Tuning the Čerenkov second harmonic contrast from ferroelectric domain walls via anomalous dispersion

Second harmonic (SH) microscopy is a flexible imaging and investigation technique into crystalline systems ranging from 2D materials (2DMs) to ferroelectrics and their domain walls (DWs).^1–7 The application of SH microscopy as an imaging process allows for real-time three-dimensional (3D) observation of domain walls (DWs) subjected to external stimuli such as external electric fields^2,5,8 which, combined with no need for special sample preparation, makes it a go-to method for investigations into novel DW structures.⁹ Most recently, SH polarimetry has revealed unique symmetries and inherent substructures^6,8,10 previously thought not to be present in ferroelectric DWs and that are believed to be connected to the high conductivity observable in certain DWs.^11,12

In SH imaging, DWs can appear both as bright lines in a lower signal background, and the other way around, as dark lines in a bright background. As shown many times before, the contrast depends on various conditions, such as the sample orientation, its thickness, or the focusing geometry.^13,14 The contrast for DWs can be explained by three distinct mechanisms that may be present at the same time, making signal interpretation challenging, especially when analyzing the symmetry of DWs.¹⁰ The three broadly accepted mechanisms are:¹⁵ 

• $∙$

  (1) interference of the SH signal in the far field from domains of different orientations rather than the DW itself;

• $∙$

  (2) Čerenkov-type phase matching at the DW, termed CSHG; and

• $∙$

  (3) local changes in the nonlinear properties at the DW, which are a central feature in novel symmetries of DWs.^6,10

In general, contrast mechanisms (1) and (3) can explain both bright and dark DW contrasts depending on the local changes in the SH response,^6,10 as well as the focus placement in relation to the material surface, the light polarization, and the numerical aperture.^7,13 In contrast, the Čerenkov contrast (2) only results in increased signals stemming from the DWs, i.e., bright DWs.

The Čerenkov SHG effect has been investigated extensively since its first description of this phase-matching scheme in second harmonic generation from thin-film waveguides (optical second harmonic generation in the form of coherent Čerenkov radiation from a thin-film waveguide). In the recent decade, the Čerenkov SHG effect was discovered to be present in ferroelectric DWs and is used as a primary contrast mechanism allowing the visualization of ferroelectric DWs. In this context, various aspects, such as the wavelength dependent emission angle,^15–18 the emission as a function of DW roughness¹⁹ or the existence of collinear CSHG,³ have been studied. While the DW contrast as a result of (1) interference or (3) local changes in the nonlinear response has been extensively studied in recent years and quantitative models have been developed,^6,7,9,10,13 the strength of CSHG in relation to the other contrast mechanisms is not established, although multiple contrast mechanisms can be present at the same time making separation necessary.^7,15 This is particularly relevant for studies of polarimetry, which mostly rely on interpreting local changes of the SH response. In the present paper, we aim to use the interference contrast as a reference point with which to study the actual contrast and intensity of the Čerenkov effect using the anomalous dispersion regime, which has been shown by Ren et al.²⁰ to block the Čerenkov second harmonic signal. Additionally, this alternative approach will be compared with our previous work,¹⁵ where we applied geometric means to separate the Čerenkov contrast and estimate relative signal strenghts enabling a more complete picture.

The bulk SH Intensity $I$ from a material slab between $z0$ and $z1$ along the z axis [sketch in Fig. 1(a)] can be calculated in a Gaussian beam picture by

Here, $Δk=2kω−k2ω$ represents the phase mismatch between the waves at the fundamental (⁠$kω$⁠) and SH frequency (⁠$k2ω$⁠) [see also Fig. 1(b)], while $b$ is the confocal parameter of the Gaussian beam. Assuming an infinitely thick medium, i.e., setting the boundaries of the integral in Eq. (1) to $±∞$⁠, this integral then converges to two well known results^4,7,21 for either a negative $Δk<0$ or a positive $Δk>0$⁠. For $Δk<0$⁠, this integral yields zero, which is typical for almost all experimental scenarios because the refractive index at the SH wavelength $2ω$ is larger as compared to the refractive index at the fundamental frequency [$n(2ω)>n(ω)$]. This condition is also called normal dispersion and underlines that SH generation in bulk media using focused beams is not permitted. Under this condition, bulk media only generate signals when the focus is placed close to the surface, i.e., the integration in Eq. (1) is performed only over a semi-infinite range (i.e., from $0$ to $∞$⁠).

