The sides of the element of solid angle are dθ and sin(θ)dφ. For the flux in the direction of angle θ relative to the surface normal we have,
where L is the Luminance. The total flux for the hemisphere is,
The assumptions we have made are those of geometrical optics which just considers the rays involved in the illumination process. The more general case which treats light as waves and takes interference into account is physical optics. It is used by astronomical optical interferometry to study the emission from stellar sources.