Diffraction & Interference

📐 Physics Equations & Derivations

Every quantity computed and displayed in this simulator — with the full derivation logic.

1 · Single-Slit Diffraction

Intensity at angle θ

I(θ) = I₀ · sinc²(β)
where sinc(β) = sin(β)/β

Each point in the slit is a Huygens source. Integrating all contributions across slit width a gives the sinc² envelope.

β = π·(a/λ)·sinθ (half phase difference across slit width)
a = slit width | λ = wavelength
I₀ = intensity at θ = 0 (central max)

Circular Aperture — Airy Disk

I(θ) = I₀ · [2·J₁(β)/β]²

For a circular hole, the integral over the aperture gives a Bessel function J₁ instead of a sine. The pattern is concentric rings (Airy disk).

β = π·(a/λ)·sinθ
J₁ = Bessel function of the first kind, order 1
First dark ring: sinθ = 1.22·λ/a (Rayleigh criterion)

Positions of Minima (dark fringes)

a · sinθ = m·λ (m = ±1, ±2, ...)

Where sin(β) = 0 (and β ≠ 0), intensity = 0. This occurs when the slit width equals an integer multiple of λ projected along sinθ.

m = order of dark fringe (m ≠ 0)
Central max width = 2·arcsin(λ/a) ≈ 2λ/a for small angles

Why central fringe is wider

Width_central = 2 × Width_secondary

From m=−1 to m=+1 spans twice the angle of any secondary fringe (m to m+1). This asymmetry is a direct consequence of the sinc² shape.

Narrowing the slit (↓ a) → broader pattern
→ analog of Heisenberg uncertainty: Δx·Δp_x ≥ ℏ/2

2 · Double-Slit Interference + Diffraction

Full double-slit intensity

I(θ) = 4·I₀ · sinc²(β) · cos²(α)

Product of the single-slit envelope (sinc²) and the two-beam interference term (cos²). The factor 4 arises because two in-phase amplitudes add to 2A, giving I = (2A)² = 4A².

β = π·(a/λ)·sinθ (slit-width phase)
α = π·(d/λ)·sinθ (slit-separation phase)
d = slit separation | a = slit width

Bright fringe positions (principal maxima)

d · sinθ = m·λ (m = 0, ±1, ±2, ...)

Constructive interference when the path difference between the two slits equals an integer number of wavelengths. All N slit waves arrive exactly in phase.

Path difference Δ = d·sinθ
At order m: Δ = m·λ → exactly m wavelengths fit
Fringe spacing on screen: Δy = λ·D/d

Missing Orders

Missing when: d/a = integer
Missing order: m = d/a

When a principal maximum of the interference pattern (cos²α = 1) coincides with a zero of the diffraction envelope (sinc²β = 0), the fringe is suppressed entirely.

e.g. d/a = 3 → orders m = ±3, ±6, ±9 ... missing
Envelope zero at a·sinθ = λ, grating max at d·sinθ = 3λ → same sinθ

One slit closed

I(θ) = I₀ · sinc²(β)
(single-slit only)

Blocking one slit removes interference. Intensity at centre drops from 4I₀ to I₀ — a 4× reduction. Energy is not destroyed; dark fringe regions had been stealing from bright ones.

Two slits open: I_max = 4·I₀
One slit open: I_max = I₀
Ratio = 4 (not 2!) because intensity ∝ amplitude²

3 · N-Slit Diffraction Grating

General N-slit intensity

I(θ) = I₀ · sinc²(β) · [sin(N·α)/sin(α)]²

The grating function [sin(Nα)/sin(α)]² gives N² at each principal maximum and rapid oscillations between them. The sinc² envelope modulates the grating peaks.

N = number of slits
Peak intensity = N²·I₀
Secondary maxima between principals: N − 2

Spectral Resolving Power

R = λ/Δλ = m·N

The minimum wavelength difference Δλ that the grating can separate at order m. More slits → sharper peaks → better resolution. This is why spectroscopes use many thousands of slits.

m = diffraction order
N = total number of slits
At m=1: R = N, so Δλ_min = λ/N

4 · Path Difference & Wave Counting

Geometric path difference

Δ = d · sinθ (far-field approx.)

For a screen far from the slits (D ≫ d), the two rays to any point P are nearly parallel. The extra distance the lower ray travels equals d·sinθ — the projection of the slit separation onto the ray direction.

d = slit separation
θ = angle from the central axis
Valid when D ≫ d (Fraunhofer / far-field condition)

Constructive vs Destructive

Constructive: Δ = m·λ (bright)
Destructive: Δ = (m+½)·λ (dark)

When exactly m wavelengths fit in the path difference, crests from both slits arrive together → bright fringe. When m+½ wavelengths fit, a crest meets a trough → complete cancellation → dark fringe.

Number of λ fitting in gap = d·sinθ/λ = (d/λ)·sinθ
The simulator shows this count for each clicked peak

5 · Coherence & Multiple Wavelengths

Two-wavelength superposition

I_total(θ) = I(θ,λ₁) + I(θ,λ₂)

Incoherent sources of different wavelengths add intensities (not amplitudes). Each wavelength produces its own fringe pattern; they overlap on the screen.

d/λ₁ ≠ d/λ₂ → different fringe spacings
Fringe beats appear when λ₁ ≈ λ₂ (closely-spaced doublet)

White light — chromatic fringes

I_screen(θ) = ∫₃₈₀⁷⁰⁰ I(θ,λ) dλ

Each wavelength from 380–700 nm creates its own pattern. At θ=0 all wavelengths are in phase → white central fringe. At larger θ, violet (short λ) diffracts less than red (long λ) → colored fringes with violet inner, red outer.

Fringe spacing: Δy = λ·D/d (depends on λ)
Violet (400 nm) spacing < Red (700 nm) spacing
Beyond ~3rd order: spectra from adjacent orders overlap

6 · Screen Geometry

Fringe spacing on screen

Δy = λ·D / d

For small angles, the physical distance between adjacent bright fringes on a screen at distance D. Doubling D doubles the fringe spacing; doubling d halves it.

D = screen distance (m)
d = slit separation (m) = (d/λ) × λ
Valid for sinθ ≈ tanθ ≈ θ (small angles, θ < ~10°)

Fraunhofer vs Fresnel condition

Fraunhofer: D ≫ a²/λ
Fresnel: D ~ a²/λ

This simulator uses Fraunhofer (far-field) diffraction where the pattern depends only on angle θ, not on distance D. D only affects the physical scale of fringes, not their angular positions.

Near-field (Fresnel) diffraction requires a different integral (Kirchhoff) and produces curved wavefronts near the slit

Note on normalisation: All intensities are shown as I/I₀ where I₀ is the intensity from a single slit at θ = 0. With N slits all in phase, the maximum intensity = N²·I₀. For double slit, maximum = 4·I₀. The single-slit envelope always has maximum = N²·I₀ (the grating function approaches N² at principal maxima). Missing orders occur where sinc²(β) = 0 exactly coincides with a principal maximum — the grating constructive condition and the single-slit destructive condition cancel each other.