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Abstract
We experimentally investigate the propagation of Airy surface plasmon polaritons (Airy SPPs) on a gold film by multiphoton Photoemission Electron Microscopy (PEEM) at different excitation wavelengths and compare the result with rigorous numerical simulations. The typical bent trajectories of the excited two-dimensional beams are observed and analyzed over a wide range of wavelengths. We furthermore investigate the generation bandwidth of the diffraction grating from modal overlap calculations and evaluate the possibility of creating ultrashort Airy plasmon pulses. This provides a viable route to engineer two-dimensional ultrashort non-diffracting pulsed beams in the field of ultrafast nanophotonics.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The concept of Airy waves was first introduced in the seminal work of Berry and Balazs [1]. Thereafter, it became popular in various fields of physics including optics [2–7]. Airy beams are a solution of the 1D paraxial wave equation that describes the propagation of light in a homogeneous dielectric medium, where diffraction is limited to one transverse spatial dimension. Such Airy beams are self accelerating, non-diffracting beams that travel along a parabolic trajectory in a medium without any refractive index gradient. They have the interesting property to regain their Airy wave profile even after colliding with a finite sized obstacle. In contrast to other non-diffracting beams such as transverse Bessel, Mathieu, and Weber beams [8], Airy beams can exist only in a planar system, which renders them particularly attractive for plasmonics or other systems supporting surface waves.
Surface plasmon polaritons (SPPs) are coupled electromagnetic oscillations of photons and electrons, which are bound to a metal-dielectric interface [9]. SPPs with the properties of Airy beams have been proposed and experimentally verified [10–13]. These so called Airy surface plasmon polaritons (Airy SPPs or just Airy plasmons) propagate diffraction free along a parabolic trajectory on a metal-dielectric interface.
2. Experimental characterization of Airy surface plasmons using PEEM
In our work, we have investigated the spatial dynamics of Airy plasmons using multiphoton Photoemission-Electron Microscopy (PEEM) over a wide range of wavelengths. PEEM is an experimental tool to directly visualize and measure the electromagnetic fields at a metal surface through photoemission [14,15]. The detected photoemitted electrons provide a map of the electromagnetic field at the surface. PEEM can be further combined with ultrafast laser excitation to study the spatio-temporal evolution of pulsed Airy plasmons. In this context, we have further investigated the propagation dynamics of pulsed Airy plasmons and their temporal evolution in space based on a numerical model. This concept has potential applications in ultrafast nanophotonics [16] and can be employed in combination with 2D materials [17,18]. One such application can be pulsed Airy plasmons on graphene surface in which active control over the propagation dynamics can be achieved by tuning the Fermi energy of graphene [19].
A diffraction grating (Fig. 1) based on the idea of Minovich et al [12] has been used to excite the Airy plasmons. The grating couples the exciting plane wave, which is incidenting from the free space on the grating, to SPPs and simultaneously imprints the Airy wave profile. In z direction, the grating structure consists of 11 periods of slit arrays whose period $p$ in the z direction has been adjusted to 745 nm. The slit size in the z direction is fixed to 200 nm. In x direction the slit size and therefore also the local period varies and is determined by the distance between the roots of the Airy function. For the slits of the first column the size in x direction is set to 2${x_0}$, where ${x_0}$ is the full width of the main lobe of the Airy profile and is set to $x_0= 0.7\mu$m. The grating pattern was milled by a Focused Ion Beam into a 200 nm thick polycrystalline gold film, which was deposited on a fused silica substrate by evaporation (see inset in Fig. 2(a) for an SEM micrograph).
Fig. 1. Schematic of the sample layout: The grating is excited from top (air side) at an angle of incidence of $4^{\circ }$ from the normal along the propagation direction (z-axis). Airy plasmons are generated at the air-gold interface. An image of an analytically calculated non-paraxial Airy plasmon is overlaid with the sample layout in the section of the homogeneous gold surface after the grating. The grating is composed of 11 periods of a slit pattern of which only 3 periods are shown in the layout. The slits are 200 nm thick (in the z direction) and have varying length (in the x direction) in accordance with the roots of the Airy SPP’s field profile. The grating period in z direction is $p=745\,nm$. Each period of the grating is composed of two elements to imprint the initial phase profile of Airy wave.
Fig. 2. (a) Logarithmic plot of the experimentally measured PEEM electron yield of the Airy plasmons at inclined ($4^{\circ }$) incidence illumination with a wavelength of $\lambda =700$ nm. The image is overlaid with an SEM micrograph of the fabricated gold grating structure, which was used for exciting the Airy plasmons (b) Logarithmic plot of the numerically calculated distribution of total intensity of the Airy plasmons 5 nm above the gold-air interface at a wavelength of 700 nm. Here, the total intensity is obtained by interference of incident plane wave and scattered surface plasmon polaritons from the grating.
