Diffraction and Polarisation

Why this chapter

Why this chapter exists

Two facts about light are hidden in plain sight. Light bends round every edge, although only a fine slit makes the bending easy to see. And light has a direction of vibration across its path, which polaroid sunglasses can pick out. This chapter follows both. A single slit makes a broad bright band with faint bands beside it, and its width tells the wavelength. The same bending blurs every telescope, microscope and eye, and sets the finest detail each can see. Then a polaroid, Malus's law and Brewster's angle show that light is a transverse wave. The chapter ends with glare on the sea, the blue sky and the red setting Sun.

You will learn

By the end of this chapter you will

Explain diffraction, and why it is noticeable only when an opening is comparable with the wavelength.
Derive the minima of a single slit, \(a\sin\theta = n\lambda\), by pairing the zones of the slit.
Use \(y_1 = \lambda D/a\) and the width \(2\lambda D/a\) of the central maximum, and predict how the pattern changes with \(a\), \(\lambda\), \(D\) and the medium.
Locate the secondary maxima, explain why the first has only about 4.5 % of the central intensity, and use the Fresnel distance \(a^2/\lambda\).
Compare interference with diffraction, and find the fringes inside the envelope of a real double slit and its missing orders.
State Rayleigh's criterion, and use \(1.22\lambda/D\) for a telescope or the eye and \(1.22\lambda/(2\mu\sin\theta)\) for a microscope.
Explain polarisation as a property of transverse waves, and describe unpolarised, plane polarised and partially polarised light.
Use \(I_0/2\) for unpolarised light through a polaroid, Malus's law \(I = I_0\cos^2\theta\), and three polaroids.
Use Brewster's law \(\tan i_p = \mu\), show that the reflected and refracted rays are then at right angles, and explain polaroid sunglasses.
Explain the blue sky, the red sunset and the polarisation of skylight by scattering that goes as \(1/\lambda^4\).

Before you begin

From the chapter on Young's double slit, you should know Huygens' wavelets, path difference and the fringe width \(\lambda D/d\). You should also know that light is a transverse electromagnetic wave. From the chapters on waves, you should know transverse and longitudinal waves. From the chapter on refraction, you should know Snell's law and the critical angle. From the chapter on optical instruments, you should know the telescope and the microscope. You should be comfortable with small-angle approximations.

Already know this?

Lessons

Light Bends Round Corners

Waves bend round the edges of an opening, by an angle of about \(\lambda/a\). For light the bending is tiny except through fine slits. Pairing the zones of a slit gives the dark bands, \(a\sin\theta = n\lambda\), and a central maximum \(2\lambda D/a\) wide.

Diffraction of lightMinima of a single slitWidth of the central maximum

15 min0%

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The Single-Slit Pattern in Detail

Faint secondary maxima near \(a\sin\theta = (2n + 1)\lambda/2\), the first about 4.5 % of the centre. A wider slit, or water, narrows the pattern. How diffraction differs from interference, and why two real slits make fringes inside one diffraction envelope with some orders missing.

Secondary maximaA wider slit gives a narrower patternInterference and diffraction comparedTwo slits inside one envelope

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Resolving Power

A round lens turns every point into a small disc with rings. Rayleigh's criterion says when two discs can still be told apart, at an angle of \(1.22\lambda/D\). The limits of a telescope, a microscope and the eye, and when two headlights become two.

Rayleighs criterionThe telescope limitThe microscope limitWhen two headlights become two

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Polarisation and Malus's Law

Only a transverse wave can be polarised, and light is one. Unpolarised, plane polarised and partially polarised light; a polaroid passes half of unpolarised light; Malus's law \(I = I_0\cos^2\theta\); testing a beam with one polaroid; and the surprise of three polaroids.

Only transverse waves polariseUnpolarised light through a polaroidMalus lawTesting light with one polaroidThree polaroids

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Polarisation by Reflection and Scattering

Light reflected at the angle \(\tan i_p = \mu\) is plane polarised, and then the reflected and refracted rays are at right angles. Glare and polaroid sunglasses. Scattering by air goes as \(1/\lambda^4\), which makes the sky blue and the setting Sun red, and skylight at 90° to the Sun is polarised.

