Plotting Feynman’s Amplitude Spiral after 50 years

I’m tickled pink to have the honor to be (so far as I know) the first person to publish a 3D rendering of one of Feynman’s favourite experiments, and hopefully make a key concept in quantum physics much more tangible to people.

First, here’s the shapes, and then I’ll explain what they actually mean. I’ll also do a follow up post with various ways for making your own 3D models of these:

Feynman Amplitude Spirals – The double slit experiment in 3D

A bit of background. In the early 1960s the kickass physicist Richard Feynman (just a couple of years before he was awarded the Nobel prize) was given the task of teaching undergraduate physics at Caltech. Many years later and the lectures are still remarkably up to date, as well as now being available freely online. (http://www.feynmanlectures.caltech.edu/)

When the class got up to the subject of Quantum physics, in the very first lecture, Feynman wisely decided to teach it in the reverse order to how it was previously done.

The double slit experiment was supposed to be an advanced topic not covered till later, but he felt that it laid a bedrock that was useful and understandable to someone with no previous background in quantum physics.

Feynman introduced it as as:

“…we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics.”

(emphasis mine)

Feynman went on to describe the double slit experiment, and explain it in a hugely vivid and memorable way. First by comparing what would happen if electrons were ordinary particles (simulated by firing a machine gun randomly at the wall), and what would happen if they were waves, etc. His lecture is well worth a read. (http://www.feynmanlectures.caltech.edu/III_01.html)

Side note: Any good explanation should involve noticing a feeling of confusion about something, examining all parts of it closely, and then not only finding ways to resolve that confusion, but also seeing what other things get explained at the same time. If you’re not confused by something, you don’t need an explanation. If you’re still confused at the end, that’s your brain giving you a hint that there’s stuff left unresolved, and you haven’t found the actual answer.

Anyway. I’m not planning to re-explain Feynman’s awesome intro. Instead, what I want to focus on his diagram below. This blog post should still hopefully be understandable for anyone that hasn’t read his lecture, but be a nice add-on for anyone that has.

I’m also redrawing his experiment just a little (to use photons rather than electrons), but the same principles work equally well for both.

Here’s the setup. It’s super simple, and can be made with about $5 in parts nowadays:

  • A “laser”,
  • two slits in a wall,
  • and a screen.

Light goes from the laser to the wall, and through the slits to the screen.

feynman double slit layout v01.png

By opening and closing the slits, you change the pattern that appears on the screen. Here’s the results you get:

  • If slit 1 is open and slit 2 is closed, you get a smooth curve (P1)
  • If slit 1 is closed and slit 2 is open, you get another smooth curve (P2)
  • If both slits are open, then you get a really lumpy curve (P12)

Notice how some of the valleys of P12 are lower than any point on P1 or P2. What. the. hell.? If one slit lets some amount of light through, then how does opening a second slit, which presumably can only let more light through, cause some parts of the screen to become darker?

I mean, we’d expect that if both slits are open then the amount detected would be the sum of P1 and P2. Or, to put it as bluntly as possible:  How the hell can two smooths add together to make a lumpy?

In other words, we’d expect the system to behave like this:

OK, let’s double check our intuition in another situation. I’m walking through the woods at night, finding my way with a torch. If I ask my friend to turn on a second torch, do we see any parts of the path suddenly get darker? Err.. hell no, everything just seems brighter. Hmmm… OK, I notice I’m confused, so I’m going to ‘bookmark’ this confusion to think about later.

Oh, and there’s one last bit of confusion to throw into the mixture. Back in the lab, we can add more slits in the screen and see the pattern change yet again. We’d want whatever explanation we come up with for the 2 slit case to also be able to explain situations like these:

feynman multislit mosaic v01.png

Now, back to the lecture. Feynman goes on to explain how the flat curves (made of real numbers) make no sense and to get the right answer we must let the contributions from each slit be composed of complex numbers (e.g. be a vector with a both a direction and a length, rather than a bar graph that only has a length.)

Or, to put it graphically, we can explain our graphs by making a bunch of vectors. So, if at each point the green is the sum of the orange and blue:

feynman vector annotation v01.png

OK, well that sounds like it has the potential to explain some things. But how exactly do we figure out the phase at each location?

Here’s the simple version of the equations for calculating it. It neglects phase changes over time, etc. but it’s good enough to give us the shape we need.

