Physical optics: Class 11 Physics Notes, Short Questions and Numericals

Wave nature of light, Huygens' principle, interference, Young's double slit experiment, thin films, diffraction by a single slit and a grating, and polarization with Malus' law and Brewster's law.

Key points

Physical optics treats light as a wave. Visible light is an electromagnetic wave with wavelengths of about 400 nm (violet) to 700 nm (red). It travels at c = 3 × 10⁸ m/s in vacuum. Interference, diffraction and polarization cannot be explained by rays, and they show that light behaves as a wave. Polarization shows that light waves are transverse.

Huygens' principle says that every point on a wavefront acts as a source of secondary wavelets that spread in the forward direction with the speed of light. The new wavefront is the surface that touches all these wavelets. A wavefront is the line or surface joining points in the same phase. The principle can explain reflection, refraction and diffraction.

Interference is the combination of two or more waves that gives a pattern of bright and dark regions. Sustained interference needs coherent sources, which have the same frequency and a constant phase difference. Two separate lamps are not coherent, so in practice both beams come from the same source. Constructive interference (bright) happens when the path difference is nλ. Destructive interference (dark) happens when the path difference is (n + ½)λ, with n = 0, 1, 2 and so on.

Young's double slit experiment:

Light of one wavelength passes through a narrow slit, then through two close parallel slits S₁ and S₂ separated by d. They act as coherent sources.
A screen is placed at a distance L from the slits, with L much greater than d.
Bright and dark fringes of equal width are seen on the screen. The central fringe is bright.
The path difference at a point y from the centre is d y / L. Bright fringes are at y = nλL/d and dark fringes at y = (n + ½)λL/d.
The fringe spacing is Δy = λL/d. It is larger for red light than for blue light, and it increases when the slits are brought closer.
With white light the central fringe is white, and the others are coloured.

Thin film interference gives the colours seen in soap bubbles and in oil on water. Light reflected from the top and the bottom of the film interferes. A reflection from a denser medium gives a phase change of 180° (half a wavelength), while a reflection from a rarer medium gives none. For a film of thickness t and refractive index n in air, the reflected light is bright when 2nt = (m + ½)λ and dark when 2nt = mλ. A quarter wavelength coating on a lens reduces reflection in the same way.

Diffraction is the bending and spreading of waves around the edges of an obstacle or through a narrow opening. It is clear when the size of the opening is similar to the wavelength. For a single slit of width a, the central maximum is wide and bright. The minima are at a sin θ = mλ, with m = 1, 2, 3. A narrower slit gives a wider pattern. Sound diffracts around corners much more than light because its wavelength is much longer.

The diffraction grating:

A grating has a large number of equally spaced parallel slits (for example 5000 lines per cm). The spacing d between neighbouring slits is the grating element, d = 1/N, where N is the number of lines per unit length.
Principal maxima appear at angles given by d sin θ = mλ, where m = 0, 1, 2 is the order.
With white light, each order (except the zeroth) gives a spectrum from violet to red.
The highest possible order is less than d/λ, because sin θ cannot exceed 1.
Gratings give sharper and brighter lines than a double slit and are used to find wavelengths.

Polarization is the restriction of the vibrations of a transverse wave to one plane. Ordinary light is unpolarised, meaning the electric field vibrates in all directions perpendicular to the ray. A polaroid sheet lets through only the component of the vibration along its transmission axis. Light can also be polarised by reflection, by scattering, and by double refraction in calcite. Longitudinal waves such as sound cannot be polarised.

Malus' law: if plane polarised light of intensity I₀ falls on an analyser with its axis at angle θ to the plane of polarization, the transmitted intensity is I = I₀ cos²θ. Unpolarised light loses half its intensity on passing through a polariser. Brewster's law: reflected light is completely plane polarised when light strikes a surface at the polarising angle θp, for which tan θp = n. At that angle, the reflected and refracted rays are at 90° to each other. Polaroid sunglasses reduce glare because the reflected glare from water and roads is partly polarised horizontally. Polarisation is also used in stress analysis of materials and in liquid crystal displays.

Study tip: For fringe problems, list λ, d and L with units in metres, then use Δy = λL/d. For grating problems, always find d first. Include a neat labelled diagram when you derive the fringe spacing, with S₁, S₂, the screen and the path difference marked.

Definitions and flashcards

Say the meaning first, then open the card to check.

Wavefront

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

Coherent sources

Sources with the same frequency and a constant phase difference.

Interference

The combination of waves giving regions of reinforcement and cancellation.

Fringe spacing

The distance between two successive bright or dark fringes.

Diffraction

The spreading of waves around obstacles and through narrow openings.

Grating element

The distance between two adjacent slits of a grating.

Polarization

Restriction of the vibrations of a transverse wave to a single plane.

