Wave Optics: Class 12 Physics · Chapter 10

10.1 Introduction

Look at a soap bubble in sunlight, or at the sheen of spilt oil on a wet road, and you see bands of colour that no prism made. Hold two fingers almost touching in front of a tube light and squint through the gap: thin dark lines appear in it. Ray optics, which treats light as straight lines, explains neither. In this chapter we treat light as a wave, and that single idea accounts for reflection and refraction as well as for these new effects.

Is light a stream of particles or a wave? Both pictures explain why a ray reflects at equal angles and why it bends at a water surface, so for well over a century neither side could prove the other wrong. Fig. 10.1 shows the one place where they part company: how fast light travels inside water.

In the particle picture, a particle of light reaching water is tugged towards the surface. The tug adds to the part of its velocity perpendicular to the surface and leaves the part along the surface alone, so the particle swings towards the normal and speeds up. In the wave picture, the end of a wavefront that enters the water first is slowed down, the rest of the front is still moving fast, and so the whole front pivots towards the normal. The bending is the same; the predictions about speed are opposite.

Timing light over a few metres of water was impossible until 1850, when Léon Foucault bounced a beam off a mirror spinning hundreds of times a second, once through air and once through a tube of water. The returning spot was displaced more for the water path, so light had taken longer there. Light is slower in water: about $2.25\times {10}^8\mathrm{m}\mathrm{/}\mathrm{s}$ against $3.00\times {10}^8\mathrm{m}\mathrm{/}\mathrm{s}$ in air. The wave model had passed a test that the particle model failed.

By then the wave model had other evidence. From 1801 Thomas Young showed that light from one source, split into two beams that overlap again, gives bright and dark bands, the hallmark of waves (Section 10.5). Such experiments also measured the wavelength of light, and it turned out to be tiny: visible light spans roughly $400$ to $700\mathrm{n}\mathrm{m}$, so about 150 wavelengths of green light fit across a human hair. Next to a mirror or a lens a few centimetres wide, a wave this short behaves almost exactly like a bundle of straight rays. Ray optics is what wave optics becomes when the wavelength is treated as vanishingly small, and a ray is then simply the direction in which the wave carries its energy.

One objection remained. Every wave known at the time, from ripples to sound, travels through some material, so how could light cross the empty space between the Sun and the Earth? Maxwell's equations answered it in the 1860s: light is an electromagnetic wave, a pattern of electric and magnetic fields that sustain each other and need no medium (Part I, Chapter 8). Hertz produced such waves at radio wavelengths in 1887, and J. C. Bose and G. Marconi soon put them to practical use.

Four questions organise the rest of the chapter. How does a wave model reproduce the laws of reflection and refraction (Section 10.2, Section 10.3)? What happens where two waves overlap, in general (Section 10.4) and in Young's experiment (Section 10.5)? What does a wave do when it squeezes through a narrow opening (Section 10.6)? And how can we tell that light waves are transverse (Section 10.7)?

10.2 Huygens principle

Imagine a small loudspeaker playing a steady note in the middle of a large empty hall. At one instant, mark every point where the air pressure has just reached its peak. These points lie on a sphere around the speaker: they are all the same distance away, so the disturbance reached them together and they oscillate in step.

A surface like this, on which every point is at the same stage (phase) of its oscillation, is called a wavefront (तरंगाग्र). As time passes the wavefront moves outward at the wave speed $v$, and the wave carries its energy at right angles to it. That is why rays, the lines along which energy flows, are drawn perpendicular to the wavefronts in any medium in which light has the same speed in every direction.

