in a non-contact mode with high sensitivity and precision. 2. Experimental setup 2.1. Pockels linear electro-optical microscope (PLEOM) We have recently developed a state-of-the-art microscope based on the Pockels linear electro-optical effect (PLEOM) [15–17] (Fig. 1A). This microscope was conceived with the aim of accessing the structure of nano-objects, by way of measuring the phase change experienced by a laser beam in interaction with such nano-objects under application of an external electric field. The high sensitivity could be achieved by means of interference measurements supple- mented by homodyne and synchronous detection. A He-Ne laser having high spatial and temporal coherence was used as the light source. The He-Ne laser beam was split into two channels, a weaker signal beam and a much stronger reference beam, with corresponding amplitudes αs and αr , using a polarization beam splitter (PBS) to allow for the im- plementation of homodyne detection [16]. The sample was placed on the piezo actuator of a 3D scanner (PiezoJena, Jena, Germany) between two long working distance micro- scope objectives (WD = 3.7 mm, NA = 0.6, 40×), adapted to transparent materials of thicknesses up to 2 mm. Because of its very weak intensity, the laser beam could be focused down to a diffraction-limited spot without causing optical damage to the sample. Two half-wave plates were placed on each side of the microscope to allow rotation of the polar- ization of light propagating through the sample, thus giving access to different coefficients of the electro-optical tensor of the sample. The signal and reference beams reached a second PBS with horizontal polarization using a combina- tion of half-wave plates. The reference beam propagating through the second PBS was then reflected with a mirror fixed on a piezo-electric actuator, directed back to the PBS with vertical polarization and then reflected onto the third mixing beam splitter. The mirror on the piezoelectric actuator was part of a feedback loop system dedicated to the stabilization of the optical setup. The polarizations of the reference and signal beams were rotated by 45◦ with respect to their original polarizations using a half-wave plate, and then the light from these two beams was mixed in a third PBS cube, lead- ing to the following amplitudes along the two orthogonal polarizations: α1 = 1√ 2 (αr + αs), (1) www.lpr-journal.org C© 2015 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim LASER &PHOTONICS REVIEWS 216 D. T. Trinh et al.: Electro-optical interferometric microscopy of periodic and aperiodic ferroelectric structures α2 = 1√ 2 (αr − αs). (2) The photocurrents generated by the photodetectors (Hamamatsu, Japan) were proportional to the intensities of the incident beams: i1 = ρ I1 = 12ρ|αr + αs | 2, (3) i2 = ρ I2 = 12ρ|αr − αs | 2, (4) where ρ is a proportionality factor depending on the de- tector responsivity. After amplification, the differentiated signal represents the interference of the two beams and can be expressed as Iint = 2Re[αrα∗s ]. (5) Variation in the interference term due to possible pertur- bation in the signal pathway due to the refractive-index modulation is given by Iint = 2Re [αrα∗s ]. (6) The difference in the geometric lengths between the two op- tical pathways introduces an inherent phase shift φr , which can be additionally influenced by drifts due to thermal ex- pansion and mechanical vibrations. The amplitude of the reference beam can be written as αr = αr exp (iφr ). (7) The electro-optical effect introduces an additional phase shift ϕEO into the signal beam, αs = αs exp (−iϕEO). (8) Consequently, the interference term then can be rewritten as Iint = 2Re {αrαs exp [i(φr − ϕEO)]} = 2αrαs cos (φr − ϕEO). (9) In our working configuration, the low-frequency compo- nents were used to lock the mean relative phase φr at π/2, thereby cancelling the difference between the mean pho- tocurrents via an optical phase-locked loop. This was per- formed by the use of a piezoelectric actuator compensating for the thermal dilatation of the optical components (involv- ing large optical path variations at low frequencies) and a piezoelectric stack compensating for mechanical and acous- tic noises (small optical path variations at higher frequen- cies). The cut-off frequency for this optical phase-locked loop was set at 20 kHz. As a consequence, Iint = 2αrαs sin(ϕEO). (10) Due to a small phase perturbation from the Pockels effect, the previous expression can be simplified to Iint ≈ 2αrαsϕEO. (11) This