Unit 3 | Engineering Physics Notes | AKTU Notes

UNIT 3: Wave Optics

3.1 Coherent Sources

For interference of light to occur, we need coherent sources.

Coherent sources are sources that emit light waves of:

• Same frequency (or wavelength)
• Same or constant phase difference
• Preferably same amplitude

Examples:

• Two slits illuminated by the same light source (Young's double slit).
• Two reflections from top and bottom surfaces of a thin film.
• Laser light is highly coherent.

Why coherence is necessary:

• If sources are not coherent, the phase difference keeps changing randomly.
• This causes the bright and dark fringes to shift rapidly — no stable pattern is seen.

3.2 Interference in Uniform and Wedge-Shaped Thin Films

When light falls on a thin film (like a soap bubble or oil film on water), it reflects from both the top and bottom surfaces. These two reflected rays interfere with each other, producing bright and dark bands.

Conditions for Interference in Thin Films:

• When light reflects from a denser medium, it undergoes a phase change of π (half-wavelength path difference).
• Condition for constructive interference (bright band): 2μt cos r = (2n−1)λ/2
• Condition for destructive interference (dark band): 2μt cos r = nλ
• Where μ = refractive index of film, t = thickness, r = angle of refraction, n = integer.

Uniform Thin Film:

• Film has same thickness everywhere.
• Produces uniform color — the entire film appears one color (like a single-colored soap bubble).

Wedge-Shaped Thin Film:

• Film has varying thickness (like two glass plates slightly tilted, touching at one end).
• Different thicknesses produce different interference conditions.
• Result: Alternating bright and dark fringes (straight, parallel bands).
• Fringe width: β = λ / 2μθ, where θ = wedge angle.
• At the thin edge (t = 0), a dark fringe always appears (due to phase reversal).

3.3 Necessity of Extended Sources

• For observing interference in thin films, we need an extended source (a source that illuminates a large area, like a broad lamp).
• A point source only illuminates one specific point on the film — we would see only one spot.
• An extended source illuminates the entire film — we can see the full interference pattern (all the fringes at once).
• Example: When you see rainbow colors on a soap bubble in sunlight, sunlight acts as an extended source.

3.4 Newton's Rings and Its Applications

Setup:

• A plano-convex lens (flat on one side, curved on other) is placed with its curved surface on a flat glass plate.
• An air film of varying thickness is trapped between the curved lens surface and the flat plate.
• When monochromatic light is incident from above, circular interference fringes are formed — these are called Newton's Rings.

Why circular fringes?

• The air film thickness is the same at all points equidistant from the point of contact — forming circles.

Important Results:

• The center is always a dark spot (because at the point of contact t = 0 and there is phase reversal).
• Radius of nth dark ring: rₙ = √(nλR)
• Radius of nth bright ring: rₙ = √((2n−1)λR/2)
• Where R = radius of curvature of lens, λ = wavelength of light.

Applications of Newton's Rings:

• To measure the wavelength of monochromatic light.
• To measure the radius of curvature of a lens.
• To test the flatness of optical surfaces.
• To determine refractive index of liquids (by filling liquid between lens and plate).

3.5 Introduction to Diffraction

Diffraction is the bending of light around the edges of an obstacle or the spreading of light when it passes through a narrow opening (slit).

• It proves the wave nature of light.
• Diffraction is significant when the size of the obstacle or slit is comparable to the wavelength of light.

Types of Diffraction:

• Fresnel Diffraction: Source and screen are at finite distances from the obstacle. No lenses needed.
• Fraunhofer Diffraction: Source and screen are at infinite distances (or effectively so, using lenses). Parallel rays are used. Simpler to analyze mathematically.

3.6 Fraunhofer Diffraction at Single Slit and Double Slit

Single Slit Diffraction:

• Light passes through a single narrow slit of width 'a'.
• The diffracted light is focused by a lens onto a screen.
• A central bright maximum is formed, flanked by alternating dark and bright bands.

Conditions for Single Slit:

• Minima (dark bands): a sinθ = nλ where n = ±1, ±2, ±3...
• Maxima (bright bands): a sinθ = (2n+1)λ/2
• Central maximum is the widest and brightest — its width = 2λ/a.
• Intensity falls off rapidly on either side of center.

Double Slit Diffraction:

• Light passes through two slits, each of width 'a', separated by distance 'd'.
• The pattern is a combination of interference (from two slits) and diffraction (from each slit).
• We see multiple bright interference fringes, but their intensity is modulated by the single-slit diffraction envelope.
• Some fringes may be "missing" — when an interference maximum coincides with a diffraction minimum.
• Condition for missing orders: d/a = n/m (where n and m are integers).

3.7 Absent Spectra

• In a double slit pattern, certain diffraction maxima are expected but do not appear — these are called absent spectra or missing orders.
• This happens when an interference maximum falls exactly at a diffraction minimum — they cancel each other.
• Condition: d/a = n/p, where n = order of interference maximum, p = order of diffraction minimum.
• Example: If d = 2a, then every 2nd order interference fringe is absent.

3.8 Diffraction Grating

A diffraction grating is an optical element with a large number of equally spaced parallel slits (or rulings) on a glass or metal surface.

• Typical grating: 500 to 15,000 lines per cm.
• When light passes through a grating, each slit diffracts light, and all diffracted beams interfere.
• Result: Very sharp, bright maxima at specific angles for each wavelength.

Grating Equation:

• (a + b) sinθ = nλ
• Where (a+b) = grating element (distance between consecutive slits), n = order of diffraction, θ = angle of diffraction, λ = wavelength.

Types of Grating:

• Transmission grating: Light passes through the slits.
• Reflection grating: Light reflects from ruled lines (like a CD surface).

3.9 Dispersive Power of Grating

Dispersive power measures how well a grating separates light of different wavelengths (colors).

• Definition: Rate of change of angle of diffraction with wavelength.
• Formula: dθ/dλ = n / (a+b)cosθ
• Higher dispersive power means the grating can separate colors more widely — like spreading a rainbow wider.
• Dispersive power increases with: higher order (n), smaller grating element (a+b), larger angle θ.

3.10 Resolving Power and Rayleigh's Criterion

Resolving Power is the ability of an optical instrument to distinguish between two closely spaced objects or spectral lines.

Rayleigh's Criterion:

• Two sources are said to be just resolved when the central maximum of one falls on the first minimum of the other.
• If they are closer than this, they cannot be distinguished — they appear as one.

Limit of Resolution:

• For a telescope: dθ = 1.22 λ / D (where D = diameter of objective lens)
• Smaller wavelength → better resolution.
• Larger aperture → better resolution.

3.11 Resolving Power of Grating

• The resolving power of a diffraction grating is its ability to separate two spectral lines of nearly equal wavelengths.
• Formula: R = λ/dλ = nN
• Where n = order of diffraction, N = total number of lines on the grating, dλ = minimum wavelength difference that can be resolved.
• Higher N and higher order n → better resolving power.
• Example: A grating with 10,000 lines in 2nd order can resolve: R = 2 × 10,000 = 20,000.