Physics
Wave optics calculator: polarisation, diffraction and interference
Pick a relation and clear the quantity you are after. Here light is a wave: these are the effects geometrical optics cannot describe, from crossed polarisers to the resolving power of a telescope.
Where rays stop being enough
Geometrical optics describes light as rays, and on that basis explains lenses, mirrors and shadows. It does not explain why two crossed polarisers go black, why a narrow slit spreads a beam instead of trimming it, why a soap bubble is coloured, or why a telescope cannot be made arbitrarily sharp. All of these have the same cause: light is a wave, and waves add with their phase.
Polarisation is about the direction of the electric field. Malus's law, I = I₀cos²θ, applies between two polarisers and gives the result that always surprises: two crossed polarisers pass nothing, yet slide a third between them at 45° and the light comes back. It is not a trick. Each polariser projects the field onto its own axis, and a projection can rotate the wave's direction as well as dim it.
Diffraction and interference, by contrast, are matters of optical path. Single slit, grating and thin film all obey a condition on path difference measured in wavelengths; what changes is whether that condition picks out maxima or minima, and whether the half-order from the reflection phase flip enters. The Rayleigh criterion closes the circle: because every aperture diffracts, an instrument's resolving power is set by its aperture and no amount of magnification improves it.
Common mistakes
- Using the single-slit minima formula as though it gave maxima: a·sin θ = m·λ locates the dark fringes. For a grating the same form gives the bright ones instead.
- Looking for a zeroth order in the single slit: there isn't one. The centre holds the principal maximum, twice as wide as the side fringes.
- Dropping the half-order in thin films: reflection at the front face, passing into a denser medium, flips the phase by half a wavelength. Without that term bright and dark swap places.
- Applying Malus to unpolarised light: the first polariser halves it regardless of angle, and only from there on does the cosine squared apply.
Frequently asked questions
What does Malus's law say?
That the intensity passed by a polariser is I = I₀·cos²θ, where θ is the angle between the polariser's axis and the incoming light's polarisation direction. It applies to already-polarised light; for natural light the first polariser simply halves the intensity.
Why do three polarisers pass light when two do not?
Because a polariser does not merely filter: it projects the field onto its own axis, rotating the polarisation direction. With axes at 90° the projection is zero. Insert a third at 45° and the light reaches the last one polarised at 45° to its axis, so a fraction gets through.
What is the difference between a single slit and a grating?
The condition a·sin θ = m·λ has the same form but the opposite meaning: for a single slit it marks the minima, for a grating the maxima. A grating, having many slits, also produces far narrower maxima and so separates nearby wavelengths much better.
What is the Rayleigh criterion?
It is the smallest angular separation at which two point sources remain distinguishable: θ = 1.22·λ/D, with D the aperture diameter. It depends only on wavelength and aperture, not on magnification, which is why a telescope's sharpness is fixed by its mirror.
Why are soap bubbles coloured?
Because light reflected from the film's two faces interferes. At any given thickness some wavelengths reinforce and others cancel, and the colour you see is what survives. Change the thickness and the colour changes, which is why a bubble shows bands that drift as it flows.
How this calculation works
Malus's law: I = I₀·cos²θ between two polarisers; natural light through the first polariser gives I = I₀/2. Brewster's angle: tan θᴃ = n₂/n₁; at that incidence the reflected and refracted rays are perpendicular, so the refraction angle is 90° − θᴃ. Single slit, intensity minima: a·sin θ = m·λ with m a non-zero integer; the central maximum's angular width is twice the first minimum's angle. Grating, principal maxima: d·sin θ = m·λ; the number of observable orders is the integer part of d/λ. Rayleigh criterion for a circular aperture: θ = 1.22·λ/D. Thin film of index n on a less dense medium, constructive reflection: 2·n·t = (m + ½)·λ, the half-order coming from the π phase flip on reflection at the front face; the destructive condition is 2·n·t = m·λ.
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