In contrast, for a positive phase mismatch $Δk>0$⁠, the integral will yield a signal even if the focus is placed within the bulk. This so called anomalous dispersion is only possible when the refractive index at the SH wavelength is smaller than the refractive index at the fundamental wavelength. This can be achieved in birefringent crystals, where an SH process with a switch from one branch of the refractive index to the other is possible. In congruent lithium niobate (cLN), for example, this is achieved for a fundamental wavelength beyond 1078 nm via an SH process involving the nonlinear tensor element $d31$⁠, which allows a fundamental beam in the ordinary polarization (refractive index $no$⁠) to generate signals in the extraordinary polarization (refractive index $ne$⁠). For fundamental wavelengths beyond 1078 nm, it holds $ne<no$ fulfilling the condition of anomalous dispersion. Alike, for other dopants in LN, the condition of $ne<no$ may be fulfilled at different wavelengths, as seen from Table I. At the respective threshold wavelength, the refractive indices for both optical directions are equal, i.e., $ne=no$⁠.

Figure 1(c) shows a vertical line scan [red arrow in Fig. 1(d)] through a periodically-poled cLN sample using both normal and anomalous dispersion, that is, the laser beam, polarized parallel to the crystal’s x axis, is incident along the crystallographic y axis using a fundamental wavelength above and below 1078 nm, respectively. As previously shown by Kaneshiro et al.,⁴ for normal dispersion [black curve in Fig. 1(c)], the surfaces of the crystal generate a SH signal, while the bulk remains relatively dark by comparison. The red curve shows the same location of the sample but with a fundamental wavelength set above 1078 nm, where we can see noticeable signal contribution throughout the entire sample, with no pronounced maxima at the surfaces.

As discussed, for normal dispersion, the SH signals in the bulk are zero. Therefore, DWs can only be observed if they have locally different linear or nonlinear properties.^6,24 However, observing DWs in normal dispersion is also possible via the Čerenkov SH process. In a simple picture, the DW acts as an extended defect associated with a lattice momentum $Δk$ that allows for improved (non-collinear) phase matching. As a result, more signal is observed whenever the focus is placed near the DW. This situation is schematically depicted in Fig. 1(b) through the respective wave vector matching. The Čerenkov SHG is generally generated at an oblique angle compared to the fundamental beam, although collinear CSHG has also been reported.³ This, however, requires further analysis of additional DW properties such as the inclination angle and roughness, we will, therefore, focus on the non-collinear case and the influence of anomalous dispersion on the DW contrast as a whole. The oblique emission is only possible for normal dispersion, as $k2ω$ needs to be larger than $2kω$⁠. Therefore, choosing an SH process under the conditions of anomalous dispersion will also prohibit the generation of Čerenkov SH generation. For imaging of DWs in bulk crystals, this has important consequences. As Čerenkov is forbidden in this case, any changes of the DW signal can only be attributed to either interference effects or changes in the local nonlinear susceptibility, which now can be studied systematically.

In this work, we show that the choice of the fundamental wavelength allows us to selectively turn on or off the Čerenkov mechanism by switching from normal (⁠$Δk<0$⁠) to anomalous dispersion (⁠$Δk>0$⁠) for a fixed focal position at a DW within the crystal. While in the past, experiments either well below or well above this threshold have been performed,^4,20 in this work, we particularly focus on the transition region. This enables us not only to change the contrast from bright to dark, but also to get a quantitative insight into the Čerenkov contrast mechanism as well, which has eluded quantitative predictions of the contrast so far, and compare it with the much better understood interference contrast. The experimental work performed here can serve as a reference for quantitative analysis of the CSHG contrast, in a similar manner as the interference contrast has been analyzed.

