RL 744 A NOTE ON THE EXTINCTION EFFICIENCY Thomas B.A. Senior Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan 48109 An important parameter in electromagnetic scattering is the extinction efficiency Qext defined as the ratio of the sum of the scattering and absorption cross sections and the geometrical cross section. It is accepted that in the limit as the electrical size of the particle becomes infinite, Qext = 2, and in a recent communication [1] an attempt was made to prove this result for a perfectly reflecting sphere by approximating the individual terms in the Mie series expression for Qext' There are, however, rigorous proofs based on the application of a Watson transformation already available in the microwave literature, and the purpose of this note is to summarize one of these. If x = 27ra/x where a is the radius of the perfectly reflecting sphere and A is the wavelength, Q 4 ReS (1) exte 2 with 00 = (n + -(an+ b (2) n=i The a and b are the Mie series coefficients defined as n n (1) (1)' an = nn(x)/n (x) bn = n(x)/ n (x), where n(x) and 1)(x) are proportional to the spherical Bessel and Hankel functions of the first kind: RL-744 = RL-744

-2 - (1) (i) Pn(X) = xjn(x), (x) = xh (x) and the prime denotes differentiation. By application of a Watson transformation to (2) we have S =f C c + v-l/2( v dv 1 (1) 1 + e27iv 2 V1/2 (x). where C is a path in the complex v plane running from v = c - iC to v = 0 and thence to v = ~ + ic with c > 0. The path encloses in a clockwise sense the zeros of 1 + e7 for v > 1/2, and the term -1/2 serves to cancel the residue at v = 1/2. If the lower half of the path is now reflected in the origin, the resulting integrals can be manipulated to give S = S~ + SC(i) + SC(2) (3) where x 0 O - 0 2, +i 2-rfiv 1 0 e- 2e 2 vdv - + -dv - 2v vdv (4) -i 1 + -2riv 1 + e2ri CO.iE 1 +e o 1 +e sC(1) - co+ic 1 1 -+i cE e27iv vdv 1 + e27Ti (5) sC(l) 2 2 + { + vdv. (6) (1) (1)1 (x E^ 1) ' (x) v-1/2 v-1/2

-3 - As split up in this manner, S~ contains terms which are geometrical in character, whereas SC(1) and Sc(2) represent the contributions from the creeping waves and the vestiges thereof. The evaluation of the expression for S is straightforward. The last two integrals in (4) are equal and opposite in sign, and can be computed by expanding the denominators in powers of the exponential. 111l I - U I L 1 - ~ = X2 (7) Sc( 1 can be evaluated as a sum of residues at the poles of the Hankel function ratios in the upper half plane, leading to the familiar expression in terms of creeping waves which have travelled at least once around the sphere, and Sc(l) decreases exponentially with increasing x. For the integrals comprising S (2), no closure of the paths is possible. SC(2) represents the contribution of rays which have merely grazed the sphere, and by introducing the Airy integral representations of the functions involved, the integrals can be reduced to known forms. The details of the calculations are given in [2], and the result is 5c(2) = x/3 (0.082972 + iO.144019) + O(x2/3). (8) From (1), (2), (3), (7) and (8) it now follows immediately that

-4 - 1 im = 2 x +-> co Qext 2 In principle at least a similar approach is possible (see [3]) for a homogeneous dielectric sphere whose Mie coefficients are Grnin(X) - n(X) bn n a (x) - p (x) bn - _n_ where m'(mx) a = m23 = m II n (mx) and m is the refractive index, and the only real difficulty is the asymptotic estimate of the terms corresponding to Sc(1) and Sc(2) above. This work was supported by the U.S. Army Chemical Systems Laboratory under Contract DAAK11-81-K-0004. REEFERENCES 1. A. Cohen, C. Acquista and J. A. Cooney, Appl. Opt. 19, 2264 (1980). 2. T.B.A. Senior and R. F. Goodrich, Proc. IEE (Lond.) 111, 9q07 (1964). 3. R. G. Newton, Scattering Theory of Waves and Particles (McGrawHill, New York, 1966): p. 84 et seq.