1. Introduction

Recently, there has been a burgeoning interest in exploiting nonlinear interactions involving counter-propagating photons because of the unique spectral, tuning and coherence properties that such three-wave mixing processes have. An eminent example of such interactions is found in the backward-wave optical parametric oscillator (BWOPO) [1], which offers radically different features compared to its co-propagating optical parametric oscillator counterpart. In a BWOPO, it is the counter-propagating nature of the interaction that establishes positive feedback by itself without the need of a cavity, making the parametric down-conversion process highly efficient in a single-pass configuration. Furthermore, the forward wave inherits the phase modulation of the pump, while the backward generated wave is inherently narrowband [2,3]. Another well-known example is backward second harmonic generation (BSHG) [4–6], for which optical bistability, and self-pulsing are expected [7]. Furthermore, counter-propagating nonlinear interactions have also been proposed for other compelling applications such as tunable photonic bandgaps [8], the slowing and stopping of light [9], and biphoton sources [10].

All these interactions present a large photon-momentum mismatch that cannot be compensated by birefringent phase-matching, since it would require the nonlinear material to have an unnaturally large birefringence, at least for the interactions ranging from the visible to the mid-IR spectral range. Therefore, quasi-phase-matching (QPM) has been practically the only viable approach to achieving momentum conservation in counter-propagating nonlinear parametric processes, although it requires the QPM period in the sub-micrometer range. This demand is technologically challenging to implement, and for a long time, it has been the main obstacle in exploiting the full potential of such interactions. QPM structures with periods in the range of few to tens of µm- are routinely fabricated in ferroelectric oxides, like KTiOPO₄ (KTP), LiNbO₃ (LN) or in semiconductor materials such as GaAs. Semiconductors require even shorter QPM periods than ferroelectric oxide materials, making the technology to structure them far from ready for counter-propagating interactions. For ferroelectrics, periodic poling with sub-micrometer periodicities over the device aperture demands extreme control over the lateral domain growth. LN, due to its trigonal crystal structure, presents isotropic domain-growth along the two non-polar directions, hampering control of the sideways domain growth. For that reason, in LN counter-propagating nonlinear interactions have been limited to high-order QPM [11–14] and sub-µm periods have only been achieved in small aperture devices [15–20].

On the other hand, bulk Rb-doped KTP (RKTP) has proven to be more suitable for the fabrication of fine-pitch domain structures, as it exhibits anisotropic domain growth as a consequence of its orthorhombic crystal structure [21]. The as-grown material has a low bulk Rb-doping (<0.3%), and presents the same attractive linear and nonlinear properties as those of regular flux-grown KTP. However, RKTP has two orders of magnitude lower ionic conductivity compared to its undoped isomorph, which facilitates domain growth along the polar direction with reduced lateral domain broadening. Despite these advantages, fabrication of sub-µm QPM gratings with standard electric-field poling techniques is still challenging, primarily due to strong fringing fields originating as a result of the geometry of the periodic metal electrodes and local crystal inhomogeneities and defects, which lead to inconsistent poling results.

Nevertheless, recently a breakthrough in consistent poling of high-quality, bulk sub-µm QPM gratings in RKTP has been achieved with the development of an alternative periodic poling technique [22] based on coercive-field engineering [23]. This method is based on creating a grating of periodically alternating low- and high- coercive field regions in the crystal, which is obtained via periodic ion-exchange (IE) through one of the polar faces of the crystal. The IE process leads to a coercive field increase in the exchanged regions. This allows domain inversion in the low-coercive field regions via an external electric field, while the orientation of spontaneous polarization remains unaltered in the IE regions. As a result, periodic poling can be performed with planar electrodes, bypassing the large fringing field problem of periodic metal-electrodes and the associated domain-broadening issues. Several sub-µm PPRKTP crystals with different periodicities have been produced by this method, and have been used for demonstrating several highly-efficient BWOPO devices [22,24–26].

