arXiv:physics/0601163v1 [physics.optics] 21 Jan 2006 rmasa fntrlmtra htcnspotsur- support per- can negative that either material with natural polaritons[3][10][11] wave of built face be slab can periodically a superlens a from of Also consisting material. dielectric varying crystals dielectric[6][7][8]or photonic media[4][5] and metallic[9] resonators, index metallic of from negative of constructed broadband consisting effective be a can with evanescent within superlens metamaterial of A occurs enhancement im- frequency. superlens the good spatial the A if by obtained plane[2][3]. waves image be the radi- can in components ability age object evanescent the the by the has ated recover superlens a effectively surface transmit- Therefore to the be from excitation. resulting can wave amplitude waves enhanced evanescent decays, with waves ted superlens, evanescent con- a which with using for slab contrast the In material along ventional waves excitation. of of slab surface electromagnetic a support from proposal of can made recent that is superlens material the a with Basically superlenses[1]. emerged limit diffraction waves. evanescent by nmdlto ftefil mle than information smaller hand, field the other of the modulation On on medium. and surrounding vacuum in length h o pta eouinifraino h edmod- field the to on up information carry ulation resolution waves spatial Propagating propa- low both waves. the scatters evanescent illumination near- and the under gating In object field. far an the evanes- field, to of travel cannot loss that the waves from cent originates which limit diffraction † ∗ a niern ulig C2410 etGenSre Urb ‡ Street Green 61801 West Me IL 1206 158 MC-244 Building, Urbana-Champaign Engineering at cal Illinois of University neering, rsn drs eateto ehncladIdsra En Industrial and Mechanical of Department address Present lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic n rmsn praho mgn eodthe beyond imaging of approach promising One ovninlotclln mgn uesfo the from suffers imaging lens optical Conventional hoyo pia mgn eodtedffato ii iha with limit diffraction the beyond imaging optical of Theory f4 mlnswt 0n a a eotie rmfrfil dat far-field 73.20.Mf from 78.66.Bz, obtained 42.30.Lr, 78.66.-w, be numbers: can PACS demo gap we nm example, 30 wavelength. an a 376nm i As with at placed lines working far-field. is nm in object 40 formed an of be when can that composed image theory is unique in which show waveleng device We of (FSL) fraction corrugation. superlens a far-field or only, a superle field so-called propose near with the obtained in however, be are, can limit diffraction the λ eettertcladeprmna tde aesontha shown have studies experimental and theoretical Recent 0 .INTRODUCTION I. /n where , 10Eceer al S aocl cec n Engineerin and Science Nanoscale NSF Hall, Etcheverry 5130 St´ephane Durant, n sterfatv ne fthe of index refractive the is λ 0 steilmnto wave- illumination the is nvriyo aiona eklyC 42-70 USA 94720-1740, CA Berkeley California, of University Dtd a 3d20,ls eiinOtbr3,2018) 30, October revision last 2005, 23rd May (Dated: λ 0 ∗ /n hoe i,Ncoa Fang, Nicholas Liu, Zhaowei r carried are chani- ana, gi- pc mg ftelclfil itiuinaoeteob- the above distribution real field a local obtain the can to of signal order image far-field in ob- space numerically The the processed domain. easily on spectral be information spatial the in provides i.e. ject rather spectrum[23], but image, image angular space The field real a unique. not is demon- is image we pattern the Moreover, that evanes- theoretically near-field. the strate other the of in of image an is, component far-field, cent that in project superlens, to a able words, of limitation near-field the as termed vice speed impor- scan practical slow of tance. the im- often and imaging[22], physical dynamic does, a a lens prevents project conventional a not as do age possi- NSOM demonstrated Nevertheless, been with have ble. images 10nm to and down configuration, resolution NSOM signals[17][18][19][20][21] the far-field of the depending de- interpret been have to studies voted the Considerable used. or of are nanoparticles where tips principle plasmonic metallic (NSOM) fundamental as such microscopy the nanoantennas optical is optical this scanning far fact, the near-field into In radiates interacts then that field. and field waves evanescent near the the into with antenna an introduce the to beyond resolution a with to field effect far limit. superlens diffraction in the image use to an how form as Therefore, remains challenge superlens. the a transmission outside quickly after very vanish evanescent and as enhanced remain zone the waves since near-field al.