Large Nanoparticles and Higher-Order Modes

For larger nanoparticles the excitation processes are more complex than the simple electronic cloud oscillating all over the nanoparticle volume parallel to the incident field (dipole or / = 1 plasmonic mode) higher-order modes must be included into the theoretical treatment to explain the extinction spectra. 

By coming back to Sec. 1.4.1, we can generalize the quasi-static solution of a nanosphere for an external potential of the type  

Solving the Laplace equation with the appropriate boundary 

Here results are obtained in the discrete dipole approximation, so they should give the exact solution to the electromagnetic problem and spectra identical to the Mie theory. An3way, like already shown in literature [1], the quasistatic approach is able to retain the most important features in the spectra for spheres of these sizes. 

While all the spectra are still dominated by the dipolar plasmon mode, the contribution from the quadrupolar mode increases as nanoparticle size increases the quadrupolar polarizability in fact increases with the fifth power of the particles size, against the third power for the dipolar one. 

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