The Forward Scattering of Light by Spheres according to Geometrical Optics

If the light scattering by transparent spheres is calculated according to geometrical optics a large part of the scattered flux falls within a cone of half-angle | m - 1| radians around the forward direction, where m is the refractive index of the sphere relative to the surrounding medium. For many purposes it is this light which is of principal interest. If I (θ) is the intensity scattered at angle theta and Δ E (θ) the flux scattered within this angle, the light externally reflected ( I 1 ) and that transmitted without internal reflection ( I 2 ) normally predominate in the forward-scattered light. For 0.75 ⩽ m ⩽ 2 graphical methods of obtaining I 1 and I 2 , and Δ E 1 and Δ E 2 , are given: this is made possible by plotting I 1 , I 2 , and Δ E 1 , Δ E 2 , against functions of single variables which themselves depend on m and θ. For light internally reflected once ( I 3 ) or twice ( I 4 ) analytical expressions are not given, but the values over which they range are indicated. Light internally reflected more than twice is shown to be negligible. The accuracy obtainable is within about 1% in each case, and the angular range over which approximation is possible increases as | m - 1| increases as long as m < 1.65: if m > 1.65 the range decreases as m increases. For more accurate computation exact expressions for I 1 and I 2 , in terms of m and θ only, are given. Unless m >> 1, I 2 (θ) >> I (θ) - I 2 (θ) for small values of θ, and consequently if accurate values of I 1 and I 2 are combined with approximations to I 3 and I 4 the overall accuracy is fairly high. To simplify the computation, from the approximations given, of I and Δ E respectively two tables show the procedures suggested in each case.