Maxwell's equations in 2D

Set-ups

Each puts a shape in the box and aims the source at it; the drawer (Experiments & help) says what to look for.

Source

Paint

Show

More: polarisation, meters, probe, grid

Polarisation

Light in the plane comes in two kinds that do not mix. TM: the electric field points out of the screen (the picture shows it as E_z). TE: the electric field lies in the picture and the magnetic field points out of it (shown as H_z); only TE has a Brewster angle.

Measure

A meter is a green line on the picture that reads the power crossing it (in the direction of its small arrow) as a share of the beam's; drag its ends. The probe is a yellow cross that records the field where it is, on a chart under the picture.

Grid and background

Experiments

Start here

This is light, computed from Maxwell's equations in a box a few wavelengths across. The colours are the electric field, red one way and blue the other, and a beam is crossing a glass prism.

Drag the beam's white dot to move it, and the ring at the tip of its arrow to aim it: the dashed line is where Snell's law says the beam should go, and the line under the picture gives the angles. Drag anywhere else to paint glass in the way (or metal, or an absorber, chosen under Paint).

Look for the shorter waves inside the glass, and for the beam turning at each face. The buttons under Set-ups put a lens, two slits, a waveguide or a crystal in the box instead; each is one of the experiments below.

1. A beam through a prism

A beam of light meets a glass prism ($n=1.5$) at $38^\circ$ to the normal of its first face. Inside the glass the waves are shorter, $\lambda/n$, and slower, and to keep in step with the waves outside all along the face the beam has to turn: towards the normal on the way in, away from it on the way out, by Snell's law, $\sin\theta_1=n\sin\theta_2$, at each face. The dashed line is the central ray that Snell's law gives, and the line under the picture says how much the prism turns it: $39.3^\circ$ here. Drag the ring at the tip of the beam's arrow to change the angle. No ray is turned by less than $37.2^\circ$, and that happens when the ray crosses the prism parallel to its base (the minimum deviation, which is how a prism's index is measured). Aim the beam a little downwards (the ring below the dot) and the ray inside reaches the prism's base, which it meets too steeply to get out, beyond the critical angle $\sin\theta_c=1/n$: it is totally reflected there (the dashed line bounces) and leaves by the third face. A little light is reflected at every face (Fresnel's formulas).

2. Circular waves

A wire along $z$ carrying an oscillating current sends out circular waves, one $\lambda$ apart. Drag it around. The amplitude falls off as $1/\sqrt r$: in two dimensions the energy spreads over a circle, whose length grows as $r$. (The compressed colours hide it: pause, and point at a crest $1\lambda$ from the wire and one $4\lambda$ from it; the second is about half as high.) Choose Pulse and press Fire for a single ring. Paint some glass in the way: inside it the waves are shorter, $\lambda/n$, and slower.

3. Refraction: Snell's law

A beam from vacuum strikes glass ($n=1.5$) at $30^\circ$. In the glass the waves are shorter, and to keep in step with the waves outside all along the surface they must turn towards the normal: $\sin\theta_1=n\sin\theta_2$. The dashed lines are the directions Snell's law and the law of reflection give; drag the beam's arrow and they follow. A little light is reflected: Fresnel's formula gives 5.8% of the power at $30^\circ$. Turn on Energy flow (under Show) to see where the energy goes.

4. Total internal reflection

Now the beam is inside the glass and meets the surface at $55^\circ$, beyond the critical angle $\sin\theta_c=1/n$ ($41.8^\circ$). It is reflected: meter A (the green line above the surface, which reads the power crossing it) catches only 0.2% of it. (A beam is a spread of directions, here about $\pm5^\circ$, and that 0.2% comes from the few nearer the normal than $41.8^\circ$. Narrow the beam and its spread widens: at a width of $0.6\lambda$, 6% gets out.) Yet the field reaches a little way out: the evanescent wave, which runs along the surface and falls off as $e^{-\kappa y}$ with $\kappa=\frac{2\pi}{\lambda}\sqrt{n^2\sin^2\theta-1}$, by $1/e$ in $0.22\lambda$ here (the notes give both numbers for the beam as it is). It carries no energy away from the surface. Switch to Intensity to see it. Aim the beam closer to the normal than $41.8^\circ$ and light escapes.

