Far field subwavelength imaging and focusing using a wire medium based resonant metalens
FFar field subwavelength imaging and focusing using a wire medium based resonant metalens
Fabrice Lemoult, ∗ Geoffroy Lerosey, and Mathias Fink
Institut Langevin, ESPCI ParisTech & CNRS, 10 rue Vauquelin, 75231 Paris Cedex 05, France (Dated: October 29, 2018)This is the second article in a series of two dealing with the concept of ”resonant metalens” we introducedrecently [Phys. Rev. Lett. 104, 203901 (2010)]. It is a new type of lens capable of coding in time andradiating efficiently in the far field region sub-diffraction information of an object. A proof of concept of sucha lens is performed in the microwave range, using a medium made out of a square lattice of parallel conductingwires with finite length. We investigate a sub-wavelength focusing scheme with time reversal and demonstrateexperimentally spots with focal widths of λ/ . Through a cross-correlation based imaging procedure we showan image reconstruction with a resolution of λ/ . Eventually we discuss the limitations of such a lens whichreside essentially in losses. PACS numbers: 41.20.-q, 81.05.Xj, 78.67.Pt
I. INTRODUCTION
Conventional imaging devices, such as optical lenses, cre-ate images by capturing the waves emitted by an object andthen bending them. The image obtained is diffraction lim-ited by the Rayleigh criterion at best equal to half the oper-ating wavelength. Indeed, the information of the object on asmaller scale is carried by evanescent waves that decrease ex-ponentially from the surface of the object: these waves neverreach the image plane explaining the Rayleigh limit.In order to increase the image resolution, the first idea thatarised was to measure the evanescent field: Synge proposedin the early 20th century the first near field imaging proce-dure. This first work has lead to the development of variousnear field scanning microscopes based on the measurement ofthe evanescent wavefield . These methods require multiplescans of the object and they cannot be used for living cells thatrequires real-time imaging.In 2000, J. Pendry suggested that a material with a nega-tive index of refraction could make a perfect lens. Such a ”su-perlens” has first interested the microwaves community, untilit reached the optical range in 2005, when Zhang’s group verified experimentally that optical evanescent waves couldindeed be enhanced as they passed through a silver super-lens. They imaged objects as small as nm with their super-lens, five times the resolution limit of classical optical devices.Nevertheless, even if the ”superlens” creates a perfect imagethe need of a near field scan is still required.There is obviously a need of far field methods for imag-ing living cells in real time. The most recent ideas for perfectimaging reside in a projection of the near field profile by theuse of a diffraction grating or by a first step of magnifica-tion before conventional imaging . In the former case theresolution is limited by the short range of wave numbers ac-cessible in the far field, and in the latter by the geometry of themagnifying lens. Moreover, the intrinsic losses of real mate-rials diminish the resolution ability of these devices. Nowa-days, many works concentrate on surface plasmons polaritonssince their dispersion relation allows propagation of waveswith high wave numbers that carry informations smaller thanthe operating wavelength.In a recent letter , we proposed the concept of a resonant metalens: a cluster of strongly coupled resonators illuminatedwith broadband wave fields. The concept is the following:the small details of the object, usually carried by evanescentwaves, are in this case converted onto propagating ones: mea-suring the far field one can obtain sub-wavelength informationabout the object. We achieved far field imaging and focusingexperiments with resolutions of respectively λ/ and λ/ ,well below the diffraction limit. Like the initial developmentof the Pendry’s ”superlens”, we performed a proof of conceptin the microwave range because of experimental easiness butthe transposition toward optical range is not speculative sincesub-wavelength optical resonators are designable.We focus on conducting wires with finite length used asresonators, and we scope on the resonant metalens introducedinitially consisting of a square lattice of N × N ( N = 20 )identical metallic wires, aligned along the vertical axis (defin-ing the longitudinal z -direction). The length of the wiresalong z is equal to cm and their diameter to mm. The pe-riod lattice in both transverse directions ( xy plane) is . cm,which is roughly equals