Physics · Optics · Grade 7

Laser, water and optical fibre

Why does the internet run as light inside glass fibre?

The situation

You point a laser pointer at an aquarium: the beam bends at the water surface and lands on the bottom at a different spot. A beam aimed upwards from under the water, beyond a certain angle, cannot get out of the water and is reflected as if from a mirror. It is this phenomenon that lets the internet run through glass fibres on the ocean floor, and makes a diamond sparkle.

Topic

Refraction of light. Laser in an aquarium, a diamond and an optical fibre

Key ideas

Snell's law
Total internal reflection
Dispersion

Formulas

n_{1} \sin α = n_{2} \sin γ

Snell's law: $α$ is the angle of incidence, $γ$ the angle of refraction (measured from the normal).

n = \frac{c}{v}

Refractive index: how many times slower light travels in the medium than in a vacuum. In water $v \approx 2.25 \cdot 10^{8}$ m/s.

\sin α_{c h} = \frac{n_{2}}{n_{1}} (n_{1} > n_{2})

Critical angle: at a larger angle the light is totally reflected. Water 48.6°, glass 41.8°, diamond 24.4°.

R_{0} = {(\frac{n_{1} - n_{2}}{n_{1} + n_{2}})}^{2}

Fresnel formula (normal incidence): water reflects 2 % of the light, glass 4 %. At oblique incidence $R$ increases.

n (λ) = A + \frac{B}{λ^{2}}

Cauchy's formula: for short-wavelength (violet) light $n$ is larger — it is refracted more strongly.

Theory

Why does the beam bend?

Light travels 1.33 times more slowly in water than in air. One edge of an obliquely incident wavefront enters the water first and slows down — the front turns, like a cart with one wheel in sand. Watch the pulses: in water they are packed closer together.

Total internal reflection

When passing from a dense medium to a less dense one, $γ > α$ . When the angle of incidence reaches the critical angle, $γ = 90 °$ — the beam skims along the surface. At a larger angle no light gets out at all: $R = 100 %$ . This is a better “mirror” than a mirror.

Optical fibre

A beam inside the fibre hits the wall at an angle larger than the critical angle and keeps reflecting without loss. If you bend the fibre too sharply, the angle of incidence on the outer wall becomes smaller and the light leaks out. That is why optical cables have a minimum permitted bend radius.

Dispersion and the rainbow

The refractive index depends on wavelength: for violet light $n$ is larger, so it is deviated more. A prism splits white light into a spectrum. Raindrops do the same and form a rainbow. A beam passing through a prism is turned through the angle of deviation $δ$ ; $n$ can also be found from its minimum value: $n = \sin \frac{A + δ_{m i n}}{2} / \sin \frac{A}{2}$ ( $A$ is the refracting angle).

Fermat's principle, phase and group velocities

Light travels between two points along the path that takes the least time (more precisely, a path whose time is stationary) — Fermat's principle. The laws of rectilinear propagation, reflection and refraction follow from it: the “Fermat” graph in the “Aquarium” mode shows the travel times of paths through different points of the water surface — the minimum is exactly at the point where the laser is refracted. The $v$ in $n = c / v$ is the phase velocity (the speed of the wave crests). A pulse (signal, energy) travels at the group velocity: $u = v - λ d v / d λ$ (Rayleigh's formula), that is, $u = c / n_{g}$ , $n_{g} = n - λ d n / d λ$ . In normal dispersion $n_{g} > n$ : in BK7 glass, for yellow light $n \approx 1.517$ , $n_{g} \approx 1.54$ (see it in the prism result). For simplicity the pulses in the scene are drawn moving at the speed $c / n$ .

Dispersion theory and refractometers

In the classical electron theory the light field makes the electrons in matter oscillate like oscillators with natural frequency $ω_{j}$ : $n^{2} (ω) = 1 + \frac{N e^{2}}{ε_{0} m} \sum_{j} \frac{f_{j}}{ω_{j}^{2} - ω^{2} - i γ_{j} ω}$ . So the permittivity depends on frequency (temporal dispersion); if it also depends on the wave vector, there is spatial dispersion (for example, optical activity). Far from resonances $n$ grows with frequency — normal dispersion (glass and water in the visible range); inside an absorption line it decreases — anomalous dispersion, where $n$ is complex and its imaginary part gives the absorption (Bouguer's law — “Scattering of light”). For glasses the Sellmeier formula $n^{2} = 1 + \sum_{i} B_{i} λ^{2} / (λ^{2} - C_{i})$ is used; Cauchy's formula is its approximation far from resonances. If the absorption over the whole spectrum is known, $n (ω)$ can be calculated — the Kramers–Kronig relations. A refractometer measures $n$ from the critical angle (the Pulfrich refractometer in the “Laboratory” scene: $n = \sqrt{N^{2} - \sin^{2} β}$ ); the mean dispersion $n_{F} - n_{C}$ and the Abbe number $ν = (n_{D} - 1) / (n_{F} - n_{C})$ characterise a glass. In the microscope method $n = d / h$ : $d$ is the thickness of the plate, $h$ the apparent thickness; a beam passing through a plane-parallel plate emerges parallel but displaced: $Δ = d \sin (α - γ) / \cos γ$ .

Did you know?

More than 95 % of intercontinental internet traffic is carried by optical-fibre cables on the ocean floor — the light travels thousands of kilometres inside the glass thanks to total internal reflection.
The critical angle of diamond is only 24.4°. Light that enters is totally reflected from the lower facets and leaves through the top — that is why a well-cut diamond “burns”.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Car mirrors and the periscope — Grade 7
The eye and spectacles — Grade 7
Shadow and eclipses — Grade 7
Measuring the speed of light — Grade 9
Optical instruments — Grade 9
Interference and diffraction — Grade 11
Polarisation of light — Grade 11
Laser physics and holography — Grade 11
Luminous flux and illuminance — Grade 11
Infrared, UV and X-rays — Grade 11