Snell's Law & Critical Angle Guide

Overview

Snell's law links direction changes of a wave to the refractive indices of two media. Whenever light, ultrasound, or microwave radiation crosses an interface, the propagation speed jumps to a new value, forcing the ray to bend. Designers use this rule to shape lenses, prisms, photonic chips, and even aquarium windows. This calculator keeps every relevant quantity—incident angle, refracted angle, and critical angle—in one panel so you can iterate quickly while sketching optical layouts.

Inputs & Usage

Provide three inputs: the refractive index of the incident medium $n_{1}$ , the refractive index of the second medium $n_{2}$ , and the incident angle $θ_{1}$ measured from the surface normal. Typical reference values are 1.000 for dry air, 1.33 for water, 1.46 for fused silica, and 1.90 for dense flint glass. Stick to angles between 0° and 90° for a physically meaningful scenario. After each keystroke the tool recomputes everything and formats the output with your locale, so commas or dots appear automatically in the numbers you read.

How It Works

The computation begins with the canonical relation

n_{1} sin θ_{1} = n_{2} sin θ_{2} .

Solving for the refracted angle yields

θ_{2} = arcsin (\frac{n _{1}}{n _{2}} sin θ_{1}) .

If the argument of $arcsin$ lies between $- 1$ and $1$ the wave refracts normally. When $n_{1} > n_{2}$ and the argument rises above 1, the expression no longer has a real solution, and total internal reflection (TIR) occurs. The boundary between transmission and TIR is the critical angle

θ_{c} = arcsin (\frac{n _{2}}{n _{1}}) .

These formulas are evaluated with high-precision floating point math and rounded only at presentation time, preserving accuracy for downstream design steps.

Interpretation

The refracted angle describes how steeply the ray continues in the new material. Values closer to 0° indicate that the ray is bending toward the normal, typical when entering a denser medium. The critical angle is only defined when $n_{1}$ exceeds $n_{2}$ ; it signals the tipping point where every larger incident angle reflects rather than transmits. The calculator also emits a clear Yes/No flag for total internal reflection so you can immediately tell whether a waveguide mode remains confined.

Example

Imagine a beam traveling from air $(n_{1} = 1.00)$ into borosilicate glass $(n_{2} = 1.52)$ with $θ_{1} = 40°$ . The refracted angle is

θ_{2} = arcsin (\frac{1.00}{1.52} sin 40°) \approx 25.4°.

If we reverse the direction—light inside the glass heading toward air—the critical angle becomes

θ_{c} = arcsin (\frac{1.00}{1.52}) \approx 41.1°.

Any incident angle inside the glass that exceeds 41.1° will reflect entirely, which is exactly how right-angle prisms redirect laser beams without metallic mirrors.

Limitations

The model assumes homogeneous, isotropic, lossless media and an abrupt, perfectly flat boundary. Polarization effects, thin-film coatings, birefringence, graded-index profiles, absorption, and surface roughness are outside its scope. Dispersion is ignored, so users must pick refractive indices that already match the wavelength of interest. Treat the results as first-order guidance; when tolerances are tight, verify them against material catalogs or electromagnetic simulation software.