natureARTICLESnanotechnologyhtps/∥dol.org/10.1038/s41565017-0034-6A broadband achromatic metalens for focusingand imaging in the visibleWei Ting Chen1,Alexander Y.Zhu',Vyshakh Sanjeev1.2,Mohammadreza Khorasaninejad,A key goal of metalens research is to achieve wavefront shaping of light using optical elements with thicknesses on the orderof the wavelength.Such miniaturization is expected to lead to compact,nanoscale optical devices with applications in cameras,lighting,displays and wearable optics.However,retaining functionality while reducing device size has proven particularly chal-lenging.For example,so far there has been no demonstration of broadband achromatic metalenses covering the entire visiblespectrum.Here,we show that by judicious design of nanofins on a surface,it is possible to simultaneously control the phase,group delay and group delay dispersion of light,thereby achieving a transmissive achromatic metalens with large bandwidth.We demonstrate diffraction-limited achromatic focusing and achromatic imaging from 470 to 670 nm.Our metalens comprisesonly a single layer of nanostructures whose thickness is on the order of the wavelength,and does not involve spatial multiplex-ing or cascading.While this initial design (numerical aperture of 0.2)has an efficiency of about 20%at 500nm,we discussways in which our approach may be further optimized to meet the demand of future applications.onventional refractive optical components are generallydispersion,including achromatic metalenses with diffraction-bulky costly and time consuming to manufacture with highlimited focusing covering nearly the entire visible (from 470nmprecision'.These are significant limitations,particularly forto 670nm).The achromatic metalens is also capable of perform-applications such as portable and wearable devices.In recent years,ing white-light imaging.Finally,we design and model a metasur-metasurfaces have emerged as a versatile platform for wavefrontface that when patterned over a commercial spherical lens rendersshaping.Since the phase is accurately controlled by subwavelength-it achromatic and diffraction-limited across the visible spectrum.spaced structures with thicknesses at the wavelength scale or below,many compact optical devices based on metasurfaces have beenPrinciple of achromatic metalensesdemonstrated.These include flat lenses,polarimeters,axicons,As an initial example,consider the achromatic metalens shown inFig.1a.The relative phase provided by the metalens elements withare highly chromatic despite consisting of weakly dispersive materi-respect to the centre follows':als.This can be attributed to two separate factors:dispersion arisingfrom a periodic lattice(see Supplementary Fig.1 for a detailed discus-sion),as well as light confinement in either a resonant or guided man-ner.Previous works have addressed this challenge by using multiplecoupled resonances to tailor phase profiles at several discrete frequen-where @c,r and F are the angular frequency,light speed,radialcies,stacking or stitching several layers of metasurfaces",increascoordinate and focal length,respectively.This spatial-and fre-ing the phase modulation to be more than 2t radians (so-calledquency-dependent phase profile (r,w)implies that at a given r,themulti-order diffractive lenses)or engineering the dispersionmetalens provides different transverse wavevectors k,=d(r,)/drRecently,achromatic focusing at green wavelengths (with a 60 nmso that different wavelengths are deflected by the same angle.bandwidth)in a reflective metalens was achieved".Other groupsEquation (1)can be expanded as a Taylor series near a designhave experimentally demonstrated achromatic metalenses in the nearfrequency a asinfrared with bandwidths of tens of terahertz(THz)s-27.However,none of these works demonstrated achromatic imaging.Here,we demonstrate the ability to engineer the frequency-dependent phase profile(r,),and thereby achieve arbitrary con-trol of metalens dispersion over alarge continuous bandwidth in thevisible.This is made possible by separately engineering the groupdelay and group delay dispersion ofeach constituent nanostructure,independent of its phase,at a given frequency.This is distinct fromother approaches,particularly that described in ref.2,where theEquation (2)indicates that to achieve achromatic focusingauthors utilized plasmonic resonances without considering groupwithin a given bandwidth A around an optical element placeddelay dispersion,and therefore did not demonstrate a system-at a radial coordinate r needs to satisfy not only the required relativeatic,general way to implement metalenses of different dispersion.phase((r,@))but also the higher-order derivative terms,whichAs a proof of concept,we demonstrate metalenses with tailoreddetermine the metalens dispersion.