Angle of Incidence, Reflectance, Fresnel, Snells Law

[salm]

Hi,

I would like to know if it is possible to draw a diagram in which the angle of incidence is plotted against the Reflectance for specific Index of Refractions (IOR).

The following page probably explains better what I mean, than I could ever do in words.

http://refractiveindex.info/

On this website you can chose a certain material, for example Aluminium. Then you enter a wavelength. The website then spits out a whole bunch of numbers and curves. I am interested in the last one which plots Angle of incidence against Reflectance. And here only in the non-polarized curve.

The problem is, that the amount of materials on the website is very limited. On the other hand there are plenty of websites that show the IOR for all kinds of materials. For example:
http://vray.info/topics/t0077.asp

From research I´ve done and talking to drunk physics phds at parties I gather that this must be possible somehow with Fresnel Equations and Snells law but It is too complicated for me to understand on my own.
So doesn anybody know if and how this works?

I´m assuming that the medium the object is in is air so there are no changes of medium or anything like that.

[Simon_Jester]

I am quite sure such a thing could be plotted.

Knowing exactly what happens to electromagnetic waves at the interface between two different media is a well understood problem in classical physics. Given the physical constants of each medium (e.g. dielectric constant), you can calculate the amplitude of the EM wave that is refracted into the material, and the amplitude of the wave that is reflected out of the material at the surface.

For someone who's up to speed on electromagnetism, working out a general equation to relate the reflectance of the surface to the angle of incidence would be a relatively simple exercise. I haven't worked with the relevant math in several years so I'd be rusty, but I'm confident I could do it if I dug through my old graduate electrodynamics textbook.

You can google 'reflectance as a function of angle' and graphs that look promising pop up, at least for certain materials. Obviously, for any specific combination of two mediums the result will be a specific graph- the exact shape of the curve depends on material properties.

[salm]

Thanks. I think I got a step further.
The following page lets me enter the Index of Refraction (IOR) of the media from which the ray is comming (~1 for air in this case) and refractive index of the medium it is entering.
http://hyperphysics.phy-astr.gsu.edu/hb ... reseq.html

As a test I used PMMA (some sort of plastic) from the http://refractiveindex.info/ website which states that the IOR is 1.4872 at a wavelength of 0.685 (which is red).

The website uses this data for calculations in the two Fresnel equations. The result has to be squared in order to get the results for the two polarized values.
The non-polarized value is simply the average of the two polarized values.

Now, this works great for plastics but not for metals. I guess this has to do with metals not being dielectric?
I assume that this is because metals have a large extinction coefficient which I´m not quite sure what it means but I assume it is some sort of value that indicates the amount of light the material "swallows".
According to Wikipedia I can use the "complex Index of Refraction" which is n = n + iK
It is stated that n is the IOR and K is the extinction coefficient but I have no clue what i is supposed to be.

[Simon_Jester]

k is a constant which is only nonzero in opaque materials. Its effect on an electromagnetic wave is to introduce a damping term. Instead of reading

"some number times the sine of how far you go into the material,"

the function will read

"some number times the sine of how far you go, times the number e to the power of -(k times how far you go)"

The effect of this is that the e^-kx term becomes very small after you penetrate far enough into the material, resulting in the electromagnetic wave being damped away to nothing.

Metals are conductors, so any electromagnetic field exerted inside them will result in the electrons inside the metal 'sloshing' around. So an EM wave oscillating inside the metal will start the electrons IN the metal moving.

This takes energy, so the energy of the electromagnetic wave is converted into energy of moving currents inside the metal. Which is precisely how an antenna works. Incoming radio waves are absorbed by a piece of metal and converted into electric current.

The consequence of this is that the electromagnetic waves are very bad at penetrating a conducting material (like metal, or salt water). Dielectric materials transmit light because all their electrons are tightly bound to some particular atom, so there is no tendency for the EM wave to set up these 'sloshing' currents that rob it of energy.

[jwl]

Thanks. I think I got a step further.
The following page lets me enter the Index of Refraction (IOR) of the media from which the ray is comming (~1 for air in this case) and refractive index of the medium it is entering.
http://hyperphysics.phy-astr.gsu.edu/hb ... reseq.html

As a test I used PMMA (some sort of plastic) from the http://refractiveindex.info/ website which states that the IOR is 1.4872 at a wavelength of 0.685 (which is red).

The website uses this data for calculations in the two Fresnel equations. The result has to be squared in order to get the results for the two polarized values.
The non-polarized value is simply the average of the two polarized values.