last equation shows that the detected photocurrent difference, i−, is directly proportional to the phase shift induced by the external electric field, E = E cos ( t + φE ), (12) where is the modulation frequency, much lower than the frequency of the incident beam at ω, and φE is the phase of the applied electric field. Due to the Pockels electro-optical effect, this electric field induces a time-varying phase shift ϕEO = ϕ cos ( t + φE ), (13) where ϕ describes the amplitude of the phase shift induced by the sample and is proportional to the relevant coeffi- cient (or combination of these) of the electro-optical tensor and to the applied electric field. The detected photocurrent difference becomes i− = 2ραrαsϕ cos ( t + φE ). (14) Using a lock-in detector ( EG&G Princeton Applied Re- search, Oak Ridge, TN, USA), we measured the electro- optical signal proportional to 2αrαsϕ, converting the dif- ferential current signal i− into a voltage which was further multiplied by a reference lock-in voltage function, R = A cos ( t + φR). (15) As a result of this operation followed by signal filtering, we obtained two values: the voltage amplitude proportional to 2Aαrαsϕ and the phase φ = φE − φR . The amplitude is thus proportional to the electric field-induced electro- optical phase shift. On the other hand, the measured phase reflects the relative phase shift between the phase of the applied electric field and the phase of the induced EO effect. The latter is a clear signature of the absolute orientation of the EO tensor with respect to the applied electric field. The amplitude of the generated voltage varied from 12.5 V to 150 V, while the frequency was changed from 20 kHz to 1 MHz. As mentioned above, the modulation frequency was higher than 20 kHz to avoid cut-off by the phase-control loop. Both amplitude and phase control characteristics of the electro-optical phase retardation induced in the sample were measured in our experiments. 2.2. Sample preparation In our experiments, we used planar gold electrodes de- posited on a 170 μm thick glass coverslip (Fig. 1B) using vacuum evaporation and soft lithography techniques. We C© 2015 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.lpr-journal.org ORIGINAL PAPER Laser Photonics Rev. 9, No. 2 (2015) 217 Figure 2 (A) Configuration of the PPKTP sample (1) on gold electrodes (2) deposited on a glass coverslip (3). (B) Electrode structure cross section in the YZ plane for the LiNbO3 sample. The thickness and width of the gold electrodes were respectively 10 nm and 70 μm and the distance between electrodes alternated between 100 μm and 50 μm. first deposited a 10 nm thick chromium adhesion layer and then a 50 (respectively 10 nm) thick gold layer using a vac- uum chamber for PPKTP (respectively LiNbO3). In the next step, photolithography and etching techniques were used to pattern the electrodes on the gold layer. These electrodes were connected by electrical wires to a function generator. The electric field was induced inside the crystal by applying a voltage signal to the electrodes. We note that the 50 nm and 10 nm thicknesses of the electrodes are negligible with respect to their 70 μm width. The distance between the electrodes was set at 100 or 50 μm (Fig. 2B). Likewise, the electrode lengths were much greater than their widths and separations. The electrically poled 10 × 2 × 0.5 mm3 KTP crystal [3,18] with a 36 μm period was set on top of the gold electrodes (Fig. 2A). The PPKTP sample was oriented so that the poled zones are perpendicular to the gold electrodes and parallel to the Y axis (Fig. 2A). The top and bottom Z planes of the crystal were optically polished. A detailed cross section of the fabricated gold electrodes is presented in Fig. 2B. The Y Z cross section shown herein corresponds to the poled zone section for the PPKTP sample. In this case, the electric field induced from the electrodes was aligned along the poled zone in the Y Z plane. Within this configu- ration, the electric field therefore displays only Ey and Ez components (Ex = 0). The same electrode configuration was used in the case of the two-dimensional quasi-periodic decagonal LiNbO3 crystal used for red–green–blue lasers [14], but with an electrode thickness down to 20 nm, comprising a 10 nm thick chromium layer and a 10 nm gold layer. This thinner electrode layer limits the attenuation of the transmitted sample beam, so as to be compatible with the characterization of the crystal where the beam goes through the electrodes, as further detailed in Sect. 4.2. The LiNbO3 sample was oriented so that Y -polarized light propagates along the crystallographic Z axis, as in the case of PPKTP. The beam was focused on the bottom surface of the crystal, close to the gold electrodes, where the electric field is the strongest. 