Negative Refraction Makes a Perfect Lens
J.B.Pendry
Condend Matter Theory Group,The Blackett Laboratory,Imperial College,London SW72BZ,United Kingdom
(Received 25April 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 tho that do not propagate in a radiative manner.Such “superlens”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 lens 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.Eq
ually 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 propo a class of “superlens,”and to suggest a prac-tical scheme for implementing such a lens.
Let us look more cloly 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,
E ͑r ,t ͒X
s ,k x ,k y
E s ͑k x ,k y ͒
3exp ͑ik z z 1ik x x 1ik y y 2i v t ͒,(1)
where we choo the axis of the lens to be the z axis.Maxwell’s equations tell us that
k z 1q
v 2c 222k 2x 2k 2y
,v 2c 22.k 2x 1k 2y .(2)The function of the lens is to apply a pha correction to each of the Fourier components so that at some distance beyond the lens the fields reasmble to a focus,and an image of the dipole source appears.However,something is missing:for larger values of the transver wave vector,
k z 1i q
k 2x 1k 2y
2v 2c 22,v 2c 22,k 2x 1k 2y .(3)The evanescent waves decay exponentially with z and no pha c
orrection will restore them to their proper ampli-tude.They are effectively removed from the image which generally compris only the propagating waves.Since the propagating waves are limited to
k 2x
1
k 2y
,v 2c
22
,(4)
the maximum resolution in the image can never be greater than D ഠ
2p k max
2p c
v
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 21.
(6)
A moments thought will show that the figure obeys Snell’s 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 2located a distance d 1from the source comes to a cond focus when
z d 22d 1.
(7)
The underlying cret 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
chon
FIG.1.A negative refractive index medium bends light to a negative angle with the surface normal.Light formerly diverging from a point source is t in rever and converges back to a point.Relead from the medium the light reaches a focus for a cond time.
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©2000The American Physical Society
´21,m21.(8) Atfirst sight this simply implies that the refractive index is that of vacuum,
np
´m,(9)
but further consideration will reveal that when both´and m are negative we must choo the negative square root in (9).However,the other relevant quantity,the impedance
of the medium,
Zr mm
´´0
,(10)
retains its positive sign so that,when both´21and m21,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 the1z direction requires that,in the medium,
k0z2q
v2c222k2x2k2y.(11)
Overall the transmission coefficient of the medium is
Ttt0exp͑ik0z d͒exp͑2i q
同声传译费用v2c222k2x2k2y d͒,
(12)
removewhere d is the slab thickness and the negative pha results from the choice of wave vector forced upon us by causality. It is this pha reversal that enables the medium to refocus light by canceling the pha acquired by light as it moves away from its source.
All this was pointed out by Velago[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 pha,as they propagate away from the object plane. Therefore to focus them we need to amplify them rather than to correct their pha.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 conrvation becau evanescent waves trans-port no energy,but nevertheless it is a surprising result.
The proof is not difficult.Let us assume S-polarized light in vacuum.The electricfield is given by
E0S1͓0,1,0͔exp͑ik z z1ik x x2i v t͒,(13) where the wave vector,
k z1i
q
k2x1k2y2v2c22,v2c22,k2x1k2y,
(14) implies exponential decay.At the interface with the medium some of the light is reflected,
E0S2r͓0,1,0͔exp͑2ik z z1ik x x2i v t͒,
(15) and some transmitted into the medium,
E1S1t͓0,1,0͔exp͑ik0z z1ik x x2i v t͒,(16) where
k0z1i
q
k2x1k2y2´mv2c22,
´mv2c22,k2x1k2y.
(17)
Causality requires that we choo this form of the wave in the medium:it must decay away exponentially from the interface.By matching wavefields at the interface,we show that
t
2m k z
m k z1k0z
,r
m k z2k0z
m k z1k0z
.(18)
Converly a wave inside the medium incident on the inter-face with vacuum experiences transmission and reflection as follows:
t0
2k0z
k0z1m k z
,r0
k0z2m k z
k0z1m k z
.(19)
To calculate transmission through both surfaces of the slab we must sum the multiple scattering events,
T Stt0exp͑ik0z d͒1tt0r02exp͑3ik0z d͒
1tt0r04exp͑5ik0z d͒1...