5. Frustrated total internal reflection

Two blocks of glass with a gap of $0.3\lambda$ between them. The beam meets the gap at $50^\circ$, beyond the critical angle, so one surface alone would reflect nearly all of it; but the evanescent wave reaches across the gap and starts a beam on the other side. Meter A (the green line above the gap) reads the power that gets through, 32%; meter B, most of the rest, reflected. Widen the gap (the slider under Set-ups) and the transmission falls off nearly exponentially, as $e^{-2\kappa d}$, from 84% at $0.1\lambda$ to 10% at $0.5\lambda$: this is tunnelling, as for a particle through a barrier. The notes under the picture give the exact plane-wave answer averaged over the beam's spread of angles; meter A agrees with it to within 1%.

6. Brewster's angle

This is TE: $\vec E$ lies in the picture, in the plane of incidence (p polarisation). At Brewster's angle, $\tan\theta_B=n$ ($56.3^\circ$ for glass), nothing is reflected, and there is no reflected beam to see. Meter A, in the glass, reads 98.8%. Not quite all: a beam is a spread of angles around $56.3^\circ$, and the notes give the 0.8% it should reflect (the last few tenths of a per cent of the beam spread past the meter's ends). The charges in the glass oscillate along $\vec E$, and at this angle that is the direction the reflected beam would take, in which they do not radiate. Switch to TM ($E_z$, s polarisation, under More): the reflected beam appears, and meter A drops to 83.4%.

7. A waveguide and a bend

A strip of glass half a wavelength wide guides light by total internal reflection. The light can only travel along it in a few patterns across it, its modes; this one has two (the notes count them), one with a single hump in the middle and one that changes sign there. A beam is sent straight into the middle of its end, so it starts only the first, which has a tail outside the glass, in the vacuum: meter A finds 84% of the beam in the guide. Round a bend of radius $2.5\lambda$, meter B finds 75% (wait for it: the light takes about 40 periods to get there). Tighten the bend (the radius, under Set-ups): the tail on the outside of the bend would have to go faster than light in vacuum to keep up, and it is shed. At $1\lambda$ meter B reads 40%, at $0.5\lambda$ 19%.

8. A metal waveguide below cutoff

A channel $0.4\lambda$ wide through a metal wall. In TM the electric field runs along the walls and must vanish on them, so across the channel it has to fit at least half a wavelength: nothing narrower than $\lambda/2$ carries a wave. Here the field decays along the channel as $e^{-\kappa x}$. At first a trickle gets out (meter A): switching the source on makes some higher frequencies, above cutoff, which crawl through; then meter A falls below 0.01%. Widen the channel past $0.5\lambda$: about 40% gets through. Switch to TE (under More): now $\vec E$ points across the channel, can be the same all the way across, and passes at any width (17% at $0.4\lambda$; most of the rest is reflected at the entrance). (Waveguide books name the modes by the direction the wave travels along the guide instead: to them the TM wave here is the TE₁ mode, whose $\vec E$ is across its travel, and the TE wave is the TEM mode, both fields across it.)

9. A lens

A wide beam passes a glass lens and comes to a focus; the picture shows the intensity. The cross marks the paraxial focus, $5.4\lambda$ from the lens's centre, from Snell's law at both surfaces for rays close to the axis. The brightest point is nearer, $3.6\lambda$ from the centre: a lens only a few wavelengths across pulls the brightest point of its focus towards itself (the focal shift, a diffraction effect), and the rim of a strong lens focuses shorter than its middle (spherical aberration). The focus is about a wavelength wide.

10. A lens without surfaces

The index can be a formula (under More, Background; the dotted lines are contours of $n$). Here it is $n=\sqrt{2-(r/R)^2}$ inside a circle of radius $R$, the Luneburg lens. Light bends towards higher index, and this profile brings a parallel beam to a focus on the far edge of the circle. Try the other backgrounds: in a graded-index fibre the light swings back and forth about the axis instead of bouncing.

11. Two slits

A wide beam on a metal wall with two slits $2.5\lambda$ apart. Behind it two circular waves overlap, and they add where their paths differ by a whole number of wavelengths: $\sin\theta=m\lambda/d$, at $23.6^\circ$ and $53.1^\circ$ (dashed). Switch to Intensity. Move the slits closer together and the fringes spread.

