1. Introduction

Ferroelectric materials like lithium niobate (LiNbO₃) see widespread use for applications ranging from integrated optics (e.g., as optical frequency converters, single photon sources or electro-optic modulators), in electronics (e.g. as RF-frequency filters, for pyroelectric detector) or for micro-mechanical devices [1–5]. One of the key aspects of ferroelectrics is that their various material properties are directly connected to the structure and orientation of ferroelectric domains and, hence, can be controlled by engineering the domain structure.

In recent years, it was demonstrated that the domain walls (DWs) separating areas of uniform dielectric polarization with different orientation of the order parameter, show properties that significantly differ from the properties of the bulk domains [6,7]. This is surprising, because ferroelectric DWs – in contrast to ferromagnetic DWs – are considered small, i.e. measuring only a few unit cells in width, and can be described by the simple Ising model [8,9]. Hence, DWs are not expected to possess intrinsic properties that are different from the bulk. Nevertheless, clear evidence have been given in the last two decades that DWs in ferroelectrics, equally in LiNbO₃, are accompanied by a multitude of nanoscopic effects, ranging from strain fields that extend over several µm, to defect accumulation at or within the DW, to deviations of the order parameter from the ideal Ising model, i.e. Néel- and Bloch-type DWs [10–13]. In this regard, one particular aspect for potential applications is the fact that DWs show an intrinsic electrical conductivity under certain conditions achieving values that is orders of magnitude larger as compared to the surrounding, reaching even 1 mA per DW across bulk LiNbO₃ [14]. This is extremely surprising as most ferroelectrics are wide bandgap, insulating oxides. Given this large electronic conductivity combined with the possibility to erase and rewrite domains and DWs practically at will, this promises a new type of reconfigurable nanoelectronic device, e.g. ferroelectric memristors [14,15].

In this context, LiNbO₃ is a strong candidate for optoelectronic applications. On the one hand, it was demonstrated that DWs show exceptionally high conductivity contrast of at least 5 orders of magnitude, as well as this conductivity can be tuned, i.e. switched, by electric fields [14,15]. On the other hand, LiNbO₃ is among the most studied dielectric single-crystalline materials. It is commercially available in the form of large-scale single-crystalline wafers up to a 6-inch diameter, in various qualities, e.g. for acoustic or optical applications, and in the form of bulk crystals, as well as single-crystalline thin films down to a 100 nm thickness [15,16]. Furthermore, recent work demonstrates that it can readily be integrated into existing CMOS technology, for example by wafer bonding [17,18]. Therefore, LiNbO₃ represents an ideal model system to study DWs.

So far, a full understanding of a DW’s conductive behavior in various ferroelectrics, its intrinsic properties and the mechanism of the conductivity remain elusive [19]. In this regard, there is a continued interest and need in studying the electric, mechanical, optical, and nonlinear optical properties of DWs and their difference to the bulk. The most common method for investigating ferroelectrics and their conductivity are methods of scanning force microscopy, most prominent piezo-response force microscopy (PFM) and conductive atomic force microscopy (c-AFM). Both methods offer a high-resolution analysis (<50 nm) of domain structures or their conductivity, respectively, however only in a surface-near layer. In particular, conductivity measurements may be heavily influenced by interface effects, e.g. Schottky barriers, which carefully need to be taken into account, when interpreting results [6,20,21].

In contrast, far field optical methods, while usually diffraction-limited, allow for the three-dimensional visualization and analysis of linear or nonlinear optical properties and, hence, of the dielectric properties over a wide frequency range [22]. For the analysis and study of domain walls, the currently applied two main techniques are second harmonic (SH) microscopy [23,24] and µ-Raman spectroscopy [25,26]. Both methods provide valuable insight into the symmetry and local crystal properties at DWs. However, both methods rely on nonlinear and second-order effects, respectively, and hence provide limited information about the linear optical response. The linear optical response of DWs, i.e. the refractive index profile, has been measured with high resolution in the visible range by scanning near-field optical microscopy [27], however, due to the limitations of scanning probe techniques only near to the surface. In contrast, Chen et al. have demonstrated that the periodically-poled LiNbO₃ (PPLN) can act as a diffraction grating and calculated the refractive index change from the efficiency of the grating [28], which, however, due to the nature of the experiment provided no insight into the local index profile of individual walls. Similar experiments were done by Pandiyan [29,30]. In this regard, optical coherence tomography (OCT) offers the ability for the non-destructive three-dimensional analysis of the refractive index, closing a gap in current analysis of domain structures.

