1. Introduction

Ferroelectric crystals like lithium niobate (LiNbO₃) are used as frequency doublers [1] and for many other optical applications. Recently, periodically poled lithium niobate (PPLN) found new applications in nonlinear optics and in waveguide structures [2–5]. Domains in these crystals can be poled which makes phase matching for efficient nonlinear processes possible or at least much easier. MgO doped lithium niobate (LNO) can withstand optical intensities at least one hundred times larger as undoped material and is therefore preferred. Moreover, ferroelectric domain walls (DWs) have been shown to be able to serve as reconfigurable nm-thin conductive channels embedded in an insulating matrix, which holds great prospects for future nanoelectronic applications [6–9]. This domain wall conductivity has also been demonstrated to occur in LNO [10,11].

LiNbO₃ and LNO is poled by applying a large electric field across the crystal [12]. Patterning suitable domains can be achieved by coating lithographically an insulating layer and applying the voltage by lithium chloride liquid electrodes for one or several electric pulses. The domain structure of PPLN has been studied by etching techniques [13], micro-Raman imaging [14], multi-photon laser scanning luminescence microscopy [15–17], Čerenkov-type second harmonic generation [15,18,19], scanning force microscopy [20,21] and more. In order to provide a quality control for the manufacturing of quasi-phase matching devices, optical methods as second harmonic generation or diffraction methods [22] are used.

Etching and AFM methods provide information about the domain configuration at the surfaces only. However, a thorough investigation of DW conductivity requires the analysis of full 3D profiles of DWs throughout the crystal bulk. While such data can be provided by Raman and non-linear techniques with diffraction limited resolution, they usually require optical access through the crystal's polar surfaces. This limits the choice of suitable electrodes for simultaneous DW current measurements and/or DW geometry modifications. Moreover, collecting such data by 3D scanning is time consuming, thereby also increasing the risk to introduce additional errors due to sample drift.

Optical coherence tomography (OCT) is an alternative technique for direct visualization of domain walls in PPLN, which offers the possibility to rapidly collect time-resolved 3D DW data through a side face of the crystal, requiring only 2D scanning. To the best of our knowledge, domain walls in lithium niobate have been visualized by OCT before only in one study [23]. With a polarization sensitive OCT system the group showed that the DWs are only visible in the extraordinary polarization. Although not stated in the publication, the OCT system used seems to be a time-domain OCT system, complicating dispersion compensation [24]. Therefore, the group was able to characterize only the top DWs in the crystal, whereas the evaluation of the near-surface domain walls was hampered by the strong reflection from the air-crystal interface due to the high refractive index of LNO (Fresnel reflex). The images provided show no deviations from uniform domain walls. Because Fourier domain OCT not only allows much higher speed but enables numerically dispersion correction in the post processing, images even from larger depth can be dispersion corrected and yield a high axial resolution information on the 3D structure of the DWs. The transversal resolution depends in both approaches on the numerical aperture used.

This paper presents the capability of high-resolution Fourier domain OCT for visualizing and evaluating the domain wall topography in PPLN. Applying numerical dispersion compensation, domain walls could be visualized over a depth range of several hundred micrometers, allowing precise measurements of the domain period. Depth profiling is presented down to a resolution of 50 nm and better by using a phase analysis.

2. Experimental setup

The OCT system utilized in this study is an enhanced version of the dual band OCT system described earlier [25]. The line scan camera for the short wavelength band was replaced by a Basler sprint spL4096-140km with 4096 pixels and the spectral range was extended to 559-938 nm, leading to an axial resolution of 1.4 µm in air (Hann-window) or 600 nm for the extraordinary polarization in lithium niobate (see below). The central wavelength (reciprocal of mean wavenumber) is 700 nm. The A-scan rate was set to 12 kHz with an exposure time of 74 µs. Because of the higher resolution, only the short wavelength band was evaluated. The fiber coupled scanner head has two galvanometer mirrors for beam deflection and a 20% beam splitter for a reference arm. With the numerical aperture of NA ≈0.056 the transversal resolution is about 7 µm. In the images shown, only the deflection in the long direction (x) of the crystal was used. The orientations of the axis are chosen as usual in OCT. The crystal axis, often called the z-axis in crystallography, is the y-direction in our coordinate system.

