制作一个负折射的完美透镜外文文献翻译、中英文翻译
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VOLUME 85, NUMBER 18 PHYSICAL REVIEW LETTERS 30OCTOBER 2000 v that a k z H33527 These phase tude. generally propagating the maximum than to a erging to a for 3966 1i k 2 x 1 k 2 y 2v 2 c 22 , v 2 c 22 , k 2 x 1 k 2 y . (3) evanescent waves decay exponentially with z and no correction will restore them to their proper ampli- They are effectively removed from the image which comprises only the propagating waves. Since the waves are limited to k 2 x 1 k 2 y ,v 2 c 22 , (4) resolution in the image can never be greater FIG. 1. A negative refractive index medium bends light negative angle with the surface normal. Light formerly div from a point source is set in reverse and converges back point. Released from the medium the light reaches a focus a second time. Negative Refraction Mak J. B. Pendr Condensed Matter Theory Group, The Blackett Laborator (Received 25 With a conventional lens sharpness of the image unconventional alternative to a lens, a slab of negati all Fourier components of a 2D image, even those “superlenses” can be realized in the microwave band a version of the lens operating at the frequency of visible of silver. This optical version resolves objects only PACS numbers: 78.20.Ci, 42.30.Wb, 73.20.Mf, 78.66.Bz Optical lenses have for centuries been one of scientists prime tools. Their operation is well understood on the ba- sis of classical optics: curved surfaces focus light by virtue of the refractive index contrast. Equally their limitations are dictated by wave optics: no lens can focus light onto an area smaller than a square wavelength. What is there new to say other than to polish the lens more perfectly and to invent slightly better dielectrics? In this Letter I want to challenge the traditional limitation on lens performance and propose a class of “superlenses,” and to suggest a prac- tical scheme for implementing such a lens. Let us look more closely at the reasons for limitation in performance. Consider an infinitesimal dipole of fre- quency v in front of a lens. The electric component of the field will be given by some 2D Fourier expansion, EH20849r,tH20850 H33527 X s,k x ,k y E s H20849k x ,k y H20850 3 expH20849ik z z 1 ik x x 1 ik y y 2 ivtH20850, (1) where we choose the axis of the lens to be the z axis. Maxwells equations tell us that k z H33527 1 q v 2 c 22 2 k 2 x 2 k 2 y , v 2 c 22 . k 2 x 1 k 2 y . (2) The function of the lens is to apply a phase correction to each of the Fourier components so that at some distance beyond the lens the fields reassemble to a focus, and an image of the dipole source appears. However, something is missing: for larger values of the transverse wave vector, q 0031-9007H2086200H2086285(18)H208623966(4)$15.00 es a Perfect Lens y y, Imperial College, London SW7 2BZ, United Kingdom April 2000) is always limited by the wavelength of light. An e refractive index material, has the power to focus do not propagate in a radiative manner. Such with current technology. Our simulations show that light can be realized in the form of a thin slab few nanometers across. D H33360 2p k max H33527 2pc v H33527 l, (5) and this is true however perfect the lens and however large the aperture. There is an unconventional alternative to a lens. Material with negative refractive index will focus light even when in the form of a parallel-sided slab of material. In Fig. 1, I sketch the focusing action of such a slab, assuming that the refractive index n H33527 21. (6) A moments thought will show that the figure obeys Snells laws of refraction at the surface as light inside the medium makes a negative angle with the surface normal. The other characteristic of the system is the double focusing effect re- vealed by a simple ray diagram. Light transmitted through a slab of thickness d 2 located a distance d 1 from the source comes to a second focus when z H33527 d 2 2 d 1 . (7) The underlying secret of this medium is that both the di- electric function, , and the magnetic