Complex Amplitude of a Single Slit

For those that are mathematically inclined, you can see the length is given by the sinc function, and the phase is just a complex rotation based on the ‘flight time’ it takes for the photon to get from the slit to the screen.

So when we graph these functions, here’s what appears on the screen. There’s a contribution due to slit 1, a contribution due to slit 2, and the green is the sum of the two:

Feynman Amplitude Spirals

It looks like we’re pretty close to understanding the lumpiness of our 2D graphs. There’s just one last trick we need to use before we start plotting everything. We need a way to ‘smush’ our 3D graphs back down into 2D. To do that we use the rule that the height of the 2D graph at any point is the absolute square of the vector length. This is known as the Born Rule and is a keystone of quantum physics. It’s written like this:

Now we have all the tricks needed to calculate the whole thing, and see if it matches the flat graphs that started us off.

Here’s my animation showing the process. I’d recommend watching it a couple of times to get the full  effect:

Does it explain our data? Yes, it’s a perfect match!

I also notice I’m no longer confused about how the ‘lumps’ in our graph came about. They’re simply the result of the two spirals being added together, mostly cancelling out at some points, and strongly reinforcing at others points.

Let’s go back to the multiple slit case. Do we still get the right result for the 4-slit case?

feynman multislit mosaic 4slit v01.png

4 slit case & the resulting pattern on the screen

Yep, another win. So once we found the right representation for what each slit provided (a complex number), we’re able to tackle situations with any number of slits (or possible paths the photon could take) just by using ordinary addition. 

So, thinking of things as having complex probability amplitudes rather than (real) probabilities was the key to making it work. I want to quickly cover the rules for doing this, since it’s the source of a lot of confusion about quantum physics. Here are Feynman’s rules for probability amplitudes:

1. The probability of an event is the absolute square of a complex quantity called probability amplitude.

2. When an event can occur in several alternative ways, the probability amplitude is the sum of each probability amplitude for each way considered separately.

3. If an experiment capable of determining which alternative is actually taken is performed, the interference is lost and the prob becomes the sum of the prob for each alternative.

So in other words, we have to be super careful where we apply the Born rule. Here’s a summary:

Guide to the Born Rule

So we’d use the first method  in any situation where we have two sources interfering, and where there is no way, even in principle, to tell which path the photon took. The interference is clear as day, and you’ll see a lumpy curve.

And we use the second method in any situation where there’s some way to ‘know’ which slit the photon went through. For example by putting different polarizing filters on each slits (which is sometimes described as ‘labelling’ the photons with their path information), we could then put another filter on our detectors, and tell which slit the photon came through. If we do that, there’s no interference pattern, and we only see a smooth curve.

Big, important point here. It doesn’t matter if you actually bother to measure the polarization of the photon. If your detector was switched off, or broken, or you forgot to look at the dial or something,  it still doesn’t matter. As long as there’s any way, even just in principle, to tell the which path the photon took, then the complex amplitudes don’t flow to the same final configuration, and the interference pattern disappears.

(Side note: Those two scenarios are the two extremes of something which is actually continuous.  If we had a sensor that performed a ‘shitty labelling’ of the photons, we would see a pattern somewhere in between the smooth and lumpy curves. Feynman explains this better in the lecture, but here’s my graph:)

feynman path detection v01.png

I apologise for my drawing skills

 

There was one final piece of confusion we haven’t tackled yet. Remember the night walk in the woods we talked about before? We noticed that turning on second torch didn’t make any parts of the walking track darker. So what’s up with that? Why did that situation show no apparent interference, but the one in the lab did?

The answer is that double slit experiment only shows the dramatic interference effects when the light sources are coherent, i.e. they have the same frequency and a fixed phase relationship. If we (for some reason) tried to walk along the dark path using only a handheld laser pointer, we’d immediately see the light was speckly, and that the beam had lots of lumpiness to it. And if our friend turned on their laser, we actually would see that some parts of the dirt track did indeed get darker! (as well as some other parts getting lighter, so that the average brightness increased)

(OK, side note for anyone as pedantic as me. While you can see laser speckle really easily with a single laser, strictly speaking any two handheld lasers are almost certainly not reliable enough frequency sources to see that kind of interference effect. It’s still doable in theory, but kinda tricky in practice. Rather than a $3 keychain laser pointer, you’d have to be hiking through the woods with something more like, say, an atomic clock strapped to your back, with a special mechanism that used the clock reference to tune a lab grade laser cavity, which had been modified to take advantage of Zeeman splitting of electron levels in a magnetic field, and has its cavity length controlled through servo feedback in order to guarantee a fixed phase relationship in the output beam. Easy, no?)

