9.4 Total internal reflection

Principles of total reflection

Total internal reflection happens when light traveling through a denser medium hits the boundary with a less dense medium and, instead of passing through, bounces entirely back. This is the principle behind fiber optics, certain prism designs, and natural phenomena like mirages. It builds directly on Snell's law and the concept of refractive index, so make sure you're comfortable with those before diving in.

Critical angle definition

The critical angle ($\theta_c$) is the smallest angle of incidence at which total internal reflection occurs. At this exact angle, the refracted ray skims along the boundary between the two media (the refraction angle equals 90°). Any angle of incidence larger than $\theta_c$ produces total internal reflection.

The critical angle depends entirely on the refractive indices of the two materials. A larger difference in refractive index means a smaller critical angle, which means total internal reflection kicks in sooner.

Snell's law application

Snell's law relates the angles and refractive indices at a boundary:

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

To find the critical angle, set $\theta_2 = 90°$ (the refracted ray just barely grazes the surface). Since $\sin(90°) = 1$, Snell's law simplifies to:

$n_1 \sin(\theta_c) = n_2$

Solving for $\theta_c$:

$\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)$

This only works when $n_1 > n_2$ (light going from a denser medium to a less dense one). If $n_1 < n_2$, there's no critical angle and total internal reflection can't happen.

Conditions for occurrence

Two conditions must both be met for total internal reflection to occur.

Refractive index requirements

Light must be traveling from a medium with a higher refractive index into one with a lower refractive index. Common examples:

• Water ($n = 1.33$) to air ($n = 1.00$)
• Glass ($n \approx 1.50$) to air ($n = 1.00$)
• Diamond ($n = 2.42$) to air ($n = 1.00$)

The bigger the difference in refractive indices, the smaller the critical angle. Diamond's large refractive index gives it a critical angle of only about 24.4°, which is why diamonds sparkle so intensely: light gets trapped inside and bounces around many times before escaping.

Angle of incidence vs critical angle

Here's how the angle of incidence determines what happens at the boundary:

• Below $\theta_c$: Light partially reflects and partially refracts into the second medium. Snell's law governs the refraction angle normally.
• At $\theta_c$: The refracted ray travels exactly along the interface ($\theta_2 = 90°$). This is the threshold.
• Above $\theta_c$: No light transmits into the second medium. All of it reflects back. This is total internal reflection.

As you increase the angle of incidence toward $\theta_c$, the refracted ray bends closer and closer to the surface until it can't escape at all.

Optical phenomena

Mirages and light bending

Mirages happen because air near a hot surface (like a road on a summer day) is less dense and has a slightly lower refractive index than the cooler air above it. Light traveling downward through these layers gradually bends upward as it passes through the density gradient, eventually undergoing something similar to total internal reflection.

• Inferior mirages (the "puddle" on a hot road) occur when hot air near the ground bends light upward, making the sky appear to be reflected on the surface.
• Superior mirages (objects appearing above their actual position) occur in cold regions where a warm air layer sits above cold air, bending light downward.

Strictly speaking, mirages involve continuous bending through a gradient rather than a sharp reflection at a single interface, but the underlying physics is the same: light curves away from regions of lower refractive index.

Fiber optics applications

Optical fibers are the most important technological application of total internal reflection. A fiber consists of:

• A core with a higher refractive index
• A cladding surrounding the core with a lower refractive index

Light entering the core at a steep enough angle hits the core-cladding boundary above the critical angle and reflects back inward. This repeats thousands of times per meter, guiding light over long distances with very little loss. Applications include high-speed internet, medical endoscopes, and submarine communication cables.

Mathematical treatment

Critical angle calculation

Starting from Snell's law with $\theta_2 = 90°$:

$\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)$

Some values worth knowing:

Notice that higher $n_1$ values produce smaller critical angles. This means total internal reflection is easier to achieve in materials with high refractive indices.

Reflectance vs angle of incidence

Below the critical angle, some light reflects and some transmits. The fraction that reflects is described by the Fresnel equations (you don't need to memorize them for this course, but know they exist).

The key behavior:

• At small angles of incidence, most light transmits through the boundary.
• As the angle increases toward $\theta_c$, reflectance gradually rises.
• Right at $\theta_c$, reflectance jumps sharply.
• Beyond $\theta_c$, reflectance is exactly 100%.

If you graph reflectance vs. angle of incidence, you'll see a smooth curve that suddenly shoots up to 1.0 at the critical angle and stays there.

Practical applications

Optical fibers in communications

Fiber optic cables transmit data as pulses of light. Here's how the system works:

1. A laser or LED converts electrical signals into light pulses.
2. Light enters the fiber core at an angle above the critical angle.
3. The light bounces along the core via repeated total internal reflections.
4. At the receiving end, a photodetector converts the light back into electrical signals.

