Numerical study on the limit of quasi-static approximation for plasmonic nanosphere

Plasmonic nanospheres are often employed as resonant substrates in many nanophotonic applications, like in enhanced spectroscopy, near-field microscopy, photovoltaics, and sensing. Accurate calculation and tuning of optical responses of such nanospheres are essential to achieve optimal performance. Mie theory is widely used to calculate optical properties of spherical particles. Although, an approximated version of Mie approach, the quasi-static approximation (QSA) can also be used to determine the very same properties of those spheres with a lot simpler formulations. In this work, we report our numerical study on the limit and accuracy of QSA with respect to the rigorous Mie approach. We calculated scattering, absorption and extinction spectra of silver and gold nanospheres in air with varying sizes using both QSA and Mie theory. Then, we extracted spectral positions of the resonance peaks from their calculated optical responses and defined the error present in QSA as the difference between the spectral positions of the resonance peaks calculated by QSA and Mie method. Our error analysis reveals that QSA approach yields nonlinear increment in error with linear increment in size of the nanosphere and that the amount of error is significantly less in the case of gold spheres compared to the silver ones. We also provide a polynomial-fitted error function that resembles the qualitative trend in error.