permeability, m, hap- pen to be negative. In that instance we have chosen 2000 The American Physical Society VOLUME 85, NUMBER 18 PHYSICAL REVIEW LETTERS 30OCTOBER 2000 H33527 21, m H33527 21. (8) m are negative we must choose the negative square root in The proof is not difficult. Let us assume S-polarized 0 z k 3967 (9). However, the other relevant quantity, the impedance of the medium, Z H33527 r mm 0 0 , (10) retains its positive sign so that, when both H33527 21 and m H33527 21, the medium is a perfect match to free space and the interfaces show no reflection. At the far boundary there is again an impedance match and the light is perfectly transmitted into vacuum. Calculations confirm that all of the energy is perfectly transmitted into the medium but in a strange manner: trans- port of energy in the 1z direction requires that, in the medium, k 0 z H33527 2 q v 2 c 22 2 k 2 x 2 k 2 y . (11) Overall the transmission coefficient of the medium is T H33527 tt 0 H33527 expH20849ik 0 z dH20850 H33527 expH208492i q v 2 c 22 2 k 2 x 2 k 2 y dH20850, (12) where d is the slab thickness and the negative phase results from the choice of wave vector forced upon us by causality. It is this phase reversal that enables the medium to refocus light by canceling the phase acquired by light as it moves away from its source. All this was pointed out by Veselago 1 some time ago. The new message in this Letter is that, remarkably, the medium can also cancel the decay of evanescent waves. The challenge here is that such waves decay in amplitude, not in phase, as they propagate away from the object plane. Therefore to focus them we need to amplify them rather than to correct their phase. We shall show that evanescent waves emerge from the far side of the medium enhanced in amplitude by the transmission process. This does not vio- late energy conservation because evanescent waves trans- port no energy, but nevertheless it is a surprising result. lim m!21 !21 T S H33527 lim m!21 !21 tt 0 expH20849ik 0 z dH20850 1 2 r 02 expH208492ik H33527 lim m!21 !21 2mk z mk z 1 k 0 z 2 k 0 z 1m H33527 expH208492ik 0 z dH20850 H33527 expH208492 (14) implies exponential decay. At the interface with the medium some of the light is reflected, E 0S2 H33527 rH208510, 1, 0H20852 expH208492ik z z 1 ik x x 2 ivtH20850, (15) and some transmitted into the medium, E 1S1 H33527 tH208510, 1, 0H20852 expH20849ik 0 z z 1 ik x x 2 ivtH20850, (16) where k 0 z H33527 1i q k 2 x 1 k 2 y 2 mv 2 c 22 , mv 2 c 22 , k 2 x 1 k 2 y . (17) Causality requires that we choose this form of the wave in the medium: it must decay away exponentially from the interface. By matching wave fields at the interface, we show that t H33527 2mk z mk z 1 k 0 z , r H33527 mk z 2 k 0 z mk z 1 k 0 z . (18) Conversely a wave inside the medium incident on the inter- face with vacuum experiences transmission and reflection as follows: t 0 H33527 2k 0 z k 0 z 1mk z , r 0 H33527 k 0 z 2mk z k 0 z 1mk z . (19) To calculate transmission through both surfaces of the slab we must sum the multiple scattering events, T S H33527 tt 0 expH20849ik 0 z dH20850 1 tt 0 r 02 expH208493ik 0 z dH20850 1 tt 0 r 04 expH208495ik 0 z dH20850 1 . H33527 tt 0 expH20849ik 0 z dH20850 1 2 r 02 expH208492ik 0 z dH20850 . (20) By substituting from (19) and (20) and taking the limit, dH20850 0 z k z expH20849ik 0 z dH20850 1 2 H20849 k 0 z 2mk z k 0 z 1mk z H20850 2 expH208492ik 0 z dH20850 ik z dH20850. (21) At first sight this simply implies that the refractive index is that of vacuum, n H33527 p m, (9) but further consideration will reveal that when both and light in vacuum. The electric field is given by E 0S1 H33527 H208510, 1, 0H20852 expH20849ik z z 1 ik x x 2 ivtH20850, (13) where the wave vector, k z H33527 1i q k 2 x 1 k 2 y 2v 2 c 22 , v 2 c 22 , k 2 x 1 k 2 y , VOLUME 85, NUMBER 18 PHYSICAL REVIEW LETTERS 30OCTOBER 2000 The reflection coefficient is given by by tuning the design parameters it is certainly possible to of light. In this system we can neglect radiative effects conclude that