Birefringent Crystal

How it works, without the PhD

1. What you are looking at

Put a clear calcite crystal on a newspaper and every letter appears twice. The crystal has no mirrors inside it and no cracks doing this. Each tiny bit of light coming up from the paper splits into two beams inside the crystal. The beams take slightly different paths, so they leave the top at two different places. Your eye traces each beam straight back and sees two copies of the page. One copy sits where you would expect; the other is slid sideways.

Try this in the simulation: turn on Spin crystal and use the top view. One copy stays put. The other copy circles around it as the crystal turns. The copy that stays put comes from the ordinary ray. The one that moves comes from the extraordinary ray. Tick Colour-code the two images to paint them red and blue.

2. Light slows down inside materials

Light travels at its top speed c only in empty space. In glass, water or a crystal it moves more slowly. The refractive index n says by how much:

n = c / vv is the speed of light inside the material. Glass has n ≈ 1.5, so light moves about 1.5 times slower in glass than in vacuum.

When light hits a surface at a slant, the slow-down bends it. Think of a marching band walking at an angle from pavement onto mud: the players who reach the mud first slow down first, so the whole row swings around. That bending rule is Snell's law:

n₁ · sin(angle₁) = n₂ · sin(angle₂)If light goes straight in (angle 0) it does not bend at all.

Hold on to that last line: light that goes straight in does not bend. In calcite, one of the two beams bends anyway.

3. Light has a sideways wiggle

Light is a wave. Its electric field wiggles sideways, across the direction the light travels, like a skipping rope you shake up and down while it carries waves away from you. The direction of that wiggle is called polarization. Light from a lamp or from paper is a random mix of all wiggle directions ("unpolarized").

4. Calcite has a grain

Inside calcite, the atoms are stacked in flat layers of carbonate groups. That gives the crystal one special direction, a bit like the grain in wood, called the optic axis. How fast light moves depends on how its wiggle lines up with that grain:

A wiggle that is perpendicular to the plane containing the grain always feels the same material, so it moves at one fixed speed. This is the ordinary ray, with index n_o = . It obeys Snell's law like light in glass.
The other wiggle direction points partly along the grain, so its speed depends on the direction it travels. This is the extraordinary ray. Its index slides between n_o and n_e = .

Unpolarized light is a mix of both wiggles, so it splits into two beams with two speeds. Each copy of the page carries half of the light. That's why the doubled letters look grey instead of black: each copy is only half dark, and where the two copies overlap you get full black.

The difference between the two indices, Δn = n_e − n_o = , is the birefringence ("double refraction"). Calcite's is unusually large, which is why it shows the effect so dramatically.

5. Why the second beam leans sideways

Christiaan Huygens explained this in 1690 with a lovely picture. Think of every point on a wavefront as a tiny source that sends out its own ripple. A moment later, the new wavefront is the surface that touches all the ripples. Light travels from the centre of each ripple to the point where the new wavefront touches it.

For the ordinary ray the ripple is a perfect sphere, because the speed is the same in every direction. A flat wavefront touches the sphere directly below its centre, so the beam goes straight down.
For the extraordinary ray the ripple is a squashed sphere (an ellipsoid) that is tilted along the grain. A flat wavefront touches it off to one side. The wavefront still moves straight down, but the energy, and so the beam you see, slides sideways at the walk-off angle ρ.

exaggerate the squashing 4× so it's easier to see

A side cut through the crystal along the grain, driven by the current settings. The flat wavefront enters straight down. The red circle is the ordinary ripple and the blue ellipse is the extraordinary ripple. Dashed lines show the grain (optic axis).

The walk-off angle depends only on the two indices and on the angle θ between the beam and the grain:

tan ρ = ½ · n(θ)² · | 1/n_e² − 1/n_o² | · sin 2θ where 1/n(θ)² = cos²θ / n_o² + sin²θ / n_e². Right now: θ = , ρ = .

You don't need to work this out by hand. Just notice two things. The sin 2θ part is zero when the beam runs along the grain (θ = 0°) or straight across it (θ = 90°), and it is largest near 45°. Nature cuts calcite into its rhomb shape with the grain about 44.6° from the face normal, which is almost exactly the best angle. Try the Optic axis tilt slider: at 0° or 90° the doubling vanishes, even though the crystal is still birefringent.

6. How far apart are the two images?

The sideways beam keeps leaning by the same angle all the way through, so the shift grows with thickness:

shift = thickness × tan ρNow: × = . For calcite that is about 11% of the thickness: a 3 cm crystal shifts the image about 3.3 mm.

Try the Lines background: a line that runs along the shift direction just slides along itself, so it looks single. Lines across the shift direction clearly double.