to λ/ for the first intrinsic reso-nance of a single wire. The experimental sample is made outof copper wires and a Teflon structure.During the first article we have demonstrated that a sub-wavelength broadband source placed at the input of the met-alens excites the sub-wavelength varying eigenmodes therein.Then we have shown that each mode radiates in the far field,and thanks to dispersion, the sub-wavelength information ofthe source reaches the far field region stored in the spectrum.In the figure 1 we sketch the physical mechanisms that per-mit the super-resolution properties of this system. The ra-diated field contains the details of the object as a frequencysignature, and exploiting this signature with the dispersionrelation permits to build an image. We will show how onecan exploit these phenomena to achieve sub-wavelength imag-ing/focusing from the far field. A time reversal scheme forsub-wavelength focusing is presented. The reciprocal coun-terpart of focusing, the imaging is then presented. The imag-ing procedure is a derivative of time reversal because it con-sists in cross-correlation imaging with knowledge of the initialGreen’s functions. Finally, we discuss the impact of losses in-herent to experimental devices and link them to the obtainableresolution of our device. a r X i v : . [ phy s i c s . op ti c s ] N ov + + + + ...+ + + + ...resonancesdecomposition onto the sub- modes of the resonant metalens f f f f f n radiation toward farfielda 2D object to be imagedimage obtainedfrom spectrum /3reconstruction from spectrum Frequency F a r fi e l d A m p l i t u d e + ...+ ... FIG. 1. (color online) Schematic representation of the resonant met-alens mechanism. A sub-wavelength object is placed at the input ofthe resonant metalens. The broadband near field emitted by the ob-ject decomposes onto the sub-wavelength eigenmodes supported bythe metalens. Each mode experiments its own resonance and radiatesfield toward far-field at its own resonance frequency: the resonantmetalens has colored the emission. Then by exploiting the disper-sion relation and the radiated spectrum, a super-resolved image canbe built. The resolution of the image will be discussed in the last partof this article.
II. TIME REVERSAL SUBWAVELENGTH FOCUSING
The resonant metalens radiates fields that contain the tem-poral signature of the near field profile, and the step is nowthe exploitation of this temporal code for focusing on a deepsubwavelength scale. Based on works made at the laboratory,we have proposed to use the time reversal technique to exploitthe radiation. Time reversal is a good candidate because it uses broadband waves and it has been proved to be a powerfulprinciple for the study of waves in complex media , theiruse for focusing , or even telecommunication . A. Theory
In a typical time reversal scheme, a dipole source emits ashort pulse. The wave field propagates and is recorded withone antenna or a set of antennas, usually referred to as thetime reversal mirror (TRM). Second, the recorded signals aredigitized, flipped in time and transmitted back by the same setof antennas. The resulting wave is found to converge back tothe initial source.For a narrowband signal of oscillating pulsation ω , the timereversal focusing on position r is equivalent to phase con-jugation, and the electric field generated at position r simplywrites : E rt ( r , ω ) = − iµ ω Im ↔ G ( r , r , ω ) p ∗ (1)Where ↔ G stands for the dyadic green function and p repre-sents the initial dipole source. When the dyadic green func-tion is the free space one, it results in a cardinal sine function:the focal spot is diffraction limited to λ/ . When the time re-versal mirror is placed in the near field of the initial source thegreen function takes into account the evanescent componentof the field generated by the source and a smaller spot can beobtained .Note that the field amplitude at the focal point is propor-tional to the imaginary part of the Green’s function which isitself proportional to the so-called local density of states (LDOS). In the special case of the resonant metalens pre-sented here, the LDOS depends on the TEM Bloch modesthat resonate at the operating angular frequency ω . However,for broadband excitation, the resulting field takes advantageof the frequency diversity. For an excitation with a flat band-width ∆ ω , all frequencies add up in phase at a given time, thecollapse time (t = 0). At this specific time, the time reversedfield writes : E rt ( r , t = 0) ∝ µ ω (cid:90) ∆ ω Im ↔ G ( r , r , ω ) p ∗ d ω (2)Thus, the time reversed field at the source point and at thefocal time is directly proportional to the number of indepen-dent TEM Bloch modes excited by the source inside the met-alens. In the presence of the metalens, the near field of thewaves inside it allows the Greens tensor to fluctuate on a scalesmaller than the wavelength, by the way of the TEM Blochmodes, and therefore to reach a sub-wavelength focusing withtime reversal . B. Experiment
