Resonant Metalenses for Breaking the Diffraction Barrier
Fabrice Lemoult, Geoffroy Lerosey, Julien de Rosny, Mathias Fink
RResonant Metalenses for Breaking the Diffraction Barrier
Fabrice Lemoult, Geoffroy Lerosey, ∗ Julien de Rosny, and Mathias Fink
Institut Langevin, ESPCI ParisTech & CNRS, Laboratoire Ondes et Acoustique, 10 rue Vauquelin, 75231 Paris Cedex 05, France (Dated: 14 April 2010)We introduce the resonant metalens, a cluster of coupled subwavelength resonators. Dispersion allows theconversion of subwavelength wavefields into temporal signatures while the Purcell effect permits an efficientradiation of this information in the far-field. The study of an array of resonant wires using microwaves providesa physical understanding of the underlying mechanism. We experimentally demonstrate imaging and focusingfrom the far-field with resolutions far below the diffraction limit. This concept is realizable at any frequencywhere subwavelength resonators can be designed.
PACS numbers: 41.20.-q, 81.05.Xj, 78.67.Pt
Within all areas of wave physics it is commonly believedthat a subwavelength wavefield cannot propagate in the farfield. This restriction arises from the fact that details withphysical dimensions much smaller than the wavelength arecarried by waves whose phase velocity exceeds that of lightin free space, which forbids their propagation. Such waves,usually referred to as evanescent waves, possess an exponen-tially decreasing amplitude from the surface of an object [1].Numerous works have been devoted to overcome thisdiffraction limit, starting from the early 20th century andthe proposal by Synge of the first near-field imaging method[2]. Since this seminal work, near-field microscopes havebeen demonstrated from radio frequencies up to optical wave-lengths, achieving resolutions well below the diffraction limit[3–8]. Fluorescence based imaging methods have also beenproposed, which allow deep subwavelength imaging of liv-ing tissues [9]. Such concepts, however, employ several mea-surement of the same sample in order to beat the diffractionlimit through image reconstruction procedures. Finally, var-ious new concepts have been proposed such as far-field su-perlens and hyperlens [10–12], demonstrating moderate sub-diffraction imaging down to a quarter of the optical wave-length.In this Letter, we introduce the concept of resonant metal-ens, a lens composed of strongly coupled subwavelength res-onators, and prove that it permits sub-diffraction imaging andfocusing from the far-field using a single illumination. Ourconcept resides in exploiting time coded far-field signals forspacial sub wavelength resolution [13, 14]. Studying the spe-cific case of an array of resonant wires, we explain theoreti-cally, prove numerically, and demonstrate experimentally howthis lens converts the subwavelength spatial profile of an ob-ject into a temporal signature and allow efficient propagationof this information towards the far-field. We achieve far-fieldimaging and focusing experiments with resolutions of respec-tively λ/ and λ/ , well below the diffraction limit.The notion of evanescent waves finds its roots in a math-ematical formalism which perfectly fits that of infinite inter-faces. Indeed, the projection of an infinitely extended sub-wavelength varying field onto a basis of free-space radiationsresults in a null value: the field sticks to the object, the wavesare evanescent. However, this does not hold anymore when considering objects of finite dimensions. In these cases thesub-diffraction details of the latter contribute to the far-fielddue to finite size effects. Practically, the efficiency of thisconversion decreases dramatically with the diminishing sizeof the subwavelength spatial variation. Hence, any measure-ment of sub-diffraction details in the far-field appears very te-dious since their contribution to the total field is much weakerthan those of propagating diffraction-limited waves.Such a monochromatic approach seems limited since allspatial information (subwavelength or not) propagating awayfrom an object mixes in a unique wavefield. Another diffi-culty is that the smaller the detail to be resolved, the weakerits contribution to the