Now, this works great for plastics but not for metals. I guess this has to do with metals not being dielectric?
I assume that this is because metals have a large extinction coefficient which I´m not quite sure what it means but I assume it is some sort of value that indicates the amount of light the material "swallows".
According to Wikipedia I can use the "complex Index of Refraction" which is n = n + iK
It is stated that n is the IOR and K is the extinction coefficient but I have no clue what i is supposed to be.

In this context i is sqrt(-1). As the guy above said, this manifests itself as absorbence. This is because sine/cosine waves can be written in terms of e^ix, and the two imaginary terms inside the exponential multiply to make a negative number, so you get an exponentially decaying amplitude.

[jwl]

On working out the refractive incidences: for transparent materials you can use the Cauchy or Sellmeier equation to to work the refractive index. For metals, you can use the Drude model, which says that n=sqrt(1-((ρ*e^2/(ε0*m))^2/(ω^2-i*γ*ω))) (technically this should be multiplied by the square root of the relative permeability, but this is almost one in most materials). ρ is the density of the electrons, e is the electron charge, ε0 is the permittivity of free space, m is the electron mass, ω is the angular frequency of the EM radiation, i=sqrt(-1), γ is the inverse of the relaxation time of the electrons.

However, since this model neglects almost all of quantum theory and special relativity (it pre-dates the theories), it doesn't work in all materials. The citation of the website you give says it uses the Brendel–Bormann model in these cases, you could look into that.

[jwl]

Oh, I forgot another problem with the drude model: it doesn't include electron-electron interactions, it assumes the electrons in the metal act as a gas.

Also, wiki pages on cauchy, sellmeier, and drude models:
https://en.m.wikipedia.org/wiki/Cauchy%27s_equation
https://en.m.wikipedia.org/wiki/Sellmeier_equation
https://en.m.wikipedia.org/wiki/Drude_model

[salm]

Thank you everybody for the explanations. It helped a lot in understanding what is going on.
I found a thread on a physics forum in which a similar question was asked.
https://www.physicsforums.com/threads/f ... -n.512034/
The guy shows how to use the fresnel equations with the Complex IOR and states that if you assume that n2+k2≫1 you can use a much simpler derivation.
The thread then goes on to explain how to get to this derivation but I don´t understand it.
However, I tested the derivation with several materials and crossreferenced it with values on http://refractiveindex.info/ and it appears to work.

This is awesome and has resulted in me being able to create a couple of great looking materials for physically based render engines.

@jwl: I looked into the different models (Sellmeier, Cauchy, Drude) briefly and the way I understand it I´d have to get certain kinds of data by measuring it. Unfortunately the equipment, refractometers, are rather expensive.

Some sources state that there are databases out ther somewhere for Sellmeier numbers but i wasn´t able to find any.

[jwl]

Thank you everybody for the explanations. It helped a lot in understanding what is going on.
I found a thread on a physics forum in which a similar question was asked.
https://www.physicsforums.com/threads/f ... -n.512034/
The guy shows how to use the fresnel equations with the Complex IOR and states that if you assume that n2+k2≫1 you can use a much simpler derivation.
The thread then goes on to explain how to get to this derivation but I don´t understand it.
However, I tested the derivation with several materials and crossreferenced it with values on http://refractiveindex.info/ and it appears to work.

This is awesome and has resulted in me being able to create a couple of great looking materials for physically based render engines.

@jwl: I looked into the different models (Sellmeier, Cauchy, Drude) briefly and the way I understand it I´d have to get certain kinds of data by measuring it. Unfortunately the equipment, refractometers, are rather expensive.

Some sources state that there are databases out the somewhere for Sellmeier numbers but i wasn´t able to find any.

In terms of a sellmeir data table, I found a pretty large one by looking at the citation in the wikipedia page. Here: http://www.schott.com/advanced_optics/e ... ng=english#

With the drude model, the two constants specific to the material are it's scatting rate and its electron density. Which I can't find a table for scattering rate, it doesn't matter because you can work it out in terms of more easily-accessible parameters. If you divide the mass density by it's atomic mass, you get the molar density. Multiply the molar density by the valence, and you have the electron density. And as mentioned in wikipedia (but using different notation) in the drude model conductivity σ=(ρe^2)/(mγ), where ρ is the electron density, e is the electron charge, m is the electron mass, and γ is the scattering rate. So if you know the mass density, atomic mass, valance, and conductivity of a material, you can use these to work out the scattering rate and electron density. (Also, a table of electron densities might be easier to find).