SpinâOrbit Interaction of Light in Plasmonic Lattices - Nano Letters

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Spin-orbit interaction of light in plasmonic lattices Shai Tsesses, Kobi Cohen, Evgeny Ostrovsky, Bergin Gjonaj, and Guy Bartal Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.9b01343 • Publication Date (Web): 02 May 2019 Downloaded from http://pubs.acs.org on May 4, 2019

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Spin-orbit interaction of light in plasmonic lattices Shai Tsesses†,*, Kobi Cohen†, Evgeny Ostrovsky†, Bergin Gjonaj †,§ and Guy Bartal†

† Andrew and Erna Viterbi Department of Electrical Engineering, Technion – Israel Institute of Technology, 3200003 Haifa, Israel KEYWORDS: Surface plasmon polaritons, topology, spin-orbit interaction, near-field scanning optical microscopy

ACS Paragon Plus Environment

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Abstract

In the past decade, the spin-orbit interaction (SOI) of light has been a driving force in the design of metamaterials, metasurfaces and schemes for light-matter interaction. A hallmark of the spin-orbit interaction of light is the spin-based plasmonic effect, converting spin angular momentum of propagating light to near-field orbital angular momentum. Although this effect has been thoroughly investigated in circular symmetry, it has yet to be characterized in a non-circular geometry, where whirling, periodic plasmonic fields are expected. Using phase-resolved near-field microscopy, we experimentally demonstrate the SOI of circularly polarized light in nanostructures possessing dihedral symmetry. We show how interaction with hexagonal slits results in four topologically different plasmonic lattices, controlled by engineered boundary conditions, and reveal a cyclic nature of the spin-based plasmonic effect which does not exist for circular symmetry. Finally, we calculate the optical forces generated by the plasmonic lattices, predicting that light with mere spin angular momentum can exert torque on a multitude of particles in an ordered fashion to form an optical nano-motor array. Our findings may be of use in both biology


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and chemistry, as a mean for simultaneous trapping, manipulation and excitation of multiple objects, controlled by the polarization of light.


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In recent years, the spin-orbit interaction (SOI) of light1,2 has become an area of avid research, owing to its fundamental significance and practical implementations. Among its many applications, the SOI of light has greatly influenced the design of optical metamaterials3 and particularly metasurfaces4–7, while producing new mechanisms for light-matter interactions8,9. Interestingly, many variants of the SOI of light involve an evanescent electromagnetic field, such as the one created by total internal reflection10 (TIR) or surface plasmon polaritons11–14 (SPPs). The unique relation between the momentum and polarization of light in this regime15,16 enables fundamentally different optical SOI phenomena than in free space. A prime example of a spin-orbit interaction with SPPs is the spin-based plasmonic effect17 (SBPE), converting free-space propagating light with spin angular momentum (i.e., carrying circular polarization) to an SPP mode with orbital angular momentum (OAM). The OAM of the mode manifests in its out-of-plane electric field distribution as an azimuthally accumulated phase of 2 m around the mode's center, where the integer m is known as the topological charge (TC).


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Since its discovery, the SBPE has been examined using circularly symmetric excitation slits18–23 and utilized for several functionalities, including control over optical singularities24, microparticle manipulation25 and generation of plasmonic vortex lattices26. The latter are a variant of two-dimensional periodic plasmonic fields27, created by impinging light with optical angular momentum on polygonal excitation slits28. However, no spin-based effect controlling the lattice topology – the topological charge of each lattice site – has been observed thus far. Here, we discover a new plasmonic SOI in systems with dihedral symmetry, converting the spin angular momentum of light into a lattice of plasmonic topological defects. Using hexagonal symmetry, we demonstrate four distinct lattice configurations, measured via phase-resolved near-field microscopy, and switch between two lattice topologies by changing the incident polarization. We further show the general, cyclic response of the SBPE in a non-circularly symmetric system. Calculating the optical forces each lattice topology exerts, we illustrate how a hexagonal array of optical nano-motors may be created by light with mere spin angular momentum. Our findings could propel a plethora of applications in biology and chemistry, including simultaneous near-field trapping and


