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in order to define the SPP dispersion relations. From this equation it follows that 

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As for the single interface one has an important relation between the real and imaginary parts of the SPP wave vectors in the different medium: 

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Since we are interested only into the solutions which are decaying along the interface we (in agreement with our previous treatment of the single interface) suppose the condition k ′ x · k ′′ x ≥ 0 satisfied. A second important point is that due to the arbitrariness in the sign of k$_{j}$ there are in fact apriori 8 possibilities for writing Eq. 15. However, Eq. 15 is invariant under the transformation k$_{1}$ → − k$_{1}$ . This means that the number of apriori possibilities for the sign of k$_{j}$ is reduced from 8 to 4. This multiplicity was studied by Burke et al. [11] however since for the present purpose we are looking for SPP waves leaking from the air/metal interface into the glass substrate we will consider only the possibility 

The sign of k$_{1}$ is however arbitrary as explained above and we choose it here positive by definition. In order to define a SPP wave leaking into the glass substrate one has thus to substitute Eq. 17 into Eq.15 and find numerically (i. e., by a minimization procedure [18, 19]) the zeros of the implicit equation with variable k ′ x and k ′′ $_{x}$. This has to be done only in the quarter of the complex plane corresponding to k ′ x > 0, k ′′ x > 0. The quarter k ′ x < 0, k ′′ x < 0 must be equivalent due to symmetry and corresponds actually to decaying SPP waves propagating in the negative x direction. The two other quarters of the complex plane correspond to growing SPP waves along the interface and will be rejected on a physical ground (compare [11]). 

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Figs. 3A and 3B show numerical calculations of dispersion relations corresponding to a SPP wave leaking through a gold film from the air side to the glass side. The thickness is taken to be D =70, 50, 20, and 10 nm respectively. For the value D ≥ 70 nm the 

