Diffraction and interference

Superposition, interference and diffraction

Water ripples, sound, light and microwaves are all waves. When two waves meet, they do not bounce off each other like balls. They pass through each other, and while they overlap their displacements simply add. This one idea, the principle of superposition, explains both topics of this unit: interference and diffraction.

Principle of superposition

When two or more waves overlap at a point, the resultant displacement is the sum of the displacements of the individual waves at that point.

Interference

The effect produced when two or more waves overlap and combine to give a resultant wave of larger or smaller amplitude.

Diffraction

The spreading (bending) of waves around the edges of an obstacle or through a narrow opening into the region of geometrical shadow.

Here $y_1$ and $y_2$ are the displacements that each wave alone would produce at a point, and $y$ is the resultant displacement.

Constructive and destructive interference

If the waves arrive in phase (crest on crest, trough on trough), they add to give a wave of larger amplitude. This is constructive interference: brighter light, louder sound, bigger ripples.

If the waves arrive in anti-phase (phase difference $\pi$ rad or $180^{\circ }$, crest on trough), they cancel. This is destructive interference: darkness, silence or calm water.

Diffraction depends on the size of the gap

When a wave passes through a gap or past an edge, it spreads into the region behind the obstacle. Every point on a wavefront acts as a source of new wavelets (Huygens' principle), and at the edges nothing cancels the sideways-moving wavelets. The spreading is large when the gap width $a$ is about the same as the wavelength $\lambda$, and small when the gap is much wider than $\lambda$.

18.1Two-source interference and conditions for fringesOften examined

Two-source interference (also called double-slit interference) happens when waves from two coherent sources overlap. At some points the waves always reinforce each other and at other points they always cancel, so a fixed pattern of strong and weak regions, called fringes, appears. It can be shown with water waves in a ripple tank, sound waves, light waves and microwaves.

Coherent sources

Two sources that emit waves of the same frequency (and so the same wavelength) with a constant phase difference between them.

Monochromatic light

Light of a single wavelength, that is, of a single colour.

Path difference

The difference between the distances travelled by two waves from their sources to the same point, $\Delta x=S_{2}P-S_{1}P$. SI unit: metre.

Phase difference

The difference in the phases of two waves at a point, measured in radians or degrees.

Path difference and phase difference

Two coherent sources that start in phase can still produce waves that arrive out of phase, because the waves travel different distances. The book says this simply: the phase difference is produced by the path difference.

Conditions for constructive and destructive interference

Conditions for observing interference fringes

A steady (sustained) pattern that we can see or measure needs the conditions below. The first four are the book's list; the last two make the fringes clear and easy to see.

1. Coherent sources. The two sources must have the same frequency and a constant phase difference. If the phase difference keeps changing, the bright and dark fringes swap places millions of times per second and the eye sees only even illumination. For light, two coherent sources are made from one source by splitting its wavefront (Young's double slit), or a laser is used.
2. Narrow slits. Each slit must be narrow (its width comparable to the wavelength), so that light spreads out from it and the beams from the two slits overlap.
3. Monochromatic waves. The waves should have a single wavelength. Different wavelengths give fringes of different spacing, which overlap and blur the pattern.
4. Stable environment. The set-up must be free of vibrations and air currents, which would change the path difference and make the fringes move.
5. Small slit separation and a distant screen. The fringe spacing $\Delta y=\lambda L/d$ is large enough to see only if the slit separation $d$ is small (a fraction of a millimetre for light) and the screen distance $L$ is large (about $1\mathrm{m}$).
6. Nearly equal amplitudes. When the two waves have equal amplitudes, destructive interference gives complete darkness, so the fringes have good contrast.

18.1.1Interference of water waves

A ripple tank is a shallow, glass-bottomed tray of water with a lamp above it. The lamp throws a shadow of the ripples onto white paper below the tank: each crest acts like a converging lens and makes a bright line, and each trough makes a dark line. So the moving water pattern can be seen clearly on the paper.

To make two coherent sources, two small dippers are fixed to the same vibrating bar. Because one motor drives both, they vibrate with the same frequency and stay in phase. Each dipper sends out circular waves, and where the two sets of waves overlap they interfere.

