Constructive and Destructive Interference

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Superposition of waves

When two waves meet, their displacements add. If the peaks of both waves line up, the result is a larger wave. If a peak meets a trough, the result is smaller or zero. This principle is called superposition, and it is the basis of interference.

Path difference

Two light waves from the same source can take different paths and meet again. The difference in path length is called the path difference. Where the path difference is a whole number of wavelengths, the waves reinforce each other: this is constructive interference. Where it is an odd number of half wavelengths, they cancel: this is destructive interference.

Thin films and coatings

Light reflected from the top and bottom surfaces of a thin film interferes. The film thickness and refractive index set the path difference. A quarter-wavelength film can be designed so that reflections from the two surfaces cancel at a chosen wavelength, which is the basis of a single-layer anti-reflection coating.

Because the cancellation depends on wavelength, a single-layer coating works best at one design wavelength. Multi-layer coatings extend the range of good performance.

Diffraction and interference

The Airy pattern described in the diffraction articles is also an interference pattern. Light from different parts of the aperture arrives at a point with different path lengths, and the sum of those waves produces the bright disk and dark rings.

Common mistakes

Assuming that anti-reflection coatings remove all reflection at all wavelengths. Confusing the physical film thickness with the optical path, which depends on refractive index. Forgetting that the simple thin-film treatment assumes a single wavelength and near-normal incidence.

Two-beam interference in the lab

A classic demonstration uses two slits or a beam splitter to produce two coherent beams. Where the path difference is a whole number of wavelengths, bright fringes appear. Where it is a half-integer, dark fringes appear. The pattern spacing depends on the wavelength and the geometry, which is why interference is also used to measure small distances and surface shapes precisely.

Coherence and everyday light

Interference requires light that is coherent, meaning the waves keep a steady phase relationship. Sunlight and most white light sources are only partly coherent, so interference fringes are weak or confined to small path differences. Lasers are highly coherent, which is why they produce strong interference patterns. Thin-film coatings work with broadband light because the effect is confined to small path differences near the film.

Designing a coating in principle

A designer chooses film materials with different refractive indices, and a thickness that gives the desired phase relationship at the design wavelength. Multi-layer stacks use several such films to reduce reflection over a wider range of wavelengths and angles. The design is a calculation of how the reflected waves add, and it is checked against the measured reflectance of the coated surface.

Phase and how it is measured

A wave is described by its phase, an angle that tells you where the wave is in its cycle. Two waves with the same phase reinforce each other. Two waves half a cycle apart cancel. Path difference converts directly to phase difference: a path difference of one wavelength corresponds to a phase difference of 2π radians.

Interferometry

Interferometers use interference to measure distance, surface shape or refractive index with very high precision. A small change in path length shifts the fringe pattern, and the shift can be counted or measured. Optical testing of lens surfaces often uses interferometry to check flatness and curvature to a fraction of a wavelength.

Anti-reflection coating thickness

A single-layer anti-reflection coating uses a film of about a quarter wavelength of the design colour, measured in the film material. Because the optical path is the film thickness multiplied by its index, the physical thickness depends on the index. Designers choose a film material with an index between that of the glass and of air, which reduces reflection more effectively than a film matched to either.

Common coating misconceptions

A coating does not destroy the energy of the reflected light; it redistributes it through interference. A coating designed for green light will reduce reflection around green, but reflection at blue and red may be higher. Coatings also have angle limits, so their performance at the edge of a wide lens can differ from the centre.

Why multi-layer coatings are wider

A single film works well at one wavelength. Several films with alternating indices can be designed so that reflections cancel across a broader band, which is why modern lenses use multi-layer coatings. The design calculation adds the reflections from each interface, and the thickness of each layer is chosen to give the right phase.

A simple check

Look at the reflection colour of a coated lens against a dark background under white light. The residual reflection often shows a colour, which indicates the wavelength where the coating works least well. This is a quick way to judge a coating without measurement equipment.

Going further

Interference is one of the most useful ideas in optics. The same principle that makes a coating work also produces the diffraction pattern of a lens, so understanding it helps with several topics in this guide.

Summary

The key point of this article is the one stated at its start. Check the formula, use the stated assumptions, and confirm any result with a test before relying on it in practice.

Sources

Eugene Hecht, Optics, Pearson (any current edition).
Max Born and Emil Wolf, Principles of Optics, Cambridge University Press (any current edition).
ISO 12233: Photography, Electronic still picture imaging, Resolution and spatial frequency responses.

Next step

Return to the pillar guide or the Wave Physics summary to read the related articles.

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General information only. This page explains optical principles and does not replace lens design work, lens testing or advice from a qualified optical engineer.