Boneyard Tools

Total internal reflection and the critical angle

How the critical angle arises from Snell's law, why it needs a denser to rarer boundary, and where total internal reflection shows up in real optics.

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From Snell's law to the critical angle

Snell's law states that n1 times the sine of the incidence angle equals n2 times the sine of the refraction angle. As you increase the incidence angle inside a denser medium, the refracted ray bends further from the normal and hugs the boundary more closely. At one special angle the refracted ray would lie flat along the surface, meaning its refraction angle reaches 90 degrees. Setting sin(90) to one and rearranging gives theta_c = asin(n2 / n1), which is exactly what this calculator evaluates.

Why the light gets trapped

Once the incidence angle passes the critical angle, Snell's law would demand a sine larger than one for the refracted ray, which no real angle can satisfy. Because the light cannot cross the boundary, all of its energy reflects back into the denser medium with almost no loss. This is different from an ordinary mirror, which absorbs a little light at every bounce, and it is why total internal reflection is so useful for guiding beams efficiently.

Everyday and technical examples

Optical fibers keep light inside a glass core by keeping every bounce steeper than the core to cladding critical angle, letting data travel for kilometers. A cut diamond has a critical angle near 24 degrees, so light entering the top reflects many times inside before leaving through a facet, producing its trademark sparkle. Prisms in binoculars and cameras fold the light path using total internal reflection instead of coated mirrors, and a swimmer looking up sees a circular window of sky rimmed by a mirror-like surface for the same reason.

Common mistakes to avoid

The most frequent error is swapping n1 and n2, which either flips the geometry or removes the critical angle entirely. Remember that n1 is always the medium the light starts in and must be the denser of the two. It also helps to keep the angle measured from the normal, the imaginary line perpendicular to the surface, rather than from the surface itself. Finally, the critical angle marks the threshold for full reflection, so partial reflection and refraction still happen at every angle below it.

Frequently asked questions

Does total internal reflection lose any energy?

In an ideal boundary it reflects essentially all of the light, which is why fibers and prisms can be more efficient than metal mirrors. In practice tiny losses come from surface roughness and absorption in the material, not from the reflection itself.

What is the evanescent wave?

Even under total internal reflection a faint field, the evanescent wave, leaks a very short distance into the rarer medium and decays fast. It carries no net energy across a clean gap, but bringing a second surface within a wavelength lets some light tunnel through, an effect used in some sensors.

Can total internal reflection happen going from air into glass?

No. Light entering a denser medium bends toward the normal and always refracts through, so there is no critical angle in that direction. It can only occur on the way from the denser medium back out to the rarer one.