Physics · Optics · Grade 11

Interference and diffraction

Why do a CD and a soap bubble shimmer in rainbow colours?

The situation

If you hold a compact disc up to the light, a rainbow glows on its surface; a soap bubble shimmers blue, yellow and pink — yet neither the disc nor the soapy water is coloured. If you point a laser pointer at a CD, several sharp spots appear on the wall. These colours do not come from pigment: light waves reinforce or cancel one another. How can an ordinary CD and a ruler be used to measure the wavelength of a laser — a millionth of a metre?

Topic

Interference and diffraction. Soap bubble and the colours of a compact disc

Key ideas

Interference
Diffraction grating
Thin film

Formulas

d \sin θ = m λ, \tan θ_{m} = \frac{x_{m}}{L}

Diffraction grating: $d$ is the period (on a CD the track spacing is 1.6 µm, on a DVD 0.74 µm), $m$ is the order, $x_{m}$ is the distance on the screen from the central spot to the $m$ -th spot, and $L$ is the distance from the grating to the screen.

I = I_{0} \cos^{2} (\frac{π d \sin θ}{λ}) {sinc}^{2} (\frac{π a \sin θ}{λ})

Young's experiment: $d$ is the distance between the slits, $a$ is the slit width.

Δ x \approx \frac{λ L}{d}

Fringe spacing (for small angles). For the Fresnel biprism, $d = 2 a (n - 1) β$ is the distance between the two virtual sources, $L = a + b$ .

Δ = 2 n h \cos θ_{t} + \frac{λ}{2}

Thin film: the path difference of the two reflected waves; $λ / 2$ — on reflection at the front surface the phase shifts by 180°.

r_{m} = \sqrt{m λ R}, R = \frac{D_{m}^{2} - D_{k}^{2}}{4 λ (m - k)}

Newton's rings: the radius of the $m$ -th dark ring in reflected light; from the diameters $D$ of two rings the radius of curvature of the lens is found.

Theory

Interference

When two coherent waves meet, if the path difference is $Δ = m λ$ they reinforce (a bright fringe), and if $Δ = (m + \frac{1}{2}) λ$ they cancel (a dark fringe): these are the conditions for maxima and minima. In a medium the optical path $n l$ is used, and the phase difference is $Δ φ = 2 π Δ / λ$ . In Young's experiment the waves from the two slits give alternating bright and dark fringes on the screen; the diffraction of each slit itself wraps the fringes in a ${sinc}^{2}$ “envelope”.

Coherence and the Huygens–Fresnel principle

A steady pattern is seen only with coherent waves — waves whose phase difference does not change in time. A real source does not emit an endless sine wave but “pieces” of wave of finite length: the coherence time $τ \approx 1 / Δ ν$ and the coherence length $l = c τ \approx λ^{2} / Δ λ$ . If the path difference exceeds $l$ , the pattern disappears. In white light ( $Δ λ \approx 300$ nm) $l \approx 1$ µm — which is why the lower part of a soap film, thicker than ≈ 1 µm, is almost colourless; in a multimode He–Ne laser $l$ reaches tens of centimetres. Spatial coherence depends on the size of the source: Young put a single small hole in front of the two slits, while laser light is coherent across its whole cross-section. Diffraction is explained by the Huygens–Fresnel principle: every point of a wavefront is a source of secondary waves, and the oscillation at the observation point is their interference. That is why a wave enters the geometric shadow at the edge of a slit; waves on a water surface passing through a narrow gap spread out in the same way. Fresnel zones and diffraction at an aperture — in the “Resolving power” experiment.

Diffraction grating: CD and DVD

The spiral track on a CD is a great many parallel lines: the rays reflected from neighbouring tracks reinforce one another in the directions $d \sin θ = m λ$ . Because there are many lines, the maxima are narrow, bright spots. The tracks of a DVD are denser, so the spots are deflected through a larger angle. In white light each colour goes to its own angle — the grating gives a diffraction spectrum: unlike a prism, red light is deflected the most and violet the least. That is the rainbow on a disc.

Colours of a soap film

The two waves reflected from the front and back surfaces of the film interfere. The wave reflected at the front surface (air → water) shifts phase by 180°, while at the back it does not. Depending on the thickness, some colours are cancelled and the rest are seen. Places of the same thickness have the same colour — these are fringes of equal thickness (a vertical film is a wedge, so the fringes are horizontal). In a thick plate, on the other hand, the path difference depends on the slope of the ray, giving fringes of equal inclination (rings). Colours are calculated over the 380–780 nm spectrum and converted to colour using the eye's sensitivity curves (CIE 1931); to make them visible, the brightness has been boosted ≈ 12 times.

Newton's rings, biprism and applications

If a plano-convex lens is placed on a glass plate, an air layer of thickness $h = r^{2} / (2 R)$ forms between them. In reflected light there are dark rings where $2 h = m λ$ : $r_{m} = \sqrt{m λ R}$ ; the centre is dark because the phase shifts by 180° at the lower surface. In the laboratory the ring diameter is measured from the microscope scale at its two edges, and $R = (D_{m}^{2} - D_{k}^{2}) / (4 λ (m - k))$ is found — taking the difference removes the error caused by dust at the contact point. The Fresnel biprism (a double prism with a very small refracting angle $β$ ) splits the source beam in two and makes two virtual coherent sources a distance $d = 2 a (n - 1) β$ apart; on the screen $Δ x = λ (a + b) / d$ . The Fresnel double mirror and Billet's split lens also make two such sources. Applications: the anti-reflection coating on lenses (thickness $λ / (4 n)$ , for example MgF₂, $n \approx 1.38$ — the waves reflected from the two surfaces cancel), checking the smoothness of a surface with a reference glass and Newton's rings, interferometers (“Interferometers and LIGO”), and holography (“Laser physics and holography”).

The black film and the limits of the model

A vertical film drains downward under gravity: its top becomes thinner and the colours slide down in bands. When the thickness is $h ≪ λ / (4 n)$ the two waves cancel almost completely — the film looks black (for a film thinner than ≈ 25 nm, less than 1 % of white light is reflected). The law by which the film thins is empirical in the model (it depends on the type of soap); for the laser grating only the positions of the orders are exact, and the brightness of the spots is approximate. For the biprism, diffraction at the edges of the region is not taken into account, and for Newton's rings reflections inside the lens are not; the ring numbers in the eyepiece are added to make counting easier.

Did you know?

The spacing between data tracks is 1.6 µm on a CD, 0.74 µm on a DVD and 0.32 µm on a Blu-ray disc — which is why Blu-ray is read with a blue-violet (405 nm) laser.
Thomas Young, in the early 19th century (1801–1803), showed the wave nature of light with his interference experiments and was one of the first to estimate the wavelengths of different colours.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Shadow and eclipses — Grade 7
Measuring the speed of light — Grade 9
Optical instruments — Grade 9
Polarisation of light — Grade 11
Laser physics and holography — Grade 11
Luminous flux and illuminance — Grade 11
Infrared, UV and X-rays — Grade 11
Scattering of light — Year 1