Light and Optics

In a uniform material, light travels in straight lines. We draw it as rays: lines with arrows showing the direction the light goes. When a ray hits a surface and bounces back, it follows the law of reflection:

angle of incidence = angle of reflection

Both angles are measured from the normal, and the incoming ray, the normal and the reflected ray all lie in the same flat plane.

A mirror or a calm lake is smooth, so parallel rays stay parallel after reflecting and you see a clear image: specular reflection. Paper, walls and your T-shirt are rough on a tiny scale. Each ray still obeys the law, but the normals point every which way, so the light scatters in all directions: diffuse reflection. That's how you see most objects, from any angle.

When you look into a flat (plane) mirror, the reflected rays reach your eye as if they came from behind the glass. Your brain traces them back in straight lines and sees an image there. Nothing is really behind the mirror, so it is a virtual image.

• The image is the same distance behind the mirror as the object is in front.
• It is upright and the same size as the object.
• It is reversed front to back, which is why writing looks backward. That's why "AMBULANCE" is printed backward on the front of ambulances: it reads correctly in a driver's mirror.

Light travels more slowly in water or glass than in a vacuum. The index of refraction tells you how much slower:

n = c ÷ v

Some values: vacuum 1 exactly, air 1.0003 (we round to 1.00), water 1.33, acrylic 1.49, crown glass 1.52, diamond 2.42. In diamond, light travels at 3.00 × 10⁸ ÷ 2.42 ≈ 1.24 × 10⁸ m/s.

When light crosses a boundary at an angle, the change of speed makes it change direction. This bending is refraction. Think of a marching band walking at an angle from pavement onto mud: the side that reaches the mud first slows first, and the whole row swings around. The rule is Snell's law:

n₁ sin θ₁ = n₂ sin θ₂

• Into a higher n (slower), light bends toward the normal.
• Into a lower n (faster), it bends away from the normal.
• Along the normal (θ₁ = 0), it slows down but doesn't bend.

The frequency never changes at a boundary; the waves arrive and leave at the same rate. Since v = fλ, the wavelength gets shorter in the slower material.

Going from a slower material into a faster one (higher n to lower n), light bends away from the normal. Increase the angle of incidence and the refracted ray gets closer and closer to the surface. At the critical angle θ_c, it would skim along the surface at 90°. Put sin 90° = 1 into Snell's law:

sin θ_c = n₂ ÷ n₁   (only when n₁ > n₂)

Beyond the critical angle, no light gets out at all. All of it reflects back inside. This is total internal reflection (TIR), and it reflects more light than an ordinary silvered mirror.

Critical angles into air: water 48.8°, glass with n = 1.50 about 41.8°, diamond just 24.4°. Diamond's small critical angle traps light that enters it, bouncing it around inside until it leaves through the top facets. Cutters choose the angles of the facets to make this happen, which is part of why diamonds sparkle. Binoculars use glass prisms instead of mirrors to fold the light path by TIR.

Optical fibers use TIR to guide light along thin strands of very pure glass. A ray that enters the end at a shallow angle meets the side wall beyond the critical angle, reflects completely, and zigzags along, even around gentle bends. Real fibers have a glass core wrapped in cladding, a glass with a slightly lower index, so TIR happens at the core's surface and scratches or dirt on the outside don't leak light.

Pulses of infrared light carry internet data through fibers under the oceans and into many homes, and doctors use bundles of fibers in endoscopes to look inside the body.

A concave mirror curves inward like the inside of a spoon. It reflects parallel rays together through a focal point in front of it, so it is a converging mirror. For a mirror shaped like part of a sphere of radius R, the focal length is about f = R ÷ 2. The center of the sphere is called C.

The mirror equation is the same as the thin lens equation, 1/f = 1/d_o + 1/d_i, with m = −d_i ÷ d_o. The difference is that a mirror's real images form in front of it, on the same side as the object, because the light bounces back.

Concave mirrors are used in shaving and makeup mirrors (your face inside F gives a magnified, upright virtual image), in flashlights and headlights (a bulb at F sends out a parallel beam), and in big telescopes, which gather starlight with huge concave mirrors.

A convex mirror bulges outward. It is diverging (negative f) and always gives a smaller, upright, virtual image, but shows a wide view. That's why it's used for store security mirrors and car passenger-side mirrors, which carry the warning "objects in mirror are closer than they appear."

Your retina has three kinds of color-sensing cone cells, most sensitive to red, green and blue light. Every color you see is your brain's reading of how strongly each kind is stimulated.

Adding light (additive color). Screens and stage lights mix red, green and blue light, the additive primaries. Red + green = yellow, green + blue = cyan, blue + red = magenta, and all three = white. Look at a phone screen with a magnifier and you'll see tiny red, green and blue dots, not yellow or white ones.

Mixing pigments (subtractive color). Paints, inks and dyes absorb some colors and reflect the rest. A blue shirt looks blue because its dye absorbs most of the red, orange, yellow and green in white light. Printers use cyan, magenta and yellow inks (plus black, the K in CMYK). Each ink subtracts one primary: cyan absorbs red, magenta absorbs green, yellow absorbs blue. Cyan + yellow ink leaves only green to reflect.

Newton thought light was a stream of tiny particles. Others, like Christiaan Huygens, thought it was a wave. The strongest early evidence for waves came from Thomas Young. In a lecture to the Royal Society in 1801 he explained colors in thin films and other effects by interference, waves adding and canceling.

The famous version is the double-slit experiment. Light passes through two narrow slits close together. Each slit acts as a new source of waves, and on a screen beyond, the two sets of waves overlap. Where crest meets crest they add up (a bright fringe); where crest meets trough they cancel (a dark fringe). Particles flying in straight lines would give just two bright strips, not a row of evenly spaced fringes. Young described an experiment like this in 1803 and drew a two-slit diagram in his 1807 lectures; historians aren't sure exactly which arrangements he actually carried out, but his explanation was right.

By the late 1800s the wave picture seemed complete: Maxwell's theory said light is an EM wave, and in 1887–88 Heinrich Hertz made and detected radio waves, confirming it.

But one experiment didn't fit. In the photoelectric effect, light shining on a metal knocks electrons out of it. Experiments, especially Philipp Lenard's in 1902, found that whether electrons come out depends on the light's frequency, not its brightness. Below a certain frequency, no electrons are freed, however bright the light. Above it, electrons come out immediately, even in dim light, and their maximum energy grows with frequency. A smooth wave spreading its energy evenly couldn't explain that.

In 1905 Albert Einstein proposed that light's energy comes in separate packets, now called photons, building on Max Planck's 1900 idea that energy is exchanged in quanta. Each photon's energy depends only on its frequency:

E = hf    h = 6.626 × 10⁻³⁴ J·s (Planck's constant)

One electron absorbs one photon. If the photon's energy is less than the energy needed to escape the metal, the work function φ, the electron stays put. Otherwise it leaves with at most KE_max = hf − φ. Brighter light means more photons, so more electrons, but not more energetic ones. Robert Millikan's careful experiments (1914–1916) confirmed Einstein's equation, and Einstein's Nobel Prize in Physics for 1921 was awarded "especially for his discovery of the law of the photoelectric effect."