Define and explain Numerical aperture?

Figure shown below shows an incident ray to the fibre,

The light bends at two interfaces, one at the air- core interface and the other is the core cladding interface.
At the first interface, we can write
n sin O0 = n1 sin O ---------(1)
but minimum value of Ø is Øc where Ømax = 90 - Øc, then (1) becomes
n sin Omax = n1 sin Ømax = n1 sin (90 - Øc)
n sin Omax = n1 cos Øc.............(2)
at the second interface, we can write
n1 sin Oc = n2 sin 90
sin Oc = n2/n1 ........(3)
therefore cos Oc = (1- sin² Oc)^1/2
cos Oc = [1- (n1/n2)2]^1/2
(2) becomes
n sin Omax = (n12 - n22)^1/2
this is called numerical aperture (NA) of the fibre.
NA = (n12 - n22)^1/2