Vortex surface plasmon polaritons on a cylindrical waveguide: generation, propagation, and diffraction

The study and application of beams with orbital angular momentum, or vortex beams, are the most rapidly developing areas in modern optics [1, 2]. In particular, the use of vortex beams of different orders is considered as a way to create multiplex transmission lines [3, 4]. Another rapidly developing area is plasmonics, the interest in which has increased significantly recently, especially in connection with the prospects for the generation of spoof plasmons and their application.

To date, a number of studies have been carried out in which vortex beams were used to generate surface plasmon polaritons (SPPs). In work [5], Bessel beams with a wavelength of 141 µm and the topological charges $\ell = \pm 1$ and $\ell = \pm 2$ were used to excite SPPs at a convex gold—zinc-sulfide—air interface by the end-fire coupling technique. In that case, conventional plasmons were generated, and the rotation of the initial vortex beams did not affect the plasmon characteristics, albeit, possibly, the efficiency of their generation depended on the Poynting vector direction at the surface edge.

Illumination of spiral slits in a metal sheet by vortex beams enables the generation on its backside of surface plasmon polariton vortices (see, e.g. paper [6] and review [7]). In this case, the value of the spiral phase of the resultant surface plasmons, which converge to the center, originates from both the geometric structure of the slit pattern and the topological charge and polarization of the illuminating beam.

In papers [8, 9] we proposed three methods for generating plasmons on a simply-connected cylindrical transmission line using vortex beams created by binary phase axicons [10] and speculated on the opportunity of creating multiplex plasmonic lines. To date, a number of studies on the transmission of plasmons and spoof plasmons, mostly with the terahertz time domain technique, were carried out with smooth or corrugated cylinders and cones [11, 12]. Fernández-Domínguez et al [13] experimentally and numerically studied excitation of plasmons with orbital momentum on wires helically grooved at the subwavelength scale. These studies were carried out with wires whose diameter was close to the radiation wavelength.

In the experiments reported in this paper, we generated terahertz vortex surface plasmon polaritons with the topological charges $\ell = \pm 3$ and $\ell = \pm 9$ by the end-fire coupling technique [14] using coherent radiation of a free electron laser, investigated their propagation along an axis-symmetric line, and demonstrated that the waves diffracted from the rear end of the transmission line possessed the same topological charge as the initial wave. It should be emphasized that in our study we used a line whose diameter exceeded greatly the radiation wavelength, and, in contrast to the experiments with nanowires (see, e.g. [15]), no azimuthal electromagnetic modes existed, as it will be shown below.

The experimental setup is shown in figure 1. The Novosibirsk free electron laser [16] generated radiation as a continuous train of 100-ps pulses with a frequency of 5.6 MHz and an average power of up to 30 W at a wavelength of 141 µm, which corresponds to an atmospheric window. The laser beam was linearly polarized and had the Gaussian profile $I(r) = {I_0}\,\exp (-2r^2/w^2)$ with the mode radius w = 12 mm. Rotating a photolithographic polarizer installed in front of a segmented half-wave plate about the optical axis, we were able to transform the Gaussian beam into a vector beam with radial or azimuthal polarization [17]. A binary spiral axicon transformed a beam with a given polarization into a Bessel beam with the orbital angular momentum $E(r) = {E_0}{J_\ell }(\kappa r)\,\exp (i\ell \varphi )$, where ϕ was the azimuthal angle and $\left| \ell \right|$ was equal to 3 or 9 (the axicons were described in [9]). The transverse wavenumber of all beams was κ = 3.1 mm⁻¹. The polarization state of the beam passing through the axicon was conserved. The sign of the topological charge was changed by the rotation of the axicon about the vertical axis.

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We used a polypropylene Fresnel lens having a diameter of 80 mm and a focal length of 75 mm to transform the Bessel beam into an annular beam with a diameter of 10 mm, which is commonly called a 'perfect beam' [18–23]. The characteristics of perfect beams created by different types of axicons were studied in detail analytically and numerically in [24], where it was shown that an annular beam created by a binary spiral axicon and a lens split into a series of spiral segments, the number of which was twice the value of the topological charge.

Figure 1 presents images of the perfect beams recorded in the present experiment with the Pyrocam IV camera in the focal plane of the lens for topological charges equal to −3 and −9. They coincide with high accuracy with the results of simulations. The only difference is that in the experiment we observe an additional peak at the optical axis, which is caused by internal reflections in the axicon. For a silicon binary axicon with the refractive index n = 3.42, by occasional coincidence, the phase difference of the secondary rays reflected from the front and back facets of the axicon in areas of different levels was $\Delta {\Phi _{12}} = \pi (3n - 1)/(n - 1) = 3.84~\pi$ [10]. These rays interfered constructively and part of the energy of the Gaussian beam passed through the axicon as plane wave, which was focused by a lens on the axis of the system.