• Antinodal lines (A): crest meets crest and trough meets trough, so the water moves up and down strongly and the paper shows sharp bright and dark bands. Path difference $=m\lambda$.
• Nodal lines (N): crest meets trough, so the water stays almost still and the paper shows an even grey. Path difference $=\left(m+\frac{1}{2}\right)\lambda$.
• The central line $A_0$, the same distance from both sources, is always antinodal, because its path difference is zero.

What changes the pattern

• Higher frequency (shorter wavelength): more nodal and antinodal lines, packed closer together.
• Larger source separation $d$: more lines, closer together.
• Only one dipper: plain circular waves, with no nodal lines, so no interference.
• Dippers out of step by half a cycle: the whole pattern shifts, and the central line becomes nodal.

18.1.2Interference of sound waves

Two loudspeakers connected to the same signal generator and amplifier give out sound of the same frequency, in phase, so they are coherent sources. Sound is a longitudinal wave made of compressions (high pressure) and rarefactions (low pressure).

Where a compression from one speaker arrives together with a compression from the other (or a rarefaction with a rarefaction), the waves interfere constructively and the sound is loud. Where a compression arrives with a rarefaction, they interfere destructively and the sound is quiet. In the book's Fig. 18.4, points $P_1$, $P_4$, $P_5$, $P_6$ and $P_9$ are loud and points $P_2$, $P_3$, $P_7$ and $P_8$ are quiet.

A person walking slowly across the room in front of the speakers hears the sound grow louder and softer. A microphone connected to a meter or an oscilloscope (CRO) shows the same thing as a rise and fall of the signal.

Beats: interference in time

If two sources have slightly different frequencies (for example tuning forks of $256\mathrm{Hz}$ and $260\mathrm{Hz}$), they are not coherent. At one place the two waves drift in and out of step. When they are in phase the sound is loud; a short time later they are in anti-phase and the sound is soft. This regular rise and fall of loudness is called beats. The number of beats heard per second equals the difference between the two frequencies, so forks of $256\mathrm{Hz}$ and $260\mathrm{Hz}$ give $4$ beats per second.

18.1.3Interference of light: Young's double-slit experimentOften examined

Light waves are very short (about $4\times 10^{-7}$ to $7\times 10^{-7}\mathrm{m}$), so light interference is much harder to see than ripple-tank interference. In 1801 the English physicist and physician Thomas Young found a way to do it. At that time scientists disagreed about light: Huygens described it as a wave, while Newton's followers explained it as a stream of particles. Young's experiment showed clearly that light behaves as a wave.

Apparatus

• Source: monochromatic light (a sodium lamp, or a laser in modern versions). Young used sunlight.
• Single slit $S_0$: a narrow slit that lets through light from one small part of the source. Light spreading from $S_0$ reaches both slits of the double slit from the same wavefront.
• Double slit $S_1$, $S_2$: two narrow parallel slits, a small distance $d$ apart (about $0.1$ to $1\mathrm{mm}$), equidistant from $S_0$. Because their light comes from the same wavefront, $S_1$ and $S_2$ act as two coherent sources in phase.
• Screen: placed a distance $L$ (about $1\mathrm{m}$) from the double slit.

Working

Light from $S_0$ spreads (diffracts) and falls on both slits. Each slit diffracts the light again and sends out semicircular wavefronts. The two sets of waves overlap in the space in front of the slits and interfere. On the screen we see alternate bright and dark bands called fringes, parallel to the slits (book Fig. 18.6).

• At the centre O the paths $S_{1}O$ and $S_{2}O$ are equal, so the waves arrive in phase: the central fringe is bright (the central maximum, $m=0$).
• Where the path difference is $\lambda ,2\lambda ,3\lambda ,\dots$ the waves again arrive in phase: bright fringes.
• Where the path difference is $\frac{1}{2}\lambda ,\frac{3}{2}\lambda ,\frac{5}{2}\lambda ,\dots$ the waves arrive in anti-phase: dark fringes.
• The fringes are equally spaced, and the bright fringes are all nearly equally bright.