Several 5 mol% MgO doped congruent periodically poled lithium niobate crystals (PPLN) were supplied by Crystal Technology Inc. Lithium niobate is birefringent with refractive indices of n_o = 2.27 and n_e = 2.19 for 693.4 nm (data from manufacturer). With this high refractive index, the system would be saturated by the Fresnel reflex even at low exposure times. Therefore, the front faces of the crystals were polished under an angle α of 4 degrees. Due to manufacturing tolerances, α measures between 3 to 5 degrees. The incident angle β was chosen approximately twice as large as α in order to obtain an incident beam perpendicular to the DWs inside the LNO crystal. This effectively reduced the collected light from the Fresnel reflex and allowed to visualize the DW inside the crystal. The size of the crystals was 5 mm by 5 mm with a thickness of 0.5 mm. Each crystal contains 3 columns of inverted domains with a nominal period of 6.96 µm, 6.93 µm and 6.90 µm. The first two columns of inverted domains have a width of 0.75 mm, the last a width of 0.85 mm. A gap of 0.1 mm separates the poled strips (see Fig. 1).

 

Fig. 1 Structure of the PPLN-crystal showing the inverted domains and the inclined surface. The entrance surface is ground under an angle of α ≈4 degrees. The OCT beam is tilted by β relative to the crystal surface, so that the domain walls are perpendicular to the OCT beam (red) inside the crystal. The pitch and width of the domains are enlarged in this sketch.

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3. Dispersion compensation

As mentioned before, lithium niobate has a large refractive index (n) of around 2.2. Moreover, the dispersion is quite large with an Abbe number around 20. Therefore, dispersion inside the material broadens the OCT signal strongly especially for a high-resolution OCT system associated with a large spectral bandwidth [26]. Dispersion compensation in FD-OCT system can be easily achieved for any dispersion by introducing a wavelength dependent phase factor [27]. The phase-function can be calculated from the dispersion function (Sellmeier equation) [28] for any thickness (d) of the material. In order to get this phase function, the optical path n·d of that layer of material multiplied by 2π/λ has to be considered.

Because the linear part of this function will shift the image, the linear part is subtracted to get the phase function Δφ_disp without shifting the image (please see Eq. (34) in [27]). By scaling this function, compensation of the dispersion can be achieved for any depth. As a function of wavenumber, the phase function is almost parabolic with only minor contributions of higher orders (see Fig. 5 in [27]). Recently, a similar algorithm has been described [29]. In order to keep the absolute value of the phase, we subtract from the phase function the value from the center wavelength. This has no influence on the signal intensity, but allows evaluating the phase over areas with different dispersion compensation.

In Fig. 2(a), an OCT image from the DWs of lithium niobate is shown without any additional phase compensation (due to slight asymmetries between reference and sample arm, there is some inherent phase mismatch in the instrument, which is numerically compensated). The image consists of 959 vertical lines (A-scans), each consisting of 2048 pixels associated to the Fourier bins. Figure 2(a) shows that in a band near the tilted surface the image of the DWs is very sharp, while in deeper layers the image is blurred. Figure 2(b) shows the same measurement with a dispersion correction using an amplitude of 5 π. There, a stripe approximately 250 points (75 µm) beneath the surface is sharp (see blue arrows in Figs. 2(a) and 2(b), while parts above and below are blurred. In order to automatically find the best dispersion, we looked in each A-scan for the highest signal in a depth range containing at least one DW (100 pixels or 30 µm) in a set of images processed with phase functions of different amplitude. To eliminate some misclassifications (missing DWs), a median filter with a width of 100 pixels was applied afterwards. The result is color-coded in Fig. 2(d). In some areas containing no DWs the algorithm gets arbitrary results, visible in the spotted parts. Actually, this has not much influence on the images, but we added a second, now horizontally smoothing, to fill these areas with reasonable data shown in Fig. 2(e). The resulting OCT image is shown in Fig. 2(c). Steps of 2 π between the amplitudes of the phase function can sometimes lead to small discontinuities in the DWs, so we used steps of only π. In agreement with calculations based on the Sellmeier equation for LNO and the spectral range of this OCT instrument, the amplitude of the phase function increases by π approximately every 50 pixels in depth.

 

Fig. 2 Image of the PPLN-crystal with (a) optimal dispersion compensation for the surface of the crystal, (b) additional dispersion compensation of 5 π, in both images the stripe with optimal dispersion is indicated by blue arrows. (c) Overlay of parts with optimal dispersion compensation. (d) Color map of optimal dispersion correction. The additional phase correction has an amplitude corresponding to the color scale in units of π. (e) The dispersion correction is smoothed horizontally to have reasonable values in areas without signal. (Scale bar calculated for PPLN area.). All images show the same area.

