1. Introduction

Domain walls are ultra-thin boundary regions [1] separating magnetic, ferroelectric or ferroelastic domains [2] with opposite order. They can be polar in ferroelastic materials [3,4], show electronic conductivity [5–14] or superconductivity [15] in otherwise insulating oxides, and display enhanced local optical responses such as photovoltaic effect [16–18] or localized nonlinear optical emission [19–25] in ferroelectrics offering new perspectives for photonic applications [26]. The discovery of their extraordinary functional properties [27–30] has recently shifted the interest in their study from a scientific curiosity to a quest for original device concepts based on domain walls [31–35]. Ferroelectric domain walls were long believed to be of Ising type. In this idealized configuration, the polarization decreases to zero at the centre of the wall, and it reverses its direction while remaining strictly parallel to the polarization of the adjacent domains. On the other hand, deviations from the Ising configuration have been predicted by several theoretical studies [36–41] showing a non-zero polarization which can be either parallel (Bloch-type) or perpendicular (Néel-type) to the walls. These non-Ising polar domain walls were named after the fundamental magnetic domain walls by Lee et al. [37]. Nowadays, it has become of common knowledge that a ferroelectric domain wall has a non-zero polarization and holds its own symmetry and properties which significantly differ from those of the adjacent domains. Yet, the exploration of their internal structure is still in its infancy stage. This is because of the difficulty to detect such embedded 2D nanostructures. A priori, only aberration-corrected transmission electron microscopy with atomic resolution should be able to provide a detailed description of the polarization state at domain walls [42]. Surprisingly, and despite their limited lateral resolution, optical methods are able to reveal important local properties arising at domain walls [43]. Among these optical techniques, second-harmonic generation (SHG) microscopy has emerged as a unique method allowing the observation of the three-dimensional morphology of domain walls in technology-relevant ferroelectric photonic crystals [44,45].

Second-harmonic microscopy with polarization analysis has recently led to important scientific breakthroughs by shedding light on previously unknown local symmetry aspects in ferroelectric domain walls [46] and twin boundaries [47]. Moreover, SHG polarimetry has confirmed the existence of non-Ising and chiral domain walls [48] exhibiting Néel [49] or Bloch-type internal structures [48], in agreement with the theoretical predictions. In these studies, the domain wall type (Néel or Bloch) is obtained by comparing the experimental results and the simulated SHG response assuming a local symmetry lowering with respect to the parent material [50] (see tutorial [46] for more details). For the sake of simplicity, the polarization analysis of the domain walls’ SHG is often conducted assuming a scalar optical field. This model is known to be correct as long as objectives with low to medium numerical apertures (NA) are used (typically, NA $\leq 0.7$ [51]). Strong focusing is known to induce a depolarization of the incident laser beam, apodization, and spherical aberrations [51]. The vector character of the focused field should obviously affect the light-matter interaction and, consequently, impact the SHG response [52] and contrast mechanism [53]. It is therefore important to clarify the extent by which the conclusions on the domain wall types derived from SHG polarimetry analysis can be affected by focusing. This is however a complex question given that the light-matter interaction does not only depend on the relative weight of the electric field components (i.e., $|E_x|$:$|E_y|$:$|E_z|$) at the focus for a given incident polarization of the fundamental wave (FW). It is also influenced by the way how these components are "mixed" through the nonlinear optical susceptibility tensor. Previously reported studies on molecular and biological systems have shown that the vector character of the electric field, per se, induces rather small distortions in SHG polarimetry, while these distortions can be strongly amplified by the measurement geometry [54,55] or by the birefringence of the material [56,57]. It is therefore important to take into account the local symmetry, the measurement geometry, as well as the structural and optical properties of the material.