OCT is a depth-resolving imaging technique based on low-coherence interferometry, originally developed for the examination of biological tissue. The method allows for fast imaging of refractive index contrast within materials and is, in particular, well established and commercialized in ophthalmology for examining the retina. While some applications also consider quantitative measures, e.g. assessing the cone photoreceptor density based on the relation of reflectivity and refractive index discontinuities [31], it is mainly used for qualitative, i.e. structural, imaging. As optical rather than geometric path length is measured in OCT, considering the refractive index of bulk tissue is, in turn, necessary for correctly determining the layer thickness, which has been comprehensively addressed [32–34].

In this work, we demonstrate by measuring domain walls in PPLN that OCT is able to not only image but also quantify refractive index changes with respect to origin and profile. Previous work has initially used time-domain OCT to resolve the domain structure [35], but was limited in penetration depth due to the high LiNbO₃ dispersion in the spectral range of the system. By applying Fourier-domain OCT, this dispersion could be successfully corrected in post-processing, thus achieving axial resolution better than 1 µm in the crystalline sample over a depth range of several 100 µm [36,37]. Additional phase analysis improved the depth profiling down to a resolution of 50 nm, but further investigations by applying a polarizer in the sample arm did not elucidate the origin of the signals, which were assumed to be limited to the extraordinary polarization [36].

Here, we investigate the linear optical properties of the domain walls in PPLN by polarization-sensitive Fourier-domain OCT and observe light travelling in both ordinary and extraordinary polarization when illuminating with circular polarization. By carefully treating polarization-dependent amplitude and phase information, we derive the origin of different DW signal and compare the results with simulations of the underlying refractive index profile, thus deducing parameters for width and refractive index profile of the DWs.

2. Material and methods

All PPLN measurements were performed using a commercial spectrometer-based polarization-sensitive OCT system (TEL220PSC2-SP1, Thorlabs Inc., New Jersey, USA) (see Fig. 1). It incorporates a broadband linearly-polarized light source in the 1300 nm range, a bulk optics Michelson interferometer hosting a 2D scanning in the sample arm, a polarization-sensitive spectrometer module equipped with two InGaAs cameras, i.e. one for each linear polarization, and a polarization-maintaining circulator connecting these components. The acquisition of each depth scan, called A-scan, for both polarizations is synchronized by hardware. The A-scan rate can be set between 5.5 and 76 kHz, thus resulting in corresponding spectrometer integration times and sensitivities between 109 and 94 dB, as specified by the manufacturer. Depending on the reflectivity of the sample, the A-scan rate was set to 28 kHz in most cases. The power on sample position was measured to be 2.5 mW. With the used objective (LSM03, Thorlabs, f’ = 36 mm) this results in a power density at the focal spot of about 3·10⁷ W/m². Within the Michelson interferometer, the light is divided into a reference and a sample arm, the latter containing two galvanometer-based scanners for x-y-beam steering and a telecentric mounted objective. Both arms contain a quarter-wave plate, being rotated by 22.5° or 45° relative to the incident polarization in the reference and sample arm, respectively. Therefore, the light in the reference arm is linearly polarized under 45°, when returning to the beam splitter, while reflected light from the sample would be linearly polarized under an angle of 90° in case of no further sample-induced polarization changes, i.e. birefringence, diattenuation or depolarization. The combined optical wave from reference and sample arm is guided by a polarization-maintaining fiber via a circulator to a spectrometer with two cameras, one for each polarization. We will call the signal from the camera that is sensitive to the returning light without birefringence, the co-polarization channel and the signal from the camera sensitive to the orthogonal polarization, the cross-polarization channel. Note, that the light incident on the sample is circularly polarized and, in return, each detection channel is thereby sensitive for returning circular light, too. Thus, the co-polarization channel is receiving the helicity that has changed due to reflection and the cross-polarization the helicity of the input beam. According to the Jones algorithm, the signals for linear polarization in direction ϕ, hence can be calculated by combining the amplitudes of both circular polarizations (a_r, a_l) as:

Due to non-perfect wave plates and asymmetry in the system, e.g. causing polarization mode dispersion of the broadband light, the result will not reach the quality of a real polarizer. The wavelength range covered by the spectrometer was measured to be 1176 nm to 1414 nm.