[16], evanescent in et still Podolskiy are by images demonstrated superlens The lens. shown. been beyond has well limit inherent imaging diffraction in- to the material[15], the due dispersive by materials strongly limited the in still of is losses superlens Although absorption ternal the film[3][13][14]. of silver resolution a the and using crystal regime photonic optical dimensional in two a microwave in using both regime[12] imaging superlens studies demonstrated Experimental have permeability. negative or mittivity nti etr eso hoeial htanwde- new a that theoretically show we letter, this In is waves evanescent the recover to approach Another super- planar the to drawback one is there However, ss mgsfre ysc superlenses such by formed Images nses. haa rmteln.I hsppr we paper, this In lens. the from away th mgn ihrslto elbeyond well resolution with imaging t fapaa ueln ihperiodical with superlens planar a of ls rxmt fsc S,a FSL, a such of proximity close n † srt ueial htimages that numerically nstrate n in Zhang Xiang and ihpoel eindFSL designed properly with a a-edsuperlens far-field etr(NSEC) Center g ulse nArXiv/Physics/Optics in Published a-edsuperlens far-field ‡ FL a overcome can (FSL) 2 ject with a resolution beyond the diffraction limit. As along the y axis. The H-field transmitted above the grat- an example, a realistic design of an optical FSL made of ing with and its angular spectrum[23] are noted Ht(x, z) ′ ′ metal/dielectric is proposed from exact numerical calcu- and Ht(k ,z). Only plane waves with |k | < nk0 that can lation. reache the far-field are considered. The near field radi- ated by the object under the grating with z0
(a) conventional superlens and (3) reduces to:
far-field zone ′ ′ z Ht(k ,z2)= t−1(k)Hobj (k,z) where k = k + G, (5) e e ′ kk' = for the positive half-space 0 < k < nk0; and […] ′ ′ Ht(k ,z1)= t+1(k)Hobj (k,z) where k = k − G, (6) kk' = near-field zone e e It follows from this result that any propagating wave z 2 transmitted in far-field by a FSL has a unique origin. d e ′ c nt a eme For a positive k for instance, the origin is the incident y anc enh evanescent wave that has been transmitted through the z1 Amplitude diffraction order −1 with k = k′ + G. This property is true for any k′ so that there is a unique one to one rela- z0 near-field zone tionship between the near-field angular spectrum under the FSL and the transmitted angular spectrum in far- field above the FSL. This results means that when an ob- (b) far-field superlens ject is placed in close proximity of a FSL, a unique image far-field zone of the near-field distribution can be projected in far-field. z Moreover, using Eq. (5) and (6) and the rigorous diffrac- kk' = k ' =−k G tion theory[23], the near-field angular spectrum radiated order 0 order -1 by the object can be retrieved unambiguously from mea- surement of the far-field transmitted angular spectrum ′ Ht(k ,z). z e If both amplitude and phase of the angular spectrum 2 Sub-λ periodic can be measured in far-field, then a real space image of d corrugation e nt ca me y nce the near-field Hobj (k,z) above the object can be recon- nha e structed from He obj (x, z) using a simple inverse Fourier z1 Amplitude transform. However, the measurement of the phase is a practical difficulty. This difficulty appears also in diffrac- z0 near-field zone tion optical microscopy[24] where both amplitude and phase of the angular spectrum have to be measured. For incident propagating plane wave k< nk 0 this purpose, an experimental set-up such as the one use
incident evanescent plane wave k> nk 0 by Lauer[24] based on interferometry may be a good ap- proach. Alternatively, a direct real space image might be FIG. 1: Schematic picture of the transmission properties of a obtained using the Fourier transform transmission prop- conventional superlens versus a far-field superlens. Through erties of lens[23] and other optical devices. a conventional superlens (a), incident evanescent waves are In principle, the maximum spatial frequency of the enhanced in transmission and vanish quickly in the near-field electromagnetic field that a FSL can image in far-field zone. In contrast, a FSL (b) both enhances and converts them is (n + λ0/d)k0. Consequently, the best transverse res- into propagating waves by diffraction while blocking incident olution ∆l that could be obtained on the image of the propagating waves. local density of electromagnetic energy is:
λ0 ∆l = . (7) dent propagating waves are expected to be very poorly 2(n + λ0/d) transmitted in far-field because of a lack of surface wave By comparison, the best resolution that could be excitation, This property may be written: achieved with a diffraction limited microscope is λ0/2n assuming a numerical aperture NA=n. |t0(k − G)| << |t−1(k)|, (4) Using a FSL, we have demonstrated that the near-field angular spectrum and subsequently the local near-field with |t0(k − G)| << |t−1(k)| within the bandwidth of distribution can be measured. However, the electromag- evanescent waves for which the superlens effect occurs. netic distribution of the field above the object depends Let us note that a similar relation occurs for negative on how the object is exposed. For instance, in normal in- transverse wave numbers if the grating has a -axis sym- cidence or with a grazing angle exposure by a plane wave, metry grating. If in addition, the superlens is designed the FSL would provide accordingly different images. A with a large transmission within selective bandwidth model is needed if one wants to image an intrinsic prop- k ∈ [G; nk0 + G], then the relationship between the far- erty of the object that does not depend on the exposure field angular spectrum above the FSL and the near-field condition such as the local polarizability or the local ab- angular spectrum below the superlens given by Eq (2) sorptivity. 4
z far-field optical FSL made of silver/glass with the proper trans- (a) fer functions satisfying to Eq (4), a necessary condition for imaging purpose. Details on the design of this FSL glass are provided in Ref.[26]. Feature sizes of this nanos- z2 tructure are shown in Fig 2a. This FSL has been de- signed to work at λ0 = 376nm with TM polarized waves. We have computed the transfer functions of this struc- a c glass Ag ture from z = z1 to the top z = z2 by solving numer- ically Maxwell’s equations using the Rigorous Coupled Ag b d Wave Analysis (RCWA) algorithm[27][28] with experi- mental permittivity data of glass ǫ = 2.31 and silver[29] z1 x ǫ = −3.16+0.2i. The numerical solution provided has glass near-field been tested using the theorem of reciprocity of electro- magnetic waves and was applied for both propagating propagating waves (b) and evanescent waves[30][31]. The results of order 0 and 3.0 k'=k-G ±1of the amplitude transfer functions are plotted in Fig. 2b. With a periodicity d = 150nm and the wavelength 2.5 λ0 = 376nm, incident transverse wave numbers trans- k'=k+G transmitted k' incident: k mitted through the order −1 of the grating are shifted 2.0 by −2.5k0.
| Fig. 2b clearly shows that Eq (4) is satisfied with p
| t | 1.5 k ∈ [2.5; 4]k0, demonstrating that using a superlens pe- riodically corrugated, a large bandwidth of evanescent 1.0 waves can be both enhanced and converted into propa- gating waves with large amplitude, while incident propa- order 0 0.5 order -1 gating waves are poorly transmitted. Consequently, this order +1 FSL could be used for imaging with resolution well be-
0.0 low the diffraction limit. Fig 2b shows similarly that -4 -3 -2 -1 0 1 2 3 4 |t0(k + G)| << |t+1(k)| with k ∈ [−4; −2.5]k0 . transverse wave numbers (unit k= ) 0 In a superlens made of silver, surface plasmon polaritons[32] (SPP) play a key role on the enhancement FIG. 2: (a) Design of an optical FSL working at λ0 = 376nm of evanescent waves [2][3][13]. At a metal/dielectric inter- with a = 45nm ; b = 35nm ; c = 55nm ; d = 150nm. (b) face, SPP are surface waves resulting from the coupling Amplitude of transmission factor through order p = 0 (red) between p-polarized electromagnetic waves and the in- − and order p = 1 (black) from near-field z = z1 to far-field duced collective excitation of free electrons in the metal. (z >> λ0) of the optical FSL shown in (a). This FSL satis- In a silver film superlens, the wavelength and the thick- fies the requirement for imaging purpose: it provides a strong ness are chosen so that SPP can be excited within a large transmission of evanescent waves and convert them into prop- agating waves through the order -1 while the transmission of bandwidth of transverse wave numbers[3][13]. How the propagating waves through the order 0 is comparatively small. optical FSL presented in this letter has been designed in close connection to SPP behavior, is detailed in Ref.