This study provides a general analytic model of the vectorial optical field and its effect on the local SHG at non-Ising domain walls. The distortion of the second-harmonic polarisation response due to focusing is evaluated in terms of rotation and broadening of the SHG polar plots as compared to those obtained based on the scalar approximation. These deviations are systematically studied at different focus values, while analyzing the effect of the domain walls’ intrinsic (local symmetry, optical anisotropy factors, birefringence), and extrinsic parameters (geometry). As an application example for the as-derived analytic model, the effect of geometry is simulated for Néel and Bloch-type domain walls in hexagonal-shape c-domains which are commonly observed in trigonal uniaxial ferroelectrics such as LiTaO$_3$ or LiNbO$_3$. The relative orientation between the polarization of the incident light and the six domain walls surrounding the hexagon is found to strongly affect the SHG polar plot distortion, irrespective of the Néel or Bloch character of the walls. The orthogonal orientation appears to be the most suitable condition to minimize SHG polarization distortions in non-oblique domain walls, even in the case of strong focusing. Yet, selecting the appropriate measurement geometry is not always sufficient to minimise SHG distortions. We show that domain walls exhibiting high intrinsic optical anisotropy factors or strong birefringence lead to larger SHG polarimetry distortions due to focusing, even if the optimum geometry is chosen. The presented analytic study and the attendant simulations provide the necessary precision to derive the internal structure of non-Ising domain walls from SHG polarimetry experiments. This approach can be extended to the exploration of newly discovered ferroelectric topologic nanostructures such as Bloch lines [48] or flexons [58].

2. Methodology

2.1 Second-harmonic polarimetry response at non-Ising domain walls

Second-harmonic microscopy experiments conducted on ferroelectric domain walls have recently confirmed the non-Ising characters of polar domain walls [48,49] predicted by theory. Furthermore, a specific symmetry is derived at such polar domain walls [46,47]: point group symmetry 2 is often found in Bloch walls, and Néel-type walls belong to point group symmetry m, while Ising-type DWs are centrosymmetric with a point group symmetry 2/m.

In the following, we present the general analytic form of the second-harmonic polarimetry accounting for the non-Ising character of Néel and Bloch-type domain walls and their local nonlinear optical susceptibility tensors. The SHG intensity variation with the FW polarization $\varphi$ and analyzer angle $\alpha$ is given by $I^{SHG}(\varphi,\alpha ) = |\boldsymbol P^{2\omega }(\varphi,\alpha )|^{2} = \sum _{i}|P_i(\varphi,\alpha )|^{2}$. The tensorial form of the second order polarization induced by the light-matter interaction $P(\varphi,\alpha )$ reads:

The domain wall orientation (horizontal, vertical or oblique) is identified with respect to the laboratory coordinates system by the angle $\delta$ (taken from $x$-axis). The related susceptibility tensor is derived using the rotation and transformation matrices assuming a Néel or Bloch character and an arbitrary orientation $\delta$ of the walls [46]:

Figure 1 displays the nonlinear susceptibility tensors derived for Bloch (left panel) and Néel (right panel) configurations in the case of a domain wall aligned with $y$-axis ($\delta = \pi /2$). Similarly, the optical tensor of any wall orientation and type can be derived by replacing $\delta$ by its appropriate value in Eqs. (3 and 4). The complete SHG polarimetry response can be modelled knowing the variation of the electric field with the laser’s polarisation $\varphi$. This means that the modelling of the electric field at the focal point (at which the second-harmonic signal is generated) is crucial for the interpretation of the SHG response, and the appraisal of the hypothetical susceptibility tensors assumed for domain walls or unknown materials.

 

Fig. 1. Non-Ising domain walls with Bloch (left) and Néel (right) internal polarization structure. The corresponding point group symmetry and the nonlinear optical susceptibility tensor are presented for each configuration in the case of a domain wall along $y$-axis.

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Let us consider for instance a scalar field approximation (valid at low focus values) and a colinear SHG propagation along the $z$-axis. In this case, the electric field components are within the ($xy$) plane ($E_{x} = E_{0}\cos \varphi$, $E_{y} = E_{0}\sin \varphi$, $E_{z} = 0$) and the SHG polarization dependence takes a straightforward form. In this case, e.g., an $x$-polarized light should induce a nonlinear response exclusively dependent on the $E_{x}$ component of the incident electric field. Whereas, under tight focusing conditions, which are often necessary to achieve high spatial resolution in two-photon microscopy, the fundamental light contains wave vector components that are tilted with respect to the original direction of propagation. In this case, the three field components $E_x$, $E_y$, and $E_z$ appear at the focal region, regardless of the linear $x$ or $y$ polarization of the incident FW as it will be demonstrated below.

