Origin of the looped trajectories of photons

Under the scalar wave approximation, the propagation of light is described by the Helmholtz equation

subject to the boundary conditions specifying the physical setup. This equation can be solved by computing the propagation from any point r 1 to any other point r 2 via the Green’s function kernel, which according to Rayleigh-Sommerfeld theory is given by

where χ is an obliquity factor16. This equation satisfies equation (4) and the Fresnel-Huygens principle in the form of the following propagator relation

If one repeatedly applies equation (6), the path-integral formulation of the propagation kernel is obtained in the form17

where is the functional integration over paths x(s). The boundary conditions can be included by restricting the possible paths x(s). If one is concerned only with diffraction from slits in a single plane, then equation (7) can be perturbatively expanded as14

where K n represents the nth application of equation (6) and each integration is carried over the plane containing the slits (see Supplementary Note 1).

Solving the wave equation taking K=K 1 is equivalent to considering only direct paths, such as the paths in Fig. 1b. These paths propagate from the source and through one of the slits to the detector. We call these wavefunctions , and . The higher-order terms in equation (8) are responsible for the looped trajectories of photons that propagate from the source to a slit, and to at least one other slit before propagating to the detector (see Fig. 1c). It follows that the wavefunction of a photon passing through the three slits is given by

where represents the contribution of the looped trajectories to the wavefunction . Note that in general , as defined by equation (3), is not zero because of the existence of these looped trajectories. Thus, the presence of looped paths leads to an apparent deviation of the superposition principle14.

Occurrence of looped trajectories of photons

The conclusion that is not simply the superposition of the wavefunctions , and is a consequence of the actual boundary conditions in a three-slit structure. Changing the boundary conditions affects the near-field components around the slits, but it typically does not affect the far-field distribution because of the short range extension of the near fields18. As shown below, the looped trajectories of photons are physically due to the near-field components of the wavefunction. Therefore, by controlling the strengths and the spatial distributions of the near-fields around the slits, it is possible to drastically increase the probability of photons to undergo looped trajectories, thereby allowing a straightforward visualization of their effect in the far-field interference pattern. To demonstrate this phenomenon, a three-slit structure was designed such that it supports surface plasmons, which are strongly confined electromagnetic fields that can exist at the surface of metals19,20. The existence of these surface waves results in near fields that extend over the entire region covering the three slits6,21, thereby increasing the probability of looped trajectories.

As a concrete example, we consider the situations depicted in Fig. 1d,e. First, we assume a situation in which the incident optical field is a Gaussian beam polarized along the long axis of the slit (y polarization) and focused to a 400-nm spot size onto the left-most slit. For this polarization, surface plasmons are not appreciably excited and the resulting far-field distribution is the typical envelope, with no fringes, indicated by the dashed curve in Fig. 1e. This intensity distribution is described by the quantity . The presented results were obtained through a full-wave numerical analysis based on the finite-difference-time-domain (FDTD) method, on a structure with dimensions w=200 nm, p=4.6 μm, and t=110 nm and at a wavelength λ=810 nm (see Methods). The height of the slit, h, was assumed to be infinite. Interestingly, the situation is very different when the incident optical field is polarized along the x direction. The Poynting vector for this situation is shown in Fig. 1d. This result shows that the Poynting vector predominantly follows a looped trajectory such as that schematically represented by the solid path in Fig. 1c. The resulting far-field interference pattern, shown as the solid curve in Fig. 1e, is an example of the interference between a straight trajectory and a looped trajectory. Thus, it is clear that the naive formulation of the superposition principle does not provide an accurate description for the case where near fields are strongly excited.

Experimental implementation

First, we experimentally verify the role that looped trajectories have in the formation of interference fringes. For this purpose we exclusively illuminate one of the three slits. This experiment is carried out in the set-up shown in Fig. 2a. As shown in Fig. 1f, no interference fringes are formed when heralded single-photons illuminating the slit are y-polarized. Remarkably, when the illuminating photons are polarized along the x direction the visibility of the far-field pattern is dramatically increased, see Fig. 1g,h. This effect unveils the presence of looped trajectories. In our experiment, the contributions from looped trajectories are quantified through the Sorkin parameter by measuring the terms in equation (3). To this end, we measured the interference patterns resulting from the seven arrangements of slits depicted in Fig. 2b, thus the illumination field fills each arrangement of slits. In our experiment, we use heralded single-photons with wavelength of 810 nm produced via degenerate parametric down-conversion (see Methods). The single photons were weakly focused onto the sample, and the transmitted photons were collected and collimated by an infinity-corrected microscope objective (see Fig. 2c). The resulting interference pattern was magnified using a telescope and recorded using an intensified charge-coupled device (ICCD) camera, which was triggered by the detection event of the heralding photon22. The strength of the near fields in the vicinity of the slits was controlled by either exciting or not exciting surface plasmons on the structure through proper polarization selection of the incident photons.

Figure 2: Experimental set-up utilized to measure exotic trajectories of light. (a) Sketch of the experimental set-up used to measure the far-field interference patterns for the various slit configurations. (b) The seven different slit arrangements used in our study. This drawing is not to scale; in the actual experiment each slit structure was well separated from its neighbors to avoid undesired cross talk. (c) Detail of the structure mounted on the set-up. The refractive index of the immersion oil matches that of the glass substrate creating a symmetric index environment around the gold film. Full size image

The scanning electron microscope images of the fabricated slits are shown in the first row of Fig. 3. The dimensions of the slits are the same as those used for the simulation in Fig. 1, with h=100 μm being much larger than the beam spot size (∼15 μm). The interference patterns obtained when the contribution from near-field effects is negligible (y polarization) are shown in the second row, while those obtained in the presence of a strong near fields in the vicinity of the slits (x polarization) are shown in the third row. These interference patterns are obtained by adding 60 background-subtracted frames, each of which is captured within a coincidence window of 7 nsec over an exposure time of 160 s (see insets in Fig. 3). Only the pattern for P AB is shown in Fig. 3 because P AB and P BC produce nearly identical patterns in the far field, a similar situation occurs for P A , P B and P C . The bottom panels show detail views of the interference patterns measured along an horizontal line.

Figure 3: Experimental results. (a–d) Measured interference patterns corresponding to the various probability terms in equation (3) (indicated as a label within each panel of the bottom). In this case the illumination field fills each arrangement of slits. The first row shows scanning electron microscope images of the slits used for the measurements, the scale bar represents 500 nm. The second and third panels show, respectively, the background-subtracted interference patterns formed when 60 frames, such as those in the insets are added, for the situations in which the probabilities of looped trajectories are negligible (using y-polarized illumination), and when such probabilities are increased due to the enhancement of near fields (using x-polarized illumination). Each of the frames shown in the insets was taken with an ICCD camera using heralded single-photons as a source. The bottom show the intensity dependence of the interference pattern measured along a horizontal line on the second and third panels. The ratio of the average probabilities obtained using x-polarized illumination to those obtained using y-polarized illumination, P x /P y , is shown at the bottom. All the measurements are conducted at a wavelength λ=810 nm, and using structures with dimensions w=200 nm, h=100 μm and p=4.6 μm Full size image