Ray Model of Light

Punch a pinhole in a shoebox, point it at a sunlit window, and a small inverted picture of the scene glows on the far wall. No lens, no electronics — just a hole and the fact that light, on the everyday scale, travels in straight lines. That single fact is the whole content of the ray model: replace the electromagnetic wave with an abstract directed line, the ray, that marks the local direction of energy flow, and an enormous range of optics — shadows, mirrors, lenses, prisms, fibres, the camera obscura on your wall — falls out of pure geometry. The ray is not a thread of light you could isolate; it is a bookkeeping device for where the energy goes. It is defined against the wave it stands in for: a ray is the curve everywhere perpendicular to the wavefronts, the surfaces of constant phase. Where the wavefronts are flat the rays run parallel; where they are spheres expanding from a source the rays fan out radially. In a uniform medium a ray is straight, and it bends only where the refractive index changes — at a mirror, a glass surface, a temperature gradient in the air. Crucially, the ray model is not a rival to wave optics but its short-wavelength limit. It emerges when the wavelength is negligible against every length that matters — aperture, mirror, lens radius. In that limit interference and diffraction switch off and Fermat's principle takes over: the true ray path is the one whose optical path length is stationary. Master this picture and you own the entry point to all of imaging optics — while never forgetting the three things it deliberately throws away: phase, interference, and the diffraction limit that sets the sharpness of every real instrument.

Key takeaways

The ray model treats light as directed geometric rays: straight in a uniform medium, bending only at interfaces where the refractive index $n$ changes.
It is the geometrical-optics ( $\lambda \to 0$ ) limit of wave optics — valid when every feature (aperture, radius, path difference) is far larger than the wavelength $\lambda$ .
A ray is everywhere perpendicular to the wavefronts (surfaces of constant phase), and ray paths make the optical path length $\int n\,\mathrm{d}s$ stationary (Fermat's principle).
From these premises alone follow reflection, Snell's law, total internal reflection, and the entire machinery of mirrors and lenses.
Its failure is sharp and quantitative: shrink an aperture toward $\lambda$ and diffraction spreads the light by an angle $\theta \sim \lambda/a$ — a pinhole cannot be made arbitrarily small before it blurs.
Key idea: a ray is the trajectory perpendicular to the wavefronts; in the short-wavelength limit light runs along such rays, and that one picture generates all of geometrical optics — but only where features dwarf the wavelength.

The core insight

A ray is the geometric line perpendicular to the wavefronts, pointing along the flow of energy. In the limit where the wavelength is negligible, light propagates along such rays — straight in uniform media, bent only at index changes — and their paths make the optical path length stationary. This single picture generates reflection, refraction, and all of imaging optics.

Where this lives today

Every camera, telescope, and microscope is designed by ray tracing: software launches thousands of rays through each surface, bending them by Snell's law, to predict focal spots, aberrations, and field coverage before a single piece of glass is ground. In the paraxial regime this collapses to multiplying $2\times2$ ray-transfer matrices, so a designer can place images and compute magnifications by hand. The same rays, followed through a graded-index profile, guide light in optical fibres and explain why the setting Sun looks flattened. And the pinhole itself never left: astronomers still project the Sun through one to watch eclipses safely, and the "coded aperture" cameras on X-ray and gamma-ray space telescopes — where no lens exists — are pinhole cameras scaled up to see the high-energy sky. The ray model's boundary, the $\sqrt{\lambda L}$ pinhole, is also the doorway to the wave optics that follows: diffraction, interference, and the resolution limits of every real instrument.

The physical picture

Think of a wavefront as a rank of soldiers marching in step, and the rays as the individual straight tracks each soldier walks — always at right angles to the rank. Send the rank across ground that slows one flank and the whole line pivots; that pivot is refraction, and every law of geometrical optics is just the geometry of turning ranks. A ray carries no width and no phase. That is exactly why it can tell you where the light lands with beautiful accuracy while staying silent about how bright a fringe is or how blurred an edge becomes.

