How Light Stays Trapped Inside an Optical Fibre

A dentist's small mirror magnifies a tooth, a coin in a bucket of water looks closer than it is, and internet data races through glass threads as light. All three follow from two simple laws — reflection and refraction — and one special case, total internal reflection.

This part covers spherical mirrors, refraction at plane surfaces, total internal reflection, and optical fibres.

**For a spherical mirror, the object distance $u$, image distance $v$ and focal length $f=\frac{R}{2}$ are related by $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$, and the magnification $m=-\frac{v}{u}$ gives the image's size and orientation.

Sign convention:**

- Distances are measured from the pole; those along the incident light are positive and those against it negative
- Heights above the axis are positive, below it negative
- A concave mirror has negative $f$; a convex mirror has positive $f$

Worked example 1 — concave mirror. An object is $30$ cm in front of a concave mirror of focal length $20$ cm, so $u=-30$ cm and $f=-20$ cm:

$$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}=-\frac{1}{20}+\frac{1}{30}=-\frac{1}{60}\Rightarrow v=-60\text{ cm},m=-\frac{v}{u}=-2$$

The image is real, inverted and twice the size, $60$ cm in front of the mirror.

Worked example 2 — convex mirror. An object $20$ cm in front of a convex mirror with $f=+15$ cm gives $\frac{1}{v}=\frac{1}{15}+\frac{1}{20}=\frac{7}{60}$, so $v\approx +8.6$ cm and $m\approx +0.43$ — a virtual, erect, diminished image.

An everyday example. Rear-view mirrors on scooters are convex, because they always give an erect, diminished view of a wide stretch of road behind.

The substance. A negative magnification means an inverted image, and a magnitude above $1$ means an enlarged one.

**Light bending at a boundary obeys two laws — the incident ray, refracted ray and normal lie in one plane, and $n_1\sin i=n_2\sin r$ — where the refractive index $n=\frac{c}{v}$ measures how much a medium slows light.

Snell's law:**

$$n_1\sin i=n_2\sin r,n_{21}=\frac{n_2}{n_1}$$

Light bends towards the normal entering a denser medium and away from it on leaving.

Worked example 1. Light enters water ($n=1.33$) from air at $45^{\circ }$:

$$\sin r=\frac{\sin 45^{\circ }}{1.33}=\frac{0.707}{1.33}\approx 0.532\Rightarrow r\approx 32^{\circ }$$

Real and apparent depth. Seen from directly above, an object at real depth $h$ in a medium of index $n$ appears at depth

$$h^{\prime }=\frac{h}{n}$$

Worked example 2. A coin at the bottom of a bucket of water $40$ cm deep appears at $\frac{40}{1.33}\approx 30$ cm, raised by about $10$ cm.

An everyday example. A straw in a glass of nimbu pani looks bent at the surface, because light from the submerged part refracts as it leaves the water.

The substance. The frequency of light does not change on refraction — its speed and wavelength change together.

**When light travels from a denser to a rarer medium at an angle of incidence greater than the critical angle $i_c$, where $\sin i_c=\frac{n_2}{n_1}$, no light refracts out and all of it is reflected back — total internal reflection.

Conditions:

- Light must go from a denser to a rarer medium
- The angle of incidence must exceed the critical angle

Critical angle.** At $i=i_c$ the refracted ray grazes the surface, so $r=90^{\circ }$:

$$n_1\sin i_c=n_2\sin 90^{\circ }\Rightarrow \sin i_c=\frac{n_2}{n_1}$$

For a medium surrounded by air, $\sin i_c=\frac{1}{n}$.

Worked example. For glass with $n=1.5$ in air:

$$\sin i_c=\frac{1}{1.5}\approx 0.667\Rightarrow i_c\approx 41.8^{\circ }$$

For diamond, $n=2.42$ gives $\sin i_c\approx 0.413$ and $i_c\approx 24.4^{\circ }$.

Applications: the sparkle of diamonds, mirages on hot roads, and right-angled prisms that turn light through $90^{\circ }$ or $180^{\circ }$ in periscopes and binoculars.

An everyday example. The shimmering puddle that seems to lie ahead on a hot highway is a mirage, produced by total internal reflection in layers of heated air.

The substance. Total internal reflection returns essentially all the light, more completely than an ordinary mirror.

An optical fibre has a glass core surrounded by cladding of lower refractive index, so light in the core strikes the boundary at angles above the critical angle and is totally reflected again and again, travelling along even curved fibres with very little loss.

Structure:

- Core — very thin glass of higher refractive index
- Cladding — glass of lower refractive index
- Protective jacket — the outer coating

Worked example. For a core of $n_1=1.50$ and cladding of $n_2=1.45$:

$$\sin i_c=\frac{1.45}{1.50}\approx 0.967\Rightarrow i_c\approx 75^{\circ }$$

Light striking the boundary at more than about $75^{\circ }$ to the normal stays trapped in the core.

Uses:

- Telecommunication and internet — signals sent as pulses of light over long distances
- Medical endoscopes — viewing inside the stomach or a knee joint

An everyday example. Fibre broadband connections in homes carry internet data as flashes of light through exactly such cables.

The substance. The cladding is essential — without a lower-index layer, light would leak out wherever the fibre touched anything.