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I would like to know if anyone can tell me what we meant by non-radiative Surface plasmons, when we look at the dispersion relations?

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Consider a single film, consisting of two dielectric materials of dielectric constants $\epsilon_1>\epsilon_2$ sandwiching a layer of gold, which we model using a Drude dielectric function, $$ \epsilon_{\textrm{g}}=1-\frac{\omega_{\textrm{p}}^2}{\omega^2}\,, $$ where $\omega_{\textrm{p}}$ is the plasma frequency in the gold. After solving the electromagnetic wave equation in all three layers, applying the source-free condition that the divergence of the displacement field is zero, and applying electromagnetic boundary conditions at the interfaces, we arrive at an expression for the plasmonic dispersion relation.

enter image description here

The dispersion relation (shown above) consists of two regions, one which is to the right of all the light lines (defined by $\omega = \frac{c}{\sqrt{\epsilon_i}}k_x$), and the other which is between the two light lines (shaded in blue). The modes to the right of the light line are localized to the structure. That is, the electric fields decay exponentially away from the device. These modes are therefore considered non-radiative modes, because no fields can "escape" to infinity. All the propagation occurs along the interfaces. An example mode is shown below.

enter image description here

On the other hands, in between the light lines$-$also known as the "leaky" region$-$the modes are not localized to the structure. Indeed, the field amplitudes grow exponentially away from the device into the material with the higher dielectric constant (in this in the negative $z$ direction). In addition, these modes can propagate away from the material, depending on the whether or not damping is included, etc. For these reasons, these modes are called radiative modes. An example mode is shown below (the exponential growth into material 1 is very slow, but it's there!).

enter image description here

The possible modes supported by a single film were basically completely worked out in this paper:

Surface-polariton-like waves guided by thin, lossy metal films , Phys. Rev. B 33, 5186 – Published 15 April 1986.

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