Simulation of tip-sample interaction in s-SNOM

Here we describe how to simulate or calculate near-field scattering signal using different type of methods: Full-wave simulations, Electrostatic simulations, and Analytical modelings. Full-wave simulations include several different approaches: finite-element method (FEM), finite-difference time-domain (FDTD), boundary element method (BEM), and so on. Analytical modelings include Point-dipole model, Finite-dipole model, Lightning-rod model, Spectral method, and cone model. Please see details below. Our review in Advanced Materials, 1804774 (2019) is also a good resource. More studies using the machine Learning methods are also coming! (contact Xinzhong Chen and Mengkun Liu for more information.)

Full-wave simulations (including FEM, FDTD, BEM etc.)

•Full-wave solvers solve the full Maxwell’s equations.

•Different approaches: finite-element method (FEM), finite-difference time-domain (FDTD), boundary element method (BEM), and so on.

•Solutions are exact but time-efficiency usually is low.

•Supported in popular commercial solvers such as COMSOL Multiphysics, CST Microwave Studio, Lumerical FDTD, FEKO, and so on.

•Examples of using full-wave simulations for s-SNOM:

P. McArdle, D. J. Lahneman, A. Biswas, F. Keilmann, and M. M. Qazilbash, "Near-field infrared nanospectroscopy of surface phonon-polariton resonances," Phys. Rev. Res. 2(2), 023272 (2020).

F. Mooshammer, M. Plankl, T. Siday, M. Zizlsperger, F. Sandner, R. Vitalone, R. Jing, M. A. Huber, D. N. Basov, and R. Huber, "Quantitative terahertz emission nanoscopy with multiresonant near-field probes," Opt. Lett. 46(15), 3572 (2021).

F. Mooshammer, M. A. Huber, F. Sandner, M. Plankl, M. Zizlsperger, and R. Huber, "Quantifying Nanoscale Electromagnetic Fields in Near-Field Microscopy by Fourier Demodulation Analysis," ACS Photonics 7(2), 344–351 (2020).

X. Chen, C. F. B. Lo, W. Zheng, H. Hu, Q. Dai, and M. Liu, "Rigorous numerical modeling of scattering-type scanning near-field optical microscopy and spectroscopy," Appl. Phys. Lett. 111(22), 223110 (2017).

C. Maissen, S. Chen, E. Nikulina, A. Govyadinov, and R. Hillenbrand, "Probes for Ultrasensitive THz Nanoscopy," ACS Photonics 6(5), 1279–1288 (2019).

S. Mastel, A. A. Govyadinov, C. Maissen, A. Chuvilin, A. Berger, and R. Hillenbrand, "Understanding the Image Contrast of Material Boundaries in IR Nanoscopy Reaching 5 nm Spatial Resolution," ACS Photonics 5(8), 3372–3378 (2018).

Finite-element method (FEM)

•FEM is a popular and versatile method for solving PDEs.

•For electromagnetic simulations, FEM is usually performed in frequency domain. To obtain a spectrum, simulations have to be performed at each discrete frequency point so can be time-consuming for spectroscopic simulations.

C. Maissen, S. Chen, E. Nikulina, A. Govyadinov, and R. Hillenbrand, "Probes for Ultrasensitive THz Nanoscopy," ACS Photonics 6(5), 1279–1288 (2019).

Finite-difference time-domain (FDTD)

•FDTD solves the Maxwell’s equations in time-domain (propagation of a pulse). Therefore, a broad spectrum can be simulated at once.

Boundary element method (BEM)

•BEM only uses discretization on the surfaces of the objects. For electromagnetic simulations it’s often referred to as the Method of Moment (MoM).

•For s-SNOM simulations this is especially beneficial since the surface/volume ratio is usually low in the tip-sample setting. Therefore, good time-efficiency can be achieved.

•However, MoM suffers from certain constraints. When the sample surface gets complicated the time-efficiency suffers as well.

P. McArdle, D. J. Lahneman, A. Biswas, F. Keilmann, and M. M. Qazilbash, "Near-field infrared nanospectroscopy of surface phonon-polariton resonances," Phys. Rev. Res. 2(2), 023272 (2020).

R. Ren, X. Chen, M. Liu, High-efficiency scattering probe design for s-polarized near-field microscopy. Appl. Phys. Express. 14, 022002 (2021).

M. Dapolito, X. Chen, C. Li, M. Tsuneto, S. Zhang, X. Du, M, Liu, A. Gozar, Scattering-type scanning near-field optical microscopy with Akiyama piezo-probes. Appl. Phys. Lett. 120, 000000 (2022).

Electrostatic simulations

•(Quasi) Electrostatic simulations only solve Gauss’s law, making the simulation much easier.

•Far-field effect is not considered. (e.g. Metamaterial mode not applicable)

•Only strictly valid when the object of interest is much smaller than the wavelength.

X. Chen, Z. Yao, S. G. Stanciu, D. N. Basov, R. Hillenbrand, and M. Liu, "Rapid simulations of hyperspectral near-field images of three-dimensional heterogeneous surfaces," Opt. Express 29(24), 39648 (2021).

X. Chen, Z. Yao, Z. Sun, S. G. Stanciu, D. N. Basov, R. Hillenbrand, and M. Liu, "Rapid simulations of hyperspectral near-field images of three-dimensional heterogeneous surfaces – part II," (not published yet)

Analytical modeling

•Analytical modeling relies on physical laws instead of numerical schemes. Computation time is mostly instantaneous.

•Due to the complexity, tip is usually approximated with simple geometries.

•Difficult to consider arbitrary structures.

B. Knoll and F. Keilmann, "Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy," Opt. Commun. 182(4–6), 321–328 (2000). (Point-dipole model)

A. Cvitkovic, N. Ocelic, and R. Hillenbrand, "Analytical model for quantitative prediction of material contrasts in scattering-type near-field optical microscopy," Opt. Express 15(14), 8550 (2007). (Finite-dipole model)

J. Aizpurua, T. Taubner, F. J. García de Abajo, M. Brehm, and R. Hillenbrand, "Substrate-enhanced infrared near-field spectroscopy," Opt. Express 16(3), 1529–1545 (2008). (Point-dipole model for multi-layer)

B. Hauer, A. P. Engelhardt, and T. Taubner, "Quasi-analytical model for scattering infrared near-field microscopy on layered systems," Opt. Express 20(12), 13173 (2012). (Finite-dipole model for multi-layer)

A. S. McLeod, P. Kelly, M. D. Goldflam, Z. Gainsforth, A. J. Westphal, G. Dominguez, M. H. Thiemens, M. M. Fogler, and D. N. Basov, "Model for quantitative tip-enhanced spectroscopy and the extraction of nanoscale-resolved optical constants," Phys. Rev. B 90(8), 085136 (2014). (Lightning-rod model)

B.-Y. Jiang, L. M. Zhang, A. H. Castro Neto, D. N. Basov, and M. M. Fogler, "Generalized spectral method for near-field optical microscopy," J. Appl. Phys. 119(5), 054305 (2016). (Spectral method) --> highly recommended for more experienced readers/researchers in the field of s-SNOM

S. T. Chui, X. Chen, M. Liu, Z. Lin, and J. Zi, "Scattering of electromagnetic waves from a cone with conformal mapping: Application to scanning near-field optical microscope," Phys. Rev. B 97(8), 081406 (2018). (Cone model)