Using white light

Young used sunlight, which is a mixture of wavelengths. Each colour forms its own set of fringes with its own spacing (spacing is proportional to $\lambda$). At the centre every colour has zero path difference, so the central fringe is white. On each side the fringes are coloured, with violet (short $\lambda$) on the inside and red (long $\lambda$) on the outside. After a few fringes the colours overlap and the pattern fades to even white. For a clear pattern with many fringes, monochromatic light is used.

The book's Fig. 18.6 (c) shows that the double-slit pattern of water waves looks almost the same as that of light. This close match is why Young's experiment is taken as evidence that light is a wave.

18.1.3Path difference, fringe positions and fringe spacingOften examined

We now find exactly where the bright and dark fringes form. The symbols are: $d$, the separation of the slits; $L$, the distance from the slits to the screen; $y$, the distance of a point P on the screen from the centre O; and $\theta$, the angle between the central line MO and the direction MP.

Using the conditions for interference with path difference $d\sin \theta$:

The integer $m$ is the order of the fringe. $m=0$ is the central maximum; $m=\pm 1$ are the two first-order bright fringes, one on each side of O, and so on. In the book's numbering the first dark fringe has $m=0$ and is called the first-order minimum.

Fringe spacing (fringe width)

The distance between the centres of two consecutive bright fringes, or of two consecutive dark fringes: $\Delta y=\lambda L/d$. SI unit: metre.

Factors that change the fringe spacing

18.1.4Interference of microwaves

Microwaves are electromagnetic waves with wavelengths of a few centimetres (3\,\mathrm{cm} at a frequency of 10\,\mathrm{GHz}). Because the wavelength is large, the slits can be gaps a few centimetres wide between metal plates, and the interference pattern is spread over a large region, so it is easy to measure.

• Transmitter: sends a beam of microwaves of one frequency towards the metal plates.
• Metal plates with two slits: metal reflects microwaves, so waves pass only through the two gaps. Both gaps receive the same wavefront, so they act as two coherent sources in phase.
• Detector (receiver probe) connected to a meter: moved along an arc or a line in front of the slits to measure the intensity.

As the detector moves, the meter reading rises and falls: it is high where the waves from the two gaps interfere constructively and very low (almost zero) where they interfere destructively. The pattern can also be shown on a recording device connected to the detector.

Finding the wavelength of microwaves

18.2Diffraction grating and grating elementOften examined

A diffraction grating is a plate of glass or plastic with a very large number of equally spaced, parallel slits (lines) ruled or etched on it, for example 600 lines per millimetre. The clear gaps between the ruled lines act as slits. When light falls on the grating, each slit becomes a source of secondary waves, and the waves from all the slits interfere.

Diffraction grating

An optical device with a large number of closely and equally spaced parallel slits (lines or grooves). It diffracts light into sharp bright maxima at definite angles.

Grating element (grating constant) d

The distance between the centres of two adjacent slits of a grating. It equals the width of one slit plus the width of one opaque line, d=a+b. SI unit: metre.

If N lines are ruled on each unit length of the grating, the grating element is

In a transmission grating light passes through the clear gaps. In a reflection grating light reflects from many fine, parallel grooves; the surface of a CD or DVD behaves like one.

The number of lines per unit length decides how far apart the orders are. The total number of lines covered by the beam decides how sharp the maxima are, and so how well the grating separates two wavelengths that are close together (its resolving power).

18.2.1Diffraction of light through a gratingOften examined

When monochromatic light falls normally (at right angles) on a grating, the light that comes through is not spread evenly. It leaves only in a few sharp, bright beams at definite angles. In between, the waves from the many slits cancel almost completely. The principle of the grating is therefore interference of the waves diffracted by all the slits.

Determining the wavelength of light with a grating

1. Let a parallel beam of monochromatic light (for example from a sodium lamp) fall normally on a grating with a known number of lines per unit length N, so that d=1/N.
2. Find the central maximum (\theta=0). Then find the first-order maxima on both sides and measure the angle between them; half of this angle is \theta_1.
3. Repeat for the second order if it can be seen.
4. Calculate \lambda=d\sin\theta/m for each order and take the average.