 

Fig. 1. Sketch of the PS-OCT system used in this experiment. In the base unit, light from a superluminescent diode (SLD) is linearly polarized (Pol) and transferred via a circulator and polarization-maintaining fibers to the scanning unit. The light from the fiber is collimated into a Michelson interferometer containing a 50/50 beamsplitter (BS). Within the reference arm, a quarter wave plate (QWP) at an angle of 22.5° results in a 45° rotation with respect to the incoming linear polarization. An iris within the reference arm allows to attenuate the reference signal, which is reflected by a retroreflector. Within the sample arm, a quarter wave plate (QWP) turned by 45° relative to the incoming polarization forms circular light that is send to the sample. Two galvanometer scanners and an objective lens (LSM03) allow beam steering over the sample. A camera behind the partially transparent galvanometer scanner allows a view on the sample (VIS). Light reflected from the sample arm is send via the QWP back to the BS and interferes with the light from the reference arm. The light is then coupled via the polarization- maintaining fiber and the circulator to a spectrometer containing a polarizing beam splitter (PBS) and two line scan cameras for co-polarization and cross-polarization.

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The 2D or 3D measurements were acquired as raw camera data using the system’s commercial interface, and subsequently processed by self-developed software. Initially, the sample-independent background signal, as acquired from the reference arm only, was subtracted from all measurements. Based on the envelope of this signal, spectral shaping was applied to the measurement data. As our data consists of peaks with very different amplitudes, we use a Hann window because of its small side lobes of less than 3%. With the available spectral bandwidth the Hann window leads to an axial resolution of ≈7 µm. The typically stepwise OCT processing of resampling to k-space, compensation of dispersion, zero-padding and Fourier transform is combined by initially calculating complex-valued exponential functions considering the dispersion of the sample to correspond to each camera pixel, i.e. wavelength. A single A-scan is then calculated by taking the scalar products with the pre-processed measured data. Advantageously, the depth-dependent dispersion of the sample can be included without the need for several Fourier transformations, as usually necessary [36,37]. As the dispersion compensation for LiNbO₃ in this spectral range is only 1.5% of the correction needed with the ultra-high resolution system used in our previous study [37], different dispersion corrections over the field of view and depth were not necessary. The dominating quadratic phase-shift [37] has a maximum of approximately 3 rad for a sample thickness of 1 mm LiNbO₃, which is already more than 2 mm optical depth of the 3.5 mm depth range available. The phase change is slightly larger for the extraordinary polarization than for the ordinary polarization. Depth-dependent amplitudes and phases are calculated for phase changes of multiples of π over the mentioned spectral range, corresponding to a twofold zero-padding in the normal processing. This doubled number of data points provides a smoother visualization and allows for an improved differentiation of neighboring peaks. Geometric depth within LiNbO₃ was calculated using the group refractive indices n_egr = 2.183 and n_ogr = 2.271 at the spectrometer’s central wavelength of 1300 nm using the Sellmeier data from Zelmon [38].

For this study, samples of periodically-poled LiNbO₃ with different periodicities were used. Samples with a period of 6.9 µm were not suitable, because the spacing of the DWs is in the range of the resolution of the PS-OCT system (data shown in the Supplement 1.1). Therefore, all data shown here are from a sample of pure LiNbO₃ containing two columns with 28.4 µm period. To mitigate a strong surface reflection in comparison to the DWs, the entrance side was polished under an angle α of about 4° and imaging was performed by tilting the angled surface by β with respect to the optical axis of the microscope objective, so that the refracted beam was perpendicular to the DWs within the sample (see Fig. 2). This prevents oversaturation of the cameras due to the strong surface reflection of the highly reflective interface or the surface signal would overlap adjacent weak signals in the sample.

 

Fig. 2. Sketch of the sample orientation for measuring the domain structure of PPLN. The entrance face is polished under an angle of α = 4° relative to the orientation of the DWs. The circular polarized OCT beam is incident under an angle β with respect to the entrance face so that the refracted beam inside the sample is perpendicular to the DWs. The E-field for the ordinary polarized beam is in the y-direction and for the extraordinary beam in the z-direction. The polar-axis of the bulk and two inverted domains is indicated by green arrows on the entrance face.