We used a PEEM from Focus GmbH, Germany to study the spatial dynamics of plasmon polariton propagation from the grating by imaging the photoelectrons emitted from the multiphoton interference of the plasmon polaritons with the exciting electromagnetic field. For the excitation light source we used a home-built tunable optical parametric chirped pulse amplifier, which provides an average output power of 100 mW at the sample position with a repetition rate of 1 MHz, corresponding to a pulse energy of $0.1\,\mu$J. A TM polarized wide Gaussian beam of diameter $80\,\mu$m is used to illuminate the grating with it’s wavevector placed in yz-plane and having an angle $4^{\circ }$ with the y axis. The central wavelength of the excitation was varied between 670 nm and 840 nm. The pulses had a spectral bandwidth of 15 nm and a Gaussian envelope with a pulse length of 500 fs [FWHM]. The total exposure time for one image was 20 minutes. The rate of emitted electrons per laser pulse is small enough in order to suppress the detrimental interaction between the photoemitted electrons and was optimized to give the best signal-to-noise ratio in the experiments.
Large scale 3D simulations have been performed to optimize and theoretically verify the design. Computation is done using a Finite Difference Time Domain (FDTD) method [20]. We have used the commercially available Lumerical FDTD solver [21] for our simulation purpose. We have used a TM polarized plane wave inclined by $4^{\circ }$ from the normal in the yz-plane and the fields are extracted in air 5 nm above the gold-air interface. The dielectric permittivity of gold is taken from Johnson and Christy [22]. The plots in Fig. 2 are shown on a logarithmic scale. Qualitatively, there is a good agreement between the electron yield distribution in the experiment and the simulated total light intensity distribution. Both plots capture similarly the details of the generated Airy plasmons and the interference pattern with inherent phase information.
Figure 2 displays the qualitative comparison of experimental findings and numerically simulated results for an excitation wavelength of $\lambda =700$ nm. An overlaid SEM micrograph of the fabricated grating structure with the experimentally obtained image by the PEEM is shown in Fig. 2(a). Airy plasmons are launched and propagate along the air-gold interface. The generated plasmons simultaneously interfere with the incident inclined plane wave at $4^{\circ }$ with respect to surface normal in the yz-plane. The resulting field is mapped by the photoemitted electrons from the gold surface. The measured electron yield by the multiphoton PEEM identifies the regions of constructive and destructive interference and provides a clear contrast between the higher and lower photoemission yield. The map clearly shows the non-diffracting, self accelerating property of the Airy plasmons. The main lobe of the Airy plasmon propagates along a parabolically curved trajectory over a distance of at least $20\,\mu$m.
Furthermore, it is important to comprehend the effect of a change in the excitation wavelength on the propagation dynamics. The wavelength dependence of the Airy plasmon’s excitation and the spatial propagation dynamics is shown in Fig. 3 over the wavelength range from 670 nm to 770 nm at 20 nm wavelength intervals for the same excitation grating as in Fig. 2. The qualitative comparison of the measured PEEM yield and numerical simulations shows a good agreement. The wavelength tuning changes the trajectory of the generated plasmon polaritons.
Fig. 3. Qualitative comparison of experimental and simulated Airy plasmon polaritons at $4^{\circ }$ incidence: (a-c, g-i) Experimentally observed PEEM yield at wavelengths from 670 to 770 nm for wavelength steps of 20 nm. The log-scaled color bars are individually adjusted for the different wavelengths due to the nonlinear n-photon process in the photoemission. (d-f, j-l) Numerically calculated total light intensity 5 nm above the gold-air interface at the same wavelengths as for the PEEM experiments. The experimental electron yield distributions show a good agreement with the numerically calculated total intensity distributions at the different wavelengths, both indicating a travelling of the Airy SPPs along curved trajectories. At 670 nm, we observe generation of a weak lobe indicated by an arrow (Fig. 3(a),(d)), due to the excitation of second order modes in the slits of the grating. The effect diminishes at the longer wavelengths.
Experiments and simulations even agree in minor details. As an example a careful observation near the first column of the grating shows the generation of a weak lobe at a wavelength 670 nm (indicated by an arrow (Fig. 3(a), (d))). It is a signature of a localized second order resonance in the individual slits of the first column of the excitation grating. This second order resonance diminishes for longer excitation wavelengths. We also observe a spatial shift of the Airy plasmon generation towards the longer slits as the wavelength increases. Furthermore, at longer wavelengths the effect of non-paraxiality on beam propagation becomes more evident. This is clearly manifested by the breaking of the main lobe along the propagation direction at wavelength 750 and 770 nm (Fig. 3(k), l) [23].
