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The majority of first-principle, homogenisation techniques makes use of the dipole terms of a small particle radiation, and, consequently, the respective dipole polarisabilities. This paper aims to take the next step and propose a new systematic technique for extracting the quadrupolarisability of planar metamaterial scatterers.

Firstly, it is assumed that the particle, under study, can be modelled as a set of dipole and quadrupole moments, and by utilising the respective polarisabilities, the far-field response of the scatterer is calculated. Then, the far-field scattering field of the particle is constructed in terms of the dipole and quadrupole moments, which, in turn, are expressed as a function of the unknown polarisabilities. Finally, the desired polarisabilities are retrieved by a system of equations, which involves numerically derived electric field values at specific positions around the scatterer.

The quadrupolarisability of planar metamaterial particles is extracted, through an easy to use, yet very accurate and efficient methodology. Moreover, the proposed technique is verified via comprehensive comparisons of consequently computed and simulated total radiated power values, which reveal its advantages and applicability limits. Finally, the total radiation power contribution of each calculated, individual multipole is provided, to further investigate the radiation mechanism of all nano-particles under study.

The initial and most important step of extracting a single quadrupolarisability of a planar realistic nano-particle has been performed, herein, for the first time. The addition of the respective quadrupole in the scattering model, shifts the multipole approximation limit upwards in terms of frequency, and, therefore, nano-particles with quadrupole resonances can, now, be precisely represented via polarisabilities for various metamaterial or metasurface applications.