CBSE Class 12 › Unit VI: Optics

Wave Optics

NCERT Chapter 10 • Huygens’ Principle, Interference & Diffraction

NCERT 2025–26 Unit VI • ~14 Marks JEE Main • 2 Questions

1. Huygens’ Principle

Christiaan Huygens proposed the wave theory of light. To understand this, we define a Wavefront: the locus of all particles vibrating in the same phase.

Diagram illustrating Huygens' principle of secondary wavelets. — Every point on a wavefront acts as a source of secondary spherical wavelets.

The Principle states:

1. Each point on a wavefront acts as a source of new disturbance, called secondary wavelets.
2. These wavelets spread out in all directions with the speed of light.
3. The new wavefront is the forward envelope (tangent) to these secondary wavelets at any instant.

Derivation: Law of Reflection using Huygens’ Principle 3 Marks

Setup:
Consider a plane wavefront $AB$ incident on a reflecting surface $MN$ at angle $i$ .
Let $v$ be the speed of light. Time taken for wavelet from $B$ to reach $C$ is $t$ .
So, $BC = vt$ .

Secondary Wavelet:
In the same time $t$ , the wavelet from point $A$ travels distance $AE = vt$ after reflection.
Draw a tangent from $C$ to the sphere of radius $vt$ centered at $A$ . This tangent $CE$ is the reflected wavefront.

Geometry:
In $\triangle ABC$ and $\triangle AEC$ :
$BC = AE = vt$ (by construction)
$AC$ is common
$\angle ABC = \angle AEC = 90^\circ$
So, $\triangle ABC \cong \triangle AEC$ (RHS congruence).

Result:
Therefore, $\angle i = \angle r$ .
The angle of incidence equals the angle of reflection.

Wavefronts through Optical Elements

Using Huygens’ principle, we can understand how wavefronts transform when passing through prisms, lenses, and mirrors:

Thin Prism: Lower part of wavefront (travelling through more glass) slows down more, causing the wavefront to tilt.
Convex Lens: Central part of wavefront (through thickest glass) is delayed the most, forming a converging spherical wavefront that meets at focus $F$ .
Concave Mirror: Reflects plane wavefront into a converging spherical wavefront focused at $F$ .

Wavefront transformation through a prism, convex lens, and concave mirror. — Huygens’ construction explains focusing by lenses and mirrors (NCERT Fig 10.7).

Why all paths take equal time:

In a convex lens, though the central ray travels a shorter geometric path, it spends more time in glass (slower speed). The optical path length (and thus time) is equal for all rays from object to image — this is Fermat’s principle.

2. Derivation: Snell’s Law using Wave Theory

Geometric proof of Snell's Law using wave theory.

Derivation: Snell’s Law of Refraction 3 Marks

Setup:
Consider a plane wavefront $AB$ incident on a surface $PP'$ separating medium 1 (speed $v_1$ ) and medium 2 (speed $v_2$ ).
Let $i$ be the angle of incidence and $r$ be the angle of refraction.
Let $t$ be the time taken for the wavelet from $B$ to reach $C$ .

Distances:
Distance $BC = v_1 t$ .
In the same time $t$ , the wavelet from $A$ travels distance $AE = v_2 t$ into the second medium.

Geometry:
In $\triangle ABC$ : $\sin i = \dfrac{BC}{AC} = \dfrac{v_1 t}{AC}$ .
In $\triangle AEC$ : $\sin r = \dfrac{AE}{AC} = \dfrac{v_2 t}{AC}$ .

Ratio:
$\dfrac{\sin i}{\sin r} = \dfrac{\dfrac{v_1 t}{AC}}{\dfrac{v_2 t}{AC}} = \dfrac{v_1}{v_2}$ .
By definition, refractive index $n_{21} = \dfrac{v_1}{v_2}$ .
Thus: $\dfrac{\sin i}{\sin r} = n_{21}$ (Snell’s Law).

3. Interference and Young’s Double Slit Experiment

Interference is the redistribution of light energy due to the superposition of waves from two coherent sources (sources with constant phase difference).

Coherent vs Incoherent Sources

For sustained interference fringes, the light sources must be coherent:

Coherent Sources: Have a constant phase difference and same frequency (e.g., light from Young’s two slits, derived from a single source).
Incoherent Sources: Phase difference changes randomly with time (e.g., two independent sodium lamps). They produce uniform illumination (no fringes).

Intensity for Arbitrary Phase Difference 2 Marks

For two coherent waves with same amplitude $a$ and phase difference $\phi$ :
$y_1 = a \cos \omega t$
$y_2 = a \cos (\omega t + \phi)$
Resultant: $y = y_1 + y_2 = 2a \cos(\phi/2) \cos(\omega t + \phi/2)$

Resultant Amplitude: $A = 2a \cos(\phi/2)$
Intensity: $I \propto A^2$ → $I = 4I_0 \cos^2(\phi/2)$
where $I_0$ is intensity due to a single source.

