Ydse Diffraction Polarisation

10.5 Young's Double-Slit Experiment (YDSE)

In 1801, Thomas Young did what Newton's corpuscles could not explain: he demonstrated the interference of light. A narrow source S illuminates a single slit (to create coherence), which then lights two close, parallel slits \(S_1\) and \(S_2\). These two slits act as coherent sources, and their waves overlap on a screen placed a distance \(D\) away, producing alternating bright and dark bands called fringes.

Path Difference and Fringe Width

For a point \(P\) at height \(y\) on the screen (with \(d,y\ll D\)), the path difference between the two rays is approximately:

\[\Delta x=S_2P-S_1P\approx\frac{y\,d}{D}\]

Bright fringes (maxima) occur where \(\Delta x=n\lambda\):

\[y_n^\text{br}=\frac{n\lambda D}{d},\qquad n=0,\pm1,\pm2,\dots\]

Dark fringes (minima) occur where \(\Delta x=(n+\tfrac12)\lambda\):

\[y_n^\text{dk}=\left(n+\tfrac12\right)\frac{\lambda D}{d}\]

The spacing between consecutive bright (or dark) fringes — the fringe width — is a hallmark result:

\[\boxed{\;\beta=\frac{\lambda D}{d}\;}\]

The intensity variation across the screen is

\[I=4I_0\cos^2\!\left(\frac{\pi y d}{\lambda D}\right)\]

so fringes are equally spaced and have equal brightness (ideal case).

10.6 Diffraction

Diffraction is the bending of waves around obstacles or through apertures comparable in size to the wavelength. It is another direct proof of the wave nature of light.

Single-Slit Diffraction

When monochromatic light of wavelength \(\lambda\) passes through a slit of width \(a\), Huygens' wavelets from every point of the slit interfere. The screen shows a wide central maximum flanked by much weaker secondary maxima.

• Minima: \(a\sin\theta=n\lambda\), \(n=\pm1,\pm2,\dots\)
• Secondary maxima: \(a\sin\theta\approx\left(n+\tfrac12\right)\lambda\)
• Central maximum width (between first minima on either side) on a screen distance \(D\) away: \(\;W=\dfrac{2\lambda D}{a}\).

YDSE vs Single-Slit Diffraction

Feature	YDSE (interference)	Single slit (diffraction)
Fringe spacing	Equal (\(\beta=\lambda D/d\))	Central max twice as wide as side fringes
Intensities	All bright fringes ~equal	Central max dominates; side max much weaker
Condition for minima	\((n+\tfrac12)\lambda\)	\(a\sin\theta=n\lambda\)
Number of fringes	Many observable	Only a few side bands visible

Resolving Power — Rayleigh's Criterion

Two point sources are just resolved by a circular aperture of diameter \(D\) when their angular separation equals

\[\Delta\theta_\text{min}=\frac{1.22\lambda}{D}\]

Larger apertures (telescopes, microscopes) and shorter wavelengths give finer resolution.

10.7 Polarisation

Light is a transverse EM wave — the electric field \(\vec E\) vibrates perpendicular to the direction of propagation. Ordinary sunlight is unpolarised: \(\vec E\) rapidly and randomly changes direction in the plane perpendicular to the ray.

In polarised light, \(\vec E\) vibrates in only one fixed plane.

Malus's Law

If already polarised light of intensity \(I_0\) falls on a polariser whose axis makes an angle \(\theta\) with the light's polarisation, the transmitted intensity is:

\[\boxed{\;I=I_0\cos^2\theta\;}\]

Polarisation by Reflection — Brewster's Law

When unpolarised light falls on glass at a special angle \(\theta_B\), the reflected ray is completely polarised (perpendicular to the plane of incidence). The condition is that the reflected and refracted rays are perpendicular to each other, giving:

\[\tan\theta_B=n,\qquad \theta_B+\theta_r=90°\]

Applications of polarisation: polaroid sunglasses cut horizontal glare from roads and water; LCDs use polarising layers to switch pixels; 3-D cinema glasses use perpendicular polarisations per eye; photographers use polarising filters to deepen sky colour; engineers use "photo-elastic" polarisation to map stress in transparent models.

Worked Examples