So we’re pretty much done. Hopefully the double slit seems a bit less mysterious now, and I hope my plots made it a bit clearer.

Oh yeah, and since (so far as I can see from checking a bunch of textbooks, talking to a couple of quantum physicists, and google image searching various keyword combinations) I’m the first person to plot this 3D shape, I’m going to take the opportunity to name it. I hereby dub it:

“The Feynman Amplitude Spiral”

If anyone knows of somewhere it’s been properly plotted before, please drop me an email? It’s obviously related to the Cornu spiral, but Cornu/Euler spirals are usually plotted as the sum of hundreds of vectors, where this is the sum of only two, and also we’re sweeping this curve in 3D as we vary the screen height.

I still can’t quite believe it took 50+ years for someone to get around to plotting it, advances in computer & display technology notwithstanding. I’m never going to be able to think about the double-slit experiment without seeing this shape.

Edit: There’s now a part 2, on making your own spiral

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Making the Braided Galton Board

Here’s the process for making the wire & bead Galton boards.

First up, cutting the wire to length. I screwed two wood screws right into the bench, then wrapped the wire back and forth till I had a bundle of 64.

wire measuring.JPG

Don’t cut the wire yet. Instead, wrap a length of copper around all the wires near one end. That’ll be the head of the display. Now all the wires are safely cinched at one end, you can cut them all at the other end. We now have 64 wires in a sort of ‘cat-of-nine-tails’ arrangement.

I used the same layout trick to make a bunch of little lengths to act as as the ties. A paint tin was about the right circumference. No idea how many I cut of these. Many?

wire for ties v01.jpg

Then, I used the lasercut frames to hold the Perler beads in place while they were being threaded. A red bead is used to separate the two sections, and also as something to tie the copper off on at the end.   threading jig v01.JPG

Oh yeah, huge tip here I almost forgot to mention. Put masking tape on the bottom of the bead frame. That stops the beads from  rolling away or getting knocked loose before you’ve finished threading them.

I think I decided it would be too fiddly to do all 64 at once, so I broke it up into groups of 16:

midway point v01.JPG

At this stage I questioned my sanity, and whether it was worth it. Thankfully I persevered. Deep breaths were needed.

Another useful trick was to tie off the first two steps of the sequence (i.e. groups of 16 in each) so that it would be easy to separate the loose strands later. Keeping 64 wires in any sort of left-to-right order on the table is pretty much impossible.

weaving in progress v01.JPG

48 of 64 threaded, and a whole lot of wrangling left to go

One thing I remember having to do is use copper wire as temporary ties. Once I had the rough structure tied together it was easy to neaten it up later. Trying to get it all cinched up and neat the first time is way too hard.

Around now it’s finally it started to come together, and I can stand back and enjoy it:

Finished work pair v01.JPG

32 state almost finished, 64 state version completed

Although it still took a very solid evening’s work, (I seem to remember leaving the makerspace after midnight?) I’m happy with the technique now. If you’re interested in maths or stats art, I’d highly recommend making one for yourself, you won’t regret it.

Files are here for anyone that wants to make their own:

https://www.thingiverse.com/thing:3189020

Oh yeah. It has not escaped our notice that the specific method (rigid copper wire, cinched with same) immediately suggests a possible construction mechanism for making a  freestanding sculpture without the need for a support board at all to stand in free space. Hmm…. I might revisit this in the future and see if I can make something nifty looking using that. If anyone else wants to try it in the meantime, I’d love to see how it goes. 😀

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Improved Braided Galton Board

I’ve played with the Galton Board before a couple of times. It’s a lovely demonstration of the ‘Central Limit Theorem’ in statistics. That is, it shows in a really intuitive way how repeated binomial events (like coin flips) average out the number of the results into a bell curve.

Finished work overview v01.JPG

In the Galton board, every way from the start to the finish is equally likely.  However, since there are lots more ways to reach the middle than there are to reach the ends, the middle results become more likely.

For example, if we flip a coin exactly 4 times, there’s precisely one way to get all heads:

                                                  HHHH

But there are six ways to get two heads:

                  HH-T-T                  H-T-H-T                  H-T-T-H

T-HH-T                  T-H-T-H                  T-T-HH

So if we flip a coin four times, we must expect that getting a total of two heads is six times more likely than getting a total four heads.

