Beaming photons with spin and orbital angular momentum via a dipole-coupled plasmonic spiral antenna

Guanghao Rui; Robert L. Nelson; Qiwen Zhan

doi:10.1364/OE.20.018819

1. Introduction

When the optical field interacts with a metallic structure with size comparable to the wavelength of the light, unusual optical behaviors, such as extraordinary optical transmission [1], beaming [2], plasmon focusing [3] and superresolution [4], can be observed with many prospective applications. These anomalous effects have been explained by the mechanisms involving the coupling of light to surface plasmon polaritons (SPPs). SPPs are electromagnetic wave propagating along a metal/dielectric interface caused by collective oscillation of free electrons. Due to their high local field enhancement and short effective wavelength, SPPs have been widely explored to manipulate light at subwavelength scale. It is well known that a plasmonic system is resonantly excited when the linear momentum selection rule of SPPs excitation is fulfilled. However, with the rapid progress of spin optics, additional selection rule associated with angular momentum (AM) attributed to the conservation of total AM in a closed physical system received increasing attention in recent years. The total AM of an optical beam can be divided into spin angular momentum (SAM) and orbital angular momentum (OAM). The intrinsic SAM is associated with the polarization helicity, where $σ_{\pm} = \pm 1$ correspond to the right and left handed circular polarizations (RHC and LHC), respectively; while the orbital angular momentum (OAM) arises from the spatial structure of the optical field which can take arbitrary integer values (l = 0, 1, 2…). The handedness of the circular polarization provides additional degree of freedom to a plasmonic system. When the incident spin photons encounter a metallic structure with anisotropic inhomogeneous boundaries, a spin-dependent behavior of plasmonic field could be observed [5, 6]. This phenomenon is due to spin-orbit interaction that is manifested by a geometric Berry’s phase. As one phenomenon predicted by the spin-orbit interaction, the optical spin-Hall effect has been observed experimentally [7, 8].

As another demonstration of the spin-orbit interaction, it has been shown that the Archimedes’ spiral slot plasmonic structure with pitch of one SPP wavelength focuses the RHC and LHC illuminations into spatially separated plasmonic fields as a receiving antenna [9–11]. Reciprocally, a spiral plasmonic structure can serve as a transmitting antenna that couples the radiation of a point emitter into a highly directional circular polarized emission in the far field [12]. However, the emitted photons generated by this specific structure do not carry OAM because only SAM transferring between the SPPs and photons is explored. To generate free-space optical vortex, several plasmonic techniques has also been proposed [13, 14]. However, these techniques do not deal with SAM. In addition, the requirement of a plane wave illumination restricts the perspective of device miniaturization with these techniques. In this work, we extend this concept and show the feasibility of constructing a nanoscale circular polarized source with variable OAM states through coupling an electric dipole emitter to a spiral optical antenna with different pitches. In order to provide more insight into its characteristics, a one-turn spiral structure is used to derive an analytical expression for the far field radiation of the coupled system based on the Huygens–Fresnel principle. Similar approach was adopted by others to study the far field radiation of plasmonic structures [15, 16]. The radiation characteristics predicted by the analytical expression are confirmed by finite element method numerical modeling. It is demonstrated that the SAM carried by emitted photons is determined by the spiral antenna handedness and the OAM of the emitted photons can be varied by changing the pitch of the spiral structure.

2. Analytical expression of the far field emission

The proposed structure and the coordinates for calculation are illustrated in Fig. 1 . A right-hand Archimedes’ spiral (RHS) slot is etched into a thin metal film deposited on glass substrate. An electric dipole is located in the geometrical center with the oscillating direction normal to the spiral surface. In its local cylindrical coordinates, the RHS structure can be described as

r = r_{0} + \frac{m λ_{s p p}}{2 π} θ,

where

r_{0}

is a constant and

λ_{s p p}

is the wavelength of SPPs on the metal/air interface, θ is the azimuthal angle and m can take arbitrary integer value. The SPPs excited by the field radiated from the electric dipole propagate to the spiral slit and are guided through the slot opening to the other side of the metal film and then re-radiate into the free space. Owing to the waveguide effect of the slot opening, the electric field at the exit aperture of the slot is mostly transversely polarized. Considering the orientation of the electric dipole, this electric field is approximately polarized in the radial direction

