## Abstract

Compressing the optical field to the atomic scale opens up possibilities for directly observing individual molecules, offering innovative imaging and research tools for both physical and life sciences. However, the diffraction limit imposes a fundamental constraint on how much the optical field can be compressed, based on the achievable photon momentum^{1,2}. In contrast to dielectric structures, plasmonics offer superior field confinement by coupling the light field with the oscillations of free electrons in metals^{3,4,5,6}. Nevertheless, plasmonics suffer from inherent ohmic loss, leading to heat generation, increased power consumption and limitations on the coherence time of plasmonic devices^{7,8}. Here we propose and demonstrate singular dielectric nanolasers showing a mode volume that breaks the optical diffraction limit. Derived from Maxwell’s equations, we discover that the electric-field singularity sustained in a dielectric bowtie nanoantenna originates from divergence of momentum. The singular dielectric nanolaser is constructed by integrating a dielectric bowtie nanoantenna into the centre of a twisted lattice nanocavity. The synergistic integration surpasses the diffraction limit, enabling the singular dielectric nanolaser to achieve an ultrasmall mode volume of about 0.0005 *λ*^{3} (*λ*, free-space wavelength), along with an exceptionally small feature size at the 1-nanometre scale. To fabricate the required dielectric bowtie nanoantenna with a single-nanometre gap, we develop a two-step process involving etching and atomic deposition. Our research showcases the ability to achieve atomic-scale field localization in laser devices, paving the way for ultra-precise measurements, super-resolution imaging, ultra-efficient computing and communication, and the exploration of light–matter interactions within the realm of extreme optical field localization.

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## Data availability

We declare that the data supporting the findings of this study are available within the paper.

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## Acknowledgements

This work is supported by the National Natural Science Foundation of China (grant numbers 12225402, 62321004 and 92250302), the national Key R&D Program of China (grant numbers 2022YFA1404700 and 2018YFA0704401), the Beijing Natural Science Foundation (Z180011) and the New Cornerstone Science Foundation through the XPLORER PRIZE. We thank Peking Nanofab and National Center for Nanoscience and Technology for fabrication assistance.

## Author information

### Authors and Affiliations

### Contributions

R.-M.M. conceived the concept and supervised the project. Y.-H.O., H.-Y.L. and W.-Z.M. performed the optical characterization. H.-Y.L. carried out the theoretical analysis. Z.-W.Z. fabricated the devices. H.-Y.L. and Y.-H.O. conducted the numerical simulations. R.-M.M., H.-Y.L. and Y.-H.O. did the data analysis. R.-M.M. wrote the paper.

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## Extended data figures and tables

### Extended Data Fig. 1 Theoretical analysis of the infinite singularity in a singular dielectric nanolaser.

**a**, Schematic of a dielectric bowtie nanoantenna with a negligible gap between its two apices. **b**, Theoretical calculated electric field intensity of the dielectric bowtie nanoantenna. **c**, Intensity profiles along \(\varphi ={\rm{\pi }}/2\) in (**b**), where the intensity is normalized to the position 0.1 nm away from the singularity. The field diminishes to a very small magnitude over a distance substantially shorter than the free-space wavelength due to the large \({k}_{\rho }\) originated from infinite singularity. **d,e**, Phases \({\varPhi }_{+}\) (**d**) and \({\varPhi }_{-}\) (**e**) of theoretical calculated electric field components \({{\bf{E}}}_{\pm }\) represented as \({{\bf{E}}}_{\pm }={E}_{\pm }\left({{\bf{e}}}_{x}\pm {\rm{i}}{{\bf{e}}}_{y}\right)\), where \({E}_{\pm }=\left|{E}_{\pm }\right|{{\rm{e}}}^{{\rm{i}}{\varPhi }_{\pm }}\). The non-integral value of the topological charge *l* of 0.62 stems from angular discontinuities in the dielectric constant, inducing phase jumps at the boundaries between the dielectric and air. **f**, Phase changes of \({{\bf{E}}}_{\pm }\) along a circle enclosing the apices. The integral of the phase change along the circle for both \({{\bf{E}}}_{\pm }\) yields an integer value of zero, fulfilling the periodic boundary condition.

### Extended Data Fig. 2 Fabrication procedure of singular dielectric nanolasers with atomic-scale nanoantenna gap.

**a**, Six essential steps in fabrication singular dielectric nanolaser with atomic-scale nanoantenna gap. **b-d**, Schematic diagram (**b**), and two zoomed-in version (**c,d**) of the final device.

### Extended Data Fig. 3 Dark-field scanning transmission electron microscopy (STEM) images of two fabricated nanoantennas.

