Wave optics: interference, diffraction and polarisation notes — Unit 1

Light travels in the form of waves. A wavefront is defined as the locus of all points in a medium that are vibrating in the same phase. Depending on the source, wavefronts can be spherical (point source), cylindrical (line source), or plane (source at infinity).

⇒Huygens' Principle

• Every point on a given wavefront acts as a fresh source of new secondary wavelets.
• These secondary wavelets travel in all directions with the speed of light in that medium.
• The forward envelope (tangent) of these secondary wavelets at any instant gives the new position of the wavefront.

When two or more light waves travel simultaneously in the same medium, they cross each other without being disturbed. The resultant displacement at any point is the vector sum of the individual displacements.

y = y₁ + y₂ + ... + yₙ
Where y₁, y₂, etc., are individual wave displacements.

This principle is the absolute foundation for both Interference and Diffraction.

Interference is the phenomenon of non-uniform distribution of energy in the medium due to the superposition of two light waves. The energy is redistributed into regions of maximum intensity (Bright fringes) and minimum intensity (Dark fringes).

⇒Coherent Sources

For sustained interference, the two sources must be coherent. Coherent sources emit light waves of the same frequency and wavelength, with a zero or constant phase difference.

Let two coherent waves be represented by:

y₁ = a sin(ωt)
y₂ = b sin(ωt + φ)

Where:
a, b = amplitudes
φ = phase difference

By superposition, the resultant intensity $I$ is given by:

I = I₁ + I₂ + 2√(I₁I₂) cos(φ)
Since I ∝ (Amplitude)²:
R² = a² + b² + 2ab cos(φ)

⇒Constructive Interference (Maxima)

Occurs when $\cos(\phi) = +1$, which means $\phi = 2n\pi$ (where $n = 0, 1, 2...$). Path difference $\Delta x = n\lambda$.

I_max = (√I₁ + √I₂)²
If I₁ = I₂ = I₀, then I_max = 4I₀

⇒Destructive Interference (Minima)

Occurs when $\cos(\phi) = -1$, which means $\phi = (2n-1)\pi$. Path difference $\Delta x = (2n-1)\frac{\lambda}{2}$.

I_min = (√I₁ - √I₂)²
If I₁ = I₂, then I_min = 0

Thomas Young demonstrated interference experimentally in 1801. Light from a single monochromatic source passes through two narrow slits $S_1$ and $S_2$ separated by distance $d$. An interference pattern forms on a screen at distance $D$.

Path difference $\Delta x$ at a point $P$ on the screen (at distance $y$ from center) is geometrically proven to be:

Δx = y·d / D

Fringe Width ($\eta$) is the distance between two consecutive bright or dark fringes:

β = (λ·D) / d

When light hits a thin transparent film (like soap bubbles or oil on water), it is partially reflected from the top surface and partially refracted and then reflected from the bottom surface. These two reflected waves interfere.

⇒Stokes' Treatment (Phase Change on Reflection)

For a film of thickness $t$ and refractive index $\mu$, viewed at angle of refraction $r$:

The geometric path difference is $2\mu t \cos(r)$. Because of the reflection from the denser medium at the top boundary, we must add an extra $\lambda/2$.

Effective Path Diff = 2μt cos(r) - λ/2

Constructive (Bright): 2μt cos(r) = (2n-1)λ/2
Destructive (Dark): 2μt cos(r) = nλ

Notice that these conditions are reversed compared to the standard interference conditions!

A wedge-shaped film has zero thickness at one end and progressively increases in thickness towards the other, forming a wedge angle $\theta$.

The path difference at a point where thickness is $t$ becomes:

Path Diff = 2μt cos(r + θ) - λ/2

For normal incidence ($r=0$) and very small $\theta$, this simplifies to $2\mu t - \lambda/2$. The fringes produced are straight, parallel, and equidistant, localized near the film.

β = λ / (2μθ)

Newton's rings are a special case of interference in a wedge-shaped film. An air film is trapped between a plano-convex lens (large radius of curvature $R$) and a plane glass plate.

• The thickness of the air film is zero at the point of contact and increases radially outwards.
• This creates a radially symmetric "wedge", resulting in concentric circular fringes.
• Light is incident normally using a glass plate tilted at 45 degrees.

At the point of contact, film thickness $t=0$. Path difference is just $\lambda/2$ (due to reflection from the lower glass plate). Thus, the central spot is always DARK in reflected light.

Using geometry of the circle, the thickness $t$ at radius $r$ from the center is:

t = r² / (2R)

Substituting this into the dark fringe condition ($2\mu t = n\lambda$):

r_n² = nλR / μ
Diameter D_n = 2r_n = 2√(nλR/μ)

⇒1. Determination of Wavelength ($\lambda$)

We measure the diameters of the $n$th and $(m+n)$th dark rings using a travelling microscope.