A perfect beam with the radius [24]

$$\begin{equation} {R_f} = \frac{{\kappa f}}{{\sqrt {{k^2} - {\kappa ^2}} }} \approx \frac{\kappa }{k}f \,, \end{equation} \tag{ 1 }$$

where k is the wavenumber, illuminated the front end of an axisymmetric conducting line, the diameter of which (10 mm) was equal to the average diameter of the beam. The line consisted of a cylindrical section 40 mm long and a 30 mm section gradually tapering to a diameter of 2 mm. The sections were CNC machined from brass. In the experiments, we used both uncoated samples and samples with a deposited zinc sulfide layer with a thickness of 1 µm. The line was kept in a fixed position by membranes transparent to terahertz radiation, made of 30 µm polypropylene films, as shown in figure 1.

The perfect beam passed through the first polypropylene film and diffracted at the cylinder edge, exciting a TM surface electromagnetic wave coupled to oscillations of electrons in the surface layer of the metal (SPPs). The surface plasmon polaritons propagated along the cylinder, 40 mm long, passed through the second film (propagation of SPPs through a dielectric film was studied in [25]), then travelled along the tapered part of the line $(\rho (z\, {\textrm{[mm]} }) = 1.36 {\textrm{acot}} (0.67z) + 1 - 0.13)$, whose length was equal to 30 mm, and reached the rear line end.

The main characteristics of a plasmon are the propagation length along the trajectory L_z and the decay length of the plasmon field L_r . Experimental data for these values for a brass-ZnS-air interface are not available, but they are known for a gold-ZnS-air flat interface [26], for which the SPP attenuation is described by the expression $I(z,x) = I_0\, \exp{[-(z/L_z+x/L_x)]}$, where x is the normal to the surface. The decay length L_x was found to be 10 mm for bare gold and 0.3 mm for gold covered with 1 µm ZnS, which was in good agreement with the calculations by the Drude model. Since the electrical characteristics of gold and brass do not differ much, we will keep these values in mind for further evaluations. According to the theory, in the plane case, the decay length falls down exponentially, while in the cylindrical case, it is described by a modified Bessel function of the second kind. If the decay length is much less than the cylinder diameter (as for a ZnS-coated surface), it can be approximated by the expression $I(r) \approx {I_0}\exp {[ - (r - R)/L_x]}$ (see, e.g. [27]). As for the propagation length of SPPs, it can be assumed to be approximately the same as in the plane case and according to [28] is about 50 mm for a 1-µm thick ZnS-coated gold surface.

Since we could not directly detect the plasmon field on the line surface, the existence of plasmons was confirmed by the characteristics of radiation diffracted at the rear end of the line and recorded using the optical system shown in figure 1. The reliability of this method was demonstrated in experiments with SPPs on flat surfaces [29]. In the present experiments, we observed the generation of plasmons on both the bare brass line and the line with dielectric coating. Since the decay length $2{L_r}$ of the SPP electric field amplitude ($E(r) \propto I(r)^{1/2}$) in the first case is about 20 mm, which is comparable with the geometric dimensions of the line and the structures supporting it, the radiation observed at the line exit was difficult to interpret, and in this paper, we restrict ourselves to the statement that plasmons were generated in this case as well. Below we present the results of the experiments only with samples coated with zinc sulfide.

Images of the rear end of the transmission line, the input end of which was illuminated by vortex beams, recorded by the microbolometer array, are shown in figure 2. Watching the diagram in the upper right corner of figure 1, we can expect three kinds of radiation in this plane: the input free radiation, 'sliding' along the surface of the large cylinder, the radiation losses of the plasmons, and the free wave that originated from the SPPs diffracted at the rear edge. Indeed, with a wide open aperture (right column in figure 2), we observe concentric rings at the surface of the narrow cylinder, the characteristic width of which is about 0.3 mm, which corresponds to the indicated plasmon decay length. The image of the inner ring does not change when the diaphragm diameter decreases to the smallest values (left column in figure 2), when penetration of any type of free radiation into the annular slit is practically impossible. We attribute this radiation to the diffraction of plasmons at the edge of the cylinder and the formation of free waves.

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A decisive evidence that plasmons, which can exist only as a TM wave, are the source of the observed radiation was the dependence of their intensity on the polarization of the radiation illuminating the line. When the polarization of the illuminating beam changed from radial to azimuthal, the output intensity dropped by an order of magnitude. For the same reason, when the front face was illuminated with a linearly polarized beam, radiation was observed only on opposite sides of the circle.

The images obtained for $\left| \ell \right| = 9$ show (see two bottom lines in figure 2) that the observed radiation is modulated in azimuth, and the number of maxima is equal to 18, which corresponds to the number of spirals in the perfect beams illuminating the front end of the line, $\left| {2\ell } \right|$ (see figure 1). This means that plasmons are generated locally, at the points of intersection of the exciting radiation with the edge of the cylinder, and propagate along their trajectories without overlapping with neighboring plasmons (we observed recently [5] a similar behavior of plasmons during their propagation across the generatrix of a cylindrical surface). From numerical calculations, the results of which are shown in figure 6 in [24], it follows that within each lobe, the phase ramps up by π.