The shape of a wavefront depends on the source and on how far away you are (Fig. 10.2). A point source sends out spherical wavefronts, with rays spreading out radially. Far from the source, any patch you look at is a tiny piece of an enormous sphere and is practically flat: a plane wavefront, with parallel rays. Sunlight at your window is an example. The Sun is $1.5\times {10}^{11}\mathrm{m}$ away, so across a window $1\mathrm{m}$ wide its wavefront bulges by only about $8\times {10}^{-13}\mathrm{m}$, roughly a millionth of a wavelength. For a bulb $2\mathrm{m}$ from the same window the bulge is about $6\mathrm{c}\mathrm{m}$, so its wavefronts are clearly curved.

Huygens' insight was a recipe for finding where a wavefront will be a little later, if we know where it is now: treat every point on the present wavefront as a small source in its own right.

Fig. 10.3 applies the recipe to part of a spherical wavefront. Around each marked point, draw a sphere of radius $v\tau$. The surface that touches all of them on the outside is again a sphere centred on the source, larger by $v\tau$: the wave has simply moved outward. For a plane wavefront the same construction gives a parallel plane a distance $v\tau$ further on. In both cases the rays, which are normals to the wavefronts, carry on in straight lines, as they should in a uniform medium.

10.3 Refraction and reflection of plane waves using Huygens principle

Huygens' principle earns its keep when a wavefront meets the boundary between two media. The part of the front that has crossed travels at a different speed from the part that has not, and the construction shows exactly how the front turns.

10.3.1 Refraction of a plane wave

Let light travel at speed $v_1$ in medium 1 and $v_2$ in medium 2, with $v_2<v_1$ (from air into glass, say). A plane wavefront meets the flat boundary at an angle, so one end of it touches the boundary before the other (Fig. 10.4).

Equation (10.3) is the law of refraction, and it carries a prediction. A ray that bends towards the normal has $r<i$, so $\sin i\mathrm{/}\sin r>1$ and therefore $v_2<v_1$: light must travel more slowly in the medium in which it bends towards the normal. This is the wave model's side of Fig. 10.1, and Foucault's measurement agrees with it.

The refractive index of a medium compares the speed of light $c$ in vacuum with its speed in the medium:

Substituting $v_1=c\mathrm{/}{n}_1$ and $v_2=c\mathrm{/}{n}_2$ into (10.3) gives Snell's law in its familiar form:

What happens to the wavelength? Choose $\tau$ to be exactly one period $T$ of the wave. In one period each part of the wave advances by one wavelength, so $QR={\lambda }_1$ and $PS={\lambda }_2$, and (10.1) and (10.2) give

Since $v\mathrm{/}\lambda$ is the frequency $\nu$, the frequency is the same on both sides of the boundary. Only the speed and the wavelength change, and both change by the same factor $n_1\mathrm{/}{n}_2$. Light entering glass of refractive index $1.5$ from air keeps its frequency, but its speed and its wavelength drop to two-thirds of their values in air.

10.3.2 Refraction at a rarer medium

Now reverse the situation, so that light passes from glass or water into air and $v_2>v_1$. The construction is the same, but the wavelet in medium 2 now grows faster than light from $Q$ closes the gap to the boundary. The refracted wavefront tilts the other way and the ray bends away from the normal, $r>i$, with Snell's law still holding (Fig. 10.5).

As $i$ increases, $r$ reaches ${90}^{\circ }$ first. The angle of incidence at which this happens is the critical angle $i_c$ (क्रांतिक कोण). Putting $r={90}^{\circ }$ in (10.6) gives

The construction also shows why nothing can be refracted beyond this angle. By (10.2), at $i=i_c$ the wavelet from $P$ grows, in the time $\tau$, just far enough to reach $R$ itself: $v_2\tau =PR$, and the refracted wavefront stands perpendicular to the boundary, skimming along it. For $i>i_c$ the wavelet has already passed $R$ by the time light from $Q$ arrives there, and no plane through a point inside a sphere can touch the sphere. So there is no refracted wave at all, and all the light is reflected back into the denser medium. This is the total internal reflection you met in the previous chapter.