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3. Results

Figure 3 shows the OCT images obtained from the sample with 28.4 µm period. Here, we see the DWs are clearly resolved. Using the refractive index for extraordinary polarization and the optical distance of the DWs measured in the co-polarization image, the mean period agrees very well with the period obtained from light microscopy images of the selectively etched z-face. The correspondence between microscopy and OCT allowed to identify the inverted domains, whereas the OCT data showed no differences between the front and back DWs. Two DWs terminating a single inverted domain were selected to show the 3D profile of both walls and the thickness map (see Fig. 4). The position of the DW was calculated by fitting a parabola to the three data points around the peak. The mean tilt of both DWs was nearly identical and in the range of 0.2°, and numerically subtracted before visualization. The reason is probably a small misalignment when manually positioning the sample. The standard deviation of the position is 0.48 µm in both DWs, while the thickness variation was only slightly larger with 0.56 µm. The standard deviation of the period along the first 50 domains was more than twice as large (1.18 µm), so the inhomogeneity seems to be larger between different domains than within a single domain. The ratio between the thickness of the inverted domains relative to the period, i.e. the duty cycle of the periodical structure, was nearly optimal with 51,7%.

 

Fig. 3. PS-OCT images of the PPLN sample with 28.4 µm period. a) co-polarization, b) cross-polarization. The width of each column is 2 mm with a gap of 0.25 mm in between. The total width of the image is 5 mm and shows an optical depth of 3.12 mm. Due to the high refractive index of LiNbO₃ this corresponds to a geometric depth of ∼1.4 mm in the sample. The broadened structures at the top are probably artifacts caused by reflections from the bottom of the sample combined with coherence revival of the light source or aliasing effects in the spectrometer. c) Enlarged part (green rectangle) of the cross-polarization signal, showing the oscillating amplitude along the top DWs. The evenly arranged blue arrows indicate intensity maxima along this DW. In deeper parts of the image, the two echoes are clearly separated. The amplitude of the DWs at the intercept with the surface has a medium value and first declines with increasing distance from the surface. Scale bars in a) show a length of 1 mm in air.

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Fig. 4. Profile of one domain of the PPLN sample. Orientation as given in 2, length units are all in µm. Top and middle surface plots show the front and back of one inverted domain. In the bottom plot, the thickness as calculated from the difference between the front and back is color-coded. The color scale displays a range of ±2 µm adjusted to the center of gravity of each DW for the surface plots or to the mean thickness of 14.5 µm, respectively. The plots show a range of 430 µm in z-direction and 1845µm in y-direction of the sample. Isolated spots are probably caused by dirt on the entrance face of the sample. The standard deviation is 0.48 µm for both DW positions and 0.56 µm for the thickness.

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When inspecting the cross-polarization channel in more detail, two observations can be made. First, in the cross-polarization image of this sample (Fig. 3(b)), the oscillating amplitude can be observed in scanning, i.e. y-direction, over the top DWs, which are shown in more detail in Fig. 3(c), indicated for one DW by blue arrows. This feature is quite different from the reduced amplitude observed in our previous studies [37], as it is very regular and restricted to the DWs near the entrance face. Zones of reduced amplitude, similar to the features observed before, are randomly found here as well, best visible in the co-polarization channel of Fig. 3 (a), and will not be discussed here again. Secondly, in the deeper lying DWs, the oscillation is not visible anymore and a second signal becomes evident, which has a larger spacing and a smaller amplitude than the signal present in both polarizations. In order to analyze the signals in more detail, we collected many A-scans at identical position and plotted the averaged amplitude for co- and cross-polarization in Fig. 5. The origin of the depth scale was set to the position of the surface reflection, given by the strongest peak in co-polarization. In total, we found three different echoes from each DW. First, the largest signal in extraordinary (z) polarization with a period corresponding to the extraordinary refractive index; second, a signal with roughly half the amplitude, corresponding to the mean refractive index between ordinary and extraordinary polarization; and third, a signal approximately 6 to 7 times smaller with a period corresponding to the ordinary refractive index.

 

Fig. 5. Averaged amplitude of 500 A-scans at identical position for (I) co- and (II) cross-polarization as a function of x, the optical distance from the entrance face. (III) Dotted in green is the co-polarization signal stretched in depth by a factor of n_o/n_e and scaled with a 6.25 times smaller amplitude. (IV) Dashed in purple is the co-polarization signal stretched by a factor of (n_e+n_o) / 2 n_e with a 2 times smaller amplitude. The effect of noise was added to both calculated curves.

