Measurement of the Photon-Plasmon Coupling Phase Shift

Scattering processes have played a crucial role in the development of quantum theory. In the field of optics, scattering phase shifts have been utilized to unveil interesting forms of light-matter interactions. Here we investigate the phase shift experienced by a single photon as it scatters into a surface plasmon polariton and vice versa. This coupling phase shift is of particular relevance for quantum plasmonic experiments. Therefore, we demonstrate that the photon-plasmon interaction at a plasmonic slit can be modeled through a quantum-mechanical tritter, a six-port scattering element. We show that the visibilities of a double-slit and a triple-slit interference patterns are convenient observables to characterize the interaction at a slit and determine the coupling phase. Our accurate and simple model of the interaction, validated by simulations and experiments, has important implications not only for quantum plasmonic interference effects, but is also advantageous to classical applications.

Figure 1

Illustration of plasmonic tritters. (a) Sketch of a slit on a gold film that acts as a tritter and couples the photonic mode $a$ to two plasmonic modes $b_1$ and $b_3$. The elements of the scattering matrix from Eq. (1) are visualized in the circle. For clarity, we only show the coupling of one plasmonic mode. The other mode couples in a similar manner. (b) Schematic diagram of the triple-slit structure where each slit acts as a tritter. The middle slit is illuminated with single photons. The distance from slit $j$ to the screen at the far field is shown by $r_j$.

Figure 2

Scheme of the experimental setup. The idler photons are used to herald the presence of the signal photons, which are focused onto the sample. A sketch of the sample with different slit arrangements (A), (B), and (C) is shown in the inset along with a scanning electron micrograph of the triple-slit structure of arrangement (C). The polarization of the signal photons is controlled by a polarizer (P) and a half-wave plate (HWP). Photons at the far field are collected with an oil-immersion objective. A lens system images the far-field pattern onto an ICCD camera.

Figure 3

Far-field interference patterns from the three different slit structures labeled in Fig. 2. Only the photons with $y$ polarization excite SPPs (green), no interference occurs for $x$ polarization (red). The Fourier transforms (FT) in each part show the spatial frequency of the fringes. Since the slit separation is larger in slit structure (B) its fringe pattern has a higher spatial frequency than slit structure (A). In slit structure (C), the interference of the two modes emerging from the two outer slits has a small contribution in the triple-slit pattern as shown with the arrows on the Fourier transform.

Figure 4

Far-field interference pattern for different slit separations from $4.3\mu \mathrm{m}$ to $4.8\mu \mathrm{m}$ obtained from FDTD simulation and from our theoretical model. For the theoretical plots Eq. (2) is multiplied by a sinc function to account for the finite width of the slits. The theoretical result matches to the numerical simulation when we incorporate a coupling phase of ${\varphi }_{\mathrm{in}}+{\varphi }_{\mathrm{out}}=5.8\mathrm{rad}$. With other values of the coupling phase the position of the minimum visibility (indicated by the arrows) shifts, as shown in the rightmost plot. The excellent agreement between the simulation and the theoretical results confirms the validity of our theoretical model. The outer fringes in the theoretical plots are faint due to the deviation from the small angle approximation used for the sinc function.