with this new lens both propagating and evanescent waves contribute to the resolution of the image. Therefore there is no physical obstacle to perfect reconstruction of the image beyond practical limitations of apertures and perfection of the lens surface. This is the principal conclusion of this Letter. No scheme can be of much interest if the means of realizing it are not available. Fortunately several recent developments make such a lens a practical possibility, at least in some regions of the spectrum. Some time ago it was shown that wire structures with lattice spacings of the order of a few millimeters behave like a plasma with a resonant frequency, v ep , in the GHz region 2. The ideal dielectric response of a plasma is given by H33527 1 2 v 2 ep v 2 (24) and takes negative values for v,v ep . More recently we have also shown 3 that a structure containing loops of conducting wire has properties mimicking a magnetic plasma, m H33360 1 2 v 2 mp v 2 , (25) and, although the analogy is less perfect, it has been shown that 2yem has been attained in these structures 4. Thus lim !21 lim k 2 x 1k 2 x ! T P H33527 lim !21 4expH20849ik z dH20850 H2084911H20850 2 2 H2084921H20850 2 ex to obtain focusing of a quasielectrostatic field, without placing any conditions on m. It is interesting to note that H33527 21 is exactly the condition needed for a surface plas- mon 5 to exist: there is a link between focusing action and the existence of well-defined surface plasmons. Let us estimate how well we can focus an image using a layer of silver. We shall assume that the object comprises an electrostatic potential with two spikes shown in Fig. 2. In the absence of the silver the electrostatic potential is blurred at a distance z H33527 2d H33527 80 nm away from the ob- ject and we can no longer resolve the two spikes because 3968 In the electrostatic limit, v c 0 q k 2 x 1 k 2 y . (27) It follows from (14) that lim k 2 x 1k 2 x ! k z H33527 lim k 2 x 1k 2 x ! i q k 2 x 1 k 2 y 2v 2 c 22 0 H33527 i q k 2 x 1 k 2 x (28) and, from (17) lim k 2 x 1k 2 x ! k 0 z H33527 lim k 2 x 1k 2 x ! i q k 2 x 1 k 2 y 2 mv 2 c 22 0 H33527 i q k 2 x 1 k 2 x H33527 k z . (29) Hence in this limit we see that, for the P-polarized fields, dependence onmis eliminated and only the dielectric func- tion is relevant. The transmission coefficient of the slab becomes lim k 2 x 1k 2 x ! T P H33527 lim k 2 x 1k 2 x ! 2k z k z 1 k 0 z 2k 0 z k 0 z 1k z 3 expH20849ik 0 z dH20850 1 2 H20849 k 0 z 2k z k 0 z 1k z H20850 2 expH208492ik 0 z dH20850 H33527 4expH20849ik z dH20850 H2084911H20850 2 2 H2084921H20850 2 expH208492ik z dH20850 , (30) and hence, in this limit, we need only assume pH208492ik z dH20850 H33527 expH208492ik z dH20850 H33527 expH208491 q k 2 x 1 k 2 x dH20850 (31) the higher order Fourier components of the potential are reduced in amplitude, VH20849x,z H33527 2dH20850 H33527 X k x y k x expH208491ik x x 2 2k x dH20850. (32) This result is shown in Fig. 2. We wish to use a slab of silver, thickness d, as a lens to restore the amplitude of the higher order Fourier com- ponents and to focus the image. We use the following approximate dielectric function for silver: Thus, even though we have meticulously carried through a strictly causal calculation, our final result is that the medium does amplify evanescent waves. Hence we decoupling electrostatic and magnetostatic fields: the electrostatics claim ownership of the P-polarized fields, and the magnetostatics claim the S-polarized fields. lim m!21 !21 R S H33527 lim m!21 !21 r 1 tt 0 r 0 expH208492ik 0 z dH20850 1 2 r 02 expH208492ik 0 z dH20850 H33527 0. (22) A similar result holds for P-polarized evanescent waves: lim m!21 !21 T P H33527 lim m!21 !21 2k z k z 1 k 0 z 2k 0 z k 0 z 1k z 3 expH20849ik 0 z dH20850 1 2 H20849 k 0 z 2k z k 0 z 1k z H20850 2 expH208492ik 0 z dH20850 H33527 expH208492ik z dH20850. (23) produce a structure closely approaching the ideal of H33527 21, m H33527 21, (26) at least at a single frequency. At optical frequencies several metals