The investigated samples were bars cut from a z-cut wafer of both periodically-poled congruent (cLN) and (5%) magnesium-oxide-doped lithium niobate (MgO:LN). The bars are then rotated 90$°$ onto their y face. This allows access to both the ordinary refractive index (n$o$⁠) using a beam polarized parallel to the crystallographic x axis as well as to the extraordinary refractive index (n$e$⁠) with a beam polarized parallel to the z axis. The utilized setup is a Zeiss LSM980MP microscope operated in transmission, equipped with a linearly-polarized, tunable Ti:Sa laser (Spectra Physics InSight X3, 690–1300 nm, 3,5 W, $<120$ fs pulse width) for excitation, using a focusing NA of 0.8 and collection NA of 0.55 in the forward direction.

Figure 1(d) depicts a sketch of the measurement geometry including a visualization of the $xy$ plane scanned through the sample. The DWs investigated in this work lay within the $zy$ plane of the sample, and our scans will, therefore, depict them as continuous vertical lines in the images.

To display whether or not the proposed switching in contrast is observed, images were recorded for the cLN sample at 1050 and 1100 nm fundamental wavelengths, i.e., above and below the theoretical threshold wavelength. Two scattering geometries are chosen:

• $∙$

  (1) The fundamental laser is polarized parallel to the x-direction of the crystal (⁠$no$⁠). Here, the SH signal will be dominantly generated via the bulk $d31$ tensor element and be polarized in z-direction (⁠$ne$⁠). In this geometry, a switch from normal to anomalous dispersion is expected.

• $∙$

  (2) In the second geometry, the fundamental laser is polarized parallel to the z axis (⁠$ne$⁠). Here, SH light is generated via the largest tensor element $d33$ and is also polarized along the z-direction of the crystal. Hence, no effect is expected.

The results of these four experiments are depicted in Fig. 2. Figures 2(a) and 2(b) depict the first scattering geometry for 1050 and 1100 nm, respectively, while Figs. 2(c) and 2(d) show the second scattering geometry without a switch to anomalous dispersion. The dashed white lines in Figs. 2(a) and 2(b) represent the location from which the signal profile was extracted at a depth of 500 $μ$m within the crystal for later analysis, as then displayed in Fig. 3. During measurements using geometry (1), a small amount of ordinarily polarized SHG light will also be generated via the $d22$ element. However, due to the smaller magnitude of this tensor element as well as its shorter coherent interaction length, the process via the $d31$ will largely dominate, as has been shown previously.^14,15

As mentioned above, for normal dispersion, we expect the DWs to appear as bright lines on a dark background due to the 20$×$ stronger signal generation from the Čerenkov mechanism combined with the lack of bulk signal.¹⁵ On the other hand, for the anomalous dispersion regime, we expect a negative contrast between DW and domains, i.e., the DWs are dark lines, as well as a noticeable signal from the domains themselves.

Below the threshold wavelength, both geometries qualitatively display the same behavior, i.e., the domain walls are visible as bright lines on the dark background of the bulk material. However, the signal obtained from the second geometry is noticeably larger likely due to the incident polarization being parallel to the largest non-linear tensor element $d33$ in lithium niobate. Please note, while the same crystal was analyzed, the region that is visualized is different, as the crystal needed to be physically rotated due to the fixed polarization of the laser in the setup (to maintain the exact same illumination conditions, the sample is rotated by 90$°$ rather than the incident laser polarization).

The difference between both geometries becomes apparent for the images recorded at 1100 nm. Here, the second geometry recreates a similar image, while for the first geometry, however, we see that the domain walls appear as dark lines on a white background, i.e., the CSHG light previously generated at the lower wavelength is suddenly no longer part of the emitted signal, while the bulk material generates significant SH signal in agreement with theory.⁴ Additionally, the measurable signal generated by the domain regions opens the door toward analysis performed on the bulk domains themselves, rather than an analysis limited to the domain walls within the bulk. Further images recorded at intermediate wavelengths in the present cLN crystal as well as cross sections for periodically-poled MgO:LN are included in Figs. S1 and S2 in the supplementary material.

So far, the analysis was limited to single wavelengths above or below the threshold. However, for an in-depth analysis of the competing mechanisms, the behavior in the vicinity of the threshold wavelength requires investigation. We calculated the theoretical threshold wavelengths as the wavelength for which $no(ω)=ne(2ω)$⁠, which are listed in Table I.