Special Cases:
– $\phi = 0, 2\pi, 4\pi...$ : $\cos^2(0) = 1$ → $I = 4I_0$ (Bright Fringe)
– $\phi = \pi, 3\pi...$ : $\cos^2(\pi/2) = 0$ → $I = 0$ (Dark Fringe)

Interference vs Diffraction:

As Feynman noted: “There is no physical difference between interference and diffraction. It’s just a question of usage.”
– Interference: Superposition from a few distinct sources (e.g., YDSE).
– Diffraction: Superposition from many coherent sources across a single aperture.

Setup of Young's Double Slit Experiment showing path difference. — Constructive interference creates Bright Fringes; Destructive creates Dark Fringes.

Derivation: Fringe Width ( $\beta$ ) 5 Marks

Path Difference:
Let $S_1$ and $S_2$ be two slits separated by $d$ . Screen distance is $D$ .
Point $P$ is at distance $x$ from center $O$ .
Path difference $\Delta p = S_2P - S_1P$ .
From geometry, for $D \gg d$ : $\Delta p \approx \dfrac{x d}{D}$ .

Constructive Interference (Bright Fringe):
Path difference must be integral multiple of $\lambda$ .
$\dfrac{x d}{D} = n\lambda$ $\Rightarrow$ $x_n = n \dfrac{\lambda D}{d}$ (where $n = 0, 1, 2...$ ).

Destructive Interference (Dark Fringe):
Path difference must be odd integral multiple of $\lambda/2$ .
$\dfrac{x d}{D} = \left(n + \dfrac{1}{2}\right)\lambda$ $\Rightarrow$ $x_n = \left(n + \dfrac{1}{2}\right) \dfrac{\lambda D}{d}$ .

Fringe Width ( $\beta$ ):
Separation between two consecutive bright (or dark) fringes.
$\beta = x_{n+1} - x_n$ .
$\beta = \dfrac{\lambda D}{d}$ .

4. Diffraction (Single Slit)

Diffraction is the bending of light around corners of an obstacle. Unlike interference (which uses two sources), diffraction occurs due to wavelets from different parts of the same wavefront.

Intensity graphs comparing interference and diffraction patterns. — Notice the central maximum is twice as wide in diffraction.

Condition for Minima (Dark Bands):
$a \sin \theta = n\lambda$
(Here $a$ is slit width, $\theta$ is diffraction angle).

Derivation: Single-Slit Minima Condition 3 Marks

Setup:
Slit width = $a$ . Divide the slit into two equal halves (width $a/2$ ).
Consider waves from the top of the first half and the top of the second half.

Path Difference:
If the path difference between these two waves is $\lambda/2$ , they interfere destructively.
Generalizing: For every point in the first half, there’s a corresponding point in the second half with path difference $\lambda/2$ .

Geometry:
Path difference = $\dfrac{a}{2} \sin \theta$ .
For destructive interference: $\dfrac{a}{2} \sin \theta = \dfrac{\lambda}{2}$ → $a \sin \theta = \lambda$ (first minima).

General Minima:
For complete cancellation, divide slit into $2n$ equal parts.
Path difference between successive parts = $\lambda/2$ .
→ $\dfrac{a}{2n} \sin \theta = \dfrac{\lambda}{2}$ → $a \sin \theta = n\lambda$ ( $n = \pm 1, \pm 2...$ )

Energy is Conserved:

In interference and diffraction, light energy is redistributed — it decreases in dark regions and increases in bright regions. Total energy remains constant, consistent with the law of conservation of energy.

Width of Central Maximum:

The central maximum extends from the first minimum on one side to the first minimum on the other ( $n = \pm 1$ ).
Angular width: $2\theta \approx \dfrac{2\lambda}{a}$ .
Linear width: $\beta_0 = \dfrac{2\lambda D}{a}$ .

5. Polarization

Polarization restricts the vibration of light vectors to a single plane. This proves the transverse nature of light.

Diagram demonstrating polarization and Malus's Law.

A. Malus’s Law

$I = I_0 \cos^2 \theta$ Where $I_0$ is the intensity of polarized light incident on the analyzer.

B. Brewster’s Law

At a specific angle of incidence ( $i_p$ ), the reflected light is completely polarized.

$\tan i_p = n$ (Reflected and refracted rays are perpendicular).

Practice Time!

Test your concepts on YDSE and Diffraction: Important Numericals for Wave Optics →

← Previous: Ray Optics Next: Dual Nature →