I was pretty happy with my previous Galton boards, but I was still itching for a display with a record of the path each string takes. I realised that if I marked each string twice with the binary code for its route, (once as it progresses through the maze, once as it reaches the bottom) then you’d be able to see that all the final points it ends up in have the same number of colours in each.

The question was just how to be able to make it without having to spend crazy amounts of time assembling it. I tried out a couple of construction methods, first craft beads & builder’s twine:

unfinished early prototype v01.jpg

So much threading of string. Make the hurt stop.

Then  I actually made a rig to spray paint string with binary patterns through a mask. Note the spacing on the right is larger, to give more room for string to be used up bending through the maze. spraypaint_mask_v02.JPG

The spray painting worked fairly well, but it wasn’t as easy as I thought to wrangle all the string together once it was done.

I put that aside for a while, until I got playing with Perler Beads for another project. I realised that since those beads are flatter cylinders than the ones I first tried, that I could make a jig to hold them in place that would make the threading much easier:

threading jig v01.JPG

Binary in motion

All that remained to make it practical was to find some stiffer thread than builder’s cord. That’s when I remembered we had a huge roll of copper wire at Robots and Dinosaurs that hadn’t been touched in a few years. That made it so much easier to put together, and has a beautiful coppery aesthetic to boot!

weaving closeup v02.JPG

Aww yeah.

And here’s a closeup showing that yes, all the bins for a given final state have the same number of white and black beads in each. The sheer ‘heft’ of the middle states is a very nice tangible demonstration of the central limit theorem. final state closeup v01.JPG

And the amount of copper at the start compared to the end is a lovely way to demonstrate the flow of probability mass. It starts off almost as thick as a finger, and tapers down to a single strand.

bifurcating v01.jpeg

For those that want to make their own, I’ll do a second post shortly with detailed instructions. This method is much improved over my first attempts, but it’s still an evening’s work to put together a board.

Still, I’m pretty darn happy with that, and I can safely say after spending so many hours with beads, the centra limit theorem is something I’m unlikely ever to forget.

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I’m on EEVBlog

At the Electronex trade show last week I bumped into Dave Jone from EEVBLOG. We got chatting about the polarization stuff I was working on, and after a few minutes Dave turned to me and said “Wait, why aren’t we filming this?”. So I toodled off to my car and grabbed a bunch of bits and pieces and we did an impromptu filming session outside:

Apologies to anyone I confused, everything was totally off the cuff and improvised, and I didn’t have a chance to print out supporting diagrams which might have made it clearer.

I should also mention that I’m just following in the footsteps of David Prutchi, and if you want to get into polarized imaging please read his excellent whitepaper on the subject.

http://www.diyphysics.com/wp-content/uploads/2015/10/DOLPi_Polarimetric_Camera_D_Prutchi_2015_v5.pdf

and and check out the page for his Dolpi Imager.

The relevant blog posts for the bits I was playing with are here:

A couple of minor things to correct in what I said:

  • Single mode fibres aren’t narrower than the wavelength of light they carry, just close-ish to it. (As opposed to multi-mode fibre that might be a hundred times larger) E.g. an IR laser might have a wavelength of 1.5um, a single mode fibre for it might be 8um, and a multi-mode fibre for it might be 100um.
  • Glucose is the sugar I was talking about, in right handed (dextrose or D-glucose) and left handed (L-glucose) and which is capable of optical rotation. Sucrose is more complicated.

And a couple of questions people asked me after:

  • Why not use sugar to do the optical rotation and determine the difference between left and right handedness?
    • Good question. A sugar rotator’s axis is through the R/L points on the sphere, so it won’t move things of the equator. To do that you need something like a waveplate (the green arrow in my model)
  • Why the hell didn’t you clean the window on your LCD device. It’s dirty!
    • That’s not dirt, that science! Or, rather I was avoiding the need for an expensive professional waveplate by using adhesive cellophane which has been stretched and oriented at an exact angle to the LCDs optical axes. It’s not super pretty, but the device would have gone from costing maybe $20 to more than $100 if I hadn’t used it. When I’ve finished my next round of testing I’ll publish a full set of design files and plans for people to make their own.
  • So what about your ipad screen?
    • Yeah, OK, that dirt is actual dirt.
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Collatz Dominos

Warning: Level 5 cognitohazard ahead. Mathematicians especially susceptible. Proceed at your own risk:

(More seriously, if you like puzzles/math, haven’t heard of the Collatz before, and have a big exam in the next couple of days, maybe skip this post?)