12. A photonic crystal

Rods of index 3 on a square lattice $0.37\lambda$ apart. The waves scattered by all the rows add up in step only going back: this frequency lies in a band gap, the light cannot travel in the crystal, and it is reflected. Meter A, behind the crystal, reads almost nothing. Change the lattice constant: the gap of this small crystal runs from about $0.27\lambda$ to $0.45\lambda$, and outside it the light gets through (44% at $0.26\lambda$, 39% at $0.46\lambda$). Electrons in a crystal have band gaps for the same reason.

13. Light round a sharp corner

The same crystal with rods missing: a channel with a right-angled bend. Light at a frequency in the band gap cannot go into the crystal, so it follows the channel, even round a corner that would make a glass waveguide leak.

14. A cavity rings down

A metal box $1/\sqrt2$ wavelengths square inside, whose lowest mode is at the source's frequency, with an opening $0.4\lambda$ wide. A short pulse sets it ringing, and the probe (the yellow cross) records the field (below the picture) and fits its decay. The quality factor $Q$ is $2\pi$ times the energy stored over the energy lost in a period: the energy falls by $1/e$ in $Q/2\pi$ periods. Here $Q$ is about 85, and the opening pulls the frequency down to 0.93. Narrow the opening to $0.3\lambda$ and $Q$ rises to about 600; close it and the box rings on at 0.9995 (the grid's value for this box) without losing anything.

15. Cherenkov radiation

A wire carrying a steady current, moving through glass at $\beta=0.9$: faster than light travels in glass, $c/n=0.67$. The waves it leaves behind cannot keep up and pile up on a V, like the bow wave of a boat (a cone, for a charge in three dimensions). In a time $t$ the source goes $\beta ct$ and its waves $ct/n$, so the V's wings are at $\sin\alpha=1/n\beta$ to the path ($47.8^\circ$, dashed), and the light leaves across them, at $\cos\theta=1/n\beta$ to the motion ($42.2^\circ$, arrows). Slow it below $\beta=1/n$ and the V is gone. Then, at $\beta=0.5$ say, switch the moving source to Oscillating for the Doppler effect: the waves crowd together ahead of it and spread out behind. (Faster than $c/n$, it outruns its own waves, and there is nothing ahead of it at all.)

How it works

Maxwell's equations are solved on a grid of cells in the plane, with everything uniform along $z$, out of the picture. Light then comes in two separate polarisations, named for the field that lies flat in the picture, transverse to $z$. In TM ($\vec H$ in the plane) the electric field points out of the picture, and the picture shows $E_z$: red positive, blue negative. In TE ($\vec E$ in the plane) it shows $H_z$. The colours are compressed, so that weak waves show beside strong ones: point at the picture to read the field there, under it. Lengths are in vacuum wavelengths $\lambda$ of the source (the bar at the bottom right of the picture is one), time is in its periods, and $c=1$. Around the picture is a layer that absorbs whatever reaches it, so the waves leave as if into open space.

Drag on the picture to paint with the material chosen under Paint (the eraser takes it off). Drag the source, or the tip of its arrow to aim it. A meter (a green line, under More) measures the power crossing it, in the direction of its little arrow, as a share of the beam's power; drag its ends. The probe (a yellow cross) records the field at a point.

How it is computed

The finite-difference time-domain method (Yee, 1966): $\vec E$ and $\vec H$ on interleaved grids, stepped alternately, with $c\,\Delta t=\Delta x/2$. It conserves energy exactly, but waves on a grid travel a little slower than they should, more so the fewer cells per wavelength (0.3% slower at 20 cells): the readout warns when there are fewer than 15 cells per wavelength in the highest index. A cell that an interface cuts gets the average permittivity of its parts. At 20 cells per wavelength in the glass, a glass surface reflects within 2% of Fresnel's value ($r=0.203$ for $0.200$). The metal is a perfect conductor, placed to within half a cell. The absorber is a lossy material matched to vacuum: it reflects 0.02% and lets $e^{-12}$ of the power through a wavelength of it. The layer around the picture is a convolutional perfectly matched layer, 16 cells thick, whose echo of a wave arriving head-on has $10^{-5}$ of its amplitude. Beams are Gaussian, injected along a line by the total-field / scattered-field method. The meters measure the flow of energy exactly as the scheme conserves it, so their readings add up, and a beam's power is reckoned the same way.

New in 2026 for BrainFlux. Source: github.com/gallicchio/brainflux, MIT licence.