Discrete diffraction in equally spaced DLSPPWs

We start with the simplest case, the investigation of the field evolution of propagating SPPs in two DLSPPW arrays with equally spaced, identical DLSPPWs. In both cases, the sample is placed in the optical setup in such a way that the laser is focused onto one of the grating couplers on the array. By using the knife edge placed in BFP’, the left propagating SPPs as well as the directly transmitted laser beam, are filtered out.

Real space images

As a first step, the resulting real space intensity distribution in a DLSPPW array with dsmall = d_large = d_gap = 600 nm, is imaged on the camera. The known length of the markers is used to calibrate the length scale, leading to an uncertainty of the axes’s values of approximately 2%. Based on this, the coordinate system is defined in a way that the origin coincides with the position of the grating coupler and the SPPs propagate in positive z-direction. The resulting normalized intensity distribution in the array is shown in Fig. 4.32 (a), on a logarithmic color scale.

Consistent with the theoretical considerations, directly at the input (z = 0), most of the intensity is confined in the central DLSPPW. Due to coupling to the neighboring DLSPPWs, the intensity distribution broadens in y-direction with increasing propagation distance. It exhibits two outer lobes of high intensity and a regular intensity pattern in between. Based on the theoretical considerations, this corresponds to the well-known discrete diffraction [28, 31, 101].

In order to compare the experimental image to the theoretical calculation based on the CMT in detail, the theoretically determined course of the main lobes is extracted from Fig. 4.15 (a) and included in Fig. 4.32 (a) as dotted black lines. Several deviations of the experimental intensity distribution can be observed. Firstly, the opening angle of the outer lobes in the experimental image is larger. This is mainly related to an increased coupling as it was predicted by the Comsol Multiphysics based eigenmode expansions. Secondly, the abrupt ending of the intensity distribution in the top part of the experimental image (indicated by the red arrow), as it simply originates from the end of the fabricated DLSPPW array, leading to a reflection of propagating SPPs. Finally, the discrete type of the patterning is blurred out in the experimental image. This stems from the fact, that here the intensity distribution of the total field on the surface is imaged,

z [µm] z [µm]

y [µm]

-3

10 1

Norm. int. [arb.u.]

(a) (b) 0 20 40 -20 -10 0 10 20 0 20 40 -20 -10 0 10 20 y [µm]

Figure 4.32: Discrete diffraction of propagating SPPs in DLSPPW arrays with center to center distances of 600 nm (a), as well as 1000 nm (b). The dotted black lines indicate the form of the calculated intensity distributions, as extracted from Fig. 4.15. The red arrow in (a) marks the edge of the DLSPPW array.

in contrast to the discrete amplitude distribution of the single DLSPPWs, which is displayed in Fig. 4.15 (a).

Based on the theoretical considerations it is further expected that increasing the center to center distances, leads to a reduced coupling and hence a narrowing of the main lobes’ opening angle. In order to test this dependency, the same measurement is repeated on a DLSPPW array with the increased separations given by dsmall = dlarge = dgap = 1000 nm. The resulting intensity distribution is shown in Fig. 4.32 (b), together with the course of the theoretically expected main lobes, extracted from Fig. 4.15 (b).

In comparison to the smaller DLSPPW separations, in the present case the opening angle between both main lobes, generated by the propagating SPPs, decreased. Furthermore, originating from the larger separations, the intensity distribution is less blurred and resembles again the one expected for discrete diffraction [28, 31, 101]. The fact that the opening angle of the main lobes in the experimental image deviates less from the theoretically expected courses, demonstrates the higher accuracy of the CMT at larger DLSPPW separations, due to the smaller coupling constants.

Fourier space images

The following procedure is performed for both systems to investigate the band structure of the excited eigenmodes in the DLSPPW arrays. Without moving the sample, the real space imaging arm of the optical setup is replaced by the single lens, used for the Fourier space imaging. Subsequently, the Fourier space is calibrated, by manually fitting a circle to the circular border of the objective’s back focal plane. The radius of this structure corresponds to the numerical aperture of the objective [34]. Therefore, it can be used together with the center of the circle to define the coordinate system of the Fourier space, by considering (4.48) and (4.49). The error, induced by this procedure, leads to an uncertainty of the coordinate system axes of approximately 2 %. Fig. 4.33 shows the resulting normalized Fourier space intensity distributions of both cases on a logarithmic color scale.

4.5 Experimental results k [1/µm]y k [1/µm]z -3 10 1

Norm. intensity [arb.u.]

k [1/µm]z (a) (b) 6 7 8 -6 -4 -2 0 2 4 6 6 7 8

Figure 4.33: Fourier space intensity distribution corresponding of the plasmonic discrete diffraction, shown in Fig. 4.32 (a) and (b), respectively. The red dotted circles correspond to the border of the objective’s BFP, while the black dotted lines correspond to the result of the Comsol Multiphysics based eigenmode expansions.

Both Fourier space images of the discrete diffraction in a plasmonic system, as depicted in Fig. 4.32, exhibit one band of excited modes, symmetrically arranged around ky = 0. In comparison to the CMT calculations both results resemble the expected behavior as period and band width of the structures decrease with increasing DLSPPW separations. However, in comparison to Fig. 4.16, a clear deviation of the absolute amplitude values is obvious, related to the previously neglected non next neighbor coupling effects.

In contrast, the experimental data is in quantitative agreement with the band structure, calculated with the help of Comsol Multiphysics (see black dotted lines in Fig. 4.33). The remaining small distortion and shift can be explained with deviations of the fabricated sample from the shape assumed in the calculations. Based on the AFM measurements, presented in the last section, the denser packed array has a slightly higher DLSPPW thickness compared to the simulated environment, leading to a higher effective refractive index. In case of the second array, the contrary explanation takes place.