I’ve previously played around with unusual ways to evaluate the Collatz sequence, but I wanted something more tangible to play with.

The prompt for this is that last week in Robots and Dinosaurs we had a few people seized by the Collatz bug, and working it out on butcher’s paper. Even with a huge canvas it’s surprisingly hard to not run out of space.

If you haven’t tried sketching out the sequence before, such as on a whiteboard or paper, it’s not that easy to do. Because of the branching nature of the sequence, knowing where to ‘budget’ your remaining empty space is difficult. I’d frequently find myself having to erase & redraw large areas, in order to insert some new branch.

I wanted something much more tangible and flexible to play with. I considered making simple tiles (like scrabble), but I thought they’d be very frustrating to move long chains of, while trying to hold them in place.

I considered using M3 bolts as pivots, and kind of hanging the tree, but I though that might be a bit unwieldy. I also thought about using string to join tiles together, but that had a bit too much of ‘A Beautiful Mind‘ vibe…

After a while I had the idea to make a captive swivel, using a two dimensional version of a ball-and-socket joint. The left hand socket is the N/2 side, and the right hand side is the 3N+1 side. After some prototyping, I came up with this geometry, which works well:

branches closeup 01.jpeg

Left hand side is N/2, right hand side is 3N+1. 

And scales up to large chains:

tiles closeup 01.JPG

Flexible chains – The key to mechanical Collatz 

I also made a snazzy case to hold everything. There’s a stack of trays which hold the numbers:

all trays 01.JPG

And each tray has spaces for the tiles, with room for a finger to reach in and remove them easily. The places are labelled to make it easy to pack away:

tray engraving closeup.JPG

Some 8mm wooden dowels glued to the base locate the trays securely, and the lid is held down with a cord, which self-centres on indentations on each side:

wrapped and finished collatz board 01.JPG

Collatz Dominos. The timeless, perfect gift for any discerning mathematician

(Note to self – that title font is called ‘Cooper standard’ and behaves really nicely for lasercutting. Remember that for future projects)

There’s many ways to assemble the tree, but here’s the method I’ve been using. First I arranged the odd numbers in columns, and then make chains by connecting all the even numbers.

playing - all evens.JPG

Now we can take each odd chain, and figure out where it joins up to the rest of the sequence. The tree is starting to take shape:

playing - all under 50.JPG

The first tray of the kit contains the numbers 1-50, whereas the second tray contains all the extra numbers needed to make the first set reach 1. (However I’ve left out the extras needed for numbers 27, 31, 41 and 47, since these four “Bastard” numbers have a chain length around three times the others, and would require having an extra two trays just to connect them up). Once the extra numbers are added, we can see the whole structure:

playing - final collatz tree 01.JPG

The completed Collatz for numbers starting from one to fifty. (With four omissions for simplicity)

I’m very happy with it so far. It’s quite fun to play with, with none of the “whitespace irritation” that comes with pen and paper methods.

If I had to do it again, I’d stain or colour the odd numbers differently, just to make it obvious which method of branching happens at each step. I might still do that if I have time.

Files here for anyone that wants to make their own:

https://www.thingiverse.com/thing:3089372

 

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Icosahedron

Here’s a model to demonstrate a little known trick of geometry. The verticies of an icosahedron (20 sided shape) can actually be created using only three rectangles at right angles to each other.

What proportion do the rectangles have to have to make it work? (sigh) The golden ratio.

(I must be jaded from so many people saying anything that’s slightly-larger-than another thing is a golden ratio, that when I actually see a legitimate usage of it I’m disappointed)

Finished cropped 01.JPG

The verticies are strung together with builder’s cord and it’s quite fun to wind it on.

parts and assembled 01.JPG

There are holes for a cable tie, which locks the plates in place. I also ended up putting a seam of hot melt glue on the edges to make sure everything stays nice and cartesian when the cord is tied on it:

cable tie closeup 01.JPG

Files are here for anyone that wants to make their own:

https://www.thingiverse.com/thing:3089076

 

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Clockwork Waveplates – Polarization Calculations With Gears

I’ve been playing around with tangible versions of the Bloch/Poincare sphere previously, and I think breaking polarization down into 3 orthogonal dimensions (Stokes vectors) yields such a useful mental model of what’s happening with various complex systems.

