Neat huh?

# Background

I was about to write "Please refill me when I'm empty" on the clear plastic tupperware container of pretzels at work when I started wondering if it would be possible to write something on one side of a transparent material that is invisible from the other side.

Thinking back to what I learned about polarizers and birefringent materials, I came up with what you see above.  I think it's a neat "magic" trick that demonstrates some of the weird properties of light, so I thought it'd be fun to write about it.

# Light

Light is caused by a disturbance in an electromagnetic field that causes it to propagate through space.  When considered in classical mechanics, it's a phenomena that falls out of Maxwell's equations.  When you move current through a wire, you generate a magnetic field (electromagnet), and when you pass a loop of wire through a magnetic field, you create a current (electric generator).  It just so happens that if you do this fast enough, you don't even need the wire or the current; the electric and magnetic fields involved will continue to oscillate and propagate outwards at the speed of light.  This is a really poor description of what's happening, but all you really need to take away from it for now is that light is a propagating sinusoidal electric and magnetic field.

If you were to draw a doodle representing the E and M fields contributing to light, it might look something like this:

This looks kind of complicated, so let me break it down:

• The E and M waves are are always at 90$^{\circ}$ angles to each other.
• The ray of light propagates in the direction of the cross product of the E and M fields.  You can determine this using the right hand rule.  If you point your hand in the E direction and then curl your fingers in the M direction, the light will travel in the direction of your thumb.  It's indicated in the above image with the black arrow.
• M is shown in this diagram, but for most of the rest of this discussion, I will be leaving M out for simplicity.

Now you might be asking questions like "where's the photon?".  This discussion will be focusing on classical physics which is fine because this phenomenon can be explained without going into the particle nature of light.

From the front, the wave looks like this:

Imagine taking a cross section at a fixed location of the wave pictured above as it moves towards you and it should make sense.

# Linear Polarization

The orientation of the E field in the above diagram describes the "polarization" of the light.  Typically, light is made up of a superposition or sum of different polarizations bunched together.  For the most part, the polarization of light doesn't matter; your eyes will interpret two different polarizations of light exactly the same way.  Below is a diagram of one such case where three different polarizations travel together.

Try not to mistake the similar shape of this diagram with the diagram above.  There is no physical connection between these three light beams besides the fact that they are traveling through the same space at the same time.  Most of the light you interact with every day is comprised of light of all different polarizations added together.

While you can't filter out specific polarizations of light as easily as you can combine them, special types of materials called "polarizers" will only allow specific polarizations of light to pass through.  All the other polarizations are either totally blocked or partially blocked so that only the parallel components of their polarization pass.  Here's an example with a vertical polarizer viewed from the front.

The vertically polarized beam is passed through the vertical polarizing filter untouched while the horizontally polarized beam is entirely blocked.

With the diagonal beam however, things get a little more complicated.  The polarizer passes all light that matches its polarization and blocks all light that is polarized at 90$^{\circ}$, but the diagonal beam is nether.  Because of this, we need to think of the diagonal beam as the sum of a smaller vertical and horizontal beam:

The horizontal component is blocked and only the vertical component is passed through.  This little trick will be especially useful in the next section.

These polarizing filters are fairly cheap and have tons of practical applications.  One famous example is IMAX 3D glasses.  The goal of 3D media is to present a different 2D image to each eye so that your brain can piece together a 3D image. IMAX 3D movie theaters project two different images on the screen using two perpendicular orientations of light.  The lenses on the 3D glasses match these perpendicular polarizations and only allow one version of the scene to pass through to each eye.

Of course, if the viewer tilts his head, a portion of each image will pass through both eyes and the image will become confusing.

# Circular Polarization

Some very unique materials will cause different polarizations of light to travel at slightly different speeds.  These are called "birefringent" materials.

If linearly polarized light is passed through a birefringent material at a certain angle, a portion of the light will travel slower than the rest.  Remember how we split the diagonal polarized light into two components? The same thing happens here, but one of those components travels slower than the other.

As soon as they leave the birefringent material, all polarizations travel at the same speed, but depending on how thick the material is, they will leave at different phases.  Consider this example where light is passed through a birefringent material (don't worry about the geometry, I flattened everything to make it easier to see/draw):

As it passes through the birefringent material, the vertical component is unaffected.  The horizontal component however is slowed down.  Due to this minor slow down, the horizontal component spends more time in the birefringent material and progresses 1/4 wave farther than the vertical component before leaving.  A material of this type and thickness is called a "quarter wave plate".

