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Wavefront shaping of plasmonic beams by selective coupling Yuval Tsur, Itai Epstein, Roei Remez, and Ady Arie ACS Photonics, Just Accepted Manuscript • Publication Date (Web): 08 Jun 2017 Downloaded from http://pubs.acs.org on June 8, 2017
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Wavefront shaping of plasmonic beams by selective coupling Yuval Tsur*,1, Itai Epstein1,¥, Roei Remez1 and Ady Arie1 1
Department of Physical Electronics, School of Electrical Engineering, Fleischman Faculty of Engineering, TelAviv University, Tel-Aviv 6997801, Israel ¥ Current affiliation - ICFO – The Institute of Photonic Sciences, Av. Carl Friedrich Gauss 08860, Barcelona, Spain *Corresponding author:
[email protected] Keywords: Two-level system, Plasmon, Wavefront shaping, Coupled modes.
Abstract Custom plasmonic beams are advantageous for numerous scientific and technological aspects. While plasmonic wavefront shaping had traditionally been a truly planar process, taking place on a single surface, here we explore a new method for plasmonic shaping by selectively coupling plasmonic waves between different surfaces of an insulator-metal-insulator structure. In contrary to most previous shaping techniques that rely on free-space illumination, here the plasmonic beam in the buried surface acts as the light source. We demonstrate, both experimentally and numerically, a way to tailor the amplitude and phase of the wavefront using this new technique. The proposed method can be used to efficiently shape the plasmonic beam, for potential applications in sensing, interferometry and communications.
S
urface-plasmon-polaritons (SPPs) are electromagnetic surface waves that propagate at the boundary between a metal and a dielectric, and are coupled with charge oscillations of free carriers1. As they feature an increased spatial confinement yielding very intense electromagnetic fields, they have drawn a considerable attention for applications such as sensing2, trapping3, sub-wavelength optics4–6 and more. Moreover, they offer interesting possibilities for on-chip communication and modulation7,8 as highbandwidth components with physical dimensions smaller than the diffraction limit. For these applications, it is often required to control the shape of the plasmonic beam, and subsequently various methods were demonstrated for this purpose, including shaping the coupling of a free-space light beam to the plasmonic mode9,10, shaping the free space illuminating beam itself11, or using metallic and dielectric elements to mold the structure of the propagating plasmonic beam12,13. However, some of these methods involve undesired excess energy at the plane of propagation acting as noise and reducing the device’s efficiency, while others lack complete amplitude and phase manipulation abilities. Here we explore a different concept, in which the plasmonic beam is shaped by selectively coupling two parallel metal-dielectric interfaces of an insulator-metal-insulator (IMI) structure. If the dielectric permittivity at the top layer is significantly different than that of the bottom layer, the plasmonic waves that propagate at the two boundaries are practically uncoupled. However, by placing a small ridge at the top layer having the same permittivity as the bottom layer, (See Figure 1A), the two plasmonic modes become coupled over the ridge’s span. Specifically, assuming we start with a plasmonic beam that propagates only at the bottom interface, the length of the top ridge controls the amplitude of the plasmonic beam at the top interface, whereas its relative position can induce some local phase shift, as illustrated in Figures 1B and 1C, respectively. We show how this concept can be generalized to shape the coupled plasmonic wavefront arbitrarily, starting with amplitude-only shaping, followed by phase-only and finally holographic amplitude and phase shaping. This demonstration opens the possibility for diverse Figure 1. (A) Cross-sectional view of the plasmonic thin layer selective coupler. Schematic examples for (B) amplitude and (C) phase modulations, where red lines depict equi-phase lines with intensity color-scaling.
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applications such as power splitting14, active switching15, interferometry16, and generation of unique accelerating beams9,10,12,17. The method we present here has important advantages for shaping plasmonic beams, among them are its high efficiency, the support of very high phase gradients, the ability to shape the wavefront in buried layers, and the clean generation of the desired beam, without undesired in-plane scattered signals manifested as diffraction orders. The principle upon which the coupling behaves in the symmetric IMI structure is that of the directional coupler. Although plasmonic directional couplers such as two adjacent, in-plane waveguides15,16,18–21, wide metal-insulator-metal waveguides22 and vertical, hybrid insulator-metal-insulator directional couplers23 were previously researched, here it is used for the first time in order to shape the plasmonic wavefront. Consider a thin metal layer vertically sandwiched between two thick dielectric layers. One can distinguish between two cases – the symmetric case in which the two dielectric permittivities are identical, hence a plasmonic wave that propagates at the bottom layer can be efficiently transferred to the top layer after a suitable coupling length24; and the asymmetric case, in which the two dielectric permittivities are significantly different, hence the coupling is essentially negligible. For the symmetric case, it is convenient to describe the system using its two eigenmodes: an anti-symmetric and a symmetric one 25,26. By assuming that the imaginary parts of the mode indices are much smaller than the real parts, given an initial planar wavefront at the bottom interface, the normalized plasmonic field amplitude at the top interface is described as follows23: =
≅ exp − sin
&'' &'' !"#$% ($) *
,
=
&'' &'' ,-#$% .$) *
, (1)
Figure 2. Amplitude only modulation. (A) Left ( < 0): SEM image of the Super-Gaussian selective coupler, right ( > 0): measured nearfield optical intensity. (B,C) A three-beam wavefront. Ag objects appear in gray (grating not to scale), SU-8 objects appear in blue and the nearfield intensity is visualized as a height map. where /0 is the free-space wavelength, 12 and 14 are the effective indices of the anti-symmetric and the symmetric modes, respectively. To derive the design equations for the arbitrary amplitude shaping, under the design constraint that coupling losses are relatively small ≫ , we can analytically invert Equation 1 to solve for any amplitude Ax (Noting that the intensity measured is ∝A ). This yields the needed ridge length (in the z direction) as a function of the transverse x coordinate: "33
8 =
"33
sin(9:8;.
