Wave Optics & Dual Nature of Radiation and Matter

Huygens · interference · diffraction · the photoelectric effect · the photon · matter waves · Class 12 Physics

1The big idea: two faces of light

This chapter tells the strangest true story in physics. In the first half, light behaves as a wave — it bends around corners, overlaps with itself and produces bright and dark bands that only waves can make. In the second half, that same light behaves as a stream of particles — packets of energy called photons that knock electrons out of metals one at a time. Neither picture is wrong; light is both, and which face it shows depends on the experiment you do.

Then comes the twist that won Louis de Broglie a Nobel Prize: if waves can act like particles, then particles can act like waves. Electrons, which everyone had counted as tiny bullets, turn out to diffract like light. So wave–particle duality is not a quirk of light — it is a property of everything. Hold that sentence; it is the whole chapter.

2Huygens' principle

Before we can talk about interference we need a way to predict where a wave goes next. Christiaan Huygens gave the recipe in 1678, and it is beautifully simple:

A wavefront is simply the set of all points vibrating in step — the crest of a ripple, say. Draw a wavefront, sprout a little expanding sphere from every point on it, wait a moment, and the smooth surface wrapping the front edges of those spheres is where the wave has advanced to. That single construction explains the two facts you already know: a wavefront far from a point source is flat (a plane wave), and light travelling in a straight line is just a plane wavefront marching forward, ray after ray perpendicular to it.

3Interference & coherent sources

When two waves arrive at the same place, their displacements simply add — this is the principle of superposition. If two crests meet, you get a bigger crest (constructive interference, extra bright). If a crest meets a trough, they cancel (destructive interference, dark). What decides which happens at a given point is the path difference — the extra distance one wave travelled compared with the other.

• Constructive when the path difference is a whole number of wavelengths: Δ = nλ (the waves arrive in step).
• Destructive when it is an odd number of half-wavelengths: Δ = (n + ½)λ (they arrive exactly out of step).

But there is a catch, and it is why you do not see interference bands from two ordinary bulbs. To get a steady pattern the two sources must be coherent: they must keep a constant phase difference in time (and hence the same frequency). Two independent bulbs flicker in phase randomly billions of times a second, so their bright and dark regions reshuffle far too fast to see and just average to a uniform glow. The trick Thomas Young used to beat this is the key to the whole experiment: split one source in two. Light from a single slit falls on two nearby slits, so whatever phase wobble the source has is shared identically by both — their phase difference stays locked, and the pattern stands still.

4Young's double slit & the fringe width

Two slits S₁ and S₂, a small distance d apart, are lit by coherent light of wavelength λ. A screen sits a distance D away, with D ≫ d. Consider a point P on the screen a height x above the centre.

The wave from S₂ has to travel slightly further than the wave from S₁. Because the screen is far away, the two rays are almost parallel and the extra distance — the path difference — is

using sin θ ≈ tan θ = x/D for the small angles involved. Now impose the two conditions from the previous section:

with n = 0, 1, 2, … The centre of the screen (x = 0, zero path difference) is always bright — the central maximum. The fringe width β is the spacing between two neighbouring bright fringes (or two dark ones). Subtract consecutive bright positions, x_n+1 − x_n:

Read what this formula is telling you. The fringes are evenly spaced (β does not depend on n). They get wider if you use a longer wavelength or move the screen back, and they get narrower if you push the slits further apart. That inverse dependence on d is the single most tested fact here — and it is exactly what you will feel in the activity below.

5Single-slit diffraction, in brief

Diffraction is the bending of a wave around an obstacle or through an aperture — the reason a wave can reach into the "shadow". Send monochromatic light through a single narrow slit of width a and, instead of a sharp bright line, you get a broad central bright band flanked by dimmer bands on each side.

The trick to locating the dark bands is to imagine the single slit sliced into many tiny Huygens sources across its width a. At an angle θ where the light from the top edge is exactly one wavelength ahead of the light from the bottom edge — that is, a sin θ = λ — the slit pairs up top-half against bottom-half and every wavelet finds a partner half a wavelength out of step, so they cancel completely. The general condition for a minimum is therefore

Watch the trap here: for the double slit, d sin θ = nλ gave the bright fringes, but for the single slit, a sin θ = nλ gives the dark ones. The central maximum runs between the first dark band on either side, so its angular half-width is θ ≈ λ/a and its width on a screen a distance D away is

The central bright band is twice as wide as the others and carries most of the light. And notice: the narrower the slit, the wider the spread — squeeze a wave and it fans out. That is diffraction in one sentence, and it is what ultimately limits the sharpness of every lens and telescope.

6The photoelectric effect

Now light changes its coat. Shine light on a clean metal surface and, above a certain colour, electrons are ejected. This is the photoelectric effect, and the details refused to fit the wave picture — which is exactly why it forced physics to invent the photon.

