Snell's Law Calculator

About the Snell's Law Calculator

Snell's law — n₁ sin θ₁ = n₂ sin θ₂ — is the cornerstone of geometrical optics, governing how light bends when crossing the boundary between two materials with different refractive indices. Discovered independently by Ibn Sahl (984 AD), Willebrord Snellius (1621), and René Descartes, it applies universally from fiber optics to diamond gemstone design to underwater photography.

When light travels from a denser medium to a less dense one (n₁ > n₂), there exists a critical angle beyond which all light is reflected — total internal reflection (TIR). This phenomenon is the operating principle of optical fibers, prisms, and retroreflectors. At the Brewster angle, reflected light becomes perfectly polarized, a fact exploited in polarizing filters and laser windows.

This calculator computes the refracted angle, detects total internal reflection, calculates Brewster's and critical angles, evaluates Fresnel reflectance for both s- and p-polarizations, and determines the wavelength and speed of light in the second medium. Comprehensive angle and material comparison tables let you explore refraction across all common optical materials and incident angles.

When This Page Helps

Use this calculator when you need refraction, critical-angle, and reflectance values from the same interface without juggling separate formulas.

It is useful for fiber optics, prism design, underwater imaging, and quick checks of whether a beam will transmit, partially reflect, or hit total internal reflection. It also keeps the wavelength and speed adjustments visible when you want to compare the same interface at a different incident angle or material pair.

How to Use the Inputs

1. Select a preset material pair or enter custom refractive indices.
2. Input the incident angle θ₁ in degrees.
3. Optionally enter the vacuum wavelength for in-medium wavelength calculation.
4. Review the refracted angle, Fresnel reflectances, and special angles.
5. Use the angle table to see refraction across all incident angles.
6. The material comparison table shows refraction into different media at your angle.

Example Calculation

Tips & Best Practices

• Use refractive indices for the actual wavelength if your application is sensitive to dispersion.
• Check whether light is traveling from higher index to lower index before expecting a critical angle or total internal reflection.
• Brewster angle only eliminates p-polarized reflection, not total reflection from the surface.
• If the interface is coated, the Fresnel result from the bare-material formula becomes only a baseline estimate.

Practical Guidance

Snell's law is most useful when you connect angle change to the broader interface behavior. The same pair of refractive indices determines the refracted angle, whether total internal reflection can occur, and how strongly the interface reflects.

Common Pitfalls

The most common mistake is applying the formula without checking the direction of travel or the refractive-index ordering. Another is assuming a single index value is exact across all wavelengths, because real materials disperse and red and blue light do not bend by exactly the same amount. Coatings and polarization effects can also change the practical reflection result at a real surface.

Frequently Asked Questions

• What is total internal reflection?

  When light hits a less dense medium (n₁ > n₂) at an angle greater than the critical angle, 100% is reflected. This is how optical fibers guide light over kilometers.

• What is the Brewster angle?

  The angle at which reflected light is completely p-polarized (zero p-reflectance). θ_B = arctan(n₂/n₁). Laser windows are tilted at Brewster's angle to minimize reflection losses.

• Does Snell's law work for all wavelengths?

  Yes, but the refractive index changes with wavelength (dispersion). This calculator uses a single n value; for precise work, use the Sellmeier equation for your wavelength.

• Why is the critical angle only for n₁ > n₂?

  When going from less dense to more dense media, sin θ₂ < sin θ₁, so θ₂ < θ₁ — refraction always produces a valid angle. TIR only occurs going from dense to less dense media.

• What are the Fresnel equations?

  They give the exact fraction of light reflected and transmitted at an interface, separately for s-polarization (perpendicular) and p-polarization (parallel). At normal incidence, both give R = ((n₁−n₂)/(n₁+n₂))².

• How does light speed change in a medium?

  Light slows down: v = c/n. In glass (n≈1.5), light travels at about 200,000 km/s instead of 300,000 km/s. The wavelength also shrinks: λ_medium = λ_vacuum/n.