7. Proving the two images are polarized differently

A polarizing filter (like polarized sunglasses) only lets through the wiggle that lines up with its slits. The fraction that gets through is Malus's law (1808):

brightness = cos²(angle between the wiggle and the filter)

The two beams wiggle at right angles to each other. As you turn the filter, one image fades while the other gets stronger. At one angle only the ordinary image remains; 90° later only the extraordinary one does; at 45° they are equal. Turn on the Polarizer and drag its angle.

8. Why one copy looks closer

A coin at the bottom of a pool looks raised because light bends at the surface. Something under a block of index n looks raised the same way: viewed from straight above, it seems to lie at depth thickness / n. The two beams see different indices, so the two images seem to float at slightly different depths. You'll notice this if you look at the angled view: the copies don't just slide sideways, they also look like they sit at different heights. The ordinary image appears above the paper.

9. Other crystals

Quartz and ice are birefringent too, but only slightly. You'll only see their doubling with a thick crystal and a fine pattern, so try 80 mm and the single-dot background. Glass is the control: it bends light but never splits it.

10. Stacking crystals: 1, 2, 3 or 4 images

Each beam that leaves a calcite crystal is fully polarized: the ordinary one wiggles one way, the extraordinary one at right angles to it. Put a second crystal on top and each beam splits again, but not always evenly. Malus's law from section 7 decides the split, with the second crystal acting as the filter:

share that stays ordinary = cos²(angle between the arriving wiggle and the second crystal's ordinary wiggle)The rest goes extraordinary. Right now the second crystal is . So of the light arriving as the first crystal's ordinary beam, .

Tick Stack a second crystal on top in the panel, or use the four preset buttons there. With two crystals of the same thickness:

Same way round: the wiggles already match, so ordinary stays ordinary and extraordinary stays extraordinary. The extraordinary beam gets pushed sideways twice, as if through one crystal of double thickness. You see 2 images, each half as bright, twice as far apart.
Turned 180°: the wiggles still match, but now the second crystal's grain leans the other way. It pushes the extraordinary beam back by exactly the amount the first one pushed it out. Both beams land in the same place, and you see 1 image, as if there were no crystal at all.
Turned 45°: every beam arrives at 45° to the second crystal's wiggles, and cos²45° = ½, so each splits in half again. That makes four beams, each a quarter as bright. They form 4 images at the corners of a parallelogram, with sides along the two crystals' grains.
Same way round, with a half-wave plate in between: a half-wave plate is a thin clear sheet that flips each wiggle around its own "fast" direction. Set at 22.5° to the grain, it turns both wiggles by 45°. So each beam splits in half again, as in the 45° case. But because the crystals line up, two of the four paths ("ordinary below, extraordinary above" and the reverse) get pushed by the same amount and land in the same place. You see 3 images in a row, with brightness ¼, ½, ¼.

The two beams that share the middle image wiggle at right angles to each other. Light like that can't interfere, so the brightnesses simply add. With the current settings, looking straight down, you get with brightness . Turn on Colour-code the two images to see which path made which image, and turn the polarizer on the three images: the outer two fade in turn, while the middle one stays steady because it holds both wiggles.

11. A little history

Rasmus Bartholin described the double image in 1669, using clear calcite from Iceland ("Iceland spar"). Huygens explained it with his ellipsoid ripples in 1690. Étienne-Louis Malus discovered polarization by reflection in 1808 while looking through a calcite crystal at sunlight bouncing off a window. Some historians think Viking navigators used a calcite "sunstone" to find the sun through clouds by watching how the two images changed in brightness.

12. What the simulation does (and doesn't do)

Every pixel sends a ray from the camera into the scene on the graphics card. Where it hits the crystal, it splits into an ordinary ray (Snell's law) and an extraordinary ray. For the extraordinary ray, the program keeps the wave's direction along the surface and solves the tilted-ellipsoid equation, then walks off along the energy direction. Rays that reach the bottom face read the paper right there, because the crystal touches it. Rays that hit a side face either leave, or bounce back in if the angle is too steep (total internal reflection). Surface reflections use a standard approximation. The optional dispersion traces red, green and blue separately with slightly different indices, which gives the colour fringes. With a second crystal, each beam that leaves one crystal through the thin gap is traced into the other and split again, weighted by Malus's law; a ray passes through each crystal at most once. The half-wave plate and rotator are idealised: they have no thickness, work the same for every colour, and act only on the flat part of the wiggle. Left out: partial reflections inside the crystal, light focusing into bright patterns (caustics) on the paper, and crystals with two optic axes.