The IE process is performed by immersing the RKTP crystal in a molten nitrate salt containing Rb⁺-, K⁺- and Ba²⁺-ions. The larger Rb⁺-ions are exchanged with the smaller K⁺-ions via a diffusion mechanism. Each divalent Ba²⁺-ion replaces two monovalent K⁺-ions in order to maintain charge neutrality, and consequently, creates extra vacancies when diffusing into the crystal. The formation of extra vacancies allows easier in-diffusion of larger Rb⁺-ions into the crystal and increases the depth of the IE. Finally, adding K⁺-ions in the melt allows to control the amount of Rb⁺-ions incorporated into the crystal, which is beneficial in order to decrease the stress created in the crystal lattice accommodating the larger Rb⁺-ions. The Rb⁺-ion concentration along the polar direction is usually expressed as follows [27]:

Here ${c_S}$ is the Rb⁺-ion concentration at the surface, ${c_{KTP}}$ is the $\textrm{R}{\textrm{b}^ + }$-ion concentration in the bulk and $D(T)$ is the diffusion coefficient dependent on temperature T, and t is time.

However, the ion diffusion process in KTP is a rather complex process with a time-varying diffusion rate, which also depends on the ionic conductivity of the specific crystal. The process is usually considered highly anisotropic with ion-diffusion happening almost exclusively along the polar axis, which is ascribed to the fact that the ionic conductivity is orders of magnitude larger along this direction compared to the other two orthogonal axes [28]. Usually, the IE through a rectangular aperture volume is therefore modeled as a rectangular cuboid [29], where the Rb⁺-concentration along the polar direction varies according to Eq. 1. While this simple assumption is sufficient to model waveguides in KTP [30], where the critical dimensions are on the micrometer level, the nanoscale features of the IE can no longer be disregarded when it is used for periodic poling with sub-µm periodicities. Unfortunately, to date, there are no comprehensive studies on the morphology of the IE regions in KTP isomorphs with nm-resolution. The most common techniques used to study the Rb⁺-content such as time of flight secondary ion mass spectrometry (ToF-SIMS), energy dispersive X-ray spectroscopy (EDX), microprobe electron microscopy and confocal Raman imaging [31] lack the necessary lateral resolution. Nevertheless, characterization of the IE regions with nanometer-resolution is crucial in order to understand their interplay with domain switching. Thus, it is of paramount importance to study the domain dynamics and the resulting domain morphology when coercive field gratings are used for periodic poling. These insights will be imperative to be able to fully exploit the potential of IE for domain engineering and will be the key to further downscale the QPM structures to smaller periods and/or upscale to larger apertures.

In this work, we study the bulk domain structure in RKTP crystals periodically poled with coercive-field gratings with periods ranging from 755 to 433 nm by employing piezo force microscopy (PFM). We correlate the ferroelectric domain morphology with the Rb⁺-profile via atomic force microscopy (AFM) studies on the cross-section of IE regions with nm resolution. We show that IE regions do not present a perfect cuboid at the nanoscale, and that it is their shape as well as the ionic-concentration distribution that determine the domain dynamics and the resulting domain duty-cycle. We demonstrate that this behavior is independent of the poling period. Our results indicate that with the IE-conditions used here, the coercive-field engineering technique should be further scalable to shorter periodicities.

2. Periodic poling

For our studies, we fabricated several sub-µm PPRKTP crystals using coercive-field gratings with periods ranging from 755 down to 433 nm. For all the crystals, the coercive field gratings were implemented using the same recipe in order to be able to determine the period scalability of the coercive-field structuring technique. All RKTP crystals used in this work had a poling volume of approx. 7 × 3 × 1 mm³ and were processed as follows.

First, a photoresist grating with an appropriate period and a duty cycle of 20-25% for the openings were created on the Z^--faces of the crystals using an in-house built UV-laser interference lithography system. Next, an IE stop layer was created in the photoresist openings as well as on the entire Z⁺-faces by oxygen plasma etching. Consequently, an IE was performed by placing the crystals into a molten salt bath containing 20mol% KNO₃, 73mol% RbNO₃ and 7mol% Ba(NO₃)₂, at a temperature of 330 °C for 4 hours. In order to characterize the Rb⁺-profile obtained under these exchange conditions, a single-domain crystal was subjected to planar IE and studied with EDX (FEI Nova 200 dual beam system equipped with an EDX detector Aztec Ultim, Oxford Instruments). The measured relative surface concentration was ${c_s}(R{b^ + })/{c_s}({K^ + }) = 12\%.$ As expected, the Rb⁺-concentration follows an inverse error function, with an erfc(1)-depth of 13 µm.

