Is a rope the correct analogy for light to understand what polarization is?

Question

I want to understand the essence of the CHSH experiment, and for this I need to understand what the polarization of light is. When light passes through a polarizer, a slightly mysterious process occurs: if you put two perpendicular polarizers, then the light will not pass through them, but if you put another polarizer between the first two ones at an angle of 45 degrees, then the light will again begin to pass through.

I have heard that rope is the model for these phenomena. You can start up a linear or circular wave through a rope. If you pass the rope through a slot, then the circular wave will turn into a linear one. If you put two perpendicular slots, then the wave will not pass; but if one more slot is placed between them at an angle of 45 degrees, then again part of the wave will pass. This illustrates that the slot, like the polarizer, “not only absorbs but also rotates”, I hope you understand what I mean. Is the written correct?

I tried to find a rope and experiment with it, but there was no suitable one.

I want to program a mathematical model that describes a polarizer with light or a slot with a rope, and so far I still don’t understand how to set a diagonal slot in this model. The point is obvious that if the vector [X=1;Y=0] is absorbed by the gap [1;1], then in the basis of this gap one of the components in the vector is set to zero, and in the original basis the component Y does not decrease, but increases, as the vector [1;0] is projected onto [1;1]. That is, as I understand it, one can say that "quantum magic is a consequence of mathematical magic" - in two-dimensional space, when a vector is absorbed, its individual components can increase. But I still don't understand how to program it.

Answer 1

The analogy still works, but instead of thinking of a single wave travelling end to end along the rope through the filters, that is absorbed or not as a whole, you need to think of the rope as a sequence of short segments, each segment receiving a driving force at one end, causing it to move, and thus apply a driving force to the next segment in the sequence.

When we come to a slot, the motion of the rope is constrained. It can move freely along the slot, and it held fixed in the direction across the slot. It is this motion that gets passed on to the next section after the slot. So if the rope is held fixed by the slot, the rope after the slot doesn't move. If the rope is unconstrained by the slot because the wave is aligned with it, then the wave passes undiminished. But if the slot is at $45^\circ$ to the incoming wave, then the rope in the slot moves at $45^\circ$ with diminished amplitude, driven by the force from the rope, but partly cancelled by the normal reaction forces from the edges of the slot. These forces cancel the component across the slot, but leave unchanged the component in the direction of the slot.

The following section of rope only sees the rope moving in the slot at $45^\circ$, it cannot see anything of what happened leading up to that. And if you shake the end of a rope at $45^\circ$, the wave that results will be at $45^\circ$, as you might naturally expect. Hence the polariser "not only absorbs but also rotates".

Another way to look at it is to say that the normal reaction forces applied by the edges of the slot add a component to the wave perpendicular to its original polarisation direction, which is what passes through the final filter. The forces from the original wave trigger a reaction force in a different direction, transferring some of the energy from one polarisation component to the other.

In a polarising filter, which consists of long chain molecules all aligned parallel to one another, the electrons are more free to wiggle in one direction than the other. When an electromagnetic wave passes through, it pushes and pulls on the electrons, and the motion of the charged electrons emits new electromagnetic waves, polarised in their direction of motion, which combine with the original. If the electrons are free to move, these act like normal reaction forces and cancel the wave. If the electrons are constrained from moving along the direction in which they are being pushed, the original wave is unaffected.

The rope/slot analogy is a reasonable way of understanding the effect.

Regarding how to program it, the thing to do is to take your incoming polarisation direction as a vector, rotate it into along-slot/cross-slot coordinates, set the cross-slot component to zero, and then rotate back. Something like:

$$\pmatrix{p_x' \\ p_y'}=\pmatrix{\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta} \pmatrix{1 & 0 \\ 0 & 0} \pmatrix{\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta} \pmatrix{p_x \\ p_y}$$

This all works fine for linear polarisation. If you want to handle circular and elliptical polarisations, the rope analogy doesn't work so well. For more information, look up the Jones Calculus: https://en.wikipedia.org/wiki/Jones_calculus