Controlling Surface Plasmons Through Covariant Transformation of the Spin-Dependent Geometric Phase Between Curved Metamaterials

General relativity uses curved space-time to describe accelerating frames. The movement of particles in different curved space-times can be regarded as equivalent physical processes based on the covariant transformation between different frames. In this Letter, we use one-dimensional curved metamaterials to mimic accelerating particles in curved space-times. The different curved shapes of structures are used to mimic different accelerating frames. The different geometric phases along the structure are used to mimic different movements in the frame. Using the covariant principle of general relativity, we can obtain equivalent nanostructures based on space-time transformations, such as the Lorentz transformation and conformal transformation. In this way, many covariant structures can be found that produce the same surface plasmon fields when excited by spin photons. A new kind of accelerating beam, the Rindler beam, is obtained based on the Rindler metric in gravity. Very large effective indices can be obtained in such systems based on geometric-phase gradient. This general covariant design method can be extended to many other optical media.

Figure 1

(a) The transformation sharing the same caustic. (b) The geometric relationship between $\tau$, $\mathrm{\Phi }$, and $l$. (c) The transformation that occurs on the caustic through Cherenkov radiation, which can be mapped to special relativity.

Figure 2

Transformation between two trajectories. (a) The theoretic calculation. SPP rays marked in red lead to a caustic marked in orange. A transformation occurs between the white dashed line and the green dashed line. (b)–(d) Figures for the case of the white dashed line and (e)–(g) for the case of the green dashed line. (b),(e) SEM images of our samples. (c),(f) Full wave simulations by comsol with the green dashed line labeling the corresponding caustic. (d),(g) Experimental results with the white dashed line labeling the corresponding caustic. Legends on the right part of simulations and experiments mark the normalized amplitude with different colors.

Figure 3

Geometric picture describing the same event in different coordinates through mimicking Bremsstrahlung radiation of moving particles (blue ball and purple ball) in flat space-time ($g=|\nabla \tau |=1$) and curvilinear space-time ($g=|\nabla \tau |$), respectively.

Figure 4

Transformations analogous to the Rindler transformation in general relativity. (a) Shows SPP rays in red and caustic in orange. The dashed white line, green line, and cyan line of nanoslots represent different corresponding coordinates. (b),(e),(h) SEM images of samples of (b) white dashed line, (e) green dashed line, and (h) cyan dashed line with inclination angle of 52°. (c),(d),(f),(g),(i),(j) The simulations and experiments of these three cases. (c),(f),(i) Full wave simulations by comsol with the green dashed line labeling the corresponding caustic. (d),(g),(j) Experimental results with the white dashed line labeling the corresponding caustic. Legends on the right part of simulations and experiments describe the normalized amplitude with different colors.