Thin Films, Newton's Rings and Other Interference
A wave reflected from a denser medium comes back turned over. That half wave explains the colours of soap films and oil on a puddle, the anti-reflection coating of a camera lens, Newton's rings, and the dark edge of Lloyd's mirror. Fresnel's biprism, the hyperbolic and circular shapes of fringes, and a tilted beam or a raised source that moves the central fringe.
Why this chapter
A soap bubble at a mela and a film of oil on a monsoon puddle shine with colours, though neither holds any dye. The glint of a phone's camera lens is faintly purple, and the centre of Newton's rings in a college lab is perfectly black. All of these are interference of light reflected from the two surfaces of a thin film. The key is a single rule. A wave reflected from a denser medium comes back turned over, as if half a wavelength had been added to its path. This chapter proves that rule, then uses it for films, coatings and rings. It then makes two coherent sources out of one with a mirror and with a prism. Last, it asks what shape fringes really have and where the central fringe lies.
You will learn
- State Stokes' relations \(r' = -r\) and \(tt' = 1 - r^2\), and decide which reflections of a film gain a phase of \(\pi\).
- Derive the path difference \(2\mu t\cos r\) between the two waves reflected by a film, and add the half wave where it belongs.
- Use the conditions for a film to look bright or dark in reflected light.
- Find which wavelengths a film reflects strongly and which are missing.
- Explain why a very thin film looks black, and why reflected and transmitted colours are complementary.
- Find the thickness of an anti-reflection coating, and explain why a coated lens looks faintly purple.
- Derive \(t = r^2/2R\) and \(r_n = \sqrt{n\lambda R}\) for Newton's rings, and explain their dark centre and their crowding.
- Find a wavelength from the diameters of two rings.
- Find the fringe width in Lloyd's mirror and Fresnel's biprism, explain the dark fringe at the mirror's edge, and find where fringes can be seen.
- Explain why fringes are hyperbolas that look straight, and why sources one behind the other give rings.
- Find how far a tilted beam, a source off the axis or a slab moves the central fringe.
Before you begin
From the chapter on Young's double slit, you should know coherent sources, the path difference \(yd/D\) and the fringe width \(\beta = \lambda D/d\). From the chapters on waves, you should know that a pulse reflected from a fixed end comes back inverted. You should also know that a path difference \(\Delta x\) gives a phase difference \(2\pi\Delta x/\lambda\). From the chapter on refraction, you should know Snell's law, the optical path \(\mu d\) and the principle of reversibility. You should also know the thin prism, \(\delta = (\mu - 1)A\). You should be comfortable with Pythagoras' theorem and small-angle approximations.
Lessons
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%Glossary
The key words of this chapter, with the meaning this chapter gives them.
- anti-reflection coating
- A layer about a quarter of a wavelength thick, laid on a lens so that its two reflections cancel. Its refractive index lies between those of air and the glass.
- central fringe
- The fringe at which the whole optical path from the source is the same through both slits. It is bright when no half wave separates the two waves.
- complementary
- Said of the reflected and transmitted patterns of a film: where one is bright the other is dark, and together they hold all the light.
- Fresnel biprism
- Two thin prisms of small angle joined base to base. It bends the light of one slit into two beams that seem to come from two coherent virtual sources \(2a(\mu - 1)\alpha\) apart.
- fringe width
- The distance \(\beta\) between two neighbouring bright, or dark, fringes. For two coherent sources \(d\) apart and a screen \(D\) away, \(\beta = \lambda D/d\).
- hyperbola
- The curve of points whose distances from two fixed points, its foci, differ by a fixed amount.
- hyperboloid
- The surface swept out by a hyperbola turned about the line through its foci. The points of one path difference from two sources lie on one.
- Lloyd's mirror
- Light from a slit a height \(h\) above a plane mirror interfering with its own reflection, as if from two sources \(2h\) apart. The fringe at the mirror's edge is dark.
- Newton's rings
- The circular fringes formed by the air film between a convex lens and a flat plate. In reflected light the centre is dark and the dark rings have radii \(\sqrt{n\lambda R}\).
- optical path
- The distance \(\mu d\) that light would travel in vacuum while it crosses a thickness \(d\) of a medium of refractive index \(\mu\).
- optically denser
- Of two media, the one with the larger refractive index, in which light travels more slowly. It need not have the larger mass density.
- phase change on reflection
- The phase of \(\pi\) that a wave gains when it reflects from the surface of an optically denser medium. It counts as an extra path of \(\lambda/2\). Reflection from a rarer medium adds none.
- principle of reversibility
- If the direction of a ray of light is reversed, it retraces its path exactly, through every reflection and refraction.
- Stokes' relations
- The relations \(r' = -r\) and \(tt' = 1 - r^2\) between the fractions of amplitude reflected and transmitted at a boundary from its two sides, found by reversing the rays.
- thin film
- A transparent layer, such as a soap film, a layer of oil or a lens coating, up to a few micrometres thick. The light reflected from its two surfaces interferes.
- virtual source
- A point from which light seems to come after a reflection or a refraction, though no light starts there. The image of a slit in a mirror is one.