Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric Bragg mirror

Conventional surface plasmons have a wave vector exceeding that of light in vacuum, and therefore cannot be directly excited by light that is simply incident on the surface. However, we propose that a plasmon-polariton state can be formed at the boundary between a metal and a dielectric Bragg mirror that can have a zero in-plane wave vector and therefore can be produced by direct optical excitation. In analogy with the electronic states at a crystal surface proposed by Tamm, we call these excitations Tamm plasmons, and predict that they may exist in both the TE and TM polarizations and are characterized by parabolic dispersion relations.

Figure 1

(a) The profile of the electric (solid line) and magnetic (dashed line) fields for a Tamm plasmon at the interface between a semi-infinite metal layer and a 14-period Bragg reflector. The real and imaginary parts of the eigenenergy are $\mathrm{Re}\left(\hslash {\omega }_{\mathrm{TP}}\right)=0.9572\mathrm{eV}$, $\mathrm{Im}\left(\hslash {\omega }_{\mathrm{TP}}\right)=-1.5\mathrm{meV}$ which correspond to $Q\approx 320$. The profile of the real part of the refractive index is shown by the dotted line where $n_A=3.7$ and $n_B=3.0$ for the $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ structure. (b) The profile of the electric (solid line) and magnetic (dashed line) fields for a Tamm plasmon at the interface between a $30\text{−}\mathrm{nm}$-thick metal film and a 14-period Bragg reflector. $\mathrm{Re}\left(\hslash {\omega }_{\mathrm{TP}}\right)=0.95271\mathrm{eV}$, $\mathrm{Im}\left(\hslash {\omega }_{\mathrm{TP}}\right)=-2.4\mathrm{meV}$ $(Q\approx 200)$. The profile of the real part of the refractive index is shown by the dotted line. (c) The general layered structure used in the derivation of Eqs. (1a, 1b, 1c, 1d). The virtual interfaces in a single uniform layer, with amplitude reflection coefficients $r_{\text{left}}$ and $r_{\text{right}}$, are shown by dotted lines. The wave gains a phase $\Phi$ in propagating from the left to the right virtual interface.

Figure 2

The dispersion relation for Tamm plasmons at the interface between a semi-infinite gold layer and a semi-infinite $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ Bragg reflector for the TE (solid line) and TM (dashed line) polarizations. The squares and circles show the positions of the features in optical spectra for the TM and TE polarizations, respectively. Solid symbols correspond to the interface of semi-infinite gold and a 14-period Bragg reflector (see Fig. 3), while the open symbols correspond to the thin gold film on a 14-period Bragg reflector (Fig. 4).

Figure 3

Reflection spectra of the $\mathrm{Al}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ Bragg reflector on a semi-infinite gold layer for TE- and TM-polarized light for various angles of incidence: normal incidence (solid line); 30° (dashed line); 45° (dotted line); and 60° (dash-dotted line). Light is incident from the right side of the structure.

Figure 4

Transmission spectra of the free-standing $\mathrm{Al}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ Bragg reflector covered by a $30\text{−}\mathrm{nm}$-thick gold film for TE and TM polarized light for various angles of incidence: normal incidence (solid line); 30° (dashed line); 45° (dotted line); and 60° (dash-dotted line). Thin solid lines show absorption spectra for each case. Light is incident from the left side of the structure.

Figure 5

Dependence on the thickness of the first layer of a 14-period Bragg reflector of the eigenenergy of the TP formed at the interface of a semi-infinite metal and the Bragg reflector. Horizontal dashed lines show the top and bottom edges of the photonic band gap of the infinite Bragg mirror. The vertical dotted line indicates the quarter-wavelength thickness of the layer.