Simulation and Optimisation of the Coupling Efficiency

For coupling of devices, the figure of merit is the maximum coupling efficiency η that can be reached. Here, optimisation of the coupling efficiency for the coupler designed in Section 8.3.1, which especially suites the needs of experiments with sin-gle SPPs and sinsin-gle emitters, is desired. The operation wavelength is λ= 780 nm, a wavelength where single photon emitters exist, but which also is sufficiently long to reduce losses in the metal (see Section 8.1.5).

In the following, FEM simulations using the commercial JCMSuite (JCMwave) code are carried out. The geometry simulated is the coupler shown in Figure 8.9 (b) with the parameters:

• distance of dielectric waveguide to the metal arms d_dist,

• gap between taper and arms dgap,

• width of the metal arms darm,

• length of the tapered region d_taper,

• height of the dielectric materialh_diel,

• height of the metal h_metal,

• width of the dielectric waveguide wdiel,

• width of the metal waveguide w_metal.

Of these parameters, the geometry of the incoming dielectric waveguide, i.e., its heighth_dieland its widthw_diel, are chosen to beh_diel= 200 nm andw_diel= 510 nm.

Furthermore, the geometry of the metal waveguide is set to have a height h_metal = 50 nm and a width of w_metal = 510 nm. Note that these choices are arbitrary, only constrained by fabrication capabilities and the requirement for single mode operation. No mode matching between incoming dielectric and outgoing metal waveguide has to be taken into account, since the working principle of the coupler does not rely on that. In fact, simulations with a propagation mode solver lead to the effective refractive indices nef f,T E = 1.597 for the transverse electric mode of the dielectric waveguide and n_{ef f,T M} = 1.537 for the transverse magnetic mode of the dielectric waveguide, compared to n_{ef f,m1} = 1.604 +i0.0186 and n_{ef f,m2} = 1.681 +i0.0156 for the two modes of the SPP waveguide [277]. For this simulations

8.3. A Dielectric Waveguide to SSP Coupler

400 nm

a b c

Figure 8.10.:Field distributions of waveguides and coupler. (a) shows the electric field’s intensity of the transversal electric mode in the dielectric waveguide while in (b) the mode of the SPP waveguide is shown. Note the high field concentration at the edges of the SPP waveguide (the colour scale is normalised to the dielectric mode). In (c), the in-plane electric field component perpendicular to the propaga-tion direcpropaga-tion at the substrate-air interface is shown for the optimised coupler. In the upper half in the overlay shows the dielectric (blue) and plasmonic (red) part.

Scalebars are 400 nm. (adapted from [277])

and for the following, the values of the dielectric function for the materials involved are _Silica = 2.37 [328], _{SiN i} = 3.99 [329], and _Au =−22.46 +i1.39 [287]. Since the coupler is designed for single mode operation, a single input mode has to be selected. Here, as input mode the TE mode is chosen. Its transverse electric field are able to excite the SPPs in the metal arms more efficiently than the electric field of the TM mode. On the SPP output side of the coupler, single mode operation then is ensured due to symmetry – the second SPP mode exhibits a breathing mode character, which does not fit to the incoming TM mode [277]. Simulated field distributions of the two modes used are shown in Figure 8.10 (a,b).

Now, fully three-dimensional simulations with the obtained TE mode as input field distribution are performed. One example of the electric field from such sim-ulations can be found in Figure 8.10 (c). However, before looking into the details, the method of obtaining the coupling efficiency η is introduced. Directly after the coupler, there are two different kinds of contributions to the electromagnetic fields, namely the bound SPP mode one wants to excite efficiently and scattered fields.

