Imaging Metasurfaces based on Graphene-Loaded Slot Antennas
IImaging Metasurfaces based on Graphene-Loaded Slot Antennas
Jordan A. Goldstein ∗ , † and Dirk R. Englund ‡ † Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts,United States ‡ Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, UnitedStates
E-mail: [email protected]
Abstract
Spectral imagers, the classic example being the color cam-era, are ubiquitous in everyday life. However, most such im-agers rely on filter arrays that absorb light outside each spec-tral channel, yielding ∼ /N efficiency for an N -channel im-ager. This is especially undesirable in thermal infrared (IR)wavelengths, where sensor detectivities are low, as well as inhighly compact systems with small entrance pupils. Diffrac-tive optics or interferometers can enable efficient spectralimagers, but such systems are too bulky for certain applica-tions. We propose an efficient and compact thermal infraredspectral imager comprising a metasurface composed of sub-wavelength-spaced, differently-tuned slot antennas coupledto photosensitive elements. Here, we demonstrate this ideausing graphene, which features a photoresponse up to ther-mal IR wavelengths. The combined antenna resonances yieldbroadband absorption in the graphene exceeding the 1 /N ef-ficiency limit. We establish a circuit model for the antennas’optical properties and demonstrate consistency with full-wave simulations. We also theoretically demonstrate broad-band ∼
36% free space-to-graphene coupling efficiency fora six-spectral-channel metasurface. This research paves theway towards compact CMOS-integrable thermal IR spectralimagers.
Keywords
Graphene, Infrared, Optical Antenna, Thermal Imaging,Multispectral Imaging, Plasmonics
Gold Dielectri cGraphene dwh a) b)Unitcell xy z
Figure 1: a) Illustration of a broadband absorbing slot antennametasurface consisting of six differently tuned slot antennas tiledwith subwavelength periodicity. The graphene patches are color-coded by antenna length, and the diagram is drawn to proportionbased on the device dimensions used to produce Figure 5. b)Depiction of a single graphene-coupled slot antenna-based pho-todetector based on the photothermoelectric effect.
We take spectral imaging for granted in daily life. Our eyesare spectral imagers, providing information about the com-position of what we see. The infrared electromagnetic bandcovers many chemical absorption resonances and thus alsoreveals compositional information. In particular, infraredspectral imaging is applied in areas such as gas emissionmonitoring, ecological monitoring, food quality con-trol, waste sorting, biological research and oceanog-raphy. Spectral imaging aims to measure a “data cube” repre-senting light intensity over two spatial dimensions x and y and one spectral dimension λ with N channels. Scanningspectral imagers sequentially measure different portions ofthe data cube over multiple exposures to form the full datacube. A common example is the pushbroom scanner whichmeasures x × λ data cube slices while scanning y and isthus typically associated with satellites and conveyor beltswhere either the camera or subject is gradually moving inone direction. The spectral axis may also be scannedsuch as in tunable filter-based imagers, which feature atmost 1 /N light utilization efficiency, or Fourier transforminterferometer-based imagers which are bulky and have mov-ing parts. In contrast, snapshot spectral imagers (SSIs)capture a data cube with a single exposure. This maybe achieved using a color filter array similar to that of acolor camera, thus limiting efficiency to 1 /N . Anothercategory of SSIs uses dichroic or dispersive optics to breakup incoming light by wavelength before arriving on a focalplane array (FPA). There are many variations of this ap-proach, but they all require increasing the etendue ofthe incoming light beam by a factor of N , leading to an un-favorable tradeoff between total FPA area, input acceptanceangle and spectral resolution. Compared to these technologies, the category of imagersbased on inherently multispectral pixels is less explored. Onesuch example is the Foveon RGB sensor, which extractsthree different electrical signals from different depths in theoptically active silicon region, as shorter wavelengths are ab-sorbed closer to the sensor surface. Another approach usesnanoantennas, optical resonators of subwavelength dimen-sions which nevertheless feature absorption cross sectionsof order λ if the antenna is conjugate impedance matchedwith its load; or, equivalently, if the antenna is criticallycoupled to the vacuum. Here, we propose a thermal IRmultispectral imager where N differently sized metallic slotantennas with infrared-sensitive loads targeting N spectralchannels are tiled to form a metasurface featuring efficientfree space-to-load optical energy transfer. Figure 1a showssuch a metasurface for N = 6. We model graphene as the1 a r X i v : . [ phy s i c s . op ti c s ] A p r hotosensitive load because its broadband absorption in themid-IR and processing flexibility make it suitable forthis platform. Not only do the antennas sort incident lightby spectral channel, but they also enhance the absorption ofthe graphene load, bridging the gap between the impedanceof free space and graphene, which, when undoped, has anoptical sheet resistance no lower than 16 . Figure 1bshows in detail a single such antenna-coupled graphene pho-todetector. This detector is designed for a strong photother-moelectric response, in which absorbed light heats up theelectron gas in the graphene, resulting in an electromotiveforce due to the Seebeck effect. The graphene channel isassumed to be isolated from the metasurface by a several-nanometer layer of dielectric, thin enough to not impact theoptical properties of the system. The asymmetric positionof the graphene channel with respect to the slot allows halfof the graphene channel to be gated by metal underneath,yielding the asymmetric graphene Fermi level profile neededfor a nonzero net photoresponse.