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At the very top of Figs. 2(a)-2(c), faint horizontal lines are visible. These signals have an increasing dispersion, but are never as sharp as the DWs within the crystal. These lines are interpreted as the interferences of the different DWs. As all DWs have more or less the same distance, these low signals sometimes add up to be above the noise level of the system. Similar faint lines are visible in Fig. 4(d) and Figs. 5(a) and 5(b), too.

4. Period and thickness of the inverted domains

Because the crystal shown in Fig. 2 has many defects in the domain structure, we selected a different crystal for a precise measurement of the domain width and period (see Fig. 3(a). To increase the SNR, the amplitude of 100 adjacent A-scans was averaged to form a signal as a function of depth. From this, the positions of the DWs were calculated with sub-pixel precision by a peak detection algorithm. As each domain leads to two peaks, the peak positions were ordered into two columns, indicating the beginning and end of the domains, respectively. The average period of the domains was analyzed by subtracting the peak positions from 35 domains. The width was analyzed by taking the difference between the two columns. Assuming a group refractive index of 2.31, we found for the first 35 domains of the middle column a period of (6.93 ± 0.11) µm and a mean width of (3.82 ± 0.11) µm, which is (55.1 ± 1,8)% of the period (values are mean and standard deviation). While the mean period is in perfect agreement with the data from the manufacturer, the width is slightly, but significantly higher than 50%.

 

Fig. 3 (a) Image of a PPLN crystal with few defects. In each column the amplitude from 100 A-scans was averaged. From this the position of the DWs was calculated by a peak detection algorithm. (Scale bar calculated for PPLN area.) In (b) the difference between each second DW is plotted as a function of the domain number for the central column starting from the gap in the domain structure. From domain number 35 to 40 the variation of the data increases because of the low SNR.

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Averaging over a number of domains

While the data supplied above show the variance in period and width, the mean period can be analyzed to a much higher precision. Taking a large number of adjacent domains, the mean period can be determined by subtracting the positions of the beginning and the end and dividing by the number of domains. The error in the total width is given by the precision when measuring the two limiting DWs, which can be estimated by the standard deviation of the period. As long as incorrect counting can be avoided, the error in the mean period decreases by the number of domains used. Because the signal decreases with depth, the uncertainty in determining the position of the DWs will increase in deeper domains (see Fig. 3(b). We found that in our images a number of 30 to 40 domains gives optimal precision for the period measurement. We found a mean period of (6928 ± 5) nm for the central column, (6899 ± 5) nm for the left and (6959 ± 8) nm for the right column, which agrees well with the data of 6.93 µm, 6.90 µm and 6.96 µm, respectively, supplied by the manufacturer. Actually, our error estimation gives only the statistical error and does not account for the error of the wavelength scale influenced by the spectral width on the detector and the refractive index of LNO. While the relative period of the 3 columns is not influenced by these estimations, the good agreement in absolute data might be fortune.

5. Flatness of the domain walls

For an analysis of the fine structure of the domain walls, a measurement with an x-pitch of only 0.8 µm, showing only the central column of domains, was acquired (see Fig. 4(d)). This measurement allowed analyzing deviations from flat DWs in much more detail. While there may be many kinds of imperfections as seen in Fig. 2, we found in many DWs dark spots of different depth (reduced amplitude) with a width in the order of the transversal resolution of 7 µm. Analyzing the interpolated peak position and the amplitude (parabolic fit to the point of maximal amplitude and the two neighbor-points) in the vicinity of such a dark spot revealed one possible scenario (see Fig. 4(a)). The peak position between left and right side yields a step of about 0.3 pixel, corresponding to about 90 nm in LNO or 210 nm in air, which is a little more than a quarter of the mean wavelength. So we assume that there is a step or steep slope separating two flat regions with a step height of about a quarter wavelength. Therefore, light from both regions will interfere almost destructively leading to the observed decrease of the backscattered light. If this border between both regions is not perpendicular to the direction of the scan (x), the width of the transition will be larger than the transversal resolution of the system. The strong deviation of the peak-position near the minimum of the amplitude is interpreted as an artifact due to the low amplitude. On a microscopic scale DWs will preferentially be oriented along the crystallografic c-axis, which means in our case that they will be oriented under 90° or ± 30° relative to the OCT beam. Hence, the flat regions, separated by 90 nm are most probably connected by a ramp with a width perpendicular to thebeam of about $\sqrt{3}/2\approx 0.866$ of this distance value, which is much smaller than the width of the OCT-beam. Therefore, the optical properties of such a profile can be almost viewed as a step function.