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manipulation of nanoparticles or a concurrent tailored interaction with a large number of quantum emitters. We begin by describing the SBPE in a circularly symmetric excitation slit, which results in an electric field of the form17,24:

Ez   Ae 

 kz z

J l   kPr  e j l

 

(1)

Where z is the out-of-plane direction; r ,  are the radial and azimuthal coordinates;

Ez  is the out-of-plane electric field for z  0 and r  r0 , in the frequency domain ( r0 is the minimal slit radius); A is the field amplitude; kP, k z are the transverse and longitudinal components of the wave vector ( kP is real and k z is imaginary), such that kP2  k z2  k02 ( k0 is the free-space wavenumber);   1 is the handedness of the incident light, where a positive (negative) contribution represents left (right) circular polarization; l is an integer, representing the TC value of a spiral excitation slit with a varying radius R    r0  l / kP ; and J l  is the  l    -order Bessel function of the first kind.


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Figure 1. The spin-based plasmonic effect in circular and polygon excitation slits. Above are calculated phase maps of the out-of-plane plasmonic electric field, excited by circularly polarized light (   1 ), using (a) a circular excitation slit ( l  0 ), (b) a hexagonal excitation slit ( L  0 ) and (c) a hexagonal excitation slit with shifted edge positions ( L  1 ), to create a different relative phase between them. Color bar is inset (a), scale bar is inset (b) and is the size of a single plasmonic wavelength. It is quite clear that in (a) there is a single 2 phase accumulation around the center, whereas in (b) there is a 2 phase accumulation around the center of each lattice site. In (c), a 4 phase is accumulated around each lattice site.


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Examining Eq. 1, it is evident that at the center of the excitation slit, the TC of the generated field is exactly l   (see Fig. 1a), resulting in each circular polarization having its own topological charge, with an ever present variance of 2 between them. In contrast, a polygon excitation slit creates in its center an interference of N surface plane waves, where N is the number of polygon edges. Therefore, the electric field acquires the form28:

Ez   e

 kz z

N

E e  e n 1

j

n

n

 jkPcos n  x  sin  n  y 

Where x, y are the Cartesian in-plane axes;  n  

(2)

 n is the angle of propagation of N

the nth surface wave, relative to the x axis; n is the relative phase of the nth surface wave; and En is the amplitude of each surface wave. For cyclic polygon slits (with their vertices lying on a circle) and a uniform, circularly polarized illumination, En  E0 and n 

26,

 n  N

as a result of the orientation-associated geometric phase of each slit edge under the

circular polarization. As such, the phase accumulated in a round trip about the edges of the polygon slit is translated into a phase accumulation around collective nodes at its central area, forming


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lattice sites with TC  

at their center (Fig. 1b). To generate any arbitrary phase

difference between the surface waves, one can simply shift the location of some polygon edges29, so as to create any required phase n (provided that the propagation losses are small enough). The most interesting case is for n 

 qn , where q is an integer, denoted N

as the lattice topology, so that a lattice is formed with TC  q at the center of each site (Fig. 1c). The integer L appearing in Fig. 1 is the inherent topological charge of the polygon slit, created by shifting the position of its edges in a similar manner to Archimedean spiral slit17. Contrary to the SBPE in circular symmetry, where the number of eigenmodes (Bessel functions of the first kind) is infinite, the number of different possible lattice topologies is limited to the number of edges. Namely, every edge can be thought of as a source for a single surface plane wave, and the number of such plane waves defines the number of the system’s eigenmodes. That said, due to the dihedral symmetry of the cyclic polygon, some eigenmodes may be degenerate (in N  4 , for example29) or be merely mirror images of other eigenmodes (for q  0 ). Eq. 2 is also found to be the general equation for


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the SBPE in an arbitrary cyclic symmetry, as it degenerates exactly to Eq. 1 in the case of N   . The polygon excitation slit imposes its symmetry onto every lattice site, thus requiring that only polygons tessellating the two-dimensional space ( N  3, 4, 6 ) may generate a periodic lattice. By using other polygons ( N  5, 7,8,... ), one can generate a quasi-periodic lattice, which will rapidly resemble the circularly symmetric case with an increasing number of edges. As the dihedral symmetry permits only 1 and 2 non-degenerate eigenmodes for polygons of N  3 and N  4 , respectively, N  6 constitutes the most topologically-diverse case of a periodic lattice - 4 non-degenerate eigenmodes. Therefore, we opt to experimentally address in this work only hexagonal excitation slits.