radiation of the object. The solutionto both issues lies in the concept of resonant metalens. Wedefine the latter as a cluster of resonators arranged on a sub-wavelength scale forming a lens in the near-field of an object,and illuminated with broadband wavefields. Its mechanism,which will be developed in detail later for a special case madeout of conducting wires, can be explained intuitively.On one hand, placing N identical resonators on a sub-wavelength scale introduces a strong coupling between them,which splits the original resonance frequency into a band of N different ones, analogous to Kronig-Penney potential wellsin solid state physics [15]. The cluster of oscillators can bedescribed by a set of N eigenmodes and eigenfrequencies,the latter being distributed on an interval which depends onthe strength of the coupling. Illuminated by a broad range ofenergies, the near-field on any subwavelength object, at theresonant metalens input, decomposes onto the modes of thesystem with a unique set of phases and amplitudes. Since allthose modes are excited at a different frequency (ignoring anydegeneracy due to symmetry), the information of the objectgets translated in the spectrum of the field generated in thelens. In the temporal domain, harnessing the modal disper-sion of the resonant metalens permits the conversion of thesubwavelength details of the object into a temporal signature.On the other hand, our lens grants an efficient propagationof subwavelength information towards the far-field. Ignoringthe intrinsic losses of the material, the energy stored in a givenmode of the lens be dissipated through radiative decay only.Any eigenmode of the system can be defined by a wavevec-tor, which will be referred to as a transverse wavevector → k ⊥ , a r X i v : . [ phy s i c s . op ti c s ] J un xyz N o r m a l i z e d A m p l i t u d e
280 300 320 340 36000.20.40.60.81 Frequency (MHz) N o r m a l i z e d A m p l i t u d e a) b) c) E x,y B x,y E z FIG. 1. (a) Three dimensional representation of the medium described in the text. Superimposed: amplitude of E x TEM Bloch modes (1,1),(2,3), (5,6) and (19,19). (b) Longitudinal profile of electric field E xy (blue), E z (green) and magnetic field B xy (red). (c) Results of thetransient 3D simulations: inside the structure and in the far-field. Red arrows: resonance frequencies of the 4 modes mapped in (a). and the higher its norm k ⊥ , the lower the efficiency of theconversion from this mode to propagating waves. This ineffi-ciency enhances the lifetime of the mode in the structure. Dueto the Purcell effect [16], the higher the lifetime of a givenmode, the better the coupling from the illuminated object tothis mode: surprisingly this counterbalances the effect of theweak coupling of the mode to the far-field. Ignoring the in-trinsic losses and due to the resonant nature of the lens, ev-ery subwavelength mode, independently of k ⊥ , radiates in thefar-field an equivalent amount of energy over time. Becauseof their higher lifetimes, deeper subwavelength modes tend toescape the lens tardily. The intrinsic losses diminish the life-times of the modes proportionally to their localization on theresonators (the higher k ⊥ , the shorter the lifetime): this, inturn, will limit the resolution of the resonant metalens.We focus now on the peculiar resonant metalens consist-ing of an ordered collection of parallel conducting wires. In-terestingly, a wire array forms in the transverse plane a sub-wavelength arrangement of resonators thanks to the resonanceoccurring along the longitudinal dimension. We numericallystudy a medium (Fig. 1.a) made out of a square periodic latticeof N × N ( N = ) perfect electric conductor wires of diame-ter d ( mm), with equal length of L ( cm), and a periodof a ( . cm) between the wires in both transverse directions( xy plane). The array lies in air and thus the first resonancefrequency of a single wire occurs if its length matches half awavelength ( f = MHz). At this frequency, the spacing be-tween the resonators corresponds roughly to λ/ , meaningthat the wires are strongly coupled. Fortunately, this lens canbe analyzed using the theory of the ”wire media” [17, 18] in-stead of calculating