Note that while perfectabsorption in this wavelength range has been demonstratedin heavily doped graphene accompanied with metal nanos-tructures, we limit our consideration to undoped grapheneas the peak Seebeck coefficient occurs at very low dopinglevels. For optical absorption, slot antennas offer a few advan-tages over planar designs such as dipole or bowtie antennas.They have unidirectional radiation patterns, and thus an ar-ray of them can perfectly absorb an incident beam, whereasplanar antennas require a quarter-wave back-reflector to doso.
The wavelength dependence of the back-reflectorphase complicates design of broadband absorbing meta-surfaces and exacerbates undesirable antenna-antenna cou-pling. Additionally, since planar antennas must be sup-ported by transparent dielectric, they cannot be embeddedin a CMOS process as the inter-layer dielectric strongly ab-sorbs thermal infrared radiation, whereas for slot an-tennas the dielectric on top and in the perforations may beetched away without sacrificing the mechanical integrity ofthe antenna.
Results and Discussion Z A V A Z gr Z wg V A Z wg b) a ) Z gr Z A Figure 2: a) Graphene-loaded slot antenna with physical fea-tures corresponding to the components in circuit b) labelled. b)Circuit schematic of a graphene-coupled slot antenna. V A repre-sents incoming light, Z A is the radiation impedance of the slotaperture, Z gr is the impedance of the graphene sheet and Z wg isthat of the slot, effectively a short-circuit waveguide stub. We model the slot antenna depicted in Figure 2a as anaperture antenna fed by a rectangular waveguide terminatedin a short circuit a distance d behind the aperture. We rep-resent the graphene sheet as a shunt impedance connected in parallel with the waveguide stub. Figure 2b illustratesthis circuit with a Th´evenin equivalent radiation impedance Z A and source V A representing the aperture antenna. Therectangular waveguide stub presents an impedance Z wg = Z e jn eff k d + re − jn eff k d e jn eff k d − re − jn eff k d , (1)where Z and n eff are the characteristic impedance and ef-fective index of the TE mode of the slot and k is thevacuum wavenumber. r is the Fresnel reflection coefficientbetween vacuum and metal for an s -polarized plane wave atan incident angle of arccos( n eff ), which describes the TE mode.The graphene presents a mostly resistive impedance of Z gr = π w hσ gr , (2)using power/current impedance normalization. w and h are defined in Figure 1b and σ gr is the sheet conductanceof the graphene, modeled here as intrinsic. We calculate Z A using finite element simulations for various h and w ; weprovide more details on these calculations in the Methodssection. See Supplementary Figure 1 for an example of thefrequency dependence of the impedances in the circuit.Define η gr as the fraction of available power from Th´eveninsource V A , Z A that is dissipated in Z gr . Solving the abovecircuit, we arrive at η gr = 4 (cid:12)(cid:12)(cid:12)(cid:12) Z gr k Z wg Z gr (cid:12)(cid:12)(cid:12)(cid:12) Re( Z A ) Re( Z gr ) | Z A + ( Z gr k Z wg ) | , (3)where k represents reciprocal addition. Define A gr = P gr I inc asthe partial absorption cross section of light of intensity I inc coupled into the graphene and P gr as the power absorbedin the graphene. For a lossless, conjugate matched antenna,antenna theory predicts A gr,max = D ( θ,φ )4 π λ where D ( θ, φ )is the antenna’s directivity at the given incident angle andpolarization, which we calculate using finite element sim-ulations. θ here is the polar angle and φ is the azimuthalangle. We omit the polarization angle in our notation, im-plicitly setting it to maximize D . Additional ohmic lossesand impedance mismatch then reduce the actual absorptioncross-section into graphene by a factor η gr , i.e. A gr = η gr D ( θ, φ )4 π λ . (4)We obtain A gr from FDTD simulations of plane waves in-cident on the graphene-loaded antennas, which we then useto calculate η gr via Equation 4.Fig. 3 compares η gr between the model described byEquation 3 and FDTD results for antennas of various di-mensions. The data show that the model is accurate towithin 10% of the η gr peak amplitude and 2% of the res-onance wavenumber. We attribute these discrepancies toaspects not captured by the quasi-analytical model, such asour assumption of a perfectly conducting outer antenna faceand finite meshing. Despite these shortcomings, the presentmodel allows us to predict slot antenna absorption prop-erties to tolerances comparable to the uncertainty due tovariations in metal quality.To further validate our model, we artificially scale thesheet conductance of the graphene load by factors rangingfrom 0 .