Brewsters lawReflected and refracted rays at right anglesPolaroid sunglasses and glareBlue sky and red sunset

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

Airy disc: The small bright disc, surrounded by faint rings, that a round lens or mirror makes as the image of a point. Its first dark ring is at \(1.22\lambda/D\).
analyser: A second polaroid placed after a polariser, used to test the polarisation. It passes \(I_0\cos^2\theta\) by Malus's law.
Brewster's angle: Another name for the polarising angle: 56.3° for glass of \(\mu = 1.5\) and 53.1° for water.
central maximum: The broad bright band at the centre of a single-slit pattern, between the first minima on either side. It is \(2\lambda D/a\) wide.
crossed polaroids: Two polaroids with their transmission axes at 90°. Together they pass no light.
diffraction: The bending of a wave round the edges of an obstacle or an opening into the region of the geometric shadow. It is noticeable when the opening is comparable with the wavelength.
diffraction envelope: The single-slit pattern of each slit of a real double slit. The two-slit fringes are bright only inside it.
Fraunhofer diffraction: Diffraction observed with parallel light, on a distant screen or in the focal plane of a lens. The rays that reach one point leave the opening parallel.
Fresnel distance: The distance \(a^2/\lambda\) from an opening of width \(a\) at which the spread by diffraction becomes as large as the opening. Well within it, ray optics holds.
geometric shadow: The region that straight rays from the source cannot reach, behind an obstacle or beside an opening.
limit of resolution: The smallest angle between two objects, or the smallest distance between two points, at which an instrument still shows them as separate.
minimum: A dark band of a diffraction pattern. For a single slit of width \(a\), the minima are at \(a\sin\theta = n\lambda\), with \(n = \pm 1, \pm 2\), and so on.
missing order: A two-slit fringe that falls on a minimum of the diffraction envelope and so does not appear. When d/a is a whole number, the orders d/a, 2d/a, and so on are missing.
numerical aperture: The quantity \(\mu\sin\theta\) of a microscope objective. Here \(\theta\) is the half-angle of the cone of light it collects, and \(\mu\) the refractive index in front of it.
oil immersion: Filling the space between a microscope slide and the objective with oil of refractive index about 1.5. It raises the numerical aperture and the resolving power.
partially polarised light: Light in which every direction of the electric field is present but one is stronger than the rest. It is a mixture of unpolarised and plane polarised light.
plane of incidence: The plane that contains the incident ray and the normal to the surface; the plane of the page in a ray diagram.
plane polarised light: Light whose electric field vibrates along one fixed line across the ray.
polarisation: The confining of the vibrations of a transverse wave, all across the direction of travel, to one fixed direction. A wave so confined is polarised.
polariser: A polaroid that turns unpolarised light into plane polarised light, passing half its intensity.
polarising angle: The angle of incidence \(i_p\) at which the light reflected from a transparent surface is completely plane polarised. It satisfies \(\tan i_p = \mu\).
polaroid: A thin plastic sheet whose aligned long molecules absorb the component of the electric field along them. It passes only the component along its transmission axis.
Rayleigh's criterion: Two point sources are just resolved when the central maximum of one's pattern falls on the first minimum of the other's.
resolving power: The inverse of the limit of resolution. A telescope has \(D/1.22\lambda\), in rad⁻¹; a microscope has \(2\mu\sin\theta/1.22\lambda\), in m⁻¹.
scattering: The sending out of light in all directions by particles whose electrons are shaken by the incident light. For particles much smaller than the wavelength, it goes as \(1/\lambda^4\).
secondary maximum: One of the faint bright bands beside the central maximum of a single slit, near \(a\sin\theta = (n + \tfrac12)\lambda\). The first has about 4.5 % of the central intensity.
transmission axis: The direction of the electric field that a polaroid passes, at right angles to its aligned molecules.
unpolarised light: Light whose electric field points in every direction across the ray, changing at random many millions of times a second. No direction is favoured; a lamp and the Sun give it.