Interference Phenomena

Wavefronts and Rays

A wavefront is a surface on which all points have the same phase of vibration. A ray is a line drawn perpendicular to the wavefront, showing the direction of wave propagation.

Spherical Wavefront: Produced by a point source — concentric spheres of increasing radii, separated by one wavelength.

Plane Wavefront: At large distances from the source, a small region of a spherical wavefront appears flat. Sunlight reaching Earth has plane wavefronts.

Producing Plane Waves: Place a point source at the focus of a convex lens; the emerging rays form plane wavefronts.

Huygens' principle allows us to predict the new position of a wavefront from its current position after a time interval $\Delta t$.

Secondary Wavelets: Every point on an existing wavefront acts as a source of secondary wavelets that spread forward at speed $c$.

Envelope Construction: The new wavefront is the tangent surface (envelope) touching all secondary wavelets of radius $c\Delta t$.

Application: This principle explains both interference and diffraction by treating each slit point as a new source.

Coherence and Conditions for Interference

Coherent sources maintain a constant phase difference, which is mandatory for producing a stable, stationary interference pattern.

Monochromaticity: Light must consist of a single wavelength to avoid overlapping different patterns.

Method: Coherence is typically achieved by splitting a single wavefront into two (e.g., passing light through two narrow slits).

Why Two Bulbs Fail: Independent sources emit light in random bursts; the phase difference changes ~10⁸ times per second, washing out any pattern.

Huygens' Link: Points on the same wavefront passing through two slits are coherent secondary sources.

Conditions for Constructive and Destructive Interference

The nature of interference at a point depends on the path difference between waves arriving from the two sources.

$$d\sin\theta = m\lambda$$

Path difference condition for bright fringes (maxima) at angle θ from the central axis.

$d$=Slit separation(m)

$\theta$=Angle from central axis to the point(radians)

$m$=Order of the fringe(integer (0, ±1, ±2...))

$\lambda$=Wavelength(m)

$m = 0$

→

Central bright fringe — zero path difference, always bright regardless of wavelength.

Constructive: Path difference = $0, \lambda, 2\lambda...$ → peaks align with peaks → bright maxima.

Destructive: Path difference = $0.5\lambda, 1.5\lambda...$ → peaks align with troughs → dark minima. Condition: $d\sin\theta = (m + \frac{1}{2})\lambda$.

Phase–Path Link: A path difference of $\lambda$ corresponds to a phase difference of $2\pi$ rad. General relation: $\phi = \frac{2\pi}{\lambda}\Delta L$.

Young's Double Slit Experiment (YDSE)

The spacing between adjacent bright or dark bands, known as fringe width, depends on geometry and wavelength. This is derived using the small-angle approximation $\sin\theta \approx \tan\theta = y/L$.

$$\Delta y = \frac{\lambda L}{d}$$

Fringe spacing — the distance between consecutive bright (or dark) fringes on the screen.

$\Delta y$=Fringe width (spacing between adjacent fringes)(m)

$\lambda$=Wavelength of light(m)

$L$=Distance from slits to screen(m)

$d$=Distance between slits(m)

$L \gg d$

→

Small angle approximation holds: $\sin\theta \approx \tan\theta \approx y/L$.

Apparatus immersed in medium of refractive index $n$

→

$\lambda' = \lambda/n$, so fringe width shrinks by factor $n$: $\Delta y' = \Delta y / n$.

Proportionality Shortcut: $\Delta y \propto \lambda L / d$ — doubling the screen distance doubles fringe width; doubling slit separation halves it.

Equal Spacing: Bright and dark fringes have the same width and are equally spaced.

Fringe Position: The $m^{th}$ bright fringe is at $y_m = m\lambda L / d$; the $m^{th}$ dark fringe is at $y_m = (m + \frac{1}{2})\lambda L / d$.

When two coherent waves of different amplitudes interfere, the resultant intensity depends on the amplitude ratio.

$$\frac{I_{max}}{I_{min}} = \left(\frac{A_1 + A_2}{A_1 - A_2}\right)^2$$

Ratio of maximum to minimum intensity in an interference pattern from sources of unequal amplitude.

$I_{max}$=Maximum intensity (constructive)(W/m²)

$I_{min}$=Minimum intensity (destructive)(W/m²)

$A_1, A_2$=Amplitudes of the two waves(m)

$A_1 = A_2$

→

$I_{min} = 0$ — complete destructive interference. Dark fringes are truly dark.

Intensity–Amplitude Link: $I \propto A^2$, so if intensity ratio is given, take the square root to get the amplitude ratio first.

Resultant Amplitude: $A_{res} = \sqrt{A_1^2 + A_2^2 + 2A_1 A_2 \cos\phi}$, where $\phi$ is the phase difference.

Limiting Case: When $\phi = 0$ (constructive), $A_{res} = A_1 + A_2$; when $\phi = \pi$ (destructive), $A_{res} = |A_1 - A_2|$.

Interference in Thin Films

A thin film is a transparent layer whose thickness is comparable to the wavelength of light. Colours in soap bubbles and oil on water arise from interference between light reflected from the film's two surfaces.

How It Works: An incident ray partly reflects from the top surface and partly refracts into the film. The refracted ray reflects off the bottom surface and emerges parallel to the first reflected ray. These two coherent reflected rays interfere.

Path Difference Depends On: (i) thickness of the film, (ii) refractive index of the film, and (iii) angle of incidence.

Phase Reversal on Reflection: When light reflects from a denser medium (low-n to high-n), it undergoes a phase shift of $\pi$ (equivalent to an extra $\lambda/2$ path difference). This is critical for determining constructive vs destructive conditions.

White Light Colours: Different thicknesses satisfy destructive interference for different colours. The remaining colours are seen — this is why oil films show rainbow patterns.

Dimensional Insight: The film thickness must be on the order of wavelengths (~400–700 nm) for visible interference effects.

Newton's Rings

When a plano-convex lens of long focal length is placed on a flat glass plate, the air gap between them forms a thin film of varying thickness, producing concentric circular interference fringes called Newton's rings.

Air Wedge: The air film thickness is zero at the contact point and increases outward. Points of equal thickness lie on circles centred at the contact point.

Dark Centre: At the contact point the film thickness is effectively zero, but reflection from the denser glass plate introduces an extra $\lambda/2$ path difference, causing destructive interference — the centre is always dark.

Ring Pattern: Alternating bright and dark rings appear. The rings get closer together as you move outward (ring spacing decreases with increasing order).

Michelson's Interferometer

The Michelson interferometer splits a beam into two perpendicular paths using a half-silvered mirror, reflects them back, and recombines them to produce interference fringes.

$$L = m\frac{\lambda}{2}$$

Relates the mirror displacement to the number of fringes shifted.

$L$=Displacement of the movable mirror(m)

$m$=Number of fringes shifted past the reference(integer)

$\lambda$=Wavelength of light used(m)

$m = 1$

→

One fringe shift corresponds to a mirror displacement of $\lambda/2$.

Beam Splitter: A half-silvered plate ($G_1$) divides the incoming beam into two perpendicular paths.

Compensator Plate: A second plate ($G_2$) equalises the optical path through glass for both beams.

Factor of 2: Moving the mirror by $\lambda/2$ changes the round-trip path by $\lambda$, shifting one full fringe.

Precision: A displacement of $\lambda/4$ produces a visible change from bright to dark. For $\lambda = 400$ nm, this means ~100 nm precision ($10^{-7}$ m).