# Polarization true or false: Repeating circular filters is different from repeating linear filters?

Begin with standard abbreviations for polarized filters:

• H = Horizontal,
• V = Vertical,
• D = Diagonal,
• R = Right circular.

HHH means three horizontal filters in a row. Start with totally unpolarized light.

• H transmits 50% as does HH, HHH, HHHH, etc.
• Similarly, V, VV, VVV, etc. And D,DD, DDD, etc.
• HD transmits 25%. As do DV, HR, and RD.

But HV transmits 0%, and HDV counterintuitively transmits 12.5% (I call HDV the famous three-filter trick).

Now, if R consists of H followed by a quarter wave plate, then R transmits 50%, but RR transmits 25%, RRR transmits 12.5% RRRR transmits 6.25%, etc., unlike V, VV, VVV, VVVV etc.

Is this true? Or do RR, RRR, RRRR, etc. all transmit 50%? Is there another form of circular polarizer that does work like linear polarizers when repeated?

You are exactly correct in your first assertion: Each R stage in and RRR... sequence will diminish the light further.

That's not because there is anything inherently different about circular basis states versus linear states, but because we cheat a bit in how we make circular polarizers, since they start with linear polarizers that are then followed by quarter wave plates. It's a bit like a mismatch of hose connections, since you keep converting back and forth between the two basis sets (linear versus circular).

Can you build a full-pass circular polarizer? Sure.

The first thing to keep in mind is that like linear polarizers, quarter plates have a definite orientation to them in how they operate on light. Along one side-to-side axis of a quarter plate, light moves faster and in one linear polarization, while along the other perpendicular axis light move slower and with the opposite polarization. They call it a "quarter plate" because its thickness is just right for delaying the slower-moving light by exactly one fourth of a wavelength of light. The shift remains (mostly) proportional to wavelength for different lengths of light, so the neat thing about a quarter plate is that it works even with white light of many different frequencies.

A circular polarizer consists of two parts. The initial linear polarizer, say $V$ for vertical, that absorbs 50% of non-polarized light and transmits the other 50% as polarized light.

The second part is a quarter plate whose fast axis is rotated 45$^\circ$ away from the vertical axis of the polarizer. That 45$^\circ$ angle allows the quarter plate to split half of the polarized light into fast mode, and the other half into slow mode. For an observer looking into the oncoming light, a quarter plate that is rotated 45$^\circ$ to the right of the polarization axis of the linear polarizer produces light that rotates clockwise as seen by the observer. That is the right circular polarized light. Call that case the $Q$ quarter plate.

If instead the fast axis of the quarter plate is rotated 45$^\circ$ to the left of the linear polarizer axis, the resulting light rotates left. Call that left circular polarized light (and if I got that switched, someone please flag me!), and call that orientation of the quarter plate $Q^-$.

So, with that in mind, here's the trick for creating true cascading circular polarizers: Simply add a $Q^-$ in front of the linear $V$, in addition to the original $Q$ plate that comes after the polarizer. The resulting sandwich is $Q^-VQ$, versus $VQ$ for ordinary circular polarizers.

The reason this works is that the new plate is 90$^\circ$ out-of-phase with the quarter plate from the circular polarizer in front of it, and so cancel out the action of that plate, since both components are slowed and sped up by the proportions by the two out-of-synch quarter plates.

(As the asker Jim Graber neatly noticed, another way to do this is to add a three-quarters plate in front of the polarizer, that is, to create a $QQQVQ$ sandwich. That also gives a null action by changing all the phases by one full wavelength! Neat trick, Jim.)

So here's how it works. For ordinary mixed polarization light entering $Q^-VQ$ from the left, the first $Q^-$ stage just shifts random phase and so has no major impact on the light. The remaining two stages $VQ$ then act like a conventional circular polarizer that, as expected, diminishes the light by 50%. So, the $Q^-VQ$ sandwich clearly qualifies as a true circular polarizer.

