Light propagation; 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.

In a step-index fibre the rays follow zig-zag paths; in a graded-index fibre the index decreases gradually from the axis, so the rays bend continuously (curved paths) instead of reflecting sharply. In a single-mode fibre the core is so small that only one mode, travelling almost along the axis, exists.

Relationship between numerical aperture and acceptance angle

          n0 (air)  |  n2 cladding
                    |-----------------------------
        θa  ray     |   φ  \  /  φ  (φ ≥ φc: TIR)
   ---------------->*-------\/----------------->
     (acceptance)   |θ1  n1 core   axis
                    |-----------------------------
                    |      n2 cladding

Let a ray enter the core from a medium of index n0 at angle θa to the axis, refract to angle θ1 inside the core, and meet the core–cladding boundary at angle φ (measured from the normal), where φ = 90° − θ1.

1. Snell's law at the input face:

n0 sin θa = n1 sin θ1

2. For total internal reflection at the core–cladding boundary, φ must be at least the critical angle φc, where

sin φc = n2 / n1

3. The limiting (maximum) entry angle θa corresponds to φ = φc, i.e. θ1 = 90° − φc:

n0 sin θa = n1 sin(90° − φc) = n1 cos φc
          = n1 √(1 − sin²φc)
          = n1 √(1 − n2²/n1²)
          = √(n1² − n2²)

4. The numerical aperture is defined as NA = n0 sin θa, so

NA = n0 sin θa = √(n1² − n2²)

5. With the relative index difference Δ = (n1² − n2²)/(2n1²) ≈ (n1 − n2)/n1:

NA = n1 √(2Δ)

For launching from air (n0 = 1): θa = sin⁻¹(NA), and the full acceptance cone angle is 2θa.

So the NA is the sine of the acceptance angle (for launching from air) and depends only on the core and cladding indices, not on the fibre diameter. A larger NA gives a wider acceptance cone and easier coupling, but more modes and more modal dispersion.

Example: n1 = 1.50, n2 = 1.48: NA = √(2.25 − 2.1904) = 0.244, θa = 14.1°.