33 to 5 and compare the resulting η gr between themodel and FDTD for a single such antenna. The results,2
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Wavenumber [cm -1 ] E ff i c i en cy η g r [ un i t l e ss ] d = 1.75 µ m, h = 6.5 µ md = 1.75 µ m, h = 4.5 µ md = 2.25 µ m, h = 6.5 µ m d = 2.25 µ m, h = 4.5 µ m Figure 3:
Comparison of simulated and modeled η gr . Mod-eled antennas are 400 nm wide. Dashed lines represent the full-wave FDTD absorption results, while the solid lines represent theimpedance model results. Wavenumber [cm -1 ] E ff i c i en cy η g r [ un i t l e ss ] σ gr σ gr σ gr σ gr Figure 4: η gr versus wavenumber for various values of the loadsheet conductance, where σ gr is the optical conductivity of intrin-sic graphene. Dashed lines represent the fullwave FDTD absorp-tion results, while the solid lines represent the model results. Theantenna featured here has dimensions d = 2 . µ m, h = 5 . µ m,and w = 0 . µ m. shown in Figure 4, show that our model accurately predictsthe sublinear scaling of η gr with respect to the load conduc-tivity, with the η gr peak amplitude reaching 0 . σ gr .Having modeled the individual components, we now dis-cuss broadband absorbing metasurfaces incorporating dif-ferently tuned antennas tiled in a periodic array. We usea three-step process to design such metasurfaces. We firstconstrain d and w to be the same for all antennas in themetasurface, and choose their values to yield high peak η gr for antennas resonant in the targeted wavelength range. Sec-ondly, we choose the values of h for the antennas, followingthe heuristic that at the wavelength where one antenna’s η gr falls to half its peak amplitude, the next antenna’s η gr should have risen to half its peak amplitude. Finally, wechoose the arrangement and pitch of the antennas to be asclosely packed as possible while satisfying qualitative fabri-cability constraints. We also avoid juxtaposing antennas ofadjacent wavelength channels to minimize antenna-antennacrosstalk. The antenna pitch, more accurately described by the Bra-vais lattice vectors of the array, is a critical parameter indetermining the potential absorption efficiency of the array.Light incident from a given direction can only be scatteredby the two-dimensional diffraction orders of the lattice. Thearray can only perfectly absorb an incoming light beam ifno nonzero diffraction orders fall within the light cone, bar-ring the event that all higher diffraction orders overlap withnodes in the individual unit cell radiation pattern. By “lightcone”, we refer here to the region in the Fourier transformspace of the xy –plane for which radiation can occur, namely k x + k y < k . For a square lattice, if the lattice pitch a < λ min /
2, no higher diffraction orders are within the lightcone for any incident angle. In practice, the limited numer-ical aperture of imaging systems relaxes this constraint.
Wavenumber [cm -1 ] G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] Figure 5:
Graphene absorption efficiency of the six-antennametasurface. The colored curves represent the contributions ofeach antenna to the overall graphene absorption of the metasur-face, which is represented by the black dashed curve. The curvecolors here match the antenna colors in Figure 1a. Shorter anten-nas yield higher resonance wavenumbers.