2.2 Vectorial modeling of the electric field in the case of strongly focused light

Obviously, the second-harmonic response resulting from the focused light-matter interaction is affected by the vector field distribution. In this case, the understanding of the nonlinear optical response requires a precise modelling involving the angular spectrum representation describing the vectorial focal field distribution, and the derivation of the related SHG signal [52,59,60]. In the following, we model the electric field distribution at the focus following the theoretical treatment originally established by Wolf and Richards [61,62], followed by Mansuripur [63], Kant [64], Sheppard and Török [65], Novotny and Hecht [66]. An advanced mathematical description of vectorial beams can be found in the recent review article by Nevy, Silberberg, and Davidson [67].

We assume a linearly polarized FW propagating in the $z$ direction $\textbf E_{inc} = E_{0}(\cos \varphi, \sin \varphi,0)$. The polarization angle $\varphi$ is taken from the $x$-axis as depicted in Fig. 2(a). The FW is focused by an objective lens characterized by a focal distance $f$ and a numerical aperture NA $= \sin \theta _{max}$. We assume that the electromagnetic field close to the focal point is formed by a superposition of plane waves. Each of them corresponds to a point on the reference sphere touching the exit pupil. The position vector of a point in the unit circle is given by $\boldsymbol \rho = (\rho \cos \phi, \rho \sin \phi )$, where $\rho =\sin \theta /\sin \theta _{max}$ is the distance between the optical axis and the point on the reference sphere characterized by the polar and azimuth angles $\theta$ and $\phi$, respectively. We consider here both the beam focused by the objective lens as well as the refracted beam at surface of the material (see position $z=-d$ in Fig. 2(b)). The focal field distribution is given by the Debye-Wolf integral in polar coordinates ($\rho,z,\beta$):

The Fresnel transmission coefficients for $s$- and $p$-polarized light are defined as $t_s = \frac {2\sin \theta _2 \cos \theta _1}{\sin (\theta _1 +\theta _2)}$, and $t_p = \frac {2\sin \theta _2 \cos \theta _1}{\sin (\theta _1+\theta _2)\cos (\theta _1-\theta _2)}$ [68]. The term $e^{ik_{\omega }\Psi }$ in Eq. (6) represents the aberration function. These spherical aberrations are typically caused by the mismatch of the refractive index $n_1$ and $n_2$ (see Fig. 2). We use the phase term $\Psi = d(n_2\cos \theta ^{max}_2 - n_1\cos \theta ^{max}_1)$ representing the maximum defocus in a scanning confocal microscope due to the propagation in air (in the case of a dray objective) and in the sample [69].

 

Fig. 2. Geometrical representation of (a) the three-dimensional focus region including the focusing optics as well as the sample region, and (b) a zoomed view showing the path of the fundamental wave (FW) in the sample across different cut planes. The parameters used for the angular spectrum representation to calculate the electric field distribution at the focus ($E_{Focus}$) are depicted. The $z$-axis is along the optical axis and $d$ is the distance between the surface of the sample and the focal point. The convergence angle in air $\theta _1$, and the divergence angle in the sample $\theta _2$ are linked by the Snell’s law $n_{1}\sin \theta _{1} = n_2 \sin \theta _{2}$ , where $n_{i=1,2}$ is the fraction index of the environment. $n_\rho$, $n_\phi$, $n_{\theta _{1}}$ and $n_{\theta _2}$ are unit vectors. They can be expressed in terms of the Cartesian unit vectors $n_x$, $n_y$, $n_z$ using the spherical coordinates $\theta _1$, $\theta _2$ and $\phi$.