Why grating maxima are sharper than double-slit fringes

A double slit and a grating with the same slit spacing d give maxima at the same angles. But with many slits, a very small change of angle away from d\sin\theta=m\lambda puts the waves from distant slits out of step, and they cancel. So the maxima become very narrow and bright, and the regions between them become darker, as the number of slits increases.

White light and spectra

With white light the central maximum (m=0) is white, because \theta=0 for every wavelength. In every other order each colour appears at its own angle, \sin\theta=m\lambda/d, so red (longest wavelength) is diffracted most and violet least. Each order is a spectrum with violet on the inside and red on the outside. Higher orders are more spread out, and the second- and third-order spectra overlap.

How the pattern changes

• More lines per unit length (smaller d): each order moves to a larger angle, so the maxima spread farther apart; fewer orders are possible, because \sin heta=m\lambda/d cannot be more than 1; the maxima become sharper.
• More slits illuminated (same d): the maxima stay at the same angles but become narrower and brighter, with darker regions between them.
• Narrower slits (same d): each slit spreads its light more widely, so the higher orders are brighter. With wider slits most of the light stays near the centre and the higher orders become faint.
• Longer wavelength: every order appears at a larger angle.

18.2.2Applications of diffraction gratings

1. Wavelength measurement: a diffraction grating is used to determine the wavelength of light accurately.
2. Atomic spectra analysis: every element emits its own set of wavelengths (its line spectrum). A grating separates them, so elements can be identified in a laboratory sample or in a distant star.
3. Biomedical imaging: gratings select particular wavelengths, for example to examine biopsy samples for signs of disease.
4. Spectroscopy: gratings disperse light into its wavelengths for detailed analysis in spectrometers used in chemistry, astronomy and environmental monitoring.

Gratings in nature and daily life

Australian opal contains tiny silica spheres packed in regular rows, and butterfly wings carry regular rows of microscopic scales and ridges. These act like reflection gratings: they send different colours in different directions, so the colours change as the opal or the wing is tilted (book Fig. 18.11). Such colours are called structural colours.

A CD or DVD stores data along a spiral track. On a CD neighbouring turns of the track are 1.6\,\mu\mathrm{m} apart, close to the wavelength of visible light, so the surface acts as a reflection grating and shows rainbow colours. These colours are only a side effect; the track exists to store data.

X-ray diffraction by crystals

X-rays have wavelengths of about 10^{-10}\,\mathrm{m}, similar to the spacing of atoms in a crystal. A crystal therefore acts as a three-dimensional diffraction grating for X-rays. This is the basis of X-ray crystallography, mentioned in the book's introduction.

Diffraction at a single slit

When light passes through one narrow slit of width a, the screen shows a broad, bright central band with much weaker bands on each side. By Huygens' principle every point in the slit sends out wavelets, and the wavelets from different parts of the same slit interfere.

• The central maximum is twice as wide as the other bright bands and much brighter.
• Its edges are at \sin\theta=\pm\lambda/a. On a screen at distance L its full width is about 2\lambda L/a.
• Increasing the slit width a makes all the bands narrower and closer together. When a is much larger than \lambda the pattern shrinks to a sharp image of the slit.
• A first dark band exists only if \lambda\le a, because \sin\theta cannot exceed 1.

18.3Electron double-slit experiment

The electron double-slit experiment shows that electrons, which we normally picture as tiny particles, also behave like waves. It is a direct demonstration of wave-particle duality and of the nature of reality at the quantum level.

• Electron gun: a heated filament emits electrons, which are accelerated into a narrow beam.
• Double slit: two very narrow parallel slits. They must be far narrower than slits for light, because the wavelength of an electron is tiny.
• Screen: a fluorescent screen or detector that shows where each electron lands.

What we expect and what we see

If electrons were classical particles, each would go through one slit or the other, and the screen would show just two bands, one behind each slit. Instead the screen shows an interference pattern of many bright and dark bands, just like light in Young's experiment.