The result is that the two polarizations of light are 90 degrees out of phase (note the black dotted lines).  The resulting light might look something like this:

To recap, we're taking a single polarization of light, decomposing it into two different polarizations, slowing one down by 1/4 phase, and adding them back together.  The result of this sum is represented by the gray line above.

Looking at this line, you'll notice that its length never changes, but it seems to rotate clockwise as time progresses.  This is "circularly polarized light".  The resulting wave looks something like a corkscrew.

Circularly polarized light has many uses.  One of which is in RealD 3D movie theaters.  Light leaving the projectors is passed through a linear polarizer and then a quarter wave plate to generate circularly polarized light.  This light is then passed through another quarter wave plate and linear polarizer in the glasses.  The quarter wave plate brings the two components back into phase to make linearly polarized light which is then passed through a linear polarizer on its way to the viewer's eye.

The orientation of the quarter wave plates and linear polarizers in the projectors can be changed to cause the light to have clockwise or counter clockwise polarization.  The corresponding orientation of the components of the 3D glasses cause them to pass or block clockwise or counterclockwise circularly polarized light.

The advantage of RealD over IMAX 3D is that the filters in the glasses that pass or block circularly polarized light don't need to be in any particular orientation.  If you rotate the filter by 90 degrees, the incoming clockwise light is still clockwise.

In other words, if you tilt your head while wearing circular polarizer 3D glasses, each eye will continue to see the correct image.

# There was promise of a magic trick?

The important thing to remember in the above 3D example is the significance of the ordering: Linear -> 1/4 wave -> 1/4 wave -> linear.  Mixing up the order dramatically changes the results.

Passing unpolarized light through a 1/4 wave plate doesn't do anything: you still have a random jumble of light polarizations coming out.  Likewise if you pass circularly polarized light through a linear polarizing filter, a portion of it passes through just like a portion of unpolarized light passes through.  There is no significance of circularly polarized light to a linear filter.  It treats the horizontal and vertical components of circularly polarized light just like it treats the light of every other orientation.

So let's consider an example with two linear polarizers:

These two polarizers came off the same roll and are currently oriented so that they will only pass horizontally oriented light.  Now I'm going to rotate one by 90 degrees:

You'll note that it got a little darker.  This is a result of the polarizing effect of the table's surface.  When light bounces up off of reflective surfaces, it tends to have a horizontal polarization.  This is why polarized sunglasses have vertical polarizers; they are trying to block light reflecting off the surface of water/slick pavement/etc.

So let's overlap these a bit:

As you can see, the light that passes through the bottom horizontal polarizer does not make it through the top vertical polarizer.  Since no light is passing through, you see a black rectangle.

Now, if we could change this horizontally polarized light into something that would pass through a vertical filter, we could make a translucent window through the black rectangle.  A properly oriented 1/4 wave plate will do the trick!  This changes the linearly polarized light into circularly polarized light which passes through without incident:

Weird, huh?

# The secret

What I didn't reveal in the above video is that I had a linear polarizer over the lens of my camera.  This is impossible to detect looking at just the footage alone.  A portion of the light passing through the linear filter is blocked making the image look a little dimmer, but adjusting the camera settings easily compensated for that.

So what about the magical cards?

The cards are all made with three layers: two 1/4 wave plates and a horizontal linear polarizer.  The desired designs are made from the negative space cut out of the quarter wave plates.  The whole thing is sandwiched inside a no-heat adhesive business card lamination pouch.  With this configuration, there are only three possible outcomes for the light:

A: Unpolarized light is passed through the horizontal polarizer creating horizontally polarized light that is entirely blocked by the vertical polarizer.  The observer cannot see this light.

B: Unpolarized light is passed through a 1/4 wave plate which does nothing and then through a horizontal polarizer.  It is then entirely blocked by the vertical polarizer just like A.  The observer cannot see this light.

C: Unpolarized light passes through the 1/4 wave plate which does nothing and then through the horizontal polarizer which linearly polarizes it.  It then passes through a 1/4 wave plate which circularly polarizes it and allows it to pass through the vertical filter.  The observer can see this light.

