How a Glass Fiber Knows Its Own Shape

Here’s a thing that still feels like magic to me. Take a glass fiber about as thick as a hair, run it through a flexible robot arm or a catheter, and plug one end into a box. The box tells you the full 3D shape of the fiber, every millimetre of it, many times a second. Nothing is attached at the tip. No camera, no X-ray, no magnets.

The whole trick rests on one sentence: the colour of reflected light is a ruler. If you’ve never done any optics, that sentence means nothing. That’s fine. By the end of this post it should feel obvious. Everything else (the stretch, the bending, the 3D shape) is bookkeeping on top of it.

Here’s the end result first. Drag to rotate.

Red and blue on the cores mean stretched and squeezed. The teal tube is what the box thinks the fiber looks like, recomputed once per laser sweep. Let’s build up to that from zero.

1. Colour is a length

Light is a wave. Not as a metaphor: an electric field wiggles up and down as it travels, like a wave running along a rope. The distance from one crest to the next is the wavelength, and that’s all colour is. Red light has crests about 650 nanometres apart. Violet, about 400. Your eye is a wavelength meter with three coarse bins, and your brain paints the answer as colour.

Keep going past red and the wave doesn’t stop existing. You just stop seeing it. That’s infrared. Fiber sensors work around 1550 nm, where glass is extremely transparent and the telecom industry has made the lasers cheap. Since nobody can see 1550 nm, every demo below paints it in false colour: shorter is bluer, longer is redder, same order as the rainbow.

2. Two echoes either add up or cancel

Now the one piece of physics we actually need. Put two very weak mirrors in the light’s way, the kind that let almost everything through and bounce back only a sliver. Each sends back a small echo. The echo from the second mirror has travelled an extra round trip: twice the spacing.

If that extra trip is a whole number of wavelengths, the two echoes come back crest on crest and add up. If it’s half a wavelength off, crest lands on trough and they cancel. Same two mirrors, different colour, completely different answer.

You’ve seen this before. It’s why a soap bubble is coloured: light bounces off the front and the back of a film thinner than a wavelength, and the two bounces add up for some colours and cancel for others. As the film drains and gets thinner, the colours change. Spacing picks colour.

3. Many mirrors pick out one colour

Two mirrors give a soft preference. Now line up thousands of them, evenly spaced, each one ridiculously weak. That’s a fiber Bragg grating: a pattern of tiny steps in the glass of the core, written in with a UV laser.

Here’s how I like to think about it. Every echo is a clock hand, and its angle says how far through its wave cycle it is when it arrives. To get the total reflection, lay all the hands head to tail. For one special colour, each mirror’s extra round trip is exactly one wavelength, so every hand points the same way and the chain shoots out in a straight line. Big reflection. Change the colour slightly and each hand is turned a little more than the one before. The chain curls up into a circle and ends near where it started. Almost nothing comes back.

So a grating reflects one colour and lets everything else through, and which colour is fixed by the spacing:

$\Lambda$ is the mirror spacing, $n \approx 1.447$ says how much slower light moves in glass, and $\lambda_B$ is the colour that comes back. For 1550 nm you need $\Lambda \approx 535.6$ nm. More mirrors make the peak sharper. A real 10 mm grating has about 18,700 of them.

This is the answer to “why does the reflected colour mean anything”. The grating is a ruler. The reflected colour is the reading on that ruler.

4. Stretch the ruler and the colour moves

So what happens when you stretch the ruler?

Stretch the fiber and the spacing grows, so the reflected colour slides longer. Measure the colour, get the stretch. The numbers are tiny. One microstrain (µε), one part per million of stretch, moves the colour by 1.21 picometres. That’s about a thousandth of a nanometre, and good instruments resolve it. (Why 1.21 and not 1.55? Stretching glass also lowers its refractive index a little, which cancels about 22% of the effect.)

Here’s the trap. Warm the fiber by one degree and the colour moves about 10 pm too, in the same direction. A single grating can’t tell “warmer” from “stretched”. Hold that thought; section 7 fixes it.

5. You don’t even need to write the mirrors

Writing gratings is an extra manufacturing step. It turns out you can skip it.

