its funny,you got problems with Si over AL and my approximation looks ok,and now your Ge over Au approximation results looking close but my Ge over Au result is looking strange.

thanks for posting your results and links.

about your -2 in the equation.

i have looked in the Si over Al paper and found the reference paper, that is listed for the equation.

here it is

https://ir.nctu.edu.tw/bitstream/115...CK60100002.pdf

there is all about the k coeffiecent and how he used it .

since my "code" based on the dielectric fresnel equation from the developer Bacterius,i just added the k value,like in the formula i posted before.but i guess this isnt sufficent enough.maybe for some materials yes,but not for all.i have looked into the optics book from born and wolf,that is used as reference in many papers.and in this metal chapter are different scenarios explaned,ie thin metal over transparent substrate and a few more.and for every case it was needed to fitting the equation to it.

my error with the Ge over Au seems to be the energy conservation at the end ,where all the equations come together.with out the k value in the energy con, it is not possible to become the colorpattern result, like in the Si over Al paper/posting.but i get strange results(a blue color with narrow fresnel circles,like if it was used a fresnel smaller than 1) ,with the Ge/Au render with the k.if i take the k out,i see the gold color but no color patterns at the different thickness values.and i am in stuck at this point.because it works and works not,i dont no how to fix it.

here from a similar Ge over Au paper.from top to buttom no Ge,7,10,15,100nm over Au.

edit,in the paper that i have posted the link above,i think have found the answer for your -2.

look at the page 203 the equation A3 looks like yours.and at A4 the part of the phaseshift term is there.notice that the phase shift term uses only one way of the refraction in the layer.but for reflection you need it multiplyed by 2 ,because the refraced light into the film and back .i guess this is the reason for the -2.(the phase term for refraction and reflection(x-2)).the term shows only one thickness distance calculated.

the phase term calculate the phase from A to B .for reflection you simply multiply it by 2 to get the phase from B to C as well(the whole phase in the film from A to B to C).

edit,wait i have looked at your equation again,the exp looks a little bit different,so i am maybe wrong.i dont know your full code,so i guess the d in the exp is for distance,so it means twice the thickness too.

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