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The far-field $\textbf {E}_{\infty }$ at the reference sphere is defined as the incident field $E_{inc}$ refracted by the objective lens, and it writes as:

After focusing, the wavefront behind the objective shows a spherical surface. The distribution of the light field on the spherical surface is characterized by the angle of convergence $\theta _1$ through the apodization function $f_{w}(\theta _1) =e^{-\frac {1}{f_{0}^{2}}\frac {\sin ^{2}{\theta _1}}{\sin ^{2}{\theta _{max}}}}$. The filling factor of the objective is defined as $f_{0} = {w_{0}}/{f\sin \theta _{max}}$ where $w_0$ is the beam waist of the fundamental Hermite-Gaussian mode.

The general form of the focal electric field induced by the incident wave $\boldsymbol E_{inc}$ with linear polarization angle $\varphi$ can be expressed as:

Equation (8) can be further simplified by defining the integral ratios $I_{1}= \frac {I_{01}}{I_{00}}$, $I_{2}= \frac {I_{02}}{I_{00}}$. Table 1 displays the electric field vector components at the focus of an objective of NA $0.95$. The results are presented at different polarization angles $\varphi$ of the FW ($\varphi =0^{\circ }$ corresponds to $x$-polarized light). We observe that the three orthogonal components electric fields are comprised in the beam focus, in agreement with previously reported results [70], irrespective of FW polarization. Furthermore, the transverse component $|E_z|$ is quasi-constant at all $\varphi$ values, while the relative weight of the field components $|E_x|$:$|E_y|$:$|E_z|$ may vary from one configuration to another. It is also worth noting that a tilted FW polarization (see, e.g., $\varphi =\pm {60}^{\circ }$ or $\pm {30}^{\circ }$ in Table 1) results in a contribution of the same order for the three components. This means that the strongest focusing effects are anticipated at this configuration.

Table 1. Focal electric field calculated for NA 0.95 at different fundamental wave (FW) polarization angles $\varphi$ .

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2.3 Analytic form of the second-harmonic polarimetry at non-Ising domain walls in the case of strong focusing

The vectorial nature of the focused light field discussed above is obviously expected to have a direct effect on the SHG polarization response through the squared electric field components $E^{2}_{x}$, $E^{2}_{y}$, $E^{2}_{z}$, $2E_{z}E_{y}$, $2E_{x}E_{z}$, $2E_{x}E_{y}$ at play in the second-harmonic process (see Eq. (1)). Their simplified form accounting for the strongest electric field integrals (i.e., $1\gg I_{1} \gg I_{2}$) reads:

Figure 3 displays the variation of the intensity maps of the focal field components involved in the SHG response. The results are displayed for the typical FW polarization used for the experimental study of domain walls surrounding hexagonal shape domains ($\varphi =0^{\circ }$ or $90^{\circ }$ for horizontal domain walls, and $\varphi =+60^{\circ }$ or $-30^{\circ }$ for oblique domain walls). These results reveal that all the six squared components of the electric field are expected to contribute to the SHG signal when the polarisation of the FW is either parallel or perpendicular to the oblique walls of the hexagon (see $\varphi =60^{\circ }$ or $-30^{\circ }$ displayed in panels c,d of Fig. 3). In contrast to this, only four components overs six are non-zero when the FW polarization is parallel or perpendicular to domain wall aligned with $x$-axis (see $\varphi =0^{\circ }$ or $90^{\circ }$ in Fig. 3(a,b). This result suggests that the domain wall orientation is an important aspect and that oblique walls are particularly sensitive to focusing effects in SHG polarimetry.

