Waves and optics

Review a wave and choose a description

A periodic wave has frequency f, period T=1/f and wavelength λ. Propagation speed satisfies v=λf. Frequency counts oscillations per second, in hertz; λ is the distance between in-phase points and is measured in metres. Propagation speed is not a medium particle’s velocity. In vacuum, light propagates at c, approximately 3,00×10⁸ m/s.

Rays represent the local direction of energy propagation in a geometric approximation. They are useful for mirrors and lenses when relevant apertures and distances allow diffraction effects to be neglected. They are not material threads and do not describe the field’s entire structure. Passing through an aperture comparable to λ spreads propagation, and the description using isolated straight paths can fail. No model should be used outside the scale justifying its approximations.

Reflection and refraction

In specular reflection from a smooth surface, the incident ray, normal and reflected ray lie in the same plane; incidence and reflection angles measured from the normal are equal. A rough surface may distribute reflected directions, although each small region obeys the local relationship. The normal is the straight line perpendicular to the surface at the meeting point.

In refraction, speed changes between media. Refractive index is n=c/v for the model considered. Snell’s law is n₁ sinθ₁=n₂ sinθ₂. If n₂>n₁, the refracted ray moves towards the normal. Frequency remains determined by the source across a stationary boundary; wavelength changes, λ=v/f. Thus a lower speed in the second medium implies lower λ, not necessarily lower frequency. Indices may depend on frequency, explaining dispersion.

Total internal reflection: two conditions

For total internal reflection, light must start in a higher-index medium and enter a lower-index one: n₁>n₂. As θ₁ increases, the refracted angle reaches 90°. At this limit, Snell gives sinθc=n₂/n₁. Angle θc is the critical angle. For incidence greater than θc, there is no propagating refracted ray in the second medium in the ideal geometric model.

It is not enough to say the angle is large: the index order must be checked and the angle compared with θc. Going from air into ordinary glass, n₂/n₁ would exceed 1 and no real critical angle exists. Optical fibres use an index structure to guide light by internal reflection. The full explanation of fields near the boundary is wave-based; rays provide a useful approximation for introducing the mechanism.

Thin lenses and a sign convention

A converging lens brings approximately axis-parallel rays together at a focus; a diverging lens makes rays spread as if coming from a focus. The thin-lens equation is 1/f=1/dₒ+1/dᵢ in the paraxial approximation: a thin lens and rays at small angles to the axis. It is not an exact description of every real lens or large aperture.

We use f>0 for a converging lens and f<0 for a diverging one; dₒ>0 for a real object on the entrance side; dᵢ>0 for a real image on the exit side and dᵢ<0 for a virtual image on the object side. Transverse magnification is m=hᵢ/hₒ=−dᵢ/dₒ. Negative m indicates inversion; |m| indicates the size ratio. A real image can be captured on a screen placed at its position; a virtual one is located by extending rays.

Rays and equations must agree

For a converging lens, a ray parallel to the axis leaves towards the focus on the other side; a ray near the optical centre is approximated as undeviated in the thin-lens model. Their intersection locates the image. If they do not meet on the exit side, extend them backwards to locate a virtual image. The drawing helps interpret the sign found in the equation.

When the object is between the focus and a converging lens, the image is virtual, upright and enlarged. At the focus, outgoing rays are parallel and no image forms at a finite distance in the model. For an object beyond the focus, the image is real and inverted. These are not rules to memorise without geometry: they arise as the meeting of rays changes. Use all distances in the same unit and translate the final sign into a location.

Superposition, interference and diffraction

In the linear wave description, disturbances superpose. Two coherent waves with equal frequency and stable relative phase can reinforce or cancel locally. For two slits illuminated in phase by monochromatic light, maxima satisfy d sinθ=mλ and minima d sinθ=(m+1/2)λ, with integer m and slit separation d. Path difference, not merely total distance, determines the result.

On a distant screen L≫d and at small angles, neighbouring maxima are separated by approximately Δy=λL/d. Each slit’s diffraction influences the pattern’s overall intensity; two real slits do not produce equally intense maxima everywhere. Destructive interference at one point does not destroy energy globally: the spatial distribution changes, with reinforcement elsewhere. A phenomenon requiring waves does not invalidate rays at another scale; it shows we need the description appropriate to the question.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

Snell angles are measured to the surface.

They are measured to the normal.

A virtual image corresponds to a meaningless distance.

The negative sign locates an image formed by extensions of rays.

Destructive interference removes energy from the system.

Energy is redistributed spatially; the global analysis includes regions of reinforcement.