Wave Optics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Wave Optics — Key Facts

Wave optics (physical optics) deals with wave phenomena like interference, diffraction, and polarisation. These effects occur when the wavelength of light is comparable to the size of obstacles or apertures.

Huygens’ Principle:

Every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the envelope of all these wavelets.

This principle explains reflection, refraction, and diffraction.

Interference:

Coherent sources: same frequency, constant phase difference.

Young’s Double Slit Experiment: $$d\sin\theta = m\lambda \quad \text{(bright fringes)}$$ $$d\sin\theta = \left(m + \frac{1}{2}\right)\lambda \quad \text{(dark fringes)}$$

Fringe width: $\beta = \frac{\lambda D}{d}$

where D = distance to screen, d = slit separation.

⚡ JEE Exam Tip: In double slit, the intensity of bright fringes is four times the intensity from a single slit (I = 4I₀ cos²δ/2). The central maximum has maximum intensity.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Diffraction:

Single Slit: $$a\sin\theta = m\lambda \quad \text{(minima, } m = \pm1, \pm2, …)$$

Note: m = 0 gives the central maximum (not a minimum).

Angular width of central maximum: $2\lambda/a$ Linear width on screen: $2\lambda D/a$

Difference from Interference:

• Double slit: many sharp, equally spaced bright fringes
• Single slit: one broad central maximum with fainter side maxima

Thin Film Interference:

When light reflects from thin films (soap bubble, oil slick):

Phase change on reflection:

• From denser medium: π phase shift (equivalent to λ/2 path difference)
• From rarer medium: no phase change

Constructive interference (bright): $$2\mu t \cos r = \left(m + \frac{1}{2}\right)\lambda \quad \text{(when phase change occurs)}$$

Destructive interference (dark): $$2\mu t \cos r = m\lambda \quad \text{(when phase change occurs)}$$

where μ = refractive index of film, t = film thickness, r = angle of refraction.

⚡ JEE Exam Tip: A film with μ > 1 (oil, soap) has a phase change at the top surface but not at the bottom. This affects the interference condition. For air film between glass plates, there are phase changes at both surfaces, so conditions swap.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Newton’s Rings:

Circular interference patterns formed by air film between a convex lens and flat glass plate.

Reflected light (dark rings): $$r_m^2 = m\lambda R \quad (m = 0, 1, 2, …)$$

Transmitted light: $$r_m^2 = \left(m + \frac{1}{2}\right)\lambda R$$

Applications:

• Testing flatness of surfaces
• Determining wavelength of light
• Measuring refractive index of liquids

Diffraction Grating:

Multiple slits (N) with spacing d = grating constant.

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

Principal maxima: very sharp because contributions from many slits add constructively.

Resolving Power: $$R = \frac{\lambda}{\Delta\lambda} = mN$$

Higher order and more slits give better resolution.

Polarisation:

Light is unpolarised when electric vectors vibrate in all directions perpendicular to propagation.

Polarised light has electric vectors vibrating in a single plane.

Methods of Polarisation:

1. Polaroid filters (absorb one polarisation)
2. Reflection at Brewster’s angle
3. Refraction through crystals (double refraction/dichroism)

Brewster’s Law:

When light reflects from a surface at Brewster’s angle, reflected light is completely polarised. $$\tan\theta_B = \frac{n_2}{n_1}$$

At this angle, reflected and refracted rays are perpendicular.

Malus’s Law: $$I = I_0 \cos^2\theta$$

where θ = angle between polariser and analyser axes.

Optical Rotation:

Some substances (sugar solution, quartz) rotate the plane of polarisation.

Angle of rotation: $\theta = \alpha l c$

where α = specific rotation, l = path length, c = concentration.

Circularly Polarised Light:

Two perpendicular vibrations with π/2 phase difference → helical motion of electric vector.

Produced by quarter-wave plate (path difference = λ/4).

Diffraction of X-rays by Crystals:

Bragg’s Law: $2d\sin\theta = n\lambda$

X-rays diffract from crystal planes with spacing d comparable to X-ray wavelength.

This is used to determine crystal structure.

Rayleigh Scattering:

Molecules scatter light with intensity $I \propto 1/\lambda^4$

This explains:

• Blue sky (shorter wavelengths scatter more)
• Red sunset (sunlight travels through atmosphere, blue scattered away, red remains)

⚡ JEE Advanced 2023 Analysis: Questions on Brewster’s angle, Young’s double slit with multiple wavelengths, and thin film interference appeared in recent papers. For thin films with multiple layers, calculate the effective phase change at each interface.

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