Ferroics, materials with switchable spontaneous ordering, have seen the emergence of ferroaxial order—a rotational structural distortion in crystals. This distortion acts as a vortex of electric dipole moments, defined by the ferroaxial moment A = Σi ri × pi, where ri is the position vector of electric dipole pi. While A is symmetric under time reversal and space inversion, it breaks mirror symmetry containing the rotation axis, creating domains with opposite A signs. Unlike chirality, which uses optical activity to identify domain states, identifying ferroaxial domain states has been challenging. Recent advancements used rotational anisotropy second harmonic generation (SHG) and electrogyration (EG) to demonstrate ferroaxial order domains. Ferroaxial order shows potential in unconventional physics and functionalities, including transverse responses where external fields induce conjugate quantities perpendicular to A and the field, and thermally-induced electric polarization perpendicular to a thermal gradient. This study uses CID-SHG to image ferroaxial domains. SHG, sensitive to symmetry breaking, usually involves electric-dipole (ED) contributions, forbidden in centrosymmetric systems. However, higher-order multipole contributions (magnetic-dipole (MD) and electric-quadrupole (EQ)) allow SHG in centrosymmetric materials. Previous work used rotational anisotropy SHG (EQ contribution) to study ferroaxial order. This study uses CID-SHG in NiTiO3, a centrosymmetric order-disorder ferroaxial crystal, drawing inspiration from SHG observations in MnTiO3 (MD contribution). The goal is to achieve three-dimensional ferroaxial domain imaging with sub-micrometer resolution.

The introduction extensively reviews the literature on ferroaxial order, highlighting its novelty as a ferroic order, its characterization through the ferroaxial moment (A), and the challenges in its experimental detection compared to chiral systems. The authors cite key works demonstrating the experimental observation of ferroaxial domains using rotational anisotropy SHG and electrogyration. They also mention the potential applications of ferroaxial materials, focusing on transverse responses and thermally induced polarization. The review of SHG emphasizes its sensitivity to symmetry breaking and its common reliance on ED contributions. The limitations of ED-based SHG in centrosymmetric systems are discussed, along with the allowance of higher-order multipole contributions (MD and EQ). Prior studies utilizing rotational anisotropy SHG in relation to ferroaxial order are referenced, setting the stage for the authors’ proposed method of using CID-SHG.

The study focuses on NiTiO3, an ilmenite-type crystal with ferroaxial point group 3. The authors derive the CID-SHG formulation for this point group, considering MD and EQ contributions. They simplify the derivation by focusing on the MD contribution, resulting in equations for the nonlinear magnetization (M) induced by MD and the source term (S) in the electric field wave equation. The expressions are derived for SHG intensity considering complex susceptibility tensors near optical resonance. The influence of ferroaxial domains (A+ and A−) on SH intensity is then discussed, showing that the mirror operation converting A+ and A− domains changes the sign of χ(2) but not χ(1), leading to different SH intensities for A+ and A− domains depending on the circular polarization of incident light. The methodology then details the CID-SHG measurements using SHG microscopy with circularly polarized fundamental light (1200 nm wavelength) on NiTiO3 samples. Multi-domain states were prepared by annealing. The experimental setup involved a femtosecond optical parametric oscillator, an attenuator, a quarter-wave plate for circular polarization control, confocal microscopy, and a point detector for SH intensity measurements. Two-dimensional images were obtained by scanning, and three-dimensional images were constructed by depth scanning. The spatial resolution is around 0.56 μm laterally and 3 μm in depth. A comparison with electrogyration (EG) measurements is also included to validate the CID-SHG results.

The key findings center around the successful three-dimensional visualization of ferroaxial domains in NiTiO3 using CID-SHG microscopy. Two-dimensional SHG images obtained using right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) light show clear contrast reversal between bright and dark regions, corresponding to A+ and A− domains. The normalized CID-SHG (ΔI/IAVE) is significant (0.6–1.3), indicating strong circular polarization dependence. The wavelength dependence of ΔI/IAVE reveals a decrease with increasing wavelength, related to the Ni2+ d-d transition. Comparison with EG measurements confirms that the bright and dark regions in CID-SHG images accurately reflect the ferroaxial domain states. The CID-SHG method exhibits a large signal difference between A+ and A− domains (32%), facilitating easy domain distinction. Three-dimensional imaging reveals nearly constant SH activity with depth, ruling out surface effects. Analysis of domain boundaries reveals SH intensity enhancement regardless of light polarization, possibly due to the flexoelectric effect or nonlinear Čerenkov SHG. Fringe-like patterns at some boundaries are linked to tilted domain boundaries. These findings establish CID-SHG as an effective technique for three-dimensional ferroaxial domain visualization.

The observed CID-SHG in NiTiO3 is qualitatively explained by the MD and EQ multipole transitions. The large CID-SHG contrast facilitates three-dimensional visualization of ferroaxial domains, superior to techniques like rotational anisotropy SHG and EG due to higher spatial resolution and the elimination of electrode requirements. The intensity enhancement at domain boundaries is discussed, with flexoelectricity and nonlinear Čerenkov SHG as possible explanations, warranting further investigation. The study’s limitations include the need for more detailed investigation into the origin of SHG enhancement at domain boundaries and the current qualitative interpretation of the experimental results.

This study successfully demonstrates the use of CID-SHG microscopy for three-dimensional visualization of ferroaxial domains in NiTiO3. The significant CID-SHG contrast between A+ and A− domains, along with the three-dimensional imaging capability, showcases the method's advantages over existing techniques. Future work should focus on a more quantitative analysis of the observed phenomena, including detailed investigation of the domain boundary effects and exploring applications of CID-SHG to other ferroaxial materials.

While the study successfully visualizes ferroaxial domains, a more quantitative model is needed to fully explain the observed CID-SHG. The origin of the SHG enhancement at domain boundaries requires further investigation. The study focuses on NiTiO3; the generalizability of the findings to other ferroaxial materials needs to be explored.