Wave (lambda) plate above vs. below the subject?

[Zondar]

Hello Forum Members,

I'm trying to understand if the position of a full wave plate (lambda plate) matters relative to the subject (assume it's birefringent). Are the positions of the subject and wave plate commutative? In other words, is there any optical difference between the following orientations:

Polarizer -> wave plate -> subject -> analyzer
vs.
Polarizer -> subject -> wave plate -> analyzer

I've seen both orientations used. If there is a difference, then what optical effects or analysis tasks are one or the other orientations suited to?

Thank you very much.

[rjlittlefield]

Zondar, welcome aboard!

I look forward to seeing answers to this question, because I do not know them either.

In the meantime I just wanted to assure you that your question has not been totally overlooked.

--Rik

[g4lab]

I don't think they are commutative. If they were they would allow the wave plate to be inserted in the substage of petrological scopes something I have never seen.

The definitive and technical answer could be obtained by asking Dr. Olaf Medenbach at Ruhr Bochum University. He seems very happy to help people who are interested in this very arcane subject.

https://homepage.rub.de/olaf.medenbach/eng.html

olaf.medenbach@rub.de

[hans2]

I have no authoritative answers, just a few observations that might be of interest, maybe with some details wrong:

In the stuff I've read about using Jones matrices to represent waveplates it is often pointed out that generally they don't commute. However if the material axes are parallel (4 different rotations 90 deg apart do this) then the matrices are diagonalized by the same rotation and therefore do commute. In the context of polarizing microscopes I have seen those simple cases referred to as "additive" (ordinary parallel with ordinary) and "subtractive" (ordinary parallel with extrordinary) alignments of the specimen relative to the waveplate since the combined effect is predicted by simply adding or subtracting optical path differences. When the axes are not parallel adding/subtracting optical path differences no longer makes physical sense, I think.

It may be useful to clarify what exactly is meant by "optical difference" in the question, as in: First, do they commute physically in the sense that order has no effect on output polarization state for any given input state? Then, if they don't commute, is order dependence observable between crossed polarizer/analyzer? I have not found any arrangement where order dependence is observable with polarizer/analyzer crossed at 90 degrees but would be curious to hear ideas for either an argument why order dependence can never be observed with polarizer/analyzer 90 deg apart or a counterexample of an arrangement where order dependence is observable.

With polarizer/analyzer less than 90 deg apart (45 deg for example) order dependence can be seen fairly easily. Calling the pieces of tape "a", "b", and "c" by analogy with the usual way of labeling a right triangle with a and b perpendicular and c the hypotenuse, c crosses on opposite sides of a and b:

a and b aligned with polarizer, analyzer at 90 deg:

c aligned with polarizer, analyzer at 90 deg. Where a and b cross the optical path differences subtract:

a and b aligned with polarizer, analyzer at 45 deg. I think the explanation is that where c comes second (upper left in the photo) a and b don't affect the polarization state so the color is the same as c alone. But where c comes first (lower right in the photo) the polarization is no longer linear and aligned with the axes when it gets to a/b so they can now further alter the polarization state.

[rjlittlefield]

I have not found any arrangement where order dependence is observable with polarizer/analyzer crossed at 90 degrees but would be curious to hear ideas for either an argument why order dependence can never be observed with polarizer/analyzer 90 deg apart or a counterexample of an arrangement where order dependence is observable.

That's a very interesting observation! If it's true in general, then I expect an argument can be made using Jones matrices. Following the treatment at https://en.wikipedia.org/wiki/Jones_calculus , I suggest symbolically multiplying horizontally polarized light (1,0) with the matrices for two "general waveplates (linear phase retarders)", followed by a vertical polarizer ((0,0),(0,1)), then calculating the resulting light intensity as the "sum of the squares of the absolute values of the two components" of the product, and demonstrating that the final number must the same for both orderings of the waveplates. I have not, however, made a serious attempt to carry out that analysis.

I think the explanation is that where c comes second (upper left in the photo) a and b don't affect the polarization state so the color is the same as c alone. But where c comes first (lower right in the photo) the polarization is no longer linear and aligned with the axes when it gets to a/b so they can now further alter the polarization state.

I agree with that explanation. I had earlier come up with a thought experiment that involved using quarter wave plates to introduce circular polarization, followed by what turned out to be a 45 degree rotation of one waveplate and its neighboring polarizer, to construct a configuration that either did or did not pass light depending on the order of the wave plates. The concept is much like what you've done, but I like your demo a lot better!

--Rik

[hans2]

Following the treatment at https://en.wikipedia.org/wiki/Jones_calculus , I suggest symbolically multiplying horizontally polarized light (1,0) with the matrices for two "general waveplates (linear phase retarders)", followed by a vertical polarizer ((0,0),(0,1)), then calculating the resulting light intensity as the "sum of the squares of the absolute values of the two components" of the product...