[26]. Due to the position of the selective bandwidth of en- III. CASE OF A SILVER FAR-FIELD hancement as shown Fig. 2b, waves transmitted into SUPERLENS order −1 and +1 can be substantially overlapped with k ∈ [−0.2;0.2]. For this reason, this small band- How to design such a FSL for which the transmission width has to be omitted from the measurement in or- properties satisfy to Eq (4) is a crucial question. One der to retrieve the near-field angular spectrum unam- may start from the design of a superlens slab that en- biguously. Finally, it can be deduced using Eq (5) and hances strongly incident evanescent waves within a large (6) that the near field angular spectrum Hobj (k,z) with e bandwidth. The enhancement can be provided by the k ∈ [−4; −2.7] ∪ [2.7; 4]k0 can be retrieved from the mea- ′ excitation of surface waves mode of the slab based su- surement of the far-field angular spectrum Ht(k ,z) with ′ perlens. When a periodic corrugation is added on a su- k ∈ [−1.5; −0.2] ∪ [0.2;1.5]k0. Because thise specific FSL perlens, the surface modes supported by the slab become can resolve a transverse field modulation with a maxi- leaky modes. As a result, as it was demonstrated by mum spatial frequency 4k0 , the transverse resolution on Smith et al[25] in case of a superlens made of metamate- the image of the local density of electromagnetic energy is rial with a negative refractive index, corrugations at the λ0/8. By comparison, the resolution of diffraction limited interfaces of the superlens lead to smaller values of the en- microscope is λ0/3 with a numerical aperture NA =1.5. hancement of evanescent waves by the superlens. Despite As an example, we provide the result of the image this expected difficulty, we have successfully designed an reconstruction using a FSL, of an object constituted of 5
object the image of the density of electromagnetic energy 5nm diffraction limited image (NA=1.5) far-field superlens image (NA=1.5) above the object. We have successively obtained faithful 1.4 images reconstruction from 120nm down to 30nm gap. The case of 40nm gap is reported by the red curve in 1.2 Fig. 3 where the separation of the two lines source is very clear. Let us note the formation of some artefacts[26] in 40nm line source width 1.0 40nm gap the image may appear because of the missing band in the near-field angular spectrum. The image in case of 30nm 0.8 gap (not shown) is the smallest gap between sources that we have been able to demonstrate following the Rayleigh 0.6 criterion[33].
0.4 IV. CONCLUSION 0.2
Density (a.u.) energy of electromagnetic We have demonstrated theoretically how to overcome 0.0 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 the limitation of a conventional superlens for which only images in the near-field can be obtained[3][16]. We have position x (nm) shown that when the object is positioned close to a new device termed as the far-field superlens (FSL), a unique FIG. 3: Electromagnetic energy 5nm above the object and image of evanescent waves radiated by the object can be corresponding images with and without FSL assuming in both formed in far-field. In contrast to conventional near-field case a numerical aperture NA=1.5. Image of the density of scanning optical microscope (NSOM), the FSL does not electromagnetic energy with the FSL is reconstructed using require scanning. In this sense, a FSL is similar with con- Eq 5 and 6 and from the rigorous computation of the trans- ventional lenses imaging with which a whole and unique mitted angular spectrum in far-field z >> λ0. This result image of an object can be recorded in a single snap-shot. computed rigorously directly demonstrates the optical imag- From the measurement of the far-field image pattern and ing method with resolution below the diffraction limit from a simple inversion of the linear and scalar Eqs. (5) and Far-field data, using a FSL made of silver/glass without any (6), the near-field electromagnetic distribution above the scanning object can be obtained with a resolution beyond the diffraction limit. By combining the superlens effect and the diffraction modes of a grating, the unique transmis- two 40nm lines sources of coherent TM waves separated sion properties of a FSL lies in a broadband excitation by a deep subwavelength gap. A unity value of the H of surface wave leaky modes used to convert the incident component is assumed on the two-line sources and van- near-field angular spectrum into a transmitted far-field ished everywhere else. The near field computed at 5nm angular spectrum, related by a one to