Ordinary fiber glass isn’t perfectly uniform. When it froze out of the melt, tiny random density fluctuations got locked in, much smaller than a wavelength. Each one scatters a vanishingly small amount of light backwards. That’s Rayleigh backscatter, the same Rayleigh as in “why the sky is blue”.

A random set of specks is just a random grating. Instead of one clean peak, it reflects a messy spectrum. But it isn’t noise: the specks never move, so the messy spectrum is exactly repeatable. It’s a fingerprint, and every millimetre has its own.

Here’s the nice part. Stretch that millimetre and every gap between specks grows by the same fraction, so the whole fingerprint slides along the colour axis by exactly as much as a Bragg peak would. The system records every millimetre’s fingerprint once, with the fiber straight. Later it slides the stored copy over the new one until they line up best. How far it had to slide is the strain.

6. Which millimetre is talking?

Every millimetre is echoing at the same time, into the same detector. How do you know which one you’re hearing?

First guess: timing. Shout down a long hallway and the far echoes come back later. Good idea, wrong scale. Two points 1 mm apart answer about 10 picoseconds apart, and ordinary electronics can’t slice time that finely.

The real trick is cuter. The laser doesn’t pulse. It shines continuously while gliding its colour smoothly. Light echoing back from far away left the laser earlier, so it carries an older colour than the one the laser is making right now. The detector mixes the two, and two slightly different colours make a beat: a slow wobble in brightness, like two guitar strings slightly out of tune. A nearby point gives a small mismatch and a slow beat. A far point gives a big mismatch and a fast beat.

All the beats arrive mixed together, but a Fourier transform pulls them apart, the way a prism splits white light into colours. Each beat speed is a position. Cut one millimetre out of that position picture, transform back, and you get that millimetre’s own fingerprint from section 5. One sweep of the laser reads every millimetre at once. This is called OFDR (optical frequency domain reflectometry), and a sweep of about 40 nm tells apart points about 20 µm apart.

7. Bending is stretching on one side

We can now read the stretch at every millimetre. But shape is about bending, not stretching.

Bend anything and the outside of the curve stretches while the inside squeezes. The line down the middle keeps its length. So a shape-sensing fiber carries several cores in one strand: one in the centre and three around it, 120° apart, about 35 µm off-centre. Bend it and the cores on the outside of the bend read stretch, the ones on the inside read squeeze. Which cores stretch tells you the direction. How much tells you how sharp. A bend with a 10 mm radius puts about 3,500 µε on the outermost core.

The centre core sits on that middle line, so bending doesn’t touch it. But heat and pulling stretch every core equally. So subtract the centre core from the outer three, and the temperature problem from section 4 simply cancels. What’s left is pure bend.

The outer cores also wind around in a slow spiral, so each one takes turns on every side. That lets the system measure twist too, which turns out to matter a lot in the next section.

8. From bends to a shape

Now we know, for every millimetre, how much it bends and which way. Turning that into a 3D shape is like following driving directions without a map. Start at the connector, where the position and heading are known. Then repeat: go forward 1 mm, turn by that millimetre’s bend. Do it a few hundred times and the shape draws itself.

That also explains the one real weakness. Every step builds on every step before it, so tiny errors compound. A slightly wrong turn near the start swings the entire rest of the fiber, so the error tends to grow with distance and the tip is usually the least accurate part. And the worst offender usually isn’t strain noise, it’s twist: if the system misjudges how far the cores have rotated, it turns by the right amount in the wrong direction, and the error runs away fast. Turn up the twist error and watch.

The whole chain

Put together:

1. A laser glides its colour, and every millimetre of glass echoes a little.
2. Each echo’s beat speed says which millimetre it came from, and a Fourier transform sorts them.
3. Each millimetre’s reflected colour pattern is a ruler for its own spacing, so a colour shift is a stretch.
4. Comparing the three outer cores with the centre one turns stretch into bend amount and direction, with temperature cancelled.
5. Walk along the fiber, turning by each bend, and you have the shape.

Do that many times a second and the fiber reports its own shape live. Robotic bronchoscopes that steer a thin instrument deep into the lung use this kind of fiber to know where their tip is without an X-ray running the whole time.

Every step above is fairly mechanical except the first idea, which took me a while to really accept: the colour bouncing back from a piece of glass is a measurement of length, good to about a millionth. Once that clicks, the rest is bookkeeping.