In the experiments, it can be seen that the electron yield starts decreasing significantly beyond 750 nm (Fig. 3(h), i). This can be explained by the fact that the photoemission in multiphoton PEEM is a nonlinear process and is very sensitive to the surface properties of the material. The work function of the evaporated gold used in our facility is about 5 eV. The incident photon energy is 1.65 eV at 750 nm. This clearly requires a 3 photon absorption process for the electrons to reach the vacuum state. The photoemission intensity at 3 photon process is proportional to the 6th power of the instantaneous total electric field integrated over time [24]. This nonlinear photoemission process results in low electron yields in the experiment. For even longer wavelengths, the photon energy becomes smaller and 4 photon absorption is required for photoemission. As a result, the electron counts in the photoemission becomes so small that we do only faintly observe the propagating plasmons for a wavelength of $\lambda =770$nm (Fig. 3(i)).
3. Generation bandwidth of Airy plasmons
Figure 3 clearly shows that the experimentally observable generation bandwidth is limited at the long wavelength side, which, as explained above, is due to the lower photon energy being insufficient to extract photoelectrons in the PEEM. This sets a limit to our experimental studies of ultra-short pulse excitation of Airy plasmons, which would require a broad bandwidth of the investigated Airy plasmon pulses. However the good correspondence of our experimental finding to the rigorous numerical simulations in Figure 3 justify to extrapolate the studies to a broader excitation bandwidth based on numerical simulation. Thus, we are now going to estimate numerically the bandwidth of the Airy plasmon’s efficient excitation. In addition, we also study the effect of the angle of incidence on the generation efficiency, which can be further utilized in pulsed Airy plasmons investigation in a later section. For this purpose, we calculated quantitatively the generation of Airy plasmons over the wavelength range of 600 nm to 1100 nm under normal and oblique ($4^{\circ }$) incidence. An overlap integral analysis is performed to quantify the Airy plasmon’s generation efficiency ($\eta$). It measures the overlap of our numerical solution with an analytically derived Airy wave profile. This is done using the following overlap integral formula
Analytical electric and magnetic fields are calculated by developing a simple formulation. We assume a TM polarized plane surface wave bound to the gold-air interface in the xz-plane at y=0 and decaying exponentially away from the interface along the the y-direction. The material above the interface (air) has the permittivity $\varepsilon _{I}$ and the material below the interface (gold) has the permittivity $\varepsilon _{II}$. The entire distribution of the TM-polarized surface plasmons in real space is then described as
Figure 4 depicts the resulting dependence of Airy plasmon’s generation efficiency on the wavelength of excitation under normal incidence (blue curve) and $4^{\circ }$ oblique incidence (red curve) calculated by the above model. For the given angles of incidence we investigated the effect of the grating scattering on the Airy plasmon formation. The scattered field from the grating can be assumed to be made of Airy plasmon modes and non-Airy plasmon modes. The idea here is to calculate the Airy plasmons mode’s contribution to the scattered fields. We did the analysis using forward propagated analytical fields [25–27]. For wavelengths from 600 to 1100 nm the overlap is calculated between the numerically simulated fields at a propagation distance of $z=10\,\mu$m and the analytically evolved fields. We find that for normal incidence the maximum generation efficiency is $\eta \approx 58\%$ at a wavelength of 790 nm. For oblique incidence the efficiency increases to $\eta \approx 64\%$ for a wavelength shifted to 820 nm.
Fig. 4. Generation efficiency $\eta$ of Airy plasmons under normal incidence (blue curve) and $4^{\circ }$ oblique incidence (red curve), calculated by the developed semianalytical model in equation (1) taking the overlap between numerical plasmons after a propagation distance of $10\,\mu$m and analytically evolved Airy fields. For normal incidence the maximum generation efficiency is found to be $\eta \approx 58\%$ at 790 nm. Vertical dotted lines show the mean (815.81 nm) and standard deviation (124.11 nm). At $4^{\circ }$ incidence the efficiency is $\eta \approx 62\%$ and the maximum is shifted to 820 nm.
4. Spatio-temporal evolution of Airy plasmon pulses
From the discussion above it is clear that the generation bandwidth of the Airy plasmon grating is large. This opens up the possibility to proceed with the investigation of short pulse excitation and to study the spatio-temporal dynamics of Airy plasmons and their evolution over time [2,4,28,29].