Targeting the 6 − µ m wavelength range, we fol-low the above methodology and arrive at an array ofsix antennas with d = 2 . µ m, w = 400 nm, and h = 3 . , . , . , . , . , . µ m, uniformly spacedand tiled as shown in Figure 1a with a 7 µ m by 12 . µ munit cell. Figure 5 shows the absorption efficiency of nor-mally incident light into the graphene load of each antennaas well as their sum, simulated with FDTD. With an av-erage efficiency of 36% across the 1050 cm − to 1600 cm − band, this structure improves upon the 1 /N limit of filterarray-based imagers by roughly a factor of two. Note thatthe unit cell highlighted in Figure 1a is not the primitiveunit cell of the lattice, although it was used as the FDTDsimulation region due to software constraints.To further understand the physics of these metasurfaces,we vary the pitch of the unit cell while keeping all otherparameters constant. The resulting total absorption effi-ciency spectra for six different cases are shown in Figure 6a,and their mean efficiencies averaged between 1050 cm − and1600 cm − are plotted in Figure 6b. We show the individualabsorption contributions from each antenna for each meta-surface pitch in Supplementary Figure 2, and we list theunit cell widths and heights in Figure 6c. The data showthat the absorption efficiencies are roughly constant, andcomparable with the peak efficiencies obtained in Figure 3,up to the 7 µ m–wide unit cell. For larger unit cells, the mean3
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Wavenumber [cm -1 ] G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] M ean E ff i c i en cy a)b) c) Unit celldimensions Normal inc.diffraction threshold μ m cm -1 μ m cm -1 μ m cm -1 μ m9 × μ m
11 × 14 μ m 1156 cm -1 cm -1 cm -1 Figure 6: a) Total graphene absorption efficiency versuswavenumber for metasurfaces with varied unit cell pitch. Thelegend indicates the width of the unit cell as illustrated in Figure1a. b) Mean absorption efficiency for each of the curves in a),averaged between 1050 cm − and 1600 cm − . c) Unit cell dimen-sions used to obtain the results in a) and b) and the correspondingmaximum wavenumbers for which normally incident light expe-riences no higher order diffraction. The unit cell x and y pitchwere varied together to attain a roughly uniform antenna-antennaproximity. efficiency decreases with increasing unit cell size. This canbe understood by analyzing the diffraction characteristics ofthe various metasurfaces. Sparser metasurfaces have tighterreciprocal lattices and thus more diffraction orders are avail-able within the light cone. For normally incident illumina-tion as we are using here, the minimum wavelength for whichno higher-order diffraction occurs (i.e., completely specularreflection) is given by λ c = max (cid:16)(cid:0) a − + a − (cid:1) − / , a , a (cid:17) ,where a and a are the unit cell dimensions. The corre-sponding wavenumbers for the various unit cells used hereare listed in Figure 6c. For the 9 µ m and 11 µ m widthunit cells, the threshold wavenumber falls in the middleof the range of interest, resulting in reduced efficiency asdiffracted light cannot participate in destructive interferencewith specularly scattered light. This is especially appar-ent for the 9 µ m unit cell, which exhibits a clear spectraltransition between high and low efficiency at the diffractionthreshold. Note that choosing an excessively tight antennaspacing is also detrimental, not only due to fabrication diffi-culty, but also because graphene detectors feature minimumJohnson noise-dominated noise-equivalent power for chan-nel lengths comparable to the hot carrier cooling length,which can range from 100 nm to over 1 µ m depending on thesubstrate and graphene quality. Exceedingly tight an-tenna spacings may not provide room for such long graphenechannels. For a spectrally sensitive metasurface to be practical, notonly must it maintain a high absorption efficiency for a rea-sonable range of incoming light directions, but also the ab-sorption spectra of the individual antennas must not shiftor distort too strongly as the incoming light angle varies.The directional dependence of our metasurface arises fromtwo factors: The directivity profile D ( θ, φ ) of the individualantennas, and array effects resulting from interference andantenna-antenna coupling. Although full angle-dependentsimulation results for our gold metasurfaces are outside thescope of this paper due to the extreme computational over-head of off-angle periodic structure simulations, we canstill provide insight by elaborating on the aformentioned fac-tors, and we also perform off-angle simulations of a simpli-fied metasurface constructed of Perfect Electrical Conductor(PEC) which permits a much larger mesh size. Supplemen-tary Figures 3a, b and c display the directivity profile ofa 6 . µ m ×
400 nm antenna on resonance. This profile issimilar to those of the other antenna lengths used in themetasurface. Intuitively, the directivity decreases as the in-cident angle approaches the long axis of the antenna, andwe thus expect a similar trend in the directional depen-dence of the array. In Supplementary Figure 3d, we plotfor three wavenumbers the sets of incident light directions,represented as components k x , k y of the incident wavevector k , for which only specular reflection from the metasurfaceoccurs. For λ − = 1050 cm − , all light incident within 45 ◦ of normal is only specularly reflected. This range decreaseswith increasing wavenumber until normally incident light ispinched off at 1579 cm − . As in Figure 6, we expect effi-ciency to suffer when the specular reflection-only conditionis not met. Supplementary Figure 4 shows the results of off-angle simulations of the simplified PEC metasurface. Thedata show that for light incident off-angle but perpendicu-lar to the antennas’ long axes, antenna resonances fallingoutside the specular reflection-only region are subject to de-creased absorption efficiency as well as blue-shifting. Forlight incident off-angle and perpendicular to the antennas’short axes, we obtain similar results, except that the peaksred-shift instead of