all of the coupling coefficients. The fieldpropagating in the structure can be expanded in Bloch modesbecause of the periodic nature of the medium, and inside thestructure [18], the boundary conditions and the deep subwave-length period impose a transverse electromagnetic nature ofthe field (TEM, E z = B z =0). Due to the transverse finiteness of the system, the → k ⊥ are quantified, → k ⊥ = πD ( m. → e x + n. → e y ) , withintegers ( m, n ) ∈ [[1; N ]] and D the size of the medium in the x and y dimensions, D = a ( N - .Those modes present a constant longitudinal wavevector k z independent of → k ⊥ : the longitudinal propagation is dispersion-less and the phase velocity equals that of plane waves in thehost matrix [17, 18]. The resonant behavior of the modes canbe revisited at the light of the TEM approach: due to theirfinite length, the wires constitute Fabry-Perot cavities for themodes [17]. This specific kind of resonant metalens presentsthe great advantage of being analyzable within two differentand complementary frames: the subwavelength coupled oscil-lators and the Fabry-Perot like TEM Bloch modes.We have performed numerical simulations whose detailsare presented in [19]. The structure is excited with a smallelectric dipole mm away from the lower interface, which wedefine as the input of this resonant metalens. The emitted sig-nal is a ns pulse centered around MHz. As expected, thefields are transverse electromagnetic (Fig. 1.b). We underlinehere that taking advantage of the Fabry-Perot resonance per-mits an efficient electrical coupling to the eigenmodes sincethe electric field is maximum at both ends of the wire medium(Fig. 1.b). Naturally, the field generated by the source ex-pands on the eigenmodes, and in figure 1.a, superimposedon the structure, we map the electric field for four differentmodes chosen among the N ones. The time varying fieldsin the structure and in the far-field are plotted alongside theirspectra in Figure 1.c. In fact, the unique decomposition of thesource onto the eigenmodes manifests itself in the spectrumand the temporal evolution of the near-field. Probing now thefar-field in the ( xy ) plane, which is vertically polarized since E x and E y are odd while the boundary conditions impose aneven E z (Fig. 1.b), the spectrum looks very similar to thatof the near-field. This proves that the near-field converts ef-ficiently to propagating waves, as predicted. Thanks to themodal dispersion of the lens, subwavelength details ranging f m / f ⊥ /k L i f e t i m e ( µ s ) FIG. 2. The dispersion relation (solid blue line) in terms of the modalresonance frequencies f m (normalized to f ) versus k ⊥ (normalizedto k ). Symbols: dispersion relation extracted from the simulations(the 4 symbols represent the 4 radiation patterns). Red symbols: life-times extracted from simulations as a function of k ⊥ (normalized to k ). Red solid curve: linear fit of slope ( Dk ⊥ ) − . from k to k ( λ /a) decompose onto the modes, are con-verted into temporal information and the profile of the sourcepropagates toward the far-field stored in the spectrum of thefield. To exemplify this, we point with arrows the exact reso-nance frequencies of the 4 modes mapped in Fig. 1.a. on thenear-field and far-field spectra.The dispersion of the wire medium, a key issue for our res-onant metalens, can be derived and understood quite easily:even though TEM Bloch modes share the same phase veloc-ity, their penetration depth in air at the z = − L/ and z = L/ interfaces depends on their transverse wavevector k ⊥ . Conse-quently, the longitudinal extension of the mode depends on k ⊥ , which modifies the effective length of the Fabry-Perotcavity, and gives the following dispersion relation [20]: f f = 1 + 2 π (cid:112) ( k ⊥ /k ) − ( f /f ) (1)This theoretical dispersion relation is plotted in terms of themodes resonance frequency f m versus k ⊥ (both normalizedto the original Fabry-Perot resonance frequency f and wavenumber k ) in Figure 2. We also extracted from the simula-tions the modes and calculated their transverse wavenumber k ⊥ . Those data are superposed with the theoretical curve, andthe good agreement proves the validity of the model.Equally