All the measurements are realized in an anechoic chamber,in order to isolate the response of the resonant lens from the N o r m a l i z e d A m p l i t u d e − λ − λ − λ − λ
80 0 λ λ λ λ − λ − λ − λ − λ
80 0 λ λ λ λ − λ − λ − λ − λ
80 0 λ λ λ λ N o r m a l i z e d A m p l i t u d e N o r m a l i z e d A m p l i t u d e − λ − λ − λ − λ
80 0 λ λ λ λ FIG. 2. Signals (top) and spectra (middle) received in the far field after emission of a 10 ns pulse from the central monopole with the lensfor distinct position of the far field receiving antenna. From left to right it corresponds respectively to an azimuthal angle of 0 ˚ , 45 ˚ and 90 ˚ .(bottom) The focal spot obtained for each one-channel time reversal experiment. λ/ widths are demonstrated in the presence of the resonantmetalens for each angle. Taking advantage of the spatial diversity by doing multi-channels time reversal reduces the spatial side lobes as shownon the bottom. one of the host medium, namely the cavity used in or atthe laboratory. The experimental resonant metalens, is placedvertically on a 1 by 1 meter ground plane in order to screenthe cable that would otherwise induce parasitic effects. Thisis a convenient way to realize a near field experiment withmicrowaves. Another approach would have consisted in usinga wax or glass prism used in total internal reflection in order toilluminate a small dipole at the input of the lens, but given theoperating wavelength here, we did not choose it for practicalreasons.In the time reversal experiment, we use two 8 channels mul-tiplexers between the generator and the sources. In this con-figuration, 16 small identical monopoles are placed betweenthe ground plane and the lens, and linked to the multiplex-ers through the ground plane acting as a shield for the ca-bles, by means of soldered connectors. The monopoles arelinearly placed, and the separation between two consecutivemonopoles is . cm, ie. one period of the wire array.We have a single receiving antenna, that will then be ourtime reversal mirror, namely it will be the source that focuses the waves at the output of the resonant metalens, back to theoriginal source. First, we emit a ns pulse using one of themiddle monopoles from the 16 monopoles array. Then, werecord the field generated by this source using the verticallypolarized antenna 6 wavelengths apart in the anechoic cham-ber. The signal is then digitized at a sampling rate of GHzusing an oscilloscope, and flipped in time using Matlab anda computer connected to the oscilloscope. We now exchangethe cables, and use the receiving antenna as a transmitting an-tenna, acting as a 1 channel Time Reversal mirror. The timereversed signal is sent to an arbitrary waveform generator, andused for the emission from the vertically polarized antenna.After propagation in the anechoic chamber, the signal ar-rives on the resonant metalens, and we record the fields gener-ated on the 16 monopoles, which are now receivers, switchingfrom one monopole to the others using multiplexers. We ac-quire the fields on other positions using the oscilloscope andstore the 16 temporal waveforms. The focal spot obtained us-ing time reversal is then defined as the maximum of the squareamplitude of each waveform across the temporal window (Fig.2). First, it confirms the theoretical explanation of the wiremedium since a short pulse transforms onto a signal that ex-tends in time due to resonances. The spectra also show manydistinct resonance peaks, each one corresponding to its ownsub-wavelength scale as stated by the theoretical dispersionrelation. With a single isotropic antenna placed in free space,a sub-wavelength spot of width λ/ is obtained at the inputof the wire medium after time reversal. Here, we show thatthis experiment is formally the same, but simplified, as theone conducted initially .To demonstrate the spatial diversity of the radiation, wehave measured the fields produced by the structure in 8 dis-tinct directions, tilting the ground plane and the wire mediumof 45 ˚ each time. We show in Figure 2 the results for threemeasurements taken at 0 ˚ , 45 ˚ and 90 ˚ in the far field. Thetemporal signals as well as their Fourier transforms receivedin the far field are plotted. The result of the same one channeltime reversal scheme has been experimented for the three an-gles. One can notice that the 3 spectra show resonance peaksat distinct positions revealing the spatial diversity. For thescope of imaging, this property will permit to fight againstmodes degeneracy: at a given frequency where multiple Blochmodes resonate it is actually possible to recover the weight ofeach one thanks to the spatial degrees of freedom. Eventually,we performed a three channels time reversal experiment. Thefocal spot obtained has the same width as previous ones, butthe side lobes obtained have a lower amplitude. III. SUBWAVELENGTH IMAGING
