# Torque on Quarter-Wave Plate

When circularly polarized light is passed through a quarter-wave plate, the plate experiences a torque. I understand this is true because angular momentum must be conserved, but I don't understand what is happening in the plate. From a classical perspective, how is the light interacting with the electrons in the plate in order to create the torque? I prefer a physical description over a mathematical one. Thanks.

• It may be helpful to think about the reversed case. How exactly do you need to move a charged particle (or dipole moment) in order to create circularly polarized light? Once you get that understood, the time-reversed situation may be more approachable. Sep 29, 2020 at 8:56

Classically, what's happening with refraction is that the induced dipole moment in the object is oscillating either ahead or behind of the electric field; this creates a torque based on how strong the induced dipole moment is and how much it lags/is ahead of the circularly polarized wave. Remember - an electric field produces a torque on a dipole if the two are not aligned.

This is not special to quarter-wave plates - for example, anything with attenuative effects will have a torque as well.

Quarter-wave plates have the added complication that the induced dipole/phase lag is different in different directions.

I made a small animation in Desmos to illustrate this. Change the $$\phi$$ slider to change the phase difference of the induced dipole moment, and change $$a$$ for a simple illustration of what happens when the magnitude of the induced moment is different for different directions. The main things this is missing that the magnitude of the induced moment is usually related to phase differences and that the phase difference can be different for the two axes as well.

https://www.desmos.com/calculator/0pbjya5gqe

• waveplates are typically thought of as changing the polarization by having a "slow" and fast axis that have different speeds. It's not obvious how this is connected to your answer. Oct 1, 2020 at 19:38

We say light can have "angular momentum" because they can cause charged particles to spin.

If I put an ion in the path of a wave that's horizontally polarized, the existance of the E-field will make the ion experience a force along the direction of the polarization:

$$E= Cos(\omega t) \hat{V} \implies \hat{F} = q Cos(\omega t) \hat{V}$$

So the particle will wiggle "up-and-down". If it is horizontally polarized it'll wiggle "left-and-right".
If I give it a combination of H and V it will go "up+left AND right+down" in a diagonal position. We can see that like this mathematically:

$$\hat{F_H} = q Cos(\omega t) \hat{H}$$

$$\hat{F_V} = q Cos(\omega t) \hat{V}$$

$$\hat{F_H}+\hat{F_V} = q Cos(\omega t)( \hat{V} + \hat{H})$$ which appears as shown in the first picture on the left:

Now if I add these two forces, but I change the phase of one of the two fields then I end up with circular light, as shown in the picture.

If you put an ion in the beam of diagonal light, it will move back-and-forth along the diagonal direction (which can be represented as the points in the picture when the wave hits the "wall" in the back).

If you put an ion in the beam of circular light, it will spin in a circle! So if a stationary ion can be made to spin in a circle, then the light must have imparted angular momentum.

(Now, as we've shown, all you need to do to accomplish this is to slow down a part of the light that's moving in a particular direction! This can be made to be very mathematical if you look up on wikipedia circular polarization.)

So the torque is imparted simply because one component of the light is slowed down relative to the other component. The plate technically recieves an equal and opposite torque in the opposite direction (than the direction that the ion would have been pushed by), but it's typically so large that you don't observe that.

(if you're still confused, maybe think about what kind of "force" is imparted on a plate that simply slows light down. it will be just the change in momentum due to the change in speed)

The global polarization state of the light is the result of the spins of individual photons. Spin is an angular momentum ($\pm \hbar$). Passing through a medium, the polarization changes if the spin of individual photons -and the associated angular momentum- is affected. This can happen by birefringence (visualize it as a interference-dependent "twist" of the photon from $\hbar$ to $-\hbar$ if you like) or even absorption ($\hbar$ to just $0$). The change in angular momentum is the torque applied or transferred to the plate.

• Thanks. Can you provide a description of this process in terms of waves and electric fields? I understand the concept of massless particles exchanging momentum with the plate, but it's a little abstract for me. I'm trying to understand where exactly there is a force and moment-arm that creates the torque. Jan 27, 2018 at 16:11
• This does not address the core of the question as posed. Sep 28, 2020 at 9:52
• @EmilioPisanty I don't really agree, the answer can be incomplete maybe, but down voting it as if it was wrong seems really extreme. Sep 29, 2020 at 7:03
• @scrx2 The downvote button signifies "This answer is not useful", as communicated by the mouse-over tooltip. I feel that it accurately describes this answer (everything in this answer is implicit in the OP's premise - there is nothing here that actually adds to the thread), so my vote stands. Keep in mind that this is ultimately just one vote - if the answer is OK you should get additional votes to counteract it. Sep 29, 2020 at 9:32