Propagation of light

A point-like torch produces a sharp shadow, while a large light source produces a blurred edge. Can a few lines on a diagram predict where light will still reach?

Place an opaque object between a small source and a screen. The shadow's geometry suggests a simple model: light travels in straight lines in a uniform transparent medium. A line showing its direction of travel is called a light ray. It is a line in a drawing, not a thin material wire. The model is useful when wave effects are unimportant; ‘uniform’ means the medium's properties do not vary from place to place.

A torch emits a beam of finite width, represented by several rays. Rays spread apart in a diverging beam, draw together in a converging beam and stay nearly equally spaced in an ideal parallel beam. A single ray indicates a direction within the beam; it is not a separately visible object. Seeing a beam from the side requires light to enter the eye, for example after scattering from dust in the air.

Choose a point on the screen and imagine looking towards the source from it. If the opaque object blocks every direct path from the source, the point lies in its umbra. Boundary rays just graze the obstacle's edges. A shadow need not be absolutely black: reflected light from walls or another source may reach it. The simple diagram excludes these extra paths to isolate the geometric cause of the shadow.

A large source emits light from many points. For a chosen screen point, the object may block only part of the source; rays from the remaining parts still arrive. This is the penumbra. Other conditions being equal, hiding more of the source reduces its illumination of that point. Shrinking the source towards a point narrows the penumbra and sharpens the edge. Here the blur comes from source size, not rays bending around the object.

In the chosen cross-section, take the source's extreme points. From each, draw straight lines through the opaque object's edges to the screen. Check the regions between the boundaries: the whole source hidden means umbra; part hidden means penumbra; the whole source visible means full illumination. A more distant screen or a changed source size can alter both penumbra width and whether an umbra exists, so one stock diagram cannot replace the actual geometry.

A straight path does not mean instantaneous travel. In vacuum, c ≈ 3·10⁸ м/с; in an ordinary transparent medium the speed is denoted v and, for visible light, is less than c. In equal times, greater speed means greater distance travelled. Across a room, the delay is too small to notice directly; over large distances it matters. The value c applies to vacuum, not automatically to glass or water.

The ray model describes large objects well when their dimensions greatly exceed the wavelength of light. Make a slit very narrow and a prediction using only straight lines may no longer match observations: light reveals its wave properties. This marks a limit without undoing successful calculations of ordinary shadows. Here we consider large obstacles and keep the penumbra of an extended source distinct from wave effects.

From one screen point the source is entirely hidden by an object; from another, half the source is visible. Which point is in umbra and which in penumbra?

At the first point, the source is completely hidden, so no direct path from it reaches the point: this is umbra. The model excludes additional scattered illumination.

At the second point, half the source is visible, leaving some direct paths open: this is penumbra. Seeing ‘half’ does not make it umbra, nor does it specify exact illuminance without information about the source's brightness.

The first point is in umbra and the second in penumbra: the whole source is hidden in the first case, but only part in the second.

A ray models direction and describes straight-line propagation in a uniform medium within the geometric approximation.

Umbra and penumbra differ in how much of the chosen source is visible from the point considered.

A real beam has width, and light takes a finite time to travel a distance.

A ray can reflect when it meets a surface. Next we will construct the reflected ray and the image in a mirror.