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Figure 2. Samples and experimental system. (a)-(d) SEM images of the different hexagonal excitation slits used in this work. (a) no edges are shifted ( L  0 ), (b) the bottom edge is shifted by half the plasmonic wavelength ( L  1 ), (c) the top-left and bottom right edges are each shifted by a third of the plasmonic wavelength ( L  2 ) (d) the top and bottom edges are shifted by five and three quarters of the plasmonic wavelength, respectively ( L  3 ). The shift direction is marked by the yellow (dashed) arrow, and a scale bar is inset (a). (e) Illustration of the experimental setup. Surface waves are excited from the slit on the gold side and propagate towards its center. The near-field signal is


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then scattered by the s-NSOM system and is interfered with the incident light on the detector. BS1,BS2 – beam splitter; L – weakly focusing lens; M – mirror; PM – parabolic mirror; /4 – quarter waveplate.

Figure 3. Experimental results and relevant simulations. (a)-(d) simulated and measured amplitude (top) and phase (bottom) of plasmonic lattices created by the SBPE in hexagonal symmetry, for lattice topologies (a) q  0 , (b) q  1 , (c) q  2 and (d) q  3 . In (a),(c), the impinging light handedness was   1 , whereas in (b),(d), the handedness was   1 . Each figure also contains an SEM image of the corresponding excitation slit


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used to generate the plasmonic lattice. Vortex points are encircled for convenience, scale bars are given in (a) and the presented area of the lattices is marked on each slit image (square).

The experimental setup is presented in fig. 2. The samples under investigation consist of four hexagonally-shaped excitation slits carved in a 200 nm Au layer. The hexagons differ by the shifting of one or two of their edges, determining the topology of the resultant plasmonic lattice. While a pure circular polarization requires a gradual shift of every edge of the hexagonal slit to generate the desired topological charge, the illumination in our system was not perfectly circularly polarized (~80% circularly polarized), hence the design was modified to support the actual polarizations, as detailed in fig. 2a-d. The gold layer is deposited by e-gun evaporation (EVATEC ltd.) atop a 1 mm glass cover slip, using a 3-nm Ti adhesion layer. The slits are comprised of six gratings, 5 m wide and 2.7 m long, carved using focused ion beam milling (FEI Helios NanoLab DualBeam G3 UC). The grating lines are 150 nm broad and an SPP wavelength (636 nm) apart. The shift in relative distance between each pair of opposite gratings causes only


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minimal amplitude asymmetry thanks to the relatively long SPP propagation length in this system. The measurement system used is illustrated in fig. 2e – a scattering near-field scanning optical microscope (neaSNOM, Neaspec ltd.), which is capable of extracting both the amplitude and phase of the out-of-plane electric field, via a pseudo-heterodyne measurement30. The samples are illuminated with a circularly polarized 660 nm continuous wave laser beam (Cobolt Flamenco). The resolution limit of the measurement is determined by the size of the scattering tip apex (8-15 nm in our case). Fig. 3 presents all the possible lattice topologies in hexagonal geometry (barring q  1, 2 , which are mirror images of q  1, 2 , respectively). The figure shows both the

measured near-field of each lattice topology (achieved using the appropriate excitation slit appearing in fig. 2a-d) and its corresponding simulation via the Huygens principle method31. Fig. 3a presents the q  0 hexagonal lattice topology, having a constant phase at the center of every lattice site. As recently shown32, this topology constitutes a lattice of optical skyrmions, exhibiting periodic polarization singularities33 in the orientation of its in-