3. Simulation results Simulation of the electric field distribution created by the electrodes enabled us to understand the response of the PPKTP and LiNbO3 structures to an applied voltage. The electric field distributions and accumulated phase in peri- odically poled KTP and quasi-periodically poled LiNbO3 crystals were evaluated by use of COMSOL software (ver- sion 4.2). This software was used to simulate the static electric field distribution for a 150 V DC voltage applied to the electrodes. As expected, the electric field exhibits strong gradients near the electrode edges (shown in red color in Fig. 3A), where the free-electron density peaks. For the mediator plane between the electrodes (e.g. perpendicular to the observed image), the electric field is reduced to its Ey component. We chose this position as the origin on the Y axis (see Fig. 2B). In the vicinity of the plane at the middle of each electrode, Ez is the only non-negligible component of the applied electric field (see Fig. 3A). The variation of Ez , the sole component contributing to the Pockels effect, is plotted in Fig. 3B at various y positions. A strong value of the electric field is visible in the vicinity of the gold electrodes together with a rapid decay of the electric field with distance along the z axis. We can see that the magnitude of Ez becomes insignificant at distances exceeding 150 μm along the z direction. It should be noted that the electric field is lower than the coercive field (∼20 kV/mm for LiNbO3) and, in addition, is alternating with a 20 kHz period, so that changes of the domain pattern can be safely ruled out. This property simplifies the general expression of the Pockels effect in the present case for PPKTP and LiNbO3 crystals. The general expression for the nonlinear polarization is Pi (ω ± ) = 20 ∑ i j χ (2) i jk(ω ± ; ω, )E j (ω)Ek( ). (16) Here E j (ω) stands for the amplitude of the laser beam and Ek( ) represents the quasi-static electric field inside the crystal, as generated by the electrodes. In this study, the gold electrodes sustain an AC voltage applied to the electrodes, which induces a low-frequency modulating field given by Eq. (12). The third-rank second-order nonlinear susceptibility tensor χ (2)i jk is linked to the Pockels electro-optical tensor ri jk by the following relation: ri jk(−ω; ω, 0) = − 2 i i j j χ (2) i jk(−ω; ω, 0). (17) www.lpr-journal.org C© 2015 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim LASER &PHOTONICS REVIEWS 218 D. T. Trinh et al.: Electro-optical interferometric microscopy of periodic and aperiodic ferroelectric structures Figure 3 Electric field distribution around the gold electrodes. (A) 2D map of the electric field in the YZ plane. The orientation is shown by black vectors and the amplitude is color coded. The electrodes are not visible due to their small thickness but their edges can be recognized due to the strong electric field gradients in their close vicinity. (B) Profile of the Ez component as a function of y. The origin of the z axis corresponds to the electrode surface level or the crystal bottom surface. The origin of the y axis is set at mid distance from the two electrodes. The electro-optical tensor ri jk is symmetric in its first two indices, allowing us to contract it into a two-dimensional array, following conventional notation. The refractive-index variation can then be expressed in terms of the applied electric field as  ( 1 n2 ) i = ∑ j ri j E j . (18) The phase shift experienced by light passing through the crystal is then given by ϕi = πn 3 i λ ∫ ∑ j ri j Ej dl. (19) The integration is performed along the optical pathway in the crystal. In the case of PPKTP, in the present configu- ration, the phase shift is proportional to the integral of Ez along z, leading to ϕy = πn3y λ ∫ r23 Ez dz, (20) where the refractive index ny equals 1.7713 for 632.8 nm, calculated using the Sellmeier equation [19]. A member of the mm2 point group, the KTP electro-optical matrix has only five different non-zero coefficients, out of which our configuration uses only r23 = 15.7 pm/V at 632.8 nm. We plot in Fig. 4A the calculated phase change of the laser beam passing through the PPKTP crystal under a quasi-static electric field created by th