**a,b**, Dark-field STEM image (**a**) and the enlarged image (**b**) of a dielectric bowtie nanoantenna with gap size that is near closure. **c,d**, Dark-field STEM image (**c**) and the enlarged image (**d**) of a dielectric bowtie nanoantenna with gap size that is close to a single nanometer. In the images, the distinction between TiO_{2} and InGaAsP materials is clearly noticeable; TiO_{2} appears darker in contrast to InGaAsP.

### Extended Data Fig. 4 Proportion of energy, field distribution and mode volume with varied air gap size by TiO_{2} deposition.

**a**, Proportion of energy within the air gap and TiO_{2} spacer area, with a diameter of 50 nm, relative to the total mode energy for varied gap sizes. **b**, Electric field intensity at varied gap sizes, demonstrating that a decrease in gap size results in more localized electric field intensity. **c**, Mode volume at varied gap sizes. The reduction of mode volume with the decrease of the gap size is due to the redistribution of energy around the air gap area caused by the alteration in gap size as shown in (**b**). **d**, Proportion of energy within the gain area relative to the total mode energy for varied gap sizes.

### Extended Data Fig. 5 Gap sizes determination.

The two edges of the air gap are modeled using two Gaussian functions that describe the blur of the edges induced by SEM resolution, which is: \(\frac{{\rm{d}}I}{{\rm{d}}x}=\sqrt{\frac{2}{{\rm{\pi }}}}\frac{A}{\sigma }\left(-\exp \left(-\frac{{\left(x-{x}_{1}\right)}^{2}}{2{\sigma }^{2}}\right)+\exp \left(-\frac{{\left(x-{x}_{2}\right)}^{2}}{2{\sigma }^{2}}\right)\right)\), where \({x}_{1}\), \({x}_{2}\) denote the position of the two edges, and *σ* is the standard deviation of the edge positions of \({x}_{1}\), \({x}_{2}\). By integrating the first derivative of the intensity profile, we obtain the expression for the intensity profile, *I*, as: \(I={I}_{0}+A\left(-{\rm{erf}}\left(\frac{x-{x}_{1}}{\sqrt{2}\sigma }\right)+{\rm{erf}}\left(\frac{x-{x}_{2}}{\sqrt{2}\sigma }\right)\right)\). This function was employed to fit the SEM image intensity profile to ascertain the values of \({x}_{1}\), \({x}_{2}\), and *σ*. **a,b**, For the device shown in Fig. 2g, the gap size, calculated as \({x}_{2}-{x}_{1}\), with an associated standard deviation of \(\sqrt{2}\sigma \), is determined to be (1.7 ± 1.0) nm. **c,d**, For the device shown in Fig. 2h, the gap size is determined to be (3.7 ± 1.2) nm.

### Extended Data Fig. 6 Nanocavities with tilt angles and point defects.

**a,b**, Field distributions of nanocavities with tilt angles of 0.26 degrees (**a**) and 1.3 degrees (**b**), respectively. The corresponding mode volumes are 0.00052 *λ*^{3} and 0.00074 *λ*^{3}. **c,d**, Field distributions of nanocavities with point defects with characteristic sizes of 1 nm (**c**) and 2 nm (**d**), respectively. The corresponding mode volumes are 0.00013 *λ*^{3} and 0.00030 *λ*^{3}.

### Extended Data Fig. 7 Fabrication precision and nanocavity with surface roughness.

**a,b**, Surface roughnesses induced by ICP etching and ALD as observed in the nanoantenna shown in Extended Data Fig. 3c,d. The root mean square (RMS) roughness from ICP etching is 1.41 nm. The surface roughness characteristic post-ALD follows the trend set by ICP etching, indicating that the ALD process forms a smooth, conformal TiO_{2} layer without introducing additional roughness. **c,d**, Field distributions of nanocavities with (**c**) and without (**d**) surface roughnesses. The corresponding mode volumes are 0.00041 *λ*^{3} and 0.00037 *λ*^{3}.

### Extended Data Fig. 8 Deterministic relationship between nanoantenna’s orientation and lasing emission polarization.

Panels (**a–e**) present a progression from left to right, showing the influence of five distinct nanoantenna orientations on the emission polarization of singular dielectric nanolasers. The top row shows SEM images, the middle row shows full-wave simulated electric field intensity distributions and polarization directions (indicated by arrows) within the cavity, and the bottom row shows experimentally measured lasing emission polarizations. Both the full-wave simulation and experimental findings demonstrate that the polarization direction of the lasing mode adjusts with the nanoantenna’s rotation angle.

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Ouyang, YH., Luan, HY., Zhao, ZW. *et al.* Singular dielectric nanolaser with atomic-scale field localization.
*Nature* **632**, 287–293 (2024). https://doi.org/10.1038/s41586-024-07674-9

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DOI: https://doi.org/10.1038/s41586-024-07674-9

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