in order to define the SPP dispersion relations. From this equation it follows that 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: 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]). 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 8 k$_{0}$ = − √ (( ω/c ) $^{2}$ϵ$_{0}$ − k 2 $_{x}$) k$_{1}$ = + (( ω/c ) $^{2}$ϵ$_{1}$ − k 2 $_{x}$) k$_{2}$ = − √ (( ω/c ) $^{2}$ϵ$_{2}$ − k 2 $_{x}$) . (17) − k ′ 0 · k ′′ 0 = ( ω/c ) $^{2}$ϵ ′′ $_{1}$/ 2 − k ′ 1 · k ′′ 1 = − k ′ 2 · k ′′ = k ′ x · k ′′ $_{x}$. (16) ( k$_{0}$/ϵ$_{0}$ + k$_{1}$/ϵ$_{1}$ )( k$_{2}$/ϵ$_{2}$ − k$_{1}$/ϵ$_{1}$ ) e ik 1 D +( k$_{1}$/ϵ$_{1}$ − k$_{0}$/ϵ$_{0}$ )( k$_{2}$/ϵ$_{2}$ + k$_{1}$/ϵ$_{1}$ ) e − ik 1 D = 0 . (15)