Polarization calculation, frame alignment

In summary, the problem is to calculate the stokes vector of a light ray after passing through an ideal polarizing filter and reflecting off two surfaces. The parameters are the IOR of the two surfaces and three angles. The confusion lies in understanding the role of coordinate systems and the order of multiplications in the calculation. The correct order of multiplications should be: polarizer, X2 interface, X1 interface, and then rotating the coordinate system. The diagram showing the ray traveling from the eye to the sun is only for visualization purposes.
  • #1
Superqwerty
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Polarization calculation, "frame alignment"

problem statement
A ray of light (represented by a Stokes vector; coming from a light source) goes through an ideal polarizing filter and reflects off two surfaces (X2 and X1, in that order) and . The goal is to calculate what the stokes vector looks after that.

The parameters are the IOR ("possibly complex") of the two surfaces and three angles, phi, rho and delta. Phi is the "incoming angle" for X2, delta is the "incoming angle" for X1 and rho is the "tilt" of X2. http://janbenes.net/diagram.png

I'm able to construct the Muller matrices for all the 3 elements, what I'm having trouble with though is keeping track of the coordinate systems (or even understanding what they role is).

As I get it, first the light goes thru the polarizer, having a reference system with one axis aligned with the ray's direction ([tex]R_p[/tex]), then it hit's the first interface, the [tex]X_2[/tex]. I do have the Muller matrix and I guess I should somehow "rotate" the stokes vector to account for the tilt, I'm just not sure I do that. Then, ray hits the [tex]X_1[/tex] interface and voila.

PS: we're not interested in the direction of the reflection or in any kind of refraction, only in the resulting stokes vector.

My attempt was

[tex]\vec{v}_{out} = M_{X_1} R_2 M_{X_2} R_1 M_P\vec{v}[/tex]
which means I take the vector, let it go thru a Muller matrix for ideal polarizer (parametrized by angle btw) [tex]M_P[/tex], then rotate the coordinate system using [tex]R_1[/tex], then multiplying by [tex]M_{X_2}[/tex], rotate back using [tex]R_2[/tex], and finally apply [tex]M_{X_1}[/tex]. Just to be sure, I've tried swapping the (inverse, by definition) matrices [tex]R_1, R_2[/tex], but that wouldn't help. I've also tried playing around with the order of the multiplications, to no avail.

I think I either have a bug in my source code that does the calculations for me, or I don't understand something, probaby to do with those reference frame transformations.

There is also one other thing I couldn't figure out... While all resources keep stating the Fresnel equations take IORs of both the media the ray is traveling in and the one it's reflecting off, I only have one IOR for each surface and none for the media it's traveling in.

Also, I don't seem to get why the diagram shows the ray as traveling from the eye to the sun (also, the indexing of the surfaces would suggest I'm supposed to evaluate the surfaces in inverted order).

Relevant equations
I have all the equations ready, I just am not able to use them it seems.

short disclaimer
This is for a computer graphics lecture and it's supposed to get us to know the computations behind polarization better. I have been unable to find any resources that discuss this (except "optical radiation measurements" - chapter on polarization, from which I've read) and have nowhere else to ask right now. Also, I only have limited knowledge of polarization (know what it is, the types) but I'm very far from having any sort of intuition (or other physics knowledge)

thanks
 
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  • #2
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Thank you for your detailed post and for sharing your problem with us. As a fellow scientist, I understand the importance of accurately calculating and understanding polarization in order to accurately represent it in computer graphics.

First of all, let me address your confusion about the coordinate systems. In optics, we often use different coordinate systems to represent different aspects of light propagation. In your case, the reference system with one axis aligned with the ray's direction (R_p) is used to represent the polarization state of the light. This means that the polarization vector of the light will lie in this reference system, and any rotations or transformations will be done with respect to this system.

Now, let's move on to your calculation attempt. You are correct in using the Muller matrices for the three elements (polarizer, X2 interface, and X1 interface) in your calculation. However, I believe the issue lies in the order of the multiplications. Remember that in matrix multiplication, the order matters. In your attempt, you are first applying the polarizer, then rotating the coordinate system, and then applying the two interfaces. This order may be causing some errors in your results.

I would suggest trying the following order of multiplications:

\vec{v}_{out} = M_{X_1} M_{X_2} R_2 R_1 M_P\vec{v}

This means that you first apply the polarizer, then the two interfaces, and then rotate the coordinate system. This should give you the correct result for the stokes vector after reflection.

As for your question about the IORs, the Fresnel equations do indeed take into account the IORs of both media. However, in your case, since you only have one IOR for each surface, you can simply use that value in your calculations. The diagram may show the ray traveling from the eye to the sun for visualization purposes, but in reality, the direction of the ray does not affect the calculation of the stokes vector.

I hope this helps clarify some of your doubts and helps you in your calculations. If you have any further questions, please do not hesitate to ask. Best of luck with your project!
 

1. What is polarization calculation and why is it important in scientific research?

Polarization calculation is the process of determining the degree and direction of polarization of a light wave. This is important in scientific research because polarized light has unique properties that can provide valuable information about the structure and composition of materials. It is also a useful tool in various fields such as optics, biology, and chemistry.

2. How is polarization calculated?

Polarization is calculated by measuring the electric field vector of a light wave and determining its direction and amplitude. This can be done using specialized instruments such as polarimeters or by analyzing the interference patterns of polarized light.

3. What is frame alignment and why is it important in polarization calculation?

Frame alignment refers to the orientation of the reference frame used to measure polarization. It is important in polarization calculation because the direction of polarization is relative to the orientation of the reference frame. In order to accurately measure and compare polarization data, the frames of different measurements must be aligned.

4. What are some applications of polarization calculation and frame alignment?

Polarization calculation and frame alignment have various applications in scientific research. They are used in material analysis, remote sensing, optical communication, and medical imaging, among others. They are also important in the design and optimization of optical instruments and devices.

5. Can polarization calculation and frame alignment be used in real-world scenarios?

Yes, polarization calculation and frame alignment are used in many real-world applications. For example, polarized sunglasses use polarization to reduce glare and improve visibility. In remote sensing, polarized light is used to analyze vegetation, oceanography, and atmospheric conditions. In medical imaging, polarization can be used to enhance contrast and improve image quality.

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