Total Internal Reflection

What this subtopic is examined on

• The definition of total internal reflection and the two conditions for it.
• The definition of critical angle and the derivation of $\sin i_c = 1/n$ from Snell's law (a standard 2- or 3-mark part).
• Applications: mirage, sparkle of diamond, totally reflecting prisms, optical fibres — with the working of an optical fibre and its uses.
• Numericals: critical angle from refractive index or from speeds; the area of the surface through which light escapes from a source under water; whether a ray in a prism is totally internally reflected.

Internal reflection and total internal reflection

• When light in a denser medium meets the boundary with a rarer medium, part is reflected back into the denser medium (internal reflection) and part is refracted into the rarer medium, bending away from the normal.
• As the angle of incidence increases, the angle of refraction increases faster, until at one particular angle the refracted ray grazes along the interface ($r = 90^\circ$).
• Beyond that angle refraction is impossible and the whole of the light is reflected back into the denser medium. This is total internal reflection.
• Ordinary reflection is always partial — some light is always transmitted, so the reflected ray is always less intense than the incident ray, however smooth the surface. In total internal reflection nothing is transmitted, so it is the only perfectly efficient reflection there is.

Conditions (state both)

• Light must travel from an optically denser medium towards an optically rarer medium.
• The angle of incidence in the denser medium must be greater than the critical angle for that pair of media.

Critical angle and its relation to refractive index

• The critical angle $i_c$ is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is $90^\circ$.
• Apply Snell's law at that instant, with $n_b$ the denser and $n_a$ the rarer medium:

$$n_b\sin i_c = n_a\sin 90^\circ = n_a$$

$$\Rightarrow\quad \sin i_c = \dfrac{n_a}{n_b} = n_{ab}, \qquad \dfrac{n_b}{n_a} = \dfrac{1}{\sin i_c}$$

• If the rarer medium is air or vacuum, $n_a = 1$ and the absolute refractive index of the denser medium is

$$n = \dfrac{1}{\sin i_c}, \qquad \sin i_c = \dfrac{1}{n}$$

• Snell's law cannot be satisfied for $i > i_c$, since $\sin r$ cannot exceed 1 — that is the algebraic reason no refraction is possible.
• A larger refractive index gives a smaller critical angle.

Applications

• Mirage. On a hot day the layers of air near the ground are hotter and optically rarer than those above. Light from a distant object travelling downwards bends progressively away from the normal, and where the angle of incidence exceeds the critical angle it is totally internally reflected upwards, so an inverted image of the sky is seen and looks like a sheet of water.
• Brilliance of diamond. The critical angle of diamond is only about $24.4^\circ$, so light entering it is totally internally reflected many times before finding a face at less than the critical angle; the faces are cut so that the emerging light is concentrated, and the stone sparkles.
• Totally reflecting prisms. A right-angled isosceles prism of glass ($i_c \approx 41^\circ < 45^\circ$) turns light through $90^\circ$ by one total internal reflection at its hypotenuse, through $180^\circ$ by two, and can invert an image without changing its size. Such prisms are used in periscopes and binoculars, and are better than mirrors because no light is lost.

Optical fibres

Construction and working

• An optical fibre is a fine, flexible thread of high-quality composite glass or quartz. It has a core of higher refractive index sheathed in a cladding of slightly lower refractive index (typically core about 1.7, cladding about 1.5).
• A light signal launched into one end of the fibre at a suitable angle strikes the core–cladding boundary at an angle greater than the critical angle for that pair, and so is totally internally reflected. It is reflected again and again along the length of the fibre and finally emerges at the far end.
• Because every reflection is total, there is no appreciable loss of intensity, and the light follows the fibre even when it is bent — the fibre acts as a "light pipe".
• Diagram in words: a long thin cylinder (core) sheathed by a slightly thicker cylinder (cladding); a ray entering at one end zig-zags between the two boundaries along the whole length and emerges at the other end.
• The main fabrication requirement is very low absorption. In silica glass fibres more than 95% of the light is transmitted through 1 km of fibre.

Uses

• Transmitting audio, video and data signals over long distances (electrical signals converted to light by transducers).
• A bundle of fibres carries a whole image, each fibre carrying one small element of it: used in endoscopy to examine the oesophagus, stomach and intestines.
• Decorative lamps in which each fibre carries light from a bulb to the tip of its free end.

Numerical patterns

• Critical angle from speeds. If $v_{air} = 3\times10^8$ and $v_{glass} = 2\times10^8$ m s$^{-1}$, then $n = 3/2 = 1.5$ and $\sin C = 1/1.5 = 0.667$, so $C \approx 41.3^\circ$.
• Critical angle between two media. From $\mu = 2.0$ into $\mu = 1.0$: $\sin C = 1.0/2.0 = 1/2$, so $C = 30^\circ$.
• Speed from critical angle. If $C = 30^\circ$ then $\mu = 1/\sin 30^\circ = 2$, and $v = c/\mu = 1.5\times10^8$ m s$^{-1}$.
• Which medium has the largest critical angle? The one with the smallest refractive index, since $C = \sin^{-1}(1/\mu)$ — among water, crown glass, flint glass and diamond, water.
• Escape cone from a source under water. A point source at depth $d$ under a liquid of index $\mu$ lights a circle of radius $r = d\tan i_c$ on the surface. With $\sin i_c = 1/\mu$, $\tan i_c = \dfrac{1}{\sqrt{\mu^2-1}}$, so

$$r = \dfrac{d}{\sqrt{\mu^2-1}}, \qquad A = \pi r^2 = \dfrac{\pi d^2}{\mu^2 - 1}$$

• Light going from a denser medium into water at $18^\circ$ when the critical angle is $20^\circ$ simply refracts, bending away from the normal — no total internal reflection, because $18^\circ < 20^\circ$.

Standard results

• $\sin i_c = \dfrac{n_{rarer}}{n_{denser}}$; with air outside, $n = 1/\sin i_c$.
• Total internal reflection needs denser to rarer, and $i > i_c$. Both, every time.
• No light at all is transmitted in total internal reflection.
• Prisms used for $90^\circ$ or $180^\circ$ turning need $i_c < 45^\circ$, i.e. $n > \sqrt2 \approx 1.414$.