 

Fig. 4 In (a), the peak position of one DW is drawn as a function of the A-scan number. From the flat areas at both sides, a step of about 0.3 pixel can be estimated. The intensity in the vicinity of this step reduces to about 10% of the maximum. The width of about 15 pixel is slightly larger than the transversal resolution of the system. (b) Phase change at one DW over 140 A-scans. Three major steps of approximately 3 rad, 2 rad and 1.5 rad can be seen. Correlated to these steps, the intensity of the peak drops. A phase change of 2 corresponds to a step of 50 nm in LNO. (c) Calculation of the relative drop of the amplitude of a DW in the middle of a step as a function of the step heights in µm. (d) High-resolution SD-OCT image of the central x,z-plane measured with a x-step of only 0.8 µm. The image shows an area of 959 pixels in both directions. The insets show the analyzed areas of (a) bottom and (b) top. Blue arrows indicate the position of reduced amplitude. In the top image, the reduction is hardly visible in the false color map due to the logarithmic scale.

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Phase analysis

As shown by several groups before, even smaller steps may be measured by analyzing the phase [30]. Due to the tilted surface between LNO and air, there is a phase shift even for horizontal DWs. The reason is the difference between peak-position affected by the group refractive index and the phase change affected by the (phase) refractive index. Replacing a layer of air by a layer of LNO leading to the same position in the OCT signal leads to a phase shift:

where φ_air and φ_LNO are the phases measured after the layer of air respectively LNO, Peak is the depth of the exchanged layer in pixels (Fourier bins) of the OCT-signal, λ₀ the center wavelength of the system and Sp the spacing of the axial points of the OCT system in air (see appendix). For the system used this is Δφ ≅ 0.627 rad·Peak. This additional phase shift due to the tilted surface was subtracted before evaluating the flatness of the DWs.

In Fig. 4(b), the intensity of one DW over 140 A-scans is shown exhibiting three pronounced minima. Furthermore, the figure shows the phase at the peak position in green. Correlated to the intensity drops, phase changes can be noted. The phase change around A-scan 80 has a height of about 2 rad, which corresponds to a step in LNO of about 50 nm. Simulating the interference of two signals caused by a step in the specimen shows for the parameters of this high-resolution OCT system oscillations of the resulting intensity drop as a function of the step height (Fig. 4(c)). The highest reduction in intensity to about 10% is caused by a step of approximately 80 nm, which is a quarter wavelength of λ₀ in LNO. At a step heights of about 0.6 µm, the signals are separated and do not overlap anymore.

6. Fan shaped LNO Crystal

In a further experiment, a PPLN crystal from a frequency doubled laser (supplied by Toptica Photonic AG, Munich, FF-Pro) was examined. In order to get efficient frequency-doubling at any desired wavelength within the tuning range, the PPLN has fan shaped domains (fan-out structured crystal) with a pitch increasing from 23.3 to 28 µm (see Fig. 5(c)). So by shifting the crystal across the beam, any pitch within this range can be chosen. The images (Figs. 5(a) and 5(b) display the last 2 mm (total width 10 mm) from the side with the large pitch, where the inverted domains are close to the surface of the crystal. Because the measurement range of the high-resolution OCT- system is rather limited, we positioned the crystal so near to the OCT scanner that the surface was near or partly in front of the zero path position leading to mirror artifacts. In order to illustrate the full geometry, we show part of the mirror image and indicate the position of the true surface in blue. Compared to the PPLN crystals investigated before, we noted a reduced intensity of the DWs, not much above the noise limit of the system. Therefore, we used complex averaging of 500 A-scans, thereby reducing the noise at least one order of magnitude [31]. The color scale of the images in Fig. 5 has a range of 30 dB, with one order (20 dB) higher sensitivity compared to the images shown before. While we expected the intensity of the DWs to increase when tilting the crystal further, the intensity dropped beneath the noise level much before the DWs were perpendicular to the beam, forming horizontal lines. At the images with the larger entrance angle and lower inclination of the domains, we noted that every second DW has a reduced intensity. Asymmetry of DWs was also noted at the normal PPLN crystal when the beam was not perpendicular to the DWs. Asymmetric effects at the DWs have been noted with conductivity measurements (unpublished measurement) but it is not clear if these effects have the same origin. The decrease in intensity of the DWs when orientating them perpendicular to the OCT beam might be caused by the microscopic structure of inclined DWs. Due to the crystalline structure, DWs will be oriented mostly under zero and 60°, exhibiting a large number of small inclined steps (see Fig. 5 (d)). If the step width is in the order of the wavelength, the facets will reflect the light only in the OCT system if the majority of the facets are oriented more or less perpendicular to the OCT beam.