Now send that light through another $Q^-VQ$ sandwich. This time the initial $Q^-$ plate has something meaningful to work with! It cancels the actions of the $Q$ plate from the first sandwich, and so converts the light back to vertical linear polarized light. Since that light is now fully matched to the $V$ plate of the second sandwich, all of it gets through. The final $Q$ plate in the second sandwich then recreates circular polarized light, again at nominally 100% efficiency.

You can repeat this indefinitely, with the internal pairs of $Q$ and $Q^-$ plates in successive sandwiches always canceling out and ensuring 100% transmission of vertically polarized light, but always with the last plate converting back to circular.

Now at this point you may be thinking "yeah, but don't you have to keep all of these supposedly true circular polarizes oriented just like vertical linear polarizers?" The cool answer is no, you don't. What's important is that the initial $Q^-$ plate is properly oriented relative to the $V$ polarizer, not to the circular polarized light that is entering into if from the front. That light is rotationally symmetric and will convert into linear polarized light at any angle, with that angle depending only on how the $Q^-$ plate is oriented.

So, the bottom line to your question is this: Yes, you can make true circular polarizers quite easily using the same materials normally used, the only difference being the addition of that initial $Q^-$ plate. They don't do that in ordinary circular polarizers because it would a very rare event to need cascaded circular polarizers. So instead of adding the cost of another quarter plate in front that would serve no real purpose, they build it in a way that provides perfectly serviceable circular polarized light at a significantly lower cost.

• What a neat trick! – Jim Graber Jul 27 '12 at 22:10
• But now I am curious, is the Qminus a three-quarter wave plate? Which I surmise is also a Left circular polarizer? – Jim Graber Jul 27 '12 at 22:15
• You could indeed make $Q^-$ that way, but the easier method is to recall that direction of the circular polarization depends on whether the fast plane of the quarter plate is rotated plus or minus $45^\circ$ relative to the linear polarizer. That's how they make theater 3D glasses, for example. So, as you suggested, you could have 3 quarter plates all with the same orientation as the trailing plate, giving a full (and thus cancelling) rotation of phase, or you could just rotate the initial quarter plate $90^\circ$ to get $Q^-$. Either method works fine. – Terry Bollinger Jul 28 '12 at 3:30

Half-Quarter-wave-plate + linear polarizer + half-quarter-wave-plate can (if the components are oriented properly) act as a "stackable" circular polarizer (i.e. a stack of any number of them will ideally be 50% transmissive to unpolarized light.) The first half-quarter-wave-plate converts R to H, the second converts H back to R.

Alternatively, there is nothing impossible (in principle) about a "true circular polarizer", a single material that by itself absorbs right-circular-polarized light but passed left-circular-polarized. If it existed, it would also be stackable. In practice, I'm not sure whether or not they exist. Well, any chiral molecule or material will absorb one helicity light over the other preferentially (circular dichroism), but usually the ratio is only slightly different than 1. I don't know anything about this field, but it seems that researchers are working on it.

• It should be quarter-wave plate, not half-wave. – straups Jul 28 '12 at 9:25

For readily-available circular polarizers, this is true and can easily be observed with a pair of 3D cinema glasses, which are a bit of a puzzle and quite fun to play with; as far as I can make out they are made with a quarter-wave plate followed by a linear polarizer. Thus flipping or turning a lens will have a large effect on transmission when both lenses are stacked.

However, in general, saying "a right-circular polarizer" is pretty ambiguous if all you require is for the device to project on right-circular polarizations and you don't ask about the polarization state afterwards. For the filters in use in 3D-cinema glasses, which are generally achieved by putting a quarter-wave plate before a linear polarizer at 45° orientation, the polarization output will be linearly polarized, so it cannot be fairly described as a "true" projective filter. Instead, to achieve that, you'd need to add an additional quarter-wave plate after the linear polarizer, so that the polarization output was the same as the filter selected on.

And, of course, once you did that, combinations like RR, RRR, RRRR and so on would be identical to repetitions of linear polarizers.