15.3 Solutions
725
15.3 Solutions
15.3.1 Geometrical Optics
General
15.1 Let the point source be located at the centre O of a sphere of radius R, located
within the medium. Light proceeding within a cone of semi-angle equal to the
critical angle C can alone escape from a plane surface on the top, Fig. 15.14.
Fig. 15.14 Light from a point
source escaping through a
plane surface on the top
Consider a circular strip of radius r and width dr , symmetrical over the
sphere’s surface. The angle θ is measured with respect to the z-axis.
Area of the circular strip = 2πr dr
Surface area of the sphere = 4 π R 2
Fraction d f =
area of the strip
area of the sphere
=
2πr dr
4π R 2
=
1
2
(R sin θ)(Rdθ)
R 2
=
1
2
sin θ dθ
Fraction of light escaping within the semi-angle θ is then given by
f =
d f =
θ
0
1
2
sin θ dθ =
1
2
(1 − cos θ)