one relationship. above the object using rigorous diffraction theory[23], A realistic design of an optical FSL was given made of is shown as the black curve in Fig. 3. For compari- silver/glass with such a transmission properties, owing to son, the computed diffraction limited image using con- the excitation of surface plasmon polariton (SPP) leaky ventional optical microscope assuming a numerical aper- modes. This new imaging approach has the potential ture NA = 1.5 is shown in blue in Fig 3. The optical to reach similar or better resolution than NSOM after FSL described in Fig 2a is placed 20nm above the ob- more development. Such a far field superlens could have ′ great impact not only in nano-scale imaging but also in ject. The angular spectrum Ht(k ,z) transmitted by the FSL in far-field (z >> λ0)e is computed rigorously us- nanolithography and photonic devices. ′ ing RCWA and Eq. (2-3). Values of Ht(k ,z) provide a complete set of data simulating a measurede signals in Acknowledgments an experiment. These known data are used subsequently for the image reconstruction. Because the designed sil- ver FSL processes a unique one to one k′ → k relation, We are very grateful to Dr Q.-H. Wei for the criti- it allows us to use this ”experimental” data to retrieve cal reading of the manuscript. This research was sup- the angular spectrum of the near-field 5nm above the ported by NSF Nano-scale Science and Engineering Cen- object unambiguously only by using Eq.(5-6) and the ter (DMI-0327077) and ONR/DARPA Multidisciplinary rigorous diffraction theory. By combining this angular University Research Initiative (MURI) (Grant#N00014- spectrum with its propagating component, we obtain the 01-1-0803). near-field angular spectrum 5nm above the object with k ∈ [−4; −2.7] ∪ [−1.5;1.5] ∪ [2.7; 4]k0. By applying a simple inverse Fourier transform, we finally obtain suc- cessively a reconstruction of the H-field distribution and 6
[1] J. B. Pendry, Phys. Rev. Lett. 85, 183966 (2000). [18] R. Carminati and J.-J. S´aenz, Phys. Rev. Lett. 84, 5156 [2] D. R. Smith, Science 308, 502 (2005). (2000). [3] N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 [19] J.-J. Greffet, A. Sentenac, and R. Carminati, Optics (2005). Communications 116, 20 (1995). [4] V. Vesalgo, Sov. Phys. Usp. 10, 509 (1968). [20] J. Porto, R. Carminati, and J.-J. Greffet, J. Appl. Phys. [5] R. Shelby, D. Smith, and S. Schultz, Science 292, 77 88, 4845 (2000). (2001). [21] P. Carney and et al, Phys. Rev. Lett. 92, 163903 (2004). [6] C. Luo, S. Johnson, and J. Joannopoulos, Phys. Rev B [22] B. Hetch, B. Sick, U. Wild, V. Deckert, R. Zenobi, 68, 045115 (2003). O. Martin, and D. Pohl, J. Chem. Phys. 112, 7761 [7] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and (2000). C. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003). [23] J. Goodman (McGraw-Hill, New York, 1996), 2nd ed. [8] C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, [24] V. Lauer, J. of Microsc. 205, 165 (2001). Phys. Rev. B 65, 201104 (2002). [25] D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, [9] C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, Op- S. Ramakrishna, and J. Pendry, App. Phys. Lett 82, 1506 tics Express 11, 746 (2003). (2003). [10] I. Larkin and M. Stockman, Nano Lett. 5, 339 (2005). [26] S. Durant, Z. Liu, J. Steele, and X. Zhang, JOSA B (in [11] V. Shalaev, W. Cai, U. Chettiar, H.-K. Yuan, press) 23 (2006). A. Sarychev, V. Drachev, and A. Kildishev, Opt. lett. [27] M. Moharam, E. Grann, D. Pommet, and T. Gaylord, 30, 3356 (2005). JOSA A 12, 1068 (1995). [12] P. Parimi, W. Lu, P. Vodo, and S. Sridhar, Nature (Lon- [28] M. Moharam, D. Pommet, E. Grann, and T. Gaylord, don) 426, 404 (2002). JOSA A 12, 1077 (1995). [13] H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, [29] P. Johnson and R. Christy, Phys Rev. B 6, 4370 (1972). M. Ambati, C. Sun, and X. Zhang, New J. of Phys. [30] R. Carminati, J. Saenz, J.-J. Greffet, and M. Neito- 7-255 (2005). Vesperinas, Phys. Rev A 62, 012712 (1998). [14] D. Melville and R. Blaikie, Opt. Express 13, 2127 (2005). [31] R. Carminati, M. Neito-Vesperinas, and J.-J. Greffet, [15] N. Garcia and M. Nieto-Vesperinas, Phys. Rev. Lett. JOSA A 15, 706 (1998). 11, 746 (2002). [32] W. Barnes, A. Dereux, and W. Ebbesen, Nature (Lon- [16] V. Podolskiy and E. Narimanov, Opt. Lett. 30, 76 don) 424, 824 (2003). (2005). [33] M. Born and E. Wolf (Pergamon Press, New York, 1980), [17] J.-J. Greffet and R. Carminati, Progress in Surface Sci- 6th ed. ence 56, 133 (1997).