Acceptance Angle in Optical Fibre

To ensure light entering an optical fibre will be guided by total internal reflection once inside, it must be launched at no more than a maximum incidence angle at the fibre's end face, called the acceptance angle $i_a$. It depends on the refractive indices of the core $n_1$, the cladding $n_2$, and the outer medium $n_3$ the light is entering from: applying Snell's law at the entry face and at the core-cladding boundary (where the internal ray must strike at exactly the critical angle, the borderline TIR case) together gives $i_a=\sin^{-1}\!\left[\dfrac{\sqrt{n_1^2-n_2^2}}{n_3}\right]$. If the outer medium is air ($n_3=1$), this simplifies to $i_a=\sin^{-1}\sqrt{n_1^2-n_2^2}$; the quantity $\sqrt{n_1^2-n_2^2}$ (or $NA=n_3\sin i_a$ more generally) is called the numerical aperture $NA$ of the fibre. Every ray entering the flat end face at an angle from $0$ up to $i_a$ traces out a cone shape, the acceptance cone, and is successfully guided; rays entering at a steeper angle leak out through the cladding and are lost. If there is no cl …