Vector Slepian basis functions with optimal energy concentration in high numerical aperture focusing

A series expansion method is proposed for high numerical aperture focusing problems. The angular spectrum of the focused field is expressed using orthogonal vector basis functions that are obtained by solving Slepian's concentration problem for a spherical cap and expressed in terms of the vector spherical harmonics. This newly obtained Slepian-type basis automatically satisfies the transversality condition and a subset exhibits excellent energy concentration inside the solid angle of illumination. This property makes the Slepian-type basis very useful in inverse focusing problems. The corresponding vector basis for focused fields is constructed as well. Examples are presented for linearly and radially polarized illumination, revealing the fact that, compared to an expansion using the vector spherical harmonics themselves, a lower number of terms is sufficient to achieve the same accuracy, suggesting that the Slepian-type basis is preferable even in direct focusing problems.