EC-5.1 Waves, Interference and Optics

What this is and why it exists

Wave behaviour turns up in almost every later subject here, and each of those subjects tends to re-derive it in a hurry.

A signal down a transmission line, light guided along a fibre, a radio wave crossing into a wall, an electron confined inside a crystal. All the same mathematics, wearing different units. Learn it once, properly, and four later courses get shorter.

There is a second reason. Interference sounds like a laboratory curiosity. It is actually how an optical fibre stays single-mode and why an antenna has a beam rather than a glow. It is also why a measurement can resolve a distance far smaller than anything you could machine.

The vocabulary

• Amplitude — the size of the oscillation, measured from the middle to the peak.
• Frequency — how many cycles pass a fixed point each second.
• Wavelength — the distance between two points doing the same thing at the same time.
• Phase — where in its cycle a wave is, at a chosen place and moment.
• Superposition — the rule that overlapping waves add rather than collide.
• Path difference — how much further one wave travelled than another before they met.
• Diffraction — the spreading of a wave after it passes an edge or an opening.
• Refraction — the change of direction when a wave crosses into a medium where it travels at a different speed.
• Total internal reflection — what happens above a critical angle: none of the wave crosses the boundary.
• Polarisation — the orientation of the oscillation, across the direction of travel.

The mental model

A travelling wave is fully described by four numbers. Amplitude, frequency, wavelength and phase. Speed ties two of them together: speed equals frequency multiplied by wavelength, so fixing the medium and the frequency fixes the wavelength.

Get fluent moving between period and frequency, and between wavelength and its reciprocal. Most arithmetic errors later are conversions, not physics.

Superposition is the whole of interference. Waves do not collide. Where two overlap, the displacements add. Two arriving in step reinforce. Two arriving exactly opposed cancel. Everything with a name — Young's fringes, an antenna array pattern, an anti-reflection coating — is that one rule applied to a particular geometry.

What decides whether they arrive in step is path difference, measured in wavelengths. A difference of a whole number of wavelengths gives reinforcement. A difference of a half wavelength gives cancellation.

Pause on how small that is. Visible light has a wavelength under a micrometre, so half a wavelength is a few hundred nanometres. Move one mirror by that and bright becomes dark. This is why optical measurement can beat any ruler, and it is the basis of an entire family of instruments.

Diffraction is the wave refusing to travel in a straight line. Pass a wave through an opening and it spreads on the far side, and it spreads more when the opening is comparable to the wavelength. A large opening gives a narrow beam and a small one gives a wide spray. That single relationship sets the beam width of an antenna and the resolution limit of a lens, and no amount of engineering escapes it.

At a boundary between two media, part of the wave crosses and part comes back. The part that crosses changes direction, because it travels at a different speed on the other side. Going from a slower medium to a faster one there is a critical angle beyond which nothing crosses at all and everything reflects. That is total internal reflection, and it is the reason an optical fibre works.

Last, polarisation. A wave that oscillates across its direction of travel also has an orientation. That orientation can be held fixed, made to rotate, or left random. Filters, antennas and display panels all work by preferring one orientation over another.

What you should now be able to explain or do

• Write down a travelling wave from its four numbers, and convert between them.
• Predict reinforcement or cancellation from a path difference in wavelengths.
• Say why a narrow opening spreads a wave more than a wide one.
• Explain refraction, and say when total internal reflection happens.
• Say what polarisation is and name something that depends on it.

Check yourself