Finally, the stop layer was removed, and the RKTP samples were poled using planar liquid electrodes at room temperature, applying single triangular electrical field pulses between 1.25-5 ms duration with a peak magnitude between 5.5 and 9 kV/mm. Subsequently, the periodically poled RKTP crystals were selectively etched in order to reveal the domain structure. Figure 1 shows the AFM-topography images of the (a) former patterned, and (b) non-patterned polar faces of a representative sample with a poling period of 505 nm. The AFM images were taken using a Bruker Dimension Icon system.

 

Fig. 1. Topography of a periodically poled (Λ=505 nm) and selectively etched RKTP crystal on the (a) patterned polar face and (b) non-patterned polar face. The white bars represent 2.0 µm distance.

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3. Experimental results

Since the nonpolar faces of KTP are not charged and selective etching cannot be used to visualize the domains, their morphology along the polar axis was studied by doing PFM scans (Bruker Dimension Icon system) on the Y-faces of the crystals. For this purpose, the crystals were cut in the middle of the grating along the XZ-plane and the Y-faces were polished to an optical finish. Consecutive PFM scans were taken throughout the entire crystal thickness in order to follow the same sets of domains, thus, unveiling the domain propagation from the former Z^--towards former Z⁺-faces. Several locations along the X-axis were scanned in this manner for each PPRKTP crystal, in order to increase the statistical significance of the obtained data. Figure 2(a) shows a selection of such subsequent PFM scans of a representative PPRKTP sample with a period of 505 nm. This set of images illustrates how the inverted domains change along the polar direction, with the first scan taken at a depth of 4 µm and the last one at a depth of 950 µm relative to the IE face. The bright color corresponds to the inverted domains and the dark one to the original spontaneous polarization. The width of each individual domain was measured for all the scans taken. Figure 2(b) shows the average inverted domain-width together with the standard deviation versus the distance from the patterned surface. The following observations can be made: close to the patterned polar face (first 5 µm), the inverted domains maintain the widths defined by the lithographic/IE grating. Subsequently, their widths lessen slightly, exhibiting a minimum at a depth of 12 µm. Between 12 µm and 30 µm in depth, the inverted domains broaden uniformly at a similar pace, whereas from 30 µm to 50 µm, the average rate of domain broadening is still the same, but there are larger individual variations, as it can be seen from the increased standard deviation. From a depth of 50 µm, the width of each individual domain remains fairly stable for the remaining sample thickness.

 

Fig. 2. (a) PFM images of the same domains along the polar direction of a periodically poled RKTP crystal with a period of 505 nm at distances from the patterned surface of 4, 10, 20, 30, 50, 350, 650 and 950 µm (from top to bottom). The bright regions correspond to the inverted domains and (b) corresponding average inverted domain width and standard deviation.

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Similar measurements were performed in the PPRKTP crystals with poling periods of 433 nm and 755 nm, and the results are summarized in Fig. 3. Figure 3(a) shows the average domain-width as a function of the depth in the crystal for the three different periods. Here, the widths of the inverted domains have been normalized to the poling period and are presented as the duty-cycle (DC). Remarkably, the domain-growth dynamics present the same features as discussed for Fig. 2, regardless of the period and the poling conditions. Note that from Fig. 3(a) it would appear that the domain broadening is largest for the longest period. However, we believe that this is merely a result of the choice of the applied field magnitude used for domain switching in combination with individual crystal properties, such as ionic conductivity and defect concentration, and that it is not related to the periodicity itself. For instance, the 433 nm crystals were poled with an electric field magnitude slightly smaller than what would have been ideal for these specific crystals, which naturally led to a smaller DC in the bulk. On the other hand, the 755 nm were slightly “over-poled”, and therefore a larger DC was obtained. Figure 3(b) compares the average domain duty cycle in different locations in the same crystal and illustrates the variation in domain-broadening existing in the same crystal. Such variations are common while periodically poling KTP isomorphs and will be more pronounced with smaller poling periods. Indeed, one would expect more domain merging for shorter periods, contrary to what we observe here.

