# How does a quarter wave plate convert linear light to circular light?

I understand that a quarter wave plate converts linearly polarised light (single axis) to circularly polarised light (two axes) with one wave delayed by quarter of a wave length. I understand that this result can be achieved by linear light being polarised through a lens angled at 45 degrees:

It makes sense that linearly polarised light can be formed by light of multiple axes being "filtered", converting it into to a single axis, however I do not understand how an axis can be "added back" through another filter/plate. How does this work (slow axis, fast axis, etc.) and what is the importance of 45 degrees? I would like to simply understand the process and logic behind how this occurs, rather than the math.

I have read this post on Stack Exchange, but the reasoning is fairly complicated.

I've had a look at a visualisation of what's happening when a light wave passes through the plate, but it, nor it's accompanying article seem to clearly explain how the spinning transformation is occurring for someone without experience with the concept:

• Read this: en.wikipedia.org/wiki/Waveplate – my2cts May 23 at 9:44
• Thanks for that @my2cts. I've had a read of that before, and noticed that the wave get transformed into a spiral (upload.wikimedia.org/wikipedia/commons/7/77/…). What I'm not getting however is what's causing the spinning. – Lachlan Etherton May 23 at 9:48
• It might help if you consider linear & circular polarization as special cases of elliptical polarization. Also, (from polarization) "one can use as basis functions any pair of orthogonal polarization states, not just linear polarizations" So we could choose right and left circular polarizations as the basis functions. Also see the Bloch sphere article, although that article is admittedly rather technical & very mathematical. – PM 2Ring May 23 at 10:47

Depends how deep you want to understand it. Mathematicaly, even a vertically linearly polarized light can be described by 2 diagonally lineraly polarized light beams. In that sense, it was always there, just "cancelled out". The retarder retards one diagonal component but not the other.

Quantumly... i struggle with it a bit but this video from 3blue1brown is excellent for visualising and explaining quantum entanglement and superposition. https://youtu.be/MzRCDLre1b4?t=480

Edit 1:

Example: All of these are vertically polarized light:

Edit 2: On the top, a horizontal and a vertical polarized beams create a diagonally polarized light when superpositioned together.

On the bottom, a horizontal and a vertical polarized beams create a circularly polarized light when superpositioned together. The difference here is, that the vertical and horizontal components are not in phase.

Follow the end of the arrays. It makes a circle at the bottom, but only a line on the top.

• So, we could correctly state that what we call a vertical wave is the addition of two vectors of other axes, similar to how a circularly polarised wave is the addition of the X and Y axis, but represented as a spiral? If so, is there a way to effectively draw this? For example, you can draw both the X and Y axis of a circularly polarised light wave. – Lachlan Etherton May 23 at 9:44
• You can imagine it in anyway that makes you solve the problem easiest. You can have 42 arrows if you wish. A vertically polarized light can also be imagined to be a sum of 2 circularly polarized light beams. (One left + one right). I've edited my answer to add a picture. The difference is that the 2 arrows ar at it's maximum at the same time and at their minimum at the same time with linear polarization (they are in phase), while with circular they are delayed between themselves. Again - this says nothing what is actually happening in real life, would need quantum physics for that. – KrNeki May 23 at 10:18
• I found this website that has good animations for when the position where the beam of light is traveling straight at you. cddemo.szialab.org – KrNeki May 23 at 10:35
• Thanks for that. Great, that diagram clarifies things. Starting to make a bit of sense, but I feel like I'm contradicting myself a bit. Let's say we have polarised the light such that has been linearly polarised. I understand that we can describe this using a number of vectors, but still, it's vertical IRL. As you've said, you need quantum physics to understand what is actually happening. Is there a simple way of describing the physical event causing the spinning transformation? – Lachlan Etherton May 23 at 11:09
• I've had a look at the visualisation. I certainly understand what the circular polarisation is, but it's the transformation part I'm not getting. I understand that energy is conserved and the wave is being transforming from linear to circular, I just don't understand what is causing it. The diagrams I've seen all seem to show the result as instantaneous. – Lachlan Etherton May 23 at 11:14

