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# Polarization - Muller Matrix

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1. How to model and partial polarized light as source?

2. We want to know how to enter parameter values of muller matrix since muller matrix is more generalised. In opticStudio we have option for Jones vector. Converting Jones parameter into muller is easy by using some formula but converting muller matrix into Jones needs lots of calculation and it is tedious. So how to enter parameter values of muller matrix?

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Best answer by Mark.Nicholson 29 September 2022, 20:29

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Hello Zemax team,

Many greetings!

I have got the above question. I referred user manual as well as knowledgebase articles but couldn’t able to solve these doubts. I seek your support here. Can you please help me to resolve this questions.

1. How to model an partial polarized light as source?

2. We want to know how to enter parameter values of muller matrix since muller matrix is more generalised. In opticStudio we have option for Jones vector. Converting Jones parameter into muller is easy by using some formula but converting muller matrix into Jones needs lots of calculation and it is tedious. So how to enter parameter values of muller matrix?

Userlevel 6
+3

Hi Neha,

The short answer is to define two co-located sources, usually identical except that one is polarized and the other in unpolarized. You would set the relative intensity of the two sources to achieve the desired degree of partial polarization.

It’s an interesting question, and it all comes down to how the software models light. Different programs do it in different ways, and I think that (for example) TracePro uses Stokes parameters and Mueller matrices. OpticStudio does not, and I think the reasoning is as follows.

OpticStudio defines a ‘ray’ as having direction and electric field E such that

k.E = Ex.l + Ey.m + Ez.n = 0

Ex and Ey are the complex Jones matrix elements and l,m,n are direction cosines. Using this ‘Jones’ definition gives us rays that contain phase information, and this is necessary for imaging system design, diffraction and interference.

Stokes/Mueller calculations start from a slightly different premise. We ignore the phase of rays and consider the light to be some kind of bundle that has macroscopic quantities like a degree of polarization. Stokes vectors and Mueller matrices operate on intensities and their differences, i.e. incoherent superpositions of light; they are not enough to describe either interference or diffraction effects. This is why you can transform a Jones into a Mueller matrix, but not the other way around. The coherent phase information is lost in the Stokes/Mueller calculus.

So how do you model a partially polarized source in OpticStudio? In non-sequential mode you would define two co-located sources, usually identical except that one is polarized and the other in unpolarized. You would set the relative intensity to achieve the desired degree of partial polarization.

In sequential mode you would use configurations to achieve the same end, with the configuration weights acting as relative intensities.

Hope this helps,

• Mark

Hi Mark,

Thank you very much for your response and thanks for explaining it in detail.

So you mean to say that there is no direct option available to convert muller into Jones vector. But we can do conversation manually by doing some calculations. So my idea over here is to use a script. We will write a script for steps involve in converting muller matrix into Jones vector and save it and run it whenever we need. Is it possible? Can we do it this way?

Userlevel 6
+3

Well, you can do anything you like with your own calculations, but somehow you have to get the coherent phase information which is not included in the Mueller matrix, as it is based on incoherent properties (intensities) only.

The purpose of Mueller calculus is to allow you to describe the partial polarization of a beam of light, and to transform this easily, through matrix multiplication, to simulate polarizing components. You simply cannot get the phase retardance of a coating out of a Mueller matrix as the data is just not in there.

I don’t think you should conflate the ‘generality’ of the Mueller matrix with it being a superior method. It definitely has its role, but the Jones matrix representation of a ray contains more data. You then have to make up the bulk properties of a beam of light by mixing rays of different polarization states together. There is nothing you cannot do with an ensemble of Jones rays, whereas the Mueller approach limits you only to incoherent properties of the beam. I suspect (without proof) that the codes that use Mueller matrices only do illumination calculations with them. That’s not a knock against them, it’s perfectly valid, but it is a subset of the things you can do with the Jones matrix.

Thanks Mark. Nice explanation.

Thank you very much for your efforts.

Can you provide any supporting demo or doc or literature regarding statement “Jones matrix representation of a ray contains more data. You then have to make up the bulk properties of a beam of light by mixing rays of different polarization states together. There is nothing you cannot do with an ensemble of Jones rays, whereas the Muller approach limits you only to incoherent properties of the beam."

Jones matrix is only valid transformation operator only when the input light is fully polarized.

Regarding first question: I am not sure as to how will you combine the results together. In multi-configuration, the different configs will generate theor own data and the transformation through a surface will change whether the light has full DOP or partial DOP. Can you provide a demo file.
Userlevel 6
+3

Hi Neha,

Check, for example, https://en.wikipedia.org/wiki/Mueller_calculus. But you do agree that the Stokes/Mueller approach is based on intensities and not complex fields? That’s the fundamental difference.

Now, as for Jones matrix is only valid transformation operator only when the input light is fully polarized, this is true. But, when you ask the polarization ray trace to trace an unpolarized ray, it traces two rays: one at some arbitrary polarization and another orthogonal to it. The averages of these two give the same results as an unpolarized ray.

I think the thing you may be struggling with is the concept that every ray is fully polarized. Even the unpolarized rays are traced as two orthogonally polarized rays 😀 and then averaged.

The ray model in OpticStudio is that every ray is polarized, and that bulk properties of a ‘beam’ is made up by summing over many rays. If you imagine freezing time and looking at a bunch of rays, each one would have its E-vector pointing at some angle, and if you let time speed up slowly you’d see each ray’s E-vector oscillate somewhere between in-plane and circularly around the direction of propagation. In OS you don’t define ‘beams’ as such. Beams are made up by summing over many rays.

I don’t want to sound like I’m ragging on Stokes and Mueller because they’re very useful in some cases. But they are limited to the illumination of homogenized sources. Beyond losing the coherent information, they don’t describe the spatial structure of the degree of polarization, so a beam with 50% polarized and 50% unpolarized intensities is assumed to be spatially uniform. But what about a beam that is fully polarized at the center, and falls off to be fully  unpolarized at the edge? Or a beam that is fully polarized over half the aperture, and fully unpolarized over the other half? Or a beam that is polarized at one wavelength and unpolarized at another? A beam where the polarized and unpolarized light propagate at different angles?

The Stokes approach can’t distinguish these cases. But by tracing a large number of rays, you can specify any starting distribution you want, and see how the state of polarization of the beam evolves ray-by-ray. You can find out the polarization state of the beam overall using detectors and polarizing optics, or by interrogating the ray database directly.

This is why I’m pushing back on your initial statement that the Stokes approach is ‘more general’ than Jones. It’s really not. It’s purely incoherent and has no concept of spatial structure, wavelength, angular distribution etc.

Here’s another way to think of it. Think about a gas. At one level, we can talk about and measure the temperature and pressure of the gas as macroscopic quantities. But if we zoom down to the level of individual molecules, little billiard balls flying around with speed and direction, you can’t talk about the temperature or pressure of one molecule. The molecules have mass, speed and direction. Only by averaging over many many molecules do quantities like temperature and pressure emerge.

That’s exactly like the ‘degree of polarization’ of a beam. 😎

Thanks, Mark.

I’ve been working on the modeling of a litho projection lens, and can’t decide between Mueller or Jones matrix approach. any advice on this kind lens modeling in terms of polarization aberration control? I was told that polarimeter or polarization measurement should be took in account.

Userlevel 6
+3

Use Jones if you’re looking at polarization aberrations. These are ray-based calculations.

Thanks, Mark.