Here is what experiment stubbornly showed:

• Below a certain threshold frequency f₀, no electrons come out — no matter how intense (bright) the light or how long you wait.
• Above f₀, electrons appear instantly, and their maximum kinetic energy (KE) depends only on the frequency of the light, not on its intensity.
• Turning up the intensity ejects more electrons (a larger current) but does not make them faster.

Wave theory predicted the opposite on every count: a bright enough light of any colour should, given time, shake an electron loose, and brighter light should give faster electrons. Einstein (1905) resolved it by proposing that light delivers its energy in indivisible packets — photons — each carrying energy E = h f, where h = 6.63 × 10⁻³⁴ J s is Planck's constant. One electron absorbs one photon, all or nothing. Part of that energy pays the work function φ₀ — the minimum energy binding the electron to the metal — and whatever is left over becomes kinetic energy:

Every stubborn fact now falls out for free. If h f < φ₀ the photon simply cannot pay the entrance fee, so nothing happens below the threshold f₀ = φ₀/h — and no amount of piling up too-weak photons helps, because the electron only ever gets one at a time. Above it, the leftover energy hf − φ₀ is set purely by frequency. Brighter light just means more photons, hence more electrons, but each still carries the same energy.

To measure that maximum KE you apply a reverse voltage that just stops the fastest electron from reaching the collector — the stopping potential V₀. Then

A graph of V₀ against frequency f is a straight line of slope h/e — the same slope for every metal — with an intercept fixed by that metal's work function. Millikan measured exactly this line and pinned down h, sealing the argument.

7The photon

So the photon is a real particle of light. It has:

• Energy E = h f = h cλ — higher frequency (bluer) means more energetic.
• Momentum p = hλ = Ec — yes, light carries momentum, which is how it can push on things (radiation pressure).
• Zero rest mass, and it always travels at c = 3 × 10⁸ m s⁻¹ in vacuum.

A convenient shortcut for numbers: hc ≈ 1240 eV·nm, so a photon's energy in electron-volts is just 1240 / λ with λ in nanometres. Green light at 500 nm carries about 2.5 eV — right in the range of typical metal work functions, which is why the photoelectric effect happens with visible and ultraviolet light.

8Matter waves & the Davisson–Germer experiment

De Broglie asked the daring reverse question. If a wave (light) carries momentum p = h/λ, then perhaps anything with momentum has a wavelength. He simply turned the relation around:

Because h is minuscule, the wavelength of any everyday object is absurdly small — a thrown cricket ball has λ ≈ 10⁻³⁴ m, far too tiny to ever notice, which is why we never see footballs diffract. But an electron is light enough for its wave to matter. Speed it up through a voltage V, so its kinetic energy is eV = p²/2m, and the wavelength becomes

(with V in volts). At a modest 100 V this gives λ ≈ 0.123 nm — comparable to the spacing between atoms in a crystal, and therefore something a crystal can diffract.

Davisson and Germer (1927) did precisely that. They fired a beam of electrons, accelerated through 54 V, at a nickel crystal and measured how many bounced off at each angle. Instead of a featureless scatter they found a sharp peak at a scattering angle of 50° — a diffraction maximum, the unmistakable signature of a wave interfering off the regular rows of atoms. The wavelength they backed out of the crystal geometry was about 0.165 nm, and de Broglie's formula for a 54 V electron predicts 0.167 nm. The two agree. Electrons are waves, matter is dual, and the story that opened with Huygens closes with a beam of particles bending like light.

9Common confusions to clear up

• Coherent ≠ merely "same colour". Two lasers of identical wavelength are still incoherent because their phase difference drifts randomly. Coherence needs a constant phase relationship — which is why the pattern comes from one source split in two, not from two separate lamps.
• Single-slit vs double-slit conditions are mirror images. For two slits, d sin θ = nλ gives a bright fringe; for one slit, a sin θ = nλ gives a dark band. Memorising one and assuming the other matches will cost you marks.
• Intensity does not change electron speed. In the photoelectric effect, brighter light means more electrons, never faster ones. Only raising the frequency raises the maximum KE and the stopping potential.
• Below the threshold, nothing happens — ever. There is no "wait long enough and it builds up". One electron absorbs one photon; if that photon is too weak, no waiting or brightening helps.
• Everything has a de Broglie wavelength, but only light things show it. A cricket ball's wavelength is real but ~10⁻³⁴ m, so it never diffracts. Electrons work because their wavelength lands near atomic spacings.
• β is inverse in d, direct in λ and D. Push the slits apart → fringes crowd together. Longer wavelength or a further screen → fringes spread out.

10Check yourself

Class 12 Physics · Wave Optics & Dual Nature of Radiation and Matter · aligned to NCERT Chapters 10–11 · SmartStudy.School