Maximizing energy coupling to complex plasmonic devices by injecting light into eigenchannels

Surface plasmon polaritons have attracted broad attention in the optoelectronics field due to their ability to merge nanoscale electronics with high-speed optical communication. As the complexity of optoelectronic devices increases to meet various needs, this integration has been hampered by the low coupling efficiency of light to plasmonic modes. Here we present a method to maximize the coupling of far-field optical waves to plasmonic waves for arbitrarily complex devices. The method consists of experimentally identifying the eigenchannels of a given nanostructure and shaping the wavefront of incident light to a particular eigenchannel that maximizes the generation of plasmonic waves. Our proposed approach increases the coupling efficiency almost four-fold with respect to the uncontrolled input. Our study will help to facilitate the integration of electronics and photonics.


I. Extended coupled-mode method
In this section, we are going to describe the extend coupled-mode method (ECMM) that is derived from the coupled-mode method (CMM). The CMM is a general framework to analyze the interaction among modes in the nanostructures and it is well described in the previous studies 1 . Therefore, we will briefly introduce the CMM and mainly describe its detailed extension to incorporation the wavefront shaping of the illumination. Figure S1 is the schematic figure to explain the CMM. Incident light coming from the region I enters region II and exits to region III. The region I and III are filled with the dielectric materials and semiinfinite space. The metal film (region II) has plural holes (or slit) having a waveguide mode. The thickness of the metal film is set h. Figure S1. The schematic diagram of the sample.
The below are the equations resulting from the CMM, and they can be a good starting point to explain the ECMM.
The equation S1a and S1b are for the boundary matching at = 0 and = ℎ, respectively. Here the terms in equations are defined as follows.
The represents the coupling between the incident field and the waveguide mode . The and ′ are the electric field at the input and output interfaces of the holes, respectively. and ′ represent the coupling between the waveguide modes and of two different holes at the = 0 and = ℎ, respectively. Therefore, the and ′ can be considered the self-interaction coupling of the waveguide mode at the = 0 and = ℎ, respectively. The is the net of the waveguide mode between up (positive z) and down (negative z) directions, and the the effect of the field on the one side to the field on the other side in waveguide with mode .
In general, two equations are needed to obtain the and ′ for one waveguide mode . Therefore, we need the 2 equations for holes. In our study, the metal film (region II) has only two slits with same width, and the width of slits are very narrow such that it is safe to assume that there is a single mode per each slit. Furthermore, if the incident field is normal to the metal film, only two equations are enough because of the symmetry of the given system. However, we inject different fields into two slits in our study, and the following four equations are required as a consequence.
Here, the subscript and indicate the left and right slits, respectively. In the main text, we assumed ′ = and ′ = as the refractive indices of the materials in the region I and II are the same.

II. Derivation of a transmission matrix
We can rewrite Eq. S3 as the following matrix equation. ( Here = ( − Σ) and = − , and ( ) = ( ) , where ( ) means the metallic loss decaying exponentially. The higher order terms of ( ) is ignored in this calculation as they are extremely smaller than the linear term. From this equation, we obtain and , which are SPP field resided between the region I and II (S5). (S5) For the sake of simplicity, we replace the common denominator of and , ( 2 − 2 ) 2 , with 1/ in Eq. (S6).

III. Finding an incident wave maximizing SPPs
In this section, we derive the equation (6)  ) .
To obtain the total intensity of SPPs we take the absolute squares of the and and obtained the summation of the two.