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plane field. Fig. 3b depicts the q  1 hexagonal lattice topology, giving rise to a closelypacked array of optical vortices34 featuring periodic phase singularities. In this manner, this lattice is actually a periodic implementation of photonic spin skyrmions35, which are topological defects in the local optical spin field. The higher order lattices visibly show a Kagome lattice for the q  2 topology (fig. 3c) and a honeycomb lattice for the q  3 topology (fig. 3d)36, instead of a hexagonal lattice. However, the phase accumulation around the center of each lattice site persists (  in fig. 3c and 6 in fig. 3d), becoming ever more discrete, to the point where the phase changes by steps of  for q  3 . It is also evident that the topological charge in these lattices splits due to the inherent instability of higher order phase singularities37. All images shown in Fig. 3 present about a quarter of the lattice area created in our samples, with its maximal size limited only by the plasmonic propagation length. To demonstrate the generalization of the SBPE, we excite the hexagonal slits using either left or right circularly polarized light. In doing so, the resulting lattice topology takes the following form, which exhibits some similarity to the SBPE in circular symmetry:


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 L    modN q  L    modN  N

 L    modN  N / 2  L    modN  N / 2

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(3)

Where L is the topological charge of the polygon slit;  is the handedness of incident circular polarization; and N is the number of polygon edges. Fig. 4 illustrates this result experimentally, for L  1 with the proper modifications compensating for the non-perfect circular polarization. Intriguingly, Eq. 3 and our results clearly state that the SBPE, in general, is a cyclic effect. In the case of q  N / 2 , negative lattice topologies are excited instead, due to the finite number of eigenmodes in the system. As such, a polygon slit with L 

N exhibits a possible lattice topology difference of N  2 2

between the fields excited by opposite circular polarizations, unlike a constant difference of 2 for Archimedean spiral with any topological charge17,18. The limitation in circular systems stems from the conservation of angular momentum, as the excitation slit imprints a certain orbital angular momentum on the plasmonic field. However, in dihedral symmetry, where a cyclic conservation of the topological charge is required instead of angular momentum conservation, this limitation is lifted.


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Figure 4. Amplitude measurements of a plasmonic lattice of topology q  0 (right) and

q  2 (left), created by impinging an L  1 hexagonal excitation slit with right and left circular polarization, respectively. Brightness in the amplitude measurement represents higher intensity.

Owing to the single, transverse magnetic polarization of SPPs, the in-plane electric field may be derived from the out-of-plane one (Eq. 2), using Maxwell’s equations32:   N k z  cos  n  e jn   jkPcosn  x sin n  y  1  Ex   kz z    jE0 e   jn   e 2  E y   n 1 k||  sin  n  e 

(4)


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Where Ex  , E y  are the Cartesian in-plane electric field components of the plasmonic lattice. However, this representation is not the most appropriate for this problem, as



constructing right circular ( Ex   iE y 



 / 2 ) and left circular (  E    iE    / 2 ) in-plane 

x



y

components is a far more natural selection for photonic spin-based phenomena:     j   N k z  e  n n    jkPcosn  x sin n  y  1  Ex  iE y   kz z    jE0 e   j    e   n n  2  Ex   iE y   2 k n 1 ||  e 

Recalling that  n  

(5)

2 n 2 qn , n  , it is clearly evident that the right (left) circular N N

component has a lattice topology that is larger (smaller) by 1 compared to that of the outof-plane field. Fig. 5 exemplifies this notion, presenting the in-plane field components extracted from a measurement of an out-of-plane field with q  1 . It is important to note that the cyclic behavior of the SBPE, emphasized by Eq. 3, also persists for the in-plane field components.


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Figure 5. Amplitudes of the total in-plane field and of both circular in-plane electric field components, as extracted from the measurement of a q  1 lattice topology in the out-ofplane electric field. The fields were numerically extracted using Maxwell's equations, while filtering the measurement for high-frequency noise to allow derivation. Brightness in the amplitude represents higher intensity.


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Figure 6. Optical forces exerted by plasmonic lattices in a hexagonal slit. (a) out-of-plane force, ubiquitous to all lattices and normalized to the force at the gold-air interface. (b)-(e) vector representation of the in-plane force (larger arrow signifies a larger force), overlaid on the amplitude of the out-of-plane electric field (color bar is inset (b) ). The figures present (b) a hexagonal trapping lattice, (c) a hexagonal rotating lattice, (d) a trapping


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Kagome lattice and (e) a trapping honeycomb lattice. The results in all panels were calculated using the experimental data of Fig. 3. Trapping positions and trajectories are marked for convenience.