{\overset{⇀}{e}}_{r}

for a spiral structure that is relatively large compared with the SPP wavelength. Thus, the electric field at point

(r, θ)

at the slot opening on the re-radiation side can be expressed as

g (r, θ) = {\overset{⇀}{e}}_{r} E_{0} e^{i {\overset{⇀}{k}}_{r} \cdot \overset{⇀}{r}}

, where E₀ is a constant and

{\overset{⇀}{k}}_{r}

is the wave vector of the SPP. The subwavelength slot opening can be regarded as an array of secondary sources for the calculation of far field radiation. Consequently, in the Fraunhofer diffraction regime, the electric field at an observation point

(z, ρ, ϕ)

in the far field can be expressed as

\vec{E} (z, ρ, ϕ) = \frac{\exp [i k_{0} (z + \frac{ρ^{2}}{2 z})]}{i λ_{0} z} \int_{0}^{\infty} \int_{0}^{2 π} g (r, θ) \exp [- i \frac{2 π}{λ_{0} z} r ρ \cos (θ - ϕ)] r {\vec{e}}_{r} d θ d r,

where λ₀ is the emission wavelength of the electric dipole and k₀ is its wave vector in air. To study the polarization of the emission in the far field, the total field is decomposed into orthogonal components in the Cartesian coordinate system. The x component of the electric field is

E_{x} (z, ρ, ϕ) = {\vec{e}}_{x} \frac{\exp [i k_{0} (z + \frac{ρ^{2}}{2 z})]}{i λ_{0} z} E_{0} Δ r \int_{0}^{2 π} \cos θ \cdot r e^{i {\vec{k}}_{r} \cdot \vec{r}} \exp [- i \frac{2 π}{λ_{0} z} (r_{0} + \frac{m λ_{s p p}}{2 π} θ) ρ \cos (θ - ϕ)] d θ,

where the slit width

Δ r

is used to simplified the expression to single integral. In principle, the local field re-radiated from the spiral slot depends on the distance from the dipole source due to the propagation loss. For simplicity, we neglect the propagation loss of the SPPs, i.e. Im(k_r) = 0, and then

{\vec{k}}_{r} \cdot \vec{r} = {\vec{e}}_{r} 2 π r / λ_{s p p}

. In addition, the spiral structure is assumed to be large enough with respect to

λ_{s p p}

, so that the size r can be approximated with

r_{0}

when it is not in a phase term. With these assumptions, Eq. (3) can be simplified as:

Fig. 1 Illustration of a right hand spiral structure and the coordinates setup for the far field calculation. The inset illustrates the spiral structure viewed from the left and the coordinates used in the analytical calculation.

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\begin{array}{l} E_{x} (z, ρ, ϕ) = {\vec{e}}_{x} \frac{\exp [i k_{0} (z + \frac{ρ^{2}}{2 z})]}{i λ_{0} z} E_{0} Δ r \times \\ \int_{0}^{2 π} r_{0} \cos θ \cdot e^{i \frac{2 π}{λ_{s p p}} (r_{0} + \frac{m λ_{s p p}}{2 π} θ)} \exp [- i \frac{2 π}{λ_{0} z} (r_{0} + \frac{m λ_{s p p}}{2 π} θ) ρ \cos (θ - ϕ)] d θ \\ = {\vec{e}}_{x} r_{0} \frac{\exp [i k_{0} (z + \frac{ρ^{2}}{2 z})]}{i λ_{0} z} E_{0} Δ r e^{i \frac{2 π r_{0}}{λ_{s p p}}} \int_{0}^{2 π} \cos θ \cdot e^{- i \frac{2 π r_{0} ρ}{λ_{0} z} \cos (θ - ϕ)} e^{i m θ [1 - \frac{λ_{s p p}}{λ_{0}} \frac{ρ}{z} \cos (θ - ϕ)]} d θ . \end{array}