The radiation observed outside the first ring is obviously the radiation losses of the plasmons passing the beginning of the tapering part of the cylinder. The loss radiation consists of eighteen sheet-like free waves. They are observed in the output plane as the second ring of 18 points with spiral-like 'tails', twisting in different directions for the exciting beams with different signs of the topological charge. This indicates (see equation (1) below) that we observe vortex beams, and, therefore, we can assume that the linear momentum of plasmons also has a transverse component.

To test this assumption, we measured directly the plasmon rotation using the optical system shown in figure 1 by placing in front of the right edge of the cylinder a metal foil with a rectangular slit 1 mm wide (see figure 3). The segmented waveplate was removed, and the slit was illuminated by a perfect beam, linearly polarized across the slit, which excited plasmons in a limited section of the cylinder. The plasmons propagated along the line, and their radiation from the output end, inverted by the optical system, is presented in figure 3. The images obtained show that the plasmons propagate along the transmission line, rotating in the same direction as the beams that excited them. Now let us compare the numerical results with the calculations.

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The rotation of a well-collimated vortex beam in the paraxial approximation can be described by following equation [30–32]:

$$\begin{equation} \frac{{d\varphi (r,z)}}{{dz}} \approx \frac{\ell }{{{r^2}k}}. \end{equation} \tag{ 2 }$$

Since the speed of propagation of plasmons in the terahertz range is practically equal to the speed of light, we can assume that the pitch angle of the Poynting vector of the generated plasmon is equal to that in the incident wave. Assuming that the azimuthal component of the plasmon velocity is conserved, ${v_\varphi } = c\sin \alpha = \textrm{const}$, and taking into account the geometric constraints on the coordinates of a plasmon propagating along a transmission line of varying diameter ρ (see supplement 1 for details), after simple transformations, we derive the expression for calculating the angle of rotation of the plasmon on the line:

$$\begin{equation} {\Phi _{\textrm{{SPP}}}}(z) = \int_0^z {\frac{{{\upsilon _\varphi }\sqrt {1 + {{[d\rho (z)/dz]}^2}} }}{{\rho (z)\sqrt {{c^2} - \upsilon _\varphi ^2} }}\,} dz\,. \end{equation} \tag{ 3 }$$

Since $\alpha = r d\varphi/dz$, using equations (2) and (3), we find (see supplement 1 for details) that the expected angles of rotation of the trajectory of plasmons with topological charges of 3 and 9 are equal to 18.4^∘ and 55.3^∘, respectively, the rotation through 6^∘ and 18^∘ of them occuring in the initial cylindrical section. It can be seen that the calculated values of the angle of rotation shown in figure 3 are in good agreement with the rotation observed in the experiment.

To study characteristics of the free waves decoupled from the rear end of the transmission line, we recorded images in a number of planes for different distances δ. The results obtained from three independent measurements performed with or without the knife are presented with dots in figure 4. In the experiment without the knife, we detected the rotation from the non-uniformity of intensity over the annular beam. The measured beam divergence was $\gamma = 0.086 = 4.9^{\circ}$ with a half-width of 1.7^∘. The result of integrating (2) with $r(z) = 1.27+0.086\cdot z$ (in mm) is shown in figure 4 with the solid line. Having measured the powers of the input and output beams with the Ophir 3A-P-THz detector, we found that the total efficiency of the conversion of a perfect vortex beam into an output vortex beam, via intermediate rotating SPPs, was about $0.6\%$ for both topological charges, $|\ell| = 3$ and $|\ell| = 9$. Most of the losses are probably the radiation losses on the convex surface of the transmission line, as shown in [28].

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To conclude, in this work, we have experimentally shown for the first time that a vortex beam can be used to create vortex plasmons and transform them again into a vortex beam with the same topological charge. The results of the work can be useful, for example, in creation of multiplex plasmonic communication systems.

All data that support the findings of this study are included within the article (and any supplementary files).

This work was supported by the Russian Science Foundation, Grant 19-12-00103. We are grateful to G N Kulipanov, V A Soifer, and N A Vinokurov for sustaining this work and useful discussions, V S Cherkassky, Yu Yu Choporova, and A K Nikitin for the great contribution in our previous investigations of vortex beams and plasmons, that were used in the interpretation of the results obtained, as well as to I A Azarov, I Sh Khasanov, A G Lemzyakov, Ya V Getmanov, Ya I Gorbachev, M A Scheglov, D A Skorokhod, and O A Shevchenko for technical support. The experiments were carried out at the Novosibirsk Free Electron Laser facility, which is a part of the Shared Research Facility 'Siberian Synchrotron and Terahertz Radiation Center.'