Snell's Law Calculator
Calculate the angle of refraction when light passes between two media using Snell's Law. Also determines the critical angle for total internal reflection.
Snell's Law describes how light bends when it crosses between transparent media of different optical density. Discovered by Ibn Sahl in 984 AD (rediscovered by Willebrord Snellius in 1621), it is the foundation of every optical system — lenses, prisms, fiber optics, microscopes, eyeglasses, cameras, and the human eye. The law is mathematically simple: n₁sin(θ₁) = n₂sin(θ₂), where n is the refractive index and θ the angle measured from the surface normal.
The refractive index n quantifies how much light slows down in a medium relative to vacuum. For vacuum n = 1. Air n ≈ 1.0003 (treated as 1.0). Water 1.33. Glass 1.45-1.95. Diamond 2.42. Higher n means light travels slower, refracts more, and bends more sharply when crossing into the medium from a lower-n medium.
When light enters a denser medium, it bends *toward* the normal (smaller refraction angle). Entering a less dense medium, it bends *away* from the normal. The reverse case has a critical angle: at certain angles, light entering a less dense medium can't refract out at all — it reflects entirely (total internal reflection). This is how optical fibers trap light: cladding has lower n than core, so light bounces back inside.
Refraction also explains everyday phenomena: a straw in a glass of water appearing bent, the apparent shifting position of an object underwater, fish looking closer than they are, rainbow colors from prism dispersion (n varies with wavelength), and mirages on hot roads (air density gradient bends light).
Common applications: lens and prism design, fiber optic engineering, eyeglass prescriptions, camera optics, microscope design, gemstone identification (refractive index varies by gem), and any optics problem.
Inputs
Results
Refracted Angle
19.47°
Critical Angle
N/A
TIR
No
Snell's Law Results
| Parameter | Value |
|---|---|
| Refractive Index n₁ | 1 |
| Refractive Index n₂ | 1.5 |
| Angle of Incidence | 30° |
| Angle of Refraction | 19.4712° |
| sin(θ₂) | 0.333333 |
| Critical Angle | N/A (n₁ ≤ n₂) |
| Total Internal Reflection | No |
| Formula | n₁sin(θ₁) = n₂sin(θ₂) |
Formula
How to use this calculator
- Enter refractive index of incoming medium (air = 1.0, water = 1.33, glass = 1.5).
- Enter refractive index of second medium.
- Enter angle of incidence in degrees (measured from normal to surface).
- Calculator returns angle of refraction.
- If going to less dense medium, calculator also returns critical angle for total internal reflection.
- For prism/dispersion problems, use n appropriate to wavelength of interest.
Worked examples
Light into water
**Scenario:** Sunlight enters a swimming pool at 30° from vertical. Refraction angle? **Calculation:** n₁ = 1.0 (air), n₂ = 1.33 (water). θ₂ = arcsin(sin(30°)/1.33) = arcsin(0.376) ≈ 22.1°. **Result:** Light bends from 30° to 22.1° upon entering water — closer to vertical (toward normal). This is why a straw appears bent in a glass of water — light from underwater bends as it exits to air, creating displaced apparent position.
Total internal reflection in fiber
**Scenario:** Optical fiber core n = 1.46, cladding n = 1.44. Critical angle? **Calculation:** θ_c = arcsin(1.44/1.46) = arcsin(0.9863) ≈ 80.5°. **Result:** Light hitting core-cladding interface at >80.5° (from normal) stays inside via total internal reflection. This high critical angle means most light entering the fiber is trapped and propagates along the core for kilometers. Foundation of all modern long-distance fiber communications.
Diamond sparkle
**Scenario:** Diamond (n = 2.42) cut with specific facet angles. Critical angle? **Calculation:** θ_c = arcsin(1.0/2.42) = arcsin(0.413) ≈ 24.4°. **Result:** Very low critical angle. Light entering the top of a diamond is largely trapped because the steep facets below reflect at angles >24°. After bouncing several times internally, light exits through specific facets giving the "brilliant" sparkle. Diamond cutters precisely tune facet angles to maximize sparkle.