Put simply, all that matters here is whether or not a quarter wave plate is between the vertical and horizontal filter.  If you attach the vertical filter to the observer, they will see different designs on either side of the card.  The quarter wave plate portion will be transparent and the rest will be black.  Everything on the back side of the card will be similarly transparent.

You need to be careful though.  If the card is rotated by 90 degrees, the linear polarizers will be oriented the same way and the entire card will be transparent.

# So what's the application?

I really thought long and hard about this one.  I guess maybe if you were a guy who sold polarized sunglasses, you might be able to make some neat business cards?  Kind of a stretch.

Besides that, it's a pretty cool science demo that will certainly gee-whiz anyone who is unfamiliar with polarized light.  If you'd like to make your own, you can order materials from a number of places online.  Be careful though, quarter wave plates are very easy to tear, so make sure you cut them with extremely sharp tools or your design might get ruined.

Outside of the world of neat cards, there are tons of applications for polarized light.  Anything from the linear polarizers in the LCD of the display you're looking at right now to the circular polarized light used in RADAR.

## 11 thoughts on “A Magic Trick”

1. That's really cool. 🙂 Kind of disappointed it can't be for much though, because of the requirment on the viewer. I suppose you could make "If you can read this, you're wearing polarized sunglasses" signs, but that's all I can really think of.

Then again... you could always make an interesting "magic trick" box, in a wood & glass display case, with polarizing filters on all the glass. Then have a "magic" card rotating (or just fixed) inside the box.

• That's a pretty neat idea too. 🙂
If I'm understanding this all correctly, then a piece of birefringence material placed in that display case would create a "portal" that allowed you to see out the other side?
Integrating that into the piece to be displayed could make for some interesting interactions, particularly of the "romantic chance encounter" sort. All in the name of science, of course. 😉

2. By the way, what kind of shops do you these materials in? I'd love to pick up some to play around with!

(captcha word was "Circular", rather appropriate.)

• I got all my stuff from http://polarization.com/polarshop/

The minimum order quantities are a little steep though (3sq feet), and it looks like they're out of the quarter wave retarders at the moment.

If you just want some to play around with, I'd be willing to part with some of my extra for a fair price. Email me if you're interested.

3. What I don't understand is how linearly polarized light is turned to circular by the 1/4 wave retarder.

Furthermore, in your initial diagram, the different polarizations have the same phase. Is this necessarily the case in natural light? I thought it was a jumble of phases and polarizations, and only laser light has everything in the same phase.

Back to the subject of generating circular polarized light: I'd expect all light out of the first linear polarization filter to have a single linear polarization. The 1/4 wave retarder will slow down light of a given polarization axis, but you still have a single linear polarization out, no?

• All great questions.

You're absolutely right that white light is notmally a combination of all polarizations, frequencies, and phases. In this case I chose a single phase and frequency to simplify the diagram. Filters like the ones I used don't discriminate based on phase, so you can model the incoming light as a whole bunch of different phases, pass them through the model, and recombine them at the end.

Similarly, light polarity is really representing the orientation of the electric field in the wave. Electric fields are vector fields, and all vectors can be described as any number of vectors added together as long as their sum creates the original vector.

So for example, a diagonally polarized beam of light can be considered as the sum of a vertically and horizontally polarized beam. So while it is just a single linear polarization, it has multiple *components* in other directions.

For a great example of this, you can conduct an experiment with three linear polarizers. Orient two at perpendicular angles so that they 100% block the incoming light. The light polarized by the first filter (let's say vertically) has a horizontal polarization vector of zero, so when it hits the second filter, it's blocked.

Now if you pass the third polarizer between the other two at 45 degrees, the vertically polarized light will be partially filtered. The component of the vertically polarized light in the diagonal filter direction will pass through while the component perpendicular will be blocked.

Now the light hitting the last filter is actually diagonal which means it has a nonzero horizontal component, so some light will pass through.

If you stack enough filters between the two filters at very slightly different angles, you can pass more and more light through as you're losing a smaller and smaller portion of the light in each filter. This is actually how Twisted Nematic LCD displays work.

I'm typing this on my phone, so let me know if I wasn't clear and I'll try to follow up with a "reader mail" post.

• OK, I think I'm getting my mind blown at "...but insert a filter at 45° between the other two, and some light will pass through the stack".

I'm working through some understanding of this, but indeed a "reader mail" post might be excellent for other people as well 🙂