Scattering of partially coherent surface plasmon polariton fields by metallic nanostripe
Abstract
Background
Surface plasmon polaritons (SPPs) are electromagnetic waves that propagate at a metal-dielectric interface. Until recently, monochromatic, fully coherent SPPs have mainly been considered.
Methods
We investigate by numerical simulations the generation and properties of polychromatic, partially coherent SPPs and their scattering from a nanostripe. We use both in-house and commercial codes.
Results
A standing SPP field is created in Kretschmann’s setup between the excitation point and the nanostripe. From the scattered far-field spectrum, all spatiotemporal coherence properties of the SPP field pattern can be deduced. Illustrative examples of such coherence variations are analyzed.
Conclusions
Plasmonic nanostructures produce strong confined SPP fields of widely controllable coherence.
Keywords
Surface plasmon polaritons Partial coherence NanoscatteringBackground
Surface plasmon polaritons (SPPs) are evanescent electromagnetic waves that propagate along an interface between a metal and a dielectric medium [1, 2]. Characterized by strong lateral confinement and large propagation lengths, SPPs have numerous applications in near-field optics and nanophotonics, including plasmonic waveguides and switches, biosensors, and data storage devices [3, 4, 5]. They allow to scale down optical systems to nanometer dimensions [6]. SPPs can be generated at a metal-dielectric interface by several techniques, among them the Kretschmann and Otto configurations [1, 2], metallic grating coupling [7], and confined fields [2]. Fulfilment of a wave vector phase-matching condition is required in most setups. On interaction with subwavelength features, such as grooves or nanoslits, the SPPs convert back to freely propagating optical waves [8]. Over the years, a vast amount of research has been conducted to investigate the fundamental problem of SPP scattering by small structures and defects (see, e.g., [9, 10, 11]).
The SPPs, customarily treated as monochromatic (fully coherent), greatly modify the statistical properties of optical near fields [12, 13, 14, 15, 16]. However, there is an emerging recognition that partial coherence can be an important degree of freedom in controlling SPP distributions [17, 18]. A crucial step was taken quite recently by introducing a general theoretical framework to customize the coherence features of polychromatic SPPs in the Kretschmann setup [19] and a protocol based on point-dipole scattering to recover statistically stationary SPP correlations from far-field spectral information was put forward [20]. Further, a general coherent pseudo-mode representation of partially coherent SPP fields was advanced [21] and planar axicon-like and lattice-type SPP fields of varying coherence states were analyzed [22, 23]. Such plasmon coherence engineering is instrumental for synthesizing structured SPP fields with desired spatiotemporal coherence and polarization properties.
In this work we investigate numerically the scattering of SPPs by a metallic nanoparticle for three slightly different wavelengths of light. The analysis is carried out in two spatial dimensions and the nanoparticle has a square shape, so we call it a nanostripe (the system may be viewed as uniform perpendicular to the plane of analysis). The SPPs are excited onto a metal-air interface in a Kretschmann geometry and subsequently interact with the metallic nanostripe a short distance away. The nanostripe has a dual physical effect: firstly, back-reflection creates standing spectral wave patterns between the SPP origin and the nanostripe, and secondly, part of the SPP energy is scattered into the far zone of the nanoscatterer. A square nanostripe is relatively easy to handle numerically. Moreover, it must be appropriately small to yield forward scattering (yet large enough for sufficient scattering efficiency). From the far-field spectrum we may deduce the relative strengths of the spectral SPP components, which in turn determine the spatiotemporal coherence properties of the polychromatic standing SPP pattern. It is demonstrated that the nanostripe reflections profoundly alter the spatial and temporal coherence, even though the SPP field is fully coherent at each frequency.
Methods
Theory
Surface plasmon polaritons
where c is the speed of light in free space.
where r(ω) is the field reflection coefficient. In it \(n(\omega) = \sqrt {\epsilon _{\mathrm {r}}(\omega)}\) is the (complex) refractive index of the metal nanostripe [24]. The coefficient r(ω) corresponds to reflection of a normally incident plane wave, as is typically the case with SPPs [22].
The relative sizes of these components depend on the metal and the frequency, which determine the SPP wave vector k(ω) and polarization vector \(\hat {\mathbf {p}}(\omega)\).
Spatiotemporal coherence
in which Γ(r_{1},r_{2},τ) is the mutual coherence matrix and τ is a time difference. Expression (15), with E_{SPP}(r,ω) given by Eq. (11), is valid everywhere between the SPP creation point and the nanostripe N, for statistically stationary excitations of any spectral distribution.
