Exact solutions for reflectance, transmittance, absorption?

[M_1]

I'm measuring transmittance through and reflectance from thick glass samples, using spectrophotometers. Does anyone know if there's an exact solution for the surface transmittance r, surface reflectance t, and absorption coefficient A (as in Beer-Lamberts law). For a specific wavelength the equation for total reflected light from the sample, R, is

R=r(1+t²e^-2Ad)/(1-r²e^-2Ad)

and the equation for the total transmitted light through the sample, T, is

T=t²e^-Ad/(1-r²e^-2Ad)

R and T are measured, thus known variables. So we have the unknown material properties r, t, and A, and two equations. I can measure a sample of the same material with another thickness d and then I have four equations and three unknowns. This should theoretically be enough but the algebraic becomes very complicated and I wonder if there actually is an analytic solution.

Of course I can approximate 1>>r^re^-2Ad in the second equation and obtain A by dividing the second equation for one thickness with the same equation for another thickness, but then it is no longer the exact solution.

Any help is most appreciated, as always. Thanks!

 

[blue_leaf77]

Does anyone know if there's an exact solution for the surface transmittance r, surface reflectance t, and absorption coefficient A

For surface reflectance and transmittance, you can use Fresnel equations. For the absorption, I think it will be easier to just look up at the table of the corresponding material.

 

[M_1]

Thanks! But I don't know the materials properties. Therefore I don't know the complex refraction index and cannot use the Fresnel equations.

 

[blue_leaf77]

For Fresnel equations, you only need to know the used wavelength, the refractive index corresponding to this wavelength, and the angle of incidence (which I think should be normal given the form of the total reflectance and transmittance). Try typing the name of that glass in your browser to see if there is refractive index table already available for that material.

 

[M_1]

Ok I see what you mean. But I also sometimes measure on semitransparent polymers which I have now knowledge about, and sometimes I don't know which glass I have either! What I'm looking for is an exact method to use experimental measurements of spectral reflectance and transmittance in order to calculate the material properties r, t, and A.

 

[DrDu]

Does not seem too complicated. Let R, T be the reflectance and transmission for thickness d, and R', T' the corresponding ones at thickness d'.
Then form RT'/(R'T) and solve for ##t^2##. Insert into the equation for T and solve for ##r^2##. Insert both ##t^2## and ##r^2## into the equation for R and you will get an expression which only contains a in terms of R, T, d and d'.