When to use this calculator
**Use Snell's Law for:**
- **Lens design**: telescopes, microscopes, eyeglasses. - **Prism analysis**: dispersion, beam steering. - **Fiber optics**: numerical aperture, mode design. - **Gemstone evaluation**: refractive index identifies gems. - **Optical instruments**: cameras, projectors. - **Atmospheric optics**: rainbows, halos, mirages. - **Underwater visibility**: photography, periscopes. - **Light steering**: total internal reflection prisms.
**Two key principles:**
1. **Light bends toward normal when entering denser medium** (n increases). 2. **Critical angle exists only when going from dense to less dense**.
**Refractive index measurement:**
- **Abbe refractometer**: gemology and chemistry lab. - **Critical angle method**: most accurate for liquids. - **Direct measurement of refraction angle**: simple but less precise. - **Interferometry**: very precise (gas analysis).
**Gemstone identification:**
Each gem has characteristic n: - Quartz: 1.54. - Topaz: 1.62. - Emerald: 1.58. - Ruby/sapphire: 1.77. - Cubic zirconia: 2.16. - Diamond: 2.42.
Combined with other properties (specific gravity, hardness), n identifies gems definitively.
**Dispersion phenomena:**
White light contains multiple wavelengths. Since n varies with λ: - Prisms separate colors into a spectrum. - Rainbows form from droplet dispersion. - Camera lenses have "chromatic aberration" — corrected with multi-element designs. - Telescopes use achromatic doublets (crown + flint) to correct chromatic aberration.
**Atmospheric refraction:**
Earth's atmosphere has n that varies with altitude (denser air at low altitudes): - Sun appears ~0.5° higher than actual position near horizon. - Stars "twinkle" due to atmospheric turbulence varying n. - Mirages form when air temperature gradients create steep n gradients.
**Lens equations:**
Thin lens formula: 1/f = 1/v − 1/u (or 1/f = 1/d_image − 1/d_object, sign conventions vary)
Magnification m = v/u.
For a thin lens of glass: 1/f = (n − 1) × (1/R₁ + 1/R₂)
Where R = radii of curvature.
**Fiber optic design:**
Numerical aperture NA = sin(θ_max) = √(n_core² − n_clad²)
Standard single-mode fiber: NA ≈ 0.14 (small acceptance cone). Standard multi-mode fiber: NA ≈ 0.275 (larger cone).
Smaller NA = lower coupling efficiency but better signal quality.
**Common applications:**
- **Eyeglasses**: refraction prescription corrects vision. - **Camera lenses**: complex multi-element designs. - **Microscopes**: high-NA objectives for resolution. - **Telescope objectives**: large aperture for light gathering. - **Binoculars**: prisms for image erection. - **Periscopes**: total internal reflection prisms. - **Light fixtures**: lenses focus or diffuse light. - **Solar concentrators**: lenses focus sunlight. - **Endoscopes**: medical fiber optic imaging.
**Special angles:**
- **Brewster's angle**: tan(θ_B) = n₂/n₁ — fully polarized reflection. - **Critical angle**: arcsin(n₂/n₁) — onset of total internal reflection. - **Achromatic angle**: balance for dispersion in prisms.
**Software:**
- **Zemax / OpticStudio**: industry-standard optical design. - **Code V**: pro-grade optical design. - **Comsol**: full electromagnetic simulation. - **POV-Ray / Blender**: ray-traced rendering uses Snell's law.
**Pitfalls:**
- **Angle from normal vs surface**: convention is from normal (perpendicular). - **Wavelength dependence**: n varies with λ; use right value. - **Critical angle only one way**: only dense → less dense. - **Total internal reflection**: 100% (no transmission), unlike Fresnel partial reflection. - **Polarization effects**: Fresnel equations needed for intensities (not just angles). - **Curved surfaces**: Snell still applies but need to find normal at each point.
Common mistakes to avoid
- Measuring angle from surface instead of normal.
- Forgetting refractive index varies with wavelength (dispersion).
- Looking for critical angle when going to denser medium (doesn't exist).
- Using arcsin of value > 1 (signals total internal reflection condition).
- Confusing reflection with refraction.
- Ignoring Fresnel intensity equations for partial reflection.
- Treating curved surfaces with flat-surface formulas (need to find local normal).
- Mixing degrees and radians in trig functions.