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To highlight these additional signals, the co-polarization signal (I in Fig. 5) was stretched in depth according to the ratio of the refractive indices and scaled to fit the peak amplitudes as close as possible (dotted green (III) and dashed purple (IV) graphs in Fig. 5). Especially, at the first DWs, up to a depth range of ≈300 µm, these signals overlap and interfere. Therefore, the signals for cross-polarization (II) vary transversally, as seen in Fig. 3, and in depth, as seen in Fig. 5, in amplitude much stronger than the signals in co-polarization. The analysis of the phase of the signals (not shown) revealed an additional phase shift of approximately 90° between first (I) and second (IV) signal (see DW in Fig. 3(c) starting at the interface with a medium value and decreasing amplitude with increasing distance from the interface). To highlight the three contributions, the signals for linear polarization, calculated according to Eq. (1), are shown in supplementary Fig. S 2.

In order to find the absolute reflectivity of the DWs, we compared the signal of the DWs to the signal of a glass plate, positioned at the same depth as the first DWs. While the positioning for a glass plate for optimal signal is relatively simple, the optimal positioning of the DW inside the PPLN sample is much more challenging. As visible in Fig. 4, the surface of the DWs is not flat and not identical for all DWs. By comparing the signal of the glass plate (-14 dB) with the signal from the DW, the amplitude of the main signal in co-polarization was estimated to be -73 dB. Therefore, the signal in cross-polarization with about 50% of the main signal amplitude is -79 dB. Hence, the third signal corresponding to the ordinary refractive index has an amplitude of -89 dB. All signals need to be corrected due to the round-trip transmittance of the crystal surface, which accounts for 1.2 to 1.3 dB loss, depending on the polarization. Moreover, the signals with linear polarization need to be corrected by 6 dB, as only one half of the light is linearly polarized and this signal is split into both polarization channels. Consequently, the signals are -66 dB for the first, -78 dB for the component reflecting circular light and -82 dB for the third, very week signal. Despite the challenge to position the DWs orthogonal to the OCT beam, we estimate the uncertainty of the signal level not larger than 2 dB. In the case of the weakest signal, which is considerably affected by noise, this ambiguity might be even higher.

4. Discussion

4.1 Structure and intensity

As similarly shown in our previous publications [36,37] using OCT systems operating at 500 to 900 nm, OCT systems at 1300 nm can analyze the domain structure of periodically poled LiNbO₃, too. Due to the lower axial resolution achievable in this wavelength range, small periods of less than 10 µm can hardly be resolved, so findings about the topography of DWs will be difficult in case of such low domain width. Beneficial for the use of 1300 nm OCT systems is the much lower dispersion of LiNbO₃ in this spectral range, making the depth-dependent correction less crucial, especially avoiding the need for different corrections below the tilted surface. When comparing the structure of the domains, we found that the thickness and flatness of the DWs in the crystal with 28.4 µm is worse compared to the previously acquired data of the crystal with 6.9 µm period [36,37].

As observed before, the main signal of the DWs is found in extraordinary polarization. Unlike previous studies, we obtained three signals from each DW. As it will be discussed in the following, we attribute these additional signals to the use of circularly polarized light and the detection in both circular polarizations with a PS-OCT system, which was not done before.

Unfortunately, the signal level of the reflection from the DWs is not provided by Pei et al. [35]. Considering the specified noise level of -77 dB and signals plotted in Fig. 2 in this publication, a signal level in the range of -62 dB can be estimated. This value, as well as the value given by Haussmann et al. [36] of -57 dB is larger than the signals observed here. Both publications do not mention correcting the transmittance of the crystal surface, thus, the difference could be even 1 dB larger. As the previous measurements with higher signals were done at shorter wavelength, we assume that our lower values are caused by the longer wavelength used here, which will be discussed in detail below.

4.2 Origin of the three observed signals from the DWs

While there have been assumptions about the origin of the reflection from the DWs in extraordinary polarization, this topic is still unclear. Pei [35] assumed a step in the refractive index between inverted and virgin domains. In contrast, in our own previous experiments [36] we could not find evidences for a step index contrast between domains of opposing signs. For this we performed OCT measurements during the application of electric field of up to ±1.5 kV in z-direction, where the contrast of the DWs was not affected despite the induced refractive index contrast between domains of opposing direction of up to Δn ≈ 10⁻³ via the Pockels effect. This indicates that an intrinsic contrast in the DW was still the main contributor to the signal. Other measurements of the refractive index difference for extraordinary polarization between virgin and inverted domains in PPLN are not known.

