Refraction

Snell's law

When light crosses from medium 1 (refractive index n₁) to medium 2 (n₂):

n₁ · sin θ₁ = n₂ · sin θ₂

Both angles measured from the surface normal. The incident ray, normal, and refracted ray are coplanar.

For air to water (n_air = 1.00, n_water = 1.33), incident at 45°:

1.00 · sin 45° = 1.33 · sin θ_water
sin θ_water = 0.707/1.33 = 0.532
θ_water = 32.1°

Light bends TOWARD the normal entering denser medium.

Refractive indices of common materials

Material	n (visible)
Vacuum	1.00 (exactly)
Air (sea level)	1.0003
Ice	1.31
Water	1.33
Ethanol	1.36
Quartz	1.46
Soda-lime glass	1.52
Crown glass	1.52
Flint glass (heavy)	1.65
Sapphire	1.77
Cubic zirconia	2.16
Diamond	2.42
Silicon	3.42 (in IR)
Some metamaterials	negative (synthetic, exotic)

Dispersion — different colors, different angles

Color	Glass n	Water n
Red (700 nm)	1.513	1.331
Yellow (590 nm)	1.518	1.333
Green (550 nm)	1.520	1.334
Blue (450 nm)	1.527	1.336
Violet (400 nm)	1.532	1.342

Difference is small (~1%) but visible — prisms split white light, raindrops form rainbows, and chromatic aberration in lenses creates color fringing.

JavaScript — refraction calculations

// Snell's law: find refracted angle
function refract(theta1_deg, n1, n2) {
  const theta1 = theta1_deg * Math.PI / 180;
  const sin_theta2 = (n1 / n2) * Math.sin(theta1);
  if (Math.abs(sin_theta2) > 1) return null;  // total internal reflection
  return Math.asin(sin_theta2) * 180 / Math.PI;
}

// Air → water at 45°
console.log(`Air to water 45°: ${refract(45, 1, 1.33).toFixed(2)}°`); // 32.1
console.log(`Air to glass 30°: ${refract(30, 1, 1.5).toFixed(2)}°`);   // 19.5

// Critical angle for total internal reflection
function criticalAngle(n_dense, n_rare) {
  // sin θ_c = n_rare / n_dense (when refracted angle = 90°)
  return Math.asin(n_rare / n_dense) * 180 / Math.PI;
}

console.log(`Glass-air critical: ${criticalAngle(1.5, 1).toFixed(2)}°`);     // 41.8
console.log(`Water-air critical: ${criticalAngle(1.33, 1).toFixed(2)}°`);   // 48.8
console.log(`Diamond-air critical: ${criticalAngle(2.42, 1).toFixed(2)}°`);  // 24.4

// Ray-tracing: where does refraction take a ray going from air to water?
function refractRay(rayDir2D, normal2D, n1, n2) {
  const cos_i = -rayDir2D[0]*normal2D[0] - rayDir2D[1]*normal2D[1];
  const sin_i_squared = 1 - cos_i * cos_i;
  const eta = n1 / n2;
  const sin_t_squared = eta * eta * sin_i_squared;
  if (sin_t_squared > 1) return null;  // total internal reflection
  const cos_t = Math.sqrt(1 - sin_t_squared);
  return [
    eta * rayDir2D[0] + (eta * cos_i - cos_t) * normal2D[0],
    eta * rayDir2D[1] + (eta * cos_i - cos_t) * normal2D[1]
  ];
}

const rayIn = [0.7071, -0.7071];  // 45° down-right
const surfNormal = [0, 1];         // upward
console.log(refractRay(rayIn, surfNormal, 1, 1.33));

// Refraction in lens (thin lens, paraxial approximation)
function lensFocalLength(R1, R2, n_lens) {
  // 1/f = (n - 1) · (1/R1 - 1/R2)
  return 1 / ((n_lens - 1) * (1/R1 - 1/R2));
}

// Plano-convex lens, R1 = 0.5 m, R2 = ∞ (plano), n = 1.5
console.log(`Lens focal length: ${lensFocalLength(0.5, Infinity, 1.5)} m`);
// f = 1/(0.5 · 1/0.5) = 1 m

// Apparent depth in water (vertical viewing)
function apparentDepth(realDepth, n_water = 1.33) {
  // Apparent depth = real_depth / n
  return realDepth / n_water;
}

console.log(`2 m pool: looks ${apparentDepth(2).toFixed(2)} m deep`); // ~1.5

// Rainbow angle: primary rainbow occurs at ~42°
function rainbowAngle(n_water = 1.33) {
  // Approximate; exact requires solving refraction + reflection
  // For visible light, primary at 42° (single internal reflection)
  return 42;
}

Where refraction shows up

• Optics — lenses. Eyeglasses, contact lenses, cameras, microscopes, telescopes.
• Optical fibers. Total internal reflection guides light along fiber. Long-distance communications, medical endoscopes.
• Photography. Lens design controls depth of field, perspective, focal length.
• Astronomy. Atmospheric refraction shifts star positions; refractive telescopes use lens-based focusing.
• Medical. Eye corrects vision via refraction; eyewear corrects focus errors.
• Atmospheric optics. Mirages, rainbows, halos, sun pillars all involve refraction in atmosphere.
• Geology. Refractometry identifies minerals by their refractive indices.

Common mistakes

• Wrong direction of bending. Going to denser medium → bends TOWARD normal. Going to less dense → bends AWAY from normal.
• Forgetting frequency stays constant. Wave speed and wavelength change at boundary; frequency does NOT. Color (frequency) doesn't change going from air to water.
• Treating apparent depth as real depth. A 2-m pool looks ~1.5 m deep when viewed from above (refraction at water surface). Don't dive into it as if it's truly 2 m!
• Confusing dispersion with refraction. Refraction = light bending. Dispersion = different colors bend by different amounts. Both happen together; dispersion is a special property of materials.
• Ignoring critical angle. Light going from dense to rare medium has total internal reflection beyond critical angle. Don't apply Snell's law beyond this.
• Treating refractive index as constant. n varies with wavelength (dispersion), and slightly with temperature. For precision optics, this matters.