Diffraction

Diffraction basics

Diffraction is the bending of waves around edges or through openings. Most pronounced when:

• Wavelength is comparable to feature size.
• Wave hits a small aperture or sharp edge.

Single-slit diffraction

Light passes through a slit of width a. On a screen far away, intensity pattern has a bright central maximum, with weaker maxima alternating with minima at angles where:

a · sin θ = m · λ  (m = 1, 2, 3, ... = positions of dark minima)

Central maximum width is approximately 2λ/a (in angle from center). Smaller slit → wider diffraction pattern.

Double-slit diffraction

Two slits separated by d. Light from both slits interferes:

d · sin θ = m · λ  (constructive — bright fringes)

Pattern: equally-spaced bright fringes (from interference), modulated by single-slit envelope (from diffraction). Foundation of Young's experiment showing light's wave nature.

Diffraction grating

Many parallel slits, separation d. Same equation as double-slit, but with many slits, fringes are MUCH sharper:

d · sin θ = m · λ

Used in spectrometers — different wavelengths diffract at different angles. CDs/DVDs (regularly spaced tracks) act as gratings — that's the rainbow on their surface.

Resolution and Rayleigh criterion

An aperture of diameter D produces a diffraction-limited image. The smallest angular separation it can resolve:

Δθ_min = 1.22 · λ / D  (Rayleigh criterion, circular aperture)

Instrument	D	λ	Δθ_min
Human eye (pupil 5 mm)	5 mm	500 nm	~25 arcsec
Hubble Space Telescope	2.4 m	500 nm	~0.05 arcsec
James Webb Space Telescope	6.5 m	2 µm (IR)	~0.07 arcsec
Very Large Telescope (VLT)	8.2 m	500 nm	~0.015 arcsec
VLBI (radio interferometry)	~thousands km	~cm	~µas

JavaScript — diffraction calculations

// Single-slit minimum positions
function singleSlitMinima(slitWidth, wavelength, m_max = 5) {
  return Array.from({length: m_max}, (_, i) => {
    const m = i + 1;
    const angle = Math.asin(m * wavelength / slitWidth);
    return { m, angleRadians: angle, angleDegrees: angle * 180/Math.PI };
  });
}

// 0.1 mm slit, 633 nm laser
console.log(singleSlitMinima(1e-4, 633e-9, 3));

// Double-slit fringe spacing
function fringeSpacing(slitSeparation, wavelength, screenDistance) {
  // Δy = λ·L/d
  return wavelength * screenDistance / slitSeparation;
}

console.log(`Double slit: ${fringeSpacing(0.5e-3, 633e-9, 1) * 1000} mm fringes`);  // 1.27 mm

// Grating: angles of bright fringes
function gratingMaxima(d, wavelength, max_orders = 3) {
  const orders = [];
  for (let m = 0; m <= max_orders; m++) {
    const sin_theta = m * wavelength / d;
    if (Math.abs(sin_theta) <= 1) {
      orders.push({ m, angleDeg: Math.asin(sin_theta) * 180/Math.PI });
    }
  }
  return orders;
}

// Diffraction grating with 600 lines/mm = d = 1.67e-6 m, 633 nm light
console.log(gratingMaxima(1/600 * 1e-3, 633e-9));

// Rayleigh resolution
function rayleighResolution(diameter, wavelength) {
  return 1.22 * wavelength / diameter * 180/Math.PI * 3600;  // arcseconds
}

console.log(`Hubble at 500 nm: ${rayleighResolution(2.4, 500e-9).toFixed(3)} arcsec`);
console.log(`Webb at 2 µm: ${rayleighResolution(6.5, 2e-6).toFixed(3)} arcsec`);

// Bragg's law for X-ray diffraction
function braggAngle(latticeSpacing, wavelength, order = 1) {
  // 2d·sin θ = n·λ
  const sin_theta = order * wavelength / (2 * latticeSpacing);
  if (Math.abs(sin_theta) > 1) return null;
  return Math.asin(sin_theta) * 180/Math.PI;
}

// X-ray (1.54 Å, copper Kα), NaCl crystal (d = 2.82 Å)
console.log(`NaCl X-ray Bragg angle: ${braggAngle(2.82e-10, 1.54e-10).toFixed(2)}°`);

// Sound diffraction around a building
function soundDiffractionAngle(wavelength, building_width) {
  // Rough rule: significant diffraction when λ > building_width/3
  return wavelength > building_width / 3
    ? 'significant'
    : 'minimal';
}

// 100 Hz bass (λ ≈ 3.4 m) around a 10 m wide building
console.log(soundDiffractionAngle(3.4, 10));  // 'significant' — bass bends well
// 1000 Hz (λ ≈ 0.34 m) around same building
console.log(soundDiffractionAngle(0.34, 10));  // 'minimal' — high freq goes mostly straight

Where diffraction matters

• Astronomy. Telescope resolution limited by diffraction. Spectroscopy uses gratings.
• Microscopy. Diffraction limit for optical microscopy ~ λ/2 ~ 200 nm. Electron microscopy uses much shorter wavelengths.
• X-ray crystallography. Diffraction patterns reveal atomic structures. DNA, proteins, materials.
• Photography. Lens diffraction limits sharpness at small apertures (high f-numbers).
• Acoustics. Sound bending around buildings, in concert halls (good for live sound!).
• Antennas. Pattern of broadcast depends on diffraction; large dishes have narrow beams.
• Lasers. Diffraction of laser through aperture defines beam divergence.

Common mistakes

• Confusing diffraction with refraction. Refraction is angle change at boundary (Snell). Diffraction is bending around edges/aperture.
• Treating wave optics as ray optics for small apertures. Geometrical (ray) optics fails when wavelength approaches feature size. Need wave optics for slits, edges, small apertures.
• Wrong slit-width formula. Single-slit minima at a·sin θ = m·λ. Double-slit MAXIMA at d·sin θ = m·λ. Different formulas for similar setups.
• Forgetting grating order limits. Higher orders may exceed sin θ = 1 → no diffracted ray. Limited orders for given λ/d ratio.
• Ignoring polarization in some configurations. Some gratings polarization-dependent. For very precise spectroscopy, account for it.
• Treating Rayleigh as a hard limit. Beyond it, you can sometimes still resolve via deconvolution, super-resolution methods. But it's a useful rule of thumb.