10.3.3 Reflection of a plane wave by a plane surface

With reflection and refraction explained, we can follow a wavefront through any optical device. One rule covers them all: a part of the front that spends longer in slow glass, or that has further to travel before a mirror sends it back, falls behind the rest, and the front changes shape accordingly (Fig. 10.7, Table 10.1). Because rays are perpendicular to the front, its new shape tells us at once where the rays go.

10.4 Coherent and incoherent addition of waves

What happens when two waves reach the same point? For ripples, sound and light alike, experiment gives a simple answer, the superposition principle (अध्यारोपण का सिद्धांत):

For light, the 'displacement' is the electric field of the wave. Interference is this principle applied to two waves of the same frequency, and whether we see anything interesting depends on whether the two waves keep in step.

Fix two prongs to one vibrating arm so that they dip into a ripple tank together. Each prong is a source of circular ripples, and because a single motor drives both, they rise and fall in step indefinitely. Sources whose phase difference stays constant like this are called coherent sources (कला-संबद्ध स्रोत). Fig. 10.8 shows a snapshot of their ripples: crests as solid circles and troughs as dashed ones.

Along some lines crest always meets crest and trough meets trough, and the water heaves twice as high as with one source. Along others crest always meets trough, and the water hardly moves. Which happens at a point $P$ depends only on the path difference $\mathrm{\Delta }=S_{2}P-S_{1}P$ (पथांतर), measured in wavelengths.

If $\mathrm{\Delta }=0$, as anywhere on the perpendicular bisector of $S_1{S}_2$, the two waves set off in step and arrive in step. If $\mathrm{\Delta }=\lambda$ or $3\lambda$, the wave from the farther source arrives one or three whole cycles late, which puts it in step again. But if $\mathrm{\Delta }=\lambda \mathrm{/}2$ or $3.5\lambda$, the late wave is behind by an odd number of half cycles, so crest arrives with trough. A path difference $\mathrm{\Delta }$ corresponds to a phase difference $\varphi =2\pi \mathrm{\Delta }\mathrm{/}\lambda$. With $n=0,1,2,\dots$, the two conditions are

Each condition picks out a curve. Points with a fixed difference of distances from two fixed points lie on a hyperbola, so the lines of strong motion (antinodal lines) and of stillness (nodal lines) in Fig. 10.8 are hyperbolas. They fan out from between the sources and become nearly straight far away, and a screen placed across them shows a row of strong and weak spots.

Now suppose the two prongs are driven by separate motors that drift in and out of step. The phase difference $\varphi$ at any point then wanders through all values, and the pattern of Fig. 10.10 slides back and forth across the tank. If $\varphi$ changes much faster than we can observe, we see only the time average of ${\cos }^2(\varphi \mathrm{/}2)$, which is $\frac{1}{2}$:

Such sources are incoherent: their intensities simply add, and no fringes appear. The two headlights of a car shining on a garage door behave like this. Where both beams fall, the door is twice as bright as with one, evenly, without a trace of a pattern.

10.5 Interference of light waves and Young's experiment

Try the ripple-tank experiment with light: put a small green LED behind each of two pinholes and look at a screen beyond them. It is evenly lit, with no fringes. The reason lies in how such sources make light. The light is a jumble of short wave trains from countless atoms, and its phase wanders at random; for a green LED it is effectively scrambled every few times ${10}^{-14}\mathrm{s}$. In the twentieth of a second your eye needs to register a picture, any fringe pattern would be replaced more than a million million times. Two independent sources are always incoherent.

Young's way round this was not to look for better lamps but to split one source in two. In Fig. 10.11 a single narrow slit $S$ passes light from a small region of the lamp, and that light reaches two slits $S_1$ and $S_2$ a fraction of a millimetre apart. The two slits are now like two loudspeakers wired to the same amplifier: however erratically the lamp behaves, both receive the same wave, so the difference between their phases never changes. Today a laser, whose light stays in step over long distances, makes the experiment easy. Beyond the slits, bright and dark bands called interference fringes (व्यतिकरण) appear on a screen.

