Simple math model for a Particle Image Velocimetry system

[TT_003]

TL;DR

I am developing a simple mathematical model for a Particle Image Velocimetry (PIV) system. My goal is to calculate the resulting radiance L_r emitted by a particle illuminated by a 532 nm laser.

Hello togehter,

I am new to this forum and hope this post followed all the guidelines here (I tried to summarized my issue as clean as possible, two pictures are attached). I would appreciate every help:

I am doing research on a Particle Image Velocimetry (PIV) system. For this I want to set a simple math model for the system. I hope you can help me out. Regarding this I have 2 main Questions.

1. I am trying to find a math model which is describing what is happening in a simple Particle Image Velocimetry System. So I have a laser with a wavelength of 532nm (the laser has a power of P_L and the beam radius is r_b), which is for example hitting a particle in air with a radius of r_P = 50um (the whole sphere is completely in the beam). What is the resulting radiance L_r [in W·sr⁻¹·m⁻²] emitted by the illuminated particle (the radiance L_r I want to use later to calculate the amount of light that hits my camera system)?
The camera is in a distance d_c to the particle and is in a position where the laser light can not reach the camera directly (only the light which comes from the illuminated particle).

So my thoughts on the steps are (the equations are in attachement 1, and description of all variables are in attachement 2):
A. [Equation 1.1] Calculate irradiance I_L of my laser -> used simple top hat model for the laser
B. [Equation 1.2] Calculate power P_P which hits the particle (modeling as a sphere) -> using cross section of the sphere as the area
C. [Equation 1.3] Calculate the power P_r which leaves the particle -> Alpha is the Albedo of the particle
D. [Equation 1.4 - 1.6] Calculate radiance coming from the particle (assuming that the light is going in all direction equally [isotrop]) -> using the sphere surface as the area (A_r). W is the solid angle (using the distance d_c to the camera as r and the "big" sphere surface area in the distance d_c as A)
E. [Equation 1.7] Result is in equation 1.7

My biggest uncertainty is in step C and D -> Did I have the correct thought here?

2. So the effect on the sphere: is it scattering or body reflection? Or are both the same (I am a little confused about that)? Can I use the Albedo alpha here?

Thanks in advance for the help! I would appreciate every help.

P.S. if the calculation steps A-E are not correct or are confusing you can also ignore it and suggest a new mathematical way for the 1st question. Thank you so much!

 

[TT_003]

I thought that, to get a better overview of the scenario, I add another image showing the setup and the problem statement. I hope the scenario is clearer that way. I would appreciate every help. Thank you in advance!

 

[boneh3ad]

I suspect the concept you are seeking is Mie scattering.

 

[berkeman]

I suspect the concept you are seeking is Mie scattering.

Even though his particle size is about 100x the laser wavelength?

https://en.wikipedia.org/wiki/Mie_scattering

 

[boneh3ad]

Even though his particle size is about 100x the laser wavelength?

https://en.wikipedia.org/wiki/Mie_scattering

Mie scattering applies to spherical particles with a size on the order of the wavelength of the incident light or larger. If they get sufficiently large, it turns into just a typical reflection rather than a true scattering event. I would think 50 microns still falls within the Mie scattering model.

 

[TT_003]

Thank you for the replies. So the particle size is 50microns or greater. The value is an example, but to keep the math model as simple as possible I wanted to consider the typical geometric case (so that the particle emits the light isotropically).

Any idea whether my calculations are correct, especially steps C and D? Thank you again for any help.

 

[tech99]

The radar cross section of a sphere is simply pi r^2, its cross sectional area. The energy hitting this area is then reflected isotropically. Mie scattering will occur when the sphere is around its resonant frequency, which does not seem applicable in this case.