Fiber optics offer huge advantages over copper wire: higher bandwidth, lower signal loss over distance, and immunity to electromagnetic interference. Modern fibers also use wavelength-division multiplexing, sending multiple signals simultaneously at different wavelengths through the same fiber.

Prisms and light guides

Total internal reflection in prisms is often more efficient than using mirrors, since no metallic coating is needed and reflection losses are essentially zero.

• Right-angle prisms reflect light by 90° or 180° and are used in binoculars and periscopes.
• Retroreflectors (corner-cube prisms) send light back exactly the way it came, regardless of the incoming angle. You'll find these on road signs, bicycle reflectors, and even on the Moon's surface (placed by Apollo astronauts for laser ranging experiments).
• Light guides in displays and illumination panels use total internal reflection to distribute light evenly across a surface.

Limitations and exceptions

Frustrated total internal reflection

If you bring a third medium very close to the reflecting surface (within a few wavelengths of light), some light can "leak" through the gap and enter that third medium. This is called frustrated total internal reflection.

The transmitted intensity drops off exponentially as the gap widens. Think of it as the optical version of quantum tunneling: the light wave doesn't abruptly stop at the boundary but instead has a decaying "tail" that can couple into a nearby material.

Applications include certain touch-sensitive screens and optical tunneling microscopes.

Evanescent waves

Even during perfect total internal reflection, the electromagnetic field doesn't just vanish at the boundary. An evanescent wave extends a short distance into the second medium, decaying exponentially with distance (typically within a fraction of a wavelength).

Evanescent waves don't carry energy across the boundary on their own, but they can interact with materials placed very close to the surface. This property is exploited in:

• Surface plasmon resonance sensors (used in biochemistry)
• Near-field scanning optical microscopy (imaging below the diffraction limit)
• Coupling light between closely spaced optical waveguides

Experimental demonstrations

Laser beam in water tank

This classic lab demo makes total internal reflection visible:

1. Fill a transparent tank with water and add a small amount of milk or fluorescent dye so you can see the beam path.
2. Shine a laser into the water at a shallow angle to the surface. You'll see the beam refract out into the air.
3. Gradually increase the angle of incidence.
4. At about 48.8°, the refracted beam skims along the water surface.
5. Beyond that angle, the beam reflects entirely back into the water.

You can use the observed critical angle to calculate the refractive index of water by rearranging the critical angle formula: $n_1 = \frac{n_2}{\sin(\theta_c)}$.

Optical fiber transmission efficiency

This experiment measures how well a fiber transmits light:

1. Couple a light source (laser or LED) into one end of an optical fiber.
2. Measure the output intensity at the other end using a photodetector.
3. Repeat for fibers of different lengths.
4. Compare input and output intensities to calculate the attenuation coefficient (signal loss per unit length, typically measured in dB/km).

Well-designed fibers show remarkably low loss, confirming that total internal reflection preserves nearly all the light energy over long distances.

Historical context

Discovery and early observations

People have observed effects of total internal reflection for centuries (the shimmering appearance of water surfaces viewed from below, mirages in deserts). Johannes Kepler provided the first scientific description in the early 1600s. Newton's prism experiments explored how light interacts with glass boundaries. In the early 19th century, Augustin-Jean Fresnel developed the mathematical framework describing reflection and transmission at interfaces, including the sharp transition to total reflection at the critical angle.

Evolution of scientific understanding

The 19th-century wave theory of light explained why total internal reflection occurs (wave behavior at boundaries). Maxwell's equations then unified optics with electromagnetism, providing a complete theoretical foundation. In the 20th century, quantum mechanics offered deeper insight into evanescent waves and tunneling effects. The invention of the laser in 1960 enabled precise experiments, and the development of low-loss optical fibers in the 1970s turned total internal reflection into the backbone of global telecommunications.

Related optical concepts

Brewster's angle comparison

Brewster's angle ($\theta_B$) is the angle of incidence at which reflected light becomes completely polarized. It's calculated as:

$\theta_B = \arctan\left(\frac{n_2}{n_1}\right)$

Don't confuse this with the critical angle. At Brewster's angle, light still partially transmits and partially reflects; only the polarization of the reflected light is special (entirely s-polarized). At the critical angle, all light reflects regardless of polarization. Brewster's angle exists for light going in either direction across a boundary, while the critical angle only exists when going from higher to lower refractive index.

Total internal reflection vs refraction

Both phenomena are governed by Snell's law. Total internal reflection is really just the extreme case of refraction: when Snell's law would require $\sin(\theta_2) > 1$ (which is impossible), the light has no choice but to reflect entirely.