The Ray Model & Reflection

A wavefront is a surface of constant phase, and a ray is the line perpendicular to it, drawn in the direction the energy moves. Far from a small source the wavefronts are spheres and the rays are radii; a collimated beam has plane wavefronts and parallel rays. The picture is a limit, not a physical object: no experiment isolates a single ray, and trying to make one by narrowing an aperture defeats itself. It earns its keep because visible light is short — λ = 550 nm against a 5.0 mm aperture is about 9.1 × 10³ wavelengths across — so over laboratory distances the wave's spreading is invisible and light does travel in straight lines within a uniform medium. Two consequences follow at once. Rays cross without deflecting one another, so a crowded ray diagram is legitimate. And the model tracks direction only: it says where light goes, never how much of it goes there, and never with what phase it arrives.

Take a plane wavefront meeting a plane surface with its normal at angle θᵢ. One end of the wavefront, at A, reaches the surface first; the other end still has a distance ct to travel before it lands at B. While it covers that distance, A re-emits a hemispherical wavelet that grows to radius ct. The new wavefront is the plane through B tangent to that wavelet, touching it at C. The triangle ABC and the incident triangle are right angled, share the hypotenuse AB, and have equal opposite sides ct, so they are congruent and the angles at A and B match: θᵣ = θᵢ. The wavelet is centred on the surface, so its tangent plane puts the reflected ray in the plane of the incident ray and the normal, on the opposite side of it. Read the angles from the normal, never from the surface: a ray arriving 28° from the normal leaves 28° from the normal, its direction of travel turned through 180° − 56° = 124°.

A rough surface does not break the law of reflection; it applies the law independently on each microscopic facet, and the facets have different normals, so the reflected rays fan out. What decides which behaviour you see is the height of the roughness against the wavelength. The Rayleigh criterion puts the boundary at Δh ≈ λ/(8 cos θᵢ): at λ = 550 nm and normal incidence a surface must be flat to 69 nm to act as a mirror, which is why float glass with a few nanometres of waviness returns a sharp image while a matte wall with 10 μm of texture, about 150 times the limit, returns none. The cos θᵢ matters too. Take a paper surface with Δh = 0.60 μm: at normal incidence the limit is 69 nm and it looks matte, but at θᵢ = 85° the limit rises to 550 nm ÷ (8 × 0.0872) = 789 nm, above its roughness, and the same paper throws a glossy sheen. That is the low-sun glare off a dry road.

Take an object point O a perpendicular distance d in front of the mirror and draw any two rays from it. Each reflects with θᵣ = θᵢ about the local normal, and the two reflected rays diverge, so no light converges anywhere. Extend them backwards and they meet at a single point I on the normal through O, a distance d behind the mirror; the two right triangles sharing the mirror line are congruent, which fixes d′ = d whichever pair of rays you chose. The image is virtual: put a screen at I and nothing appears on it. The eye cannot tell, because it samples only the diverging cone that reaches it. Stand 1.8 m from a wall mirror and your eyes accommodate as though the object were 3.6 m away, and what they see is erect and life-size, m = +1. The reversal people notice is along the normal rather than sideways: the mirror swaps front and back, which is why the image of a right hand is a left hand.

Hold the incident ray fixed and rotate the mirror through α about an axis lying in its surface, perpendicular to the plane of incidence. The normal rotates with the mirror, through the same α, so the angle of incidence changes by α. The reflected ray must stay at that angle on the other side of the normal, so it moves twice over: once because the normal moved through α, and again because its angle from that normal changed by α. The total is 2α, and the original angle of incidence never enters, so the factor of two is exact rather than a small-angle approximation. That is the optical lever. Send the reflected beam onto a scale a distance L away, arranged so the beam meets the scale perpendicularly, and the spot moves L tan(2α): with L = 1.50 m a mirror rotation of 0.25° shows as 1.50 × tan 0.50° = 13.1 mm. Galvanometers, atomic-force-microscope cantilevers and laser scanners all live on this factor of two.

The ray model is the assumption that apertures are huge compared with the wavelength. Push the other way and it fails in a measurable, predictable manner: light passing through a width a spreads through a half-angle of roughly λ/a. At λ = 550 nm a 2.0 mm beam carries θ ≈ 2.8 × 10⁻⁴ rad, which is 0.28 mm of spreading per metre — nothing against a 2.0 mm beam on a benchtop, so ray tracing is safe. Shrink the aperture to 20 μm and the same estimate gives 2.8 × 10⁻² rad, or 28 mm per metre: the beam is now a fan and no straight line describes it. The discarded phase returns whenever two beams overlap as well, since rays are drawn as though they ignore each other while real light interferes. So trace rays for mirrors, lenses and instrument layouts, and abandon them for slits, thin films and any question about resolution.