This is what the plotting code I posted in the other thread is doing numerically (or supposed to be doing, at least) except it doesn't use the symbolic result from the table, just the diagonal form with similarity transformation applied numerically. (I did check symbolically that "general waveplate" in the table is transpose(Q)*J_diag*Q where Q is rotation by theta.) The "sum of the squares of the absolute values of the two components" to get intensity is done by multiplying the Jones vectors with their conjugate transpose. I haven't tried working the whole thing out symbolically either but I did check a large number of cases like so:

analyzer at 90 deg, quarter wave plate
analyzer at 45 deg, quarter wave plate
analyzer at 90 deg, half wave plate
analyzer at 45 deg, half wave plate
analyzer at 90 deg, full wave plate
analyzer at 45 deg, full wave plate

(Not that it would be desirable in this case, but is there a way to get video files to show up inline similar to using the [img] tag with externally hosted images?)

[rjlittlefield]

I haven't tried working the whole thing out symbolically either but I did check a large number of cases like so:

analyzer at 90 deg, quarter wave plate
analyzer at 45 deg, quarter wave plate
analyzer at 90 deg, half wave plate
analyzer at 45 deg, half wave plate
analyzer at 90 deg, full wave plate
analyzer at 45 deg, full wave plate

This is strong evidence.

When I used to teach algebra, I required that students check their symbolic work by plugging in numbers and confirming that supposedly equivalent expressions really did compute the same values. The rules for picking test values were no zeroes, no ones, no two values the same, no value the same as any constant that appears in the symbols. As I read your tests, you've ticked all those boxes numerous times across the suite of tests.

So, all that remains is to formally verify the result. I expect that one of the modern symbolic math packages could knock this off pretty painlessly, at least relative to the pain of doing it by hand. But I have no current experience in that area so I'm happy to let somebody else pursue the details.

(Not that it would be desirable in this case, but is there a way to get video files to show up inline similar to using the [img] tag with externally hosted images?)

There is a tag for videos hosted at YouTube, described HERE. I do not know of one that works for video hosted elsewhere.

Probably somebody has written a plugin for phpBB, but history has made me leery of plugins because they often rot quickly after the creator moves on to other things.

--Rik

[Macro_Cosmos]

I have tested this for transmitted light and I could not see a difference.

The universal condenser I use offers a slot for a waveplate. The nosepiece is able to take a waveplate as well. I could not see a difference. The matter of not having the waveplate inserted into the stage could just be a convenience thing, however, there are systems to contest that. Electrophysiology microscope frames often has this option. A waveplate can be inserted at the light port. The condenser used usually offers a built-in rotatable waveplate (de Sénarmont compensator).

None of my objectives are strain free and I have no knowledge in quantitative polarised light microscopy applications. Maybe I was not able to find something where differences may show.

What could be different is the image quality. Surely the objective would appreciate one less piece of glass in its pathway?

[hans2]

Interesting, I didn't know there were microscopes where it was easy to insert a waveplate on either side. I don't have a real polarizing microscope, just a kit for the Reichert Microstar IV which I believe was sold mainly to identify gout crystals based on the sign of the birefringence. There is not much adjustability. The polarizer and full wave plate fit over the light port/field lens below the condenser and there is an analyzer only in infinity space under the head.

I did find a relatively simple example with 3 waveplates where order dependence can be seen with polarizer/analyzer crossed at 90 degrees but it is not directly relevant to the original question because it is not a simple swap of a single waveplate with a group of one or more waveplates representing the sample. This one is some plastic sheet 0.6 mm / 25 mil thick with 1/6 wave path difference and arrows drawn to show the axes. Measuring from arrow pointing up the cases are 45° CCW -> 0° -> 45° CW on the left in the image and 45° CCW -> 45° CW -> 0° on the right. In the 45°/-45°/0° case the first two are in subtractive alignment and have no effect then the third is aligned with the initial linear polarization and also has no effect. But when the 0° piece is between the +/-45° pieces it disrupts the subtraction:

[Macro_Cosmos]

I will leave this to someone specialising in polarised microscopy.

I did briefly own a Berek Compensator, this was supposed to be inserted into the nosepiece. It wrecked the image quality, my good objectives looked worse than a field microscope, easily outdone by the achromats on one of my toys. I was able to place the compensator above the condenser, the images returned to normal but I bet any quantitative use vanished.

[g4lab]

Like @MacroCosmos above I will leave the conceptual and technical questions to those with higher degrees.

However @hans2 post above reminded me of a clock a wealthy uncle of mine had.

His was a cube and he bought it at a store like Haverhills or Hammacher and Schlemmer. It was pleasing to your eye. Here is a modern version or two.


https://clockforward.com/aurora-clock/

https://www.chronoart.com/qty_clk.html

https://www.chronoart.com/history.html