(r,@)/do and d'(r,)/do2Harvard John A.Paulson School of Engineering and Applied Sciences,Harvard University,Cambridge,MA,USA2University of Waterloo,Waterloo,ON,Canada.3Department of Physics,Harvard University,Cambridge,MA,USA."e-mail:capasso@seas.harvard.eduNATURE NANOTECHNOLOGY www.nature.com/naturenanotechnology2018 Macmillan Publishers Limited,part of Springer Nature.All rights reserved.ARTICLESNATURE NANOTECHNOLOGYsubstituting different values for n.We refer to the metalenses withn=0 and n=1 as achromatic and diffractive metalenses hereaf.ter.The diffractive metalens possesses a focal length shift similarto Fresnel lenses.From equation(3),the positive (negative)valuesof n imply that shorter (longer)wavelengths are focused fartherfrom (closer to)the metalens,respectively.The larger the absolutevalue of n,the farther the separation between the focal spots of twowavelengths,resulting in stronger dispersion.Figure 1b,c showsthe required relative group delays and group delay dispersions as afunction of radial coordinate for metalenses with numerical aper-ture (NA)=0.2 at=530nm.For n=2 and -1,they require non-negligible group delay dispersion to precisely control the focal lengthshift and achieve diffraction-limited focusing.Note that the required0.0group delay and group delay dispersions are relatively smallfor n=1.This agrees with our previous observation that a diffractive metalens05(NA=0.8)implemented using geometric phase can still focus lightwith a focal spot size approximately equal to a wavelengthIndependent control of phase and dispersionn=2n=2To increase the degrees of freedom in our design,we utilized cou--15pled phase-shift elements:two nanofins in close proximity,actingn=0(achromatic)■n=0(achromatic)as coupled waveguides.Their geometric parameters are defined in-20Fig.2a,and scanning electron microscopy images of a fabricated-10-505-10-50510metalens are provided in Fig.2b and Supplementary Fig.2.It hasRadial coordinate (um)Radial coordinate (um)been previously shown that coupled waveguides can support tun-able dispersion,for example,near-zero group delay dispersions forFig.1|Dispersion engineering of metalenses.a,Schematic of anachromatic metalens.To realize achromatic focusing,the phase profileity,we first consider the optical properties of a single TiO,nanofin,of (r)must satisfy equation (1).The metalens is designed to providewhich can be fabricated using electron-beam lithography followedspatially dependent group delays such that wavepackets from differentby atomic layer deposition.When a left-handed circularly polar-locations arrive simultaneously at the focus.The yellow line shows theized beam passes through the nanofin,the transmitted light can bespherical wavefront.b,Required relative group delays as a function ofdescribed by the Jones vector:metalens coordinate.The focal length is parametrized as F()=kor.Notethat the NA is a function of wavelength for nO due to the change in focalt-texp(i2a】length.Depending on the value of n,the metalens can be designed as2achromatic (n=0)or chromatic with focal length inversely proportional towavelength (n=1,dispersion similar to Fresnel lenses),or proportional towhere t and fs represent complex transmission coefficients whenwavelength(n=-1).The case of n=2 exhibits stronger dispersion.Thesethe incident light is polarized along the long and short axis of themetalenses have a diameter of 20um and a focal length ofnanofin,and a is the rotation angle.The second term in equation49 um at =530nm.c,Required relative group delay dispersion of the(4)is cross-polarized;we refer to its normalized amplitude squaredsame metalenses.as the polarization conversion efficiency hereafter.The phase shiftis determined by the product(-is)exp (i2a),where 2a is a fre-quency-independent geometric phase equal to twice the rotationare the relative group delay and group delay dispersion,and areangle.This allows us to decouple the target phase profile fromtypically ofthe order of femtoseconds(fs)and femtoseconds squarethe required group delay and group delay dispersion(controlled(fs")in the visible.Conventional diffractive lenses only satisfy theby i-fs).Figure 2c shows phase spectra for a nanofin with dif-required phase,that is,the phase profile at a design frequency.Theferent rotation angles.The slope is approximately linear within aneglect of these derivative terms results in chromatic effects.Angiven bandwidth,and is independent of the rotation angle of theintuitive interpretation of each term of equation (2)is shown innanofin.This property allows us to design achromatic metalensesFig.la.The first term leads to a spherical wavefront(yellow linein Fig.la).The group delay term compensates for the difference inTo gain physical insight into the dispersion design,each TiO,the wavepackets'arrival times at the focus,while the higher-ordernanofin can be regarded as a truncated waveguide.Neglecting endderivative terms(group delay dispersion,and so on)ensure that thereflections,the phase of the transmitted light after passing throughoutgoing wavepackets are identical.The net effect is the minimiza-the structure at a given coordinate r is (r,)="nh,where netion of the spread in the arrival times of wavepackets at the focus toand h represent the effective index and the height of the nanofin,ensure they constructively interfere.The smaller the time spread,respectively.The derivative with respect to angular frequencythe larger the bandwidth achievable.Therefore,to realize diffrac-tion-limited focusing for a broad bandwidth,both phase and group(5)delay,as well as higher-order terms,need to be considered.To account for the dispersion of a metalens,the focal length Finequation (1)can be parametrized as:yields the group delay:this is the ratio of the nanofin height to groupvelocity,which can be controlled by the nanofin dimensions and/or(3)material used.Figure 2d shows a comparison of polarization