Figure

Rays are the lines perpendicular to the wavefronts — parallel for plane waves, radial from a point source

The picture hands you three habits worth keeping. First, rays follow energy, not phase: in an isotropic medium the ray points along $\nabla\phi$ , the wavefront normal, which is also the time-averaged Poynting vector. Second, bending lives only at index changes: in uniform glass or air a ray is dead straight, and all the action — reflection, refraction, focusing — happens at surfaces where $n$ jumps or in regions where it varies. Third, and most important, the model has a validity coupon stamped on it: it works only while features dwarf the wavelength. Push a feature down toward a few wavelengths and the wave "remembers" its wavelength — diffraction reappears, shadows soften, and no point can be focused to a point. That last habit is not a vague warning; it is a sharp, computable trade-off, and the humble pinhole camera puts it on a single graph. A pinhole makes an image by geometry: each point of the scene projects a little disc the size of the hole, so a smaller hole means a sharper geometric image. But a small hole is also a small aperture, and diffraction spreads the light passing through it by an angle $\sim\lambda/d$ , smearing the spot by $\sim 2.44\,\lambda L/d$ at screen distance $L$ . Shrink the hole and geometry improves while diffraction worsens. The two effects cross, and there is a sharpest possible pinhole.

Figure

The sharpest pinhole — geometry wants a small hole, diffraction wants a large one; they meet at d ≈ 0.45 mm

The blue curve is the whole ray model in one U: on its right shoulder, where the hole is comfortably larger than a wavelength, geometry rules and light behaves like rays; on its left shoulder, where the hole approaches the wavelength, diffraction takes over and the ray picture collapses. The bottom of the valley is where the two meet — the sharpest a lensless hole can ever be. That crossover point is the boundary of the ray model, drawn in real millimetres.

The core equations

The ray model rests on one geometric definition and one variational principle; everything else is consequence. Start with what a ray is. The wave carries a phase $\phi(\mathbf{r})$ , and its surfaces of constant phase are the wavefronts. The direction of propagation at any point is the steepest-ascent direction of the phase, so the ray's unit tangent is:

\hat{\mathbf{s}} = \frac{\nabla \phi}{|\nabla \phi|}, \qquad \text{ray} \perp \text{wavefront} \qquad (1)

where: $\hat{\mathbf{s}}$ — unit vector along the ray; $\phi$ — optical phase; $\nabla\phi$ — its gradient, normal to the surfaces $\phi=\text{const}$ and parallel to the local wavevector. Read it physically: a ray is not an independent object but the normal congruence of the wavefronts. In isotropic media this normal also points along the energy flow, which is why "the direction of the ray" and "the direction the light goes" are the same thing. To decide which path a ray actually takes, weight geometric distance by the local index. The optical path length counts wavelengths, not metres:

\mathrm{OPL} = \int_A^B n(\mathbf{r})\,\mathrm{d}s \qquad (2)

where: $n(\mathbf{r})$ — local refractive index; $\mathrm{d}s$ — element of geometric arc length. Since the wavelength in a medium is $\lambda_0/n$ , the OPL is exactly $\lambda_0$ times the number of wave crests along the path — the natural currency of phase. Fermat's principle then selects the true ray as the one whose OPL is stationary against nearby paths, $\delta\!\int_A^B n\,\mathrm{d}s = 0$ ; reflection, Snell's law, and lens focusing are all just this stationarity worked out at a surface. Where does the straight-line premise come from? From the wave equation itself. Writing a field of slowly varying amplitude and rapidly varying phase and keeping the leading order in the short-wavelength limit collapses the wave equation to the eikonal equation:

|\nabla S|^2 = n^2, \qquad \phi = k_0 S, \qquad k_0 = \frac{2\pi}{\lambda_0} \qquad (3)

where: $S$ — the eikonal (optical path function, units of length); $k_0$ — vacuum wavenumber. Its characteristic curves — the trajectories that follow $\nabla S$ — are precisely the rays, and differentiating along one gives the ray equation $\frac{\mathrm{d}}{\mathrm{d}s}\!\left(n\,\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s}\right)=\nabla n$ . When $n$ is constant this reads $\mathrm{d}^2\mathbf{r}/\mathrm{d}s^2 = 0$ : rays are straight. When $n$ varies smoothly — hot air over a road, a graded-index fibre — rays curve, and the same equation predicts mirages. The full development of $(3)$ is the job of the eikonal-equation article; here we take its two headline consequences and build.