 

Fig. 3. Contour plots representing the spacial distribution of the squared focal field components $E_iE_j$ calculated at different FW polarization (a) $\varphi =0^{\circ }$, (b) $\varphi =90^{\circ }$, (c) $\varphi =60^{\circ }$, (d) $\varphi =-30^{\circ }$. The intensity is normalized to the maximum of the $E{_i}E{_j}$ series at a given FW polarization

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It is however worth noting that a discussion exclusively based on the vectorial form of the electric field is not sufficient to understand how the SHG polarimetry is affected by focusing. Additional factors are expected to also enhance or reduce the impact of the vector field on the polarimetry response. Among these factors, the susceptibility tensor can be regarded as a "mixing desk", capable of eliminating or enhancing the $E_iE_j$ terms at play in the SHG process by accounting for the material structure, symmetry, and optical properties. As previously stated, non-Ising domain walls are optical nanomaterials holding a symmetry different from that of the parent material in which they develop (i.e., different from that of the adjacent domains). Let us then see how the susceptibility tensors associated to Néel and Bloch type domain walls can modify the effect of focusing on SHG polarimetry distortions. We assume for instance domain walls oriented parallel to $x$-axis of the laboratory coordinates system (corresponding to $\delta =0^{\circ }$ in Eqs. (3 and 4). For $x$-polarized light, P$^{2\omega }$ reads:

It is important to bear in mind that the $d$-tensor given above for Bloch and Néel walls is not only an intrinsic material property but, in addition to this, it accounts for the geometry via the $\delta$ angle in Eqs. (3 and 4). Crystallographically equivalent horizontal, vertical or oblique walls (e.g., in hexagonal-shape domains) may thus show different SHG distortions, depending on their orientation with respect to the laboratory coordinates system.

3. Results and discussion

Having delineated the main factors at play in SHG polarimetry, we will now move on to the quantitative analysis of the factors affecting the SHG distortion at non-Ising domain walls in the case of focused light. Here, we emphasize the effect of focusing on the SHG polarimetry anisotropy in $xy$-plane. Even if quantitative SHG analysis is beyond the scope of this work, it is worth noting that tight focusing is also expected to affect the intensity of the SHG and its dependence on the phase mismatch between the fundamental and the second-harmonic waves. Such quantitative analysis would, however, require a three-dimensional analysis of the focusing characteristics taking into account the thickness of the studied sample (bulk versus film), the focus position, and the length of the focal region [71]. In the following, the polar plots presenting the variation of the SHG intensity with the analyzer angle $\alpha$ at a given FW polarization $\varphi$ are referred to as SHG polarimetry. The SHG polarimetry distortion due to focusing is characterized by the rotation and the broadening of these polar plots with respect to the non-focusing (scalar field approximation) reference polar plot. We study the effect of the intrinsic parameters related to the optical properties of the domain walls (anisotropy factors and birefringence), and those controlled by extrinsic parameters related to the measurement geometry. We consider that there is no second-harmonic emission from the adjacent domains which is often the case in the experimental conditions chosen for polarimetry experiments on domain walls.

3.1 Role of extrinsic parameters

The polarization of the incident laser beam is certainly the most obvious extrinsic parameters affecting the SHG polarization response. As shown above, it controls the weight of the vector field components (see Table 1) as well as the number of the mixed electric field terms $E_iE_j$ causing distortions in SHG polarimetry (see Fig. 3). Yet, to study the internal dipole perpendicular to the domain wall (Néel) or parallel to it (Bloch), it is usually necessary to align the FW polarized either parallel or perpendicular to the wall in SHG experiments. Therefore, at a fixed FW polarization, the domain wall orientation becomes the main extrinsic parameter describing the measurement geometry. We use as a model system non-Ising $180^{\circ }$ domain walls surrounding hexagonal shaped c-domains which are typically found in trigonal uniaxial ferroelectrics. This system is a rich playground for geometry studies as it combines horizontal (HDWs) and oblique domain walls (ODWs) at $\pm 60^{\circ }$, or vertical (VDWs) and oblique domain walls (ODWs) at $\pm 30^{\circ }$, depending on the crystal orientation $(X,Y,Z)$ with respect to the laboratory coordinates system $(x,y,z)$ (see Fig. 4). The SHG intensity at each wall type and geometry is calculated based on the analytic form given in Table S2. We use the following susceptibility tensor coefficients $d_{22}:d_{21}:d_{16}:d_{14}:d_{25}:d_{23}=1:0.6:-6:6:6:6$ for Bloch walls; and $d_{12}:d_{11}:d_{26}:d_{24}:d_{15}:d_{13}=1:0.6:-6:6:6:6$ for Néel walls.

