VOLUME 85, NUMBER 18 PHYSICAL REVIEW LETTERS 30OCTOBER 2000 Negative Refraction Makes a Perfect Lens J. B. Pendry Condensed Matter Theory Group, The Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom (Received 25 April 2000) With a conventional lens sharpness of the image is always limited by the wavelength of light. An unconventional alternative to a lens, a slab of negative refractive index material, has the power to focus all Fourier components of a 2D image, even those that do not propagate in a radiative manner. Such “superlenses” can be realized in the microwave band with current technology. Our simulations show that a version of the lens operating at the frequency of visible light can be realized in the form of a thin slab of silver. This optical version resolves objects only a few nanometers across. PACS numbers: 78.20.Ci, 42.30.Wb, 73.20.Mf, 78.66.Bz Optical lenses have for centuries been one of scientists’ 2p 2pc D ഠ ෇ ෇ l , (5) prime tools. Their operation is well understood on the ba- k v sis of classical optics: curved surfaces focus light by virtue max of the refractive index contrast. Equally their limitations and this is true however perfect the lens and however large are dictated by wave optics: no lens can focus light onto the aperture. an area smaller than a square wavelength. What is there There is an unconventional alternative to a lens. Material new to say other than to polish the lens more perfectly and with negative refractive index will focus light even when to invent slightly better dielectrics? In this Letter I want in the form of a parallel-sided slab of material. In Fig. 1, to challenge the traditional limitation on lens performance I sketch the focusing action of such a slab, assuming that and propose a class of “superlenses,” and to suggest a prac- the refractive index tical scheme for implementing such a lens. n ෇ 21. (6) Let us look more closely at the reasons for limitation in performance. Consider an infinitesimal dipole of fre- A moments thought will show that the figure obeys Snell’s quency v in front of a lens. The electric component of the laws of refraction at the surface as light inside the medium field will be given by some 2D Fourier expansion, makes a negative angle with the surface normal. The other X ͑ ͒ ෇ ͑ ͒ characteristic of the system is the double focusing effect re- E r, t Es kx, ky vealed by a simple ray diagram. Light transmitted through s,kx ,ky ͑ ͒ a slab of thickness d2 located a distance d1 from the source 3 exp ikzz 1 ikxx 1 ikyy 2 ivt , (1) comes to a second focus when where we choose the axis of the lens to be the z axis. ෇ Maxwell’s equations tell us that z d2 2 d1 . (7) q ෇ 2 22 2 2 2 22 2 2 kz 1 v c 2 kx 2 ky , v c . kx 1 ky . The underlying secret of this medium is that both the di- (2) electric function, ´, and the magnetic permeability, m, hap- pen to be negative. In that instance we have chosen 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 ෇ 2 2 2 22 2 22 2 2 kz 1i kx 1 ky 2vc , v c , kx 1 ky . (3) These evanescent waves decay exponentially with z and no phase correction will restore them to their proper ampli- tude. They are effectively removed from the image which generally comprises only the propagating waves. Since the propagating waves are limited to FIG. 1. A negative refractive index medium bends light to a 2 2 2 22 kx 1 ky ,vc , (4) negative angle with the surface normal. Light formerly diverging from a point source is set in reverse and converges back to a the maximum resolution in the image can never be greater point. Released from the medium the light reaches a focus for than a second time. 3966 0031-9007͞00͞85(18)͞3966(4)$15.00 © 2000 The American Physical Society VOLUME 85, NUMBER 18 PHYSICAL REVIEW LETTERS 30OCTOBER 2000 ´ ෇ 21, m ෇ 21. (8) The proof is not difficult. Let us assume S-polarized light in vacuum. The electric field is given by At first sight this simply implies that the refractive index is that