Using Maxwell’s equations further enables the extraction of the entire electromagnetic field information (including the magnetic field). Hence, it is possible to calculate the optical forces and torques plasmonic lattices may exert on small particles38,39. Substituting equation 14 of ref. 38 into equations 3,4 of ref. 39 and taking the Rayleigh approximation40 in a non-magnetic medium, the resulting force is:

r 1 r r 2 1       E    Re F   0 Re  medium 2 2





 j   0

 

r



r

    E     H   *

0 medium



(5)

Where  0 , 0 are the vacuum permittivity and permeability;  is the angular frequency   of the SPPs;  medium is the frequency-dependent relative permittivity of the medium in 

which particles reside;  



r r is the particle polarizability; and E   , H   are the electric and

magnetic fields, in the frequency domain.


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Fig. 6 shows the optical forces of the plasmonic lattices in our system, as extracted from the measurements in fig. 3, assuming polystyrene particles in vacuum. Fig. 6a shows how the out-of-plane force, ubiquitous to all lattices, drives particles towards the interface due to the large field gradient and the power dissipation into the metal. Fig. 6b-e present the in-plane optical forces, clearly showing a hexagonal trapping lattice (fig. 6b); a hexagonal rotating lattice (fig. 6c); a Kagome trapping lattice (fig. 6d); and a trapping honeycomb lattice (fig. 6e). The incident optical power required for the aforementioned trapping and rotation is calculated to be on the order of 100 mW or less, such that it is readily available using common lasers. The calculation takes into account the possible coupling efficiency to plasmons41 and a stable trap for particles with sub-micron diameter42. In the circularly symmetric SBPE, the angular momentum in the out-of-plane electric field determines the total transferrable angular momentum of light and the in-plane electric field must preserve it for each handedness (as shown in35,43). Such is the case for the SBPE in non-circular systems, with two exceptions: first, each in-plane handedness in this case must preserve the lattice topology of the out-of-plane field. Second, the lattice topology is not readily transferable to matter and such a transfer only occurs in part of the


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topologies, as evident from Fig. 6. Even so, Fig. 6c shows that a non-zero topological charge can be transferred from each lattice site to matter, allowing the association of the lattice topology with angular momentum density in this case44. In summary, we theoretically investigated and experimentally demonstrated the spinorbit interaction of light in polygon slits. Using hexagonal slits, we converted the circular polarization of propagating light to four different plasmonic lattices, each having a different topological charge at the center of every lattice site. By changing the incident polarization, we switched between two lattice topologies in accordance with the cyclic, general spinbased plasmonic effect. Especially interesting is the unique ability of our system, in certain conditions, to transfer the topological charge of each lattice site from light to matter. In this respect, we show that spin angular momentum alone can suffice to generate an array of optical nano-motors45–47, limited in size only by the propagation length of the SPPs creating it. Our system also supports switching between two different lattices, making it a very diverse tool for applications in biology, chemistry and physics. For example, it could be used to simultaneously trap or rotate many particles, forming new types of metamaterials


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and photonic crystals48; selectively excite different quantum emitters, organized in interleaved arrays49; or interact simultaneously with an electron wavepacket for novel light-matter interaction phenomena50.


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AUTHOR INFORMATION

Corresponding Author * [email protected]

Present Addresses § Faculty of Medical Sciences, Albanian University, Durrës st., Tirana 1000, Albania

Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Funding Sources This research was supported by “Circle of Light,” Israeli Centers for Research Excellence (I-CORE), through the Israel Science Foundation (ISF), grant no. 1802/12.

ACKNOWLEDGMENT The authors wish to thank S. Dolev, G. Ankonina and G. Atiya for their help in sample fabrication; and M. Makhoul and H. Zoabi for their aid in optical force simulation. S.T. also


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acknowledges the generous support of the Jacobs foundation and thanks G. Spektor and M. Orenstein for stimulating discussions.

ABBREVIATIONS SOI, spin-orbit interaction; TIR, total internal reflection; SPP, surface plasmon polaritons; SBPE, spin-based plasmonic effect; OAM orbital angular momentum; TC, topological charge.


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SpinâOrbit Interaction of Light in Plasmonic Lattices - Nano Letters

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