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

Abstract

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 here calculate the series expansion of the T -matrix for a spheroidal particle in the small-size/longwavelength limit, up to third lowest order with respect to the size parameter,X, which we will define rigorously for a non-spherical 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(X 3 ) and equivalent to the quasi-static limit or Rayleigh approximation. Expressions to order O(X 5 ) 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(X 6 ). 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. arXiv:1810.06107v1 [physics.optics]

Quasistatic limit of the electric-magnetic coupling blocks of the T-matrix for spheroids

“…In a recent paper [17] the T -matrix was found to 3 rd lowest order, i.e. O(X 6 ), which involves only up to multipolarity n = 3.…”

“…As per Ref. [17] we shall define a size parameter X = k o c, where for small particles relative to the wavelength, X 1 and the fields may be considered as asymptotic expansions in powers of X, and here we consider just the lowest two orders. This approximation also requires k i c = sX 1, so the relative refractive index s must not be too large.…”

“…This definition for X is convenient here, but as shown in Ref. [17], it is usually more relevant to define the size parameter in terms of the radius of the sphere of equivalent volume. One should also note that spherical Bessel functions of higher order n can be well approximated by their dominant term (small X approximation) for up to X n [16], so one can expect that our final expressions for matrix elements for large n, k will be valid for relatively large X.…”

“…In the near field, X ≈ k o r: (15) and E 0 = O(X 0 ) while H 0 = O(X 1 ). The quasistatic limit of the Q-matrix and R-matrix elements take a special form for a spheroidal particle [18,19] and the order of the dominant terms for all matrices are summarized below [17,18]:…”

“…[11], expressions were derived for the T 22 block, while in Ref. [17], expressions were obtained for all blocks but only for n, k ≤ 3.…”