behave like a nearly perfect plasma with a dielectric function modeled by (24): silver, gold, and copper are perhaps the best examples. The magnetic properties of known materials are less obliging. However we can still make some progress even in this case. Consider the electrostatic limit: a system in which all dimensions are smaller than the wavelength VOLUME 85, NUMBER 18 PHYSICAL REVIEW LETTERS 30OCTOBER 2000 40nm (a) 80nm This result is also plotted in Fig. 2. Evidently only the finite imaginary part of the dielectric function prevents ideal reconstruction. However, considerable focusing is achieved. object plane image plane silver slab z-axis 0 +100-100 x-axis (nanometers) object intensity - 2 V 0 +100-100 image with silver slab image without silver slab x-axis (nanometers) image intensity - 2 V (b) (c) FIG. 2. (a) Plan view of the new lens in operation. A quasi- electrostatic potential in the object plane is imaged by the action of a silver lens. (b) The electrostatic field in the object plane. (c) The electrostatic field in the image plane with and without the silver slab in place. The reconstruction would be perfect were it not for finite absorption in the silver. H33360 5.7 2 9 2 v 22 1 0.4i . (33) Evidently the imaginary part of the dielectric function will place some practical limitations on the focusing ef- fect and, by choosing the optimum frequency for focusing of 3.48 eV, the “focused” image becomes V f H20849x,z H33527 2dH20850 H33527 X k x y k x expH208491ik x x 2 2k x dH20850 0.04 1 expH2084922k x dH20850 . (34) Intense focusing of light by exploiting surface plas- mons can also be achieved via a completely different route as Ebbesen et al. 6 and Porto et al. 7 have recently demonstrated. The quasistatic limit also considerably eases design cri- teria at microwave frequencies. For example we could make a near field electrostatic lens operating in the GHz band by using a slab of material containing thin gold wires oriented normal to the surface and spaced in a square lat- tice cell side 5 mm. Perhaps the most interesting possibil- ity for imaging in the GHz band is the magnetostatic limit. A structure comprising a set of metallic rings as described in an earlier paper would give m H33527 21 at an appropri- ate frequency, and would focus sources of magnetic fields into sharp images. Since many materials are transparent to magnetic fields, this would make an interesting imaging device for peering inside nonmagnetic objects. We have given a prescription for bringing light to a per- fect focus without the usual constraints imposed by wave- length. This is achieved by recognizing that the recently discovered negative refractive index material restores not only the phase of propagating waves but also the ampli- tude of evanescent states. For very short distances the electrostatic or magnetostatic limits apply, enabling a prac- tical implementation to be simulated in the form of a slab of silver. This device focuses light tuned to the surface plasma frequency of silver and is limited only by the re- sistive losses in the metal. We do not doubt that there are many further practical consequences of this concept. I thank David Smith, Sheldon Schultz, and Mike Wilt- shire for valuable correspondence on the concept of nega- tive refractive index. 1 V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968). 2 D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, Phys. Rev. Lett. 76, 2480 (1996); J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 (1996); J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, J. Phys. Condens. Matter 10, 4785 (1998). 3 J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999). 4 D. R. Smith, Willie J. Padilla, D. C. Vier, S. C. Nemat- Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000). 5 R. H. Ritchie, Phys. Rev. 106, 874 (1957). 6 T. W. Ebbesen et al., Nature (London) 391, 667 (1998). 7 J. A. Porto, F. J. Garcia Vidal, and J. B. Pendry, Phys. Rev. Lett. 83, 2845 (1999). 3969
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