of vacuum, E0S1 ෇ ͓0, 1, 0͔ exp͑ikzz 1 ikxx 2 ivt͒ , (13) p n ෇ ´m , (9) where the wave vector, q ෇ 2 2 2 22 2 22 2 2 but further consideration will reveal that when both ´ and kz 1i kx 1 ky 2vc , v c , kx 1 ky , m are negative we must choose the negative square root in (14) (9). However, the other relevant quantity, the impedance of the medium, implies exponential decay. At the interface with the r mm medium some of the light is reflected, Z ෇ 0 , (10) ´´0 E S2 ෇ r͓0, 1, 0͔ exp͑2ikzz 1 ikxx 2 ivt͒ , ෇ 0 retains its positive sign so that, when both ´ 21 and (15) m ෇ 21, the medium is a perfect match to free space and the interfaces show no reflection. At the far boundary and some transmitted into the medium, there is again an impedance match and the light is perfectly transmitted into vacuum. E ෇ t͓0, 1, 0͔ exp͑ik0 z 1 ik x 2 ivt͒ , (16) Calculations confirm that all of the energy is perfectly 1S1 z x transmitted into the medium but in a strange manner: trans- where port of energy in the 1z direction requires that, in the q medium, k0 ෇ 1i k2 1 k2 2 ´mv2c22, q z x y (17) 2 0 ෇ 2 22 2 2 ´mv2c 2 , k2 1 k2 . kz 2 v c 2 kx 2 ky . (11) x y Causality requires that we choose this form of the wave in Overall the transmission coefficient of the medium is q the medium: it must decay away exponentially from the interface. By matching wave fields at the interface, we T ෇ tt0 ෇ ͑ik0 d͒ ෇ ͑2i v2c22 2 k2 2 k2 d͒ exp z exp x y , show that (12) 0 ෇ 2mkz ෇ mkz 2 kz t 0 , r 0 . (18) where d is the slab thickness and the negative phase results mkz 1 kz mkz 1 kz from the choice of wave vector forced upon us by causality. Conversely a wave inside the medium incident on the inter- It is this phase reversal that enables the medium to refocus face with vacuum experiences transmission and reflection light by canceling the phase acquired by light as it moves as follows: away from its source. All this was pointed out by Veselago [1] some time ago. 2k0 k0 2mk t0 ෇ z r0 ෇ z z The new message in this Letter is that, remarkably, the 0 , 0 . (19) kz 1mkz kz 1mkz medium can also cancel the decay of evanescent waves. The challenge here is that such waves decay in amplitude, To calculate transmission through both surfaces of the slab not in phase, as they propagate away from the object plane. we must sum the multiple scattering events, Therefore to focus them we need to amplify them rather 0 0 0 02 0 TS ෇ tt exp͑ik d͒ 1 tt r exp͑3ik d͒ than to correct their phase. We shall show that evanescent z z 0 04 ͑ 0 ͒ waves emerge from the far side of the medium enhanced in 1 tt r exp 5ikzd 1 ... amplitude by the transmission process. This does not vio- tt0 exp͑ik0 d͒ ෇ z . (20) late energy conservation because evanescent waves trans- 1 2 r02 exp͑2ik0 d͒ port no energy, but nevertheless it is a surprising result. z By substituting from (19) and (20) and taking the limit, 0 ͑ 0 ͒ tt exp ikzd lim TS ෇ lim m!21 m!21 1 2 r02 exp͑2ik0 d͒ ´!21 ´!21 z 0 ͑ 0 ͒ 2mkz 2kz exp ikzd ෇ lim 0 m!21 0 0 kz 2mkz 2 0 mkz 1 kz kz 1mkz 1 2 ͑ 0 ͒ exp͑2ik d͒ ´!21 kz 1mkz z ෇ ͑ 0 ͒ ෇ ͑ ͒ exp 2ikzd exp 2ikzd . (21) 3967 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 0 0 ͑ 0 ͒ produce a structure closely approaching the ideal of ෇ tt r exp 2ikzd ෇ lim RS lim r 1 0. (22) ෇ ෇ m!21 m!21 1 2 r02 exp͑2ik0 d͒ ´ 21, m 21, (26) ´!21 ´!21 z A similar result holds for P-polarized evanescent waves: at least at a single frequency. At optical frequencies several metals behave like a 2´k 2k0 ෇ z z nearly perfect plasma with a dielectric function modeled lim TP lim 0 0 m!21 m!21 ´kz 1 k k 1´kz ´!21 ´!21 z z by (24): silver, gold, and copper are perhaps the best ͑ 0 ͒ exp ikzd examples. The magnetic properties of known materials are 3 k0 2´k ͑ z z ͒2 ͑ 0 ͒ less obliging. However we can still make some progress 1 2 k0 1´k exp 2ikzd z z even in this case. Consider the electrostatic limit: a system ෇ exp͑2ikzd͒ . (23) in which all dimensions are smaller than the wavelength of light.