Snell's Law Calculator
Calculate the refraction angle when light passes between two media using Snell's law: n₁ sin(θ₁) = n₂ sin(θ₂). Includes critical angle and Fresnel reflectance.
About this calculator
This calculator solves Snell's law, n₁ sin(θ₁) = n₂ sin(θ₂), for the refraction angle a light ray takes when it crosses the boundary between two media of refractive index n₁ and n₂. Raising Refractive Index n₂ alone always bends the ray closer to the surface normal (a smaller refraction angle), because the ray is slowing down more as it enters a denser second medium. Raising the Incident Angle or Refractive Index n₁ instead always increases the refraction angle, since both raise the left-hand side of the equation the second medium has to match.
When n₁ is greater than n₂ -- light moving from a denser to a less-dense medium, such as glass into air -- there is a Critical Angle beyond which no refraction is possible at all, and the calculator reports "Total internal reflection" instead of an angle. The Reflectance and Transmittance outputs come from the unpolarized Fresnel equations, averaging the reflectance of s- and p-polarized light at the interface; they rise toward 1 and 0 respectively as the incident angle approaches total internal reflection. What this model does not account for: dispersion (refractive index varying by wavelength, the cause of a prism's rainbow), absorption within either medium, or surface roughness -- it assumes a perfectly smooth, non-absorbing interface between two ideal media.
Inputs
Results
Refraction Angle
48.59
How to Use This Calculator
- Enter Refractive Index n₁, Incident Angle, and Refractive Index n₂.
- Review the Refraction Angle result.
- Use Critical Angle and Reflectance to inform your decision.
- Use the chart to visualize the results and explore different scenarios by adjusting inputs.
How the result changes with Refractive Index n₂
| Refractive Index n₂ | Refraction Angle |
|---|---|
| 1 | 48.59 |
| 1.5 | 30 |
| 2.5 | 17.46 |
What each input means
- Refractive Index n₁
- Refractive index of the first medium. Glass ≈ 1.5, water ≈ 1.33, diamond ≈ 2.42.
- Incident Angle
- Angle of the incident ray measured from the surface normal.
- Refractive Index n₂
- Refractive index of the second medium. Air ≈ 1.0003, vacuum = 1.
How this is calculated
Worked example, using the default values
- Identify Input ParametersRefractive Index n₁ = 1.5, Incident Angle = 30, Refractive Index n₂ = 1 = 3 input(s) provided
- Calculate Refraction Angle48.59 = 48.59
- Calculate Critical Angle41.81 = 41.81
- Calculate ReflectanceReflectance0.0552 = 0.0552
Engine last updated . Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Does a larger n₂ always shrink the refraction angle?
Yes -- because the equation solves sin(θ₂) = n₁ sin(θ₁) / n₂, and n₂ sits alone in the denominator, raising it always pulls sin(θ₂) down and bends the ray closer to the surface normal, regardless of what the incident angle or n₁ happen to be. Refractive Index n₁ and the Incident Angle move the result the opposite way: raising either one always increases the refraction angle instead.
Why does my result say "Total internal reflection" instead of an angle?
That happens when n₁ sin(θ₁) / n₂ works out to a value greater than 1, which is only possible when light travels from a higher-index medium into a lower-index one (n₁ greater than n₂) at an incident angle beyond the Critical Angle. Past that angle, no refracted ray exists -- all the light reflects back into the first medium, which is how fiber-optic cables and prism binoculars work.
What is the Critical Angle output, and when is it shown?
The Critical Angle is the incident angle at which the refraction angle reaches exactly 90°, calculated as the arcsine of n₂ divided by n₁. It only exists when n₁ is greater than n₂; if the first medium is less dense than the second, the calculator reports "N/A" because a ray moving into a denser medium can never hit total internal reflection, at any incident angle.
Does the reflectance output mean 10% of the light disappears?
No -- Reflectance and Transmittance always add up to 1 (or 100%): whatever fraction of light intensity is not reflected at the interface is transmitted into the second medium instead. A Reflectance of 0.04, for example, means about 4% of the light bounces back and about 96% continues on as the refracted ray, with no light actually lost from the system.
Why doesn't the calculator account for color or wavelength?
Real materials have a refractive index that varies slightly with wavelength -- a property called dispersion, and the reason a prism splits white light into a rainbow. This calculator takes n₁ and n₂ as single fixed values, so it models one wavelength at a time; to see the spread of colors you would need to run it once per wavelength using that material's dispersion curve.
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