Snell’s Law Calculator

Snell’s Law Calculator computes refraction angles, critical angles, refractive indices and transmitted light paths for accurate optics, photonics and physics calculations in media

Calculate Refraction Angle, Critical Angle, and Fresnel Reflectance with This Snell’s Law Calculator

This tool computes how a light ray bends when it crosses the boundary between two optical media, along with related values such as wave velocity, critical angle, Brewster’s angle, numerical aperture, and Fresnel reflectance. Optical engineers, physics students, and lens or fiber-optic designers use it to check refraction and polarization behavior without working through the trigonometry by hand.

Setting Up the Refractive Indices and Incident Angle

Enter refractive index for Medium 1 (n₁) and Medium 2 (n₂) — both unitless — plus the Incident Angle (θ₁) in degrees. Choose which variable to solve for. The tool returns the Refraction Angle (θ₂), wave velocity in each medium (Mm/s), angular deflection, Brewster’s angle, critical angle and TIR status, numerical aperture, and Fresnel reflectance (%).

How Snell’s Law Determines the Refraction Angle

Snell’s law relates the incident and refraction angles to the refractive indices of the two media, per Georgia State University’s HyperPhysics reference:

$$n_1 \sin\theta_1 = n_2 \sin\theta_2$$

Solving for the refraction angle:

$$\theta_2 = \arcsin\left(\frac{n_1 \sin\theta_1}{n_2}\right)$$

Using $n_1 = 1.0003$ (air), $n_2 = 1.52$ (glass), and $\theta_1 = 30°$ gives $\theta_2 \approx 19.21°$. A common input mistake is swapping $n_1$ and $n_2$, which inverts the ratio and returns an angle that doesn’t match the physical setup.

The remaining outputs follow standard optics relations. Wave velocity in each medium is $v = c/n$. The critical angle, which only exists when $n_1 > n_2$, is $\theta_c = \arcsin(n_2/n_1)$ per HyperPhysics.

Brewster’s angle, per the RP Photonics Encyclopedia, is $\theta_B = \arctan(n_2/n_1)$. Numerical aperture for a two-index boundary is $NA = \sqrt{n_1^2 – n_2^2}$, also per the RP Photonics Encyclopedia, and is only real-valued when $n_1 > n_2$. Fresnel reflectance for each polarization state is derived from the Fresnel equations, documented in the same encyclopedia.

Visualizing Incident and Refracted Rays at the Boundary

The diagram below shows the incident ray, the surface normal, and the refracted ray bending toward the normal as light moves from the lower-index medium into the higher-index medium.

Refractive Index Values for Common Optical Media

The table below lists refractive index values for common transparent media at approximately 589 nm, per Georgia State University’s HyperPhysics index of refraction table. Measured values shift slightly with temperature, pressure, and light wavelength, so treat these as starting points rather than fixed constants.

Common Questions About Refraction, Critical Angle, and Total Internal Reflection

What does the “Solve For” option change?

By default the calculator solves for the refraction angle (θ₂). Selecting a different target rearranges Snell’s law algebraically to solve for n₁, n₂, or θ₁ instead, using the same equation with the other three values held fixed.

Why do Critical Angle, TIR Status, or Numerical Aperture sometimes show “None,” “Impossible,” or “N/A”?

Total internal reflection, the critical angle, and numerical aperture only exist when light travels from a higher-index medium into a lower-index one (n₁ > n₂). Entering air-to-glass values (n₁ < n₂) makes these outputs undefined — that’s mathematically correct, not a calculator error.

What are the most common input mistakes?

Swapping n₁ and n₂ flips the refraction direction; entering an angle in radians when the field expects degrees skews every downstream result; and assuming critical angle, Brewster’s angle, or numerical aperture always return a value, when some only apply for specific n₁/n₂ orderings.

Does the calculator account for the wavelength of light?

No. Refractive index varies slightly with wavelength (dispersion), so results depend entirely on the n₁ and n₂ values entered. For wavelength-specific accuracy, source refractive indices measured at your light’s actual wavelength rather than a generic table value.

What’s the difference between the critical angle and Brewster’s angle?

The critical angle marks where refraction stops and total internal reflection begins. Brewster’s angle is the angle at which reflected light becomes fully polarized, with no total internal reflection involved — the two describe different optical phenomena entirely.