interesting, the far-field propagation of the sub-wavelength modes deserves explanation. The spectra in Fig-ure 1.c show a diminution of the linewidth of the modes forthe high transverse wavenumbers, consistently with the res-onant metalens intuitive description we gave: the more sub-wavelength a wave mode, the higher its lifetime in the struc-ture. To estimate the lifetime theoretically, we evaluate theefficiency of the conversion of the subwavelength modes tofree-space waves [20]: it is roughly proportional to ( Dk ⊥ ) − . The Purcell effect results in an increase of the coupling be-tween the source (our ”object”) and the high k ⊥ modes of theresonant metalens. The measure of the source’s return lossdemonstrates the impedance matching in the presence of themetalens [20]. The resonant nature of the modes matches theimpedance of the small electric dipole, or equivalently, thePurcell effect compensates for the weak radiation of the deepsubwavelength modes.We estimate the lifetimes of the modes in the metalens re-sulting from our simulations, through a time/frequency anal-ysis. As we stopped the simulation after µ s for calcula-tion time issues, the lifetimes of the high k ⊥ could not beextracted. We plot the data in Figure 2, as well as a linear fit.An important remark concerns the harmony between the life-times of the modes and the corresponding dispersion relation.Indeed, the coding of subwavelength information in time re-quires that a maximum of the modes can be resolved. Here,nicely enough, the increase of the lifetime for modes of high k ⊥ counterbalances the flattening of the dispersion relation.In Fig. 2, the four types of radiation pattern generatedby this structure are also presented. Depending on the m and n indexes of → k ⊥ , monopolar, dipolar ( x or y oriented)or quadrupolar patterns coexist [20]. This underlines thatthe lens, although subwavelength ( λ/ ), also possesses fourspatial degrees of freedom, representing as many informationchannels exploitable for imaging and focusing [21].In order to seal the validity of the concept for real mate-rials, we performed an experimental verification of the lens.The realistic lens presented in Figure 3.a replicates the simu-lated one, except that we use copper wires and a Teflon sup-port ( ε T = . ). Also, to avoid radiation leakage on the sourcecable as well as parasitic effects, we screen it by placing thestructure mm on top of a 1 meter square ground plane. Wemeasure in 8 directions the far-field generated by a small elec-tric monopole located between the ground plane and the res-onant metalens in an anechoic chamber [19]. The signal andspectrum for one direction, plotted in Figure 3.b.c, proves avery good agreement with the simulation results. Two remarksarise; First, the resonance frequencies show a redshift due tothe Teflon structure. Second, the signals spread over a shortertime due to both the skin effect on the copper wires, whosepermittivity is finite, and the losses in the Teflon; this effectmanifests itself more clearly at high frequencies. Indeed, thehigher k ⊥ , the more localized on the wires the modes, whichdecreases their lifetime due to ohmic losses.In Figure 3.d, we plot the result of a time reversal focus-ing experiment like in [22, 23] achieved from the far-field,in an anechoic chamber in order to measure the effect of thelens only. The focal spot obtained plotted alongside the con-trol experiment (without the resonant metalens, no focusingat all) is roughly λ/ wide (at the central frequency of theexcitation pulse). This means that subwavelength informa-tion of the sources have been converted in the far-field, andvice versa. Using time reversal, the green function betweena source and the far-field antenna is ”flipped” in time andreemitted. At each frequency, the signal is phase-conjugated, N o r m a l i z e d A m p l i t u d e
200 250 300 35000.20.40.60.81 Frequency (MHz) N o r m a l i z e d A m p l i t u d e a) b) c)d) e) −1−0.500.51 − λ − λ − λ − λ
80 0 λ λ λ λ − λ − λ − λ − λ