Derivation AWhy light goes straight — rays as the short-wavelength limit

We want to show, not merely assert, that the ray model follows from wave optics: that the wavefront normals propagate along straight lines in a uniform medium, so the "directed geometric line" of the ray model is a genuine limit of Maxwell's waves rather than a separate postulate. The strategy is to feed a rapidly-oscillating-phase field into the scalar wave (Helmholtz) equation, keep only the leading order as the wavelength shrinks, and read off the geometry of the resulting phase fronts.

Step 1. Write the field as a slowly varying amplitude times a fast phase: $U(\mathbf{r}) = A(\mathbf{r})\,e^{\,i k_0 S(\mathbf{r})}$ , with $k_0 = 2\pi/\lambda_0 \to \infty$ the large parameter. Why: "geometrical optics" is precisely the regime where the phase turns over in a wavelength while $A$ and $n$ barely change over the same distance — separating the two lets us keep the dominant one.

Step 2. Insert this into the Helmholtz equation $\nabla^2 U + k_0^2 n^2 U = 0$ and collect powers of $k_0$ . The leading term, of order $k_0^2$ , is $\big(n^2 - |\nabla S|^2\big)U = 0$ , giving the eikonal equation $|\nabla S|^2 = n^2$ . Why: at short wavelength the $k_0^2$ term dominates the amplitude derivatives (order $k_0^0$ ) and the mixed term (order $k_0^1$ ); balancing the biggest terms fixes the phase geometry first.

Step 3. The ray is the curve tangent to $\nabla S$ : set $n\,\mathrm{d}\mathbf{r}/\mathrm{d}s = \nabla S$ , which is a unit tangent because $|\nabla S| = n$ . Differentiate along the ray, $\frac{\mathrm{d}}{\mathrm{d}s}(\nabla S) = (\hat{\mathbf{s}}\cdot\nabla)\nabla S = \tfrac{1}{n}(\nabla S\cdot\nabla)\nabla S = \tfrac{1}{2n}\nabla|\nabla S|^2 = \tfrac{1}{2n}\nabla(n^2) = \nabla n$ . Why: the eikonal identity $|\nabla S|^2=n^2$ lets the awkward second-derivative term collapse to a gradient of $n$ — the only place the medium can steer the ray.

Step 4. This is the ray equation $\frac{\mathrm{d}}{\mathrm{d}s}\!\left(n\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s}\right)=\nabla n$ . In a uniform medium $\nabla n = 0$ , so $\mathrm{d}^2\mathbf{r}/\mathrm{d}s^2 = 0$ and the ray is a straight line; a smooth gradient $\nabla n \ne 0$ curves it toward higher index. Why: rectilinear propagation, the founding assumption of ray optics, is now a theorem — and the same equation quantitatively predicts mirages and graded-index lenses. *Verified against Born & Wolf Ch. 3 and reproduced symbolically in verify_ray-model-of-light.py (the $\lambda/a$ diffraction scaling and the constant- $n$ straight-ray reduction).*

Go deeper · Derivation B · 1 minThe sharpest pinhole — where geometry meets diffraction

Derivation BThe sharpest pinhole — where geometry meets diffraction

The ray model's boundary is not philosophical but numerical, and the pinhole camera lets us compute it. A hole of diameter $d$ at distance $L$ from the screen blurs a point of the scene two ways at once: geometrically, each scene point paints a disc the size of the hole (blur $\approx d$ ), and by diffraction, the finite aperture spreads the light into an Airy patch (blur $\approx 2.44\,\lambda L/d$ ). We add the two and minimise the total spot to find the single hole size that is objectively sharpest — the point where ray optics hands over to wave optics.

Step 1. Model the total spot width as the sum of the two blurs, $w(d) = d + \dfrac{2.44\,\lambda L}{d}$ . Why: shrinking $d$ improves geometry (first term falls) but worsens diffraction (second term rises), so the two pull in opposite directions and a minimum must exist.

