Wave interference

When waves overlap, their displacements combine at each point. This is superposition: the resultant displacement is the sum of the individual displacements. The change in amplitude caused by this overlap is called interference.

If the waves arrive in phase, their oscillations match: a crest of one meets a crest of the other, and a trough meets a trough. Their displacements reinforce each other, producing a larger resultant amplitude. This is constructive interference.

If the waves arrive in antiphase, a crest of one meets a trough of the other. Their displacements oppose each other, reducing the resultant amplitude. This is destructive interference. If the waves have equal amplitudes, they can cancel completely where they overlap.

Path difference and interference

The path difference is the difference between the distances travelled by two waves from their sources to the same point. It affects whether the waves arrive in phase or in antiphase.

For sources that start in phase, the interference at a point depends on the path difference, $$\Delta x$$, measured in wavelengths, $$\lambda$$:

Here $$n = 0, 1, 2, \ldots$$. A path difference of a whole number of wavelengths leaves the waves in phase. A path difference of an odd number of half-wavelengths, such as $$\frac{1}{2}\lambda$$ or $$\frac{3}{2}\lambda$$, puts them in antiphase.

For example, if one wave travels $$5\lambda$$ to a point and the other travels $$3.5\lambda$$, their path difference is $$1.5\lambda$$. If their sources are in phase, the waves arrive in antiphase and interfere destructively.

Coherent sources and stable interference

A stable interference pattern requires coherent sources: sources with the same frequency and a constant phase difference. Their waves then maintain the same phase relationship at each point, so the positions of constructive and destructive interference remain fixed.

Coherent sources do not have to be in phase. The path-difference conditions above apply specifically when the sources are in phase. Both sound waves and electromagnetic waves can produce interference.

Diffraction of light

Light spreads out when it passes through an opening or around the edge of an obstacle. This spreading is called diffraction.

The amount of spreading depends on the size of the opening relative to the wavelength of the light. Diffraction is more pronounced when the opening is narrower relative to the wavelength. For a fixed wavelength, narrowing the opening produces greater spreading.

Diffraction through a single slit

A laser provides monochromatic light: light of a single wavelength. When a laser beam passes through a narrow slit, the diffracted light forms a pattern on a screen.

Light from different parts of the slit combines on the screen. At some positions the waves reinforce each other, producing bright regions; at others they cancel, producing dark regions. The pattern has a central bright maximum, with weaker bright maxima on either side, separated by dark regions.

Laser safety

Laser light can damage the eyes. Never direct a laser beam into anyone’s eyes, and do not look directly into the beam or at its specular reflections from mirror-like surfaces.

Young’s double-slit experiment

When light from two narrow slits reaches a screen, the waves overlap. At some positions they reinforce one another; at others they cancel. This produces an interference pattern of alternating bright and dark bands, called fringes.

For a steady pattern, the waves from the slits must be coherent: they must have the same frequency and maintain a constant phase difference. This can be achieved by illuminating both slits with light from a single source. The slits then act as coherent sources.

Fringes with monochromatic light

Monochromatic light has a single wavelength, $$\lambda$$. Consider a point on the screen: light reaches it from both slits, but the two paths may have different lengths. This path difference determines how the waves arrive relative to one another.

• A path difference of a whole number of wavelengths, such as $$0$$, $$\lambda$$ or $$2\lambda$$, gives constructive interference and a bright fringe.
• A path difference of an odd number of half-wavelengths, such as $$\lambda/2$$ or $$3\lambda/2$$, gives destructive interference and a dark fringe.

At the centre of the screen, the paths from the slits are equal. With the slits illuminated in phase, the path difference is zero, so the central fringe is bright.

Fringe spacing

The fringe spacing $$w$$ is the distance between the centres of adjacent bright fringes. For slit separation $$s$$, screen distance $$D$$ and wavelength $$\lambda$$:

This approximation applies when the screen is far from the slits relative to their separation, for fringes near the centre of the pattern. Increasing the wavelength or the screen distance increases the fringe spacing. Increasing the slit separation decreases it.

Fringes with white light

White light contains a range of wavelengths. At the centre of the screen, the path difference is zero for all wavelengths, so they reinforce there and form a white central fringe.

Away from the centre, each wavelength has a different fringe spacing. Their bright fringes therefore occur at different positions, producing coloured fringes. Further from the centre, the fringes for different wavelengths increasingly overlap.

Development of models of electromagnetic radiation

Scientific models of light have changed as new observations tested what existing theories could explain. Early accounts competed: one treated light as a stream of particles, while another treated it as a wave.

Evidence for wave behaviour

Interference occurs when waves overlap. Depending on how the waves meet, they can reinforce or cancel one another, producing a pattern of bright and dark regions. Diffraction is the spreading of waves when they pass through an opening or around an obstacle. Observations of both effects in light provided strong evidence for a wave model.

Maxwell’s theory later described light as an electromagnetic wave, made up of changing electric and magnetic fields that propagate together. This connected light with other electromagnetic radiation within a single wave theory.

Evidence for quantised energy

The photoelectric effect occurs when light causes electrons to be released from a material. Experiments showed that, for a given material, light below a certain frequency could not release electrons, however intense it was. A classical wave model, in which light transfers energy continuously, could not fully explain this observation.

The quantum explanation treats light energy as being exchanged in discrete packets called photons. A photon’s energy increases with the frequency of the radiation. Below the threshold frequency, an individual photon does not have enough energy to release an electron, even if the light is made more intense.

The modern quantum description

The modern description incorporates both wave-like and particle-like behaviour. Interference and diffraction reveal wave-like behaviour, while the exchange of energy in photons reveals particle-like behaviour. Neither early account alone explains all the observations.

The development of these models shows how scientific explanations can be extended or revised when new evidence exposes their limits. The quantum description does not discard the evidence for waves; it also accounts for observations that the classical wave model could not fully explain.