Scattered far field
implying that the scattered far-zone intensity multiplied by distance s is a constant that, in general, depends on the scattering direction and the frequency.
Simulation
The simulations of vectorial SPP fields on the metallic surface and their scattering from the nanostripe are performed in 2D by utilizing in-house numerical codes based on the Fourier modal method [29], as well as COMSOL Multiphysics software that employs the finite element method. The SPP excitation in the Kretschmann geometry takes place by means of a perfectly phase-matched focused beam at each frequency. In all our simulations the two different computational methods lead to substantially similar results.
Results and discussion
Spectral intensity determination
SPP excitation and characteristics at three wavelengths on an air-Au interface
j | λ [nm] | θ [ ^{∘}] | λ_{SPP} [nm] | l_{SPP} [ μm] | t_{SPP} [ps] |
---|---|---|---|---|---|
1 | 633 | 45.83 | 603.1 | 12.9 | 0.045 |
2 | 642 | 45.70 | 613.4 | 14.9 | 0.052 |
3 | 650 | 45.60 | 622.3 | 17.0 | 0.059 |
Normalized spectral intensities at SPP excitation in the Kretschmann geometry, obtained by direct simulation and from far-field scattering by a nanostripe
Wavelength [nm] | Direct simulation | Far-field scattering |
---|---|---|
633 | 1.0 | 1.0 |
642 | 0.880 | 0.879 |
650 | 0.790 | 0.781 |
According to Eq. (17), the scattered far-field intensity multiplied by the distance s from the nanostripe N acquires a constant value, independent of s. This provides, in principle, an experimental means of recovering the spectral SPP intensities at the excitation point. With reference to Fig. 2, we evaluate the scattered field intensities along a straight line from N into the far zone, in the center of the second scattering cone. Multiplied by s, this gives the left-hand side of Eq. (17) for the three wavelengths, and their behavior is shown in the inset to Fig. 3. Taking the scattering coefficients |r(ϕ,ω)|^{2} equal for all wavelengths, we thereby obtain the spectral SPP intensity ratios at the nanostripe N. On further accounting for the different SPP propagation losses we again recover the relative intensities I(λ_{j})/I(λ_{1}), j=1,2,3, at the excitation point. These values, obtained via far-field scattering, are likewise given in Table 2. We observe that the agreement between the results from direct simulations and from far-field scattering is excellent, thus confirming the validity of the approach.
Spatial coherence
for the total SPP intensity at point r=(x,0). Using Eq. (20), we consider first the spatial coherence function γ(x_{1},x_{2},0) on the metal surface.
Temporal coherence
where G(x,ω_{j}) is obtained from Eq. (22).
Conclusions
In summary, we have studied by numerical simulations the scattering of SPPs from a metallic nanostripe. The SPPs constitute a polychromatic, statistically stationary, electromagnetic surface field, excited onto an air-metal interface in the Kretschmann configuration through exact phase matching at each frequency. For the simulation we use both in-house codes and commercial software. By comparison with direct calculations we demonstrate, first of all, that detection of the spectrum in the far-zone of the nanostripe allows one to recover the intensities of the spectral SPP constituents and thereby deduce the complete spatiotemporal coherence properties of the SPP field. Secondly, our analyses and simulations show that the presence of the nanostripe leads to the creation of a standing SPP pattern whose spatial and temporal coherence properties differ significantly from those in the absence of the nanoscatterer. Characteristic features of coherence include near periodicity originating from statistical similarity and variations of maxima and minima due to spectrally dependent SPP survival lengths. And finally, although we explicitly assessed red spectral components in gold only, the method is general, i.e., the materials and excitation spectra can be arbitrary providing an opportunity to judiciously tailor the SPP standing-field coherence properties. Such plasmon coherence engineering may find uses in controlled excitation of particles and clusters and in emerging applications of nanoplasmonics technology.
Notes
Acknowledgements
Not applicable.
Funding
This work was supported by the Natural Science and Engineering Research Council (NSERC) of Canada (Grant No. RGPIN-2018-05497), the Academy of Finland (Project No. 310511), and the Joensuu University Foundation.
Availability of supporting data
Not applicable.
Authors’ contributions
The original ideas and results emerged from discussions among all the authors. SD and KS performed the calculations and assisted SAP and ATF in writing the manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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