In the same vein, I’ve been meaning to come up with a tangible model of what waveplates do to polarization of light. Waveplates are a crucial building block of so much in physics, from microscopes to cameras to quantum computers.

While the Poincare models I made before were handy as a reference, to model a system with multiple filters you have to pick up and rotate the sphere several times, simulating axes at various points. I specifically wanted to find something more direct, and that could be continuously played with to ‘steer’ to a given result. Something you could put in front of a student and say “OK, given horizontal polarization as an input, exactly how would you set up the equipment to convert that to 45 degrees?”

Here’s what I’ve got for the Half-Wave Plate. The two gears rotate around each other, and the coordinate labels on the outer gear swivel as it does so. The input coordinate system is on the base, and the outer gear shows the output frame.

By playing with it for a second, you can see how it shows how Right and Left circular polarizations are always swapped, but by rotating the “fast axis” of the waveplate, you can choose how the linear polarizations are converted.

Annotated halfwaveplate v02.jpeg

Half-wave plate model with gears

Along the same lines, here’s what I came up with for the Quarter-Wave Plate:

Quarter wave plate v01.JPG

Quarter-wave plate model with gears

Here’s a quick application of quarter-wave plates, using three of them as a polarization controller that can put any polarization state into any other polarization state. (Side note: polarization controllers are more commonly made using a quarter-half-quarter arrangement of plates, not a quarter-quarter-quarter as shown here, but this has sufficient degrees of freedom to do this job)

Polarization controller sketch v01.jpeg

Polarization controller, in this case consisting of three linked quarter-wave plates. By adjusting the three angles, any polarization can be converted to any other polarization

Before we test the bold claim about three QWPs doing everything, let’s do a sanity check on the behaviour of the lasercut device. A waveplate shouldn’t affect polarization of light which perfectly matches either the fast or slow axis, so let’s set up the plate with theta=0, and see what happens:

Quarter wave plate sketch 01.jpeg

Quarter wave plate test

As you can see vertical (V) light is unaffected, and left circular (L) light is converted to +45deg diagonal (D) light, as expected.

 

What is we combine two waveplates, and see if we can convert Left-Circular (L) light to Right-Circular (R) light? After playing around with the gears for a few seconds, this configuration seems to work nicely:

Quarter wave plate sketch 02.jpeg

Two Quarter-Wave Plates acting as a Half-Wave Plate, only because their angles match

The nifty thing to note here is that the angle of the first waveplate (120*) matches the second (120*), indeed that’s the only way to make the gears flat. This makes sense, as having them at the same angle is how to turn two QWPs into a HWP.

Alternately if we put the two plates at 90* to each other, then the second plate would ‘undo’ all the rotations of the first plate, and the polarization would be back where it started.

OK, now for a challenge. Here’s a task I picked at random, including at least one ‘weird’ direction to make it harder. Can three QWPs do this? Let’s define our task:

  1. Flip left and right circular polarizations. (LHCP <–> RHCP )
  2. Convert 67.5 degree light (halfway between Diagonal and Vertical) into Horizontal

First, I bolted three QWP models together:IMG_1371.JPG

and then spent a while fiddling until I got it lined up with the goal:

Quarterwaveplate challenge 01.jpeg

Calculating the correct waveplate settings to do an arbitrary rotation of the polarization.

I’m eyeballing it, but I’m going to call the three angles 60 degrees, 15 degrees and 55 degrees respectively. (The camera parallax is a bit misleading too, they look more parallel in real life. )

So that would mean we set up the parts on the optical table like so:

Calculated configuration sketch 01.jpeg

How to double check our answer? It kinda looks right sitting on the table, but would it work if you tried it in the lab with real optics? Let’s do Mueller calculus with those numbers and predict what would happen if we shine LHCP and D/V light in there.

Mueller calculus 01.png

Predicting the result with QWPs oriented at 60, 15 and 55 degrees

The expected number for right-hand circular is [1,0,0,1], which is pretty close to what we got, with just a smidge too much linear polarization. And the expected number for horizontal is [1,0,1,0], which is extremely close.

For lasercut plywood and hastily bent pieces of aluminium, I’m pretty happy with that.

Files are here for the Half Waveplate: https://www.thingiverse.com/thing:3076339

and the Polarization Controller: https://www.thingiverse.com/thing:3076410

 

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