Step 2. Set $\mathrm{d}w/\mathrm{d}d = 1 - 2.44\,\lambda L/d^2 = 0$ , giving $d_{\text{opt}} = \sqrt{2.44\,\lambda L}$ , and $\mathrm{d}^2w/\mathrm{d}d^2 = 2(2.44\lambda L)/d^3 > 0$ confirms it is a minimum. Why: the balance point is exactly where the geometric and diffraction blurs are equal — the physical signature of the ray/wave crossover.

Step 3. Evaluate for visible light and a shoebox: $\lambda = 550\ \mathrm{nm}$ , $L = 15\ \mathrm{cm}$ give $d_{\text{opt}} = \sqrt{2.44(550\times10^{-9})(0.15)} = 0.449\ \mathrm{mm}$ , with a residual spot $w_{\text{min}} = 2d_{\text{opt}} = 0.90\ \mathrm{mm}$ . Why: the sharpest lensless image is set by $\sqrt{\lambda L}$ — a wavelength leaking into a geometry problem, the fingerprint of the model's edge.

Step 4. Read the two regimes off the result: for $d \gg d_{\text{opt}}$ the geometric term wins and light behaves like rays; for $d \lesssim d_{\text{opt}}$ diffraction wins and the ray picture fails. The Fresnel number $N_F = a^2/(\lambda L)$ with $a = d/2$ measures which side you are on — $N_F \approx 76$ for a 5 mm hole (deep ray regime), $N_F \approx 0.6$ at the optimum (the crossover). Why: this gives the promised sharp validity criterion — ray optics is trustworthy precisely when $N_F \gg 1$ . *Verified in verify_ray-model-of-light.py: $d_{\text{opt}} = \sqrt{2.44\lambda L}$ symbolically, the minimum, and the numbers $0.449\ \mathrm{mm}$ , $0.90\ \mathrm{mm}$ , and the Fresnel numbers.*

Limiting cases

Case	Result	Meaning
$a \gg \lambda$ (large features)	$N_F = a^2/\lambda L \gg 1$	diffraction spread $\sim\lambda/a$ negligible; rays run straight — ray model excellent
$a \sim \lambda$ (tiny aperture)	$N_F \sim 1$	light spreads by $\theta \sim \lambda/a$ ; no sharp shadow, no point focus — use wave optics
$\nabla n = 0$ (uniform medium)	$\mathrm{d}^2\mathbf{r}/\mathrm{d}s^2 = 0$	rays are exactly straight; all bending confined to interfaces
$\nabla n \ne 0$ (graded index)	$\frac{\mathrm{d}}{\mathrm{d}s}(n\,\mathrm{d}\mathbf{r}/\mathrm{d}s) = \nabla n$	rays curve smoothly toward higher $n$ — mirages, GRIN lenses, atmospheric refraction
isotropic vs birefringent	ray $\parallel \nabla\phi$ (isotropic only)	in anisotropic crystals the energy ray and the wavefront normal split apart

Common trap“A smaller pinhole always makes a sharper picture”

It does not — and believing so is the ray model applied past its expiry date. Geometry alone says the image blur equals the hole size, so shrink the hole and the picture sharpens without limit. But the ray model has no phase and therefore no diffraction, and diffraction is exactly what stops you: below $d_{\text{opt}} = \sqrt{2.44\lambda L}$ the Airy spread $2.44\lambda L/d$ grows faster than the hole shrinks, and the image gets worse. The same blindness means the ray model can never predict a microscope's resolution limit or the smallest focal spot of a lens — those are wave phenomena. Whenever an answer implies "arbitrarily sharp" or "arbitrarily small," the ray model has been pushed into the regime $a \sim \lambda$ where it simply does not apply.

Worked examples

1 · The sharpest pinhole (confidence builder)

You are building a pinhole camera with the screen $L = 15\ \mathrm{cm}$ behind the hole, using visible light $\lambda = 550\ \mathrm{nm}$ . What hole diameter gives the sharpest image, and how sharp is it?

Step 1. The total blur is geometry plus diffraction, $w(d) = d + 2.44\lambda L/d$ . Minimising gives $d_{\text{opt}} = \sqrt{2.44\,\lambda L}$ .

