Module 4: Wave Optics

Complete Notes with All Formulas and Clear Physical Explanation

1. Coherent Sources

Two sources are coherent if they maintain a constant phase difference with time.
Only coherent sources can produce sustained (stable) interference pattern.

Methods: Division of wavefront (Young’s double slit, Fresnel’s mirrors) or Division of amplitude (thin films, Newton’s rings)

2. Interference in Thin Films (Uniform Thickness & Wedge)

Path difference = 2μt cos r ± λ/2 (due to reflection phase change)

Case	Condition for Bright Fringe	Condition for Dark Fringe
Air film (reflection)	2μt = nλ	2μt = (2n−1)λ/2 λ
Thin glass/oil film in air (reflection)	2μt = (2n−1)λ/2	2μt = nλ

Wedge-Shaped Thin Film (Air Wedge)

Fringe width β = λ / (2μθ) (θ in radians, small angle)

At contact (t=0) → dark fringe (phase change of π at denser surface)

3. Newton’s Rings (Circular Interference)

Air film between plano-convex lens and glass plate

Radius of nth dark ring (reflected light): r_n = √(n λ R)
Radius of nth bright ring: r_n = √((n + 1/2) λ R)

Central spot is dark in reflected light.
Applications: Measure radius of curvature R, test flatness of surfaces, measure wavelength.

4. Necessity of Extended Sources

Point source gives non-localised fringes. To see clear fringes with naked eye, we need extended source + compensating plate → broad source method (fringes become localised at infinity or on screen).

5. Fraunhofer Diffraction

Pattern	Central Maxima Width	Position of Minima
Single slit	—	a sinθ = nλ (n = ±1,±2,...)
Angular width of central maxima	2λ/a
Double slit	Interference: d sinθ = mλ Diffraction envelope: a sinθ = nλ

Intensity in single slit: I = I₀ [sin²(β)/β²] where β = (π a sinθ)/λ

6. Absent Spectra in Double-Slit Diffraction

Minima of diffraction envelope fall exactly on certain interference maxima → those orders disappear.

Missing orders when d/a = integer (e.g., d = 2a → 2nd, 4th, … orders absent)

7. Diffraction Grating

N slits, each of width a, grating element (a+b) = constant

Principal maxima: (a+b) sinθ = nλ n = 0, ±1, ±2,...

Minima (between two principal maxima): N(a+b) sinθ = mλ (m ≠ multiple of N)

8. Dispersive Power of Grating

Dispersive power = dθ/dλ = n / ((a+b) cosθ)

Higher order → better dispersion

9. Resolving Power of Grating

Resolving power R = λ/Δλ = n N

n = order, N = total number of lines illuminated
Example: 6000 lines/mm, 2 cm width → N = 1,20,000 → R = 1,20,000 in 1st order

10. Rayleigh’s Criterion for Resolution

Two wavelengths (or images) are just resolved when the central maximum of one falls on the first minimum of the other.

Circular aperture (telescope/microscope): Minimum angular separation δθ = 1.22 λ/D
Grating Resolving power = nN (derived from Rayleigh criterion)

Summary of All Important Formulas

Topic	Formula	Meaning
Thin film (reflection, air)	2t = nλ (bright), 2t = (2n−1)λ/2 (dark)
Wedge fringe width	β = λ/(2θ)	θ in rad
Newton’s rings dark (refl.)	rₙ² = n λ R	Central dark
Single slit minima	a sinθ = nλ	n ≠ 0
Double slit interference	d sinθ = mλ
Grating principal maxima	(a+b) sinθ = nλ
Dispersive power	dθ/dλ = n/((a+b)cosθ)
Resolving power of grating	R = λ/Δλ = n N	Most important
Rayleigh criterion (circular)	δθ = 1.22 λ/D

All formulas are standard university-level results — remember the conditions (reflected/transmitted, air/glass film) and the meaning of n, N, a, b, d, R. Perfect for exams and practicals!