Diffraction and Polarization

Single Slit Diffraction

Diffraction is the bending of light around obstacles and its spreading into geometrical shadow regions, fundamentally explained by Huygens' Principle.

$$\sin \theta \approx \frac{\lambda}{b}$$

Determines the angular position of the first minimum in a single-slit pattern.

$\theta$=Angle to the first dark fringe(radians)

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

$b$=Slit width (aperture size)(m)

$b \gg \lambda$

→

Negligible diffraction (light travels in straight lines)

$b \approx \lambda$

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Significant spreading of the Central Maximum

Huygens' Principle: Every point on the wavefront within the slit acts as a source of secondary wavelets.

Central Maximum: The bright broad peak directly opposite the slit, containing most of the wave energy.

Secondary Maxima: Fainter bright fringes on either side of the center caused by partial constructive interference.

General Minima Condition: Dark fringes occur at $d \sin \theta = m\lambda$ where $m = \pm 1, \pm 2, \pm 3, ...$

Proportionality Shortcut: $\theta \propto \lambda / b$ — doubling wavelength doubles the spread; doubling slit width halves it.

Diffraction Prominence: The effect is significant only when $\lambda \sim b$. For $b \gg \lambda$, light essentially travels in straight lines.

Diffraction Gratings

A Diffraction Grating uses thousands of parallel slits to produce extremely sharp and localized interference maxima through Constructive Interference.

$$d \sin \theta = n\lambda$$

Predicts the angles of principal maxima for multi-slit interference.

$d$=Grating element (spacing between slits)(m)

$n$=Order of diffraction(integer)

$\theta$=Angular deviation of the nth order(degrees)

$n = 0$

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Zero-order maximum (central spot)

$n \lambda > d$

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Order not observable (sin θ cannot exceed 1)

Grating Element: The slit spacing $d$ is the reciprocal of line density: $d = 1/N$. Convert units carefully — 5000 lines/cm means $d = 1/500{,}000$ m.

Maximum Observable Order: Since $\sin \theta \leq 1$, the highest order is $n_{\text{max}} = \lfloor d/\lambda \rfloor$ (the greatest integer ≤ $d/\lambda$).

Angular Dispersion: Higher orders ($n$) separate different wavelengths more effectively, producing wider spectra.

White Light: Each wavelength diffracts at a different angle, producing a spectrum at each order. Higher orders show more spread-out spectra.

Bragg's Law — X-ray Diffraction by Crystals

X-rays have wavelengths (~$10^{-10}$ m) comparable to interplanar spacing in crystals, so crystal lattice planes act as a natural Diffraction Grating for X-rays.

$$2d \sin \theta = n\lambda$$

Bragg's equation: gives the condition for constructive interference of X-rays reflected from parallel crystal planes.

$d$=Interplanar spacing between crystal planes(m)

$\theta$=Bragg angle (angle of incidence measured from the crystal plane, not the normal)(degrees)

$n$=Order of reflection(integer)

$\lambda$=Wavelength of incident X-rays(m)

$n = 1$

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First-order reflection — strongest and most commonly observed

$\theta$ increases

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Higher-order reflections become possible, but intensity decreases

Path Difference: The extra distance traveled by the beam reflecting from the lower plane is $2d \sin \theta$, leading to the factor of 2.

Applications: Determining crystal structure (X-ray crystallography), used to find the structure of DNA and haemoglobin.

Grating vs Bragg: The grating equation $d \sin \theta = n\lambda$ has no factor of 2 because it involves transmission through slits, not reflection from parallel planes.

Polarization

Polarization proves light is a Transverse Wave by restricting oscillations to a single plane using a Polarizer.

$$I = I_0 \cos^2 \theta$$

Malus's Law: Relates transmitted intensity to the alignment of two polarizers.

$I$=Transmitted intensity(W/m²)

$I_0$=Intensity of polarized light entering the analyzer(W/m²)

$\theta$=Angle between polarizer and analyzer axes(degrees)

$\theta = 0^\circ$

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Parallel polarizers; full intensity $I_0$ transmitted

$\theta = 45^\circ$

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Half intensity transmitted ($\cos^2 45° = 1/2$)

$\theta = 90^\circ$

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Crossed polarizers; zero intensity transmitted

Unpolarized Light: Random oscillations in all planes perpendicular to propagation.

Selective Absorption: Polaroid sheets (dichroic materials) absorb E-field components perpendicular to the transmission axis.

Half-Rule: When unpolarized light passes through the first polarizer, transmitted intensity is always $I_0/2$, regardless of polarizer orientation.

Crossed Polarizers: Two polaroids at $90°$ block all light. Inserting a third at $45°$ between them allows light through — a classic demonstration.

Proof of Transverse Nature: Longitudinal waves cannot be polarized. The fact that light can be polarized proves it is a transverse wave.

Optical Rotation: Certain substances (quartz, sugar solutions) rotate the plane of polarization. This property is used to measure concentration of optically active solutions.

Brewster's Law

Light reflected from a surface is perfectly plane-polarized at Brewster's Angle, where the reflected and refracted rays are perpendicular.

$$n = \tan \theta_p$$

Calculates the specific angle of incidence for complete polarization by reflection.

$n$=Refractive index of the medium(dimensionless)

$\theta_p$=Brewster angle of incidence(degrees)

Water ($n=1.33$)

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$\theta_p \approx 53^\circ$

90° Rule: At Brewster's angle, the reflected and refracted rays are exactly perpendicular ($\theta_p + \theta_r = 90°$).

Glare Reduction: Polaroid sunglasses block horizontally polarized light reflected from roads, water, and glass surfaces.

Partial Polarization: At angles other than Brewster's, the reflected ray is only partially polarized.

Common Values: Glass ($n \approx 1.5$) → $\theta_p \approx 56.3°$; Water ($n = 1.33$) → $\theta_p \approx 53°$.