Wave Optics

🎯 Key Points

• YDSE fringe width β=λD/d; n-th bright fringe at y=nλD/d, n-th dark fringe at y=(2n−1)λD/2d
• Single slit diffraction: first minimum at sinθ=λ/a; central maximum is TWICE as wide as secondary maxima (unlike YDSE's equal-width fringes)
• Malus's Law: I=I₀cos²θ for light through a polariser/analyser pair
• Brewster's angle: tanθ_B=n - at this angle, reflected light is fully polarised and reflected+refracted rays are perpendicular
• Coherent sources need constant phase difference + same frequency; two independent bulbs are always incoherent

Huygens Principle

• Every point on a wavefront acts as a source of secondary wavelets
• New wavefront is the envelope of all secondary wavelets
• Explains reflection, refraction, and diffraction

Interference of Light

• Condition: coherent sources (same frequency and constant phase difference)
• Constructive: path difference = nλ (bright fringe)
• Destructive: path difference = (2n-1)λ/2 (dark fringe)

Young's Double Slit Experiment (YDSE)

• Fringe width: β = λD/d (D = screen distance, d = slit separation)
• Path difference for P at distance y from centre: Δ = yd/D
• Position of n-th bright fringe: y_n = nλD/d
• Position of n-th dark fringe: y_n = (2n-1)λD/2d

Young’s double slit: with both slits open the screen shows fringes, matching wave theory and not particle theory. Fringe width β = λD/d. Image: Inductiveload, CC BY-SA 3.0, via Wikimedia Commons.

Diffraction

• Single slit: first minimum at sinθ = λ/a (a = slit width)
• Central maximum width = 2λD/a
• Diffraction grating: d·sinθ = nλ (d = grating spacing)
• Resolving power of telescope: θ_min = 1.22λ/D

Polarization

• Transverse waves can be polarized; longitudinal (sound) cannot
• Malus Law: I = I₀·cos²(θ) (transmitted intensity through analyser)
• Brewster angle: tan(θ_B) = n₂/n₁ (reflected light is completely polarized)
• Polaroids used in sunglasses, LCD screens, 3D movies

Coherent and Incoherent Sources

• Coherent sources maintain a constant phase difference and emit the same frequency, essential for sustained, observable interference
• Two independent sources (like two separate bulbs) are incoherent because their phase relationship changes randomly; in YDSE, both slits are illuminated by the same source to ensure coherence

Intensity Distribution in Interference

• Resultant intensity: I = I₁ + I₂ + 2√(I₁I₂)·cos(δ), where δ is the phase difference
• If I₁ = I₂ = I₀: I_max = 4I₀ (constructive), I_min = 0 (destructive)
• Ratio of maximum to minimum intensity: I_max/I_min = (√I₁ + √I₂)² / (√I₁ - √I₂)²

Diffraction vs Interference

• Interference occurs between waves from two (or more) distinct coherent sources; diffraction occurs due to superposition of secondary wavelets from different parts of the same wavefront
• In single-slit diffraction, the central maximum is twice as wide as secondary maxima and is much brighter than the fringes seen in YDSE

Polarization by Scattering and Reflection

• Scattered light from the sky is partially polarized due to scattering by atmospheric molecules (basis of polarized sunglasses reducing glare)
• At Brewster's angle, reflected ray is completely polarized and the reflected and refracted rays are perpendicular to each other

Wavefront and its Types

• Wavefront: the locus of all points of a wave that are in the same phase; the direction of propagation is always perpendicular to the wavefront (along the rays)
• Spherical wavefront: produced by a point source (nearby)
• Cylindrical wavefront: produced by a linear (slit) source
• Plane wavefront: a spherical wavefront at very large distance from the source appears plane (e.g. light from the Sun or a distant star)

Reflection and Refraction using Huygens Principle

• Treating each point of a wavefront as a source of secondary wavelets and drawing the new envelope reproduces the laws of reflection and refraction
• Reflection: the geometry gives angle of incidence = angle of reflection, and the incident ray, reflected ray, and normal lie in one plane
• Refraction: because wave speed changes across the boundary, the wavefront bends, giving Snell's law n₁sinθ₁ = n₂sinθ₂ with n = c/v
• Frequency stays the same on refraction; wavelength changes as λ_medium = λ_vacuum/n, since v = fλ and speed changes

