Total Internal Reflection

Critical angle

For light traveling from a denser medium (n₁) to a less dense medium (n₂ < n₁):

sin θ_c = n₂ / n₁

For incidence angles greater than θ_c, all light reflects internally — no transmission.

Boundary	θ_c (degrees)
Water → air (n_water = 1.33)	48.8°
Glass → air (n = 1.5)	41.8°
Diamond → air (n = 2.42)	24.4°
Optical fiber core/cladding (n = 1.46/1.45)	~83° (very near grazing)

Real-world TIR applications

System	How TIR is used
Optical fiber	Light bounces along fiber core via TIR
Binoculars	TIR prisms invert image and shorten optical path
SLR cameras	Pentaprism uses TIR for viewfinder image inversion
Periscope	Two right-angle TIR prisms (more efficient than mirrors)
Diamond's "fire"	Multiple TIRs inside the diamond create dazzle
Plant leaf optics	TIR channels light into deeper cells
Bicycle reflectors	Cube-corner geometry uses 3 TIRs to retro-reflect
Endoscopes	Optical fiber bundles for medical imaging

JavaScript — TIR calculations

// Critical angle
function criticalAngle(n_dense, n_rare = 1.0) {
  return Math.asin(n_rare / n_dense) * 180 / Math.PI;
}

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

// Check if TIR will occur
function isTIR(angle_incident_deg, n_dense, n_rare) {
  return angle_incident_deg >= criticalAngle(n_dense, n_rare);
}

console.log(`Glass at 50°: ${isTIR(50, 1.5, 1) ? 'TIR' : 'refracts'}`); // TIR
console.log(`Glass at 30°: ${isTIR(30, 1.5, 1) ? 'TIR' : 'refracts'}`); // refracts

// Acceptance angle for optical fiber
function fiberNA(n_core, n_cladding) {
  // Numerical aperture: NA = √(n_core² - n_cladding²)
  return Math.sqrt(n_core*n_core - n_cladding*n_cladding);
}

function acceptanceAngle(NA, n_outside = 1) {
  // sin θ_max = NA / n_outside
  return Math.asin(NA / n_outside) * 180 / Math.PI;
}

const NA = fiberNA(1.46, 1.45);
console.log(`Fiber NA: ${NA.toFixed(3)}`);  // ~0.171
console.log(`Acceptance angle: ${acceptanceAngle(NA).toFixed(2)}°`); // ~9.84°

// Light path in a fiber: how many bounces per meter?
function bouncesPerMeter(coreRadius_m, angleFromAxis_deg) {
  // For small angles, distance per bounce ≈ 2·R/tan(θ_axis)
  const angleRad = angleFromAxis_deg * Math.PI / 180;
  const distancePerBounce = 2 * coreRadius_m / Math.tan(angleRad);
  return 1 / distancePerBounce;
}

// 50 µm core, 5° from axis
console.log(`Bounces per meter: ${bouncesPerMeter(50e-6, 5).toFixed(0)}`);  // ~875

// Diamond simulation: probability of light escaping after N internal reflections
function diamondLight(angleEntry_deg) {
  // Simplified: each reflection has 4% probability of refracting out (at small angles)
  // After N bounces, P(still inside) ~ 0.96^N
  return Math.pow(0.96, 10);  // 10 bounces typical
}

console.log(`Light surviving 10 internal bounces: ${(diamondLight() * 100).toFixed(0)}%`);
// ~66%, but each bounce off TIR adds dispersion → "fire"

Where TIR matters

• Telecommunications. Optical fiber networks rely on TIR — internet, phone, TV.
• Medical. Endoscopes use fiber bundles for imaging inside the body.
• Photography. SLR/DSLR pentaprisms use TIR for viewfinder image.
• Optics. Right-angle prisms in lab equipment, microscopes, periscopes.
• Gem industry. Cuts of diamonds and gems designed to maximize TIR for sparkle.
• Bioluminescence. Some animals use TIR to channel light from internal sources.
• Architecture. "Light pipes" use TIR to channel daylight from rooftop to interior.

Common mistakes

• Wrong direction. TIR occurs when light goes from DENSE to RARE. Light from rare to dense never has TIR (smaller refracted angles).
• Forgetting angle is from normal. Critical angle is measured from normal, not from surface. 41.8° critical angle for glass means anything STEEPER (i.e., closer to grazing) than 41.8° from normal causes TIR.
• Treating below-critical as no reflection. Some reflection always occurs (Fresnel coefficients). Below critical, partial reflection + transmission. At and above critical, 100% reflection.
• Ignoring evanescent waves. TIR has 100% reflection of energy, but a small evanescent field penetrates the rarer medium. Important for FTIR (frustrated TIR) and biosensors.
• Confusing with regular reflection. TIR is a special case at the boundary. Regular reflection happens at any angle (just partial). TIR is total — only beyond critical.
• Treating TIR as wavelength-independent. Critical angle depends on n, which depends on wavelength (dispersion). Different colors have slightly different θ_c.