Snell's Law Calculator, Refraction
Calculate refraction angles and critical angles for light passing between media.
MEDIUM 1 (incident side)
MEDIUM 2 (transmitted side)
What Is the Snell's Law Calculator, Refraction?
The Snell's Law Calculator finds the refracted angle when light (or any wave) crosses an interface between two media with different refractive indices. It detects total internal reflection (TIR) when the incident angle exceeds the critical angle, computes the Fresnel reflectivity, and draws a ray diagram showing the incident, reflected, and refracted beams. Ten material presets with accurate refractive indices are included.
- ›Snell's law: n₁ sin θ₁ = n₂ sin θ₂, angles measured from the normal
- ›Entering a denser medium (n₂ > n₁): ray bends toward the normal (θ₂ < θ₁)
- ›Entering a less dense medium: ray bends away from normal; TIR possible
- ›Critical angle: incident angle above which all light is reflected (no refraction)
Formula
Snell's Law Formulas
Snell's law
n₁ sin θ₁ = n₂ sin θ₂
Refracted angle
θ₂ = arcsin((n₁/n₂) sin θ₁)
Critical angle
θ_c = arcsin(n₂/n₁), if n₁ > n₂
TIR condition
|n₁/n₂ × sin θ₁| > 1
Fresnel Rs
((n₁cosθ₁−n₂cosθ₂)/(n₁cosθ₁+n₂cosθ₂))²
Index definition
n = c / v (speed of light ratio)
How to Use
- 1Select a material for Medium 1 (incident side) from the preset dropdown
- 2Select a material for Medium 2 (transmitted side)
- 3Enter the incident angle θ₁ in degrees (measured from the interface normal)
- 4Click Apply Snell's Law
- 5If TIR occurs, a red alert shows, increase n₁ or decrease θ₁ to get refraction
- 6The ray diagram shows incident (blue), reflected (yellow dashed), and refracted (green) beams
Example Calculation
Air (n₁=1.000) → Crown Glass (n₂=1.520), θ₁=45°:
sin θ₂ = (1.000/1.520) × sin 45°
sin θ₂ = 0.6579 × 0.7071 = 0.4651
θ₂ = arcsin(0.4651) = 27.70°
Light bends toward normal (θ₂ < θ₁), entering denser medium
Diamond (n₁=2.418) → Air (n₂=1.000), θ₁=25°:
Total Internal Reflection, no refracted ray!
Critical angle = arcsin(1/2.418) = arcsin(0.4135) = 24.42°
θ₁ = 25° > 24.42° → TIR (this is why diamonds sparkle)
Why diamonds sparkle, TIR at work
Diamond's very high refractive index (2.418) gives it an extremely low critical angle of 24.4°. Gem cutters shape diamonds so most internal rays strike facets above the critical angle, causing TIR. Light bounces internally many times, creating the characteristic brilliance and fire.
Understanding Snell's Law, Refraction
Refractive Indices of Common Materials
| Material | n (589 nm) | Critical angle (air) | Application |
|---|---|---|---|
| Vacuum / Air | 1.000 | — | Reference |
| Ice | 1.309 | 49.8° | Ice optics, halos |
| Water (20°C) | 1.333 | 48.6° | Underwater optics |
| Fused silica | 1.458 | 43.2° | Optical fibres |
| Crown glass | 1.520 | 41.1° | Camera lenses |
| Acrylic (PMMA) | 1.490 | 42.2° | Contact lenses, displays |
| Flint glass | 1.620 | 38.1° | High-dispersion lenses |
| Sapphire | 1.762 | 34.6° | Watch crystal, LEDs |
| Zircon | 1.923 | 31.3° | Gemstones |
| Diamond | 2.418 | 24.4° | Gems, high TIR brilliance |
Frequently Asked Questions
What is Snell's law and where is it used?
Willebrord Snell derived the law empirically in 1621; Descartes published it. It applies to any wave (sound, seismic, radio) crossing a boundary between media with different wave speeds.
- ›Camera lenses: multi-element designs use Snell's law at every glass-air interface
- ›Fibre optics: TIR keeps light inside the fibre core
- ›Corrective lenses: lens curvature and refractive index corrects focal point
- ›Mirages: air density gradient acts like layered media with different indices
What is the refractive index?
The refractive index quantifies how much a medium slows light. A higher index means slower propagation and greater bending at the interface.
- ›Air: n ≈ 1.0003 (treated as 1.000 for most calculations)
- ›Water: n = 1.333 at 20°C, explains why objects appear closer underwater
- ›Crown glass: n ≈ 1.52, Flint glass: n ≈ 1.62 (used in camera lenses)
- ›Diamond: n = 2.418, highest of common transparent materials
What is total internal reflection?
TIR is an all-or-nothing phenomenon: below the critical angle, light partially refracts and partially reflects. At and above the critical angle, 100% of light reflects internally.
- ›Water→air critical angle: arcsin(1/1.333) = 48.6°
- ›Glass→air critical angle: arcsin(1/1.52) = 41.1°
- ›Diamond→air critical angle: arcsin(1/2.418) = 24.4°
- ›Optical fibres use TIR to transmit data as light over long distances with minimal loss
What are Fresnel equations?
The Fresnel reflectivity shown in this calculator is the average of s-polarised and p-polarised reflection coefficients, an approximation for unpolarised natural light.
- ›Normal incidence, air-glass: R ≈ ((1−1.52)/(1+1.52))² ≈ 0.042 (4.2% reflected)
- ›Multi-element camera lens: 4% loss per surface × 14 surfaces = significant light loss
- ›Anti-reflection coatings use destructive interference to reduce Fresnel loss to <0.5%
- ›Brewster's angle: p-polarised reflection = 0; used in photography polarising filters
How does Snell's law apply to sound and seismic waves?
The universality of Snell\'s law makes it one of the most broadly applied principles in wave physics, from earthquake geophysics to medical ultrasound.
- ›Seismic refraction: P-waves refract at rock layer boundaries; time-distance curves reveal depth
- ›Ultrasound imaging: beam steering and focusing use wave refraction principles
- ›Underwater acoustics: sound bends toward slower layers (deep sound channel effect)
- ›Radio waves: ionospheric refraction allows AM radio to propagate beyond the horizon
Why do objects appear bent or shifted underwater?
The apparent depth of an underwater object is approximately real depth × (n_air / n_water) = real depth / 1.333 ≈ 0.75 × real depth.
- ›1 metre deep object appears to be at 0.75 m (viewed straight down)
- ›At oblique angles: apparent position shifts further, fish are even more displaced
- ›Spearfishing: aim below the apparent fish position to account for refraction
- ›Same principle: pool appears shallower than it is when viewed from outside
What is the relationship between refractive index and wavelength (dispersion)?
The wavelength dependence of n is described by the Cauchy or Sellmeier equations. Telescope and camera lens designers use multiple glass types to cancel dispersion (achromatic doublets).
- ›Crown glass: n_red ≈ 1.514, n_blue ≈ 1.523 (Abbe number ≈ 64)
- ›Flint glass: n_red ≈ 1.605, n_blue ≈ 1.634 (Abbe number ≈ 36, high dispersion)
- ›Rainbows: water droplets disperse and TIR-reflect sunlight at 40–42°
- ›Achromatic doublet: crown+flint combination cancels colour fringing in telescopes