 

Fig. 4. Simulation of the local SHG intensity variation at non-Ising $180^{\circ }$ domain walls for (a) $Y-$axis of the walls aligned along the $x-$axis of the laboratory coordinates system and (b) Bloch, or (c) Néel internal structure. The results are also displayed in the case where (e) the $Y-$axis of the walls is aligned along the $y-$axis of the laboratory coordinates system and (f) Bloch, or (g) Néel internal structure. All the walls are cristallographically equivalent and of $Y-$type. The red arrows represent the polarization of the FW. It is either parallel or perpendicular to the walls. The simulations are conducted using the analytic form of the SHG signal in the common scalar model approximation (black curve), and using a vectorial modeling accounting for the vector character of the focused light in the case of low focusing (NA 0.65, blue curve) and high focusing (N.A. 0.95, red curve).

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Oblique domain walls (labeled ODW$1$ and ODW$2$ in Fig. 4) show much stronger SHG distortions (i.e., rotation and widening of the blue and red curves w.r.t. the scalar model) than HDWs and VDWs, irrespective of their Néel or Bloch internal structure, and regardless of the FW polarization. This configuration should thus be avoided, even in the case of low NA. The distortions are clearly less pronounced for HDW and VDWs at which the scalar model (black curves in Fig. 4) and the low focus case (blue curves in Fig. 4) coincide. Surprisingly, the walls parallel to $x-$axis (HDWs) or $y-$axis (VDWs) exhibit distortions only when the FW polarization is parallel to the walls (see red curves in 4). Interestingly, quasi-identical SHG polarimentry responses are obtained at HDWs and VDWs when the input laser polarization is perpendicular to the domain walls (i.e., $\varphi =90^{\circ }$ in HDWs and $\varphi =0^{\circ }$ in VDW), independently on the focusing conditions and the Bloch or Néel character of the domain walls.

These results show that the measurement geometry has a significant impact on focusing effects. Nevertheless, it is possible to identify specific domain wall orientations and polarizer angles at which focusing effects are negligible. It is also worth noting that we have deliberately presented the variation of the SHG intensity as a function of the analyzer instead of a variation with the polarizer angle as it is often presented in the literature. As indicated previously, the relative weight of the focal field components changes with the polarization of the fundamental wave. A rotation of the input polarization angle, $\varphi$, results in strong variations of the degree of the SHG distortion within the same curves. Therefore, we recommend the use of SHG polar plots based on the output SHG analyzer rotation at a fixed—and wisely chosen—input polarization.