Measurements similar to Figs. 2(a) and 2(b) were then recreated using a range of wavelengths for periodically-poled cLN and MgLN crystals around the calculated threshold in y steps of 10 nm. As we are interested in the bulk behavior neglecting any influence of surface effects, signal profiles were extracted at a depth of 500 $μ$m within the crystal, as indicated by the dashed lines in Figs. 2(a) and 2(b).

Due to the stark difference in the overall signal between the experiments at different fundamental wavelengths, for example, due to the largely different coherent interaction lengths, each profile was normalized to achieve comparability. For this work, we chose two types of normalization. First, each profile was normalized to the interval [0,1]. This was specifically used in Figs. 2 and 3 as well as Figs. S1 and S2 in the supplementary material. This normalization particularly highlights the nature of the line shapes and qualitative behavior of the contrast. Second, the data were normalized to the bulk intensity, which directly highlights the contrast as defined below. Here, after dark count subtraction, where necessary, the intensity was normalized to the intensity value measured in the bulk in between two domain walls, which consequently has an intensity of unity. This type of normalization was also used on previous works, where the DW contrast was analyzed numerically.^7,9 The respective data can be found in Figs. S3 and S4 in the supplementary material for cLN and MgO:LN, respectively.

The resulting data are plotted in Fig. 3, with data recorded at the same fundamental wavelength displayed at the same height in the image. As with the images in Fig. 2, on the high or low end of the recorded wavelength ranges, the DWs appear solely as a positive or negative peak, respectively. More clearly, for MgO:LN, DWs at a 1000 nm wavelength give a positive SH response (black), whereas they appear “negative” at 1080 nm (red). In fact, as the wavelength in each material closes in on the threshold wavelength, we observe a decreasing contrast between DW and the neighboring domain signal, although the profile is still dominated by a single positive or negative peak.

At approximately the calculated threshold wavelength, a flip in contrast is observed. However, this flip is not sudden, but continuous over a range of $>30$ nm in wavelength, which is due to the spectral width of the exciting ultra short laser pulse (⁠$FWHM≈15$ nm at 1000 nm). A range of wavelengths makes up the incident pulse, of which a certain fraction may still be below the threshold. The result should be a superposition of the two peak types. This can be seen for 1060 nm in the case of MgO:LN and 1090 nm in the case of cLN, where a central peak belonging to the Čerenkov contrast emerges from the dip belonging to the interference contrast dominating in the anomalous dispersion region. Interestingly, in both materials, we observe this superposition at a larger wavelength than expected theoretically for monochromatic incident light.

The determined thresholds are summarized in Table I. Here, we observe that the measured thresholds are off from the expected wavelength in both materials by approximately $+26$ nm (MgO:LN) and $+12$ nm (sLN), respectively. In contrast, the relative order in threshold wavelength is reproduced, i.e., threshold wavelength in sLN is larger compared to MgO:LN). Here, likely reasons are deviations from ideal conditions, e.g., temperature, as well as stoichiometric variations in the different compositions of LN, which is known to have an impact on the dispersion. Indeed, in previous work, we also observed slight systematic shifts of thin-film resonances compared to theoretical calculations, which likely also indicated variations of stoichiometry or conditions.¹⁴ 

Previous work has shown that the Čerenkov SHG responsible for the bright domain walls within the bulk is approximately 20 times stronger than the other contrast mechanisms in SH microscopy.¹⁵ As a result, it would require less incident light below the anomalous threshold to generate a signal comparable to “traditional” SHG from the surrounding domain, which is caused by the wavelengths above the threshold. This would explain the apparently higher threshold wavelength, as the vast majority of the spectral components of the pulse would have to be far beyond the actual threshold. For further analyis of the influence of anomalous dispersion, the DW signal contrast was calculated. Within this work here, we define the contrast $C$ as the quotient of the difference in signal magnitude between DW (⁠$SDW$⁠) and domain (⁠$SDomain$⁠), and the average domain signal, i.e.,

The sign of the contrast $C$ now indicates the type of DW contrast, as a negative contrast value infers that the DWs are darker than the average domain signal and vice versa for a positive contrast. Additionally, the absolute magnitude gives an indication of the strength thereof. Figure 4(a) depicts the extraction of the contrast as shown in Eq. (2).