Even when electrons are sent one at a time, each electron arrives as a single dot at one point: particle-like behaviour. But after thousands of electrons have arrived, the dots build up the interference pattern: wave-like behaviour. So the motion of each electron is described by a wave that passes through both slits and interferes with itself.

Why the experiment matters

• Quantum mechanics: it laid the foundation for quantum mechanics, which describes the behaviour of particles at the atomic and subatomic level.
• Limits of classical physics: classical physics cannot explain the pattern. Particles at the quantum level do not follow definite paths.
• Philosophical implications: it raises questions about the nature of reality, about observation, and about the role of the observer. If a detector is placed at the slits to find which slit each electron passes through, the interference pattern disappears.

Physicists agree on what is observed but not on what it means. The two best-known interpretations are the Copenhagen interpretation and the many-worlds interpretation.

18.3 (i)Copenhagen interpretation

The Copenhagen interpretation is a widely held interpretation of quantum mechanics, developed by Niels Bohr and Werner Heisenberg. It states that a quantum system, such as an electron before it is measured, does not have a definite position or momentum. Instead it exists in a superposition of all its possible states at the same time, described by a wave function.

How it explains the electron double-slit experiment

• Before detection the electron is in a superposition of states, so it passes through both slits at once (wave-like behaviour).
• The act of measurement, detecting the electron on the screen, forces the wave function to collapse into one definite position. That is why we see a single electron hit at one point on the screen (particle-like behaviour).
• The interference pattern is a pattern of probability. Bright fringes are where the probability of detecting the electron is high; dark fringes are where it is low.

Criticisms

• The 'collapse of the wave function' on measurement seems like an arbitrary break from the smooth, deterministic laws of physics, and it is not clear what exactly counts as a measurement.
• It does not explain what happens to the 'other possibilities' in the superposition after the measurement.

The Copenhagen interpretation remains widely held and influential, but its implications and limits are still debated among physicists and philosophers. Its key ideas are indeterminism (only probabilities can be predicted), complementarity (the wave and particle pictures are both needed, but not at the same time) and the central role of measurement.

18.3 (ii)Many-worlds interpretation

The many-worlds interpretation was proposed by Hugh Everett III in 1957. It says that the wave function never collapses. Instead, every quantum interaction with several possible outcomes splits the universe into several realities (worlds), one for each outcome. Each branch of reality carries forward one possibility from the superposition.

How it explains the electron double-slit experiment

• The electron starts in a superposition, going through both slits.
• When it reaches the screen the universe branches. In each world the electron interacts with the detector screen and leaves its mark at one particular place (particle-like behaviour).
• The interference pattern reflects the combined effect of electrons hitting the screen in all the parallel universes. We experience only one world, the one in which the electron struck our screen at a particular point, but the pattern shows that the other possibilities exist too.

Criticisms

• The other universes cannot be observed, so the idea is very difficult to test by experiment.
• It does not give a clear explanation of the change from a superposition to the one definite result that we observe.

Neither interpretation is universally accepted. Both try to explain the puzzling behaviour of quantum systems; the experiment itself clearly shows wave-particle duality, but the underlying reality is still being explored.

Formula sheet

Key terms

Principle of superposition

When waves overlap, the resultant displacement at any point is the sum of the individual displacements at that point.

Interference

The combination of two or more overlapping waves to give a resultant wave of larger or smaller amplitude.

Constructive interference

Interference in which the waves arrive in phase (crest on crest) and give a larger amplitude: brighter light or louder sound. Path difference =m\lambda.

Destructive interference

Interference in which the waves arrive in anti-phase (crest on trough) and give a smaller or zero amplitude: darkness or quieter sound. Path difference =\left(m+\tfrac12\right)\lambda.

Coherent sources

Sources that emit waves of the same frequency with a constant phase difference.

Monochromatic light

Light of a single wavelength (a single colour).

Path difference

The difference between the distances travelled by two waves from their sources to the same point.

Phase difference

The difference in phase between two waves at a point. It is produced by the path difference between them.

Wavelength

The distance between two consecutive crests (or troughs) of a wave, \lambda.

Frequency

The number of wave cycles passing a point per second, measured in hertz (\mathrm{Hz}).