With the time reversal experiment, the focusing feasibilityon a sub-wavelength scale has been demonstrated. In this part,we consider the reciprocal operation, namely the imaging. Inmost of broadband imaging techniques, images are created byapplying filters to the received RF data. The simplest filter isthe delay line filter known as beamforming that only considersfree space propagation of short pulses. Its resolution is givenby the Rayleigh criterion δr = 1 . λF/D , where F is thefocal length and D corresponds to the aperture length. Whenthe propagation occurs in complex media the Green’s functionbecomes more complicated, the beamforming shows its limi-tations and more complicated filters are required. The imagingprocedure consists in cross-correlations between these filtersand a set of received signals that supposedly carry an imageinformation (Fig. 3).As in the time reversal scheme, we propose to use filtersthat exploit the whole duration of the Green’s functions, anda calibration step will consist in recording all of the Green’sfunctions. Typically, with N receiving antennas placed in thefar field region, we first record a set of L × N Green’s func-tions g ij ( t ) . With this knowledge, we are presumably able toget L distinct pixels in the image plane. The time reversedGreen’s functions could have been used to build the image,but it is not the best signals in terms of resolution since timereversal is an operation that maximizes the energy, and notminimizing the side lobes level.The idea is thus to design signals that are more like an in- −1−0,500.51 − λ − λ − λ − λ
80 0 λ λ λ λ t ttsub- λ objects Farfield antennasResonant Metalens Scope TDS 6604Bcross-correlationwith predesignedsignalssub- λ imagegenerationsub- λ object /80 FIG. 3. (top) Schematic representation of the subwavelength imag-ing procedure. When a subwavelength source, placed near the reso-nant metalens, emits a broadband pulse, the radiated field is recordedwith 8 antennas placed in distinct azimuthal directions. With an a pri-ori knowledge of all of the impulse responses, a bank of filters werepreviously built. The image is reconstructed by cross correlation be-tween the received signals and the filters. (bottom) The experimentalsub-wavelength object (left) consists in a simultaneous emission ofthe 16 small monopoles (shown on picture) with a given amplitudeprofile (green line). An image reconstruction (right) with a resolutionof λ/ can be achieved. verse filter which has better resolution capabilities . Inver-sion procedure suffers from matrix inversion and noise, andat the laboratory inversion procedures based on iterations oftime reversal have been proposed . Using the signals g ij ( t ) , we then construct a bank of L × N pseudo-inverse sig-nals h ij ( t ) based on numerical iterations of time reversal. Inthis peculiar case, we applied 50 iterations of time reversal in order to diminish the spatio-temporal lobes, followed byan iterative spatial inversion permitting a better discrimina-tion between sources at the focusing time. The designed fil-ters generate smaller spatial side lobes than the time reversedGreen’s functions. Undoubtedly one can design more robustsignals, especially if the image area grows, and this is onlya proof of concept. At this step the system is calibrated andready for the imaging procedure.In the experiment presented in the original paper , we ap-plied different weights to each monopolar antenna, placed inthe near field of the resonant metalens, and make them emit si-multaneously a short pulse. We record with the far field anten-nas a set of N = 8 RF signals s j ( t ) (This step was done by lin-earity of the wave equation). These signals supposedly carrythe subwavelength information of the object. The image re-construction starts with cross-correlations between these sig-nals and the filters h ij ( t ) . The final image is the result ofthis operation at the origin time. The result presented in thepaper demonstrated that the Green’s functions of two pointsseparated by λ/ are sufficiently uncorrelated, and an imageresolution