3 . . lor After the coefficients al{3, I = 1,2, ... , N, and (3 = 1,2, ... , N b are solved, the far field scattered field in the direction (e s , cPs) is expressed asVz 1 d-r -f 1{3 (-r) E me (-r) = {volfl(3 . Eine(rl) 0ikr Es(r) = k 2 :7fr (VsVs + hs!"s) .L L aja (Erj - 1) lv j=la=l dr' e-iks.r7ja(r')(10.3.98) where Erj = Ej / E. Under the small spheroid assumption, only the dipole fields will contribute to the far field radiation in (10.3.98). Thus, we haveikr Es(r) ::::: k 2 :7fr (vsv s + hsh s) .L L aja (Erj j=la=l 1) VOjfjae-iks rj (10.3.99)We next illustrate the results of the numerical simulations by using N = 2000 spheroids and up to f = 30% by volume fraction. The relative permittivity used for the spheroids is 3.2 and the size parameter of the10 DENSE MEDIA MODELS AND THREE-DIMENSIONAL SIMULATIONS spheroids used is such that ka = 0.2. At this volume fraction, permittivity, and size, we did not include the quadrupole effects in the simulations. For dipole interactions, we replace the integral in the last term of (10.3.93) as follows.dr fl(Ar) . 7ijaJr) =(Erj -1) VOj vOl k2 fl(3' G(rl' rj) fja (10.3.100)In the simulations, all the spheroids are prolate and are identical in size with c = ea, where e is the elongation ratio of the prolate spheroid. The size of the box in which the spheroids are placed is Nv(10.3.101)where f is the fractional volume, and v = 47fa 2 c/3 is the volume of one spheroid. An incident electric field of Einc(r) = ye ikz (10.3.102) is launched onto the box containing the N spheroids. The matrix equation of (10.3.96) is solved by iteration. After the matrix equation is solved, the scattered field is calculated by (10.3.99). The scattered field is decomposed into vertical and horizontal polarization (10.3.103) We performed N r = 50 realizations in the numerical illustrations. Let (J be the realization index. Decomposition of the field into coherent and incoherent scattered fields is also made as in Section 3.2. The incoherent scattered field is decomposed into vertical and horizontal polarization