Polarising angle

The angle of incidence at which the reflected light is completely polarised.

Formula sheet

Name	Formula	What it means
Constructive interference	path difference = nλ	Bright fringe.
Destructive interference	path difference = (n + ½)λ	Dark fringe.
Fringe spacing	Δy = λL/d	Young's double slit.
Single slit minima	a sin θ = mλ	m = 1, 2, 3 …
Diffraction grating	d sin θ = mλ	d = 1/N.
Malus' law	I = I₀ cos²θ	θ is the angle between the polariser and analyser axes.
Brewster's law	tan θp = n	n is the refractive index.

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

What are coherent sources? Why are they needed for interference? (2 marks)

Coherent sources emit waves of the same frequency with a constant phase difference. If the phase difference changed randomly, the bright and dark positions would move rapidly and the eye would see a uniform illumination instead of a fringe pattern.

State Huygens' principle. (2 marks)

Every point on a wavefront acts as a source of secondary spherical wavelets which spread forward with the speed of the wave. The new wavefront after a time t is the surface that is tangent to all these wavelets.

Derive the fringe spacing in Young's double slit experiment. (5 marks)

Let the slits S₁ and S₂ be separated by d and the screen be at distance L, with L much greater than d. For a point P at distance y from the central point, the path difference is S₂P − S₁P = d y/L (for small angles). A bright fringe occurs when this equals nλ, so y_n = nλL/d. The next bright fringe is at y_(n+1) = (n + 1)λL/d. The spacing between successive bright fringes is Δy = y_(n+1) − y_n = λL/d. The dark fringes are equally spaced by the same amount.

Why do we not see a diffraction pattern when light passes through a doorway? (2 marks)

Diffraction is noticeable only when the width of the opening is comparable with the wavelength. The wavelength of light is about 10⁻⁷ m, which is very small compared with a doorway, so the bending is too small to notice.

State Malus' law and Brewster's law. (3 marks)

Malus' law: when plane polarised light of intensity I₀ passes through an analyser, the transmitted intensity is I = I₀ cos²θ, where θ is the angle between the plane of polarization and the axis of the analyser. Brewster's law: light reflected from a transparent surface is completely plane polarised at the angle of incidence θp for which tan θp = n, the refractive index of the surface.

How does a diffraction grating produce a spectrum with white light? (3 marks)

The positions of the maxima are given by d sin θ = mλ. For a given order m, a longer wavelength gives a larger angle. So each colour is deviated by a different angle, and the white light is spread into a spectrum with violet nearest the centre and red farthest. The central maximum (m = 0) stays white.

Numericals with worked steps

In a Young's experiment λ = 600 nm, d = 0.5 mm and L = 1.5 m. Find the fringe spacing.

Δy = λL/d.
λ = 600 × 10⁻⁹ m, d = 0.5 × 10⁻³ m.
Δy = (600 × 10⁻⁹ × 1.5)/(0.5 × 10⁻³) = 9 × 10⁻⁷/5 × 10⁻⁴ = 1.8 × 10⁻³ m.

Answer: 1.8 mm.

The fringe spacing is 2 mm in a double slit experiment with d = 0.3 mm and L = 1.2 m. Find the wavelength of light.

λ = Δy d/L.
= (2 × 10⁻³ × 3 × 10⁻⁴)/1.2 = 6 × 10⁻⁷/1.2.
= 5 × 10⁻⁷ m = 500 nm.

Answer: 500 nm.

A grating has 5000 lines per cm. Find the angles of the first and second order maxima for light of wavelength 500 nm.

N = 5000 per cm = 5 × 10⁵ per m, so d = 1/N = 2 × 10⁻⁶ m.
First order: sin θ = λ/d = 5 × 10⁻⁷/2 × 10⁻⁶ = 0.25, so θ = 14.5°.
Second order: sin θ = 2λ/d = 0.5, so θ = 30°.

Answer: 14.5° and 30°.

Find the polarising angle for glass of refractive index 1.5.

tan θp = n = 1.5.
θp = tan⁻¹(1.5) = 56.3°.

Answer: 56.3°.

Plane polarised light of intensity 80 W/m² falls on an analyser whose axis makes 60° with the plane of polarization. Find the transmitted intensity.

I = I₀ cos²θ.
cos 60° = 0.5, so cos²60° = 0.25.
I = 80 × 0.25 = 20 W/m².

Answer: 20 W/m².

Common mistakes

Using the slit width instead of the slit separation d in Young's formula.
Not converting nm, mm and cm to metres before substituting.
Forgetting that d = 1/N for a grating, and using N itself.
Saying that interference creates or destroys energy. It only redistributes energy between bright and dark regions.
Thinking that polarization can occur for sound or other longitudinal waves.

Practise this chapter

Practice questions and timed test Printable worksheet