Transmission by pure scatterer layer

[snorkack]

Beer´s law does not apply if there is any scattering.

In a pure absorber, if 1 layer absorbs 50 % of incident light and scatters 0 % then 2 consecutive layers transmit 25 % of light and absorb 75 %. And so on exponentially. 9 layers of pure absorbers would transmit 0,195% and absorb 99,8 %.

But say that there is a pure scatterer layer which reflects 50 % of incident light back to the direction it came from.

Then, assuming that 2 such layers consecutively are followed by darkness where transmitted light escapes or is absorbed, that double layer would transmit 33,3 % of incident light. 25% would be transmitted directly - but 6,25 % would be transmitted after double scattering (50 % reaches second layer, 25 % is reflected back to first layer, 12,5 % is reflected back forward to second layer, so 6,25% passes through second layer on second try), 1,5625% would be transmitted after quadruple scattering, etc. totalling 33,33%. And 66,666% would be reflected.

Likewise, from 9 layers of pure scatterer, 0,195% would pass straight through - but 9,8% would be transmitted after various numbers of scatterings.

At the limit of infinite optical depth, there would be no transmission - but unlike the case with absorber, in a scatterer the transmission at large optical depth undergoes harmonic decrease, not exponential decrease.

But my mathematic above was simplified. I counted discrete scatterers, and only transmission or reflection straight back.

Now imagine real scattering, in absence of any absorption. Like Rayleigh scattering, which can happen to any direction - light would be free to be scattered sideways till it is again Rayleigh scattered down again or back up.

Does the qualitative conclusion hold - that in absence of absorption, scattering causes only harmonic decrease of transmission and, in the limit of infinite depth, complete reflection?

 

[Alkim]

Hi,
You need to define better your system before going to the appropriate model. First of all, the thickness of the layers matters. If it is comparable to the wavelength of the light traveling in them you would have a sort of 1D photonic crystal. If it is much larger, then it is just reflection. If the layers are much thinner you have a metamaterial, i .e. a material which is homogeneous to light, but having a graded refractive index. Then it is important to know the optical properties of each layer, particularly their complex refractive index. If what you call "the scattering layer" contains particles smaller than the wavelength of your radiation in this medium, then an effective refractive index can be calculated. If the particles are larger, you will have an intermediate layer which is actually a 3D photonic crystal, and this would greatly complicate the problem since in this case your entire system becomes a 3D photonic crystal. Any configuration different from a simple stack of homogeneous dielectric layers is quite difficult to treat analytically and then numerical simulations become very useful.

 

[snorkack]

Hi,
You need to define better your system before going to the appropriate model. First of all, the thickness of the layers matters. If it is comparable to the wavelength of the light traveling in them you would have a sort of 1D photonic crystal. If it is much larger, then it is just reflection. If the layers are much thinner you have a metamaterial, i .e. a material which is homogeneous to light, but having a graded refractive index.

No defined layers - the layers would be arbitrary divisions of single thick layer.

Then it is important to know the optical properties of each layer, particularly their complex refractive index. If what you call "the scattering layer" contains particles smaller than the wavelength of your radiation in this medium, then an effective refractive index can be calculated.

Particles far smaller than wavelength - Rayleigh scattering limit. And no absorption.

 

[mfb]

The scattering direction does not matter - components perpendicular to the incoming light do not change the distribution of light intensity in the relevant dimension (assuming your material is homogeneous in perpendicular to the incoming light). Light would perform a random walk in your material.
I would expect that the fraction of transmitted light is given by $\frac{1}{1+d/d_0}$ where d is the thickness of your material and d₀ is some material constant.

 

[Alkim]

No defined layers - the layers would be arbitrary divisions of single thick layer.


Particles far smaller than wavelength - Rayleigh scattering limit. And no absorption.

In such case, the layer containing the particles can be considered to be homogeneous and one can calculate an effective refractive index using some of the numerous effective medium theories. You still need to know the particle concentration (or filing factor) and size, refractive index of the particles and refractive index of the suspending medium. Also you need to know the thickness of each layer, since this is crucial for the properties of the stack. You might want to read about dielectric mirrors.