blue-shift, and we observe an overall re-duction of the absorption at steep incident angles due to thereduced directivity.We also investigate metasurfaces comprising numbers ofspectral channels besides N = 6. Supplementary Figure 5shows the geometric details and simulation results for meta-surfaces with N = 3, 4, 5, and 8 as well as the default valueof 6. We achieve good results with uniformly high absorp-tion efficiency for N = 5 and 6. For lower values of N , thewide frequency spacing between the individual resonancesyields deep troughs in the overall absorption efficiency curve,whereas for higher values of N , excessive overlap between theantenna resonances causes the overall metasurface efficiencyto suffer.With realistic metal, the efficiency of these devices is ul-timately limited by ohmic losses. However, more advancedantenna designs can be used to improve η gr . As it turns out,not only can the copper layers in the back end of a CMOSprocess be designed to incorporate slot antennas, but theyalso provide a convenient medium to realize those antennadesigns that would be exceedingly difficult to fabricate inan academic setting. Figure 7a introduces such a designin which the slot inlet is narrower than the internal cavitywidth. This design concentrates the electric field around thegraphene, which reduces Z gr without increasing the TE mode loss. Figure 7b shows a CMOS adaptation of this de-4
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Wavenumber [cm -1 ] E ff i c i en cy η g r [ un i t l e ss ] Open,
Evap , w =
400 nm
Open, SC , w =
400 nm
Slitted,
Evap , w =
600 nm
Slitted, SC , w =
600 nm
Open,
Evap , w =
200 nm
Open, SC , w =
200 nm c) a) w dw i d i b) Figure 7: a) Cross-section of slot antenna with narrowed inputslit, as well as dimension definitions. b) Illustration of how suchan antenna could be implemented in the wiring layers of a CMOSchip. c) η gr versus wavenumber for various antenna geometriesand gold optical models. “Open” refers to the the normal slotantenna design in Figure 1b), while “slitted” refers to the slitteddesign in a). “Evap” refers to evaporated gold as used throughoutthe paper, whereas “SC” refers to single crystal gold. For theopen antennas plotted here, d = 2 . µ m. For the slitted inletantennas, d = 1 . µ m, w i = 0 . µ m, and d i = 0 . µ m. Thesesimulations use solid, not perforated sidewalls. sign, in which the walls of the cavity are perforated to com-ply with via design rules. If the perforations are sub-cutofffor the resonant wavelength and sufficiently deep, they donot leak light. Additionally, it would be necessary to etchaway the inter-layer dielectric to prevent light absorption.Besides different antenna designs, material quality also af-fects efficiency considerably. We explore both of these vari-ations in Figure 7c, which plots η gr versus wavenumber forintrinsic graphene as simulated by FDTD for narrowed in-let (“slitted”) antennas and normal slot antennas, as well aswith a single-crystal gold model in addition to our defaultevaporated gold model. The data show that adopting a slit-ted antenna design increases the peak η gr from 0 . . . η gr ≈ . Whileone can achieve high efficiencies with clever antenna designsand more opaque loads than monolayer graphene, the Q isultimately bounded by that of a sealed metal cavity whichwe simulate to be about 40 for the present gold model andantenna shapes. Silver has less mid-IR optical loss than cop-per or gold, but unlike copper it is not considered CMOS-compatible and thus may be difficult to integrate. Polarmaterials supporting optical phonons in the mid-IR are re-ported to have high Q plasmonic resonances and are thusworth investigating if higher Q is necessary, although plas-monic behavior only occurs in narrow wavelength bands,
Table 1:
Estimated sensitivity figures for graphene imaging ar-ray. R c [Ω µ m] R [A / W] NEP [pW Hz − ] D ∗ [Jones]0 0 .
83 9 . . × .
42 12 . . × limiting applicability. We can apply data describing the resistivity of gatedgraphene to our model to estimate the room temperature de-tectivity of the spectral imager described by Figures 1 and 5.Using the methods described in Song et al. and the elec-trical properties of polycrystalline, non-annealed grapheneachieved by de Fazio et al., we arrive at the values inTable 1 with and without accounting for graphene contactresistance for normally incident x -polarized light. Usingthe more advanced antenna designs shown in Figure 7, thedetectivities would reach the 10 Jones range. We discussthese calculations in the Methods section. For comparison,more conventional bolometer-based thermal IR FPAs oper-ating at room temperature have been reported to achievedetectivities in the 1–2 × Jones range, although such de-vices do not feature spectral resolution and are limited tomillisecond-range response times.
Finally, we would like to emphasize the general applicabil-ity of the antenna metasurface concept to other wavelengthranges, photosensitive elements and antenna designs. In-deed, Tamang et al. have proposed a similar concept ap-plied to RGB imaging where silicon nanorods act as boththe antenna and sensitive element. We propose that be-sides graphene and other 2D materials, III-V or HgCdTesemiconductor photosensitive elements could also be incor-porated by a transfer printing heterointegration process. The slot antenna-based metasurface imager approach couldalso scale to terahertz, where Ohmic losses are much lessthan in the mid-IR and the antennas could be fabricateddirectly in a circuit board-like platform. Conclusion
In conclusion, we introduced a six-spectral-channelgraphene-coupled slot antenna metasurface with 36% ef-ficiency functioning as a spectral imager in the thermal IR,as well as a model for estimating the optical properties ofthe individual antennas therein. This device is appropri-ate for integration in the wiring layers of a CMOS processwith suitable post-processing to remove inter-layer dielectricwithin the cavity and transfer graphene. We have shownthat more sophisticated antenna designs can improve theefficiency of optical energy transfer to the load to above η gr = 0 .
