Light Waves

Is light a stream of tiny particles or a wave? The question kept physicists arguing for three centuries, and the answer turned out to be stranger than either side expected.

A long argument#

Newton pictured light as a stream of tiny corpuscles fired out by a source and travelling in straight lines. Sharp shadows seemed to support this. Reflection became corpuscles bouncing elastically off a surface; refraction became corpuscles being pulled by the particles of glass or water as they entered.

Huygens, working at the same time, proposed that light is a wave. Its straight-line travel would then just mean that its wavelength is far smaller than ordinary openings and obstacles. Newton’s reputation kept the particle picture on top for about a century, until a run of results settled the matter:

• Young’s double-slit experiment (1801) showed light from two coherent sources producing bright and dark bands: interference.
• Fresnel’s experiments explained diffraction in detail.
• Foucault (1850) found light travels slower in water than in air, as the wave theory predicts and the corpuscle theory doesn’t.
• Maxwell’s theory of electromagnetic waves (1860s) predicted the speed of light correctly.

Then came a twist. Around 1900 it was found that light falling on a metal knocks out electrons whose energy doesn’t depend on how bright the light is: the photoelectric effect. Einstein explained it in 1905 only by treating light as particles. Today we accept that light has a dual character: it behaves as a wave in some situations and as particles in others. (Even electrons turn out to show interference.) This chapter is about the wave side.

What is waving?#

A wave on a string moves the string; a sound wave changes the pressure of air. Light is different: it needs no medium and crosses the empty space between the sun and the earth. Early physicists, unable to imagine a wave without a medium, invented an all-filling elastic “ether”. It was never found.

What oscillates in a light wave is the electric field, a vector quantity you’ll meet properly in later chapters. It points across the direction of travel, so light is a transverse wave and can be polarized. For a plane wave moving along x with the field along a fixed direction:

The speed of light in vacuum, $c \approx 3\times10^8\ \text{m/s}$, is the same for every observer, however they move. (This surprising fact is the starting point of the special theory of relativity.) In a transparent material light is slower:

For a point source the waves spread as spheres, $E = \dfrac ar\sin\omega(t - r/v)$: the amplitude falls as 1/r, so the intensity falls as 1/r².

Colours and the visible spectrum#

The eye responds to frequencies from about 3.8 × 10¹⁴ Hz to 7.8 × 10¹⁴ Hz, which is wavelengths from about 780 nm (red) down to 380 nm (violet). The colour we sense is linked to the wavelength:

Monochromatic and white light

Light of a single wavelength is monochromatic. No real source is perfectly monochromatic (even a laser has a tiny spread), so we use the word for light with one dominant wavelength. Most sources emit a mixture, and the eye blends the mixture into one of a huge range of colours. White light is a mixture of all visible wavelengths in suitable proportions.

When can we use rays?#

Waves bend round obstacles and spread from openings, strongly when these are comparable to the wavelength. Light’s wavelength is thousands of times smaller than everyday objects, so for most purposes diffraction is negligible. Light then travels in straight rays and casts sharp shadows. This geometrical optics approximation rests on three rules:

1. Rectilinear propagation: light travels in straight lines in a uniform medium.
2. Reflection: angle of incidence = angle of reflection, and the incident ray, reflected ray and normal are coplanar.
3. Refraction: $\dfrac{\sin i}{\sin r} = \dfrac{v_1}{v_2}$, with the incident ray, refracted ray and normal coplanar.

Huygens had a simple way to predict where a wave goes next: treat every point the wave reaches as a tiny new source. From that one idea the laws of reflection and refraction drop out.

Secondary wavelets#

Draw any surface around a light source. Light reaching points beyond it must cross that surface. Huygens argued that each point of the surface, once the wave arrives, starts vibrating and acts as a new source of spherical waves. He called these points secondary sources and their waves secondary wavelets.

The new wavefront is the envelope

Take a spherical wavefront around a point source and let every point on it send out a short wavelet. After a time t each wavelet is a sphere of radius vt. Just inside the outer surface touching all of them (their envelope), the wavelets meet in step and reinforce. Deeper inside, wavelets from neighbouring points arrive with opposite signs and cancel. So the disturbance ends up on the envelope:

In a uniform medium we only take the envelope in the forward direction. If the wave meets a new medium, wavelets going backwards (the reflected wave) must be included too.

