Light propagation; define NA and acceptance angle

Principle of light propagation in an optical fibre

Light travels in an optical fibre by total internal reflection (TIR). The core has a higher refractive index (n1) than the cladding (n2). When a ray inside the core meets the core–cladding boundary at an angle (from the normal) greater than the critical angle φc = sin⁻¹(n2/n1), it is totally reflected back into the core with no refraction loss. Repeated TIR guides the ray along the fibre, even around gentle bends.

 cladding n2 ___________________________________
             \        /\        /\        /
 core n1  ray \  φ   /  \      /  \      /
         ----->\    /    \    /    \    /  --->
                \  /      \  /      \  /
 cladding n2 ____\/________\/________\/_________
            φ > φc at every reflection (TIR)

Conditions for TIR:

1. Light must travel from a denser medium to a rarer medium (n1 > n2).
2. The angle of incidence at the core–cladding boundary must exceed the critical angle φc = sin⁻¹(n2/n1).

To meet condition 2, light must enter the fibre end within the acceptance angle θa. Rays entering at larger angles strike the boundary below φc, partly refract into the cladding and are lost after a short distance.

Acceptance angle

The acceptance angle θa is the maximum angle to the fibre axis at which light can enter and still be guided by TIR. Rays within the cone of half-angle θa are guided.

            \
         θa  \     acceptance cone
   axis ------*==========================
             /  core
            /

θa = sin⁻¹( √(n1² − n2²) / n0 )

Numerical aperture

Numerical aperture (NA) measures the light-gathering ability of the fibre. It is the sine of the acceptance angle times the index of the outside medium:

NA = n0 sin θa = √(n1² − n2²) ≈ n1 √(2Δ)
Δ  = (n1 − n2)/n1

(This follows from Snell's law n0 sin θa = n1 cos φc and sin φc = n2/n1.)

Typical values: 0.1–0.15 for single-mode fibre, 0.2–0.3 for multimode fibre.

Example: n1 = 1.48, n2 = 1.46: NA = 0.2425, θa = 14.0°.