Far-field intensity from scattering of small particles

Howdy, I'm building a simulation for looking at the light field underwater. In order to verify my simulation, I'm looking for some data showing the far-field intensity that comes from single scattering from many small particles in suspension. I suspect Mie theory plays a part here, but I'm having a hard time finding some results, rather than doing all the derivations myself.

In other words, I want to know the power distribution on a plane after a beam of light has been scattered by a bunch of small particles through a volume. I know Oregon Medical has a nice online simulation that produces scattering phase functions (http://omlc.ogi.edu/calc/mie_calc.html), but that doesn't give me the power on a plane - only the scattering profile from individual particles. I'm fine with only a single scattering result.

I want to do initial verification using a fixed particle size. Having a hard time finding a reference with this data. Help?

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The main problem about a rigorous solution to such a scattering proplem is that computations are extremely demanding. Just imagine you have a wavelength $\lambda$ of some $400$nm to $700$nm for visible light (from here):

Now, to do physically meaningful simulations, you will need a sub-wavelength lattice which makes any computational cell above, say $10\,\mu m^3$ not accessible since you have in the order of one million grid points.

Approximative Approaches

But of course there can be ways out of it if you are willing to make some approximations which will largely depend on the characteristics of the particles you are looking at. It is best to assume that we only have spherical particles since we can apply Mie theory in this case.

Large Particles

First of all, let us consider particles which are much larger than the wavelength. Then, the radius $R$ times the wave vector $k=2\pi/\lambda$ is much bigger than one, $$kR\gg1$$ which basically means that one observes reflection at a plane interface. You can implement these particles using geometrical optics (mixed with Fresnel reflection if you like) since nothing really wave-like will happen as in this image (taken from here):

Small Particles

Second, the particles should be much smaller than the wavelength, $$kR\ll1\,.$$ Then, everything what is observed is a sum of dipolar responses of the particles in the so-called Rayleigh-scattering. Then,

the intensity of light scattered by a single small particle from a beam of unpolarized light of wavelength $\lambda$ and intensity $I_0$ is given by:

$$I=I_0(1+\cos^2\theta)\frac{(kR)^6}{2(kr)^2}\left(\frac{n_p^2-1}{n_p^2+2}\right)$$

where I have chosen the variables to be consistent with the used terminology and $r$ is the distance to the object, $\theta$ is the scattering angle and $n_p$ is the sphere's refractive index. Here is an image of such a situation with some metal particles also having quadrupolar excitation (from here):

A Mean Field Approach - Effective Permittivity

If you have a lot of these small objects, you may use the Clausius-Mossotti relation which gives you an effective permittivity $\epsilon_p=n_p^2$ depending on the concentration of the particle in some volume: $$\epsilon_{eff} = \epsilon_p + \frac{n\alpha}{1-\frac{n\alpha}{3\epsilon_p}}$$ where $\alpha$ is the polarizability of the sphere, for details see e.g. Electromagnetic mixing formulas and applications by Sihvola. This would be something like a mean-field approach. You can make some very neat effects using this effective approach since it allows you to calculate a continuous refraction around some particle streams under water.

However, if the particles size is in the order of the wavelength, $$kR\approx 1$$ then you may have to take higher multipole moments into account which may be a very demanding task.

For much more on the subject I would recommend Bohren & Huffmanns classic Absorption and Scattering of Light by Small Particles.

Sincerely

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Very nice answer. –  Colin K Aug 8 '11 at 21:04
Thank you @Colin :) –  Robert Filter Aug 9 '11 at 9:14
Excellent answer, @Robert. I posted an answer showing the experimental data I found and was ultimately able to use, but I'm choosing your response as the best answer. –  gallamine Aug 10 '11 at 13:51
@gallamine: Thank you very much. You may also cross-check the information in the thesis to those given in this answer. There shouldn't be too much contradiction :) Greets –  Robert Filter Aug 11 '11 at 8:50

Well, the Oregon Medical site does give you (almost) the power on a plane. There's a linear plot of Magnitude vs. angle that you can convert to a plane using x=arctan(angle). On the other hand, calculating Mie scattering is rather simple. Just check Boren & Huffman "Absorption and Scattering of Light by Small Particles", where they give the explicit formulas and several approximations.

Anyway, I was wondering if what you're trying to do is right. Are you trying to calculate the transmission through clear, calm water? Because in that case, you are using an incorrect approach. Mie scattering is the electromagnetic solution for a single spherical (or elliptical) particle. It works ok if you have many particles, but not too many. If multiple scattering (waves that scatter on more than one particle) becomes important, it's not longer valid. I'm not sure what is the right approach for liquids or solids, but calculating individual particles isn't. Possibly mean-field theory or effective medium, depending on what are you exactly after.

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I'll check out the book. For my simulation, I'm computing the power distribution on a plane from a laser propagating through scattering medium. Looking at the Oregon Medical site, it's unclear what the input light field looks like for their computations. Do you happen to know? I'm using Mie scattering to verify my Monte Carlo-type model. The actual simulation will be for underwater light scattering. I was trying to choose something easy to check against first. –  gallamine May 24 '11 at 18:48

I was able to find experimental and simulated data for the plane intensity from multiple scattering of small (1, 5 and 10 $\mu$m spheres in the Thesis of Edouard Berrocal. His thesis can be downloaded here.

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Seems an interesting work, especially the introduction of some dynamics into the system by an assumed random walk which lead to the Monte-Carlo approach. –  Robert Filter Aug 11 '11 at 8:55