Figures 4(b) and 4(c) display the DW contrast as a function of depth and wavelength for the example of the CLN sample. For each parameter setting, the DW contrast values were averaged across the DW signatures in profiles such as those shown in Fig. 3.

In Fig. 4(b), the contrast is plotted as a function of depth for 1050, 1090, and 1110 nm, i.e., at wavelengths below, approximately at, and above the anomalous dispersion threshold, respectively. The contrast values for 1050 nm were scaled down to a factor of 0.05 to allow for comparison within the same range. Here, the contrast at the outermost wavelengths is dominated by a single type of contrast throughout the sample thickness, negative above the threshold with $C≈−0.6$⁠, and $C≈16$ below the threshold, while close to the threshold, an overlapping of both mechanisms is present.

For the wavelength well above the threshold, it is interesting to note that the contrast value for 1110 nm is centered around $C≈−0.6$⁠, i.e., the DW signal is 0.4 times the domain signal. In simulations of SH microscopy of ferroelectric domain walls, Rüsing et al. showed that the expected ratio between DW and domain signal is approximately $−0.6$ in thin films inspected in reflection geometry, and is a result of phase interference between overlapping SH waves from neighboring domains.⁹ The positive contrast for Čerenkov SHG of $C≈16$ is comparable to our previous experiments with blocking apertures.¹⁵ The error bars in Fig. 4(b) were calculated from the standard deviation of the contrast value. Generally, we see a low error within both regimes (orange and blue), while the transition region can show a slightly higher variance due to the competing mechanisms and low contrast in itself.

When examined closer, the lines approximately at (1090 nm) and below the threshold (1050 nm) show a general decrease in contrast, in particular, for a larger depth. This behavior resembles the overall decrease in intensity with depth as shown in Fig. 1(c) and likely relates to the decrease in focus power density due to the temporal broadening of the pulse and spatial broadening of the focal spot in larger depth. This observation provides a hint, that the temporal shape and/or focus spot shape also influences the CSHG contrast. For a fundamental wavelength of 1050 nm close to the threshold a negative contrast is observed close to the surface, which then flips to a slightly positive $C≈0.6$ at increasing depth. The negative contrast close to the surface can be explained by strong interference type contrast, which dominates due to the strong SHG from interfaces also in the case of normal dispersion.^4,7 It is similarly observed for a fundamental wavelength of 1050 nm, where the contrast first shows a low value of $C≈1$ close to the surface due to overlapping of interference contrast. For increasing depth at this fundamental wavelength, the contrast rapidly grows to a value of almost 20 at depth of 30 $μ$m. However, the contrast stays not constant but shows another local minimum in a depth of about 70 $μ$m with $C≈12$⁠. As can be seen in Fig. S5 in the supplementary material, such a minimum is also present for 1060 nm and almost disappears for larger wavelengths. The origin of this local minimum is not clear. One potential explanation for this effect could be an influence of axial phase matching, which explains the slight shift to larger depth for increasing wavelengths and the disappearance close to the threshold wavelength, where the coherent interaction length converges to infinity. Similar minima were also observed in previous work.⁴ However, in contrast to Kaneshiro et al., in our case, only negligible bulk SHG signal for a fundamental wavelength of 1050 and 1060 nm is observed. Therefore, this minimum is only present in the Cerenkov signal. For further insight, more theoretical work is indicated.

Figure 4(c) shows the average contrast as a scatterplot vs fundamental wavelength for four selected depths, these being shortly (⁠$−5$ $μ$m) before the surface, exactly at the surface (0 $μ$m), at an intermediate depth (20 $μ$m), and far within the bulk (500 $μ$m). Generally, we see that the average DW contrast decreases with increasing wavelength, eventually showing a fully negative contrast for 1100 nm and above, regardless of the extraction depth. When comparing the chosen depths over the entire wavelength range, it is possible to group them into a surface-near and surface-far behavior. The monotonous decrease of the contrast, instead of a step-like switching at the threshold wavelength may be explained by the spectral width of the pulse, which has an FWHM of approximately 15 nm at 1000 nm, as discussed for Fig. 3 as well. This means that certain parts of the pulse are already at or beyond the threshold, which will result in a contrast below 1 due to the interference contrast, which will decrease in the overall contrast. The highest values of $C≈13$ in Fig. 4(c) at the lowest wavelength and a depth of 500 $μ$m are of similar order as observed at even lower wavelength in our previous experiment.¹⁵ It can be expected that for lower wavelengths, a constant contrast of the order of $C≈15−20$ will be observed.