Four things follow from (10.13) and (10.14):

• The fringes are equally spaced, bright and dark alternating, with a bright fringe at the centre $O$, where the path difference is zero.
• $\beta \propto \lambda$: red fringes are wider than blue ones. In white light the central fringe is white, since every colour has zero path difference there, and it is flanked by a few coloured fringes that soon blur together.
• $\beta \propto 1\mathrm{/}d$: bringing the slits closer spreads the fringes out. That is why the slits must be very close, typically a fraction of a millimetre apart.
• Immersing the whole apparatus in a liquid of refractive index $n$ shortens the wavelength to $\lambda \mathrm{/}n$, so the fringes close up by the same factor.

10.6 Diffraction

Remember the dark lines you saw between two nearly touching fingers held up to a tube light (Section 10.1)? Light squeezing through the narrow gap does not go straight on as ray optics says. It spreads out, and the spread-out light forms bright and dark bands. With a small enough source you can see the same thing at the edge of any shadow, which is never perfectly sharp. This spreading of waves around obstacles and through openings is called diffraction (विवर्तन).

Every kind of wave diffracts. Sea waves passing the end of a harbour wall curl round into the sheltered water behind it, and the band playing in the next street reaches you round the corner of a building. Light diffracts just as readily, but its wavelength is about a million times shorter than that of sound, so the effects become noticeable only for openings and obstacles a fraction of a millimetre across. That is why everyday shadows look sharp, and why ray optics works so well.

Diffraction also sets a hard limit on optical instruments. A lens or mirror of finite size cannot focus a point of light to a perfect point: it makes a tiny diffraction blur, and two objects closer together than about the size of that blur cannot be told apart, however well the lens is made. The finest detail a microscope or telescope can show is therefore set by the wavelength of the light. You can see diffraction by a regular pattern of openings, too: look at a distant street light through the cloth of an umbrella, and each gap between the threads spreads the light into a grid of spots fringed with colour.

10.6.1 The single slit

Let a parallel beam of monochromatic light fall straight onto a narrow slit of width $a$, and look at the light on a distant screen (or at the focus of a lens placed after the slit). What appears is not a sharp strip the width of the slit. The middle of the screen holds a wide bright band, and on each side lies a row of much dimmer bands, separated by dark gaps and fading with distance. The narrower the slit, the more the whole pattern spreads.

Treat the slit as a row of many tiny wavelet sources (Huygens again). The incoming wavefront lies flat across the slit, so every source starts at the same phase; what differs is how far each one is from the point where we look. In a direction $\theta$, the wavelet from the lower edge travels $a\sin \theta \approx a\theta$ further than the one from the upper edge, and the wavelets in between are spread evenly over this range (Fig. 10.13).

The central maximum runs from $-\lambda \mathrm{/}a$ to $+\lambda \mathrm{/}a$, so its angular width is $2\lambda \mathrm{/}a$, twice that of the other bands. On a screen at a distance $D$ it is $2\lambda D\mathrm{/}a$ wide. Making the slit narrower spreads the pattern out; making it much wider than the wavelength shrinks the pattern to a sharp image of the slit, and ray optics takes over.

Both effects are superposition at work, and there is no deep physical line between them. Richard Feynman remarked that the difference is only a matter of usage: when a few sources are combined we tend to say interference, and when very many are combined we say diffraction.

In a real double-slit experiment each slit has a width of its own, so both effects act at once. Each slit spreads its light into a single-slit pattern, and within that spread the two slits interfere. The result is Young's fringes, spaced $\lambda D\mathrm{/}d$, whose brightness follows the single-slit pattern as an envelope (Fig. 10.14). Where an interference maximum falls on a diffraction minimum, that fringe is missing.

10.6.2 Seeing the single slit diffraction pattern

You can see a single-slit pattern with things found at home or in the school laboratory. All you need is a narrow, adjustable slit and a thin, bright line of light.