(2)
For numerical Finite Discrete Time Domain (FDTD) simulations, the permittivities of silver27 and SU-828 at the free-space wavelength /0 = 1064 1@ were taken as A- = −57.9 + 0.61G and AH = 2.51, respectively. Solving for these permittivities, for 45 1@ Ag sandwiched between SU-8, we get close anti"33 "33 symmetric and symmetric mode indices: 12 = 1.637 − 0.00098G and 14 = 1.596 − 0.00015G, respectively. Plugging these into Equation 1 yields the coupling length = 13 L@ and the propagation length = 300 L@, fulfilling the losses design constraint. In accordance, the power coupling efficiency ACS Paragon Plus Environment
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derived from Equation 1, expO−2 / Q, is therefore 92%. The mode indices outside in the coupler ridge "33 "33 air region were found to be 1R- = 1.618 − 0.00050G, and 1 = 1.013 − 0.00026G. This large difference ensures that the bottom and the top plasmonic waves are practically uncoupled without the ridge. First, we demonstrate straight-forward amplitude-only selective coupling of a plasmonic beam having a Super-Gaussian profile, which is mathematically expressed as: 9 S U
8 = exp − , (3) T
where a beam width V = 6 L@ and a super-Gaussian power W = 6 represent a good tradeoff between a nearly flat-top wavefront retaining smooth boundaries. Figure 2A depicts the resulting plasmonic beam. A sharp spike is observed near = 0 owing to decoupling to free-space at the abrupt interface change, as further discussed in the following example. The selectively coupled wavefront can be easily extended beyond Gaussian-like beams, and an example of a 3-beam shape is defined in Equation 4. This beam features multiple (three in this case) power transmission channels, and is seen in Figure 2(B-C). Its amplitude is expressed as follows: 8 = X
0.5 :1 + YZ[ 2\8/Λ; , − 3/2 < 8/Λ < 3/2 , (4) 0, ^[^_ℎ^a^
where Λ = 16.7 L@ was chosen. A good correspondence between simulation and experiment is seen along 80 L@ propagation, which is the NSOM scanning limit. A fast decay followed by a slow decay can be seen in the longitudinal profile. The FDTD simulations show that the fast signal decay (“spikes” seen e.g. in Figure 2B at = 0) in the first few L@ is caused by two processes: (1) decoupling to free-space from the abrupt medium change. (2) The Ag/Air (top) mode is less confined than the SU-8/Ag (bottom) mode, leading to a decrease in the peak intensity (though not in total energy). Simulations show that 89% of the energy (in a good agreement with the 92% analytical prediction) couples from the bottom to the top interface, splitting to 51% in the intended plasmonic Ag/Air mode and 38% mostly decoupled to freespace. These non-optimized efficiency is similar to reports of highly optimized grating couplers29 and are inherently higher compared with beam shaping techniques relying on binary gratings. In the latter case, most of the energy is diffracted outside of the desired 1st order30. Better termination of the coupler, by means of a subwavelength dielectric grating should boost the efficiency even further. Next we consider the non-trivial case of phase modulation, which exploits the different plasmonic phase velocities at the Ag/air mode with respect to the Ag/SU-8 mode. Let us revisit the scheme shown in Figure 1A, where the bottom SU-8/Ag and top Ag/Air modes are abruptly replaced by the symmetric and the anti-symmetric SU-8/Ag/SU-8 modes. The bottom plasmonic wave does not feel the medium change at the ridge start plane as only a tiny fraction of the mode energy resides above the Ag layer. On the other hand, the ridge ends where much (if not all) of the energy had risen to the top, and as seen in Figure 1C, it now accumulates a phase difference according to the following local displacement of the ridge’s end relative to the = 0 plane: "$H 8 =
bS cd
, (5)
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Figure 3. Nano-focusing lens, having a y = 47 L@ aperture and u = 55 L@ focal length (NA 0.39). (A) FDTD Simulations, with the ridge topography overlaid in blue, and 6 equi-phase lines overlaid in white. (B) Experimental measurements along with SEM image of the fabricated ridge.
where Φ8 is offset such that its maximal value is zero, and Δk = O1R- − 1 Q ⋅ 2\//0 . We note that the same equation would follow by means of applying Snell’s law at the ridge's end interface, and remembering that the wave does not refracts at the starting surface of the coupler. In an analogous manner to the inspiring work by Lee 31, Equation 2 and 5 may be combined to form a binary equation for the ridge's shape: "33
ℎ0 Φ8 2 + sin(9 :8;op k1 + sign m − 2 \ Δn h8, = j 0
