Derive the equation for acceptanc angle and numerical aperture, of optical fiber. Acceptance angle in optical fibre:

Explanation

To ensure the cirtical 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 while entering in to it. This angle is called acceptance angle. It depends on the refractive incidens of the core `n_(1),` cladding `n_(2)` and the outer medium `n_(3)`, Assume the light is incident at an angle called acceptance angle `i_(a)` at the outer medium and core bounadary at A.
The Snell.s law in the product form, equation for this refraction at the point A.


`n_(3) sin i_(a) = n_(1) sin r_(a)`
To have the total interal reflection inside optical, fibre, the angle of incidence at the core-cadding interface at B should be atleast critical angle `i_(c)`,. Snell.s law in the product form, equaiton for the refraction at point B is,
`n_(1) sin i_(c) = n_(2) 90^(@)`
`n_(1) sin i_(c) n_(2) " " therefore sin 90^(@) = 1`
`therefore sin i_(c) n (n_(2))/(n_(1))`
From the right angle triangle `DeltaABC`,
`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))`
`sin r_(a) = sqrt(1 - cos^(2) r_(a))`
Substituting for `cos r_(a)`
`sinr_(a)=sqrt(1-((n_(2))/(n_(1)))^(2))=sqrt((n_(1)^(2)-n_(2)^(2))/(n_(1)^(2)))`
Substituting this in equation (1) `n_(3)sini_(a)=n_(1)=n_(1)sqrt((n_(1)^(2)-n_(2)^(2))/(n_(1)^(2)))=sqrt(n_(1)^(2)-n_(2)^(2))`
On further simoplificaiton,
`sini_(a)=sqrt((n_(1)^(2)-n_(2)^(2))/(n_(3)))(or)sini_(a)=sqrt((n_(1)^(2)-n_(2)^(2))/(n_(3)^(2))`
`i_(a)=sini^(-1)(sqrt((n_(1)^(2)-n_(2)^(2))/(n_(3)^(2))))`
If outer medium is air, then `n_(3) = 1`. The acceptance angle `i_(a)` becomes,
`i_(a)=sini^(-1)(sqrt(n_(1)^(2)-n_(2)^(2)))`
Light can have any angle of incidence from o to `i_(a)` with the normal at the end of the optical fibre forming a conical shape called acceptance cone. In the equation (6), the term `(n_(3)sini_(a))`
`NA=n_(3)sini_(a)sqrt(n_(1)^(2)-n_(2)^(2))`