Question

Derive an expression for nhmerical aperture in terms of factional refractive jndex

Answer 1

Numerical Aperture is the ability of fiber to collect the light from the source and save the light inside it by maintaining the condition of total internal reflection.



Consider a light ray entering from a medium air of refractive index nonointo the fiber with a core of refractive index n1n1 which is slightly greater than that of the cladding n2n2.

Applying Snell’s law of reflection at point A,

sinӨ1sinӨ2=n1no=n1  as  n0=1sin⁡Ө1sin⁡Ө2=n1no=n1  as  n0=1

In right angled Δabc,

θ2=π2−∅cθ2=π2−∅c

sinθ1=n1sin(π2−∅c)=n1cos∅csinθ1=n1sin(π2−∅c)=n1cos∅c

cos∅c=(1−sin∅2c)1/2cos∅c=(1−sin⁡∅c2)1/2

From the above equation

sinθ1=n1(1−sin∅2c)1/2