For Fraunhofer diffraction, $z > > ρ$ , so the term $\exp {i m θ [1 - λ_{s p p} / λ_{0} \cdot ρ \cos (θ - ϕ) / z]}$ can be approximated with $\exp (i m θ)$ . We have

\begin{array}{l} E_{x} (z, ρ, ϕ) = {\vec{e}}_{x} r_{0} \frac{\exp [i k_{0} (z + \frac{ρ^{2}}{2 z})]}{i λ_{0} z} E_{0} Δ r e^{i \frac{2 π r_{0}}{λ_{s p p}}} \int_{0}^{2 π} \cos θ \cdot e^{- i \frac{2 π r_{0} ρ}{λ_{0} z} \cos (θ - ϕ)} e^{i m θ} d θ \\ = {\vec{e}}_{x} π r_{0} \frac{\exp [i k_{0} (z + \frac{ρ^{2}}{2 z})]}{i λ_{0} z} E_{0} Δ r e^{i \frac{2 π r_{0}}{λ_{s p p}}} \times \\ [i^{m + 1} J_{m + 1} (- \frac{2 π r_{0} ρ}{λ_{0} z}) e^{i (m + 1) ϕ} + i^{m - 1} J_{m - 1} (- \frac{2 π r_{0} ρ}{λ_{0} z}) e^{i (m - 1) ϕ}] . \end{array}

Similarly, the y component of the electric field in the far field can be found as

\begin{array}{l} E_{y} (z, ρ, ϕ) = - {\vec{e}}_{y} i π r_{0} \frac{\exp [i k_{0} (z + \frac{ρ^{2}}{2 z})]}{i λ_{0} z} E_{0} Δ r e^{i \frac{2 π r_{0}}{λ_{s p p}}} \times \\ [i^{m + 1} J_{m + 1} (- \frac{2 π r_{0} ρ}{λ_{0} z}) e^{i (m + 1) ϕ} - i^{m - 1} J_{m - 1} (- \frac{2 π r_{0} ρ}{λ_{0} z}) e^{i (m - 1) ϕ}] . \end{array}

From Eqs. (5) and (6), both the x and y components of the electric field in the far-field have two terms. Therefore, the total electric field is a superposition of two different modes proportional to the (m-1)^th and (m + 1)^th order Bessel functions, respectively, except for the case of m = 0 when the structure depicted by Eq. (1) is reduced to a ring. For the ring structure, the far-field radiation is a first-order Bessel beam with radial polarization. For a spiral structure with m > 0, it is interesting to notice that both of the two modes possess SAM and OAM simultaneously, but with reversed spin and different topological charge. Using these analytical expressions, intensity and phase distributions corresponding to m = 0, 1, 2, 3 are calculated and illustrated in Fig. 2 as examples. The topological charge can be visualized by inspecting the phase along closed contours within the main lobes shown in Fig. 2. Due to the difference between transverse profiles of Bessel functions with lower (m-1) and higher (m + 1) orders, the main lobe of the emitted source is mostly attributed to the mode obtained RHC polarization and OAM of m-1. However, owing to the overlap between these two modes away from the main lobe, it is hard to separate the mode with LHC and OAM of m + 1 and the total field exhibits hybrid polarization distribution and complicated phase patterns in the side lobes.

Fig. 2 Analytical calculation results for the intensity distribution (a, b, c, d) and E_x phase distribution (e, f, g, h) and E_y phase distribution (i, j, k, l) of the spiral structure with different geometric topological charge m. The charge m varies from 0 to 3 with increment of 1. The green colored closed contours shown on both intensity and phase plots illustrate the topological charge.