6. Further research on this concept may focus onexperimental demonstration of the absorption enhancementfunctionality, or on optimizing the device design to meetcertain engineering goals.
Methods
Simulation details
Unless otherwise specified, we use the evaporated gold opti-cal model described in Palik et al. throughout the paper. We model graphene as an infinitely thin conductive sheet us-ing the optical conductivity model described in Hanson. As input parameters to the model, we use a temperature of300 K, intrinsic graphene (zero Fermi level), and a scatteringrate Γ = 0 .
514 meV.5e use the Lumerical FDTD package for our FDTD sim-ulations. For simulations of individual antennas, we use x -, y - and z -meshes of 8 nm in the vicinities of the slot apertureand slot bottom, as well as a z -mesh of 40 nm within theslot. As such, the finest meshes coincide with metallic sur-faces and corners, allowing us to capture the nonzero skindepth of the metal, whereas the more gradual z -dependenceof the fields inside the slot permits a coarser mesh. We findthat a minimum mesh size of 8 nm yields converged resultsfor these simulations. We use PML boundary conditions onall sides except for the − z side, where we use a metallicboundary condition as the light does not penetrate beyondthe slots anyway. We also apply symmetry conditions acrossthe xz and yz planes. For the metasurface simulations, weuse the same meshing scheme, but with a fine mesh of 15 nmand a z mesh of 100 nm within the slot due to computationalresource availability limits. To illustrate the error associatedwith this choice of mesh, we plot mesh-dependent absorptionefficiency curves for a 9 µ m by 13 . µ m unit cell, N = 6metasurface in Supplementary Figure 6, with the mean effi-ciency averaged between 1050 cm − and 1600 cm − plottedin the inset. The results validate our qualitative conclusionsand put an approximately 3% relative error bound on thespectrally averaged efficiencies of the metasurfaces, althoughthe coarse mesh does somewhat distort the actual spectra.For the metasurface simulations, we change the x - and y -boundary conditions to Bloch boundary conditions to re-flect the periodic nature of the metasurface, and we applysymmetry across only the xz plane. Impedance model details
We calculate Z and n eff for our impedance model usingthe Lumerical MODE waveguide mode solver with an 8 nmmesh. We use the graphene model from Falkovsky, whichgives almost identical results to the Hanson model used byLumerical for the parameters we use. We use Ansys HFSSfinite element software to calculate Z A . These simulationsexcite the aperture from within by its TE mode yieldingthe S scattering coefficient of the internally reflected light,from which the software calculates Z A . In the finite elementsimulations, we model the aperture and slot as perfect elec-trical conductors, as we expect the real part of the antennaimpedance to be dominated by radiative loss (rather thanohmic loss) and the imaginary part by energy storage in thenear field of the aperture (rather than plasmonically withinthe metal). From these same simulations we also extract theantenna directivity D ( θ, φ ). Detectivity estimation
We base our estimation of the detectivity D ∗ on the formu-lation put forth in Song et al. To calculate the electronictemperature profile of graphene suspended over the slot, wesolve the 2-dimensional partial differential equation: − ∇ · ( κ ∆ T el ) + γC el ∆ T el = α(cid:15) ˙ N − j · ∇ Π (5)Here κ represents the electronic planar thermal conductivityof the graphene; ∆ T el is the difference between the thermallyexcited electronic temperature T el and the lattice tempera-ture T = 300 K, γ represents the electron-phonon thermaldecay rate, and C el represents the electronic heat capacityof graphene. α is the efficiency with which optical energyabsorbed by the graphene is deposited into the electronicsystem on a sub-picosecond timescale, taken to be unitysince the incident photon energy is below graphene’s op-tical phonon energy. (cid:15) ˙ N represents the intensity profile of absorbed light. j is the electrical current density, and Πrefers to the Peltier coefficient. We choose an antenna with w = 400 nm and h = 5 . µ m in these