Snell's Law Calculator

Understanding Snell's Law of Refraction

When light travels from one medium to another, it bends or refracts. This calculator lets you explore this topic in detail and understand the principles of refraction. Read on to discover how Snell's law of refraction is formulated, calculate the angle of refraction, and identify critical angles.

Snell's Law Formula

Snell's law describes how exactly refraction works. When a light ray enters a different medium, its speed and wavelength change. The ray bends either towards the normal of the two media boundaries (when its speed decreases) or away from it (when its speed increases). The mathematical relationship is dictated by the refractive indices of both media:

• $n_1$ is the refractive index of the first medium (where the ray originates).
• $n_2$ is the refractive index of the second medium (into which the ray travels).
• $\theta_1$ is the angle of incidence — the angle between the incoming ray and the perpendicular normal line.
• $\theta_2$ is the angle of refraction — the angle between the normal line and the ray traveling through the second medium.

Finding the Angle of Refraction: An Example

Let's calculate the angle of refraction for a light beam traveling from air to glass with an angle of incidence of $30^\circ$.

1. Determine the refractive index of air: $n_1 \approx 1.000$
2. Determine the refractive index of glass: $n_2 \approx 1.500$
3. Transform the Snell's Law equation to solve for $\theta_2$: $$ \sin(\theta_2) = \frac{n_1 \sin(\theta_1)}{n_2} $$
4. Perform the calculation: $$ \sin(\theta_2) = \frac{1.000 \times \sin(30^\circ)}{1.500} = \frac{0.500}{1.500} \approx 0.333 $$
5. Calculate the arcsine to find the exact angle: $$ \theta_2 = \arcsin(0.333) = 19.48^\circ $$

The Critical Angle & Total Internal Reflection (TIR)

When applying Snell's law to light moving from a denser medium to a rarer medium (e.g., from glass to air), you might calculate a $\sin(\theta_2)$ value greater than $1$. Since the sine function mathematically maxes out at $1$, this calculation is physically impossible.

When this happens, light does not cross the boundary. Instead, all of it reflects backward into the original medium. This is called Total Internal Reflection (TIR).

The highest angle of incidence at which light still refracts (traveling exactly along the boundary at $\theta_2 = 90^\circ$) is known as the critical angle ($\theta_c$). You can find the critical angle by simplifying Snell's law when $\sin(90^\circ) = 1$:

If the incident angle $\theta_1$ exceeds $\theta_c$, TIR occurs. Our calculator automatically flags this event to keep your optic designs mathematically sound.