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These simple analytical expressions also allow us to examine the directivity of the emission by calculating the radiation angular width of the main lobe. If the main lobe is described by a J₀–Bessel function, the angular width will be evaluated at its full-width-half-maximum (FWHM). For the cases where the main lobe is given by higher order Bessel functions, the angular width will be evaluated by the peak-to-peak intensity angular separation across the dark center. From Eqs. (5) and (6), this calculations will give the values of ρ/z as a function of r₀/λ₀, from which the emission angular width can be calculated. Apparently the directivity depends on the size of the device, with larger r₀ leading to higher directivity and narrower angular width. Assuming λ₀ = 633 nm and a gold/air interface with the SPP wavelength calculated to be λ_spp = 598.8 nm, for a spiral device with $r_{0} = 2 λ_{s p p}$ , the calculated angular width for m = 0, 1, 2, 3 are 17.59°, 14.57°, 17.59°, and 28.68°, respectively. If we double the size so that $r_{0} = 4 λ_{s p p}$ , the corresponding angular width for m = 0, 1, 2, 3 then become 8.85°, 5.44°, 8.85°, and 14.57°, respectively.

3. Numerical modeling with three-dimensional finite element method

The analytical derivation above provides valuable insights into the emission characteristics of the propose dipole coupled plasmonic spiral antenna. However, it is only suitable for a single turn spiral structure that is relatively large compared with $λ_{s p p}$ , and the propagation loss of SPPs is ignored. In addition to making r₀ larger, increasing the number of turns of the spiral structure also offers the opportunity to beam the emitted photons with higher directivity and higher intensity, which is another important characteristic of an optical antenna that could not be studied with the analytical expressions derived above. To take all the effects into consideration and validate the analytical derivation, we perform numerical simulation with a three-dimensional finite element method model (COMSOL). To demonstrate the beaming capability without making the structure excessively large for the numerical modeling, a 4-turn spiral structure with a 200 nm slot width and pitch of $m \cdot λ_{s p p}$ is etched through a 150 nm gold film (n = 0.197 + i3.0908) deposited on a glass substrate (n = 1.5). An electric dipole is placed 5 nm above the center of the spiral with oscillating direction normal to the surface and emission wavelength of 633 nm (shown in Fig. 3 ). The SPPs wavelength is calculated to be 598.8 nm. The spiral parameters in Eq. (1) are $r_{0} = 2 λ_{s p p}$ . Simulation results of far-field emission on the other side of the dipole source for different m are shown in Fig. 4 .

Fig. 3 Diagram of the spiral antenna structure (m = 0, 1, 2, 3) used in the numerical simulation. An electric dipole is located 5 nm above the center of the spiral surface. The oscillating direction of the electric dipole is illustrated by the red arrow. The spiral re-radiates the emission of electric dipole into the free space on the other side of the dipole source.

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Fig. 4 Numerical simulation results for 4-turn spiral structures with different geometric topological charge m. (Top panel: intensity distribution, middle panel: polarization map superimposed on zoom-in intensity distribution, bottom panel: phase distribution of E_x for the main lobe, the area corresponding to the dark center of the intensity is blocked with a solid circle). The topological charge m varies from 0 to 3 with increment of 1. The green colored closed contours shown in both of the intensity and phase plots illustrate the topological charge within the main lobe.

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In this work we focus on the characteristics of main lobe because it dominates the total intensity of the emitted source. Besides intensity distribution, the polarization maps of selected points superimposed on a zoom-in intensity distribution and zoom-in phase map are also shown in Fig. 4. Good agreements have been obtained between the numerical simulation and analytical prediction in terms of the AM conversion for the main lobe. First we consider the 4-ring bulls eye structure, it is a special case when m = 0. The far-field radiation is a donut-shaped first-order Bessel beam with radial polarization, no transfer of AM occurs in this case because of the lack of chirality due to structural symmetry. The angular width between the two peaks of the main lobe is 11°. The dark center of the intensity distribution is due to the π phase-jump shown in phase map. For m = 1, it represents the standard spiral structure that has been recently studied [12]. The far field pattern shows a solid spot with RHC polarization and zero OAM within the main lobe. The emission pattern is highly directional, with a FWHM of 4.9° in the x-z plane and 6.1° in the y-z plane. More details could be found in [12]. For m = 2, the main lobe of radiation in the far field still hold the RHC polarization because the chirality of the spiral antenna is transferred to the spin carried by the emitted photons. Moreover, a donut-shaped first-order Bessel vortex beam with topological charge of 1 has been resulted in the main lobe. The peak-to-peak angular width of the main lobe is estimated to be 8.95°. In order to better illustrate the topological charge, the corresponding area of dark center is blocked with a solid circle in the phase map. The asymmetry of the intensity pattern is due to the anisotropic spiral structure and propagation loss of SPPs. With the value of m further increased, a higher-order Bessel vortex radiation with RHC polarization and larger topological charge is obtained in the far field, which is confirmed by both the analytical derivation and numerical simulation. For example, for the case of spiral structure with charge 3, the emitted photons carry SAM of 1 and OAM of 2 in the main lobe, with the peak-to-peak angular width of the main lobe of 9.63° and 8.95° in the x-z plane and y-z plane, respectively.