Reflection from Huygens’ principle#

A plane wavefront AB meets a mirror obliquely. Its edge touches the mirror at A at t = 0; the other edge reaches C at time t, after travelling BC = vt. Meanwhile the wavelet from A has grown to a hemisphere of radius vt. Every point between A and C is reached a little later and sends a smaller wavelet. All of them touch one line through C, tangent to A’s wavelet at D. That line CD is the reflected wavefront.

The right-angled triangles ABC and ADC share the side AC and have AD = BC = vt, so they are congruent, and ∠BAC = ∠DCA. These are the angles of incidence and reflection. So i = r, and the rays and the normal all lie in the plane of the drawing.

Refraction from Huygens’ principle#

Now the wave crosses into medium 2, where light is slower (speed v₂ instead of v₁). While the far edge of the wavefront covers BC = v₁t in medium 1, the wavelet from A grows only to radius AD = v₂t. The tangent CD from C is the refracted wavefront. In the two right-angled triangles:

In 1801 Thomas Young let sunlight through two pinholes and saw something particles could never produce: stripes of light and dark where the two beams overlapped. Light plus light can make darkness.

Young’s pinholes#

Young let sunlight through one pinhole in a card, and the light from it fall on a second card with two pinholes side by side. On a screen beyond, the light from the two holes overlapped and formed a pattern of bright and dark patches. At any point the two waves arrive with different phases and interfere. With white sunlight the pattern was coloured, but Young still estimated the wavelengths of extreme red and violet light surprisingly well.

The double-slit experiment#

In the modern version, two long narrow parallel slits S₁ and S₂ replace the pinholes, and they are lit by a parallel beam of monochromatic light. A screen parallel to the slits shows a row of bright and dark stripes called fringes.

At a point P on the screen, the waves have travelled S₁P = x and S₂P = x + Δx. Their fields are

so they differ in phase by

This is exactly the problem of two waves on a string from chapter 15: the result is a wave of amplitude $E_0 = \sqrt{E_{01}^2 + E_{02}^2 + 2E_{01}E_{02}\cos\delta}$.

How bright?

For identical slits each giving intensity I′ alone, E₀₁ = E₀₂, and since intensity ∝ amplitude²:

Fringe width#

Let the slits be d apart and the screen a distance D away, with D ≫ d. Then S₁P and S₂P are almost parallel, at a small angle θ to the centre line, and the extra path is

where y is the distance of P from the centre of the pattern. Bright fringes need yd/D = nλ:

Why fringes are hard to see with wide slit spacing

w shrinks as d grows. If d is many thousands of wavelengths, the fringes crowd so closely that the screen looks evenly lit. This is a general rule: wave effects are hard to notice when the wavelength is small compared with the openings involved.

Unequal slits

If one slit lets through more light, the dark fringes aren’t completely dark. With amplitudes A₁ and A₂, bright fringes have amplitude A₁ + A₂ and dark ones |A₁ − A₂|, so

For interference, what matters is not how far two waves travel but how many wavelengths they fit in along the way. Inside glass a wavelength is shorter, so a centimetre of glass counts for more.

Phase change in a medium#

In a medium of refractive index μ, light travels at v = c/μ, so a light wave there can be written

Travelling a distance Δx changes the phase by $\omega\mu\,\Delta x/c$. In vacuum, the same phase change would need a distance μΔx. So a path Δx in the medium is equivalent, for phase, to μΔx in vacuum.

In air (μ ≈ 1) the optical and geometrical paths are practically equal.

The same idea through wavelength#

A wave’s frequency is fixed by its source and doesn’t change when it enters a new medium. Its speed drops by μ, so its wavelength drops by μ too:

Points one wavelength apart have the same phase. In the medium that is λ₀/μ; in vacuum, λ₀. So a length λ₀/μ of medium is “worth” λ₀ of vacuum, which is again the factor μ.

