Question

An optical fibre of glass core (refractive index ${\mu }_2$) is surrounded by a cladding (refractive index ${\mu }_3<{\mu }_2$), suppose a beam of light enters the fibre from air (refractive index ${\mu }_1$) at an angle $\theta$ with the fibre axis, as shown. The greatest value of angle $\theta$ for which a ray can be propagated along the fibre, is

• A. ${\sin }^1\frac{1}{{\mu }_1}\sqrt{{\mu }_2^2{\mu }_3^2}$
• B. ${\cos }^1\frac{1}{{\mu }_1}\sqrt{{\mu }_2^2{\mu }_3^2}$
• C. ${\cos }^1\frac{1}{{\mu }_3}\sqrt{{\mu }_2^2{\mu }_1^2}$
• D. ${\sin }^1\frac{1}{{\mu }_3}\sqrt{{\mu }_2^2{\mu }_3^2}$

Answer 1

The correct option is A ${\sin }^1\frac{1}{{\mu }_1}\sqrt{{\mu }_2^2{\mu }_3^2}$

At 1-2 interface ${\mu }_1\sin \theta ={\mu }_2\sin r$ …(i)

At 2-3 interface $\angle i>c_{23}$

$\Rightarrow$ $\sin i\le \sin c_{23}$

$\Rightarrow$ $\sin i\ge \frac{{\mu }_3}{{\mu }_2}$

$\Rightarrow$ $\sin \left(90r\right)\ge \frac{{\mu }_3}{{\mu }_2}$

$\Rightarrow$ $\sin r\le \frac{1}{{\mu }_2}\sqrt{{\mu }_2^2{\mu }_3^2}$

From (i),
$\frac{{\mu }_1}{{\mu }_2}\sin \theta \le \frac{1}{{\mu }_2}\sqrt{{\mu }_2^2{\mu }_3^2}$

$\Rightarrow$ $\sin \theta \le \frac{1}{{\mu }_1}\sqrt{{\mu }_2^2{\mu }_3^2}$

$\Rightarrow$ ${\theta }_{max}={\sin }^1\frac{1}{{\mu }_1}\sqrt{{\mu }_2^2{\mu }_3^2}$