Reflection and Transmission at a Boundary

Raise the angle of incidence to reach total internal reflection

Light travels from glass below (n = 1.5) into air above (n = 1.0). Sky blue is the incoming light, gold the reflected ray, and green the refracted ray; as you raise the angle of incidence, the green refracted ray lies down closer and closer to the boundary. Past about 41.8°, the refracted ray vanishes and the light bounces back whole. The arrows in the picture always have the same length, but in reality the reflected share grows as the angle nears the critical angle and becomes everything at it. Where that critical angle comes from is the question of this lesson.

Light splits at the boundary

When light reaches the boundary between two media, as when it passes from air into glass or from water into air, one beam splits into two. Part bounces back as the reflected ray and part bends in as the refracted ray, and with no absorption, the energies of the two add up to the energy of the original light. A mirror showing an image and your faint reflection in a window are due to reflection, and a straw looking bent in water is due to refraction. The share each gets depends on the angle of incidence and the polarization: looking straight at a window, almost all the light goes through, but the more slanted the view, the stronger the reflection.

Refractive index and slowing light

Light is fastest in vacuum and slows down inside a medium. How much it slows is given by the refractive index n, and the speed in the medium is v = c/n. Glass has a refractive index of about 1.5, so light in glass travels at 2/3 of c. In a nonmagnetic medium the refractive index equals the square root of the relative permittivity ε_r, but here ε_r is the value measured at the frequency of light, so for some materials, water among them, the permittivity measured in a static field and the refractive index disagree badly. Crossing a boundary leaves the frequency unchanged and only cuts the speed, so the wavelength gets shorter, and this very difference in how much light slows in each medium is why light bends.

Snell's law

The angle of refraction follows Snell's law n₁ sinθ₁ = n₂ sinθ₂. When light goes from a dense medium (large n) to a rare one (small n), it bends away from the normal, and in the opposite direction it bends toward the normal. The reflected ray is simpler: its angle always equals the angle of incidence (θ_r = θ₁). These rules come about because the crests of the incoming and transmitted waves must keep in step along the boundary. Since the wavelength differs in each medium, keeping the crest spacing along the boundary the same requires the angle to change exactly as Snell's law says.

Total internal reflection

Going from a dense medium to a rare one, raising the angle of incidence makes the angle of refraction grow even faster, so it reaches 90° first. The angle of incidence at that moment is the critical angle θ_c, with sinθ_c = n₂/n₁. Past the critical angle, Snell's law would demand sinθ₂ > 1, and there is no such angle, so the light cannot get through and is all reflected. This is total internal reflection. Even then, an evanescent wave seeps beyond the boundary to a depth of about a wavelength, and bringing another piece of glass within that distance lets some of the light cross over. Optical fibers trapping light and carrying it far, and diamonds sparkling, owe it to total internal reflection; a desert mirage, by contrast, is light bending gradually back through layers of air whose refractive index varies a little with temperature, which is a bit different from total internal reflection at a single boundary.

Back to the first screen

On the first screen, the higher you pushed the angle of incidence, the closer the green refracted ray lay to the boundary, until past one angle it vanished and all the light went back as the gold reflected ray. That angle is the critical angle θ_c, the one satisfying sinθ_c = n₂/n₁. At smaller angles, Snell's law n₁ sinθ₁ = n₂ sinθ₂ set the angle of refraction, and the angle of reflection always matched the angle of incidence. Because light runs at a different speed in each medium, it splits and bends at the boundary.

Up to now, light has been a wave spreading freely. From the next lesson (EM-26), electromagnetic waves are trapped between two conductors and sent in one direction: the transmission line. Voltage and current run as waves inside a cable, a new quantity called the characteristic impedance appears, and the lessons turn to how high-frequency signals travel through circuits.