Shortly before the surface (⁠$−5$ $μ$m), the contrast is close to 0, because there is only a low signal generation, as only a fraction of the focus is placed within the sample. The comparably low contrast directly at the surface (0 $μ$m) can also be explained by the strong surface-SHG signal, which is generated regardless of which dispersion regime the experiment is performed in. Independent of the fundamental wavelength, simulations predict a negative contrast for the surface in the investigated geometry due to a strong contribution from interference contrast,^7,9 which in this experiment appears to be overlapped by a positive contrast from the Čerenkov effect. Despite the lower contrast magnitude, however, the contrast for both depths is positive far within the normal regime, i.e., at 1050 nm, and negative only far within the anomalous regime, i.e., at 1110 nm. This could indicate a lower energy density requirement at the DWs for generating CSHG.

The contrast values extracted at depths of 20 and 500 $μ$m show a strong positive value of up to 13 and 10, respectively, for wavelengths below the threshold. The contrast rapidly decreases with increasing fundamental wavelength before transitioning to a negative contrast at 1090 nm and higher. The notably large contrast within the normal dispersion regime is caused by the CSHG signal generated at the DWs as well as the absence of bulk signals from the domain regions. As seen in the blue data points of Fig. 4(b), in the anomalous regime, the contrast settles into a value of approximately $C≈−0.6$ regardless of depth, which is a result of the absence of CSHG. Here, the contrast can only be attributed to interference or local changes in non-linearity.

In conclusion, we have investigated the influence of the dispersion on contrast in SH microscopy of ferroelectric DWs. It is shown that, in continuation of Kaneshiro et al.,⁴ above a certain threshold wavelength, it is possible to generate SH signal from the bulk rather than the DWs of a periodically-poled lithium niobate crystal. Far from the aforementioned threshold, we observe a signal dominated by either CSHG or another SH mechanism, such as interference contrast, respectively. Examining the contrast close to the threshold, it becomes obvious that multiple processes are present at the same time. However, if SH generation from bulk is forbidden, the strong CSHG signal¹⁵ overpowers the other contributions. This potentially has implications for the study of DW substructures via SHG polarimetry, which is gaining recent interest in bulk and thin films alike.^6,10

The choice of wavelength and process is shown to be a valid parameter when performing SH microscopy to selectively disable CSHG, granting access to the signal stemming from phase interference and changes to the local non-linearity without the dominating contribution of CSHG. Additionally, the generation of SH signals within bulk domains allows for analyzing the domain regions themselves rather than being limited to the DWs, e.g., the addition of a known reference crystal could allow for the determination of the local polarization orientation via interference imaging.⁴ For example, from studies with Raman spectroscopy of ferroelectric DWs, it is known that long range strain fields exist, which extend at least several micrometers around the domain transition.²⁵ This may explain the large FWHM of the domain transitions observed in Fig. 3 in the anomalous regime but requires further studies.

Ideally, this investigation into anomalous dispersion in SH microscopy can serve as a novel approach in separating the participating contrast mechanisms and developing a more advanced understanding of ferroelectric materials and their DWs.

See the supplementary material for a full compilation of the cross-section images shown in Fig. 2 for both cLN and MgO:LN as well as further plots of the extracted signal profiles which retain the relative peak magnitudes between wavelengths for both crystals.

The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through projects CRC1415 (ID: 417590517), EN 434/41-1 (TOP-ELEC), INST 269/656-1 FUGG, and FOR5044 (ID: 426703838), as well as the Würzburg-Dresden Cluster of Excellence on “Complexity and Topology in Quantum Matter”—ct.qmat (EXC 2147; ID 39085490). Also, we would like to acknowledge the excellent support by the Light Microscopy Facility, a Core Facility of the CMCB Technology Platform at TU Dresden, where the SHG analysis was performed.

The authors have no conflicts to disclose.

Peter A. Hegarty: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Lukas M. Eng: Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Michael Rüsing: Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.