Although it seems to be most puzzling, one might consider initially the origin of the second signals (II in Fig. 5(b)), visible in the cross-polarization channel only and corresponding to the medium refractive index (n_e+n_o)/2. The optically induced change of the refractive index in LiNbO₃ is known since the late 60s [39–41]. As described by Chen et al. [39], light induces an electric current in LiNbO₃, thus inducing an electric field, which in turn changes the refractive index. Simultaneously with the induced electric current, the photoconductivity is enhanced and limits the electric field. Kanaev et al. [42] found over a large range of optical power densities electric fields varying from 8·10⁶ V/m to less than 20·10⁶ V/m, which is only a factor of 2.5. The power density used in our study should give a field in the lower range. Such electric fields can cause a reduction of the refractive index, mostly for the extraordinary polarization and up to 1.5·10⁻³. Here, circularly polarized light had field components in y- and z-direction of the crystal. This induces electric fields in both directions. Under the field E_y in y-direction the uniaxial crystal becomes biaxial and the direction of the axes change. The new z-direction becomes tilted by an angle θ [43] with:

In [47], Chen et al. report a value of 2θ = (m π ± 0.991)/2692 rad, with m any integer. Assuming similar electric fields in both experiments, despite the different wavelength and power levels, this would result in m = 5 or 6. Unfortunately, no clue about m is given in this publication [47].

The largest signal (I) from the DWs was observed for extraordinary polarization. As discussed above, the OCT beam with circular polarization will cause a field in z-direction. In the inverted domain, the sign will change. But as the coefficient y₃₃, responsible for the effect of the electric field on the refractive index in z-direction, changes its sign, too, the refractive index change in both domains will be the same. In the vicinity of the DW, the electric field in z-direction has to change its sign. So near to the DW this field will be almost zero and, consequently, there will be no change of the refractive index for the extraordinary polarization near the DW. Depending on the type of the DW (Ising, Bloch or Néel) there may be components of the electric field in normal or transverse directions, but these components will be small [51,52].

Though the photovoltaic effect at wavelength of 1500 nm has been described before [46–48], there are doubts because the absorption of pure LN is extremely small at this wavelength. Chen et al. [48] do not describe a mechanism causing the PVE effect at 1500 nm but mention that the observed effect takes several minutes to become stable, meaning that the effect is extremely small. Here, it should be noted that while our material is not intentionally iron-doped, iron defects (in low concentrations) are common defects in the LN system and increase the absorption in the applied wavelength range.

If the PVE effect at 1300 nm is excluded from the explanation, the question therefore arises, wherefrom the refractive index difference centered around DWs do originate. In this regard, we can identify several other potential mechanisms, whereas each one alone or a combination may explain the observed behavior:

The assumed refractive index jump around the DW could also be explained by internal electric fields. According to experiments by Gopalan et al. [10], intrinsic electric fields of the order of 5-10 kV/mm have been detected in the vicinity of DWs based on spectroscopic analysis of Erbium doped LN. However, the direction of those fields were not presented. Nevertheless, based on the Pockels effect electric fields of that magnitude would yield refractive index changes on the order of 10⁻³ yielding similar orders of magnitude as our assumptions.

Likewise, as mentioned above [14,24], DWs in LN have been observed to show significantly higher electrical conductivity compared to the surrounding bulk [10,13,14,24]. The reason for the observed conductivity is not yet fully understood. Several major models are discussed in literature including local band bending of the conduction and valence band or defect moderated hoping mechanisms [10,13]. Free or weakly bound charges, local defects or a change in the (local) band structure may result directly in a local change of the dielectric function – and hence – the refractive index.

Furthermore, recent investigations with polarization resolved SHG microscopy have provided evidence, that ferroelectric DWs can show a deviation from the ideal Ising-like domain wall behavior, i.e. having polarization components drastically deviating from the predominant z-direction. Hence, Néel- or Bloch-type DWs may result that give rise to unexpected novel topologies and local (electric) field distributions [10,13]. These effects are also suspected to play a key role in observed DW conductivities [19,22], even in LN, and hence play a significant role in the observed conductivity. The observation of a linear optical response (refractive index change) hence may be directly linked to this behavior as well.

Besides the PVE at 1300 nm, the electric fields inside the investigated crystal sample could be caused by illumination already during the manufacturing process (e.g. for device inspection) or due to previous experiments, using light of shorter wavelength.

As of now, a conclusive answer to the underlying mechanism of the refractive index of the DW remains elusive and is beyond the scope of the present work. If the PVE effect is not caused by 1300 nm light this would make the method even more attractive as it allows measuring the refractive index change caused by other radiation without perturbation.