 

Fig. 5 OCT images of a PPLN crystal with variable pitch. The simplified structure of the domains inside the crystal and the orientation of the analyzing OCT- beam (red arrow) are shown in (c). The crystal was tilted until the Fresnel reflex from the surface did not saturate the detector. As the domains are separated from the surface, the crystal was moved closely to the scanner leading to mirror artifacts of the crystal surface at larger entrance angles. Therefore, the mirror image is partly shown at the top. The position of zero path is the horizontal bright line. In (a), the true surface is behind the zero path position while in (b) the surface crosses zero path near the middle of the image. In both images the true surface is marked by a blue line showing the larger entrance angle in (b). Further tilting of the crystal in order to get the DWs horizontal reduced the intensity under the noise level. At the left border of the domains, the period of the DWs was measured from 12 domains to be (28.05 ± 1.00) µm in perfect agreement to the data from the manufacturer (28 µm). Again, above the DWs faint echoes are the results of interference between different DWs. This shows, better than the strongly inclined DWs itself, the increasing width of the domains from left to right. Especially at the stronger inclined image (b), it is apparent that every second DW results in a weaker echo. This effect is not understood. Within the LNO crystal the horizontal scale is compressed by a factor of ~10. (d) Scheme explaining the reduction of the intensity of inclined DWs. The microscopic structure of the inclined DW is mostly oriented in the direction of the crystallographic axis and therefore has facets under zero and 60°. Although the mean surface plane (violet line) is oriented perpendicular to the OCT beam, the light is reflected at the facets leading to a reduction of the intensity when tilting the crystallographic axis against the direction of the OCT beam.

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In Fig. 5(a) within the crystal, but above the DWs spots, mostly arranged in vertical pairs, are visible. We suppose these are the DWs of small, probably hexagonal domains, where only the top and bottom facets are visible, which were unintentionally produced during the poling process.

7. Conclusion and outlook

The DWs in PPLN were visualized by Fourier domain OCT over several hundred µm with high resolution. The large dispersion of lithium niobate had to be compensated numerically to yield sharp images over this large depth range. OCT is able to show deviations from the ideal structure making it a tool to optimize the fabrication of PPLN. We could measure the average period of the domains with an accuracy of better 10 nm, differentiating structures with only 30 nm distinct periods. The ratio between domain thickness and period could be measured and deviations from the ideal 50% level were identified. Therefore, PPLN-crystals could be used for the calibration of the length scale and resolution of OCT systems, too.

Steps in DWs lead to a reduction in the amplitude of the reflected signal. Steps of only 90 nm could be identified by a drop in signal amplitude and a step in the interpolated peak position. Even smaller steps down to 50 nm (and probable lower) could be recognized by analyzing the phase of the OCT signal at the peak position. We could show that due to the difference between phase and group refractive index a reflex from a horizontal line must not have the same phase. Furthermore, the fan-shaped structure of a variable frequency doubler with DWs oblique to the crystal axis could be visualized. Interestingly, the intensity of the reflected signal declined when turning the crystal in a position that the DWs are perpendicular to the OCT beam.

Measurements of the DW structure of PPLN crystals can be used to optimize the fabrication and to analyze nondestructively the quality and kind of imperfections of a certain crystal. Because OCT systems can be quite fast with A-scan rates in the MHz region in future the process of domain formation can be visualized with high temporal resolution. The transversal resolution of the used system was about 7 µm, which is much less than the typical diameter of laser beams for frequency doubling. By increasing the numerical aperture, the resolution could be improved to about 1 µm but then the spherical aberration has to be compensated depth-dependently. With the numerical aperture of the system used (0.056) and the high refractive index of lithium niobate (2.2) DWs can be analyzed down to a depth of nearly 10 mm in the center of the crystal without vignetting the beam entrance. With higher numerical aperture for higher transversal resolution the depth range will decrease. Due to the high refractive index both, focus position and the zero path position, have to be adjusted correctly. Probably, the large dispersion would allow dispersion encoded full-range OCT [27] without additional dispersion, but the algorithm for a continuous dispersion needs to be programmed. Therefore, we avoided DWs on both sides of the zero path position.

The quadratic part of the dispersion compensation could probably be achieved by fractional Fourier transform [32] as well, but we preferred the method presented here because of simplicity and compensation of the higher order terms. Beside our adaptive dispersion compensation algorithm, one could identify the surface of the crystal in a first step and then calculate with the known dispersion all signals inside the crystal saving memory and computing time. But even our method could be accelerated by execution on a GPU.