I went through the book "Surface Plasmons on Smooth and rough surface and on Gratings by Raether." If you look at the equations 2.4, 2.5, and 2.6 in the link below:
https://ui.adsabs.harvard.edu/link_gateway/1988STMP..111.....R/doi:10.1007/BFb0048317
Going through the 2nd page, we have the resonance condition of the surface plasmons:
kx=ωc(ϵ1ϵ2/ϵ1+ϵ2)^1/2
Here,
ϵ1,ϵ2ϵ1,ϵ2
are the dielectric permittivity of the metal and dielectric medium.
Where the dielectric medium of the metal is given by :
ϵ1=ϵ1′+iϵ1′′
and the dielectric medium permittivity is given as:
ϵ2
Are we considering the medium to be lossless as an assumption
Now, if we have a real
ω and ϵ2
we consider that the wave vector to have a real and imaginary part where:
kx=kx′ + ikx"
Real part of the wave vector is given by following expression:
kx′=ωc(ϵ1′ϵ2/ϵ1′+ϵ2)^1/2
and the imaginary part for the wave vector is given by the following expression:
kx"=ωc(ϵ1′ϵ2/ϵ1′+ϵ2)^3/2 (ϵ1"/2(ϵ1′2))
I would like to know how the mathematical expression values of
kx′ and kx′′
are obtained.
Can anyone please helo and suggest how these expressions are obtained.
I can't get the paper from that page, the links there give 404 page not found.
This is because researchgate has only recognised half the link (and underlined it). When I copied the whole link it worked but went to a page where I would have to pay to see the paper.
kx=ωc(ϵ1ϵ2/ϵ1+ϵ2)^1/2 is unlikely to be correct because it is the same as kx=ωc(2ϵ2)^1/2, unless ϵ1ϵ2 is not the same as ϵ1 x ϵ2
k is usually proportional to ω, the proportionality is usually 1/v, where v is some kind of velocity. Often v=c/(ϵr)^1/2. Without more idea of the original I can't do more.
Should ωc be ω/c?
I suggest you read https://en.wikipedia.org/wiki/Surface_plasmon_polariton, where the equations look like I would expect them to.
hello!
I apologize for this : it is a book surface plasmons on smooth, rough surfaces by Raether
yes, it is w/c
Malcolm White : Hi, I did check, it is just the same thing, but I don't know if the real and the imaginary parts are obtained mathematically or assumed with the respective equations.
I was trying to substitute the dielectric permittivity to obtain the wave number equation, also I was trying to figure if there was a specific reason to assume |ε1′| >> ε1″, or it is just generally considered.
Thank you for your time and consideration
regards
Ketan
It just generally is. If you look up wikipedia you will see the equations and some values for |ε1′| and ε1″ Actually >> is not always true, so the approximate formulae are not very accurate in those cases.
You should be able to work them out for yourself. It is very simple matematics. Look at the formulae in the wikipedia article.
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