3.2 Role of intrinsic parameters

3.2.1 Effect of the domain walls optical anisotropy factors

In the following, we investigate the effect of the nonlinear optical parameters of the domain walls on the distortion of the SHG polarimetry response at different focus conditions. We take as an example the case of a Bloch-type domain wall oriented parallel to $x-$axis as depicted in Fig. 5(a,e). The SHG polar plots are simulated at different NA values using both the scalar approximation (reference signal) and the vectorial model accounting for focusing effects. The validity of the scalar description of the focused light field at the domain walls is characterised by the rotation of the SHG polar plots accounting for the vector character of the electric field with respect to those derived from scalar model. Given that the polar plots expected at Néel and Bloch-type walls are $90^{\circ }$ away from each other [46,48], a rotation of $10^{\circ }$ or smaller between the scalar and vectorial models does not affect the conclusions of the SHG polarimetry study. We can thus consider that the scalar model is a good approximation, and it can be used safely. To simplify the discussion, we introduce the intrinsic anisotropy factors $D_i$ corresponding to the normalized nonlinear optical susceptibility factors: $D_1= \frac {d_{21}}{d_{22}}$, $D_2=\frac {d_{16}}{d_{22}}$, $D_3 = \frac {d_{14}}{d_{22}}$, $D_4=\frac {d_{25}}{d_{22}}$, and $D_5=\frac {d_{23}}{d_{22}}$. The rotation angles are presented in Fig. 5(b-d,f-h) as contour plots (red and blue colors correspond to the maximum and minimum rotation, respectively) as a function of the anisotropy factor values, $D_i$. The analytic form of SHG accounting for the vector character of the focal field involves $D_2$ and $D_3$ at a FW polarisation along $x-$axis, while for a polarization perpendicular to the wall, $D_2$ and $D_4$ are involved (see Table S2). Figure 5 illustrates the rotation of the SHG polar plots due to the vector character of the electric field at different optical anisotropy factors $0\leq D_i\leq 10$ and focusing conditions $0.65\leq NA\leq 0.95$. The lowest NA $0.65$ leads to a negligible rotation of less than $3^{\circ }$ (see Fig. 5(b,f)) independently of the intrinsic optical anisotropy factors. This result was expected given that the scalar approximation is known to apply at NA $\leq 0.7$ [51]. Interestingly, the regions presented in blue-greenish colour in Fig. 5(c,d,g,h) are characteristic for low variations with respect to the scalar model in spite of the high focusing (NA 0.85 and 0.95). When the FW polarization is along the HDW (see Fig. 5 a), an acceptable rotation of 10$^{\circ }$ or less is obtained at NA $0.85$ when $D_3 < 8$, and at NA $0.95$ when $D_3 < 6$. Even less restrictive values ($D_2 < 9$ for NA $0.85$, and $D_2 < 8$ for NA $0.95$) are found when the FW polarization is perpendicular to the domain walls (see Fig. 5(h,g)). This result is better supported by the SHG polar plots (see Fig. 5(j,k,m,n)) corresponding to the points designated by the letters "A" $(|D_3|=|D_2|=6)$, "B" $(|D_3|=|D_2|=8)$, "C" $(|D_4|=|D_2|=6)$, and "D" $(|D_4|=|D_2|=8)$ in the contour plots. Note that both senses of rotation of the polar plots, clockwise and counterclockwise, are possible. The senses of the rotation depends on the sign of the optical anisotropy factors as shown in Figure S1.

 

Fig. 5. Contour plots showing the rotation of the SHG anisotropy due to focusing as a function of the intrinsic optical anisotropy factors of Bloch-type domain walls. We consider horizontal domain walls oriented along $x-$axis. The results are displayed in the case a FW polarization parallel to the wall as schematically shown in panel (a), for: (b) NA $0.65$, (c) NA $0.85$, and (d) NA $0.95$; and in the case a FW polarization perpendicular as shown in panel (e), for: (f) NA $0.65$, (g) NA $0.85$, and (h) NA $0.95$. The SHG polar plots are displayed for selected points labeled (j)"A", (k) "B", (m) "C", and (n) "D" in the contour plots. Red arrows represent the polarization of the FW.

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3.2.2 Focusing effects at birefringence non-Ising domain walls

Birefringence, $\Delta$n, is an intrinsic property that makes the material capable of splitting light into ordinary and extraordinary rays. This can cause both defocusing (i.e., aberrations causing distortion and broadening of the focal point) [72,73], as well as a phase shift of the polarized light propagating in the media with tilted optical axes [56]. The effect of birefringence on the electric field distribution at the focal point is often neglected for sake of simplicity.

Here we show that strong focusing have two major effects in birefringent media. First, the electric field integrals given by Eqs. (9), (10, and 11) are significantly affected by the aberrations induced by birefringence. Second, strong focusing results in a rotation of the polarization of the focused light with respect to the optical axes. The local SHG emission resulting from the as-modified focal electric field will experience a phase shifted between the optical axes. Birefringence is thus expected to induce supplementary distortions of the SHG polarimetry in the case of strong focusing. Both effects are expected to modify the SHG polarimetry distortions.