Fresnel Distance and Validity of Ray Optics

• A beam of width a travelling a distance z spreads by diffraction through an angle ≈ λ/a
• Fresnel distance z_F = a²/λ is the distance over which the diffraction spread becomes comparable to the beam width itself
• For distances much smaller than z_F, spreading is negligible and light effectively travels in straight lines - this is why ray (geometrical) optics is a valid approximation
• Beyond z_F, diffraction dominates and the beam can no longer be treated as a ray

Resolving Power and Doppler Effect in Light

• Limit of resolution: two close objects are just resolved when the central maximum of one diffraction pattern falls on the first minimum of the other (Rayleigh criterion)
• Telescope: smallest resolvable angle θ_min = 1.22λ/D; a larger objective aperture D gives higher resolving power
• Microscope: smallest resolvable distance d_min = 1.22λ/(2n·sinβ); the term n·sinβ is the numerical aperture, increased using immersion oil
• Doppler effect in light: relative motion shifts the observed frequency; source receding gives a redshift (Δλ/λ = v/c for v ≪ c), source approaching gives a blueshift - used to measure the speeds of stars and galaxies

🚀 JEE Advanced Edge

YDSE with a thin film/glass slab inserted: Inserting a thin transparent sheet of thickness t and refractive index n in front of one slit introduces an extra path difference (n−1)t, shifting the entire fringe pattern by (n−1)tD/d - without changing the fringe width itself.

YDSE in a different medium: If the whole apparatus is immersed in a medium of refractive index n (instead of air), the wavelength inside that medium becomes λ/n, so fringe width becomes β'=β/n - fringes get closer together.

Worked problem: In a YDSE setup with d=1mm, D=1m, and λ=500nm, find the fringe width, then find the new fringe width if the apparatus is immersed in water (n=4/3). Approach: β=λD/d=(500×10⁻⁹×1)/10⁻³=0.5mm. In water: β'=β/n=0.5/(4/3)=0.375mm.

Superposition Principle and Path Difference

• The principle of superposition states that the resultant displacement at a point is the algebraic sum of the displacements due to individual waves.
• Constructive interference occurs when the path difference is an integral multiple of the wavelength: path difference = n(lambda).
• Destructive interference occurs when the path difference equals an odd multiple of half the wavelength: (n + 1/2)(lambda).
• Phase difference and path difference are related by phase difference = (2 pi / lambda) x path difference.
• The resultant intensity depends on the phase difference and can range between (a₁ + a₂) squared and (a₁ - a₂) squared for amplitudes a₁ and a₂.

Conditions for Sustained Interference

• The two sources must be coherent, that is, they must maintain a constant phase difference over time.
• The sources should emit light of the same frequency (and ideally the same wavelength).
• The amplitudes should be nearly equal so that dark fringes are truly dark and contrast is maximum.
• The sources must be narrow and close together to produce observable fringe widths.
• Two independent light sources cannot produce sustained interference because their phase difference fluctuates randomly and rapidly.
• In Young's experiment a single source split into two ensures coherence.

Malus' Law and Plane Polarised Light

• An unpolarised beam has vibrations in all directions perpendicular to propagation; a polaroid transmits vibrations along one axis only.
• Malus' law: when plane polarised light of intensity I₀ passes through an analyser, the transmitted intensity is I = I₀ cos squared (theta), where theta is the angle between the polarisation and analyser axis.
• When unpolarised light passes through a polaroid, the intensity is reduced to half (I₀/2), independent of orientation.
• If the analyser axis is perpendicular to the polariser (theta = 90 degrees), no light is transmitted (crossed polaroids).
• Polarisation confirms that light is a transverse wave.

Brewster's Law and Polarising Angle

• When unpolarised light strikes a transparent surface at the polarising angle, the reflected light is completely plane polarised.
• Brewster's law: n = tan(i_p), where i_p is the polarising angle and n is the refractive index of the medium.
• At the polarising angle the reflected and refracted rays are mutually perpendicular.
• The reflected ray is polarised with vibrations perpendicular to the plane of incidence.
• Since n depends on wavelength, the polarising angle differs slightly for different colours.