of λ/ has been achieved.Actually, as all of the near fields techniques the object to beimaged disturbs the near field of the resonant metalens, andthis why we chose to use a simultaneous emission of smallmonopoles on the top of ground of plane. Thus, the exper-imental setup did not permit us to perform 2D imaging be-cause of the original arrangement of the monopoles. But, forthese ends we could use a wax prism (and a contrast objectplaced on the top of it) illuminated in total internal reflectionto reach the full 2D superresolved image. For the range of fre-quency considered here this prism would be very heavy andwe will wait to design a smaller lens to perform this exper-iment. Eventually, the design of the resonant metalens withits high symmetries is not a break to achieve 2D imaging. Asstated in the original paper , the square geometry gives fourdistinct radiation directivities for the Bloch modes. Thus, ata given frequency where several Bloch modes are degenerate,the spatial degrees of freedom permit to recover independentlythe information carried by each of them. IV. INFLUENCE OF LOSSES
Ignoring losses, the limitation for the metalens seems tobe the periodicity of the medium: the mode with the highesttransverse wavenumber that experiences a resonance has theperiodicity of the medium. Thus in the far field region, onecan measure information of the object as thin as the latticeparameter.Nevertheless, when considering losses, a new limitation ap-pears. Losses deteriorate the resonance Q -factor. We haveseen that the energy stored in a given mode of the wiremedium decreases through radiative decay only when consid-ering perfect conductors. It permitted to express a radiativequality factor and from now we refer to it as Q rad . Introducinglosses in metals (and/or in the dielectric structure) results in adegradation of the resonance that we will quantify with a loss Q -factor Q loss . The resonance quality factor Q is related tothese two quantities by: Q = 1 Q rad + 1 Q loss (3)From the previous article, we know that Q rad is roughly pro-portional to ( DLk ⊥ x k ⊥ y ) . The Q loss can be evaluated in asimple way when considering only the ohmic loss. First, us-ing the fact that the modes are TEM inside the structure, thedensity of energy along the wire is constant. In the transversedirections we have sine functions, thus the energy stored in-troduces a / constant. Thus, the energy stored in a givenmode can be estimated by: w stored = D L E µ c (4)Second, the surface currents on the conductor surfaces canbe estimated from the TEM fields inside the wire medium, andthe ohmic losses per unit conductor area are then integrated over the whole area. The surface resistance of lossy metalsdepends on the skin depth δ and the metal resistivity ρ andis equal to ρ/δ . Assuming that the magnetic field inside thewire medium evolves as cos( πz/ L ) , we can finally expressthe lost energy as: w loss = 1 ω ρδ L π ) r µ c (cid:88) wires E m (5)where E m refers to the amplitude of the electric in the m thwire which is linked to the Bloch mode considered. The reso-nant eigenmodes inside the wire medium correspond to sinefunctions thus the sum can be approximated by N E / where N corresponds to the number of wires in one direction.After simplification the Q loss writes: Q loss = D δN (2 + π ) r (6)One can notice that Q loss does not depend on the mode con-sidered (assuming the skin depth to be constant for the rangeof frequency used) and depends only on the design of the met-alens. The impact of losses manifest essentially when Q rad be-comes comparable to the loss constant which occurs for largetransverse wavenumbers, and losses tend to decrease the Q -factor. This decrease of the resonance quality has two conse-quences. First, losses have an influence on the amplitude ra-diated by a given mode. Second, the decrease of the Q-factorsresults in an increase of the resonance linewidth. A. Emission decrease
In order to illustrate this impact on imaging, we describe ametalens with two parameters: the length of its side D and itsnumber of resonators N +1 in this direction (the lattice period a is simply given by D/N ). The projection of a field sourceplaced at the input onto the eigenmodes write: P ( x, y ) = N (cid:88) m,n =1 A m,n sin (cid:16) mπD x (cid:17) sin (cid:16) nπD y (cid:17) (7)Each mode experiences its own cavity resonance describedby its own Q-factor Q m,n and its own resonance frequency f m,n . Then it propagates toward the far field region with anefficiency proportional to the square root of the radiative Q factor. The far field records permit to have a reconstruction ofthe point source that can be written: A ( x, y ) ∝ N (cid:88) m,n =1 A m,n (cid:115) Q m,n Q rad m,n sin (cid:16) mπD x (cid:17) sin (cid:16) nπD y (cid:17) (8)If there is no loss, A ( x, y ) gives the exact projection P ( x, y ) of the source onto the eigenmodes (figure 4). But, /3 objectimage without loss image with loss-110