Step 2. Put in the numbers: $d_{\text{opt}} = \sqrt{2.44 \times (550\times10^{-9}\,\mathrm{m}) \times (0.15\,\mathrm{m})} = \sqrt{2.01\times10^{-7}\,\mathrm{m}^2} = \boxed{0.449\ \mathrm{mm}}$ .

Step 3. At that hole the two blurs are equal, so the residual spot is $w_{\text{min}} = 2\,d_{\text{opt}} = \boxed{0.90\ \mathrm{mm}}$ . A finer hole would diffract worse; a coarser hole would project a bigger disc.

Step 4. Sanity check with the Fresnel number: $N_F = (d_{\text{opt}}/2)^2/(\lambda L) = 0.61 \approx 1$ , right at the ray/wave crossover — exactly where a purely geometric model should be at the limit of its validity. Halve the screen distance to $L = 7.5\ \mathrm{cm}$ and the best hole shrinks only as $\sqrt{L}$ , to $0.32\ \mathrm{mm}$ .

2 · The Sun on the wall (exam level)

The same pinhole ( $d = 0.45\ \mathrm{mm}$ , $L = 15\ \mathrm{cm}$ ) is aimed at the Sun, whose angular diameter is $0.53^\circ$ . How big is the Sun's image, and is it a sharp point or a disc?

Step 1. A distant extended source images to a patch set by its angular size: each point of the Sun sends a nearly parallel bundle through the hole, and the bundles from opposite edges land $\theta_\odot L$ apart. Convert the angle: $\theta_\odot = 0.53^\circ = 9.25\times10^{-3}\ \mathrm{rad}$ .

Step 2. Geometric image diameter: $\theta_\odot L = (9.25\times10^{-3})(0.15\,\mathrm{m}) = \boxed{1.39\ \mathrm{mm}}$ .

Step 3. Compare with the hole and its diffraction: the $1.39\ \mathrm{mm}$ solar disc is three times the $0.45\ \mathrm{mm}$ hole and far larger than the diffraction spread, so the image is a genuine round disc of the Sun, not a point — which is why pinhole projection safely shows sunspots and the crescent Sun during an eclipse.

Step 4. The lesson of the two examples together: for a nearby point the sharpness is limited by hole size and diffraction (Example 1), while for a distant extended source the image size is set by the source's angular size (Example 2). Ray optics handles both — as long as the hole stays comfortably above $\sqrt{\lambda L}$ .

Physics of the portrait

The ray model is often taught as if it were obvious, but turning "light travels in straight lines" into a science required insisting on experiment over authority — and that insistence has a name.

Origins · 1 minIbn al-Haytham and how this was discovered

Ibn al-Haytham

c. 965 – c. 1040 · Basra and Cairo

Ibn al-Haytham (Latinised Alhazen) wrote the seven-volume Kitāb al-Manāẓir (Book of Optics) around 1021, and with it turned optics into an experimental science. He settled a centuries-old debate by arguing that vision works by light entering the eye from objects (intromission), not by rays emitted from the eye — and backed the claim with the camera obscura, showing that light from separate sources crosses through a small hole and forms an inverted image without mixing, direct evidence that light travels in straight, independent rays. He studied reflection from spherical and parabolic mirrors (the reflection problem on a sphere still bears his name), measured refraction, and above all insisted that claims be tested against controlled observation. Translated into Latin in the twelfth century, his work shaped Roger Bacon, Kepler, and the whole European tradition that became geometrical optics.

History

300 BCE

— Euclid's Optica treats vision with straight rays and states the equal-angle law of reflection geometrically.

150

CE — Ptolemy tabulates angles of incidence and refraction, the first quantitative refraction data.

1021

— Ibn al-Haytham's Kitāb al-Manāẓir founds experimental optics: intromission, the camera obscura, and rectilinear propagation.

1604

— Kepler analyses the pinhole image and the optics of the eye in Ad Vitellionem Paralipomena.

1657

— Fermat states the principle of least time, giving the ray model its variational foundation.

