Mie/Rayleigh Phase Function Differences

[Steleo]

Good Day,

Understanding that Rayleigh scattering is a limiting case of Mie scattering why physically do we see such a change in phase function (i.e. what's happening in between)? I am thinking that we are seeing more destructive interference in the side/back directions and more constructive in the forward direction, but it's not completely clear to me why physically this is happening.

Thanks

Max

 

[Andy Resnick]

Can you be a little more specific about 'phase function'? Mie scattering has a lot of interesting features, including infinities (caustics/rainbows) and scattering efficiencies > 1.

 

[Steleo]

Andy,

I guess I'm more interested in the difference in the gross features between the Rayleigh and Mie "scattering patterns", especially in the regime of atmospheric scattering. I guess what it boils down to is why should increasing particle size as in Mie scattering give the increase in forward scattering and decrease in other directions as compared to Rayleigh.

Cheers

Max

 

[DrDu]

If the scattering particles are smaller than the wavelength of the light, it makes sense to expand the electric fields of the incident and scattered light into a multipole series. The electric field at the surface of the particle is of the order of r^(-l) where r is the radius of the particle and l is the angular momentum of the spherical harmonic involved. For very small particles only the s-type harmonic with l=0 is relevant (dipole approximation) which leads to isotropic scattering (Rayleigh) for larger diameters, more and more harmonics have to be taken into account and the scattering becomes more anisotropic till the classical regime is reached.

 

[Steleo]

Thanks DrDru I appreciate your response.

I guess part of my question is why are those other harmonics important as the particle size increases. Is there more 'room' for the higher order moments to develop within the sphere?

I was really just after some sort of "physical intuition" as to why the scattering pattern should change with particle size in the way it does but I am suspecting its just the complex interaction between the higher order moments that are induced that gives the resulting patterns?

Regards,

Max

 

[DrDu]

I remember the original article by Mie to be very informative (however it is in German).
Alternatively I think most intuition can be gained from an explicity calculation, e.g. for spherical metallic particles, which is the easiest situation.

 

[Andy Resnick]

Andy,

I guess I'm more interested in the difference in the gross features between the Rayleigh and Mie "scattering patterns", especially in the regime of atmospheric scattering. I guess what it boils down to is why should increasing particle size as in Mie scattering give the increase in forward scattering and decrease in other directions as compared to Rayleigh.

Cheers

Max

Rayleigh (or Rayleigh-Gans) scattering is a limit of Mie scattering: when the product of the wavenumber and particle size is much less than 1 (ka<<1). For atmospheric scattering (as opposed to particulate scattering), this is a good approximation.

If you start with the multipole expansion of scattering and take the limit ka -> 0, all the Bessel and Neuman functions reduce to simple expressions, and the multipole expansion is dominated by the dipole term.

Heuristically, the scattering particle sees a constant E and B at any instant of time, so it acquires a simple polarization state which oscillates in time, producing dipole radiation. Becasue the induces polarization is parallel to E, there is no scattering in the direction of the incident E.

Mie scattering is an exact solution to the scattering of a plane wave by a spherical particle, and so contains many features which are 'smoothened' by the Rayleigh approximation- interference between the transmitted and specularly reflected light, rainbows, Glory scattering, morphology-dependent resonances, internal reflections, polarization effects, etc.