calculations, extract-ing the spatial dependence of (cid:15) ˙ N from Ansys HFSS finiteelement simulations. For the graphene’s conductivity σ asa function of Fermi level, we use the data measured by deFazio et al. for unannealed, polycrystalline graphene; thisis then used to calculate κ via the Wiedemann-Franz law andΠ as well as the Seebeck coefficient S via the Mott formula.The value of γC el is estimated by assuming a electronic ther-mal cooling length of p κ/γC el = 1 µ m, an empirical value. As shown in Figure 1b, the graphene is assumed to be ter-minated at the slot edge on one side, and is taken to extend400nm past the slot edge on the other side, where its Fermilevel is gated through the metal to the n-type Seebeck co-efficient peak. The graphene Fermi level in the suspendedregion is simply taken to be the zero-gate-voltage Fermi levelfrom de Fazio et al. as it cannot be controlled. For simplic-ity we assume a sharp jump in the values of σ , κ , Π and S between the n- and p-doped sides of the device, neglect-ing fringing fields from the gate. The graphene channel isassumed to be short-circuited with graphene-metal contactresistances R c of either 0 or 1000 Ω µ m per contact, the lat-ter being consistent with 1-dimensional contacts to graphenenear the Dirac point. The average ∆ T el at the graphenep-n junction and the Seebeck coefficients on either side de-termine the thermal electromotive force via E = − S ∇ T , which in turn determines j via the total device resistance.Thus, Equation 5 including the Peltier term may be solveddirectly, as j is a linear functional of ∆ T el . Solving for ∆ T el over two spatial dimensions, we find linear thermal decayprofiles along the + x and − x directions away from the ∆ T el peak which indicates that the device is in the short-channelregime where carrier cooling is dominated by the ∆ T el = 0boundary conditions, and p κ/γC el is large enough for theelectron-phonon interaction term to be inconsequential.Having obtained the short-circuit responsivity R underzero bias as such, we calculate the noise-equivalent powerof the device assuming Johnson noise at 300 K as the domi-nant noise source, a reasonable assumption for an unbiaseddevice. The detectivity for the array is calculated incor-porating the antenna pitch, noting that there are two an-tennas per unit cell. To account for the decreased opticalabsorption efficiency of a metasurface loaded with graphenedoped to the Seebeck coefficient peaks of roughly ± .
05 eV,we redo the simulation used to generate Figure 5 with thegraphene doped as such. We plot the resulting absorptionspectra in Supplementary Figure 7, which shows a mean ab-sorption efficiency of 33% averaged between 1050 cm − and1600 cm − . We finally calculate the external values of re-sponsitivity, NEP, and detectivity by scaling the internalvalues by this efficiency factor. Acknowledgement
The authors would like to thank Chris Panuski and Dr.Laura Kim of MIT as well as Sebastian Castilla of Insti-tut de Ci`encies Fot`oniques (ICFO) for helpful feedback andinsight in the course of preparing this article. This researchwas funded in part by a grant from the Institute of SoldierNanotechnologies of MIT. This material is based upon worksupported by the National Science Foundation Graduate Re-search Fellowship Program under Grant No. 1122374. Anyopinions, findings, and conclusions or recommendations ex-pressed in this material are those of the author(s) and do notnecessarily reflect the views of the National Science Founda-6ion.
Supporting Information
Additional plots to supplement the main text, including:1. Example wavenumber dependence of the impedancesused in Eqn. 3;2. Graphene absorption efficiency spectra and individ-ual antenna components for metasurfaces of variousunit cell pitches;3. Antenna radiation pattern and diagram of specularreflection-only condition for off-angle excitation;4. Off-angle graphene absorption spectra for simplifiedPEC metasurface model;5. Graphene absorption spectra for metasurfaces withvaried numbers N of different antenna sizes;6. Graphene absorption spectra for a metasurface sim-ulated with different FDTD mesh sizes;7. Graphene absorption spectra for a metasurfaceloaded with graphene doped to 0 .
05 eV.