Thus, whatever the reasons for the electric fields are, all explanations result in a changed refractive index profile in the vicinity of the DW. We assume that the refractive index profile along the x-direction will schematically look as shown in Fig. 6. While there are calculations of the electric field and the refractive index in the vicinity of a DW for BaTiO₃, the width of this zone is still questionable, especially in the presence of a high intensity electromagnetic wave [52]. Assumptions for the width of this zone in [52] are between 0.5 and 2 nm.

 

Fig. 6. Model of the proposed refractive index profile (n) for extraordinary polarization in PPLN as a function of the coordinate x in units of the domain thickness p. Due to the electric field, the refractive index decreases in both domains. In the vicinity of the DWs, the electric field in z-direction will vanish and therefore the refractive index will have the value without electric field. Note that the range of the refractive index increase around the DW will be very small compared to the width of the domains and is not true to scale for the analyzed PPLN sample.

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If the reason for the reflection would be a change in the refractive index between virgin and inverted domains, we would expect a phase change alternating between 0° and 180° each second DW, which was not observed experimentally. The reflection in extraordinary polarization was shifted to the signal in circular polarization by 90° for all domains. Diamant et al. [53] have shown that a reflection by a discontinuity in the first derivative of the refractive index results in a phase change of ±π/2 (i.e. ±90°), where the sign depends on the sign of the discontinuity. In Supplement 1.4 we show that a symmetric refractive index change centered at the DW will always lead to a phase shift of ±90°, with the sign depending on the sign of the refractive index change. Moreover, we calculate in Supplement 1.4, based on the work of Grafström [54], the complex reflection coefficient ρ for several refractive index changes shown in Fig. 7(a) with the results shown in Fig. 7(b):

 

Fig. 7. a) Different modelled changes of the refractive index profile in the vicinity of a DW. b) Normalized reflection coefficient r·2n/Δn as a function of the normalized half width a = n·d/λ for the refractive index profiles from a). Gaussian with 1/e half width of 1, exponential with 1/e half width of 1, tanh(x/√2), square with half width of 1, triangle with half width of 1.

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The absolute value of the reflection coefficient $r = {\rm abs}(\rho )$ for this profile is:

For the case of a refractive index change with a width much smaller than the wavelength the general result of Eq. (3) simplifies to:

For the special case of the exponential refractive index profile of Eq. (4):

This approximation would be valid for ${\rm d} < 0.05\cdot \displaystyle{\lambda \over n}$, so in our case up to ∼30 nm, which is larger than many estimates of the domain wall width.

Unfortunately, neither the width nor the refractive index change near the DW is known, but the results can be used to restrict possible values. First, there must be a minimal refractive index change to cause the observed reflectivity. Secondly, the measured reflectivity can be used to give a value for the integral over the reflective index change in the form of Eq. (3), respectively Eq. (6) if a small width is assumed.

Using the measured reflectivity of -66 dB for the signal in extraordinary polarization and that ${\rm r}\cdot 2{\rm n}/\Delta {\rm n} < 2$ (see Fig. 7(b)), this leads to a minimal Δn of 1$\cdot 10^{-3}$, which is in the range of the assumed refractive index change caused by the electric field in z-direction. Using the values of Haussmann [36] or Pei [35], $\Delta {\rm n}$ must be at least $3\cdot 10^{-3}$ or $2\cdot 10^{-3}$, respectively. Actually, for smooth refractive index profiles $\Delta {\rm n}$ must be even larger, as the normalized reflection coefficient is smaller than 2 in this cases, i.e. ${\rm r}\cdot 2{\rm n}/{\rm n}\ll 2$. This again reminds that the assumption of taking just the bulk values of the electric field and the electro-optic constants in the vicinity of the DWs is not valid.

The results for the integral $\mathop \smallint \nolimits_{-\infty }^\infty \delta n\left( x \right){\rm \; }dx$, assuming negligible width relative to the wavelength, of Pei [35] at 560 nm, Haussmann [36] at 600 nm and ours at 1300 nm are 0.07 nm, 0.135 nm and 0.1 nm, respectively. Beside dispersion, which is probably only a minor contribution, the reason for this discrepancy could be different material compositions, differences in the setup (e.g. circular polarization vs. linear polarization, different power levels and beam radii) or just uncertainties in measuring the reflectivity. The difference corresponds to just 2.4 dB in reflectivity with an estimated uncertainty of 2 dB for our measurement.

