Optics: Light, Lenses & Refraction

How light bends to make pictures

Junior level — plain language, no maths

Light almost always travels in dead-straight lines — which is why you get sharp shadows and why you can't see around corners. But the instant light crosses from one clear material into another, from air into water or glass, it bends. That bending is called refraction, and it's the reason a straw in a glass of water looks snapped in half, and why a coin at the bottom of a pool sits higher than it really is.

Shape a piece of glass just right and you can steer that bending on purpose — that's a lens. A magnifying glass bulges in the middle, so it gathers stray rays of light and funnels them to meet at a single point. Hold it at the right distance and those rays cross over and rebuild a picture of whatever they came from — often flipped upside down. That crossing-over is exactly why a camera, a telescope and your own eyeball can all take the light bouncing off the world and fold it into a clear image.

Your eye does it with a soft, living lens that muscles squeeze to change its shape, focusing near then far in a fraction of a second. A camera does the same job with a sliding glass lens. In the simulation below, move the object and reshape the lens, and watch the rays cross to build the image — real, flipped, and resized.

Snell's law and the thin-lens equation

Student level — the core equations

Refraction happens because light travels at different speeds in different materials, and each material is rated by its refractive index \(n = c/v\) — how many times slower light goes inside it than in a vacuum (water is 1.33, glass about 1.5). At a boundary the ray bends by exactly the amount that keeps its wavefronts in step, captured by Snell's law \(n_1\sin\theta_1 = n_2\sin\theta_2\). Going into a denser medium bends the ray toward the normal; coming out bends it away.

Push that exit angle far enough and something dramatic happens: past the critical angle \(\theta_c = \arcsin(n_2/n_1)\) the ray can't escape at all and reflects entirely back inside — total internal reflection, the trick that pipes light down a fibre-optic cable for thousands of kilometres with barely any loss. A lens simply refracts a ray twice, once at each curved surface, and its focusing power is set by the lensmaker's equation from those curvatures and the glass index.

Where a lens forms its image follows the beautifully simple thin-lens equation \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), linking focal length \(f\), object distance \(d_o\) and image distance \(d_i\). The image's size and orientation come from the magnification \(m = -d_i/d_o\): a negative \(m\) means inverted. Put the object beyond \(f\) and you get a real, flipped image you can catch on a screen (a camera); bring it inside \(f\) and the image turns virtual, upright and enlarged (a magnifying glass).

Wave optics, aberrations and the diffraction limit

Scholar level — full mathematical depth

01Rays are shortcuts; light is a wave

Ray optics — straight lines that bend at surfaces — is a superb approximation, but it is only the short-wavelength limit of something deeper. Fermat's principle reframes every path light takes as the one of stationary optical path length, \(\delta\!\int n\,ds = 0\). Snell's law, the law of reflection, the very existence of focal points all fall out of this single variational statement — light behaves as if it sniffs out the extremal route. It is the same mathematics that reappears as the principle of least action in mechanics.

02Refraction as wavefronts changing gear

Why does a ray bend toward the normal in glass? Picture the wavefront as a marching row: when one end enters the slower medium first, it lags while the other end keeps pace, and the whole front pivots — Huygens' construction made literal. Speed and wavelength drop by the factor \(n\) while frequency is conserved, so \(\lambda_{\text{medium}} = \lambda_0/n\). The ray picture and the wave picture agree exactly, but only the wave picture survives when the lens gets small.

03Why no lens is ever perfect: aberrations

The tidy thin-lens equation assumes rays hug the axis. Real lenses violate it. Spherical aberration focuses edge rays and central rays at different points because a sphere isn't the ideal shape; chromatic aberration smears colour because \(n(\lambda)\) means blue light bends more than red, so a lens has a different focal length for every colour. Designers fight back by cementing crown and flint glasses into achromatic doublets and grinding aspheric surfaces — a whole craft devoted to undoing the sphere's imperfections.

04The wall you cannot climb: diffraction

Even a flawless lens cannot focus light to a point. Because light is a wave passing through a finite aperture, it spreads into an Airy disc, and two points blur together once they are closer than the Rayleigh criterion \(\theta_{\min} = 1.22\,\lambda/D\). Resolution is set by the numerical aperture, \(d_{\min} = 0.61\,\lambda/\text{NA}\): the smallest thing a microscope can cleanly separate is roughly half a wavelength, a few hundred nanometres. This is a limit of physics, not engineering.

05A lens computes a Fourier transform

There is a startling identity at the heart of optics: the light field in the back focal plane of a lens is the spatial Fourier transform of the field in its front focal plane. Fine detail in the object becomes high spatial frequency far off-axis, so an aperture that clips those rays is literally a low-pass filter. This is the foundation of Fourier optics, of spatial filtering, and of the modulation transfer function that quantifies exactly how much contrast a lens preserves at each level of detail.

06Cheating the diffraction limit

For a century the Rayleigh limit looked absolute — and then microscopy simply went around it. STED shrinks the glowing spot with a doughnut of depletion light; PALM/STORM switch single molecules on and off and locate each far more precisely than it blurs, winning the 2014 Nobel Prize in Chemistry. Meanwhile metamaterials and near-field probes exploit the evanescent waves that ordinary lenses throw away. The diffraction limit still holds for far-field propagating light — but cleverness about which light you use has pushed optical imaging down to the scale of individual molecules.