References
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NatureNanotechnology , , 814–819. upplementary Information for Imaging Metasurfacesbased on Graphene-Loaded Slot Antennas Jordan A. Goldstein and Dirk R. EnglundApril 16, 2020 a r X i v : . [ phy s i c s . op ti c s ] A p r xample of impedances in antenna properties calculation Wavenumber [cm -1 ] -1000-5000500100015002000 I m pedan c e [ Ω ] Z A Z wg Z wg || Z gr Z gr Wavenumber [cm -1 ] I m pedan c e [ Ω ] Z A Z wg Z wg || Z gr Z gr Supplementary Figure 1: Wavenumber dependence of the impedances in the circuit in MainFigure 2a. Above: Linear scaling. Below: Log-abs scaling. The antenna featured here hasdimensions d = 2 . µ m, h = 5 . µ m, and w = 0 . µ m.2 raphene absorption spectra and individual antenna contributions for metasur-faces of varied pitch G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] Wavenumber [cm -1 ] G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] Wavenumber [cm -1 ] Supplementary Figure 2: Graphene absorption efficiency and individual antenna contribu-tions for metasurfaces of varied pitch. Full unit cell dimensions for each case are given inMain Figure 6. 3 ntenna radiation pattern and off-angle excitation specular reflection condition -90 -60 -30 0 30 60 9001234 D i r e c t i v i t y [ un i t l e ss ] a)b) c ) uv -2000 -1000 0 1000 2000 k x [cm -1 ] -2000-1000010002000 k y [ c m - ] d ) (cid:20)(cid:19)(cid:24)(cid:19) cm -1 (cid:3)(cid:20)
300 cm -1 (cid:3)(cid:20)
550 cm -1 (cid:3) Light cone edge and specular reflection region:
Radiation angle θ [degrees]
Supplementary Figure 3: Different representations of the on-resonance far-field radiationpattern of a 6 . µ m-long slot antenna, which is qualitatively very similar to those of antennasof other lengths. a) 3D-representation of the far-field directivity pattern. D ( θ, φ ) goes tozero for light incident along the long axis of the aperture. b) Directivity vs. elevation angle θ along the azimuth of the antenna’s long axis, φ = 90 ◦ . c) Directivity in directional cosinespace. Note also that k x = k u and k y = k v where k x and k y are the x - and y -componentsof the incident wavevector. d) Depiction in k x k y –space of the light cone edges (colored rings)and the specular reflection-only incident light directions (colored patches) for three differentwavenumbers for the 7 µ m × . µ m 6-antenna unit cell design in Main Figure 1a. Theblack dots represent the reciprocal lattice points of the metasurface. For each wavelength,the specular reflection-only region is the set of remaining { k x , k y } after subtracting from thelight cone all copies of the light cone transposed by all nonzero reciprocal lattice vectors.4 ff-angle absorption spectra for simplified, perfectly conducting metasurface ab-sorber G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] Wavenumber [cm -1 ] = 15, = 0 = 30, = 0 = 45, = 0 = 15, = 90 = 30, = 90 = 45, = 90 = 0 Supplementary Figure 4: Graphene absorption efficiency and individual antenna contribu-tions for a perfectly conducting metasurface with various off-angle excitation directions.Using Perfect Electrical Conductor (PEC) as the metasurface material allows us to use acoarser mesh of 40 nm, as the electric field does not penetrate the conductor (the “skindepth” which requires careful modelling for realistic metal). We also adjust the metasurfacedimenisons to align all features to the mesh: The unit cell is 7 . × . µ m, and the antennalengths are 3.4, 3.8, 4.4, 5.04, 5.8, and 7 . µ m. The topmost plot shows the absorptionspectra for normally incident light. The second row depicts the case of light incident per-pendicular to the antennas’ long axes (along the line of maximum directivity), and the thirdrow depicts light incident perpendicular to the antennas’ short axes.5 raphene absorption spectra and individual antenna contributions for metasur-faces with different numbers of antennas G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] Wavenumber [cm -1 ] G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] Wavenumber [cm -1 ] Wavenumber [cm -1 ] G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] Antenna arrangement Antenna Lengths [μm] N Supplementary Figure 5: Graphene absorption efficiency and individual antenna contribu-tions for metasurfaces with varied numbers N of antennas of different lengths. The lower-right table shows the arrangements of the antennas within the unit cell, with 1, 2... N cor-responding to the shortest, second shortest, etc. up to longest antenna. The table also liststhe lengths of the antennas in each case, chosen roughly to maintain a constant degree offrequency-space overlap between adjacent spectral channels as simulated on an individualantenna basis. Antennas are placed on a 1 . µ m × . µ m grid as in Main Figure 5.6 raphene absorption spectra and individual antenna contributions for metasur-faces with different FDTD mesh Wavenumber [cm -1 ] G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ]
20 nm mesh18 nm mesh16.5 nm mesh15 nm mesh13.5 nm mesh12.5 nm mesh
10 15 20 25
Minimum mesh step [nm] M ean e ff i c i en cy Supplementary Figure 6: Graphene absorption efficiency and individual antenna contribu-tions for a 9 µ m by 13 . µ m metasurface with varied minimum FDTD mesh size as discussedin the Methods section of the main paper. The inset plots the mesh dependence of the meangraphene absorption efficiency in the 1050 cm − to 1600 cm − range.7 raphene absorption spectra and individual antenna contributions for a meta-surface with graphene doped to 0.05 eV Wavenumber [cm -1 ] G r aphene ab s o r p t i on e ff i c i en cy [ un i t l e ss ] Supplementary Figure 7: Graphene absorption efficiency and individual antenna contribu-tions for a 7 µ m by 12 . µ m metasurface with the same geometric parameters used to gen-erate Main Figure 5, but with the graphene doped to 0.05 eV which decreases the graphenesheet conductivity to a degree, especially for longer wavelengths. Here, the mean efficiencyaveraged between 1050 cm − and 1600 cm −1