Figure 1
From: Creating a zero-order resonator using an optical surface transformation
(a) Two arbitrarily shaped surfaces S1 and S2 as the terminated surfaces of two conformal smooth curves Γ1 and Γ2 (in accordance with the normal direction of these two smooth curves) will perform equivalently, if the ONM given in Eq. (2) is filled inside the each sub-region between Γ1 and Γ2 (the yellow region). (b) The coordinate transformation relation in each small sub-region. The left is the reference space (x0, y0, z0) and the right is the real space (x, y, z) which corresponds to the local coordinate (xi, yi, zi) in (a). When Δ → 0, the blue volume in the reference space reduces to a surface which corresponds to the yellow region (i.e. the ONM) in the real space. If there is some scaling along the y direction (i.e. M ≠ 0), the main axis of the ONM is not along the x direction (see Eq. (2)). If there is no compression and extension along y direction (i.e. the normal direction of the conformal curves in (a)), we have M = 1, which means the main axis of the ONM is along the x direction (i.e. the tangential direction of the conformal curves in (a) and see Eq. (3)). (c) A closed loop is formed if we connect S1 and S2 together. (d) another way to design an optical resonator by the OST: S1 and S2 have been linked by the ONM. S1’ and S2’ have exactly the same shape as S1 and S2, respectively. S1’ and S2’ are also linked by the ONM. If we connect S1 and S1’, S2 and S2’ together, respectively, a closed loop filled with the ONM has been created. Such closed loop performs like an optical open resonator. The orange and green regions labeled by ‘X’ and ‘Y’ stand for the ONM with main axis along the x and y directions, respectively.