Compared with the directivities predicted by the analytical model, we notice that the directivities calculated by the numerically modeling are rather close to the analytical results for the single-turn spiral with $r_{0} = 4 λ_{s p p}$ . This is due to the fact that the 4-turn spiral we used in the numerical modeling increases the effective radiation aperture. The largest differences occurred for m = 3, where the numerical modeling predicts a narrower angular width. This is caused by the large pitch ( $3 λ_{s p p}$ ) of the spiral, which leads to a larger effective radiation aperture and a narrower angular width. Thus the simple analytical expressions we derived can be used to estimate the directivity of the device with the use a suitable effective aperture size.

4. Discussions

In the spiral optical antenna, the spiral corrugation couples the emitted field from the electric dipole into a plasmonic wave and induces a dynamic spiral phase according to the spiral grating pitch. This phase leads to two radiation modes in the far field with the total OAM to be $l = m - σ_{\pm}$ , respectively. Consequently, two AM selection rules exist in this simple plasmonic system as shown in the analytical derivation of Eqs. (5) and (6). The chirality of the spiral structure is converted into SAM of the emitted photons, while the extra topological charge would be converted into OAM. However, the analytical expressions also indicate that, as far as the main lobe is concerned, only one AM selection rule in which the emitted photons possess circular polarization with the same handedness with respect to the spiral antenna structure needs to be considered. This is confirmed by the numerical modeling.

The numerical modeling also confirms the beaming capabilities of the spiral plasmonic antenna, which enables directional emission of photons with both spin and orbital angular momentum. The far field radiation patterns obtained in the numerical modeling do not show perfect rotational symmetry as predicted by the analytical expressions based on a very simple model mainly due to the propagation losses of SPPs. Several optimization methods could be explored in the future to make the intensity distribution of the emitted radiation more symmetric: by enlarging the device size $r_{0}$ or choosing a metallic material with low SPPs propagation loss, the impact raised by the asymmetric spiral structure would be reduced and the directivity can be further increased.

5. Conclusions

In conclusion, we introduced and studied a method to beam photons carrying SAM and OAM through coupling a point source to a spiral optical antenna. Due to the interaction between the spiral antenna and the excited SPPs, a circularly polarized source with various topological charges could be obtained in the far field with narrow angular width. An AM selection rule is proposed that represents a matching of the angular momentum of the emitted photons and the spiral pitch in the units of SPPs wavelength. The SAM and OAM carried by the emitted light are determined by the chirality and pitch of the spiral, respectively. This miniature plasmonic antenna device could control the polarization and the topological charge of the emitted source simultaneously. In addition, it is known that the spontaneous emission from a dipole source could be modified by the nearby metal/air interface due to the Purcell Effect [17], which can enhance the photon generation efficiency. This interesting phenomenon could be very beneficial for the realization of high brightness nanoscale sources and deserves further studies in the future. Such a nanoscale vortex spin photon source proposed in this work may find applications in quantum optical information processing, single molecule sensing, and integrated photonic circuits.

Acknowledgments

G. Rui acknowledges the support by the National Basic Research Program of China under Grant No. 2011CB301802. R. Nelson and Q. Zhan acknowledge the support of the Air Force Research Laboratory’s (Materials and Manufacturing Directorate) Metamaterials Program.

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Beaming photons with spin and orbital angular momentum via a dipole-coupled plasmonic spiral antenna

Abstract

1. Introduction

2. Analytical expression of the far field emission

3. Numerical modeling with three-dimensional finite element method

4. Discussions

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (6)

Optics Express