Wavefront

A surface joining points of a wave that are in the same phase.

Fringes

The alternate bright and dark bands produced by interference or diffraction.

Fringe spacing

The distance between two consecutive bright or dark fringes, \Delta y=\lambda L/d.

Order of a fringe

The whole number m that labels a bright fringe or maximum; m=0 is the central maximum.

Antinodal and nodal lines

In a two-source pattern, lines of constructive interference (antinodal) and of destructive interference (nodal).

Beats

The regular rise and fall in loudness heard when two sounds of slightly different frequencies overlap; f_b=\lvert f_1-f_2\rvert.

Diffraction

The bending of a wave around the edges of an obstacle or through a narrow slit into the region of geometrical shadow.

Slit width

The width of the opening through which a wave passes. The smaller it is compared with the wavelength, the stronger the diffraction.

Diffraction pattern

The pattern of intensity observed after a wave diffracts around an obstacle or through a slit: alternate bright and dark bands.

Diffraction grating

A plate with a large number of closely and equally spaced parallel slits or grooves that diffracts light into sharp maxima.

Grating element (grating constant)

The distance between the centres of two adjacent slits of a grating, d=1/N.

Wave-particle duality

The fact that matter and light show both wave properties (interference, diffraction) and particle properties (arriving in single lumps).

Superposition of states

In quantum mechanics, the condition of a system existing in all its possible states at once until it is measured.

Copenhagen interpretation

The interpretation of quantum mechanics (Bohr and Heisenberg) in which a measurement collapses the wave function to one definite outcome; it stresses indeterminism, complementarity and the role of measurement.

Many-worlds interpretation

The interpretation of quantum mechanics (Everett, 1957) in which the wave function never collapses and every possible outcome happens in its own branch of the universe.

FBISE 2026 questions

Questions from the FBISE HSSC-II 2026 (1st Annual) papers that belong to this unit, answered the way an examiner expects.

Exam focus

1. Explain the working of Young's double-slit experiment with a labelled diagram and derive \Delta y=\lambda L/d. (long)
   Asked as Q.4 (7 marks) in both new-syllabus 2026 sets, B and D.
   Go to answer
2. Derive the grating equation d\sin\theta=m\lambda and describe how a grating is used to find the wavelength of light. (long)
   A core learning outcome and the basis of most grating numericals and MCQs.
   Go to answer
3. State the conditions for observing sustained interference fringes. (short)
   2026 Set D Q.2 (ii) OR, and the same as book Short question 10.
   Go to answer
4. Explain constructive and destructive interference with their path-difference conditions. (short)
   2026 Set B Q.2 (ii) OR.
   Go to answer
5. Define grating element and write its formula; describe the effect of more lines per unit length. (short)
   2026 Set B Q.2 (xiii) and MCQ 6, Set D Q.2 (xiii); book Short question 7.
   Go to answer
6. How do d, L and \lambda change the fringe spacing in Young's experiment? (short)
   2026 Set B Q.2 (xiv) and Set D Q.2 (xiv); book exercise MCQ 9.
   Go to answer
7. Numericals on \Delta y=\lambda L/d and fringe positions y=m\lambda L/d, y=\left(m+\tfrac12\right)\lambda L/d. (numerical)
   Learning outcome 3. Convert nm, \mum and mm to metres carefully: three of the book's printed answers are wrong because of such slips.
   Go to answer
8. Numericals on the grating equation and lines per cm. (numerical)
   Learning outcome 4; a typical 3-mark numerical.
   Go to answer
9. Explain the electron double-slit experiment and the Copenhagen and many-worlds interpretations. (long)
   Learning outcome 6; new in the 2022-23 curriculum and likely to appear as a short or long question.
   Go to answer
10. Use the single-slit condition a\sin\theta=m\lambda in MCQs. (mcq)
    Book exercise MCQs 4 and 14 and Assignment 18.2 need it, although the text does not explain it.
    Go to answer
11. Describe the ripple tank experiment for two-source interference. (long)
    Learning outcome 1; book Comprehensive question 1.
    Go to answer