tt
0exp͑ik0
z
d͒
grimm
12r02exp͑2ik0z d͒
.(20) By substituting from(19)and(20)and taking the limit,
lim m!21´!21T Slim
m!21
´!21
tt0exp͑ik0z d͒
12r02exp͑2ik0z d͒
lim
m!21
´!21
2m k z
m k z1k0z
2k0z
k0z1m k z
exp͑ik0z d͒
12͑z z
k0z1m k z
͒2exp͑2ik0z d͒
exp͑2ik0z d͒exp͑2ik z d͒.(21)
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The reflection coefficient is given by
lim m!21´!21R Slim
m!21
´!21
r1
tt0r0exp͑2ik0z d͒
12r02exp͑2ik0z d͒
0.(22)
A similar result holds for P-polarized evanescent waves:
lim m!21´!21T Plim
m!21
´!21
2´k z
´k z1k0z
2k0z
k0z1´k z
3
exp͑ik0z d͒
12͑k0z2´k z
k0z1´k z
͒2exp͑2ik0z d͒
exp͑2ik z d͒.(23)
Thus,even though we have meticulously carried through a strictly causal calculation,ourfinal result is that the medium does amplify evanescent waves.Hence we 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 veral 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 respon of a plasma is given by
´12v2ep
v2
(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,
mഠ12v2mp
v2
,(25)
and,although the analogy is less perfect,it has been shown that2y e m has been attained in the structures[4].Thus by tuning the design parameters it is certainly possible to produce a structure cloly approaching the ideal of
´21,m21,(26) at least at a single frequency.
At optical frequencies veral metals behave like a nearly perfect plasma with a dielectric function m
odeled 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 ca.Consider the electrostatic limit:a system in which all dimensions are smaller than the wavelength of light.In this system we can neglect radiative effects decoupling electrostatic and magnetostaticfields:the electrostatics claim ownership of the P-polarizedfields, and the magnetostatics claim the S-polarizedfields.
In the electrostatic limit,
vøc0
q
k2x1k2y.(27) It follows from(14)that
lim
k2x1k2x!`
k zlim
k2x1k2x!`
i
q
k2x1k2y2v2c220
i
q
k2x1k2x(28) and,from(17)
lim
k2x1k2x!`
k0zlim
k2x1k2x!`
lastmonthi
q
酒水销售k2x1k2y2´mv2c220
i
q
k2x1k2xk z.(29) Hence in this limit we e that,for the P-polarizedfields, dependence on m is eliminated and only the dielectric func-tion is relevant.The transmission coefficient of the slab becomes
lim
k2x1k2x!`
T Plim
k2x1k2x!`
2´k z
´k z1k0z
2k0z
k0z1´k z
3
exp͑ik0z d͒
12͑k0z2´k z
k0z1´k z
͒2exp͑2ik0z d͒
4´exp͑ik z d͒
͑´11͒22͑´21͒2exp͑2ik z d͒,(30) and hence,in this limit,we need only assume
lim ´!21中式英语
lim
火车男k2x1k2x!`
T Plim
´!21
4´exp͑ik z d͒
͑´11͒22͑´21͒2exp͑2ik z d͒
exp͑2ik z d͒exp͑1专业用英语怎么说
q
k2x1k2x d͒(31)
to obtain focusing of a quasielectrostaticfield,without
placing any conditions on m.It is interesting to note that ´21is 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 compris an electrostatic potential with two spikes shown in Fig.2. In the abnce of the silver the electrostatic potential is blurred at a distance z2d80nm away from the ob-ject and we can no longer resolve the two spikes becau the higher order Fourier components of the potential are reduced in amplitude,
V͑x,z2d͒
X
k x
y k
x
exp͑1ik x x22k x d͒.(32)
This result is shown in Fig.2.
We wish to u 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 u the following approximate dielectric function for silver:
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x-axis (nanometers)
with slab without slab
x-axis (nanometers)
(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.
´ഠ5.7292v 2210.4i .
(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 3.48eV ,the “focud ”image becomes
2011年四级真题V f ͑x ,z 2d ͒X k x
y k x
exp ͑1ik x x 22k x d ͒nickvujicic
0.041exp ͑22k x d ͒.(34)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.
Inten focusing of light by exploiting surface plas-mons can also be achieved via a completely different route as Ebben et al.[6]and Porto et al.[7]have recently demonstrated.
The quasistatic limit also considerably eas 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 5mm.Perhaps the most interesting possibil-ity for imaging in the GHz band is the magnetostatic limit.A structure comprising a t of metallic rings as described in an earlier paper would give m 21at 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 impod by wave-length.This is achieved by recognizing that the recently discovered negative refractive index material restores not only the pha 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 focus light tuned to the surface plasma frequency of silver and is limited only by the re-sistive loss in the metal.We do not doubt that there are many further practical conquences 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.Velago,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-Nasr,and S.Schultz,Phys.Rev.Lett.84,4184(2000).[5]R.H.Ritchie,Phys.Rev.106,874(1957).
[6]T.W.Ebben 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).
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