25.4 Total Internal Reflection

Properties of Light

Total Internal Reflection Fundamentals

Total internal reflection happens when light traveling from a denser medium into a less dense medium gets completely bounced back instead of passing through. This only occurs above a specific angle called the critical angle, and it's the principle behind fiber optics, the sparkle of diamonds, and many optical instruments.

For total internal reflection to occur, two conditions must be met:

• Light must travel from a medium with a higher refractive index to one with a lower refractive index (e.g., water to air, glass to air).
• The angle of incidence must be greater than the critical angle.

The critical angle is the angle of incidence at which the refracted ray skims right along the boundary between the two media (angle of refraction = 90°). You can find it using Snell's law:

$n_1 \sin \theta_c = n_2 \sin 90°$

Since $\sin 90° = 1$, this simplifies to:

$\sin \theta_c = \frac{n_2}{n_1}$

where $n_1$ is the refractive index of the denser medium (the one light starts in) and $n_2$ is the refractive index of the less dense medium.

Here's what happens as you increase the angle of incidence step by step:

1. At small angles, most light passes through into the second medium, bending away from the normal.
2. As the angle increases, the refracted ray bends further from the normal.
3. At the critical angle, the refracted ray travels exactly along the interface (90° from the normal).
4. Beyond the critical angle, no light passes through. All of it reflects back into the original medium. That's total internal reflection.

Example: For light going from glass ($n_1 = 1.50$) into air ($n_2 = 1.00$):

$\sin \theta_c = \frac{1.00}{1.50} = 0.667$

$\theta_c \approx 41.8°$

Any light hitting the glass-air boundary at an angle greater than 41.8° from the normal will be totally internally reflected.

Note that the wavelength of light slightly affects the refractive index of a material, which means the critical angle varies a tiny bit for different colors. This contributes to dispersion effects.

Fiber Optic Technology Applications

Fiber optics are one of the most important real-world uses of total internal reflection. An optical fiber has two layers: a core with a high refractive index and a surrounding cladding with a lower refractive index. Light entering the core at a steep enough angle hits the core-cladding boundary above the critical angle and bounces along the fiber, traveling long distances with very little energy loss.

Communications: Fiber optic cables carry internet, telephone, and television signals. A laser or LED encodes data as pulses of light, which propagate through the fiber via repeated total internal reflections. Compared to copper wires, optical fibers offer much higher bandwidth and far less signal degradation over distance.

Medical imaging: Endoscopes use bundles of optical fibers to see inside the body without major surgery. One set of fibers carries light from an external source to illuminate internal structures. A second set captures the reflected light and sends it back to a camera or display. This is how procedures like colonoscopies and arthroscopies work.

Properties of Materials

Diamond Brilliance Through Reflection

Diamonds have a refractive index of about 2.42, which is unusually high. This means light bends sharply when entering a diamond and, more importantly, the critical angle for the diamond-air interface is very small:

$\sin \theta_c = \frac{1.00}{2.42} \approx 0.413$

$\theta_c \approx 24.4°$

That low critical angle means light striking internal surfaces at most angles will be totally internally reflected rather than escaping. In a well-cut diamond, light enters through the top, bounces off multiple internal facets (each time exceeding the critical angle), and eventually exits back through the top toward your eyes.

Cuts like the round brilliant cut are specifically designed to maximize this effect. The facet angles and proportions are calculated so that light undergoes several total internal reflections before leaving the stone. Because the refractive index varies slightly with wavelength, different colors of light exit at slightly different angles. This separates white light into spectral colors, producing the "fire" that makes diamonds distinctive.

Advanced Concepts in Total Internal Reflection

These topics go beyond the basics but are worth knowing at an introductory level:

• Evanescent waves: Even during total internal reflection, the electromagnetic field doesn't stop perfectly at the boundary. It penetrates a tiny distance into the less dense medium, decaying exponentially. This "evanescent wave" doesn't carry energy away on its own, but it matters in advanced applications like frustrated total internal reflection and certain types of microscopy.
• Fresnel equations: These equations describe exactly how much light is reflected and how much is transmitted at a boundary between two media, for any angle of incidence. At angles below the critical angle, they tell you the split between reflected and refracted light. At and above the critical angle, they confirm 100% reflection.
• Dispersion: Different wavelengths of light have slightly different refractive indices in a given material. This means each color refracts at a slightly different angle, which is why prisms and diamonds spread white light into a rainbow of colors.