 

Fig. 3. Average duty cycle of the inverted domains versus distance from patterned surface for (a) periods 433 nm, 505 nm and 755 nm; (b) for 3 different locations along the X-axis in the same crystal (period of 505 nm).

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As discussed above, the average bulk domain width gets determined close to the patterned face, within the first 40-50 µm depth. This region corresponds to the volume in the crystal where the IE grating is present, and the domain growth may be obviously related to the IE grating properties. Therefore, it is important to characterize the morphology of the IE regions along the polar direction. To do so, we took advantage of differential etching between the exchanged and non-exchanged regions. Using the KOH:KNO₃ solution, the IE regions on polar faces etch at a slower rate than the non-exchanged ones, with a differential etch rate of 4 nm/min. However, as mentioned before, the non-polar faces of KTP are not charged, offering virtually no etch contrast. Thus, the crystal's Y-face was polished at a small angle relative to the XZ-plane to obtain selective etching between the exchanged and non-exchanged regions; a sketch is shown in Fig. 4(a). In order to characterize the morphology of the IE regions, we used a crystal with a 580 nm-period IE grating. In this case the sample had not been periodically poled, i.e., the sample was single domain, and therefore, the etching contrast originates solely from the IE pattern. After 10 min etching, the height difference between exchanged and non-exchanged regions was 8 nm. Subsequent topographic AFM scans were taken along the polar direction to study the profile of the IE regions in a similar manner as the PFM scans were used to track the domain morphology. A representative set of images is displayed in Fig. 4(b). Here the brighter color corresponds to higher topography, i.e., the IE regions. Note that the first scan was taken at a depth of 7 µm since etching considerably damages the corner edge of the crystal, impeding AFM-scanning closer to the polar face. The width of the IE regions broadens slightly during the first 10 µm from the polar face. Afterwards, it monotonously decreases until a depth of 32 µm, at which the Rb⁺-concentration, if any, is too small to give any etching contrast. In contrast with the wide-spread model for IE in KTP [29]; the present results show that, when the features are on the nanoscale, the IE boundaries exhibit a small angle with respect to the polar axis, which manifests itself in a dagger shape in the XZ-plane, as illustrated in Fig. 4(c). It is also worth noting that the IE is deeper than what is detectable with EDX. We attribute this discrepancy to the limited detection sensitivity of EDX.

 

Fig. 4. (a) Ion-exchanged RKTP crystal prior to poling. The crystal is polished at a small angle in the XZ-plane to allow for selective etching. (b) Consecutive topography scans of the nonpolar surface of a periodically ion-exchanged (580 nm) and etched sample prior to poling. The distance from the patterned surface reads from top to bottom as 7, 10, 13, 16, 21, 24, 27 and 30 µm. The bright areas correspond to the ion-exchanged regions. The white reference bar corresponds to 1.0 µm distance. (c) Illustration of the ion-exchange shape and its effect on lateral domain growth. (d) The average duty cycle of inverted domains with different periods (black, red and blue) and average ion-exchange duty cycle (green) versus Z-position from the patterned surface.

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The observed morphology of the IE regions can be understood by considering that there is a small, but non-zero Rb⁺-diffusion in the X-direction of the crystal. This diffusion rate is directly related to the ionic conductivity along the X-direction in KTP, which is estimated to be 3 to 4 orders of magnitude smaller than along the polar direction [28]. Indeed, Wellendorf et al. observed diffusion in the non-polar direction and reported a large counter-diffusion ratio of Rb⁺-and K⁺-ions along Z- and Y-direction, with a value that depends on the Ba²⁺-concentration in the melt [32,33]. This in turn, may result in a gradient of Rb⁺-concentration perpendicular to the main diffusion flow along the polar axis and consequently the IE regions get narrower with depth. In fact, a similar observation was made by Kriegel et al. [33] for optical phase gratings in KTP fabricated with a mixture of Rb/K/Ba, although with quite different concentrations. Together with the fact that the volume closest to the surface is experiencing IE for a longer time, this can explain the observed IE broadening in the first few micrometers.

