Describing resonance in multi-layered spherical particles

New mathematical expressions make it far easier to model the resonant frequencies of multi-layer spherical particles

New York | Heidelberg, 19 August 2025

Multi-layered spherical particles feature an intricate relationship between their physical structure and resonant frequency. Named the ‘morphology-dependent resonance’ (MDR), this relationship is highly sensitive to a variety of properties, making it especially useful in measurement.

However, since these relationships are so complex, most existing models of MDRs will only focus on the outermost layers of these particles, severely limiting their accuracy.

Through new research published in EPJ D, Lufang Guo and colleagues at the University of Shanghai for Science and Technology have created new mathematical expressions for the MDRs of multi-layered particles, which make the relationship far easier to model.

Since these MDRs can vary with factors including particle size, temperature, and composition, the team’s approach could lead to advanced new measurement techniques in fields including spectroscopy, biological sensing, and high-resolution optical imaging.

Compared with uniform particles, the MDRs of multi-layered particles are far more challenging to model. This complexity emerges since the resonant frequencies of these particles are highly sensitive to the sizes and resonant frequencies of each layer – leading to multiple, overlapping frequency modes.

In their study, Guo’s team defined new mathematical expressions for describing the likelihood that incoming electromagnetic waves will be scattered at each interface within a 3-layer spherical particle. This redefinition enabled the team to examine the resonance characteristics of each layer, allowing them to identify the exact locations within the sphere where resonance occurs.

In turn, they could make a straightforward analysis of the particles’ different resonant frequencies – and predict how they will be affected by their physical structures.

Altogether, the team’s approach lays a valuable theoretical foundation for modelling the MDRs of multi-layered spherical particles, and could pave the way for their more widespread use across a diverse array of applications.

Reference: Guo, L., Yu, H. & Shen, J. Morphology-dependent resonances in three-layered spherical particle. Eur. Phys. J. D 79, 83 (2025). https://doi.org/10.1140/epjd/s10053-024-00947-y