Why all rays through a lens arrive together#

At a boundary, the incident wavefront and the refracted wavefront each have a single phase along their length. The phase change from one wavefront to the next must therefore be the same along every ray. Using Snell’s law one can show that a path in vacuum and the matching shorter path in glass are related by exactly the factor μ.

The same holds for a lens forming an image. Rays from a point to its image travel different geometrical distances, but they have equal optical paths: a ray along the edge has a longer route through air but crosses only a thin part of the glass, while the central ray goes straight but through the thickest glass. All arrive in phase, which is why they reinforce at the image.

A drop of oil on a wet road, a soap bubble, the wing of a dragonfly: all shimmer with colours that shift as you tilt your head. None of them contain any dye. The colours come from interference.

Many reflections#

Take a thin transparent film of thickness d and refractive index μ, with parallel faces. Light falling on it splits at the top face: part reflects, part enters. The part inside splits again at the bottom face: part leaves, part reflects back up, and so on. The film sends out a whole series of reflected waves and a series of transmitted waves, each weaker than the last. The film can be viewed by either set.

Take nearly normal incidence. Each transmitted wave has made one more round trip inside the film than the one before: an extra geometrical path 2d, through a medium of index μ. So neighbouring waves differ in optical path by

Seen by transmitted light#

Seen by reflected light: a twist#

You might expect the same conditions for reflected light. But think about energy: if a film both transmitted and reflected a wavelength strongly, where would the extra energy come from? Whatever the film transmits strongly it must reflect weakly, and vice versa. Experiment confirms that the conditions swap:

The reason: a phase flip on reflection

Recall a pulse on a light string running into a heavier one: the reflected pulse is inverted, a phase change of π. Reflected from a lighter string, it isn’t. Light behaves the same way: a medium of higher refractive index is optically denser.

So even when 2μd is a whole number of wavelengths, the first two reflected waves end up half a cycle apart and cancel.

Where the colours come from#

In white light, a given thickness satisfies the bright condition for some wavelengths and the dark condition for others. Those colours are reflected strongly or weakly, and the eye sees the mixture. Where the thickness varies, as in a draining soap film or a spreading oil slick, the colours vary in bands. A film much thinner than a wavelength reflects almost nothing (2μd ≈ 0, so the reflection is dark): the top of a soap film turns black just before it bursts.

Put two light bulbs side by side and their light overlaps everywhere, yet you never see a single fringe. Interference needs a special partnership between sources, and that partnership is rare.

The Fresnel biprism#

Fresnel found another way to make two sources from one. A biprism is two very thin prisms joined base to base. Their refracting angles are tiny, about half a degree, so it looks like a single glass plate with a barely raised ridge. A narrow slit S, lit by monochromatic light, is set parallel to the ridge.

Light through the upper half is bent slightly down; light through the lower half slightly up. To an observer beyond, the two beams seem to come from two virtual images of the slit, S₁ and S₂, very close together. Where the two cones of light overlap on a screen, fringes form, just as with two slits a distance d apart:

Coherent and incoherent sources#

Why ordinary sources are incoherent

In a lamp, a candle or the sun, light is emitted by atoms in separate bursts. An atom emits a short pulse lasting roughly 10⁻⁸ s (a wavetrain a few metres long), falls quiet, gets re-energized and emits again. Different atoms act independently, so each new wavetrain starts with a random phase.

With two lamps in place of the slits, the phase difference between them changes randomly millions of times a second. Fringes form and shift faster than any eye or instrument can follow, so we just see uniform light.

How a single slit fixes it

In Young’s experiment, a narrow first slit picks out light that reaches both slits from the same wavetrain. When the lamp emits a new wavetrain with a new random phase, the change reaches both slits at once, so their phase difference stays constant. The biprism works the same way: S₁ and S₂ are images of one slit.

Keep the path difference small

Wavetrains are only a few metres long, and in practice only a few centimetres of path difference can be tolerated. If one path is much longer, a wavetrain arriving by the short path meets a different wavetrain arriving by the long one, and the fixed phase relation is lost.