260 280 300 320 340 360 38000.20.40.60.81
260 280 300 320 340 360 38000.20.40.60.81
Frequency [MHz] Frequency [MHz]unresolved modes(a)(b) (c)(e) (f)radiation(d)
Q = loss Q < Q loss Q Q lossrad rad FIG. 4. Illustration of the image resolution. (a) A 2 dimensional sub-wavelength field profile corresponding to our object. (b-c) The fieldspectrum radiated in a given direction by the object placed near theresonant metalens respectively without and with considering losses.(d) Illustration of the impact of losses: the losses tend to decrease theresonant Q -factor of a given mode which manifests in a decrease ofthe resonance amplitude and a decrease in the resonance linewidth.When the resonance linewidth becomes higher than the spectral dis-tance between two consecutive modes they cannot be resolved. (e-f)According to the criterion we plot the super-resolved image obtainedrespectively without and with losses in the wires. Without loss, allof the modes can be resolved and the resolution is given by the pe-riod lattice, here λ/ . While in the presence of losses the mostsub-wavelength modes are lost, and the resolution decreases down to λ/ which is still a deeply sub-wavelength resolution. when considering losses, there is a diminution of the ampli-tude received in the far field due to Q loss . It explains why thefocal spot obtained in the time reversal procedure is not asthin as the period lattice. Indeed, time reversal only matchesphases of signals and it cannot compensate for the attenua-tion. For imaging scope, the decrease of Q does not appeardramatic: by applying inversion procedures one can expect to compensate this decrease. This decrease may have an impactwhen considering noise: if the amplitude radiated becomeslower than the average amplitude of the noise, the inversionprocedure will not work. B. Resonance linewidth increase
The influence of the Q -factor also manifests in the reso-nance bandwidth. It results in a mixing of the resonance peaksand it becomes impossible to resolve all of the peaks. To quan-tify this aspect, we need the dispersion relation linking the res-onance frequency to the Bloch wave number. From this lawwe can extract the spectral distance δω between two consecu-tive modes: δω = ∂ω∂k ⊥ δk ⊥ (9)where the spectral distance between two consecutive modes δk ⊥ has been introduced. Now we have to estimate the groupvelocity of the medium of interest: this quantity depends onthe type of resonators used to build the resonant metalens. Inthe case of the wire medium, from the theoretical law obtainedin the previous article, we can express the group velocity as: ∂ω∂k ⊥ = ωk ⊥
11 + L (cid:113) k ⊥ − (cid:0) ωc (cid:1) (10)Combining the two previous equations, we estimate thespectral distance (normalized to the resonance frequency) be-tween two consecutive modes, calculated from the dispersionlaw. Assuming that the modes are strongly sub-wavelength,the opposite of this quantity writes in the case of the wiremedium based resonant metalens: (cid:18) δωω (cid:19) m,n ≈ δk ⊥ ) m,n Lk m,n (11)where ( δk ⊥ ) m,n corresponds to the k -space distance betweenthe mode with transverse wavenumber k m,n and its nearestlower neighbor mode radiating in the same direction. Forexample, if we consider the radiation in the direction (0 x ), ( δk ⊥ ) m,n = k m,n − max( k m − ,n , k m,n − ) .The spectral distance needs to be compared to the resonancelinewidth, a quantity contained in the resonance Q -factor. Wewill be able to discriminate two consecutive modes if thisspectral length is higher than the resonance linewidth of the ( m, n ) th resonance. Introducing the Q -factors already calcu-lated for the wire medium, it gives the following criterion: δk ⊥ ) m,n Lπ ( m + n ) (cid:62) π (cid:18) Lmn (cid:19) + 1 δN (2 + π ) r (12)When ignoring losses (ie. the skin depth δ is null) this rela-tion is always true since the left hand operand of the inequal-ity essentially decreases as / ( n + m ) while the right onedecreases as / ( m.n ) , and the trueness does not depend on N . It means that we can increase N as well as we like inorder to achieve the best resolution possible. Without con-sidering losses, the resolution that can be achieved is at bestequal to the period of the metalens. As an illustration of thisphenomenon, we have performed on figure 4 the projection ofa two dimensional