For the calculation of η, only the SPP mode has to be taken into account, so it is important to distinguish the different contributions. This is done by adding 5µm of straight SPP waveguide to the simulated coupler and evaluating the energy flux Φ through surfaces perpendicular to the waveguide at different distancesz. For the guided SPPs, this energy flux is decaying mono-exponentially:

A=A0e^−αz, (8.48)

while the scattered fields decay faster. Furthermore, the decay constant α = 4π^{Im[nef f,m1}_λ ^]of the guided SPP mode can be calculated with the propagation mode

1 2 3 4

Figure 8.11.: Calculation of the coupling efficiency. In (a), the power flux through a surface perpendicular to the SPP waveguide is shown. The line shown is an exponential fit of the decay at long distances according to Equation 8.48. In (b), the length of the taper is varied. The coupling efficiency shows the oscillatory behaviour expected from the working principle of the coupler. . (c) shows the coupling efficiency of the coupler when the wavelength is varied. The lines in (b,c) are guides to the eye. (adapted from [277])

solver, so that the amplitude of the exponential A₀ directly leads to the coupling efficiency. Figure 8.11 (a) shows this behaviour with normalisation to the source energy Φ₀, i.e., A= ^Φ(z)_Φ

0 .

Optimising the four free parameters of the coupler d_dist, d_gap, d_arm, and d_taper using the Taguchi method [277, 330] yields an optimal coupler with a coupling efficiencyη = 60 % for the parametersd_dist = 80 nm,dgap= 20 nm,darm= 120 nm, and d_taper = 800 nm. Figure 8.10 (c) shows the field distribution obtained for this optimal coupler.

A value of η = 60 % is comparable to other couplers found in literature, where theoretical values of up to 90 % are reached [331] for a dielectric loaded coupler in at a wavelength of 1550 nm (see Reference [289] for a review). In contrast to the other schemes, the coupler introduced here satisfies the needs specified in Section 8.3.1. In addition many values found in literature might be overestimated, due to only performing two dimensional simulations (e.g., in References [332–334]).

The coupling efficiency of the coupler can be also easily enhanced by changing the design restrictions and changing the materials involved. With silver as SPP waveguide material for example, a coupling efficiency of η_silver = 68 % is found. In addition, by changing the refractive indices of the involved media, e.g., by using magnesium fluoride (n = 1.38 [335]) instead of glass as substrate, the efficiency possibly can be enhanced further due to lower absorption of the SPP mode (see Section 8.1.5).

To further evaluate the properties of the coupler and to gain more insight into its working principle, parameter scans are performed starting from the optimal coupler found with the Taguchi method. Figure 8.11 (b) shows a variation of the taper lengthd_taperas well as a variation of the operating wavelength with unchanged

8.3. A Dielectric Waveguide to SSP Coupler

geometry is shown. The coupling efficiencyηshows a damped oscillatory behaviour for a variation ofdtaperas it is expected from the considerations in Section 8.3.1. The wavelength dependency shown in Figure 8.11 (c) also gives an important result: the coupler can be operated over a broad wavelength range of at least 150 nm without suffering from significant additional losses. Hence, this type of coupler can be used to interface different emitters without the need to perform an extra optimisation run each time.

In summary, the coupler simulated and optimised in this section satisfies the requirements for doing quantum plasmonic experiments while having a reasonable high coupling efficiency and is easy to fabricate. The possibility to couple sin-gle emitters to it using the nanomanipulation techniques described in Section 5.2 makes it a good interface between integrated waveguide and quantum photonic or plasmonic structures.

Chapter Summary: Plasmonics

In this chapter, surface plasmon polaritons were introduced. At first, a review on macroscopic electrodynamics and the optics of metals was given. Then, the prop-erties of propagating SPPs were examined and their applications discussed. It was reported on an experiment on the generation of single SPPs using a hybrid quantum system. The pick-and-place technique was used to place a nanodiamond with single NV centre near a silver nanowire, which served as waveguide. Last, to address the problem of damping for propagating SPPs, a coupler from SPPs to photons was designed and investigated via numerical calculations. Such a coupler enables for efficient conversion and therefore makes it possible to use the unique properties of SPPs, e.g., the possibility for nanofocussing, just where they are needed, while the photons are used to guide excitations over long distances.

Having shown that concepts for hybrid integration also work with plasmonic structures, what is needed next is a technique to quantitatively measure their prop-erties on the nanoscale. Such a technique will be introduced in the next chapter.

9. Quantum Emitter Fluorescence