(i) Optical fibres consists of inner part called core and outer part called cladding (or) sleeving. (ii) To ensure the critical angle incidence in the core-cladding boundary inside the optical fibre, the light should be incident at a certain angle at the end of the optical fiber while entering in to it. (iii) This angle is called acceptance angle. It depends on the refractive indices of the core `n_1` cladding `n_2` and the outer medium `n_3`. (iv) Assume the light is incident at an angle called acceptance angle `i_a` at the outer medium and core boundary at A. (v) The Snell's law in the product form, equation `n_1 sin i = n_2` sin r for this refraction at the point A is as shown in the Figure .
` n_3 sin i_a = n_1 sin r_a` ...(1)
(vi) To have the total internal reflection inside optical fibre, the angle of incidence at the core-cladding interface at B should be atleast critical angle `i_c` . Snell's law in the product form, equation `(n_1 sin i = n_2 sin r)` for the refraction at point B is,
` n_1 sin i_c = n_2 sin 90^@ " " ...(2)`
`n_1 sin i_c = n_2 " " because sin 90^@ = 1`
` sin i_c = n_2/n_1 " " ..(3)`
From the right angle triangle `triangle ABC` ,
` i_c = 90^@ - r_a`
Now, equation (3) becomes,
` sin (90^@ - r_a ) = n_2/n_1`
Using trigonometry, `cos r_a = n_2 / n_1 " " ...(4)`
` sin r_a = sqrt( 1 - cos^2 r_a)`
Substituting for cos `r_a`
`sin r_a = sqrt( 1 - (n_2/n_1)^2) = sqrt((n_1^2 - n_2^2)/(n_1^2)) " " ...(5)`
Substituting this in equation (1).
` n_3 sin i_a = n_1 sqrt( (n_1^2 - n_2^2)/(n_1^2)) = sqrt(n_1^2 - n_2^2) " " .... (6)`
On further simplification,
` sin i_a = (sqrt(n_1^2 - n_2^2))/(n_3) " or " sin i_a = sqrt( (n_1^2 - n_2^2)/( n_3^2) ) " " ...(7)`
` i_a = sin^(-1) (sqrt((n_1^2 - n_2^2)/(n_3^2)) ) " " ...(8)`
If outer medium is air, then `n_3= 1`. The acceptance angle `i_a` becomes,
` i_a = sin^(-1) ( (sqrt(n_1^2 - n_2^2)) " " .....(9)`
(vii) Light can have any angle of incidence from 0 to `i_a` with the normal at the end of the optical fibre forming a conical shape called acceptance cone as shown in Figure
In the equation (6) the term `(n_3sin i_a)` is called numerical aperture NA of the optical fibre.
` NA = n_3 sin i_a = sqrt(n_1^2 - n_2^2) " " ....(10)`
If outer medium is air, then `n_3 = 1` . The numerical aperture NA becomes,
` NA = sin i_a = sqrt( n_1^2 - n_2^2) " " ....(11)`