Now hold two fingers almost touching in front of the same lamp, as at the start of the chapter. The dark lines you see in the gap are this same diffraction pattern, made by a slit whose width you can change by squeezing your fingers together.

Here is a check worth making. For two slits, the intensity swings between $0$ and $4I_0$, and its average over a fringe is $2I_0$: exactly what the two slits would deliver separately. A dark fringe is therefore not light destroyed but light sent elsewhere, into the neighbouring bright fringes. The same bookkeeping holds for a single slit, where the energy spread over the whole pattern equals the energy that passed through the opening, as conservation of energy requires.

10.7 Polarisation

Tie one end of a long rope to a pole and shake the free end up and down regularly. A wave runs along the rope (call this the $x$ direction), and each bit of rope moves only up and down, along $y$:

Here $a$ is the amplitude, $\omega =2\pi \nu$ is the angular frequency, and

is the angular wave number. Watch any one bit of the rope: it moves up and down along a fixed line, always at right angles to the direction in which the wave travels, so this is a transverse wave. A transverse wave in which every particle keeps to one fixed line is linearly polarised (रैखिकतः ध्रुवित). Here the line is the $y$ axis, so this is a $y$-polarised wave; because the whole rope stays in the $x$–$y$ plane, such a wave is also called plane polarised. Shake the rope from side to side instead and you get a $z$-polarised wave:

If you keep changing the direction of shaking at random, many times a second, the vibration is still at right angles to the rope but has no fixed direction: the wave is unpolarised. Now pass the rope through a vertical slot in a board (Fig. 10.15). An up-and-down wave goes through, a side-to-side wave is stopped, and an unpolarised wave comes out vertically polarised, with only its up-and-down part left.

Light behaves in the same way, and that is our evidence that light waves are transverse. In a light wave the electric field $𝐄$ is always perpendicular to the direction of travel, and the direction in which $𝐄$ oscillates is called the direction of polarisation (ध्रुवण). Light from the Sun, a bulb or a candle is unpolarised: its atoms emit independently, so the direction of $𝐄$ changes rapidly and at random, though always within the plane perpendicular to the ray.

A polaroid acts on light as the slot acts on the rope. Every polaroid has one special direction in its plane, its pass-axis: the component of $𝐄$ along this direction goes through, and the component at right angles to it is absorbed. What makes the direction special is the material. A typical sheet is polyvinyl alcohol, stretched so that its long-chain molecules line up, and stained with iodine so that electrons can move along the chains. An electric field along the chains drives small currents in them and loses its energy; a field across the chains cannot, and passes. So the pass-axis is perpendicular to the chains. Edwin Land invented polarising sheets in 1929, and this stretched, iodine-stained kind has been in use since 1938.

Put one polaroid in front of a lamp. Of the randomly oriented $𝐄$, only the component along the pass-axis gets through. Averaged over all directions, that is half the intensity, whichever way you turn the sheet, so rotating it changes nothing you can see. The light that emerges, though, is now linearly polarised along the pass-axis.

Add a second polaroid. If its pass-axis is parallel to the first one's, the polarised light goes straight through. Turn it, and the light dims; at ${90}^{\circ }$ (crossed polaroids) almost nothing gets through. Fig. 10.16 shows why. Let the light leaving the first sheet have amplitude $E_0$ and intensity $I_0$, and let the second pass-axis make an angle $\theta$ with the first. Only the component $E_0\cos \theta$ along the second axis is transmitted, and intensity goes as amplitude squared.

Two sheets therefore make a dimmer: turning one of them sets the transmitted intensity anywhere between half the lamp's intensity (axes parallel) and zero (axes crossed). As the second sheet turns through a full circle, the light passes through two maxima, at $\theta =0^{\circ }$ and ${180}^{\circ }$, and two minima, at ${90}^{\circ }$ and ${270}^{\circ }$.