When trying to find the dispersion relationship for plasmon supported by a noble metal thin film surrounded by an asymmetric dielectric environment, it is customary to postulate no incident field. Then, normally solving for a TM mode where the z direction is normal to the thin film, one writes down the appropriate fields with a purely imaginary Kz (field exponential decay away from the interfaces) and complex Kx. Finally applying the appropriate boundary conditions the dispersion relation for Kx and w (omega) is found, and Kx is usually complex, in particular for real metals.
In the same problem is solved with a transfer matrix approach, and the thin film is illuminated by a plane wave, one is able to launch a plasmon when the plane wave has appropriate frequency and momentum, i.e. Kx_pw=Re[Kx] and w_pw=w. In this approach Kx_pw is bound to be purely real; the consequences are twofold: (i) Kz_pw will be different from Kz since the characteristic equation to compute Kz is the same in both cases and Kx_pw is different from Kx (one real the other complex) (ii) the information about the plasmon propagation length (imaginary part of Kx) is completely lost.
The question is: how is it possible to reconcile these two different views of the same problem? Can you point me to some paper or book that tackle this problem?
Yes i did. I also tried Principle of Nano Optics (Novotny and Hecht) and Plasmonics Fundamentals and Applications (Maier) as well as many papers. Nevertheless, in each of them plasmon modes are calculated as solution of the homogenous equation (eigenmodes, Kx complex). When speaking about launching the plasmons by means of a plane wave, they simply suppose to be in a material without losses and that is it. Thanks for your suggestion anyway.
Giovanni, wouldn't it be so that, in the case of the SP being excited by an incident plane wave, the difference in the Kz_pw (from that obtained in the first approach) is in the additional real part (compared with a purely imaginary Kz_pw in the first approach), which physically comes from a transmitted or reflected plane wave with Kx = Kx_pw=Re[Kx_incident_wave]? Indeed, an incident plane wave should result in three waves: reflected with real Kz, transmitted with real Kz (unless the condition of total internal reflection is met), and the surface plasmon propagating along the surface, with imaginary Kz.
...so if you are solving for a TM wave in this case, it will be a superposition of all the three waves, i.e., it would have a complex Kz, with both real and imaginary parts.
Yet in the first approach, you are just asking a question "is there a natural mode of the system that propagates along the film?", not considering how it is excited. Hence in this approach you only get ONE surface mode (instead of a superposition of a surface and a volume modes) with purely imaginary Kz.
And on the question of why in case of the SP's excitation by an incident plane wave the information about the plasmon propagation length is lost: there is no plasmon propagation length in this setting, as the plasmon is excited uniformly across the entire infinite surface, by definition of the plane wave. However, if you consider an incident narrow RAY (i.e., a laser beam) instead of a planar wave, you would get a locally excited surface plasmon, which WILL have a finite propagation length.
It is quite impossible to excite a plasmon by a plane wave if we have translation invariance along the interface. As it is in the case of metal film or metal semi-space. That occurs because the plasmon has wave number (kx), which is greater than that of a plane wave (kw) in the surrounding space. That is why the plasmon is localized at the interface. Indeed, in the surrounding space on has (kx)2+ (kz) 2= (kw) 2. It follows from this relation that if there are losses in the metal both kx and kz of the plasmon are complex.
In the Raether-Kretschmann experiments the kx is fixed by the incident, say top-down, wave. The ATR condition are as follows: (kx)< (kw) in the upper semi-space but (kx)> (kw) in the bottom semi-space. This Raether-Kretschmann solution differs from a plasmon on the metal film. The real plasmon is evanescent on both sides of the film, whereas in the upper semi-space the Raether-Kretschmann solution is a sum of incident and reflected plane waves. In bottom semi-space kx should be greater than the local (kw) and due to TR (kz) is purely imaginary quantity. The Poynting vector is parallel to the interface. Going out of the illuminated region the solution will attenuate due to radiation of a leaky wave into the upper semi-space. Going out of the prism the solution will transform in the real plasmon with complex (kx).
Nevertheless, the reflection coefficient has a minimum at some (kx), that is of the order of (kx) of plasmon. You are quite right the first quantity is real and the second is complex. But, these are quite different, independent solutions.
If you use the complex plane, you can find the theoretical link between the dispersion relation of the surface plasmon (well of the whole structure) and the reflection coefficient. In the case of a left-handed slab (something which probably won't ever exist, well not with losses that would be low enough to actually see all the phenomena I have in mind), the physics is even more complicated. I guess you will find some answers here:
https://www.jeos.org/index.php/jeos_rp/article/view/08032
A paper that is anyway completely ignored. Don't bother citing it ;)
I really hope this can help.
There is good analisys of the Otto and the Kretschmann configuration. See attachment.
Best regards, Evgeny Vinogradov
There is good analisys of the Otto and the Kretschmann configuration. See attachment.
Best regards, Evgeny Vinogradov
Thanks everybody for the answers! I have to chew on them for a little bit! The otto-barchiesi paper seems especially useful! If any of you has more insights, please do not esitate to share them with me. Thanks a lot!
I have simulated a plamsonic structure in COMSOL.
COMSOL calculates the absolute value of propagation constant, suppose K=kx+ky,
Comsol calculate the emw.k=sqrt(kx^2+ky^2).
as you know what is important for calculating the dispersion curve of plasmonic wave is just kx or ky depending on the direction of propagation. I am now wondering how to extract kx or ky form K.
can any one help me or give any suggestion.
I am thinking of poynting vector as it is parallel to direction of propagation to find the angle of K with respect to interface, but the desired result is not achieved yet.
does any one have any suggestion?
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