Now we will study the spatio-temporal propagation characteristics of Airy plasmon pulses generated at metal-dielectric interfaces by ultrashort pulse excitation with 20 fs (FWHM) Gaussian pulses having a bandwidth of $\approx 40$ nm (FWHM), which easily fits into the excitation bandwidth demonstrated in Fig. 4. These simulations are carried out using again the Lumerical FDTD solver. Since the efficient FDTD simulation of ultrashort pulses with broad spectral width and spatial inclination imposed numerical problems, we investigated the normal incidence case. Nevertheless, from the close correspondence of the results derived from the bandwidth estimation above for inclined and normal incidence we expect that the influence of the inclined excitation on the observed principle physics will be negligible also for short pulses. Thus we set the central wavelength of the 20 fs excitation pulses to 745 nm, which is the center of the plateau of high excitation efficiency for normal incidence in Fig. 4.
The time-integrated intensity of the simulated pulsed Airy SPP excitation and propagation is plotted in Fig. 5(a). It clearly shows that the Airy plasmon pulse preserves the properties of an Airy wave e.g. self accelerating bending of the beam and a non-diffracting nature of the main lobe. The full temporal evolution of such a spatio-temporal Airy wave packet is visualized in the supporting information.
Fig. 5. Simulated propagation dynamics of an Airy plasmon pulse: (a) Time-integrated spatial intensity evolution of the Airy plasmon excited by a 20 fs (FWHM) pulse centered at $\lambda =745$ nm having a bandwidth 40 nm (FWHM). It is noteworthy that despite the different acceleration trajectories of the different spectral components, the Airy plasmon pulse shows a common trajectory. A corresponding movie for the time-resolved spatial intensity evolution is provided in Visualization 1. (b) Time traces of the excitation source (dashed line) and Airy plasmon pulses (solid lines) at different spatial locations along the main lobe. The numbers correspond to the positions indicated in (a). ‘S’ stands for source. (c) Spectral intensity of source and pulsed Airy plasmons. The generation bandwidth of the grating is clearly of the same order as the incident source bandwidth.
We have taken several spatial location along the main lobe to study the temporal evolution of the Airy plasmon pulse. We also observe a high intensity spot along the main lobe at $z=13.41\,\mu$m along the propagation direction (shown as point 5 in Fig. 5(a)). Beyond this distance the side lobes vanish and the main lobe starts to diffract significantly.
Figure 5(b) shows the time traces of the intensity over time for the positions indicated in Fig. 5(a). The dotted curve is the source signal with a 50.24 fs offset. At position 1 ($0.5\,\mu$m from the grating edge), we see an enhancement in intensity over the source signal due to local resonances in the grating holes. This near-field effect vanishes at position 2 and the intensity peak diminishes. After this location, we see an increase in the peak intensity up to the maximum intensity position of the main lobe (position 5). The appearance of a maximum intensity along the propagation direction can be attributed to an initial phase mismatch of the grating period. After this maximum position the main lobe starts diffracting spatially, which is evident from the decrease in peak intensity at positions 6 and 7 in Fig. 5(a). We can also observe this intensity decreases in the transverse direction from point 7 to point 8, characterizing the divergence of the main lobe.
Furthermore, the spectral intensity distribution shown in the Fig. 5(c) does not change significantly. We have also calculated the time bandwidth product ($\Delta t\cdot \Delta {\nu }$), which becomes minimum at the maximum intensity position 5 (see Table 1).
Table 1. Time-bandwidth product of Airy plasmon pulse at different positions
5. Conclusion
In summary, we investigated the Airy plasmon propagation dynamics for a broad range of wavelengths using multiphoton PEEM and rigorous numerical simulations. The experimental and numerical results show good agreement. The photoelectron yield by PEEM clearly maps the constructive and destructive interference of the total electromagnetic field. We have also developed a semianalytical formulation to quantify the Airy like plasmon generation efficiency from the diffraction grating and it is estimated to be $\eta \,\approx 58\%$ under normal incidence and $\eta \,\approx 62\%$ for $4^{\circ }$ inclined incidence. We, furthermore demonstrated the creation of a spatio-temporal Airy plasmon pulse and estimated the time bandwidth product. The Airy plasmon pulse preserves its non-diffracting property along the propagation. These findings can be further utilized to engineer complexly shaped pulsed surface plasmon polaritons for applications in e.g. nanoparticle sensing or manipulation as well as in high speed on-chip optical signal processing.
Funding
German Academic Exchange Service New Delhi; Deutsche Forschungsgemeinschaft (278747906, SFB 1375).
Acknowledgments
The authors thank Matthias Zilk for his contribution to analytical modeling and numerical analysis in this manuscript.
Disclosures
The authors declare no conflicts of interest.
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