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We here calculate the series expansion of the T-matrix for a spheroidalparticle in the small-size/long-wavelength limit, up to third lowest order withrespect to the size parameter, X, which we will define rigorously for anon-spherical particle. T is calculated from the standard extended boundarycondition method with a linear system involving two infinite matrices P and Q,whose matrix elements are integrals on the particle surface. We show that thelimiting form of the P- and Q-matrices, which is different in the special caseof spheroid, ensures that this Taylor expansion can be obtained by consideringonly multipoles of order 3 or less (i.e. dipoles, quadrupoles, and octupoles).This allows us to obtain self-contained expressions for the Taylor expansion ofT(X). The lowest order is O(X^3) and equivalent to the quasi-static limit orRayleigh approximation. Expressions to order O(X^5) are obtained by Taylorexpansion of the integrals in P and Q followed by matrix inversion. We thenapply a radiative correction scheme, which makes the resulting expressionsvalid up to order O(X^6). Orientation-averaged extinction, scattering, andabsorption cross-sections are then derived. All results are compared to theexact T-matrix predictions to confirm the validity of our expressions andassess their range of applicability. For a wavelength of 400nm, the newapproximation remains valid (within 1% error) up to particle dimensions of theorder of 100-200nm depending on the exact parameters (aspect ratio andmaterial). These results provide a relatively simple andcomputationally-friendly alternative to the standard T-matrix method forspheroidal particles smaller than the wavelength, in a size range much largerthan for the commonly-used Rayleigh approximation.

Approximate T-matrix and optical properties of spheroidal particles to third order in size parameter