Figure 4(d) compares the duty-cycle of the IE regions along the polar axis with that of the inverted domains. It is clear that there is a strong anti-correlation relationship between the width of the IE regions and that of the domains. This is supported by the estimated Pearson correlation coefficients of −0.94, −0.96 and −0.99 between IE data and domain width data for 433 nm, 505 nm and 755 nm periods, respectively. It becomes obvious that the shape of the IE regions is responsible for the morphology of the inverted domains. Furthermore, even though the data was obtained in crystals with different periodicities, the strong correlation suggests that the interplay between IE and domain growth is the same for all studied periods. Given the amount of broadening observed here for the IE-regions, these results indicate that this technique should be scalable to periods smaller than 200 nm. This should satisfy most of the nonlinear counter-propagating interactions in the transparency range of the material.

To clarify the role of the IE regions in preventing the sideways expansion of the inverted domains we have studied crystals that have been poled with an electric field large enough to cause bulk domain merging (so-called overpoled samples). Figure 5(a) shows a PFM image of the Y-face taken underneath the IE face in one of such crystals together with (b) the domain width versus depth for the first 23 µm. It can be clearly seen that up to the 26 µm depth, the inverted domains propagate following the same features as discussed above. However, at 26 µm depth, the domains merge abruptly. This sharp merging was observed for all overpoled samples at typical depth of $25 \pm 2{\; }$µm, and regardless of the grating period, indicating that the critical depth is solely a function of the exchange conditions, i. e., melt composition, exchange- temperature and duration. This implies that IE prohibits domain merging and constrains the domain growth primarily to the polar direction until a critical depth is reached, at which this constraint cannot longer be sustained. Interestingly, this critical depth coincides with the depth at which larger individual domain-widths variations are starting to be observed. However, this critical depth does not correspond to the depth at which the differential etch contrast between exchanged and non-exchanged regions ceases, and it is deeper than the depth at which the presence of Rb⁺-ions can be detected by EDX, suggesting that the critical depth rather corresponds to a critical Rb+-concentration, related to the minimum exchange required to prevent sideways domain growth. This critical depth is the same for all the periods tested, and it is independent of the pattern duty-cycle, since for crystals patterned with different DC, domain merging was observed at the same depth. Furthermore, it appears to be independent of the magnitude of the applied poling voltage for a large field window of 5.5 to 9 kV/mm. These observations point to the fact that it is solely the IE conditions that determine the critical depth.

 

Fig. 5. (a) PFM image of a Y-face of an overpoled sample. The bright regions correspond to the inverted domains. The domains abruptly merge 26 µm from the patterned surface and (b) corresponding average inverted domain width and standard deviation in the overpoled sample.

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4. Conclusions

We have demonstrated that the domain dynamics of sub-µm periodic domain structures, when coercive field engineering is used, are governed by the properties of the IE regions, specifically by their morphology and Rb⁺-concentration distribution. The observed anti-correlation behavior between the widths of switched domains and the widths of IE regions indicates a strong confinement of polarization switching to non-exchanged regions, and consequently, the domains grow coherently along the polar direction within the first 25 µm of depth. Beyond this critical depth, the domain broadening still reflects the morphology of the IE regions, although the domain growth is less homogeneous, with more variations from domain to domain. Outside the IE region, the domains maintain their width throughout the rest of the crystal thickness. The fact that this behavior is observed for all the periods studied here, together with the observed broadening behavior, suggests that this technique should be scalable to poling periods below 200 nm (aspect ratios of 10⁴). This should be enough to enable most of the nonlinear counter-propagating interactions within the transparency window of KTP. Finally, we also believe that this method should be scalable to larger aperture QPM devices with sub-µm periodicities; however, the IE should be modified in order to have a deeper critical depth that can accommodate higher electric fields and large domain aspect ratios.