Surface magneto-plasmons in magnetic multilayers - Walther ...

Section 2.4
Simulation of the reflectivity of surface plasmons 23
θB = arctan
n2
n1
. (2.48)
n1 and n2 are the refractive indices of the two media. For BK7 the Brewster angle is
about 33.4 ◦ [46] what matches with the simulated value.
With increasing θ the reflectivity increases and reaches 1 at θ ≈ 41.4 ◦ what is quite
similar to the literature value of 41.34 ◦ (Eq. (2.23)). Since the simulation is done
with a step size of 0.1 ◦ the difference between both values can be explained by the
lack of resolution in the simulation.
When the prism is covered by a Ag film of dAg = 20 nm (Fig. 2.9(b)) then the
reflectivity for θ = 0 ◦ goes up to 0.67. With increasing θ, R p slowly decreases and
goes through a minimum at θB = 38.6 ◦ and then increases and reaches a maximum
of 0.98 at θc, i.e. for total internal reflection. At somewhat larger θ, a minimum in
R p develops at the resonance angle (θSPP = 44.2 ◦ ) of the surface plasmons. This can
be explained with destructive interference (Sect. 2.3.1).
For silver films with dAg = 50 nm the surface plasmon resonance dip gets sharper
and θSPP shifts to 43 ◦ (Fig. 2.9(c)). While for dAg = 100 nm, a vanishing resonance
appears at θSPP = 42.9 ◦ (Fig. 2.9(d)).
The change of R p at θSPP can be explained by radiation damping [9]. As explained
in Sect. 2.3.1 the light at the metal-air interface is radiated light into the metal
which interferes with the incident light at the prism-metal interface. Since, the metal
has a complex dielectric function, this back-radiated light is damped. This effect is
called radiation damping. On increasing the thickness of the metal films the radiation
damping increases, so that the destructive interference between the incident and back-
radiated light at the prism-metal interface decreases and thus R p increases. As the
surface plasmon dip in R p is due to interference the depth of the surface plasmon dip
(the minimal R p value) thus decreases. Now, decreasing the metal film thickness at a
certain metal film thickness dopt the amplitude of the back-reflected light equals the
amplitude of the incident light and the phase of the back-reflected light is shifted by
π, so that R p = 0. When the thickness is decreased further d < dopt the effect of the
radiation damping increases again which yields an increase of R p .
For a single layer the optimal layer thickness dopt for which the R p goes to zero at
θ = θSPP can be calculated with a first order approximation equation reported by
Kretschmann [34].