80 0 λ λ λ λ λ FIG. 3. (a) The experimental resonant metalens on the ground copper plane. Experiments are performed in an anechoic chamber. (b-c) Signalsand spectra received in the far-field after emission from central monopole with the lens (blue) and without as a control curve (red). (d) Focalspot obtained after one channel Time Reversal of (b) from the far-field: a λ/ width is demonstrated in the presence of the resonant metalens(blue), no focusing without the lens (red). (e) An imaging experiment. 16 monopoles generate a subwavelength phase and amplitude profilein the near field of the lens (black points). The far-field is acquired on 8 antennas. We plot the result of the image reconstruction: a true λ/ resolved image of the initial pattern is reconstructed in the presence of the resonant metalens (blue) while it is impossible without (red). meaning here that all of the TEM Bloch modes generated inthe lens add up in phase at a deterministic time, hence allow-ing the Time Reversal focusing. Since the decomposition ofa point-like source onto the eigenmodes of the resonant met-alens is unique, the modes add up incoherently at other po-sitions. We point out that our precedent results [22] can beinterpreted at the light of the resonant metalens concept. Wenote that losses limit our focal spot sizes to λ/ , but usingother focusing techniques may shrink the spots even further.Finally, we prove the imaging capabilities of the resonantmetalens through a simple experiment: a subwavelength pro-file is generated at the input of the lens using simultaneously16 monopoles [19], and the far-field recorded in the anechoicchamber.An inversion procedure with predesigned filters [20]is used to reconstruct the profile, using the knowledge of eachmonopole temporal signature (Fig. 3.e). The subwavelengthprofile is perfectly reconstructed and an imaging resolution ofabout λ/ is demonstrated through this basic experiment.To conclude, this specific studied lens is scalable towardsnear-IR and in this range, the losses will increase, limitingthe resolution. Using gain media in the matrix may counterthis problem. More generally, we are currently working ona criterion linking the resolution achievable to the losses andtypical size of the metalens. This lens presents degeneratedmodes because of the symmetry: adding some disorder in thespatial or resonant frequency distribution of the resonators, aswell as in the matrix should lift this degeneracy and enhancedispersion. Finally, we would like to emphasize that the con-cept of resonant metalens should be realizable in any part ofthe electromagnetic spectrum, with any subwavelength res-onator, such as split-rings [24], nanoparticles [25], resonantwires [26], and even bubbles in acoustics [27].We thank David F.P. Pile for his help with writing the manuscript and A. Souilah for the fabrication of the exper-imental prototype. F. Lemoult acknowledges funding fromFrench ”Direction G´en´erale de l’Armement”. ∗ [email protected][1] J. Goodman, Introduction to Fourier optics (Roberts & Com-pany Publishers, 2005).[2] E. H. Synge, Philos. Mag. , 356 (1928).[3] D. W. Pohl et al. , App. Phys. Lett. , 651 (1984).[4] A. Lewis et al. , Ultramicroscopy , 227 (1984).[5] E. Betzig and J. Trautman, Science , 189 (1992).[6] F. Zenhausern et al. , Science , 1083 (1995).[7] T. Taubner et al. , Science , 1595 (2006).[8] C. P. Vlahacos et al. , App. Phys. Lett. , 3272 (1996).[9] S. W. Hell and J. Wichmann, Opt. Lett. , 780 (1994).[10] S. Durant et al. , J. Opt. Soc. Am. B , 2383 (2006).[11] Z. Liu et al. , Nano Letters , 403 (2007).[12] J. B. Pendry, Phys. Rev. Lett. , 3966 (Oct 2000).[13] A. Grbic, L. Jiang, and R. Merlin, Science , 511 (2008).[14] X. Li and M. I. Stockman, Phys. Rev. B , 195109 (2008).[15] R. Kronig et al. , Proc. Roy. Soc.(London) A , 499 (1931).[16] E. Purcell, Phys. Rev. , 681 (1946).[17] P. Belov et al. , Phys. Rev. B , 33108 (2006).[18] G. Shvets et al. , Phys. rev. Lett. , 53903 (2007).[19] See supplementary material at http://link.aps.org/... [20]
A more detailed study will be published elsewhere. [21] F. Lemoult et al. , Phys. Rev. Lett. , 173902 (2009).[22] G. Lerosey et al. , Science , 1120 (2007).[23] G. Lerosey et al. , Phys. Rev. Lett. , 193904 (2004).[24] D. R. Smith et al. , Phys. Rev. Lett. , 4184 (May 2000).[25] J. L. West et al. , Ann. Rev. Biomed. Eng. , 285 (2003).[26] O. Muskens et al. , Nano Letters9