Approximate T matrix and optical properties of spheroidal particles to third order with respect to size parameter

© 2019 American Physical Society. In electromagnetic scattering, the so-called T matrix encompasses the optical response of a scatterer for any incident excitation and is most commonly defined using the basis of multipolar fields. It can therefore be viewed as a generalization of the concept of polarizability of the scatterer. We calculate here the series expansion of the T matrix for a spheroidal particle in the small-size, long-wavelength limit, up to third lowest order with respect to the size parameter, X, which we will define rigorously for a nonspherical particle. T is calculated from the standard extended boundary condition method with a linear system involving two infinite matrices P and Q, whose matrix elements are integrals on the particle surface. We show that the limiting form of the P and Q matrices, which is different in the special case of spheroid, ensures that this Taylor expansion can be obtained by considering only multipoles of order 3 or less (i.e., dipoles, quadrupoles, and octupoles). This allows us to obtain self-contained expressions for the Taylor expansion of T(X). The lowest order is O(X3) and equivalent to the quasistatic limit or Rayleigh approximation. Expressions to order O(X5) are obtained by Taylor expansion of the integrals in P and Q followed by matrix inversion. We then apply a radiative correction scheme, which makes the resulting expressions valid up to order O(X6). Orientation-averaged extinction, scattering, and absorption cross sections are then derived. All results are compared to the exact T-matrix predictions to confirm the validity of our expressions and assess their range of applicability. For a wavelength of 400 nm, the new approximation remains valid (within 1% error) up to particle dimensions of the order of 100-200 nm depending on the exact parameters (aspect ratio and material). These results provide a relatively simple and computationally friendly alternative to the standard T-matrix method for spheroidal particles smaller than the wavelength, in a size range much larger than for the commonly used Rayleigh approximation.