Chaib et al. [52] calculated for BaTiO₃ a change of the refractive index up to 0.1 for a DW thickness of 1 nm, which would be in agreement with our measurements. Similar calculations for LiNbO₃ would allow an experimental proof. The assumption of an additional step in the refractive index profile between the DWs would alter the phase jump considerably and, therefore, can be excluded for the explanation of the high reflectivity. An additional smooth change of the refractive index over a distance in the range of λ/n or larger does not cause or alter the reflectivity of the boundary and can therefore not be excluded by our measurements.

Comparing the reflectivity measured by Haussmann [36] with our result, shows almost an inverse proportionality with the wavelength, which is assumed from our calculations only for the range of $a < 0.1$ (see Fig. 7(b)), which gives a maximal value for the width d of less than 30 nm. In Tab. 1, the necessary refractive index change for domain full width (2d), reported by other measurements, is given. As the range of reported width is nearly 3 orders of magnitude, the estimated refractive index change varies accordingly. Measurements under the same conditions over a large wavelength range could narrow the possible range considerably.

Table 1. Estimated refractive index change for different domain width for results from other studies^a

View Table

The third signal (III in Fig. 5) affected by the ordinary refractive index with an amplitude of -82 dB is probably caused by the effect of the z-component of the electric field as well. The electric field in y-direction does not vary within the inverted domains nor does the influence of this field change the refractive index between the domains as y₂₂ does not change its sign. Again, this signal has a phase of 90° to the signal in circular polarization and therefore is probably caused by a similar change of the refractive index as before. The linear electro-optic coefficient y₃₁ describes the impact of the electric field in z-direction on the refractive index in y- (and x-) direction and is approximately 30% of the value of y₃₃. As the reflected amplitude is proportional to the refractive index change, we would expect an amplitude that is 30% of the signal in extraordinary polarization, while we find a signal which is only about 15% of the signal in z-direction. The minimal refractive index change needed for the observed amplitude would be only Δn₀ = 3.4·10⁻⁴, which is quiet reasonable. The other discussion on the refractive index profile for the extraordinary polarization is valid here, too.

Contrary to previous measurements performed with linear polarized light, we observed three distinct reflections from each DW. Linearly polarized light in z-direction produces a field in z-direction and therefore an altered refractive index profile, which leads to reflected light in the extraordinary polarization. But this polarization does not cause a field in y-direction, so the angle θ between the domains will be zero (or 180°). As there is no light polarized in y-direction there is no reflection in this polarization, regardless if there is a change in the refractive index for ordinary polarization or not. This situation is similar to the Šolc-Filter with light only in the z-direction [48]. Linearly polarized light in y-direction (ordinary polarization) does not generate a field in z-direction, consequently the first and the third signal will vanish. The angle θ will not be zero, consequently there will be light reflected at the DW, but this light changes from ordinary to extraordinary polarization and therefore, with a polarizer in the sample arm or interfering with reference light of the same linear polarization, will not reach the detector or contribute an interference signal, respectively. For that reason, the newly spotted signals here were not observed using a polarizer in the sample arm transmitting only ordinary or extraordinary polarized light, respectively.

5. Conclusion

Using a PS-OCT system that operates with circularly-polarized light at a 1300 nm center wavelength, we were able to visualize and reconstruct the domain and domain wall (DW) structures in periodically-poled lithium niobate (PPLN) in full 3D and down to a depth of >500 µm. Surprisingly, we recorded three distinct reflections from every DW, significantly differing in both amplitude and polarization with respect to the PPLN coordinate system. In fact, when carefully evaluating these interference signals, we were able to associate the aforementioned three signals governed by the ordinary n_o and extraordinary n_e refractive index, as well as to the mean-value between them, i.e. ½ (n_e + n_o). Moreover, these results allowed us to quantify the refractive index jump for one single PPLN DW to measure approximately $10^{-3}$, and being confined to a narrow region of < 30 nm in width, which agrees reasonably well with previous reports. These findings clearly prove, that every such DW in PPLN per se must possess its own refractive index, a fact that is of uttermost importance, both when modeling and simulating DW properties and equally for assembling prospective DW devices. We believe that PS-OCT has the power to quantify local electric fields in general, by applying a sophisticated analysis of the local polarization reflections in 3D, hence reveling valuable insights into a manifold of potential sample systems, such as biological nanosystems or inorganic electro-optical devices.