Going further

Texts. Hecht, Optics, 5th ed., Ch. 4–5 (geometrical optics, Fermat's principle) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 1 (Ray Optics) · Born & Wolf, Principles of Optics, Ch. 3 (foundations of geometrical optics, the eikonal) · Goodman, Introduction to Fourier Optics (where the ray model gives way to diffraction).

Provenance. All results are standard geometrical optics. A verifier should reproduce: the eikonal equation $|\nabla S|^2 = n^2$ and ray equation $\frac{\mathrm{d}}{\mathrm{d}s}(n\,\mathrm{d}\mathbf{r}/\mathrm{d}s)=\nabla n$ from the short-wavelength expansion of the Helmholtz equation (Born & Wolf Ch. 3); the constant- $n$ reduction to straight rays; the optimal-pinhole result $d_{\text{opt}}=\sqrt{2.44\lambda L}$ and the numbers $0.449\ \mathrm{mm}$ , $0.90\ \mathrm{mm}$ ( $\lambda=550$ nm, $L=15$ cm); the Fresnel number $N_F = a^2/\lambda L$ ; and the Sun-image size $\theta_\odot L = 1.39\ \mathrm{mm}$ ( $\theta_\odot = 0.53^\circ$ ).

Prerequisites. Wavefronts and the wave nature of light · Refractive Index and Optical Path Length. Leads to. Fermat's Principle of Least Time · Law of Reflection · Snell's Law of Refraction · Thin Lens Equation and Lensmaker's Equation · Image Formation and Magnification.

Is a ray a real, physical thing I could isolate with a narrow enough slit?

No — and trying reveals exactly why. A ray is a mathematical line marking the direction of energy flow, with no width and no phase. If you try to isolate one by shrinking a slit, the slit becomes an aperture comparable to $\lambda$ , and diffraction spreads the light into a fan of angle $\sim\lambda/a$ : the harder you squeeze, the wider it spreads. The ray is an idealisation valid in bundles over large apertures, not a thread you can pull out one at a time.

What exactly is the difference between a ray and a wavefront?

They are perpendicular partners. A wavefront is a surface of constant phase — think of the expanding sphere of crests from a dropped pebble. A ray is a line that crosses every wavefront at right angles, pointing along the direction of travel. Wavefronts tell you when the phase arrives; rays tell you where the energy goes. In isotropic media the two descriptions are equivalent; in birefringent crystals they can point in slightly different directions.

Why does light "choose" the stationary optical path — does it try all paths?

Fermat's principle is a description, not a mechanism: the ray does not sample alternatives. The deep reason comes from the wave picture, where the field really does spread over all paths, but contributions from paths near the stationary one add in phase (they have nearly equal OPL) while others cancel. What survives is the neighbourhood of the stationary path — which the ray model then reports as "the" path. The full story is the domain of wave optics and the path integral.

If the ray model ignores diffraction, why is it still worth learning?

Because for the overwhelming majority of optical engineering, features are millions of wavelengths across and diffraction is a tiny correction. Camera lenses, telescopes, microscopes, prisms, and fibres are all designed first with rays — reflection, refraction, focal lengths, magnifications, aberrations — and diffraction is added only at the very end to set the ultimate resolution. Rays give the intuition and the fast arithmetic; wave optics supplies the fine print.

How can a lensless pinhole form an image at all?

By selection, not by focusing. A pinhole does not bend light; it simply blocks all but a narrow bundle of rays from each scene point, so each point maps to a small spot on the screen and the spots assemble into a (dim, inverted) image. A smaller hole selects a narrower bundle and sharpens the geometric image — until diffraction, ignored by the ray model, spreads the transmitted light and sets the limit computed in this article.

When exactly does the ray model break down — is there a number?

Yes: the Fresnel number $N_F = a^2/(\lambda L)$ , where $a$ is the aperture radius and $L$ the propagation distance. When $N_F \gg 1$ the geometry dominates and ray optics is trustworthy; when $N_F \sim 1$ or smaller, diffraction takes over and you must switch to wave optics. For a 5 mm aperture and a 15 cm distance $N_F \approx 76$ (safely ray-like); at the optimal pinhole $N_F \approx 0.6$ (the crossover). It is the single number that tells you which optics to use.