Deriving Photon Rotation Formula for Monte Carlo Simulation - Step-by-Step Guide

In summary, the conversation discusses the simulation of photons in tissues using a Monte Carlo simulation. The focus is on the computation of the new photon direction after scattering in an isotropic or non-isotropic medium. The formula used for this calculation is derived using a coordinate system and is explained in detail in a textbook or review article on x-ray scattering.
  • #1
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Hello and apologies if the title of the question is not very precise.

Question: I am reading the document talking about the simulation of photons in tissues using a Monte Carlo simulation. The exact title is "MCNP - A general Monte Carlo N-Partcle Transport Code".

Link to MCNP - A general Monte Carlo N-Partcle Transport Code

When photons are scattered, a new direction for the photon is sampled. When the medium is isotropic the direction is random so the math for this case are not hard, but when the medium is not isotropic, the angle of deflection needs to be computed from a function such as the Heney-Greenstein function.

Anyway to be short, this function returns a cos(θ) which is the angle of deflection between the existing photon direction and the new desired direction. In the paper, this term is called μlab. In the document I am referring to, they also compute a ϕ angle by sampling two random uniformly distributed variables, which are inscribed in the unit disk. These are called ϵ1 and ϵ2 in the paper.

So the formula they use in this paper (see reference below page 2-38) to compute the new photon direction (using the three aforementioned variables) is:

Formula for rotating photon direction

And where u0v0w0 are the coordinates of the "old photon direction" and uvw the new photon direction after scattering. My problem, is that I have no idea how they derived this formula.

So the way I understand how this can be done is by "reconstructing" a coordinate system in which the old direction is the z unit vector, and express the coordinates of the new direction within this frame? Is that correct. But could someone put me on the right track so that I can derive this formula (understand how they get there)? I don't need a full answer, I am happy to make an effort, I just need someone to put me not the right track (and I will publish the answer when I have one).
 
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  • #2
The classical situation is described for x-ray scattering; a review of that method may be useful.

Something like: http://www.helsinki.fi/~serimaa/xray-luento/xray-scattering.html
though a full textbook derivation would be better.
 

What is the purpose of deriving the photon rotation formula for Monte Carlo simulation?

The purpose of deriving the photon rotation formula is to accurately simulate the behavior of photons in a Monte Carlo simulation. This formula allows for more realistic and accurate results in simulations involving the scattering and absorption of light.

What is the process for deriving the photon rotation formula?

The process involves using mathematical equations and principles, such as Snell's law and the law of reflection, to model the behavior of photons as they interact with different materials. This involves breaking down the steps of photon rotation into smaller mathematical components and combining them to create the overall formula.

What are the key variables and assumptions used in the photon rotation formula?

The key variables include the angle of incidence, the refractive index of the materials, and the angle of rotation. The assumptions made include the assumption of a single photon interacting with a single material at a time, and the assumption of a perfectly smooth surface for reflection.

How does the photon rotation formula contribute to the accuracy of Monte Carlo simulations?

The photon rotation formula takes into account the different factors that affect the rotation of photons, such as the refractive index and angle of incidence, which allows for more accurate simulations of light scattering and absorption. This leads to more realistic and precise results in the simulation.

Are there any limitations to the photon rotation formula in Monte Carlo simulations?

Like any mathematical model, the photon rotation formula has its limitations. It may not accurately capture the behavior of photons in all scenarios, such as in highly complex or dynamic environments. Additionally, it may not take into account all possible variables, leading to some degree of error in the simulation results.

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