D_{m+n}² - D_n² = 4(m+n)λR - 4nλR = 4mλR
∴ λ = (D_{m+n}² - D_n²) / (4mR)

⇒2. Determination of Refractive Index ($\mu$) of a Liquid

Place the liquid between the lens and plate. The rings shrink. Measure diameter in air ($D_{air}$) and in liquid ($D_{liquid}$).

μ = (D_{m+n}² - D_n²)_{air} / (D_{m+n}² - D_n²)_{liquid}

Diffraction is the bending of light around the sharp corners of an obstacle or aperture, causing light to encroach into the geometrical shadow region. It occurs prominently when the size of the obstacle is comparable to the wavelength of light.

A plane wavefront is incident on a narrow slit of width $e$. According to Huygens' principle, every point on the slit acts as a source of secondary wavelets. These wavelets interfere to produce a diffraction pattern.

Path difference between wavelets from top and bottom edges diffracted at angle $\theta$ is $e \sin\theta$.

⇒Condition for Minima (Dark Fringes)

e sinθ = nλ   (where n = 1, 2, 3...)

Notice that this looks like the formula for interference maxima, but in single-slit diffraction, this gives minima!

By integrating the contributions from all infinitesimal elements of the slit, the resultant amplitude $R$ and intensity $I$ are derived:

Let α = (π·e·sinθ) / λ
R = A₀ [sin(α) / α]
I = I₀ [sin(α) / α]²

• Central Maximum: At $\theta = 0$, $\alpha = 0$, $I = I_0$ (Maximum intensity).
• Minima: Occur when $\sin(\alpha) = 0$ (but $\alpha \neq 0$), so $\alpha = n\pi$, leading to $e \sin\theta = n\lambda$.
• Secondary Maxima: Occur roughly halfway between minima, at $\alpha = (2n+1)\pi/2$. Their intensities are $I_0/22$, $I_0/61$, etc., dropping off very rapidly.

A plane transmission diffraction grating consists of a large number of parallel, equidistant, narrow slits. Let $e$ be the width of each slit and $d$ be the opaque spacing between them. The term $(e+d)$ is the grating element.

If $N$ is the number of lines per inch, grating element $(e+d) = 2.54 / N$ cm.

⇒Principal Maxima Condition

The wavelets from corresponding points in adjacent slits have a path difference $(e+d)\sin\theta$. Constructive interference yields the extremely sharp principal maxima:

(e+d) sinθ = nλ   (where n = 0, 1, 2...)

Resolving Power is the ability of an optical instrument to form separate, distinct images of two closely spaced objects or spectral lines.

⇒Rayleigh's Criterion of Resolution

Two spectral lines of wavelengths $\lambda$ and $\lambda + d\lambda$ are "just resolved" if the principal maximum of one coincides with the first minimum of the other.

R.P. = λ / dλ = nN

Where:
n = order of the spectrum
N = total number of rulings illuminated on the grating

Light is a transverse electromagnetic wave. In unpolarised light, the electric field vector vibrates symmetrically in all directions perpendicular to the direction of propagation.

Plane Polarised Light: The electric field vibrates strictly in a single plane. Polarisation definitively proves the transverse nature of light.

⇒Malus's Law

When completely plane polarised light of intensity $I_0$ passes through an analyzer, the transmitted intensity $I$ depends on the angle $\theta$ between the transmission axes of the polarizer and analyzer.

I = I₀ cos²θ

When an unpolarised ray of light enters an anisotropic crystal (like Calcite or Quartz), it splits into two refracted rays: the Ordinary (O) ray and the Extraordinary (E) ray. Both are plane polarised, in perpendicular directions.

• O-ray: Obeys Snell's laws, velocity is constant in all directions.
• E-ray: Does not obey Snell's laws, velocity changes with direction.

⇒Nicol Prism

An optical device made of calcite used to produce and analyze plane polarised light. It works by eliminating the O-ray via Total Internal Reflection at a layer of Canada Balsam cement, allowing only the polarised E-ray to pass through.

Retardation plates introduce a specific phase or path difference between the O-ray and E-ray by exploiting their different velocities in a crystal.

• Half-Wave Plate: Introduces path diff of $\lambda/2$. Thickness $t = \lambda / (2(\mu_o - \mu_e))$.
• Quarter-Wave Plate: Introduces path diff of $\lambda/4$. Thickness $t = \lambda / (4(\mu_o - \mu_e))$. Used to produce circularly polarised light.

⇒Unit 1 Mastery Checklist

• Can you derive the expression for fringe width in YDSE?
• Can you prove that the center of Newton's Rings is dark?
• How do you determine $\lambda$ using Newton's Rings and a Grating?
• State Rayleigh's criterion for resolution.
• Explain Brewster's Law and Malus's Law.
• How does a Nicol prism work?