Let me start by the meaning of the fast and the slow axes of the quarter half plate, and to do so let me compare it with a regular piece of glass. As usual let the speed of light in vacuum be $$c$$. Light travel in glass with a speed $$v$$ that is less than $$c$$, we write $$v=c/n$$, with $$n$$ is the refractive index and $$n>1$$ for glass. For a regular piece of glass this $$n$$ doesn’t depend on the polarization, that is, send in light with any polarization whatsoever and they all slow down by the same amount. Now, the special thing about the quarter half plat is that its refractive index does depend on polarization. Send light polarized in the direction of the fast axis and it slows down to some $$v_f = c/n_f$$, but send light polarized in the direction of the slow axis and it slows down to $$v_s = c/n_s$$, such that $$c>v_f>v_s$$. The fast and the slow axes are perpendicular to each other. Such a behavior is not uncommon, and it requires the atoms of the material to have some asymmetry. So for example some rod like atoms.

Now how does this behavior lead to turning linearly polarized light to circularly polarized light? Well sending linearly polarized light in the direction of either the fast or the slow axis does affect the speed but it leaves the polarization unchanged. Let’s think about the situation where we send in light linearly polarized but at $$45^{\circ}$$ from the fast axis. We think about this situation by decomposing the oscillating electric fields of light along the directions of the fast and the slow axes. Before the light hit the plate these two components of the electric field are in phase, that this they are mirrors of each other. In another words, these two components together oscillating in phase give us the $$45^{\circ}$$ relative to the fast axis polarized light. As these two component enter the quarter half plate, the two components move at different speeds, causing them to have different phases when emerge from the plate.

Let’s see how to calculate this change in phase from the two different refractive indices. The two components of the electric field move at different speeds meaning they spend different times inside the plate. This means that the slower components oscillate more inside the plate. Let $$t_s = d/v_s$$ and $$t_f = d/v_f$$ be the time the slow and fast components spend inside the plate, with $$d$$ being the thickness of the plate. Let the frequency of light be $$f$$, then the number of oscillations each component does is $$f t_s$$ and $$f t_f$$. In order to get the a circularly polarized light we need this difference between the number of oscillations to be quarter, so we need $$f(t_s - t_f) = 1/4$$.

I believe your problem is that you want it to make sense. Pictures of it happening don't make sense of it. Mathematical descriptions of it happening don't make it make sense.

You are asking for an explanation of something in terms of waves, and for the last 100+ years there has been no emphasis on looking for explanations.

Still, you ask the question. You and I would both like a picture of light waves that fits the measurements. That would be satisfying. The picture we make would not have to be truly correct. Still, if it kind of fit the data and if it made sense that would be pleasant.

We have models of light as water waves, but those do us no good because they can only give us vertical waves -- linear waves.

You have a string model. Circular waves in the string turn linear when the string must pass through a slot which allows only vertical motion. So we want a way for linear waves in the string to turn circular.

The slot which absorbs motion in one direction but not the other is easy to understand.

To go the other way, what we want is a kind of slot which doesn't absorb the wave in the second direction, but delays it.

I'm not sure how to make a physical model of that. Maybe you have a rope that for some distance is thicker in one direction than the other? Would waves in the thicker dimension travel slower? I'm not sure they would. Ideally it would be denser in one dimension than the other.

How about this. Your waves travel through a springy metal cable, and for a section the cable has an asymmetric core which is a different heavier metal, some sort of alloy which makes it just as springy, but denser.

MAYBE if the cable was waving back and forth, it would go slower if it was up-and-down than if it was left-to-right.

And if so, if the special part of the cable was just the right length, then a linear wave oriented at a 45 degree angle going in, would become a circular wave coming out. A linear wave oriented to 0 degrees or 90 degrees would be unaffected. (The 90 degree wave would be delayed but you wouldn't notice that, it would still be linear.)

That's the best I can do.