No light is perfectly monochromatic#

A perfect single wavelength would be a sine wave going on forever. A wavetrain of finite length is equivalent to a spread of sine waves of slightly different wavelengths: the shorter the wavetrain, the larger the spread. A sodium lamp gives two yellow lines, 589.0 nm and 589.6 nm, each spread by about ±0.01 nm.

Squint at a street light through your eyelashes and it sprouts streaks and fringes. Light squeezed through narrow gaps doesn’t keep going straight: it spreads.

Diffraction#

When part of a wavefront is blocked by an obstacle, only the uncovered part sends out Huygens wavelets. Their superposition gives a new wavefront with a different shape, so the light bends round edges into regions that would otherwise be in shadow. This is diffraction. With light it produces patterns of bright and dark fringes.

A diffraction set-up has three parts: a narrow source, the diffracting element (an obstacle or an opening), and a screen. Wavelets from the uncovered part of the wavefront interfere on the screen.

A single slit#

A parallel beam falls normally on a long slit of width b. Every point of the slit sends out wavelets. A lens just after the slit collects the waves leaving at angle θ and brings them together at one point P on a screen at its focal plane.

The centre is bright

At θ = 0 every wavelet travels the same optical path to P₀, so all arrive in phase: the brightest point.

Where the first dark fringe is

At angle θ, the wave from the top edge and the wave from the middle of the slit differ in path by $\tfrac b2\sin\theta$. When that equals λ/2 they cancel. And then every point in the top half cancels with the point b/2 below it in the bottom half. The whole slit cancels in pairs: darkness, at $b\sin\theta = \lambda$.

Dividing the slit into 4, 6, … strips in the same way gives the other dark fringes:

The bright fringes lie roughly midway between them. A full calculation gives the intensity:

A wide central band

Most of the light falls between the first dark fringes, sin θ = ±λ/b. The first side band is only about 1/22 as bright as the centre. The spread is set by λ/b: a narrower slit spreads the light more widely; a slit much wider than λ lets the light through almost undiffracted. Once again, diffraction shows up clearly only when the opening is comparable to the wavelength.

Even a perfect lens cannot focus starlight to a perfect point. Diffraction at its rim always blurs the image a little, and that sets the finest detail any telescope, microscope or eye can see.

Diffraction by a circular hole#

Send a parallel beam through a small round hole in an opaque board and look at a distant screen. You see a bright disc surrounded by fainter dark and bright rings. The analysis (harder than for a slit) puts the first dark ring at

On a screen a distance D away (D ≫ b), or at the focal plane of a lens of focal length f placed after the hole, the radius of this ring is

Most of the light falls inside this ring, so R is called the radius of the diffraction disc.

Every lens is a hole

A lens only collects the part of a wavefront that falls within its rim, which is usually circular. So a lens diffracts light too, and then focuses that diffracted light. The image of a distant point is therefore a tiny bright disc with faint rings, never a point. The same is true for any point source: the image disc has radius 1.22λD/b, where D is the distance at which the light is focused.

Diffraction at a straight edge#

Now a Fresnel case. Light from a narrow slit meets an opaque sheet with a sharp straight edge, and falls on a screen. Ray optics predicts a sharp boundary between light and shadow. Instead:

• inside the geometrical shadow, the light fades away gradually rather than stopping abruptly;
• outside it, the brightness rises and falls in fringes, which get weaker further out until the illumination is uniform.

The limit of resolution#

Two point sources in front of a lens produce two disc images. If the discs’ centres are much closer than their radii, they overlap into what looks like a single blob: the points are not resolved. Move the sources apart and at some point you can tell there are two discs: they are just resolved.

Rayleigh’s criterion

Whether two blurry discs look like two depends a little on who is looking. Rayleigh gave a definite rule: two images are just resolved when the centre of one disc falls on the first dark ring of the other. Then the combined intensity dips noticeably between the two peaks. The smallest angular separation the lens can resolve is therefore about

Why is the sky blue and not violet, white or black? And why does the same sun turn orange at dusk? Both answers come from how strongly tiny molecules redirect different colours.

