Design Parameters and Operation Principle

When trying to build advanced quantum optical elements using SPPs [325], SPPs need to be interfaced with lossless channels, since a main obstacle for routing of SPPs is their loss (see Section 8.1.5). The obvious choice for this channels are photonic waveguides, so SPPs have to be converted to photons and vice versa. This conversion has to be highly efficient when working at the level of single quanta. At this level, no classical amplification is possible, so loss will dramatically reduce the fidelity of any experiment. Also, scalability of the experiment (and future applications) favors an integrated approach which can be produced using standard clean-room processes. As introduced in Section 8.1.6, excitation of SPPs with photons requires phase matching of photons and SPPs. To achieve this efficiently, a special coupler has to be designed.

In the following, a dielectric waveguide to SPP coupler, which satisfies the re-sulting experimental requirements, is introduced. The coupler should:

• interface a mono-mode dielectric waveguide with a single plasmonic mode,

• work in a wavelength range, where single photon emitters like molecules, defect centres, or quantum dots emit, since interfacing emitters is crucial in quantum plasmonics [325],

• be accessible with single emitters,

• be easy and reliable to fabricate in standard processes.

With these demands in mind, the material system and geometrical boundaries of the coupler are chosen.

For the incoming dielectric waveguide, a rectangular silicon nitride waveguide lying on silica is used, as both materials are transparent in the red spectral range, where the planned wavelength of operation (λ= 780 nm) is situated. As material for the plasmonic part, gold is chosen, which is done not for efficiency (silver would have had a lower loss; see Section 8.1.5), but for stability, since silver nanostructures may degrade under ambient environment [326]. The SPP waveguide part of the coupler is a strip waveguide. Such a waveguide allows for easy coupling of emitters to the SPP modes and also for nanofocussing by tapering down the waveguide (see Section 8.1.8). To facilitate nanopositioning of emitters near the structures, the

Si₃N₄ Au

Figure 8.8.: Constraints and possibilities for SPP coupler design. (a) and (b) show top and side view of the geometry considered, respectively. The waveguides are defined by their widthsd_d/mand their heightsh_d/m. The coupler itself is shown as black box, which is a placeholder for different possible designs. In (c) i-iii), some of the possible designs are shown. Butt coupling in i), directional coupling in ii), and a design similar to the one used in the remainder of this section in iii).

structures must not be capped, so the upper layer of material is air. To avoid additional fabrication steps, the metal layer is designed to directly lie on the silica.

This leads to a vertical offset of the centres of the modes in dielectric and SPP waveguide. The basic design parameters are visualised in Figure 8.8.

For a fast and efficient design process, it is important to have some tools to un-derstand what happens for different geometries. It can not be overestimated how just "getting a feeling" can speed up development. For this reason, many coupling structures (some are shown in Figure 8.8) are simulated with a finite difference time domain code (FDTD Solutions, Lumerical; for FDTD see Appendix C) and the spa-tial and temporal distributions of them are monitored. The insights obtained were then used to develop new ideas and designs. Note that to save time, these simulation were carried out an a relative coarse three-dimensional mesh (≈ ₁₈^λ) and conver-gence of the simulations is not checked. After a design which works reasonably well is found, it is parameterised and an optimisation is carried out using a finite element method (FEM) [327] code for further simulations (JCMSuite, JCMWave).

The parameterised design found in this way is shown in Figure 8.8 (c iii).

When the dielectric and SPP waveguide’s geometries are given as boundary

con-8.3. A Dielectric Waveguide to SSP Coupler

b

darm dtaper

ddist dgap

Si3N4 Au

a _Si

3N4Au c

d e f g

Figure 8.9.: Design and operation principle of the SPP coupler. (a) and (b) show the layout of the coupler, which is fully described by the properties of incoming and outgoing waveguide and the four parameters (ddist,dgap,darm,dtaper). (c) clarifies the geometry used in (d-g), where the operation principle of the coupler is shown via electric field distributions in a FDTD simulation. In (d), a pulse propagates downwards along the dielectric waveguide. In (e), high field strengths start to build up in the gap, which subsequently couple to the outside of the metal arm as shown in (f). In (g), the excited SPP mode is shown. ((a) courtesy of G. Kewes)

ditions, the coupler design finally used for optimisation has four free design param-eters, as shown in Figure 8.9 (b). Before showing the FEM simulations performed in order to get quantitative results for the coupling efficiency, the working principle of the coupler is introduced. Figure 8.9 (d-g) show the operation mechanism of the coupler. The images are taken from the FDTD simulations used to find a good coupler design. As source, a pulse is sent in the dielectric waveguide’s mode and the electric field strengths are shown for different time steps. At first, the pulse is guided towards the coupling region by the dielectric waveguide (Figure 8.9 (d)).

Inside the coupling region, the field is concentrated in the gap between the dielectric taper region and the coupler’s metal arm (Figure 8.9 (e)), which acts as an SPP waveguide. On the outside of the coupler’s metal arm, also an SPP mode exists.

Mode beating between the mode on the inside and the one on the outside of the arm leads to an energy transfer from mode to mode. When the tapered region has the right length, it can be achieved that almost all of the energy is on the outside of the arm at the end of the taper (Figure 8.9 (f)). In this case, the SPP starts propagating on the outside of the SPP waveguide, as shown in Figure 8.9 (f). From

this working principle, it can also be concluded that it is possible to excite SPPs in a gap waveguide with this coupling scheme just by elongating the taper until the mode beating has its maximum on the inside of the coupler’s arms.