speculative sub-wavelength source (a) ontothe modes of the structure and determinate the image achiev-able (e). One can notice that the image obtained is not pre-cisely the original object, but it gives an approximation of theobject with a resolution far below the diffraction limit.But, when introducing losses, the Q -factor characterizingthe resonance linewidth takes into account losses and a max-imal value for the couple ( m, n ) appears. Thus, it definesa new resolution limit because the sum in equation (8) willstop at ( N max , N max ) instead of ( N, N ) . This implies that theimaging resolution is now limited to D/N max : the resolutionhas become higher than the lattice parameter.The figure 4 shows the impact of losses on resolution: theradiated amplitude per mode decreases as well as the reso-nance linewidth. Thus it becomes impossible to distinguishthe resonances near f meaning that the modes associated tothem are lost. Finally, the image reconstruction that can beachievable when considering losses with the geometric pa-rameters of our metalens shows a lower resolution than thecase without loss, but a sub-wavelength image is still obtained.Another interesting remark concerns the design of such alens. We have seen that without losses, the smaller the latticeparameter the thinner the resolution: one can argue that it is interesting to increase N as high as possible in order to getthe thinnest details. But when considering losses in metals,increasing the number N of wires will drastically increase theohmic losses. Undoubtedly, an optimal number of resonatorsfor a given transverse dimension exists. V. CONCLUSION
In this article, we have demonstrated that it is indeed pos-sible to control wave on a scale that is smaller than the wave-length from the far field. As we stated in the previous article,the broadband radiation emanated from the resonant mediumcontains sub-wavelength information of a source. Through atime reversal focusing scheme, which has already proved tobe a powerful technique dealing with broadband signals, wedemonstrated focal spots of width λ/ . As time reversal can-not compensate for losses, a more accurately method has beeninvestigated for a cross-correlation based imaging procedure.We have performed the proof of concept that an image recon-struction with a resolution of λ/ can be achieved. Here, wehave used half a wavelength long wires as a single resonator,but we would like to emphasize that the concept of resonantmetalens should be realizable with any subwavelength res-onator, such as split-rings or nanoparticles. We believe thatthe concept could probably be a good candidate for real timeimaging of living cells, with a resolution far better than clas-sical microscopes. ∗ [email protected] E. H. Synge, Philos. Mag. , 356 (1928) E. A. Ash and G. Nicholls, Nature , 510 (Jun. 1972) A. Lewis, M. Isaacson, A. Harootunian, and A. Murray, Ultrami-croscopy , 227 (1984) D. W. Pohl, W. Denk, and M. Lanz, App. Phys. Lett. , 651(1984) J. B. Pendry, Phys. Rev. Lett. , 3966 (Oct 2000) V. G. Veselago, Physics-Uspekhi , 509 (1968) N. Fang, H. Lee, C. Sun, and X. Zhang, Science , 534 (2005) Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun,and X. Zhang, Nano Letters , 403 (2007) Z. Jacob, L. V. Alekseyev, and E. Narimanov, Opt. Express ,8247 (2006) A. Salandrino and N. Engheta, Phys. Rev. B , 075103 (Aug2006) F. Lemoult, G. Lerosey, J. de Rosny, and M. Fink, Phys. Rev. Lett. , 203901 (2010) A. Derode, A. Tourin, and M. Fink, Phys. Rev. E , 036606(2001) G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, andM. Fink, Phys. Rev. Lett. , 193904 (2004) M. Fink, Physics Today , 34 (1997) G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, Ap-plied Physics Letters , 154101 (2006) B. E. Henty and D. D. Stancil, Phys. Rev. Lett. , 243904 (2004) G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, andM. Fink, Radio Science , 6 (2005) G. Montaldo, G. Lerosey, A. Derode, A. Tourin, J. de Rosny, andM. Fink, Waves in Random Media , 287 (2004) M. Fink, J. de Rosny, G. Lerosey, and A. Tourin, Comptes RendusPhysique , 447 (2009) R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, Opt. Lett. ,3107 (2007) J. de Rosny and M. Fink, Phys. Rev. A , 065801 (Dec 2007) R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M.de Sterke, and N. A. Nicorovici, Phys. Rev. E , 016609 (Jan2004) G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science ,1120 (2007) M. Tanter et al. , J. Acoust. Soc. Am. , 37 (2001) G. Montaldo, M. Tanter, and M. Fink, The Journal of the Acous-tical Society of America , 768 (2004) F. Lemoult, G. Lerosey, J. de Rosny, and M. Fink, Phys. Rev. Lett.103