What scattering is#

When a beam of light crosses a gas, some of it appears in other directions. Molecules absorb a little of the light and re-radiate it every which way. This is scattering. It differs from absorption: in absorption, light energy becomes internal energy of the medium; in scattering, the energy is sent off in new directions. Either way, the beam gets weaker.

Rayleigh’s law#

How strongly light scatters depends on the size of the particles and on the wavelength. For particles much smaller than the wavelength, such as air molecules:

Red for danger

Red light is scattered least, so a red signal is seen from the furthest away through air, mist and haze. That’s why danger and stop signals are red.

The blue sky#

When you look at the sky away from the sun, the light reaching your eye has been scattered towards you by air. Short wavelengths are scattered most. Violet scatters even more than blue, but sunlight contains less violet, and our eyes are less sensitive to it, so the sky looks blue.

Red sunrise and sunset

When the sun is low, its light crosses a much longer stretch of atmosphere to reach you. Along the way most of the blue and nearby colours are scattered out of the beam, and what arrives straight from the sun is mostly orange and red.

A black sky

Without an atmosphere there would be nothing to scatter sunlight towards you, so the sky would be black, with stars visible by day. About 20 km up, where the air is very thin, the sky does turn black, as high-altitude pilots and astronauts report.

Haze and pale skies#

Water droplets, dust and smoke particles also scatter light. Many are larger than the wavelength, and for them the 1/λ⁴ law doesn’t hold: they scatter all colours more evenly. So a humid sky before rain looks pale blue, a clear dry sky deep blue, and the sky over smoky industrial cities greyish and hazy until rain or wind clears the particles.

Tilt your head while wearing polarizing sunglasses and the glare from a wet road brightens and fades. Light has a hidden property that our eyes can’t sense directly: the direction in which its electric field points.

Linearly polarized light#

Light travelling along x has its electric field somewhere in the y–z plane, perpendicular to the motion. If the field at a point always stays along one fixed direction, say y, the light is linearly polarized (also called plane polarized) along y.

Unpolarized light#

One atom in one burst emits light with a fixed field direction. But in a lamp, a candle or the sun, countless atoms emit independently with random field directions, so the resulting field changes direction randomly and rapidly. This is unpolarized light, and it is what ordinary sources give.

Describing every case with two components

Split the field into y and z components with some phase difference δ:

• δ changes randomly: the direction of E wanders randomly. Unpolarized.
• δ = 0 or π: E_z/E_y is constant, so E stays along one line. Linearly polarized.
• δ = π/2 and E₁ = E₂: the field has constant size and its tip turns in a circle at angular speed ω. Circularly polarized.
• δ = π/2 and E₁ ≠ E₂: the tip traces an ellipse. Elliptically polarized.

Polaroids#

A device that produces polarized light from unpolarized light is a polarizer. A common one is the polaroid, a plastic sheet containing long, aligned chains of hydrocarbon molecules. At optical frequencies these chains conduct: the field component along them drives currents and is absorbed, while the component perpendicular to them passes. That perpendicular direction is the sheet’s transmission axis. Light leaving a polaroid is linearly polarized along its axis.

Malus’ law

Send linearly polarized light into a polaroid. Along the axis, it all passes; perpendicular to it, none does. At an angle θ, only the component E cos θ gets through, so the intensity (∝ E²) is

Unpolarized light averages cos²θ over all directions, so a single polaroid passes exactly half of it.

Polarization by reflection#

Light from air strikes a transparent surface. The incident, reflected and refracted rays lie in one plane, the plane of incidence. Split the field of unpolarized incident light into a part in this plane and a part perpendicular to it. The perpendicular part is reflected more strongly, so reflected light is partly polarized, and the refracted light is correspondingly richer in the in-plane part.

At one special angle the in-plane part is not reflected at all:

The refracted light is never completely polarized.

Polarization by scattering#

Light scattered by small particles is partially polarized, so the blue light of the sky is partially polarized. Our eyes can’t tell, but bees can: Karl von Frisch showed over years of experiments that bees sense both whether light is polarized and its direction of polarization, and use the sky’s pattern to navigate.