Let us first address the effect of birefringence on the alteration of the focusing properties of light. In uniaxial birefringent media, light with $s$- or $p$-polarization propagating along the ordinary and extraordinary axes (associated to the refractive indices $n_o$ and $n_e$, respectively) experience different speeds. This induces a phase shift between them expressed as [72]:

Table S3 summarizes the focused field integrals calculated at different birefringence values. The obtained results show that the approximation $I_{00}\gg I_{01} \gg I_{02}$, generally assumed to simplify the expression of the vectorial field [52], is not valid in the case of strongly birefringent media. Besides the alteration of the focusing properties of light through aberrations, birefringence is also expected to modify the polarization of the electric field at focal point. The SHG polarization will be consequently affected by both effects.

 

Fig. 6. Second-harmonic polar plots derived for a horizontal wall (parallel to x-axis) as schematically displayed in panel (a) with: (b) Bloch, or (c) Néel internal structure. The results are displayed in the case of low focus (NA 0.65) and high focus (NA 0.95) at different values of the birefringence $0\leq \Delta n\leq 0.015$.

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Domain walls are expected to show a uniaxial birefringence with the extraordinary axis parallel to the walls, and the ordinary axis perpendicular to them [36,74]. Let us consider for instance a FW polarization along the optical axes, i.e., parallel or perpendicular to the walls as depicted in Fig. 6(a). The incident light as well as the SHG should not split into ordinary and extraordinary rays in this configuration. However, due to the focused vector field causing a polarization rotation of the FW with respect to the optical axes, the resulting second-harmonic polarization $\boldsymbol P^{\Delta n}$ shows a phase shift $\Phi (z) = zk_\omega \Delta n$ at the focal position $z$ between the extraordinary axis, characterized by the wall angle $\delta$, and the ordinary axis. The new expression of the SHG polarization state accounting for the birefringence in the case of focused light reads as:

The integral ratios $I_{1}= \frac {I_{01}}{I_{00}}$, $I_{2}= \frac {I_{02}}{I_{00}}$ are calculated based on Eqs. (17), (18, and 19) to take into account the aberrations due to birefringence. The related analytic form of SHG is derived for Bloch and Néel-type birefringent domain walls (see Table S2).

Figure 6 displays the variation of the SHG polar plots with the birefringence, $\Delta n$, at low and high focus (NA $0.65$ and NA $0.95$) conditions in horizontal Bloch and Néel walls. We consider a fixed phase in this comparative analysis corresponding to a fixed depth $d=60\mu m$. We observe that at low focus conditions, SHG polarimetry is not strongly affected by birefringence. Note that this is the case only if the incident polarization of the FW is aligned with the optical axes. Strong variations are observed otherwise, as demonstrated by Aït-Belkacem et al. [56]. In the case of strong focusing, the simulations show a large rotation at $\Delta n\geq 0.015$, even if an optimum geometry is chosen (e.g., domain wall along $x$-axis and FW polarized along $y$-axis), and irrespective of the Néel or Bloch-type of the domain wall.

4. Conclusions

We provide an analytic modeling of the SHG response of non-Ising polar domain walls taking into account the vector character of focused light as well as the local symmetry of the Néel and Bloch-type walls. The as-derived model is applied to assess the validity of the paraxial approximation to determine the internal structure of non-Ising domain walls from SHG polarimetry data. This evaluation is conducted by comparing the results derived from the scalar and vectorial models. We show that the deviation of the two models is not only dependent on the strength of the focusing parameters. The domain walls’ extrinsic and intrinsic parameters are found to play a major role. Oblique walls produce large distortions, even at low focusing, while a perfect alignment of the walls with $x$ or $y$-axis combined to an orthogonal polarization of the FW annuls the distortions. It is shown that the combination of optimum geometry with low birefringence and nonlinear optical anisotropy factors can sensibly decrease focusing effects in SHG polarimetry. In summary, this study offers a complete methodology to determine the internal structure of non-Ising domain walls, and it emphasis the precautions to be taken into account in the case of confined optical fields. Besides, we anticipate that polar domain walls can be applied as versatile diagnostic tool to map the focusing characteristic of complex vector beams in a similar way as single molecules [75,76]. The polarization of the incident light is also capable of enhancing the performance of focusing characteristics. In particular, radial polarization has been proven to produce sharper focus [77] providing new capabilities for nonlinear microscopy [78].