Quantum theory of surface-plasmon polariton scattering

We introduce the quantum mechanical formalism for treating surface plasmon polariton scattering at an interface. Our developed theory—which differs fundamentally from the analogous photonic scenario—is used to investigate the possibility of plasmonic beam splitters at the quantum level. Remarkably, we find that a wide range of splitting ratios can be reached. As an application, we characterize a 50:50 plasmonic beam splitter and investigate first-order quantum interference of surface plasmon polaritons. The results of this theoretical study show that surface plasmon beam splitters are able to reliably and efficiently operate in the quantum domain.

Figure 1

Scattering of SPPs at an interface. (a) An SPP excitation ${\mathrm{â}}^f$ in region $i$ is scattered into an SPP excitation ${\mathrm{b̂}}^f$ in region $j$ and SPP excitation ${\mathrm{â}}^b$ in region $i$. The superscript $f$ ($b$) denotes a forward- (backward-) propagating excitation with respect to the $x$ axis. Near- and far-field radiation excitations are also excited in the process (represented by the central white haze and red jagged arrows). The symbols ${\mathrm{Â}}_k^b$ and ${\mathrm{B̂}}_k^f$ are used for the radiation in regions $i$ and $j$, respectively. Similar scattering occurs for the SPP excitation ${\mathrm{b̂}}^b$ in region $j$. Further details on the angles are given in Fig. 2b and related discussion is provided in Sec. 5. (b) Dispersion relations for the SPP’s and radiation in regions $i$ and $j$. Shaded regions correspond to a continuum spanned by the wave numbers $q_i$ and $q_j$, respectively. Examples of lines corresponding to set $q_i$ and $q_j$ are shown.

Figure 2

Transfer matrix and SPP beam-splitter angles. (a) Input-output transfer matrix $T$ linking all the excitations at the interface. (b) Angles involved in the scattering process: the incident angle ${\theta }_{i_i}$ (${\theta }_{j_i}$) of the SPP excitation ${\mathrm{â}}^f$ (${\mathrm{b̂}}^b$) in region $i$ ($j$) and the transmitted angle ${\theta }_{i_t}$ (${\theta }_{j_t}$) of the SPP excitation ${\mathrm{b̂}}^f$ (${\mathrm{â}}^b$) in region $j$ ($i$). The relations ${\theta }_{i_r}={\theta }_{j_t}$ and ${\theta }_{j_r}={\theta }_{i_t}$ must hold for the excited outputs to be indistinguishable. This can always be achieved; see text for details.

Figure 3

Scattering interface as an SPP beam splitter for ${\lambda }_0=790$ nm shown in (a) and (b) and for $1500$ nm in (c) and (d). [(a) and (c)] Forward SPP transmission $\tau$ (solid lines), reflection $\rho$ (dashed lines), and scattering $\sigma$ (dotted lines) coefficients versus the incidence angle ${\theta }_{i_i}$. In all plots ${\epsilon }_{d,i}$ increases in steps of 0.25 from 1 to 3 going from right to left (bottom to top for the scattering coefficient). [(b) and (d)] Coefficients $|\tau -{\tau }^{\prime }|$, $|\rho -{\rho }^{\prime }|$, and $|\sigma -{\sigma }^{\prime }|$. In all cases, ${\epsilon }_{d,j}=1$ and ${\omega }_{p,i/j}={\omega }_p^{\mathrm{silver}}=1.402\times {10}^{16}$ rad/s are chosen. Note that in the plots of (a) and (c) the coefficients $\tau$, $\rho$, and $\sigma$ are limited at large angles of ${\theta }_{i_i}$. This is due to a reduction in numerical precision close to plasmon TIR for which we do not include the corresponding values.

Figure 4

Transmission angle ${\theta }_{i_t}$ as a function of the incidence angle ${\theta }_{i_i}$. (a) ${\lambda }_0=790$ nm. (b) ${\lambda }_0=1500$ nm. In both plots ${\epsilon }_{d,i}$ increases in steps of 0.25 from 1 to 3 going from right to left.

Figure 5

Results of the optimization procedure for achieving a 50:50 SPP beam splitter with the scattering interface over a range of incidence angles ${\theta }_{i_i}$ for ${\lambda }_0=790$ nm ($1500$ nm). (a) [(g)]: SPP transmission $\tau$ (solid), reflection $\rho$ (dash), and scattering $\sigma$ (dotted) coefficients. (b) [(h)] Values of ${\epsilon }_{d,i}$ (solid), ${\theta }_{i_t}$ (dash), ${\omega }_{p,i}$ (dotted), ${\omega }_{p,j}$ (dash-dot). (c) [(i)] Coefficients $|\tau -{\tau }^{\prime }|$ (solid), $|\rho -{\rho }^{\prime }|$ (dash), $|\sigma -{\sigma }^{\prime }|$ (dotted). (d–f) [(j–l)] Same as (a–c) [(g–i)] but with the restriction ${\omega }_{p,i}={\omega }_{p,j}=1$. In all cases, ${\epsilon }_{d,j}=1$ is chosen and values of ${\omega }_{p,i/j}$ are measured in units of ${\omega }_p^{\mathrm{silver}}=1.402\times {10}^{16}$ rad/s. The vertical dotted lines designate optimized angles and the corresponding material parameters for the 50:50 beam splitter.

Figure 6

Scattered TM radiation plots resulting from the scattering of an incoming SPP from region $i$. (a) 50:50 beam splitter for ${\lambda }_0=790$ nm. Here, the parameters chosen correspond to those defined by the dotted line of Fig. 5a, where the metal dielectric is allowed to vary. (b) 50:50 beam splitter for ${\lambda }_0=1500$ nm. Here, the parameters chosen correspond to those defined by the dotted line of Fig. 5g. (c) 50:50 beam splitter for ${\lambda }_0=1500$ nm. Here, the parameters chosen correspond to those defined by the dotted line of Fig. 5j, where the metal is the same on both sides of the interface. As one cannot reach 50:50 for ${\lambda }_0=790$ nm in this regime, such a plot has not been included.

Figure 7

First-order quantum interference of SPPs where the parameters chosen correspond to the optimization procedure used for the 50:50 SPP beam splitter described in Fig. 5 for ${\lambda }_0=790$ nm ($1500$ nm). (a) [(c)] Twofold coincidence probabilities. (b) [(d)] Twofold coincidence probabilities with the restriction ${\omega }_{p_i}={\omega }_{p_j}={\omega }_p^{\mathrm{silver}}$. In all cases, ${\epsilon }_{d_j}=1$ is chosen. (e) Optical beam-splitter analogy for the plasmonic beam splitter with two detectors measuring the presence of an SPP at both ports at the same time ($C$ represents a coincidence detection circuit).