VOLUME85, NUMBER 18 PHYSICAL REVIEW LETTERS 30OCTOBER2000Negative Refraction Makes a Perfect LensJ. B. PendryCondensed 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. Anunconventional alternative to a lens, a slab of negative refractive index material, has the power to focusall 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 thata version of the lens operating at the frequency of visible light can be realized in the form of a thin slabof silver. This optical version resolves objects only a few nanometers across.PACS numbers: 78.20.Ci, 42.30.Wb, 73.20.Mf, 78.66.BzOptical 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 virtueof the refractive index contrast. Equally their limitationsare dictated by wave optics: no lens can focus light ontoan area smaller than a square wavelength. What is therenew to say other than to polish the lens more perfectly andto invent slightly better dielectrics? In this Letter I wantto challenge the traditional limitation on lens performanceand 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 limitationin performance. Consider an infinitesimal dipole of fre-quency v in front of a lens. The electric component of thefield will be given by some 2D Fourier expansion,E共r, t兲 苷Xs,kx,kyEs共kx, ky兲3 exp共ikzz 1 ikxx 1 ikyy 2 ivt兲 , (1)where we choose the axis of the lens to be the z axis.Maxwell’s equations tell us thatkz苷 1qv2c222 k2x2 k2y, v2c22. k2x1 k2y.(2)The function of the lens is to apply a phase correction toeach of the Fourier components so that at some distancebeyond the lens the fields reassemble to a focus, and animage of the dipole source appears. However, somethingis missing: for larger values of the transverse wave vector,kz苷 1iqk2x1 k2y2v2c22, v2c22, k2x1 k2y.(3)These evanescent waves decay exponentially with z and nophase correction will restore them to their proper ampli-tude. They are effectively removed from the image whichgenerally comprises only the propagating waves. Since thepropagating waves are limited tok2x1 k2y,v2c22, (4)the maximum resolution in the image can never be greaterthanD 艐2pkmax苷2pcv苷 l , (5)and this is true however perfect the lens and however largethe aperture.There is an unconventional alternative to a lens. Materialwith negative refractive index will focus light even whenin the form of a parallel-sided slab of material. In Fig. 1,I sketch the focusing action of such a slab, assuming thatthe refractive indexn 苷 21. (6)A moments thought will show that the figure obeys Snell’slaws of refraction at the surface as light inside the mediummakes a negative angle with the surface normal. The othercharacteristic of the system is the double focusing effect re-vealed by a simple ray diagram. Light transmitted througha slab of thickness d2located a distance d1from the sourcecomes to a second focus whenz 苷 d22 d1. (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 chosenFIG. 1. A negative refractive index medium bends light to anegative angle with the surface normal. Light formerly divergingfrom a point source is set in reverse and converges back to apoint. Released from the medium the light reaches a focus fora second time.3966 0031-9007兾 00兾85(18)兾3966(4)$15.00 © 2000 The American Physical SocietyVOLUME85, NUMBER 18 PHYSICAL REVIEW LETTERS 30OCTOBER2000´ 苷 21, m 苷 21. (8)At first sight this simply implies that the refractive indexis that of vacuum,n 苷p´m , (9)but further consideration will reveal that when both ´ andm are negative we must choose the negative square root in(9). However, the other relevant quantity, the impedanceof the medium,Z 苷rmm0´´0, (10)retains its positive sign so that, when both ´ 苷 21 andm 苷 21, the medium is a perfect match to free spaceand the interfaces show no reflection. At the far boundarythere is again an impedance match and the light is perfectlytransmitted into vacuum.Calculations confirm that all of the energy is perfectlytransmitted into the medium but in a strange manner: trans-port of energy in the 1z direction requires that, in themedium,k0z苷 2qv2c222 k2x2 k2y. (11)Overall the transmission coefficient of the medium isT 苷 tt0苷 exp共ik0zd兲 苷 exp共2iqv2c222 k2x2 k2yd兲 ,(12)where d is the slab thickness and the negative phase resultsfrom the choice of wave vector forced upon us by causality.It is this phase reversal that enables the medium to refocuslight by canceling the phase acquired by light as it movesaway from its source.All this was pointed out by Veselago [1] some time ago.The new message in this Letter is that, remarkably, themedium 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 ratherthan to correct their phase. We shall show that evanescentwaves emerge from the far side of the medium enhanced inamplitude 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.The proof is not difficult. Let us assume S-polarizedlight in vacuum. The electric field is given byE0S1苷 关0, 1, 0兴 exp共ikzz 1 ikxx 2 ivt兲 , (13)where the wave vector,kz苷 1iqk2x1 k2y2v2c22, v2c22, k2x1 k2y,(14)implies exponential decay. At the interface with themedium some of the light is reflected,E0S2苷 r关0, 1, 0兴 exp共2ikzz 1 ikxx 2 ivt兲 ,(15)and some transmitted into the medium,E1S1苷 t关0, 1, 0兴 exp共ik0zz 1 ikxx 2 ivt兲 , (16)wherek0z苷 1iqk2x1 k2y2 ´mv2c22,´mv2c22, k2x1 k2y.(17)Causality requires that we choose this form of the wave inthe medium: it must decay away exponentially from theinterface. By matching wave fields at the interface, weshow thatt 苷2mkzmkz1 k0z, r 苷mkz2 k0zmkz1 k0z. (18)Conversely a wave inside the medium