con-version efficiency using the eigenmode solver and finite-differencewhere k is a positive constant and n is a real number.The disper-time-domain (FDTD)methods(see Methods).The good agreementsion of a metalens can thus be designed to arbitrary specification byverifies the validity of treating the nanofins as short waveguides.AtNATURE NANOTECHNOLOGY www.nature.com/naturenanotechnology2018 Macmillan Publishers Limited,part of Springer Nature.All rights reservedNATURE NANOTECHNOLOGYARTICLESWavelength (nm)666600545500461(110.70.150.50)(170.70.230.50)0230.110.110.50Wavelength(nm)Wavelength(nm)0666600545500461666600545500461=0100-60210.130.230.50-a=6080600⑦1=200nmMODE20230.170.210.70FDTD8、0450500550600650450500550600650450500550600650Frequency (THz)Frequency (THz)Fig.2|Optical properties of nanofins and scanning electron micrograph.a,Schematic of a metalens element.The element consists of one or moreTiO,nanofins of varying dimensions but equal height h=600 nm,evenly spaced by a distance p=400 nm.The gap between nanofins is g=60nm.Thelength I,width w,height h and rotation angle a are also shown;the subscripts denote the left and right nanofin,respectively.The nanofins are rotatedwith respect to the centre of the square (400 x 400 nm2).b,Scanning electron micrograph of a region of a fabricated metalens.Scale bar,500nm.Formore images of the metalens,see Supplementary Fig.2.c,d,Simulation results for a single nanofin.c,Phase plots as a function of frequency for differentrotation angles for nanofins with /=250nm and w=80nm.d,A comparison of polarization conversion efficiency for different nanofin lengths from FDTDcalculations (solid lines)versus MODE Solutions (dashed lines).The lengths of the nanofins are labelled:they have a constant width w=80 nm.e,Phasespectra and polarization conversion efficiencies for five different elements showing the tunability of the group delay by changing the lengths and widths ofnanofins.The shaded region marks the design bandwidth of 120 nm.Each coloured curve corresponds to itselement schematically shown on the right.Theparameters (h wi.,w2)of each nanofin are labelled;all dimensions are in nanometres.The elements in the coloured squares are located at different radialpositions from the edge of the metalens(red square)to the centre(purple square),such that the corresponding group delay (slope of the phase versusangular frequency plot)increases from the edge to the centre.This ensures achromatic focusing as illustrated in Fig.1a.higher frequencies,the observed deviations result from the excita-imparted by identical nanofins using the geometric phase.Thetion of higher-order modes and resonances within the nanofins2.achromatic metalenses were designed by digitizing the requiredFigure 2e shows the phases and polarization conversion efficienciesphase and group delay,which were then implemented by select.of five different nanofin elements.Their group delays were obtaineding elements from a library of various nanofin parameters.Forusing linear fitting of the phase spectra within a bandwidth ofn=2,we also selected elements with group delay dispersion close120nm,centered at 530nm (see Methods for details).This ensuresto the required ones (see Methods for details).Supplementarythat the group delay of an element fulfils the requirement shown inVideos 1 and 2 show the imparted phases for the n=0 and n=2Fig.1b and its group delay dispersion is close to zero for at least themetalenses.The measured normalized focal length shifts and120nm bandwidth being considered.However,as seen later in thetheir theoretically predicted values from =470nm to 670nmtext,our simulations and experimental results show that the met-are shown in Fig.3a.The latter was calculated by propagatingalens focal length is only weakly dependent on wavelength beyondthe fields generated by the nanofins using Fresnel-Kirchhoffthis bandwidth,up to 670nm.Note that for low NA,the requiredintegration,neglecting the actual coupling between metalens ele-range of group delay is proportional to the product of lens radiusments.However,we also solved Maxwell's equations using FDTDand NA(see Methods).We designed and implemented metalensessimulations for a full metalens with NA=0.6 and 6um diameterwith nanofin dimensions corresponding to a group delay range of(due to the limitation ofcomputational resources)to confirm ach-about 5 fs;see Supplementary Fig.3 for a plot of polarization con-romatic focusing.Note that the effect of larger NA compensatesversion efficiencies versus group delays.for the smaller diameter.The results are shown in SupplementaryVideo 3.Experimentally,the focal lengths at different wavelengthsAchromatic focusing and imaging.To demonstrate the versatil-were obtained by measuring their intensity profiles(point spreadity of our approach,we designed and fabricated an achromaticfunctions)along the propagation direction(zaxis)of the incidentmetalens (n=0)as well as two other metalenses with n=1 andbeam in steps of 1 um,as shown in Fig.3b,d.The z coordinate cor-2.They all possess NA=0.2 at wavelength=530 nm.For n=1,responding to the peak intensity value gives the focal length for athat is,a regular diffractive metalens,the phase profile wasgiven wavelength.NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnology2018 Macmillan Publishers Limited,part of Springer Nature.All rights reserved